LMFDB - Embedded newform 2500.2.c.b.1249.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2500,2,Mod(1249,2500)] 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("2500.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2500, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2500 = 2^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2500.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$19.9626005053$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.58140625.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 $ x^8 - 3x^7 + 4x^6 - 7x^5 + 11x^4 + 5x^3 - 10x^2 - 25*x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.8
Root			$-0.357358 + 1.86824i$ of defining polynomial
Character	$\chi$	$=$	2500.1249
Dual form			2500.2.c.b.1249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30927i q^{3} +4.21139i q^{7} -2.33275 q^{9} -0.558282 q^{11} +4.02999i q^{13} +3.96225i q^{17} +6.93090 q^{19} -9.72525 q^{21} -0.381287i q^{23} +1.54087i q^{27} +8.04746 q^{29} -0.248356 q^{31} -1.28923i q^{33} +7.17542i q^{37} -9.30636 q^{39} -4.55828 q^{41} -0.207272i q^{43} -9.25893i q^{47} -10.7358 q^{49} -9.14992 q^{51} -2.36102i q^{53} +16.0054i q^{57} -3.62857 q^{59} -3.90332 q^{61} -9.82411i q^{63} -6.90040i q^{67} +0.880496 q^{69} +14.3709 q^{71} +2.25392i q^{73} -2.35114i q^{77} -7.63616 q^{79} -10.5565 q^{81} -7.59857i q^{83} +18.5838i q^{87} +16.0334 q^{89} -16.9719 q^{91} -0.573522i q^{93} -3.27102i q^{97} +1.30233 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{9} - 10 q^{11} + 12 q^{19} - 22 q^{21} + 32 q^{29} - 2 q^{31} + 2 q^{39} - 42 q^{41} + 14 q^{49} - 14 q^{51} + 24 q^{59} - 34 q^{61} + 36 q^{69} + 4 q^{71} - 4 q^{79} - 28 q^{81} + 58 q^{89}+ \cdots + 20 q^{99}+O(q^{100}) $ 8 * q + 2 * q^9 - 10 * q^11 + 12 * q^19 - 22 * q^21 + 32 * q^29 - 2 * q^31 + 2 * q^39 - 42 * q^41 + 14 * q^49 - 14 * q^51 + 24 * q^59 - 34 * q^61 + 36 * q^69 + 4 * q^71 - 4 * q^79 - 28 * q^81 + 58 * q^89 - 18 * q^91 + 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times$.

$n$	$1251$	$1877$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0	0
$2$	0	0
$3$	2.30927i	1.33326i	0.745389	+	0.666630i	$0.232264\pi$
$3$	2.30927i	1.33326i	−0.745389	+	0.666630i	$0.767736\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.21139i	1.59175i	0.605458	+	0.795877i	$0.292990\pi$
$7$	4.21139i	1.59175i	−0.605458	+	0.795877i	$0.707010\pi$
$8$	0	0
$9$	−2.33275	−0.777583
$10$	0	0
$11$	−0.558282	−0.168328	−0.0841641	−	0.996452i	$-0.526822\pi$
$11$	−0.558282	−0.168328	−0.0841641	+	0.996452i	$0.526822\pi$
$12$	0	0
$13$	4.02999i	1.11772i	0.829263	+	0.558859i	$0.188761\pi$
$13$	4.02999i	1.11772i	−0.829263	+	0.558859i	$0.811239\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.96225i	0.960987i	0.876998	+	0.480493i	$0.159542\pi$
$17$	3.96225i	0.960987i	−0.876998	+	0.480493i	$0.840458\pi$
$18$	0	0
$19$	6.93090	1.59006	0.795029	−	0.606572i	$-0.207456\pi$
$19$	6.93090	1.59006	0.795029	+	0.606572i	$0.207456\pi$
$20$	0	0
$21$	−9.72525	−2.12222
$22$	0	0
$23$	− 0.381287i	− 0.0795038i	−0.999210	−	0.0397519i	$-0.987343\pi$
$23$	− 0.381287i	− 0.0795038i	0.999210	−	0.0397519i	$-0.0126568\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.54087i	0.296540i
$28$	0	0
$29$	8.04746	1.49438	0.747188	−	0.664612i	$-0.231403\pi$
$29$	8.04746	1.49438	0.747188	+	0.664612i	$0.231403\pi$
$30$	0	0
$31$	−0.248356	−0.0446061	−0.0223030	−	0.999751i	$-0.507100\pi$
$31$	−0.248356	−0.0446061	−0.0223030	+	0.999751i	$0.507100\pi$
$32$	0	0
$33$	− 1.28923i	− 0.224425i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.17542i	1.17963i	0.807538	+	0.589816i	$0.200800\pi$
$37$	7.17542i	1.17963i	−0.807538	+	0.589816i	$0.799200\pi$
$38$	0	0
$39$	−9.30636	−1.49021
$40$	0	0
$41$	−4.55828	−0.711884	−0.355942	−	0.934508i	$-0.615840\pi$
$41$	−4.55828	−0.711884	−0.355942	+	0.934508i	$0.615840\pi$
$42$	0	0
$43$	− 0.207272i	− 0.0316086i	−0.999875	−	0.0158043i	$-0.994969\pi$
$43$	− 0.207272i	− 0.0316086i	0.999875	−	0.0158043i	$-0.00503088\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.25893i	− 1.35055i	−0.737565	−	0.675277i	$-0.764024\pi$
$47$	− 9.25893i	− 1.35055i	0.737565	−	0.675277i	$-0.235976\pi$
$48$	0	0
$49$	−10.7358	−1.53368
$50$	0	0
$51$	−9.14992	−1.28125
$52$	0	0
$53$	− 2.36102i	− 0.324312i	−0.986765	−	0.162156i	$-0.948155\pi$
$53$	− 2.36102i	− 0.324312i	0.986765	−	0.162156i	$-0.0518447\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	16.0054i	2.11996i
$58$	0	0
$59$	−3.62857	−0.472399	−0.236200	−	0.971705i	$-0.575902\pi$
$59$	−3.62857	−0.472399	−0.236200	+	0.971705i	$0.575902\pi$
$60$	0	0
$61$	−3.90332	−0.499769	−0.249884	−	0.968276i	$-0.580393\pi$
$61$	−3.90332	−0.499769	−0.249884	+	0.968276i	$0.580393\pi$
$62$	0	0
$63$	− 9.82411i	− 1.23772i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 6.90040i	− 0.843018i	−0.906824	−	0.421509i	$-0.861501\pi$
