LMFDB - Embedded newform 1575.2.d.e.1324.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1324,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1575.1324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1575 = 3^{2} \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1575.d (of order $2$ , degree $1$ , 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:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 9x^{2} + 16$ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1324.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	1575.1324
Dual form			1575.2.d.e.1324.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155i q^{2} -4.56155 q^{4} -1.00000i q^{7} -6.56155i q^{8} +1.56155 q^{11} -0.438447i q^{13} +2.56155 q^{14} +7.68466 q^{16} +0.438447i q^{17} +7.12311 q^{19} +4.00000i q^{22} +3.12311i q^{23} +1.12311 q^{26} +4.56155i q^{28} +6.68466 q^{29} +6.56155i q^{32} -1.12311 q^{34} +6.00000i q^{37} +18.2462i q^{38} -5.12311 q^{41} -0.876894i q^{43} -7.12311 q^{44} -8.00000 q^{46} +8.68466i q^{47} -1.00000 q^{49} +2.00000i q^{52} -5.12311i q^{53} -6.56155 q^{56} +17.1231i q^{58} -4.00000 q^{59} +15.3693 q^{61} -1.43845 q^{64} +10.2462i q^{67} -2.00000i q^{68} -8.00000 q^{71} +12.2462i q^{73} -15.3693 q^{74} -32.4924 q^{76} -1.56155i q^{77} +2.43845 q^{79} -13.1231i q^{82} +4.00000i q^{83} +2.24621 q^{86} -10.2462i q^{88} -1.12311 q^{89} -0.438447 q^{91} -14.2462i q^{92} -22.2462 q^{94} +5.80776i q^{97} -2.56155i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 10 q^{4} - 2 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{19} - 12 q^{26} + 2 q^{29} + 12 q^{34} - 4 q^{41} - 12 q^{44} - 32 q^{46} - 4 q^{49} - 18 q^{56} - 16 q^{59} + 12 q^{61} - 14 q^{64} - 32 q^{71} - 12 q^{74}+ \cdots - 56 q^{94}+O(q^{100})$ 4 * q - 10 * q^4 - 2 * q^11 + 2 * q^14 + 6 * q^16 + 12 * q^19 - 12 * q^26 + 2 * q^29 + 12 * q^34 - 4 * q^41 - 12 * q^44 - 32 * q^46 - 4 * q^49 - 18 * q^56 - 16 * q^59 + 12 * q^61 - 14 * q^64 - 32 * q^71 - 12 * q^74 - 64 * q^76 + 18 * q^79 - 24 * q^86 + 12 * q^89 - 10 * q^91 - 56 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$ .

$n$	$127$	$451$	$1226$
$\chi(n)$	$-1$	$1$	$1$

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

)

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

$2$	2.56155i	1.81129i	0.424035	+	0.905646i	$0.360613\pi$
$2$	2.56155i	1.81129i	−0.424035	+	0.905646i	$0.639387\pi$
$3$	0	0
$3$	0	0
$4$	−4.56155	−2.28078
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	− 6.56155i	− 2.31986i
$9$	0	0
$10$	0	0
$11$	1.56155	0.470826	0.235413	−	0.971895i	$-0.424356\pi$
$11$	1.56155	0.470826	0.235413	+	0.971895i	$0.424356\pi$
$12$	0	0
$13$	− 0.438447i	− 0.121603i	−0.998150	−	0.0608017i	$-0.980634\pi$
$13$	− 0.438447i	− 0.121603i	0.998150	−	0.0608017i	$-0.0193657\pi$
$14$	2.56155	0.684604
$15$	0	0
$16$	7.68466	1.92116
$17$	0.438447i	0.106339i	0.998586	+	0.0531695i	$0.0169324\pi$
$17$	0.438447i	0.106339i	−0.998586	+	0.0531695i	$0.983068\pi$
$18$	0	0
$19$	7.12311	1.63415	0.817076	−	0.576530i	$-0.195593\pi$
$19$	7.12311	1.63415	0.817076	+	0.576530i	$0.195593\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	3.12311i	0.651213i	0.945505	+	0.325606i	$0.105568\pi$
$23$	3.12311i	0.651213i	−0.945505	+	0.325606i	$0.894432\pi$
$24$	0	0
$25$	0	0
$26$	1.12311	0.220259
$27$	0	0
$28$	4.56155i	0.862052i
$29$	6.68466	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$29$	6.68466	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	6.56155i	1.15993i
$33$	0	0
$34$	−1.12311	−0.192611
$35$	0	0
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	18.2462i	2.95993i
$39$	0	0
$40$	0	0
$41$	−5.12311	−0.800095	−0.400047	−	0.916494i	$-0.631006\pi$
$41$	−5.12311	−0.800095	−0.400047	+	0.916494i	$0.631006\pi$
$42$	0	0
$43$	− 0.876894i	− 0.133725i	−0.997762	−	0.0668626i	$-0.978701\pi$
$43$	− 0.876894i	− 0.133725i	0.997762	−	0.0668626i	$-0.0212989\pi$
$44$	−7.12311	−1.07385
$45$	0	0
$46$	−8.00000	−1.17954
$47$	8.68466i	1.26679i	0.773830	+	0.633394i	$0.218339\pi$
$47$	8.68466i	1.26679i	−0.773830	+	0.633394i	$0.781661\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	2.00000i	0.277350i
$53$	− 5.12311i	− 0.703713i	−0.936054	−	0.351856i	$-0.885551\pi$
$53$	− 5.12311i	− 0.703713i	0.936054	−	0.351856i	$-0.114449\pi$
$54$	0	0
$55$	0	0
$56$	−6.56155	−0.876824
$57$	0	0
$58$	17.1231i	2.24837i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	15.3693	1.96784	0.983920	−	0.178611i	$-0.0571605\pi$
$61$	15.3693	1.96784	0.983920	+	0.178611i	$0.0571605\pi$
$62$	0	0
$63$	0	0
$64$	−1.43845	−0.179806
$65$	0	0
$66$	0	0
$67$	10.2462i	1.25177i	0.779914	+	0.625887i	$0.215263\pi$
$67$	10.2462i	1.25177i	−0.779914	+	0.625887i	$0.784737\pi$
