LMFDB - Embedded newform 625.2.a.f.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(1,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	625.a (trivial)

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:	yes
Analytic conductor:	$4.99065012633$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.14884000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.08529$ of defining polynomial
Character	$\chi$	$=$	625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.08529 q^{2} +2.19849 q^{3} +2.34841 q^{4} +4.58448 q^{6} -0.992398 q^{7} +0.726543 q^{8} +1.83337 q^{9} +O(q^{10})$	\(q+2.08529 q^{2} +2.19849 q^{3} +2.34841 q^{4} +4.58448 q^{6} -0.992398 q^{7} +0.726543 q^{8} +1.83337 q^{9} +2.00000 q^{11} +5.16297 q^{12} +3.37406 q^{13} -2.06943 q^{14} -3.18178 q^{16} -2.89451 q^{17} +3.82309 q^{18} -2.58448 q^{19} -2.18178 q^{21} +4.17057 q^{22} +4.54963 q^{23} +1.59730 q^{24} +7.03588 q^{26} -2.56484 q^{27} -2.33056 q^{28} -5.38430 q^{29} +0.136538 q^{31} -8.08800 q^{32} +4.39698 q^{33} -6.03588 q^{34} +4.30550 q^{36} +2.14910 q^{37} -5.38938 q^{38} +7.41785 q^{39} +8.63318 q^{41} -4.54963 q^{42} -4.64398 q^{43} +4.69683 q^{44} +9.48728 q^{46} -9.92630 q^{47} -6.99512 q^{48} -6.01515 q^{49} -6.36356 q^{51} +7.92369 q^{52} -7.56521 q^{53} -5.34841 q^{54} -0.721020 q^{56} -5.68196 q^{57} -11.2278 q^{58} +4.91775 q^{59} -2.76972 q^{61} +0.284720 q^{62} -1.81943 q^{63} -10.5022 q^{64} +9.16896 q^{66} -2.18577 q^{67} -6.79751 q^{68} +10.0023 q^{69} +9.64254 q^{71} +1.33202 q^{72} -0.775929 q^{73} +4.48150 q^{74} -6.06943 q^{76} -1.98480 q^{77} +15.4683 q^{78} +15.8508 q^{79} -11.1389 q^{81} +18.0026 q^{82} +1.77110 q^{83} -5.12372 q^{84} -9.68401 q^{86} -11.8373 q^{87} +1.45309 q^{88} +14.5080 q^{89} -3.34841 q^{91} +10.6844 q^{92} +0.300177 q^{93} -20.6992 q^{94} -17.7814 q^{96} +17.0291 q^{97} -12.5433 q^{98} +3.66673 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9}+O(q^{10}) $ 8 * q + 6 * q^4 + 6 * q^6 + 4 * q^9	$ 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9} + 16 q^{11} + 12 q^{14} - 2 q^{16} + 10 q^{19} + 6 q^{21} + 20 q^{24} + 6 q^{26} + 20 q^{29} + 16 q^{31} + 2 q^{34} - 12 q^{36} + 18 q^{39} + 26 q^{41} + 12 q^{44} + 6 q^{46} - 14 q^{49} - 4 q^{51} - 30 q^{54} + 10 q^{56} + 30 q^{59} + 6 q^{61} - 44 q^{64} + 12 q^{66} + 8 q^{69} + 46 q^{71} + 12 q^{74} - 20 q^{76} + 10 q^{79} - 32 q^{81} - 18 q^{84} - 14 q^{86} + 30 q^{89} - 14 q^{91} - 68 q^{94} - 54 q^{96} + 8 q^{99}+O(q^{100}) $ 8 * q + 6 * q^4 + 6 * q^6 + 4 * q^9 + 16 * q^11 + 12 * q^14 - 2 * q^16 + 10 * q^19 + 6 * q^21 + 20 * q^24 + 6 * q^26 + 20 * q^29 + 16 * q^31 + 2 * q^34 - 12 * q^36 + 18 * q^39 + 26 * q^41 + 12 * q^44 + 6 * q^46 - 14 * q^49 - 4 * q^51 - 30 * q^54 + 10 * q^56 + 30 * q^59 + 6 * q^61 - 44 * q^64 + 12 * q^66 + 8 * q^69 + 46 * q^71 + 12 * q^74 - 20 * q^76 + 10 * q^79 - 32 * q^81 - 18 * q^84 - 14 * q^86 + 30 * q^89 - 14 * q^91 - 68 * q^94 - 54 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	2.08529	1.47452	0.737260	−	0.675610i	$-0.236119\pi$
$2$	2.08529	1.47452	0.737260	+	0.675610i	$0.236119\pi$
$3$	2.19849	1.26930	0.634650	−	0.772800i	$-0.281144\pi$
$3$	2.19849	1.26930	0.634650	+	0.772800i	$0.281144\pi$
$4$	2.34841	1.17421
$5$	0	0
$5$	0	0
$6$	4.58448	1.87161
$7$	−0.992398	−0.375091	−0.187546	−	0.982256i	$-0.560053\pi$
$7$	−0.992398	−0.375091	−0.187546	+	0.982256i	$0.560053\pi$
$8$	0.726543	0.256872
$9$	1.83337	0.611122
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	5.16297	1.49042
$13$	3.37406	0.935796	0.467898	−	0.883782i	$-0.345011\pi$
$13$	3.37406	0.935796	0.467898	+	0.883782i	$0.345011\pi$
$14$	−2.06943	−0.553079
$15$	0	0
$16$	−3.18178	−0.795445
$17$	−2.89451	−0.702022	−0.351011	−	0.936371i	$-0.614162\pi$
$17$	−2.89451	−0.702022	−0.351011	+	0.936371i	$0.614162\pi$
$18$	3.82309	0.901111
$19$	−2.58448	−0.592921	−0.296460	−	0.955045i	$-0.595806\pi$
$19$	−2.58448	−0.592921	−0.296460	+	0.955045i	$0.595806\pi$
$20$	0	0
$21$	−2.18178	−0.476103
$22$	4.17057	0.889169
$23$	4.54963	0.948664	0.474332	−	0.880346i	$-0.342690\pi$
$23$	4.54963	0.948664	0.474332	+	0.880346i	$0.342690\pi$
$24$	1.59730	0.326047
$25$	0	0
$26$	7.03588	1.37985
$27$	−2.56484	−0.493603
$28$	−2.33056	−0.440435
$29$	−5.38430	−0.999839	−0.499919	−	0.866072i	$-0.666637\pi$
$29$	−5.38430	−0.999839	−0.499919	+	0.866072i	$0.666637\pi$
$30$	0	0
$31$	0.136538	0.0245229	0.0122614	−	0.999925i	$-0.496097\pi$
$31$	0.136538	0.0245229	0.0122614	+	0.999925i	$0.496097\pi$
$32$	−8.08800	−1.42977
$33$	4.39698	0.765417
$34$	−6.03588	−1.03515
$35$	0	0
$36$	4.30550	0.717584
$37$	2.14910	0.353311	0.176655	−	0.984273i	$-0.443472\pi$
$37$	2.14910	0.353311	0.176655	+	0.984273i	$0.443472\pi$
$38$	−5.38938	−0.874273
$39$	7.41785	1.18781
$40$	0	0
$41$	8.63318	1.34828	0.674138	−	0.738605i	$-0.264515\pi$
$41$	8.63318	1.34828	0.674138	+	0.738605i	$0.264515\pi$
$42$	−4.54963	−0.702024
$43$	−4.64398	−0.708200	−0.354100	−	0.935208i	$-0.615213\pi$
$43$	−4.64398	−0.708200	−0.354100	+	0.935208i	$0.615213\pi$
$44$	4.69683	0.708073
$45$	0	0
$46$	9.48728	1.39882
$47$	−9.92630	−1.44790	−0.723950	−	0.689853i	$-0.757675\pi$
$47$	−9.92630	−1.44790	−0.723950	+	0.689853i	$0.757675\pi$
$48$	−6.99512	−1.00966
$49$	−6.01515	−0.859306
$50$	0	0
$51$	−6.36356	−0.891077
$52$	7.92369	1.09882
$53$	−7.56521	−1.03916	−0.519581	−	0.854421i	$-0.673912\pi$
$53$	−7.56521	−1.03916	−0.519581	+	0.854421i	$0.673912\pi$
$54$	−5.34841	−0.727827
$55$	0	0
$56$	−0.721020	−0.0963503
$57$	−5.68196	−0.752594
$58$	−11.2278	−1.47428
$59$	4.91775	0.640237	0.320118	−	0.947378i	$-0.396277\pi$
$59$	4.91775	0.640237	0.320118	+	0.947378i	$0.396277\pi$
