LMFDB - Embedded newform 54.6.c.b.37.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,6,Mod(19,54)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("54.19");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	54.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.66072626990$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.47347183152.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 $ x^6 - 3x^5 + 118x^4 - 231x^3 + 3700x^2 - 3585*x + 32331
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{9} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.3
Root			$0.500000 + 4.03013i$ of defining polynomial
Character	$\chi$	$=$	54.37
Dual form			54.6.c.b.19.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 + 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(39.2420 - 67.9691i) q^{5} +(-110.566 - 191.505i) q^{7} -64.0000 q^{8} +O(q^{10})$	\(q+(2.00000 + 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(39.2420 - 67.9691i) q^{5} +(-110.566 - 191.505i) q^{7} -64.0000 q^{8} +313.936 q^{10} +(115.144 + 199.436i) q^{11} +(385.793 - 668.213i) q^{13} +(442.262 - 766.020i) q^{14} +(-128.000 - 221.703i) q^{16} +769.880 q^{17} -383.367 q^{19} +(627.872 + 1087.51i) q^{20} +(-460.577 + 797.743i) q^{22} +(193.178 - 334.594i) q^{23} +(-1517.37 - 2628.15i) q^{25} +3086.34 q^{26} +3538.10 q^{28} +(-394.883 - 683.958i) q^{29} +(-1609.97 + 2788.54i) q^{31} +(512.000 - 886.810i) q^{32} +(1539.76 + 2666.94i) q^{34} -17355.2 q^{35} +2465.33 q^{37} +(-766.733 - 1328.02i) q^{38} +(-2511.49 + 4350.02i) q^{40} +(-4621.73 + 8005.07i) q^{41} +(-5315.76 - 9207.16i) q^{43} -3684.62 q^{44} +1545.42 q^{46} +(976.032 + 1690.54i) q^{47} +(-16046.0 + 27792.4i) q^{49} +(6069.46 - 10512.6i) q^{50} +(6172.68 + 10691.4i) q^{52} +32589.2 q^{53} +18074.0 q^{55} +(7076.19 + 12256.3i) q^{56} +(1579.53 - 2735.83i) q^{58} +(-11916.1 + 20639.2i) q^{59} +(-18804.4 - 32570.2i) q^{61} -12879.7 q^{62} +4096.00 q^{64} +(-30278.5 - 52444.0i) q^{65} +(11525.6 - 19962.9i) q^{67} +(-6159.04 + 10667.8i) q^{68} +(-34710.5 - 60120.3i) q^{70} +66050.4 q^{71} +65130.0 q^{73} +(4930.66 + 8540.16i) q^{74} +(3066.93 - 5312.08i) q^{76} +(25462.0 - 44101.5i) q^{77} +(35433.7 + 61373.0i) q^{79} -20091.9 q^{80} -36973.8 q^{82} +(27643.5 + 47880.0i) q^{83} +(30211.6 - 52328.0i) q^{85} +(21263.0 - 36828.6i) q^{86} +(-7369.24 - 12763.9i) q^{88} -10598.6 q^{89} -170622. q^{91} +(3090.85 + 5353.50i) q^{92} +(-3904.13 + 6762.15i) q^{94} +(-15044.1 + 26057.1i) q^{95} +(41409.9 + 71724.1i) q^{97} -128368. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} - 48 q^{4} + 54 q^{5} - 132 q^{7} - 384 q^{8}+O(q^{10}) $ 6 * q + 12 * q^2 - 48 * q^4 + 54 * q^5 - 132 * q^7 - 384 * q^8	$ 6 q + 12 q^{2} - 48 q^{4} + 54 q^{5} - 132 q^{7} - 384 q^{8} + 432 q^{10} + 315 q^{11} - 744 q^{13} + 528 q^{14} - 768 q^{16} - 2898 q^{17} + 2262 q^{19} + 864 q^{20} - 1260 q^{22} + 3168 q^{23} - 2883 q^{25} - 5952 q^{26} + 4224 q^{28} + 5148 q^{29} - 8610 q^{31} + 3072 q^{32} - 5796 q^{34} - 2700 q^{35} + 39936 q^{37} + 4524 q^{38} - 3456 q^{40} - 5049 q^{41} - 31389 q^{43} - 10080 q^{44} + 25344 q^{46} - 12924 q^{47} - 52857 q^{49} + 11532 q^{50} - 11904 q^{52} + 96048 q^{53} + 126252 q^{55} + 8448 q^{56} - 20592 q^{58} - 62955 q^{59} - 75966 q^{61} - 68880 q^{62} + 24576 q^{64} - 108702 q^{65} - 32991 q^{67} + 23184 q^{68} - 5400 q^{70} + 129672 q^{71} - 8466 q^{73} + 79872 q^{74} - 18096 q^{76} - 88740 q^{77} + 89202 q^{79} - 27648 q^{80} - 40392 q^{82} - 32634 q^{83} + 71388 q^{85} + 125556 q^{86} - 20160 q^{88} - 66132 q^{89} - 301836 q^{91} + 50688 q^{92} + 51696 q^{94} + 82944 q^{95} + 46245 q^{97} - 422856 q^{98}+O(q^{100}) $ 6 * q + 12 * q^2 - 48 * q^4 + 54 * q^5 - 132 * q^7 - 384 * q^8 + 432 * q^10 + 315 * q^11 - 744 * q^13 + 528 * q^14 - 768 * q^16 - 2898 * q^17 + 2262 * q^19 + 864 * q^20 - 1260 * q^22 + 3168 * q^23 - 2883 * q^25 - 5952 * q^26 + 4224 * q^28 + 5148 * q^29 - 8610 * q^31 + 3072 * q^32 - 5796 * q^34 - 2700 * q^35 + 39936 * q^37 + 4524 * q^38 - 3456 * q^40 - 5049 * q^41 - 31389 * q^43 - 10080 * q^44 + 25344 * q^46 - 12924 * q^47 - 52857 * q^49 + 11532 * q^50 - 11904 * q^52 + 96048 * q^53 + 126252 * q^55 + 8448 * q^56 - 20592 * q^58 - 62955 * q^59 - 75966 * q^61 - 68880 * q^62 + 24576 * q^64 - 108702 * q^65 - 32991 * q^67 + 23184 * q^68 - 5400 * q^70 + 129672 * q^71 - 8466 * q^73 + 79872 * q^74 - 18096 * q^76 - 88740 * q^77 + 89202 * q^79 - 27648 * q^80 - 40392 * q^82 - 32634 * q^83 + 71388 * q^85 + 125556 * q^86 - 20160 * q^88 - 66132 * q^89 - 301836 * q^91 + 50688 * q^92 + 51696 * q^94 + 82944 * q^95 + 46245 * q^97 - 422856 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/54\mathbb{Z}\right)^\times$.

$n$	$29$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	2.00000	+	3.46410i	0.353553	+	0.612372i
$2$	2.00000	+	3.46410i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−8.00000	+	13.8564i	−0.250000	+	0.433013i
$5$	39.2420	−	67.9691i	0.701982	−	1.21587i	−0.265788	−	0.964032i	$-0.585632\pi$
$5$	39.2420	−	67.9691i	0.701982	−	1.21587i	0.967770	−	0.251837i	$-0.0810346\pi$
$6$		0			0
$7$	−110.566	−	191.505i	−0.852854	−	1.47719i	−0.878622	−	0.477518i	$-0.841536\pi$
$7$	−110.566	−	191.505i	−0.852854	−	1.47719i	0.0257678	−	0.999668i	$-0.491797\pi$
$8$	−64.0000			−0.353553
$9$		0			0
$10$	313.936			0.992752
$11$	115.144	+	199.436i	0.286920	+	0.496960i	0.973073	−	0.230497i	$-0.0740353\pi$
$11$	115.144	+	199.436i	0.286920	+	0.496960i	−0.686153	+	0.727457i	$0.740702\pi$
$12$		0			0
$13$	385.793	−	668.213i	0.633134	−	1.09662i	−0.353773	−	0.935331i	$-0.615101\pi$
$13$	385.793	−	668.213i	0.633134	−	1.09662i	0.986907	−	0.161289i	$-0.0515652\pi$
$14$	442.262	−	766.020i	0.603059	−	1.04453i
$15$		0			0
$16$	−128.000	−	221.703i	−0.125000	−	0.216506i
$17$	769.880			0.646101			0.323051	−	0.946382i	$-0.395292\pi$
$17$	769.880			0.646101			0.323051	+	0.946382i	$0.395292\pi$
$18$		0			0
$19$	−383.367			−0.243630			−0.121815	−	0.992553i	$-0.538871\pi$
$19$	−383.367			−0.243630			−0.121815	+	0.992553i	$0.538871\pi$
$20$	627.872	+	1087.51i	0.350991	+	0.607934i
$21$		0			0
$22$	−460.577	+	797.743i	−0.202883	+	0.351404i
$23$	193.178	−	334.594i	0.0761444	−	0.131886i	−0.825439	−	0.564491i	$-0.809072\pi$
$23$	193.178	−	334.594i	0.0761444	−	0.131886i	0.901583	+	0.432605i	$0.142406\pi$
$24$		0			0
$25$	−1517.37	−	2628.15i	−0.485557	−	0.841009i
$26$	3086.34			0.895387
$27$		0			0
$28$	3538.10			0.852854
$29$	−394.883	−	683.958i	−0.0871914	−	0.151020i	0.819131	−	0.573606i	$-0.194456\pi$
$29$	−394.883	−	683.958i	−0.0871914	−	0.151020i	−0.906323	+	0.422586i	$0.861123\pi$
$30$		0			0
$31$	−1609.97	+	2788.54i	−0.300893	+	0.521163i	−0.976339	−	0.216247i	$-0.930618\pi$
$31$	−1609.97	+	2788.54i	−0.300893	+	0.521163i	0.675445	+	0.737410i	$0.263952\pi$
$32$	512.000	−	886.810i	0.0883883	−	0.153093i
$33$		0			0
$34$	1539.76	+	2666.94i	0.228431	+	0.395655i
$35$	−17355.2			−2.39475
$36$		0			0
$37$	2465.33			0.296054			0.148027	−	0.988983i	$-0.452708\pi$
$37$	2465.33			0.296054			0.148027	+	0.988983i	$0.452708\pi$
$38$	−766.733	−	1328.02i	−0.0861361	−	0.149192i
$39$		0			0
$40$	−2511.49	+	4350.02i	−0.248188	+	0.429874i
$41$	−4621.73	+	8005.07i	−0.429383	+	0.743713i	−0.996819	−	0.0797047i	$-0.974602\pi$
$41$	−4621.73	+	8005.07i	−0.429383	+	0.743713i	0.567436	+	0.823418i	$0.307936\pi$
$42$		0			0
$43$	−5315.76	−	9207.16i	−0.438424	−	0.759372i	0.559145	−	0.829070i	$-0.311130\pi$
$43$	−5315.76	−	9207.16i	−0.438424	−	0.759372i	−0.997568	+	0.0696983i	$0.977796\pi$
$44$	−3684.62			−0.286920
$45$		0			0
$46$	1545.42			0.107684
$47$	976.032	+	1690.54i	0.0644495	+	0.111630i	0.896450	−	0.443146i	$-0.146138\pi$
