LMFDB - Embedded newform 525.2.d.e.274.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$525 = 3 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	525.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:	$4.19214610612$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.3
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.2.d.e.274.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.381966i q^{2} -1.00000i q^{3} +1.85410 q^{4} +0.381966 q^{6} -1.00000i q^{7} +1.47214i q^{8} -1.00000 q^{9} +3.47214 q^{11} -1.85410i q^{12} -5.23607i q^{13} +0.381966 q^{14} +3.14590 q^{16} +5.70820i q^{17} -0.381966i q^{18} -1.23607 q^{19} -1.00000 q^{21} +1.32624i q^{22} -5.00000i q^{23} +1.47214 q^{24} +2.00000 q^{26} +1.00000i q^{27} -1.85410i q^{28} +8.70820 q^{29} -4.47214 q^{31} +4.14590i q^{32} -3.47214i q^{33} -2.18034 q^{34} -1.85410 q^{36} -3.47214i q^{37} -0.472136i q^{38} -5.23607 q^{39} -8.00000 q^{41} -0.381966i q^{42} -3.76393i q^{43} +6.43769 q^{44} +1.90983 q^{46} +2.76393i q^{47} -3.14590i q^{48} -1.00000 q^{49} +5.70820 q^{51} -9.70820i q^{52} +8.47214i q^{53} -0.381966 q^{54} +1.47214 q^{56} +1.23607i q^{57} +3.32624i q^{58} +5.23607 q^{59} +11.4164 q^{61} -1.70820i q^{62} +1.00000i q^{63} +4.70820 q^{64} +1.32624 q^{66} +10.7082i q^{67} +10.5836i q^{68} -5.00000 q^{69} -9.47214 q^{71} -1.47214i q^{72} +3.23607i q^{73} +1.32624 q^{74} -2.29180 q^{76} -3.47214i q^{77} -2.00000i q^{78} -6.23607 q^{79} +1.00000 q^{81} -3.05573i q^{82} -3.52786i q^{83} -1.85410 q^{84} +1.43769 q^{86} -8.70820i q^{87} +5.11146i q^{88} -7.70820 q^{89} -5.23607 q^{91} -9.27051i q^{92} +4.47214i q^{93} -1.05573 q^{94} +4.14590 q^{96} +3.52786i q^{97} -0.381966i q^{98} -3.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 6 q^{4} + 6 q^{6} - 4 q^{9} - 4 q^{11} + 6 q^{14} + 26 q^{16} + 4 q^{19} - 4 q^{21} - 12 q^{24} + 8 q^{26} + 8 q^{29} + 36 q^{34} + 6 q^{36} - 12 q^{39} - 32 q^{41} + 66 q^{44} + 30 q^{46} - 4 q^{49}+ \cdots + 4 q^{99}+O(q^{100})$ 4 * q - 6 * q^4 + 6 * q^6 - 4 * q^9 - 4 * q^11 + 6 * q^14 + 26 * q^16 + 4 * q^19 - 4 * q^21 - 12 * q^24 + 8 * q^26 + 8 * q^29 + 36 * q^34 + 6 * q^36 - 12 * q^39 - 32 * q^41 + 66 * q^44 + 30 * q^46 - 4 * q^49 - 4 * q^51 - 6 * q^54 - 12 * q^56 + 12 * q^59 - 8 * q^61 - 8 * q^64 - 26 * q^66 - 20 * q^69 - 20 * q^71 - 26 * q^74 - 36 * q^76 - 16 * q^79 + 4 * q^81 + 6 * q^84 + 46 * q^86 - 4 * q^89 - 12 * q^91 - 40 * q^94 + 30 * q^96 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$ .

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

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

)

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

$2$	0.381966i	0.270091i	0.990839	+	0.135045i	$0.0431180\pi$
$2$	0.381966i	0.270091i	−0.990839	+	0.135045i	$0.956882\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.85410	0.927051
$5$	0	0
$5$	0	0
$6$	0.381966	0.155937
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	1.47214i	0.520479i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.47214	1.04689	0.523444	−	0.852060i	$-0.324647\pi$
$11$	3.47214	1.04689	0.523444	+	0.852060i	$0.324647\pi$
$12$	− 1.85410i	− 0.535233i
$13$	− 5.23607i	− 1.45222i	−0.687576	−	0.726112i	$-0.741325\pi$
$13$	− 5.23607i	− 1.45222i	0.687576	−	0.726112i	$-0.258675\pi$
$14$	0.381966	0.102085
$15$	0	0
$16$	3.14590	0.786475
$17$	5.70820i	1.38444i	0.721685	+	0.692221i	$0.243368\pi$
$17$	5.70820i	1.38444i	−0.721685	+	0.692221i	$0.756632\pi$
$18$	− 0.381966i	− 0.0900303i
$19$	−1.23607	−0.283573	−0.141787	−	0.989897i	$-0.545285\pi$
$19$	−1.23607	−0.283573	−0.141787	+	0.989897i	$0.545285\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	1.32624i	0.282755i
$23$	− 5.00000i	− 1.04257i	−0.853382	−	0.521286i	$-0.825452\pi$
$23$	− 5.00000i	− 1.04257i	0.853382	−	0.521286i	$-0.174548\pi$
$24$	1.47214	0.300498
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000i	0.192450i
$28$	− 1.85410i	− 0.350392i
$29$	8.70820	1.61707	0.808536	−	0.588446i	$-0.200260\pi$
$29$	8.70820	1.61707	0.808536	+	0.588446i	$0.200260\pi$
$30$	0	0
$31$	−4.47214	−0.803219	−0.401610	−	0.915811i	$-0.631549\pi$
$31$	−4.47214	−0.803219	−0.401610	+	0.915811i	$0.631549\pi$
$32$	4.14590i	0.732898i
$33$	− 3.47214i	− 0.604421i
$34$	−2.18034	−0.373925
$35$	0	0
$36$	−1.85410	−0.309017
$37$	− 3.47214i	− 0.570816i	−0.958406	−	0.285408i	$-0.907871\pi$
$37$	− 3.47214i	− 0.570816i	0.958406	−	0.285408i	$-0.0921290\pi$
$38$	− 0.472136i	− 0.0765906i
$39$	−5.23607	−0.838442
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	− 0.381966i	− 0.0589386i
$43$	− 3.76393i	− 0.573994i	−0.957931	−	0.286997i	$-0.907343\pi$
$43$	− 3.76393i	− 0.573994i	0.957931	−	0.286997i	$-0.0926570\pi$
$44$	6.43769	0.970519
$45$	0	0
$46$	1.90983	0.281589
$47$	2.76393i	0.403161i	0.979472	+	0.201580i	$0.0646078\pi$
$47$	2.76393i	0.403161i	−0.979472	+	0.201580i	$0.935392\pi$
$48$	− 3.14590i	− 0.454071i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	5.70820	0.799308
$52$	− 9.70820i	− 1.34629i
$53$	8.47214i	1.16374i	0.813283	+	0.581869i	$0.197678\pi$
$53$	8.47214i	1.16374i	−0.813283	+	0.581869i	$0.802322\pi$
$54$	−0.381966	−0.0519790
$55$	0	0
$56$	1.47214	0.196722
$57$	1.23607i	0.163721i
$58$	3.32624i	0.436756i
$59$	5.23607	0.681678	0.340839	−	0.940122i	$-0.389289\pi$
$59$	5.23607	0.681678	0.340839	+	0.940122i	$0.389289\pi$
