LMFDB - Embedded newform 2000.2.c.b.1249.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2000,2,Mod(1249,2000)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2000.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2000 = 2^{4} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2000.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-12,0,-18,0,0,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$15.9700804043$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 250)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	2000.1249
Dual form			2000.2.c.b.1249.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607i q^{3} -1.61803i q^{7} -7.47214 q^{9} -3.38197 q^{11} -2.61803i q^{13} +2.47214i q^{17} -3.61803 q^{19} -5.23607 q^{21} -0.145898i q^{23} +14.4721i q^{27} -2.76393 q^{29} +5.23607 q^{31} +10.9443i q^{33} +4.38197i q^{37} -8.47214 q^{39} +7.32624 q^{41} -1.52786i q^{43} -6.61803i q^{47} +4.38197 q^{49} +8.00000 q^{51} +8.56231i q^{53} +11.7082i q^{57} -12.5623 q^{59} -12.4721 q^{61} +12.0902i q^{63} +9.23607i q^{67} -0.472136 q^{69} -13.7082 q^{71} -15.7082i q^{73} +5.47214i q^{77} +4.47214 q^{79} +24.4164 q^{81} +4.00000i q^{83} +8.94427i q^{87} +3.09017 q^{89} -4.23607 q^{91} -16.9443i q^{93} -8.18034i q^{97} +25.2705 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9} - 18 q^{11} - 10 q^{19} - 12 q^{21} - 20 q^{29} + 12 q^{31} - 16 q^{39} - 2 q^{41} + 22 q^{49} + 32 q^{51} - 10 q^{59} - 32 q^{61} + 16 q^{69} - 28 q^{71} + 44 q^{81} - 10 q^{89} - 8 q^{91}+ \cdots + 34 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 - 18 * q^11 - 10 * q^19 - 12 * q^21 - 20 * q^29 + 12 * q^31 - 16 * q^39 - 2 * q^41 + 22 * q^49 + 32 * q^51 - 10 * q^59 - 32 * q^61 + 16 * q^69 - 28 * q^71 + 44 * q^81 - 10 * q^89 - 8 * q^91 + 34 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2000\mathbb{Z}\right)^\times$.

$n$	$501$	$751$	$1377$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 3.23607i	− 1.86834i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	− 3.23607i	− 1.86834i	0.356822	−	0.934172i	$-0.383860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.61803i	− 0.611559i	−0.952102	−	0.305780i	$-0.901083\pi$
$7$	− 1.61803i	− 0.611559i	0.952102	−	0.305780i	$-0.0989171\pi$
$8$	0	0
$9$	−7.47214	−2.49071
$10$	0	0
$11$	−3.38197	−1.01970	−0.509851	−	0.860263i	$-0.670299\pi$
$11$	−3.38197	−1.01970	−0.509851	+	0.860263i	$0.670299\pi$
$12$	0	0
$13$	− 2.61803i	− 0.726112i	−0.931767	−	0.363056i	$-0.881733\pi$
$13$	− 2.61803i	− 0.726112i	0.931767	−	0.363056i	$-0.118267\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.47214i	0.599581i	0.954005	+	0.299791i	$0.0969168\pi$
$17$	2.47214i	0.599581i	−0.954005	+	0.299791i	$0.903083\pi$
$18$	0	0
$19$	−3.61803	−0.830034	−0.415017	−	0.909814i	$-0.636224\pi$
$19$	−3.61803	−0.830034	−0.415017	+	0.909814i	$0.636224\pi$
$20$	0	0
$21$	−5.23607	−1.14260
$22$	0	0
$23$	− 0.145898i	− 0.0304218i	−0.999884	−	0.0152109i	$-0.995158\pi$
$23$	− 0.145898i	− 0.0304218i	0.999884	−	0.0152109i	$-0.00484197\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	14.4721i	2.78516i
$28$	0	0
$29$	−2.76393	−0.513249	−0.256625	−	0.966511i	$-0.582610\pi$
$29$	−2.76393	−0.513249	−0.256625	+	0.966511i	$0.582610\pi$
$30$	0	0
$31$	5.23607	0.940426	0.470213	−	0.882553i	$-0.344177\pi$
$31$	5.23607	0.940426	0.470213	+	0.882553i	$0.344177\pi$
$32$	0	0
$33$	10.9443i	1.90515i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.38197i	0.720391i	0.932877	+	0.360195i	$0.117290\pi$
$37$	4.38197i	0.720391i	−0.932877	+	0.360195i	$0.882710\pi$
$38$	0	0
$39$	−8.47214	−1.35663
$40$	0	0
$41$	7.32624	1.14417	0.572083	−	0.820196i	$-0.306135\pi$
$41$	7.32624	1.14417	0.572083	+	0.820196i	$0.306135\pi$
$42$	0	0
$43$	− 1.52786i	− 0.232997i	−0.993191	−	0.116499i	$-0.962833\pi$
$43$	− 1.52786i	− 0.232997i	0.993191	−	0.116499i	$-0.0371670\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.61803i	− 0.965339i	−0.875802	−	0.482670i	$-0.839667\pi$
$47$	− 6.61803i	− 0.965339i	0.875802	−	0.482670i	$-0.160333\pi$
$48$	0	0
$49$	4.38197	0.625995
$50$	0	0
$51$	8.00000	1.12022
$52$	0	0
$53$	8.56231i	1.17612i	0.808816	+	0.588062i	$0.200109\pi$
$53$	8.56231i	1.17612i	−0.808816	+	0.588062i	$0.799891\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	11.7082i	1.55079i
$58$	0	0
$59$	−12.5623	−1.63547	−0.817736	−	0.575593i	$-0.804771\pi$
$59$	−12.5623	−1.63547	−0.817736	+	0.575593i	$0.804771\pi$
$60$	0	0
$61$	−12.4721	−1.59689	−0.798447	−	0.602066i	$-0.794345\pi$
$61$	−12.4721	−1.59689	−0.798447	+	0.602066i	$0.794345\pi$
$62$	0	0
$63$	12.0902i	1.52322i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.23607i	1.12837i	0.825650	+	0.564183i	$0.190809\pi$
$67$	9.23607i	1.12837i	−0.825650	+	0.564183i	$0.809191\pi$
$68$	0	0
$69$	−0.472136	−0.0568385
