LMFDB - Embedded newform 4225.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.219687$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.219687 q^{2} +1.60020 q^{3} -1.95174 q^{4} -0.351542 q^{6} -0.332247 q^{7} +0.868145 q^{8} -0.439374 q^{9} +5.37182 q^{11} -3.12316 q^{12} +0.0729902 q^{14} +3.71276 q^{16} +5.06430 q^{17} +0.0965246 q^{18} -2.26795 q^{19} -0.531659 q^{21} -1.18012 q^{22} +2.83918 q^{23} +1.38920 q^{24} -5.50367 q^{27} +0.648458 q^{28} -2.90348 q^{29} -5.46410 q^{31} -2.55193 q^{32} +8.59596 q^{33} -1.11256 q^{34} +0.857542 q^{36} +5.97201 q^{37} +0.498239 q^{38} -3.73205 q^{41} +0.116799 q^{42} +5.06430 q^{43} -10.4844 q^{44} -0.623730 q^{46} +8.34285 q^{47} +5.94114 q^{48} -6.88961 q^{49} +8.10387 q^{51} +1.56063 q^{53} +1.20908 q^{54} -0.288438 q^{56} -3.62916 q^{57} +0.637855 q^{58} -2.70732 q^{59} +14.1039 q^{61} +1.20039 q^{62} +0.145980 q^{63} -6.86488 q^{64} -1.88842 q^{66} +10.3322 q^{67} -9.88418 q^{68} +4.54324 q^{69} -12.7973 q^{71} -0.381440 q^{72} -9.68922 q^{73} -1.31197 q^{74} +4.42644 q^{76} -1.78477 q^{77} +4.51851 q^{79} -7.48883 q^{81} +0.819883 q^{82} -4.26371 q^{83} +1.03766 q^{84} -1.11256 q^{86} -4.64613 q^{87} +4.66351 q^{88} -3.22584 q^{89} -5.54133 q^{92} -8.74363 q^{93} -1.83281 q^{94} -4.08359 q^{96} -2.50791 q^{97} +1.51356 q^{98} -2.36023 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{6} + 10 q^{7} + 6 q^{8} + 4 q^{9} + 10 q^{12} + 2 q^{14} + 2 q^{16} + 2 q^{17} + 20 q^{18} - 16 q^{19} - 4 q^{21} - 12 q^{22} + 10 q^{23} + 24 q^{24} + 2 q^{27}+ \cdots - 24 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^6 + 10 * q^7 + 6 * q^8 + 4 * q^9 + 10 * q^12 + 2 * q^14 + 2 * q^16 + 2 * q^17 + 20 * q^18 - 16 * q^19 - 4 * q^21 - 12 * q^22 + 10 * q^23 + 24 * q^24 + 2 * q^27 + 8 * q^28 + 8 * q^29 - 8 * q^31 + 4 * q^32 + 18 * q^33 + 4 * q^34 + 20 * q^36 - 2 * q^37 - 8 * q^38 - 8 * q^41 + 4 * q^42 + 2 * q^43 - 12 * q^44 - 16 * q^46 + 8 * q^47 + 28 * q^48 + 12 * q^49 + 4 * q^51 + 12 * q^53 + 16 * q^54 + 12 * q^56 - 14 * q^57 + 22 * q^58 - 12 * q^59 + 28 * q^61 - 4 * q^62 + 4 * q^63 + 4 * q^64 + 6 * q^66 + 30 * q^67 - 14 * q^68 - 16 * q^69 - 4 * q^71 + 12 * q^72 - 8 * q^73 - 10 * q^74 - 20 * q^76 - 18 * q^77 - 8 * q^79 - 8 * q^81 - 4 * q^82 - 12 * q^83 + 28 * q^84 + 4 * q^86 + 22 * q^87 + 18 * q^88 + 12 * q^89 - 22 * q^92 + 8 * q^93 - 32 * q^94 - 4 * q^96 + 2 * q^97 - 24 * q^98 - 24 * q^99

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 $

$2$	−0.219687	−0.155342	−0.0776710	−	0.996979i	$-0.524748\pi$
$2$	−0.219687	−0.155342	−0.0776710	+	0.996979i	$0.524748\pi$
$3$	1.60020	0.923873	0.461937	−	0.886913i	$-0.347155\pi$
$3$	1.60020	0.923873	0.461937	+	0.886913i	$0.347155\pi$
$4$	−1.95174	−0.975869
$5$	0	0
$5$	0	0
$6$	−0.351542	−0.143516
