LMFDB - Embedded newform 162.6.c.c.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,6,Mod(55,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, 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("162.55");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	162.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:	$25.9821788097$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	162.109
Dual form			162.6.c.c.55.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(-12.0000 + 20.7846i) q^{5} +(-38.5000 - 66.6840i) q^{7} +64.0000 q^{8} +O(q^{10})$	\(q+(-2.00000 - 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(-12.0000 + 20.7846i) q^{5} +(-38.5000 - 66.6840i) q^{7} +64.0000 q^{8} +96.0000 q^{10} +(-204.000 - 353.338i) q^{11} +(-44.5000 + 77.0763i) q^{13} +(-154.000 + 266.736i) q^{14} +(-128.000 - 221.703i) q^{16} +2088.00 q^{17} -2617.00 q^{19} +(-192.000 - 332.554i) q^{20} +(-816.000 + 1413.35i) q^{22} +(-876.000 + 1517.28i) q^{23} +(1274.50 + 2207.50i) q^{25} +356.000 q^{26} +1232.00 q^{28} +(3648.00 + 6318.52i) q^{29} +(-1174.00 + 2033.43i) q^{31} +(-512.000 + 886.810i) q^{32} +(-4176.00 - 7233.04i) q^{34} +1848.00 q^{35} -4993.00 q^{37} +(5234.00 + 9065.55i) q^{38} +(-768.000 + 1330.22i) q^{40} +(3264.00 - 5653.41i) q^{41} +(3116.00 + 5397.07i) q^{43} +6528.00 q^{44} +7008.00 q^{46} +(14916.0 + 25835.3i) q^{47} +(5439.00 - 9420.62i) q^{49} +(5098.00 - 8830.00i) q^{50} +(-712.000 - 1233.22i) q^{52} +22608.0 q^{53} +9792.00 q^{55} +(-2464.00 - 4267.77i) q^{56} +(14592.0 - 25274.1i) q^{58} +(-9804.00 + 16981.0i) q^{59} +(11022.5 + 19091.5i) q^{61} +9392.00 q^{62} +4096.00 q^{64} +(-1068.00 - 1849.83i) q^{65} +(-24065.5 + 41682.7i) q^{67} +(-16704.0 + 28932.2i) q^{68} +(-3696.00 - 6401.66i) q^{70} +51120.0 q^{71} +30737.0 q^{73} +(9986.00 + 17296.3i) q^{74} +(20936.0 - 36262.2i) q^{76} +(-15708.0 + 27207.1i) q^{77} +(-19109.5 - 33098.6i) q^{79} +6144.00 q^{80} -26112.0 q^{82} +(-4056.00 - 7025.20i) q^{83} +(-25056.0 + 43398.3i) q^{85} +(12464.0 - 21588.3i) q^{86} +(-13056.0 - 22613.7i) q^{88} -44280.0 q^{89} +6853.00 q^{91} +(-14016.0 - 24276.4i) q^{92} +(59664.0 - 103341. i) q^{94} +(31404.0 - 54393.3i) q^{95} +(68325.5 + 118343. i) q^{97} -43512.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 16 q^{4} - 24 q^{5} - 77 q^{7} + 128 q^{8}+O(q^{10}) $ 2 * q - 4 * q^2 - 16 * q^4 - 24 * q^5 - 77 * q^7 + 128 * q^8	$ 2 q - 4 q^{2} - 16 q^{4} - 24 q^{5} - 77 q^{7} + 128 q^{8} + 192 q^{10} - 408 q^{11} - 89 q^{13} - 308 q^{14} - 256 q^{16} + 4176 q^{17} - 5234 q^{19} - 384 q^{20} - 1632 q^{22} - 1752 q^{23} + 2549 q^{25} + 712 q^{26} + 2464 q^{28} + 7296 q^{29} - 2348 q^{31} - 1024 q^{32} - 8352 q^{34} + 3696 q^{35} - 9986 q^{37} + 10468 q^{38} - 1536 q^{40} + 6528 q^{41} + 6232 q^{43} + 13056 q^{44} + 14016 q^{46} + 29832 q^{47} + 10878 q^{49} + 10196 q^{50} - 1424 q^{52} + 45216 q^{53} + 19584 q^{55} - 4928 q^{56} + 29184 q^{58} - 19608 q^{59} + 22045 q^{61} + 18784 q^{62} + 8192 q^{64} - 2136 q^{65} - 48131 q^{67} - 33408 q^{68} - 7392 q^{70} + 102240 q^{71} + 61474 q^{73} + 19972 q^{74} + 41872 q^{76} - 31416 q^{77} - 38219 q^{79} + 12288 q^{80} - 52224 q^{82} - 8112 q^{83} - 50112 q^{85} + 24928 q^{86} - 26112 q^{88} - 88560 q^{89} + 13706 q^{91} - 28032 q^{92} + 119328 q^{94} + 62808 q^{95} + 136651 q^{97} - 87024 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 - 16 * q^4 - 24 * q^5 - 77 * q^7 + 128 * q^8 + 192 * q^10 - 408 * q^11 - 89 * q^13 - 308 * q^14 - 256 * q^16 + 4176 * q^17 - 5234 * q^19 - 384 * q^20 - 1632 * q^22 - 1752 * q^23 + 2549 * q^25 + 712 * q^26 + 2464 * q^28 + 7296 * q^29 - 2348 * q^31 - 1024 * q^32 - 8352 * q^34 + 3696 * q^35 - 9986 * q^37 + 10468 * q^38 - 1536 * q^40 + 6528 * q^41 + 6232 * q^43 + 13056 * q^44 + 14016 * q^46 + 29832 * q^47 + 10878 * q^49 + 10196 * q^50 - 1424 * q^52 + 45216 * q^53 + 19584 * q^55 - 4928 * q^56 + 29184 * q^58 - 19608 * q^59 + 22045 * q^61 + 18784 * q^62 + 8192 * q^64 - 2136 * q^65 - 48131 * q^67 - 33408 * q^68 - 7392 * q^70 + 102240 * q^71 + 61474 * q^73 + 19972 * q^74 + 41872 * q^76 - 31416 * q^77 - 38219 * q^79 + 12288 * q^80 - 52224 * q^82 - 8112 * q^83 - 50112 * q^85 + 24928 * q^86 - 26112 * q^88 - 88560 * q^89 + 13706 * q^91 - 28032 * q^92 + 119328 * q^94 + 62808 * q^95 + 136651 * q^97 - 87024 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

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
$5$	−12.0000	+	20.7846i	−0.214663	+	0.371806i	−0.953168	−	0.302441i	$-0.902198\pi$
$5$	−12.0000	+	20.7846i	−0.214663	+	0.371806i	0.738506	+	0.674247i	$0.235532\pi$
$6$		0			0
$7$	−38.5000	−	66.6840i	−0.296972	−	0.514371i	0.678470	−	0.734628i	$-0.262644\pi$
$7$	−38.5000	−	66.6840i	−0.296972	−	0.514371i	−0.975442	+	0.220258i	$0.929310\pi$
$8$	64.0000			0.353553
$9$		0			0
$10$	96.0000			0.303579
$11$	−204.000	−	353.338i	−0.508333	−	0.880459i	−0.999953	−	0.00964920i	$-0.996929\pi$
$11$	−204.000	−	353.338i	−0.508333	−	0.880459i	0.491620	−	0.870810i	$-0.336405\pi$
$12$		0			0
$13$	−44.5000	+	77.0763i	−0.0730301	+	0.126492i	−0.900228	−	0.435419i	$-0.856600\pi$
$13$	−44.5000	+	77.0763i	−0.0730301	+	0.126492i	0.827198	+	0.561911i	$0.189934\pi$
$14$	−154.000	+	266.736i	−0.209991	+	0.363715i
$15$		0			0
$16$	−128.000	−	221.703i	−0.125000	−	0.216506i
$17$	2088.00			1.75230			0.876149	−	0.482040i	$-0.160104\pi$
$17$	2088.00			1.75230			0.876149	+	0.482040i	$0.160104\pi$
$18$		0			0
$19$	−2617.00			−1.66311			−0.831553	−	0.555446i	$-0.812548\pi$
$19$	−2617.00			−1.66311			−0.831553	+	0.555446i	$0.812548\pi$
$20$	−192.000	−	332.554i	−0.107331	−	0.185903i
$21$		0			0
$22$	−816.000	+	1413.35i	−0.359446	+	0.622578i
$23$	−876.000	+	1517.28i	−0.345290	+	0.598061i	−0.985406	−	0.170218i	$-0.945553\pi$
$23$	−876.000	+	1517.28i	−0.345290	+	0.598061i	0.640116	+	0.768278i	$0.278886\pi$
$24$		0			0
$25$	1274.50	+	2207.50i	0.407840	+	0.706400i
$26$	356.000			0.103280
$27$		0			0
$28$	1232.00			0.296972
$29$	3648.00	+	6318.52i	0.805489	+	1.39515i	0.915960	+	0.401269i	$0.131431\pi$
$29$	3648.00	+	6318.52i	0.805489	+	1.39515i	−0.110471	+	0.993879i	$0.535236\pi$
$30$		0			0
$31$	−1174.00	+	2033.43i	−0.219414	+	0.380036i	−0.954629	−	0.297798i	$-0.903748\pi$
$31$	−1174.00	+	2033.43i	−0.219414	+	0.380036i	0.735215	+	0.677834i	$0.237081\pi$
$32$	−512.000	+	886.810i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−4176.00	−	7233.04i	−0.619531	−	1.07306i
$35$	1848.00			0.254995
$36$		0			0
$37$	−4993.00			−0.599594			−0.299797	−	0.954003i	$-0.596919\pi$
$37$	−4993.00			−0.599594			−0.299797	+	0.954003i	$0.596919\pi$
$38$	5234.00	+	9065.55i	0.587996	+	1.01844i
$39$		0			0
$40$	−768.000	+	1330.22i	−0.0758947	+	0.131453i
$41$	3264.00	−	5653.41i	0.303243	−	0.525232i	−0.673626	−	0.739073i	$-0.735264\pi$
$41$	3264.00	−	5653.41i	0.303243	−	0.525232i	0.976869	+	0.213841i	$0.0685973\pi$
$42$		0			0
$43$	3116.00	+	5397.07i	0.256996	+	0.445130i	0.965436	−	0.260641i	$-0.0839340\pi$
$43$	3116.00	+	5397.07i	0.256996	+	0.445130i	−0.708440	+	0.705771i	$0.750601\pi$
$44$	6528.00			0.508333
$45$		0			0
$46$	7008.00			0.488314
$47$	14916.0	+	25835.3i	0.984935	+	1.70596i	0.642223	+	0.766518i	$0.278012\pi$
$47$	14916.0	+	25835.3i	0.984935	+	1.70596i	0.342712	+	0.939441i	$0.388654\pi$
$48$		0			0
$49$	5439.00	−	9420.62i	0.323615	−	0.560518i
$50$	5098.00	−	8830.00i	0.288386	−	0.499500i
$51$		0			0
$52$	−712.000	−	1233.22i	−0.0365150	−	0.0632459i
$53$	22608.0			1.10553			0.552767	−	0.833336i	$-0.313572\pi$
$53$	22608.0			1.10553			0.552767	+	0.833336i	$0.313572\pi$
$54$		0			0
$55$	9792.00			0.436480
$56$	−2464.00	−	4267.77i	−0.104995	−	0.181858i
