LMFDB - Embedded newform 4000.2.c.d.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1249,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4000 = 2^{5} \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4000.c (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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{6} + x^{4} - x^{2} + 1$ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{8}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$0.951057 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	4000.1249
Dual form			4000.2.c.d.1249.8

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.90211i q^{3} +1.17557i q^{7} -0.618034 q^{9} +2.35114 q^{11} -3.23607i q^{13} +2.00000i q^{17} -1.45309 q^{19} +2.23607 q^{21} +1.17557i q^{23} -4.53077i q^{27} -1.38197 q^{29} +9.95959 q^{31} -4.47214i q^{33} -3.70820i q^{37} -6.15537 q^{39} -5.85410 q^{41} +1.00406i q^{43} +7.15942i q^{47} +5.61803 q^{49} +3.80423 q^{51} -4.00000i q^{53} +2.76393i q^{57} +2.35114 q^{59} +3.61803 q^{61} -0.726543i q^{63} -6.15537i q^{67} +2.23607 q^{69} +10.8576 q^{71} -11.2361i q^{73} +2.76393i q^{77} +13.2088 q^{79} -10.4721 q^{81} -2.62866i q^{83} +2.62866i q^{87} -8.09017 q^{89} +3.80423 q^{91} -18.9443i q^{93} -8.18034i q^{97} -1.45309 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 4 q^{9} - 20 q^{29} - 20 q^{41} + 36 q^{49} + 20 q^{61} - 48 q^{81} - 20 q^{89}+O(q^{100})$ 8 * q + 4 * q^9 - 20 * q^29 - 20 * q^41 + 36 * q^49 + 20 * q^61 - 48 * q^81 - 20 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$ .

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 1.90211i	− 1.09819i	−0.835761	−	0.549093i	$-0.814973\pi$
$3$	− 1.90211i	− 1.09819i	0.835761	−	0.549093i	$-0.185027\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.17557i	0.444324i	0.975010	+	0.222162i	$0.0713114\pi$
$7$	1.17557i	0.444324i	−0.975010	+	0.222162i	$0.928689\pi$
$8$	0	0
$9$	−0.618034	−0.206011
$10$	0	0
$11$	2.35114	0.708896	0.354448	−	0.935076i	$-0.384669\pi$
$11$	2.35114	0.708896	0.354448	+	0.935076i	$0.384669\pi$
$12$	0	0
$13$	− 3.23607i	− 0.897524i	−0.893651	−	0.448762i	$-0.851865\pi$
$13$	− 3.23607i	− 0.897524i	0.893651	−	0.448762i	$-0.148135\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	−1.45309	−0.333361	−0.166680	−	0.986011i	$-0.553305\pi$
$19$	−1.45309	−0.333361	−0.166680	+	0.986011i	$0.553305\pi$
$20$	0	0
$21$	2.23607	0.487950
$22$	0	0
$23$	1.17557i	0.245123i	0.992461	+	0.122562i	$0.0391109\pi$
$23$	1.17557i	0.245123i	−0.992461	+	0.122562i	$0.960889\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.53077i	− 0.871947i
$28$	0	0
$29$	−1.38197	−0.256625	−0.128312	−	0.991734i	$-0.540956\pi$
$29$	−1.38197	−0.256625	−0.128312	+	0.991734i	$0.540956\pi$
$30$	0	0
$31$	9.95959	1.78880	0.894398	−	0.447272i	$-0.147604\pi$
$31$	9.95959	1.78880	0.894398	+	0.447272i	$0.147604\pi$
$32$	0	0
$33$	− 4.47214i	− 0.778499i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 3.70820i	− 0.609625i	−0.952412	−	0.304812i	$-0.901406\pi$
$37$	− 3.70820i	− 0.609625i	0.952412	−	0.304812i	$-0.0985938\pi$
$38$	0	0
$39$	−6.15537	−0.985648
$40$	0	0
$41$	−5.85410	−0.914257	−0.457129	−	0.889401i	$-0.651122\pi$
$41$	−5.85410	−0.914257	−0.457129	+	0.889401i	$0.651122\pi$
$42$	0	0
$43$	1.00406i	0.153117i	0.997065	+	0.0765586i	$0.0243932\pi$
$43$	1.00406i	0.153117i	−0.997065	+	0.0765586i	$0.975607\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.15942i	1.04431i	0.852851	+	0.522155i	$0.174872\pi$
$47$	7.15942i	1.04431i	−0.852851	+	0.522155i	$0.825128\pi$
$48$	0	0
$49$	5.61803	0.802576
$50$	0	0
$51$	3.80423	0.532698
$52$	0	0
$53$	− 4.00000i	− 0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	− 4.00000i	− 0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.76393i	0.366092i
$58$	0	0
$59$	2.35114	0.306092	0.153046	−	0.988219i	$-0.451092\pi$
$59$	2.35114	0.306092	0.153046	+	0.988219i	$0.451092\pi$
$60$	0	0
$61$	3.61803	0.463242	0.231621	−	0.972806i	$-0.425597\pi$
$61$	3.61803	0.463242	0.231621	+	0.972806i	$0.425597\pi$
$62$	0	0
$63$	− 0.726543i	− 0.0915358i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 6.15537i	− 0.751998i	−0.926620	−	0.375999i	$-0.877300\pi$
$67$	− 6.15537i	− 0.751998i	0.926620	−	0.375999i	$-0.122700\pi$
$68$	0	0
$69$	2.23607	0.269191
$70$	0	0
$71$	10.8576	1.28857	0.644283	−	0.764787i	$-0.277156\pi$
$71$	10.8576	1.28857	0.644283	+	0.764787i	$0.277156\pi$
$72$	0	0
$73$	− 11.2361i	− 1.31508i	−0.753419	−	0.657541i	$-0.771597\pi$
$73$	− 11.2361i	− 1.31508i	0.753419	−	0.657541i	$-0.228403\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.76393i	0.314979i