$7$	−0.332247	−0.125577	−0.0627887	−	0.998027i	$-0.519999\pi$
$7$	−0.332247	−0.125577	−0.0627887	+	0.998027i	$0.519999\pi$
$8$	0.868145	0.306936
$9$	−0.439374	−0.146458
$10$	0	0
$11$	5.37182	1.61966	0.809832	−	0.586662i	$-0.199558\pi$
$11$	5.37182	1.61966	0.809832	+	0.586662i	$0.199558\pi$
$12$	−3.12316	−0.901579
$13$	0	0
$13$	0	0
$14$	0.0729902	0.0195074
$15$	0	0
$16$	3.71276	0.928189
$17$	5.06430	1.22827	0.614136	−	0.789200i	$-0.289505\pi$
$17$	5.06430	1.22827	0.614136	+	0.789200i	$0.289505\pi$
$18$	0.0965246	0.0227511
$19$	−2.26795	−0.520303	−0.260152	−	0.965568i	$-0.583773\pi$
$19$	−2.26795	−0.520303	−0.260152	+	0.965568i	$0.583773\pi$
$20$	0	0
$21$	−0.531659	−0.116018
$22$	−1.18012	−0.251602
$23$	2.83918	0.592010	0.296005	−	0.955186i	$-0.404346\pi$
$23$	2.83918	0.592010	0.296005	+	0.955186i	$0.404346\pi$
$24$	1.38920	0.283570
$25$	0	0
$26$	0	0
$27$	−5.50367	−1.05918
$28$	0.648458	0.122547
$29$	−2.90348	−0.539162	−0.269581	−	0.962978i	$-0.586885\pi$
$29$	−2.90348	−0.539162	−0.269581	+	0.962978i	$0.586885\pi$
$30$	0	0
$31$	−5.46410	−0.981382	−0.490691	−	0.871334i	$-0.663256\pi$
$31$	−5.46410	−0.981382	−0.490691	+	0.871334i	$0.663256\pi$
$32$	−2.55193	−0.451122
$33$	8.59596	1.49636
$34$	−1.11256	−0.190802
$35$	0	0
$36$	0.857542	0.142924
$37$	5.97201	0.981793	0.490896	−	0.871218i	$-0.336669\pi$
$37$	5.97201	0.981793	0.490896	+	0.871218i	$0.336669\pi$
$38$	0.498239	0.0808250
$39$	0	0
$40$	0	0
$41$	−3.73205	−0.582848	−0.291424	−	0.956594i	$-0.594129\pi$
$41$	−3.73205	−0.582848	−0.291424	+	0.956594i	$0.594129\pi$
$42$	0.116799	0.0180224
$43$	5.06430	0.772298	0.386149	−	0.922436i	$-0.373805\pi$
$43$	5.06430	0.772298	0.386149	+	0.922436i	$0.373805\pi$
$44$	−10.4844	−1.58058
$45$	0	0
$46$	−0.623730	−0.0919640
$47$	8.34285	1.21693	0.608465	−	0.793581i	$-0.291786\pi$
$47$	8.34285	1.21693	0.608465	+	0.793581i	$0.291786\pi$
$48$	5.94114	0.857529
$49$	−6.88961	−0.984230
$50$	0	0
$51$	8.10387	1.13477
$52$	0	0
$53$	1.56063	0.214369	0.107184	−	0.994239i	$-0.465817\pi$
$53$	1.56063	0.214369	0.107184	+	0.994239i	$0.465817\pi$
$54$	1.20908	0.164536
$55$	0	0
$56$	−0.288438	−0.0385442
$57$	−3.62916	−0.480694
$58$	0.637855	0.0837545
$59$	−2.70732	−0.352463	−0.176232	−	0.984349i	$-0.556391\pi$
$59$	−2.70732	−0.352463	−0.176232	+	0.984349i	$0.556391\pi$
$60$	0	0
$61$	14.1039	1.80582	0.902908	−	0.429835i	$-0.141428\pi$
$61$	14.1039	1.80582	0.902908	+	0.429835i	$0.141428\pi$
$62$	1.20039	0.152450
$63$	0.145980	0.0183918
$64$	−6.86488	−0.858111
$65$	0	0
$66$	−1.88842	−0.232448
$67$	10.3322	1.26228	0.631142	−	0.775667i	$-0.282586\pi$
$67$	10.3322	1.26228	0.631142	+	0.775667i	$0.282586\pi$
$68$	−9.88418	−1.19863
$69$	4.54324	0.546942
$70$	0	0
$71$	−12.7973	−1.51876	−0.759382	−	0.650645i	$-0.774498\pi$