$60$	0	0
$61$	11.4164	1.46172	0.730861	−	0.682527i	$-0.239119\pi$
$61$	11.4164	1.46172	0.730861	+	0.682527i	$0.239119\pi$
$62$	− 1.70820i	− 0.216942i
$63$	1.00000i	0.125988i
$64$	4.70820	0.588525
$65$	0	0
$66$	1.32624	0.163249
$67$	10.7082i	1.30822i	0.756401	+	0.654108i	$0.226956\pi$
$67$	10.7082i	1.30822i	−0.756401	+	0.654108i	$0.773044\pi$
$68$	10.5836i	1.28345i
$69$	−5.00000	−0.601929
$70$	0	0
$71$	−9.47214	−1.12414	−0.562068	−	0.827091i	$-0.689994\pi$
$71$	−9.47214	−1.12414	−0.562068	+	0.827091i	$0.689994\pi$
$72$	− 1.47214i	− 0.173493i
$73$	3.23607i	0.378753i	0.981905	+	0.189377i	$0.0606467\pi$
$73$	3.23607i	0.378753i	−0.981905	+	0.189377i	$0.939353\pi$
$74$	1.32624	0.154172
$75$	0	0
$76$	−2.29180	−0.262887
$77$	− 3.47214i	− 0.395687i
$78$	− 2.00000i	− 0.226455i
$79$	−6.23607	−0.701612	−0.350806	−	0.936448i	$-0.614092\pi$
$79$	−6.23607	−0.701612	−0.350806	+	0.936448i	$0.614092\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 3.05573i	− 0.337449i
$83$	− 3.52786i	− 0.387233i	−0.981077	−	0.193617i	$-0.937978\pi$
$83$	− 3.52786i	− 0.387233i	0.981077	−	0.193617i	$-0.0620218\pi$
$84$	−1.85410	−0.202299
$85$	0	0
$86$	1.43769	0.155031
$87$	− 8.70820i	− 0.933617i
$88$	5.11146i	0.544883i
$89$	−7.70820	−0.817068	−0.408534	−	0.912743i	$-0.633960\pi$
$89$	−7.70820	−0.817068	−0.408534	+	0.912743i	$0.633960\pi$
$90$	0	0
$91$	−5.23607	−0.548889
$92$	− 9.27051i	− 0.966517i
$93$	4.47214i	0.463739i
$94$	−1.05573	−0.108890
$95$	0	0
$96$	4.14590	0.423139
$97$	3.52786i	0.358200i	0.983831	+	0.179100i	$0.0573186\pi$
$97$	3.52786i	0.358200i	−0.983831	+	0.179100i	$0.942681\pi$
$98$	− 0.381966i	− 0.0385844i
$99$	−3.47214	−0.348963
$100$	0	0
$101$	−17.7082	−1.76203	−0.881016	−	0.473086i	$-0.843140\pi$
$101$	−17.7082	−1.76203	−0.881016	+	0.473086i	$0.843140\pi$
$102$	2.18034i	0.215886i
$103$	3.70820i	0.365380i	0.983171	+	0.182690i	$0.0584805\pi$
$103$	3.70820i	0.365380i	−0.983171	+	0.182690i	$0.941520\pi$
$104$	7.70820	0.755852
$105$	0	0
$106$	−3.23607	−0.314315
$107$	12.9443i	1.25137i	0.780076	+	0.625685i	$0.215180\pi$
$107$	12.9443i	1.25137i	−0.780076	+	0.625685i	$0.784820\pi$
$108$	1.85410i	0.178411i
$109$	−20.4164	−1.95554	−0.977769	−	0.209687i	$-0.932756\pi$
$109$	−20.4164	−1.95554	−0.977769	+	0.209687i	$0.932756\pi$
$110$	0	0
$111$	−3.47214	−0.329561
$112$	− 3.14590i	− 0.297259i
$113$	11.7639i	1.10666i	0.832963	+	0.553329i	$0.186643\pi$
$113$	11.7639i	1.10666i	−0.832963	+	0.553329i	$0.813357\pi$
$114$	−0.472136	−0.0442196
$115$	0	0
$116$	16.1459	1.49911
$117$	5.23607i	0.484075i
$118$	2.00000i	0.184115i
$119$	5.70820	0.523270
$120$	0	0
$121$	1.05573	0.0959753
$122$	4.36068i	0.394797i
$123$	8.00000i	0.721336i
$124$	−8.29180	−0.744625
$125$	0	0
$126$	−0.381966	−0.0340282
$127$	− 7.76393i	− 0.688938i	−0.938798	−	0.344469i	$-0.888059\pi$
$127$	− 7.76393i	− 0.688938i	0.938798	−	0.344469i	$-0.111941\pi$
$128$	10.0902i	0.891853i
$129$	−3.76393	−0.331396
$130$	0	0
$131$	−17.7082	−1.54717	−0.773586	−	0.633691i	$-0.781539\pi$
$131$	−17.7082	−1.54717	−0.773586	+	0.633691i	$0.781539\pi$
$132$	− 6.43769i	− 0.560329i
$133$	1.23607i	0.107181i
$134$	−4.09017	−0.353337
$135$	0	0
$136$	−8.40325	−0.720573
$137$	8.47214i	0.723823i	0.932212	+	0.361912i	$0.117876\pi$
$137$	8.47214i	0.723823i	−0.932212	+	0.361912i	$0.882124\pi$
$138$	− 1.90983i	− 0.162576i
$139$	9.70820	0.823439	0.411720	−	0.911311i	$-0.364928\pi$
$139$	9.70820	0.823439	0.411720	+	0.911311i	$0.364928\pi$
$140$	0	0
$141$	2.76393	0.232765
$142$	− 3.61803i	− 0.303619i
$143$	− 18.1803i	− 1.52032i
$144$	−3.14590	−0.262158
$145$	0	0
$146$	−1.23607	−0.102298
$147$	1.00000i	0.0824786i
$148$	− 6.43769i	− 0.529175i
$149$	−3.76393	−0.308353	−0.154177	−	0.988043i	$-0.549272\pi$
$149$	−3.76393	−0.308353	−0.154177	+	0.988043i	$0.549272\pi$
$150$	0	0
$151$	14.7082	1.19694	0.598468	−	0.801146i	$-0.295776\pi$
$151$	14.7082	1.19694	0.598468	+	0.801146i	$0.295776\pi$
$152$	− 1.81966i	− 0.147594i
$153$	− 5.70820i	− 0.461481i
$154$	1.32624	0.106871
$155$	0	0
$156$	−9.70820	−0.777278
$157$	− 0.944272i	− 0.0753611i	−0.999290	−	0.0376806i	$-0.988003\pi$
$157$	− 0.944272i	− 0.0753611i	0.999290	−	0.0376806i	$-0.0119969\pi$
$158$	− 2.38197i	− 0.189499i
$159$	8.47214	0.671884
$160$	0	0
$161$	−5.00000	−0.394055
$162$	0.381966i	0.0300101i
$163$	− 9.52786i	− 0.746280i	−0.927775	−	0.373140i	$-0.878281\pi$
$163$	− 9.52786i	− 0.746280i	0.927775	−	0.373140i	$-0.121719\pi$
$164$	−14.8328	−1.15825
$165$	0	0
$166$	1.34752	0.104588
$167$	− 19.7082i	− 1.52507i	−0.646949	−	0.762533i	$-0.723955\pi$
$167$	− 19.7082i	− 1.52507i	0.646949	−	0.762533i	$-0.276045\pi$
$168$	− 1.47214i	− 0.113578i
$169$	−14.4164	−1.10895
$170$	0	0
$171$	1.23607	0.0945245
$172$	− 6.97871i	− 0.532122i
$173$	26.1803i	1.99045i	0.0975850	+	0.995227i	$0.468888\pi$
$173$	26.1803i	1.99045i	−0.0975850	+	0.995227i	$0.531112\pi$
$174$	3.32624	0.252161
$175$	0	0
$176$	10.9230	0.823351
$177$	− 5.23607i	− 0.393567i
$178$	− 2.94427i	− 0.220683i
$179$	12.9443	0.967500	0.483750	−	0.875206i	$-0.339274\pi$
$179$	12.9443	0.967500	0.483750	+	0.875206i	$0.339274\pi$
$180$	0	0
$181$	−26.6525	−1.98106	−0.990531	−	0.137286i	$-0.956162\pi$
$181$	−26.6525	−1.98106	−0.990531	+	0.137286i	$0.956162\pi$
$182$	− 2.00000i	− 0.148250i
$183$	− 11.4164i	− 0.843925i
$184$	7.36068	0.542637