$78$	0	0
$79$	13.2088	1.48610	0.743052	−	0.669233i	$-0.233377\pi$
$79$	13.2088	1.48610	0.743052	+	0.669233i	$0.233377\pi$
$80$	0	0
$81$	−10.4721	−1.16357
$82$	0	0
$83$	− 2.62866i	− 0.288532i	−0.989539	−	0.144266i	$-0.953918\pi$
$83$	− 2.62866i	− 0.288532i	0.989539	−	0.144266i	$-0.0460822\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.62866i	0.281821i
$88$	0	0
$89$	−8.09017	−0.857556	−0.428778	−	0.903410i	$-0.641056\pi$
$89$	−8.09017	−0.857556	−0.428778	+	0.903410i	$0.641056\pi$
$90$	0	0
$91$	3.80423	0.398791
$92$	0	0
$93$	− 18.9443i	− 1.96443i
$94$	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
$99$	−1.45309	−0.146041
$100$	0	0
$101$	6.09017	0.605995	0.302997	−	0.952991i	$-0.402013\pi$
$101$	6.09017	0.605995	0.302997	+	0.952991i	$0.402013\pi$
$102$	0	0
$103$	10.8576i	1.06984i	0.844904	+	0.534918i	$0.179657\pi$
$103$	10.8576i	1.06984i	−0.844904	+	0.534918i	$0.820343\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6.60440i	− 0.638471i	−0.947675	−	0.319235i	$-0.896574\pi$
$107$	− 6.60440i	− 0.638471i	0.947675	−	0.319235i	$-0.103426\pi$
$108$	0	0
$109$	−2.90983	−0.278711	−0.139356	−	0.990242i	$-0.544503\pi$
$109$	−2.90983	−0.278711	−0.139356	+	0.990242i	$0.544503\pi$
$110$	0	0
$111$	−7.05342	−0.669481
$112$	0	0
$113$	− 17.7082i	− 1.66585i	−0.553388	−	0.832924i	$-0.686665\pi$
$113$	− 17.7082i	− 1.66585i	0.553388	−	0.832924i	$-0.313335\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−2.35114	−0.215529
$120$	0	0
$121$	−5.47214	−0.497467
$122$	0	0
$123$	11.1352i	1.00402i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.5640i	1.46982i	0.678167	+	0.734908i	$0.262775\pi$
$127$	16.5640i	1.46982i	−0.678167	+	0.734908i	$0.737225\pi$
$128$	0	0
$129$	1.90983	0.168151
$130$	0	0
$131$	14.3188	1.25104	0.625522	−	0.780207i	$-0.284886\pi$
$131$	14.3188	1.25104	0.625522	+	0.780207i	$0.284886\pi$
$132$	0	0
$133$	− 1.70820i	− 0.148120i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.41641i	− 0.462755i	−0.972864	−	0.231377i	$-0.925677\pi$
$137$	− 5.41641i	− 0.462755i	0.972864	−	0.231377i	$-0.0743232\pi$
$138$	0	0
$139$	−5.60034	−0.475014	−0.237507	−	0.971386i	$-0.576330\pi$
$139$	−5.60034	−0.475014	−0.237507	+	0.971386i	$0.576330\pi$
$140$	0	0
$141$	13.6180	1.14685
$142$	0	0
$143$	− 7.60845i	− 0.636251i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 10.6861i	− 0.881378i
$148$	0	0
$149$	−0.381966	−0.0312919	−0.0156459	−	0.999878i	$-0.504980\pi$
$149$	−0.381966	−0.0312919	−0.0156459	+	0.999878i	$0.504980\pi$
$150$	0	0
$151$	−16.1150	−1.31142	−0.655708	−	0.755014i	$-0.727630\pi$
$151$	−16.1150	−1.31142	−0.655708	+	0.755014i	$0.727630\pi$
$152$	0	0
$153$	− 1.23607i	− 0.0999302i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.52786i	0.600789i	0.953815	+	0.300394i	$0.0971183\pi$
$157$	7.52786i	0.600789i	−0.953815	+	0.300394i	$0.902882\pi$
$158$	0	0
$159$	−7.60845	−0.603390
$160$	0	0
$161$	−1.38197	−0.108914
$162$	0	0
$163$	− 16.7355i	− 1.31083i	−0.755271	−	0.655413i	$-0.772495\pi$
$163$	− 16.7355i	− 1.31083i	0.755271	−	0.655413i	$-0.227505\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.34708i	0.104240i	0.998641	+	0.0521202i	$0.0165979\pi$
$167$	1.34708i	0.104240i	−0.998641	+	0.0521202i	$0.983402\pi$
$168$	0	0
$169$	2.52786	0.194451
$170$	0	0
$171$	0.898056	0.0686761
$172$	0	0
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.47214i	− 0.336146i
$178$	0	0
$179$	25.1765	1.88178	0.940890	−	0.338713i	$-0.109991\pi$
$179$	25.1765	1.88178	0.940890	+	0.338713i	$0.109991\pi$
$180$	0	0
$181$	19.5623	1.45405	0.727027	−	0.686608i	$-0.240901\pi$
$181$	19.5623	1.45405	0.727027	+	0.686608i	$0.240901\pi$
$182$	0	0
$183$	− 6.88191i	− 0.508725i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.70228i	0.343865i
$188$	0	0
$189$	5.32624	0.387427
$190$	0	0
$191$	−6.15537	−0.445387	−0.222693	−	0.974889i	$-0.571485\pi$
$191$	−6.15537	−0.445387	−0.222693	+	0.974889i	$0.571485\pi$
$192$	0	0
$193$	− 22.1803i	− 1.59658i	−0.602276	−	0.798288i	$-0.705739\pi$
$193$	− 22.1803i	− 1.59658i	0.602276	−	0.798288i	$-0.294261\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.05573i	0.645194i	0.946536	+	0.322597i	$0.104556\pi$
$197$	9.05573i	0.645194i	−0.946536	+	0.322597i	$0.895444\pi$
$198$	0	0
$199$	10.8576	0.769678	0.384839	−	0.922984i	$-0.374257\pi$
$199$	10.8576	0.769678	0.384839	+	0.922984i	$0.374257\pi$
$200$	0	0
$201$	−11.7082	−0.825833
$202$	0	0
$203$	− 1.62460i	− 0.114024i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 0.726543i	− 0.0504982i
