LMFDB - Embedded newform 7225.2.a.bp.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7225,2,Mod(1,7225)] 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("7225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,0,12,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 30x^{10} + 343x^{8} - 1860x^{6} + 4823x^{4} - 5230x^{2} + 1681 $ x^12 - 30x^10 + 343x^8 - 1860x^6 + 4823x^4 - 5230*x^2 + 1681
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.27039$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.232389 q^{2} -2.39435 q^{3} -1.94600 q^{4} +0.556420 q^{6} -2.06570 q^{7} +0.917007 q^{8} +2.73289 q^{9} +0.480046 q^{11} +4.65939 q^{12} +4.07073 q^{13} +0.480046 q^{14} +3.67889 q^{16} -0.635095 q^{18} +4.00000 q^{19} +4.94600 q^{21} -0.111558 q^{22} -8.15123 q^{23} -2.19563 q^{24} -0.945995 q^{26} +0.639548 q^{27} +4.01984 q^{28} -1.03647 q^{29} +6.06053 q^{31} -2.68895 q^{32} -1.14940 q^{33} -5.31820 q^{36} +1.29684 q^{37} -0.929557 q^{38} -9.74675 q^{39} +10.7832 q^{41} -1.14940 q^{42} +7.45685 q^{43} -0.934167 q^{44} +1.89426 q^{46} +3.60596 q^{47} -8.80853 q^{48} -2.73289 q^{49} -7.92163 q^{52} +6.14969 q^{53} -0.148624 q^{54} -1.89426 q^{56} -9.57738 q^{57} +0.240864 q^{58} +6.00000 q^{59} -5.65685 q^{61} -1.40840 q^{62} -5.64533 q^{63} -6.73289 q^{64} +0.267107 q^{66} +3.14118 q^{67} +19.5169 q^{69} -1.81789 q^{71} +2.50608 q^{72} -12.1711 q^{73} -0.301373 q^{74} -7.78398 q^{76} -0.991630 q^{77} +2.26504 q^{78} +10.2268 q^{79} -9.72998 q^{81} -2.50590 q^{82} +2.23672 q^{83} -9.62488 q^{84} -1.73289 q^{86} +2.48166 q^{87} +0.440206 q^{88} -9.37220 q^{89} -8.40891 q^{91} +15.8623 q^{92} -14.5110 q^{93} -0.837986 q^{94} +6.43827 q^{96} -16.7189 q^{97} +0.635095 q^{98} +1.31191 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{4} + 28 q^{9} + 4 q^{16} + 48 q^{19} + 24 q^{21} + 24 q^{26} + 68 q^{36} - 28 q^{49} + 72 q^{59} - 76 q^{64} + 8 q^{66} + 88 q^{69} + 48 q^{76} + 60 q^{81} - 40 q^{84} - 16 q^{86} - 16 q^{89}+ \cdots + 96 q^{94}+O(q^{100}) $ 12 * q + 12 * q^4 + 28 * q^9 + 4 * q^16 + 48 * q^19 + 24 * q^21 + 24 * q^26 + 68 * q^36 - 28 * q^49 + 72 * q^59 - 76 * q^64 + 8 * q^66 + 88 * q^69 + 48 * q^76 + 60 * q^81 - 40 * q^84 - 16 * q^86 - 16 * q^89 + 96 * q^94

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 $

$2$	−0.232389	−0.164324	−0.0821620	−	0.996619i	$-0.526183\pi$
$2$	−0.232389	−0.164324	−0.0821620	+	0.996619i	$0.526183\pi$
$3$	−2.39435	−1.38238	−0.691188	−	0.722675i	$-0.742912\pi$
$3$	−2.39435	−1.38238	−0.691188	+	0.722675i	$0.742912\pi$
$4$	−1.94600	−0.972998
$5$	0	0
$5$	0	0
$6$	0.556420	0.227158
$7$	−2.06570	−0.780760	−0.390380	−	0.920654i	$-0.627656\pi$
$7$	−2.06570	−0.780760	−0.390380	+	0.920654i	$0.627656\pi$
$8$	0.917007	0.324211
$9$	2.73289	0.910964
$10$	0	0
$11$	0.480046	0.144739	0.0723697	−	0.997378i	$-0.476944\pi$
$11$	0.480046	0.144739	0.0723697	+	0.997378i	$0.476944\pi$
$12$	4.65939	1.34505
$13$	4.07073	1.12902	0.564509	−	0.825427i	$-0.309065\pi$
$13$	4.07073	1.12902	0.564509	+	0.825427i	$0.309065\pi$
$14$	0.480046	0.128298
$15$	0	0
$16$	3.67889	0.919722
$17$	0	0
$17$	0	0
$18$	−0.635095	−0.149693
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	4.94600	1.07930
$22$	−0.111558	−0.0237842
$23$	−8.15123	−1.69965	−0.849825	−	0.527065i	$-0.823292\pi$
$23$	−8.15123	−1.69965	−0.849825	+	0.527065i	$0.823292\pi$
$24$	−2.19563	−0.448182
$25$	0	0
$26$	−0.945995	−0.185525
$27$	0.639548	0.123081
$28$	4.01984	0.759678
$29$	−1.03647	−0.192467	−0.0962335	−	0.995359i	$-0.530680\pi$
$29$	−1.03647	−0.192467	−0.0962335	+	0.995359i	$0.530680\pi$
$30$	0	0
$31$	6.06053	1.08850	0.544251	−	0.838922i	$-0.316814\pi$
$31$	6.06053	1.08850	0.544251	+	0.838922i	$0.316814\pi$
$32$	−2.68895	−0.475343
$33$	−1.14940	−0.200084
$34$	0	0
$35$	0	0
$36$	−5.31820	−0.886366
$37$	1.29684	0.213200	0.106600	−	0.994302i	$-0.466004\pi$
$37$	1.29684	0.213200	0.106600	+	0.994302i	$0.466004\pi$
$38$	−0.929557	−0.150794
$39$	−9.74675	−1.56073
$40$	0	0
$41$	10.7832	1.68405	0.842027	−	0.539435i	$-0.181362\pi$
$41$	10.7832	1.68405	0.842027	+	0.539435i	$0.181362\pi$
$42$	−1.14940	−0.177356
$43$	7.45685	1.13716	0.568580	−	0.822628i	$-0.307493\pi$
$43$	7.45685	1.13716	0.568580	+	0.822628i	$0.307493\pi$
$44$	−0.934167	−0.140831
$45$	0	0
$46$	1.89426	0.279293
$47$	3.60596	0.525983	0.262991	−	0.964798i	$-0.415291\pi$
$47$	3.60596	0.525983	0.262991	+	0.964798i	$0.415291\pi$
$48$	−8.80853	−1.27140
$49$	−2.73289	−0.390413
$50$	0	0
$51$	0	0
$52$	−7.92163	−1.09853
$53$	6.14969	0.844725	0.422362	−	0.906427i	$-0.361201\pi$
$53$	6.14969	0.844725	0.422362	+	0.906427i	$0.361201\pi$
$54$	−0.148624	−0.0202252
$55$	0	0
$56$	−1.89426	−0.253131
$57$	−9.57738	−1.26856
$58$	0.240864	0.0316270
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−5.65685	−0.724286	−0.362143	−	0.932123i	$-0.617955\pi$
$61$	−5.65685	−0.724286	−0.362143	+	0.932123i	$0.617955\pi$
$62$	−1.40840	−0.178867
$63$	−5.64533	−0.711245
$64$	−6.73289	−0.841612
$65$	0	0
$66$	0.267107	0.0328787
$67$	3.14118	0.383756	0.191878	−	0.981419i	$-0.438542\pi$
$67$	3.14118	0.383756	0.191878	+	0.981419i	$0.438542\pi$
$68$	0	0
$69$	19.5169	2.34956
$70$	0	0
$71$	−1.81789	−0.215743	−0.107872	−	0.994165i	$-0.534404\pi$
$71$	−1.81789	−0.215743	−0.107872	+	0.994165i	$0.534404\pi$
$72$	2.50608	0.295345