$185$	0	0
$186$	−1.70820	−0.125252
$187$	19.8197i	1.44936i
$188$	5.12461i	0.373751i
$189$	1.00000	0.0727393
$190$	0	0
$191$	15.4164	1.11549	0.557746	−	0.830012i	$-0.311666\pi$
$191$	15.4164	1.11549	0.557746	+	0.830012i	$0.311666\pi$
$192$	− 4.70820i	− 0.339785i
$193$	26.4164i	1.90149i	0.309967	+	0.950747i	$0.399682\pi$
$193$	26.4164i	1.90149i	−0.309967	+	0.950747i	$0.600318\pi$
$194$	−1.34752	−0.0967466
$195$	0	0
$196$	−1.85410	−0.132436
$197$	− 10.2361i	− 0.729290i	−0.931147	−	0.364645i	$-0.881190\pi$
$197$	− 10.2361i	− 0.729290i	0.931147	−	0.364645i	$-0.118810\pi$
$198$	− 1.32624i	− 0.0942516i
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	10.7082	0.755298
$202$	− 6.76393i	− 0.475909i
$203$	− 8.70820i	− 0.611196i
$204$	10.5836	0.741000
$205$	0	0
$206$	−1.41641	−0.0986858
$207$	5.00000i	0.347524i
$208$	− 16.4721i	− 1.14214i
$209$	−4.29180	−0.296870
$210$	0	0
$211$	−4.94427	−0.340378	−0.170189	−	0.985411i	$-0.554438\pi$
$211$	−4.94427	−0.340378	−0.170189	+	0.985411i	$0.554438\pi$
$212$	15.7082i	1.07884i
$213$	9.47214i	0.649020i
$214$	−4.94427	−0.337983
$215$	0	0
$216$	−1.47214	−0.100166
$217$	4.47214i	0.303588i
$218$	− 7.79837i	− 0.528173i
$219$	3.23607	0.218673
$220$	0	0
$221$	29.8885	2.01052
$222$	− 1.32624i	− 0.0890113i
$223$	− 3.23607i	− 0.216703i	−0.994113	−	0.108352i	$-0.965443\pi$
$223$	− 3.23607i	− 0.216703i	0.994113	−	0.108352i	$-0.0345572\pi$
$224$	4.14590	0.277009
$225$	0	0
$226$	−4.49342	−0.298898
$227$	− 9.23607i	− 0.613019i	−0.951868	−	0.306510i	$-0.900839\pi$
$227$	− 9.23607i	− 0.613019i	0.951868	−	0.306510i	$-0.0991612\pi$
$228$	2.29180i	0.151778i
$229$	22.3607	1.47764	0.738818	−	0.673905i	$-0.235384\pi$
$229$	22.3607	1.47764	0.738818	+	0.673905i	$0.235384\pi$
$230$	0	0
$231$	−3.47214	−0.228450
$232$	12.8197i	0.841652i
$233$	− 20.7082i	− 1.35664i	−0.734767	−	0.678320i	$-0.762708\pi$
$233$	− 20.7082i	− 1.35664i	0.734767	−	0.678320i	$-0.237292\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	9.70820	0.631950
$237$	6.23607i	0.405076i
$238$	2.18034i	0.141330i
$239$	−16.3607	−1.05828	−0.529142	−	0.848533i	$-0.677486\pi$
$239$	−16.3607	−1.05828	−0.529142	+	0.848533i	$0.677486\pi$
$240$	0	0
$241$	14.7639	0.951028	0.475514	−	0.879708i	$-0.342262\pi$
$241$	14.7639	0.951028	0.475514	+	0.879708i	$0.342262\pi$
$242$	0.403252i	0.0259220i
$243$	− 1.00000i	− 0.0641500i
$244$	21.1672	1.35509
$245$	0	0
$246$	−3.05573	−0.194826
$247$	6.47214i	0.411812i
$248$	− 6.58359i	− 0.418059i
$249$	−3.52786	−0.223569
$250$	0	0
$251$	7.23607	0.456737	0.228368	−	0.973575i	$-0.426661\pi$
$251$	7.23607	0.456737	0.228368	+	0.973575i	$0.426661\pi$
$252$	1.85410i	0.116797i
$253$	− 17.3607i	− 1.09146i
$254$	2.96556	0.186076
$255$	0	0
$256$	5.56231	0.347644
$257$	0.472136i	0.0294510i	0.999892	+	0.0147255i	$0.00468745\pi$
$257$	0.472136i	0.0294510i	−0.999892	+	0.0147255i	$0.995313\pi$
$258$	− 1.43769i	− 0.0895069i
$259$	−3.47214	−0.215748
$260$	0	0
$261$	−8.70820	−0.539024
$262$	− 6.76393i	− 0.417877i
$263$	− 19.9443i	− 1.22982i	−0.788599	−	0.614908i	$-0.789193\pi$
$263$	− 19.9443i	− 1.22982i	0.788599	−	0.614908i	$-0.210807\pi$
$264$	5.11146	0.314588
$265$	0	0
$266$	−0.472136	−0.0289485
$267$	7.70820i	0.471734i
$268$	19.8541i	1.21278i
$269$	19.5279	1.19063	0.595317	−	0.803491i	$-0.297026\pi$
$269$	19.5279	1.19063	0.595317	+	0.803491i	$0.297026\pi$
$270$	0	0
$271$	18.7639	1.13983	0.569914	−	0.821704i	$-0.306977\pi$
$271$	18.7639	1.13983	0.569914	+	0.821704i	$0.306977\pi$
$272$	17.9574i	1.08883i
$273$	5.23607i	0.316901i
$274$	−3.23607	−0.195498
$275$	0	0
$276$	−9.27051	−0.558019
$277$	19.8885i	1.19499i	0.801874	+	0.597493i	$0.203837\pi$
$277$	19.8885i	1.19499i	−0.801874	+	0.597493i	$0.796163\pi$
$278$	3.70820i	0.222403i
$279$	4.47214	0.267740
$280$	0	0
$281$	−17.6525	−1.05306	−0.526529	−	0.850157i	$-0.676507\pi$
$281$	−17.6525	−1.05306	−0.526529	+	0.850157i	$0.676507\pi$
$282$	1.05573i	0.0628677i
$283$	− 13.4164i	− 0.797523i	−0.917055	−	0.398761i	$-0.869440\pi$
$283$	− 13.4164i	− 0.797523i	0.917055	−	0.398761i	$-0.130560\pi$
$284$	−17.5623	−1.04213
$285$	0	0
$286$	6.94427	0.410623
$287$	8.00000i	0.472225i
$288$	− 4.14590i	− 0.244299i
$289$	−15.5836	−0.916682
$290$	0	0
$291$	3.52786	0.206807
$292$	6.00000i	0.351123i
$293$	− 4.65248i	− 0.271801i	−0.990723	−	0.135900i	$-0.956607\pi$
$293$	− 4.65248i	− 0.271801i	0.990723	−	0.135900i	$-0.0433927\pi$
$294$	−0.381966	−0.0222767
$295$	0	0
$296$	5.11146	0.297097
$297$	3.47214i	0.201474i
$298$	− 1.43769i	− 0.0832834i
$299$	−26.1803	−1.51405
$300$	0	0
$301$	−3.76393	−0.216949
$302$	5.61803i	0.323282i
$303$	17.7082i	1.01731i
$304$	−3.88854	−0.223023
$305$	0	0
$306$	2.18034	0.124642
$307$	− 6.65248i	− 0.379677i	−0.981815	−	0.189838i	$-0.939204\pi$
$307$	− 6.65248i	− 0.379677i	0.981815	−	0.189838i	$-0.0607964\pi$
$308$	− 6.43769i	− 0.366822i
$309$	3.70820	0.210952
$310$	0	0
$311$	8.76393	0.496957	0.248478	−	0.968637i	$-0.420069\pi$
$311$	8.76393	0.496957	0.248478	+	0.968637i	$0.420069\pi$
$312$	− 7.70820i	− 0.436391i
$313$	− 7.70820i	− 0.435693i	−0.975983	−	0.217847i	$-0.930097\pi$
$313$	− 7.70820i	− 0.435693i	0.975983	−	0.217847i	$-0.0699033\pi$
$314$	0.360680	0.0203543
$315$	0	0
$316$	−11.5623	−0.650431
$317$	− 13.6525i	− 0.766799i	−0.923583	−	0.383400i	$-0.874753\pi$
$317$	− 13.6525i	− 0.766799i	0.923583	−	0.383400i	$-0.125247\pi$
