LMFDB - Embedded newform 378.4.f.b.253.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(127,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("378.127");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.f (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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 2x^{6} + 53x^{5} + 38x^{4} - 166x^{3} + 7x^{2} + 1543x + 2707 $ x^8 - 2x^7 - 2x^6 + 53x^5 + 38x^4 - 166x^3 + 7x^2 + 1543*x + 2707
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{6} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			253.1
Root			$-1.30108 + 1.53317i$ of defining polynomial
Character	$\chi$	$=$	378.253
Dual form			378.4.f.b.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-9.75052 + 16.8884i) q^{5} +(3.50000 + 6.06218i) q^{7} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-9.75052 + 16.8884i) q^{5} +(3.50000 + 6.06218i) q^{7} -8.00000 q^{8} -39.0021 q^{10} +(18.9451 + 32.8139i) q^{11} +(-5.34044 + 9.24992i) q^{13} +(-7.00000 + 12.1244i) q^{14} +(-8.00000 - 13.8564i) q^{16} -56.7222 q^{17} +37.5846 q^{19} +(-39.0021 - 67.5536i) q^{20} +(-37.8902 + 65.6277i) q^{22} +(-31.1370 + 53.9310i) q^{23} +(-127.645 - 221.088i) q^{25} -21.3618 q^{26} -28.0000 q^{28} +(28.8504 + 49.9703i) q^{29} +(115.744 - 200.475i) q^{31} +(16.0000 - 27.7128i) q^{32} +(-56.7222 - 98.2457i) q^{34} -136.507 q^{35} -368.657 q^{37} +(37.5846 + 65.0984i) q^{38} +(78.0041 - 135.107i) q^{40} +(250.703 - 434.230i) q^{41} +(187.349 + 324.497i) q^{43} -151.561 q^{44} -124.548 q^{46} +(-280.327 - 485.541i) q^{47} +(-24.5000 + 42.4352i) q^{49} +(255.290 - 442.176i) q^{50} +(-21.3618 - 36.9997i) q^{52} -155.867 q^{53} -738.898 q^{55} +(-28.0000 - 48.4974i) q^{56} +(-57.7008 + 99.9407i) q^{58} +(-175.518 + 304.006i) q^{59} +(203.460 + 352.403i) q^{61} +462.977 q^{62} +64.0000 q^{64} +(-104.144 - 180.383i) q^{65} +(-383.155 + 663.644i) q^{67} +(113.444 - 196.491i) q^{68} +(-136.507 - 236.437i) q^{70} +268.881 q^{71} -1110.56 q^{73} +(-368.657 - 638.533i) q^{74} +(-75.1691 + 130.197i) q^{76} +(-132.616 + 229.697i) q^{77} +(120.928 + 209.453i) q^{79} +312.016 q^{80} +1002.81 q^{82} +(251.054 + 434.838i) q^{83} +(553.071 - 957.946i) q^{85} +(-374.697 + 648.994i) q^{86} +(-151.561 - 262.511i) q^{88} +1059.36 q^{89} -74.7662 q^{91} +(-124.548 - 215.724i) q^{92} +(560.654 - 971.081i) q^{94} +(-366.469 + 634.743i) q^{95} +(519.835 + 900.380i) q^{97} -98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8	$ 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8} - 4 q^{10} - 5 q^{11} + 21 q^{13} - 56 q^{14} - 64 q^{16} + 46 q^{17} - 188 q^{19} - 4 q^{20} + 10 q^{22} - 374 q^{23} - 41 q^{25} + 84 q^{26} - 224 q^{28} - 271 q^{29} + 243 q^{31} + 128 q^{32} + 46 q^{34} - 14 q^{35} - 362 q^{37} - 188 q^{38} + 8 q^{40} + 213 q^{41} + 238 q^{43} + 40 q^{44} - 1496 q^{46} - 675 q^{47} - 196 q^{49} + 82 q^{50} + 84 q^{52} - 108 q^{53} - 2828 q^{55} - 224 q^{56} + 542 q^{58} - 202 q^{59} + 1212 q^{61} + 972 q^{62} + 512 q^{64} - 549 q^{65} - 139 q^{67} - 92 q^{68} - 14 q^{70} + 2590 q^{71} - 4000 q^{73} - 362 q^{74} + 376 q^{76} + 35 q^{77} + 1545 q^{79} + 32 q^{80} + 852 q^{82} + 142 q^{83} + 793 q^{85} - 476 q^{86} + 40 q^{88} + 264 q^{89} + 294 q^{91} - 1496 q^{92} + 1350 q^{94} - 1244 q^{95} + 638 q^{97} - 784 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8 - 4 * q^10 - 5 * q^11 + 21 * q^13 - 56 * q^14 - 64 * q^16 + 46 * q^17 - 188 * q^19 - 4 * q^20 + 10 * q^22 - 374 * q^23 - 41 * q^25 + 84 * q^26 - 224 * q^28 - 271 * q^29 + 243 * q^31 + 128 * q^32 + 46 * q^34 - 14 * q^35 - 362 * q^37 - 188 * q^38 + 8 * q^40 + 213 * q^41 + 238 * q^43 + 40 * q^44 - 1496 * q^46 - 675 * q^47 - 196 * q^49 + 82 * q^50 + 84 * q^52 - 108 * q^53 - 2828 * q^55 - 224 * q^56 + 542 * q^58 - 202 * q^59 + 1212 * q^61 + 972 * q^62 + 512 * q^64 - 549 * q^65 - 139 * q^67 - 92 * q^68 - 14 * q^70 + 2590 * q^71 - 4000 * q^73 - 362 * q^74 + 376 * q^76 + 35 * q^77 + 1545 * q^79 + 32 * q^80 + 852 * q^82 + 142 * q^83 + 793 * q^85 - 476 * q^86 + 40 * q^88 + 264 * q^89 + 294 * q^91 - 1496 * q^92 + 1350 * q^94 - 1244 * q^95 + 638 * q^97 - 784 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	−9.75052	+	16.8884i	−0.872113	+	1.51054i	−0.0123055	+	0.999924i	$0.503917\pi$
$5$	−9.75052	+	16.8884i	−0.872113	+	1.51054i	−0.859807	+	0.510619i	$0.829416\pi$
$6$		0			0
$7$	3.50000	+	6.06218i	0.188982	+	0.327327i
$7$	3.50000	+	6.06218i	0.188982	+	0.327327i
$8$	−8.00000			−0.353553
$9$		0			0
$10$	−39.0021			−1.23335
$11$	18.9451	+	32.8139i	0.519287	+	0.899432i	0.999749	+	0.0224162i	$0.00713590\pi$
$11$	18.9451	+	32.8139i	0.519287	+	0.899432i	−0.480461	+	0.877016i	$0.659531\pi$
$12$		0			0
$13$	−5.34044	+	9.24992i	−0.113936	+	0.197343i	−0.917354	−	0.398072i	$-0.869679\pi$
$13$	−5.34044	+	9.24992i	−0.113936	+	0.197343i	0.803418	+	0.595416i	$0.203013\pi$
$14$	−7.00000	+	12.1244i	−0.133631	+	0.231455i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	−56.7222			−0.809244			−0.404622	−	0.914484i	$-0.632597\pi$
$17$	−56.7222			−0.809244			−0.404622	+	0.914484i	$0.632597\pi$
$18$		0			0
$19$	37.5846			0.453815			0.226908	−	0.973916i	$-0.427138\pi$
$19$	37.5846			0.453815			0.226908	+	0.973916i	$0.427138\pi$
$20$	−39.0021	−	67.5536i	−0.436056	−	0.755272i
$21$		0			0
$22$	−37.8902	+	65.6277i	−0.367192	+	0.635995i
$23$	−31.1370	+	53.9310i	−0.282284	+	0.488930i	−0.971947	−	0.235201i	$-0.924425\pi$
$23$	−31.1370	+	53.9310i	−0.282284	+	0.488930i	0.689663	+	0.724130i	$0.257759\pi$
$24$		0			0
$25$	−127.645	−	221.088i	−1.02116	−	1.76870i
$26$	−21.3618			−0.161130
$27$		0			0
$28$	−28.0000			−0.188982
$29$	28.8504	+	49.9703i	0.184737	+	0.319974i	0.943488	−	0.331407i	$-0.107523\pi$
$29$	28.8504	+	49.9703i	0.184737	+	0.319974i	−0.758751	+	0.651381i	$0.774190\pi$
$30$		0			0
$31$	115.744	−	200.475i	0.670590	−	1.16150i	−0.307147	−	0.951662i	$-0.599374\pi$
$31$	115.744	−	200.475i	0.670590	−	1.16150i	0.977737	−	0.209833i	$-0.0672922\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	−56.7222	−	98.2457i	−0.286111	−	0.495559i
$35$	−136.507			−0.659255
$36$		0			0
$37$	−368.657			−1.63802			−0.819011	−	0.573777i	$-0.805477\pi$
$37$	−368.657			−1.63802			−0.819011	+	0.573777i	$0.805477\pi$
$38$	37.5846	+	65.0984i	0.160448	+	0.277904i
$39$		0			0
$40$	78.0041	−	135.107i	0.308338	−	0.534058i
$41$	250.703	−	434.230i	0.954957	−	1.65403i	0.220489	−	0.975389i	$-0.429235\pi$
$41$	250.703	−	434.230i	0.954957	−	1.65403i	0.734467	−	0.678644i	$-0.237432\pi$
$42$		0			0
$43$	187.349	+	324.497i	0.664428	+	1.15082i	0.979440	+	0.201735i	$0.0646580\pi$
$43$	187.349	+	324.497i	0.664428	+	1.15082i	−0.315012	+	0.949088i	$0.602009\pi$
$44$	−151.561			−0.519287
$45$		0			0
$46$	−124.548			−0.399209
$47$	−280.327	−	485.541i	−0.869998	−	1.50688i	−0.861998	−	0.506912i	$-0.830787\pi$
$47$	−280.327	−	485.541i	−0.869998	−	1.50688i	−0.00800029	−	0.999968i	$-0.502547\pi$
$48$		0			0
$49$	−24.5000	+	42.4352i	−0.0714286	+	0.123718i
$50$	255.290	−	442.176i	0.722070	−	1.25066i
$51$		0			0
$52$	−21.3618	−	36.9997i	−0.0569682	−	0.0986717i
$53$	−155.867			−0.403962			−0.201981	−	0.979389i	$-0.564738\pi$
$53$	−155.867			−0.403962			−0.201981	+	0.979389i	$0.564738\pi$
$54$		0			0
$55$	−738.898			−1.81151
$56$	−28.0000	−	48.4974i	−0.0668153	−	0.115728i
$57$		0			0