$60$	0	0
$61$	−2.76972	−0.354626	−0.177313	−	0.984155i	$-0.556740\pi$
$61$	−2.76972	−0.354626	−0.177313	+	0.984155i	$0.556740\pi$
$62$	0.284720	0.0361595
$63$	−1.81943	−0.229227
$64$	−10.5022	−1.31278
$65$	0	0
$66$	9.16896	1.12862
$67$	−2.18577	−0.267035	−0.133517	−	0.991046i	$-0.542627\pi$
$67$	−2.18577	−0.267035	−0.133517	+	0.991046i	$0.542627\pi$
$68$	−6.79751	−0.824319
$69$	10.0023	1.20414
$70$	0	0
$71$	9.64254	1.14436	0.572179	−	0.820128i	$-0.306098\pi$
$71$	9.64254	1.14436	0.572179	+	0.820128i	$0.306098\pi$
$72$	1.33202	0.156980
$73$	−0.775929	−0.0908157	−0.0454078	−	0.998969i	$-0.514459\pi$
$73$	−0.775929	−0.0908157	−0.0454078	+	0.998969i	$0.514459\pi$
$74$	4.48150	0.520963
$75$	0	0
$76$	−6.06943	−0.696212
$77$	−1.98480	−0.226189
$78$	15.4683	1.75144
$79$	15.8508	1.78336	0.891679	−	0.452667i	$-0.149527\pi$
$79$	15.8508	1.78336	0.891679	+	0.452667i	$0.149527\pi$
$80$	0	0
$81$	−11.1389	−1.23765
$82$	18.0026	1.98806
$83$	1.77110	0.194404	0.0972019	−	0.995265i	$-0.469011\pi$
$83$	1.77110	0.194404	0.0972019	+	0.995265i	$0.469011\pi$
$84$	−5.12372	−0.559044
$85$	0	0
$86$	−9.68401	−1.04425
$87$	−11.8373	−1.26909
$88$	1.45309	0.154899
$89$	14.5080	1.53785	0.768923	−	0.639341i	$-0.220793\pi$
$89$	14.5080	1.53785	0.768923	+	0.639341i	$0.220793\pi$
$90$	0	0
$91$	−3.34841	−0.351009
$92$	10.6844	1.11393
$93$	0.300177	0.0311269
$94$	−20.6992	−2.13496
$95$	0	0
$96$	−17.7814	−1.81481
$97$	17.0291	1.72904	0.864522	−	0.502595i	$-0.167621\pi$
$97$	17.0291	1.72904	0.864522	+	0.502595i	$0.167621\pi$
$98$	−12.5433	−1.26706
$99$	3.66673	0.368520
$100$	0	0
$101$	−2.54716	−0.253452	−0.126726	−	0.991938i	$-0.540447\pi$
$101$	−2.54716	−0.253452	−0.126726	+	0.991938i	$0.540447\pi$
$102$	−13.2698	−1.31391
$103$	−10.1654	−1.00163	−0.500815	−	0.865555i	$-0.666966\pi$
$103$	−10.1654	−1.00163	−0.500815	+	0.865555i	$0.666966\pi$
$104$	2.45140	0.240380
$105$	0	0
$106$	−15.7756	−1.53226
$107$	4.81720	0.465697	0.232848	−	0.972513i	$-0.425195\pi$
$107$	4.81720	0.465697	0.232848	+	0.972513i	$0.425195\pi$
$108$	−6.02330	−0.579592
$109$	16.2743	1.55879	0.779397	−	0.626531i	$-0.215526\pi$
$109$	16.2743	1.55879	0.779397	+	0.626531i	$0.215526\pi$
$110$	0	0
$111$	4.72479	0.448457
$112$	3.15759	0.298365
$113$	−6.75704	−0.635649	−0.317825	−	0.948150i	$-0.602952\pi$
$113$	−6.75704	−0.635649	−0.317825	+	0.948150i	$0.602952\pi$
$114$	−11.8485	−1.10971
$115$	0	0
$116$	−12.6446	−1.17402
$117$	6.18589	0.571886
$118$	10.2549	0.944041
$119$	2.87251	0.263322
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−5.77565	−0.522903
$123$	18.9800	1.71137
$124$	0.320647	0.0287950
$125$	0	0
$126$	−3.79403	−0.337999
$127$	1.49081	0.132288	0.0661441	−	0.997810i	$-0.478930\pi$
$127$	1.49081	0.132288	0.0661441	+	0.997810i	$0.478930\pi$
$128$	−5.72414	−0.505948
$129$	−10.2097	−0.898918
$130$	0	0
$131$	14.1147	1.23320	0.616602	−	0.787275i	$-0.288509\pi$
$131$	14.1147	1.23320	0.616602	+	0.787275i	$0.288509\pi$
$132$	10.3259	0.898757
$133$	2.56484	0.222399
$134$	−4.55796	−0.393748
$135$	0	0
$136$	−2.10299	−0.180330
$137$	0.689447	0.0589035	0.0294517	−	0.999566i	$-0.490624\pi$
$137$	0.689447	0.0589035	0.0294517	+	0.999566i	$0.490624\pi$
$138$	20.8577	1.77553
$139$	−16.5719	−1.40561	−0.702803	−	0.711384i	$-0.748069\pi$
$139$	−16.5719	−1.40561	−0.702803	+	0.711384i	$0.748069\pi$
$140$	0	0
$141$	−21.8229	−1.83782
$142$	20.1074	1.68738
$143$	6.74812	0.564307
$144$	−5.83337	−0.486114
$145$	0	0
$146$	−1.61803	−0.133909
$147$	−13.2242	−1.09072
$148$	5.04699	0.414860
$149$	3.21156	0.263101	0.131551	−	0.991309i	$-0.458004\pi$
$149$	3.21156	0.263101	0.131551	+	0.991309i	$0.458004\pi$
$150$	0	0
$151$	17.6863	1.43929	0.719647	−	0.694340i	$-0.244304\pi$
$151$	17.6863	1.43929	0.719647	+	0.694340i	$0.244304\pi$
$152$	−1.87774	−0.152305
$153$	−5.30670	−0.429021
$154$	−4.13887	−0.333519
$155$	0	0
$156$	17.4202	1.39473
$157$	−1.65512	−0.132093	−0.0660465	−	0.997817i	$-0.521039\pi$
$157$	−1.65512	−0.132093	−0.0660465	+	0.997817i	$0.521039\pi$
$158$	33.0535	2.62960
$159$	−16.6321	−1.31901
$160$	0	0
$161$	−4.51505	−0.355836
$162$	−23.2277	−1.82494
$163$	−0.892934	−0.0699400	−0.0349700	−	0.999388i	$-0.511134\pi$
$163$	−0.892934	−0.0699400	−0.0349700	+	0.999388i	$0.511134\pi$
$164$	20.2743	1.58316
$165$	0	0
$166$	3.69325	0.286652
$167$	−5.19558	−0.402046	−0.201023	−	0.979587i	$-0.564427\pi$
$167$	−5.19558	−0.402046	−0.201023	+	0.979587i	$0.564427\pi$
$168$	−1.58516	−0.122297
$169$	−1.61570	−0.124285
$170$	0	0
$171$	−4.73830	−0.362347
$172$	−10.9060	−0.831573
$173$	5.76465	0.438278	0.219139	−	0.975694i	$-0.429675\pi$
$173$	5.76465	0.438278	0.219139	+	0.975694i	$0.429675\pi$
$174$	−24.6842	−1.87130
$175$	0	0
$176$	−6.36356	−0.479671
$177$	10.8116	0.812652
$178$	30.2534	2.26758
$179$	8.66887	0.647942	0.323971	−	0.946067i	$-0.394982\pi$
$179$	8.66887	0.647942	0.323971	+	0.946067i	$0.394982\pi$
$180$	0	0
$181$	14.2909	1.06223	0.531116	−	0.847299i	$-0.321773\pi$
$181$	14.2909	1.06223	0.531116	+	0.847299i	$0.321773\pi$
$182$	−6.98240	−0.517570
$183$	−6.08920	−0.450127
$184$	3.30550	0.243685
$185$	0	0
$186$	0.625955	0.0458972
$187$	−5.78902	−0.423335
$188$	−23.3111	−1.70013
$189$	2.54534	0.185146
$190$	0	0
$191$	−1.26636	−0.0916305	−0.0458153	−	0.998950i	$-0.514589\pi$
$191$	−1.26636	−0.0916305	−0.0458153	+	0.998950i	$0.514589\pi$
$192$	−23.0891	−1.66631
$193$	−21.1730	−1.52406	−0.762031	−	0.647540i	$-0.775798\pi$
$193$	−21.1730	−1.52406	−0.762031	+	0.647540i	$0.775798\pi$
$194$	35.5105	2.54951
$195$	0	0
$196$	−14.1261	−1.00900
$197$	12.2013	0.869308	0.434654	−	0.900597i	$-0.356871\pi$
$197$	12.2013	0.869308	0.434654	+	0.900597i	$0.356871\pi$