$68$	− 2.00000i	− 0.242536i
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	12.2462i	1.43331i	0.697428	+	0.716655i	$0.254328\pi$
$73$	12.2462i	1.43331i	−0.697428	+	0.716655i	$0.745672\pi$
$74$	−15.3693	−1.78665
$75$	0	0
$76$	−32.4924	−3.72714
$77$	− 1.56155i	− 0.177955i
$78$	0	0
$79$	2.43845	0.274347	0.137173	−	0.990547i	$-0.456198\pi$
$79$	2.43845	0.274347	0.137173	+	0.990547i	$0.456198\pi$
$80$	0	0
$81$	0	0
$82$	− 13.1231i	− 1.44920i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	2.24621	0.242215
$87$	0	0
$88$	− 10.2462i	− 1.09225i
$89$	−1.12311	−0.119049	−0.0595245	−	0.998227i	$-0.518958\pi$
$89$	−1.12311	−0.119049	−0.0595245	+	0.998227i	$0.518958\pi$
$90$	0	0
$91$	−0.438447	−0.0459618
$92$	− 14.2462i	− 1.48527i
$93$	0	0
$94$	−22.2462	−2.29452
$95$	0	0
$96$	0	0
$97$	5.80776i	0.589689i	0.955545	+	0.294845i	$0.0952679\pi$
$97$	5.80776i	0.589689i	−0.955545	+	0.294845i	$0.904732\pi$
$98$	− 2.56155i	− 0.258756i
$99$	0	0
$100$	0	0
$101$	16.2462	1.61656	0.808279	−	0.588799i	$-0.200399\pi$
$101$	16.2462	1.61656	0.808279	+	0.588799i	$0.200399\pi$
$102$	0	0
$103$	− 5.56155i	− 0.547996i	−0.961730	−	0.273998i	$-0.911654\pi$
$103$	− 5.56155i	− 0.547996i	0.961730	−	0.273998i	$-0.0883462\pi$
$104$	−2.87689	−0.282103
$105$	0	0
$106$	13.1231	1.27463
$107$	− 13.3693i	− 1.29246i	−0.763142	−	0.646230i	$-0.776345\pi$
$107$	− 13.3693i	− 1.29246i	0.763142	−	0.646230i	$-0.223655\pi$
$108$	0	0
$109$	−5.31534	−0.509117	−0.254559	−	0.967057i	$-0.581930\pi$
$109$	−5.31534	−0.509117	−0.254559	+	0.967057i	$0.581930\pi$
$110$	0	0
$111$	0	0
$112$	− 7.68466i	− 0.726132i
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	0	0
$115$	0	0
$116$	−30.4924	−2.83115
$117$	0	0
$118$	− 10.2462i	− 0.943240i
$119$	0.438447	0.0401924
$120$	0	0
$121$	−8.56155	−0.778323
$122$	39.3693i	3.56433i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.24621i	− 0.554262i	−0.960832	−	0.277131i	$-0.910616\pi$
$127$	− 6.24621i	− 0.554262i	0.960832	−	0.277131i	$-0.0893835\pi$
$128$	9.43845i	0.834249i
$129$	0	0
$130$	0	0
$131$	0.876894	0.0766146	0.0383073	−	0.999266i	$-0.487803\pi$
$131$	0.876894	0.0766146	0.0383073	+	0.999266i	$0.487803\pi$
$132$	0	0
$133$	− 7.12311i	− 0.617652i
$134$	−26.2462	−2.26733
$135$	0	0
$136$	2.87689	0.246692
$137$	17.1231i	1.46293i	0.681881	+	0.731463i	$0.261162\pi$
$137$	17.1231i	1.46293i	−0.681881	+	0.731463i	$0.738838\pi$
$138$	0	0
$139$	15.1231	1.28273	0.641363	−	0.767238i	$-0.278369\pi$
$139$	15.1231	1.28273	0.641363	+	0.767238i	$0.278369\pi$
$140$	0	0
$141$	0	0
$142$	− 20.4924i	− 1.71969i
$143$	− 0.684658i	− 0.0572540i
$144$	0	0
$145$	0	0
$146$	−31.3693	−2.59614
$147$	0	0
$148$	− 27.3693i	− 2.24974i
$149$	12.2462	1.00325	0.501624	−	0.865086i	$-0.332736\pi$
$149$	12.2462	1.00325	0.501624	+	0.865086i	$0.332736\pi$
$150$	0	0
$151$	−6.93087	−0.564026	−0.282013	−	0.959411i	$-0.591002\pi$
$151$	−6.93087	−0.564026	−0.282013	+	0.959411i	$0.591002\pi$
$152$	− 46.7386i	− 3.79100i
$153$	0	0
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	20.2462i	1.61582i	0.589303	+	0.807912i	$0.299402\pi$
$157$	20.2462i	1.61582i	−0.589303	+	0.807912i	$0.700598\pi$
$158$	6.24621i	0.496922i
$159$	0	0
$160$	0	0
$161$	3.12311	0.246135
$162$	0	0
$163$	7.12311i	0.557925i	0.960302	+	0.278962i	$0.0899905\pi$
$163$	7.12311i	0.557925i	−0.960302	+	0.278962i	$0.910010\pi$
$164$	23.3693	1.82484
$165$	0	0
$166$	−10.2462	−0.795260
$167$	6.93087i	0.536327i	0.963373	+	0.268163i	$0.0864167\pi$
$167$	6.93087i	0.536327i	−0.963373	+	0.268163i	$0.913583\pi$
$168$	0	0
$169$	12.8078	0.985213
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	− 4.43845i	− 0.337449i	−0.985663	−	0.168724i	$-0.946035\pi$
$173$	− 4.43845i	− 0.337449i	0.985663	−	0.168724i	$-0.0539648\pi$
$174$	0	0
$175$	0	0
$176$	12.0000	0.904534
$177$	0	0
$178$	− 2.87689i	− 0.215632i
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−17.6155	−1.30935	−0.654676	−	0.755910i	$-0.727195\pi$
$181$	−17.6155	−1.30935	−0.654676	+	0.755910i	$0.727195\pi$
$182$	− 1.12311i	− 0.0832501i
$183$	0	0
$184$	20.4924	1.51072
$185$	0	0
$186$	0	0
$187$	0.684658i	0.0500672i
$188$	− 39.6155i	− 2.88926i
$189$	0	0
$190$	0	0
$191$	13.5616	0.981280	0.490640	−	0.871363i	$-0.336763\pi$
$191$	13.5616	0.981280	0.490640	+	0.871363i	$0.336763\pi$
$192$	0	0
$193$	− 19.3693i	− 1.39423i	−0.716957	−	0.697117i	$-0.754466\pi$
$193$	− 19.3693i	− 1.39423i	0.716957	−	0.697117i	$-0.245534\pi$
$194$	−14.8769	−1.06810
$195$	0	0
$196$	4.56155	0.325825
$197$	− 1.12311i	− 0.0800180i	−0.999199	−	0.0400090i	$-0.987261\pi$
$197$	− 1.12311i	− 0.0800180i	0.999199	−	0.0400090i	$-0.0127387\pi$
$198$	0	0
$199$	1.75379	0.124323	0.0621614	−	0.998066i	$-0.480201\pi$
$199$	1.75379	0.124323	0.0621614	+	0.998066i	$0.480201\pi$