$58$	−57.7008	+	99.9407i	−0.130629	+	0.226256i
$59$	−175.518	+	304.006i	−0.387296	+	0.670817i	−0.992085	−	0.125569i	$-0.959924\pi$
$59$	−175.518	+	304.006i	−0.387296	+	0.670817i	0.604789	+	0.796386i	$0.293258\pi$
$60$		0			0
$61$	203.460	+	352.403i	0.427055	+	0.739681i	0.996610	−	0.0822724i	$-0.0262177\pi$
$61$	203.460	+	352.403i	0.427055	+	0.739681i	−0.569555	+	0.821953i	$0.692884\pi$
$62$	462.977			0.948357
$63$		0			0
$64$	64.0000			0.125000
$65$	−104.144	−	180.383i	−0.198731	−	0.344211i
$66$		0			0
$67$	−383.155	+	663.644i	−0.698654	+	1.21010i	0.270280	+	0.962782i	$0.412884\pi$
$67$	−383.155	+	663.644i	−0.698654	+	1.21010i	−0.968933	+	0.247322i	$0.920449\pi$
$68$	113.444	−	196.491i	0.202311	−	0.350413i
$69$		0			0
$70$	−136.507	−	236.437i	−0.233082	−	0.403710i
$71$	268.881			0.449441			0.224721	−	0.974423i	$-0.427853\pi$
$71$	268.881			0.449441			0.224721	+	0.974423i	$0.427853\pi$
$72$		0			0
$73$	−1110.56			−1.78057			−0.890286	−	0.455402i	$-0.849495\pi$
$73$	−1110.56			−1.78057			−0.890286	+	0.455402i	$0.849495\pi$
$74$	−368.657	−	638.533i	−0.579128	−	1.00308i
$75$		0			0
$76$	−75.1691	+	130.197i	−0.113454	+	0.196508i
$77$	−132.616	+	229.697i	−0.196272	+	0.339953i
$78$		0			0
$79$	120.928	+	209.453i	0.172220	+	0.298295i	0.939196	−	0.343382i	$-0.111573\pi$
$79$	120.928	+	209.453i	0.172220	+	0.298295i	−0.766975	+	0.641676i	$0.778239\pi$
$80$	312.016			0.436056
$81$		0			0
$82$	1002.81			1.35051
$83$	251.054	+	434.838i	0.332009	+	0.575056i	0.982906	−	0.184110i	$-0.0589403\pi$
$83$	251.054	+	434.838i	0.332009	+	0.575056i	−0.650897	+	0.759166i	$0.725607\pi$
$84$		0			0
$85$	553.071	−	957.946i	0.705752	−	1.22240i
$86$	−374.697	+	648.994i	−0.469821	+	0.813754i
$87$		0			0
$88$	−151.561	−	262.511i	−0.183596	−	0.317997i
$89$	1059.36			1.26171			0.630854	−	0.775901i	$-0.282705\pi$
$89$	1059.36			1.26171			0.630854	+	0.775901i	$0.282705\pi$
$90$		0			0
$91$	−74.7662			−0.0861277
$92$	−124.548	−	215.724i	−0.141142	−	0.244465i
$93$		0			0
$94$	560.654	−	971.081i	0.615181	−	1.06553i
$95$	−366.469	+	634.743i	−0.395778	+	0.685508i
$96$		0			0
$97$	519.835	+	900.380i	0.544136	+	0.942472i	0.998661	+	0.0517376i	$0.0164760\pi$
$97$	519.835	+	900.380i	0.544136	+	0.942472i	−0.454524	+	0.890734i	$0.650191\pi$
$98$	−98.0000			−0.101015
$99$		0			0
$100$	1021.16			1.02116
$101$	−509.276	−	882.091i	−0.501731	−	0.869023i	−0.999998	−	0.00199973i	$-0.999363\pi$
$101$	−509.276	−	882.091i	−0.501731	−	0.869023i	0.498267	−	0.867024i	$-0.333970\pi$
$102$		0			0
$103$	161.367	−	279.497i	0.154369	−	0.267375i	−0.778460	−	0.627694i	$-0.783999\pi$
$103$	161.367	−	279.497i	0.154369	−	0.267375i	0.932829	+	0.360319i	$0.117332\pi$
$104$	42.7235	−	73.9993i	0.0402826	−	0.0697714i
$105$		0			0
$106$	−155.867	−	269.970i	−0.142822	−	0.247375i
$107$	−252.766			−0.228372			−0.114186	−	0.993459i	$-0.536426\pi$
$107$	−252.766			−0.228372			−0.114186	+	0.993459i	$0.536426\pi$
$108$		0			0
$109$	−552.267			−0.485299			−0.242650	−	0.970114i	$-0.578017\pi$
$109$	−552.267			−0.485299			−0.242650	+	0.970114i	$0.578017\pi$
$110$	−738.898	−	1279.81i	−0.640465	−	1.10932i
$111$		0			0
$112$	56.0000	−	96.9948i	0.0472456	−	0.0818317i
$113$	535.675	−	927.816i	0.445947	−	0.772404i	−0.552170	−	0.833731i	$-0.686200\pi$
$113$	535.675	−	927.816i	0.445947	−	0.772404i	0.998118	+	0.0613277i	$0.0195335\pi$
$114$		0			0
$115$	−607.205	−	1051.71i	−0.492366	−	0.852803i
$116$	−230.803			−0.184737
$117$		0			0
$118$	−702.071			−0.547720
$119$	−198.528	−	343.860i	−0.152933	−	0.264887i
$120$		0			0
$121$	−52.3333	+	90.6439i	−0.0393188	+	0.0681021i
$122$	−406.920	+	704.805i	−0.301973	+	0.523033i
$123$		0			0
$124$	462.977	+	801.900i	0.335295	+	0.580748i
$125$	2540.79			1.81804
$126$		0			0
$127$	1449.67			1.01289			0.506447	−	0.862271i	$-0.330959\pi$
$127$	1449.67			1.01289			0.506447	+	0.862271i	$0.330959\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	208.288	−	360.766i	0.140524	−	0.243394i
$131$	−981.232	+	1699.54i	−0.654432	+	1.13351i	0.327604	+	0.944815i	$0.393759\pi$
$131$	−981.232	+	1699.54i	−0.654432	+	1.13351i	−0.982036	+	0.188694i	$0.939574\pi$
$132$		0			0
$133$	131.546	+	227.844i	0.0857630	+	0.148546i
$134$	−1532.62			−0.988046
$135$		0			0
$136$	453.778			0.286111
$137$	1476.50	+	2557.38i	0.920774	+	1.59483i	0.798221	+	0.602365i	$0.205775\pi$
$137$	1476.50	+	2557.38i	0.920774	+	1.59483i	0.122553	+	0.992462i	$0.460892\pi$
$138$		0			0
$139$	−90.1925	+	156.218i	−0.0550362	+	0.0953255i	−0.892231	−	0.451579i	$-0.850861\pi$
$139$	−90.1925	+	156.218i	−0.0550362	+	0.0953255i	0.837195	+	0.546905i	$0.184194\pi$
$140$	273.014	−	472.875i	0.164814	−	0.285466i
$141$		0			0
$142$	268.881	+	465.716i	0.158902	+	0.275225i
$143$	−404.701			−0.236663
$144$		0			0
$145$	−1125.22			−0.644447
$146$	−1110.56	−	1923.55i	−0.629527	−	1.09037i
$147$		0			0
$148$	737.314	−	1277.07i	0.409506	−	0.709285i
$149$	−958.042	+	1659.38i	−0.526751	+	0.912359i	0.472763	+	0.881189i	$0.343257\pi$
$149$	−958.042	+	1659.38i	−0.526751	+	0.912359i	−0.999514	+	0.0311696i	$0.990077\pi$
$150$		0			0
$151$	720.359	+	1247.70i	0.388225	+	0.672425i	0.992211	−	0.124570i	$-0.0397550\pi$
$151$	720.359	+	1247.70i	0.388225	+	0.672425i	−0.603986	+	0.796995i	$0.706422\pi$
$152$	−300.677			−0.160448
$153$		0			0
$154$	−530.463			−0.277571
$155$	2257.13	+	3909.47i	1.16966	+	2.02591i
$156$		0			0
$157$	−190.466	+	329.896i	−0.0968204	+	0.167698i	−0.910367	−	0.413802i	$-0.864201\pi$
$157$	−190.466	+	329.896i	−0.0968204	+	0.167698i	0.813546	+	0.581500i	$0.197534\pi$
$158$	−241.855	+	418.905i	−0.121778	+	0.210926i
$159$		0			0
$160$	312.016	+	540.428i	0.154169	+	0.267029i
$161$	−435.919			−0.213386
$162$		0			0
$163$	−184.568			−0.0886900			−0.0443450	−	0.999016i	$-0.514120\pi$
$163$	−184.568			−0.0886900			−0.0443450	+	0.999016i	$0.514120\pi$
$164$	1002.81	+	1736.92i	0.477478	+	0.827017i
$165$		0			0
$166$	−502.108	+	869.676i	−0.234766	+	0.406626i
$167$	−1029.01	+	1782.29i	−0.476808	+	0.825855i	−0.999647	−	0.0265763i	$-0.991540\pi$
$167$	−1029.01	+	1782.29i	−0.476808	+	0.825855i	0.522839	+	0.852431i	$0.324873\pi$
$168$		0			0
$169$	1041.46	+	1803.86i	0.474037	+	0.821056i
$170$	2212.28			0.998084
$171$		0			0
$172$	−1498.79			−0.664428
$173$	579.503	+	1003.73i	0.254675	+	0.441110i	0.964807	−	0.262958i	$-0.0846982\pi$
$173$	579.503	+	1003.73i	0.254675	+	0.441110i	−0.710132	+	0.704068i	$0.751365\pi$
$174$		0			0
$175$	893.516	−	1547.61i	0.385962	−	0.668507i
$176$	303.122	−	525.022i	0.129822	−	0.224858i
$177$		0			0
$178$	1059.36	+	1834.87i	0.446081	+	0.772635i
$179$	−2830.87			−1.18206			−0.591030	−	0.806649i	$-0.701279\pi$
$179$	−2830.87			−1.18206			−0.591030	+	0.806649i	$0.701279\pi$
$180$		0			0
$181$	−1576.02			−0.647207			−0.323604	−	0.946193i	$-0.604894\pi$
$181$	−1576.02			−0.647207			−0.323604	+	0.946193i	$0.604894\pi$
$182$	−74.7662	−	129.499i	−0.0304508	−	0.0527423i
$183$		0			0
$184$	249.096	−	431.448i	0.0998023	−	0.172863i
$185$	3594.60	−	6226.02i	1.42854	−	2.47430i
$186$		0			0
$187$	−1074.61	−	1861.27i	−0.420230	−	0.727860i
$188$	2242.62			0.869998
$189$		0			0
$190$	−1465.88			−0.559715
$191$	311.810	+	540.070i	0.118124	+	0.204597i	0.919024	−	0.394201i	$-0.128979\pi$
$191$	311.810	+	540.070i	0.118124	+	0.204597i	−0.800900	+	0.598798i	$0.795645\pi$
$192$		0			0