$198$	7.64618	0.543390
$199$	10.4065	0.737695	0.368848	−	0.929490i	$-0.379752\pi$
$199$	10.4065	0.737695	0.368848	+	0.929490i	$0.379752\pi$
$200$	0	0
$201$	−4.80540	−0.338947
$202$	−5.31156	−0.373720
$203$	5.34337	0.375031
$204$	−14.9443	−1.04631
$205$	0	0
$206$	−21.1978	−1.47692
$207$	8.34114	0.579749
$208$	−10.7355	−0.744375
$209$	−5.16896	−0.357545
$210$	0	0
$211$	−8.65769	−0.596020	−0.298010	−	0.954563i	$-0.596323\pi$
$211$	−8.65769	−0.596020	−0.298010	+	0.954563i	$0.596323\pi$
$212$	−17.7662	−1.22019
$213$	21.1990	1.45253
$214$	10.0452	0.686679
$215$	0	0
$216$	−1.86346	−0.126793
$217$	−0.135500	−0.00919832
$218$	33.9365	2.29847
$219$	−1.70587	−0.115272
$220$	0	0
$221$	−9.76626	−0.656950
$222$	9.85253	0.661259
$223$	−28.3434	−1.89801	−0.949007	−	0.315256i	$-0.897909\pi$
$223$	−28.3434	−1.89801	−0.949007	+	0.315256i	$0.897909\pi$
$224$	8.02652	0.536295
$225$	0	0
$226$	−14.0904	−0.937277
$227$	−22.2415	−1.47622	−0.738109	−	0.674682i	$-0.764281\pi$
$227$	−22.2415	−1.47622	−0.738109	+	0.674682i	$0.764281\pi$
$228$	−13.3436	−0.883701
$229$	2.47559	0.163592	0.0817958	−	0.996649i	$-0.473934\pi$
$229$	2.47559	0.163592	0.0817958	+	0.996649i	$0.473934\pi$
$230$	0	0
$231$	−4.36356	−0.287101
$232$	−3.91192	−0.256830
$233$	−5.95605	−0.390194	−0.195097	−	0.980784i	$-0.562502\pi$
$233$	−5.95605	−0.390194	−0.195097	+	0.980784i	$0.562502\pi$
$234$	12.8993	0.843256
$235$	0	0
$236$	11.5489	0.751770
$237$	34.8479	2.26362
$238$	5.99000	0.388274
$239$	−7.03243	−0.454890	−0.227445	−	0.973791i	$-0.573037\pi$
$239$	−7.03243	−0.454890	−0.227445	+	0.973791i	$0.573037\pi$
$240$	0	0
$241$	1.17976	0.0759953	0.0379976	−	0.999278i	$-0.487902\pi$
$241$	1.17976	0.0759953	0.0379976	+	0.999278i	$0.487902\pi$
$242$	−14.5970	−0.938330
$243$	−16.7942	−1.07735
$244$	−6.50444	−0.416404
$245$	0	0
$246$	39.5787	2.52344
$247$	−8.72020	−0.554853
$248$	0.0992004	0.00629923
$249$	3.89375	0.246757
$250$	0	0
$251$	4.60867	0.290897	0.145448	−	0.989366i	$-0.453538\pi$
$251$	4.60867	0.290897	0.145448	+	0.989366i	$0.453538\pi$
$252$	−4.27277	−0.269159
$253$	9.09927	0.572066
$254$	3.10877	0.195062
$255$	0	0
$256$	9.06799	0.566750
$257$	9.75542	0.608526	0.304263	−	0.952588i	$-0.401590\pi$
$257$	9.75542	0.608526	0.304263	+	0.952588i	$0.401590\pi$
$258$	−21.2902	−1.32547
$259$	−2.13277	−0.132524
$260$	0	0
$261$	−9.87138	−0.611023
$262$	29.4331	1.81838
$263$	−0.995828	−0.0614054	−0.0307027	−	0.999529i	$-0.509775\pi$
$263$	−0.995828	−0.0614054	−0.0307027	+	0.999529i	$0.509775\pi$
$264$	3.19460	0.196614
$265$	0	0
$266$	5.34841	0.327932
$267$	31.8958	1.95199
$268$	−5.13310	−0.313554
$269$	−3.28853	−0.200506	−0.100253	−	0.994962i	$-0.531965\pi$
$269$	−3.28853	−0.200506	−0.100253	+	0.994962i	$0.531965\pi$
$270$	0	0
$271$	12.1500	0.738063	0.369031	−	0.929417i	$-0.379689\pi$
$271$	12.1500	0.738063	0.369031	+	0.929417i	$0.379689\pi$
$272$	9.20970	0.558420
$273$	−7.36146	−0.445536
$274$	1.43769	0.0868543
$275$	0	0
$276$	23.4896	1.41391
$277$	11.8666	0.712993	0.356496	−	0.934297i	$-0.383971\pi$
$277$	11.8666	0.712993	0.356496	+	0.934297i	$0.383971\pi$
$278$	−34.5571	−2.07259
$279$	0.250324	0.0149865
$280$	0	0
$281$	−24.6416	−1.47000	−0.734998	−	0.678070i	$-0.762817\pi$
$281$	−24.6416	−1.47000	−0.734998	+	0.678070i	$0.762817\pi$
$282$	−45.5069	−2.70990
$283$	3.36343	0.199935	0.0999675	−	0.994991i	$-0.468126\pi$
$283$	3.36343	0.199935	0.0999675	+	0.994991i	$0.468126\pi$
$284$	22.6447	1.34371
$285$	0	0
$286$	14.0718	0.832081
$287$	−8.56755	−0.505727
$288$	−14.8283	−0.873764
$289$	−8.62180	−0.507165
$290$	0	0
$291$	37.4383	2.19467
$292$	−1.82220	−0.106636
$293$	8.96340	0.523647	0.261824	−	0.965116i	$-0.415676\pi$
$293$	8.96340	0.523647	0.261824	+	0.965116i	$0.415676\pi$
$294$	−27.5763	−1.60828
$295$	0	0
$296$	1.56142	0.0907555
$297$	−5.12967	−0.297654
$298$	6.69702	0.387948
$299$	15.3507	0.887756
$300$	0	0
$301$	4.60867	0.265640
$302$	36.8811	2.12227
$303$	−5.59991	−0.321706
$304$	8.22325	0.471636
$305$	0	0
$306$	−11.0660	−0.632600
$307$	−9.48133	−0.541128	−0.270564	−	0.962702i	$-0.587210\pi$
$307$	−9.48133	−0.541128	−0.270564	+	0.962702i	$0.587210\pi$
$308$	−4.66112	−0.265592
$309$	−22.3486	−1.27137
$310$	0	0
$311$	29.3320	1.66327	0.831633	−	0.555325i	$-0.187406\pi$
$311$	29.3320	1.66327	0.831633	+	0.555325i	$0.187406\pi$
$312$	5.38938	0.305114
$313$	−18.8901	−1.06773	−0.533865	−	0.845570i	$-0.679261\pi$
$313$	−18.8901	−1.06773	−0.533865	+	0.845570i	$0.679261\pi$
$314$	−3.45140	−0.194774
$315$	0	0
$316$	37.2243	2.09403
$317$	−22.7893	−1.27998	−0.639988	−	0.768385i	$-0.721061\pi$
$317$	−22.7893	−1.27998	−0.639988	+	0.768385i	$0.721061\pi$
$318$	−34.6826	−1.94490
$319$	−10.7686	−0.602925
$320$	0	0
$321$	10.5906	0.591109
$322$	−9.41516	−0.524687
$323$	7.48081	0.416244
$324$	−26.1587	−1.45326
$325$	0	0
$326$	−1.86202	−0.103128
$327$	35.7789	1.97858
$328$	6.27237	0.346334
$329$	9.85084	0.543094
$330$	0	0
$331$	2.96299	0.162861	0.0814304	−	0.996679i	$-0.474051\pi$
$331$	2.96299	0.162861	0.0814304	+	0.996679i	$0.474051\pi$
$332$	4.15928	0.228270
$333$	3.94010	0.215916
$334$	−10.8343	−0.592824
$335$	0	0
$336$	6.94194	0.378714
$337$	18.8123	1.02477	0.512385	−	0.858756i	$-0.328762\pi$
$337$	18.8123	1.02477	0.512385	+	0.858756i	$0.328762\pi$
$338$	−3.36920	−0.183261
$339$	−14.8553	−0.806829
$340$	0	0
$341$	0.273075	0.0147879
$342$	−9.88071	−0.534287
$343$	12.9162	0.697410
$344$	−3.37405	−0.181916
$345$	0	0
$346$	12.0209	0.646249
$347$	22.7382	1.22065	0.610325	−	0.792151i	$-0.291039\pi$
$347$	22.7382	1.22065	0.610325	+	0.792151i	$0.291039\pi$
$348$	−27.7989	−1.49018
$349$	1.93849	0.103765	0.0518824	−	0.998653i	$-0.483478\pi$