$57$		0			0
$58$	14592.0	−	25274.1i	0.569567	−	0.986519i
$59$	−9804.00	+	16981.0i	−0.366668	+	0.635088i	−0.989042	−	0.147632i	$-0.952835\pi$
$59$	−9804.00	+	16981.0i	−0.366668	+	0.635088i	0.622374	+	0.782720i	$0.286168\pi$
$60$		0			0
$61$	11022.5	+	19091.5i	0.379276	+	0.656926i	0.990957	−	0.134179i	$-0.0428397\pi$
$61$	11022.5	+	19091.5i	0.379276	+	0.656926i	−0.611681	+	0.791105i	$0.709506\pi$
$62$	9392.00			0.310298
$63$		0			0
$64$	4096.00			0.125000
$65$	−1068.00	−	1849.83i	−0.0313536	−	0.0543061i
$66$		0			0
$67$	−24065.5	+	41682.7i	−0.654950	+	1.13441i	0.326957	+	0.945039i	$0.393977\pi$
$67$	−24065.5	+	41682.7i	−0.654950	+	1.13441i	−0.981906	+	0.189367i	$0.939357\pi$
$68$	−16704.0	+	28932.2i	−0.438075	+	0.758768i
$69$		0			0
$70$	−3696.00	−	6401.66i	−0.0901544	−	0.156152i
$71$	51120.0			1.20350			0.601748	−	0.798686i	$-0.294471\pi$
$71$	51120.0			1.20350			0.601748	+	0.798686i	$0.294471\pi$
$72$		0			0
$73$	30737.0			0.675079			0.337539	−	0.941311i	$-0.390405\pi$
$73$	30737.0			0.675079			0.337539	+	0.941311i	$0.390405\pi$
$74$	9986.00	+	17296.3i	0.211988	+	0.367175i
$75$		0			0
$76$	20936.0	−	36262.2i	0.415776	−	0.720146i
$77$	−15708.0	+	27207.1i	−0.301922	+	0.522943i
$78$		0			0
$79$	−19109.5	−	33098.6i	−0.344494	−	0.596681i	0.640768	−	0.767735i	$-0.278616\pi$
$79$	−19109.5	−	33098.6i	−0.344494	−	0.596681i	−0.985262	+	0.171054i	$0.945283\pi$
$80$	6144.00			0.107331
$81$		0			0
$82$	−26112.0			−0.428850
$83$	−4056.00	−	7025.20i	−0.0646253	−	0.111934i	0.831902	−	0.554922i	$-0.187252\pi$
$83$	−4056.00	−	7025.20i	−0.0646253	−	0.111934i	−0.896528	+	0.442988i	$0.853919\pi$
$84$		0			0
$85$	−25056.0	+	43398.3i	−0.376153	+	0.651516i
$86$	12464.0	−	21588.3i	0.181724	−	0.314754i
$87$		0			0
$88$	−13056.0	−	22613.7i	−0.179723	−	0.311289i
$89$	−44280.0			−0.592560			−0.296280	−	0.955101i	$-0.595746\pi$
$89$	−44280.0			−0.592560			−0.296280	+	0.955101i	$0.595746\pi$
$90$		0			0
$91$	6853.00			0.0867516
$92$	−14016.0	−	24276.4i	−0.172645	−	0.299030i
$93$		0			0
$94$	59664.0	−	103341.i	0.696454	−	1.20629i
$95$	31404.0	−	54393.3i	0.357006	−	0.618353i
$96$		0			0
$97$	68325.5	+	118343.i	0.737316	+	1.27707i	0.953700	+	0.300760i	$0.0972403\pi$
$97$	68325.5	+	118343.i	0.737316	+	1.27707i	−0.216384	+	0.976308i	$0.569426\pi$
$98$	−43512.0			−0.457661
$99$		0			0
$100$	−40784.0			−0.407840
$101$	−11904.0	−	20618.3i	−0.116115	−	0.201118i	0.802110	−	0.597177i	$-0.203711\pi$
$101$	−11904.0	−	20618.3i	−0.116115	−	0.201118i	−0.918225	+	0.396059i	$0.870378\pi$
$102$		0			0
$103$	−86984.5	+	150662.i	−0.807884	+	1.39930i	0.106443	+	0.994319i	$0.466054\pi$
$103$	−86984.5	+	150662.i	−0.807884	+	1.39930i	−0.914327	+	0.404977i	$0.867280\pi$
$104$	−2848.00	+	4932.88i	−0.0258200	+	0.0447216i
$105$		0			0
$106$	−45216.0	−	78316.4i	−0.390866	−	0.676999i
$107$	−188856.			−1.59467			−0.797336	−	0.603536i	$-0.793758\pi$
$107$	−188856.			−1.59467			−0.797336	+	0.603536i	$0.793758\pi$
$108$		0			0
$109$	−208654.			−1.68213			−0.841067	−	0.540931i	$-0.818072\pi$
$109$	−208654.			−1.68213			−0.841067	+	0.540931i	$0.818072\pi$
$110$	−19584.0	−	33920.5i	−0.154319	−	0.267289i
$111$		0			0
$112$	−9856.00	+	17071.1i	−0.0742430	+	0.128593i
$113$	77820.0	−	134788.i	0.573317	−	0.993015i	−0.422905	−	0.906174i	$-0.638990\pi$
$113$	77820.0	−	134788.i	0.573317	−	0.993015i	0.996222	−	0.0868407i	$-0.0276771\pi$
$114$		0			0
$115$	−21024.0	−	36414.6i	−0.148242	−	0.256762i
$116$	−116736.			−0.805489
$117$		0			0
$118$	78432.0			0.518547
$119$	−80388.0	−	139236.i	−0.520384	−	0.901331i
$120$		0			0
$121$	−2706.50	+	4687.80i	−0.0168052	+	0.0291075i
$122$	44090.0	−	76366.1i	0.268189	−	0.464517i
$123$		0			0
$124$	−18784.0	−	32534.8i	−0.109707	−	0.190018i
$125$	−136176.			−0.779517
$126$		0			0
$127$	111332.			0.612507			0.306253	−	0.951950i	$-0.400925\pi$
$127$	111332.			0.612507			0.306253	+	0.951950i	$0.400925\pi$
$128$	−8192.00	−	14189.0i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−4272.00	+	7399.32i	−0.0221704	+	0.0384002i
$131$	−66576.0	+	115313.i	−0.338953	+	0.587084i	−0.984236	−	0.176860i	$-0.943406\pi$
$131$	−66576.0	+	115313.i	−0.338953	+	0.587084i	0.645283	+	0.763944i	$0.276739\pi$
$132$		0			0
$133$	100754.	+	174512.i	0.493896	+	0.855453i
$134$	192524.			0.926239
$135$		0			0
$136$	133632.			0.619531
$137$	−142908.	−	247524.i	−0.650512	−	1.12672i	−0.982999	−	0.183612i	$-0.941221\pi$
$137$	−142908.	−	247524.i	−0.650512	−	1.12672i	0.332487	−	0.943108i	$-0.392112\pi$
$138$		0			0
$139$	−26504.5	+	45907.1i	−0.116354	+	0.201532i	−0.918320	−	0.395838i	$-0.870454\pi$
$139$	−26504.5	+	45907.1i	−0.116354	+	0.201532i	0.801966	+	0.597370i	$0.203787\pi$
$140$	−14784.0	+	25606.6i	−0.0637488	+	0.110416i
$141$		0			0
$142$	−102240.	−	177085.i	−0.425500	−	0.736988i
$143$	36312.0			0.148494
$144$		0			0
$145$	−175104.			−0.691634
$146$	−61474.0	−	106476.i	−0.238676	−	0.413400i
$147$		0			0
$148$	39944.0	−	69185.0i	0.149898	−	0.259632i
$149$	−105240.	+	182281.i	−0.388343	+	0.672629i	−0.992227	−	0.124443i	$-0.960286\pi$
$149$	−105240.	+	182281.i	−0.388343	+	0.672629i	0.603884	+	0.797072i	$0.293619\pi$
$150$		0			0
$151$	15360.5	+	26605.2i	0.0548230	+	0.0949563i	0.892134	−	0.451770i	$-0.149207\pi$
$151$	15360.5	+	26605.2i	0.0548230	+	0.0949563i	−0.837311	+	0.546726i	$0.815874\pi$
$152$	−167488.			−0.587996
$153$		0			0
$154$	125664.			0.426982
$155$	−28176.0	−	48802.3i	−0.0941998	−	0.163159i
$156$		0			0
$157$	226769.	−	392775.i	0.734234	−	1.27173i	−0.220824	−	0.975314i	$-0.570875\pi$
$157$	226769.	−	392775.i	0.734234	−	1.27173i	0.955059	−	0.296417i	$-0.0957919\pi$
$158$	−76438.0	+	132394.i	−0.243594	+	0.421917i
$159$		0			0
$160$	−12288.0	−	21283.4i	−0.0379473	−	0.0657267i
$161$	134904.			0.410166
$162$		0			0
$163$	175241.			0.516615			0.258307	−	0.966063i	$-0.416835\pi$
$163$	175241.			0.516615			0.258307	+	0.966063i	$0.416835\pi$
$164$	52224.0	+	90454.6i	0.151621	+	0.262616i
$165$		0			0
$166$	−16224.0	+	28100.8i	−0.0456970	+	0.0791495i
$167$	28356.0	−	49114.0i	0.0786781	−	0.136274i	−0.824002	−	0.566587i	$-0.808263\pi$
$167$	28356.0	−	49114.0i	0.0786781	−	0.136274i	0.902680	+	0.430313i	$0.141597\pi$
$168$		0			0
$169$	181686.	+	314689.i	0.489333	+	0.847550i
$170$	200448.			0.531961
$171$		0			0
$172$	−99712.0			−0.256996
$173$	124608.	+	215827.i	0.316542	+	0.548266i	0.979764	−	0.200156i	$-0.0641450\pi$
$173$	124608.	+	215827.i	0.316542	+	0.548266i	−0.663222	+	0.748422i	$0.730812\pi$
$174$		0			0
$175$	98136.5	−	169977.i	0.242234	−	0.419562i
$176$	−52224.0	+	90454.6i	−0.127083	+	0.220115i
$177$		0			0
$178$	88560.0	+	153390.i	0.209502	+	0.362868i
$179$	177552.			0.414184			0.207092	−	0.978322i	$-0.433600\pi$
$179$	177552.			0.414184			0.207092	+	0.978322i	$0.433600\pi$
$180$		0			0
$181$	453053.			1.02790			0.513952	−	0.857819i	$-0.328181\pi$
$181$	453053.			1.02790			0.513952	+	0.857819i	$0.328181\pi$
$182$	−13706.0	−	23739.5i	−0.0306713	−	0.0531243i
$183$		0			0
$184$	−56064.0	+	97105.7i	−0.122079	+	0.211446i
$185$	59916.0	−	103778.i	0.128710	−	0.222933i
$186$		0			0
$187$	−425952.	−	737771.i	−0.890752	−	1.54283i
$188$	−477312.			−0.984935
$189$		0			0
$190$	−251232.			−0.504883
$191$	294204.	+	509576.i	0.583533	+	1.01071i	0.995057	+	0.0993092i	$0.0316633\pi$
$191$	294204.	+	509576.i	0.583533	+	1.01071i	−0.411524	+	0.911399i	$0.635003\pi$
$192$		0			0
$193$	−293220.	+	507871.i	−0.566630	+	0.981432i	0.430266	+	0.902702i	$0.358420\pi$
$193$	−293220.	+	507871.i	−0.566630	+	0.981432i	−0.996896	+	0.0787298i	$0.974914\pi$
$194$	273302.	−	473373.i	0.521361	−	0.903024i
$195$		0			0
$196$	87024.0	+	150730.i	0.161808	+	0.280259i