$193$	−1395.68	+	2417.40i	−0.520537	+	0.901596i	0.479178	+	0.877718i	$0.340935\pi$
$193$	−1395.68	+	2417.40i	−0.520537	+	0.901596i	−0.999715	+	0.0238784i	$0.992399\pi$
$194$	−1039.67	+	1800.76i	−0.384763	+	0.666428i
$195$		0			0
$196$	−98.0000	−	169.741i	−0.0357143	−	0.0618590i
$197$	−3683.66			−1.33223			−0.666117	−	0.745848i	$-0.732045\pi$
$197$	−3683.66			−1.33223			−0.666117	+	0.745848i	$0.732045\pi$
$198$		0			0
$199$	−1511.55			−0.538448			−0.269224	−	0.963078i	$-0.586767\pi$
$199$	−1511.55			−0.538448			−0.269224	+	0.963078i	$0.586767\pi$
$200$	1021.16	+	1768.70i	0.361035	+	0.625331i
$201$		0			0
$202$	1018.55	−	1764.18i	0.354777	−	0.614492i
$203$	−201.953	+	349.792i	−0.0698241	+	0.120939i
$204$		0			0
$205$	4888.97	+	8467.94i	1.66566	+	2.88501i
$206$	645.470			0.218311
$207$		0			0
$208$	170.894			0.0569682
$209$	712.043	+	1233.30i	0.235661	+	0.408176i
$210$		0			0
$211$	268.215	−	464.562i	0.0875104	−	0.151572i	−0.818948	−	0.573868i	$-0.805442\pi$
$211$	268.215	−	464.562i	0.0875104	−	0.151572i	0.906458	+	0.422296i	$0.138776\pi$
$212$	311.734	−	539.940i	0.100991	−	0.174921i
$213$		0			0
$214$	−252.766	−	437.804i	−0.0807418	−	0.139849i
$215$	−7306.98			−2.31782
$216$		0			0
$217$	1620.42			0.506918
$218$	−552.267	−	956.555i	−0.171579	−	0.297184i
$219$		0			0
$220$	1477.80	−	2559.62i	0.452877	−	0.784406i
$221$	302.922	−	524.676i	0.0922023	−	0.159699i
$222$		0			0
$223$	−2516.10	−	4358.01i	−0.755563	−	1.30867i	−0.945094	−	0.326799i	$-0.894030\pi$
$223$	−2516.10	−	4358.01i	−0.755563	−	1.30867i	0.189531	−	0.981875i	$-0.439303\pi$
$224$	224.000			0.0668153
$225$		0			0
$226$	2142.70			0.630665
$227$	−334.050	−	578.592i	−0.0976727	−	0.169174i	0.813048	−	0.582196i	$-0.197806\pi$
$227$	−334.050	−	578.592i	−0.0976727	−	0.169174i	−0.910721	+	0.413022i	$0.864473\pi$
$228$		0			0
$229$	1425.20	−	2468.52i	0.411267	−	0.712335i	−0.583762	−	0.811925i	$-0.698420\pi$
$229$	1425.20	−	2468.52i	0.411267	−	0.712335i	0.995029	+	0.0995900i	$0.0317531\pi$
$230$	1214.41	−	2103.42i	0.348155	−	0.603023i
$231$		0			0
$232$	−230.803	−	399.763i	−0.0653145	−	0.113128i
$233$	294.918			0.0829217			0.0414608	−	0.999140i	$-0.486799\pi$
$233$	294.918			0.0829217			0.0414608	+	0.999140i	$0.486799\pi$
$234$		0			0
$235$	10933.3			3.03494
$236$	−702.071	−	1216.02i	−0.193648	−	0.335408i
$237$		0			0
$238$	397.055	−	687.720i	0.108140	−	0.187304i
$239$	1707.52	−	2957.52i	0.462136	−	0.800443i	−0.536931	−	0.843626i	$-0.680417\pi$
$239$	1707.52	−	2957.52i	0.462136	−	0.800443i	0.999067	+	0.0431832i	$0.0137499\pi$
$240$		0			0
$241$	−699.779	−	1212.05i	−0.187040	−	0.323964i	0.757222	−	0.653158i	$-0.226556\pi$
$241$	−699.779	−	1212.05i	−0.187040	−	0.323964i	−0.944262	+	0.329194i	$0.893223\pi$
$242$	−209.333			−0.0556051
$243$		0			0
$244$	−1627.68			−0.427055
$245$	−477.775	−	827.531i	−0.124588	−	0.215792i
$246$		0			0
$247$	−200.718	+	347.654i	−0.0517060	+	0.0895575i
$248$	−925.954	+	1603.80i	−0.237089	+	0.410651i
$249$		0			0
$250$	2540.79	+	4400.78i	0.642775	+	1.11332i
$251$	7348.11			1.84784			0.923922	−	0.382581i	$-0.124965\pi$
$251$	7348.11			1.84784			0.923922	+	0.382581i	$0.124965\pi$
$252$		0			0
$253$	−2359.58			−0.586345
$254$	1449.67	+	2510.90i	0.358112	+	0.620268i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−3681.17	+	6375.97i	−0.893483	+	1.54756i	−0.0578114	+	0.998328i	$0.518412\pi$
$257$	−3681.17	+	6375.97i	−0.893483	+	1.54756i	−0.835671	+	0.549230i	$0.814921\pi$
$258$		0			0
$259$	−1290.30	−	2234.86i	−0.309557	−	0.536169i
$260$	833.153			0.198731
$261$		0			0
$262$	−3924.93			−0.925507
$263$	−2406.93	−	4168.92i	−0.564326	−	0.977441i	−0.997112	−	0.0759441i	$-0.975803\pi$
$263$	−2406.93	−	4168.92i	−0.564326	−	0.977441i	0.432787	−	0.901496i	$-0.357530\pi$
$264$		0			0
$265$	1519.79	−	2632.35i	0.352301	−	0.610203i
$266$	−263.092	+	455.689i	−0.0606436	+	0.105038i
$267$		0			0
$268$	−1532.62	−	2654.57i	−0.349327	−	0.605052i
$269$	−2082.46			−0.472007			−0.236004	−	0.971752i	$-0.575838\pi$
$269$	−2082.46			−0.472007			−0.236004	+	0.971752i	$0.575838\pi$
$270$		0			0
$271$	−8580.06			−1.92325			−0.961627	−	0.274362i	$-0.911533\pi$
$271$	−8580.06			−1.92325			−0.961627	+	0.274362i	$0.911533\pi$
$272$	453.778	+	785.966i	0.101156	+	0.175207i
$273$		0			0
$274$	−2953.00	+	5114.75i	−0.651085	+	1.12771i
$275$	4836.50	−	8377.06i	1.06055	−	1.83693i
$276$		0			0
$277$	3878.94	+	6718.52i	0.841383	+	1.45732i	0.888726	+	0.458439i	$0.151591\pi$
$277$	3878.94	+	6718.52i	0.841383	+	1.45732i	−0.0473431	+	0.998879i	$0.515075\pi$
$278$	−360.770			−0.0778329
$279$		0			0
$280$	1092.06			0.233082
$281$	3486.85	+	6039.40i	0.740243	+	1.28214i	0.952385	+	0.304899i	$0.0986227\pi$
$281$	3486.85	+	6039.40i	0.740243	+	1.28214i	−0.212142	+	0.977239i	$0.568044\pi$
$282$		0			0
$283$	4283.83	−	7419.82i	0.899814	−	1.55852i	0.0720833	−	0.997399i	$-0.477035\pi$
$283$	4283.83	−	7419.82i	0.899814	−	1.55852i	0.827731	−	0.561125i	$-0.189631\pi$
$284$	−537.763	+	931.432i	−0.112360	+	0.194614i
$285$		0			0
$286$	−404.701	−	700.962i	−0.0836729	−	0.144926i
$287$	3509.84			0.721879
$288$		0			0
$289$	−1695.59			−0.345124
$290$	−1125.22	−	1948.95i	−0.227846	−	0.394641i
$291$		0			0
$292$	2221.13	−	3847.11i	0.445143	−	0.771010i
$293$	332.494	−	575.897i	0.0662953	−	0.114827i	−0.830973	−	0.556313i	$-0.812215\pi$
$293$	332.494	−	575.897i	0.0662953	−	0.114827i	0.897268	+	0.441487i	$0.145549\pi$
$294$		0			0
$295$	−3422.78	−	5928.43i	−0.675532	−	1.17006i
$296$	2949.26			0.579128
$297$		0			0
$298$	−3832.17			−0.744938
$299$	−332.571	−	576.030i	−0.0643247	−	0.111414i
$300$		0			0
$301$	−1311.44	+	2271.48i	−0.251130	+	0.434970i
$302$	−1440.72	+	2495.40i	−0.274517	+	0.475477i
$303$		0			0
$304$	−300.677	−	520.787i	−0.0567269	−	0.0982539i
$305$	−7935.35			−1.48976
$306$		0			0
$307$	226.338			0.0420776			0.0210388	−	0.999779i	$-0.493303\pi$
$307$	226.338			0.0420776			0.0210388	+	0.999779i	$0.493303\pi$
$308$	−530.463	−	918.788i	−0.0981361	−	0.169977i
$309$		0			0
$310$	−4514.26	+	7818.93i	−0.827074	+	1.43253i
$311$	2813.11	−	4872.44i	0.512915	−	0.888395i	−0.486973	−	0.873417i	$-0.661899\pi$
$311$	2813.11	−	4872.44i	0.512915	−	0.888395i	0.999888	−	0.0149780i	$-0.00476783\pi$
$312$		0			0
$313$	−307.465	−	532.544i	−0.0555237	−	0.0961699i	0.836928	−	0.547314i	$-0.184350\pi$
$313$	−307.465	−	532.544i	−0.0555237	−	0.0961699i	−0.892451	+	0.451144i	$0.851016\pi$
$314$	−761.862			−0.136925
$315$		0			0
$316$	−967.420			−0.172220
$317$	404.159	+	700.024i	0.0716083	+	0.124029i	0.899606	−	0.436702i	$-0.143854\pi$
$317$	404.159	+	700.024i	0.0716083	+	0.124029i	−0.827998	+	0.560731i	$0.810520\pi$
$318$		0			0
$319$	−1093.15	+	1893.39i	−0.191863	+	0.332317i
$320$	−624.033	+	1080.86i	−0.109014	+	0.188818i
$321$		0			0
$322$	−435.919	−	755.033i	−0.0754435	−	0.130672i
$323$	−2131.88			−0.367248
$324$		0			0
$325$	2726.72			0.465389
$326$	−184.568	−	319.681i	−0.0313566	−	0.0543113i
$327$		0			0
$328$	−2005.62	+	3473.84i	−0.337628	+	0.584789i
$329$	1962.29	−	3398.78i	0.328828	−	0.569547i
$330$		0			0
$331$	174.906	+	302.946i	0.0290444	+	0.0503065i	0.880182	−	0.474636i	$-0.157420\pi$
$331$	174.906	+	302.946i	0.0290444	+	0.0503065i	−0.851138	+	0.524942i	$0.824087\pi$
$332$	−2008.43			−0.332009
$333$		0			0
$334$	−4116.02			−0.674308
$335$	−7471.91	−	12941.7i	−1.21861	−	2.11069i