$47$	976.032	+	1690.54i	0.0644495	+	0.111630i	−0.832000	+	0.554775i	$0.812804\pi$
$48$		0			0
$49$	−16046.0	+	27792.4i	−0.954720	+	1.65362i
$50$	6069.46	−	10512.6i	0.343341	−	0.594683i
$51$		0			0
$52$	6172.68	+	10691.4i	0.316567	+	0.548310i
$53$	32589.2			1.59362			0.796809	−	0.604231i	$-0.206520\pi$
$53$	32589.2			1.59362			0.796809	+	0.604231i	$0.206520\pi$
$54$		0			0
$55$	18074.0			0.805651
$56$	7076.19	+	12256.3i	0.301529	+	0.522264i
$57$		0			0
$58$	1579.53	−	2735.83i	0.0616537	−	0.106787i
$59$	−11916.1	+	20639.2i	−0.445659	+	0.771903i	−0.998098	−	0.0616498i	$-0.980364\pi$
$59$	−11916.1	+	20639.2i	−0.445659	+	0.771903i	0.552439	+	0.833553i	$0.313697\pi$
$60$		0			0
$61$	−18804.4	−	32570.2i	−0.647046	−	1.12072i	−0.983825	−	0.179133i	$-0.942671\pi$
$61$	−18804.4	−	32570.2i	−0.647046	−	1.12072i	0.336779	−	0.941584i	$-0.390663\pi$
$62$	−12879.7			−0.425528
$63$		0			0
$64$	4096.00			0.125000
$65$	−30278.5	−	52444.0i	−0.888897	−	1.53962i
$66$		0			0
$67$	11525.6	−	19962.9i	0.313672	−	0.543295i	−0.665483	−	0.746413i	$-0.731774\pi$
$67$	11525.6	−	19962.9i	0.313672	−	0.543295i	0.979154	+	0.203118i	$0.0651075\pi$
$68$	−6159.04	+	10667.8i	−0.161525	+	0.279770i
$69$		0			0
$70$	−34710.5	−	60120.3i	−0.846673	−	1.46648i
$71$	66050.4			1.55500			0.777498	−	0.628885i	$-0.216489\pi$
$71$	66050.4			1.55500			0.777498	+	0.628885i	$0.216489\pi$
$72$		0			0
$73$	65130.0			1.43045			0.715227	−	0.698893i	$-0.246324\pi$
$73$	65130.0			1.43045			0.715227	+	0.698893i	$0.246324\pi$
$74$	4930.66	+	8540.16i	0.104671	+	0.181295i
$75$		0			0
$76$	3066.93	−	5312.08i	0.0609074	−	0.105495i
$77$	25462.0	−	44101.5i	0.489402	−	0.847669i
$78$		0			0
$79$	35433.7	+	61373.0i	0.638776	+	1.10639i	0.985702	+	0.168500i	$0.0538924\pi$
$79$	35433.7	+	61373.0i	0.638776	+	1.10639i	−0.346925	+	0.937893i	$0.612774\pi$
$80$	−20091.9			−0.350991
$81$		0			0
$82$	−36973.8			−0.607239
$83$	27643.5	+	47880.0i	0.440451	+	0.762884i	0.997723	−	0.0674462i	$-0.0214851\pi$
$83$	27643.5	+	47880.0i	0.440451	+	0.762884i	−0.557272	+	0.830330i	$0.688152\pi$
$84$		0			0
$85$	30211.6	−	52328.0i	0.453551	−	0.785574i
$86$	21263.0	−	36828.6i	0.310012	−	0.536957i
$87$		0			0
$88$	−7369.24	−	12763.9i	−0.101442	−	0.175702i
$89$	−10598.6			−0.141831			−0.0709156	−	0.997482i	$-0.522592\pi$
$89$	−10598.6			−0.141831			−0.0709156	+	0.997482i	$0.522592\pi$
$90$		0			0
$91$	−170622.			−2.15988
$92$	3090.85	+	5353.50i	0.0380722	+	0.0659430i
$93$		0			0
$94$	−3904.13	+	6762.15i	−0.0455727	+	0.0789342i
$95$	−15044.1	+	26057.1i	−0.171024	+	0.296222i
$96$		0			0
$97$	41409.9	+	71724.1i	0.446864	+	0.773991i	0.998180	−	0.0603057i	$-0.0192076\pi$
$97$	41409.9	+	71724.1i	0.446864	+	0.773991i	−0.551316	+	0.834296i	$0.685874\pi$
$98$	−128368.			−1.35018
$99$		0			0
$100$	48555.7			0.485557
$101$	10604.1	+	18366.9i	0.103436	+	0.179156i	0.913098	−	0.407740i	$-0.133683\pi$
$101$	10604.1	+	18366.9i	0.103436	+	0.179156i	−0.809662	+	0.586896i	$0.800350\pi$
$102$		0			0
$103$	−65877.4	+	114103.i	−0.611848	+	1.05975i	0.379081	+	0.925363i	$0.376240\pi$
$103$	−65877.4	+	114103.i	−0.611848	+	1.05975i	−0.990929	+	0.134388i	$0.957093\pi$
$104$	−24690.7	+	42765.6i	−0.223847	+	0.387714i
$105$		0			0
$106$	65178.4	+	112892.i	0.563429	+	0.975888i
$107$	−54796.5			−0.462693			−0.231347	−	0.972871i	$-0.574313\pi$
$107$	−54796.5			−0.462693			−0.231347	+	0.972871i	$0.574313\pi$
$108$		0			0
$109$	160570.			1.29448			0.647242	−	0.762284i	$-0.275922\pi$
$109$	160570.			1.29448			0.647242	+	0.762284i	$0.275922\pi$
$110$	36147.9	+	62610.1i	0.284840	+	0.493358i
$111$		0			0
$112$	−28304.8	+	49025.3i	−0.213213	+	0.369297i
$113$	−33217.8	+	57534.9i	−0.244723	+	0.423873i	−0.962054	−	0.272860i	$-0.912030\pi$
$113$	−33217.8	+	57534.9i	−0.244723	+	0.423873i	0.717331	+	0.696733i	$0.245364\pi$
$114$		0			0
$115$	−15161.4	−	26260.3i	−0.106904	−	0.185163i
$116$	12636.3			0.0871914
$117$		0			0
$118$	−95328.4			−0.630256
$119$	−85122.2	−	147436.i	−0.551030	−	0.954412i
$120$		0			0
$121$	54009.1	−	93546.4i	0.335354	−	0.580850i
$122$	75217.7	−	130281.i	0.457531	−	0.792467i
$123$		0			0
$124$	−25759.5	−	44616.7i	−0.150447	−	0.260581i
$125$	7084.72			0.0405553
$126$		0			0
$127$	−299603.			−1.64830			−0.824152	−	0.566368i	$-0.808348\pi$
$127$	−299603.			−1.64830			−0.824152	+	0.566368i	$0.808348\pi$
$128$	8192.00	+	14189.0i	0.0441942	+	0.0765466i
$129$		0			0
$130$	121114.	−	209776.i	0.628545	−	1.08867i
$131$	−166224.	+	287909.i	−0.846283	+	1.46581i	0.0382189	+	0.999269i	$0.487832\pi$
$131$	−166224.	+	287909.i	−0.846283	+	1.46581i	−0.884502	+	0.466536i	$0.845502\pi$
$132$		0			0
$133$	42387.1	+	73416.7i	0.207781	+	0.359887i
$134$	92204.5			0.443599
$135$		0			0
$136$	−49272.3			−0.228431
$137$	−93601.3	−	162122.i	−0.426070	−	0.737974i	0.570450	−	0.821332i	$-0.306769\pi$
$137$	−93601.3	−	162122.i	−0.426070	−	0.737974i	−0.996520	+	0.0833580i	$0.973436\pi$
$138$		0			0
$139$	105285.	−	182358.i	0.462198	−	0.800550i	−0.536872	−	0.843664i	$-0.680394\pi$
$139$	105285.	−	182358.i	0.462198	−	0.800550i	0.999070	+	0.0431133i	$0.0137276\pi$
$140$	138842.	−	240481.i	0.598688	−	1.03696i
$141$		0			0
$142$	132101.	+	228805.i	0.549774	+	0.952237i
$143$	177687.			0.726635
$144$		0			0
$145$	−61984.0			−0.244827
$146$	130260.	+	225617.i	0.505742	+	0.875970i
$147$		0			0
$148$	−19722.7	+	34160.6i	−0.0740135	+	0.128195i
$149$	242872.	−	420667.i	0.896215	−	1.55229i	0.0639212	−	0.997955i	$-0.479639\pi$
$149$	242872.	−	420667.i	0.896215	−	1.55229i	0.832294	−	0.554335i	$-0.187027\pi$
$150$		0			0
$151$	−105105.	−	182047.i	−0.375130	−	0.649744i	0.615217	−	0.788358i	$-0.289069\pi$
$151$	−105105.	−	182047.i	−0.375130	−	0.649744i	−0.990346	+	0.138614i	$0.955735\pi$
$152$	24535.5			0.0861361
$153$		0			0
$154$	203696.			0.692119
$155$	126357.	+	218856.i	0.422443	+	0.731694i
$156$		0			0
$157$	196123.	−	339696.i	0.635009	−	1.09987i	−0.351504	−	0.936186i	$-0.614330\pi$
$157$	196123.	−	339696.i	0.635009	−	1.09987i	0.986513	−	0.163682i	$-0.0523371\pi$
$158$	−141735.	+	245492.i	−0.451683	+	0.782338i
$159$		0			0
$160$	−40183.8	−	69600.4i	−0.124094	−	0.214937i
$161$	−85435.3			−0.259760
$162$		0			0
$163$	190208.			0.560738			0.280369	−	0.959892i	$-0.409543\pi$
$163$	190208.			0.560738			0.280369	+	0.959892i	$0.409543\pi$
$164$	−73947.7	−	128081.i	−0.214692	−	0.371857i
$165$		0			0
$166$	−110574.	+	191520.i	−0.311446	+	0.539440i
$167$	−200619.	+	347483.i	−0.556650	+	0.964145i	0.441123	+	0.897446i	$0.354580\pi$
$167$	−200619.	+	347483.i	−0.556650	+	0.964145i	−0.997773	+	0.0666991i	$0.978753\pi$
$168$		0			0
$169$	−112026.	−	194034.i	−0.301718	−	0.522590i
$170$	241693.			0.641418
$171$		0			0
$172$	170104.			0.438424
$173$	47391.9	+	82085.1i	0.120389	+	0.208521i	0.919921	−	0.392103i	$-0.128252\pi$
$173$	47391.9	+	82085.1i	0.120389	+	0.208521i	−0.799532	+	0.600624i	$0.794919\pi$
$174$		0			0
$175$	−335537.	+	581166.i	−0.828218	+	1.43452i
$176$	29477.0	−	51055.6i	0.0717300	−	0.124240i
$177$		0			0
$178$	−21197.1	−	36714.5i	−0.0501449	−	0.0868536i
$179$	218816.			0.510442			0.255221	−	0.966883i	$-0.417852\pi$
$179$	218816.			0.510442			0.255221	+	0.966883i	$0.417852\pi$
$180$		0			0
$181$	−344513.			−0.781645			−0.390823	−	0.920466i	$-0.627809\pi$
$181$	−344513.			−0.781645			−0.390823	+	0.920466i	$0.627809\pi$
$182$	−341243.	−	591050.i	−0.763634	−	1.32265i
$183$		0			0
$184$	−12363.4	+	21414.0i	−0.0269211	+	0.0466287i
$185$	96744.5	−	167566.i	0.207825	−	0.359963i
$186$		0			0
$187$	88647.3	+	153542.i	0.185379	+	0.321087i
$188$	−31233.0			−0.0644495
$189$		0			0
$190$	−120353.			−0.241864
$191$	−67476.5	−	116873.i	−0.133835	−	0.231809i	0.791317	−	0.611406i	$-0.209396\pi$
$191$	−67476.5	−	116873.i	−0.133835	−	0.231809i	−0.925152	+	0.379597i	$0.876062\pi$