$318$	3.23607i	0.181470i
$319$	30.2361	1.69289
$320$	0	0
$321$	12.9443	0.722479
$322$	− 1.90983i	− 0.106431i
$323$	− 7.05573i	− 0.392591i
$324$	1.85410	0.103006
$325$	0	0
$326$	3.63932	0.201563
$327$	20.4164i	1.12903i
$328$	− 11.7771i	− 0.650281i
$329$	2.76393	0.152381
$330$	0	0
$331$	18.2361	1.00234	0.501172	−	0.865347i	$-0.332902\pi$
$331$	18.2361	1.00234	0.501172	+	0.865347i	$0.332902\pi$
$332$	− 6.54102i	− 0.358985i
$333$	3.47214i	0.190272i
$334$	7.52786	0.411906
$335$	0	0
$336$	−3.14590	−0.171623
$337$	− 19.5279i	− 1.06375i	−0.846823	−	0.531875i	$-0.821488\pi$
$337$	− 19.5279i	− 1.06375i	0.846823	−	0.531875i	$-0.178512\pi$
$338$	− 5.50658i	− 0.299518i
$339$	11.7639	0.638929
$340$	0	0
$341$	−15.5279	−0.840881
$342$	0.472136i	0.0255302i
$343$	1.00000i	0.0539949i
$344$	5.54102	0.298752
$345$	0	0
$346$	−10.0000	−0.537603
$347$	− 13.0000i	− 0.697877i	−0.937146	−	0.348938i	$-0.886542\pi$
$347$	− 13.0000i	− 0.697877i	0.937146	−	0.348938i	$-0.113458\pi$
$348$	− 16.1459i	− 0.865511i
$349$	−5.23607	−0.280280	−0.140140	−	0.990132i	$-0.544755\pi$
$349$	−5.23607	−0.280280	−0.140140	+	0.990132i	$0.544755\pi$
$350$	0	0
$351$	5.23607	0.279481
$352$	14.3951i	0.767263i
$353$	− 18.9443i	− 1.00830i	−0.863616	−	0.504151i	$-0.831806\pi$
$353$	− 18.9443i	− 1.00830i	0.863616	−	0.504151i	$-0.168194\pi$
$354$	2.00000	0.106299
$355$	0	0
$356$	−14.2918	−0.757464
$357$	− 5.70820i	− 0.302110i
$358$	4.94427i	0.261313i
$359$	−8.05573	−0.425165	−0.212583	−	0.977143i	$-0.568187\pi$
$359$	−8.05573	−0.425165	−0.212583	+	0.977143i	$0.568187\pi$
$360$	0	0
$361$	−17.4721	−0.919586
$362$	− 10.1803i	− 0.535067i
$363$	− 1.05573i	− 0.0554114i
$364$	−9.70820	−0.508848
$365$	0	0
$366$	4.36068	0.227936
$367$	8.94427i	0.466887i	0.972370	+	0.233444i	$0.0749994\pi$
$367$	8.94427i	0.466887i	−0.972370	+	0.233444i	$0.925001\pi$
$368$	− 15.7295i	− 0.819956i
$369$	8.00000	0.416463
$370$	0	0
$371$	8.47214	0.439851
$372$	8.29180i	0.429910i
$373$	15.0000i	0.776671i	0.921518	+	0.388335i	$0.126950\pi$
$373$	15.0000i	0.776671i	−0.921518	+	0.388335i	$0.873050\pi$
$374$	−7.57044	−0.391458
$375$	0	0
$376$	−4.06888	−0.209837
$377$	− 45.5967i	− 2.34835i
$378$	0.381966i	0.0196462i
$379$	32.5967	1.67438	0.837191	−	0.546910i	$-0.184196\pi$
$379$	32.5967	1.67438	0.837191	+	0.546910i	$0.184196\pi$
$380$	0	0
$381$	−7.76393	−0.397758
$382$	5.88854i	0.301284i
$383$	15.1246i	0.772832i	0.922325	+	0.386416i	$0.126287\pi$
$383$	15.1246i	0.772832i	−0.922325	+	0.386416i	$0.873713\pi$
$384$	10.0902	0.514912
$385$	0	0
$386$	−10.0902	−0.513576
$387$	3.76393i	0.191331i
$388$	6.54102i	0.332070i
$389$	6.23607	0.316181	0.158091	−	0.987425i	$-0.449466\pi$
$389$	6.23607	0.316181	0.158091	+	0.987425i	$0.449466\pi$
$390$	0	0
$391$	28.5410	1.44338
$392$	− 1.47214i	− 0.0743541i
$393$	17.7082i	0.893261i
$394$	3.90983	0.196974
$395$	0	0
$396$	−6.43769	−0.323506
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	6.11146i	0.306340i
$399$	1.23607	0.0618808
$400$	0	0
$401$	−13.2918	−0.663761	−0.331880	−	0.943322i	$-0.607683\pi$
$401$	−13.2918	−0.663761	−0.331880	+	0.943322i	$0.607683\pi$
$402$	4.09017i	0.203999i
$403$	23.4164i	1.16645i
$404$	−32.8328	−1.63349
$405$	0	0
$406$	3.32624	0.165078
$407$	− 12.0557i	− 0.597580i
$408$	8.40325i	0.416023i
$409$	−14.1803	−0.701173	−0.350586	−	0.936530i	$-0.614018\pi$
$409$	−14.1803	−0.701173	−0.350586	+	0.936530i	$0.614018\pi$
$410$	0	0
$411$	8.47214	0.417900
$412$	6.87539i	0.338726i
$413$	− 5.23607i	− 0.257650i
$414$	−1.90983	−0.0938630
$415$	0	0
$416$	21.7082	1.06433
$417$	− 9.70820i	− 0.475413i
$418$	− 1.63932i	− 0.0801818i
$419$	−26.9443	−1.31631	−0.658157	−	0.752881i	$-0.728664\pi$
$419$	−26.9443	−1.31631	−0.658157	+	0.752881i	$0.728664\pi$
$420$	0	0
$421$	26.4164	1.28746	0.643728	−	0.765254i	$-0.277387\pi$
$421$	26.4164	1.28746	0.643728	+	0.765254i	$0.277387\pi$
$422$	− 1.88854i	− 0.0919329i
$423$	− 2.76393i	− 0.134387i
$424$	−12.4721	−0.605700
$425$	0	0
$426$	−3.61803	−0.175294
$427$	− 11.4164i	− 0.552479i
$428$	24.0000i	1.16008i
$429$	−18.1803	−0.877755
$430$	0	0
$431$	−1.88854	−0.0909680	−0.0454840	−	0.998965i	$-0.514483\pi$
$431$	−1.88854	−0.0909680	−0.0454840	+	0.998965i	$0.514483\pi$
$432$	3.14590i	0.151357i
$433$	25.3050i	1.21608i	0.793907	+	0.608039i	$0.208044\pi$
$433$	25.3050i	1.21608i	−0.793907	+	0.608039i	$0.791956\pi$
$434$	−1.70820	−0.0819964
$435$	0	0
$436$	−37.8541	−1.81288
$437$	6.18034i	0.295646i
$438$	1.23607i	0.0590616i
$439$	6.76393	0.322825	0.161412	−	0.986887i	$-0.448395\pi$
$439$	6.76393	0.322825	0.161412	+	0.986887i	$0.448395\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	11.4164i	0.543023i
$443$	− 39.4164i	− 1.87273i	−0.351028	−	0.936365i	$-0.614168\pi$
$443$	− 39.4164i	− 1.87273i	0.351028	−	0.936365i	$-0.385832\pi$
$444$	−6.43769	−0.305519
$445$	0	0
$446$	1.23607	0.0585295
$447$	3.76393i	0.178028i
$448$	− 4.70820i	− 0.222442i
$449$	−16.7082	−0.788509	−0.394254	−	0.919001i	$-0.628997\pi$
$449$	−16.7082	−0.788509	−0.394254	+	0.919001i	$0.628997\pi$
$450$	0	0
$451$	−27.7771	−1.30797
$452$	21.8115i	1.02593i
$453$	− 14.7082i	− 0.691052i
$454$	3.52786	0.165571
$455$	0	0
$456$	−1.81966	−0.0852134
$457$	− 20.5279i	− 0.960253i	−0.877199	−	0.480126i	$-0.840591\pi$
$457$	− 20.5279i	− 0.960253i	0.877199	−	0.480126i	$-0.159409\pi$