$71$	−12.7973	−1.51876	−0.759382	+	0.650645i	$0.774498\pi$
$72$	−0.381440	−0.0449531
$73$	−9.68922	−1.13404	−0.567019	−	0.823705i	$-0.691903\pi$
$73$	−9.68922	−1.13404	−0.567019	+	0.823705i	$0.691903\pi$
$74$	−1.31197	−0.152514
$75$	0	0
$76$	4.42644	0.507748
$77$	−1.78477	−0.203393
$78$	0	0
$79$	4.51851	0.508372	0.254186	−	0.967155i	$-0.418192\pi$
$79$	4.51851	0.508372	0.254186	+	0.967155i	$0.418192\pi$
$80$	0	0
$81$	−7.48883	−0.832092
$82$	0.819883	0.0905409
$83$	−4.26371	−0.468003	−0.234001	−	0.972236i	$-0.575182\pi$
$83$	−4.26371	−0.468003	−0.234001	+	0.972236i	$0.575182\pi$
$84$	1.03766	0.113218
$85$	0	0
$86$	−1.11256	−0.119970
$87$	−4.64613	−0.498117
$88$	4.66351	0.497132
$89$	−3.22584	−0.341938	−0.170969	−	0.985276i	$-0.554690\pi$
$89$	−3.22584	−0.341938	−0.170969	+	0.985276i	$0.554690\pi$
$90$	0	0
$91$	0	0
$92$	−5.54133	−0.577724
$93$	−8.74363	−0.906672
$94$	−1.83281	−0.189040
$95$	0	0
$96$	−4.08359	−0.416780
$97$	−2.50791	−0.254640	−0.127320	−	0.991862i	$-0.540637\pi$
$97$	−2.50791	−0.254640	−0.127320	+	0.991862i	$0.540637\pi$
$98$	1.51356	0.152892
$99$	−2.36023	−0.237213
$100$	0	0
$101$	12.4467	1.23849	0.619247	−	0.785196i	$-0.287438\pi$
$101$	12.4467	1.23849	0.619247	+	0.785196i	$0.287438\pi$
$102$	−1.78031	−0.176277
$103$	15.0247	1.48043	0.740215	−	0.672370i	$-0.234724\pi$
$103$	15.0247	1.48043	0.740215	+	0.672370i	$0.234724\pi$
$104$	0	0
$105$	0	0
$106$	−0.342849	−0.0333004
$107$	13.0643	1.26297	0.631487	−	0.775387i	$-0.282445\pi$
$107$	13.0643	1.26297	0.631487	+	0.775387i	$0.282445\pi$
$108$	10.7417	1.03362
$109$	11.2325	1.07587	0.537937	−	0.842985i	$-0.319204\pi$
$109$	11.2325	1.07587	0.537937	+	0.842985i	$0.319204\pi$
$110$	0	0
$111$	9.55639	0.907052
$112$	−1.23355	−0.116560
$113$	18.3438	1.72564	0.862821	−	0.505509i	$-0.168695\pi$
$113$	18.3438	1.72564	0.862821	+	0.505509i	$0.168695\pi$
$114$	0.797279	0.0746721
$115$	0	0
$116$	5.66682	0.526151
$117$	0	0
$118$	0.594763	0.0547524
$119$	−1.68260	−0.154243
$120$	0	0
$121$	17.8564	1.62331
$122$	−3.09843	−0.280519
$123$	−5.97201	−0.538478
$124$	10.6645	0.957700
$125$	0	0
$126$	−0.0320700	−0.00285702
$127$	−3.23996	−0.287500	−0.143750	−	0.989614i	$-0.545916\pi$
$127$	−3.23996	−0.287500	−0.143750	+	0.989614i	$0.545916\pi$
$128$	6.61199	0.584423
$129$	8.10387	0.713506
$130$	0	0
$131$	0.175664	0.0153478	0.00767390	−	0.999971i	$-0.497557\pi$
$131$	0.175664	0.0153478	0.00767390	+	0.999971i	$0.497557\pi$
$132$	−16.7771	−1.46025
$133$	0.753518	0.0653383
$134$	−2.26986	−0.196086
$135$	0	0
$136$	4.39654	0.377001
$137$	−17.9829	−1.53638	−0.768190	−	0.640221i	$-0.778843\pi$
$137$	−17.9829	−1.53638	−0.768190	+	0.640221i	$0.778843\pi$
$138$	−0.998090	−0.0849631
$139$	11.9861	1.01665	0.508325	−	0.861165i	$-0.330265\pi$