$208$	0	0
$209$	−3.41641	−0.236318
$210$	0	0
$211$	−21.9273	−1.50954	−0.754769	−	0.655991i	$-0.772251\pi$
$211$	−21.9273	−1.50954	−0.754769	+	0.655991i	$0.772251\pi$
$212$	0	0
$213$	− 20.6525i	− 1.41508i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	11.7082i	0.794805i
$218$	0	0
$219$	−21.3723	−1.44420
$220$	0	0
$221$	6.47214	0.435363
$222$	0	0
$223$	− 17.4620i	− 1.16934i	−0.811269	−	0.584672i	$-0.801223\pi$
$223$	− 17.4620i	− 1.16934i	0.811269	−	0.584672i	$-0.198777\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 24.3440i	− 1.61576i	−0.589344	−	0.807882i	$-0.700614\pi$
$227$	− 24.3440i	− 1.61576i	0.589344	−	0.807882i	$-0.299386\pi$
$228$	0	0
$229$	7.03444	0.464849	0.232425	−	0.972614i	$-0.425334\pi$
$229$	7.03444	0.464849	0.232425	+	0.972614i	$0.425334\pi$
$230$	0	0
$231$	5.25731	0.345906
$232$	0	0
$233$	− 0.472136i	− 0.0309307i	−0.999880	−	0.0154653i	$-0.995077\pi$
$233$	− 0.472136i	− 0.0309307i	0.999880	−	0.0154653i	$-0.00492296\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 25.1246i	− 1.63202i
$238$	0	0
$239$	−10.3026	−0.666421	−0.333211	−	0.942852i	$-0.608132\pi$
$239$	−10.3026	−0.666421	−0.333211	+	0.942852i	$0.608132\pi$
$240$	0	0
$241$	−19.1459	−1.23330	−0.616648	−	0.787239i	$-0.711510\pi$
$241$	−19.1459	−1.23330	−0.616648	+	0.787239i	$0.711510\pi$
$242$	0	0
$243$	6.32688i	0.405870i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.70228i	0.299199i
$248$	0	0
$249$	−5.00000	−0.316862
$250$	0	0
$251$	−27.5276	−1.73753	−0.868765	−	0.495225i	$-0.835085\pi$
$251$	−27.5276	−1.73753	−0.868765	+	0.495225i	$0.835085\pi$
$252$	0	0
$253$	2.76393i	0.173767i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.9443i	1.30647i	0.757156	+	0.653234i	$0.226588\pi$
$257$	20.9443i	1.30647i	−0.757156	+	0.653234i	$0.773412\pi$
$258$	0	0
$259$	4.35926	0.270871
$260$	0	0
$261$	0.854102	0.0528676
$262$	0	0
$263$	21.2663i	1.31133i	0.755050	+	0.655667i	$0.227613\pi$
$263$	21.2663i	1.31133i	−0.755050	+	0.655667i	$0.772387\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	15.3884i	0.941756i
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−7.60845	−0.462181	−0.231090	−	0.972932i	$-0.574229\pi$
$271$	−7.60845	−0.462181	−0.231090	+	0.972932i	$0.574229\pi$
$272$	0	0
$273$	− 7.23607i	− 0.437947i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 12.4721i	− 0.749378i	−0.927151	−	0.374689i	$-0.877749\pi$
$277$	− 12.4721i	− 0.749378i	0.927151	−	0.374689i	$-0.122251\pi$
$278$	0	0
$279$	−6.15537	−0.368512
$280$	0	0
$281$	−17.0344	−1.01619	−0.508095	−	0.861301i	$-0.669650\pi$
$281$	−17.0344	−1.01619	−0.508095	+	0.861301i	$0.669650\pi$
$282$	0	0
$283$	26.0746i	1.54997i	0.631979	+	0.774986i	$0.282243\pi$
$283$	26.0746i	1.54997i	−0.631979	+	0.774986i	$0.717757\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 6.88191i	− 0.406226i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	−15.5599	−0.912139
$292$	0	0
$293$	− 23.2361i	− 1.35747i	−0.734385	−	0.678733i	$-0.762529\pi$
$293$	− 23.2361i	− 1.35747i	0.734385	−	0.678733i	$-0.237471\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 10.6525i	− 0.618119i
$298$	0	0
$299$	3.80423	0.220004
$300$	0	0
$301$	−1.18034	−0.0680337
$302$	0	0
$303$	− 11.5842i	− 0.665494i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	23.1029i	1.31855i	0.751902	+	0.659275i	$0.229137\pi$
$307$	23.1029i	1.31855i	−0.751902	+	0.659275i	$0.770863\pi$
$308$	0	0
$309$	20.6525	1.17488
$310$	0	0
$311$	−15.7719	−0.894344	−0.447172	−	0.894448i	$-0.647569\pi$
$311$	−15.7719	−0.894344	−0.447172	+	0.894448i	$0.647569\pi$
$312$	0	0
$313$	− 13.8885i	− 0.785027i	−0.919746	−	0.392513i	$-0.871606\pi$
$313$	− 13.8885i	− 0.785027i	0.919746	−	0.392513i	$-0.128394\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.1246i	1.52347i	0.647889	+	0.761735i	$0.275652\pi$
$317$	27.1246i	1.52347i	−0.647889	+	0.761735i	$0.724348\pi$
$318$	0	0
$319$	−3.24920	−0.181920
$320$	0	0
$321$	−12.5623	−0.701160
$322$	0	0
$323$	− 2.90617i	− 0.161704i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.53483i	0.306077i
$328$	0	0
$329$	−8.41641	−0.464012
$330$	0	0
$331$	8.50651	0.467560	0.233780	−	0.972290i	$-0.424890\pi$
$331$	8.50651	0.467560	0.233780	+	0.972290i	$0.424890\pi$
$332$	0	0
$333$	2.29180i	0.125590i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 12.6525i	− 0.689224i	−0.938745	−	0.344612i	$-0.888010\pi$
$337$	− 12.6525i	− 0.689224i	0.938745	−	0.344612i	$-0.111990\pi$
$338$	0	0
$339$	−33.6830	−1.82941
$340$	0	0
$341$	23.4164	1.26807
$342$	0	0
