LMFDB - Embedded newform 3800.2.d.p.3649.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 28x^{10} + 274x^{8} + 1078x^{6} + 1385x^{4} + 478x^{2} + 49 $ x^12 + 28x^10 + 274x^8 + 1078x^6 + 1385x^4 + 478*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.3
Root			$-2.70452i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.p.3649.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70452i q^{3} +3.77934i q^{7} -4.31442 q^{9} +4.71724 q^{11} +5.44195i q^{13} -2.40944i q^{17} -1.00000 q^{19} +10.2213 q^{21} -4.38924i q^{23} +3.55488i q^{27} -9.05146 q^{29} -9.92581 q^{31} -12.7579i q^{33} +10.0429i q^{37} +14.7179 q^{39} +1.11712 q^{41} +12.1169i q^{43} +6.72432i q^{47} -7.28339 q^{49} -6.51637 q^{51} -4.34694i q^{53} +2.70452i q^{57} +4.03914 q^{59} -9.14562 q^{61} -16.3057i q^{63} +4.39672i q^{67} -11.8708 q^{69} +7.90222 q^{71} -3.07165i q^{73} +17.8280i q^{77} -11.3996 q^{79} -3.32902 q^{81} +4.37652i q^{83} +24.4798i q^{87} -2.53933 q^{89} -20.5670 q^{91} +26.8445i q^{93} -0.302105i q^{97} -20.3522 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{9} + 6 q^{11} - 12 q^{19} + 22 q^{21} - 14 q^{29} + 10 q^{31} - 16 q^{39} + 22 q^{41} + 4 q^{49} + 26 q^{51} + 8 q^{59} + 26 q^{61} - 14 q^{69} + 58 q^{71} - 56 q^{79} + 76 q^{81} + 24 q^{89}+ \cdots - 20 q^{99}+O(q^{100}) $ 12 * q - 20 * q^9 + 6 * q^11 - 12 * q^19 + 22 * q^21 - 14 * q^29 + 10 * q^31 - 16 * q^39 + 22 * q^41 + 4 * q^49 + 26 * q^51 + 8 * q^59 + 26 * q^61 - 14 * q^69 + 58 * q^71 - 56 * q^79 + 76 * q^81 + 24 * q^89 - 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 2.70452i	− 1.56145i	−0.624872	−	0.780727i	$-0.714849\pi$
$3$	− 2.70452i	− 1.56145i	0.624872	−	0.780727i	$-0.285151\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.77934i	1.42846i	0.699914	+	0.714228i	$0.253222\pi$
$7$	3.77934i	1.42846i	−0.699914	+	0.714228i	$0.746778\pi$
$8$	0	0
$9$	−4.31442	−1.43814
$10$	0	0
$11$	4.71724	1.42230	0.711150	−	0.703040i	$-0.248175\pi$
$11$	4.71724	1.42230	0.711150	+	0.703040i	$0.248175\pi$
$12$	0	0
$13$	5.44195i	1.50933i	0.656113	+	0.754663i	$0.272200\pi$
$13$	5.44195i	1.50933i	−0.656113	+	0.754663i	$0.727800\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.40944i	− 0.584375i	−0.956361	−	0.292187i	$-0.905617\pi$
$17$	− 2.40944i	− 0.584375i	0.956361	−	0.292187i	$-0.0943831\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	10.2213	2.23047
$22$	0	0
$23$	− 4.38924i	− 0.915220i	−0.889153	−	0.457610i	$-0.848706\pi$
$23$	− 4.38924i	− 0.915220i	0.889153	−	0.457610i	$-0.151294\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.55488i	0.684137i
$28$	0	0
$29$	−9.05146	−1.68081	−0.840407	−	0.541956i	$-0.817684\pi$
$29$	−9.05146	−1.68081	−0.840407	+	0.541956i	$0.817684\pi$
$30$	0	0
$31$	−9.92581	−1.78273	−0.891364	−	0.453288i	$-0.850251\pi$
$31$	−9.92581	−1.78273	−0.891364	+	0.453288i	$0.850251\pi$
$32$	0	0
$33$	− 12.7579i	− 2.22086i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.0429i	1.65105i	0.564367	+	0.825524i	$0.309120\pi$
$37$	10.0429i	1.65105i	−0.564367	+	0.825524i	$0.690880\pi$
$38$	0	0
$39$	14.7179	2.35674
$40$	0	0
$41$	1.11712	0.174465	0.0872326	−	0.996188i	$-0.472198\pi$
$41$	1.11712	0.174465	0.0872326	+	0.996188i	$0.472198\pi$
$42$	0	0
$43$	12.1169i	1.84781i	0.382625	+	0.923904i	$0.375020\pi$
$43$	12.1169i	1.84781i	−0.382625	+	0.923904i	$0.624980\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.72432i	0.980842i	0.871486	+	0.490421i	$0.163157\pi$
$47$	6.72432i	0.980842i	−0.871486	+	0.490421i	$0.836843\pi$
$48$	0	0
$49$	−7.28339	−1.04048
$50$	0	0
$51$	−6.51637	−0.912474
$52$	0	0
$53$	− 4.34694i	− 0.597098i	−0.954394	−	0.298549i	$-0.903497\pi$
$53$	− 4.34694i	− 0.597098i	0.954394	−	0.298549i	$-0.0965026\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.70452i	0.358222i
$58$	0	0
$59$	4.03914	0.525851	0.262926	−	0.964816i	$-0.415313\pi$
$59$	4.03914	0.525851	0.262926	+	0.964816i	$0.415313\pi$
$60$	0	0
$61$	−9.14562	−1.17098	−0.585488	−	0.810681i	$-0.699097\pi$
$61$	−9.14562	−1.17098	−0.585488	+	0.810681i	$0.699097\pi$
$62$	0	0
$63$	− 16.3057i	− 2.05432i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.39672i	0.537145i	0.963259	+	0.268572i	$0.0865519\pi$
$67$	4.39672i	0.537145i	−0.963259	+	0.268572i	$0.913448\pi$
$68$	0	0
$69$	−11.8708	−1.42907
$70$	0	0
$71$	7.90222	0.937821	0.468910	−	0.883246i	$-0.344647\pi$
$71$	7.90222	0.937821	0.468910	+	0.883246i	$0.344647\pi$
$72$	0	0
$73$	− 3.07165i	− 0.359510i	−0.983711	−	0.179755i	$-0.942470\pi$
$73$	− 3.07165i	− 0.359510i	0.983711	−	0.179755i	$-0.0575305\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17.8280i	2.03169i
$78$	0	0
$79$	−11.3996	−1.28256	−0.641280	−	0.767307i	$-0.721596\pi$