$73$	−12.1711	−1.42452	−0.712258	−	0.701918i	$-0.752327\pi$
$73$	−12.1711	−1.42452	−0.712258	+	0.701918i	$0.752327\pi$
$74$	−0.301373	−0.0350339
$75$	0	0
$76$	−7.78398	−0.892884
$77$	−0.991630	−0.113007
$78$	2.26504	0.256465
$79$	10.2268	1.15060	0.575302	−	0.817941i	$-0.304885\pi$
$79$	10.2268	1.15060	0.575302	+	0.817941i	$0.304885\pi$
$80$	0	0
$81$	−9.72998	−1.08111
$82$	−2.50590	−0.276731
$83$	2.23672	0.245512	0.122756	−	0.992437i	$-0.460827\pi$
$83$	2.23672	0.245512	0.122756	+	0.992437i	$0.460827\pi$
$84$	−9.62488	−1.05016
$85$	0	0
$86$	−1.73289	−0.186863
$87$	2.48166	0.266062
$88$	0.440206	0.0469261
$89$	−9.37220	−0.993451	−0.496726	−	0.867908i	$-0.665464\pi$
$89$	−9.37220	−0.993451	−0.496726	+	0.867908i	$0.665464\pi$
$90$	0	0
$91$	−8.40891	−0.881493
$92$	15.8623	1.65376
$93$	−14.5110	−1.50472
$94$	−0.837986	−0.0864316
$95$	0	0
$96$	6.43827	0.657103
$97$	−16.7189	−1.69755	−0.848774	−	0.528757i	$-0.822658\pi$
$97$	−16.7189	−1.69755	−0.848774	+	0.528757i	$0.822658\pi$
$98$	0.635095	0.0641543
$99$	1.31191	0.131852
$100$	0	0
$101$	−4.41178	−0.438989	−0.219494	−	0.975614i	$-0.570441\pi$
$101$	−4.41178	−0.438989	−0.219494	+	0.975614i	$0.570441\pi$
$102$	0	0
$103$	−17.4323	−1.71766	−0.858829	−	0.512262i	$-0.828808\pi$
$103$	−17.4323	−1.71766	−0.858829	+	0.512262i	$0.828808\pi$
$104$	3.73289	0.366040
$105$	0	0
$106$	−1.42912	−0.138809
$107$	−5.21115	−0.503781	−0.251890	−	0.967756i	$-0.581052\pi$
$107$	−5.21115	−0.503781	−0.251890	+	0.967756i	$0.581052\pi$
$108$	−1.24456	−0.119758
$109$	8.10753	0.776561	0.388280	−	0.921541i	$-0.373069\pi$
$109$	8.10753	0.776561	0.388280	+	0.921541i	$0.373069\pi$
$110$	0	0
$111$	−3.10509	−0.294722
$112$	−7.59947	−0.718082
$113$	−3.47410	−0.326816	−0.163408	−	0.986559i	$-0.552249\pi$
$113$	−3.47410	−0.326816	−0.163408	+	0.986559i	$0.552249\pi$
$114$	2.22568	0.208454
$115$	0	0
$116$	2.01696	0.187270
$117$	11.1249	1.02850
$118$	−1.39434	−0.128359
$119$	0	0
$120$	0	0
$121$	−10.7696	−0.979051
$122$	1.31459	0.119018
$123$	−25.8187	−2.32800
$124$	−11.7938	−1.05911
$125$	0	0
$126$	1.31191	0.116875
$127$	2.98341	0.264735	0.132367	−	0.991201i	$-0.457742\pi$
$127$	2.98341	0.264735	0.132367	+	0.991201i	$0.457742\pi$
$128$	6.94255	0.613640
$129$	−17.8543	−1.57198
$130$	0	0
$131$	7.19377	0.628522	0.314261	−	0.949337i	$-0.398243\pi$
$131$	7.19377	0.628522	0.314261	+	0.949337i	$0.398243\pi$
$132$	2.23672	0.194681
$133$	−8.26279	−0.716475
$134$	−0.729976	−0.0630603
$135$	0	0
$136$	0	0
$137$	−0.526852	−0.0450120	−0.0225060	−	0.999747i	$-0.507164\pi$
$137$	−0.526852	−0.0450120	−0.0225060	+	0.999747i	$0.507164\pi$
$138$	−4.53551	−0.386089
$139$	17.6756	1.49923	0.749613	−	0.661877i	$-0.230240\pi$
$139$	17.6756	1.49923	0.749613	+	0.661877i	$0.230240\pi$
$140$	0	0
$141$	−8.63391	−0.727106
$142$	0.422457	0.0354518
$143$	1.95414	0.163413
$144$	10.0540	0.837834
$145$	0	0
$146$	2.82843	0.234082
$147$	6.54349	0.539698
$148$	−2.52365	−0.207443
$149$	7.32111	0.599769	0.299884	−	0.953976i	$-0.403052\pi$
$149$	7.32111	0.599769	0.299884	+	0.953976i	$0.403052\pi$
$150$	0	0
$151$	7.46579	0.607557	0.303778	−	0.952743i	$-0.401752\pi$
$151$	7.46579	0.607557	0.303778	+	0.952743i	$0.401752\pi$
$152$	3.66803	0.297516
$153$	0	0
$154$	0.230444	0.0185697
$155$	0	0
$156$	18.9671	1.51859
$157$	−12.4571	−0.994188	−0.497094	−	0.867697i	$-0.665600\pi$
$157$	−12.4571	−0.994188	−0.497094	+	0.867697i	$0.665600\pi$
$158$	−2.37660	−0.189072
$159$	−14.7245	−1.16773
$160$	0	0
$161$	16.8380	1.32702
$162$	2.26114	0.177652
$163$	−8.56767	−0.671071	−0.335536	−	0.942027i	$-0.608917\pi$
$163$	−8.56767	−0.671071	−0.335536	+	0.942027i	$0.608917\pi$
$164$	−20.9841	−1.63858
$165$	0	0
$166$	−0.519790	−0.0403435
$167$	−9.44808	−0.731114	−0.365557	−	0.930789i	$-0.619122\pi$
$167$	−9.44808	−0.731114	−0.365557	+	0.930789i	$0.619122\pi$
$168$	4.53551	0.349922
$169$	3.57088	0.274683
$170$	0	0
$171$	10.9316	0.835958
$172$	−14.5110	−1.10645
$173$	−16.5433	−1.25777	−0.628883	−	0.777500i	$-0.716488\pi$
$173$	−16.5433	−1.25777	−0.628883	+	0.777500i	$0.716488\pi$
$174$	−0.576711	−0.0437204
$175$	0	0
$176$	1.76604	0.133120
$177$	−14.3661	−1.07982
$178$	2.17800	0.163248
$179$	19.5313	1.45984	0.729919	−	0.683534i	$-0.239558\pi$
$179$	19.5313	1.45984	0.729919	+	0.683534i	$0.239558\pi$
$180$	0	0
$181$	−10.4818	−0.779109	−0.389555	−	0.921003i	$-0.627371\pi$
$181$	−10.4818	−0.779109	−0.389555	+	0.921003i	$0.627371\pi$
$182$	1.95414	0.144851
$183$	13.5445	1.00124
$184$	−7.47474	−0.551045
$185$	0	0
$186$	3.37220	0.247262
$187$	0	0
$188$	−7.01717	−0.511780
$189$	−1.32111	−0.0960968
$190$	0	0
$191$	5.32111	0.385022	0.192511	−	0.981295i	$-0.438337\pi$
$191$	5.32111	0.385022	0.192511	+	0.981295i	$0.438337\pi$
$192$	16.1209	1.16342
$193$	−12.4820	−0.898472	−0.449236	−	0.893413i	$-0.648304\pi$
$193$	−12.4820	−0.898472	−0.449236	+	0.893413i	$0.648304\pi$
$194$	3.88529	0.278948
$195$	0	0
$196$	5.31820	0.379871
$197$	0.551763	0.0393115	0.0196558	−	0.999807i	$-0.493743\pi$
$197$	0.551763	0.0393115	0.0196558	+	0.999807i	$0.493743\pi$
$198$	−0.304875	−0.0216665
$199$	17.5992	1.24758	0.623788	−	0.781593i	$-0.285593\pi$
$199$	17.5992	1.24758	0.623788	+	0.781593i	$0.285593\pi$
$200$	0	0
$201$	−7.52106	−0.530495
$202$	1.02525	0.0721364
$203$	2.14103	0.150271
$204$	0	0
$205$	0	0