$349$	1.93849	0.103765	0.0518824	+	0.998653i	$0.483478\pi$
$350$	0	0
$351$	−8.65392	−0.461912
$352$	−16.1760	−0.862184
$353$	−5.24945	−0.279400	−0.139700	−	0.990194i	$-0.544614\pi$
$353$	−5.24945	−0.279400	−0.139700	+	0.990194i	$0.544614\pi$
$354$	22.5453	1.19827
$355$	0	0
$356$	34.0708	1.80575
$357$	6.31519	0.334235
$358$	18.0771	0.955402
$359$	22.5937	1.19245	0.596226	−	0.802817i	$-0.296666\pi$
$359$	22.5937	1.19245	0.596226	+	0.802817i	$0.296666\pi$
$360$	0	0
$361$	−12.3205	−0.648445
$362$	29.8005	1.56628
$363$	−15.3894	−0.807736
$364$	−7.86346	−0.412157
$365$	0	0
$366$	−12.6977	−0.663720
$367$	7.29872	0.380990	0.190495	−	0.981688i	$-0.438991\pi$
$367$	7.29872	0.380990	0.190495	+	0.981688i	$0.438991\pi$
$368$	−14.4759	−0.754610
$369$	15.8278	0.823961
$370$	0	0
$371$	7.50770	0.389781
$372$	0.704940	0.0365494
$373$	22.3074	1.15503	0.577516	−	0.816380i	$-0.304022\pi$
$373$	22.3074	1.15503	0.577516	+	0.816380i	$0.304022\pi$
$374$	−12.0718	−0.624216
$375$	0	0
$376$	−7.21188	−0.371924
$377$	−18.1669	−0.935645
$378$	5.30776	0.273002
$379$	−32.9466	−1.69235	−0.846177	−	0.532903i	$-0.821101\pi$
$379$	−32.9466	−1.69235	−0.846177	+	0.532903i	$0.821101\pi$
$380$	0	0
$381$	3.27754	0.167913
$382$	−2.64072	−0.135111
$383$	20.7002	1.05773	0.528865	−	0.848706i	$-0.322618\pi$
$383$	20.7002	1.05773	0.528865	+	0.848706i	$0.322618\pi$
$384$	−12.5845	−0.642199
$385$	0	0
$386$	−44.1516	−2.24726
$387$	−8.51410	−0.432796
$388$	39.9914	2.03026
$389$	−1.14446	−0.0580263	−0.0290132	−	0.999579i	$-0.509236\pi$
$389$	−1.14446	−0.0580263	−0.0290132	+	0.999579i	$0.509236\pi$
$390$	0	0
$391$	−13.1690	−0.665983
$392$	−4.37026	−0.220731
$393$	31.0310	1.56531
$394$	25.4432	1.28181
$395$	0	0
$396$	8.61100	0.432719
$397$	4.68513	0.235140	0.117570	−	0.993065i	$-0.462490\pi$
$397$	4.68513	0.235140	0.117570	+	0.993065i	$0.462490\pi$
$398$	21.7005	1.08775
$399$	5.63877	0.282292
$400$	0	0
$401$	−24.0851	−1.20275	−0.601376	−	0.798966i	$-0.705381\pi$
$401$	−24.0851	−1.20275	−0.601376	+	0.798966i	$0.705381\pi$
$402$	−10.0206	−0.499784
$403$	0.460687	0.0229484
$404$	−5.98179	−0.297605
$405$	0	0
$406$	11.1424	0.552990
$407$	4.29821	0.213054
$408$	−4.62340	−0.228892
$409$	−1.89934	−0.0939165	−0.0469583	−	0.998897i	$-0.514953\pi$
$409$	−1.89934	−0.0939165	−0.0469583	+	0.998897i	$0.514953\pi$
$410$	0	0
$411$	1.51574	0.0747661
$412$	−23.8726	−1.17612
$413$	−4.88037	−0.240147
$414$	17.3937	0.854852
$415$	0	0
$416$	−27.2894	−1.33797
$417$	−36.4331	−1.78414
$418$	−10.7788	−0.527207
$419$	−2.32806	−0.113733	−0.0568666	−	0.998382i	$-0.518111\pi$
$419$	−2.32806	−0.113733	−0.0568666	+	0.998382i	$0.518111\pi$
$420$	0	0
$421$	−23.9501	−1.16725	−0.583627	−	0.812022i	$-0.698367\pi$
$421$	−23.9501	−1.16725	−0.583627	+	0.812022i	$0.698367\pi$
$422$	−18.0537	−0.878842
$423$	−18.1985	−0.884843
$424$	−5.49645	−0.266931
$425$	0	0
$426$	44.2060	2.14179
$427$	2.74866	0.133017
$428$	11.3128	0.546824
$429$	14.8357	0.716274
$430$	0	0
$431$	1.19227	0.0574294	0.0287147	−	0.999588i	$-0.490859\pi$
$431$	1.19227	0.0574294	0.0287147	+	0.999588i	$0.490859\pi$
$432$	8.16074	0.392634
$433$	−25.6138	−1.23092	−0.615461	−	0.788167i	$-0.711030\pi$
$433$	−25.6138	−1.23092	−0.615461	+	0.788167i	$0.711030\pi$
$434$	−0.282556	−0.0135631
$435$	0	0
$436$	38.2187	1.83035
$437$	−11.7584	−0.562483
$438$	−3.55723	−0.169971
$439$	19.3741	0.924676	0.462338	−	0.886704i	$-0.347011\pi$
$439$	19.3741	0.924676	0.462338	+	0.886704i	$0.347011\pi$
$440$	0	0
$441$	−11.0280	−0.525141
$442$	−20.3654	−0.968685
$443$	2.46263	0.117003	0.0585016	−	0.998287i	$-0.481368\pi$
$443$	2.46263	0.117003	0.0585016	+	0.998287i	$0.481368\pi$
$444$	11.0958	0.526582
$445$	0	0
$446$	−59.1040	−2.79866
$447$	7.06059	0.333955
$448$	10.4224	0.492412
$449$	−14.3585	−0.677618	−0.338809	−	0.940855i	$-0.610024\pi$
$449$	−14.3585	−0.677618	−0.338809	+	0.940855i	$0.610024\pi$
$450$	0	0
$451$	17.2664	0.813041
$452$	−15.8683	−0.746384
$453$	38.8833	1.82690
$454$	−46.3798	−2.17671
$455$	0	0
$456$	−4.12819	−0.193320
$457$	−25.1964	−1.17864	−0.589319	−	0.807901i	$-0.700604\pi$
$457$	−25.1964	−1.17864	−0.589319	+	0.807901i	$0.700604\pi$
$458$	5.16231	0.241219
$459$	7.42395	0.346520
$460$	0	0
$461$	28.8255	1.34254	0.671269	−	0.741214i	$-0.265749\pi$
$461$	28.8255	1.34254	0.671269	+	0.741214i	$0.265749\pi$
$462$	−9.09927	−0.423336
$463$	31.8796	1.48157	0.740786	−	0.671742i	$-0.234453\pi$
$463$	31.8796	1.48157	0.740786	+	0.671742i	$0.234453\pi$
$464$	17.1316	0.795317
$465$	0	0
$466$	−12.4201	−0.575348
$467$	43.0996	1.99441	0.997206	−	0.0747039i	$-0.0238012\pi$
$467$	43.0996	1.99441	0.997206	+	0.0747039i	$0.0238012\pi$
$468$	14.5270	0.671512
$469$	2.16916	0.100162
$470$	0	0
$471$	−3.63877	−0.167666
$472$	3.57295	0.164459
$473$	−9.28795	−0.427060
$474$	72.6679	3.33775
$475$	0	0
$476$	6.74584	0.309195
$477$	−13.8698	−0.635054
$478$	−14.6646	−0.670744
$479$	21.0263	0.960717	0.480359	−	0.877072i	$-0.340506\pi$
$479$	21.0263	0.960717	0.480359	+	0.877072i	$0.340506\pi$
$480$	0	0
$481$	7.25121	0.330627
$482$	2.46014	0.112057
$483$	−9.92630	−0.451662
$484$	−16.4389	−0.747223
$485$	0	0
$486$	−35.0207	−1.58857
$487$	−27.9190	−1.26513	−0.632565	−	0.774507i	$-0.717998\pi$
$487$	−27.9190	−1.26513	−0.632565	+	0.774507i	$0.717998\pi$
$488$	−2.01232	−0.0910933
$489$	−1.96311	−0.0887748
$490$	0	0
$491$	14.9611	0.675183	0.337591	−	0.941293i	$-0.390388\pi$
$491$	14.9611	0.675183	0.337591	+	0.941293i	$0.390388\pi$
$492$	44.5728	2.00950
$493$	15.5849	0.701909
$494$	−18.1841	−0.818142
$495$	0	0
$496$	−0.434433	−0.0195066
$497$	−9.56924	−0.429239
$498$	8.11959	0.363847
$499$	−44.3253	−1.98427	−0.992137	−	0.125160i	$-0.960056\pi$