$70$	0	0
$71$	−13.7082	−1.62686	−0.813432	−	0.581660i	$-0.802404\pi$
$71$	−13.7082	−1.62686	−0.813432	+	0.581660i	$0.802404\pi$
$72$	0	0
$73$	− 15.7082i	− 1.83851i	−0.393667	−	0.919253i	$-0.628794\pi$
$73$	− 15.7082i	− 1.83851i	0.393667	−	0.919253i	$-0.371206\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.47214i	0.623608i
$78$	0	0
$79$	4.47214	0.503155	0.251577	−	0.967837i	$-0.419051\pi$
$79$	4.47214	0.503155	0.251577	+	0.967837i	$0.419051\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	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
$86$	0	0
$87$	8.94427i	0.958927i
$88$	0	0
$89$	3.09017	0.327557	0.163779	−	0.986497i	$-0.447632\pi$
$89$	3.09017	0.327557	0.163779	+	0.986497i	$0.447632\pi$
$90$	0	0
$91$	−4.23607	−0.444061
$92$	0	0
$93$	− 16.9443i	− 1.75704i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 8.18034i	− 0.830588i	−0.909687	−	0.415294i	$-0.863679\pi$
$97$	− 8.18034i	− 0.830588i	0.909687	−	0.415294i	$-0.136321\pi$
$98$	0	0
$99$	25.2705	2.53978
$100$	0	0
$101$	−6.94427	−0.690981	−0.345490	−	0.938422i	$-0.612287\pi$
$101$	−6.94427	−0.690981	−0.345490	+	0.938422i	$0.612287\pi$
$102$	0	0
$103$	− 2.90983i	− 0.286714i	−0.989671	−	0.143357i	$-0.954210\pi$
$103$	− 2.90983i	− 0.286714i	0.989671	−	0.143357i	$-0.0457897\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.763932i	− 0.0738521i	−0.999318	−	0.0369260i	$-0.988243\pi$
$107$	− 0.763932i	− 0.0738521i	0.999318	−	0.0369260i	$-0.0117566\pi$
$108$	0	0
$109$	8.94427	0.856706	0.428353	−	0.903612i	$-0.359094\pi$
$109$	8.94427	0.856706	0.428353	+	0.903612i	$0.359094\pi$
$110$	0	0
$111$	14.1803	1.34594
$112$	0	0
$113$	3.23607i	0.304424i	0.988348	+	0.152212i	$0.0486396\pi$
$113$	3.23607i	0.304424i	−0.988348	+	0.152212i	$0.951360\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	19.5623i	1.80854i
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	0.437694	0.0397904
$122$	0	0
$123$	− 23.7082i	− 2.13770i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.4164i	1.36798i	0.729489	+	0.683992i	$0.239758\pi$
$127$	15.4164i	1.36798i	−0.729489	+	0.683992i	$0.760242\pi$
$128$	0	0
$129$	−4.94427	−0.435319
$130$	0	0
$131$	−12.3262	−1.07695	−0.538474	−	0.842642i	$-0.680999\pi$
$131$	−12.3262	−1.07695	−0.538474	+	0.842642i	$0.680999\pi$
$132$	0	0
$133$	5.85410i	0.507615i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.47214i	− 0.552952i	−0.961021	−	0.276476i	$-0.910833\pi$
$137$	− 6.47214i	− 0.552952i	0.961021	−	0.276476i	$-0.0891666\pi$
$138$	0	0
$139$	14.7984	1.25518	0.627591	−	0.778543i	$-0.284041\pi$
$139$	14.7984	1.25518	0.627591	+	0.778543i	$0.284041\pi$
$140$	0	0
$141$	−21.4164	−1.80359
$142$	0	0
$143$	8.85410i	0.740417i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 14.1803i	− 1.16957i
$148$	0	0
$149$	6.18034	0.506313	0.253157	−	0.967425i	$-0.418531\pi$
$149$	6.18034	0.506313	0.253157	+	0.967425i	$0.418531\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	− 18.4721i	− 1.49338i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 16.4721i	− 1.31462i	−0.753620	−	0.657310i	$-0.771694\pi$
$157$	− 16.4721i	− 1.31462i	0.753620	−	0.657310i	$-0.228306\pi$
$158$	0	0
$159$	27.7082	2.19740
$160$	0	0
$161$	−0.236068	−0.0186048
$162$	0	0
$163$	− 6.00000i	− 0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$	− 6.00000i	− 0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.5066i	1.81899i	0.415711	+	0.909497i	$0.363533\pi$
$167$	23.5066i	1.81899i	−0.415711	+	0.909497i	$0.636467\pi$
$168$	0	0
$169$	6.14590	0.472761
$170$	0	0
$171$	27.0344	2.06738
$172$	0	0
$173$	− 5.90983i	− 0.449316i	−0.974438	−	0.224658i	$-0.927874\pi$
$173$	− 5.90983i	− 0.449316i	0.974438	−	0.224658i	$-0.0721265\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	40.6525i	3.05563i
$178$	0	0
$179$	9.14590	0.683597	0.341798	−	0.939773i	$-0.388964\pi$
$179$	9.14590	0.683597	0.341798	+	0.939773i	$0.388964\pi$
$180$	0	0
$181$	−15.2361	−1.13249	−0.566244	−	0.824238i	$-0.691604\pi$
$181$	−15.2361	−1.13249	−0.566244	+	0.824238i	$0.691604\pi$
$182$	0	0
$183$	40.3607i	2.98355i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 8.36068i	− 0.611393i
$188$	0	0
$189$	23.4164	1.70329
$190$	0	0
$191$	−19.8885	−1.43908	−0.719542	−	0.694449i	$-0.755648\pi$
$191$	−19.8885	−1.43908	−0.719542	+	0.694449i	$0.755648\pi$
$192$	0	0
$193$	12.1803i	0.876760i	0.898790	+	0.438380i	$0.144448\pi$
$193$	12.1803i	0.876760i	−0.898790	+	0.438380i	$0.855552\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.94427i	0.494759i	0.968919	+	0.247379i	$0.0795694\pi$
$197$	6.94427i	0.494759i	−0.968919	+	0.247379i	$0.920431\pi$
$198$	0	0
$199$	−17.2361	−1.22183	−0.610916	−	0.791695i	$-0.709199\pi$
$199$	−17.2361	−1.22183	−0.610916	+	0.791695i	$0.709199\pi$
$200$	0	0
$201$	29.8885	2.10818
$202$	0	0