$192$		0			0
$193$	−113675.	+	196892.i	−0.219671	+	0.380482i	−0.954707	−	0.297546i	$-0.903832\pi$
$193$	−113675.	+	196892.i	−0.219671	+	0.380482i	0.735036	+	0.678028i	$0.237165\pi$
$194$	−165640.	+	286896.i	−0.315980	+	0.547294i
$195$		0			0
$196$	−256736.	−	444679.i	−0.477360	−	0.826811i
$197$	−430437.			−0.790213			−0.395107	−	0.918635i	$-0.629292\pi$
$197$	−430437.			−0.790213			−0.395107	+	0.918635i	$0.629292\pi$
$198$		0			0
$199$	719704.			1.28831			0.644156	−	0.764894i	$-0.277209\pi$
$199$	719704.			1.28831			0.644156	+	0.764894i	$0.277209\pi$
$200$	97111.4	+	168202.i	0.171670	+	0.297342i
$201$		0			0
$202$	−42416.5	+	73467.5i	−0.0731402	+	0.126683i
$203$	−87321.0	+	151244.i	−0.148723	+	0.257596i
$204$		0			0
$205$	362732.	+	628269.i	0.602838	+	1.04415i
$206$	−527019.			−0.865283
$207$		0			0
$208$	−197526.			−0.316567
$209$	−44142.5	−	76457.1i	−0.0699023	−	0.121074i
$210$		0			0
$211$	−30700.0	+	53174.0i	−0.0474715	+	0.0822230i	−0.888785	−	0.458325i	$-0.848450\pi$
$211$	−30700.0	+	53174.0i	−0.0474715	+	0.0822230i	0.841313	+	0.540548i	$0.181783\pi$
$212$	−260714.	+	451569.i	−0.398405	+	0.690057i
$213$		0			0
$214$	−109593.	−	189821.i	−0.163587	−	0.283341i
$215$	−834403.			−1.23106
$216$		0			0
$217$	712027.			1.02647
$218$	321139.	+	556229.i	0.457670	+	0.792707i
$219$		0			0
$220$	−144592.	+	250440.i	−0.201413	+	0.348857i
$221$	297014.	−	514443.i	0.409069	−	0.708528i
$222$		0			0
$223$	−329013.	−	569868.i	−0.443049	−	0.767383i	0.554865	−	0.831940i	$-0.312770\pi$
$223$	−329013.	−	569868.i	−0.443049	−	0.767383i	−0.997914	+	0.0645574i	$0.979436\pi$
$224$	−226438.			−0.301529
$225$		0			0
$226$	−265742.			−0.346091
$227$	−5067.16	−	8776.57i	−0.00652679	−	0.0113047i	0.862744	−	0.505642i	$-0.168744\pi$
$227$	−5067.16	−	8776.57i	−0.00652679	−	0.0113047i	−0.869270	+	0.494337i	$0.835411\pi$
$228$		0			0
$229$	−611518.	+	1.05918e6i	−0.770584	+	1.33469i	0.166659	+	0.986015i	$0.446702\pi$
$229$	−611518.	+	1.05918e6i	−0.770584	+	1.33469i	−0.937243	+	0.348677i	$0.886631\pi$
$230$	60645.5	−	105041.i	0.0755925	−	0.130930i
$231$		0			0
$232$	25272.5	+	43773.3i	0.0308268	+	0.0533936i
$233$	1.16513e6			1.40599			0.702996	−	0.711194i	$-0.251845\pi$
$233$	1.16513e6			1.40599			0.702996	+	0.711194i	$0.251845\pi$
$234$		0			0
$235$	153206.			0.180969
$236$	−190657.	−	330227.i	−0.222829	−	0.385952i
$237$		0			0
$238$	340489.	−	589744.i	0.389637	−	0.674871i
$239$	659760.	−	1.14274e6i	0.747122	−	1.29405i	−0.202075	−	0.979370i	$-0.564769\pi$
$239$	659760.	−	1.14274e6i	0.747122	−	1.29405i	0.949197	−	0.314683i	$-0.101898\pi$
$240$		0			0
$241$	−265905.	−	460560.i	−0.294906	−	0.510792i	0.680057	−	0.733159i	$-0.261955\pi$
$241$	−265905.	−	460560.i	−0.294906	−	0.510792i	−0.974963	+	0.222367i	$0.928622\pi$
$242$	432072.			0.474262
$243$		0			0
$244$	601741.			0.647046
$245$	1.25935e6	+	2.18126e6i	1.34039	+	2.32163i
$246$		0			0
$247$	−147900.	+	256170.i	−0.154250	+	0.267169i
$248$	103038.	−	178467.i	0.106382	−	0.184259i
$249$		0			0
$250$	14169.4	+	24542.2i	0.0143385	+	0.0248349i
$251$	−475.750			−0.000476644			−0.000238322	−	1.00000i	$-0.500076\pi$
$251$	−475.750			−0.000476644			−0.000238322		1.00000i	$0.500076\pi$
$252$		0			0
$253$	88973.4			0.0873894
$254$	−599207.	−	1.03786e6i	−0.582764	−	1.00938i
$255$		0			0
$256$	−32768.0	+	56755.8i	−0.0312500	+	0.0541266i
$257$	−13603.3	+	23561.6i	−0.0128473	+	0.0222522i	−0.872378	−	0.488833i	$-0.837423\pi$
$257$	−13603.3	+	23561.6i	−0.0128473	+	0.0222522i	0.859530	+	0.511085i	$0.170756\pi$
$258$		0			0
$259$	−272581.	−	472124.i	−0.252491	−	0.437327i
$260$	968913.			0.888897
$261$		0			0
$262$	−1.32979e6			−1.19683
$263$	−69991.2	−	121228.i	−0.0623956	−	0.108072i	0.833140	−	0.553062i	$-0.186541\pi$
$263$	−69991.2	−	121228.i	−0.0623956	−	0.108072i	−0.895536	+	0.444990i	$0.853207\pi$
$264$		0			0
$265$	1.27887e6	−	2.21506e6i	1.11869	−	1.93763i
$266$	−169549.	+	293667.i	−0.146923	+	0.254478i
$267$		0			0
$268$	184409.	+	319406.i	0.156836	+	0.271648i
$269$	−1.63842e6			−1.38053			−0.690263	−	0.723558i	$-0.742505\pi$
$269$	−1.63842e6			−1.38053			−0.690263	+	0.723558i	$0.742505\pi$
$270$		0			0
$271$	−280791.			−0.232252			−0.116126	−	0.993234i	$-0.537048\pi$
$271$	−280791.			−0.232252			−0.116126	+	0.993234i	$0.537048\pi$
$272$	−98544.6	−	170684.i	−0.0807627	−	0.139885i
$273$		0			0
$274$	374405.	−	648489.i	0.301277	−	0.521827i
$275$	349432.	−	605234.i	0.278632	−	0.482605i
$276$		0			0
$277$	−346404.	−	599989.i	−0.271259	−	0.469834i	0.697926	−	0.716170i	$-0.254107\pi$
$277$	−346404.	−	599989.i	−0.271259	−	0.469834i	−0.969184	+	0.246336i	$0.920773\pi$
$278$	842277.			0.653647
$279$		0			0
$280$	1.11074e6			0.846673
$281$	−1.05464e6	−	1.82670e6i	−0.796782	−	1.38007i	−0.921701	−	0.387901i	$-0.873200\pi$
$281$	−1.05464e6	−	1.82670e6i	−0.796782	−	1.38007i	0.124919	−	0.992167i	$-0.460133\pi$
$282$		0			0
$283$	731346.	−	1.26673e6i	0.542821	−	0.940193i	−0.455920	−	0.890021i	$-0.650690\pi$
$283$	731346.	−	1.26673e6i	0.542821	−	0.940193i	0.998741	−	0.0501725i	$-0.0159771\pi$
$284$	−528403.	+	915221.i	−0.388749	+	0.673333i
$285$		0			0
$286$	355375.	+	615527.i	0.256904	+	0.444972i
$287$	2.04402e6			1.46480
$288$		0			0
$289$	−827142.			−0.582553
$290$	−123968.	−	214719.i	−0.0865595	−	0.149925i
$291$		0			0
$292$	−521040.	+	902467.i	−0.357613	+	0.619404i
$293$	−1.27745e6	+	2.21261e6i	−0.869313	+	1.50569i	−0.00661216	+	0.999978i	$0.502105\pi$
$293$	−1.27745e6	+	2.21261e6i	−0.869313	+	1.50569i	−0.862700	+	0.505715i	$0.831229\pi$
$294$		0			0
$295$	935219.	+	1.61985e6i	0.625688	+	1.08372i
$296$	−157781.			−0.104671
$297$		0			0
$298$	1.94298e6			1.26744
$299$	−149053.	−	258168.i	−0.0964192	−	0.167003i
$300$		0			0
$301$	−1.17548e6	+	2.03599e6i	−0.747822	+	1.29527i
$302$	420421.	−	728190.i	0.265257	−	0.459438i
$303$		0			0
$304$	49070.9	+	84993.3i	0.0304537	+	0.0527474i
$305$	−2.95169e6			−1.81686
$306$		0			0
$307$	−2.14982e6			−1.30183			−0.650917	−	0.759149i	$-0.725615\pi$
$307$	−2.14982e6			−1.30183			−0.650917	+	0.759149i	$0.725615\pi$
$308$	407392.	+	705623.i	0.244701	+	0.423834i
$309$		0			0
$310$	−505426.	+	875424.i	−0.298713	+	0.517385i
$311$	−183809.	+	318367.i	−0.107762	+	0.186650i	−0.914863	−	0.403763i	$-0.867702\pi$
$311$	−183809.	+	318367.i	−0.107762	+	0.186650i	0.807101	+	0.590413i	$0.201035\pi$
$312$		0			0
$313$	−91371.8	−	158261.i	−0.0527171	−	0.0913086i	0.838463	−	0.544959i	$-0.183455\pi$
$313$	−91371.8	−	158261.i	−0.0527171	−	0.0913086i	−0.891180	+	0.453650i	$0.850121\pi$
$314$	1.56899e6			0.898039
$315$		0			0
$316$	−1.13388e6			−0.638776
$317$	432482.	+	749081.i	0.241724	+	0.418679i	0.961206	−	0.275833i	$-0.0889538\pi$
$317$	432482.	+	749081.i	0.241724	+	0.418679i	−0.719481	+	0.694512i	$0.755620\pi$
$318$		0			0
$319$	90937.2	−	157508.i	0.0500339	−	0.0866613i
$320$	160735.	−	278401.i	0.0877477	−	0.151984i
$321$		0			0
$322$	−170871.	−	295956.i	−0.0918391	−	0.159070i
$323$	−295146.			−0.157409
$324$		0			0
$325$	−2.34155e6			−1.22969
$326$	380416.	+	658900.i	0.198251	+	0.343381i
$327$		0			0
$328$	295791.	−	512324.i	0.151810	−	0.262942i
$329$	215831.	−	373830.i	0.109932	−	0.190408i
$330$		0			0
$331$	1.49880e6	+	2.59600e6i	0.751923	+	1.30237i	0.946890	+	0.321559i	$0.104207\pi$
$331$	1.49880e6	+	2.59600e6i	0.751923	+	1.30237i	−0.194967	+	0.980810i	$0.562460\pi$
$332$	−884592.			−0.440451
$333$		0			0
$334$	−1.60496e6			−0.787221
$335$	−904572.	−	1.56676e6i	−0.440384	−	0.762767i
$336$		0			0
$337$	−359139.	+	622047.i	−0.172261	+	0.298365i	−0.939210	−	0.343343i	$-0.888441\pi$
$337$	−359139.	+	622047.i	−0.172261	+	0.298365i	0.766949	+	0.641708i	$0.221774\pi$
$338$	448103.	−	776136.i	0.213347	−	0.369527i