$79$	−11.3996	−1.28256	−0.641280	+	0.767307i	$0.721596\pi$
$80$	0	0
$81$	−3.32902	−0.369891
$82$	0	0
$83$	4.37652i	0.480386i	0.970725	+	0.240193i	$0.0772107\pi$
$83$	4.37652i	0.480386i	−0.970725	+	0.240193i	$0.922789\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	24.4798i	2.62451i
$88$	0	0
$89$	−2.53933	−0.269169	−0.134584	−	0.990902i	$-0.542970\pi$
$89$	−2.53933	−0.269169	−0.134584	+	0.990902i	$0.542970\pi$
$90$	0	0
$91$	−20.5670	−2.15600
$92$	0	0
$93$	26.8445i	2.78365i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 0.302105i	− 0.0306741i	−0.999882	−	0.0153371i	$-0.995118\pi$
$97$	− 0.302105i	− 0.0306741i	0.999882	−	0.0153371i	$-0.00488213\pi$
$98$	0	0
$99$	−20.3522	−2.04547
$100$	0	0
$101$	4.77080	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$101$	4.77080	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$102$	0	0
$103$	17.9014i	1.76388i	0.471364	+	0.881939i	$0.343762\pi$
$103$	17.9014i	1.76388i	−0.471364	+	0.881939i	$0.656238\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.1673i	− 1.27293i	−0.771305	−	0.636465i	$-0.780396\pi$
$107$	− 13.1673i	− 1.27293i	0.771305	−	0.636465i	$-0.219604\pi$
$108$	0	0
$109$	−0.756212	−0.0724320	−0.0362160	−	0.999344i	$-0.511530\pi$
$109$	−0.756212	−0.0724320	−0.0362160	+	0.999344i	$0.511530\pi$
$110$	0	0
$111$	27.1613	2.57804
$112$	0	0
$113$	20.6082i	1.93866i	0.245766	+	0.969329i	$0.420960\pi$
$113$	20.6082i	1.93866i	−0.245766	+	0.969329i	$0.579040\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 23.4789i	− 2.17062i
$118$	0	0
$119$	9.10608	0.834753
$120$	0	0
$121$	11.2523	1.02294
$122$	0	0
$123$	− 3.02128i	− 0.272419i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19.3178i	1.71418i	0.515168	+	0.857089i	$0.327730\pi$
$127$	19.3178i	1.71418i	−0.515168	+	0.857089i	$0.672270\pi$
$128$	0	0
$129$	32.7703	2.88527
$130$	0	0
$131$	−4.29192	−0.374986	−0.187493	−	0.982266i	$-0.560036\pi$
$131$	−4.29192	−0.374986	−0.187493	+	0.982266i	$0.560036\pi$
$132$	0	0
$133$	− 3.77934i	− 0.327710i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 21.2413i	− 1.81476i	−0.420306	−	0.907382i	$-0.638077\pi$
$137$	− 21.2413i	− 1.81476i	0.420306	−	0.907382i	$-0.361923\pi$
$138$	0	0
$139$	−1.94814	−0.165239	−0.0826197	−	0.996581i	$-0.526329\pi$
$139$	−1.94814	−0.165239	−0.0826197	+	0.996581i	$0.526329\pi$
$140$	0	0
$141$	18.1860	1.53154
$142$	0	0
$143$	25.6710i	2.14671i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	19.6981i	1.62467i
$148$	0	0
$149$	4.56655	0.374107	0.187053	−	0.982350i	$-0.440106\pi$
$149$	4.56655	0.374107	0.187053	+	0.982350i	$0.440106\pi$
$150$	0	0
$151$	−8.84246	−0.719590	−0.359795	−	0.933031i	$-0.617153\pi$
$151$	−8.84246	−0.719590	−0.359795	+	0.933031i	$0.617153\pi$
$152$	0	0
$153$	10.3953i	0.840413i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.63494i	0.449717i	0.974391	+	0.224859i	$0.0721920\pi$
$157$	5.63494i	0.449717i	−0.974391	+	0.224859i	$0.927808\pi$
$158$	0	0
$159$	−11.7564	−0.932341
$160$	0	0
$161$	16.5884	1.30735
$162$	0	0
$163$	− 3.71114i	− 0.290679i	−0.989382	−	0.145340i	$-0.953573\pi$
$163$	− 3.71114i	− 0.290679i	0.989382	−	0.145340i	$-0.0464275\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.403668i	0.0312368i	0.999878	+	0.0156184i	$0.00497169\pi$
$167$	0.403668i	0.0312368i	−0.999878	+	0.0156184i	$0.995028\pi$
$168$	0	0
$169$	−16.6148	−1.27806
$170$	0	0
$171$	4.31442	0.329932
$172$	0	0
$173$	− 13.9048i	− 1.05716i	−0.848884	−	0.528580i	$-0.822725\pi$
$173$	− 13.9048i	− 1.05716i	0.848884	−	0.528580i	$-0.177275\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 10.9239i	− 0.821093i
$178$	0	0
$179$	−3.27673	−0.244914	−0.122457	−	0.992474i	$-0.539077\pi$
$179$	−3.27673	−0.244914	−0.122457	+	0.992474i	$0.539077\pi$
$180$	0	0
$181$	13.0254	0.968173	0.484086	−	0.875020i	$-0.339152\pi$
$181$	13.0254	0.968173	0.484086	+	0.875020i	$0.339152\pi$
$182$	0	0
$183$	24.7345i	1.82843i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 11.3659i	− 0.831156i
$188$	0	0
$189$	−13.4351	−0.977260
$190$	0	0
$191$	−17.6403	−1.27641	−0.638203	−	0.769868i	$-0.720322\pi$
$191$	−17.6403	−1.27641	−0.638203	+	0.769868i	$0.720322\pi$
$192$	0	0
$193$	− 16.8445i	− 1.21250i	−0.795276	−	0.606248i	$-0.792674\pi$
$193$	− 16.8445i	− 1.21250i	0.795276	−	0.606248i	$-0.207326\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 19.8742i	− 1.41598i	−0.706224	−	0.707988i	$-0.749603\pi$
$197$	− 19.8742i	− 1.41598i	0.706224	−	0.707988i	$-0.250397\pi$
$198$	0	0
$199$	8.54003	0.605386	0.302693	−	0.953088i	$-0.402114\pi$
$199$	8.54003	0.605386	0.302693	+	0.953088i	$0.402114\pi$
$200$	0	0
$201$	11.8910	0.838727
$202$	0	0