$206$	4.05109	0.282253
$207$	−22.2764	−1.54832
$208$	14.9758	1.03838
$209$	1.92018	0.132822
$210$	0	0
$211$	−12.8302	−0.883269	−0.441634	−	0.897195i	$-0.645601\pi$
$211$	−12.8302	−0.883269	−0.441634	+	0.897195i	$0.645601\pi$
$212$	−11.9673	−0.821915
$213$	4.35265	0.298238
$214$	1.21102	0.0827833
$215$	0	0
$216$	0.586470	0.0399042
$217$	−12.5192	−0.849860
$218$	−1.88410	−0.127608
$219$	29.1418	1.96922
$220$	0	0
$221$	0	0
$222$	0.721590	0.0484300
$223$	4.07073	0.272597	0.136298	−	0.990668i	$-0.456479\pi$
$223$	4.07073	0.272597	0.136298	+	0.990668i	$0.456479\pi$
$224$	5.55456	0.371129
$225$	0	0
$226$	0.807344	0.0537037
$227$	15.7745	1.04699	0.523494	−	0.852029i	$-0.324628\pi$
$227$	15.7745	1.04699	0.523494	+	0.852029i	$0.324628\pi$
$228$	18.6375	1.23430
$229$	14.9460	0.987659	0.493830	−	0.869559i	$-0.335597\pi$
$229$	14.9460	0.987659	0.493830	+	0.869559i	$0.335597\pi$
$230$	0	0
$231$	2.37431	0.156218
$232$	−0.950447	−0.0623999
$233$	0.727332	0.0476491	0.0238246	−	0.999716i	$-0.492416\pi$
$233$	0.727332	0.0476491	0.0238246	+	0.999716i	$0.492416\pi$
$234$	−2.58530	−0.169007
$235$	0	0
$236$	−11.6760	−0.760041
$237$	−24.4865	−1.59057
$238$	0	0
$239$	9.57379	0.619277	0.309639	−	0.950854i	$-0.399792\pi$
$239$	9.57379	0.619277	0.309639	+	0.950854i	$0.399792\pi$
$240$	0	0
$241$	−8.48528	−0.546585	−0.273293	−	0.961931i	$-0.588113\pi$
$241$	−8.48528	−0.546585	−0.273293	+	0.961931i	$0.588113\pi$
$242$	2.50273	0.160882
$243$	21.3783	1.37142
$244$	11.0082	0.704729
$245$	0	0
$246$	6.00000	0.382546
$247$	16.2829	1.03606
$248$	5.55755	0.352905
$249$	−5.35548	−0.339390
$250$	0	0
$251$	6.03666	0.381031	0.190515	−	0.981684i	$-0.438984\pi$
$251$	6.03666	0.381031	0.190515	+	0.981684i	$0.438984\pi$
$252$	10.9858	0.692039
$253$	−3.91297	−0.246006
$254$	−0.693313	−0.0435023
$255$	0	0
$256$	11.8524	0.740776
$257$	27.2752	1.70138	0.850689	−	0.525670i	$-0.176185\pi$
$257$	27.2752	1.70138	0.850689	+	0.525670i	$0.176185\pi$
$258$	4.14914	0.258314
$259$	−2.67889	−0.166458
$260$	0	0
$261$	−2.83255	−0.175331
$262$	−1.67175	−0.103281
$263$	12.3700	0.762765	0.381382	−	0.924417i	$-0.375448\pi$
$263$	12.3700	0.762765	0.381382	+	0.924417i	$0.375448\pi$
$264$	−1.05400	−0.0648695
$265$	0	0
$266$	1.92018	0.117734
$267$	22.4403	1.37332
$268$	−6.11272	−0.373394
$269$	20.0758	1.22405	0.612023	−	0.790840i	$-0.290356\pi$
$269$	20.0758	1.22405	0.612023	+	0.790840i	$0.290356\pi$
$270$	0	0
$271$	−24.2102	−1.47066	−0.735332	−	0.677707i	$-0.762974\pi$
$271$	−24.2102	−1.47066	−0.735332	+	0.677707i	$0.762974\pi$
$272$	0	0
$273$	20.1338	1.21855
$274$	0.122435	0.00739655
$275$	0	0
$276$	−37.9797	−2.28611
$277$	−6.76058	−0.406204	−0.203102	−	0.979158i	$-0.565102\pi$
$277$	−6.76058	−0.406204	−0.203102	+	0.979158i	$0.565102\pi$
$278$	−4.10762	−0.246359
$279$	16.5628	0.991587
$280$	0	0
$281$	18.0367	1.07598	0.537989	−	0.842952i	$-0.319184\pi$
$281$	18.0367	1.07598	0.537989	+	0.842952i	$0.319184\pi$
$282$	2.00643	0.119481
$283$	25.2165	1.49897	0.749484	−	0.662023i	$-0.230302\pi$
$283$	25.2165	1.49897	0.749484	+	0.662023i	$0.230302\pi$
$284$	3.53760	0.209918
$285$	0	0
$286$	−0.454121	−0.0268528
$287$	−22.2749	−1.31484
$288$	−7.34861	−0.433021
$289$	0	0
$290$	0	0
$291$	40.0308	2.34665
$292$	23.6848	1.38605
$293$	26.2584	1.53403	0.767017	−	0.641627i	$-0.221740\pi$
$293$	26.2584	1.53403	0.767017	+	0.641627i	$0.221740\pi$
$294$	−1.52064	−0.0886854
$295$	0	0
$296$	1.18922	0.0691217
$297$	0.307012	0.0178147
$298$	−1.70135	−0.0985565
$299$	−33.1815	−1.91894
$300$	0	0
$301$	−15.4036	−0.887849
$302$	−1.73497	−0.0998362
$303$	10.5633	0.606847
$304$	14.7156	0.843995
$305$	0	0
$306$	0	0
$307$	13.9136	0.794088	0.397044	−	0.917800i	$-0.370036\pi$
$307$	13.9136	0.794088	0.397044	+	0.917800i	$0.370036\pi$
$308$	1.92971	0.109955
$309$	41.7390	2.37445
$310$	0	0
$311$	−8.74033	−0.495619	−0.247809	−	0.968809i	$-0.579711\pi$
$311$	−8.74033	−0.495619	−0.247809	+	0.968809i	$0.579711\pi$
$312$	−8.93784	−0.506005
$313$	−6.94820	−0.392735	−0.196368	−	0.980530i	$-0.562915\pi$
$313$	−6.94820	−0.392735	−0.196368	+	0.980530i	$0.562915\pi$
$314$	2.89491	0.163369
$315$	0	0
$316$	−19.9013	−1.11953
$317$	4.21918	0.236973	0.118486	−	0.992956i	$-0.462196\pi$
$317$	4.21918	0.236973	0.118486	+	0.992956i	$0.462196\pi$
$318$	3.42181	0.191886
$319$	−0.497552	−0.0278575
$320$	0	0
$321$	12.4773	0.696415
$322$	−3.91297	−0.218061
$323$	0	0
$324$	18.9345	1.05192
$325$	0	0
$326$	1.99103	0.110273
$327$	−19.4122	−1.07350
$328$	9.88828	0.545989
$329$	−7.44881	−0.410666
$330$	0	0
$331$	13.6760	0.751699	0.375850	−	0.926681i	$-0.377351\pi$
$331$	13.6760	0.751699	0.375850	+	0.926681i	$0.377351\pi$
$332$	−4.35265	−0.238883
$333$	3.54414	0.194217
$334$	2.19563	0.120140
$335$	0	0
$336$	18.1958	0.992660
$337$	6.48422	0.353218	0.176609	−	0.984281i	$-0.443487\pi$
$337$	6.48422	0.353218	0.176609	+	0.984281i	$0.443487\pi$
$338$	−0.829834	−0.0451370
$339$	8.31820	0.451782
$340$	0	0
$341$	2.90933	0.157549
$342$	−2.54038	−0.137368
$343$	20.1052	1.08558
$344$	6.83799	0.368679
$345$	0	0
$346$	3.84449	0.206681
$347$	−5.00578	−0.268724	−0.134362	−	0.990932i	$-0.542899\pi$
$347$	−5.00578	−0.268724	−0.134362	+	0.990932i	$0.542899\pi$
$348$	−4.82930	−0.258878
$349$	−9.21310	−0.493166	−0.246583	−	0.969122i	$-0.579308\pi$
$349$	−9.21310	−0.493166	−0.246583	+	0.969122i	$0.579308\pi$