$67$	− 6.90040i	− 0.843018i	0.906824	−	0.421509i	$-0.138499\pi$
$68$	0	0
$69$	0.880496	0.105999
$70$	0	0
$71$	14.3709	1.70551	0.852754	−	0.522313i	$-0.174931\pi$
$71$	14.3709	1.70551	0.852754	+	0.522313i	$0.174931\pi$
$72$	0	0
$73$	2.25392i	0.263801i	0.991263	+	0.131901i	$0.0421080\pi$
$73$	2.25392i	0.263801i	−0.991263	+	0.131901i	$0.957892\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.35114i	− 0.267937i
$78$	0	0
$79$	−7.63616	−0.859136	−0.429568	−	0.903035i	$-0.641334\pi$
$79$	−7.63616	−0.859136	−0.429568	+	0.903035i	$0.641334\pi$
$80$	0	0
$81$	−10.5565	−1.17295
$82$	0	0
$83$	− 7.59857i	− 0.834051i	−0.908895	−	0.417026i	$-0.863073\pi$
$83$	− 7.59857i	− 0.834051i	0.908895	−	0.417026i	$-0.136927\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	18.5838i	1.99239i
$88$	0	0
$89$	16.0334	1.69953	0.849766	−	0.527160i	$-0.176743\pi$
$89$	16.0334	1.69953	0.849766	+	0.527160i	$0.176743\pi$
$90$	0	0
$91$	−16.9719	−1.77913
$92$	0	0
$93$	− 0.573522i	− 0.0594715i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 3.27102i	− 0.332122i	−0.986115	−	0.166061i	$-0.946895\pi$
$97$	− 3.27102i	− 0.332122i	0.986115	−	0.166061i	$-0.0531049\pi$
$98$	0	0
$99$	1.30233	0.130889
$100$	0	0
$101$	−9.34504	−0.929866	−0.464933	−	0.885346i	$-0.653922\pi$
$101$	−9.34504	−0.929866	−0.464933	+	0.885346i	$0.653922\pi$
$102$	0	0
$103$	− 5.99040i	− 0.590252i	−0.955458	−	0.295126i	$-0.904638\pi$
$103$	− 5.99040i	− 0.590252i	0.955458	−	0.295126i	$-0.0953616\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.60078i	− 0.541448i	−0.962657	−	0.270724i	$-0.912737\pi$
$107$	− 5.60078i	− 0.541448i	0.962657	−	0.270724i	$-0.0872630\pi$
$108$	0	0
$109$	13.7300	1.31510	0.657548	−	0.753413i	$-0.271594\pi$
$109$	13.7300	1.31510	0.657548	+	0.753413i	$0.271594\pi$
$110$	0	0
$111$	−16.5700	−1.57276
$112$	0	0
$113$	4.05741i	0.381689i	0.981620	+	0.190845i	$0.0611227\pi$
$113$	4.05741i	0.381689i	−0.981620	+	0.190845i	$0.938877\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 9.40096i	− 0.869119i
$118$	0	0
$119$	−16.6866	−1.52966
$120$	0	0
$121$	−10.6883	−0.971666
$122$	0	0
$123$	− 10.5263i	− 0.949127i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.12920i	− 0.366407i	−0.983075	−	0.183204i	$-0.941353\pi$
$127$	− 4.12920i	− 0.366407i	0.983075	−	0.183204i	$-0.0586468\pi$
$128$	0	0
$129$	0.478647	0.0421425
$130$	0	0
$131$	−13.1746	−1.15107	−0.575533	−	0.817778i	$-0.695206\pi$
$131$	−13.1746	−1.15107	−0.575533	+	0.817778i	$0.695206\pi$
$132$	0	0
$133$	29.1887i	2.53098i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.65881i	0.141722i	0.997486	+	0.0708609i	$0.0225746\pi$
$137$	1.65881i	0.141722i	−0.997486	+	0.0708609i	$0.977425\pi$
$138$	0	0
$139$	0.701344	0.0594872	0.0297436	−	0.999558i	$-0.490531\pi$
$139$	0.701344	0.0594872	0.0297436	+	0.999558i	$0.490531\pi$
$140$	0	0
$141$	21.3814	1.80064
$142$	0	0
$143$	− 2.24987i	− 0.188144i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 24.7919i	− 2.04480i
$148$	0	0
$149$	18.3234	1.50111	0.750556	−	0.660807i	$-0.229786\pi$
$149$	18.3234	1.50111	0.750556	+	0.660807i	$0.229786\pi$
$150$	0	0
$151$	0.0521578	0.00424454	0.00212227	−	0.999998i	$-0.499324\pi$
$151$	0.0521578	0.00424454	0.00212227	+	0.999998i	$0.499324\pi$
$152$	0	0
$153$	− 9.24293i	− 0.747247i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.18979i	0.254573i	0.991866	+	0.127287i	$0.0406268\pi$
$157$	3.18979i	0.254573i	−0.991866	+	0.127287i	$0.959373\pi$
$158$	0	0
$159$	5.45225	0.432392
$160$	0	0
$161$	1.60575	0.126551
$162$	0	0
$163$	− 11.7102i	− 0.917216i	−0.888639	−	0.458608i	$-0.848348\pi$
$163$	− 11.7102i	− 0.917216i	0.888639	−	0.458608i	$-0.151652\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 18.1089i	− 1.40131i	−0.713501	−	0.700654i	$-0.752891\pi$
$167$	− 18.1089i	− 1.40131i	0.713501	−	0.700654i	$-0.247109\pi$
$168$	0	0
$169$	−3.24083	−0.249294
$170$	0	0
$171$	−16.1681	−1.23640
$172$	0	0
$173$	17.2786i	1.31367i	0.754036	+	0.656833i	$0.228104\pi$
$173$	17.2786i	1.31367i	−0.754036	+	0.656833i	$0.771896\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 8.37936i	− 0.629831i
$178$	0	0
$179$	−3.99065	−0.298275	−0.149138	−	0.988816i	$-0.547650\pi$
$179$	−3.99065	−0.298275	−0.149138	+	0.988816i	$0.547650\pi$
$180$	0	0
$181$	1.76218	0.130982	0.0654909	−	0.997853i	$-0.479139\pi$
$181$	1.76218	0.130982	0.0654909	+	0.997853i	$0.479139\pi$
$182$	0	0
$183$	− 9.01384i	− 0.666322i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.21205i	− 0.161761i
$188$	0	0
$189$	−6.48918	−0.472019
$190$	0	0
$191$	−7.46279	−0.539988	−0.269994	−	0.962862i	$-0.587022\pi$
$191$	−7.46279	−0.539988	−0.269994	+	0.962862i	$0.587022\pi$
$192$	0	0
$193$	15.6167i	1.12411i	0.827099	+	0.562057i	$0.189990\pi$
$193$	15.6167i	1.12411i	−0.827099	+	0.562057i	$0.810010\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 24.8393i	− 1.76973i	−0.465848	−	0.884865i	$-0.654251\pi$