$336$		0			0
$337$	2014.56	−	3489.32i	0.325638	−	0.564021i	−0.656003	−	0.754758i	$-0.727754\pi$
$337$	2014.56	−	3489.32i	0.325638	−	0.564021i	0.981641	+	0.190737i	$0.0610876\pi$
$338$	−2082.92	+	3607.72i	−0.335195	+	0.580574i
$339$		0			0
$340$	2212.28	+	3831.79i	0.352876	+	0.611199i
$341$	8771.14			1.39291
$342$		0			0
$343$	−343.000			−0.0539949
$344$	−1498.79	−	2595.98i	−0.234911	−	0.406877i
$345$		0			0
$346$	−1159.01	+	2007.46i	−0.180082	+	0.311912i
$347$	−4507.20	+	7806.70i	−0.697289	+	1.20774i	0.272114	+	0.962265i	$0.412277\pi$
$347$	−4507.20	+	7806.70i	−0.697289	+	1.20774i	−0.969403	+	0.245475i	$0.921056\pi$
$348$		0			0
$349$	−1349.23	−	2336.93i	−0.206942	−	0.358433i	0.743808	−	0.668393i	$-0.233018\pi$
$349$	−1349.23	−	2336.93i	−0.206942	−	0.358433i	−0.950750	+	0.309960i	$0.899684\pi$
$350$	3574.06			0.545833
$351$		0			0
$352$	1212.49			0.183596
$353$	234.358	+	405.920i	0.0353360	+	0.0612038i	0.883153	−	0.469086i	$-0.155417\pi$
$353$	234.358	+	405.920i	0.0353360	+	0.0612038i	−0.847817	+	0.530290i	$0.822083\pi$
$354$		0			0
$355$	−2621.73	+	4540.97i	−0.391963	+	0.678901i
$356$	−2118.72	+	3669.73i	−0.315427	+	0.546336i
$357$		0			0
$358$	−2830.87	−	4903.20i	−0.417922	−	0.723861i
$359$	−2045.04			−0.300650			−0.150325	−	0.988637i	$-0.548032\pi$
$359$	−2045.04			−0.300650			−0.150325	+	0.988637i	$0.548032\pi$
$360$		0			0
$361$	−5446.40			−0.794052
$362$	−1576.02	−	2729.74i	−0.228822	−	0.396332i
$363$		0			0
$364$	149.532	−	258.998i	0.0215319	−	0.0372944i
$365$	10828.6	−	18755.6i	1.55286	−	2.68963i
$366$		0			0
$367$	−3088.06	−	5348.68i	−0.439225	−	0.760759i	0.558405	−	0.829568i	$-0.311413\pi$
$367$	−3088.06	−	5348.68i	−0.439225	−	0.760759i	−0.997630	+	0.0688089i	$0.978080\pi$
$368$	996.386			0.141142
$369$		0			0
$370$	14378.4			2.02026
$371$	−545.535	−	944.895i	−0.0763417	−	0.132228i
$372$		0			0
$373$	2154.38	−	3731.50i	0.299060	−	0.517988i	−0.676861	−	0.736111i	$-0.736660\pi$
$373$	2154.38	−	3731.50i	0.299060	−	0.517988i	0.975921	+	0.218123i	$0.0699934\pi$
$374$	2149.21	−	3722.55i	0.297148	−	0.514675i
$375$		0			0
$376$	2242.62	+	3884.33i	0.307591	+	0.532763i
$377$	−616.295			−0.0841931
$378$		0			0
$379$	−6931.22			−0.939401			−0.469700	−	0.882826i	$-0.655638\pi$
$379$	−6931.22			−0.939401			−0.469700	+	0.882826i	$0.655638\pi$
$380$	−1465.88	−	2538.97i	−0.197889	−	0.342754i
$381$		0			0
$382$	−623.619	+	1080.14i	−0.0835265	+	0.144672i
$383$	2513.13	−	4352.87i	0.335287	−	0.580734i	−0.648253	−	0.761425i	$-0.724500\pi$
$383$	2513.13	−	4352.87i	0.335287	−	0.580734i	0.983540	+	0.180691i	$0.0578334\pi$
$384$		0			0
$385$	−2586.14	−	4479.33i	−0.342343	−	0.592955i
$386$	−5582.74			−0.736150
$387$		0			0
$388$	−4158.68			−0.544136
$389$	−2727.32	−	4723.85i	−0.355477	−	0.615703i	0.631723	−	0.775194i	$-0.282348\pi$
$389$	−2727.32	−	4723.85i	−0.355477	−	0.615703i	−0.987199	+	0.159491i	$0.949015\pi$
$390$		0			0
$391$	1766.16	−	3059.08i	0.228436	−	0.395664i
$392$	196.000	−	339.482i	0.0252538	−	0.0437409i
$393$		0			0
$394$	−3683.66	−	6380.29i	−0.471016	−	0.815823i
$395$	−4716.42			−0.600782
$396$		0			0
$397$	6886.06			0.870533			0.435266	−	0.900302i	$-0.356654\pi$
$397$	6886.06			0.870533			0.435266	+	0.900302i	$0.356654\pi$
$398$	−1511.55	−	2618.09i	−0.190370	−	0.329731i
$399$		0			0
$400$	−2042.32	+	3537.40i	−0.255290	+	0.442176i
$401$	−4207.03	+	7286.78i	−0.523912	+	0.907443i	0.475700	+	0.879608i	$0.342195\pi$
$401$	−4207.03	+	7286.78i	−0.523912	+	0.907443i	−0.999613	+	0.0278354i	$0.991139\pi$
$402$		0			0
$403$	1236.25	+	2141.25i	0.152809	+	0.264673i
$404$	4074.20			0.501731
$405$		0			0
$406$	−807.811			−0.0987462
$407$	−6984.24	−	12097.1i	−0.850605	−	1.47329i
$408$		0			0
$409$	−4847.71	+	8396.49i	−0.586073	+	1.01511i	0.408667	+	0.912683i	$0.365994\pi$
$409$	−4847.71	+	8396.49i	−0.586073	+	1.01511i	−0.994741	+	0.102425i	$0.967340\pi$
$410$	−9777.93	+	16935.9i	−1.17780	+	2.04001i
$411$		0			0
$412$	645.470	+	1117.99i	0.0771845	+	0.133687i
$413$	−2457.25			−0.292768
$414$		0			0
$415$	−9791.61			−1.15820
$416$	170.894	+	295.997i	0.0201413	+	0.0348857i
$417$		0			0
$418$	−1424.09	+	2466.59i	−0.166637	+	0.288624i
$419$	−2008.54	+	3478.89i	−0.234185	+	0.405621i	−0.959036	−	0.283286i	$-0.908576\pi$
$419$	−2008.54	+	3478.89i	−0.234185	+	0.405621i	0.724850	+	0.688906i	$0.241909\pi$
$420$		0			0
$421$	796.284	+	1379.20i	0.0921818	+	0.159663i	0.908429	−	0.418039i	$-0.137283\pi$
$421$	796.284	+	1379.20i	0.0921818	+	0.159663i	−0.816247	+	0.577703i	$0.803949\pi$
$422$	1072.86			0.123758
$423$		0			0
$424$	1246.94			0.142822
$425$	7240.31	+	12540.6i	0.826369	+	1.43131i
$426$		0			0
$427$	−1424.22	+	2466.82i	−0.161412	+	0.279573i
$428$	505.532	−	875.608i	0.0570931	−	0.0988881i
$429$		0			0
$430$	−7306.98	−	12656.1i	−0.819474	−	1.41937i
$431$	−5936.30			−0.663437			−0.331718	−	0.943378i	$-0.607628\pi$
$431$	−5936.30			−0.663437			−0.331718	+	0.943378i	$0.607628\pi$
$432$		0			0
$433$	3116.42			0.345879			0.172940	−	0.984932i	$-0.444673\pi$
$433$	3116.42			0.345879			0.172940	+	0.984932i	$0.444673\pi$
$434$	1620.42	+	2806.65i	0.179223	+	0.310423i
$435$		0			0
$436$	1104.53	−	1913.11i	0.121325	−	0.210141i
$437$	−1170.27	+	2026.97i	−0.128105	+	0.221884i
$438$		0			0
$439$	2961.66	+	5129.75i	0.321987	+	0.557699i	0.980898	−	0.194523i	$-0.0623158\pi$
$439$	2961.66	+	5129.75i	0.321987	+	0.557699i	−0.658911	+	0.752221i	$0.728982\pi$
$440$	5911.18			0.640465
$441$		0			0
$442$	1211.69			0.130394
$443$	5630.51	+	9752.33i	0.603868	+	1.04593i	0.992229	+	0.124423i	$0.0397080\pi$
$443$	5630.51	+	9752.33i	0.603868	+	1.04593i	−0.388361	+	0.921507i	$0.626959\pi$
$444$		0			0
$445$	−10329.3	+	17890.9i	−1.10035	+	1.90586i
$446$	5032.20	−	8716.03i	0.534264	−	0.925372i
$447$		0			0
$448$	224.000	+	387.979i	0.0236228	+	0.0409159i
$449$	17965.0			1.88824			0.944119	−	0.329605i	$-0.106916\pi$
$449$	17965.0			1.88824			0.944119	+	0.329605i	$0.106916\pi$
$450$		0			0
$451$	18998.4			1.98359
$452$	2142.70	+	3711.27i	0.222974	+	0.386202i
$453$		0			0
$454$	668.100	−	1157.18i	0.0690650	−	0.119624i
$455$	729.009	−	1262.68i	0.0751131	−	0.130100i
$456$		0			0
$457$	−5016.67	−	8689.12i	−0.513501	−	0.889409i	−0.999877	−	0.0156600i	$-0.995015\pi$
$457$	−5016.67	−	8689.12i	−0.513501	−	0.889409i	0.486377	−	0.873749i	$-0.338318\pi$
$458$	5700.81			0.581619
$459$		0			0
$460$	4857.64			0.492366
$461$	2211.40	+	3830.25i	0.223416	+	0.386968i	0.955843	−	0.293877i	$-0.0949457\pi$
$461$	2211.40	+	3830.25i	0.223416	+	0.386968i	−0.732427	+	0.680846i	$0.761612\pi$
$462$		0			0
$463$	1186.07	−	2054.33i	0.119052	−	0.206205i	−0.800340	−	0.599546i	$-0.795348\pi$
$463$	1186.07	−	2054.33i	0.119052	−	0.206205i	0.919392	+	0.393342i	$0.128681\pi$
$464$	461.606	−	799.525i	0.0461843	−	0.0799936i
$465$		0			0
$466$	294.918	+	510.814i	0.0293172	+	0.0507789i
$467$	−14269.6			−1.41396			−0.706981	−	0.707233i	$-0.749943\pi$
$467$	−14269.6			−1.41396			−0.706981	+	0.707233i	$0.749943\pi$
$468$		0			0
$469$	−5364.17			−0.528133
$470$	10933.3	+	18937.1i	1.07301	+	1.85852i
$471$		0			0
$472$	1404.14	−	2432.05i	0.136930	−	0.237170i
$473$	−7098.67	+	12295.3i	−0.690058	+	1.19522i
$474$		0			0
$475$	−4797.49	−	8309.49i	−0.463418	−	0.802664i
$476$	1588.22			0.152933
$477$		0			0
$478$	6830.09			0.653559