$339$		0			0
$340$	483386.	+	837249.i	0.226776	+	0.392787i
$341$	−741514.			−0.345329
$342$		0			0
$343$	3.37998e6			1.55124
$344$	340208.	+	589258.i	0.155006	+	0.268478i
$345$		0			0
$346$	−189567.	+	328340.i	−0.0851282	+	0.147446i
$347$	−1.87463e6	+	3.24695e6i	−0.835779	+	1.44761i	0.0576157	+	0.998339i	$0.481650\pi$
$347$	−1.87463e6	+	3.24695e6i	−0.835779	+	1.44761i	−0.893395	+	0.449273i	$0.851683\pi$
$348$		0			0
$349$	1.16679e6	+	2.02094e6i	0.512778	+	0.888157i	0.999890	+	0.0148181i	$0.00471692\pi$
$349$	1.16679e6	+	2.02094e6i	0.512778	+	0.888157i	−0.487112	+	0.873339i	$0.661950\pi$
$350$	−2.68429e6			−1.17128
$351$		0			0
$352$	235816.			0.101442
$353$	2.17285e6	+	3.76348e6i	0.928094	+	1.60751i	0.786508	+	0.617580i	$0.211887\pi$
$353$	2.17285e6	+	3.76348e6i	0.928094	+	1.60751i	0.141586	+	0.989926i	$0.454780\pi$
$354$		0			0
$355$	2.59195e6	−	4.48938e6i	1.09158	−	1.89067i
$356$	84788.5	−	146858.i	0.0354578	−	0.0614147i
$357$		0			0
$358$	437632.	+	758001.i	0.180469	+	0.312581i
$359$	2.71281e6			1.11092			0.555461	−	0.831542i	$-0.312542\pi$
$359$	2.71281e6			1.11092			0.555461	+	0.831542i	$0.312542\pi$
$360$		0			0
$361$	−2.32913e6			−0.940645
$362$	−689027.	−	1.19343e6i	−0.276353	−	0.478658i
$363$		0			0
$364$	1.36497e6	−	2.36420e6i	0.539971	−	0.935257i
$365$	2.55583e6	−	4.42682e6i	1.00415	−	1.73924i
$366$		0			0
$367$	−1.95814e6	−	3.39160e6i	−0.758889	−	1.31443i	−0.943417	−	0.331608i	$-0.892409\pi$
$367$	−1.95814e6	−	3.39160e6i	−0.758889	−	1.31443i	0.184528	−	0.982827i	$-0.440924\pi$
$368$	−98907.1			−0.0380722
$369$		0			0
$370$	773956.			0.293908
$371$	−3.60324e6	−	6.24100e6i	−1.35912	−	2.35407i
$372$		0			0
$373$	−890322.	+	1.54208e6i	−0.331341	+	0.573899i	−0.982775	−	0.184806i	$-0.940834\pi$
$373$	−890322.	+	1.54208e6i	−0.331341	+	0.573899i	0.651434	+	0.758705i	$0.274168\pi$
$374$	−354589.	+	614167.i	−0.131083	+	0.227042i
$375$		0			0
$376$	−62466.1	−	108194.i	−0.0227863	−	0.0394671i
$377$	−609373.			−0.220815
$378$		0			0
$379$	−631740.			−0.225912			−0.112956	−	0.993600i	$-0.536032\pi$
$379$	−631740.			−0.225912			−0.112956	+	0.993600i	$0.536032\pi$
$380$	−240705.	−	416913.i	−0.0855118	−	0.148111i
$381$		0			0
$382$	269906.	−	467491.i	0.0946355	−	0.163914i
$383$	66803.0	−	115706.i	0.0232701	−	0.0403050i	−0.854156	−	0.520017i	$-0.825926\pi$
$383$	66803.0	−	115706.i	0.0232701	−	0.0403050i	0.877426	+	0.479712i	$0.159259\pi$
$384$		0			0
$385$	−1.99836e6	−	3.46126e6i	−0.687102	−	1.19010i
$386$	−909404.			−0.310662
$387$		0			0
$388$	−1.32512e6			−0.446864
$389$	2.79735e6	+	4.84515e6i	0.937286	+	1.62343i	0.770505	+	0.637434i	$0.220004\pi$
$389$	2.79735e6	+	4.84515e6i	0.937286	+	1.62343i	0.166781	+	0.985994i	$0.446663\pi$
$390$		0			0
$391$	148724.	−	257597.i	0.0491970	−	0.0852117i
$392$	1.02694e6	−	1.77872e6i	0.337544	−	0.584644i
$393$		0			0
$394$	−860875.	−	1.49108e6i	−0.279383	−	0.483905i
$395$	5.56195e6			1.79364
$396$		0			0
$397$	4.19051e6			1.33441			0.667206	−	0.744873i	$-0.267490\pi$
$397$	4.19051e6			1.33441			0.667206	+	0.744873i	$0.267490\pi$
$398$	1.43941e6	+	2.49313e6i	0.455487	+	0.788927i
$399$		0			0
$400$	−388445.	+	672807.i	−0.121389	+	0.210252i
$401$	−229109.	+	396829.i	−0.0711511	+	0.123237i	−0.899406	−	0.437114i	$-0.856001\pi$
$401$	−229109.	+	396829.i	−0.0711511	+	0.123237i	0.828255	+	0.560351i	$0.189334\pi$
$402$		0			0
$403$	1.24223e6	+	2.15160e6i	0.381012	+	0.659932i
$404$	−339332.			−0.103436
$405$		0			0
$406$	−698568.			−0.210326
$407$	283869.	+	491675.i	0.0849438	+	0.147127i
$408$		0			0
$409$	−8467.95	+	14666.9i	−0.00250305	+	0.00433541i	−0.867274	−	0.497831i	$-0.834130\pi$
$409$	−8467.95	+	14666.9i	−0.00250305	+	0.00433541i	0.864771	+	0.502166i	$0.167463\pi$
$410$	−1.45093e6	+	2.51308e6i	−0.426271	+	0.738323i
$411$		0			0
$412$	−1.05404e6	−	1.82565e6i	−0.305924	−	0.529876i
$413$	5.27002e6			1.52033
$414$		0			0
$415$	4.33914e6			1.23676
$416$	−395052.	−	684250.i	−0.111923	−	0.193857i
$417$		0			0
$418$	176570.	−	305828.i	0.0494284	−	0.0856124i
$419$	2.24889e6	−	3.89520e6i	0.625798	−	1.08391i	−0.362588	−	0.931949i	$-0.618107\pi$
$419$	2.24889e6	−	3.89520e6i	0.625798	−	1.08391i	0.988386	−	0.151964i	$-0.0485598\pi$
$420$		0			0
$421$	217248.	+	376285.i	0.0597381	+	0.103469i	0.894348	−	0.447372i	$-0.147640\pi$
$421$	217248.	+	376285.i	0.0597381	+	0.103469i	−0.834610	+	0.550842i	$0.814307\pi$
$422$	−245600.			−0.0671348
$423$		0			0
$424$	−2.08571e6			−0.563429
$425$	−1.16819e6	−	2.02336e6i	−0.313719	−	0.543377i
$426$		0			0
$427$	−4.15824e6	+	7.20229e6i	−1.10367	+	1.91162i
$428$	438372.	−	759282.i	0.115673	−	0.200352i
$429$		0			0
$430$	−1.66881e6	−	2.89046e6i	−0.435246	−	0.753868i
$431$	−5.16700e6			−1.33982			−0.669908	−	0.742444i	$-0.733666\pi$
$431$	−5.16700e6			−1.33982			−0.669908	+	0.742444i	$0.733666\pi$
$432$		0			0
$433$	3.95066e6			1.01263			0.506314	−	0.862349i	$-0.331008\pi$
$433$	3.95066e6			1.01263			0.506314	+	0.862349i	$0.331008\pi$
$434$	1.42405e6	+	2.46654e6i	0.362913	+	0.628584i
$435$		0			0
$436$	−1.28456e6	+	2.22492e6i	−0.323621	+	0.560528i
$437$	−74058.0	+	128272.i	−0.0185510	+	0.0321313i
$438$		0			0
$439$	243803.	+	422279.i	0.0603778	+	0.104577i	0.894634	−	0.446799i	$-0.147436\pi$
$439$	243803.	+	422279.i	0.0603778	+	0.104577i	−0.834257	+	0.551377i	$0.814103\pi$
$440$	−1.15673e6			−0.284840
$441$		0			0
$442$	2.37611e6			0.578511
$443$	−4027.52	−	6975.87i	−0.000975053	−	0.00168884i	0.865537	−	0.500844i	$-0.166977\pi$
$443$	−4027.52	−	6975.87i	−0.000975053	−	0.00168884i	−0.866513	+	0.499155i	$0.833644\pi$
$444$		0			0
$445$	−415909.	+	720375.i	−0.0995630	+	0.172448i
$446$	1.31605e6	−	2.27947e6i	0.313283	−	0.542622i
$447$		0			0
$448$	−452876.	−	784405.i	−0.106607	−	0.184648i
$449$	−6.79644e6			−1.59098			−0.795492	−	0.605964i	$-0.792788\pi$
$449$	−6.79644e6			−1.59098			−0.795492	+	0.605964i	$0.792788\pi$
$450$		0			0
$451$	−2.12866e6			−0.492794
$452$	−531485.	−	920559.i	−0.122362	−	0.211936i
$453$		0			0
$454$	20268.6	−	35106.3i	0.00461514	−	0.00799365i
$455$	−6.69553e6	+	1.15970e7i	−1.51620	+	2.62613i
$456$		0			0
$457$	−199059.	−	344780.i	−0.0445852	−	0.0772238i	0.842872	−	0.538115i	$-0.180863\pi$
$457$	−199059.	−	344780.i	−0.0445852	−	0.0772238i	−0.887457	+	0.460891i	$0.847530\pi$
$458$	−4.89214e6			−1.08977
$459$		0			0
$460$	485164.			0.106904
$461$	−620959.	−	1.07553e6i	−0.136085	−	0.235706i	0.789926	−	0.613202i	$-0.210119\pi$
$461$	−620959.	−	1.07553e6i	−0.136085	−	0.235706i	−0.926011	+	0.377495i	$0.876785\pi$
$462$		0			0
$463$	1.21707e6	−	2.10802e6i	0.263853	−	0.457007i	−0.703409	−	0.710785i	$-0.748340\pi$
$463$	1.21707e6	−	2.10802e6i	0.263853	−	0.457007i	0.967263	+	0.253778i	$0.0816732\pi$
$464$	−101090.	+	175093.i	−0.0217979	+	0.0377550i
$465$		0			0
$466$	2.33025e6	+	4.03611e6i	0.497093	+	0.860991i
$467$	1.08309e6			0.229811			0.114906	−	0.993376i	$-0.463343\pi$
$467$	1.08309e6			0.229811			0.114906	+	0.993376i	$0.463343\pi$
$468$		0			0
$469$	−5.09732e6			−1.07006
$470$	306411.	+	530720.i	0.0639824	+	0.110821i
$471$		0			0
$472$	762627.	−	1.32091e6i	0.157564	−	0.272909i
$473$	1.22416e6	−	2.12030e6i	0.251585	−	0.435758i
$474$		0			0
$475$	581707.	+	1.00755e6i	0.118296	+	0.204895i
$476$	2.72391e6			0.551030
$477$		0			0
$478$	5.27808e6			1.05659
$479$	−1.66906e6	−	2.89090e6i	−0.332379	−	0.575698i	0.650599	−	0.759422i	$-0.274518\pi$
$479$	−1.66906e6	−	2.89090e6i	−0.332379	−	0.575698i	−0.982978	+	0.183724i	$0.941185\pi$
$480$		0			0
$481$	951107.	−	1.64737e6i	0.187442	−	0.324659i
$482$	1.06362e6	−	1.84224e6i	0.208530	−	0.361184i
$483$		0			0
$484$	864145.	+	1.49674e6i	0.167677	+	0.290425i
$485$	6.50003e6			1.25476
$486$		0			0
$487$	−5.17210e6			−0.988198			−0.494099	−	0.869406i	$-0.664502\pi$