$197$	− 24.8393i	− 1.76973i	0.465848	−	0.884865i	$-0.345749\pi$
$198$	0	0
$199$	4.27124	0.302780	0.151390	−	0.988474i	$-0.451625\pi$
$199$	4.27124	0.302780	0.151390	+	0.988474i	$0.451625\pi$
$200$	0	0
$201$	15.9349	1.12396
$202$	0	0
$203$	33.8910i	2.37868i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.889446i	0.0618208i
$208$	0	0
$209$	−3.86940	−0.267652
$210$	0	0
$211$	8.36978	0.576200	0.288100	−	0.957600i	$-0.406977\pi$
$211$	8.36978	0.576200	0.288100	+	0.957600i	$0.406977\pi$
$212$	0	0
$213$	33.1863i	2.27389i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.04592i	− 0.0710019i
$218$	0	0
$219$	−5.20492	−0.351716
$220$	0	0
$221$	−15.9678	−1.07411
$222$	0	0
$223$	8.88323i	0.594865i	0.954743	+	0.297433i	$0.0961304\pi$
$223$	8.88323i	0.594865i	−0.954743	+	0.297433i	$0.903870\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 11.3470i	− 0.753127i	−0.926391	−	0.376563i	$-0.877106\pi$
$227$	− 11.3470i	− 0.753127i	0.926391	−	0.376563i	$-0.122894\pi$
$228$	0	0
$229$	−26.6878	−1.76358	−0.881788	−	0.471645i	$-0.843660\pi$
$229$	−26.6878	−1.76358	−0.881788	+	0.471645i	$0.843660\pi$
$230$	0	0
$231$	5.42943	0.357230
$232$	0	0
$233$	− 19.4065i	− 1.27136i	−0.771953	−	0.635680i	$-0.780720\pi$
$233$	− 19.4065i	− 1.27136i	0.771953	−	0.635680i	$-0.219280\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 17.6340i	− 1.14545i
$238$	0	0
$239$	4.80194	0.310612	0.155306	−	0.987866i	$-0.450364\pi$
$239$	4.80194	0.310612	0.155306	+	0.987866i	$0.450364\pi$
$240$	0	0
$241$	−20.1634	−1.29884	−0.649419	−	0.760430i	$-0.724988\pi$
$241$	−20.1634	−1.29884	−0.649419	+	0.760430i	$0.724988\pi$
$242$	0	0
$243$	− 19.7553i	− 1.26730i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	27.9315i	1.77724i
$248$	0	0
$249$	17.5472	1.11201
$250$	0	0
$251$	−13.8723	−0.875614	−0.437807	−	0.899069i	$-0.644245\pi$
$251$	−13.8723	−0.875614	−0.437807	+	0.899069i	$0.644245\pi$
$252$	0	0
$253$	0.212865i	0.0133827i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.27079i	0.266405i	0.991089	+	0.133202i	$0.0425260\pi$
$257$	4.27079i	0.266405i	−0.991089	+	0.133202i	$0.957474\pi$
$258$	0	0
$259$	−30.2185	−1.87768
$260$	0	0
$261$	−18.7727	−1.16200
$262$	0	0
$263$	− 11.7172i	− 0.722512i	−0.932467	−	0.361256i	$-0.882348\pi$
$263$	− 11.7172i	− 0.722512i	0.932467	−	0.361256i	$-0.117652\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	37.0254i	2.26592i
$268$	0	0
$269$	10.5718	0.644571	0.322286	−	0.946642i	$-0.395549\pi$
$269$	10.5718	0.644571	0.322286	+	0.946642i	$0.395549\pi$
$270$	0	0
$271$	13.0380	0.792003	0.396001	−	0.918250i	$-0.370398\pi$
$271$	13.0380	0.792003	0.396001	+	0.918250i	$0.370398\pi$
$272$	0	0
$273$	− 39.1927i	− 2.37205i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 12.5246i	− 0.752528i	−0.926513	−	0.376264i	$-0.877209\pi$
$277$	− 12.5246i	− 0.752528i	0.926513	−	0.376264i	$-0.122791\pi$
$278$	0	0
$279$	0.579352	0.0346849
$280$	0	0
$281$	1.91494	0.114236	0.0571178	−	0.998367i	$-0.481809\pi$
$281$	1.91494	0.114236	0.0571178	+	0.998367i	$0.481809\pi$
$282$	0	0
$283$	− 8.67692i	− 0.515789i	−0.966173	−	0.257895i	$-0.916971\pi$
$283$	− 8.67692i	− 0.515789i	0.966173	−	0.257895i	$-0.0830287\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 19.1967i	− 1.13314i
$288$	0	0
$289$	1.30058	0.0765046
$290$	0	0
$291$	7.55369	0.442805
$292$	0	0
$293$	18.1632i	1.06110i	0.847653	+	0.530551i	$0.178015\pi$
$293$	18.1632i	1.06110i	−0.847653	+	0.530551i	$0.821985\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 0.860237i	− 0.0499160i
$298$	0	0
$299$	1.53658	0.0888628
$300$	0	0
$301$	0.872901	0.0503132
$302$	0	0
$303$	− 21.5803i	− 1.23975i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	23.4255i	1.33696i	0.743729	+	0.668481i	$0.233055\pi$
$307$	23.4255i	1.33696i	−0.743729	+	0.668481i	$0.766945\pi$
$308$	0	0
$309$	13.8335	0.786959
$310$	0	0
$311$	23.3473	1.32390	0.661952	−	0.749546i	$-0.269728\pi$
$311$	23.3473	1.32390	0.661952	+	0.749546i	$0.269728\pi$
$312$	0	0
$313$	5.83039i	0.329553i	0.986331	+	0.164777i	$0.0526903\pi$
$313$	5.83039i	0.329553i	−0.986331	+	0.164777i	$0.947310\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 7.48285i	− 0.420279i	−0.977671	−	0.210139i	$-0.932608\pi$
$317$	− 7.48285i	− 0.420279i	0.977671	−	0.210139i	$-0.0673918\pi$
$318$	0	0
$319$	−4.49275	−0.251546
$320$	0	0
$321$	12.9337	0.721891
$322$	0	0
$323$	27.4620i	1.52802i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	31.7064i	1.75337i
$328$	0	0
$329$	38.9929	2.14975
$330$	0	0
$331$	30.1946	1.65964	0.829821	−	0.558029i	$-0.188442\pi$
$331$	30.1946	1.65964	0.829821	+	0.558029i	$0.188442\pi$
$332$	0	0
$333$	− 16.7384i	− 0.917261i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.3367i	1.21676i	0.793646	+	0.608379i	$0.208180\pi$
$337$	22.3367i	1.21676i	−0.793646	+	0.608379i	$0.791820\pi$
$338$	0	0
$339$	−9.36968	−0.508891
$340$	0	0