$343$	14.8334i	0.800928i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5.53483i	− 0.297125i	−0.988903	−	0.148563i	$-0.952535\pi$
$347$	− 5.53483i	− 0.297125i	0.988903	−	0.148563i	$-0.0474646\pi$
$348$	0	0
$349$	−34.2705	−1.83446	−0.917229	−	0.398360i	$-0.869579\pi$
$349$	−34.2705	−1.83446	−0.917229	+	0.398360i	$0.869579\pi$
$350$	0	0
$351$	−14.6619	−0.782593
$352$	0	0
$353$	21.1246i	1.12435i	0.827018	+	0.562175i	$0.190035\pi$
$353$	21.1246i	1.12435i	−0.827018	+	0.562175i	$0.809965\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.47214i	0.236691i
$358$	0	0
$359$	12.8658	0.679029	0.339515	−	0.940601i	$-0.389737\pi$
$359$	12.8658	0.679029	0.339515	+	0.940601i	$0.389737\pi$
$360$	0	0
$361$	−16.8885	−0.888871
$362$	0	0
$363$	10.4086i	0.546311i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 22.3358i	− 1.16592i	−0.812500	−	0.582961i	$-0.801894\pi$
$367$	− 22.3358i	− 1.16592i	0.812500	−	0.582961i	$-0.198106\pi$
$368$	0	0
$369$	3.61803	0.188347
$370$	0	0
$371$	4.70228	0.244130
$372$	0	0
$373$	2.18034i	0.112894i	0.998406	+	0.0564469i	$0.0179771\pi$
$373$	2.18034i	0.112894i	−0.998406	+	0.0564469i	$0.982023\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.47214i	0.230327i
$378$	0	0
$379$	18.8091	0.966160	0.483080	−	0.875576i	$-0.339518\pi$
$379$	18.8091	0.966160	0.483080	+	0.875576i	$0.339518\pi$
$380$	0	0
$381$	31.5066	1.61413
$382$	0	0
$383$	− 17.4620i	− 0.892269i	−0.894966	−	0.446134i	$-0.852800\pi$
$383$	− 17.4620i	− 0.892269i	0.894966	−	0.446134i	$-0.147200\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 0.620541i	− 0.0315439i
$388$	0	0
$389$	20.5066	1.03972	0.519862	−	0.854250i	$-0.325983\pi$
$389$	20.5066	1.03972	0.519862	+	0.854250i	$0.325983\pi$
$390$	0	0
$391$	−2.35114	−0.118902
$392$	0	0
$393$	− 27.2361i	− 1.37388i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 31.5967i	− 1.58580i	−0.609355	−	0.792898i	$-0.708571\pi$
$397$	− 31.5967i	− 1.58580i	0.609355	−	0.792898i	$-0.291429\pi$
$398$	0	0
$399$	−3.24920	−0.162663
$400$	0	0
$401$	−7.14590	−0.356849	−0.178425	−	0.983954i	$-0.557100\pi$
$401$	−7.14590	−0.356849	−0.178425	+	0.983954i	$0.557100\pi$
$402$	0	0
$403$	− 32.2299i	− 1.60549i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.71851i	− 0.432161i
$408$	0	0
$409$	−15.5066	−0.766751	−0.383375	−	0.923593i	$-0.625238\pi$
$409$	−15.5066	−0.766751	−0.383375	+	0.923593i	$0.625238\pi$
$410$	0	0
$411$	−10.3026	−0.508191
$412$	0	0
$413$	2.76393i	0.136004i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10.6525i	0.521654i
$418$	0	0
$419$	−1.79611	−0.0877458	−0.0438729	−	0.999037i	$-0.513970\pi$
$419$	−1.79611	−0.0877458	−0.0438729	+	0.999037i	$0.513970\pi$
$420$	0	0
$421$	16.5066	0.804481	0.402241	−	0.915534i	$-0.368232\pi$
$421$	16.5066	0.804481	0.402241	+	0.915534i	$0.368232\pi$
$422$	0	0
$423$	− 4.42477i	− 0.215140i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.25325i	0.205829i
$428$	0	0
$429$	−14.4721	−0.698721
$430$	0	0
$431$	10.3026	0.496260	0.248130	−	0.968727i	$-0.420184\pi$
$431$	10.3026	0.496260	0.248130	+	0.968727i	$0.420184\pi$
$432$	0	0
$433$	18.7639i	0.901737i	0.892590	+	0.450869i	$0.148886\pi$
$433$	18.7639i	0.901737i	−0.892590	+	0.450869i	$0.851114\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.70820i	− 0.0817145i
$438$	0	0
$439$	−6.15537	−0.293780	−0.146890	−	0.989153i	$-0.546926\pi$
$439$	−6.15537	−0.293780	−0.146890	+	0.989153i	$0.546926\pi$
$440$	0	0
$441$	−3.47214	−0.165340
$442$	0	0
$443$	3.18368i	0.151261i	0.997136	+	0.0756307i	$0.0240970\pi$
$443$	3.18368i	0.151261i	−0.997136	+	0.0756307i	$0.975903\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.726543i	0.0343643i
$448$	0	0
$449$	31.8885	1.50491	0.752457	−	0.658642i	$-0.228869\pi$
$449$	31.8885	1.50491	0.752457	+	0.658642i	$0.228869\pi$
$450$	0	0
$451$	−13.7638	−0.648113
$452$	0	0
$453$	30.6525i	1.44018i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.944272i	0.0441712i	0.999756	+	0.0220856i	$0.00703063\pi$
$457$	0.944272i	0.0441712i	−0.999756	+	0.0220856i	$0.992969\pi$
$458$	0	0
$459$	9.06154	0.422956
$460$	0	0
$461$	6.09017	0.283647	0.141824	−	0.989892i	$-0.454703\pi$
$461$	6.09017	0.283647	0.141824	+	0.989892i	$0.454703\pi$
$462$	0	0
$463$	21.0948i	0.980356i	0.871622	+	0.490178i	$0.163068\pi$
$463$	21.0948i	0.980356i	−0.871622	+	0.490178i	$0.836932\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	37.2097i	1.72186i	0.508723	+	0.860930i	$0.330118\pi$
$467$	37.2097i	1.72186i	−0.508723	+	0.860930i	$0.669882\pi$
$468$	0	0
$469$	7.23607	0.334131
$470$	0	0
$471$	14.3188	0.659778
$472$	0	0