$499$	−44.3253	−1.98427	−0.992137	+	0.125160i	$0.960056\pi$
$500$	0	0
$501$	−11.4224	−0.510316
$502$	9.61040	0.428933
$503$	23.6212	1.05322	0.526609	−	0.850108i	$-0.323463\pi$
$503$	23.6212	1.05322	0.526609	+	0.850108i	$0.323463\pi$
$504$	−1.32189	−0.0588818
$505$	0	0
$506$	18.9746	0.843522
$507$	−3.55211	−0.157755
$508$	3.50105	0.155334
$509$	−26.7154	−1.18414	−0.592070	−	0.805886i	$-0.701689\pi$
$509$	−26.7154	−1.18414	−0.592070	+	0.805886i	$0.701689\pi$
$510$	0	0
$511$	0.770031	0.0340642
$512$	30.3576	1.34163
$513$	6.62877	0.292667
$514$	20.3428	0.897283
$515$	0	0
$516$	−23.9767	−1.05552
$517$	−19.8526	−0.873116
$518$	−4.44743	−0.195409
$519$	12.6735	0.556306
$520$	0	0
$521$	32.7073	1.43293	0.716466	−	0.697622i	$-0.245759\pi$
$521$	32.7073	1.43293	0.716466	+	0.697622i	$0.245759\pi$
$522$	−20.5846	−0.900966
$523$	−0.235966	−0.0103181	−0.00515904	−	0.999987i	$-0.501642\pi$
$523$	−0.235966	−0.0103181	−0.00515904	+	0.999987i	$0.501642\pi$
$524$	33.1471	1.44804
$525$	0	0
$526$	−2.07658	−0.0905434
$527$	−0.395210	−0.0172156
$528$	−13.9902	−0.608847
$529$	−2.30084	−0.100037
$530$	0	0
$531$	9.01604	0.391263
$532$	6.02330	0.261143
$533$	29.1289	1.26171
$534$	66.5117	2.87824
$535$	0	0
$536$	−1.58806	−0.0685936
$537$	19.0584	0.822432
$538$	−6.85753	−0.295649
$539$	−12.0303	−0.518181
$540$	0	0
$541$	−33.5572	−1.44274	−0.721369	−	0.692551i	$-0.756487\pi$
$541$	−33.5572	−1.44274	−0.721369	+	0.692551i	$0.756487\pi$
$542$	25.3363	1.08829
$543$	31.4183	1.34829
$544$	23.4108	1.00373
$545$	0	0
$546$	−15.3507	−0.656951
$547$	−38.5125	−1.64668	−0.823338	−	0.567552i	$-0.807891\pi$
$547$	−38.5125	−1.64668	−0.823338	+	0.567552i	$0.807891\pi$
$548$	1.61911	0.0691648
$549$	−5.07790	−0.216720
$550$	0	0
$551$	13.9156	0.592825
$552$	7.26712	0.309309
$553$	−15.7303	−0.668922
$554$	24.7452	1.05132
$555$	0	0
$556$	−38.9176	−1.65047
$557$	−4.33445	−0.183657	−0.0918283	−	0.995775i	$-0.529271\pi$
$557$	−4.33445	−0.183657	−0.0918283	+	0.995775i	$0.529271\pi$
$558$	0.521996	0.0220978
$559$	−15.6691	−0.662731
$560$	0	0
$561$	−12.7271	−0.537339
$562$	−51.3848	−2.16754
$563$	34.9018	1.47094	0.735468	−	0.677559i	$-0.236962\pi$
$563$	34.9018	1.47094	0.735468	+	0.677559i	$0.236962\pi$
$564$	−51.2492	−2.15798
$565$	0	0
$566$	7.01371	0.294808
$567$	11.0542	0.464233
$568$	7.00572	0.293953
$569$	−41.9646	−1.75925	−0.879623	−	0.475671i	$-0.842205\pi$
$569$	−41.9646	−1.75925	−0.879623	+	0.475671i	$0.842205\pi$
$570$	0	0
$571$	−13.8332	−0.578900	−0.289450	−	0.957193i	$-0.593472\pi$
$571$	−13.8332	−0.578900	−0.289450	+	0.957193i	$0.593472\pi$
$572$	15.8474	0.662613
$573$	−2.78408	−0.116307
$574$	−17.8658	−0.745704
$575$	0	0
$576$	−19.2544	−0.802268
$577$	−12.7793	−0.532008	−0.266004	−	0.963972i	$-0.585704\pi$
$577$	−12.7793	−0.532008	−0.266004	+	0.963972i	$0.585704\pi$
$578$	−17.9789	−0.747824
$579$	−46.5486	−1.93449
$580$	0	0
$581$	−1.75764	−0.0729191
$582$	78.0696	3.23609
$583$	−15.1304	−0.626638
$584$	−0.563746	−0.0233280
$585$	0	0
$586$	18.6912	0.772128
$587$	−12.1870	−0.503009	−0.251505	−	0.967856i	$-0.580925\pi$
$587$	−12.1870	−0.503009	−0.251505	+	0.967856i	$0.580925\pi$
$588$	−31.0560	−1.28073
$589$	−0.352879	−0.0145401
$590$	0	0
$591$	26.8245	1.10341
$592$	−6.83798	−0.281039
$593$	−31.2580	−1.28361	−0.641807	−	0.766866i	$-0.721815\pi$
$593$	−31.2580	−1.28361	−0.641807	+	0.766866i	$0.721815\pi$
$594$	−10.6968	−0.438896
$595$	0	0
$596$	7.54208	0.308936
$597$	22.8785	0.936356
$598$	32.0107	1.30901
$599$	−33.3707	−1.36349	−0.681746	−	0.731589i	$-0.738779\pi$
$599$	−33.3707	−1.36349	−0.681746	+	0.731589i	$0.738779\pi$
$600$	0	0
$601$	−46.8052	−1.90922	−0.954611	−	0.297854i	$-0.903729\pi$
$601$	−46.8052	−1.90922	−0.954611	+	0.297854i	$0.903729\pi$
$602$	9.61040	0.391691
$603$	−4.00732	−0.163191
$604$	41.5349	1.69003
$605$	0	0
$606$	−11.6774	−0.474362
$607$	30.7401	1.24770	0.623851	−	0.781543i	$-0.285567\pi$
$607$	30.7401	1.24770	0.623851	+	0.781543i	$0.285567\pi$
$608$	20.9033	0.847741
$609$	11.7473	0.476027
$610$	0	0
$611$	−33.4919	−1.35494
$612$	−12.4623	−0.503760
$613$	−38.2895	−1.54650	−0.773248	−	0.634103i	$-0.781369\pi$
$613$	−38.2895	−1.54650	−0.773248	+	0.634103i	$0.781369\pi$
$614$	−19.7713	−0.797904
$615$	0	0
$616$	−1.44204	−0.0581014
$617$	0.425306	0.0171222	0.00856109	−	0.999963i	$-0.497275\pi$
$617$	0.425306	0.0171222	0.00856109	+	0.999963i	$0.497275\pi$
$618$	−46.6032	−1.87466
$619$	7.51147	0.301912	0.150956	−	0.988541i	$-0.451765\pi$
$619$	7.51147	0.301912	0.150956	+	0.988541i	$0.451765\pi$
$620$	0	0
$621$	−11.6691	−0.468263
$622$	61.1656	2.45252
$623$	−14.3977	−0.576833
$624$	−23.6020	−0.944834
$625$	0	0
$626$	−39.3912	−1.57439
$627$	−11.3639	−0.453831
$628$	−3.88691	−0.155105
$629$	−6.22061	−0.248032
$630$	0	0
$631$	−11.2716	−0.448714	−0.224357	−	0.974507i	$-0.572028\pi$
$631$	−11.2716	−0.448714	−0.224357	+	0.974507i	$0.572028\pi$
$632$	11.5163	0.458094
$633$	−19.0338	−0.756528
$634$	−47.5222	−1.88735
$635$	0	0
$636$	−39.0589	−1.54879
$637$	−20.2955	−0.804136
$638$	−22.4556	−0.889025
$639$	17.6783	0.699343
$640$	0	0
$641$	26.0825	1.03020	0.515099	−	0.857131i	$-0.327755\pi$
$641$	26.0825	1.03020	0.515099	+	0.857131i	$0.327755\pi$
$642$	22.0844	0.871601
$643$	−31.9492	−1.25995	−0.629977	−	0.776614i	$-0.716936\pi$
$643$	−31.9492	−1.25995	−0.629977	+	0.776614i	$0.716936\pi$
$644$	−10.6032	−0.417825
$645$	0	0
$646$	15.5996	0.613759
$647$	−7.39433	−0.290701	−0.145351	−	0.989380i	$-0.546431\pi$
$647$	−7.39433	−0.290701	−0.145351	+	0.989380i	$0.546431\pi$
$648$	−8.09286	−0.317918
$649$	9.83550	0.386077
$650$	0	0
$651$	−0.297895	−0.0116754
$652$	−2.09698	−0.0821240