$197$	374328.			0.687206			0.343603	−	0.939115i	$-0.388353\pi$
$197$	374328.			0.687206			0.343603	+	0.939115i	$0.388353\pi$
$198$		0			0
$199$	−303415.			−0.543131			−0.271565	−	0.962420i	$-0.587541\pi$
$199$	−303415.			−0.543131			−0.271565	+	0.962420i	$0.587541\pi$
$200$	81568.0	+	141280.i	0.144193	+	0.249750i
$201$		0			0
$202$	−47616.0	+	82473.3i	−0.0821059	+	0.142212i
$203$	280896.	−	486526.i	0.478416	−	0.828640i
$204$		0			0
$205$	78336.0	+	135682.i	0.130190	+	0.225495i
$206$	695876.			1.14252
$207$		0			0
$208$	22784.0			0.0365150
$209$	533868.	+	924687.i	0.845412	+	1.46430i
$210$		0			0
$211$	−114722.	+	198705.i	−0.177395	+	0.307258i	−0.940988	−	0.338441i	$-0.890100\pi$
$211$	−114722.	+	198705.i	−0.177395	+	0.307258i	0.763592	+	0.645699i	$0.223434\pi$
$212$	−180864.	+	313266.i	−0.276384	+	0.478711i
$213$		0			0
$214$	377712.	+	654216.i	0.563802	+	0.976533i
$215$	−149568.			−0.220670
$216$		0			0
$217$	180796.			0.260639
$218$	417308.	+	722799.i	0.594724	+	1.03009i
$219$		0			0
$220$	−78336.0	+	135682.i	−0.109120	+	0.189002i
$221$	−92916.0	+	160935.i	−0.127970	+	0.221651i
$222$		0			0
$223$	−533338.	−	923769.i	−0.718192	−	1.24394i	−0.961716	−	0.274050i	$-0.911637\pi$
$223$	−533338.	−	923769.i	−0.718192	−	1.24394i	0.243524	−	0.969895i	$-0.421697\pi$
$224$	78848.0			0.104995
$225$		0			0
$226$	−622560.			−0.810793
$227$	−133944.	−	231998.i	−0.172528	−	0.298827i	0.766775	−	0.641916i	$-0.221860\pi$
$227$	−133944.	−	231998.i	−0.172528	−	0.298827i	−0.939303	+	0.343089i	$0.888527\pi$
$228$		0			0
$229$	509603.	−	882658.i	0.642160	−	1.11225i	−0.342790	−	0.939412i	$-0.611372\pi$
$229$	509603.	−	882658.i	0.642160	−	1.11225i	0.984950	−	0.172842i	$-0.0552949\pi$
$230$	−84096.0	+	145659.i	−0.104823	+	0.181558i
$231$		0			0
$232$	233472.	+	404385.i	0.284784	+	0.493260i
$233$	187488.			0.226247			0.113124	−	0.993581i	$-0.463914\pi$
$233$	187488.			0.226247			0.113124	+	0.993581i	$0.463914\pi$
$234$		0			0
$235$	−715968.			−0.845715
$236$	−156864.	−	271696.i	−0.183334	−	0.317544i
$237$		0			0
$238$	−321552.	+	556944.i	−0.367967	+	0.637337i
$239$	−520536.	+	901595.i	−0.589462	+	1.02098i	0.404841	+	0.914387i	$0.367327\pi$
$239$	−520536.	+	901595.i	−0.589462	+	1.02098i	−0.994303	+	0.106591i	$0.966006\pi$
$240$		0			0
$241$	−371792.	−	643963.i	−0.412342	−	0.714198i	0.582803	−	0.812613i	$-0.301956\pi$
$241$	−371792.	−	643963.i	−0.412342	−	0.714198i	−0.995145	+	0.0984155i	$0.968623\pi$
$242$	21652.0			0.0237662
$243$		0			0
$244$	−352720.			−0.379276
$245$	130536.	+	226095.i	0.138936	+	0.240644i
$246$		0			0
$247$	116456.	−	201709.i	0.121457	−	0.210369i
$248$	−75136.0	+	130139.i	−0.0775745	+	0.134363i
$249$		0			0
$250$	272352.	+	471728.i	0.275601	+	0.477355i
$251$	1.66608e6			1.66921			0.834606	−	0.550847i	$-0.185695\pi$
$251$	1.66608e6			1.66921			0.834606	+	0.550847i	$0.185695\pi$
$252$		0			0
$253$	714816.			0.702090
$254$	−222664.	−	385665.i	−0.216554	−	0.375082i
$255$		0			0
$256$	−32768.0	+	56755.8i	−0.0312500	+	0.0541266i
$257$	−600600.	+	1.04027e6i	−0.567221	+	0.982456i	0.429618	+	0.903011i	$0.358648\pi$
$257$	−600600.	+	1.04027e6i	−0.567221	+	0.982456i	−0.996839	+	0.0794453i	$0.974685\pi$
$258$		0			0
$259$	192230.	+	332953.i	0.178063	+	0.308414i
$260$	34176.0			0.0313536
$261$		0			0
$262$	532608.			0.479352
$263$	81624.0	+	141377.i	0.0727660	+	0.126034i	0.900113	−	0.435657i	$-0.143484\pi$
$263$	81624.0	+	141377.i	0.0727660	+	0.126034i	−0.827347	+	0.561692i	$0.810151\pi$
$264$		0			0
$265$	−271296.	+	469898.i	−0.237317	+	0.411045i
$266$	403018.	−	698048.i	0.349237	−	0.604896i
$267$		0			0
$268$	−385048.	−	666923.i	−0.327475	−	0.567203i
$269$	−970776.			−0.817972			−0.408986	−	0.912541i	$-0.634117\pi$
$269$	−970776.			−0.817972			−0.408986	+	0.912541i	$0.634117\pi$
$270$		0			0
$271$	−828601.			−0.685365			−0.342683	−	0.939451i	$-0.611336\pi$
$271$	−828601.			−0.685365			−0.342683	+	0.939451i	$0.611336\pi$
$272$	−267264.	−	462915.i	−0.219037	−	0.379384i
$273$		0			0
$274$	−571632.	+	990096.i	−0.459981	+	0.796711i
$275$	519996.	−	900659.i	0.414637	−	0.718173i
$276$		0			0
$277$	308171.	+	533768.i	0.241319	+	0.417978i	0.961090	−	0.276234i	$-0.0890865\pi$
$277$	308171.	+	533768.i	0.241319	+	0.417978i	−0.719771	+	0.694212i	$0.755753\pi$
$278$	212036.			0.164550
$279$		0			0
$280$	118272.			0.0901544
$281$	−1.30819e6	−	2.26586e6i	−0.988338	−	1.71185i	−0.626042	−	0.779790i	$-0.715326\pi$
$281$	−1.30819e6	−	2.26586e6i	−0.988338	−	1.71185i	−0.362297	−	0.932063i	$-0.618007\pi$
$282$		0			0
$283$	1.02517e6	−	1.77564e6i	0.760902	−	1.31792i	−0.181484	−	0.983394i	$-0.558090\pi$
$283$	1.02517e6	−	1.77564e6i	0.760902	−	1.31792i	0.942386	−	0.334528i	$-0.108577\pi$
$284$	−408960.	+	708339.i	−0.300874	+	0.521129i
$285$		0			0
$286$	−72624.0	−	125788.i	−0.0525007	−	0.0909339i
$287$	−502656.			−0.360219
$288$		0			0
$289$	2.93989e6			2.07055
$290$	350208.	+	606578.i	0.244529	+	0.423537i
$291$		0			0
$292$	−245896.	+	425904.i	−0.168770	+	0.292318i
$293$	822300.	−	1.42427e6i	0.559579	−	0.969219i	−0.437953	−	0.898998i	$-0.644296\pi$
$293$	822300.	−	1.42427e6i	0.559579	−	0.969219i	0.997531	−	0.0702209i	$-0.0223704\pi$
$294$		0			0
$295$	−235296.	−	407545.i	−0.157420	−	0.272659i
$296$	−319552.			−0.211988
$297$		0			0
$298$	841920.			0.549200
$299$	−77964.0	−	135038.i	−0.0504332	−	0.0873528i
$300$		0			0
$301$	239932.	−	415574.i	0.152641	−	0.264382i
$302$	61442.0	−	106421.i	0.0387657	−	0.0671442i
$303$		0			0
$304$	334976.	+	580195.i	0.207888	+	0.360073i
$305$	−529080.			−0.325665
$306$		0			0
$307$	−2.17154e6			−1.31499			−0.657493	−	0.753461i	$-0.728383\pi$
$307$	−2.17154e6			−1.31499			−0.657493	+	0.753461i	$0.728383\pi$
$308$	−251328.	−	435313.i	−0.150961	−	0.261472i
$309$		0			0
$310$	−112704.	+	195209.i	−0.0666093	+	0.115371i
$311$	84156.0	−	145762.i	0.0493383	−	0.0854564i	−0.840302	−	0.542119i	$-0.817622\pi$
$311$	84156.0	−	145762.i	0.0493383	−	0.0854564i	0.889640	+	0.456663i	$0.150955\pi$
$312$		0			0
$313$	691854.	+	1.19833e6i	0.399166	+	0.691376i	0.993623	−	0.112751i	$-0.0359663\pi$
$313$	691854.	+	1.19833e6i	0.399166	+	0.691376i	−0.594457	+	0.804127i	$0.702633\pi$
$314$	−1.81415e6			−1.03836
$315$		0			0
$316$	611504.			0.344494
$317$	−242808.	−	420556.i	−0.135711	−	0.235058i	0.790158	−	0.612903i	$-0.209999\pi$
$317$	−242808.	−	420556.i	−0.135711	−	0.235058i	−0.925869	+	0.377845i	$0.876665\pi$
$318$		0			0
$319$	1.48838e6	−	2.57796e6i	0.818914	−	1.41840i
$320$	−49152.0	+	85133.8i	−0.0268328	+	0.0464758i
$321$		0			0
$322$	−269808.	−	467321.i	−0.145016	−	0.251175i
$323$	−5.46430e6			−2.91426
$324$		0			0
$325$	−226861.			−0.119138
$326$	−350482.	−	607053.i	−0.182651	−	0.316361i
$327$		0			0
$328$	208896.	−	361818.i	0.107213	−	0.185698i
$329$	1.14853e6	−	1.98932e6i	0.584997	−	1.01324i
$330$		0			0
$331$	−571052.	−	989092.i	−0.286488	−	0.496211i	0.686481	−	0.727148i	$-0.259154\pi$
$331$	−571052.	−	989092.i	−0.286488	−	0.496211i	−0.972969	+	0.230936i	$0.925821\pi$
$332$	129792.			0.0646253
$333$		0			0
$334$	−226848.			−0.111268
$335$	−577572.	−	1.00038e6i	−0.281186	−	0.487029i
$336$		0			0
$337$	−1.19029e6	+	2.06165e6i	−0.570926	+	0.988873i	0.425545	+	0.904937i	$0.360082\pi$
$337$	−1.19029e6	+	2.06165e6i	−0.570926	+	0.988873i	−0.996471	+	0.0839354i	$0.973251\pi$
$338$	726744.	−	1.25876e6i	0.346011	−	0.599308i
$339$		0			0
$340$	−400896.	−	694372.i	−0.188076	−	0.325758i
$341$	957984.			0.446141
$342$		0			0
$343$	−2.13174e6			−0.978363
$344$	199424.	+	345413.i	0.0908618	+	0.157377i
$345$		0			0
$346$	498432.	−	863310.i	0.223829	−	0.387683i
$347$	−1.57894e6	+	2.73480e6i	−0.703948	+	1.21927i	0.263121	+	0.964763i	$0.415248\pi$