$200$	0	0
$201$	0	0
$202$	41.6155i	2.92806i
$203$	− 6.68466i	− 0.469171i
$204$	0	0
$205$	0	0
$206$	14.2462	0.992581
$207$	0	0
$208$	− 3.36932i	− 0.233620i
$209$	11.1231	0.769401
$210$	0	0
$211$	14.0540	0.967516	0.483758	−	0.875202i	$-0.339272\pi$
$211$	14.0540	0.967516	0.483758	+	0.875202i	$0.339272\pi$
$212$	23.3693i	1.60501i
$213$	0	0
$214$	34.2462	2.34102
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 13.6155i	− 0.922160i
$219$	0	0
$220$	0	0
$221$	0.192236	0.0129312
$222$	0	0
$223$	2.43845i	0.163291i	0.996661	+	0.0816453i	$0.0260175\pi$
$223$	2.43845i	0.163291i	−0.996661	+	0.0816453i	$0.973983\pi$
$224$	6.56155	0.438412
$225$	0	0
$226$	35.8617	2.38549
$227$	− 11.3153i	− 0.751026i	−0.926817	−	0.375513i	$-0.877467\pi$
$227$	− 11.3153i	− 0.751026i	0.926817	−	0.375513i	$-0.122533\pi$
$228$	0	0
$229$	−10.8769	−0.718765	−0.359383	−	0.933190i	$-0.617013\pi$
$229$	−10.8769	−0.718765	−0.359383	+	0.933190i	$0.617013\pi$
$230$	0	0
$231$	0	0
$232$	− 43.8617i	− 2.87966i
$233$	5.12311i	0.335626i	0.985819	+	0.167813i	$0.0536704\pi$
$233$	5.12311i	0.335626i	−0.985819	+	0.167813i	$0.946330\pi$
$234$	0	0
$235$	0	0
$236$	18.2462	1.18773
$237$	0	0
$238$	1.12311i	0.0728001i
$239$	19.8078	1.28126	0.640629	−	0.767851i	$-0.278674\pi$
$239$	19.8078	1.28126	0.640629	+	0.767851i	$0.278674\pi$
$240$	0	0
$241$	−4.24621	−0.273523	−0.136761	−	0.990604i	$-0.543669\pi$
$241$	−4.24621	−0.273523	−0.136761	+	0.990604i	$0.543669\pi$
$242$	− 21.9309i	− 1.40977i
$243$	0	0
$244$	−70.1080	−4.48820
$245$	0	0
$246$	0	0
$247$	− 3.12311i	− 0.198718i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.87689	0.560305	0.280152	−	0.959956i	$-0.409615\pi$
$251$	8.87689	0.560305	0.280152	+	0.959956i	$0.409615\pi$
$252$	0	0
$253$	4.87689i	0.306608i
$254$	16.0000	1.00393
$255$	0	0
$256$	−27.0540	−1.69087
$257$	10.4924i	0.654499i	0.944938	+	0.327250i	$0.106122\pi$
$257$	10.4924i	0.654499i	−0.944938	+	0.327250i	$0.893878\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	0	0
$262$	2.24621i	0.138771i
$263$	− 12.8769i	− 0.794023i	−0.917814	−	0.397012i	$-0.870047\pi$
$263$	− 12.8769i	− 0.794023i	0.917814	−	0.397012i	$-0.129953\pi$
$264$	0	0
$265$	0	0
$266$	18.2462	1.11875
$267$	0	0
$268$	− 46.7386i	− 2.85502i
$269$	−20.7386	−1.26446	−0.632228	−	0.774782i	$-0.717860\pi$
$269$	−20.7386	−1.26446	−0.632228	+	0.774782i	$0.717860\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	3.36932i	0.204295i
$273$	0	0
$274$	−43.8617	−2.64978
$275$	0	0
$276$	0	0
$277$	− 0.246211i	− 0.0147934i	−0.999973	−	0.00739670i	$-0.997646\pi$
$277$	− 0.246211i	− 0.0147934i	0.999973	−	0.00739670i	$-0.00235446\pi$
$278$	38.7386i	2.32339i
$279$	0	0
$280$	0	0
$281$	−12.4384	−0.742016	−0.371008	−	0.928630i	$-0.620988\pi$
$281$	−12.4384	−0.742016	−0.371008	+	0.928630i	$0.620988\pi$
$282$	0	0
$283$	11.3153i	0.672627i	0.941750	+	0.336314i	$0.109180\pi$
$283$	11.3153i	0.672627i	−0.941750	+	0.336314i	$0.890820\pi$
$284$	36.4924	2.16543
$285$	0	0
$286$	1.75379	0.103704
$287$	5.12311i	0.302407i
$288$	0	0
$289$	16.8078	0.988692
$290$	0	0
$291$	0	0
$292$	− 55.8617i	− 3.26906i
$293$	− 2.68466i	− 0.156839i	−0.996920	−	0.0784197i	$-0.975013\pi$
$293$	− 2.68466i	− 0.156839i	0.996920	−	0.0784197i	$-0.0249874\pi$
$294$	0	0
$295$	0	0
$296$	39.3693	2.28830
$297$	0	0
$298$	31.3693i	1.81718i
$299$	1.36932	0.0791896
$300$	0	0
$301$	−0.876894	−0.0505434
$302$	− 17.7538i	− 1.02162i
$303$	0	0
$304$	54.7386	3.13948
$305$	0	0
$306$	0	0
$307$	− 19.3153i	− 1.10238i	−0.834378	−	0.551192i	$-0.814173\pi$
$307$	− 19.3153i	− 1.10238i	0.834378	−	0.551192i	$-0.185827\pi$
$308$	7.12311i	0.405877i
$309$	0	0
$310$	0	0
$311$	−31.6155	−1.79275	−0.896376	−	0.443294i	$-0.853810\pi$
$311$	−31.6155	−1.79275	−0.896376	+	0.443294i	$0.853810\pi$
$312$	0	0
$313$	22.3002i	1.26048i	0.776400	+	0.630241i	$0.217044\pi$
$313$	22.3002i	1.26048i	−0.776400	+	0.630241i	$0.782956\pi$
$314$	−51.8617	−2.92673
$315$	0	0
$316$	−11.1231	−0.625724
$317$	− 10.4924i	− 0.589313i	−0.955603	−	0.294657i	$-0.904795\pi$
$317$	− 10.4924i	− 0.589313i	0.955603	−	0.294657i	$-0.0952053\pi$
$318$	0	0
$319$	10.4384	0.584441
$320$	0	0
$321$	0	0
$322$	8.00000i	0.445823i
$323$	3.12311i	0.173774i
$324$	0	0
$325$	0	0
$326$	−18.2462	−1.01056
$327$	0	0
$328$	33.6155i	1.85611i
$329$	8.68466	0.478801
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	− 18.2462i	− 1.00139i
$333$	0	0
$334$	−17.7538	−0.971444
$335$	0	0
$336$	0	0
$337$	− 1.50758i	− 0.0821230i	−0.999157	−	0.0410615i	$-0.986926\pi$
$337$	− 1.50758i	− 0.0821230i	0.999157	−	0.0410615i	$-0.0130740\pi$
$338$	32.8078i	1.78451i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	−5.75379	−0.310223
$345$	0	0
$346$	11.3693	0.611218