$350$	0	0
$351$	2.60343	0.138961
$352$	−1.29082	−0.0688009
$353$	−13.8600	−0.737693	−0.368847	−	0.929490i	$-0.620247\pi$
$353$	−13.8600	−0.737693	−0.368847	+	0.929490i	$0.620247\pi$
$354$	3.33852	0.177440
$355$	0	0
$356$	18.2383	0.966626
$357$	0	0
$358$	−4.53887	−0.239886
$359$	−11.5024	−0.607076	−0.303538	−	0.952819i	$-0.598168\pi$
$359$	−11.5024	−0.607076	−0.303538	+	0.952819i	$0.598168\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	2.43587	0.128026
$363$	25.7860	1.35342
$364$	16.3637	0.857691
$365$	0	0
$366$	−3.14759	−0.164527
$367$	19.4241	1.01393	0.506966	−	0.861966i	$-0.330767\pi$
$367$	19.4241	1.01393	0.506966	+	0.861966i	$0.330767\pi$
$368$	−29.9875	−1.56321
$369$	29.4694	1.53411
$370$	0	0
$371$	−12.7034	−0.659528
$372$	28.2383	1.46409
$373$	5.58922	0.289399	0.144699	−	0.989476i	$-0.453779\pi$
$373$	5.58922	0.289399	0.144699	+	0.989476i	$0.453779\pi$
$374$	0	0
$375$	0	0
$376$	3.30669	0.170529
$377$	−4.21918	−0.217299
$378$	0.307012	0.0157910
$379$	0.556420	0.0285814	0.0142907	−	0.999898i	$-0.495451\pi$
$379$	0.556420	0.0285814	0.0142907	+	0.999898i	$0.495451\pi$
$380$	0	0
$381$	−7.14332	−0.365963
$382$	−1.23657	−0.0632684
$383$	−24.8020	−1.26732	−0.633662	−	0.773610i	$-0.718449\pi$
$383$	−24.8020	−1.26732	−0.633662	+	0.773610i	$0.718449\pi$
$384$	−16.6229	−0.848282
$385$	0	0
$386$	2.90068	0.147641
$387$	20.3788	1.03591
$388$	32.5349	1.65171
$389$	−8.76664	−0.444486	−0.222243	−	0.974991i	$-0.571338\pi$
$389$	−8.76664	−0.444486	−0.222243	+	0.974991i	$0.571338\pi$
$390$	0	0
$391$	0	0
$392$	−2.50608	−0.126576
$393$	−17.2244	−0.868854
$394$	−0.128224	−0.00645983
$395$	0	0
$396$	−2.55298	−0.128292
$397$	−7.03598	−0.353126	−0.176563	−	0.984289i	$-0.556498\pi$
$397$	−7.03598	−0.353126	−0.176563	+	0.984289i	$0.556498\pi$
$398$	−4.08987	−0.205007
$399$	19.7840	0.990438
$400$	0	0
$401$	−10.6346	−0.531066	−0.265533	−	0.964102i	$-0.585548\pi$
$401$	−10.6346	−0.531066	−0.265533	+	0.964102i	$0.585548\pi$
$402$	1.74782	0.0871731
$403$	24.6708	1.22894
$404$	8.58530	0.427135
$405$	0	0
$406$	−0.497552	−0.0246931
$407$	0.622545	0.0308584
$408$	0	0
$409$	2.53421	0.125309	0.0626544	−	0.998035i	$-0.480043\pi$
$409$	2.53421	0.125309	0.0626544	+	0.998035i	$0.480043\pi$
$410$	0	0
$411$	1.26146	0.0622235
$412$	33.9232	1.67128
$413$	−12.3942	−0.609878
$414$	5.17681	0.254426
$415$	0	0
$416$	−10.9460	−0.536672
$417$	−42.3215	−2.07249
$418$	−0.446230	−0.0218258
$419$	−26.3614	−1.28784	−0.643918	−	0.765094i	$-0.722692\pi$
$419$	−26.3614	−1.28784	−0.643918	+	0.765094i	$0.722692\pi$
$420$	0	0
$421$	7.94308	0.387122	0.193561	−	0.981088i	$-0.437996\pi$
$421$	7.94308	0.387122	0.193561	+	0.981088i	$0.437996\pi$
$422$	2.98161	0.145142
$423$	9.85469	0.479151
$424$	5.63931	0.273869
$425$	0	0
$426$	−1.01151	−0.0490078
$427$	11.6854	0.565494
$428$	10.1409	0.490178
$429$	−4.67889	−0.225899
$430$	0	0
$431$	11.3396	0.546211	0.273105	−	0.961984i	$-0.411949\pi$
$431$	11.3396	0.546211	0.273105	+	0.961984i	$0.411949\pi$
$432$	2.35282	0.113200
$433$	6.24538	0.300134	0.150067	−	0.988676i	$-0.452051\pi$
$433$	6.24538	0.300134	0.150067	+	0.988676i	$0.452051\pi$
$434$	2.90933	0.139652
$435$	0	0
$436$	−15.7772	−0.755592
$437$	−32.6049	−1.55971
$438$	−6.77223	−0.323590
$439$	−10.5282	−0.502482	−0.251241	−	0.967925i	$-0.580839\pi$
$439$	−10.5282	−0.502482	−0.251241	+	0.967925i	$0.580839\pi$
$440$	0	0
$441$	−7.46870	−0.355652
$442$	0	0
$443$	29.2669	1.39051	0.695257	−	0.718761i	$-0.255291\pi$
$443$	29.2669	1.39051	0.695257	+	0.718761i	$0.255291\pi$
$444$	6.04250	0.286764
$445$	0	0
$446$	−0.945995	−0.0447942
$447$	−17.5293	−0.829106
$448$	13.9081	0.657097
$449$	32.7274	1.54450	0.772250	−	0.635318i	$-0.219131\pi$
$449$	32.7274	1.54450	0.772250	+	0.635318i	$0.219131\pi$
$450$	0	0
$451$	5.17644	0.243749
$452$	6.76058	0.317991
$453$	−17.8757	−0.839872
$454$	−3.66582	−0.172045
$455$	0	0
$456$	−8.78253	−0.411280
$457$	−31.4752	−1.47235	−0.736174	−	0.676792i	$-0.763369\pi$
$457$	−31.4752	−1.47235	−0.736174	+	0.676792i	$0.763369\pi$
$458$	−3.47329	−0.162296
$459$	0	0
$460$	0	0
$461$	26.6442	1.24094	0.620472	−	0.784228i	$-0.286941\pi$
$461$	26.6442	1.24094	0.620472	+	0.784228i	$0.286941\pi$
$462$	−0.551763	−0.0256704
$463$	−14.2040	−0.660115	−0.330058	−	0.943961i	$-0.607068\pi$
$463$	−14.2040	−0.660115	−0.330058	+	0.943961i	$0.607068\pi$
$464$	−3.81304	−0.177016
$465$	0	0
$466$	−0.169024	−0.00782990
$467$	29.1177	1.34741	0.673703	−	0.739002i	$-0.264703\pi$
$467$	29.1177	1.34741	0.673703	+	0.739002i	$0.264703\pi$
$468$	−21.6490	−1.00072
$469$	−6.48872	−0.299621
$470$	0	0
$471$	29.8267	1.37434
$472$	5.50204	0.253252
$473$	3.57963	0.164592
$474$	5.69040	0.261369
$475$	0	0
$476$	0	0
$477$	16.8064	0.769514
$478$	−2.22485	−0.101762
$479$	−1.96651	−0.0898521	−0.0449261	−	0.998990i	$-0.514305\pi$
$479$	−1.96651	−0.0898521	−0.0449261	+	0.998990i	$0.514305\pi$
$480$	0	0
$481$	5.27911	0.240707
$482$	1.97189	0.0898171
$483$	−40.3160	−1.83444
$484$	20.9575	0.952614
$485$	0	0
$486$	−4.96809	−0.225357
$487$	−4.83495	−0.219093	−0.109546	−	0.993982i	$-0.534940\pi$
$487$	−4.83495	−0.219093	−0.109546	+	0.993982i	$0.534940\pi$
$488$	−5.18738	−0.234821
$489$	20.5140	0.927673
$490$	0	0
$491$	16.8949	0.762456	0.381228	−	0.924481i	$-0.375501\pi$