$203$	− 34.2085i	− 2.40097i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	18.9370i	1.31622i
$208$	0	0
$209$	−4.71724	−0.326298
$210$	0	0
$211$	−0.0127177	−0.000875525 0	−0.000437762	−	1.00000i	$-0.500139\pi$
$211$	−0.0127177	−0.000875525 0	−0.000437762		1.00000i	$0.500139\pi$
$212$	0	0
$213$	− 21.3717i	− 1.46437i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 37.5130i	− 2.54655i
$218$	0	0
$219$	−8.30734	−0.561358
$220$	0	0
$221$	13.1120	0.882012
$222$	0	0
$223$	18.6403i	1.24825i	0.781326	+	0.624124i	$0.214544\pi$
$223$	18.6403i	1.24825i	−0.781326	+	0.624124i	$0.785456\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.75187i	0.514510i	0.966344	+	0.257255i	$0.0828180\pi$
$227$	7.75187i	0.514510i	−0.966344	+	0.257255i	$0.917182\pi$
$228$	0	0
$229$	19.3735	1.28023	0.640117	−	0.768277i	$-0.278886\pi$
$229$	19.3735	1.28023	0.640117	+	0.768277i	$0.278886\pi$
$230$	0	0
$231$	48.2162	3.17240
$232$	0	0
$233$	8.63389i	0.565625i	0.959175	+	0.282813i	$0.0912674\pi$
$233$	8.63389i	0.565625i	−0.959175	+	0.282813i	$0.908733\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	30.8306i	2.00266i
$238$	0	0
$239$	−9.47644	−0.612980	−0.306490	−	0.951874i	$-0.599155\pi$
$239$	−9.47644	−0.612980	−0.306490	+	0.951874i	$0.599155\pi$
$240$	0	0
$241$	29.3641	1.89151	0.945755	−	0.324880i	$-0.105324\pi$
$241$	29.3641	1.89151	0.945755	+	0.324880i	$0.105324\pi$
$242$	0	0
$243$	19.6681i	1.26171i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5.44195i	− 0.346263i
$248$	0	0
$249$	11.8364	0.750101
$250$	0	0
$251$	−7.57377	−0.478052	−0.239026	−	0.971013i	$-0.576828\pi$
$251$	−7.57377	−0.478052	−0.239026	+	0.971013i	$0.576828\pi$
$252$	0	0
$253$	− 20.7051i	− 1.30172i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 11.7573i	− 0.733400i	−0.930339	−	0.366700i	$-0.880488\pi$
$257$	− 11.7573i	− 0.733400i	0.930339	−	0.366700i	$-0.119512\pi$
$258$	0	0
$259$	−37.9556	−2.35845
$260$	0	0
$261$	39.0518	2.41725
$262$	0	0
$263$	− 2.20334i	− 0.135864i	−0.997690	−	0.0679319i	$-0.978360\pi$
$263$	− 2.20334i	− 0.135864i	0.997690	−	0.0679319i	$-0.0216401\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.86767i	0.420295i
$268$	0	0
$269$	−6.14642	−0.374754	−0.187377	−	0.982288i	$-0.559999\pi$
$269$	−6.14642	−0.374754	−0.187377	+	0.982288i	$0.559999\pi$
$270$	0	0
$271$	25.6620	1.55886	0.779428	−	0.626491i	$-0.215510\pi$
$271$	25.6620	1.55886	0.779428	+	0.626491i	$0.215510\pi$
$272$	0	0
$273$	55.6238i	3.36650i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 4.64202i	− 0.278912i	−0.990228	−	0.139456i	$-0.955465\pi$
$277$	− 4.64202i	− 0.278912i	0.990228	−	0.139456i	$-0.0445354\pi$
$278$	0	0
$279$	42.8241	2.56381
$280$	0	0
$281$	−0.369967	−0.0220704	−0.0110352	−	0.999939i	$-0.503513\pi$
$281$	−0.369967	−0.0220704	−0.0110352	+	0.999939i	$0.503513\pi$
$282$	0	0
$283$	1.21325i	0.0721203i	0.999350	+	0.0360602i	$0.0114808\pi$
$283$	1.21325i	0.0721203i	−0.999350	+	0.0360602i	$0.988519\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.22198i	0.249216i
$288$	0	0
$289$	11.1946	0.658506
$290$	0	0
$291$	−0.817050	−0.0478963
$292$	0	0
$293$	− 8.66265i	− 0.506077i	−0.967456	−	0.253039i	$-0.918570\pi$
$293$	− 8.66265i	− 0.506077i	0.967456	−	0.253039i	$-0.0814300\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.7692i	0.973049i
$298$	0	0
$299$	23.8860	1.38137
$300$	0	0
$301$	−45.7938	−2.63951
$302$	0	0
$303$	− 12.9027i	− 0.741242i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.26123i	0.471493i	0.971815	+	0.235747i	$0.0757535\pi$
$307$	8.26123i	0.471493i	−0.971815	+	0.235747i	$0.924246\pi$
$308$	0	0
$309$	48.4147	2.75421
$310$	0	0
$311$	7.54408	0.427786	0.213893	−	0.976857i	$-0.431386\pi$
$311$	7.54408	0.427786	0.213893	+	0.976857i	$0.431386\pi$
$312$	0	0
$313$	3.43152i	0.193961i	0.995286	+	0.0969804i	$0.0309184\pi$
$313$	3.43152i	0.193961i	−0.995286	+	0.0969804i	$0.969082\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.50784i	0.140854i	0.997517	+	0.0704270i	$0.0224362\pi$
$317$	2.50784i	0.140854i	−0.997517	+	0.0704270i	$0.977564\pi$
$318$	0	0
$319$	−42.6979	−2.39062
$320$	0	0
$321$	−35.6112	−1.98762
$322$	0	0
$323$	2.40944i	0.134065i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.04519i	0.113099i
$328$	0	0
$329$	−25.4135	−1.40109
$330$	0	0
$331$	29.1167	1.60040	0.800200	−	0.599734i	$-0.204727\pi$
$331$	29.1167	1.60040	0.800200	+	0.599734i	$0.204727\pi$
$332$	0	0
$333$	− 43.3295i	− 2.37444i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.4033i	1.11144i	0.831370	+	0.555720i	$0.187557\pi$
$337$	20.4033i	1.11144i	−0.831370	+	0.555720i	$0.812443\pi$
$338$	0	0
$339$	55.7353	3.02713
$340$	0	0
$341$	−46.8224	−2.53557
$342$	0	0