$458$	8.54102i	0.399096i
$459$	−5.70820	−0.266436
$460$	0	0
$461$	−20.6525	−0.961882	−0.480941	−	0.876753i	$-0.659705\pi$
$461$	−20.6525	−0.961882	−0.480941	+	0.876753i	$0.659705\pi$
$462$	− 1.32624i	− 0.0617022i
$463$	16.3607i	0.760345i	0.924916	+	0.380173i	$0.124135\pi$
$463$	16.3607i	0.760345i	−0.924916	+	0.380173i	$0.875865\pi$
$464$	27.3951	1.27179
$465$	0	0
$466$	7.90983	0.366416
$467$	− 11.8885i	− 0.550136i	−0.961425	−	0.275068i	$-0.911300\pi$
$467$	− 11.8885i	− 0.550136i	0.961425	−	0.275068i	$-0.0887004\pi$
$468$	9.70820i	0.448762i
$469$	10.7082	0.494459
$470$	0	0
$471$	−0.944272	−0.0435098
$472$	7.70820i	0.354799i
$473$	− 13.0689i	− 0.600908i
$474$	−2.38197	−0.109407
$475$	0	0
$476$	10.5836	0.485098
$477$	− 8.47214i	− 0.387912i
$478$	− 6.24922i	− 0.285833i
$479$	18.7639	0.857346	0.428673	−	0.903460i	$-0.358981\pi$
$479$	18.7639	0.857346	0.428673	+	0.903460i	$0.358981\pi$
$480$	0	0
$481$	−18.1803	−0.828952
$482$	5.63932i	0.256864i
$483$	5.00000i	0.227508i
$484$	1.95743	0.0889740
$485$	0	0
$486$	0.381966	0.0173263
$487$	− 29.1803i	− 1.32229i	−0.750259	−	0.661144i	$-0.770071\pi$
$487$	− 29.1803i	− 1.32229i	0.750259	−	0.661144i	$-0.229929\pi$
$488$	16.8065i	0.760795i
$489$	−9.52786	−0.430865
$490$	0	0
$491$	−9.47214	−0.427472	−0.213736	−	0.976892i	$-0.568563\pi$
$491$	−9.47214	−0.427472	−0.213736	+	0.976892i	$0.568563\pi$
$492$	14.8328i	0.668715i
$493$	49.7082i	2.23874i
$494$	−2.47214	−0.111227
$495$	0	0
$496$	−14.0689	−0.631712
$497$	9.47214i	0.424883i
$498$	− 1.34752i	− 0.0603840i
$499$	−39.7771	−1.78067	−0.890333	−	0.455309i	$-0.849529\pi$
$499$	−39.7771	−1.78067	−0.890333	+	0.455309i	$0.849529\pi$
$500$	0	0
$501$	−19.7082	−0.880498
$502$	2.76393i	0.123360i
$503$	9.41641i	0.419857i	0.977717	+	0.209928i	$0.0673231\pi$
$503$	9.41641i	0.419857i	−0.977717	+	0.209928i	$0.932677\pi$
$504$	−1.47214	−0.0655741
$505$	0	0
$506$	6.63119	0.294792
$507$	14.4164i	0.640255i
$508$	− 14.3951i	− 0.638680i
$509$	−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
$511$	3.23607	0.143155
$512$	22.3050i	0.985749i
$513$	− 1.23607i	− 0.0545737i
$514$	−0.180340	−0.00795445
$515$	0	0
$516$	−6.97871	−0.307221
$517$	9.59675i	0.422064i
$518$	− 1.32624i	− 0.0582715i
$519$	26.1803	1.14919
$520$	0	0
$521$	33.7771	1.47980	0.739901	−	0.672716i	$-0.234873\pi$
$521$	33.7771	1.47980	0.739901	+	0.672716i	$0.234873\pi$
$522$	− 3.32624i	− 0.145585i
$523$	− 35.7771i	− 1.56442i	−0.623013	−	0.782211i	$-0.714092\pi$
$523$	− 35.7771i	− 1.56442i	0.623013	−	0.782211i	$-0.285908\pi$
$524$	−32.8328	−1.43431
$525$	0	0
$526$	7.61803	0.332162
$527$	− 25.5279i	− 1.11201i
$528$	− 10.9230i	− 0.475362i
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	−5.23607	−0.227226
$532$	2.29180i	0.0993620i
$533$	41.8885i	1.81439i
$534$	−2.94427	−0.127411
$535$	0	0
$536$	−15.7639	−0.680898
$537$	− 12.9443i	− 0.558587i
$538$	7.45898i	0.321579i
$539$	−3.47214	−0.149555
$540$	0	0
$541$	−0.0557281	−0.00239594	−0.00119797	−	0.999999i	$-0.500381\pi$
$541$	−0.0557281	−0.00239594	−0.00119797	+	0.999999i	$0.500381\pi$
$542$	7.16718i	0.307857i
$543$	26.6525i	1.14377i
$544$	−23.6656	−1.01466
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	23.0689i	0.986354i	0.869929	+	0.493177i	$0.164165\pi$
$547$	23.0689i	0.986354i	−0.869929	+	0.493177i	$0.835835\pi$
$548$	15.7082i	0.671021i
$549$	−11.4164	−0.487240
$550$	0	0
$551$	−10.7639	−0.458559
$552$	− 7.36068i	− 0.313291i
$553$	6.23607i	0.265185i
$554$	−7.59675	−0.322755
$555$	0	0
$556$	18.0000	0.763370
$557$	23.6525i	1.00219i	0.865393	+	0.501094i	$0.167069\pi$
$557$	23.6525i	1.00219i	−0.865393	+	0.501094i	$0.832931\pi$
$558$	1.70820i	0.0723140i
$559$	−19.7082	−0.833568
$560$	0	0
$561$	19.8197	0.836787
$562$	− 6.74265i	− 0.284421i
$563$	27.3050i	1.15077i	0.817884	+	0.575383i	$0.195147\pi$
$563$	27.3050i	1.15077i	−0.817884	+	0.575383i	$0.804853\pi$
$564$	5.12461	0.215785
$565$	0	0
$566$	5.12461	0.215404
$567$	− 1.00000i	− 0.0419961i
$568$	− 13.9443i	− 0.585089i
$569$	7.76393	0.325481	0.162740	−	0.986669i	$-0.447967\pi$
$569$	7.76393	0.325481	0.162740	+	0.986669i	$0.447967\pi$
$570$	0	0
$571$	−27.2918	−1.14213	−0.571063	−	0.820906i	$-0.693469\pi$
$571$	−27.2918	−1.14213	−0.571063	+	0.820906i	$0.693469\pi$
$572$	− 33.7082i	− 1.40941i
$573$	− 15.4164i	− 0.644030i
$574$	−3.05573	−0.127544
$575$	0	0
$576$	−4.70820	−0.196175
$577$	6.76393i	0.281586i	0.990039	+	0.140793i	$0.0449652\pi$
$577$	6.76393i	0.281586i	−0.990039	+	0.140793i	$0.955035\pi$
$578$	− 5.95240i	− 0.247587i
$579$	26.4164	1.09783
$580$	0	0
$581$	−3.52786	−0.146360
$582$	1.34752i	0.0558567i
$583$	29.4164i	1.21830i
$584$	−4.76393	−0.197133
$585$	0	0
$586$	1.77709	0.0734108
$587$	33.1246i	1.36720i	0.729857	+	0.683600i	$0.239586\pi$
$587$	33.1246i	1.36720i	−0.729857	+	0.683600i	$0.760414\pi$
$588$	1.85410i	0.0764619i
$589$	5.52786	0.227772
$590$	0	0
$591$	−10.2361	−0.421056
$592$	− 10.9230i	− 0.448932i
$593$	37.3050i	1.53193i	0.642882	+	0.765965i	$0.277739\pi$
$593$	37.3050i	1.53193i	−0.642882	+	0.765965i	$0.722261\pi$
$594$	−1.32624	−0.0544162
$595$	0	0
$596$	−6.97871	−0.285859
$597$	− 16.0000i	− 0.654836i
$598$	− 10.0000i	− 0.408930i
$599$	28.0557	1.14633	0.573163	−	0.819441i	$-0.305716\pi$
$599$	28.0557	1.14633	0.573163	+	0.819441i	$0.305716\pi$
$600$	0	0
$601$	16.3607	0.667366	0.333683	−	0.942685i	$-0.391708\pi$