$653$	18.6853	0.731212	0.365606	−	0.930770i	$-0.380862\pi$
$653$	18.6853	0.731212	0.365606	+	0.930770i	$0.380862\pi$
$654$	74.6091	2.91745
$655$	0	0
$656$	−27.4689	−1.07248
$657$	−1.42256	−0.0554994
$658$	20.5418	0.800803
$659$	9.80157	0.381815	0.190907	−	0.981608i	$-0.438857\pi$
$659$	9.80157	0.381815	0.190907	+	0.981608i	$0.438857\pi$
$660$	0	0
$661$	−28.1585	−1.09524	−0.547619	−	0.836728i	$-0.684466\pi$
$661$	−28.1585	−1.09524	−0.547619	+	0.836728i	$0.684466\pi$
$662$	6.17868	0.240141
$663$	−21.4710	−0.833866
$664$	1.28678	0.0499368
$665$	0	0
$666$	8.21622	0.318372
$667$	−24.4966	−0.948511
$668$	−12.2014	−0.472085
$669$	−62.3127	−2.40915
$670$	0	0
$671$	−5.53943	−0.213847
$672$	17.6462	0.680718
$673$	−39.0253	−1.50432	−0.752158	−	0.658983i	$-0.770987\pi$
$673$	−39.0253	−1.50432	−0.752158	+	0.658983i	$0.770987\pi$
$674$	39.2290	1.51104
$675$	0	0
$676$	−3.79434	−0.145936
$677$	5.03533	0.193523	0.0967617	−	0.995308i	$-0.469152\pi$
$677$	5.03533	0.193523	0.0967617	+	0.995308i	$0.469152\pi$
$678$	−30.9775	−1.18969
$679$	−16.8997	−0.648549
$680$	0	0
$681$	−48.8977	−1.87376
$682$	0.569440	0.0218050
$683$	−30.3312	−1.16059	−0.580295	−	0.814406i	$-0.697063\pi$
$683$	−30.3312	−1.16059	−0.580295	+	0.814406i	$0.697063\pi$
$684$	−11.1275	−0.425470
$685$	0	0
$686$	26.9340	1.02834
$687$	5.44257	0.207647
$688$	14.7761	0.563334
$689$	−25.5255	−0.972444
$690$	0	0
$691$	−21.2329	−0.807739	−0.403869	−	0.914817i	$-0.632335\pi$
$691$	−21.2329	−0.807739	−0.403869	+	0.914817i	$0.632335\pi$
$692$	13.5378	0.514629
$693$	−3.63886	−0.138229
$694$	47.4156	1.79987
$695$	0	0
$696$	−8.60032	−0.325994
$697$	−24.9888	−0.946520
$698$	4.04230	0.153003
$699$	−13.0943	−0.495273
$700$	0	0
$701$	32.7698	1.23770	0.618849	−	0.785510i	$-0.287599\pi$
$701$	32.7698	1.23770	0.618849	+	0.785510i	$0.287599\pi$
$702$	−18.0459	−0.681098
$703$	−5.55432	−0.209485
$704$	−21.0045	−0.791636
$705$	0	0
$706$	−10.9466	−0.411981
$707$	2.52780	0.0950676
$708$	25.3902	0.954222
$709$	19.8459	0.745330	0.372665	−	0.927966i	$-0.378444\pi$
$709$	19.8459	0.745330	0.372665	+	0.927966i	$0.378444\pi$
$710$	0	0
$711$	29.0604	1.08985
$712$	10.5407	0.395029
$713$	0.621196	0.0232640
$714$	13.1690	0.492836
$715$	0	0
$716$	20.3581	0.760817
$717$	−15.4607	−0.577392
$718$	47.1144	1.75829
$719$	24.2201	0.903258	0.451629	−	0.892206i	$-0.350843\pi$
$719$	24.2201	0.903258	0.451629	+	0.892206i	$0.350843\pi$
$720$	0	0
$721$	10.0882	0.375703
$722$	−25.6917	−0.956144
$723$	2.59370	0.0964608
$724$	33.5609	1.24728
$725$	0	0
$726$	−32.0914	−1.19102
$727$	−5.86510	−0.217524	−0.108762	−	0.994068i	$-0.534689\pi$
$727$	−5.86510	−0.217524	−0.108762	+	0.994068i	$0.534689\pi$
$728$	−2.43277	−0.0901643
$729$	−3.50531	−0.129826
$730$	0	0
$731$	13.4420	0.497172
$732$	−14.3000	−0.528542
$733$	−34.5015	−1.27434	−0.637171	−	0.770723i	$-0.719895\pi$
$733$	−34.5015	−1.27434	−0.637171	+	0.770723i	$0.719895\pi$
$734$	15.2199	0.561777
$735$	0	0
$736$	−36.7974	−1.35637
$737$	−4.37155	−0.161028
$738$	33.0054	1.21495
$739$	39.5712	1.45565	0.727826	−	0.685762i	$-0.240531\pi$
$739$	39.5712	1.45565	0.727826	+	0.685762i	$0.240531\pi$
$740$	0	0
$741$	−19.1713	−0.704275
$742$	15.6557	0.574739
$743$	29.7058	1.08980	0.544900	−	0.838501i	$-0.316567\pi$
$743$	29.7058	1.08980	0.544900	+	0.838501i	$0.316567\pi$
$744$	0.218091	0.00799562
$745$	0	0
$746$	46.5172	1.70312
$747$	3.24708	0.118804
$748$	−13.5950	−0.497083
$749$	−4.78058	−0.174679
$750$	0	0
$751$	26.8870	0.981122	0.490561	−	0.871407i	$-0.336792\pi$
$751$	26.8870	0.981122	0.490561	+	0.871407i	$0.336792\pi$
$752$	31.5833	1.15172
$753$	10.1321	0.369235
$754$	−37.8833	−1.37963
$755$	0	0
$756$	5.97751	0.217400
$757$	−44.6792	−1.62389	−0.811947	−	0.583731i	$-0.801592\pi$
$757$	−44.6792	−1.62389	−0.811947	+	0.583731i	$0.801592\pi$
$758$	−68.7031	−2.49541
$759$	20.0047	0.726123
$760$	0	0
$761$	20.3080	0.736163	0.368081	−	0.929794i	$-0.380015\pi$
$761$	20.3080	0.736163	0.368081	+	0.929794i	$0.380015\pi$
$762$	6.83461	0.247592
$763$	−16.1506	−0.584690
$764$	−2.97393	−0.107593
$765$	0	0
$766$	43.1658	1.55964
$767$	16.5928	0.599131
$768$	19.9359	0.719375
$769$	26.0577	0.939665	0.469832	−	0.882756i	$-0.344314\pi$
$769$	26.0577	0.939665	0.469832	+	0.882756i	$0.344314\pi$
$770$	0	0
$771$	21.4472	0.772402
$772$	−49.7229	−1.78956
$773$	−13.7305	−0.493851	−0.246926	−	0.969034i	$-0.579420\pi$
$773$	−13.7305	−0.493851	−0.246926	+	0.969034i	$0.579420\pi$
$774$	−17.7543	−0.638166
$775$	0	0
$776$	12.3724	0.444142
$777$	−4.68887	−0.168212
$778$	−2.38652	−0.0855609
$779$	−22.3123	−0.799421
$780$	0	0
$781$	19.2851	0.690074
$782$	−27.4610	−0.982005
$783$	13.8098	0.493523
$784$	19.1389	0.683531
$785$	0	0
$786$	64.7085	2.30807
$787$	19.6660	0.701018	0.350509	−	0.936559i	$-0.386009\pi$
$787$	19.6660	0.701018	0.350509	+	0.936559i	$0.386009\pi$
$788$	28.6538	1.02075
$789$	−2.18932	−0.0779418
$790$	0	0
$791$	6.70568	0.238427
$792$	2.66404	0.0946624
$793$	−9.34520	−0.331858
$794$	9.76984	0.346719
$795$	0	0
$796$	24.4387	0.866207
$797$	−10.9441	−0.387660	−0.193830	−	0.981035i	$-0.562091\pi$
$797$	−10.9441	−0.387660	−0.193830	+	0.981035i	$0.562091\pi$
$798$	11.7584	0.416244
$799$	28.7318	1.01646
$800$	0	0
$801$	26.5985	0.939812
$802$	−50.2243	−1.77348
$803$	−1.55186	−0.0547639
$804$	−11.2851	−0.397994
$805$	0	0
$806$	0.960663	0.0338379
$807$	−7.22982	−0.254502
$808$	−1.85062	−0.0651046
$809$	16.4427	0.578096	0.289048	−	0.957315i	$-0.406661\pi$
$809$	16.4427	0.578096	0.289048	+	0.957315i	$0.406661\pi$
$810$	0	0
$811$	22.6473	0.795253	0.397627	−	0.917547i	$-0.369834\pi$
$811$	22.6473	0.795253	0.397627	+	0.917547i	$0.369834\pi$