$479$	5959.10	+	10321.5i	0.568430	+	0.984550i	0.996721	+	0.0809095i	$0.0257825\pi$
$479$	5959.10	+	10321.5i	0.568430	+	0.984550i	−0.428291	+	0.903641i	$0.640884\pi$
$480$		0			0
$481$	1968.79	−	3410.05i	0.186630	−	0.323253i
$482$	1399.56	−	2424.11i	0.132258	−	0.229077i
$483$		0			0
$484$	−209.333	−	362.576i	−0.0196594	−	0.0340511i
$485$	−20274.6			−1.89819
$486$		0			0
$487$	6743.42			0.627461			0.313730	−	0.949512i	$-0.398421\pi$
$487$	6743.42			0.627461			0.313730	+	0.949512i	$0.398421\pi$
$488$	−1627.68	−	2819.22i	−0.150987	−	0.261517i
$489$		0			0
$490$	955.550	−	1655.06i	0.0880967	−	0.152588i
$491$	3426.21	−	5934.37i	0.314914	−	0.545447i	−0.664505	−	0.747283i	$-0.731358\pi$
$491$	3426.21	−	5934.37i	0.314914	−	0.545447i	0.979419	+	0.201837i	$0.0646911\pi$
$492$		0			0
$493$	−1636.46	−	2834.43i	−0.149498	−	0.258937i
$494$	−802.873			−0.0731234
$495$		0			0
$496$	−3703.82			−0.335295
$497$	941.084	+	1630.01i	0.0849364	+	0.147114i
$498$		0			0
$499$	−896.271	+	1552.39i	−0.0804060	+	0.139267i	−0.903424	−	0.428747i	$-0.858955\pi$
$499$	−896.271	+	1552.39i	−0.0804060	+	0.139267i	0.823018	+	0.568015i	$0.192288\pi$
$500$	−5081.59	+	8801.56i	−0.454511	+	0.787236i
$501$		0			0
$502$	7348.11	+	12727.3i	0.653311	+	1.13157i
$503$	−14082.3			−1.24830			−0.624152	−	0.781303i	$-0.714556\pi$
$503$	−14082.3			−1.24830			−0.624152	+	0.781303i	$0.714556\pi$
$504$		0			0
$505$	19862.8			1.75026
$506$	−2359.58	−	4086.91i	−0.207304	−	0.359062i
$507$		0			0
$508$	−2899.34	+	5021.81i	−0.253223	+	0.438596i
$509$	2529.50	−	4381.23i	0.220271	−	0.381521i	−0.734619	−	0.678480i	$-0.762639\pi$
$509$	2529.50	−	4381.23i	0.220271	−	0.381521i	0.954890	+	0.296959i	$0.0959723\pi$
$510$		0			0
$511$	−3886.98	−	6732.44i	−0.336496	−	0.582829i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	−14724.7			−1.26358
$515$	3146.83	+	5450.47i	0.269254	+	0.466362i
$516$		0			0
$517$	10621.6	−	18397.2i	0.903558	−	1.56501i
$518$	2580.60	−	4469.73i	0.218890	−	0.379129i
$519$		0			0
$520$	833.153	+	1443.06i	0.0702619	+	0.121697i
$521$	23240.4			1.95428			0.977142	−	0.212585i	$-0.0681884\pi$
$521$	23240.4			1.95428			0.977142	+	0.212585i	$0.0681884\pi$
$522$		0			0
$523$	7945.65			0.664320			0.332160	−	0.943223i	$-0.392223\pi$
$523$	7945.65			0.664320			0.332160	+	0.943223i	$0.392223\pi$
$524$	−3924.93	−	6798.17i	−0.327216	−	0.566755i
$525$		0			0
$526$	4813.86	−	8337.85i	0.399038	−	0.691155i
$527$	−6565.27	+	11371.4i	−0.542671	+	0.939934i
$528$		0			0
$529$	4144.47	+	7178.43i	0.340632	+	0.589992i
$530$	6079.14			0.498228
$531$		0			0
$532$	−1052.37			−0.0857630
$533$	2677.73	+	4637.96i	0.217608	+	0.376909i
$534$		0			0
$535$	2464.60	−	4268.81i	0.199166	−	0.344966i
$536$	3065.24	−	5309.15i	0.247011	−	0.427836i
$537$		0			0
$538$	−2082.46	−	3606.93i	−0.166880	−	0.289044i
$539$	−1856.62			−0.148368
$540$		0			0
$541$	8668.97			0.688925			0.344462	−	0.938800i	$-0.388061\pi$
$541$	8668.97			0.688925			0.344462	+	0.938800i	$0.388061\pi$
$542$	−8580.06	−	14861.1i	−0.679973	−	1.17775i
$543$		0			0
$544$	−907.555	+	1571.93i	−0.0715278	+	0.123890i
$545$	5384.89	−	9326.91i	0.423236	−	0.733066i
$546$		0			0
$547$	9606.66	+	16639.2i	0.750916	+	1.30062i	0.947379	+	0.320114i	$0.103721\pi$
$547$	9606.66	+	16639.2i	0.750916	+	1.30062i	−0.196463	+	0.980511i	$0.562946\pi$
$548$	−11812.0			−0.920774
$549$		0			0
$550$	19346.0			1.49985
$551$	1084.33	+	1878.11i	0.0838366	+	0.145209i
$552$		0			0
$553$	−846.493	+	1466.17i	−0.0650932	+	0.112745i
$554$	−7757.88	+	13437.0i	−0.594947	+	1.03048i
$555$		0			0
$556$	−360.770	−	624.872i	−0.0275181	−	0.0476627i
$557$	5467.31			0.415902			0.207951	−	0.978139i	$-0.433321\pi$
$557$	5467.31			0.415902			0.207951	+	0.978139i	$0.433321\pi$
$558$		0			0
$559$	−4002.10			−0.302810
$560$	1092.06	+	1891.50i	0.0824069	+	0.142733i
$561$		0			0
$562$	−6973.70	+	12078.8i	−0.523431	+	0.906608i
$563$	814.939	−	1411.52i	0.0610046	−	0.105663i	−0.833910	−	0.551900i	$-0.813903\pi$
$563$	814.939	−	1411.52i	0.0610046	−	0.105663i	0.894915	+	0.446237i	$0.147236\pi$
$564$		0			0
$565$	10446.2	+	18093.4i	0.777833	+	1.34725i
$566$	17135.3			1.27253
$567$		0			0
$568$	−2151.05			−0.158902
$569$	−1339.84	−	2320.67i	−0.0987153	−	0.170980i	0.812438	−	0.583048i	$-0.198140\pi$
$569$	−1339.84	−	2320.67i	−0.0987153	−	0.170980i	−0.911153	+	0.412068i	$0.864807\pi$
$570$		0			0
$571$	−701.311	+	1214.71i	−0.0513992	+	0.0890261i	−0.890580	−	0.454826i	$-0.849701\pi$
$571$	−701.311	+	1214.71i	−0.0513992	+	0.0890261i	0.839181	+	0.543852i	$0.183035\pi$
$572$	809.401	−	1401.92i	0.0591657	−	0.102478i
$573$		0			0
$574$	3509.84	+	6079.22i	0.255223	+	0.442059i
$575$	15898.0			1.15303
$576$		0			0
$577$	18955.5			1.36764			0.683821	−	0.729650i	$-0.260317\pi$
$577$	18955.5			1.36764			0.683821	+	0.729650i	$0.260317\pi$
$578$	−1695.59	−	2936.85i	−0.122020	−	0.211344i
$579$		0			0
$580$	2250.45	−	3897.89i	0.161112	−	0.279054i
$581$	−1757.38	+	3043.87i	−0.125488	+	0.217351i
$582$		0			0
$583$	−2952.92	−	5114.61i	−0.209773	−	0.363337i
$584$	8884.51			0.629527
$585$		0			0
$586$	1329.98			0.0937557
$587$	−6706.82	−	11616.5i	−0.471584	−	0.816808i	0.527887	−	0.849314i	$-0.322984\pi$
$587$	−6706.82	−	11616.5i	−0.471584	−	0.816808i	−0.999472	+	0.0325067i	$0.989651\pi$
$588$		0			0
$589$	4350.20	−	7534.76i	0.304324	−	0.527104i
$590$	6845.56	−	11856.9i	0.477673	−	0.827354i
$591$		0			0
$592$	2949.26	+	5108.26i	0.204753	+	0.354642i
$593$	−1732.86			−0.120000			−0.0600000	−	0.998198i	$-0.519110\pi$
$593$	−1732.86			−0.120000			−0.0600000	+	0.998198i	$0.519110\pi$
$594$		0			0
$595$	7742.99			0.533499
$596$	−3832.17	−	6637.51i	−0.263375	−	0.456180i
$597$		0			0
$598$	665.142	−	1152.06i	0.0454844	−	0.0787814i
$599$	−5176.41	+	8965.80i	−0.353092	+	0.611574i	−0.986790	−	0.162007i	$-0.948203\pi$
$599$	−5176.41	+	8965.80i	−0.353092	+	0.611574i	0.633697	+	0.773581i	$0.281537\pi$
$600$		0			0
$601$	−337.209	−	584.063i	−0.0228869	−	0.0396413i	0.854355	−	0.519690i	$-0.173952\pi$
$601$	−337.209	−	584.063i	−0.0228869	−	0.0396413i	−0.877242	+	0.480048i	$0.840619\pi$
$602$	−5245.76			−0.355152
$603$		0			0
$604$	−5762.87			−0.388225
$605$	−1020.55	−	1767.65i	−0.0685808	−	0.118785i
$606$		0			0
$607$	−13783.5	+	23873.7i	−0.921673	+	1.59638i	−0.124846	+	0.992176i	$0.539844\pi$
$607$	−13783.5	+	23873.7i	−0.921673	+	1.59638i	−0.796827	+	0.604208i	$0.793490\pi$
$608$	601.353	−	1041.57i	0.0401120	−	0.0694760i
$609$		0			0
$610$	−7935.35	−	13744.4i	−0.526710	−	0.912288i
$611$	5988.28			0.396497
$612$		0			0
$613$	13165.1			0.867427			0.433713	−	0.901051i	$-0.357203\pi$
$613$	13165.1			0.867427			0.433713	+	0.901051i	$0.357203\pi$
$614$	226.338	+	392.030i	0.0148767	+	0.0257671i
$615$		0			0
$616$	1060.93	−	1837.58i	0.0693927	−	0.120192i
$617$	−2244.42	+	3887.45i	−0.146446	+	0.253651i	−0.929911	−	0.367784i	$-0.880117\pi$
$617$	−2244.42	+	3887.45i	−0.146446	+	0.253651i	0.783466	+	0.621435i	$0.213450\pi$
$618$		0			0
$619$	9987.84	+	17299.5i	0.648538	+	1.12330i	0.983472	+	0.181060i	$0.0579528\pi$
$619$	9987.84	+	17299.5i	0.648538	+	1.12330i	−0.334934	+	0.942242i	$0.608714\pi$
$620$	−18057.1			−1.16966
$621$		0			0
$622$	11252.4			0.725372
$623$	3707.76	+	6422.03i	0.238440	+	0.412991i
$624$		0			0
$625$	−8818.40	+	15273.9i	−0.564378	+	0.977531i
$626$	614.929	−	1065.09i	0.0392612	−	0.0680024i
$627$		0			0