$347$	−1.57894e6	+	2.73480e6i	−0.703948	+	1.21927i	−0.967070	+	0.254512i	$0.918085\pi$
$348$		0			0
$349$	771684.	+	1.33660e6i	0.339138	+	0.587404i	0.984271	−	0.176666i	$-0.0565314\pi$
$349$	771684.	+	1.33660e6i	0.339138	+	0.587404i	−0.645133	+	0.764070i	$0.723198\pi$
$350$	−785092.			−0.342571
$351$		0			0
$352$	417792.			0.179723
$353$	1.03850e6	+	1.79874e6i	0.443579	+	0.768302i	0.997952	−	0.0639665i	$-0.0203751\pi$
$353$	1.03850e6	+	1.79874e6i	0.443579	+	0.768302i	−0.554373	+	0.832269i	$0.687042\pi$
$354$		0			0
$355$	−613440.	+	1.06251e6i	−0.258346	+	0.447468i
$356$	354240.	−	613562.i	0.148140	−	0.256586i
$357$		0			0
$358$	−355104.	−	615058.i	−0.146436	−	0.253635i
$359$	2.46362e6			1.00888			0.504439	−	0.863448i	$-0.331699\pi$
$359$	2.46362e6			1.00888			0.504439	+	0.863448i	$0.331699\pi$
$360$		0			0
$361$	4.37259e6			1.76592
$362$	−906106.	−	1.56942e6i	−0.363419	−	0.629460i
$363$		0			0
$364$	−54824.0	+	94958.0i	−0.0216879	+	0.0375645i
$365$	−368844.	+	638857.i	−0.144914	+	0.250999i
$366$		0			0
$367$	−406892.	−	704758.i	−0.157694	−	0.273134i	0.776343	−	0.630311i	$-0.217073\pi$
$367$	−406892.	−	704758.i	−0.157694	−	0.273134i	−0.934037	+	0.357177i	$0.883739\pi$
$368$	448512.			0.172645
$369$		0			0
$370$	−479328.			−0.182024
$371$	−870408.	−	1.50759e6i	−0.328313	−	0.568655i
$372$		0			0
$373$	−1.60218e6	+	2.77506e6i	−0.596265	+	1.03276i	0.397102	+	0.917774i	$0.370016\pi$
$373$	−1.60218e6	+	2.77506e6i	−0.596265	+	1.03276i	−0.993367	+	0.114986i	$0.963318\pi$
$374$	−1.70381e6	+	2.95108e6i	−0.629857	+	1.09094i
$375$		0			0
$376$	954624.	+	1.65346e6i	0.348227	+	0.603147i
$377$	−649344.			−0.235300
$378$		0			0
$379$	−1.94680e6			−0.696182			−0.348091	−	0.937461i	$-0.613170\pi$
$379$	−1.94680e6			−0.696182			−0.348091	+	0.937461i	$0.613170\pi$
$380$	502464.	+	870293.i	0.178503	+	0.309177i
$381$		0			0
$382$	1.17682e6	−	2.03831e6i	0.412620	−	0.714679i
$383$	1.56134e6	−	2.70433e6i	0.543878	−	0.942025i	−0.454798	−	0.890594i	$-0.650289\pi$
$383$	1.56134e6	−	2.70433e6i	0.543878	−	0.942025i	0.998677	−	0.0514303i	$-0.0163780\pi$
$384$		0			0
$385$	−376992.	−	652969.i	−0.129622	−	0.224513i
$386$	2.34576e6			0.801336
$387$		0			0
$388$	−2.18642e6			−0.737316
$389$	−2.28404e6	−	3.95608e6i	−0.765298	−	1.32553i	−0.940089	−	0.340929i	$-0.889258\pi$
$389$	−2.28404e6	−	3.95608e6i	−0.765298	−	1.32553i	0.174791	−	0.984606i	$-0.444075\pi$
$390$		0			0
$391$	−1.82909e6	+	3.16807e6i	−0.605052	+	1.04798i
$392$	348096.	−	602920.i	0.114415	−	0.198173i
$393$		0			0
$394$	−748656.	−	1.29671e6i	−0.242964	−	0.420826i
$395$	917256.			0.295800
$396$		0			0
$397$	−3.27733e6			−1.04362			−0.521812	−	0.853061i	$-0.674744\pi$
$397$	−3.27733e6			−1.04362			−0.521812	+	0.853061i	$0.674744\pi$
$398$	606830.	+	1.05106e6i	0.192026	+	0.332598i
$399$		0			0
$400$	326272.	−	565120.i	0.101960	−	0.176600i
$401$	−2.72014e6	+	4.71141e6i	−0.844753	+	1.46315i	0.0410827	+	0.999156i	$0.486919\pi$
$401$	−2.72014e6	+	4.71141e6i	−0.844753	+	1.46315i	−0.885836	+	0.463999i	$0.846414\pi$
$402$		0			0
$403$	−104486.	−	180975.i	−0.0320476	−	0.0555081i
$404$	380928.			0.116115
$405$		0			0
$406$	−2.24717e6			−0.676582
$407$	1.01857e6	+	1.76422e6i	0.304794	+	0.527918i
$408$		0			0
$409$	−2.21361e6	+	3.83409e6i	−0.654325	+	1.13332i	0.327738	+	0.944769i	$0.393714\pi$
$409$	−2.21361e6	+	3.83409e6i	−0.654325	+	1.13332i	−0.982063	+	0.188555i	$0.939620\pi$
$410$	313344.	−	542728.i	0.0920580	−	0.159449i
$411$		0			0
$412$	−1.39175e6	−	2.41059e6i	−0.403942	−	0.699648i
$413$	1.50982e6			0.435561
$414$		0			0
$415$	194688.			0.0554905
$416$	−45568.0	−	78926.1i	−0.0129100	−	0.0223608i
$417$		0			0
$418$	2.13547e6	−	3.69875e6i	0.597796	−	1.03541i
$419$	−797916.	+	1.38203e6i	−0.222035	+	0.384577i	−0.955426	−	0.295231i	$-0.904603\pi$
$419$	−797916.	+	1.38203e6i	−0.222035	+	0.384577i	0.733391	+	0.679808i	$0.237937\pi$
$420$		0			0
$421$	3.02268e6	+	5.23544e6i	0.831165	+	1.43962i	0.897115	+	0.441798i	$0.145659\pi$
$421$	3.02268e6	+	5.23544e6i	0.831165	+	1.43962i	−0.0659494	+	0.997823i	$0.521008\pi$
$422$	917780.			0.250875
$423$		0			0
$424$	1.44691e6			0.390866
$425$	2.66116e6	+	4.60926e6i	0.714658	+	1.23782i
$426$		0			0
$427$	848732.	−	1.47005e6i	0.225269	−	0.390177i
$428$	1.51085e6	−	2.61687e6i	0.398668	−	0.690513i
$429$		0			0
$430$	299136.	+	518119.i	0.0780185	+	0.135132i
$431$	3.92220e6			1.01704			0.508518	−	0.861051i	$-0.330193\pi$
$431$	3.92220e6			1.01704			0.508518	+	0.861051i	$0.330193\pi$
$432$		0			0
$433$	4.74163e6			1.21537			0.607685	−	0.794178i	$-0.292098\pi$
$433$	4.74163e6			1.21537			0.607685	+	0.794178i	$0.292098\pi$
$434$	−361592.	−	626296.i	−0.0921498	−	0.159608i
$435$		0			0
$436$	1.66923e6	−	2.89119e6i	0.420533	−	0.728385i
$437$	2.29249e6	−	3.97071e6i	0.574254	−	0.994638i
$438$		0			0
$439$	−212794.	−	368570.i	−0.0526985	−	0.0912764i	0.838473	−	0.544943i	$-0.183449\pi$
$439$	−212794.	−	368570.i	−0.0526985	−	0.0912764i	−0.891171	+	0.453667i	$0.850116\pi$
$440$	626688.			0.154319
$441$		0			0
$442$	743328.			0.180978
$443$	1.30116e6	+	2.25368e6i	0.315008	+	0.545610i	0.979439	−	0.201740i	$-0.0646595\pi$
$443$	1.30116e6	+	2.25368e6i	0.315008	+	0.545610i	−0.664431	+	0.747349i	$0.731326\pi$
$444$		0			0
$445$	531360.	−	920343.i	0.127201	−	0.220318i
$446$	−2.13335e6	+	3.69507e6i	−0.507838	+	0.879602i
$447$		0			0
$448$	−157696.	−	273137.i	−0.0371215	−	0.0642963i
$449$	167256.			0.0391531			0.0195765	−	0.999808i	$-0.493768\pi$
$449$	167256.			0.0391531			0.0195765	+	0.999808i	$0.493768\pi$
$450$		0			0
$451$	−2.66342e6			−0.616594
$452$	1.24512e6	+	2.15661e6i	0.286659	+	0.496507i
$453$		0			0
$454$	−535776.	+	927991.i	−0.121995	+	0.211302i
$455$	−82236.0	+	142437.i	−0.0186223	+	0.0322548i
$456$		0			0
$457$	−590731.	−	1.02318e6i	−0.132312	−	0.229171i	0.792255	−	0.610190i	$-0.208907\pi$
$457$	−590731.	−	1.02318e6i	−0.132312	−	0.229171i	−0.924567	+	0.381018i	$0.875573\pi$
$458$	−4.07682e6			−0.908151
$459$		0			0
$460$	672768.			0.148242
$461$	2.80830e6	+	4.86412e6i	0.615448	+	1.06599i	0.990306	+	0.138905i	$0.0443581\pi$
$461$	2.80830e6	+	4.86412e6i	0.615448	+	1.06599i	−0.374858	+	0.927082i	$0.622309\pi$
$462$		0			0
$463$	562834.	−	974856.i	0.122019	−	0.211343i	−0.798545	−	0.601935i	$-0.794396\pi$
$463$	562834.	−	974856.i	0.122019	−	0.211343i	0.920564	+	0.390592i	$0.127730\pi$
$464$	933888.	−	1.61754e6i	0.201372	−	0.348787i
$465$		0			0
$466$	−374976.	−	649477.i	−0.0799905	−	0.138548i
$467$	4.74732e6			1.00729			0.503647	−	0.863910i	$-0.331991\pi$
$467$	4.74732e6			1.00729			0.503647	+	0.863910i	$0.331991\pi$
$468$		0			0
$469$	3.70609e6			0.778007
$470$	1.43194e6	+	2.48019e6i	0.299005	+	0.517892i
$471$		0			0
$472$	−627456.	+	1.08679e6i	−0.129637	+	0.224538i
$473$	1.27133e6	−	2.20200e6i	0.261279	−	0.452549i
$474$		0			0
$475$	−3.33537e6	−	5.77702e6i	−0.678281	−	1.17482i
$476$	2.57242e6			0.520384
$477$		0			0
$478$	4.16429e6			0.833626
$479$	4.89036e6	+	8.47035e6i	0.973872	+	1.68680i	0.683607	+	0.729850i	$0.260410\pi$
$479$	4.89036e6	+	8.47035e6i	0.973872	+	1.68680i	0.290265	+	0.956946i	$0.406256\pi$
$480$		0			0
$481$	222188.	−	384842.i	0.0437884	−	0.0758437i
$482$	−1.48717e6	+	2.57585e6i	−0.291570	+	0.505014i
$483$		0			0
$484$	−43304.0	−	75004.7i	−0.00840262	−	0.0145538i
$485$	−3.27962e6			−0.633096
$486$		0			0
$487$	−2.34782e6			−0.448582			−0.224291	−	0.974522i	$-0.572007\pi$
$487$	−2.34782e6			−0.448582			−0.224291	+	0.974522i	$0.572007\pi$
$488$	705440.	+	1.22186e6i	0.134094	+	0.232258i
$489$		0			0
$490$	522144.	−	904380.i	0.0982427	−	0.170161i