$203$	4.47214i	0.313882i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.09017i	0.0757720i
$208$	0	0
$209$	12.2361	0.846387
$210$	0	0
$211$	−10.0902	−0.694636	−0.347318	−	0.937747i	$-0.612908\pi$
$211$	−10.0902	−0.694636	−0.347318	+	0.937747i	$0.612908\pi$
$212$	0	0
$213$	44.3607i	3.03954i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 8.47214i	− 0.575126i
$218$	0	0
$219$	−50.8328	−3.43496
$220$	0	0
$221$	6.47214	0.435363
$222$	0	0
$223$	− 4.94427i	− 0.331093i	−0.986202	−	0.165546i	$-0.947061\pi$
$223$	− 4.94427i	− 0.331093i	0.986202	−	0.165546i	$-0.0529388\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 18.6525i	− 1.23801i	−0.785388	−	0.619004i	$-0.787536\pi$
$227$	− 18.6525i	− 1.23801i	0.785388	−	0.619004i	$-0.212464\pi$
$228$	0	0
$229$	−21.7082	−1.43452	−0.717259	−	0.696806i	$-0.754604\pi$
$229$	−21.7082	−1.43452	−0.717259	+	0.696806i	$0.754604\pi$
$230$	0	0
$231$	17.7082	1.16511
$232$	0	0
$233$	11.5279i	0.755215i	0.925966	+	0.377608i	$0.123253\pi$
$233$	11.5279i	0.755215i	−0.925966	+	0.377608i	$0.876747\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 14.4721i	− 0.940066i
$238$	0	0
$239$	10.6525	0.689051	0.344526	−	0.938777i	$-0.388040\pi$
$239$	10.6525	0.689051	0.344526	+	0.938777i	$0.388040\pi$
$240$	0	0
$241$	−4.90983	−0.316270	−0.158135	−	0.987418i	$-0.550548\pi$
$241$	−4.90983	−0.316270	−0.158135	+	0.987418i	$0.550548\pi$
$242$	0	0
$243$	− 35.5967i	− 2.28353i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9.47214i	0.602698i
$248$	0	0
$249$	12.9443	0.820310
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	0.493422i	0.0310212i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 13.7082i	− 0.855094i	−0.903993	−	0.427547i	$-0.859378\pi$
$257$	− 13.7082i	− 0.855094i	0.903993	−	0.427547i	$-0.140622\pi$
$258$	0	0
$259$	7.09017	0.440562
$260$	0	0
$261$	20.6525	1.27836
$262$	0	0
$263$	3.14590i	0.193984i	0.995285	+	0.0969922i	$0.0309222\pi$
$263$	3.14590i	0.193984i	−0.995285	+	0.0969922i	$0.969078\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	0	0
$269$	−29.5967	−1.80455	−0.902273	−	0.431166i	$-0.858102\pi$
$269$	−29.5967	−1.80455	−0.902273	+	0.431166i	$0.858102\pi$
$270$	0	0
$271$	−21.5967	−1.31191	−0.655954	−	0.754800i	$-0.727734\pi$
$271$	−21.5967	−1.31191	−0.655954	+	0.754800i	$0.727734\pi$
$272$	0	0
$273$	13.7082i	0.829658i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 19.5623i	− 1.17539i	−0.809084	−	0.587693i	$-0.800036\pi$
$277$	− 19.5623i	− 1.17539i	0.809084	−	0.587693i	$-0.199964\pi$
$278$	0	0
$279$	−39.1246	−2.34233
$280$	0	0
$281$	−11.0902	−0.661584	−0.330792	−	0.943704i	$-0.607316\pi$
$281$	−11.0902	−0.661584	−0.330792	+	0.943704i	$0.607316\pi$
$282$	0	0
$283$	− 24.9443i	− 1.48278i	−0.671073	−	0.741392i	$-0.734166\pi$
$283$	− 24.9443i	− 1.48278i	0.671073	−	0.741392i	$-0.265834\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 11.8541i	− 0.699726i
$288$	0	0
$289$	10.8885	0.640503
$290$	0	0
$291$	−26.4721	−1.55182
$292$	0	0
$293$	− 17.0902i	− 0.998418i	−0.866481	−	0.499209i	$-0.833624\pi$
$293$	− 17.0902i	− 0.998418i	0.866481	−	0.499209i	$-0.166376\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 48.9443i	− 2.84003i
$298$	0	0
$299$	−0.381966	−0.0220897
$300$	0	0
$301$	−2.47214	−0.142492
$302$	0	0
$303$	22.4721i	1.29099i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.81966i	0.332146i	0.986113	+	0.166073i	$0.0531087\pi$
$307$	5.81966i	0.332146i	−0.986113	+	0.166073i	$0.946891\pi$
$308$	0	0
$309$	−9.41641	−0.535681
$310$	0	0
$311$	17.5967	0.997820	0.498910	−	0.866654i	$-0.333734\pi$
$311$	17.5967	0.997820	0.498910	+	0.866654i	$0.333734\pi$
$312$	0	0
$313$	− 30.8328i	− 1.74277i	−0.490596	−	0.871387i	$-0.663221\pi$
$313$	− 30.8328i	− 1.74277i	0.490596	−	0.871387i	$-0.336779\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.8541i	1.33978i	0.742460	+	0.669890i	$0.233659\pi$
$317$	23.8541i	1.33978i	−0.742460	+	0.669890i	$0.766341\pi$
$318$	0	0
$319$	9.34752	0.523361
$320$	0	0
$321$	−2.47214	−0.137981
$322$	0	0
$323$	− 8.94427i	− 0.497673i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 28.9443i	− 1.60062i
$328$	0	0
$329$	−10.7082	−0.590362
$330$	0	0
$331$	5.88854	0.323664	0.161832	−	0.986818i	$-0.448260\pi$
$331$	5.88854	0.323664	0.161832	+	0.986818i	$0.448260\pi$
$332$	0	0
$333$	− 32.7426i	− 1.79429i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	30.3607i	1.65385i	0.562311	+	0.826926i	$0.309912\pi$
$337$	30.3607i	1.65385i	−0.562311	+	0.826926i	$0.690088\pi$
$338$	0	0
$339$	10.4721	0.568768
$340$	0	0
$341$	−17.7082	−0.958953
$342$	0	0
$343$	− 18.4164i	− 0.994393i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.41641i	− 0.0760368i	−0.999277	−	0.0380184i	$-0.987895\pi$
$347$	− 1.41641i	− 0.0760368i	0.999277	−	0.0380184i	$-0.0121045\pi$
$348$	0	0