$473$	2.36068i	0.108544i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.47214i	0.113191i
$478$	0	0
$479$	28.4257	1.29880	0.649402	−	0.760446i	$-0.275019\pi$
$479$	28.4257	1.29880	0.649402	+	0.760446i	$0.275019\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	0	0
$483$	2.62866i	0.119608i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 38.4913i	− 1.74421i	−0.489322	−	0.872103i	$-0.662756\pi$
$487$	− 38.4913i	− 1.74421i	0.489322	−	0.872103i	$-0.337244\pi$
$488$	0	0
$489$	−31.8328	−1.43953
$490$	0	0
$491$	16.6700	0.752306	0.376153	−	0.926558i	$-0.377247\pi$
$491$	16.6700	0.752306	0.376153	+	0.926558i	$0.377247\pi$
$492$	0	0
$493$	− 2.76393i	− 0.124481i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.7639i	0.572541i
$498$	0	0
$499$	20.2622	0.907061	0.453531	−	0.891241i	$-0.350164\pi$
$499$	20.2622	0.907061	0.453531	+	0.891241i	$0.350164\pi$
$500$	0	0
$501$	2.56231	0.114475
$502$	0	0
$503$	42.1240i	1.87822i	0.343622	+	0.939108i	$0.388346\pi$
$503$	42.1240i	1.87822i	−0.343622	+	0.939108i	$0.611654\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.80828i	− 0.213543i
$508$	0	0
$509$	16.8328	0.746101	0.373051	−	0.927811i	$-0.378312\pi$
$509$	16.8328	0.746101	0.373051	+	0.927811i	$0.378312\pi$
$510$	0	0
$511$	13.2088	0.584322
$512$	0	0
$513$	6.58359i	0.290673i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.8328i	0.740306i
$518$	0	0
$519$	−26.6296	−1.16891
$520$	0	0
$521$	−10.2148	−0.447518	−0.223759	−	0.974645i	$-0.571833\pi$
$521$	−10.2148	−0.447518	−0.223759	+	0.974645i	$0.571833\pi$
$522$	0	0
$523$	1.69011i	0.0739034i	0.999317	+	0.0369517i	$0.0117648\pi$
$523$	1.69011i	0.0739034i	−0.999317	+	0.0369517i	$0.988235\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19.9192i	0.867693i
$528$	0	0
$529$	21.6180	0.939915
$530$	0	0
$531$	−1.45309	−0.0630585
$532$	0	0
$533$	18.9443i	0.820568i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 47.8885i	− 2.06654i
$538$	0	0
$539$	13.2088	0.568943
$540$	0	0
$541$	8.72949	0.375310	0.187655	−	0.982235i	$-0.439911\pi$
$541$	8.72949	0.375310	0.187655	+	0.982235i	$0.439911\pi$
$542$	0	0
$543$	− 37.2097i	− 1.59682i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 5.53483i	− 0.236652i	−0.992975	−	0.118326i	$-0.962247\pi$
$547$	− 5.53483i	− 0.236652i	0.992975	−	0.118326i	$-0.0377528\pi$
$548$	0	0
$549$	−2.23607	−0.0954331
$550$	0	0
$551$	2.00811	0.0855485
$552$	0	0
$553$	15.5279i	0.660312i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.7771i	1.43118i	0.698520	+	0.715591i	$0.253842\pi$
$557$	33.7771i	1.43118i	−0.698520	+	0.715591i	$0.746158\pi$
$558$	0	0
$559$	3.24920	0.137426
$560$	0	0
$561$	8.94427	0.377627
$562$	0	0
$563$	− 34.7931i	− 1.46635i	−0.680039	−	0.733176i	$-0.738037\pi$
$563$	− 34.7931i	− 1.46635i	0.680039	−	0.733176i	$-0.261963\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 12.3107i	− 0.517002i
$568$	0	0
$569$	7.09017	0.297235	0.148618	−	0.988895i	$-0.452518\pi$
$569$	7.09017	0.297235	0.148618	+	0.988895i	$0.452518\pi$
$570$	0	0
$571$	15.2169	0.636808	0.318404	−	0.947955i	$-0.396853\pi$
$571$	15.2169	0.636808	0.318404	+	0.947955i	$0.396853\pi$
$572$	0	0
$573$	11.7082i	0.489117i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 22.4721i	− 0.935527i	−0.883854	−	0.467764i	$-0.845060\pi$
$577$	− 22.4721i	− 0.935527i	0.883854	−	0.467764i	$-0.154940\pi$
$578$	0	0
$579$	−42.1895	−1.75334
$580$	0	0
$581$	3.09017	0.128202
$582$	0	0
$583$	− 9.40456i	− 0.389497i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.1814i	1.65846i	0.558905	+	0.829232i	$0.311222\pi$
$587$	40.1814i	1.65846i	−0.558905	+	0.829232i	$0.688778\pi$
$588$	0	0
$589$	−14.4721	−0.596314
$590$	0	0
$591$	17.2250	0.708543
$592$	0	0
$593$	− 12.2918i	− 0.504764i	−0.967628	−	0.252382i	$-0.918786\pi$
$593$	− 12.2918i	− 0.504764i	0.967628	−	0.252382i	$-0.0812139\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 20.6525i	− 0.845250i
$598$	0	0
$599$	19.7072	0.805214	0.402607	−	0.915373i	$-0.368104\pi$
$599$	19.7072	0.805214	0.402607	+	0.915373i	$0.368104\pi$
$600$	0	0
$601$	20.9787	0.855740	0.427870	−	0.903840i	$-0.359264\pi$
$601$	20.9787	0.855740	0.427870	+	0.903840i	$0.359264\pi$
$602$	0	0
$603$	3.80423i	0.154920i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 26.0746i	− 1.05833i	−0.848518	−	0.529167i	$-0.822505\pi$
$607$	− 26.0746i	− 1.05833i	0.848518	−	0.529167i	$-0.177495\pi$
$608$	0	0
$609$	−3.09017	−0.125220
$610$	0	0
$611$	23.1684	0.937292
$612$	0	0
$613$	− 22.8328i	− 0.922209i	−0.887346	−	0.461104i	$-0.847453\pi$