$343$	− 1.07103i	− 0.0578300i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 25.9967i	− 1.39557i	−0.716305	−	0.697787i	$-0.754168\pi$
$347$	− 25.9967i	− 1.39557i	0.716305	−	0.697787i	$-0.245832\pi$
$348$	0	0
$349$	−3.32628	−0.178052	−0.0890258	−	0.996029i	$-0.528375\pi$
$349$	−3.32628	−0.178052	−0.0890258	+	0.996029i	$0.528375\pi$
$350$	0	0
$351$	−19.3455	−1.03259
$352$	0	0
$353$	15.0904i	0.803179i	0.915820	+	0.401590i	$0.131542\pi$
$353$	15.0904i	0.803179i	−0.915820	+	0.401590i	$0.868458\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 24.6276i	− 1.30343i
$358$	0	0
$359$	12.8292	0.677102	0.338551	−	0.940948i	$-0.390063\pi$
$359$	12.8292	0.677102	0.338551	+	0.940948i	$0.390063\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 30.4321i	− 1.59727i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.11198i	0.214644i	0.994224	+	0.107322i	$0.0342275\pi$
$367$	4.11198i	0.214644i	−0.994224	+	0.107322i	$0.965772\pi$
$368$	0	0
$369$	−4.81974	−0.250905
$370$	0	0
$371$	16.4285	0.852927
$372$	0	0
$373$	16.5930i	0.859153i	0.903031	+	0.429576i	$0.141337\pi$
$373$	16.5930i	0.859153i	−0.903031	+	0.429576i	$0.858663\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 49.2576i	− 2.53690i
$378$	0	0
$379$	−30.6822	−1.57604	−0.788021	−	0.615649i	$-0.788894\pi$
$379$	−30.6822	−1.57604	−0.788021	+	0.615649i	$0.788894\pi$
$380$	0	0
$381$	52.2454	2.67661
$382$	0	0
$383$	30.4909i	1.55801i	0.627017	+	0.779006i	$0.284276\pi$
$383$	30.4909i	1.55801i	−0.627017	+	0.779006i	$0.715724\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 52.2774i	− 2.65741i
$388$	0	0
$389$	−16.3448	−0.828714	−0.414357	−	0.910114i	$-0.635994\pi$
$389$	−16.3448	−0.828714	−0.414357	+	0.910114i	$0.635994\pi$
$390$	0	0
$391$	−10.5756	−0.534831
$392$	0	0
$393$	11.6076i	0.585524i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 15.5946i	− 0.782670i	−0.920248	−	0.391335i	$-0.872013\pi$
$397$	− 15.5946i	− 0.782670i	0.920248	−	0.391335i	$-0.127987\pi$
$398$	0	0
$399$	−10.2213	−0.511705
$400$	0	0
$401$	−20.3465	−1.01606	−0.508029	−	0.861340i	$-0.669626\pi$
$401$	−20.3465	−1.01606	−0.508029	+	0.861340i	$0.669626\pi$
$402$	0	0
$403$	− 54.0158i	− 2.69072i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	47.3749i	2.34829i
$408$	0	0
$409$	5.65655	0.279699	0.139849	−	0.990173i	$-0.455338\pi$
$409$	5.65655	0.279699	0.139849	+	0.990173i	$0.455338\pi$
$410$	0	0
$411$	−57.4474	−2.83367
$412$	0	0
$413$	15.2653i	0.751155i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.26879i	0.258014i
$418$	0	0
$419$	18.1490	0.886635	0.443318	−	0.896365i	$-0.353801\pi$
$419$	18.1490	0.886635	0.443318	+	0.896365i	$0.353801\pi$
$420$	0	0
$421$	5.94874	0.289924	0.144962	−	0.989437i	$-0.453694\pi$
$421$	5.94874	0.289924	0.144962	+	0.989437i	$0.453694\pi$
$422$	0	0
$423$	− 29.0115i	− 1.41059i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 34.5644i	− 1.67269i
$428$	0	0
$429$	69.4276	3.35200
$430$	0	0
$431$	1.93242	0.0930816	0.0465408	−	0.998916i	$-0.485180\pi$
$431$	1.93242	0.0930816	0.0465408	+	0.998916i	$0.485180\pi$
$432$	0	0
$433$	14.7014i	0.706505i	0.935528	+	0.353253i	$0.114924\pi$
$433$	14.7014i	0.706505i	−0.935528	+	0.353253i	$0.885076\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.38924i	0.209966i
$438$	0	0
$439$	−10.5922	−0.505540	−0.252770	−	0.967526i	$-0.581342\pi$
$439$	−10.5922	−0.505540	−0.252770	+	0.967526i	$0.581342\pi$
$440$	0	0
$441$	31.4236	1.49636
$442$	0	0
$443$	38.7357i	1.84039i	0.391459	+	0.920195i	$0.371970\pi$
$443$	38.7357i	1.84039i	−0.391459	+	0.920195i	$0.628030\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 12.3503i	− 0.584151i
$448$	0	0
$449$	29.2298	1.37944	0.689719	−	0.724078i	$-0.257734\pi$
$449$	29.2298	1.37944	0.689719	+	0.724078i	$0.257734\pi$
$450$	0	0
$451$	5.26973	0.248142
$452$	0	0
$453$	23.9146i	1.12361i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.18004i	0.195534i	0.995209	+	0.0977671i	$0.0311700\pi$
$457$	4.18004i	0.195534i	−0.995209	+	0.0977671i	$0.968830\pi$
$458$	0	0
$459$	8.56527	0.399793
$460$	0	0
$461$	35.9590	1.67478	0.837389	−	0.546607i	$-0.184081\pi$
$461$	35.9590	1.67478	0.837389	+	0.546607i	$0.184081\pi$
$462$	0	0
$463$	32.5866i	1.51443i	0.653167	+	0.757214i	$0.273440\pi$
$463$	32.5866i	1.51443i	−0.653167	+	0.757214i	$0.726560\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.66769i	0.262269i	0.991365	+	0.131135i	$0.0418620\pi$
$467$	5.66769i	0.262269i	−0.991365	+	0.131135i	$0.958138\pi$
$468$	0	0
$469$	−16.6167	−0.767287
$470$	0	0
$471$	15.2398	0.702213
$472$	0	0
$473$	57.1582i	2.62814i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	18.7545i	0.858711i
$478$	0	0
$479$	16.2096	0.740635	0.370318	−	0.928905i	$-0.379249\pi$