$349$	−27.8885	−1.49284	−0.746420	−	0.665475i	$-0.768229\pi$
$349$	−27.8885	−1.49284	−0.746420	+	0.665475i	$0.768229\pi$
$350$	0	0
$351$	37.8885	2.02234
$352$	0	0
$353$	0.472136i	0.0251293i	0.999921	+	0.0125646i	$0.00399955\pi$
$353$	0.472136i	0.0251293i	−0.999921	+	0.0125646i	$0.996000\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 12.9443i	− 0.685084i
$358$	0	0
$359$	−17.2361	−0.909685	−0.454842	−	0.890572i	$-0.650304\pi$
$359$	−17.2361	−0.909685	−0.454842	+	0.890572i	$0.650304\pi$
$360$	0	0
$361$	−5.90983	−0.311044
$362$	0	0
$363$	− 1.41641i	− 0.0743421i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 22.4721i	− 1.17304i	−0.809936	−	0.586518i	$-0.800498\pi$
$367$	− 22.4721i	− 1.17304i	0.809936	−	0.586518i	$-0.199502\pi$
$368$	0	0
$369$	−54.7426	−2.84979
$370$	0	0
$371$	13.8541	0.719269
$372$	0	0
$373$	− 22.0902i	− 1.14379i	−0.820328	−	0.571893i	$-0.806209\pi$
$373$	− 22.0902i	− 1.14379i	0.820328	−	0.571893i	$-0.193791\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.23607i	0.372676i
$378$	0	0
$379$	−20.3262	−1.04409	−0.522044	−	0.852918i	$-0.674830\pi$
$379$	−20.3262	−1.04409	−0.522044	+	0.852918i	$0.674830\pi$
$380$	0	0
$381$	49.8885	2.55587
$382$	0	0
$383$	− 9.61803i	− 0.491459i	−0.969338	−	0.245729i	$-0.920973\pi$
$383$	− 9.61803i	− 0.491459i	0.969338	−	0.245729i	$-0.0790274\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.4164i	0.580329i
$388$	0	0
$389$	−5.52786	−0.280274	−0.140137	−	0.990132i	$-0.544754\pi$
$389$	−5.52786	−0.280274	−0.140137	+	0.990132i	$0.544754\pi$
$390$	0	0
$391$	0.360680	0.0182404
$392$	0	0
$393$	39.8885i	2.01211i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.965558i	0.0484600i	0.999706	+	0.0242300i	$0.00771340\pi$
$397$	0.965558i	0.0484600i	−0.999706	+	0.0242300i	$0.992287\pi$
$398$	0	0
$399$	18.9443	0.948400
$400$	0	0
$401$	−35.6869	−1.78212	−0.891060	−	0.453886i	$-0.850037\pi$
$401$	−35.6869	−1.78212	−0.891060	+	0.453886i	$0.850037\pi$
$402$	0	0
$403$	− 13.7082i	− 0.682854i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 14.8197i	− 0.734583i
$408$	0	0
$409$	−17.0344	−0.842299	−0.421149	−	0.906991i	$-0.638373\pi$
$409$	−17.0344	−0.842299	−0.421149	+	0.906991i	$0.638373\pi$
$410$	0	0
$411$	−20.9443	−1.03310
$412$	0	0
$413$	20.3262i	1.00019i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 47.8885i	− 2.34511i
$418$	0	0
$419$	17.8885	0.873913	0.436956	−	0.899483i	$-0.356056\pi$
$419$	17.8885	0.873913	0.436956	+	0.899483i	$0.356056\pi$
$420$	0	0
$421$	−14.1803	−0.691107	−0.345554	−	0.938399i	$-0.612309\pi$
$421$	−14.1803	−0.691107	−0.345554	+	0.938399i	$0.612309\pi$
$422$	0	0
$423$	49.4508i	2.40438i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	20.1803i	0.976595i
$428$	0	0
$429$	28.6525	1.38335
$430$	0	0
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	0	0
$433$	7.70820i	0.370433i	0.982698	+	0.185216i	$0.0592986\pi$
$433$	7.70820i	0.370433i	−0.982698	+	0.185216i	$0.940701\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.527864i	0.0252512i
$438$	0	0
$439$	−7.23607	−0.345359	−0.172679	−	0.984978i	$-0.555242\pi$
$439$	−7.23607	−0.345359	−0.172679	+	0.984978i	$0.555242\pi$
$440$	0	0
$441$	−32.7426	−1.55917
$442$	0	0
$443$	19.5279i	0.927797i	0.885888	+	0.463898i	$0.153550\pi$
$443$	19.5279i	0.927797i	−0.885888	+	0.463898i	$0.846450\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 20.0000i	− 0.945968i
$448$	0	0
$449$	9.79837	0.462414	0.231207	−	0.972905i	$-0.425733\pi$
$449$	9.79837	0.462414	0.231207	+	0.972905i	$0.425733\pi$
$450$	0	0
$451$	−24.7771	−1.16671
$452$	0	0
$453$	6.47214i	0.304087i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.81966i	0.0851201i	0.999094	+	0.0425601i	$0.0135514\pi$
$457$	1.81966i	0.0851201i	−0.999094	+	0.0425601i	$0.986449\pi$
$458$	0	0
$459$	−35.7771	−1.66993
$460$	0	0
$461$	4.36068	0.203097	0.101549	−	0.994831i	$-0.467620\pi$
$461$	4.36068	0.203097	0.101549	+	0.994831i	$0.467620\pi$
$462$	0	0
$463$	21.8885i	1.01725i	0.860989	+	0.508623i	$0.169845\pi$
$463$	21.8885i	1.01725i	−0.860989	+	0.508623i	$0.830155\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.12461i	− 0.144590i	−0.997383	−	0.0722949i	$-0.976968\pi$
$467$	− 3.12461i	− 0.144590i	0.997383	−	0.0722949i	$-0.0230323\pi$
$468$	0	0
$469$	14.9443	0.690062
$470$	0	0
$471$	−53.3050	−2.45616
$472$	0	0
$473$	5.16718i	0.237587i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 63.9787i	− 2.92938i
$478$	0	0
$479$	2.36068	0.107862	0.0539311	−	0.998545i	$-0.482825\pi$
$479$	2.36068	0.107862	0.0539311	+	0.998545i	$0.482825\pi$
$480$	0	0
$481$	11.4721	0.523084
$482$	0	0
$483$	0.763932i	0.0347601i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 41.7426i	− 1.89154i	−0.324837	−	0.945770i	$-0.605310\pi$
$487$	− 41.7426i	− 1.89154i	0.324837	−	0.945770i	$-0.394690\pi$