$613$	− 22.8328i	− 0.922209i	0.887346	−	0.461104i	$-0.152547\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	46.7214i	1.88093i	0.339889	+	0.940466i	$0.389610\pi$
$617$	46.7214i	1.88093i	−0.339889	+	0.940466i	$0.610390\pi$
$618$	0	0
$619$	11.9677	0.481023	0.240511	−	0.970646i	$-0.422685\pi$
$619$	11.9677	0.481023	0.240511	+	0.970646i	$0.422685\pi$
$620$	0	0
$621$	5.32624	0.213735
$622$	0	0
$623$	− 9.51057i	− 0.381033i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.49839i	0.259521i
$628$	0	0
$629$	7.41641	0.295712
$630$	0	0
$631$	18.6781	0.743564	0.371782	−	0.928320i	$-0.378747\pi$
$631$	18.6781	0.743564	0.371782	+	0.928320i	$0.378747\pi$
$632$	0	0
$633$	41.7082i	1.65775i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 18.1803i	− 0.720331i
$638$	0	0
$639$	−6.71040	−0.265459
$640$	0	0
$641$	−25.8541	−1.02118	−0.510588	−	0.859826i	$-0.670572\pi$
$641$	−25.8541	−1.02118	−0.510588	+	0.859826i	$0.670572\pi$
$642$	0	0
$643$	− 14.9394i	− 0.589152i	−0.955628	−	0.294576i	$-0.904822\pi$
$643$	− 14.9394i	− 0.589152i	0.955628	−	0.294576i	$-0.0951784\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.13914i	− 0.0840982i	−0.999116	−	0.0420491i	$-0.986611\pi$
$647$	− 2.13914i	− 0.0840982i	0.999116	−	0.0420491i	$-0.0133886\pi$
$648$	0	0
$649$	5.52786	0.216988
$650$	0	0
$651$	22.2703	0.872843
$652$	0	0
$653$	40.0689i	1.56802i	0.620750	+	0.784008i	$0.286828\pi$
$653$	40.0689i	1.56802i	−0.620750	+	0.784008i	$0.713172\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.94427i	0.270922i
$658$	0	0
$659$	−9.06154	−0.352987	−0.176494	−	0.984302i	$-0.556476\pi$
$659$	−9.06154	−0.352987	−0.176494	+	0.984302i	$0.556476\pi$
$660$	0	0
$661$	−30.3262	−1.17955	−0.589777	−	0.807566i	$-0.700784\pi$
$661$	−30.3262	−1.17955	−0.589777	+	0.807566i	$0.700784\pi$
$662$	0	0
$663$	− 12.3107i	− 0.478109i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.62460i	− 0.0629047i
$668$	0	0
$669$	−33.2148	−1.28416
$670$	0	0
$671$	8.50651	0.328390
$672$	0	0
$673$	39.0132i	1.50385i	0.659251	+	0.751923i	$0.270874\pi$
$673$	39.0132i	1.50385i	−0.659251	+	0.751923i	$0.729126\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.8328i	0.570071i	0.958517	+	0.285036i	$0.0920055\pi$
$677$	14.8328i	0.570071i	−0.958517	+	0.285036i	$0.907995\pi$
$678$	0	0
$679$	9.61657	0.369050
$680$	0	0
$681$	−46.3050	−1.77441
$682$	0	0
$683$	24.5155i	0.938058i	0.883183	+	0.469029i	$0.155396\pi$
$683$	24.5155i	0.938058i	−0.883183	+	0.469029i	$0.844604\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 13.3803i	− 0.510490i
$688$	0	0
$689$	−12.9443	−0.493137
$690$	0	0
$691$	30.2218	1.14969	0.574846	−	0.818262i	$-0.305062\pi$
$691$	30.2218	1.14969	0.574846	+	0.818262i	$0.305062\pi$
$692$	0	0
$693$	− 1.70820i	− 0.0648893i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 11.7082i	− 0.443480i
$698$	0	0
$699$	−0.898056	−0.0339676
$700$	0	0
$701$	−2.36068	−0.0891616	−0.0445808	−	0.999006i	$-0.514195\pi$
$701$	−2.36068	−0.0891616	−0.0445808	+	0.999006i	$0.514195\pi$
$702$	0	0
$703$	5.38834i	0.203225i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.15942i	0.269258i
$708$	0	0
$709$	7.03444	0.264184	0.132092	−	0.991237i	$-0.457831\pi$
$709$	7.03444	0.264184	0.132092	+	0.991237i	$0.457831\pi$
$710$	0	0
$711$	−8.16348	−0.306154
$712$	0	0
$713$	11.7082i	0.438476i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	19.5967i	0.731854i
$718$	0	0
$719$	−17.2250	−0.642385	−0.321192	−	0.947014i	$-0.604084\pi$
$719$	−17.2250	−0.642385	−0.321192	+	0.947014i	$0.604084\pi$
$720$	0	0
$721$	−12.7639	−0.475354
$722$	0	0
$723$	36.4177i	1.35439i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 6.26137i	− 0.232221i	−0.993236	−	0.116111i	$-0.962957\pi$
$727$	− 6.26137i	− 0.232221i	0.993236	−	0.116111i	$-0.0370427\pi$
$728$	0	0
$729$	−19.3820	−0.717851
$730$	0	0
$731$	−2.00811	−0.0742728
$732$	0	0
$733$	43.2361i	1.59696i	0.602021	+	0.798480i	$0.294362\pi$
$733$	43.2361i	1.59696i	−0.602021	+	0.798480i	$0.705638\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 14.4721i	− 0.533088i
$738$	0	0
$739$	33.1280	1.21863	0.609316	−	0.792927i	$-0.291444\pi$
$739$	33.1280	1.21863	0.609316	+	0.792927i	$0.291444\pi$
$740$	0	0
$741$	8.94427	0.328576
$742$	0	0
$743$	27.1846i	0.997307i	0.866802	+	0.498653i	$0.166172\pi$
$743$	27.1846i	0.997307i	−0.866802	+	0.498653i	$0.833828\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.62460i	0.0594410i
$748$	0	0
$749$	7.76393	0.283688
$750$	0	0
$751$	−3.11817	−0.113784	−0.0568919	−	0.998380i	$-0.518119\pi$
$751$	−3.11817	−0.113784	−0.0568919	+	0.998380i	$0.518119\pi$
$752$	0	0