$601$	16.3607	0.667366	0.333683	+	0.942685i	$0.391708\pi$
$602$	− 1.43769i	− 0.0585960i
$603$	− 10.7082i	− 0.436072i
$604$	27.2705	1.10962
$605$	0	0
$606$	−6.76393	−0.274766
$607$	15.7082i	0.637576i	0.947826	+	0.318788i	$0.103276\pi$
$607$	15.7082i	0.637576i	−0.947826	+	0.318788i	$0.896724\pi$
$608$	− 5.12461i	− 0.207830i
$609$	−8.70820	−0.352874
$610$	0	0
$611$	14.4721	0.585480
$612$	− 10.5836i	− 0.427816i
$613$	− 27.9443i	− 1.12866i	−0.825550	−	0.564329i	$-0.809135\pi$
$613$	− 27.9443i	− 1.12866i	0.825550	−	0.564329i	$-0.190865\pi$
$614$	2.54102	0.102547
$615$	0	0
$616$	5.11146	0.205946
$617$	9.65248i	0.388594i	0.980943	+	0.194297i	$0.0622426\pi$
$617$	9.65248i	0.388594i	−0.980943	+	0.194297i	$0.937757\pi$
$618$	1.41641i	0.0569763i
$619$	15.1246	0.607909	0.303955	−	0.952686i	$-0.401693\pi$
$619$	15.1246	0.607909	0.303955	+	0.952686i	$0.401693\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	3.34752i	0.134223i
$623$	7.70820i	0.308823i
$624$	−16.4721	−0.659413
$625$	0	0
$626$	2.94427	0.117677
$627$	4.29180i	0.171398i
$628$	− 1.75078i	− 0.0698636i
$629$	19.8197	0.790262
$630$	0	0
$631$	0.124612	0.00496072	0.00248036	−	0.999997i	$-0.499210\pi$
$631$	0.124612	0.00496072	0.00248036	+	0.999997i	$0.499210\pi$
$632$	− 9.18034i	− 0.365174i
$633$	4.94427i	0.196517i
$634$	5.21478	0.207105
$635$	0	0
$636$	15.7082	0.622871
$637$	5.23607i	0.207461i
$638$	11.5492i	0.457235i
$639$	9.47214	0.374712
$640$	0	0
$641$	−28.7082	−1.13391	−0.566953	−	0.823750i	$-0.691878\pi$
$641$	−28.7082	−1.13391	−0.566953	+	0.823750i	$0.691878\pi$
$642$	4.94427i	0.195135i
$643$	− 44.7214i	− 1.76364i	−0.471588	−	0.881819i	$-0.656319\pi$
$643$	− 44.7214i	− 1.76364i	0.471588	−	0.881819i	$-0.343681\pi$
$644$	−9.27051	−0.365309
$645$	0	0
$646$	2.69505	0.106035
$647$	− 18.9443i	− 0.744776i	−0.928077	−	0.372388i	$-0.878539\pi$
$647$	− 18.9443i	− 0.744776i	0.928077	−	0.372388i	$-0.121461\pi$
$648$	1.47214i	0.0578310i
$649$	18.1803	0.713641
$650$	0	0
$651$	4.47214	0.175277
$652$	− 17.6656i	− 0.691840i
$653$	1.41641i	0.0554283i	0.999616	+	0.0277142i	$0.00882282\pi$
$653$	1.41641i	0.0554283i	−0.999616	+	0.0277142i	$0.991177\pi$
$654$	−7.79837	−0.304941
$655$	0	0
$656$	−25.1672	−0.982613
$657$	− 3.23607i	− 0.126251i
$658$	1.05573i	0.0411566i
$659$	−23.0557	−0.898124	−0.449062	−	0.893501i	$-0.648242\pi$
$659$	−23.0557	−0.898124	−0.449062	+	0.893501i	$0.648242\pi$
$660$	0	0
$661$	−14.1803	−0.551551	−0.275776	−	0.961222i	$-0.588935\pi$
$661$	−14.1803	−0.551551	−0.275776	+	0.961222i	$0.588935\pi$
$662$	6.96556i	0.270724i
$663$	− 29.8885i	− 1.16077i
$664$	5.19350	0.201547
$665$	0	0
$666$	−1.32624	−0.0513907
$667$	− 43.5410i	− 1.68592i
$668$	− 36.5410i	− 1.41381i
$669$	−3.23607	−0.125114
$670$	0	0
$671$	39.6393	1.53026
$672$	− 4.14590i	− 0.159931i
$673$	6.00000i	0.231283i	0.993291	+	0.115642i	$0.0368924\pi$
$673$	6.00000i	0.231283i	−0.993291	+	0.115642i	$0.963108\pi$
$674$	7.45898	0.287309
$675$	0	0
$676$	−26.7295	−1.02806
$677$	− 19.3050i	− 0.741950i	−0.928643	−	0.370975i	$-0.879024\pi$
$677$	− 19.3050i	− 0.741950i	0.928643	−	0.370975i	$-0.120976\pi$
$678$	4.49342i	0.172569i
$679$	3.52786	0.135387
$680$	0	0
$681$	−9.23607	−0.353927
$682$	− 5.93112i	− 0.227114i
$683$	− 40.8885i	− 1.56456i	−0.622929	−	0.782278i	$-0.714057\pi$
$683$	− 40.8885i	− 1.56456i	0.622929	−	0.782278i	$-0.285943\pi$
$684$	2.29180	0.0876290
$685$	0	0
$686$	−0.381966	−0.0145835
$687$	− 22.3607i	− 0.853113i
$688$	− 11.8409i	− 0.451432i
$689$	44.3607	1.69001
$690$	0	0
$691$	32.3607	1.23106	0.615529	−	0.788114i	$-0.288942\pi$
$691$	32.3607	1.23106	0.615529	+	0.788114i	$0.288942\pi$
$692$	48.5410i	1.84525i
$693$	3.47214i	0.131896i
$694$	4.96556	0.188490
$695$	0	0
$696$	12.8197	0.485928
$697$	− 45.6656i	− 1.72971i
$698$	− 2.00000i	− 0.0757011i
$699$	−20.7082	−0.783256
$700$	0	0
$701$	6.58359	0.248659	0.124329	−	0.992241i	$-0.460322\pi$
$701$	6.58359	0.248659	0.124329	+	0.992241i	$0.460322\pi$
$702$	2.00000i	0.0754851i
$703$	4.29180i	0.161868i
$704$	16.3475	0.616120
$705$	0	0
$706$	7.23607	0.272333
$707$	17.7082i	0.665986i
$708$	− 9.70820i	− 0.364857i
$709$	−8.11146	−0.304632	−0.152316	−	0.988332i	$-0.548673\pi$
$709$	−8.11146	−0.304632	−0.152316	+	0.988332i	$0.548673\pi$
$710$	0	0
$711$	6.23607	0.233871
$712$	− 11.3475i	− 0.425266i
$713$	22.3607i	0.837414i
$714$	2.18034	0.0815972
$715$	0	0
$716$	24.0000	0.896922
$717$	16.3607i	0.611001i
$718$	− 3.07701i	− 0.114833i
$719$	38.2492	1.42646	0.713228	−	0.700932i	$-0.247233\pi$
$719$	38.2492	1.42646	0.713228	+	0.700932i	$0.247233\pi$
$720$	0	0
$721$	3.70820	0.138101
$722$	− 6.67376i	− 0.248372i
$723$	− 14.7639i	− 0.549077i
$724$	−49.4164	−1.83655
$725$	0	0
$726$	0.403252	0.0149661
$727$	48.5410i	1.80029i	0.435594	+	0.900143i	$0.356538\pi$
$727$	48.5410i	1.80029i	−0.435594	+	0.900143i	$0.643462\pi$
$728$	− 7.70820i	− 0.285685i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	21.4853	0.794662
$732$	− 21.1672i	− 0.782362i
$733$	− 12.0000i	− 0.443230i	−0.975134	−	0.221615i	$-0.928867\pi$
$733$	− 12.0000i	− 0.443230i	0.975134	−	0.221615i	$-0.0711328\pi$
$734$	−3.41641	−0.126102
$735$	0	0
$736$	20.7295	0.764099
$737$	37.1803i	1.36956i
$738$	3.05573i	0.112483i
$739$	−8.23607	−0.302969	−0.151484	−	0.988460i	$-0.548405\pi$
$739$	−8.23607	−0.302969	−0.151484	+	0.988460i	$0.548405\pi$