$479$	16.2096	0.740635	0.370318	+	0.928905i	$0.379249\pi$
$480$	0	0
$481$	−54.6531	−2.49197
$482$	0	0
$483$	− 44.8637i	− 2.04137i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	19.0329i	0.862461i	0.902242	+	0.431231i	$0.141920\pi$
$487$	19.0329i	0.862461i	−0.902242	+	0.431231i	$0.858080\pi$
$488$	0	0
$489$	−10.0369	−0.453882
$490$	0	0
$491$	−16.5658	−0.747602	−0.373801	−	0.927509i	$-0.621946\pi$
$491$	−16.5658	−0.747602	−0.373801	+	0.927509i	$0.621946\pi$
$492$	0	0
$493$	21.8089i	0.982224i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	29.8652i	1.33964i
$498$	0	0
$499$	−6.36691	−0.285022	−0.142511	−	0.989793i	$-0.545518\pi$
$499$	−6.36691	−0.285022	−0.142511	+	0.989793i	$0.545518\pi$
$500$	0	0
$501$	1.09173	0.0487748
$502$	0	0
$503$	− 5.79766i	− 0.258505i	−0.991612	−	0.129252i	$-0.958742\pi$
$503$	− 5.79766i	− 0.258505i	0.991612	−	0.129252i	$-0.0412578\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	44.9352i	1.99564i
$508$	0	0
$509$	−23.0521	−1.02177	−0.510883	−	0.859650i	$-0.670681\pi$
$509$	−23.0521	−1.02177	−0.510883	+	0.859650i	$0.670681\pi$
$510$	0	0
$511$	11.6088	0.513544
$512$	0	0
$513$	− 3.55488i	− 0.156952i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	31.7202i	1.39505i
$518$	0	0
$519$	−37.6057	−1.65071
$520$	0	0
$521$	11.7470	0.514645	0.257323	−	0.966326i	$-0.417160\pi$
$521$	11.7470	0.514645	0.257323	+	0.966326i	$0.417160\pi$
$522$	0	0
$523$	24.0715i	1.05257i	0.850308	+	0.526286i	$0.176416\pi$
$523$	24.0715i	1.05257i	−0.850308	+	0.526286i	$0.823584\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23.9156i	1.04178i
$528$	0	0
$529$	3.73456	0.162372
$530$	0	0
$531$	−17.4266	−0.756248
$532$	0	0
$533$	6.07932i	0.263325i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.86198i	0.382423i
$538$	0	0
$539$	−34.3575	−1.47988
$540$	0	0
$541$	−3.93335	−0.169108	−0.0845540	−	0.996419i	$-0.526947\pi$
$541$	−3.93335	−0.169108	−0.0845540	+	0.996419i	$0.526947\pi$
$542$	0	0
$543$	− 35.2275i	− 1.51176i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 21.2622i	− 0.909105i	−0.890720	−	0.454553i	$-0.849799\pi$
$547$	− 21.2622i	− 0.909105i	0.890720	−	0.454553i	$-0.150201\pi$
$548$	0	0
$549$	39.4581	1.68403
$550$	0	0
$551$	9.05146	0.385605
$552$	0	0
$553$	− 43.0831i	− 1.83208i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 10.2899i	− 0.435996i	−0.975949	−	0.217998i	$-0.930047\pi$
$557$	− 10.2899i	− 0.435996i	0.975949	−	0.217998i	$-0.0699527\pi$
$558$	0	0
$559$	−65.9395	−2.78894
$560$	0	0
$561$	−30.7393	−1.29781
$562$	0	0
$563$	− 21.8745i	− 0.921901i	−0.887426	−	0.460950i	$-0.847509\pi$
$563$	− 21.8745i	− 0.921901i	0.887426	−	0.460950i	$-0.152491\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 12.5815i	− 0.528373i
$568$	0	0
$569$	−22.8560	−0.958172	−0.479086	−	0.877768i	$-0.659032\pi$
$569$	−22.8560	−0.958172	−0.479086	+	0.877768i	$0.659032\pi$
$570$	0	0
$571$	−5.58019	−0.233524	−0.116762	−	0.993160i	$-0.537251\pi$
$571$	−5.58019	−0.233524	−0.116762	+	0.993160i	$0.537251\pi$
$572$	0	0
$573$	47.7085i	1.99305i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 8.48419i	− 0.353201i	−0.984283	−	0.176601i	$-0.943490\pi$
$577$	− 8.48419i	− 0.353201i	0.984283	−	0.176601i	$-0.0565101\pi$
$578$	0	0
$579$	−45.5564	−1.89326
$580$	0	0
$581$	−16.5404	−0.686210
$582$	0	0
$583$	− 20.5055i	− 0.849252i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.8166i	1.06557i	0.846252	+	0.532783i	$0.178854\pi$
$587$	25.8166i	1.06557i	−0.846252	+	0.532783i	$0.821146\pi$
$588$	0	0
$589$	9.92581	0.408986
$590$	0	0
$591$	−53.7501	−2.21098
$592$	0	0
$593$	− 26.3984i	− 1.08405i	−0.840362	−	0.542025i	$-0.817658\pi$
$593$	− 26.3984i	− 1.08405i	0.840362	−	0.542025i	$-0.182342\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 23.0967i	− 0.945284i
$598$	0	0
$599$	−36.1825	−1.47838	−0.739188	−	0.673499i	$-0.764790\pi$
$599$	−36.1825	−1.47838	−0.739188	+	0.673499i	$0.764790\pi$
$600$	0	0
$601$	3.20839	0.130873	0.0654364	−	0.997857i	$-0.479156\pi$
$601$	3.20839	0.130873	0.0654364	+	0.997857i	$0.479156\pi$
$602$	0	0
$603$	− 18.9693i	− 0.772490i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 20.3390i	− 0.825534i	−0.910837	−	0.412767i	$-0.864562\pi$
$607$	− 20.3390i	− 0.825534i	0.910837	−	0.412767i	$-0.135438\pi$
$608$	0	0
$609$	−92.5176	−3.74900
$610$	0	0
$611$	−36.5934	−1.48041
$612$	0	0
$613$	− 18.9062i	− 0.763616i	−0.924242	−	0.381808i	$-0.875302\pi$
$613$	− 18.9062i	− 0.763616i	0.924242	−	0.381808i	$-0.124698\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.0702i	1.41187i	0.708275	+	0.705936i	$0.249473\pi$
$617$	35.0702i	1.41187i	−0.708275	+	0.705936i	$0.750527\pi$
$618$	0	0
$619$	45.1329	1.81404	0.907021	−	0.421085i	$-0.138350\pi$