$628$	−761.862	−	1319.58i	−0.0484102	−	0.0838489i
$629$	20911.0			1.32556
$630$		0			0
$631$	−9034.27			−0.569966			−0.284983	−	0.958533i	$-0.591988\pi$
$631$	−9034.27			−0.569966			−0.284983	+	0.958533i	$0.591988\pi$
$632$	−967.420	−	1675.62i	−0.0608891	−	0.105463i
$633$		0			0
$634$	−808.318	+	1400.05i	−0.0506347	+	0.0877019i
$635$	−14135.0	+	24482.6i	−0.883357	+	1.53002i
$636$		0			0
$637$	−261.682	−	453.246i	−0.0162766	−	0.0281919i
$638$	−4372.59			−0.271336
$639$		0			0
$640$	−2496.13			−0.154169
$641$	4083.54	+	7072.90i	0.251623	+	0.435823i	0.963973	−	0.266001i	$-0.0857024\pi$
$641$	4083.54	+	7072.90i	0.251623	+	0.435823i	−0.712350	+	0.701824i	$0.752369\pi$
$642$		0			0
$643$	−3467.11	+	6005.21i	−0.212643	+	0.368309i	−0.952541	−	0.304411i	$-0.901540\pi$
$643$	−3467.11	+	6005.21i	−0.212643	+	0.368309i	0.739898	+	0.672719i	$0.234874\pi$
$644$	871.837	−	1510.07i	0.0533466	−	0.0923990i
$645$		0			0
$646$	−2131.88	−	3692.52i	−0.129842	−	0.224892i
$647$	21350.8			1.29735			0.648676	−	0.761065i	$-0.275323\pi$
$647$	21350.8			1.29735			0.648676	+	0.761065i	$0.275323\pi$
$648$		0			0
$649$	−13300.8			−0.804472
$650$	2726.72	+	4722.83i	0.164540	+	0.284991i
$651$		0			0
$652$	369.136	−	639.361i	0.0221725	−	0.0384039i
$653$	932.769	−	1615.60i	0.0558990	−	0.0968200i	−0.836722	−	0.547628i	$-0.815531\pi$
$653$	932.769	−	1615.60i	0.0558990	−	0.0968200i	0.892621	+	0.450808i	$0.148864\pi$
$654$		0			0
$655$	−19135.0	−	33142.8i	−1.14148	−	1.97710i
$656$	−8022.49			−0.477478
$657$		0			0
$658$	7849.16			0.465033
$659$	−9713.86	−	16824.9i	−0.574201	−	0.994545i	−0.996128	−	0.0879151i	$-0.971980\pi$
$659$	−9713.86	−	16824.9i	−0.574201	−	0.994545i	0.421927	−	0.906630i	$-0.361354\pi$
$660$		0			0
$661$	−11526.4	+	19964.3i	−0.678252	+	1.17477i	0.297255	+	0.954798i	$0.403929\pi$
$661$	−11526.4	+	19964.3i	−0.678252	+	1.17477i	−0.975507	+	0.219969i	$0.929404\pi$
$662$	−349.812	+	605.893i	−0.0205375	+	0.0355720i
$663$		0			0
$664$	−2008.43	−	3478.70i	−0.117383	−	0.203313i
$665$	−5130.56			−0.299180
$666$		0			0
$667$	−3593.26			−0.208593
$668$	−4116.02	−	7129.16i	−0.238404	−	0.412928i
$669$		0			0
$670$	14943.8	−	25883.5i	0.861687	−	1.49249i
$671$	−7709.13	+	13352.6i	−0.443529	+	0.768214i
$672$		0			0
$673$	−11635.7	−	20153.7i	−0.666455	−	1.15433i	−0.978889	−	0.204394i	$-0.934478\pi$
$673$	−11635.7	−	20153.7i	−0.666455	−	1.15433i	0.312434	−	0.949939i	$-0.398856\pi$
$674$	8058.23			0.460521
$675$		0			0
$676$	−8331.67			−0.474037
$677$	12263.6	+	21241.1i	0.696200	+	1.20585i	0.969774	+	0.244003i	$0.0784608\pi$
$677$	12263.6	+	21241.1i	0.696200	+	1.20585i	−0.273574	+	0.961851i	$0.588206\pi$
$678$		0			0
$679$	−3638.84	+	6302.66i	−0.205664	+	0.356221i
$680$	−4424.57	+	7663.57i	−0.249521	+	0.432183i
$681$		0			0
$682$	8771.14	+	15192.1i	0.492470	+	0.852983i
$683$	30662.8			1.71783			0.858916	−	0.512117i	$-0.171139\pi$
$683$	30662.8			1.71783			0.858916	+	0.512117i	$0.171139\pi$
$684$		0			0
$685$	−57586.6			−3.21207
$686$	−343.000	−	594.093i	−0.0190901	−	0.0330650i
$687$		0			0
$688$	2997.58	−	5191.96i	0.166107	−	0.287706i
$689$	832.400	−	1441.76i	0.0460260	−	0.0797193i
$690$		0			0
$691$	−8934.38	−	15474.8i	−0.491866	−	0.851938i	0.508090	−	0.861304i	$-0.330352\pi$
$691$	−8934.38	−	15474.8i	−0.491866	−	0.851938i	−0.999956	+	0.00936659i	$0.997018\pi$
$692$	−4636.02			−0.254675
$693$		0			0
$694$	−18028.8			−0.986115
$695$	−1758.85	−	3046.41i	−0.0959955	−	0.166269i
$696$		0			0
$697$	−14220.4	+	24630.5i	−0.772793	+	1.33852i
$698$	2698.46	−	4673.87i	0.146330	−	0.253451i
$699$		0			0
$700$	3574.06	+	6190.46i	0.192981	+	0.334253i
$701$	−21700.1			−1.16919			−0.584594	−	0.811326i	$-0.698746\pi$
$701$	−21700.1			−1.16919			−0.584594	+	0.811326i	$0.698746\pi$
$702$		0			0
$703$	−13855.8			−0.743360
$704$	1212.49	+	2100.09i	0.0649109	+	0.112429i
$705$		0			0
$706$	−468.716	+	811.840i	−0.0249863	+	0.0432776i
$707$	3564.93	−	6174.64i	0.189636	−	0.328460i
$708$		0			0
$709$	2458.43	+	4258.12i	0.130223	+	0.225553i	0.923763	−	0.382966i	$-0.125097\pi$
$709$	2458.43	+	4258.12i	0.130223	+	0.225553i	−0.793539	+	0.608519i	$0.791764\pi$
$710$	−10486.9			−0.554320
$711$		0			0
$712$	−8474.89			−0.446081
$713$	7207.87	+	12484.4i	0.378593	+	0.655742i
$714$		0			0
$715$	3946.04	−	6834.74i	0.206397	−	0.357489i
$716$	5661.73	−	9806.41i	0.295515	−	0.511847i
$717$		0			0
$718$	−2045.04	−	3542.12i	−0.106296	−	0.184110i
$719$	−32252.1			−1.67288			−0.836440	−	0.548059i	$-0.815367\pi$
$719$	−32252.1			−1.67288			−0.836440	+	0.548059i	$0.815367\pi$
$720$		0			0
$721$	2259.14			0.116692
$722$	−5446.40	−	9433.44i	−0.280740	−	0.486255i
$723$		0			0
$724$	3152.04	−	5459.49i	0.161802	−	0.280249i
$725$	7365.22	−	12756.9i	0.377293	−	0.653491i
$726$		0			0
$727$	5991.40	+	10377.4i	0.305652	+	0.529404i	0.977406	−	0.211370i	$-0.0677924\pi$
$727$	5991.40	+	10377.4i	0.305652	+	0.529404i	−0.671755	+	0.740774i	$0.734459\pi$
$728$	598.129			0.0304508
$729$		0			0
$730$	43314.3			2.19607
$731$	−10626.8	−	18406.2i	−0.537684	−	0.931297i
$732$		0			0
$733$	6591.98	−	11417.6i	0.332169	−	0.575334i	−0.650768	−	0.759277i	$-0.725553\pi$
$733$	6591.98	−	11417.6i	0.332169	−	0.575334i	0.982937	+	0.183943i	$0.0588861\pi$
$734$	6176.12	−	10697.4i	0.310579	−	0.537938i
$735$		0			0
$736$	996.386	+	1725.79i	0.0499012	+	0.0864314i
$737$	−29035.6			−1.45121
$738$		0			0
$739$	33748.2			1.67990			0.839952	−	0.542661i	$-0.182583\pi$
$739$	33748.2			1.67990			0.839952	+	0.542661i	$0.182583\pi$
$740$	14378.4	+	24904.1i	0.714270	+	1.23715i
$741$		0			0
$742$	1091.07	−	1889.79i	0.0539817	−	0.0934991i
$743$	−8711.36	+	15088.5i	−0.430133	+	0.745012i	−0.996884	−	0.0788768i	$-0.974867\pi$
$743$	−8711.36	+	15088.5i	−0.430133	+	0.745012i	0.566752	+	0.823889i	$0.308200\pi$
$744$		0			0
$745$	−18682.8	−	32359.6i	−0.918772	−	1.59136i
$746$	8617.52			0.422935
$747$		0			0
$748$	8596.86			0.420230
$749$	−884.682	−	1532.31i	−0.0431583	−	0.0747524i
$750$		0			0
$751$	14757.3	−	25560.3i	0.717044	−	1.24196i	−0.245122	−	0.969492i	$-0.578828\pi$
$751$	14757.3	−	25560.3i	0.717044	−	1.24196i	0.962166	−	0.272464i	$-0.0878386\pi$
$752$	−4485.23	+	7768.65i	−0.217499	+	0.376720i
$753$		0			0
$754$	−616.295	−	1067.45i	−0.0297668	−	0.0515576i
$755$	−28095.5			−1.35430
$756$		0			0
$757$	28262.1			1.35694			0.678470	−	0.734628i	$-0.262643\pi$
$757$	28262.1			1.35694			0.678470	+	0.734628i	$0.262643\pi$
$758$	−6931.22	−	12005.2i	−0.332128	−	0.575263i
$759$		0			0
$760$	2931.75	−	5077.94i	0.139929	−	0.242364i
$761$	13082.4	−	22659.5i	0.623178	−	1.07938i	−0.365713	−	0.930728i	$-0.619175\pi$
$761$	13082.4	−	22659.5i	0.623178	−	1.07938i	0.988890	−	0.148647i	$-0.0474920\pi$
$762$		0			0
$763$	−1932.94	−	3347.94i	−0.0917129	−	0.158851i
$764$	−2494.48			−0.118124
$765$		0			0
$766$	10052.5			0.474167
$767$	−1874.69	−	3247.05i	−0.0882542	−	0.152861i
$768$		0			0
$769$	−18031.2	+	31231.0i	−0.845543	+	1.46452i	0.0396063	+	0.999215i	$0.487390\pi$
$769$	−18031.2	+	31231.0i	−0.845543	+	1.46452i	−0.885149	+	0.465308i	$0.845944\pi$
$770$	5172.28	−	8958.66i	0.242073	−	0.419283i
$771$		0			0
$772$	−5582.74	−	9669.59i	−0.260268	−	0.450798i
$773$	14768.8			0.687189			0.343594	−	0.939118i	$-0.388355\pi$
$773$	14768.8			0.687189			0.343594	+	0.939118i	$0.388355\pi$
$774$		0			0
$775$	−59096.7			−2.73912