$740$	0	0
$741$	6.47214	0.237760
$742$	3.23607i	0.118800i
$743$	− 50.2492i	− 1.84347i	−0.387826	−	0.921733i	$-0.626774\pi$
$743$	− 50.2492i	− 1.84347i	0.387826	−	0.921733i	$-0.373226\pi$
$744$	−6.58359	−0.241366
$745$	0	0
$746$	−5.72949	−0.209772
$747$	3.52786i	0.129078i
$748$	36.7477i	1.34363i
$749$	12.9443	0.472973
$750$	0	0
$751$	−4.36068	−0.159123	−0.0795617	−	0.996830i	$-0.525352\pi$
$751$	−4.36068	−0.159123	−0.0795617	+	0.996830i	$0.525352\pi$
$752$	8.69505i	0.317076i
$753$	− 7.23607i	− 0.263697i
$754$	17.4164	0.634268
$755$	0	0
$756$	1.85410	0.0674330
$757$	3.94427i	0.143357i	0.997428	+	0.0716785i	$0.0228356\pi$
$757$	3.94427i	0.143357i	−0.997428	+	0.0716785i	$0.977164\pi$
$758$	12.4508i	0.452235i
$759$	−17.3607	−0.630153
$760$	0	0
$761$	−53.3050	−1.93230	−0.966151	−	0.257975i	$-0.916945\pi$
$761$	−53.3050	−1.93230	−0.966151	+	0.257975i	$0.916945\pi$
$762$	− 2.96556i	− 0.107431i
$763$	20.4164i	0.739124i
$764$	28.5836	1.03412
$765$	0	0
$766$	−5.77709	−0.208735
$767$	− 27.4164i	− 0.989949i
$768$	− 5.56231i	− 0.200712i
$769$	−8.58359	−0.309532	−0.154766	−	0.987951i	$-0.549462\pi$
$769$	−8.58359	−0.309532	−0.154766	+	0.987951i	$0.549462\pi$
$770$	0	0
$771$	0.472136	0.0170036
$772$	48.9787i	1.76278i
$773$	17.0557i	0.613452i	0.951798	+	0.306726i	$0.0992335\pi$
$773$	17.0557i	0.613452i	−0.951798	+	0.306726i	$0.900766\pi$
$774$	−1.43769	−0.0516768
$775$	0	0
$776$	−5.19350	−0.186436
$777$	3.47214i	0.124562i
$778$	2.38197i	0.0853976i
$779$	9.88854	0.354294
$780$	0	0
$781$	−32.8885	−1.17684
$782$	10.9017i	0.389844i
$783$	8.70820i	0.311206i
$784$	−3.14590	−0.112354
$785$	0	0
$786$	−6.76393	−0.241261
$787$	43.2361i	1.54120i	0.637319	+	0.770600i	$0.280043\pi$
$787$	43.2361i	1.54120i	−0.637319	+	0.770600i	$0.719957\pi$
$788$	− 18.9787i	− 0.676089i
$789$	−19.9443	−0.710035
$790$	0	0
$791$	11.7639	0.418277
$792$	− 5.11146i	− 0.181628i
$793$	− 59.7771i	− 2.12275i
$794$	6.87539	0.243998
$795$	0	0
$796$	29.6656	1.05147
$797$	18.0689i	0.640033i	0.947412	+	0.320016i	$0.103688\pi$
$797$	18.0689i	0.640033i	−0.947412	+	0.320016i	$0.896312\pi$
$798$	0.472136i	0.0167134i
$799$	−15.7771	−0.558153
$800$	0	0
$801$	7.70820	0.272356
$802$	− 5.07701i	− 0.179276i
$803$	11.2361i	0.396512i
$804$	19.8541	0.700200
$805$	0	0
$806$	−8.94427	−0.315049
$807$	− 19.5279i	− 0.687413i
$808$	− 26.0689i	− 0.917100i
$809$	29.1803	1.02593	0.512963	−	0.858411i	$-0.328548\pi$
$809$	29.1803	1.02593	0.512963	+	0.858411i	$0.328548\pi$
$810$	0	0
$811$	−6.00000	−0.210688	−0.105344	−	0.994436i	$-0.533594\pi$
$811$	−6.00000	−0.210688	−0.105344	+	0.994436i	$0.533594\pi$
$812$	− 16.1459i	− 0.566610i
$813$	− 18.7639i	− 0.658080i
$814$	4.60488	0.161401
$815$	0	0
$816$	17.9574	0.628636
$817$	4.65248i	0.162770i
$818$	− 5.41641i	− 0.189380i
$819$	5.23607	0.182963
$820$	0	0
$821$	6.94427	0.242357	0.121178	−	0.992631i	$-0.461333\pi$
$821$	6.94427	0.242357	0.121178	+	0.992631i	$0.461333\pi$
$822$	3.23607i	0.112871i
$823$	− 32.0132i	− 1.11591i	−0.829872	−	0.557954i	$-0.811586\pi$
$823$	− 32.0132i	− 1.11591i	0.829872	−	0.557954i	$-0.188414\pi$
$824$	−5.45898	−0.190173
$825$	0	0
$826$	2.00000	0.0695889
$827$	14.0557i	0.488766i	0.969679	+	0.244383i	$0.0785853\pi$
$827$	14.0557i	0.488766i	−0.969679	+	0.244383i	$0.921415\pi$
$828$	9.27051i	0.322172i
$829$	−31.2361	−1.08487	−0.542437	−	0.840097i	$-0.682498\pi$
$829$	−31.2361	−1.08487	−0.542437	+	0.840097i	$0.682498\pi$
$830$	0	0
$831$	19.8885	0.689926
$832$	− 24.6525i	− 0.854671i
$833$	− 5.70820i	− 0.197778i
$834$	3.70820	0.128405
$835$	0	0
$836$	−7.95743	−0.275213
$837$	− 4.47214i	− 0.154580i
$838$	− 10.2918i	− 0.355524i
$839$	13.1246	0.453112	0.226556	−	0.973998i	$-0.427253\pi$
$839$	13.1246	0.453112	0.226556	+	0.973998i	$0.427253\pi$
$840$	0	0
$841$	46.8328	1.61492
$842$	10.0902i	0.347730i
$843$	17.6525i	0.607984i
$844$	−9.16718	−0.315547
$845$	0	0
$846$	1.05573	0.0362967
$847$	− 1.05573i	− 0.0362752i
$848$	26.6525i	0.915250i
$849$	−13.4164	−0.460450
$850$	0	0
$851$	−17.3607	−0.595116
$852$	17.5623i	0.601675i
$853$	− 7.63932i	− 0.261565i	−0.991411	−	0.130783i	$-0.958251\pi$
$853$	− 7.63932i	− 0.261565i	0.991411	−	0.130783i	$-0.0417490\pi$
$854$	4.36068	0.149219
$855$	0	0
$856$	−19.0557	−0.651311
$857$	12.5836i	0.429847i	0.976631	+	0.214924i	$0.0689503\pi$
$857$	12.5836i	0.429847i	−0.976631	+	0.214924i	$0.931050\pi$
$858$	− 6.94427i	− 0.237074i
$859$	−8.47214	−0.289066	−0.144533	−	0.989500i	$-0.546168\pi$
$859$	−8.47214	−0.289066	−0.144533	+	0.989500i	$0.546168\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	− 0.721360i	− 0.0245696i
$863$	− 19.9443i	− 0.678911i	−0.940622	−	0.339455i	$-0.889757\pi$
$863$	− 19.9443i	− 0.678911i	0.940622	−	0.339455i	$-0.110243\pi$
$864$	−4.14590	−0.141046
$865$	0	0
$866$	−9.66563	−0.328452
$867$	15.5836i	0.529247i
$868$	8.29180i	0.281442i
$869$	−21.6525	−0.734510
$870$	0	0
$871$	56.0689	1.89982
$872$	− 30.0557i	− 1.01782i
$873$	− 3.52786i	− 0.119400i
$874$	−2.36068	−0.0798512
$875$	0	0
$876$	6.00000	0.202721
$877$	6.00000i	0.202606i	0.994856	+	0.101303i	$0.0323011\pi$
$877$	6.00000i	0.202606i	−0.994856	+	0.101303i	$0.967699\pi$
$878$	2.58359i	0.0871920i
$879$	−4.65248	−0.156924
$880$	0	0
$881$	6.36068	0.214297	0.107148	−	0.994243i	$-0.465828\pi$
$881$	6.36068	0.214297	0.107148	+	0.994243i	$0.465828\pi$