$341$	0.138653	0.00750846
$342$	0	0
$343$	− 15.7328i	− 0.849494i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.1662i	0.599433i	0.954028	+	0.299716i	$0.0968920\pi$
$347$	11.1662i	0.599433i	−0.954028	+	0.299716i	$0.903108\pi$
$348$	0	0
$349$	−4.15360	−0.222337	−0.111168	−	0.993802i	$-0.535459\pi$
$349$	−4.15360	−0.222337	−0.111168	+	0.993802i	$0.535459\pi$
$350$	0	0
$351$	−6.20967	−0.331448
$352$	0	0
$353$	− 4.39639i	− 0.233996i	−0.993132	−	0.116998i	$-0.962673\pi$
$353$	− 4.39639i	− 0.233996i	0.993132	−	0.116998i	$-0.0373272\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 38.5339i	− 2.03943i
$358$	0	0
$359$	0.476894	0.0251695	0.0125848	−	0.999921i	$-0.495994\pi$
$359$	0.476894	0.0251695	0.0125848	+	0.999921i	$0.495994\pi$
$360$	0	0
$361$	29.0374	1.52828
$362$	0	0
$363$	− 24.6823i	− 1.29548i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2.63226i	− 0.137403i	−0.997637	−	0.0687015i	$-0.978114\pi$
$367$	− 2.63226i	− 0.137403i	0.997637	−	0.0687015i	$-0.0218856\pi$
$368$	0	0
$369$	10.6333	0.553549
$370$	0	0
$371$	9.94319	0.516225
$372$	0	0
$373$	− 25.0619i	− 1.29766i	−0.760935	−	0.648828i	$-0.775260\pi$
$373$	− 25.0619i	− 1.29766i	0.760935	−	0.648828i	$-0.224740\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.4312i	1.67029i
$378$	0	0
$379$	2.18401	0.112185	0.0560927	−	0.998426i	$-0.482136\pi$
$379$	2.18401	0.112185	0.0560927	+	0.998426i	$0.482136\pi$
$380$	0	0
$381$	9.53546	0.488516
$382$	0	0
$383$	23.8671i	1.21955i	0.792574	+	0.609776i	$0.208741\pi$
$383$	23.8671i	1.21955i	−0.792574	+	0.609776i	$0.791259\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.483513i	0.0245783i
$388$	0	0
$389$	−19.9796	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$389$	−19.9796	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$390$	0	0
$391$	1.51075	0.0764021
$392$	0	0
$393$	− 30.4237i	− 1.53467i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	16.8678i	0.846572i	0.905996	+	0.423286i	$0.139123\pi$
$397$	16.8678i	0.846572i	−0.905996	+	0.423286i	$0.860877\pi$
$398$	0	0
$399$	−67.4047	−3.37446
$400$	0	0
$401$	5.14817	0.257087	0.128544	−	0.991704i	$-0.458970\pi$
$401$	5.14817	0.257087	0.128544	+	0.991704i	$0.458970\pi$
$402$	0	0
$403$	− 1.00087i	− 0.0498570i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4.00590i	− 0.198565i
$408$	0	0
$409$	5.39358	0.266696	0.133348	−	0.991069i	$-0.457427\pi$
$409$	5.39358	0.266696	0.133348	+	0.991069i	$0.457427\pi$
$410$	0	0
$411$	−3.83065	−0.188952
$412$	0	0
$413$	− 15.2813i	− 0.751944i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.61960i	0.0793119i
$418$	0	0
$419$	24.8517	1.21408	0.607042	−	0.794670i	$-0.292356\pi$
$419$	24.8517	1.21408	0.607042	+	0.794670i	$0.292356\pi$
$420$	0	0
$421$	33.6907	1.64198	0.820992	−	0.570940i	$-0.193421\pi$
$421$	33.6907	1.64198	0.820992	+	0.570940i	$0.193421\pi$
$422$	0	0
$423$	21.5987i	1.05017i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 16.4384i	− 0.795509i
$428$	0	0
$429$	5.19557	0.250844
$430$	0	0
$431$	−17.9871	−0.866410	−0.433205	−	0.901295i	$-0.642617\pi$
$431$	−17.9871	−0.866410	−0.433205	+	0.901295i	$0.642617\pi$
$432$	0	0
$433$	35.8255i	1.72167i	0.508888	+	0.860833i	$0.330057\pi$
$433$	35.8255i	1.72167i	−0.508888	+	0.860833i	$0.669943\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.64266i	− 0.126416i
$438$	0	0
$439$	4.13582	0.197392	0.0986959	−	0.995118i	$-0.468533\pi$
$439$	4.13582	0.197392	0.0986959	+	0.995118i	$0.468533\pi$
$440$	0	0
$441$	25.0439	1.19257
$442$	0	0
$443$	24.8979i	1.18293i	0.806329	+	0.591467i	$0.201451\pi$
$443$	24.8979i	1.18293i	−0.806329	+	0.591467i	$0.798549\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	42.3138i	2.00137i
$448$	0	0
$449$	0.385885	0.0182111	0.00910553	−	0.999959i	$-0.497102\pi$
$449$	0.385885	0.0182111	0.00910553	+	0.999959i	$0.497102\pi$
$450$	0	0
$451$	2.54481	0.119830
$452$	0	0
$453$	0.120447i	0.00565907i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 29.8819i	− 1.39781i	−0.715212	−	0.698907i	$-0.753670\pi$
$457$	− 29.8819i	− 1.39781i	0.715212	−	0.698907i	$-0.246330\pi$
$458$	0	0
$459$	−6.10529	−0.284971
$460$	0	0
$461$	−19.9585	−0.929559	−0.464780	−	0.885426i	$-0.653866\pi$
$461$	−19.9585	−0.929559	−0.464780	+	0.885426i	$0.653866\pi$
$462$	0	0
$463$	24.9519i	1.15961i	0.814754	+	0.579806i	$0.196872\pi$
$463$	24.9519i	1.15961i	−0.814754	+	0.579806i	$0.803128\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	37.8282i	1.75048i	0.483689	+	0.875240i	$0.339297\pi$
$467$	37.8282i	1.75048i	−0.483689	+	0.875240i	$0.660703\pi$
$468$	0	0
$469$	29.0603	1.34188
$470$	0	0
$471$	−7.36611	−0.339412
$472$	0	0
$473$	0.115716i	0.00532063i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.50768i	0.252179i
$478$	0	0
$479$	12.0042	0.548486	0.274243	−	0.961660i	$-0.411573\pi$
$479$	12.0042	0.548486	0.274243	+	0.961660i	$0.411573\pi$
$480$	0	0
$481$	−28.9169	−1.31850