$347$	− 7.12311i	− 0.382388i	−0.981552	−	0.191194i	$-0.938764\pi$
$347$	− 7.12311i	− 0.382388i	0.981552	−	0.191194i	$-0.0612360\pi$
$348$	0	0
$349$	−10.4924	−0.561646	−0.280823	−	0.959760i	$-0.590607\pi$
$349$	−10.4924	−0.561646	−0.280823	+	0.959760i	$0.590607\pi$
$350$	0	0
$351$	0	0
$352$	10.2462i	0.546125i
$353$	5.80776i	0.309116i	0.987984	+	0.154558i	$0.0493954\pi$
$353$	5.80776i	0.309116i	−0.987984	+	0.154558i	$0.950605\pi$
$354$	0	0
$355$	0	0
$356$	5.12311	0.271524
$357$	0	0
$358$	51.2311i	2.70765i
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	31.7386	1.67045
$362$	− 45.1231i	− 2.37162i
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	− 8.68466i	− 0.453335i	−0.973972	−	0.226668i	$-0.927217\pi$
$367$	− 8.68466i	− 0.453335i	0.973972	−	0.226668i	$-0.0727831\pi$
$368$	24.0000i	1.25109i
$369$	0	0
$370$	0	0
$371$	−5.12311	−0.265978
$372$	0	0
$373$	− 4.63068i	− 0.239768i	−0.992788	−	0.119884i	$-0.961748\pi$
$373$	− 4.63068i	− 0.239768i	0.992788	−	0.119884i	$-0.0382522\pi$
$374$	−1.75379	−0.0906863
$375$	0	0
$376$	56.9848	2.93877
$377$	− 2.93087i	− 0.150947i
$378$	0	0
$379$	16.4924	0.847159	0.423579	−	0.905859i	$-0.360773\pi$
$379$	16.4924	0.847159	0.423579	+	0.905859i	$0.360773\pi$
$380$	0	0
$381$	0	0
$382$	34.7386i	1.77738i
$383$	6.24621i	0.319166i	0.987184	+	0.159583i	$0.0510150\pi$
$383$	6.24621i	0.319166i	−0.987184	+	0.159583i	$0.948985\pi$
$384$	0	0
$385$	0	0
$386$	49.6155	2.52536
$387$	0	0
$388$	− 26.4924i	− 1.34495i
$389$	−24.9309	−1.26405	−0.632023	−	0.774950i	$-0.717775\pi$
$389$	−24.9309	−1.26405	−0.632023	+	0.774950i	$0.717775\pi$
$390$	0	0
$391$	−1.36932	−0.0692493
$392$	6.56155i	0.331408i
$393$	0	0
$394$	2.87689	0.144936
$395$	0	0
$396$	0	0
$397$	27.5616i	1.38327i	0.722245	+	0.691637i	$0.243110\pi$
$397$	27.5616i	1.38327i	−0.722245	+	0.691637i	$0.756890\pi$
$398$	4.49242i	0.225185i
$399$	0	0
$400$	0	0
$401$	−31.5616	−1.57611	−0.788054	−	0.615606i	$-0.788911\pi$
$401$	−31.5616	−1.57611	−0.788054	+	0.615606i	$0.788911\pi$
$402$	0	0
$403$	0	0
$404$	−74.1080	−3.68701
$405$	0	0
$406$	17.1231	0.849805
$407$	9.36932i	0.464420i
$408$	0	0
$409$	−6.49242	−0.321030	−0.160515	−	0.987033i	$-0.551315\pi$
$409$	−6.49242	−0.321030	−0.160515	+	0.987033i	$0.551315\pi$
$410$	0	0
$411$	0	0
$412$	25.3693i	1.24986i
$413$	4.00000i	0.196827i
$414$	0	0
$415$	0	0
$416$	2.87689	0.141051
$417$	0	0
$418$	28.4924i	1.39361i
$419$	26.2462	1.28221	0.641106	−	0.767453i	$-0.278476\pi$
$419$	26.2462	1.28221	0.641106	+	0.767453i	$0.278476\pi$
$420$	0	0
$421$	−2.68466	−0.130842	−0.0654211	−	0.997858i	$-0.520839\pi$
$421$	−2.68466	−0.130842	−0.0654211	+	0.997858i	$0.520839\pi$
$422$	36.0000i	1.75245i
$423$	0	0
$424$	−33.6155	−1.63251
$425$	0	0
$426$	0	0
$427$	− 15.3693i	− 0.743773i
$428$	60.9848i	2.94781i
$429$	0	0
$430$	0	0
$431$	19.8078	0.954106	0.477053	−	0.878874i	$-0.341705\pi$
$431$	19.8078	0.954106	0.477053	+	0.878874i	$0.341705\pi$
$432$	0	0
$433$	− 8.24621i	− 0.396288i	−0.980173	−	0.198144i	$-0.936509\pi$
$433$	− 8.24621i	− 0.396288i	0.980173	−	0.198144i	$-0.0634913\pi$
$434$	0	0
$435$	0	0
$436$	24.2462	1.16118
$437$	22.2462i	1.06418i
$438$	0	0
$439$	−9.36932	−0.447173	−0.223587	−	0.974684i	$-0.571777\pi$
$439$	−9.36932	−0.447173	−0.223587	+	0.974684i	$0.571777\pi$
$440$	0	0
$441$	0	0
$442$	0.492423i	0.0234221i
$443$	− 2.63068i	− 0.124988i	−0.998045	−	0.0624938i	$-0.980095\pi$
$443$	− 2.63068i	− 0.124988i	0.998045	−	0.0624938i	$-0.0199054\pi$
$444$	0	0
$445$	0	0
$446$	−6.24621	−0.295767
$447$	0	0
$448$	1.43845i	0.0679602i
$449$	−1.80776	−0.0853137	−0.0426568	−	0.999090i	$-0.513582\pi$
$449$	−1.80776	−0.0853137	−0.0426568	+	0.999090i	$0.513582\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	63.8617i	3.00380i
$453$	0	0
$454$	28.9848	1.36033
$455$	0	0
$456$	0	0
$457$	− 17.1231i	− 0.800985i	−0.916300	−	0.400493i	$-0.868839\pi$
$457$	− 17.1231i	− 0.800985i	0.916300	−	0.400493i	$-0.131161\pi$
$458$	− 27.8617i	− 1.30189i
$459$	0	0
$460$	0	0
$461$	13.1231	0.611204	0.305602	−	0.952159i	$-0.401142\pi$
$461$	13.1231	0.611204	0.305602	+	0.952159i	$0.401142\pi$
$462$	0	0
$463$	− 12.4924i	− 0.580572i	−0.956940	−	0.290286i	$-0.906250\pi$
$463$	− 12.4924i	− 0.580572i	0.956940	−	0.290286i	$-0.0937505\pi$
$464$	51.3693	2.38476
$465$	0	0
$466$	−13.1231	−0.607916
$467$	− 22.4384i	− 1.03833i	−0.854675	−	0.519164i	$-0.826243\pi$
$467$	− 22.4384i	− 1.03833i	0.854675	−	0.519164i	$-0.173757\pi$
$468$	0	0
$469$	10.2462	0.473126
$470$	0	0
$471$	0	0
$472$	26.2462i	1.20808i
$473$	− 1.36932i	− 0.0629613i
$474$	0	0
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	50.7386i	2.32073i
$479$	4.87689	0.222831	0.111415	−	0.993774i	$-0.464462\pi$
$479$	4.87689	0.222831	0.111415	+	0.993774i	$0.464462\pi$