$882$	0.381966i	0.0128615i
$883$	− 29.7639i	− 1.00164i	−0.865553	−	0.500818i	$-0.833033\pi$
$883$	− 29.7639i	− 1.00164i	0.865553	−	0.500818i	$-0.166967\pi$
$884$	55.4164	1.86386
$885$	0	0
$886$	15.0557	0.505807
$887$	11.4164i	0.383325i	0.981461	+	0.191663i	$0.0613880\pi$
$887$	11.4164i	0.383325i	−0.981461	+	0.191663i	$0.938612\pi$
$888$	− 5.11146i	− 0.171529i
$889$	−7.76393	−0.260394
$890$	0	0
$891$	3.47214	0.116321
$892$	− 6.00000i	− 0.200895i
$893$	− 3.41641i	− 0.114326i
$894$	−1.43769	−0.0480837
$895$	0	0
$896$	10.0902	0.337089
$897$	26.1803i	0.874136i
$898$	− 6.38197i	− 0.212969i
$899$	−38.9443	−1.29886
$900$	0	0
$901$	−48.3607	−1.61113
$902$	− 10.6099i	− 0.353271i
$903$	3.76393i	0.125256i
$904$	−17.3181	−0.575992
$905$	0	0
$906$	5.61803	0.186647
$907$	28.3607i	0.941701i	0.882213	+	0.470850i	$0.156053\pi$
$907$	28.3607i	0.941701i	−0.882213	+	0.470850i	$0.843947\pi$
$908$	− 17.1246i	− 0.568300i
$909$	17.7082	0.587344
$910$	0	0
$911$	34.4164	1.14027	0.570133	−	0.821552i	$-0.306892\pi$
$911$	34.4164	1.14027	0.570133	+	0.821552i	$0.306892\pi$
$912$	3.88854i	0.128763i
$913$	− 12.2492i	− 0.405390i
$914$	7.84095	0.259355
$915$	0	0
$916$	41.4590	1.36984
$917$	17.7082i	0.584776i
$918$	− 2.18034i	− 0.0719619i
$919$	40.0132	1.31991	0.659956	−	0.751304i	$-0.270575\pi$
$919$	40.0132	1.31991	0.659956	+	0.751304i	$0.270575\pi$
$920$	0	0
$921$	−6.65248	−0.219207
$922$	− 7.88854i	− 0.259795i
$923$	49.5967i	1.63250i
$924$	−6.43769	−0.211785
$925$	0	0
$926$	−6.24922	−0.205362
$927$	− 3.70820i	− 0.121793i
$928$	36.1033i	1.18515i
$929$	7.81966	0.256555	0.128277	−	0.991738i	$-0.459055\pi$
$929$	7.81966	0.256555	0.128277	+	0.991738i	$0.459055\pi$
$930$	0	0
$931$	1.23607	0.0405105
$932$	− 38.3951i	− 1.25767i
$933$	− 8.76393i	− 0.286918i
$934$	4.54102	0.148587
$935$	0	0
$936$	−7.70820	−0.251951
$937$	− 40.3607i	− 1.31853i	−0.751912	−	0.659263i	$-0.770868\pi$
$937$	− 40.3607i	− 1.31853i	0.751912	−	0.659263i	$-0.229132\pi$
$938$	4.09017i	0.133549i
$939$	−7.70820	−0.251548
$940$	0	0
$941$	−8.83282	−0.287942	−0.143971	−	0.989582i	$-0.545987\pi$
$941$	−8.83282	−0.287942	−0.143971	+	0.989582i	$0.545987\pi$
$942$	− 0.360680i	− 0.0117516i
$943$	40.0000i	1.30258i
$944$	16.4721	0.536122
$945$	0	0
$946$	4.99187	0.162300
$947$	15.0557i	0.489245i	0.969618	+	0.244623i	$0.0786641\pi$
$947$	15.0557i	0.489245i	−0.969618	+	0.244623i	$0.921336\pi$
$948$	11.5623i	0.375526i
$949$	16.9443	0.550034
$950$	0	0
$951$	−13.6525	−0.442712
$952$	8.40325i	0.272351i
$953$	24.1246i	0.781473i	0.920503	+	0.390736i	$0.127780\pi$
$953$	24.1246i	0.781473i	−0.920503	+	0.390736i	$0.872220\pi$
$954$	3.23607	0.104772
$955$	0	0
$956$	−30.3344	−0.981084
$957$	− 30.2361i	− 0.977393i
$958$	7.16718i	0.231561i
$959$	8.47214	0.273580
$960$	0	0
$961$	−11.0000	−0.354839
$962$	− 6.94427i	− 0.223892i
$963$	− 12.9443i	− 0.417123i
$964$	27.3738	0.881652
$965$	0	0
$966$	−1.90983	−0.0614478
$967$	− 37.8885i	− 1.21841i	−0.793011	−	0.609207i	$-0.791488\pi$
$967$	− 37.8885i	− 1.21841i	0.793011	−	0.609207i	$-0.208512\pi$
$968$	1.55418i	0.0499531i
$969$	−7.05573	−0.226663
$970$	0	0
$971$	−37.5967	−1.20654	−0.603269	−	0.797538i	$-0.706135\pi$
$971$	−37.5967	−1.20654	−0.603269	+	0.797538i	$0.706135\pi$
$972$	− 1.85410i	− 0.0594703i
$973$	− 9.70820i	− 0.311231i
$974$	11.1459	0.357138
$975$	0	0
$976$	35.9149	1.14961
$977$	16.7082i	0.534543i	0.963621	+	0.267271i	$0.0861220\pi$
$977$	16.7082i	0.534543i	−0.963621	+	0.267271i	$0.913878\pi$
$978$	− 3.63932i	− 0.116373i
$979$	−26.7639	−0.855379
$980$	0	0
$981$	20.4164	0.651846
$982$	− 3.61803i	− 0.115456i
$983$	29.4164i	0.938238i	0.883135	+	0.469119i	$0.155428\pi$
$983$	29.4164i	0.938238i	−0.883135	+	0.469119i	$0.844572\pi$
$984$	−11.7771	−0.375440
$985$	0	0
$986$	−18.9868	−0.604664
$987$	− 2.76393i	− 0.0879769i
$988$	12.0000i	0.381771i
$989$	−18.8197	−0.598430
$990$	0	0
$991$	−40.5967	−1.28960	−0.644799	−	0.764352i	$-0.723059\pi$
$991$	−40.5967	−1.28960	−0.644799	+	0.764352i	$0.723059\pi$
$992$	− 18.5410i	− 0.588678i
$993$	− 18.2361i	− 0.578704i
$994$	−3.61803	−0.114757
$995$	0	0
$996$	−6.54102	−0.207260
$997$	− 14.5410i	− 0.460519i	−0.973129	−	0.230259i	$-0.926043\pi$
$997$	− 14.5410i	− 0.460519i	0.973129	−	0.230259i	$-0.0739575\pi$
$998$	− 15.1935i	− 0.480942i
$999$	3.47214	0.109854

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	525.2.d.e.274.3		4
3.2	odd	2		1575.2.d.f.1324.2		4
5.2	odd	4		525.2.a.e.1.2	✓	2
5.3	odd	4		525.2.a.i.1.1	yes	2
5.4	even	2	inner	525.2.d.e.274.2		4
15.2	even	4		1575.2.a.v.1.1		2
15.8	even	4		1575.2.a.l.1.2		2
15.14	odd	2		1575.2.d.f.1324.3		4
20.3	even	4		8400.2.a.cy.1.1		2
20.7	even	4		8400.2.a.da.1.1		2
35.13	even	4		3675.2.a.bh.1.1		2
35.27	even	4		3675.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.a.e.1.2	✓	2	5.2	odd	4
525.2.a.i.1.1	yes	2	5.3	odd	4
525.2.d.e.274.2		4	5.4	even	2	inner
525.2.d.e.274.3		4	1.1	even	1	trivial
1575.2.a.l.1.2		2	15.8	even	4
1575.2.a.v.1.1		2	15.2	even	4
1575.2.d.f.1324.2		4	3.2	odd	2
1575.2.d.f.1324.3		4	15.14	odd	2
3675.2.a.r.1.2		2	35.27	even	4
3675.2.a.bh.1.1		2	35.13	even	4
8400.2.a.cy.1.1		2	20.3	even	4
8400.2.a.da.1.1		2	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	525.2.d.e.274.3

Level	$525$
Weight	$2$
Character	525.274
Analytic conductor	$4.192$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$