$619$	45.1329	1.81404	0.907021	+	0.421085i	$0.138350\pi$
$620$	0	0
$621$	15.6032	0.626136
$622$	0	0
$623$	− 9.59699i	− 0.384495i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	12.7579i	0.509500i
$628$	0	0
$629$	24.1978	0.964830
$630$	0	0
$631$	35.0467	1.39519	0.697593	−	0.716494i	$-0.254254\pi$
$631$	35.0467	1.39519	0.697593	+	0.716494i	$0.254254\pi$
$632$	0	0
$633$	0.0343954i	0.00136709i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 39.6359i	− 1.57043i
$638$	0	0
$639$	−34.0935	−1.34872
$640$	0	0
$641$	−10.8470	−0.428431	−0.214215	−	0.976786i	$-0.568719\pi$
$641$	−10.8470	−0.428431	−0.214215	+	0.976786i	$0.568719\pi$
$642$	0	0
$643$	− 37.6665i	− 1.48542i	−0.669611	−	0.742712i	$-0.733539\pi$
$643$	− 37.6665i	− 1.48542i	0.669611	−	0.742712i	$-0.266461\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 37.2893i	− 1.46599i	−0.680231	−	0.732997i	$-0.738121\pi$
$647$	− 37.2893i	− 1.46599i	0.680231	−	0.732997i	$-0.261879\pi$
$648$	0	0
$649$	19.0536	0.747918
$650$	0	0
$651$	−101.455	−3.97632
$652$	0	0
$653$	5.10026i	0.199588i	0.995008	+	0.0997942i	$0.0318184\pi$
$653$	5.10026i	0.199588i	−0.995008	+	0.0997942i	$0.968182\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	13.2524i	0.517026i
$658$	0	0
$659$	−36.1180	−1.40696	−0.703479	−	0.710716i	$-0.748371\pi$
$659$	−36.1180	−1.40696	−0.703479	+	0.710716i	$0.748371\pi$
$660$	0	0
$661$	−15.5014	−0.602934	−0.301467	−	0.953477i	$-0.597476\pi$
$661$	−15.5014	−0.602934	−0.301467	+	0.953477i	$0.597476\pi$
$662$	0	0
$663$	− 35.4618i	− 1.37722i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	39.7290i	1.53831i
$668$	0	0
$669$	50.4131	1.94908
$670$	0	0
$671$	−43.1420	−1.66548
$672$	0	0
$673$	− 20.8897i	− 0.805240i	−0.915367	−	0.402620i	$-0.868100\pi$
$673$	− 20.8897i	− 0.805240i	0.915367	−	0.402620i	$-0.131900\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 39.3686i	− 1.51306i	−0.653960	−	0.756529i	$-0.726894\pi$
$677$	− 39.3686i	− 1.51306i	0.653960	−	0.756529i	$-0.273106\pi$
$678$	0	0
$679$	1.14176	0.0438166
$680$	0	0
$681$	20.9651	0.803383
$682$	0	0
$683$	41.6596i	1.59406i	0.603940	+	0.797030i	$0.293597\pi$
$683$	41.6596i	1.59406i	−0.603940	+	0.797030i	$0.706403\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 52.3959i	− 1.99903i
$688$	0	0
$689$	23.6558	0.901215
$690$	0	0
$691$	3.90123	0.148410	0.0742050	−	0.997243i	$-0.476358\pi$
$691$	3.90123	0.148410	0.0742050	+	0.997243i	$0.476358\pi$
$692$	0	0
$693$	− 76.9177i	− 2.92186i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 2.69164i	− 0.101953i
$698$	0	0
$699$	23.3505	0.883198
$700$	0	0
$701$	−24.6495	−0.931000	−0.465500	−	0.885048i	$-0.654125\pi$
$701$	−24.6495	−0.931000	−0.465500	+	0.885048i	$0.654125\pi$
$702$	0	0
$703$	− 10.0429i	− 0.378776i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.0305i	0.678106i
$708$	0	0
$709$	−9.56624	−0.359268	−0.179634	−	0.983734i	$-0.557491\pi$
$709$	−9.56624	−0.359268	−0.179634	+	0.983734i	$0.557491\pi$
$710$	0	0
$711$	49.1829	1.84450
$712$	0	0
$713$	43.5668i	1.63159i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	25.6292i	0.957141i
$718$	0	0
$719$	26.6273	0.993030	0.496515	−	0.868028i	$-0.334613\pi$
$719$	26.6273	0.993030	0.496515	+	0.868028i	$0.334613\pi$
$720$	0	0
$721$	−67.6554	−2.51962
$722$	0	0
$723$	− 79.4159i	− 2.95351i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 24.3779i	− 0.904127i	−0.891986	−	0.452063i	$-0.850688\pi$
$727$	− 24.3779i	− 0.904127i	0.891986	−	0.452063i	$-0.149312\pi$
$728$	0	0
$729$	43.2056	1.60021
$730$	0	0
$731$	29.1949	1.07981
$732$	0	0
$733$	6.53865i	0.241511i	0.992682	+	0.120755i	$0.0385316\pi$
$733$	6.53865i	0.241511i	−0.992682	+	0.120755i	$0.961468\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20.7404i	0.763981i
$738$	0	0
$739$	−11.7605	−0.432617	−0.216308	−	0.976325i	$-0.569402\pi$
$739$	−11.7605	−0.432617	−0.216308	+	0.976325i	$0.569402\pi$
$740$	0	0
$741$	−14.7179	−0.540674
$742$	0	0
$743$	40.7714i	1.49576i	0.663835	+	0.747879i	$0.268928\pi$
$743$	40.7714i	1.49576i	−0.663835	+	0.747879i	$0.731072\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 18.8822i	− 0.690863i
$748$	0	0
$749$	49.7636	1.81832
$750$	0	0
$751$	−0.578858	−0.0211228	−0.0105614	−	0.999944i	$-0.503362\pi$
$751$	−0.578858	−0.0211228	−0.0105614	+	0.999944i	$0.503362\pi$
$752$	0	0
$753$	20.4834i	0.746457i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 14.3710i	− 0.522325i	−0.965295	−	0.261162i	$-0.915894\pi$
$757$	− 14.3710i	− 0.522325i	0.965295	−	0.261162i	$-0.0841058\pi$
$758$	0	0
$759$	−55.9973	−2.03257
$760$	0	0
$761$	−16.4753	−0.597230	−0.298615	−	0.954374i	$-0.596525\pi$
$761$	−16.4753	−0.597230	−0.298615	+	0.954374i	$0.596525\pi$