$753$	52.3607i	1.90813i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.6525i	0.823318i	0.911338	+	0.411659i	$0.135051\pi$
$757$	22.6525i	0.823318i	−0.911338	+	0.411659i	$0.864949\pi$
$758$	0	0
$759$	5.25731	0.190828
$760$	0	0
$761$	38.3820	1.39135	0.695673	−	0.718359i	$-0.255106\pi$
$761$	38.3820	1.39135	0.695673	+	0.718359i	$0.255106\pi$
$762$	0	0
$763$	− 3.42071i	− 0.123838i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 7.60845i	− 0.274725i
$768$	0	0
$769$	24.3951	0.879711	0.439855	−	0.898069i	$-0.355030\pi$
$769$	24.3951	0.879711	0.439855	+	0.898069i	$0.355030\pi$
$770$	0	0
$771$	39.8384	1.43474
$772$	0	0
$773$	10.8328i	0.389629i	0.980840	+	0.194815i	$0.0624105\pi$
$773$	10.8328i	0.389629i	−0.980840	+	0.194815i	$0.937589\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.29180i	− 0.297467i
$778$	0	0
$779$	8.50651	0.304777
$780$	0	0
$781$	25.5279	0.913459
$782$	0	0
$783$	6.26137i	0.223763i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	39.0463i	1.39185i	0.718114	+	0.695925i	$0.245006\pi$
$787$	39.0463i	1.39185i	−0.718114	+	0.695925i	$0.754994\pi$
$788$	0	0
$789$	40.4508	1.44009
$790$	0	0
$791$	20.8172	0.740176
$792$	0	0
$793$	− 11.7082i	− 0.415771i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 4.18034i	− 0.148075i	−0.997255	−	0.0740376i	$-0.976412\pi$
$797$	− 4.18034i	− 0.148075i	0.997255	−	0.0740376i	$-0.0235885\pi$
$798$	0	0
$799$	−14.3188	−0.506564
$800$	0	0
$801$	5.00000	0.176666
$802$	0	0
$803$	− 26.4176i	− 0.932256i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	26.6296i	0.937406i
$808$	0	0
$809$	−52.8115	−1.85675	−0.928377	−	0.371639i	$-0.878796\pi$
$809$	−52.8115	−1.85675	−0.928377	+	0.371639i	$0.878796\pi$
$810$	0	0
$811$	20.4742	0.718947	0.359473	−	0.933155i	$-0.382956\pi$
$811$	20.4742	0.718947	0.359473	+	0.933155i	$0.382956\pi$
$812$	0	0
$813$	14.4721i	0.507560i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.45898i	− 0.0510433i
$818$	0	0
$819$	−2.35114	−0.0821555
$820$	0	0
$821$	10.4508	0.364737	0.182369	−	0.983230i	$-0.441624\pi$
$821$	10.4508	0.364737	0.182369	+	0.983230i	$0.441624\pi$
$822$	0	0
$823$	28.9807i	1.01020i	0.863060	+	0.505102i	$0.168545\pi$
$823$	28.9807i	1.01020i	−0.863060	+	0.505102i	$0.831455\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 22.0583i	− 0.767043i	−0.923532	−	0.383522i	$-0.874711\pi$
$827$	− 22.0583i	− 0.767043i	0.923532	−	0.383522i	$-0.125289\pi$
$828$	0	0
$829$	−23.2705	−0.808218	−0.404109	−	0.914711i	$-0.632418\pi$
$829$	−23.2705	−0.808218	−0.404109	+	0.914711i	$0.632418\pi$
$830$	0	0
$831$	−23.7234	−0.822956
$832$	0	0
$833$	11.2361i	0.389307i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 45.1246i	− 1.55973i
$838$	0	0
$839$	−0.212002	−0.00731913	−0.00365957	−	0.999993i	$-0.501165\pi$
$839$	−0.212002	−0.00731913	−0.00365957	+	0.999993i	$0.501165\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	0	0
$843$	32.4014i	1.11596i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 6.43288i	− 0.221036i
$848$	0	0
$849$	49.5967	1.70216
$850$	0	0
$851$	4.35926	0.149433
$852$	0	0
$853$	47.4164i	1.62351i	0.584000	+	0.811753i	$0.301487\pi$
$853$	47.4164i	1.62351i	−0.584000	+	0.811753i	$0.698513\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	51.0132i	1.74258i	0.490772	+	0.871288i	$0.336715\pi$
$857$	51.0132i	1.74258i	−0.490772	+	0.871288i	$0.663285\pi$
$858$	0	0
$859$	−38.0423	−1.29799	−0.648993	−	0.760795i	$-0.724809\pi$
$859$	−38.0423	−1.29799	−0.648993	+	0.760795i	$0.724809\pi$
$860$	0	0
$861$	−13.0902	−0.446112
$862$	0	0
$863$	9.89408i	0.336798i	0.985719	+	0.168399i	$0.0538598\pi$
$863$	9.89408i	0.336798i	−0.985719	+	0.168399i	$0.946140\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 24.7275i	− 0.839789i
$868$	0	0
$869$	31.0557	1.05349
$870$	0	0
$871$	−19.9192	−0.674936
$872$	0	0
$873$	5.05573i	0.171110i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 45.2361i	− 1.52751i	−0.645504	−	0.763757i	$-0.723353\pi$
$877$	− 45.2361i	− 1.52751i	0.645504	−	0.763757i	$-0.276647\pi$
$878$	0	0
$879$	−44.1976	−1.49075
$880$	0	0
$881$	−10.2016	−0.343702	−0.171851	−	0.985123i	$-0.554975\pi$
$881$	−10.2016	−0.343702	−0.171851	+	0.985123i	$0.554975\pi$
$882$	0	0
$883$	− 16.0494i	− 0.540107i	−0.962845	−	0.270053i	$-0.912959\pi$
$883$	− 16.0494i	− 0.540107i	0.962845	−	0.270053i	$-0.0870414\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 29.2582i	− 0.982395i	−0.871048	−	0.491198i	$-0.836559\pi$
$887$	− 29.2582i	− 0.982395i	0.871048	−	0.491198i	$-0.163441\pi$
$888$	0	0
$889$	−19.4721	−0.653074
$890$	0	0