$487$	−5.17210e6			−0.988198			−0.494099	+	0.869406i	$0.664502\pi$
$488$	1.20348e6	+	2.08449e6i	0.228765	+	0.396233i
$489$		0			0
$490$	−5.03741e6	+	8.72504e6i	−0.947800	+	1.64164i
$491$	303947.	−	526452.i	0.0568977	−	0.0985497i	−0.836174	−	0.548465i	$-0.815212\pi$
$491$	303947.	−	526452.i	0.0568977	−	0.0985497i	0.893071	+	0.449915i	$0.148546\pi$
$492$		0			0
$493$	−304013.	−	526565.i	−0.0563345	−	0.0975742i
$494$	−1.18320e6			−0.218143
$495$		0			0
$496$	824303.			0.150447
$497$	−7.30289e6	−	1.26490e7i	−1.32618	−	2.29702i
$498$		0			0
$499$	−1.75645e6	+	3.04226e6i	−0.315780	+	0.546947i	−0.979603	−	0.200943i	$-0.935599\pi$
$499$	−1.75645e6	+	3.04226e6i	−0.315780	+	0.546947i	0.663823	+	0.747890i	$0.268933\pi$
$500$	−56677.7	+	98168.7i	−0.0101388	+	0.0175610i
$501$		0			0
$502$	−951.500	−	1648.05i	−0.000168519	−	0.000291884i
$503$	8.23500e6			1.45125			0.725627	−	0.688088i	$-0.241550\pi$
$503$	8.23500e6			1.45125			0.725627	+	0.688088i	$0.241550\pi$
$504$		0			0
$505$	1.66451e6			0.290440
$506$	177947.	+	308213.i	0.0308968	+	0.0535149i
$507$		0			0
$508$	2.39683e6	−	4.15143e6i	0.412076	−	0.713737i
$509$	−5.30292e6	+	9.18492e6i	−0.907236	+	1.57138i	−0.0893495	+	0.996000i	$0.528479\pi$
$509$	−5.30292e6	+	9.18492e6i	−0.907236	+	1.57138i	−0.817887	+	0.575379i	$0.804855\pi$
$510$		0			0
$511$	−7.20113e6	−	1.24727e7i	−1.21997	−	2.11305i
$512$	−262144.			−0.0441942
$513$		0			0
$514$	−108826.			−0.0181688
$515$	5.17032e6	+	8.95525e6i	0.859012	+	1.48785i
$516$		0			0
$517$	−224769.	+	389312.i	−0.0369837	+	0.0640576i
$518$	1.09032e6	−	1.88849e6i	0.178538	−	0.309237i
$519$		0			0
$520$	1.93783e6	+	3.35641e6i	0.314273	+	0.544336i
$521$	−1.03709e7			−1.67388			−0.836939	−	0.547297i	$-0.815657\pi$
$521$	−1.03709e7			−1.67388			−0.836939	+	0.547297i	$0.815657\pi$
$522$		0			0
$523$	8.86641e6			1.41740			0.708702	−	0.705508i	$-0.249281\pi$
$523$	8.86641e6			1.41740			0.708702	+	0.705508i	$0.249281\pi$
$524$	−2.65959e6	−	4.60654e6i	−0.423142	−	0.732903i
$525$		0			0
$526$	279965.	−	484913.i	0.0441204	−	0.0764187i
$527$	−1.23948e6	+	2.14684e6i	−0.194408	+	0.336724i
$528$		0			0
$529$	3.14354e6	+	5.44476e6i	0.488404	+	0.845941i
$530$	1.02309e7			1.58207
$531$		0			0
$532$	−1.35639e6			−0.207781
$533$	3.56606e6	+	6.17660e6i	0.543714	+	0.941740i
$534$		0			0
$535$	−2.15032e6	+	3.72447e6i	−0.324802	+	0.562574i
$536$	−737636.	+	1.27762e6i	−0.110900	+	0.192084i
$537$		0			0
$538$	−3.27684e6	−	5.67566e6i	−0.488090	−	0.845397i
$539$	−7.39041e6			−1.09571
$540$		0			0
$541$	−1.22530e6			−0.179990			−0.0899949	−	0.995942i	$-0.528685\pi$
$541$	−1.22530e6			−0.179990			−0.0899949	+	0.995942i	$0.528685\pi$
$542$	−561582.	−	972688.i	−0.0821135	−	0.142225i
$543$		0			0
$544$	394178.	−	682737.i	0.0571078	−	0.0989137i
$545$	6.30107e6	−	1.09138e7i	0.908705	−	1.57392i
$546$		0			0
$547$	−1.34505e6	−	2.32969e6i	−0.192207	−	0.332913i	0.753774	−	0.657134i	$-0.228231\pi$
$547$	−1.34505e6	−	2.32969e6i	−0.192207	−	0.332913i	−0.945981	+	0.324221i	$0.894898\pi$
$548$	2.99524e6			0.426070
$549$		0			0
$550$	2.79546e6			0.394045
$551$	151385.	+	262207.i	0.0212424	+	0.0367930i
$552$		0			0
$553$	7.83549e6	−	1.35715e7i	1.08957	−	1.88718i
$554$	1.38562e6	−	2.39996e6i	0.191809	−	0.332223i
$555$		0			0
$556$	1.68455e6	+	2.91773e6i	0.231099	+	0.400275i
$557$	5.82722e6			0.795835			0.397918	−	0.917421i	$-0.369733\pi$
$557$	5.82722e6			0.795835			0.397918	+	0.917421i	$0.369733\pi$
$558$		0			0
$559$	−8.20312e6			−1.11032
$560$	2.22147e6	+	3.84770e6i	0.299344	+	0.518479i
$561$		0			0
$562$	4.21857e6	−	7.30678e6i	0.563410	−	0.975855i
$563$	4.66117e6	−	8.07338e6i	0.619760	−	1.07346i	−0.369769	−	0.929124i	$-0.620563\pi$
$563$	4.66117e6	−	8.07338e6i	0.619760	−	1.07346i	0.989529	−	0.144333i	$-0.0461035\pi$
$564$		0			0
$565$	2.60707e6	+	4.51557e6i	0.343582	+	0.595102i
$566$	5.85076e6			0.767665
$567$		0			0
$568$	−4.22722e6			−0.549774
$569$	5.38674e6	+	9.33011e6i	0.697502	+	1.20811i	0.969330	+	0.245763i	$0.0790386\pi$
$569$	5.38674e6	+	9.33011e6i	0.697502	+	1.20811i	−0.271828	+	0.962346i	$0.587628\pi$
$570$		0			0
$571$	1.66600e6	−	2.88559e6i	0.213838	−	0.370378i	−0.739075	−	0.673623i	$-0.764737\pi$
$571$	1.66600e6	−	2.88559e6i	0.213838	−	0.370378i	0.952912	+	0.303246i	$0.0980703\pi$
$572$	−1.42150e6	+	2.46211e6i	−0.181659	+	0.314642i
$573$		0			0
$574$	4.08803e6	+	7.08068e6i	0.517886	+	0.897006i
$575$	−1.17249e6			−0.147890
$576$		0			0
$577$	3.94388e6			0.493156			0.246578	−	0.969123i	$-0.420694\pi$
$577$	3.94388e6			0.493156			0.246578	+	0.969123i	$0.420694\pi$
$578$	−1.65428e6	−	2.86530e6i	−0.205964	−	0.356739i
$579$		0			0
$580$	495872.	−	858876.i	0.0612068	−	0.106013i
$581$	6.11284e6	−	1.05877e7i	0.751281	−	1.30126i
$582$		0			0
$583$	3.75246e6	+	6.49946e6i	0.457241	+	0.791965i
$584$	−4.16832e6			−0.505742
$585$		0			0
$586$	−1.02196e7			−1.22939
$587$	−817286.	−	1.41558e6i	−0.0978991	−	0.169566i	0.812916	−	0.582381i	$-0.197879\pi$
$587$	−817286.	−	1.41558e6i	−0.0978991	−	0.169566i	−0.910815	+	0.412815i	$0.864546\pi$
$588$		0			0
$589$	617208.	−	1.06904e6i	0.0733066	−	0.126971i
$590$	−3.74087e6	+	6.47939e6i	−0.442429	+	0.766309i
$591$		0			0
$592$	−315562.	−	546570.i	−0.0370068	−	0.0640976i
$593$	−9.45304e6			−1.10391			−0.551957	−	0.833873i	$-0.686119\pi$
$593$	−9.45304e6			−1.10391			−0.551957	+	0.833873i	$0.686119\pi$
$594$		0			0
$595$	−1.33614e7			−1.54725
$596$	3.88596e6	+	6.73067e6i	0.448107	+	0.776145i
$597$		0			0
$598$	596213.	−	1.03267e6i	0.0681787	−	0.118089i
$599$	−1.03628e6	+	1.79489e6i	−0.118008	+	0.204395i	−0.918978	−	0.394309i	$-0.870984\pi$
$599$	−1.03628e6	+	1.79489e6i	−0.118008	+	0.204395i	0.800970	+	0.598704i	$0.204317\pi$
$600$		0			0
$601$	−5.30513e6	−	9.18876e6i	−0.599115	−	1.03770i	−0.992952	−	0.118517i	$-0.962186\pi$
$601$	−5.30513e6	−	9.18876e6i	−0.599115	−	1.03770i	0.393838	−	0.919180i	$-0.371147\pi$
$602$	−9.40383e6			−1.05758
$603$		0			0
$604$	3.36336e6			0.375130
$605$	−4.23884e6	−	7.34189e6i	−0.470824	−	0.815492i
$606$		0			0
$607$	−3.67761e6	+	6.36980e6i	−0.405129	+	0.701704i	−0.994336	−	0.106278i	$-0.966107\pi$
$607$	−3.67761e6	+	6.36980e6i	−0.405129	+	0.701704i	0.589207	+	0.807982i	$0.299440\pi$
$608$	−196284.	+	339973.i	−0.0215340	+	0.0372980i
$609$		0			0
$610$	−5.90338e6	−	1.02250e7i	−0.642357	−	1.11259i
$611$	1.50618e6			0.163221
$612$		0			0
$613$	1.09586e7			1.17789			0.588946	−	0.808173i	$-0.299543\pi$
$613$	1.09586e7			1.17789			0.588946	+	0.808173i	$0.299543\pi$
$614$	−4.29963e6	−	7.44718e6i	−0.460268	−	0.797207i
$615$		0			0
$616$	−1.62957e6	+	2.82249e6i	−0.173030	+	0.299696i
$617$	1.88984e6	−	3.27330e6i	0.199854	−	0.346157i	−0.748627	−	0.662991i	$-0.769287\pi$
$617$	1.88984e6	−	3.27330e6i	0.199854	−	0.346157i	0.948481	+	0.316835i	$0.102620\pi$
$618$		0			0
$619$	246460.	+	426881.i	0.0258535	+	0.0447796i	0.878663	−	0.477443i	$-0.158436\pi$
$619$	246460.	+	426881.i	0.0258535	+	0.0447796i	−0.852809	+	0.522223i	$0.825103\pi$
$620$	−4.04341e6			−0.422443
$621$		0			0
$622$	−1.47048e6			−0.152399
$623$	1.17184e6	+	2.02968e6i	0.120961	+	0.209511i
$624$		0			0
$625$	5.01978e6	−	8.69452e6i	0.514026	−	0.890319i
$626$	365487.	−	633042.i	0.0372766	−	0.0645649i
$627$		0			0
$628$	3.13797e6	+	5.43513e6i	0.317505	+	0.549934i
$629$	1.89801e6			0.191281
$630$		0			0
$631$	−1.31134e6			−0.131112			−0.0655558	−	0.997849i	$-0.520882\pi$
$631$	−1.31134e6			−0.131112			−0.0655558	+	0.997849i	$0.520882\pi$
$632$	−2.26776e6	−	3.92787e6i	−0.225842	−	0.391169i
$633$		0			0
$634$	−1.72993e6	+	2.99632e6i	−0.170925	+	0.296050i
$635$	−1.17570e7	+	2.03638e7i	−1.15708	+	2.00412i
$636$		0			0
$637$	1.23808e7	+	2.14442e7i	1.20893	+	2.09393i
$638$	727497.			0.0707587
$639$		0			0