$480$	0	0
$481$	2.63068	0.119949
$482$	− 10.8769i	− 0.495429i
$483$	0	0
$484$	39.0540	1.77518
$485$	0	0
$486$	0	0
$487$	− 3.12311i	− 0.141521i	−0.997493	−	0.0707607i	$-0.977457\pi$
$487$	− 3.12311i	− 0.141521i	0.997493	−	0.0707607i	$-0.0225427\pi$
$488$	− 100.847i	− 4.56511i
$489$	0	0
$490$	0	0
$491$	41.1771	1.85830	0.929148	−	0.369708i	$-0.120542\pi$
$491$	41.1771	1.85830	0.929148	+	0.369708i	$0.120542\pi$
$492$	0	0
$493$	2.93087i	0.132000i
$494$	8.00000	0.359937
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	0	0
$499$	−41.1771	−1.84334	−0.921670	−	0.387976i	$-0.873174\pi$
$499$	−41.1771	−1.84334	−0.921670	+	0.387976i	$0.873174\pi$
$500$	0	0
$501$	0	0
$502$	22.7386i	1.01487i
$503$	38.9309i	1.73584i	0.496703	+	0.867921i	$0.334544\pi$
$503$	38.9309i	1.73584i	−0.496703	+	0.867921i	$0.665456\pi$
$504$	0	0
$505$	0	0
$506$	−12.4924	−0.555356
$507$	0	0
$508$	28.4924i	1.26415i
$509$	−11.7538	−0.520978	−0.260489	−	0.965477i	$-0.583884\pi$
$509$	−11.7538	−0.520978	−0.260489	+	0.965477i	$0.583884\pi$
$510$	0	0
$511$	12.2462	0.541740
$512$	− 50.4233i	− 2.22842i
$513$	0	0
$514$	−26.8769	−1.18549
$515$	0	0
$516$	0	0
$517$	13.5616i	0.596436i
$518$	15.3693i	0.675289i
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	− 40.4924i	− 1.77061i	−0.465011	−	0.885305i	$-0.653950\pi$
$523$	− 40.4924i	− 1.77061i	0.465011	−	0.885305i	$-0.346050\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	32.9848	1.43821
$527$	0	0
$528$	0	0
$529$	13.2462	0.575922
$530$	0	0
$531$	0	0
$532$	32.4924i	1.40873i
$533$	2.24621i	0.0972942i
$534$	0	0
$535$	0	0
$536$	67.2311	2.90394
$537$	0	0
$538$	− 53.1231i	− 2.29030i
$539$	−1.56155	−0.0672608
$540$	0	0
$541$	−37.8078	−1.62548	−0.812741	−	0.582625i	$-0.802026\pi$
$541$	−37.8078	−1.62548	−0.812741	+	0.582625i	$0.802026\pi$
$542$	− 40.9848i	− 1.76045i
$543$	0	0
$544$	−2.87689	−0.123346
$545$	0	0
$546$	0	0
$547$	− 2.24621i	− 0.0960411i	−0.998846	−	0.0480205i	$-0.984709\pi$
$547$	− 2.24621i	− 0.0960411i	0.998846	−	0.0480205i	$-0.0152913\pi$
$548$	− 78.1080i	− 3.33661i
$549$	0	0
$550$	0	0
$551$	47.6155	2.02849
$552$	0	0
$553$	− 2.43845i	− 0.103693i
$554$	0.630683	0.0267952
$555$	0	0
$556$	−68.9848	−2.92561
$557$	13.1231i	0.556044i	0.960575	+	0.278022i	$0.0896788\pi$
$557$	13.1231i	0.556044i	−0.960575	+	0.278022i	$0.910321\pi$
$558$	0	0
$559$	−0.384472	−0.0162614
$560$	0	0
$561$	0	0
$562$	− 31.8617i	− 1.34401i
$563$	− 28.0000i	− 1.18006i	−0.807382	−	0.590030i	$-0.799116\pi$
$563$	− 28.0000i	− 1.18006i	0.807382	−	0.590030i	$-0.200884\pi$
$564$	0	0
$565$	0	0
$566$	−28.9848	−1.21832
$567$	0	0
$568$	52.4924i	2.20253i
$569$	−30.9848	−1.29895	−0.649476	−	0.760382i	$-0.725012\pi$
$569$	−30.9848	−1.29895	−0.649476	+	0.760382i	$0.725012\pi$
$570$	0	0
$571$	40.4924	1.69456	0.847278	−	0.531150i	$-0.178240\pi$
$571$	40.4924	1.69456	0.847278	+	0.531150i	$0.178240\pi$
$572$	3.12311i	0.130584i
$573$	0	0
$574$	−13.1231	−0.547748
$575$	0	0
$576$	0	0
$577$	− 24.0540i	− 1.00138i	−0.865627	−	0.500690i	$-0.833080\pi$
$577$	− 24.0540i	− 1.00138i	0.865627	−	0.500690i	$-0.166920\pi$
$578$	43.0540i	1.79081i
$579$	0	0
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	− 8.00000i	− 0.331326i
$584$	80.3542	3.32508
$585$	0	0
$586$	6.87689	0.284082
$587$	26.2462i	1.08330i	0.840605	+	0.541649i	$0.182200\pi$
$587$	26.2462i	1.08330i	−0.840605	+	0.541649i	$0.817800\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	46.1080i	1.89503i
$593$	− 27.5616i	− 1.13182i	−0.824468	−	0.565909i	$-0.808525\pi$
$593$	− 27.5616i	− 1.13182i	0.824468	−	0.565909i	$-0.191475\pi$
$594$	0	0
$595$	0	0
$596$	−55.8617	−2.28819
$597$	0	0
$598$	3.50758i	0.143436i
$599$	−11.8078	−0.482452	−0.241226	−	0.970469i	$-0.577550\pi$
$599$	−11.8078	−0.482452	−0.241226	+	0.970469i	$0.577550\pi$
$600$	0	0
$601$	6.49242	0.264831	0.132416	−	0.991194i	$-0.457727\pi$
$601$	6.49242	0.264831	0.132416	+	0.991194i	$0.457727\pi$
$602$	− 2.24621i	− 0.0915487i
$603$	0	0
$604$	31.6155	1.28642
$605$	0	0
$606$	0	0
$607$	− 42.0540i	− 1.70692i	−0.521160	−	0.853459i	$-0.674500\pi$
$607$	− 42.0540i	− 1.70692i	0.521160	−	0.853459i	$-0.325500\pi$
$608$	46.7386i	1.89550i
$609$	0	0
$610$	0	0
$611$	3.80776	0.154046
$612$	0	0
$613$	− 40.7386i	− 1.64542i	−0.568463	−	0.822709i	$-0.692462\pi$
$613$	− 40.7386i	− 1.64542i	0.568463	−	0.822709i	$-0.307538\pi$
$614$	49.4773	1.99674
$615$	0	0
$616$	−10.2462	−0.412832
$617$	− 32.2462i	− 1.29818i	−0.760710	−	0.649092i	$-0.775149\pi$
$617$	− 32.2462i	− 1.29818i	0.760710	−	0.649092i	$-0.224851\pi$
$618$	0	0
$619$	−32.1080	−1.29053	−0.645264	−	0.763960i	$-0.723253\pi$
$619$	−32.1080	−1.29053	−0.645264	+	0.763960i	$0.723253\pi$
$620$	0	0
$621$	0	0
$622$	− 80.9848i	− 3.24720i