$762$	0	0
$763$	− 2.85798i	− 0.103466i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.9808i	0.793681i
$768$	0	0
$769$	18.8410	0.679425	0.339712	−	0.940529i	$-0.389670\pi$
$769$	18.8410	0.679425	0.339712	+	0.940529i	$0.389670\pi$
$770$	0	0
$771$	−31.7978	−1.14517
$772$	0	0
$773$	− 37.5917i	− 1.35208i	−0.736866	−	0.676039i	$-0.763695\pi$
$773$	− 37.5917i	− 1.35208i	0.736866	−	0.676039i	$-0.236305\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	102.652i	3.68261i
$778$	0	0
$779$	−1.11712	−0.0400251
$780$	0	0
$781$	37.2766	1.33386
$782$	0	0
$783$	− 32.1769i	− 1.14991i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 44.5235i	− 1.58709i	−0.608511	−	0.793545i	$-0.708233\pi$
$787$	− 44.5235i	− 1.58709i	0.608511	−	0.793545i	$-0.291767\pi$
$788$	0	0
$789$	−5.95897	−0.212145
$790$	0	0
$791$	−77.8854	−2.76929
$792$	0	0
$793$	− 49.7700i	− 1.76738i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3.84212i	− 0.136095i	−0.997682	−	0.0680475i	$-0.978323\pi$
$797$	− 3.84212i	− 0.136095i	0.997682	−	0.0680475i	$-0.0216769\pi$
$798$	0	0
$799$	16.2018	0.573179
$800$	0	0
$801$	10.9558	0.387103
$802$	0	0
$803$	− 14.4897i	− 0.511331i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.6231i	0.585161i
$808$	0	0
$809$	1.51537	0.0532774	0.0266387	−	0.999645i	$-0.491520\pi$
$809$	1.51537	0.0532774	0.0266387	+	0.999645i	$0.491520\pi$
$810$	0	0
$811$	−20.1768	−0.708502	−0.354251	−	0.935150i	$-0.615264\pi$
$811$	−20.1768	−0.708502	−0.354251	+	0.935150i	$0.615264\pi$
$812$	0	0
$813$	− 69.4034i	− 2.43408i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 12.1169i	− 0.423916i
$818$	0	0
$819$	88.7346	3.10064
$820$	0	0
$821$	−33.0855	−1.15469	−0.577347	−	0.816499i	$-0.695912\pi$
$821$	−33.0855	−1.15469	−0.577347	+	0.816499i	$0.695912\pi$
$822$	0	0
$823$	22.9779i	0.800958i	0.916306	+	0.400479i	$0.131156\pi$
$823$	22.9779i	0.800958i	−0.916306	+	0.400479i	$0.868844\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.9730i	1.91160i	0.294023	+	0.955798i	$0.405006\pi$
$827$	54.9730i	1.91160i	−0.294023	+	0.955798i	$0.594994\pi$
$828$	0	0
$829$	−36.6692	−1.27357	−0.636787	−	0.771039i	$-0.719737\pi$
$829$	−36.6692	−1.27357	−0.636787	+	0.771039i	$0.719737\pi$
$830$	0	0
$831$	−12.5544	−0.435508
$832$	0	0
$833$	17.5489i	0.608033i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 35.2851i	− 1.21963i
$838$	0	0
$839$	31.4619	1.08619	0.543093	−	0.839672i	$-0.317253\pi$
$839$	31.4619	1.08619	0.543093	+	0.839672i	$0.317253\pi$
$840$	0	0
$841$	52.9289	1.82513
$842$	0	0
$843$	1.00058i	0.0344619i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	42.5263i	1.46122i
$848$	0	0
$849$	3.28126	0.112613
$850$	0	0
$851$	44.0808	1.51107
$852$	0	0
$853$	− 9.45230i	− 0.323641i	−0.986820	−	0.161820i	$-0.948263\pi$
$853$	− 9.45230i	− 0.323641i	0.986820	−	0.161820i	$-0.0517365\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17.9551i	0.613335i	0.951817	+	0.306668i	$0.0992140\pi$
$857$	17.9551i	0.613335i	−0.951817	+	0.306668i	$0.900786\pi$
$858$	0	0
$859$	24.3628	0.831249	0.415624	−	0.909536i	$-0.363563\pi$
$859$	24.3628	0.831249	0.415624	+	0.909536i	$0.363563\pi$
$860$	0	0
$861$	11.4184	0.389139
$862$	0	0
$863$	− 45.1946i	− 1.53844i	−0.638982	−	0.769222i	$-0.720644\pi$
$863$	− 45.1946i	− 1.53844i	0.638982	−	0.769222i	$-0.279356\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 30.2760i	− 1.02823i
$868$	0	0
$869$	−53.7748	−1.82419
$870$	0	0
$871$	−23.9267	−0.810727
$872$	0	0
$873$	1.30341i	0.0441138i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.55736i	0.255194i	0.991826	+	0.127597i	$0.0407264\pi$
$877$	7.55736i	0.255194i	−0.991826	+	0.127597i	$0.959274\pi$
$878$	0	0
$879$	−23.4283	−0.790217
$880$	0	0
$881$	−50.5760	−1.70395	−0.851974	−	0.523584i	$-0.824595\pi$
$881$	−50.5760	−1.70395	−0.851974	+	0.523584i	$0.824595\pi$
$882$	0	0
$883$	− 6.40946i	− 0.215696i	−0.994167	−	0.107848i	$-0.965604\pi$
$883$	− 6.40946i	− 0.215696i	0.994167	−	0.107848i	$-0.0343959\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.14684i	0.0720838i	0.999350	+	0.0360419i	$0.0114750\pi$
$887$	2.14684i	0.0720838i	−0.999350	+	0.0360419i	$0.988525\pi$
$888$	0	0
$889$	−73.0085	−2.44863
$890$	0	0
$891$	−15.7038	−0.526097
$892$	0	0
$893$	− 6.72432i	− 0.225021i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 64.6003i	− 2.15694i
$898$	0	0
$899$	89.8430	2.99643
$900$	0	0
$901$	−10.4737	−0.348929
$902$	0	0
$903$	123.850i	4.12148i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 41.3186i	− 1.37196i	−0.727620	−	0.685980i	$-0.759374\pi$
$907$	− 41.3186i	− 1.37196i	0.727620	−	0.685980i	$-0.240626\pi$
$908$	0	0
$909$	−20.5833	−0.682704
$910$	0	0