$812$	12.5484	0.440364
$813$	26.7118	0.936823
$814$	8.96299	0.314153
$815$	0	0
$816$	20.2474	0.708802
$817$	12.0023	0.419906
$818$	−3.96067	−0.138482
$819$	−6.13887	−0.214509
$820$	0	0
$821$	−39.7792	−1.38830	−0.694151	−	0.719829i	$-0.744220\pi$
$821$	−39.7792	−1.38830	−0.694151	+	0.719829i	$0.744220\pi$
$822$	3.16076	0.110244
$823$	−16.5602	−0.577252	−0.288626	−	0.957442i	$-0.593198\pi$
$823$	−16.5602	−0.577252	−0.288626	+	0.957442i	$0.593198\pi$
$824$	−7.38562	−0.257290
$825$	0	0
$826$	−10.1770	−0.354102
$827$	26.0361	0.905365	0.452683	−	0.891672i	$-0.350467\pi$
$827$	26.0361	0.905365	0.452683	+	0.891672i	$0.350467\pi$
$828$	19.5885	0.680746
$829$	5.14357	0.178644	0.0893218	−	0.996003i	$-0.471530\pi$
$829$	5.14357	0.178644	0.0893218	+	0.996003i	$0.471530\pi$
$830$	0	0
$831$	26.0885	0.905002
$832$	−35.4352	−1.22849
$833$	17.4109	0.603252
$834$	−75.9734	−2.63074
$835$	0	0
$836$	−12.1389	−0.419832
$837$	−0.350197	−0.0121046
$838$	−4.85467	−0.167702
$839$	38.5664	1.33146	0.665730	−	0.746193i	$-0.268120\pi$
$839$	38.5664	1.33146	0.665730	+	0.746193i	$0.268120\pi$
$840$	0	0
$841$	−0.00936035	−0.000322771 0
$842$	−49.9427	−1.72114
$843$	−54.1744	−1.86586
$844$	−20.3318	−0.699850
$845$	0	0
$846$	−37.9491	−1.30472
$847$	6.94679	0.238694
$848$	24.0708	0.826596
$849$	7.39447	0.253777
$850$	0	0
$851$	9.77764	0.335173
$852$	49.7841	1.70558
$853$	−9.14763	−0.313209	−0.156604	−	0.987661i	$-0.550055\pi$
$853$	−9.14763	−0.313209	−0.156604	+	0.987661i	$0.550055\pi$
$854$	5.73175	0.196136
$855$	0	0
$856$	3.49990	0.119624
$857$	13.6712	0.466998	0.233499	−	0.972357i	$-0.424982\pi$
$857$	13.6712	0.466998	0.233499	+	0.972357i	$0.424982\pi$
$858$	30.9367	1.05616
$859$	−35.6556	−1.21655	−0.608277	−	0.793725i	$-0.708139\pi$
$859$	−35.6556	−1.21655	−0.608277	+	0.793725i	$0.708139\pi$
$860$	0	0
$861$	−18.8357	−0.641919
$862$	2.48621	0.0846808
$863$	−33.9333	−1.15510	−0.577552	−	0.816354i	$-0.695992\pi$
$863$	−33.9333	−1.15510	−0.577552	+	0.816354i	$0.695992\pi$
$864$	20.7444	0.705739
$865$	0	0
$866$	−53.4121	−1.81502
$867$	−18.9550	−0.643744
$868$	−0.318210	−0.0108007
$869$	31.7017	1.07541
$870$	0	0
$871$	−7.37494	−0.249890
$872$	11.8240	0.400410
$873$	31.2206	1.05666
$874$	−24.5197	−0.829391
$875$	0	0
$876$	−4.00610	−0.135354
$877$	−28.6991	−0.969099	−0.484550	−	0.874764i	$-0.661017\pi$
$877$	−28.6991	−0.969099	−0.484550	+	0.874764i	$0.661017\pi$
$878$	40.4006	1.36345
$879$	19.7060	0.664665
$880$	0	0
$881$	8.33039	0.280658	0.140329	−	0.990105i	$-0.455184\pi$
$881$	8.33039	0.280658	0.140329	+	0.990105i	$0.455184\pi$
$882$	−22.9964	−0.774331
$883$	50.3165	1.69329	0.846643	−	0.532161i	$-0.178620\pi$
$883$	50.3165	1.69329	0.846643	+	0.532161i	$0.178620\pi$
$884$	−22.9352	−0.771395
$885$	0	0
$886$	5.13529	0.172524
$887$	−12.1186	−0.406903	−0.203452	−	0.979085i	$-0.565216\pi$
$887$	−12.1186	−0.406903	−0.203452	+	0.979085i	$0.565216\pi$
$888$	3.43276	0.115196
$889$	−1.47948	−0.0496202
$890$	0	0
$891$	−22.2777	−0.746332
$892$	−66.5620	−2.22866
$893$	25.6543	0.858490
$894$	14.7233	0.492422
$895$	0	0
$896$	5.68063	0.189777
$897$	33.7485	1.12683
$898$	−29.9415	−0.999161
$899$	−0.735159	−0.0245189
$900$	0	0
$901$	21.8976	0.729514
$902$	36.0053	1.19884
$903$	10.1321	0.337176
$904$	−4.90928	−0.163280
$905$	0	0
$906$	81.0827	2.69379
$907$	31.9105	1.05957	0.529786	−	0.848132i	$-0.322272\pi$
$907$	31.9105	1.05957	0.529786	+	0.848132i	$0.322272\pi$
$908$	−52.2322	−1.73339
$909$	−4.66988	−0.154890
$910$	0	0
$911$	−24.6880	−0.817949	−0.408975	−	0.912546i	$-0.634114\pi$
$911$	−24.6880	−0.817949	−0.408975	+	0.912546i	$0.634114\pi$
$912$	18.0788	0.598647
$913$	3.54220	0.117230
$914$	−52.5416	−1.73792
$915$	0	0
$916$	5.81371	0.192090
$917$	−14.0074	−0.462565
$918$	15.4810	0.510951
$919$	−19.4850	−0.642752	−0.321376	−	0.946952i	$-0.604145\pi$
$919$	−19.4850	−0.642752	−0.321376	+	0.946952i	$0.604145\pi$
$920$	0	0
$921$	−20.8446	−0.686854
$922$	60.1094	1.97960
$923$	32.5345	1.07089
$924$	−10.2474	−0.337116
$925$	0	0
$926$	66.4781	2.18461
$927$	−18.6369	−0.612118
$928$	43.5482	1.42954
$929$	11.7642	0.385972	0.192986	−	0.981201i	$-0.438183\pi$
$929$	11.7642	0.385972	0.192986	+	0.981201i	$0.438183\pi$
$930$	0	0
$931$	15.5460	0.509501
$932$	−13.9873	−0.458168
$933$	64.4862	2.11118
$934$	89.8749	2.94080
$935$	0	0
$936$	4.49431	0.146901
$937$	21.0683	0.688271	0.344136	−	0.938920i	$-0.388172\pi$
$937$	21.0683	0.688271	0.344136	+	0.938920i	$0.388172\pi$
$938$	4.52331	0.147691
$939$	−41.5296	−1.35527
$940$	0	0
$941$	−2.24706	−0.0732521	−0.0366261	−	0.999329i	$-0.511661\pi$
$941$	−2.24706	−0.0732521	−0.0366261	+	0.999329i	$0.511661\pi$
$942$	−7.58787	−0.247226
$943$	39.2778	1.27906
$944$	−15.6472	−0.509273
$945$	0	0
$946$	−19.3680	−0.629709
$947$	6.75625	0.219549	0.109774	−	0.993957i	$-0.464987\pi$
$947$	6.75625	0.219549	0.109774	+	0.993957i	$0.464987\pi$
$948$	81.8374	2.65795
$949$	−2.61803	−0.0849850
$950$	0	0
$951$	−50.1021	−1.62467
$952$	2.08700	0.0676400
$953$	−59.9534	−1.94208	−0.971040	−	0.238918i	$-0.923207\pi$
$953$	−59.9534	−1.94208	−0.971040	+	0.238918i	$0.923207\pi$
$954$	−28.9225	−0.936400
$955$	0	0
$956$	−16.5150	−0.534135
$957$	−23.6747	−0.765293
$958$	43.8459	1.41660
$959$	−0.684206	−0.0220942
$960$	0	0
$961$	−30.9814	−0.999399
$962$	15.1208	0.487516
$963$	8.83169	0.284597
$964$	2.77057	0.0892342
$965$	0	0
$966$	−20.6992	−0.665984
$967$	9.05599	0.291221	0.145610	−	0.989342i	$-0.453485\pi$
$967$	9.05599	0.291221	0.145610	+	0.989342i	$0.453485\pi$
$968$	−5.08580	−0.163464
$969$	16.4465	0.528338
$970$	0	0
$971$	47.3508	1.51956	0.759780	−	0.650180i	$-0.225307\pi$
$971$	47.3508	1.51956	0.759780	+	0.650180i	$0.225307\pi$