$623$	1.12311i	0.0449963i
$624$	0	0
$625$	0	0
$626$	−57.1231	−2.28310
$627$	0	0
$628$	− 92.3542i	− 3.68533i
$629$	−2.63068	−0.104892
$630$	0	0
$631$	−11.8078	−0.470060	−0.235030	−	0.971988i	$-0.575519\pi$
$631$	−11.8078	−0.470060	−0.235030	+	0.971988i	$0.575519\pi$
$632$	− 16.0000i	− 0.636446i
$633$	0	0
$634$	26.8769	1.06742
$635$	0	0
$636$	0	0
$637$	0.438447i	0.0173719i
$638$	26.7386i	1.05859i
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	− 1.56155i	− 0.0615816i	−0.999526	−	0.0307908i	$-0.990197\pi$
$643$	− 1.56155i	− 0.0615816i	0.999526	−	0.0307908i	$-0.00980257\pi$
$644$	−14.2462	−0.561379
$645$	0	0
$646$	−8.00000	−0.314756
$647$	− 36.4924i	− 1.43467i	−0.696731	−	0.717333i	$-0.745363\pi$
$647$	− 36.4924i	− 1.43467i	0.696731	−	0.717333i	$-0.254637\pi$
$648$	0	0
$649$	−6.24621	−0.245185
$650$	0	0
$651$	0	0
$652$	− 32.4924i	− 1.27250i
$653$	− 33.2311i	− 1.30043i	−0.759750	−	0.650216i	$-0.774678\pi$
$653$	− 33.2311i	− 1.30043i	0.759750	−	0.650216i	$-0.225322\pi$
$654$	0	0
$655$	0	0
$656$	−39.3693	−1.53711
$657$	0	0
$658$	22.2462i	0.867248i
$659$	9.17708	0.357488	0.178744	−	0.983896i	$-0.442797\pi$
$659$	9.17708	0.357488	0.178744	+	0.983896i	$0.442797\pi$
$660$	0	0
$661$	−5.12311	−0.199266	−0.0996329	−	0.995024i	$-0.531767\pi$
$661$	−5.12311	−0.199266	−0.0996329	+	0.995024i	$0.531767\pi$
$662$	30.7386i	1.19469i
$663$	0	0
$664$	26.2462	1.01855
$665$	0	0
$666$	0	0
$667$	20.8769i	0.808357i
$668$	− 31.6155i	− 1.22324i
$669$	0	0
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	− 31.8617i	− 1.22818i	−0.789236	−	0.614090i	$-0.789523\pi$
$673$	− 31.8617i	− 1.22818i	0.789236	−	0.614090i	$-0.210477\pi$
$674$	3.86174	0.148749
$675$	0	0
$676$	−58.4233	−2.24705
$677$	− 4.93087i	− 0.189509i	−0.995501	−	0.0947544i	$-0.969793\pi$
$677$	− 4.93087i	− 0.189509i	0.995501	−	0.0947544i	$-0.0302066\pi$
$678$	0	0
$679$	5.80776	0.222882
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 6.73863i	− 0.257847i	−0.991655	−	0.128923i	$-0.958848\pi$
$683$	− 6.73863i	− 0.257847i	0.991655	−	0.128923i	$-0.0411521\pi$
$684$	0	0
$685$	0	0
$686$	−2.56155	−0.0978005
$687$	0	0
$688$	− 6.73863i	− 0.256908i
$689$	−2.24621	−0.0855738
$690$	0	0
$691$	−24.4924	−0.931736	−0.465868	−	0.884854i	$-0.654258\pi$
$691$	−24.4924	−0.931736	−0.465868	+	0.884854i	$0.654258\pi$
$692$	20.2462i	0.769645i
$693$	0	0
$694$	18.2462	0.692617
$695$	0	0
$696$	0	0
$697$	− 2.24621i	− 0.0850813i
$698$	− 26.8769i	− 1.01731i
$699$	0	0
$700$	0	0
$701$	−28.9309	−1.09270	−0.546352	−	0.837556i	$-0.683984\pi$
$701$	−28.9309	−1.09270	−0.546352	+	0.837556i	$0.683984\pi$
$702$	0	0
$703$	42.7386i	1.61192i
$704$	−2.24621	−0.0846573
$705$	0	0
$706$	−14.8769	−0.559899
$707$	− 16.2462i	− 0.611002i
$708$	0	0
$709$	−27.1771	−1.02066	−0.510328	−	0.859980i	$-0.670476\pi$
$709$	−27.1771	−1.02066	−0.510328	+	0.859980i	$0.670476\pi$
$710$	0	0
$711$	0	0
$712$	7.36932i	0.276177i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−91.2311	−3.40946
$717$	0	0
$718$	20.4924i	0.764770i
$719$	8.38447	0.312688	0.156344	−	0.987703i	$-0.450029\pi$
$719$	8.38447	0.312688	0.156344	+	0.987703i	$0.450029\pi$
$720$	0	0
$721$	−5.56155	−0.207123
$722$	81.3002i	3.02568i
$723$	0	0
$724$	80.3542	2.98634
$725$	0	0
$726$	0	0
$727$	52.4924i	1.94684i	0.229035	+	0.973418i	$0.426443\pi$
$727$	52.4924i	1.94684i	−0.229035	+	0.973418i	$0.573557\pi$
$728$	2.87689i	0.106625i
$729$	0	0
$730$	0	0
$731$	0.384472	0.0142202
$732$	0	0
$733$	− 6.68466i	− 0.246903i	−0.992351	−	0.123452i	$-0.960604\pi$
$733$	− 6.68466i	− 0.246903i	0.992351	−	0.123452i	$-0.0393964\pi$
$734$	22.2462	0.821123
$735$	0	0
$736$	−20.4924	−0.755361
$737$	16.0000i	0.589368i
$738$	0	0
$739$	−34.9309	−1.28495	−0.642476	−	0.766305i	$-0.722093\pi$
$739$	−34.9309	−1.28495	−0.642476	+	0.766305i	$0.722093\pi$
$740$	0	0
$741$	0	0
$742$	− 13.1231i	− 0.481764i
$743$	− 32.9848i	− 1.21010i	−0.796189	−	0.605048i	$-0.793154\pi$
$743$	− 32.9848i	− 1.21010i	0.796189	−	0.605048i	$-0.206846\pi$
$744$	0	0
$745$	0	0
$746$	11.8617	0.434289
$747$	0	0
$748$	− 3.12311i	− 0.114192i
$749$	−13.3693	−0.488504
$750$	0	0
$751$	17.0691	0.622861	0.311431	−	0.950269i	$-0.399192\pi$
$751$	17.0691	0.622861	0.311431	+	0.950269i	$0.399192\pi$
$752$	66.7386i	2.43371i
$753$	0	0
$754$	7.50758	0.273410
$755$	0	0
$756$	0	0
$757$	39.3693i	1.43090i	0.698663	+	0.715451i	$0.253779\pi$
$757$	39.3693i	1.43090i	−0.698663	+	0.715451i	$0.746221\pi$
$758$	42.2462i	1.53445i
$759$	0	0
$760$	0	0
$761$	−48.2462	−1.74892	−0.874462	−	0.485094i	$-0.838785\pi$
$761$	−48.2462	−1.74892	−0.874462	+	0.485094i	$0.838785\pi$
$762$	0	0
$763$	5.31534i	0.192428i
$764$	−61.8617	−2.23808
$765$	0	0
$766$	−16.0000	−0.578103
$767$	1.75379i	0.0633256i
$768$	0	0