$491$	16.8949	0.762456	0.381228	+	0.924481i	$0.375501\pi$
$492$	50.2431	2.26514
$493$	0	0
$494$	−3.78398	−0.170249
$495$	0	0
$496$	22.2960	1.00112
$497$	3.75520	0.168444
$498$	1.24456	0.0557699
$499$	4.34494	0.194506	0.0972531	−	0.995260i	$-0.468994\pi$
$499$	4.34494	0.194506	0.0972531	+	0.995260i	$0.468994\pi$
$500$	0	0
$501$	22.6220	1.01067
$502$	−1.40286	−0.0626125
$503$	−11.5078	−0.513105	−0.256553	−	0.966530i	$-0.582587\pi$
$503$	−11.5078	−0.513105	−0.256553	+	0.966530i	$0.582587\pi$
$504$	−5.17681	−0.230593
$505$	0	0
$506$	0.909332	0.0404247
$507$	−8.54992	−0.379715
$508$	−5.80570	−0.257586
$509$	−34.8177	−1.54327	−0.771634	−	0.636066i	$-0.780560\pi$
$509$	−34.8177	−1.54327	−0.771634	+	0.636066i	$0.780560\pi$
$510$	0	0
$511$	25.1418	1.11221
$512$	−16.6395	−0.735368
$513$	2.55819	0.112947
$514$	−6.33845	−0.279577
$515$	0	0
$516$	34.7443	1.52953
$517$	1.73103	0.0761304
$518$	0.622545	0.0273531
$519$	39.6105	1.73871
$520$	0	0
$521$	−22.4501	−0.983559	−0.491779	−	0.870720i	$-0.663653\pi$
$521$	−22.4501	−0.983559	−0.491779	+	0.870720i	$0.663653\pi$
$522$	0.658255	0.0288110
$523$	−16.4943	−0.721243	−0.360622	−	0.932712i	$-0.617435\pi$
$523$	−16.4943	−0.721243	−0.360622	+	0.932712i	$0.617435\pi$
$524$	−13.9990	−0.611551
$525$	0	0
$526$	−2.87465	−0.125341
$527$	0	0
$528$	−4.22850	−0.184022
$529$	43.4426	1.88881
$530$	0	0
$531$	16.3974	0.711585
$532$	16.0794	0.697128
$533$	43.8956	1.90133
$534$	−5.21488	−0.225670
$535$	0	0
$536$	2.88048	0.124418
$537$	−46.7647	−2.01805
$538$	−4.66541	−0.201140
$539$	−1.31191	−0.0565082
$540$	0	0
$541$	12.7279	0.547216	0.273608	−	0.961841i	$-0.411783\pi$
$541$	12.7279	0.547216	0.273608	+	0.961841i	$0.411783\pi$
$542$	5.62619	0.241666
$543$	25.0972	1.07702
$544$	0	0
$545$	0	0
$546$	−4.67889	−0.200238
$547$	12.5057	0.534707	0.267354	−	0.963599i	$-0.413851\pi$
$547$	12.5057	0.534707	0.267354	+	0.963599i	$0.413851\pi$
$548$	1.02525	0.0437965
$549$	−15.4596	−0.659799
$550$	0	0
$551$	−4.14587	−0.176620
$552$	17.8971	0.761752
$553$	−21.1255	−0.898346
$554$	1.57109	0.0667491
$555$	0	0
$556$	−34.3966	−1.45874
$557$	−38.9354	−1.64975	−0.824873	−	0.565318i	$-0.808753\pi$
$557$	−38.9354	−1.64975	−0.824873	+	0.565318i	$0.808753\pi$
$558$	−3.84901	−0.162942
$559$	30.3549	1.28387
$560$	0	0
$561$	0	0
$562$	−4.19153	−0.176809
$563$	−8.17844	−0.344680	−0.172340	−	0.985038i	$-0.555133\pi$
$563$	−8.17844	−0.344680	−0.172340	+	0.985038i	$0.555133\pi$
$564$	16.8015	0.707472
$565$	0	0
$566$	−5.86005	−0.246316
$567$	20.0992	0.844087
$568$	−1.66701	−0.0699463
$569$	−11.8920	−0.498538	−0.249269	−	0.968434i	$-0.580190\pi$
$569$	−11.8920	−0.498538	−0.249269	+	0.968434i	$0.580190\pi$
$570$	0	0
$571$	−3.96719	−0.166022	−0.0830109	−	0.996549i	$-0.526454\pi$
$571$	−3.96719	−0.166022	−0.0830109	+	0.996549i	$0.526454\pi$
$572$	−3.80275	−0.159001
$573$	−12.7406	−0.532246
$574$	5.17644	0.216060
$575$	0	0
$576$	−18.4003	−0.766678
$577$	−6.58601	−0.274179	−0.137090	−	0.990559i	$-0.543775\pi$
$577$	−6.58601	−0.274179	−0.137090	+	0.990559i	$0.543775\pi$
$578$	0	0
$579$	29.8862	1.24203
$580$	0	0
$581$	−4.62039	−0.191686
$582$	−9.30274	−0.385611
$583$	2.95213	0.122265
$584$	−11.1610	−0.461844
$585$	0	0
$586$	−6.10218	−0.252079
$587$	22.3369	0.921944	0.460972	−	0.887415i	$-0.347501\pi$
$587$	22.3369	0.921944	0.460972	+	0.887415i	$0.347501\pi$
$588$	−12.7336	−0.525125
$589$	24.2421	0.998879
$590$	0	0
$591$	−1.32111	−0.0543433
$592$	4.77094	0.196085
$593$	−30.1344	−1.23747	−0.618736	−	0.785599i	$-0.712355\pi$
$593$	−30.1344	−1.23747	−0.618736	+	0.785599i	$0.712355\pi$
$594$	−0.0713464	−0.00292738
$595$	0	0
$596$	−14.2468	−0.583574
$597$	−42.1387	−1.72462
$598$	7.71103	0.315327
$599$	−8.88907	−0.363198	−0.181599	−	0.983373i	$-0.558127\pi$
$599$	−8.88907	−0.363198	−0.181599	+	0.983373i	$0.558127\pi$
$600$	0	0
$601$	−24.2421	−0.988856	−0.494428	−	0.869219i	$-0.664622\pi$
$601$	−24.2421	−0.988856	−0.494428	+	0.869219i	$0.664622\pi$
$602$	3.57963	0.145895
$603$	8.58450	0.349588
$604$	−14.5284	−0.591151
$605$	0	0
$606$	−2.45480	−0.0997196
$607$	14.9118	0.605252	0.302626	−	0.953109i	$-0.402137\pi$
$607$	14.9118	0.605252	0.302626	+	0.953109i	$0.402137\pi$
$608$	−10.7558	−0.436205
$609$	−5.12636	−0.207731
$610$	0	0
$611$	14.6789	0.593844
$612$	0	0
$613$	31.3459	1.26605	0.633024	−	0.774132i	$-0.281813\pi$
$613$	31.3459	1.26605	0.633024	+	0.774132i	$0.281813\pi$
$614$	−3.23336	−0.130488
$615$	0	0
$616$	−0.909332	−0.0366380
$617$	−15.4221	−0.620869	−0.310434	−	0.950595i	$-0.600474\pi$
$617$	−15.4221	−0.620869	−0.310434	+	0.950595i	$0.600474\pi$
$618$	−9.69971	−0.390179
$619$	−45.1035	−1.81286	−0.906431	−	0.422354i	$-0.861204\pi$
$619$	−45.1035	−1.81286	−0.906431	+	0.422354i	$0.861204\pi$
$620$	0	0
$621$	−5.21310	−0.209195
$622$	2.03116	0.0814421
$623$	19.3601	0.775647
$624$	−35.8572	−1.43544
$625$	0	0
$626$	1.61469	0.0645359
$627$	−4.59759	−0.183610
$628$	24.2415	0.967343
$629$	0	0
$630$	0	0
$631$	−29.7493	−1.18430	−0.592150	−	0.805827i	$-0.701721\pi$
$631$	−29.7493	−1.18430	−0.592150	+	0.805827i	$0.701721\pi$
$632$	9.37804	0.373038
$633$	30.7200	1.22101
$634$	−0.980492	−0.0389403
$635$	0	0
$636$	28.6538	1.13620
$637$	−11.1249	−0.440784
$638$	0.115626	0.00457767
$639$	−4.96809	−0.196534
$640$	0	0
$641$	44.1420	1.74350	0.871752	−	0.489947i	$-0.162984\pi$