$491$	2.39881e6	−	4.15486e6i	0.449048	−	0.777773i	−0.549277	−	0.835641i	$-0.685097\pi$
$491$	2.39881e6	−	4.15486e6i	0.449048	−	0.777773i	0.998324	+	0.0578672i	$0.0184300\pi$
$492$		0			0
$493$	7.61702e6	+	1.31931e7i	1.41146	+	2.44472i
$494$	−931652.			−0.171766
$495$		0			0
$496$	601088.			0.109707
$497$	−1.96812e6	−	3.40888e6i	−0.357405	−	0.619044i
$498$		0			0
$499$	−3.51861e6	+	6.09441e6i	−0.632586	+	1.09567i	0.354435	+	0.935081i	$0.384673\pi$
$499$	−3.51861e6	+	6.09441e6i	−0.632586	+	1.09567i	−0.987021	+	0.160590i	$0.948660\pi$
$500$	1.08941e6	−	1.88691e6i	0.194879	−	0.337541i
$501$		0			0
$502$	−3.33216e6	−	5.77147e6i	−0.590156	−	1.02218i
$503$	−3.98858e6			−0.702908			−0.351454	−	0.936205i	$-0.614313\pi$
$503$	−3.98858e6			−0.702908			−0.351454	+	0.936205i	$0.614313\pi$
$504$		0			0
$505$	571392.			0.0997024
$506$	−1.42963e6	−	2.47620e6i	−0.248226	−	0.429941i
$507$		0			0
$508$	−890656.	+	1.54266e6i	−0.153127	+	0.265223i
$509$	−5.36387e6	+	9.29049e6i	−0.917664	+	1.58944i	−0.114710	+	0.993399i	$0.536594\pi$
$509$	−5.36387e6	+	9.29049e6i	−0.917664	+	1.58944i	−0.802954	+	0.596041i	$0.796739\pi$
$510$		0			0
$511$	−1.18337e6	−	2.04966e6i	−0.200479	−	0.347241i
$512$	262144.			0.0441942
$513$		0			0
$514$	4.80480e6			0.802172
$515$	−2.08763e6	−	3.61588e6i	−0.346845	−	0.600753i
$516$		0			0
$517$	6.08573e6	−	1.05408e7i	1.00135	−	1.73439i
$518$	768922.	−	1.33181e6i	0.125909	−	0.218081i
$519$		0			0
$520$	−68352.0	−	118389.i	−0.0110852	−	0.0192001i
$521$	5.86332e6			0.946345			0.473172	−	0.880970i	$-0.343109\pi$
$521$	5.86332e6			0.946345			0.473172	+	0.880970i	$0.343109\pi$
$522$		0			0
$523$	−2.31968e6			−0.370830			−0.185415	−	0.982660i	$-0.559363\pi$
$523$	−2.31968e6			−0.370830			−0.185415	+	0.982660i	$0.559363\pi$
$524$	−1.06522e6	−	1.84501e6i	−0.169476	−	0.293542i
$525$		0			0
$526$	326496.	−	565508.i	0.0514533	−	0.0891198i
$527$	−2.45131e6	+	4.24580e6i	−0.384478	+	0.665936i
$528$		0			0
$529$	1.68342e6	+	2.91577e6i	0.261549	+	0.453016i
$530$	2.17037e6			0.335617
$531$		0			0
$532$	−3.22414e6			−0.493896
$533$	290496.	+	503154.i	0.0442917	+	0.0767154i
$534$		0			0
$535$	2.26627e6	−	3.92530e6i	0.342316	−	0.592909i
$536$	−1.54019e6	+	2.66769e6i	−0.231560	+	0.401073i
$537$		0			0
$538$	1.94155e6	+	3.36287e6i	0.289197	+	0.500903i
$539$	−4.43822e6			−0.658017
$540$		0			0
$541$	−3.26629e6			−0.479801			−0.239901	−	0.970797i	$-0.577115\pi$
$541$	−3.26629e6			−0.479801			−0.239901	+	0.970797i	$0.577115\pi$
$542$	1.65720e6	+	2.87036e6i	0.242313	+	0.419699i
$543$		0			0
$544$	−1.06906e6	+	1.85166e6i	−0.154883	+	0.268265i
$545$	2.50385e6	−	4.33679e6i	0.361091	−	0.625428i
$546$		0			0
$547$	−2.15840e6	−	3.73846e6i	−0.308435	−	0.534225i	0.669585	−	0.742735i	$-0.266472\pi$
$547$	−2.15840e6	−	3.73846e6i	−0.308435	−	0.534225i	−0.978020	+	0.208510i	$0.933139\pi$
$548$	4.57306e6			0.650512
$549$		0			0
$550$	−4.15997e6			−0.586386
$551$	−9.54682e6	−	1.65356e7i	−1.33961	−	2.32028i
$552$		0			0
$553$	−1.47143e6	+	2.54859e6i	−0.204610	+	0.354395i
$554$	1.23268e6	−	2.13507e6i	0.170639	−	0.295555i
$555$		0			0
$556$	−424072.	−	734514.i	−0.0581772	−	0.100766i
$557$	5.78052e6			0.789458			0.394729	−	0.918798i	$-0.370839\pi$
$557$	5.78052e6			0.789458			0.394729	+	0.918798i	$0.370839\pi$
$558$		0			0
$559$	−554648.			−0.0750737
$560$	−236544.	−	409706.i	−0.0318744	−	0.0552081i
$561$		0			0
$562$	−5.23277e6	+	9.06342e6i	−0.698861	+	1.21046i
$563$	2.47862e6	−	4.29310e6i	0.329564	−	0.570821i	−0.652861	−	0.757477i	$-0.726432\pi$
$563$	2.47862e6	−	4.29310e6i	0.329564	−	0.570821i	0.982425	+	0.186656i	$0.0597649\pi$
$564$		0			0
$565$	1.86768e6	+	3.23492e6i	0.246140	+	0.426326i
$566$	−8.20134e6			−1.07608
$567$		0			0
$568$	3.27168e6			0.425500
$569$	−1.66751e6	−	2.88821e6i	−0.215917	−	0.373980i	0.737639	−	0.675196i	$-0.235941\pi$
$569$	−1.66751e6	−	2.88821e6i	−0.215917	−	0.373980i	−0.953556	+	0.301216i	$0.902607\pi$
$570$		0			0
$571$	−2.78464e6	+	4.82313e6i	−0.357419	+	0.619069i	−0.987529	−	0.157438i	$-0.949677\pi$
$571$	−2.78464e6	+	4.82313e6i	−0.357419	+	0.619069i	0.630110	+	0.776506i	$0.283010\pi$
$572$	−290496.	+	503154.i	−0.0371236	+	0.0643000i
$573$		0			0
$574$	1.00531e6	+	1.74125e6i	0.127357	+	0.220588i
$575$	−4.46585e6			−0.563293
$576$		0			0
$577$	−226861.			−0.0283675			−0.0141837	−	0.999899i	$-0.504515\pi$
$577$	−226861.			−0.0283675			−0.0141837	+	0.999899i	$0.504515\pi$
$578$	−5.87977e6	−	1.01841e7i	−0.732050	−	1.26795i
$579$		0			0
$580$	1.40083e6	−	2.42631e6i	0.172908	−	0.299486i
$581$	−312312.	+	540940.i	−0.0383838	+	0.0664827i
$582$		0			0
$583$	−4.61203e6	−	7.98827e6i	−0.561980	−	0.973378i
$584$	1.96717e6			0.238676
$585$		0			0
$586$	−6.57840e6			−0.791364
$587$	−977748.	−	1.69351e6i	−0.117120	−	0.202858i	0.801505	−	0.597988i	$-0.204033\pi$
$587$	−977748.	−	1.69351e6i	−0.117120	−	0.202858i	−0.918625	+	0.395130i	$0.870700\pi$
$588$		0			0
$589$	3.07236e6	−	5.32148e6i	0.364908	−	0.632039i
$590$	−941184.	+	1.63018e6i	−0.111313	+	0.192799i
$591$		0			0
$592$	639104.	+	1.10696e6i	0.0749492	+	0.129816i
$593$	−9.83938e6			−1.14903			−0.574514	−	0.818495i	$-0.694809\pi$
$593$	−9.83938e6			−1.14903			−0.574514	+	0.818495i	$0.694809\pi$
$594$		0			0
$595$	3.85862e6			0.446828
$596$	−1.68384e6	−	2.91650e6i	−0.194171	−	0.336315i
$597$		0			0
$598$	−311856.	+	540150.i	−0.0356616	+	0.0617677i
$599$	−2.22931e6	+	3.86128e6i	−0.253866	+	0.439708i	−0.964587	−	0.263766i	$-0.915035\pi$
$599$	−2.22931e6	+	3.86128e6i	−0.253866	+	0.439708i	0.710721	+	0.703474i	$0.248369\pi$
$600$		0			0
$601$	−6.04720e6	−	1.04741e7i	−0.682917	−	1.18285i	−0.974087	−	0.226176i	$-0.927378\pi$
$601$	−6.04720e6	−	1.04741e7i	−0.682917	−	1.18285i	0.291170	−	0.956671i	$-0.405956\pi$
$602$	−1.91946e6			−0.215867
$603$		0			0
$604$	−491536.			−0.0548230
$605$	−64956.0	−	112507.i	−0.00721491	−	0.0124966i
$606$		0			0
$607$	−1.77015e6	+	3.06598e6i	−0.195001	+	0.337752i	−0.946901	−	0.321525i	$-0.895804\pi$
$607$	−1.77015e6	+	3.06598e6i	−0.195001	+	0.337752i	0.751900	+	0.659278i	$0.229138\pi$
$608$	1.33990e6	−	2.32078e6i	0.146999	−	0.254610i
$609$		0			0
$610$	1.05816e6	+	1.83279e6i	0.115140	+	0.199429i
$611$	−2.65505e6			−0.287720
$612$		0			0
$613$	−1.30110e7			−1.39849			−0.699243	−	0.714884i	$-0.746480\pi$
$613$	−1.30110e7			−1.39849			−0.699243	+	0.714884i	$0.746480\pi$
$614$	4.34307e6	+	7.52242e6i	0.464918	+	0.805261i
$615$		0			0
$616$	−1.00531e6	+	1.74125e6i	−0.106745	+	0.184888i
$617$	1.68421e6	−	2.91714e6i	0.178108	−	0.308492i	−0.763124	−	0.646252i	$-0.776336\pi$
$617$	1.68421e6	−	2.91714e6i	0.178108	−	0.308492i	0.941233	+	0.337759i	$0.109669\pi$
$618$		0			0
$619$	−2.08139e6	−	3.60507e6i	−0.218336	−	0.378170i	0.735963	−	0.677022i	$-0.236730\pi$
$619$	−2.08139e6	−	3.60507e6i	−0.218336	−	0.378170i	−0.954300	+	0.298852i	$0.903396\pi$
$620$	901632.			0.0941998
$621$		0			0
$622$	−673248.			−0.0697749
$623$	1.70478e6	+	2.95277e6i	0.175974	+	0.304796i
$624$		0			0
$625$	−2.34870e6	+	4.06807e6i	−0.240507	+	0.416570i
$626$	2.76742e6	−	4.79331e6i	0.282253	−	0.488877i
$627$		0			0
$628$	3.62830e6	+	6.28441e6i	0.367117	+	0.635865i
$629$	−1.04254e7			−1.05067
$630$		0			0
$631$	1.65343e7			1.65315			0.826577	−	0.562823i	$-0.190285\pi$
$631$	1.65343e7			1.65315			0.826577	+	0.562823i	$0.190285\pi$
$632$	−1.22301e6	−	2.11831e6i	−0.121797	−	0.210959i
$633$		0			0
$634$	−971232.	+	1.68222e6i	−0.0959621	+	0.166211i
$635$	−1.33598e6	+	2.31399e6i	−0.131482	+	0.227734i
$636$		0			0
$637$	484071.	+	838436.i	0.0472673	+	0.0818693i
$638$	−1.19071e7			−1.15812
$639$		0			0
$640$	393216.			0.0379473
$641$	−4.05190e6	−	7.01809e6i	−0.389505	−	0.674643i	0.602878	−	0.797834i	$-0.294021\pi$