$776$	−4158.68	−	7203.04i	−0.192381	−	0.333214i
$777$		0			0
$778$	5454.63	−	9447.70i	0.251360	−	0.435368i
$779$	9422.56	−	16320.4i	0.433374	−	0.750626i
$780$		0			0
$781$	5093.98	+	8823.03i	0.233389	+	0.404242i
$782$	7064.65			0.323058
$783$		0			0
$784$	784.000			0.0357143
$785$	−3714.27	−	6433.31i	−0.168877	−	0.292503i
$786$		0			0
$787$	1405.07	−	2433.65i	0.0636409	−	0.110229i	−0.832449	−	0.554101i	$-0.813062\pi$
$787$	1405.07	−	2433.65i	0.0636409	−	0.110229i	0.896090	+	0.443872i	$0.146395\pi$
$788$	7367.32	−	12760.6i	0.333058	−	0.576874i
$789$		0			0
$790$	−4716.42	−	8169.09i	−0.212409	−	0.367903i
$791$	7499.45			0.337105
$792$		0			0
$793$	−4346.26			−0.194628
$794$	6886.06	+	11927.0i	0.307780	+	0.533090i
$795$		0			0
$796$	3023.10	−	5236.17i	0.134612	−	0.233155i
$797$	−8135.48	+	14091.1i	−0.361573	+	0.626262i	−0.988220	−	0.153041i	$-0.951093\pi$
$797$	−8135.48	+	14091.1i	−0.361573	+	0.626262i	0.626647	+	0.779303i	$0.284427\pi$
$798$		0			0
$799$	15900.8	+	27540.9i	0.704041	+	1.21943i
$800$	−8169.29			−0.361035
$801$		0			0
$802$	−16828.1			−0.740924
$803$	−21039.7	−	36441.9i	−0.924628	−	1.60150i
$804$		0			0
$805$	4250.43	−	7361.96i	0.186097	−	0.322329i
$806$	−2472.50	+	4282.50i	−0.108052	+	0.187152i
$807$		0			0
$808$	4074.20	+	7056.73i	0.177389	+	0.307246i
$809$	731.397			0.0317856			0.0158928	−	0.999874i	$-0.494941\pi$
$809$	731.397			0.0317856			0.0158928	+	0.999874i	$0.494941\pi$
$810$		0			0
$811$	−16588.4			−0.718248			−0.359124	−	0.933290i	$-0.616924\pi$
$811$	−16588.4			−0.718248			−0.359124	+	0.933290i	$0.616924\pi$
$812$	−807.811	−	1399.17i	−0.0349121	−	0.0604695i
$813$		0			0
$814$	13968.5	−	24194.1i	0.601468	−	1.04177i
$815$	1799.63	−	3117.05i	0.0773476	−	0.133970i
$816$		0			0
$817$	7041.42	+	12196.1i	0.301527	+	0.522261i
$818$	−19390.9			−0.828833
$819$		0			0
$820$	−39111.7			−1.66566
$821$	9703.95	+	16807.7i	0.412510	+	0.714487i	0.995163	−	0.0982329i	$-0.0313190\pi$
$821$	9703.95	+	16807.7i	0.412510	+	0.714487i	−0.582654	+	0.812720i	$0.697986\pi$
$822$		0			0
$823$	2655.82	−	4600.01i	0.112486	−	0.194831i	−0.804286	−	0.594242i	$-0.797452\pi$
$823$	2655.82	−	4600.01i	0.112486	−	0.194831i	0.916772	+	0.399411i	$0.130785\pi$
$824$	−1290.94	+	2235.97i	−0.0545777	+	0.0945313i
$825$		0			0
$826$	−2457.25	−	4256.08i	−0.103509	−	0.179283i
$827$	16857.2			0.708804			0.354402	−	0.935093i	$-0.384684\pi$
$827$	16857.2			0.708804			0.354402	+	0.935093i	$0.384684\pi$
$828$		0			0
$829$	2488.46			0.104255			0.0521277	−	0.998640i	$-0.483400\pi$
$829$	2488.46			0.104255			0.0521277	+	0.998640i	$0.483400\pi$
$830$	−9791.61	−	16959.6i	−0.409484	−	0.709247i
$831$		0			0
$832$	−341.788	+	591.995i	−0.0142420	+	0.0246679i
$833$	1389.69	−	2407.02i	0.0578032	−	0.100118i
$834$		0			0
$835$	−20066.7	−	34756.5i	−0.831660	−	1.44048i
$836$	−5696.35			−0.235661
$837$		0			0
$838$	−8034.16			−0.331188
$839$	12506.6	+	21662.1i	0.514632	+	0.891368i	0.999856	+	0.0169785i	$0.00540468\pi$
$839$	12506.6	+	21662.1i	0.514632	+	0.891368i	−0.485224	+	0.874390i	$0.661262\pi$
$840$		0			0
$841$	10529.8	−	18238.2i	0.431744	−	0.747803i
$842$	−1592.57	+	2758.41i	−0.0651824	+	0.112899i
$843$		0			0
$844$	1072.86	+	1858.25i	0.0437552	+	0.0757862i
$845$	−40619.1			−1.65365
$846$		0			0
$847$	−732.666			−0.0297222
$848$	1246.94	+	2159.76i	0.0504953	+	0.0874604i
$849$		0			0
$850$	−14480.6	+	25081.2i	−0.584331	+	1.01209i
$851$	11478.9	−	19882.0i	0.462387	−	0.800878i
$852$		0			0
$853$	−1456.04	−	2521.93i	−0.0584451	−	0.101230i	0.835322	−	0.549760i	$-0.185281\pi$
$853$	−1456.04	−	2521.93i	−0.0584451	−	0.101230i	−0.893768	+	0.448530i	$0.851948\pi$
$854$	−5696.87			−0.228270
$855$		0			0
$856$	2022.13			0.0807418
$857$	2452.22	+	4247.37i	0.0977435	+	0.169297i	0.910750	−	0.412957i	$-0.135504\pi$
$857$	2452.22	+	4247.37i	0.0977435	+	0.169297i	−0.813007	+	0.582254i	$0.802171\pi$
$858$		0			0
$859$	8732.18	−	15124.6i	0.346843	−	0.600750i	−0.638844	−	0.769337i	$-0.720587\pi$
$859$	8732.18	−	15124.6i	0.346843	−	0.600750i	0.985687	+	0.168587i	$0.0539203\pi$
$860$	14614.0	−	25312.1i	0.579456	−	1.00365i
$861$		0			0
$862$	−5936.30	−	10282.0i	−0.234560	−	0.406271i
$863$	−21348.9			−0.842092			−0.421046	−	0.907039i	$-0.638337\pi$
$863$	−21348.9			−0.842092			−0.421046	+	0.907039i	$0.638337\pi$
$864$		0			0
$865$	−22601.8			−0.888421
$866$	3116.42	+	5397.80i	0.122287	+	0.211807i
$867$		0			0
$868$	−3240.84	+	5613.30i	−0.126730	+	0.219502i
$869$	−4581.97	+	7936.20i	−0.178864	+	0.309801i
$870$		0			0
$871$	−4092.43	−	7088.30i	−0.159204	−	0.275750i
$872$	4418.14			0.171579
$873$		0			0
$874$	−4681.09			−0.181167
$875$	8892.78	+	15402.7i	0.343578	+	0.595094i
$876$		0			0
$877$	−18032.5	+	31233.2i	−0.694315	+	1.20259i	0.276097	+	0.961130i	$0.410959\pi$
$877$	−18032.5	+	31233.2i	−0.694315	+	1.20259i	−0.970411	+	0.241458i	$0.922374\pi$
$878$	−5923.33	+	10259.5i	−0.227680	+	0.394352i
$879$		0			0
$880$	5911.18	+	10238.5i	0.226439	+	0.392203i
$881$	39563.7			1.51298			0.756489	−	0.654006i	$-0.226913\pi$
$881$	39563.7			1.51298			0.756489	+	0.654006i	$0.226913\pi$
$882$		0			0
$883$	−41856.5			−1.59522			−0.797612	−	0.603171i	$-0.793903\pi$
$883$	−41856.5			−1.59522			−0.797612	+	0.603171i	$0.793903\pi$
$884$	1211.69	+	2098.70i	0.0461012	+	0.0798495i
$885$		0			0
$886$	−11261.0	+	19504.7i	−0.426999	+	0.739585i
$887$	1720.80	−	2980.51i	0.0651396	−	0.112825i	−0.831616	−	0.555351i	$-0.812584\pi$
$887$	1720.80	−	2980.51i	0.0651396	−	0.112825i	0.896756	+	0.442526i	$0.145917\pi$
$888$		0			0
$889$	5073.85	+	8788.16i	0.191419	+	0.331547i
$890$	−41317.3			−1.55613
$891$		0			0
$892$	20128.8			0.755563
$893$	−10536.0	−	18248.8i	−0.394818	−	0.683845i
$894$		0			0
$895$	27602.4	−	47808.8i	1.03089	−	1.78555i
$896$	−448.000	+	775.959i	−0.0167038	+	0.0289319i
$897$		0			0
$898$	17965.0	+	31116.2i	0.667593	+	1.15630i
$899$	13357.1			0.495532
$900$		0			0
$901$	8841.13			0.326904
$902$	18998.4	+	32906.1i	0.701304	+	1.21469i
$903$		0			0
$904$	−4285.40	+	7422.53i	−0.157666	+	0.273086i
$905$	15367.0	−	26616.4i	0.564437	−	0.977634i
$906$		0			0
$907$	990.530	+	1715.65i	0.0362624	+	0.0628083i	0.883587	−	0.468267i	$-0.155121\pi$
$907$	990.530	+	1715.65i	0.0362624	+	0.0628083i	−0.847325	+	0.531075i	$0.821788\pi$
$908$	2672.40			0.0976727
$909$		0			0
$910$	2916.04			0.106226
$911$	−4718.76	−	8173.13i	−0.171613	−	0.297243i	0.767371	−	0.641203i	$-0.221565\pi$
$911$	−4718.76	−	8173.13i	−0.171613	−	0.297243i	−0.938984	+	0.343961i	$0.888231\pi$
$912$		0			0
$913$	−9512.47	+	16476.1i	−0.344816	+	0.597239i
$914$	10033.3	−	17378.2i	0.363100	−	0.628907i
$915$		0			0
$916$	5700.81	+	9874.10i	0.205633	+	0.356168i
$917$	−13737.2			−0.494704
$918$		0			0
$919$	47362.3			1.70004			0.850021	−	0.526748i	$-0.176589\pi$
$919$	47362.3			1.70004			0.850021	+	0.526748i	$0.176589\pi$
$920$	4857.64	+	8413.67i	0.174078	+	0.301512i
$921$		0			0
$922$	−4422.79	+	7660.50i	−0.157979	+	0.273628i
$923$	−1435.94	+	2487.13i	−0.0512077	+	0.0886943i
$924$		0			0
$925$	47057.3	+	81505.6i	1.67268	+	2.89717i
$926$	4744.27			0.168365
$927$		0			0
$928$	1846.42			0.0653145
$929$	1192.66	+	2065.74i	0.0421203	+	0.0729546i	0.886317	−	0.463079i	$-0.153255\pi$
$929$	1192.66	+	2065.74i	0.0421203	+	0.0729546i	−0.844197	+	0.536034i	$0.819922\pi$
$930$		0			0
$931$	−920.822	+	1594.91i	−0.0324154	+	0.0561451i