$641$	44.1420	1.74350	0.871752	+	0.489947i	$0.162984\pi$
$642$	−2.89959	−0.114438
$643$	−7.58172	−0.298994	−0.149497	−	0.988762i	$-0.547765\pi$
$643$	−7.58172	−0.298994	−0.149497	+	0.988762i	$0.547765\pi$
$644$	−32.7666	−1.29119
$645$	0	0
$646$	0	0
$647$	−10.2540	−0.403128	−0.201564	−	0.979475i	$-0.564602\pi$
$647$	−10.2540	−0.403128	−0.201564	+	0.979475i	$0.564602\pi$
$648$	−8.92246	−0.350507
$649$	2.88028	0.113061
$650$	0	0
$651$	29.9753	1.17483
$652$	16.6726	0.652951
$653$	14.3128	0.560104	0.280052	−	0.959985i	$-0.409648\pi$
$653$	14.3128	0.560104	0.280052	+	0.959985i	$0.409648\pi$
$654$	4.51120	0.176402
$655$	0	0
$656$	39.6702	1.54886
$657$	−33.2622	−1.29768
$658$	1.73103	0.0674824
$659$	−43.9653	−1.71265	−0.856323	−	0.516441i	$-0.827257\pi$
$659$	−43.9653	−1.71265	−0.856323	+	0.516441i	$0.827257\pi$
$660$	0	0
$661$	20.7077	0.805438	0.402719	−	0.915324i	$-0.368065\pi$
$661$	20.7077	0.805438	0.402719	+	0.915324i	$0.368065\pi$
$662$	−3.17815	−0.123522
$663$	0	0
$664$	2.05109	0.0795977
$665$	0	0
$666$	−0.823619	−0.0319146
$667$	8.44848	0.327126
$668$	18.3859	0.711372
$669$	−9.74675	−0.376831
$670$	0	0
$671$	−2.71555	−0.104833
$672$	−13.2995	−0.513040
$673$	46.2662	1.78343	0.891715	−	0.452598i	$-0.149503\pi$
$673$	46.2662	1.78343	0.891715	+	0.452598i	$0.149503\pi$
$674$	−1.50686	−0.0580422
$675$	0	0
$676$	−6.94891	−0.267266
$677$	1.76082	0.0676739	0.0338370	−	0.999427i	$-0.489227\pi$
$677$	1.76082	0.0676739	0.0338370	+	0.999427i	$0.489227\pi$
$678$	−1.93306	−0.0742387
$679$	34.5362	1.32538
$680$	0	0
$681$	−37.7696	−1.44733
$682$	−0.676098	−0.0258891
$683$	22.9635	0.878675	0.439338	−	0.898322i	$-0.355213\pi$
$683$	22.9635	0.878675	0.439338	+	0.898322i	$0.355213\pi$
$684$	−21.2728	−0.813385
$685$	0	0
$686$	−4.67224	−0.178387
$687$	−35.7859	−1.36532
$688$	27.4329	1.04587
$689$	25.0337	0.953710
$690$	0	0
$691$	25.7831	0.980837	0.490418	−	0.871487i	$-0.336844\pi$
$691$	25.7831	0.980837	0.490418	+	0.871487i	$0.336844\pi$
$692$	32.1932	1.22380
$693$	−2.71002	−0.102945
$694$	1.16329	0.0441579
$695$	0	0
$696$	2.27570	0.0862602
$697$	0	0
$698$	2.14103	0.0810391
$699$	−1.74148	−0.0658690
$700$	0	0
$701$	25.1195	0.948751	0.474376	−	0.880323i	$-0.342674\pi$
$701$	25.1195	0.948751	0.474376	+	0.880323i	$0.342674\pi$
$702$	−0.605009	−0.0228346
$703$	5.18738	0.195646
$704$	−3.23210	−0.121814
$705$	0	0
$706$	3.22092	0.121221
$707$	9.11340	0.342745
$708$	27.9563	1.05066
$709$	33.5389	1.25958	0.629789	−	0.776766i	$-0.283141\pi$
$709$	33.5389	1.25958	0.629789	+	0.776766i	$0.283141\pi$
$710$	0	0
$711$	27.9487	1.04816
$712$	−8.59437	−0.322088
$713$	−49.4008	−1.85007
$714$	0	0
$715$	0	0
$716$	−38.0078	−1.42042
$717$	−22.9230	−0.856074
$718$	2.67305	0.0997572
$719$	−41.6927	−1.55488	−0.777438	−	0.628960i	$-0.783481\pi$
$719$	−41.6927	−1.55488	−0.777438	+	0.628960i	$0.783481\pi$
$720$	0	0
$721$	36.0099	1.34108
$722$	0.697168	0.0259459
$723$	20.3167	0.755586
$724$	20.3976	0.758071
$725$	0	0
$726$	−5.99240	−0.222399
$727$	−12.2458	−0.454172	−0.227086	−	0.973875i	$-0.572920\pi$
$727$	−12.2458	−0.454172	−0.227086	+	0.973875i	$0.572920\pi$
$728$	−7.71103	−0.285790
$729$	−21.9971	−0.814707
$730$	0	0
$731$	0	0
$732$	−26.3575	−0.974200
$733$	32.8393	1.21295	0.606473	−	0.795104i	$-0.292584\pi$
$733$	32.8393	1.21295	0.606473	+	0.795104i	$0.292584\pi$
$734$	−4.51396	−0.166613
$735$	0	0
$736$	21.9182	0.807917
$737$	1.50791	0.0555446
$738$	−6.84837	−0.252092
$739$	6.00000	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$740$	0	0
$741$	−38.9870	−1.43222
$742$	2.95213	0.108376
$743$	−22.8684	−0.838962	−0.419481	−	0.907764i	$-0.637788\pi$
$743$	−22.8684	−0.838962	−0.419481	+	0.907764i	$0.637788\pi$
$744$	−13.3067	−0.487847
$745$	0	0
$746$	−1.29887	−0.0475552
$747$	6.11272	0.223653
$748$	0	0
$749$	10.7647	0.393332
$750$	0	0
$751$	32.0742	1.17040	0.585202	−	0.810888i	$-0.301015\pi$
$751$	32.0742	1.17040	0.585202	+	0.810888i	$0.301015\pi$
$752$	13.2659	0.483758
$753$	−14.4539	−0.526728
$754$	0.980492	0.0357074
$755$	0	0
$756$	2.57088	0.0935019
$757$	36.2220	1.31651	0.658256	−	0.752794i	$-0.271294\pi$
$757$	36.2220	1.31651	0.658256	+	0.752794i	$0.271294\pi$
$758$	−0.129306	−0.00469661
$759$	9.36900	0.340073
$760$	0	0
$761$	−24.4851	−0.887584	−0.443792	−	0.896130i	$-0.646367\pi$
$761$	−24.4851	−0.887584	−0.443792	+	0.896130i	$0.646367\pi$
$762$	1.66003	0.0601366
$763$	−16.7477	−0.606308
$764$	−10.3549	−0.374626
$765$	0	0
$766$	5.76372	0.208252
$767$	24.4244	0.881914
$768$	−28.3788	−1.02403
$769$	−5.55937	−0.200476	−0.100238	−	0.994963i	$-0.531960\pi$
$769$	−5.55937	−0.200476	−0.100238	+	0.994963i	$0.531960\pi$
$770$	0	0
$771$	−65.3061	−2.35194
$772$	24.2899	0.874211
$773$	36.5376	1.31416	0.657082	−	0.753819i	$-0.271790\pi$
$773$	36.5376	1.31416	0.657082	+	0.753819i	$0.271790\pi$
$774$	−4.73581	−0.170225
$775$	0	0
$776$	−15.3314	−0.550363
$777$	6.41418	0.230108
$778$	2.03727	0.0730398
$779$	43.1329	1.54539
$780$	0	0
$781$	−0.872669	−0.0312265
$782$	0	0
$783$	−0.662870	−0.0236890
$784$	−10.0540	−0.359072
$785$	0	0
$786$	4.00276	0.142774
$787$	16.0499	0.572116	0.286058	−	0.958212i	$-0.407655\pi$
$787$	16.0499	0.572116	0.286058	+	0.958212i	$0.407655\pi$
$788$	−1.07373	−0.0382500
$789$	−29.6180	−1.05443
$790$	0	0
$791$	7.17644	0.255165
$792$	1.20303	0.0427480
$793$	−23.0276	−0.817732