$488$	0	0
$489$	−19.4164	−0.878040
$490$	0	0
$491$	−2.85410	−0.128804	−0.0644019	−	0.997924i	$-0.520514\pi$
$491$	−2.85410	−0.128804	−0.0644019	+	0.997924i	$0.520514\pi$
$492$	0	0
$493$	− 6.83282i	− 0.307735i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.1803i	0.994924i
$498$	0	0
$499$	32.0344	1.43406	0.717029	−	0.697043i	$-0.245501\pi$
$499$	32.0344	1.43406	0.717029	+	0.697043i	$0.245501\pi$
$500$	0	0
$501$	76.0689	3.39851
$502$	0	0
$503$	25.3820i	1.13173i	0.824499	+	0.565863i	$0.191457\pi$
$503$	25.3820i	1.13173i	−0.824499	+	0.565863i	$0.808543\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 19.8885i	− 0.883281i
$508$	0	0
$509$	−0.652476	−0.0289205	−0.0144602	−	0.999895i	$-0.504603\pi$
$509$	−0.652476	−0.0289205	−0.0144602	+	0.999895i	$0.504603\pi$
$510$	0	0
$511$	−25.4164	−1.12436
$512$	0	0
$513$	− 52.3607i	− 2.31178i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	22.3820i	0.984358i
$518$	0	0
$519$	−19.1246	−0.839477
$520$	0	0
$521$	−31.0902	−1.36209	−0.681043	−	0.732244i	$-0.738473\pi$
$521$	−31.0902	−1.36209	−0.681043	+	0.732244i	$0.738473\pi$
$522$	0	0
$523$	12.2918i	0.537483i	0.963212	+	0.268741i	$0.0866077\pi$
$523$	12.2918i	0.537483i	−0.963212	+	0.268741i	$0.913392\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.9443i	0.563861i
$528$	0	0
$529$	22.9787	0.999075
$530$	0	0
$531$	93.8673	4.07349
$532$	0	0
$533$	− 19.1803i	− 0.830793i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 29.5967i	− 1.27719i
$538$	0	0
$539$	−14.8197	−0.638328
$540$	0	0
$541$	19.8885	0.855075	0.427538	−	0.903998i	$-0.359381\pi$
$541$	19.8885	0.855075	0.427538	+	0.903998i	$0.359381\pi$
$542$	0	0
$543$	49.3050i	2.11588i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	46.0689i	1.96976i	0.173229	+	0.984882i	$0.444580\pi$
$547$	46.0689i	1.96976i	−0.173229	+	0.984882i	$0.555420\pi$
$548$	0	0
$549$	93.1935	3.97740
$550$	0	0
$551$	10.0000	0.426014
$552$	0	0
$553$	− 7.23607i	− 0.307709i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 27.8541i	− 1.18022i	−0.807324	−	0.590108i	$-0.799085\pi$
$557$	− 27.8541i	− 1.18022i	0.807324	−	0.590108i	$-0.200915\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−27.0557	−1.14229
$562$	0	0
$563$	− 27.7082i	− 1.16776i	−0.811839	−	0.583881i	$-0.801534\pi$
$563$	− 27.7082i	− 1.16776i	0.811839	−	0.583881i	$-0.198466\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 39.5066i	− 1.65912i
$568$	0	0
$569$	37.5623	1.57469	0.787347	−	0.616510i	$-0.211454\pi$
$569$	37.5623	1.57469	0.787347	+	0.616510i	$0.211454\pi$
$570$	0	0
$571$	−10.0902	−0.422260	−0.211130	−	0.977458i	$-0.567714\pi$
$571$	−10.0902	−0.422260	−0.211130	+	0.977458i	$0.567714\pi$
$572$	0	0
$573$	64.3607i	2.68871i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 19.8885i	− 0.827971i	−0.910283	−	0.413985i	$-0.864136\pi$
$577$	− 19.8885i	− 0.827971i	0.910283	−	0.413985i	$-0.135864\pi$
$578$	0	0
$579$	39.4164	1.63809
$580$	0	0
$581$	6.47214	0.268509
$582$	0	0
$583$	− 28.9574i	− 1.19929i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 4.58359i	− 0.189185i	−0.995516	−	0.0945925i	$-0.969845\pi$
$587$	− 4.58359i	− 0.189185i	0.995516	−	0.0945925i	$-0.0301548\pi$
$588$	0	0
$589$	−18.9443	−0.780585
$590$	0	0
$591$	22.4721	0.924380
$592$	0	0
$593$	0.472136i	0.0193883i	0.999953	+	0.00969415i	$0.00308579\pi$
$593$	0.472136i	0.0193883i	−0.999953	+	0.00969415i	$0.996914\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	55.7771i	2.28280i
$598$	0	0
$599$	−5.12461	−0.209386	−0.104693	−	0.994505i	$-0.533386\pi$
$599$	−5.12461	−0.209386	−0.104693	+	0.994505i	$0.533386\pi$
$600$	0	0
$601$	23.3820	0.953770	0.476885	−	0.878966i	$-0.341766\pi$
$601$	23.3820	0.953770	0.476885	+	0.878966i	$0.341766\pi$
$602$	0	0
$603$	− 69.0132i	− 2.81043i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 10.5623i	− 0.428711i	−0.976756	−	0.214355i	$-0.931235\pi$
$607$	− 10.5623i	− 0.428711i	0.976756	−	0.214355i	$-0.0687651\pi$
$608$	0	0
$609$	14.4721	0.586441
$610$	0	0
$611$	−17.3262	−0.700945
$612$	0	0
$613$	− 39.8541i	− 1.60969i	−0.593484	−	0.804846i	$-0.702248\pi$
$613$	− 39.8541i	− 1.60969i	0.593484	−	0.804846i	$-0.297752\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 48.1803i	− 1.93967i	−0.243767	−	0.969834i	$-0.578383\pi$
$617$	− 48.1803i	− 1.93967i	0.243767	−	0.969834i	$-0.421617\pi$
$618$	0	0
$619$	−9.27051	−0.372613	−0.186307	−	0.982492i	$-0.559652\pi$
$619$	−9.27051	−0.372613	−0.186307	+	0.982492i	$0.559652\pi$
$620$	0	0
$621$	2.11146	0.0847298
$622$	0	0
$623$	− 5.00000i	− 0.200321i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 39.5967i	− 1.58134i
$628$	0	0
$629$	−10.8328	−0.431933
$630$	0	0
$631$	2.87539	0.114467	0.0572337	−	0.998361i	$-0.481772\pi$
$631$	2.87539	0.114467	0.0572337	+	0.998361i	$0.481772\pi$
$632$	0	0