$640$	1.28588e6			0.124094
$641$	692306.	+	1.19911e6i	0.0665508	+	0.115269i	0.897381	−	0.441257i	$-0.145467\pi$
$641$	692306.	+	1.19911e6i	0.0665508	+	0.115269i	−0.830830	+	0.556526i	$0.812134\pi$
$642$		0			0
$643$	−3.97132e6	+	6.87852e6i	−0.378797	+	0.656096i	−0.990888	−	0.134692i	$-0.956996\pi$
$643$	−3.97132e6	+	6.87852e6i	−0.378797	+	0.656096i	0.612090	+	0.790788i	$0.290329\pi$
$644$	683482.	−	1.18383e6i	0.0649400	−	0.112479i
$645$		0			0
$646$	−590292.	−	1.02242e6i	−0.0556527	−	0.0963932i
$647$	2.30348e6			0.216334			0.108167	−	0.994133i	$-0.465502\pi$
$647$	2.30348e6			0.216334			0.108167	+	0.994133i	$0.465502\pi$
$648$		0			0
$649$	−5.48826e6			−0.511474
$650$	−4.68311e6	−	8.11138e6i	−0.434761	−	0.753029i
$651$		0			0
$652$	−1.52166e6	+	2.63560e6i	−0.140185	+	0.242807i
$653$	−3.96246e6	+	6.86318e6i	−0.363649	+	0.629858i	−0.988558	−	0.150839i	$-0.951802\pi$
$653$	−3.96246e6	+	6.86318e6i	−0.363649	+	0.629858i	0.624910	+	0.780697i	$0.285136\pi$
$654$		0			0
$655$	1.30459e7	+	2.25962e7i	1.18815	+	2.05794i
$656$	2.36633e6			0.214692
$657$		0			0
$658$	1.72665e6			0.155467
$659$	4.75895e6	+	8.24275e6i	0.426872	+	0.739364i	0.996593	−	0.0824739i	$-0.0262821\pi$
$659$	4.75895e6	+	8.24275e6i	0.426872	+	0.739364i	−0.569721	+	0.821838i	$0.692949\pi$
$660$		0			0
$661$	−6.42742e6	+	1.11326e7i	−0.572181	+	0.991046i	0.424161	+	0.905587i	$0.360569\pi$
$661$	−6.42742e6	+	1.11326e7i	−0.572181	+	0.991046i	−0.996342	+	0.0854592i	$0.972764\pi$
$662$	−5.99520e6	+	1.03840e7i	−0.531690	+	0.920914i
$663$		0			0
$664$	−1.76918e6	−	3.06432e6i	−0.155723	−	0.269720i
$665$	6.65342e6			0.583433
$666$		0			0
$667$	−305131.			−0.0265565
$668$	−3.20991e6	−	5.55973e6i	−0.278325	−	0.482073i
$669$		0			0
$670$	3.61829e6	−	6.26706e6i	0.311398	−	0.539358i
$671$	4.33045e6	−	7.50055e6i	0.371301	−	0.643112i
$672$		0			0
$673$	−3.27549e6	−	5.67332e6i	−0.278766	−	0.482836i	0.692313	−	0.721598i	$-0.256592\pi$
$673$	−3.27549e6	−	5.67332e6i	−0.278766	−	0.482836i	−0.971078	+	0.238762i	$0.923259\pi$
$674$	−2.87311e6			−0.243614
$675$		0			0
$676$	3.58482e6			0.301718
$677$	4.58192e6	+	7.93613e6i	0.384217	+	0.665483i	0.991660	−	0.128880i	$-0.0411382\pi$
$677$	4.58192e6	+	7.93613e6i	0.384217	+	0.665483i	−0.607443	+	0.794363i	$0.707805\pi$
$678$		0			0
$679$	9.15702e6	−	1.58604e7i	0.762219	−	1.32020i
$680$	−1.93354e6	+	3.34899e6i	−0.160355	+	0.277742i
$681$		0			0
$682$	−1.48303e6	−	2.56868e6i	−0.122092	−	0.211470i
$683$	−8.96737e6			−0.735552			−0.367776	−	0.929914i	$-0.619881\pi$
$683$	−8.96737e6			−0.735552			−0.367776	+	0.929914i	$0.619881\pi$
$684$		0			0
$685$	−1.46924e7			−1.19637
$686$	6.75995e6	+	1.17086e7i	0.548445	+	0.949935i
$687$		0			0
$688$	−1.36083e6	+	2.35703e6i	−0.109606	+	0.189843i
$689$	1.25727e7	−	2.17765e7i	1.00897	−	1.74759i
$690$		0			0
$691$	−6.85931e6	−	1.18807e7i	−0.546494	−	0.946555i	−0.998511	−	0.0545458i	$-0.982629\pi$
$691$	−6.85931e6	−	1.18807e7i	−0.546494	−	0.946555i	0.452018	−	0.892009i	$-0.350704\pi$
$692$	−1.51654e6			−0.120389
$693$		0			0
$694$	−1.49970e7			−1.18197
$695$	−8.26315e6	−	1.43122e7i	−0.648909	−	1.12394i
$696$		0			0
$697$	−3.55818e6	+	6.16294e6i	−0.277425	+	0.480514i
$698$	−4.66716e6	+	8.08376e6i	−0.362589	+	0.628022i
$699$		0			0
$700$	−5.36859e6	−	9.29866e6i	−0.414109	−	0.717258i
$701$	−3.42100e6			−0.262941			−0.131470	−	0.991320i	$-0.541970\pi$
$701$	−3.42100e6			−0.262941			−0.131470	+	0.991320i	$0.541970\pi$
$702$		0			0
$703$	−945126.			−0.0721276
$704$	471631.	+	816889.i	0.0358650	+	0.0621200i
$705$		0			0
$706$	−8.69138e6	+	1.50539e7i	−0.656262	+	1.13668i
$707$	2.34490e6	−	4.06149e6i	0.176431	−	0.305588i
$708$		0			0
$709$	−6.61125e6	−	1.14510e7i	−0.493933	−	0.855516i	0.506043	−	0.862508i	$-0.331108\pi$
$709$	−6.61125e6	−	1.14510e7i	−0.493933	−	0.855516i	−0.999976	+	0.00699191i	$0.997774\pi$
$710$	2.07356e7			1.54373
$711$		0			0
$712$	678308.			0.0501449
$713$	622020.	+	1.07737e6i	0.0458227	+	0.0793672i
$714$		0			0
$715$	6.97281e6	−	1.20773e7i	0.510085	−	0.883493i
$716$	−1.75053e6	+	3.03201e6i	−0.127611	+	0.221028i
$717$		0			0
$718$	5.42563e6	+	9.39746e6i	0.392770	+	0.680298i
$719$	6.17969e6			0.445804			0.222902	−	0.974841i	$-0.428447\pi$
$719$	6.17969e6			0.445804			0.222902	+	0.974841i	$0.428447\pi$
$720$		0			0
$721$	2.91351e7			2.08727
$722$	−4.65826e6	−	8.06834e6i	−0.332568	−	0.576025i
$723$		0			0
$724$	2.75611e6	−	4.77372e6i	0.195411	−	0.338462i
$725$	−1.19836e6	+	2.07563e6i	−0.0846728	+	0.146658i
$726$		0			0
$727$	4.83384e6	+	8.37245e6i	0.339200	+	0.587512i	0.984282	−	0.176601i	$-0.0565103\pi$
$727$	4.83384e6	+	8.37245e6i	0.339200	+	0.587512i	−0.645082	+	0.764113i	$0.723177\pi$
$728$	1.09198e7			0.763634
$729$		0			0
$730$	2.04466e7			1.42009
$731$	−4.09249e6	−	7.08841e6i	−0.283266	−	0.490631i
$732$		0			0
$733$	−1.13678e7	+	1.96895e7i	−0.781475	+	1.35355i	0.149608	+	0.988745i	$0.452199\pi$
$733$	−1.13678e7	+	1.96895e7i	−0.781475	+	1.35355i	−0.931082	+	0.364809i	$0.881134\pi$
$734$	7.83256e6	−	1.35664e7i	0.536616	−	0.929446i
$735$		0			0
$736$	−197814.	−	342624.i	−0.0134606	−	0.0233144i
$737$	5.30842e6			0.359995
$738$		0			0
$739$	2.13860e7			1.44052			0.720260	−	0.693704i	$-0.244022\pi$
$739$	2.13860e7			1.44052			0.720260	+	0.693704i	$0.244022\pi$
$740$	1.54791e6	+	2.68106e6i	0.103912	+	0.179981i
$741$		0			0
$742$	1.44130e7	−	2.49640e7i	0.961045	−	1.66458i
$743$	4.30567e6	−	7.45765e6i	0.286134	−	0.495598i	−0.686750	−	0.726894i	$-0.740963\pi$
$743$	4.30567e6	−	7.45765e6i	0.286134	−	0.495598i	0.972884	+	0.231296i	$0.0742965\pi$
$744$		0			0
$745$	−1.90616e7	−	3.30156e7i	−1.25825	−	2.17936i
$746$	−7.12257e6			−0.468587
$747$		0			0
$748$	−2.83671e6			−0.185379
$749$	6.05860e6	+	1.04938e7i	0.394610	+	0.683484i
$750$		0			0
$751$	3.62736e6	−	6.28277e6i	0.234688	−	0.406491i	−0.724494	−	0.689281i	$-0.757927\pi$
$751$	3.62736e6	−	6.28277e6i	0.234688	−	0.406491i	0.959182	+	0.282790i	$0.0912599\pi$
$752$	249864.	−	432777.i	0.0161124	−	0.0279074i
$753$		0			0
$754$	−1.21875e6	−	2.11093e6i	−0.0780701	−	0.135221i
$755$	−1.64981e7			−1.05334
$756$		0			0
$757$	−3.04305e6			−0.193005			−0.0965027	−	0.995333i	$-0.530766\pi$
$757$	−3.04305e6			−0.193005			−0.0965027	+	0.995333i	$0.530766\pi$
$758$	−1.26348e6	−	2.18841e6i	−0.0798721	−	0.138343i
$759$		0			0
$760$	962820.	−	1.66765e6i	0.0604660	−	0.104730i
$761$	−1.33530e7	+	2.31282e7i	−0.835832	+	1.44770i	0.0575199	+	0.998344i	$0.481681\pi$
$761$	−1.33530e7	+	2.31282e7i	−0.835832	+	1.44770i	−0.893352	+	0.449359i	$0.851653\pi$
$762$		0			0
$763$	−1.77535e7	−	3.07499e7i	−1.10401	−	1.91220i
$764$	2.15925e6			0.133835
$765$		0			0
$766$	534424.			0.0329089
$767$	9.19425e6	+	1.59249e7i	0.564323	+	0.977437i
$768$		0			0
$769$	−2.45064e6	+	4.24463e6i	−0.149439	+	0.258836i	−0.931020	−	0.364968i	$-0.881080\pi$
$769$	−2.45064e6	+	4.24463e6i	−0.149439	+	0.258836i	0.781581	+	0.623803i	$0.214413\pi$
$770$	7.99343e6	−	1.38450e7i	0.485855	−	0.841525i
$771$		0			0
$772$	−1.81881e6	−	3.15027e6i	−0.109836	−	0.190241i
$773$	−5.22612e6			−0.314579			−0.157290	−	0.987552i	$-0.550276\pi$
$773$	−5.22612e6			−0.314579			−0.157290	+	0.987552i	$0.550276\pi$
$774$		0			0
$775$	9.77163e6			0.584404
$776$	−2.65024e6	−	4.59034e6i	−0.157990	−	0.273647i
$777$		0			0
$778$	−1.11894e7	+	1.93806e7i	−0.662762	+	1.14794i
$779$	1.77182e6	−	3.06888e6i	0.104610	−	0.181191i
$780$		0			0
$781$	7.60532e6	+	1.31728e7i	0.446160	+	0.772771i
$782$	1.18979e6			0.0695750
$783$		0			0
$784$	8.21554e6			0.477360
$785$	−1.53925e7	−	2.66607e7i	−0.891530	−	1.54418i
$786$		0			0
$787$	272112.	−	471312.i	0.0156607	−	0.0271251i	−0.858089	−	0.513501i	$-0.828348\pi$
$787$	272112.	−	471312.i	0.0156607	−	0.0271251i	0.873750	+	0.486376i	$0.161682\pi$