$794$	1.63509	0.0580271
$795$	0	0
$796$	−34.2480	−1.21389
$797$	34.7907	1.23235	0.616175	−	0.787609i	$-0.288681\pi$
$797$	34.7907	1.23235	0.616175	+	0.787609i	$0.288681\pi$
$798$	−4.59759	−0.162753
$799$	0	0
$800$	0	0
$801$	−25.6132	−0.904999
$802$	2.47136	0.0872669
$803$	−5.84268	−0.206184
$804$	14.6360	0.516170
$805$	0	0
$806$	−5.73323	−0.201944
$807$	−48.0685	−1.69209
$808$	−4.04563	−0.142325
$809$	18.8866	0.664018	0.332009	−	0.943276i	$-0.392274\pi$
$809$	18.8866	0.664018	0.332009	+	0.943276i	$0.392274\pi$
$810$	0	0
$811$	−13.1316	−0.461113	−0.230556	−	0.973059i	$-0.574055\pi$
$811$	−13.1316	−0.461113	−0.230556	+	0.973059i	$0.574055\pi$
$812$	−4.16643	−0.146213
$813$	57.9676	2.03301
$814$	−0.144673	−0.00507078
$815$	0	0
$816$	0	0
$817$	29.8274	1.04353
$818$	−0.588924	−0.0205913
$819$	−22.9806	−0.803009
$820$	0	0
$821$	−39.3443	−1.37313	−0.686563	−	0.727070i	$-0.740882\pi$
$821$	−39.3443	−1.37313	−0.686563	+	0.727070i	$0.740882\pi$
$822$	−0.293151	−0.0102248
$823$	40.2222	1.40206	0.701028	−	0.713134i	$-0.252725\pi$
$823$	40.2222	1.40206	0.701028	+	0.713134i	$0.252725\pi$
$824$	−15.9856	−0.556884
$825$	0	0
$826$	2.88028	0.100218
$827$	33.9078	1.17909	0.589545	−	0.807736i	$-0.299307\pi$
$827$	33.9078	1.17909	0.589545	+	0.807736i	$0.299307\pi$
$828$	43.3499	1.50651
$829$	−8.18134	−0.284150	−0.142075	−	0.989856i	$-0.545377\pi$
$829$	−8.18134	−0.284150	−0.142075	+	0.989856i	$0.545377\pi$
$830$	0	0
$831$	16.1872	0.561527
$832$	−27.4078	−0.950195
$833$	0	0
$834$	9.83507	0.340561
$835$	0	0
$836$	−3.73667	−0.129235
$837$	3.87600	0.133974
$838$	6.12610	0.211623
$839$	21.9742	0.758634	0.379317	−	0.925267i	$-0.376159\pi$
$839$	21.9742	0.758634	0.379317	+	0.925267i	$0.376159\pi$
$840$	0	0
$841$	−27.9257	−0.962956
$842$	−1.84589	−0.0636135
$843$	−43.1860	−1.48741
$844$	24.9675	0.859418
$845$	0	0
$846$	−2.29012	−0.0787361
$847$	22.2466	0.764404
$848$	22.6240	0.776912
$849$	−60.3771	−2.07214
$850$	0	0
$851$	−10.5709	−0.362365
$852$	−8.47023	−0.290185
$853$	−36.9651	−1.26566	−0.632831	−	0.774290i	$-0.718107\pi$
$853$	−36.9651	−1.26566	−0.632831	+	0.774290i	$0.718107\pi$
$854$	−2.71555	−0.0929242
$855$	0	0
$856$	−4.77866	−0.163331
$857$	15.0234	0.513189	0.256594	−	0.966519i	$-0.417400\pi$
$857$	15.0234	0.513189	0.256594	+	0.966519i	$0.417400\pi$
$858$	1.08732	0.0371206
$859$	−40.2180	−1.37222	−0.686110	−	0.727498i	$-0.740683\pi$
$859$	−40.2180	−1.37222	−0.686110	+	0.727498i	$0.740683\pi$
$860$	0	0
$861$	53.3337	1.81761
$862$	−2.63521	−0.0897556
$863$	12.3865	0.421643	0.210822	−	0.977525i	$-0.432386\pi$
$863$	12.3865	0.421643	0.210822	+	0.977525i	$0.432386\pi$
$864$	−1.71971	−0.0585058
$865$	0	0
$866$	−1.45136	−0.0493192
$867$	0	0
$868$	24.3623	0.826911
$869$	4.90933	0.166538
$870$	0	0
$871$	12.7869	0.433267
$872$	7.43467	0.251770
$873$	−45.6910	−1.54640
$874$	7.57704	0.256297
$875$	0	0
$876$	−56.7097	−1.91604
$877$	45.1747	1.52544	0.762720	−	0.646728i	$-0.223863\pi$
$877$	45.1747	1.52544	0.762720	+	0.646728i	$0.223863\pi$
$878$	2.44663	0.0825699
$879$	−62.8717	−2.12061
$880$	0	0
$881$	17.0224	0.573500	0.286750	−	0.958006i	$-0.407425\pi$
$881$	17.0224	0.573500	0.286750	+	0.958006i	$0.407425\pi$
$882$	1.73565	0.0584423
$883$	13.3497	0.449254	0.224627	−	0.974445i	$-0.427884\pi$
$883$	13.3497	0.449254	0.224627	+	0.974445i	$0.427884\pi$
$884$	0	0
$885$	0	0
$886$	−6.80132	−0.228495
$887$	44.5291	1.49514	0.747571	−	0.664182i	$-0.231220\pi$
$887$	44.5291	1.49514	0.747571	+	0.664182i	$0.231220\pi$
$888$	−2.84739	−0.0955522
$889$	−6.16283	−0.206695
$890$	0	0
$891$	−4.67084	−0.156479
$892$	−7.92163	−0.265236
$893$	14.4238	0.482675
$894$	4.07362	0.136242
$895$	0	0
$896$	−14.3412	−0.479106
$897$	79.4480	2.65269
$898$	−7.60550	−0.253799
$899$	−6.28153	−0.209501
$900$	0	0
$901$	0	0
$902$	−1.20295	−0.0400538
$903$	36.8815	1.22734
$904$	−3.18577	−0.105957
$905$	0	0
$906$	4.15412	0.138011
$907$	−32.9453	−1.09393	−0.546965	−	0.837155i	$-0.684217\pi$
$907$	−32.9453	−1.09393	−0.546965	+	0.837155i	$0.684217\pi$
$908$	−30.6971	−1.01872
$909$	−12.0569	−0.399903
$910$	0	0
$911$	35.5859	1.17901	0.589506	−	0.807764i	$-0.299322\pi$
$911$	35.5859	1.17901	0.589506	+	0.807764i	$0.299322\pi$
$912$	−35.2341	−1.16672
$913$	1.07373	0.0355352
$914$	7.31450	0.241942
$915$	0	0
$916$	−29.0848	−0.960990
$917$	−14.8601	−0.490725
$918$	0	0
$919$	−28.5997	−0.943418	−0.471709	−	0.881754i	$-0.656363\pi$
$919$	−28.5997	−0.943418	−0.471709	+	0.881754i	$0.656363\pi$
$920$	0	0
$921$	−33.3139	−1.09773
$922$	−6.19183	−0.203917
$923$	−7.40013	−0.243578
$924$	−4.62039	−0.152000
$925$	0	0
$926$	3.30085	0.108473
$927$	−47.6407	−1.56473
$928$	2.78701	0.0914879
$929$	−12.7034	−0.416785	−0.208392	−	0.978045i	$-0.566823\pi$
$929$	−12.7034	−0.416785	−0.208392	+	0.978045i	$0.566823\pi$
$930$	0	0
$931$	−10.9316	−0.358268
$932$	−1.41538	−0.0463625
$933$	20.9274	0.685131
$934$	−6.76664	−0.221411
$935$	0	0
$936$	10.2016	0.333450
$937$	30.7655	1.00506	0.502532	−	0.864558i	$-0.332402\pi$
$937$	30.7655	1.00506	0.502532	+	0.864558i	$0.332402\pi$
$938$	1.50791	0.0492350
$939$	16.6364	0.542908
$940$	0	0
$941$	39.2476	1.27943	0.639717	−	0.768611i	$-0.279052\pi$
$941$	39.2476	1.27943	0.639717	+	0.768611i	$0.279052\pi$
$942$	−6.93141	−0.225837
$943$	−87.8965	−2.86230
$944$	22.0733	0.718426
$945$	0	0