$769$	42.4924	1.53232	0.766158	−	0.642652i	$-0.222166\pi$
$769$	42.4924	1.53232	0.766158	+	0.642652i	$0.222166\pi$
$770$	0	0
$771$	0	0
$772$	88.3542i	3.17994i
$773$	36.9309i	1.32831i	0.747594	+	0.664156i	$0.231209\pi$
$773$	36.9309i	1.32831i	−0.747594	+	0.664156i	$0.768791\pi$
$774$	0	0
$775$	0	0
$776$	38.1080	1.36800
$777$	0	0
$778$	− 63.8617i	− 2.28955i
$779$	−36.4924	−1.30748
$780$	0	0
$781$	−12.4924	−0.447014
$782$	− 3.50758i	− 0.125431i
$783$	0	0
$784$	−7.68466	−0.274452
$785$	0	0
$786$	0	0
$787$	− 49.1771i	− 1.75297i	−0.481426	−	0.876487i	$-0.659881\pi$
$787$	− 49.1771i	− 1.75297i	0.481426	−	0.876487i	$-0.340119\pi$
$788$	5.12311i	0.182503i
$789$	0	0
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	− 6.73863i	− 0.239296i
$794$	−70.6004	−2.50551
$795$	0	0
$796$	−8.00000	−0.283552
$797$	− 24.0540i	− 0.852036i	−0.904715	−	0.426018i	$-0.859916\pi$
$797$	− 24.0540i	− 0.852036i	0.904715	−	0.426018i	$-0.140084\pi$
$798$	0	0
$799$	−3.80776	−0.134709
$800$	0	0
$801$	0	0
$802$	− 80.8466i	− 2.85479i
$803$	19.1231i	0.674840i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 106.600i	− 3.75019i
$809$	16.5464	0.581740	0.290870	−	0.956763i	$-0.406055\pi$
$809$	16.5464	0.581740	0.290870	+	0.956763i	$0.406055\pi$
$810$	0	0
$811$	19.6155	0.688794	0.344397	−	0.938824i	$-0.388083\pi$
$811$	19.6155	0.688794	0.344397	+	0.938824i	$0.388083\pi$
$812$	30.4924i	1.07007i
$813$	0	0
$814$	−24.0000	−0.841200
$815$	0	0
$816$	0	0
$817$	− 6.24621i	− 0.218527i
$818$	− 16.6307i	− 0.581478i
$819$	0	0
$820$	0	0
$821$	21.4233	0.747678	0.373839	−	0.927494i	$-0.378041\pi$
$821$	21.4233	0.747678	0.373839	+	0.927494i	$0.378041\pi$
$822$	0	0
$823$	36.4924i	1.27205i	0.771670	+	0.636023i	$0.219422\pi$
$823$	36.4924i	1.27205i	−0.771670	+	0.636023i	$0.780578\pi$
$824$	−36.4924	−1.27127
$825$	0	0
$826$	−10.2462	−0.356511
$827$	5.36932i	0.186709i	0.995633	+	0.0933547i	$0.0297591\pi$
$827$	5.36932i	0.186709i	−0.995633	+	0.0933547i	$0.970241\pi$
$828$	0	0
$829$	−34.8769	−1.21132	−0.605662	−	0.795722i	$-0.707092\pi$
$829$	−34.8769	−1.21132	−0.605662	+	0.795722i	$0.707092\pi$
$830$	0	0
$831$	0	0
$832$	0.630683i	0.0218650i
$833$	− 0.438447i	− 0.0151913i
$834$	0	0
$835$	0	0
$836$	−50.7386	−1.75483
$837$	0	0
$838$	67.2311i	2.32246i
$839$	−28.8769	−0.996941	−0.498471	−	0.866907i	$-0.666105\pi$
$839$	−28.8769	−0.996941	−0.498471	+	0.866907i	$0.666105\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	− 6.87689i	− 0.236993i
$843$	0	0
$844$	−64.1080	−2.20669
$845$	0	0
$846$	0	0
$847$	8.56155i	0.294178i
$848$	− 39.3693i	− 1.35195i
$849$	0	0
$850$	0	0
$851$	−18.7386	−0.642352
$852$	0	0
$853$	7.26137i	0.248624i	0.992243	+	0.124312i	$0.0396724\pi$
$853$	7.26137i	0.248624i	−0.992243	+	0.124312i	$0.960328\pi$
$854$	39.3693	1.34719
$855$	0	0
$856$	−87.7235	−2.99833
$857$	15.7538i	0.538139i	0.963121	+	0.269070i	$0.0867162\pi$
$857$	15.7538i	0.538139i	−0.963121	+	0.269070i	$0.913284\pi$
$858$	0	0
$859$	16.4924	0.562714	0.281357	−	0.959603i	$-0.409215\pi$
$859$	16.4924	0.562714	0.281357	+	0.959603i	$0.409215\pi$
$860$	0	0
$861$	0	0
$862$	50.7386i	1.72816i
$863$	− 25.7538i	− 0.876669i	−0.898812	−	0.438335i	$-0.855569\pi$
$863$	− 25.7538i	− 0.876669i	0.898812	−	0.438335i	$-0.144431\pi$
$864$	0	0
$865$	0	0
$866$	21.1231	0.717792
$867$	0	0
$868$	0	0
$869$	3.80776	0.129170
$870$	0	0
$871$	4.49242	0.152220
$872$	34.8769i	1.18108i
$873$	0	0
$874$	−56.9848	−1.92754
$875$	0	0
$876$	0	0
$877$	− 40.2462i	− 1.35902i	−0.733667	−	0.679509i	$-0.762193\pi$
$877$	− 40.2462i	− 1.35902i	0.733667	−	0.679509i	$-0.237807\pi$
$878$	− 24.0000i	− 0.809961i
$879$	0	0
$880$	0	0
$881$	11.8617	0.399632	0.199816	−	0.979833i	$-0.435966\pi$
$881$	11.8617	0.399632	0.199816	+	0.979833i	$0.435966\pi$
$882$	0	0
$883$	8.49242i	0.285793i	0.989738	+	0.142896i	$0.0456416\pi$
$883$	8.49242i	0.285793i	−0.989738	+	0.142896i	$0.954358\pi$
$884$	−0.876894	−0.0294931
$885$	0	0
$886$	6.73863	0.226389
$887$	− 20.4924i	− 0.688068i	−0.938957	−	0.344034i	$-0.888206\pi$
$887$	− 20.4924i	− 0.688068i	0.938957	−	0.344034i	$-0.111794\pi$
$888$	0	0
$889$	−6.24621	−0.209491
$890$	0	0
$891$	0	0
$892$	− 11.1231i	− 0.372429i
$893$	61.8617i	2.07012i
$894$	0	0
$895$	0	0
$896$	9.43845	0.315316
$897$	0	0
$898$	− 4.63068i	− 0.154528i
$899$	0	0
$900$	0	0
$901$	2.24621	0.0748321
$902$	− 20.4924i	− 0.682323i
$903$	0	0
$904$	−91.8617	−3.05528
$905$	0	0
$906$	0	0
$907$	− 24.1080i	− 0.800491i	−0.916408	−	0.400246i	$-0.868925\pi$
$907$	− 24.1080i	− 0.800491i	0.916408	−	0.400246i	$-0.131075\pi$
$908$	51.6155i	1.71292i
$909$	0	0
$910$	0	0
$911$	28.4924	0.943996	0.471998	−	0.881600i	$-0.343533\pi$
$911$	28.4924	0.943996	0.471998	+	0.881600i	$0.343533\pi$
$912$	0	0
$913$	6.24621i	0.206719i