$139$	11.9861	1.01665	0.508325	+	0.861165i	$0.330265\pi$
$140$	0	0
$141$	13.3502	1.12429
$142$	2.81140	0.235928
$143$	0	0
$144$	−1.63129	−0.135941
$145$	0	0
$146$	2.12859	0.176164
$147$	−11.0247	−0.909304
$148$	−11.6558	−0.958101
$149$	3.41041	0.279391	0.139696	−	0.990194i	$-0.455388\pi$
$149$	3.41041	0.279391	0.139696	+	0.990194i	$0.455388\pi$
$150$	0	0
$151$	−7.96141	−0.647890	−0.323945	−	0.946076i	$-0.605009\pi$
$151$	−7.96141	−0.647890	−0.323945	+	0.946076i	$0.605009\pi$
$152$	−1.96891	−0.159700
$153$	−2.22512	−0.179890
$154$	0.392090	0.0315955
$155$	0	0
$156$	0	0
$157$	16.4329	1.31148	0.655742	−	0.754985i	$-0.272356\pi$
$157$	16.4329	1.31148	0.655742	+	0.754985i	$0.272356\pi$
$158$	−0.992658	−0.0789716
$159$	2.49731	0.198049
$160$	0	0
$161$	−0.943307	−0.0743430
$162$	1.64520	0.129259
$163$	17.8072	1.39477	0.697384	−	0.716697i	$-0.254347\pi$
$163$	17.8072	1.39477	0.697384	+	0.716697i	$0.254347\pi$
$164$	7.28398	0.568784
$165$	0	0
$166$	0.936681	0.0727006
$167$	−6.29366	−0.487018	−0.243509	−	0.969899i	$-0.578298\pi$
$167$	−6.29366	−0.487018	−0.243509	+	0.969899i	$0.578298\pi$
$168$	−0.461557	−0.0356099
$169$	0	0
$170$	0	0
$171$	0.996477	0.0762025
$172$	−9.88418	−0.753662
$173$	−15.9751	−1.21457	−0.607283	−	0.794486i	$-0.707740\pi$
$173$	−15.9751	−1.21457	−0.607283	+	0.794486i	$0.707740\pi$
$174$	1.02069	0.0773786
$175$	0	0
$176$	19.9442	1.50335
$177$	−4.33225	−0.325632
$178$	0.708674	0.0531173
$179$	23.6174	1.76525	0.882625	−	0.470079i	$-0.155774\pi$
$179$	23.6174	1.76525	0.882625	+	0.470079i	$0.155774\pi$
$180$	0	0
$181$	−2.62590	−0.195182	−0.0975909	−	0.995227i	$-0.531114\pi$
$181$	−2.62590	−0.195182	−0.0975909	+	0.995227i	$0.531114\pi$
$182$	0	0
$183$	22.5689	1.66834
$184$	2.46482	0.181709
$185$	0	0
$186$	1.92086	0.140844
$187$	27.2045	1.98939
$188$	−16.2831	−1.18756
$189$	1.82858	0.133009
$190$	0	0
$191$	−2.01582	−0.145860	−0.0729298	−	0.997337i	$-0.523235\pi$
$191$	−2.01582	−0.145860	−0.0729298	+	0.997337i	$0.523235\pi$
$192$	−10.9852	−0.792786
$193$	22.8211	1.64270	0.821348	−	0.570427i	$-0.193222\pi$
$193$	22.8211	1.64270	0.821348	+	0.570427i	$0.193222\pi$
$194$	0.550955	0.0395563
$195$	0	0
$196$	13.4467	0.960480
$197$	−0.643026	−0.0458137	−0.0229068	−	0.999738i	$-0.507292\pi$
$197$	−0.643026	−0.0458137	−0.0229068	+	0.999738i	$0.507292\pi$
$198$	0.518513	0.0368491
$199$	3.06684	0.217403	0.108701	−	0.994074i	$-0.465331\pi$
$199$	3.06684	0.217403	0.108701	+	0.994074i	$0.465331\pi$
$200$	0	0
$201$	16.5336	1.16619
$202$	−2.73438	−0.192390
$203$	0.964670	0.0677065
$204$	−15.8166	−1.10739
$205$	0	0
$206$	−3.30074	−0.229973
$207$	−1.24746	−0.0867045
$208$	0	0
$209$	−12.1830	−0.842716
$210$	0	0
$211$	−8.20039	−0.564538	−0.282269	−	0.959335i	$-0.591087\pi$
$211$	−8.20039	−0.564538	−0.282269	+	0.959335i	$0.591087\pi$