$633$	32.6525i	1.29782i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 11.4721i	− 0.454543i
$638$	0	0
$639$	102.430	4.05205
$640$	0	0
$641$	−28.4508	−1.12374	−0.561871	−	0.827225i	$-0.689918\pi$
$641$	−28.4508	−1.12374	−0.561871	+	0.827225i	$0.689918\pi$
$642$	0	0
$643$	− 5.34752i	− 0.210886i	−0.994425	−	0.105443i	$-0.966374\pi$
$643$	− 5.34752i	− 0.210886i	0.994425	−	0.105443i	$-0.0336260\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.7426i	0.422337i	0.977450	+	0.211168i	$0.0677269\pi$
$647$	10.7426i	0.422337i	−0.977450	+	0.211168i	$0.932273\pi$
$648$	0	0
$649$	42.4853	1.66769
$650$	0	0
$651$	−27.4164	−1.07453
$652$	0	0
$653$	49.7426i	1.94658i	0.229579	+	0.973290i	$0.426265\pi$
$653$	49.7426i	1.94658i	−0.229579	+	0.973290i	$0.573735\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	117.374i	4.57919i
$658$	0	0
$659$	20.3262	0.791798	0.395899	−	0.918294i	$-0.370433\pi$
$659$	20.3262	0.791798	0.395899	+	0.918294i	$0.370433\pi$
$660$	0	0
$661$	−1.41641	−0.0550919	−0.0275459	−	0.999621i	$-0.508769\pi$
$661$	−1.41641	−0.0550919	−0.0275459	+	0.999621i	$0.508769\pi$
$662$	0	0
$663$	− 20.9443i	− 0.813408i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.403252i	0.0156140i
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	42.1803	1.62835
$672$	0	0
$673$	− 40.1803i	− 1.54884i	−0.632673	−	0.774419i	$-0.718042\pi$
$673$	− 40.1803i	− 1.54884i	0.632673	−	0.774419i	$-0.281958\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 4.96556i	− 0.190842i	−0.995437	−	0.0954210i	$-0.969580\pi$
$677$	− 4.96556i	− 0.190842i	0.995437	−	0.0954210i	$-0.0304197\pi$
$678$	0	0
$679$	−13.2361	−0.507954
$680$	0	0
$681$	−60.3607	−2.31303
$682$	0	0
$683$	− 14.9443i	− 0.571827i	−0.958255	−	0.285913i	$-0.907703\pi$
$683$	− 14.9443i	− 0.571827i	0.958255	−	0.285913i	$-0.0922969\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	70.2492i	2.68018i
$688$	0	0
$689$	22.4164	0.853997
$690$	0	0
$691$	20.3607	0.774557	0.387278	−	0.921963i	$-0.373415\pi$
$691$	20.3607	0.774557	0.387278	+	0.921963i	$0.373415\pi$
$692$	0	0
$693$	− 40.8885i	− 1.55323i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.1115i	0.686020i
$698$	0	0
$699$	37.3050	1.41100
$700$	0	0
$701$	0.291796	0.0110210	0.00551049	−	0.999985i	$-0.498246\pi$
$701$	0.291796	0.0110210	0.00551049	+	0.999985i	$0.498246\pi$
$702$	0	0
$703$	− 15.8541i	− 0.597949i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.2361i	0.422576i
$708$	0	0
$709$	−30.6525	−1.15118	−0.575589	−	0.817739i	$-0.695227\pi$
$709$	−30.6525	−1.15118	−0.575589	+	0.817739i	$0.695227\pi$
$710$	0	0
$711$	−33.4164	−1.25321
$712$	0	0
$713$	− 0.763932i	− 0.0286095i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 34.4721i	− 1.28739i
$718$	0	0
$719$	3.81966	0.142449	0.0712246	−	0.997460i	$-0.477309\pi$
$719$	3.81966	0.142449	0.0712246	+	0.997460i	$0.477309\pi$
$720$	0	0
$721$	−4.70820	−0.175343
$722$	0	0
$723$	15.8885i	0.590901i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 13.9787i	− 0.518442i	−0.965818	−	0.259221i	$-0.916534\pi$
$727$	− 13.9787i	− 0.518442i	0.965818	−	0.259221i	$-0.0834658\pi$
$728$	0	0
$729$	−41.9443	−1.55349
$730$	0	0
$731$	3.77709	0.139701
$732$	0	0
$733$	− 39.8541i	− 1.47204i	−0.676957	−	0.736022i	$-0.736702\pi$
$733$	− 39.8541i	− 1.47204i	0.676957	−	0.736022i	$-0.263298\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 31.2361i	− 1.15060i
$738$	0	0
$739$	−30.8541	−1.13499	−0.567493	−	0.823378i	$-0.692087\pi$
$739$	−30.8541	−1.13499	−0.567493	+	0.823378i	$0.692087\pi$
$740$	0	0
$741$	30.6525	1.12605
$742$	0	0
$743$	35.6312i	1.30718i	0.756848	+	0.653591i	$0.226738\pi$
$743$	35.6312i	1.30718i	−0.756848	+	0.653591i	$0.773262\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 29.8885i	− 1.09356i
$748$	0	0
$749$	−1.23607	−0.0451649
$750$	0	0
$751$	19.3050	0.704448	0.352224	−	0.935916i	$-0.385426\pi$
$751$	19.3050	0.704448	0.352224	+	0.935916i	$0.385426\pi$
$752$	0	0
$753$	− 25.8885i	− 0.943431i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 10.6180i	− 0.385919i	−0.981207	−	0.192960i	$-0.938191\pi$
$757$	− 10.6180i	− 0.385919i	0.981207	−	0.192960i	$-0.0618086\pi$
$758$	0	0
$759$	1.59675	0.0579583
$760$	0	0
$761$	34.5623	1.25288	0.626441	−	0.779469i	$-0.284511\pi$
$761$	34.5623	1.25288	0.626441	+	0.779469i	$0.284511\pi$
$762$	0	0
$763$	− 14.4721i	− 0.523926i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32.8885i	1.18754i
$768$	0	0
$769$	−29.9230	−1.07905	−0.539525	−	0.841969i	$-0.681396\pi$
$769$	−29.9230	−1.07905	−0.539525	+	0.841969i	$0.681396\pi$
$770$	0	0
$771$	−44.3607	−1.59761
$772$	0	0
$773$	38.3607i	1.37974i	0.723934	+	0.689869i	$0.242332\pi$
$773$	38.3607i	1.37974i	−0.723934	+	0.689869i	$0.757668\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 22.9443i	− 0.823121i