$641$	−4.05190e6	−	7.01809e6i	−0.389505	−	0.674643i	−0.992383	+	0.123191i	$0.960687\pi$
$642$		0			0
$643$	3.29853e6	−	5.71323e6i	0.314625	−	0.544947i	−0.664733	−	0.747081i	$-0.731455\pi$
$643$	3.29853e6	−	5.71323e6i	0.314625	−	0.544947i	0.979358	+	0.202135i	$0.0647879\pi$
$644$	−1.07923e6	+	1.86928e6i	−0.102542	+	0.177607i
$645$		0			0
$646$	1.09286e7	+	1.89289e7i	1.03035	+	1.78461i
$647$	−1.06116e7			−0.996603			−0.498301	−	0.867004i	$-0.666043\pi$
$647$	−1.06116e7			−0.996603			−0.498301	+	0.867004i	$0.666043\pi$
$648$		0			0
$649$	8.00006e6			0.745558
$650$	453722.	+	785870.i	0.0421218	+	0.0729570i
$651$		0			0
$652$	−1.40193e6	+	2.42821e6i	−0.129154	+	0.223701i
$653$	5.72402e6	−	9.91430e6i	0.525313	−	0.909870i	−0.474252	−	0.880389i	$-0.657281\pi$
$653$	5.72402e6	−	9.91430e6i	0.525313	−	0.909870i	0.999565	−	0.0294804i	$-0.00938525\pi$
$654$		0			0
$655$	−1.59782e6	−	2.76751e6i	−0.145521	−	0.252050i
$656$	−1.67117e6			−0.151621
$657$		0			0
$658$	−9.18826e6			−0.827310
$659$	5.17044e6	+	8.95546e6i	0.463782	+	0.803294i	0.999146	−	0.0413276i	$-0.0131587\pi$
$659$	5.17044e6	+	8.95546e6i	0.463782	+	0.803294i	−0.535364	+	0.844622i	$0.679825\pi$
$660$		0			0
$661$	1.60533e6	−	2.78051e6i	0.142909	−	0.247526i	−0.785682	−	0.618631i	$-0.787688\pi$
$661$	1.60533e6	−	2.78051e6i	0.142909	−	0.247526i	0.928591	+	0.371105i	$0.121021\pi$
$662$	−2.28421e6	+	3.95637e6i	−0.202577	+	0.350874i
$663$		0			0
$664$	−259584.	−	449613.i	−0.0228485	−	0.0395748i
$665$	−4.83622e6			−0.424084
$666$		0			0
$667$	−1.27826e7			−1.11251
$668$	453696.	+	785825.i	0.0393390	+	0.0681372i
$669$		0			0
$670$	−2.31029e6	+	4.00154e6i	−0.198829	+	0.344381i
$671$	4.49718e6	−	7.78934e6i	0.385597	−	0.667874i
$672$		0			0
$673$	−3.62383e6	−	6.27665e6i	−0.308411	−	0.534183i	0.669604	−	0.742718i	$-0.266464\pi$
$673$	−3.62383e6	−	6.27665e6i	−0.308411	−	0.534183i	−0.978015	+	0.208535i	$0.933130\pi$
$674$	9.52236e6			0.807411
$675$		0			0
$676$	−5.81395e6			−0.489333
$677$	−1.43315e6	−	2.48229e6i	−0.120176	−	0.208152i	0.799661	−	0.600452i	$-0.205013\pi$
$677$	−1.43315e6	−	2.48229e6i	−0.120176	−	0.208152i	−0.919837	+	0.392300i	$0.871679\pi$
$678$		0			0
$679$	5.26106e6	−	9.11243e6i	0.437924	−	0.758507i
$680$	−1.60358e6	+	2.77749e6i	−0.132990	+	0.230346i
$681$		0			0
$682$	−1.91597e6	−	3.31855e6i	−0.157735	−	0.273205i
$683$	2.12852e7			1.74593			0.872964	−	0.487785i	$-0.162195\pi$
$683$	2.12852e7			1.74593			0.872964	+	0.487785i	$0.162195\pi$
$684$		0			0
$685$	6.85958e6			0.558562
$686$	4.26349e6	+	7.38458e6i	0.345904	+	0.599122i
$687$		0			0
$688$	797696.	−	1.38165e6i	0.0642490	−	0.111283i
$689$	−1.00606e6	+	1.74254e6i	−0.0807373	+	0.139841i
$690$		0			0
$691$	−104020.	−	180168.i	−0.00828747	−	0.0143543i	0.861852	−	0.507160i	$-0.169305\pi$
$691$	−104020.	−	180168.i	−0.00828747	−	0.0143543i	−0.870139	+	0.492806i	$0.835971\pi$
$692$	−3.98746e6			−0.316542
$693$		0			0
$694$	1.26315e7			0.995533
$695$	−636108.	−	1.10177e6i	−0.0499538	−	0.0865226i
$696$		0			0
$697$	6.81523e6	−	1.18043e7i	0.531372	−	0.920363i
$698$	3.08674e6	−	5.34639e6i	0.239807	−	0.415357i
$699$		0			0
$700$	1.57018e6	+	2.71964e6i	0.121117	+	0.209781i
$701$	952488.			0.0732090			0.0366045	−	0.999330i	$-0.488346\pi$
$701$	952488.			0.0732090			0.0366045	+	0.999330i	$0.488346\pi$
$702$		0			0
$703$	1.30667e7			0.997188
$704$	−835584.	−	1.44727e6i	−0.0635416	−	0.110057i
$705$		0			0
$706$	4.15402e6	−	7.19497e6i	0.313658	−	0.543272i
$707$	−916608.	+	1.58761e6i	−0.0689660	+	0.119453i
$708$		0			0
$709$	4.48216e6	+	7.76333e6i	0.334867	+	0.580006i	0.983459	−	0.181130i	$-0.0579754\pi$
$709$	4.48216e6	+	7.76333e6i	0.334867	+	0.580006i	−0.648593	+	0.761136i	$0.724642\pi$
$710$	4.90752e6			0.365356
$711$		0			0
$712$	−2.83392e6			−0.209502
$713$	−2.05685e6	−	3.56257e6i	−0.151523	−	0.262445i
$714$		0			0
$715$	−435744.	+	754731.i	−0.0318762	+	0.0552112i
$716$	−1.42042e6	+	2.46023e6i	−0.103546	+	0.179347i
$717$		0			0
$718$	−4.92725e6	−	8.53424e6i	−0.356692	−	0.617809i
$719$	1.33824e7			0.965407			0.482703	−	0.875784i	$-0.339655\pi$
$719$	1.33824e7			0.965407			0.482703	+	0.875784i	$0.339655\pi$
$720$		0			0
$721$	1.33956e7			0.959676
$722$	−8.74518e6	−	1.51471e7i	−0.624347	−	1.08140i
$723$		0			0
$724$	−3.62442e6	+	6.27769e6i	−0.256976	+	0.445096i
$725$	−9.29875e6	+	1.61059e7i	−0.657022	+	1.13799i
$726$		0			0
$727$	−3.39422e6	−	5.87896e6i	−0.238179	−	0.412539i	0.722013	−	0.691880i	$-0.243217\pi$
$727$	−3.39422e6	−	5.87896e6i	−0.238179	−	0.412539i	−0.960192	+	0.279341i	$0.909884\pi$
$728$	438592.			0.0306713
$729$		0			0
$730$	2.95075e6			0.204939
$731$	6.50621e6	+	1.12691e7i	0.450334	+	0.780001i
$732$		0			0
$733$	−2.60530e6	+	4.51251e6i	−0.179101	+	0.310212i	−0.941573	−	0.336810i	$-0.890652\pi$
$733$	−2.60530e6	+	4.51251e6i	−0.179101	+	0.310212i	0.762472	+	0.647021i	$0.223985\pi$
$734$	−1.62757e6	+	2.81903e6i	−0.111506	+	0.193135i
$735$		0			0
$736$	−897024.	−	1.55369e6i	−0.0610393	−	0.105723i
$737$	1.96374e7			1.33173
$738$		0			0
$739$	6.17470e6			0.415915			0.207958	−	0.978138i	$-0.433318\pi$
$739$	6.17470e6			0.415915			0.207958	+	0.978138i	$0.433318\pi$
$740$	958656.	+	1.66044e6i	0.0643552	+	0.111466i
$741$		0			0
$742$	−3.48163e6	+	6.03036e6i	−0.232152	+	0.402100i
$743$	154644.	−	267851.i	0.0102769	−	0.0178001i	−0.860841	−	0.508874i	$-0.830062\pi$
$743$	154644.	−	267851.i	0.0102769	−	0.0178001i	0.871118	+	0.491074i	$0.163395\pi$
$744$		0			0
$745$	−2.52576e6	−	4.37474e6i	−0.166725	−	0.288777i
$746$	1.28174e7			0.843246
$747$		0			0
$748$	1.36305e7			0.890752
$749$	7.27096e6	+	1.25937e7i	0.473573	+	0.820253i
$750$		0			0
$751$	4.71325e6	−	8.16358e6i	0.304944	−	0.528179i	−0.672305	−	0.740275i	$-0.734696\pi$
$751$	4.71325e6	−	8.16358e6i	0.304944	−	0.528179i	0.977249	+	0.212096i	$0.0680289\pi$
$752$	3.81850e6	−	6.61383e6i	0.246234	−	0.426490i
$753$		0			0
$754$	1.29869e6	+	2.24939e6i	0.0831910	+	0.144091i
$755$	−737304.			−0.0470738
$756$		0			0
$757$	−3.03790e7			−1.92679			−0.963393	−	0.268095i	$-0.913606\pi$
$757$	−3.03790e7			−1.92679			−0.963393	+	0.268095i	$0.913606\pi$
$758$	3.89359e6	+	6.74390e6i	0.246137	+	0.426323i
$759$		0			0
$760$	2.00986e6	−	3.48117e6i	0.126221	−	0.218621i
$761$	−6.44668e6	+	1.11660e7i	−0.403528	+	0.698932i	−0.994149	−	0.108018i	$-0.965550\pi$
$761$	−6.44668e6	+	1.11660e7i	−0.403528	+	0.698932i	0.590620	+	0.806949i	$0.298883\pi$
$762$		0			0
$763$	8.03318e6	+	1.39139e7i	0.499547	+	0.865240i
$764$	−9.41453e6			−0.583533
$765$		0			0
$766$	−1.24908e7			−0.769160
$767$	−872556.	−	1.51131e6i	−0.0535556	−	0.0927610i
$768$		0			0
$769$	−1.21356e6	+	2.10195e6i	−0.0740023	+	0.128176i	−0.900652	−	0.434541i	$-0.856911\pi$
$769$	−1.21356e6	+	2.10195e6i	−0.0740023	+	0.128176i	0.826650	+	0.562717i	$0.190244\pi$
$770$	−1.50797e6	+	2.61188e6i	−0.0916569	+	0.158754i
$771$		0			0
$772$	−4.69151e6	−	8.12594e6i	−0.283315	−	0.490716i
$773$	1.14496e7			0.689193			0.344597	−	0.938751i	$-0.388016\pi$
$773$	1.14496e7			0.689193			0.344597	+	0.938751i	$0.388016\pi$
$774$		0			0
$775$	−5.98505e6			−0.357943
$776$	4.37283e6	+	7.57397e6i	0.260680	+	0.451512i
$777$		0			0
$778$	−9.13618e6	+	1.58243e7i	−0.541147	+	0.937295i
$779$	−8.54189e6	+	1.47950e7i	−0.504325	+	0.873516i
$780$		0			0
$781$	−1.04285e7	−	1.80627e7i	−0.611777	−	1.05963i
$782$	1.46327e7			0.855673
$783$		0			0
$784$	−2.78477e6			−0.161808
$785$	5.44246e6	+	9.42661e6i	0.315225	+	0.545986i
$786$		0			0
$787$	−1.61460e6	+	2.79657e6i	−0.0929240	+	0.160949i	−0.908740	−	0.417362i	$-0.862955\pi$
$787$	−1.61460e6	+	2.79657e6i	−0.0929240	+	0.160949i	0.815816	+	0.578311i	$0.196288\pi$