$932$	−589.837	+	1021.63i	−0.0207304	+	0.0359061i
$933$		0			0
$934$	−14269.6	−	24715.7i	−0.499911	−	0.865871i
$935$	41911.9			1.46595
$936$		0			0
$937$	−1053.11			−0.0367169			−0.0183584	−	0.999831i	$-0.505844\pi$
$937$	−1053.11			−0.0367169			−0.0183584	+	0.999831i	$0.505844\pi$
$938$	−5364.17	−	9291.01i	−0.186723	−	0.323414i
$939$		0			0
$940$	−21866.7	+	37874.2i	−0.758736	+	1.31417i
$941$	16294.9	−	28223.6i	0.564504	−	0.977749i	−0.432592	−	0.901590i	$-0.642401\pi$
$941$	16294.9	−	28223.6i	0.564504	−	0.977749i	0.997096	−	0.0761594i	$-0.0242658\pi$
$942$		0			0
$943$	15612.3	+	27041.3i	0.539137	+	0.933813i
$944$	5616.57			0.193648
$945$		0			0
$946$	−28394.7			−0.975889
$947$	17077.2	+	29578.6i	0.585992	+	1.01497i	0.994751	+	0.102325i	$0.0326281\pi$
$947$	17077.2	+	29578.6i	0.585992	+	1.01497i	−0.408760	+	0.912642i	$0.634039\pi$
$948$		0			0
$949$	5930.90	−	10272.6i	0.202872	−	0.351384i
$950$	9594.97	−	16619.0i	0.327686	−	0.567569i
$951$		0			0
$952$	1588.22	+	2750.88i	0.0540699	+	0.0936518i
$953$	39070.4			1.32803			0.664016	−	0.747719i	$-0.268851\pi$
$953$	39070.4			1.32803			0.664016	+	0.747719i	$0.268851\pi$
$954$		0			0
$955$	−12161.2			−0.412071
$956$	6830.09	+	11830.1i	0.231068	+	0.400221i
$957$		0			0
$958$	−11918.2	+	20642.9i	−0.401941	+	0.696182i
$959$	−10335.5	+	17901.6i	−0.348020	+	0.602788i
$960$		0			0
$961$	−11898.0	−	20607.9i	−0.399381	−	0.691748i
$962$	7875.16			0.263935
$963$		0			0
$964$	5598.24			0.187040
$965$	−27217.3	−	47141.7i	−0.907933	−	1.57259i
$966$		0			0
$967$	1834.63	−	3177.68i	0.0610112	−	0.105675i	−0.833906	−	0.551906i	$-0.813901\pi$
$967$	1834.63	−	3177.68i	0.0610112	−	0.105675i	0.894918	+	0.446231i	$0.147234\pi$
$968$	418.666	−	725.151i	0.0139013	−	0.0240777i
$969$		0			0
$970$	−20274.6	−	35116.7i	−0.671113	−	1.16240i
$971$	48777.9			1.61211			0.806055	−	0.591841i	$-0.201599\pi$
$971$	48777.9			1.61211			0.806055	+	0.591841i	$0.201599\pi$
$972$		0			0
$973$	−1262.70			−0.0416034
$974$	6743.42	+	11679.9i	0.221841	+	0.384240i
$975$		0			0
$976$	3255.36	−	5638.44i	0.106764	−	0.184920i
$977$	−1722.03	+	2982.64i	−0.0563895	+	0.0976696i	−0.892842	−	0.450370i	$-0.851292\pi$
$977$	−1722.03	+	2982.64i	−0.0563895	+	0.0976696i	0.836453	+	0.548039i	$0.184626\pi$
$978$		0			0
$979$	20069.7	+	34761.7i	0.655189	+	1.13482i
$980$	3822.20			0.124588
$981$		0			0
$982$	13704.8			0.445355
$983$	−1233.22	−	2135.99i	−0.0400137	−	0.0693058i	0.845325	−	0.534252i	$-0.179407\pi$
$983$	−1233.22	−	2135.99i	−0.0400137	−	0.0693058i	−0.885339	+	0.464947i	$0.846074\pi$
$984$		0			0
$985$	35917.6	−	62211.1i	1.16186	−	2.01240i
$986$	3272.91	−	5668.85i	0.105711	−	0.183096i
$987$		0			0
$988$	−802.873	−	1390.62i	−0.0258530	−	0.0447787i
$989$	−23333.9			−0.750228
$990$		0			0
$991$	1988.78			0.0637493			0.0318746	−	0.999492i	$-0.489852\pi$
$991$	1988.78			0.0637493			0.0318746	+	0.999492i	$0.489852\pi$
$992$	−3703.82	−	6415.20i	−0.118545	−	0.205325i
$993$		0			0
$994$	−1882.17	+	3260.01i	−0.0600591	+	0.104025i
$995$	14738.4	−	25527.7i	0.469587	−	0.813349i
$996$		0			0
$997$	9448.09	+	16364.6i	0.300124	+	0.519831i	0.976164	−	0.217035i	$-0.0696385\pi$
$997$	9448.09	+	16364.6i	0.300124	+	0.519831i	−0.676040	+	0.736865i	$0.736305\pi$
$998$	−3585.08			−0.113711
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.f.b.253.1		8
3.2	odd	2		126.4.f.b.85.3	yes	8
9.2	odd	6		126.4.f.b.43.3	✓	8
9.4	even	3		1134.4.a.l.1.4		4
9.5	odd	6		1134.4.a.o.1.1		4
9.7	even	3	inner	378.4.f.b.127.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.4.f.b.43.3	✓	8	9.2	odd	6
126.4.f.b.85.3	yes	8	3.2	odd	2
378.4.f.b.127.1		8	9.7	even	3	inner
378.4.f.b.253.1		8	1.1	even	1	trivial
1134.4.a.l.1.4		4	9.4	even	3
1134.4.a.o.1.1		4	9.5	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.f (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-9.75052 + 16.8884i) q^{5} +(3.50000 + 6.06218i) q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-9.75052 + 16.8884i) q^{5} +(3.50000 + 6.06218i) q^{7} -8.00000 q^{8} -39.0021 q^{10} +(18.9451 + 32.8139i) q^{11} +(-5.34044 + 9.24992i) q^{13} +(-7.00000 + 12.1244i) q^{14} +(-8.00000 - 13.8564i) q^{16} -56.7222 q^{17} +37.5846 q^{19} +(-39.0021 - 67.5536i) q^{20} +(-37.8902 + 65.6277i) q^{22} +(-31.1370 + 53.9310i) q^{23} +(-127.645 - 221.088i) q^{25} -21.3618 q^{26} -28.0000 q^{28} +(28.8504 + 49.9703i) q^{29} +(115.744 - 200.475i) q^{31} +(16.0000 - 27.7128i) q^{32} +(-56.7222 - 98.2457i) q^{34} -136.507 q^{35} -368.657 q^{37} +(37.5846 + 65.0984i) q^{38} +(78.0041 - 135.107i) q^{40} +(250.703 - 434.230i) q^{41} +(187.349 + 324.497i) q^{43} -151.561 q^{44} -124.548 q^{46} +(-280.327 - 485.541i) q^{47} +(-24.5000 + 42.4352i) q^{49} +(255.290 - 442.176i) q^{50} +(-21.3618 - 36.9997i) q^{52} -155.867 q^{53} -738.898 q^{55} +(-28.0000 - 48.4974i) q^{56} +(-57.7008 + 99.9407i) q^{58} +(-175.518 + 304.006i) q^{59} +(203.460 + 352.403i) q^{61} +462.977 q^{62} +64.0000 q^{64} +(-104.144 - 180.383i) q^{65} +(-383.155 + 663.644i) q^{67} +(113.444 - 196.491i) q^{68} +(-136.507 - 236.437i) q^{70} +268.881 q^{71} -1110.56 q^{73} +(-368.657 - 638.533i) q^{74} +(-75.1691 + 130.197i) q^{76} +(-132.616 + 229.697i) q^{77} +(120.928 + 209.453i) q^{79} +312.016 q^{80} +1002.81 q^{82} +(251.054 + 434.838i) q^{83} +(553.071 - 957.946i) q^{85} +(-374.697 + 648.994i) q^{86} +(-151.561 - 262.511i) q^{88} +1059.36 q^{89} -74.7662 q^{91} +(-124.548 - 215.724i) q^{92} +(560.654 - 971.081i) q^{94} +(-366.469 + 634.743i) q^{95} +(519.835 + 900.380i) q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8	\( 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8} - 4 q^{10} - 5 q^{11} + 21 q^{13} - 56 q^{14} - 64 q^{16} + 46 q^{17} - 188 q^{19} - 4 q^{20} + 10 q^{22} - 374 q^{23} - 41 q^{25} + 84 q^{26} - 224 q^{28} - 271 q^{29} + 243 q^{31} + 128 q^{32} + 46 q^{34} - 14 q^{35} - 362 q^{37} - 188 q^{38} + 8 q^{40} + 213 q^{41} + 238 q^{43} + 40 q^{44} - 1496 q^{46} - 675 q^{47} - 196 q^{49} + 82 q^{50} + 84 q^{52} - 108 q^{53} - 2828 q^{55} - 224 q^{56} + 542 q^{58} - 202 q^{59} + 1212 q^{61} + 972 q^{62} + 512 q^{64} - 549 q^{65} - 139 q^{67} - 92 q^{68} - 14 q^{70} + 2590 q^{71} - 4000 q^{73} - 362 q^{74} + 376 q^{76} + 35 q^{77} + 1545 q^{79} + 32 q^{80} + 852 q^{82} + 142 q^{83} + 793 q^{85} - 476 q^{86} + 40 q^{88} + 264 q^{89} + 294 q^{91} - 1496 q^{92} + 1350 q^{94} - 1244 q^{95} + 638 q^{97} - 784 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8 - 4 * q^10 - 5 * q^11 + 21 * q^13 - 56 * q^14 - 64 * q^16 + 46 * q^17 - 188 * q^19 - 4 * q^20 + 10 * q^22 - 374 * q^23 - 41 * q^25 + 84 * q^26 - 224 * q^28 - 271 * q^29 + 243 * q^31 + 128 * q^32 + 46 * q^34 - 14 * q^35 - 362 * q^37 - 188 * q^38 + 8 * q^40 + 213 * q^41 + 238 * q^43 + 40 * q^44 - 1496 * q^46 - 675 * q^47 - 196 * q^49 + 82 * q^50 + 84 * q^52 - 108 * q^53 - 2828 * q^55 - 224 * q^56 + 542 * q^58 - 202 * q^59 + 1212 * q^61 + 972 * q^62 + 512 * q^64 - 549 * q^65 - 139 * q^67 - 92 * q^68 - 14 * q^70 + 2590 * q^71 - 4000 * q^73 - 362 * q^74 + 376 * q^76 + 35 * q^77 + 1545 * q^79 + 32 * q^80 + 852 * q^82 + 142 * q^83 + 793 * q^85 - 476 * q^86 + 40 * q^88 + 264 * q^89 + 294 * q^91 - 1496 * q^92 + 1350 * q^94 - 1244 * q^95 + 638 * q^97 - 784 * q^98

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