$778$	0	0
$779$	−26.5066	−0.949697
$780$	0	0
$781$	46.3607	1.65892
$782$	0	0
$783$	− 40.0000i	− 1.42948i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.291796i	0.0104014i	0.999986	+	0.00520070i	$0.00165544\pi$
$787$	0.291796i	0.0104014i	−0.999986	+	0.00520070i	$0.998345\pi$
$788$	0	0
$789$	10.1803	0.362430
$790$	0	0
$791$	5.23607	0.186173
$792$	0	0
$793$	32.6525i	1.15952i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.7426i	0.593055i	0.955024	+	0.296527i	$0.0958286\pi$
$797$	16.7426i	0.593055i	−0.955024	+	0.296527i	$0.904171\pi$
$798$	0	0
$799$	16.3607	0.578799
$800$	0	0
$801$	−23.0902	−0.815851
$802$	0	0
$803$	53.1246i	1.87473i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	95.7771i	3.37151i
$808$	0	0
$809$	−5.20163	−0.182879	−0.0914397	−	0.995811i	$-0.529147\pi$
$809$	−5.20163	−0.182879	−0.0914397	+	0.995811i	$0.529147\pi$
$810$	0	0
$811$	42.7984	1.50285	0.751427	−	0.659816i	$-0.229366\pi$
$811$	42.7984	1.50285	0.751427	+	0.659816i	$0.229366\pi$
$812$	0	0
$813$	69.8885i	2.45110i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.52786i	0.193395i
$818$	0	0
$819$	31.6525	1.10603
$820$	0	0
$821$	10.9443	0.381958	0.190979	−	0.981594i	$-0.438834\pi$
$821$	10.9443	0.381958	0.190979	+	0.981594i	$0.438834\pi$
$822$	0	0
$823$	− 29.2148i	− 1.01836i	−0.860659	−	0.509182i	$-0.829948\pi$
$823$	− 29.2148i	− 1.01836i	0.860659	−	0.509182i	$-0.170052\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 15.2361i	− 0.529810i	−0.964275	−	0.264905i	$-0.914659\pi$
$827$	− 15.2361i	− 0.529810i	0.964275	−	0.264905i	$-0.0853406\pi$
$828$	0	0
$829$	34.4721	1.19727	0.598633	−	0.801023i	$-0.295711\pi$
$829$	34.4721	1.19727	0.598633	+	0.801023i	$0.295711\pi$
$830$	0	0
$831$	−63.3050	−2.19602
$832$	0	0
$833$	10.8328i	0.375335i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	75.7771i	2.61924i
$838$	0	0
$839$	12.1115	0.418134	0.209067	−	0.977901i	$-0.432957\pi$
$839$	12.1115	0.418134	0.209067	+	0.977901i	$0.432957\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	0	0
$843$	35.8885i	1.23607i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.708204i	− 0.0243342i
$848$	0	0
$849$	−80.7214	−2.77035
$850$	0	0
$851$	0.639320	0.0219156
$852$	0	0
$853$	17.5066i	0.599414i	0.954031	+	0.299707i	$0.0968889\pi$
$853$	17.5066i	0.599414i	−0.954031	+	0.299707i	$0.903111\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.9443i	0.920399i	0.887816	+	0.460199i	$0.152222\pi$
$857$	26.9443i	0.920399i	−0.887816	+	0.460199i	$0.847778\pi$
$858$	0	0
$859$	14.6738	0.500662	0.250331	−	0.968160i	$-0.419461\pi$
$859$	14.6738	0.500662	0.250331	+	0.968160i	$0.419461\pi$
$860$	0	0
$861$	−38.3607	−1.30733
$862$	0	0
$863$	− 9.61803i	− 0.327402i	−0.986510	−	0.163701i	$-0.947657\pi$
$863$	− 9.61803i	− 0.327402i	0.986510	−	0.163701i	$-0.0523432\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 35.2361i	− 1.19668i
$868$	0	0
$869$	−15.1246	−0.513067
$870$	0	0
$871$	24.1803	0.819320
$872$	0	0
$873$	61.1246i	2.06875i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.72949i	0.294774i	0.989079	+	0.147387i	$0.0470863\pi$
$877$	8.72949i	0.294774i	−0.989079	+	0.147387i	$0.952914\pi$
$878$	0	0
$879$	−55.3050	−1.86539
$880$	0	0
$881$	40.0902	1.35067	0.675336	−	0.737510i	$-0.263999\pi$
$881$	40.0902	1.35067	0.675336	+	0.737510i	$0.263999\pi$
$882$	0	0
$883$	− 16.0000i	− 0.538443i	−0.963078	−	0.269221i	$-0.913234\pi$
$883$	− 16.0000i	− 0.538443i	0.963078	−	0.269221i	$-0.0867663\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.2705i	0.546310i	0.961970	+	0.273155i	$0.0880672\pi$
$887$	16.2705i	0.546310i	−0.961970	+	0.273155i	$0.911933\pi$
$888$	0	0
$889$	24.9443	0.836604
$890$	0	0
$891$	−82.5755	−2.76638
$892$	0	0
$893$	23.9443i	0.801265i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.23607i	0.0412711i
$898$	0	0
$899$	−14.4721	−0.482673
$900$	0	0
$901$	−21.1672	−0.705181
$902$	0	0
$903$	8.00000i	0.266223i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 33.1246i	− 1.09988i	−0.835203	−	0.549942i	$-0.814650\pi$
$907$	− 33.1246i	− 1.09988i	0.835203	−	0.549942i	$-0.185350\pi$
$908$	0	0
$909$	51.8885	1.72103
$910$	0	0
$911$	−38.1803	−1.26497	−0.632486	−	0.774572i	$-0.717965\pi$
$911$	−38.1803	−1.26497	−0.632486	+	0.774572i	$0.717965\pi$
$912$	0	0
$913$	− 13.5279i	− 0.447707i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.9443i	0.658618i
$918$	0	0
$919$	−49.5967	−1.63605	−0.818023	−	0.575186i	$-0.804930\pi$
$919$	−49.5967	−1.63605	−0.818023	+	0.575186i	$0.804930\pi$
$920$	0	0
$921$	18.8328	0.620562
$922$	0	0
$923$	35.8885i	1.18129i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	21.7426i	0.714122i
$928$	0	0
$929$	3.09017	0.101385	0.0506926	−	0.998714i	$-0.483857\pi$
$929$	3.09017	0.101385	0.0506926	+	0.998714i	$0.483857\pi$
$930$	0	0