$788$	−2.99462e6	+	5.18684e6i	−0.171801	+	0.297569i
$789$		0			0
$790$	−1.83451e6	−	3.17747e6i	−0.104581	−	0.181140i
$791$	−1.19843e7			−0.681037
$792$		0			0
$793$	−1.96200e6			−0.110794
$794$	6.55466e6	+	1.13530e7i	0.368977	+	0.639086i
$795$		0			0
$796$	2.42732e6	−	4.20424e6i	0.135783	−	0.235182i
$797$	1.07124e7	−	1.85544e7i	0.597367	−	1.03467i	−0.395842	−	0.918319i	$-0.629547\pi$
$797$	1.07124e7	−	1.85544e7i	0.597367	−	1.03467i	0.993208	−	0.116351i	$-0.0371196\pi$
$798$		0			0
$799$	3.11446e7	+	5.39440e7i	1.72590	+	2.98935i
$800$	−2.61018e6			−0.144193
$801$		0			0
$802$	2.17611e7			1.19466
$803$	−6.27035e6	−	1.08606e7i	−0.343165	−	0.594379i
$804$		0			0
$805$	−1.61885e6	+	2.80393e6i	−0.0880474	+	0.152503i
$806$	−417944.	+	723900.i	−0.0226611	+	0.0392501i
$807$		0			0
$808$	−761856.	−	1.31957e6i	−0.0410530	−	0.0711058i
$809$	1.55614e7			0.835942			0.417971	−	0.908460i	$-0.362741\pi$
$809$	1.55614e7			0.835942			0.417971	+	0.908460i	$0.362741\pi$
$810$		0			0
$811$	−1.91972e7			−1.02491			−0.512454	−	0.858715i	$-0.671263\pi$
$811$	−1.91972e7			−1.02491			−0.512454	+	0.858715i	$0.671263\pi$
$812$	4.49434e6	+	7.78442e6i	0.239208	+	0.414320i
$813$		0			0
$814$	4.07429e6	−	7.05687e6i	0.215522	−	0.373294i
$815$	−2.10289e6	+	3.64232e6i	−0.110898	+	0.192081i
$816$		0			0
$817$	−8.15457e6	−	1.41241e7i	−0.427411	−	0.740298i
$818$	1.77089e7			0.925355
$819$		0			0
$820$	−2.50675e6			−0.130190
$821$	1.18151e7	+	2.04644e7i	0.611760	+	1.05960i	0.990944	+	0.134278i	$0.0428715\pi$
$821$	1.18151e7	+	2.04644e7i	0.611760	+	1.05960i	−0.379184	+	0.925321i	$0.623795\pi$
$822$		0			0
$823$	838980.	−	1.45316e6i	0.0431770	−	0.0747847i	−0.843629	−	0.536926i	$-0.819585\pi$
$823$	838980.	−	1.45316e6i	0.0431770	−	0.0747847i	0.886806	+	0.462141i	$0.152919\pi$
$824$	−5.56701e6	+	9.64234e6i	−0.285630	+	0.494726i
$825$		0			0
$826$	−3.01963e6	−	5.23016e6i	−0.153994	−	0.266725i
$827$	−1.39053e7			−0.706995			−0.353497	−	0.935436i	$-0.615008\pi$
$827$	−1.39053e7			−0.706995			−0.353497	+	0.935436i	$0.615008\pi$
$828$		0			0
$829$	−1.33464e7			−0.674493			−0.337247	−	0.941416i	$-0.609496\pi$
$829$	−1.33464e7			−0.674493			−0.337247	+	0.941416i	$0.609496\pi$
$830$	−389376.	−	674419.i	−0.0196189	−	0.0339809i
$831$		0			0
$832$	−182272.	+	315704.i	−0.00912876	+	0.0158115i
$833$	1.13566e7	−	1.96703e7i	0.567070	−	0.982195i
$834$		0			0
$835$	680544.	+	1.17874e6i	0.0337785	+	0.0585060i
$836$	−1.70838e7			−0.845412
$837$		0			0
$838$	6.38333e6			0.314005
$839$	1.47613e7	+	2.55673e7i	0.723967	+	1.25395i	0.959398	+	0.282057i	$0.0910167\pi$
$839$	1.47613e7	+	2.55673e7i	0.723967	+	1.25395i	−0.235431	+	0.971891i	$0.575650\pi$
$840$		0			0
$841$	−1.63602e7	+	2.83368e7i	−0.797626	+	1.38153i
$842$	1.20907e7	−	2.09418e7i	0.587723	−	1.01797i
$843$		0			0
$844$	−1.83556e6	−	3.17928e6i	−0.0886977	−	0.153629i
$845$	−8.72093e6			−0.420166
$846$		0			0
$847$	416801.			0.0199627
$848$	−2.89382e6	−	5.01225e6i	−0.138192	−	0.239355i
$849$		0			0
$850$	1.06446e7	−	1.84370e7i	0.505339	−	0.875273i
$851$	4.37387e6	−	7.57576e6i	0.207034	−	0.358593i
$852$		0			0
$853$	−6.15414e6	−	1.06593e7i	−0.289597	−	0.501598i	0.684116	−	0.729373i	$-0.260188\pi$
$853$	−6.15414e6	−	1.06593e7i	−0.289597	−	0.501598i	−0.973714	+	0.227776i	$0.926855\pi$
$854$	−6.78986e6			−0.318578
$855$		0			0
$856$	−1.20868e7			−0.563802
$857$	8.29452e6	+	1.43665e7i	0.385780	+	0.668190i	0.991877	−	0.127201i	$-0.0405992\pi$
$857$	8.29452e6	+	1.43665e7i	0.385780	+	0.668190i	−0.606097	+	0.795390i	$0.707266\pi$
$858$		0			0
$859$	−6.56430e6	+	1.13697e7i	−0.303533	+	0.525734i	−0.976934	−	0.213543i	$-0.931500\pi$
$859$	−6.56430e6	+	1.13697e7i	−0.303533	+	0.525734i	0.673401	+	0.739278i	$0.264833\pi$
$860$	1.19654e6	−	2.07248e6i	0.0551674	−	0.0955527i
$861$		0			0
$862$	−7.84440e6	−	1.35869e7i	−0.359577	−	0.622805i
$863$	2.09802e7			0.958918			0.479459	−	0.877564i	$-0.340833\pi$
$863$	2.09802e7			0.958918			0.479459	+	0.877564i	$0.340833\pi$
$864$		0			0
$865$	−5.98118e6			−0.271798
$866$	−9.48327e6	−	1.64255e7i	−0.429698	−	0.744259i
$867$		0			0
$868$	−1.44637e6	+	2.50518e6i	−0.0651598	+	0.112860i
$869$	−7.79668e6	+	1.35042e7i	−0.350235	+	0.606626i
$870$		0			0
$871$	−2.14183e6	−	3.70976e6i	−0.0956620	−	0.165691i
$872$	−1.33539e7			−0.594724
$873$		0			0
$874$	−1.83399e7			−0.812118
$875$	5.24278e6	+	9.08075e6i	0.231495	+	0.400961i
$876$		0			0
$877$	−2.36941e6	+	4.10394e6i	−0.104026	+	0.180178i	−0.913340	−	0.407198i	$-0.866506\pi$
$877$	−2.36941e6	+	4.10394e6i	−0.104026	+	0.180178i	0.809314	+	0.587376i	$0.199839\pi$
$878$	−851176.	+	1.47428e6i	−0.0372635	+	0.0645422i
$879$		0			0
$880$	−1.25338e6	−	2.17091e6i	−0.0545600	−	0.0945008i
$881$	−3.09880e7			−1.34510			−0.672549	−	0.740053i	$-0.734801\pi$
$881$	−3.09880e7			−1.34510			−0.672549	+	0.740053i	$0.734801\pi$
$882$		0			0
$883$	−1.94710e6			−0.0840402			−0.0420201	−	0.999117i	$-0.513379\pi$
$883$	−1.94710e6			−0.0840402			−0.0420201	+	0.999117i	$0.513379\pi$
$884$	−1.48666e6	−	2.57496e6i	−0.0639852	−	0.110826i
$885$		0			0
$886$	5.20464e6	−	9.01470e6i	0.222744	−	0.385804i
$887$	9.26268e6	−	1.60434e7i	0.395301	−	0.684681i	−0.597839	−	0.801616i	$-0.703974\pi$
$887$	9.26268e6	−	1.60434e7i	0.395301	−	0.684681i	0.993140	+	0.116935i	$0.0373071\pi$
$888$		0			0
$889$	−4.28628e6	−	7.42406e6i	−0.181897	−	0.315055i
$890$	−4.25088e6			−0.179889
$891$		0			0
$892$	1.70668e7			0.718192
$893$	−3.90352e7	−	6.76109e7i	−1.63805	−	2.83719i
$894$		0			0
$895$	−2.13062e6	+	3.69035e6i	−0.0889097	+	0.153996i
$896$	−630784.	+	1.09255e6i	−0.0262489	+	0.0454644i
$897$		0			0
$898$	−334512.	−	579392.i	−0.0138427	−	0.0239763i
$899$	−1.71310e7			−0.706942
$900$		0			0
$901$	4.72055e7			1.93723
$902$	5.32685e6	+	9.22637e6i	0.217999	+	0.377585i
$903$		0			0
$904$	4.98048e6	−	8.62644e6i	0.202698	−	0.351084i
$905$	−5.43664e6	+	9.41653e6i	−0.220652	+	0.382181i
$906$		0			0
$907$	1.80616e7	+	3.12837e7i	0.729019	+	1.26270i	0.957298	+	0.289102i	$0.0933568\pi$
$907$	1.80616e7	+	3.12837e7i	0.729019	+	1.26270i	−0.228279	+	0.973596i	$0.573310\pi$
$908$	4.28621e6			0.172528
$909$		0			0
$910$	657888.			0.0263359
$911$	−1.51560e7	−	2.62509e7i	−0.605044	−	1.04797i	−0.992044	−	0.125888i	$-0.959822\pi$
$911$	−1.51560e7	−	2.62509e7i	−0.605044	−	1.04797i	0.387000	−	0.922080i	$-0.373511\pi$
$912$		0			0
$913$	−1.65485e6	+	2.86628e6i	−0.0657024	+	0.113800i
$914$	−2.36292e6	+	4.09270e6i	−0.0935587	+	0.162048i
$915$		0			0
$916$	8.15365e6	+	1.41225e7i	0.321080	+	0.556127i
$917$	1.02527e7			0.402638
$918$		0			0
$919$	1.82688e7			0.713545			0.356772	−	0.934191i	$-0.383877\pi$
$919$	1.82688e7			0.713545			0.356772	+	0.934191i	$0.383877\pi$
$920$	−1.34554e6	−	2.33054e6i	−0.0524114	−	0.0907792i
$921$		0			0
$922$	1.12332e7	−	1.94565e7i	0.435187	−	0.753766i
$923$	−2.27484e6	+	3.94014e6i	−0.0878914	+	0.152232i
$924$		0			0
$925$	−6.36358e6	−	1.10220e7i	−0.244538	−	0.423553i
$926$	−4.50267e6			−0.172561
$927$		0			0
$928$	−7.47110e6			−0.284784
$929$	4.59948e6	+	7.96653e6i	0.174852	+	0.302852i	0.940110	−	0.340872i	$-0.110722\pi$
$929$	4.59948e6	+	7.96653e6i	0.174852	+	0.302852i	−0.765258	+	0.643723i	$0.777389\pi$
$930$		0			0
$931$	−1.42339e7	+	2.46538e7i	−0.538206	+	0.932200i
$932$	−1.49990e6	+	2.59791e6i	−0.0565619	+	0.0979680i
$933$		0			0
$934$	−9.49464e6	−	1.64452e7i	−0.356132	−	0.616839i
$935$	2.04457e7			0.764844
$936$		0			0
$937$	−6.01912e6			−0.223967			−0.111983	−	0.993710i	$-0.535720\pi$
$937$	−6.01912e6			−0.223967			−0.111983	+	0.993710i	$0.535720\pi$
$938$	−7.41217e6	−	1.28383e7i	−0.275067	−	0.476430i
$939$		0			0
$940$	5.72774e6	−	9.92074e6i	0.211429	−	0.366205i