$946$	−0.831868	−0.0270464
$947$	−3.56791	−0.115941	−0.0579707	−	0.998318i	$-0.518463\pi$
$947$	−3.56791	−0.115941	−0.0579707	+	0.998318i	$0.518463\pi$
$948$	47.6506	1.54762
$949$	−49.5452	−1.60831
$950$	0	0
$951$	−10.1022	−0.327586
$952$	0	0
$953$	−53.8491	−1.74434	−0.872172	−	0.489200i	$-0.837289\pi$
$953$	−53.8491	−1.74434	−0.872172	+	0.489200i	$0.837289\pi$
$954$	−3.90564	−0.126450
$955$	0	0
$956$	−18.6306	−0.602555
$957$	1.19131	0.0385096
$958$	0.456996	0.0147649
$959$	1.08832	0.0351436
$960$	0	0
$961$	5.72998	0.184838
$962$	−1.22681	−0.0395539
$963$	−14.2415	−0.458926
$964$	16.5123	0.531826
$965$	0	0
$966$	9.36900	0.301443
$967$	14.6688	0.471716	0.235858	−	0.971788i	$-0.424210\pi$
$967$	14.6688	0.471716	0.235858	+	0.971788i	$0.424210\pi$
$968$	−9.87576	−0.317419
$969$	0	0
$970$	0	0
$971$	−14.2468	−0.457203	−0.228602	−	0.973520i	$-0.573415\pi$
$971$	−14.2468	−0.457203	−0.228602	+	0.973520i	$0.573415\pi$
$972$	−41.6020	−1.33439
$973$	−36.5125	−1.17054
$974$	1.12359	0.0360022
$975$	0	0
$976$	−20.8109	−0.666142
$977$	20.9141	0.669103	0.334551	−	0.942378i	$-0.391415\pi$
$977$	20.9141	0.669103	0.334551	+	0.942378i	$0.391415\pi$
$978$	−4.76722	−0.152439
$979$	−4.49909	−0.143791
$980$	0	0
$981$	22.1570	0.707419
$982$	−3.92620	−0.125290
$983$	−17.4000	−0.554973	−0.277486	−	0.960730i	$-0.589501\pi$
$983$	−17.4000	−0.554973	−0.277486	+	0.960730i	$0.589501\pi$
$984$	−23.6760	−0.754762
$985$	0	0
$986$	0	0
$987$	17.8350	0.567696
$988$	−31.6865	−1.00808
$989$	−60.7825	−1.93277
$990$	0	0
$991$	5.27771	0.167652	0.0838259	−	0.996480i	$-0.473286\pi$
$991$	5.27771	0.167652	0.0838259	+	0.996480i	$0.473286\pi$
$992$	−16.2964	−0.517413
$993$	−32.7450	−1.03913
$994$	−0.872669	−0.0276794
$995$	0	0
$996$	10.4217	0.330226
$997$	25.6390	0.811995	0.405997	−	0.913874i	$-0.366924\pi$
$997$	25.6390	0.811995	0.405997	+	0.913874i	$0.366924\pi$
$998$	−1.00972	−0.0319621
$999$	0.829394	0.0262409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bp.1.5		12
5.2	odd	4		1445.2.b.f.579.6		12
5.3	odd	4		1445.2.b.f.579.7		12
5.4	even	2	inner	7225.2.a.bp.1.8		12
17.8	even	8		425.2.e.d.251.4		12
17.15	even	8		425.2.e.d.276.3		12
17.16	even	2	inner	7225.2.a.bp.1.6		12
85.8	odd	8		85.2.j.c.64.4	yes	12
85.32	odd	8		85.2.j.c.4.4	yes	12
85.33	odd	4		1445.2.b.f.579.8		12
85.42	odd	8		85.2.j.c.64.3	yes	12
85.49	even	8		425.2.e.d.276.4		12
85.59	even	8		425.2.e.d.251.3		12
85.67	odd	4		1445.2.b.f.579.5		12
85.83	odd	8		85.2.j.c.4.3	✓	12
85.84	even	2	inner	7225.2.a.bp.1.7		12
255.8	even	8		765.2.t.e.64.3		12
255.32	even	8		765.2.t.e.514.3		12
255.83	even	8		765.2.t.e.514.4		12
255.212	even	8		765.2.t.e.64.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.j.c.4.3	✓	12	85.83	odd	8
85.2.j.c.4.4	yes	12	85.32	odd	8
85.2.j.c.64.3	yes	12	85.42	odd	8
85.2.j.c.64.4	yes	12	85.8	odd	8
425.2.e.d.251.3		12	85.59	even	8
425.2.e.d.251.4		12	17.8	even	8
425.2.e.d.276.3		12	17.15	even	8
425.2.e.d.276.4		12	85.49	even	8
765.2.t.e.64.3		12	255.8	even	8
765.2.t.e.64.4		12	255.212	even	8
765.2.t.e.514.3		12	255.32	even	8
765.2.t.e.514.4		12	255.83	even	8
1445.2.b.f.579.5		12	85.67	odd	4
1445.2.b.f.579.6		12	5.2	odd	4
1445.2.b.f.579.7		12	5.3	odd	4
1445.2.b.f.579.8		12	85.33	odd	4
7225.2.a.bp.1.5		12	1.1	even	1	trivial
7225.2.a.bp.1.6		12	17.16	even	2	inner
7225.2.a.bp.1.7		12	85.84	even	2	inner
7225.2.a.bp.1.8		12	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.232389 q^{2} -2.39435 q^{3} -1.94600 q^{4} +0.556420 q^{6} -2.06570 q^{7} +0.917007 q^{8} +2.73289 q^{9} +0.480046 q^{11} +4.65939 q^{12} +4.07073 q^{13} +0.480046 q^{14} +3.67889 q^{16} -0.635095 q^{18} +4.00000 q^{19} +4.94600 q^{21} -0.111558 q^{22} -8.15123 q^{23} -2.19563 q^{24} -0.945995 q^{26} +0.639548 q^{27} +4.01984 q^{28} -1.03647 q^{29} +6.06053 q^{31} -2.68895 q^{32} -1.14940 q^{33} -5.31820 q^{36} +1.29684 q^{37} -0.929557 q^{38} -9.74675 q^{39} +10.7832 q^{41} -1.14940 q^{42} +7.45685 q^{43} -0.934167 q^{44} +1.89426 q^{46} +3.60596 q^{47} -8.80853 q^{48} -2.73289 q^{49} -7.92163 q^{52} +6.14969 q^{53} -0.148624 q^{54} -1.89426 q^{56} -9.57738 q^{57} +0.240864 q^{58} +6.00000 q^{59} -5.65685 q^{61} -1.40840 q^{62} -5.64533 q^{63} -6.73289 q^{64} +0.267107 q^{66} +3.14118 q^{67} +19.5169 q^{69} -1.81789 q^{71} +2.50608 q^{72} -12.1711 q^{73} -0.301373 q^{74} -7.78398 q^{76} -0.991630 q^{77} +2.26504 q^{78} +10.2268 q^{79} -9.72998 q^{81} -2.50590 q^{82} +2.23672 q^{83} -9.62488 q^{84} -1.73289 q^{86} +2.48166 q^{87} +0.440206 q^{88} -9.37220 q^{89} -8.40891 q^{91} +15.8623 q^{92} -14.5110 q^{93} -0.837986 q^{94} +6.43827 q^{96} -16.7189 q^{97} +0.635095 q^{98} +1.31191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{4} + 28 q^{9} + 4 q^{16} + 48 q^{19} + 24 q^{21} + 24 q^{26} + 68 q^{36} - 28 q^{49} + 72 q^{59} - 76 q^{64} + 8 q^{66} + 88 q^{69} + 48 q^{76} + 60 q^{81} - 40 q^{84} - 16 q^{86} - 16 q^{89}+ \cdots + 96 q^{94}+O(q^{100}) \) 12 * q + 12 * q^4 + 28 * q^9 + 4 * q^16 + 48 * q^19 + 24 * q^21 + 24 * q^26 + 68 * q^36 - 28 * q^49 + 72 * q^59 - 76 * q^64 + 8 * q^66 + 88 * q^69 + 48 * q^76 + 60 * q^81 - 40 * q^84 - 16 * q^86 - 16 * q^89 + 96 * q^94

Introduction

L-functions

Modular forms

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Database

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