$931$	−15.8541	−0.519597
$932$	0	0
$933$	− 56.9443i	− 1.86427i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.360680i	0.0117829i	0.999983	+	0.00589145i	$0.00187532\pi$
$937$	0.360680i	0.0117829i	−0.999983	+	0.00589145i	$0.998125\pi$
$938$	0	0
$939$	−99.7771	−3.25610
$940$	0	0
$941$	−9.05573	−0.295208	−0.147604	−	0.989047i	$-0.547156\pi$
$941$	−9.05573	−0.295208	−0.147604	+	0.989047i	$0.547156\pi$
$942$	0	0
$943$	− 1.06888i	− 0.0348076i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.3607i	0.791616i	0.918333	+	0.395808i	$0.129535\pi$
$947$	24.3607i	0.791616i	−0.918333	+	0.395808i	$0.870465\pi$
$948$	0	0
$949$	−41.1246	−1.33496
$950$	0	0
$951$	77.1935	2.50317
$952$	0	0
$953$	1.52786i	0.0494924i	0.999694	+	0.0247462i	$0.00787776\pi$
$953$	1.52786i	0.0494924i	−0.999694	+	0.0247462i	$0.992122\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 30.2492i	− 0.977819i
$958$	0	0
$959$	−10.4721	−0.338163
$960$	0	0
$961$	−3.58359	−0.115600
$962$	0	0
$963$	5.70820i	0.183944i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 13.4508i	− 0.432550i	−0.976332	−	0.216275i	$-0.930609\pi$
$967$	− 13.4508i	− 0.432550i	0.976332	−	0.216275i	$-0.0693908\pi$
$968$	0	0
$969$	−28.9443	−0.929824
$970$	0	0
$971$	37.2705	1.19607	0.598034	−	0.801471i	$-0.295949\pi$
$971$	37.2705	1.19607	0.598034	+	0.801471i	$0.295949\pi$
$972$	0	0
$973$	− 23.9443i	− 0.767618i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.8885i	1.14818i	0.818794	+	0.574088i	$0.194643\pi$
$977$	35.8885i	1.14818i	−0.818794	+	0.574088i	$0.805357\pi$
$978$	0	0
$979$	−10.4508	−0.334011
$980$	0	0
$981$	−66.8328	−2.13381
$982$	0	0
$983$	56.2837i	1.79517i	0.440841	+	0.897585i	$0.354680\pi$
$983$	56.2837i	1.79517i	−0.440841	+	0.897585i	$0.645320\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	34.6525i	1.10300i
$988$	0	0
$989$	−0.222912	−0.00708820
$990$	0	0
$991$	18.0000	0.571789	0.285894	−	0.958261i	$-0.407709\pi$
$991$	18.0000	0.571789	0.285894	+	0.958261i	$0.407709\pi$
$992$	0	0
$993$	− 19.0557i	− 0.604715i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	29.5066i	0.934483i	0.884130	+	0.467241i	$0.154752\pi$
$997$	29.5066i	0.934483i	−0.884130	+	0.467241i	$0.845248\pi$
$998$	0	0
$999$	−63.4164	−2.00641

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2000.2.c.b.1249.1		4
4.3	odd	2		250.2.b.a.249.4		4
5.2	odd	4		2000.2.a.c.1.1		2
5.3	odd	4		2000.2.a.j.1.2		2
5.4	even	2	inner	2000.2.c.b.1249.4		4
12.11	even	2		2250.2.c.a.1999.2		4
20.3	even	4		250.2.a.c.1.1	yes	2
20.7	even	4		250.2.a.b.1.2	✓	2
20.19	odd	2		250.2.b.a.249.1		4
40.3	even	4		8000.2.a.s.1.2		2
40.13	odd	4		8000.2.a.f.1.1		2
40.27	even	4		8000.2.a.e.1.1		2
40.37	odd	4		8000.2.a.t.1.2		2
60.23	odd	4		2250.2.a.d.1.2		2
60.47	odd	4		2250.2.a.k.1.1		2
60.59	even	2		2250.2.c.a.1999.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
250.2.a.b.1.2	✓	2	20.7	even	4
250.2.a.c.1.1	yes	2	20.3	even	4
250.2.b.a.249.1		4	20.19	odd	2
250.2.b.a.249.4		4	4.3	odd	2
2000.2.a.c.1.1		2	5.2	odd	4
2000.2.a.j.1.2		2	5.3	odd	4
2000.2.c.b.1249.1		4	1.1	even	1	trivial
2000.2.c.b.1249.4		4	5.4	even	2	inner
2250.2.a.d.1.2		2	60.23	odd	4
2250.2.a.k.1.1		2	60.47	odd	4
2250.2.c.a.1999.2		4	12.11	even	2
2250.2.c.a.1999.3		4	60.59	even	2
8000.2.a.e.1.1		2	40.27	even	4
8000.2.a.f.1.1		2	40.13	odd	4
8000.2.a.s.1.2		2	40.3	even	4
8000.2.a.t.1.2		2	40.37	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2000 = 2^{4} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2000.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.23607i q^{3} -1.61803i q^{7} -7.47214 q^{9} -3.38197 q^{11} -2.61803i q^{13} +2.47214i q^{17} -3.61803 q^{19} -5.23607 q^{21} -0.145898i q^{23} +14.4721i q^{27} -2.76393 q^{29} +5.23607 q^{31} +10.9443i q^{33} +4.38197i q^{37} -8.47214 q^{39} +7.32624 q^{41} -1.52786i q^{43} -6.61803i q^{47} +4.38197 q^{49} +8.00000 q^{51} +8.56231i q^{53} +11.7082i q^{57} -12.5623 q^{59} -12.4721 q^{61} +12.0902i q^{63} +9.23607i q^{67} -0.472136 q^{69} -13.7082 q^{71} -15.7082i q^{73} +5.47214i q^{77} +4.47214 q^{79} +24.4164 q^{81} +4.00000i q^{83} +8.94427i q^{87} +3.09017 q^{89} -4.23607 q^{91} -16.9443i q^{93} -8.18034i q^{97} +25.2705 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9} - 18 q^{11} - 10 q^{19} - 12 q^{21} - 20 q^{29} + 12 q^{31} - 16 q^{39} - 2 q^{41} + 22 q^{49} + 32 q^{51} - 10 q^{59} - 32 q^{61} + 16 q^{69} - 28 q^{71} + 44 q^{81} - 10 q^{89} - 8 q^{91}+ \cdots + 34 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 - 18 * q^11 - 10 * q^19 - 12 * q^21 - 20 * q^29 + 12 * q^31 - 16 * q^39 - 2 * q^41 + 22 * q^49 + 32 * q^51 - 10 * q^59 - 32 * q^61 + 16 * q^69 - 28 * q^71 + 44 * q^81 - 10 * q^89 - 8 * q^91 + 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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