$482$	0	0
$483$	3.70811i	0.168725i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 13.1552i	− 0.596120i	−0.954547	−	0.298060i	$-0.903660\pi$
$487$	− 13.1552i	− 0.596120i	0.954547	−	0.298060i	$-0.0963396\pi$
$488$	0	0
$489$	27.0421	1.22289
$490$	0	0
$491$	18.8043	0.848627	0.424314	−	0.905515i	$-0.360515\pi$
$491$	18.8043	0.848627	0.424314	+	0.905515i	$0.360515\pi$
$492$	0	0
$493$	31.8861i	1.43608i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	60.5213i	2.71475i
$498$	0	0
$499$	−23.3350	−1.04462	−0.522309	−	0.852756i	$-0.674929\pi$
$499$	−23.3350	−1.04462	−0.522309	+	0.852756i	$0.674929\pi$
$500$	0	0
$501$	41.8184	1.86831
$502$	0	0
$503$	20.0691i	0.894838i	0.894325	+	0.447419i	$0.147657\pi$
$503$	20.0691i	0.894838i	−0.894325	+	0.447419i	$0.852343\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 7.48396i	− 0.332374i
$508$	0	0
$509$	38.3409	1.69943	0.849714	−	0.527243i	$-0.176774\pi$
$509$	38.3409	1.69943	0.849714	+	0.527243i	$0.176774\pi$
$510$	0	0
$511$	−9.49212	−0.419907
$512$	0	0
$513$	10.6796i	0.471515i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.16909i	0.227336i
$518$	0	0
$519$	−39.9010	−1.75146
$520$	0	0
$521$	25.2841	1.10771	0.553857	−	0.832612i	$-0.313155\pi$
$521$	25.2841	1.10771	0.553857	+	0.832612i	$0.313155\pi$
$522$	0	0
$523$	− 1.63921i	− 0.0716778i	−0.999358	−	0.0358389i	$-0.988590\pi$
$523$	− 1.63921i	− 0.0716778i	0.999358	−	0.0358389i	$-0.0114103\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 0.984049i	− 0.0428658i
$528$	0	0
$529$	22.8546	0.993679
$530$	0	0
$531$	8.46454	0.367330
$532$	0	0
$533$	− 18.3698i	− 0.795686i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9.21551i	− 0.397679i
$538$	0	0
$539$	5.99359	0.258162
$540$	0	0
$541$	11.4287	0.491358	0.245679	−	0.969351i	$-0.420989\pi$
$541$	11.4287	0.491358	0.245679	+	0.969351i	$0.420989\pi$
$542$	0	0
$543$	4.06936i	0.174633i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 18.4404i	− 0.788456i	−0.919013	−	0.394228i	$-0.871012\pi$
$547$	− 18.4404i	− 0.788456i	0.919013	−	0.394228i	$-0.128988\pi$
$548$	0	0
$549$	9.10546	0.388612
$550$	0	0
$551$	55.7762	2.37614
$552$	0	0
$553$	− 32.1588i	− 1.36753i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.1154i	0.428603i	0.976768	+	0.214301i	$0.0687475\pi$
$557$	10.1154i	0.428603i	−0.976768	+	0.214301i	$0.931253\pi$
$558$	0	0
$559$	0.835303	0.0353295
$560$	0	0
$561$	5.10823	0.215670
$562$	0	0
$563$	− 8.38115i	− 0.353223i	−0.984281	−	0.176612i	$-0.943486\pi$
$563$	− 8.38115i	− 0.353223i	0.984281	−	0.176612i	$-0.0565137\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 44.4576i	− 1.86705i
$568$	0	0
$569$	24.1369	1.01187	0.505936	−	0.862571i	$-0.331147\pi$
$569$	24.1369	1.01187	0.505936	+	0.862571i	$0.331147\pi$
$570$	0	0
$571$	−33.7164	−1.41099	−0.705494	−	0.708716i	$-0.749275\pi$
$571$	−33.7164	−1.41099	−0.705494	+	0.708716i	$0.749275\pi$
$572$	0	0
$573$	− 17.2336i	− 0.719945i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	45.2735i	1.88476i	0.334543	+	0.942380i	$0.391418\pi$
$577$	45.2735i	1.88476i	−0.334543	+	0.942380i	$0.608582\pi$
$578$	0	0
$579$	−36.0632	−1.49874
$580$	0	0
$581$	32.0005	1.32761
$582$	0	0
$583$	1.31812i	0.0545908i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.1381i	1.16139i	0.814123	+	0.580693i	$0.197218\pi$
$587$	28.1381i	1.16139i	−0.814123	+	0.580693i	$0.802782\pi$
$588$	0	0
$589$	−1.72133	−0.0709262
$590$	0	0
$591$	57.3608	2.35951
$592$	0	0
$593$	− 1.52944i	− 0.0628064i	−0.999507	−	0.0314032i	$-0.990002\pi$
$593$	− 1.52944i	− 0.0628064i	0.999507	−	0.0314032i	$-0.00999760\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.86348i	0.403685i
$598$	0	0
$599$	17.0156	0.695240	0.347620	−	0.937635i	$-0.386990\pi$
$599$	17.0156	0.695240	0.347620	+	0.937635i	$0.386990\pi$
$600$	0	0
$601$	−30.8970	−1.26031	−0.630157	−	0.776468i	$-0.717009\pi$
$601$	−30.8970	−1.26031	−0.630157	+	0.776468i	$0.717009\pi$
$602$	0	0
$603$	16.0969i	0.655517i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 29.5437i	− 1.19914i	−0.800322	−	0.599570i	$-0.795338\pi$
$607$	− 29.5437i	− 1.19914i	0.800322	−	0.599570i	$-0.204662\pi$
$608$	0	0
$609$	−78.2636	−3.17140
$610$	0	0
$611$	37.3134	1.50954
$612$	0	0
$613$	6.17580i	0.249438i	0.992192	+	0.124719i	$0.0398030\pi$
$613$	6.17580i	0.249438i	−0.992192	+	0.124719i	$0.960197\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	43.4297i	1.74841i	0.485554	+	0.874207i	$0.338618\pi$
$617$	43.4297i	1.74841i	−0.485554	+	0.874207i	$0.661382\pi$
$618$	0	0
$619$	−49.3695	−1.98433	−0.992163	−	0.124949i	$-0.960123\pi$
$619$	−49.3695	−1.98433	−0.992163	+	0.124949i	$0.960123\pi$
$620$	0	0
$621$	0.587512	0.0235760
$622$	0	0
$623$	67.5227i	2.70524i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 8.93550i	− 0.356849i
$628$	0	0
$629$	−28.4308	−1.13361
$630$	0	0
$631$	13.2454	0.527292	0.263646	−	0.964620i	$-0.415075\pi$