$911$	59.7328	1.97903	0.989517	−	0.144415i	$-0.0461300\pi$
$911$	59.7328	1.97903	0.989517	+	0.144415i	$0.0461300\pi$
$912$	0	0
$913$	20.6451i	0.683253i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 16.2206i	− 0.535651i
$918$	0	0
$919$	34.3526	1.13319	0.566594	−	0.823997i	$-0.308261\pi$
$919$	34.3526	1.13319	0.566594	+	0.823997i	$0.308261\pi$
$920$	0	0
$921$	22.3426	0.736215
$922$	0	0
$923$	43.0035i	1.41548i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 77.2342i	− 2.53670i
$928$	0	0
$929$	48.0752	1.57729	0.788647	−	0.614846i	$-0.210782\pi$
$929$	48.0752	1.57729	0.788647	+	0.614846i	$0.210782\pi$
$930$	0	0
$931$	7.28339	0.238703
$932$	0	0
$933$	− 20.4031i	− 0.667968i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 42.9537i	− 1.40324i	−0.712552	−	0.701619i	$-0.752461\pi$
$937$	− 42.9537i	− 1.40324i	0.712552	−	0.701619i	$-0.247539\pi$
$938$	0	0
$939$	9.28060	0.302861
$940$	0	0
$941$	−42.8745	−1.39767	−0.698835	−	0.715283i	$-0.746298\pi$
$941$	−42.8745	−1.39767	−0.698835	+	0.715283i	$0.746298\pi$
$942$	0	0
$943$	− 4.90332i	− 0.159674i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.1350i	0.914264i	0.889399	+	0.457132i	$0.151123\pi$
$947$	28.1350i	0.914264i	−0.889399	+	0.457132i	$0.848877\pi$
$948$	0	0
$949$	16.7158	0.542617
$950$	0	0
$951$	6.78249	0.219937
$952$	0	0
$953$	16.7612i	0.542949i	0.962446	+	0.271474i	$0.0875112\pi$
$953$	16.7612i	0.542949i	−0.962446	+	0.271474i	$0.912489\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	115.477i	3.73285i
$958$	0	0
$959$	80.2780	2.59231
$960$	0	0
$961$	67.5217	2.17812
$962$	0	0
$963$	56.8093i	1.83065i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 35.3277i	− 1.13606i	−0.823008	−	0.568030i	$-0.807706\pi$
$967$	− 35.3277i	− 1.13606i	0.823008	−	0.568030i	$-0.192294\pi$
$968$	0	0
$969$	6.51637	0.209336
$970$	0	0
$971$	54.3917	1.74551	0.872756	−	0.488156i	$-0.162330\pi$
$971$	54.3917	1.74551	0.872756	+	0.488156i	$0.162330\pi$
$972$	0	0
$973$	− 7.36269i	− 0.236037i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 47.4481i	− 1.51800i	−0.651091	−	0.759000i	$-0.725688\pi$
$977$	− 47.4481i	− 1.51800i	0.651091	−	0.759000i	$-0.274312\pi$
$978$	0	0
$979$	−11.9786	−0.382839
$980$	0	0
$981$	3.26262	0.104167
$982$	0	0
$983$	− 42.8550i	− 1.36686i	−0.730014	−	0.683432i	$-0.760487\pi$
$983$	− 42.8550i	− 1.36686i	0.730014	−	0.683432i	$-0.239513\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	68.7312i	2.18774i
$988$	0	0
$989$	53.1839	1.69115
$990$	0	0
$991$	23.4118	0.743700	0.371850	−	0.928293i	$-0.378724\pi$
$991$	23.4118	0.743700	0.371850	+	0.928293i	$0.378724\pi$
$992$	0	0
$993$	− 78.7467i	− 2.49895i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.9905i	0.411412i	0.978614	+	0.205706i	$0.0659491\pi$
$997$	12.9905i	0.411412i	−0.978614	+	0.205706i	$0.934051\pi$
$998$	0	0
$999$	−35.7014	−1.12954

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.p.3649.3		12
5.2	odd	4		3800.2.a.bb.1.2	✓	6
5.3	odd	4		3800.2.a.bd.1.5	yes	6
5.4	even	2	inner	3800.2.d.p.3649.10		12
20.3	even	4		7600.2.a.ci.1.2		6
20.7	even	4		7600.2.a.cm.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.bb.1.2	✓	6	5.2	odd	4
3800.2.a.bd.1.5	yes	6	5.3	odd	4
3800.2.d.p.3649.3		12	1.1	even	1	trivial
3800.2.d.p.3649.10		12	5.4	even	2	inner
7600.2.a.ci.1.2		6	20.3	even	4
7600.2.a.cm.1.5		6	20.7	even	4

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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.70452i q^{3} +3.77934i q^{7} -4.31442 q^{9} +4.71724 q^{11} +5.44195i q^{13} -2.40944i q^{17} -1.00000 q^{19} +10.2213 q^{21} -4.38924i q^{23} +3.55488i q^{27} -9.05146 q^{29} -9.92581 q^{31} -12.7579i q^{33} +10.0429i q^{37} +14.7179 q^{39} +1.11712 q^{41} +12.1169i q^{43} +6.72432i q^{47} -7.28339 q^{49} -6.51637 q^{51} -4.34694i q^{53} +2.70452i q^{57} +4.03914 q^{59} -9.14562 q^{61} -16.3057i q^{63} +4.39672i q^{67} -11.8708 q^{69} +7.90222 q^{71} -3.07165i q^{73} +17.8280i q^{77} -11.3996 q^{79} -3.32902 q^{81} +4.37652i q^{83} +24.4798i q^{87} -2.53933 q^{89} -20.5670 q^{91} +26.8445i q^{93} -0.302105i q^{97} -20.3522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{9} + 6 q^{11} - 12 q^{19} + 22 q^{21} - 14 q^{29} + 10 q^{31} - 16 q^{39} + 22 q^{41} + 4 q^{49} + 26 q^{51} + 8 q^{59} + 26 q^{61} - 14 q^{69} + 58 q^{71} - 56 q^{79} + 76 q^{81} + 24 q^{89}+ \cdots - 20 q^{99}+O(q^{100}) \) 12 * q - 20 * q^9 + 6 * q^11 - 12 * q^19 + 22 * q^21 - 14 * q^29 + 10 * q^31 - 16 * q^39 + 22 * q^41 + 4 * q^49 + 26 * q^51 + 8 * q^59 + 26 * q^61 - 14 * q^69 + 58 * q^71 - 56 * q^79 + 76 * q^81 + 24 * q^89 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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