$891$	−24.6215	−0.824850
$892$	0	0
$893$	− 10.4033i	− 0.348132i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 7.23607i	− 0.241605i
$898$	0	0
$899$	−13.7638	−0.459049
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	2.24514i	0.0747136i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 16.7355i	− 0.555693i	−0.960625	−	0.277847i	$-0.910379\pi$
$907$	− 16.7355i	− 0.555693i	0.960625	−	0.277847i	$-0.0896207\pi$
$908$	0	0
$909$	−3.76393	−0.124842
$910$	0	0
$911$	15.7719	0.522547	0.261274	−	0.965265i	$-0.415857\pi$
$911$	15.7719	0.522547	0.261274	+	0.965265i	$0.415857\pi$
$912$	0	0
$913$	− 6.18034i	− 0.204539i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.8328i	0.555869i
$918$	0	0
$919$	37.6992	1.24358	0.621791	−	0.783183i	$-0.286405\pi$
$919$	37.6992	1.24358	0.621791	+	0.783183i	$0.286405\pi$
$920$	0	0
$921$	43.9443	1.44801
$922$	0	0
$923$	− 35.1361i	− 1.15652i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6.71040i	− 0.220398i
$928$	0	0
$929$	16.8541	0.552965	0.276483	−	0.961019i	$-0.410831\pi$
$929$	16.8541	0.552965	0.276483	+	0.961019i	$0.410831\pi$
$930$	0	0
$931$	−8.16348	−0.267547
$932$	0	0
$933$	30.0000i	0.982156i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 20.3607i	− 0.665154i	−0.943076	−	0.332577i	$-0.892082\pi$
$937$	− 20.3607i	− 0.665154i	0.943076	−	0.332577i	$-0.107918\pi$
$938$	0	0
$939$	−26.4176	−0.862105
$940$	0	0
$941$	7.52786	0.245401	0.122701	−	0.992444i	$-0.460844\pi$
$941$	7.52786	0.245401	0.122701	+	0.992444i	$0.460844\pi$
$942$	0	0
$943$	− 6.88191i	− 0.224106i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.8294i	0.774352i	0.922006	+	0.387176i	$0.126549\pi$
$947$	23.8294i	0.774352i	−0.922006	+	0.387176i	$0.873451\pi$
$948$	0	0
$949$	−36.3607	−1.18032
$950$	0	0
$951$	51.5941	1.67305
$952$	0	0
$953$	43.1935i	1.39917i	0.714547	+	0.699587i	$0.246633\pi$
$953$	43.1935i	1.39917i	−0.714547	+	0.699587i	$0.753367\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.18034i	0.199782i
$958$	0	0
$959$	6.36737	0.205613
$960$	0	0
$961$	68.1935	2.19979
$962$	0	0
$963$	4.08174i	0.131532i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.1562i	0.648180i	0.946026	+	0.324090i	$0.105058\pi$
$967$	20.1562i	0.648180i	−0.946026	+	0.324090i	$0.894942\pi$
$968$	0	0
$969$	−5.52786	−0.177581
$970$	0	0
$971$	−53.0472	−1.70236	−0.851182	−	0.524870i	$-0.824114\pi$
$971$	−53.0472	−1.70236	−0.851182	+	0.524870i	$0.824114\pi$
$972$	0	0
$973$	− 6.58359i	− 0.211060i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.7214i	1.36678i	0.730055	+	0.683389i	$0.239495\pi$
$977$	42.7214i	1.36678i	−0.730055	+	0.683389i	$0.760505\pi$
$978$	0	0
$979$	−19.0211	−0.607918
$980$	0	0
$981$	1.79837	0.0574177
$982$	0	0
$983$	− 56.5084i	− 1.80234i	−0.433469	−	0.901168i	$-0.642711\pi$
$983$	− 56.5084i	− 1.80234i	0.433469	−	0.901168i	$-0.357289\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16.0090i	0.509571i
$988$	0	0
$989$	−1.18034	−0.0375326
$990$	0	0
$991$	14.1068	0.448119	0.224059	−	0.974575i	$-0.428069\pi$
$991$	14.1068	0.448119	0.224059	+	0.974575i	$0.428069\pi$
$992$	0	0
$993$	− 16.1803i	− 0.513468i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 9.05573i	− 0.286798i	−0.989665	−	0.143399i	$-0.954197\pi$
$997$	− 9.05573i	− 0.286798i	0.989665	−	0.143399i	$-0.0458032\pi$
$998$	0	0
$999$	−16.8010	−0.531561

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	4000.2.c.d.1249.2		8
4.3	odd	2	inner	4000.2.c.d.1249.7		8
5.2	odd	4		4000.2.a.e.1.1	✓	4
5.3	odd	4		4000.2.a.f.1.4	yes	4
5.4	even	2	inner	4000.2.c.d.1249.8		8
20.3	even	4		4000.2.a.f.1.1	yes	4
20.7	even	4		4000.2.a.e.1.4	yes	4
20.19	odd	2	inner	4000.2.c.d.1249.1		8
40.3	even	4		8000.2.a.bg.1.4		4
40.13	odd	4		8000.2.a.bg.1.1		4
40.27	even	4		8000.2.a.bh.1.1		4
40.37	odd	4		8000.2.a.bh.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.e.1.1	✓	4	5.2	odd	4
4000.2.a.e.1.4	yes	4	20.7	even	4
4000.2.a.f.1.1	yes	4	20.3	even	4
4000.2.a.f.1.4	yes	4	5.3	odd	4
4000.2.c.d.1249.1		8	20.19	odd	2	inner
4000.2.c.d.1249.2		8	1.1	even	1	trivial
4000.2.c.d.1249.7		8	4.3	odd	2	inner
4000.2.c.d.1249.8		8	5.4	even	2	inner
8000.2.a.bg.1.1		4	40.13	odd	4
8000.2.a.bg.1.4		4	40.3	even	4
8000.2.a.bh.1.1		4	40.27	even	4
8000.2.a.bh.1.4		4	40.37	odd	4

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

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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	4000.2.c.d.1249.2

Level	$4000$
Weight	$2$
Character	4000.1249
Analytic conductor	$31.940$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$