$631$	13.2454	0.527292	0.263646	+	0.964620i	$0.415075\pi$
$632$	0	0
$633$	19.3281i	0.768224i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 43.2651i	− 1.71423i
$638$	0	0
$639$	−33.5236	−1.32617
$640$	0	0
$641$	−14.7218	−0.581476	−0.290738	−	0.956803i	$-0.593901\pi$
$641$	−14.7218	−0.581476	−0.290738	+	0.956803i	$0.593901\pi$
$642$	0	0
$643$	− 27.1451i	− 1.07050i	−0.844695	−	0.535249i	$-0.820218\pi$
$643$	− 27.1451i	− 1.07050i	0.844695	−	0.535249i	$-0.179782\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.30574i	0.0906480i	0.998972	+	0.0453240i	$0.0144320\pi$
$647$	2.30574i	0.0906480i	−0.998972	+	0.0453240i	$0.985568\pi$
$648$	0	0
$649$	2.02576	0.0795182
$650$	0	0
$651$	2.41532	0.0946640
$652$	0	0
$653$	− 11.2793i	− 0.441391i	−0.975343	−	0.220696i	$-0.929167\pi$
$653$	− 11.2793i	− 0.441391i	0.975343	−	0.220696i	$-0.0708328\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5.25783i	− 0.205127i
$658$	0	0
$659$	45.9675	1.79064	0.895321	−	0.445422i	$-0.146946\pi$
$659$	45.9675	1.79064	0.895321	+	0.445422i	$0.146946\pi$
$660$	0	0
$661$	25.1564	0.978472	0.489236	−	0.872151i	$-0.337276\pi$
$661$	25.1564	0.978472	0.489236	+	0.872151i	$0.337276\pi$
$662$	0	0
$663$	− 36.8741i	− 1.43207i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.06839i	− 0.118809i
$668$	0	0
$669$	−20.5138	−0.793110
$670$	0	0
$671$	2.17915	0.0841252
$672$	0	0
$673$	− 0.823251i	− 0.0317340i	−0.999874	−	0.0158670i	$-0.994949\pi$
$673$	− 0.823251i	− 0.0317340i	0.999874	−	0.0158670i	$-0.00505083\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 23.9993i	− 0.922367i	−0.887305	−	0.461183i	$-0.847425\pi$
$677$	− 23.9993i	− 0.922367i	0.887305	−	0.461183i	$-0.152575\pi$
$678$	0	0
$679$	13.7756	0.528657
$680$	0	0
$681$	26.2033	1.00411
$682$	0	0
$683$	− 32.4123i	− 1.24022i	−0.784514	−	0.620111i	$-0.787088\pi$
$683$	− 32.4123i	− 1.24022i	0.784514	−	0.620111i	$-0.212912\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 61.6294i	− 2.35131i
$688$	0	0
$689$	9.51491	0.362489
$690$	0	0
$691$	−21.5171	−0.818550	−0.409275	−	0.912411i	$-0.634218\pi$
$691$	−21.5171	−0.818550	−0.409275	+	0.912411i	$0.634218\pi$
$692$	0	0
$693$	5.48462i	0.208344i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 18.0610i	− 0.684111i
$698$	0	0
$699$	44.8149	1.69505
$700$	0	0
$701$	−42.0612	−1.58863	−0.794314	−	0.607507i	$-0.792170\pi$
$701$	−42.0612	−1.58863	−0.794314	+	0.607507i	$0.792170\pi$
$702$	0	0
$703$	49.7321i	1.87568i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 39.3556i	− 1.48012i
$708$	0	0
$709$	−6.86278	−0.257737	−0.128869	−	0.991662i	$-0.541135\pi$
$709$	−6.86278	−0.257737	−0.128869	+	0.991662i	$0.541135\pi$
$710$	0	0
$711$	17.8133	0.668049
$712$	0	0
$713$	0.0946948i	0.00354635i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.0890i	0.414127i
$718$	0	0
$719$	38.4708	1.43472	0.717360	−	0.696703i	$-0.245350\pi$
$719$	38.4708	1.43472	0.717360	+	0.696703i	$0.245350\pi$
$720$	0	0
$721$	25.2279	0.939536
$722$	0	0
$723$	− 46.5628i	− 1.73169i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.79323i	0.363211i	0.983371	+	0.181605i	$0.0581294\pi$
$727$	9.79323i	0.363211i	−0.983371	+	0.181605i	$0.941871\pi$
$728$	0	0
$729$	13.9509	0.516700
$730$	0	0
$731$	0.821262	0.0303755
$732$	0	0
$733$	− 30.2252i	− 1.11639i	−0.829709	−	0.558196i	$-0.811494\pi$
$733$	− 30.2252i	− 1.11639i	0.829709	−	0.558196i	$-0.188506\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.85237i	0.141904i
$738$	0	0
$739$	−26.5331	−0.976035	−0.488017	−	0.872834i	$-0.662280\pi$
$739$	−26.5331	−0.976035	−0.488017	+	0.872834i	$0.662280\pi$
$740$	0	0
$741$	−64.5014	−2.36952
$742$	0	0
$743$	1.43832i	0.0527669i	0.999652	+	0.0263835i	$0.00839909\pi$
$743$	1.43832i	0.0527669i	−0.999652	+	0.0263835i	$0.991601\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	17.7256i	0.648544i
$748$	0	0
$749$	23.5871	0.861852
$750$	0	0
$751$	34.4429	1.25684	0.628420	−	0.777874i	$-0.283702\pi$
$751$	34.4429	1.25684	0.628420	+	0.777874i	$0.283702\pi$
$752$	0	0
$753$	− 32.0350i	− 1.16742i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 11.5191i	− 0.418668i	−0.977844	−	0.209334i	$-0.932870\pi$
$757$	− 11.5191i	− 0.418668i	0.977844	−	0.209334i	$-0.0671296\pi$
$758$	0	0
$759$	−0.491565	−0.0178427
$760$	0	0
$761$	−8.53421	−0.309365	−0.154682	−	0.987964i	$-0.549435\pi$
$761$	−8.53421	−0.309365	−0.154682	+	0.987964i	$0.549435\pi$
$762$	0	0
$763$	57.8224i	2.09331i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 14.6231i	− 0.528010i
$768$	0	0
$769$	−32.6044	−1.17574	−0.587872	−	0.808954i	$-0.700034\pi$
$769$	−32.6044	−1.17574	−0.587872	+	0.808954i	$0.700034\pi$
$770$	0	0
$771$	−9.86243	−0.355187
$772$	0	0
$773$	− 29.8390i	− 1.07324i	−0.843826	−	0.536618i	$-0.819702\pi$
$773$	− 29.8390i	− 1.07324i	0.843826	−	0.536618i	$-0.180298\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 69.7827i	− 2.50344i