$941$	1.61808e7	−	2.80259e7i	0.595696	−	1.03178i	−0.397752	−	0.917493i	$-0.630209\pi$
$941$	1.61808e7	−	2.80259e7i	0.595696	−	1.03178i	0.993448	−	0.114283i	$-0.0364572\pi$
$942$		0			0
$943$	5.71853e6	+	9.90478e6i	0.209414	+	0.362715i
$944$	5.01965e6			0.183334
$945$		0			0
$946$	−1.01706e7			−0.369505
$947$	−1.68154e7	−	2.91252e7i	−0.609303	−	1.05534i	−0.991355	−	0.131203i	$-0.958116\pi$
$947$	−1.68154e7	−	2.91252e7i	−0.609303	−	1.05534i	0.382052	−	0.924141i	$-0.375217\pi$
$948$		0			0
$949$	−1.36780e6	+	2.36909e6i	−0.0493010	+	0.0853919i
$950$	−1.33415e7	+	2.31081e7i	−0.479617	+	0.830721i
$951$		0			0
$952$	−5.14483e6	−	8.91111e6i	−0.183983	−	0.318669i
$953$	−2.17482e7			−0.775697			−0.387848	−	0.921723i	$-0.626782\pi$
$953$	−2.17482e7			−0.775697			−0.387848	+	0.921723i	$0.626782\pi$
$954$		0			0
$955$	−1.41218e7			−0.501050
$956$	−8.32858e6	−	1.44255e7i	−0.294731	−	0.510489i
$957$		0			0
$958$	1.95614e7	−	3.38814e7i	0.688632	−	1.19275i
$959$	−1.10039e7	+	1.90593e7i	−0.386368	+	0.669208i
$960$		0			0
$961$	1.15580e7	+	2.00191e7i	0.403715	+	0.699255i
$962$	−1.77751e6			−0.0619261
$963$		0			0
$964$	1.18974e7			0.412342
$965$	−7.03727e6	−	1.21889e7i	−0.243268	−	0.421353i
$966$		0			0
$967$	1.64077e7	−	2.84189e7i	0.564262	−	0.977330i	−0.432856	−	0.901463i	$-0.642494\pi$
$967$	1.64077e7	−	2.84189e7i	0.564262	−	0.977330i	0.997118	−	0.0758671i	$-0.0241725\pi$
$968$	−173216.	+	300019.i	−0.00594155	+	0.0102911i
$969$		0			0
$970$	6.55925e6	+	1.13610e7i	0.223833	+	0.387691i
$971$	−1.98137e6			−0.0674399			−0.0337200	−	0.999431i	$-0.510735\pi$
$971$	−1.98137e6			−0.0674399			−0.0337200	+	0.999431i	$0.510735\pi$
$972$		0			0
$973$	4.08169e6			0.138216
$974$	4.69564e6	+	8.13308e6i	0.158598	+	0.274699i
$975$		0			0
$976$	2.82176e6	−	4.88743e6i	0.0948190	−	0.164231i
$977$	−1.92564e7	+	3.33530e7i	−0.645414	+	1.11789i	0.338792	+	0.940861i	$0.389982\pi$
$977$	−1.92564e7	+	3.33530e7i	−0.645414	+	1.11789i	−0.984206	+	0.177028i	$0.943352\pi$
$978$		0			0
$979$	9.03312e6	+	1.56458e7i	0.301218	+	0.521725i
$980$	−4.17715e6			−0.138936
$981$		0			0
$982$	−1.91905e7			−0.635049
$983$	−1.57929e7	−	2.73541e7i	−0.521288	−	0.902898i	−0.999693	−	0.0247588i	$-0.992118\pi$
$983$	−1.57929e7	−	2.73541e7i	−0.521288	−	0.902898i	0.478405	−	0.878139i	$-0.341215\pi$
$984$		0			0
$985$	−4.49194e6	+	7.78026e6i	−0.147517	+	0.255507i
$986$	3.04681e7	−	5.27723e7i	0.998052	−	1.72868i
$987$		0			0
$988$	1.86330e6	+	3.22734e6i	0.0607283	+	0.105185i
$989$	−1.09185e7			−0.354953
$990$		0			0
$991$	5.92538e7			1.91660			0.958302	−	0.285758i	$-0.0922453\pi$
$991$	5.92538e7			1.91660			0.958302	+	0.285758i	$0.0922453\pi$
$992$	−1.20218e6	−	2.08223e6i	−0.0387872	−	0.0671815i
$993$		0			0
$994$	−7.87248e6	+	1.36355e7i	−0.252723	+	0.437730i
$995$	3.64098e6	−	6.30636e6i	0.116590	−	0.201939i
$996$		0			0
$997$	−2.33863e7	−	4.05062e7i	−0.745115	−	1.29058i	−0.950141	−	0.311821i	$-0.899061\pi$
$997$	−2.33863e7	−	4.05062e7i	−0.745115	−	1.29058i	0.205026	−	0.978757i	$-0.434272\pi$
$998$	2.81489e7			0.894612
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.6.c.c.109.1		2
3.2	odd	2		162.6.c.j.109.1		2
9.2	odd	6		162.6.c.j.55.1		2
9.4	even	3		54.6.a.e.1.1	yes	1
9.5	odd	6		54.6.a.b.1.1	✓	1
9.7	even	3	inner	162.6.c.c.55.1		2
36.23	even	6		432.6.a.d.1.1		1
36.31	odd	6		432.6.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.6.a.b.1.1	✓	1	9.5	odd	6
54.6.a.e.1.1	yes	1	9.4	even	3
162.6.c.c.55.1		2	9.7	even	3	inner
162.6.c.c.109.1		2	1.1	even	1	trivial
162.6.c.j.55.1		2	9.2	odd	6
162.6.c.j.109.1		2	3.2	odd	2
432.6.a.d.1.1		1	36.23	even	6
432.6.a.g.1.1		1	36.31	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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(-12.0000 + 20.7846i) q^{5} +(-38.5000 - 66.6840i) q^{7} +64.0000 q^{8} +O(q^{10})\)	\(q+(-2.00000 - 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(-12.0000 + 20.7846i) q^{5} +(-38.5000 - 66.6840i) q^{7} +64.0000 q^{8} +96.0000 q^{10} +(-204.000 - 353.338i) q^{11} +(-44.5000 + 77.0763i) q^{13} +(-154.000 + 266.736i) q^{14} +(-128.000 - 221.703i) q^{16} +2088.00 q^{17} -2617.00 q^{19} +(-192.000 - 332.554i) q^{20} +(-816.000 + 1413.35i) q^{22} +(-876.000 + 1517.28i) q^{23} +(1274.50 + 2207.50i) q^{25} +356.000 q^{26} +1232.00 q^{28} +(3648.00 + 6318.52i) q^{29} +(-1174.00 + 2033.43i) q^{31} +(-512.000 + 886.810i) q^{32} +(-4176.00 - 7233.04i) q^{34} +1848.00 q^{35} -4993.00 q^{37} +(5234.00 + 9065.55i) q^{38} +(-768.000 + 1330.22i) q^{40} +(3264.00 - 5653.41i) q^{41} +(3116.00 + 5397.07i) q^{43} +6528.00 q^{44} +7008.00 q^{46} +(14916.0 + 25835.3i) q^{47} +(5439.00 - 9420.62i) q^{49} +(5098.00 - 8830.00i) q^{50} +(-712.000 - 1233.22i) q^{52} +22608.0 q^{53} +9792.00 q^{55} +(-2464.00 - 4267.77i) q^{56} +(14592.0 - 25274.1i) q^{58} +(-9804.00 + 16981.0i) q^{59} +(11022.5 + 19091.5i) q^{61} +9392.00 q^{62} +4096.00 q^{64} +(-1068.00 - 1849.83i) q^{65} +(-24065.5 + 41682.7i) q^{67} +(-16704.0 + 28932.2i) q^{68} +(-3696.00 - 6401.66i) q^{70} +51120.0 q^{71} +30737.0 q^{73} +(9986.00 + 17296.3i) q^{74} +(20936.0 - 36262.2i) q^{76} +(-15708.0 + 27207.1i) q^{77} +(-19109.5 - 33098.6i) q^{79} +6144.00 q^{80} -26112.0 q^{82} +(-4056.00 - 7025.20i) q^{83} +(-25056.0 + 43398.3i) q^{85} +(12464.0 - 21588.3i) q^{86} +(-13056.0 - 22613.7i) q^{88} -44280.0 q^{89} +6853.00 q^{91} +(-14016.0 - 24276.4i) q^{92} +(59664.0 - 103341. i) q^{94} +(31404.0 - 54393.3i) q^{95} +(68325.5 + 118343. i) q^{97} -43512.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 16 q^{4} - 24 q^{5} - 77 q^{7} + 128 q^{8}+O(q^{10}) \) 2 * q - 4 * q^2 - 16 * q^4 - 24 * q^5 - 77 * q^7 + 128 * q^8	\( 2 q - 4 q^{2} - 16 q^{4} - 24 q^{5} - 77 q^{7} + 128 q^{8} + 192 q^{10} - 408 q^{11} - 89 q^{13} - 308 q^{14} - 256 q^{16} + 4176 q^{17} - 5234 q^{19} - 384 q^{20} - 1632 q^{22} - 1752 q^{23} + 2549 q^{25} + 712 q^{26} + 2464 q^{28} + 7296 q^{29} - 2348 q^{31} - 1024 q^{32} - 8352 q^{34} + 3696 q^{35} - 9986 q^{37} + 10468 q^{38} - 1536 q^{40} + 6528 q^{41} + 6232 q^{43} + 13056 q^{44} + 14016 q^{46} + 29832 q^{47} + 10878 q^{49} + 10196 q^{50} - 1424 q^{52} + 45216 q^{53} + 19584 q^{55} - 4928 q^{56} + 29184 q^{58} - 19608 q^{59} + 22045 q^{61} + 18784 q^{62} + 8192 q^{64} - 2136 q^{65} - 48131 q^{67} - 33408 q^{68} - 7392 q^{70} + 102240 q^{71} + 61474 q^{73} + 19972 q^{74} + 41872 q^{76} - 31416 q^{77} - 38219 q^{79} + 12288 q^{80} - 52224 q^{82} - 8112 q^{83} - 50112 q^{85} + 24928 q^{86} - 26112 q^{88} - 88560 q^{89} + 13706 q^{91} - 28032 q^{92} + 119328 q^{94} + 62808 q^{95} + 136651 q^{97} - 87024 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 - 16 * q^4 - 24 * q^5 - 77 * q^7 + 128 * q^8 + 192 * q^10 - 408 * q^11 - 89 * q^13 - 308 * q^14 - 256 * q^16 + 4176 * q^17 - 5234 * q^19 - 384 * q^20 - 1632 * q^22 - 1752 * q^23 + 2549 * q^25 + 712 * q^26 + 2464 * q^28 + 7296 * q^29 - 2348 * q^31 - 1024 * q^32 - 8352 * q^34 + 3696 * q^35 - 9986 * q^37 + 10468 * q^38 - 1536 * q^40 + 6528 * q^41 + 6232 * q^43 + 13056 * q^44 + 14016 * q^46 + 29832 * q^47 + 10878 * q^49 + 10196 * q^50 - 1424 * q^52 + 45216 * q^53 + 19584 * q^55 - 4928 * q^56 + 29184 * q^58 - 19608 * q^59 + 22045 * q^61 + 18784 * q^62 + 8192 * q^64 - 2136 * q^65 - 48131 * q^67 - 33408 * q^68 - 7392 * q^70 + 102240 * q^71 + 61474 * q^73 + 19972 * q^74 + 41872 * q^76 - 31416 * q^77 - 38219 * q^79 + 12288 * q^80 - 52224 * q^82 - 8112 * q^83 - 50112 * q^85 + 24928 * q^86 - 26112 * q^88 - 88560 * q^89 + 13706 * q^91 - 28032 * q^92 + 119328 * q^94 + 62808 * q^95 + 136651 * q^97 - 87024 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more