$788$	3.44350e6	−	5.96432e6i	0.197553	−	0.342172i
$789$		0			0
$790$	1.11239e7	+	1.92672e7i	0.634147	+	1.09837i
$791$	1.46910e7			0.834852
$792$		0			0
$793$	−2.90184e7			−1.63867
$794$	8.38101e6	+	1.45163e7i	0.471786	+	0.817158i
$795$		0			0
$796$	−5.75763e6	+	9.97251e6i	−0.322078	+	0.557855i
$797$	−8.60486e6	+	1.49041e7i	−0.479842	+	0.831110i	−0.999733	−	0.0231224i	$-0.992639\pi$
$797$	−8.60486e6	+	1.49041e7i	−0.479842	+	0.831110i	0.519891	+	0.854233i	$0.325973\pi$
$798$		0			0
$799$	751427.	+	1.30151e6i	0.0416409	+	0.0721241i
$800$	−3.10756e6			−0.171670
$801$		0			0
$802$	−1.83287e6			−0.100623
$803$	7.49935e6	+	1.29892e7i	0.410426	+	0.710878i
$804$		0			0
$805$	−3.35265e6	+	5.80696e6i	−0.182347	+	0.315834i
$806$	−4.96891e6	+	8.60640e6i	−0.269416	+	0.466642i
$807$		0			0
$808$	−678664.	−	1.17548e6i	−0.0365701	−	0.0633413i
$809$	9.72387e6			0.522357			0.261179	−	0.965290i	$-0.415889\pi$
$809$	9.72387e6			0.522357			0.261179	+	0.965290i	$0.415889\pi$
$810$		0			0
$811$	1.04900e6			0.0560045			0.0280023	−	0.999608i	$-0.491085\pi$
$811$	1.04900e6			0.0560045			0.0280023	+	0.999608i	$0.491085\pi$
$812$	−1.39714e6	−	2.41991e6i	−0.0743616	−	0.128798i
$813$		0			0
$814$	−1.13548e6	+	1.96670e6i	−0.0600644	+	0.104035i
$815$	7.46414e6	−	1.29283e7i	0.393628	−	0.681784i
$816$		0			0
$817$	2.03788e6	+	3.52972e6i	0.106813	+	0.185006i
$818$	−67743.6			−0.00353985
$819$		0			0
$820$	−1.16074e7			−0.602838
$821$	−2.86412e6	−	4.96080e6i	−0.148297	−	0.256858i	0.782301	−	0.622901i	$-0.214046\pi$
$821$	−2.86412e6	−	4.96080e6i	−0.148297	−	0.256858i	−0.930598	+	0.366042i	$0.880713\pi$
$822$		0			0
$823$	−1.60427e7	+	2.77868e7i	−0.825617	+	1.43001i	0.0758307	+	0.997121i	$0.475839\pi$
$823$	−1.60427e7	+	2.77868e7i	−0.825617	+	1.43001i	−0.901447	+	0.432889i	$0.857494\pi$
$824$	4.21615e6	−	7.30259e6i	0.216321	−	0.374679i
$825$		0			0
$826$	1.05400e7	+	1.82559e7i	0.537517	+	0.931006i
$827$	5.90474e6			0.300218			0.150109	−	0.988669i	$-0.452037\pi$
$827$	5.90474e6			0.300218			0.150109	+	0.988669i	$0.452037\pi$
$828$		0			0
$829$	−1.24236e6			−0.0627859			−0.0313929	−	0.999507i	$-0.509994\pi$
$829$	−1.24236e6			−0.0627859			−0.0313929	+	0.999507i	$0.509994\pi$
$830$	8.67829e6	+	1.50312e7i	0.437259	+	0.757355i
$831$		0			0
$832$	1.58021e6	−	2.73700e6i	0.0791418	−	0.137078i
$833$	−1.23535e7	+	2.13968e7i	−0.616846	+	1.06841i
$834$		0			0
$835$	1.57454e7	+	2.72718e7i	0.781516	+	1.35363i
$836$	1.41256e6			0.0699023
$837$		0			0
$838$	1.79912e7			0.885012
$839$	−9.84453e6	−	1.70512e7i	−0.482825	−	0.836278i	0.516980	−	0.855997i	$-0.327056\pi$
$839$	−9.84453e6	−	1.70512e7i	−0.482825	−	0.836278i	−0.999806	+	0.0197194i	$0.993723\pi$
$840$		0			0
$841$	9.94371e6	−	1.72230e7i	0.484795	−	0.839690i
$842$	−868994.	+	1.50514e6i	−0.0422412	+	0.0731639i
$843$		0			0
$844$	−491201.	−	850785.i	−0.0237357	−	0.0411115i
$845$	−1.75844e7			−0.847201
$846$		0			0
$847$	−2.38862e7			−1.14403
$848$	−4.17142e6	−	7.22511e6i	−0.199202	−	0.345028i
$849$		0			0
$850$	4.67276e6	−	8.09345e6i	0.221833	−	0.384226i
$851$	476247.	−	824885.i	0.0225428	−	0.0390454i
$852$		0			0
$853$	−1.43299e7	−	2.48201e7i	−0.674327	−	1.16797i	−0.976665	−	0.214767i	$-0.931101\pi$
$853$	−1.43299e7	−	2.48201e7i	−0.674327	−	1.16797i	0.302339	−	0.953201i	$-0.402233\pi$
$854$	−3.32659e7			−1.56083
$855$		0			0
$856$	3.50698e6			0.163587
$857$	3.69269e6	+	6.39594e6i	0.171748	+	0.297476i	0.939031	−	0.343832i	$-0.111725\pi$
$857$	3.69269e6	+	6.39594e6i	0.171748	+	0.297476i	−0.767283	+	0.641308i	$0.778392\pi$
$858$		0			0
$859$	−3.84152e6	+	6.65370e6i	−0.177631	+	0.307667i	−0.941069	−	0.338215i	$-0.890177\pi$
$859$	−3.84152e6	+	6.65370e6i	−0.177631	+	0.307667i	0.763437	+	0.645882i	$0.223510\pi$
$860$	6.67522e6	−	1.15618e7i	0.307765	−	0.533065i
$861$		0			0
$862$	−1.03340e7	−	1.78990e7i	−0.473696	−	0.820466i
$863$	−3.40976e7			−1.55846			−0.779232	−	0.626735i	$-0.784391\pi$
$863$	−3.40976e7			−1.55846			−0.779232	+	0.626735i	$0.784391\pi$
$864$		0			0
$865$	7.43900e6			0.338045
$866$	7.90133e6	+	1.36855e7i	0.358018	+	0.620106i
$867$		0			0
$868$	−5.69622e6	+	9.86614e6i	−0.256618	+	0.444476i
$869$	−8.15998e6	+	1.41335e7i	−0.366556	+	0.634893i
$870$		0			0
$871$	−8.89296e6	−	1.54031e7i	−0.397193	−	0.687958i
$872$	−1.02765e7			−0.457670
$873$		0			0
$874$	−592464.			−0.0262351
$875$	−783325.	−	1.35676e6i	−0.0345877	−	0.0599077i
$876$		0			0
$877$	631848.	−	1.09439e6i	0.0277405	−	0.0480479i	−0.851822	−	0.523832i	$-0.824502\pi$
$877$	631848.	−	1.09439e6i	0.0277405	−	0.0480479i	0.879562	+	0.475784i	$0.157835\pi$
$878$	−975211.	+	1.68912e6i	−0.0426936	+	0.0739474i
$879$		0			0
$880$	−2.31347e6	−	4.00704e6i	−0.100706	−	0.174428i
$881$	−1.89395e7			−0.822108			−0.411054	−	0.911611i	$-0.634839\pi$
$881$	−1.89395e7			−0.822108			−0.411054	+	0.911611i	$0.634839\pi$
$882$		0			0
$883$	3.16414e7			1.36570			0.682848	−	0.730560i	$-0.260741\pi$
$883$	3.16414e7			1.36570			0.682848	+	0.730560i	$0.260741\pi$
$884$	4.75223e6	+	8.23110e6i	0.204534	+	0.354264i
$885$		0			0
$886$	16110.1	−	27903.5i	0.000689467	−	0.00119419i
$887$	1.39213e7	−	2.41124e7i	0.594116	−	1.02904i	−0.399555	−	0.916709i	$-0.630835\pi$
$887$	1.39213e7	−	2.41124e7i	0.594116	−	1.02904i	0.993671	−	0.112330i	$-0.0358314\pi$
$888$		0			0
$889$	3.31258e7	+	5.73756e7i	1.40576	+	2.43485i
$890$	−3.32727e6			−0.140803
$891$		0			0
$892$	1.05284e7			0.443049
$893$	−374178.	−	648096.i	−0.0157018	−	0.0271963i
$894$		0			0
$895$	8.58678e6	−	1.48727e7i	0.358321	−	0.620631i
$896$	1.81151e6	−	3.13762e6i	0.0753824	−	0.130566i
$897$		0			0
$898$	−1.35929e7	−	2.35436e7i	−0.562498	−	0.974275i
$899$	2.54300e6			0.104941
$900$		0			0
$901$	2.50898e7			1.02964
$902$	−4.25733e6	−	7.37391e6i	−0.174229	−	0.301774i
$903$		0			0
$904$	2.12594e6	−	3.68224e6i	0.0865227	−	0.149862i
$905$	−1.35194e7	+	2.34163e7i	−0.548701	+	0.950377i
$906$		0			0
$907$	4.48735e6	+	7.77232e6i	0.181122	+	0.313713i	0.942263	−	0.334874i	$-0.108694\pi$
$907$	4.48735e6	+	7.77232e6i	0.181122	+	0.313713i	−0.761141	+	0.648587i	$0.775360\pi$
$908$	162149.			0.00652679
$909$		0			0
$910$	−5.35642e7			−2.14423
$911$	−4.43865e6	−	7.68797e6i	−0.177196	−	0.306913i	0.763723	−	0.645544i	$-0.223369\pi$
$911$	−4.43865e6	−	7.68797e6i	−0.177196	−	0.306913i	−0.940919	+	0.338631i	$0.890036\pi$
$912$		0			0
$913$	−6.36599e6	+	1.10262e7i	−0.252749	+	0.437773i
$914$	796235.	−	1.37912e6i	0.0315265	−	0.0546055i
$915$		0			0
$916$	−9.78428e6	−	1.69469e7i	−0.385292	−	0.667346i
$917$	7.35146e7			2.88702
$918$		0			0
$919$	−1.48072e7			−0.578343			−0.289172	−	0.957277i	$-0.593380\pi$
$919$	−1.48072e7			−0.578343			−0.289172	+	0.957277i	$0.593380\pi$
$920$	970327.	+	1.68066e6i	0.0377962	+	0.0654650i
$921$		0			0
$922$	2.48384e6	−	4.30213e6i	0.0962268	−	0.166670i
$923$	2.54818e7	−	4.41357e7i	0.984521	−	1.70524i
$924$		0			0
$925$	−3.74081e6	−	6.47927e6i	−0.143751	−	0.248984i
$926$	9.73654e6			0.373145
$927$		0			0
$928$	−808721.			−0.0308268
$929$	−7.13029e6	−	1.23500e7i	−0.271062	−	0.469492i	0.698072	−	0.716027i	$-0.254041\pi$
$929$	−7.13029e6	−	1.23500e7i	−0.271062	−	0.469492i	−0.969134	+	0.246535i	$0.920708\pi$
$930$		0			0
$931$	6.15149e6	−	1.06547e7i	0.232598	−	0.402872i
$932$	−9.32100e6	+	1.61444e7i	−0.351498	+	0.608812i
$933$		0			0
$934$	2.16618e6	+	3.75193e6i	0.0812506	+	0.140730i
$935$	1.39148e7			0.520532
$936$		0			0
$937$	−3.54753e7			−1.32001			−0.660006	−	0.751261i	$-0.729446\pi$
$937$	−3.54753e7			−1.32001			−0.660006	+	0.751261i	$0.729446\pi$
$938$	−1.01946e7	−	1.76576e7i	−0.378325	−	0.655278i
$939$		0			0
$940$	−1.22565e6	+	2.12288e6i	−0.0452424	+	0.0783621i
$941$	1.30008e7	−	2.25180e7i	0.478624	−	0.829001i	−0.521076	−	0.853511i	$-0.674469\pi$