$778$	0	0
$779$	−31.5930	−1.13194
$780$	0	0
$781$	−8.02299	−0.287085
$782$	0	0
$783$	12.4001i	0.443142i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 52.9437i	− 1.88724i	−0.331031	−	0.943620i	$-0.607397\pi$
$787$	− 52.9437i	− 1.88724i	0.331031	−	0.943620i	$-0.392603\pi$
$788$	0	0
$789$	27.0582	0.963296
$790$	0	0
$791$	−17.0873	−0.607556
$792$	0	0
$793$	− 15.7303i	− 0.558601i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 13.3155i	− 0.471659i	−0.971794	−	0.235830i	$-0.924219\pi$
$797$	− 13.3155i	− 0.471659i	0.971794	−	0.235830i	$-0.0757808\pi$
$798$	0	0
$799$	36.6862	1.29786
$800$	0	0
$801$	−37.4018	−1.32153
$802$	0	0
$803$	− 1.25832i	− 0.0444052i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.4131i	0.859382i
$808$	0	0
$809$	22.8693	0.804041	0.402021	−	0.915631i	$-0.368308\pi$
$809$	22.8693	0.804041	0.402021	+	0.915631i	$0.368308\pi$
$810$	0	0
$811$	−39.3749	−1.38264	−0.691320	−	0.722549i	$-0.742970\pi$
$811$	−39.3749	−1.38264	−0.691320	+	0.722549i	$0.742970\pi$
$812$	0	0
$813$	30.1084i	1.05595i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.43658i	− 0.0502595i
$818$	0	0
$819$	39.5911	1.38342
$820$	0	0
$821$	32.3238	1.12811	0.564053	−	0.825738i	$-0.309241\pi$
$821$	32.3238	1.12811	0.564053	+	0.825738i	$0.309241\pi$
$822$	0	0
$823$	− 44.9994i	− 1.56858i	−0.620393	−	0.784291i	$-0.713027\pi$
$823$	− 44.9994i	− 1.56858i	0.620393	−	0.784291i	$-0.286973\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.40055i	− 0.118249i	−0.998251	−	0.0591243i	$-0.981169\pi$
$827$	− 3.40055i	− 0.118249i	0.998251	−	0.0591243i	$-0.0188308\pi$
$828$	0	0
$829$	−18.3613	−0.637716	−0.318858	−	0.947802i	$-0.603299\pi$
$829$	−18.3613	−0.637716	−0.318858	+	0.947802i	$0.603299\pi$
$830$	0	0
$831$	28.9226	1.00332
$832$	0	0
$833$	− 42.5379i	− 1.47385i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 0.382683i	− 0.0132275i
$838$	0	0
$839$	37.9119	1.30886	0.654432	−	0.756121i	$-0.272908\pi$
$839$	37.9119	1.30886	0.654432	+	0.756121i	$0.272908\pi$
$840$	0	0
$841$	35.7617	1.23316
$842$	0	0
$843$	4.42212i	0.152306i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 45.0127i	− 1.54665i
$848$	0	0
$849$	20.0374	0.687681
$850$	0	0
$851$	2.73589	0.0937851
$852$	0	0
$853$	1.81056i	0.0619923i	0.999520	+	0.0309962i	$0.00986796\pi$
$853$	1.81056i	0.0619923i	−0.999520	+	0.0309962i	$0.990132\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.3384i	1.44625i	0.690717	+	0.723126i	$0.257295\pi$
$857$	42.3384i	1.44625i	−0.690717	+	0.723126i	$0.742705\pi$
$858$	0	0
$859$	−37.6958	−1.28616	−0.643082	−	0.765797i	$-0.722345\pi$
$859$	−37.6958	−1.28616	−0.643082	+	0.765797i	$0.722345\pi$
$860$	0	0
$861$	44.3304	1.51078
$862$	0	0
$863$	− 54.7715i	− 1.86444i	−0.361886	−	0.932222i	$-0.617867\pi$
$863$	− 54.7715i	− 1.86444i	0.361886	−	0.932222i	$-0.382133\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.00339i	0.102001i
$868$	0	0
$869$	4.26313	0.144617
$870$	0	0
$871$	27.8086	0.942257
$872$	0	0
$873$	7.63048i	0.258253i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 3.11646i	− 0.105235i	−0.998615	−	0.0526177i	$-0.983244\pi$
$877$	− 3.11646i	− 0.105235i	0.998615	−	0.0526177i	$-0.0167565\pi$
$878$	0	0
$879$	−41.9437	−1.41473
$880$	0	0
$881$	42.8870	1.44490	0.722450	−	0.691423i	$-0.243016\pi$
$881$	42.8870	1.44490	0.722450	+	0.691423i	$0.243016\pi$
$882$	0	0
$883$	6.18458i	0.208128i	0.994571	+	0.104064i	$0.0331846\pi$
$883$	6.18458i	0.208128i	−0.994571	+	0.104064i	$0.966815\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.84660i	0.263463i	0.991285	+	0.131732i	$0.0420537\pi$
$887$	7.84660i	0.263463i	−0.991285	+	0.131732i	$0.957946\pi$
$888$	0	0
$889$	17.3897	0.583231
$890$	0	0
$891$	5.89352	0.197440
$892$	0	0
$893$	− 64.1727i	− 2.14746i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.54839i	0.118477i
$898$	0	0
$899$	−1.99864	−0.0666583
$900$	0	0
$901$	9.35497	0.311659
$902$	0	0
$903$	2.01577i	0.0670806i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0.385211i	0.0127907i	0.999980	+	0.00639536i	$0.00203572\pi$
$907$	0.385211i	0.0127907i	−0.999980	+	0.00639536i	$0.997964\pi$
$908$	0	0
$909$	21.7996	0.723048
$910$	0	0
$911$	−0.627615	−0.0207938	−0.0103969	−	0.999946i	$-0.503309\pi$
$911$	−0.627615	−0.0207938	−0.0103969	+	0.999946i	$0.503309\pi$
$912$	0	0
$913$	4.24214i	0.140394i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 55.4832i	− 1.83222i
$918$	0	0
$919$	−28.7375	−0.947964	−0.473982	−	0.880535i	$-0.657184\pi$
$919$	−28.7375	−0.947964	−0.473982	+	0.880535i	$0.657184\pi$
$920$	0	0
$921$	−54.0958	−1.78252
$922$	0	0
$923$	57.9145i	1.90628i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	13.9741i	0.458970i
$928$	0	0
$929$	28.6894	0.941269	0.470634	−	0.882328i	$-0.344025\pi$
$929$	28.6894	0.941269	0.470634	+	0.882328i	$0.344025\pi$
$930$	0	0
$931$	−74.4087	−2.43865
$932$	0	0