$972$	−39.4397	−1.26503
$973$	16.4459	0.527231
$974$	−58.2191	−1.86546
$975$	0	0
$976$	8.81263	0.282085
$977$	4.74467	0.151795	0.0758977	−	0.997116i	$-0.475818\pi$
$977$	4.74467	0.151795	0.0758977	+	0.997116i	$0.475818\pi$
$978$	−4.09364	−0.130900
$979$	29.0160	0.927357
$980$	0	0
$981$	29.8367	0.952613
$982$	31.1981	0.995570
$983$	−18.5656	−0.592150	−0.296075	−	0.955165i	$-0.595678\pi$
$983$	−18.5656	−0.592150	−0.296075	+	0.955165i	$0.595678\pi$
$984$	13.7898	0.439601
$985$	0	0
$986$	32.4990	1.03498
$987$	21.6570	0.689350
$988$	−20.4786	−0.651513
$989$	−21.1284	−0.671843
$990$	0	0
$991$	−39.7199	−1.26174	−0.630871	−	0.775887i	$-0.717302\pi$
$991$	−39.7199	−1.26174	−0.630871	+	0.775887i	$0.717302\pi$
$992$	−1.10432	−0.0350621
$993$	6.51411	0.206719
$994$	−19.9546	−0.632921
$995$	0	0
$996$	9.14414	0.289743
$997$	−30.2914	−0.959338	−0.479669	−	0.877449i	$-0.659243\pi$
$997$	−30.2914	−0.959338	−0.479669	+	0.877449i	$0.659243\pi$
$998$	−92.4309	−2.92585
$999$	−5.51210	−0.174395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.a.f.1.7		8
3.2	odd	2		5625.2.a.x.1.2		8
4.3	odd	2		10000.2.a.bj.1.2		8
5.2	odd	4		625.2.b.c.624.7		8
5.3	odd	4		625.2.b.c.624.2		8
5.4	even	2	inner	625.2.a.f.1.2		8
15.14	odd	2		5625.2.a.x.1.7		8
20.19	odd	2		10000.2.a.bj.1.7		8
25.2	odd	20		125.2.e.b.24.2		8
25.3	odd	20		625.2.e.a.249.1		8
25.4	even	10		625.2.d.o.376.4		16
25.6	even	5		625.2.d.o.251.1		16
25.8	odd	20		625.2.e.i.374.2		8
25.9	even	10		125.2.d.b.26.1		16
25.11	even	5		125.2.d.b.101.4		16
25.12	odd	20		25.2.e.a.19.1	yes	8
25.13	odd	20		125.2.e.b.99.2		8
25.14	even	10		125.2.d.b.101.1		16
25.16	even	5		125.2.d.b.26.4		16
25.17	odd	20		625.2.e.a.374.1		8
25.19	even	10		625.2.d.o.251.4		16
25.21	even	5		625.2.d.o.376.1		16
25.22	odd	20		625.2.e.i.249.2		8
25.23	odd	20		25.2.e.a.4.1	✓	8
75.23	even	20		225.2.m.a.154.2		8
75.62	even	20		225.2.m.a.19.2		8
100.23	even	20		400.2.y.c.129.1		8
100.87	even	20		400.2.y.c.369.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.e.a.4.1	✓	8	25.23	odd	20
25.2.e.a.19.1	yes	8	25.12	odd	20
125.2.d.b.26.1		16	25.9	even	10
125.2.d.b.26.4		16	25.16	even	5
125.2.d.b.101.1		16	25.14	even	10
125.2.d.b.101.4		16	25.11	even	5
125.2.e.b.24.2		8	25.2	odd	20
125.2.e.b.99.2		8	25.13	odd	20
225.2.m.a.19.2		8	75.62	even	20
225.2.m.a.154.2		8	75.23	even	20
400.2.y.c.129.1		8	100.23	even	20
400.2.y.c.369.1		8	100.87	even	20
625.2.a.f.1.2		8	5.4	even	2	inner
625.2.a.f.1.7		8	1.1	even	1	trivial
625.2.b.c.624.2		8	5.3	odd	4
625.2.b.c.624.7		8	5.2	odd	4
625.2.d.o.251.1		16	25.6	even	5
625.2.d.o.251.4		16	25.19	even	10
625.2.d.o.376.1		16	25.21	even	5
625.2.d.o.376.4		16	25.4	even	10
625.2.e.a.249.1		8	25.3	odd	20
625.2.e.a.374.1		8	25.17	odd	20
625.2.e.i.249.2		8	25.22	odd	20
625.2.e.i.374.2		8	25.8	odd	20
5625.2.a.x.1.2		8	3.2	odd	2
5625.2.a.x.1.7		8	15.14	odd	2
10000.2.a.bj.1.2		8	4.3	odd	2
10000.2.a.bj.1.7		8	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	625.a (trivial)

\(f(q)\)	\(=\)	\(q+2.08529 q^{2} +2.19849 q^{3} +2.34841 q^{4} +4.58448 q^{6} -0.992398 q^{7} +0.726543 q^{8} +1.83337 q^{9} +O(q^{10})\)	\(q+2.08529 q^{2} +2.19849 q^{3} +2.34841 q^{4} +4.58448 q^{6} -0.992398 q^{7} +0.726543 q^{8} +1.83337 q^{9} +2.00000 q^{11} +5.16297 q^{12} +3.37406 q^{13} -2.06943 q^{14} -3.18178 q^{16} -2.89451 q^{17} +3.82309 q^{18} -2.58448 q^{19} -2.18178 q^{21} +4.17057 q^{22} +4.54963 q^{23} +1.59730 q^{24} +7.03588 q^{26} -2.56484 q^{27} -2.33056 q^{28} -5.38430 q^{29} +0.136538 q^{31} -8.08800 q^{32} +4.39698 q^{33} -6.03588 q^{34} +4.30550 q^{36} +2.14910 q^{37} -5.38938 q^{38} +7.41785 q^{39} +8.63318 q^{41} -4.54963 q^{42} -4.64398 q^{43} +4.69683 q^{44} +9.48728 q^{46} -9.92630 q^{47} -6.99512 q^{48} -6.01515 q^{49} -6.36356 q^{51} +7.92369 q^{52} -7.56521 q^{53} -5.34841 q^{54} -0.721020 q^{56} -5.68196 q^{57} -11.2278 q^{58} +4.91775 q^{59} -2.76972 q^{61} +0.284720 q^{62} -1.81943 q^{63} -10.5022 q^{64} +9.16896 q^{66} -2.18577 q^{67} -6.79751 q^{68} +10.0023 q^{69} +9.64254 q^{71} +1.33202 q^{72} -0.775929 q^{73} +4.48150 q^{74} -6.06943 q^{76} -1.98480 q^{77} +15.4683 q^{78} +15.8508 q^{79} -11.1389 q^{81} +18.0026 q^{82} +1.77110 q^{83} -5.12372 q^{84} -9.68401 q^{86} -11.8373 q^{87} +1.45309 q^{88} +14.5080 q^{89} -3.34841 q^{91} +10.6844 q^{92} +0.300177 q^{93} -20.6992 q^{94} -17.7814 q^{96} +17.0291 q^{97} -12.5433 q^{98} +3.66673 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9}+O(q^{10}) \) 8 * q + 6 * q^4 + 6 * q^6 + 4 * q^9	\( 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9} + 16 q^{11} + 12 q^{14} - 2 q^{16} + 10 q^{19} + 6 q^{21} + 20 q^{24} + 6 q^{26} + 20 q^{29} + 16 q^{31} + 2 q^{34} - 12 q^{36} + 18 q^{39} + 26 q^{41} + 12 q^{44} + 6 q^{46} - 14 q^{49} - 4 q^{51} - 30 q^{54} + 10 q^{56} + 30 q^{59} + 6 q^{61} - 44 q^{64} + 12 q^{66} + 8 q^{69} + 46 q^{71} + 12 q^{74} - 20 q^{76} + 10 q^{79} - 32 q^{81} - 18 q^{84} - 14 q^{86} + 30 q^{89} - 14 q^{91} - 68 q^{94} - 54 q^{96} + 8 q^{99}+O(q^{100}) \) 8 * q + 6 * q^4 + 6 * q^6 + 4 * q^9 + 16 * q^11 + 12 * q^14 - 2 * q^16 + 10 * q^19 + 6 * q^21 + 20 * q^24 + 6 * q^26 + 20 * q^29 + 16 * q^31 + 2 * q^34 - 12 * q^36 + 18 * q^39 + 26 * q^41 + 12 * q^44 + 6 * q^46 - 14 * q^49 - 4 * q^51 - 30 * q^54 + 10 * q^56 + 30 * q^59 + 6 * q^61 - 44 * q^64 + 12 * q^66 + 8 * q^69 + 46 * q^71 + 12 * q^74 - 20 * q^76 + 10 * q^79 - 32 * q^81 - 18 * q^84 - 14 * q^86 + 30 * q^89 - 14 * q^91 - 68 * q^94 - 54 * q^96 + 8 * q^99

Introduction

L-functions

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Database

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