$941$	1.30008e7	−	2.25180e7i	0.478624	−	0.829001i	0.999700	+	0.0245095i	$0.00780238\pi$
$942$		0			0
$943$	1.78563e6	+	3.09280e6i	0.0653902	+	0.113259i
$944$	6.10102e6			0.222829
$945$		0			0
$946$	9.79327e6			0.355795
$947$	1.11889e7	+	1.93797e7i	0.405426	+	0.702218i	0.994371	−	0.105955i	$-0.0337899\pi$
$947$	1.11889e7	+	1.93797e7i	0.405426	+	0.702218i	−0.588945	+	0.808173i	$0.700457\pi$
$948$		0			0
$949$	2.51267e7	−	4.35207e7i	0.905669	−	1.56866i
$950$	−2.32683e6	+	4.03019e6i	−0.0836480	+	0.144883i
$951$		0			0
$952$	5.44782e6	+	9.43590e6i	0.194819	+	0.337436i
$953$	4.62557e7			1.64981			0.824903	−	0.565275i	$-0.191230\pi$
$953$	4.62557e7			1.64981			0.824903	+	0.565275i	$0.191230\pi$
$954$		0			0
$955$	−1.05916e7			−0.375798
$956$	1.05562e7	+	1.82838e7i	0.373561	+	0.647026i
$957$		0			0
$958$	6.67625e6	−	1.15636e7i	0.235028	−	0.407080i
$959$	−2.06982e7	+	3.58503e7i	−0.726750	+	1.25877i
$960$		0			0
$961$	9.13059e6	+	1.58146e7i	0.318926	+	0.552396i
$962$	7.60886e6			0.265083
$963$		0			0
$964$	8.50895e6			0.294906
$965$	8.92170e6	+	1.54528e7i	0.308411	+	0.534183i
$966$		0			0
$967$	2.04313e7	−	3.53880e7i	0.702634	−	1.21700i	−0.264905	−	0.964274i	$-0.585341\pi$
$967$	2.04313e7	−	3.53880e7i	0.702634	−	1.21700i	0.967539	−	0.252723i	$-0.0813260\pi$
$968$	−3.45658e6	+	5.98697e6i	−0.118565	+	0.205361i
$969$		0			0
$970$	1.30001e7	+	2.25168e7i	0.443625	+	0.768381i
$971$	3.69663e7			1.25822			0.629112	−	0.777315i	$-0.283419\pi$
$971$	3.69663e7			1.25822			0.629112	+	0.777315i	$0.283419\pi$
$972$		0			0
$973$	−4.65634e7			−1.57675
$974$	−1.03442e7	−	1.79167e7i	−0.349381	−	0.605145i
$975$		0			0
$976$	−4.81393e6	+	8.33797e6i	−0.161762	+	0.280179i
$977$	−2.95829e7	+	5.12390e7i	−0.991525	+	1.71737i	−0.383253	+	0.923643i	$0.625196\pi$
$977$	−2.95829e7	+	5.12390e7i	−0.991525	+	1.71737i	−0.608272	+	0.793729i	$0.708137\pi$
$978$		0			0
$979$	−1.22036e6	−	2.11373e6i	−0.0406942	−	0.0704845i
$980$	−4.02992e7			−1.34039
$981$		0			0
$982$	2.43158e6			0.0804655
$983$	−9.12629e6	−	1.58072e7i	−0.301238	−	0.521760i	0.675178	−	0.737655i	$-0.264067\pi$
$983$	−9.12629e6	−	1.58072e7i	−0.301238	−	0.521760i	−0.976417	+	0.215894i	$0.930733\pi$
$984$		0			0
$985$	−1.68912e7	+	2.92564e7i	−0.554715	+	0.960795i
$986$	1.21605e6	−	2.10626e6i	0.0398345	−	0.0689954i
$987$		0			0
$988$	−2.36640e6	−	4.09873e6i	−0.0771252	−	0.133585i
$989$	−4.10755e6			−0.133534
$990$		0			0
$991$	4.10661e7			1.32831			0.664154	−	0.747596i	$-0.268792\pi$
$991$	4.10661e7			1.32831			0.664154	+	0.747596i	$0.268792\pi$
$992$	1.64861e6	+	2.85547e6i	0.0531910	+	0.0921294i
$993$		0			0
$994$	2.92116e7	−	5.05959e7i	0.937754	−	1.62424i
$995$	2.82426e7	−	4.89176e7i	0.904372	−	1.56642i
$996$		0			0
$997$	−9.35086e6	−	1.61962e7i	−0.297930	−	0.516029i	0.677732	−	0.735309i	$-0.262963\pi$
$997$	−9.35086e6	−	1.61962e7i	−0.297930	−	0.516029i	−0.975662	+	0.219279i	$0.929629\pi$
$998$	−1.40516e7			−0.446580
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.6.c.b.37.3		6
3.2	odd	2		18.6.c.b.13.1	yes	6
4.3	odd	2		432.6.i.b.145.3		6
9.2	odd	6		18.6.c.b.7.1	✓	6
9.4	even	3		162.6.a.i.1.1		3
9.5	odd	6		162.6.a.j.1.3		3
9.7	even	3	inner	54.6.c.b.19.3		6
12.11	even	2		144.6.i.b.49.3		6
36.7	odd	6		432.6.i.b.289.3		6
36.11	even	6		144.6.i.b.97.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.6.c.b.7.1	✓	6	9.2	odd	6
18.6.c.b.13.1	yes	6	3.2	odd	2
54.6.c.b.19.3		6	9.7	even	3	inner
54.6.c.b.37.3		6	1.1	even	1	trivial
144.6.i.b.49.3		6	12.11	even	2
144.6.i.b.97.3		6	36.11	even	6
162.6.a.i.1.1		3	9.4	even	3
162.6.a.j.1.3		3	9.5	odd	6
432.6.i.b.145.3		6	4.3	odd	2
432.6.i.b.289.3		6	36.7	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	54.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 + 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(39.2420 - 67.9691i) q^{5} +(-110.566 - 191.505i) q^{7} -64.0000 q^{8} +O(q^{10})\)	\(q+(2.00000 + 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(39.2420 - 67.9691i) q^{5} +(-110.566 - 191.505i) q^{7} -64.0000 q^{8} +313.936 q^{10} +(115.144 + 199.436i) q^{11} +(385.793 - 668.213i) q^{13} +(442.262 - 766.020i) q^{14} +(-128.000 - 221.703i) q^{16} +769.880 q^{17} -383.367 q^{19} +(627.872 + 1087.51i) q^{20} +(-460.577 + 797.743i) q^{22} +(193.178 - 334.594i) q^{23} +(-1517.37 - 2628.15i) q^{25} +3086.34 q^{26} +3538.10 q^{28} +(-394.883 - 683.958i) q^{29} +(-1609.97 + 2788.54i) q^{31} +(512.000 - 886.810i) q^{32} +(1539.76 + 2666.94i) q^{34} -17355.2 q^{35} +2465.33 q^{37} +(-766.733 - 1328.02i) q^{38} +(-2511.49 + 4350.02i) q^{40} +(-4621.73 + 8005.07i) q^{41} +(-5315.76 - 9207.16i) q^{43} -3684.62 q^{44} +1545.42 q^{46} +(976.032 + 1690.54i) q^{47} +(-16046.0 + 27792.4i) q^{49} +(6069.46 - 10512.6i) q^{50} +(6172.68 + 10691.4i) q^{52} +32589.2 q^{53} +18074.0 q^{55} +(7076.19 + 12256.3i) q^{56} +(1579.53 - 2735.83i) q^{58} +(-11916.1 + 20639.2i) q^{59} +(-18804.4 - 32570.2i) q^{61} -12879.7 q^{62} +4096.00 q^{64} +(-30278.5 - 52444.0i) q^{65} +(11525.6 - 19962.9i) q^{67} +(-6159.04 + 10667.8i) q^{68} +(-34710.5 - 60120.3i) q^{70} +66050.4 q^{71} +65130.0 q^{73} +(4930.66 + 8540.16i) q^{74} +(3066.93 - 5312.08i) q^{76} +(25462.0 - 44101.5i) q^{77} +(35433.7 + 61373.0i) q^{79} -20091.9 q^{80} -36973.8 q^{82} +(27643.5 + 47880.0i) q^{83} +(30211.6 - 52328.0i) q^{85} +(21263.0 - 36828.6i) q^{86} +(-7369.24 - 12763.9i) q^{88} -10598.6 q^{89} -170622. q^{91} +(3090.85 + 5353.50i) q^{92} +(-3904.13 + 6762.15i) q^{94} +(-15044.1 + 26057.1i) q^{95} +(41409.9 + 71724.1i) q^{97} -128368. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} - 48 q^{4} + 54 q^{5} - 132 q^{7} - 384 q^{8}+O(q^{10}) \) 6 * q + 12 * q^2 - 48 * q^4 + 54 * q^5 - 132 * q^7 - 384 * q^8	\( 6 q + 12 q^{2} - 48 q^{4} + 54 q^{5} - 132 q^{7} - 384 q^{8} + 432 q^{10} + 315 q^{11} - 744 q^{13} + 528 q^{14} - 768 q^{16} - 2898 q^{17} + 2262 q^{19} + 864 q^{20} - 1260 q^{22} + 3168 q^{23} - 2883 q^{25} - 5952 q^{26} + 4224 q^{28} + 5148 q^{29} - 8610 q^{31} + 3072 q^{32} - 5796 q^{34} - 2700 q^{35} + 39936 q^{37} + 4524 q^{38} - 3456 q^{40} - 5049 q^{41} - 31389 q^{43} - 10080 q^{44} + 25344 q^{46} - 12924 q^{47} - 52857 q^{49} + 11532 q^{50} - 11904 q^{52} + 96048 q^{53} + 126252 q^{55} + 8448 q^{56} - 20592 q^{58} - 62955 q^{59} - 75966 q^{61} - 68880 q^{62} + 24576 q^{64} - 108702 q^{65} - 32991 q^{67} + 23184 q^{68} - 5400 q^{70} + 129672 q^{71} - 8466 q^{73} + 79872 q^{74} - 18096 q^{76} - 88740 q^{77} + 89202 q^{79} - 27648 q^{80} - 40392 q^{82} - 32634 q^{83} + 71388 q^{85} + 125556 q^{86} - 20160 q^{88} - 66132 q^{89} - 301836 q^{91} + 50688 q^{92} + 51696 q^{94} + 82944 q^{95} + 46245 q^{97} - 422856 q^{98}+O(q^{100}) \) 6 * q + 12 * q^2 - 48 * q^4 + 54 * q^5 - 132 * q^7 - 384 * q^8 + 432 * q^10 + 315 * q^11 - 744 * q^13 + 528 * q^14 - 768 * q^16 - 2898 * q^17 + 2262 * q^19 + 864 * q^20 - 1260 * q^22 + 3168 * q^23 - 2883 * q^25 - 5952 * q^26 + 4224 * q^28 + 5148 * q^29 - 8610 * q^31 + 3072 * q^32 - 5796 * q^34 - 2700 * q^35 + 39936 * q^37 + 4524 * q^38 - 3456 * q^40 - 5049 * q^41 - 31389 * q^43 - 10080 * q^44 + 25344 * q^46 - 12924 * q^47 - 52857 * q^49 + 11532 * q^50 - 11904 * q^52 + 96048 * q^53 + 126252 * q^55 + 8448 * q^56 - 20592 * q^58 - 62955 * q^59 - 75966 * q^61 - 68880 * q^62 + 24576 * q^64 - 108702 * q^65 - 32991 * q^67 + 23184 * q^68 - 5400 * q^70 + 129672 * q^71 - 8466 * q^73 + 79872 * q^74 - 18096 * q^76 - 88740 * q^77 + 89202 * q^79 - 27648 * q^80 - 40392 * q^82 - 32634 * q^83 + 71388 * q^85 + 125556 * q^86 - 20160 * q^88 - 66132 * q^89 - 301836 * q^91 + 50688 * q^92 + 51696 * q^94 + 82944 * q^95 + 46245 * q^97 - 422856 * q^98

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