$933$	53.9153i	1.76511i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	36.3613i	1.18787i	0.804512	+	0.593936i	$0.202427\pi$
$937$	36.3613i	1.18787i	−0.804512	+	0.593936i	$0.797573\pi$
$938$	0	0
$939$	−13.4640	−0.439380
$940$	0	0
$941$	−15.0807	−0.491617	−0.245809	−	0.969318i	$-0.579053\pi$
$941$	−15.0807	−0.491617	−0.245809	+	0.969318i	$0.579053\pi$
$942$	0	0
$943$	1.73801i	0.0565975i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.30531i	− 0.269886i	−0.990853	−	0.134943i	$-0.956915\pi$
$947$	− 8.30531i	− 0.269886i	0.990853	−	0.134943i	$-0.0430852\pi$
$948$	0	0
$949$	−9.08327	−0.294855
$950$	0	0
$951$	17.2800	0.560341
$952$	0	0
$953$	30.7249i	0.995278i	0.867384	+	0.497639i	$0.165800\pi$
$953$	30.7249i	0.995278i	−0.867384	+	0.497639i	$0.834200\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 10.3750i	− 0.335376i
$958$	0	0
$959$	−6.98589	−0.225586
$960$	0	0
$961$	−30.9383	−0.998010
$962$	0	0
$963$	13.0652i	0.421021i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	21.4577i	0.690035i	0.938596	+	0.345017i	$0.112127\pi$
$967$	21.4577i	0.690035i	−0.938596	+	0.345017i	$0.887873\pi$
$968$	0	0
$969$	−63.4172	−2.03725
$970$	0	0
$971$	−53.6420	−1.72146	−0.860728	−	0.509066i	$-0.829991\pi$
$971$	−53.6420	−1.72146	−0.860728	+	0.509066i	$0.829991\pi$
$972$	0	0
$973$	2.95363i	0.0946890i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	54.4679i	1.74258i	0.490767	+	0.871291i	$0.336717\pi$
$977$	54.4679i	1.74258i	−0.490767	+	0.871291i	$0.663283\pi$
$978$	0	0
$979$	−8.95113	−0.286079
$980$	0	0
$981$	−32.0287	−1.02260
$982$	0	0
$983$	2.46333i	0.0785679i	0.999228	+	0.0392840i	$0.0125077\pi$
$983$	2.46333i	0.0785679i	−0.999228	+	0.0392840i	$0.987492\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	90.0454i	2.86618i
$988$	0	0
$989$	−0.0790299	−0.00251300
$990$	0	0
$991$	48.5923	1.54358	0.771792	−	0.635875i	$-0.219361\pi$
$991$	48.5923	1.54358	0.771792	+	0.635875i	$0.219361\pi$
$992$	0	0
$993$	69.7275i	2.21274i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 11.0548i	− 0.350108i	−0.984559	−	0.175054i	$-0.943990\pi$
$997$	− 11.0548i	− 0.350108i	0.984559	−	0.175054i	$-0.0560100\pi$
$998$	0	0
$999$	−11.0564	−0.349807

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2500.2.c.b.1249.8		8
5.2	odd	4		2500.2.a.f.1.8		8
5.3	odd	4		2500.2.a.f.1.1		8
5.4	even	2	inner	2500.2.c.b.1249.1		8
20.3	even	4		10000.2.a.bi.1.8		8
20.7	even	4		10000.2.a.bi.1.1		8
25.3	odd	20		500.2.g.b.201.1		16
25.4	even	10		500.2.i.a.49.2		8
25.6	even	5		500.2.i.a.449.2		8
25.8	odd	20		500.2.g.b.301.1		16
25.17	odd	20		500.2.g.b.301.4		16
25.19	even	10		100.2.i.a.89.1	yes	8
25.21	even	5		100.2.i.a.9.1	✓	8
25.22	odd	20		500.2.g.b.201.4		16
75.44	odd	10		900.2.w.a.289.1		8
75.71	odd	10		900.2.w.a.109.1		8
100.19	odd	10		400.2.y.b.289.2		8
100.71	odd	10		400.2.y.b.209.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.2.i.a.9.1	✓	8	25.21	even	5
100.2.i.a.89.1	yes	8	25.19	even	10
400.2.y.b.209.2		8	100.71	odd	10
400.2.y.b.289.2		8	100.19	odd	10
500.2.g.b.201.1		16	25.3	odd	20
500.2.g.b.201.4		16	25.22	odd	20
500.2.g.b.301.1		16	25.8	odd	20
500.2.g.b.301.4		16	25.17	odd	20
500.2.i.a.49.2		8	25.4	even	10
500.2.i.a.449.2		8	25.6	even	5
900.2.w.a.109.1		8	75.71	odd	10
900.2.w.a.289.1		8	75.44	odd	10
2500.2.a.f.1.1		8	5.3	odd	4
2500.2.a.f.1.8		8	5.2	odd	4
2500.2.c.b.1249.1		8	5.4	even	2	inner
2500.2.c.b.1249.8		8	1.1	even	1	trivial
10000.2.a.bi.1.1		8	20.7	even	4
10000.2.a.bi.1.8		8	20.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2500 = 2^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2500.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.30927i q^{3} +4.21139i q^{7} -2.33275 q^{9} -0.558282 q^{11} +4.02999i q^{13} +3.96225i q^{17} +6.93090 q^{19} -9.72525 q^{21} -0.381287i q^{23} +1.54087i q^{27} +8.04746 q^{29} -0.248356 q^{31} -1.28923i q^{33} +7.17542i q^{37} -9.30636 q^{39} -4.55828 q^{41} -0.207272i q^{43} -9.25893i q^{47} -10.7358 q^{49} -9.14992 q^{51} -2.36102i q^{53} +16.0054i q^{57} -3.62857 q^{59} -3.90332 q^{61} -9.82411i q^{63} -6.90040i q^{67} +0.880496 q^{69} +14.3709 q^{71} +2.25392i q^{73} -2.35114i q^{77} -7.63616 q^{79} -10.5565 q^{81} -7.59857i q^{83} +18.5838i q^{87} +16.0334 q^{89} -16.9719 q^{91} -0.573522i q^{93} -3.27102i q^{97} +1.30233 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{9} - 10 q^{11} + 12 q^{19} - 22 q^{21} + 32 q^{29} - 2 q^{31} + 2 q^{39} - 42 q^{41} + 14 q^{49} - 14 q^{51} + 24 q^{59} - 34 q^{61} + 36 q^{69} + 4 q^{71} - 4 q^{79} - 28 q^{81} + 58 q^{89}+ \cdots + 20 q^{99}+O(q^{100}) \) 8 * q + 2 * q^9 - 10 * q^11 + 12 * q^19 - 22 * q^21 + 32 * q^29 - 2 * q^31 + 2 * q^39 - 42 * q^41 + 14 * q^49 - 14 * q^51 + 24 * q^59 - 34 * q^61 + 36 * q^69 + 4 * q^71 - 4 * q^79 - 28 * q^81 + 58 * q^89 - 18 * q^91 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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