$914$	43.8617	1.45082
$915$	0	0
$916$	49.6155	1.63934
$917$	− 0.876894i	− 0.0289576i
$918$	0	0
$919$	−40.3002	−1.32938	−0.664690	−	0.747119i	$-0.731436\pi$
$919$	−40.3002	−1.32938	−0.664690	+	0.747119i	$0.731436\pi$
$920$	0	0
$921$	0	0
$922$	33.6155i	1.10707i
$923$	3.50758i	0.115453i
$924$	0	0
$925$	0	0
$926$	32.0000	1.05159
$927$	0	0
$928$	43.8617i	1.43983i
$929$	22.1080	0.725338	0.362669	−	0.931918i	$-0.381866\pi$
$929$	22.1080	0.725338	0.362669	+	0.931918i	$0.381866\pi$
$930$	0	0
$931$	−7.12311	−0.233450
$932$	− 23.3693i	− 0.765487i
$933$	0	0
$934$	57.4773	1.88071
$935$	0	0
$936$	0	0
$937$	− 55.6695i	− 1.81864i	−0.416094	−	0.909322i	$-0.636601\pi$
$937$	− 55.6695i	− 1.81864i	0.416094	−	0.909322i	$-0.363399\pi$
$938$	26.2462i	0.856969i
$939$	0	0
$940$	0	0
$941$	−43.8617	−1.42985	−0.714926	−	0.699200i	$-0.753540\pi$
$941$	−43.8617	−1.42985	−0.714926	+	0.699200i	$0.753540\pi$
$942$	0	0
$943$	− 16.0000i	− 0.521032i
$944$	−30.7386	−1.00046
$945$	0	0
$946$	3.50758	0.114041
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	0	0
$949$	5.36932	0.174295
$950$	0	0
$951$	0	0
$952$	− 2.87689i	− 0.0932407i
$953$	− 33.1231i	− 1.07296i	−0.843912	−	0.536481i	$-0.819753\pi$
$953$	− 33.1231i	− 1.07296i	0.843912	−	0.536481i	$-0.180247\pi$
$954$	0	0
$955$	0	0
$956$	−90.3542	−2.92226
$957$	0	0
$958$	12.4924i	0.403612i
$959$	17.1231	0.552934
$960$	0	0
$961$	−31.0000	−1.00000
$962$	6.73863i	0.217262i
$963$	0	0
$964$	19.3693	0.623844
$965$	0	0
$966$	0	0
$967$	− 35.1231i	− 1.12948i	−0.825268	−	0.564741i	$-0.808976\pi$
$967$	− 35.1231i	− 1.12948i	0.825268	−	0.564741i	$-0.191024\pi$
$968$	56.1771i	1.80560i
$969$	0	0
$970$	0	0
$971$	−49.4773	−1.58780	−0.793901	−	0.608048i	$-0.791953\pi$
$971$	−49.4773	−1.58780	−0.793901	+	0.608048i	$0.791953\pi$
$972$	0	0
$973$	− 15.1231i	− 0.484825i
$974$	8.00000	0.256337
$975$	0	0
$976$	118.108	3.78054
$977$	− 33.2311i	− 1.06316i	−0.847009	−	0.531578i	$-0.821599\pi$
$977$	− 33.2311i	− 1.06316i	0.847009	−	0.531578i	$-0.178401\pi$
$978$	0	0
$979$	−1.75379	−0.0560513
$980$	0	0
$981$	0	0
$982$	105.477i	3.36591i
$983$	51.4233i	1.64015i	0.572257	+	0.820074i	$0.306068\pi$
$983$	51.4233i	1.64015i	−0.572257	+	0.820074i	$0.693932\pi$
$984$	0	0
$985$	0	0
$986$	−7.50758	−0.239090
$987$	0	0
$988$	14.2462i	0.453232i
$989$	2.73863	0.0870835
$990$	0	0
$991$	12.4924	0.396835	0.198417	−	0.980118i	$-0.436420\pi$
$991$	12.4924	0.396835	0.198417	+	0.980118i	$0.436420\pi$
$992$	0	0
$993$	0	0
$994$	−20.4924	−0.649980
$995$	0	0
$996$	0	0
$997$	− 2.68466i	− 0.0850240i	−0.999096	−	0.0425120i	$-0.986464\pi$
$997$	− 2.68466i	− 0.0850240i	0.999096	−	0.0425120i	$-0.0135361\pi$
$998$	− 105.477i	− 3.33882i
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.d.e.1324.4		4
3.2	odd	2		175.2.b.b.99.1		4
5.2	odd	4		1575.2.a.p.1.1		2
5.3	odd	4		315.2.a.e.1.2		2
5.4	even	2	inner	1575.2.d.e.1324.1		4
12.11	even	2		2800.2.g.t.449.3		4
15.2	even	4		175.2.a.f.1.2		2
15.8	even	4		35.2.a.b.1.1	✓	2
15.14	odd	2		175.2.b.b.99.4		4
20.3	even	4		5040.2.a.bt.1.1		2
21.20	even	2		1225.2.b.f.99.1		4
35.13	even	4		2205.2.a.x.1.2		2
60.23	odd	4		560.2.a.i.1.1		2
60.47	odd	4		2800.2.a.bi.1.2		2
60.59	even	2		2800.2.g.t.449.2		4
105.23	even	12		245.2.e.i.116.2		4
105.38	odd	12		245.2.e.h.226.2		4
105.53	even	12		245.2.e.i.226.2		4
105.62	odd	4		1225.2.a.s.1.2		2
105.68	odd	12		245.2.e.h.116.2		4
105.83	odd	4		245.2.a.d.1.1		2
105.104	even	2		1225.2.b.f.99.4		4
120.53	even	4		2240.2.a.bh.1.1		2
120.83	odd	4		2240.2.a.bd.1.2		2
165.98	odd	4		4235.2.a.m.1.2		2
195.38	even	4		5915.2.a.l.1.2		2
420.83	even	4		3920.2.a.bs.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.a.b.1.1	✓	2	15.8	even	4
175.2.a.f.1.2		2	15.2	even	4
175.2.b.b.99.1		4	3.2	odd	2
175.2.b.b.99.4		4	15.14	odd	2
245.2.a.d.1.1		2	105.83	odd	4
245.2.e.h.116.2		4	105.68	odd	12
245.2.e.h.226.2		4	105.38	odd	12
245.2.e.i.116.2		4	105.23	even	12
245.2.e.i.226.2		4	105.53	even	12
315.2.a.e.1.2		2	5.3	odd	4
560.2.a.i.1.1		2	60.23	odd	4
1225.2.a.s.1.2		2	105.62	odd	4
1225.2.b.f.99.1		4	21.20	even	2
1225.2.b.f.99.4		4	105.104	even	2
1575.2.a.p.1.1		2	5.2	odd	4
1575.2.d.e.1324.1		4	5.4	even	2	inner
1575.2.d.e.1324.4		4	1.1	even	1	trivial
2205.2.a.x.1.2		2	35.13	even	4
2240.2.a.bd.1.2		2	120.83	odd	4
2240.2.a.bh.1.1		2	120.53	even	4
2800.2.a.bi.1.2		2	60.47	odd	4
2800.2.g.t.449.2		4	60.59	even	2
2800.2.g.t.449.3		4	12.11	even	2
3920.2.a.bs.1.2		2	420.83	even	4
4235.2.a.m.1.2		2	165.98	odd	4
5040.2.a.bt.1.1		2	20.3	even	4
5915.2.a.l.1.2		2	195.38	even	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1575.2.d.e.1324.4

Level	$1575$
Weight	$2$
Character	1575.1324
Analytic conductor	$12.576$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$