$212$	−3.04593	−0.209196
$213$	−20.4782	−1.40314
$214$	−2.87005	−0.196193
$215$	0	0
$216$	−4.77798	−0.325101
$217$	1.81543	0.123239
$218$	−2.46762	−0.167129
$219$	−15.5046	−1.04771
$220$	0	0
$221$	0	0
$222$	−2.09941	−0.140903
$223$	10.2442	0.686002	0.343001	−	0.939335i	$-0.388557\pi$
$223$	10.2442	0.686002	0.343001	+	0.939335i	$0.388557\pi$
$224$	0.847871	0.0566508
$225$	0	0
$226$	−4.02990	−0.268065
$227$	7.04381	0.467514	0.233757	−	0.972295i	$-0.424898\pi$
$227$	7.04381	0.467514	0.233757	+	0.972295i	$0.424898\pi$
$228$	7.08317	0.469095
$229$	−1.32899	−0.0878219	−0.0439109	−	0.999035i	$-0.513982\pi$
$229$	−1.32899	−0.0878219	−0.0439109	+	0.999035i	$0.513982\pi$
$230$	0	0
$231$	−2.85598	−0.187909
$232$	−2.52064	−0.165488
$233$	1.24746	0.0817238	0.0408619	−	0.999165i	$-0.486990\pi$
$233$	1.24746	0.0817238	0.0408619	+	0.999165i	$0.486990\pi$
$234$	0	0
$235$	0	0
$236$	5.28398	0.343958
$237$	7.23050	0.469672
$238$	0.369644	0.0239605
$239$	9.94207	0.643099	0.321549	−	0.946893i	$-0.395796\pi$
$239$	9.94207	0.643099	0.321549	+	0.946893i	$0.395796\pi$
$240$	0	0
$241$	22.5869	1.45495	0.727475	−	0.686134i	$-0.240694\pi$
$241$	22.5869	1.45495	0.727475	+	0.686134i	$0.240694\pi$
$242$	−3.92282	−0.252168
$243$	4.52742	0.290434
$244$	−27.5270	−1.76224
$245$	0	0
$246$	1.31197	0.0836483
$247$	0	0
$248$	−4.74363	−0.301221
$249$	−6.82277	−0.432376
$250$	0	0
$251$	−6.76836	−0.427215	−0.213608	−	0.976920i	$-0.568521\pi$
$251$	−6.76836	−0.427215	−0.213608	+	0.976920i	$0.568521\pi$
$252$	−0.284915	−0.0179480
$253$	15.2515	0.958856
$254$	0.711777	0.0446609
$255$	0	0
$256$	12.2772	0.767325
$257$	10.2538	0.639616	0.319808	−	0.947482i	$-0.396382\pi$
$257$	10.2538	0.639616	0.319808	+	0.947482i	$0.396382\pi$
$258$	−1.78031	−0.110837
$259$	−1.98418	−0.123291
$260$	0	0
$261$	1.27571	0.0789645
$262$	−0.0385910	−0.00238416
$263$	−18.6570	−1.15044	−0.575220	−	0.817999i	$-0.695083\pi$
$263$	−18.6570	−1.15044	−0.575220	+	0.817999i	$0.695083\pi$
$264$	7.46254	0.459287
$265$	0	0
$266$	−0.165538	−0.0101498
$267$	−5.16197	−0.315907
$268$	−20.1658	−1.23182
$269$	17.9579	1.09491	0.547456	−	0.836835i	$-0.315596\pi$
$269$	17.9579	1.09491	0.547456	+	0.836835i	$0.315596\pi$
$270$	0	0
$271$	−30.8977	−1.87690	−0.938450	−	0.345415i	$-0.887738\pi$
$271$	−30.8977	−1.87690	−0.938450	+	0.345415i	$0.887738\pi$
$272$	18.8025	1.14007
$273$	0	0
$274$	3.95060	0.238665
$275$	0	0
$276$	−8.86721	−0.533744
$277$	−26.5045	−1.59250	−0.796250	−	0.604967i	$-0.793186\pi$
$277$	−26.5045	−1.59250	−0.796250	+	0.604967i	$0.793186\pi$
$278$	−2.63320	−0.157929
$279$	2.40078	0.143731
$280$	0	0
$281$	−4.97766	−0.296942	−0.148471	−	0.988917i	$-0.547435\pi$
$281$	−4.97766	−0.296942	−0.148471	+	0.988917i	$0.547435\pi$
$282$	−2.93286	−0.174649
$283$	12.5863	0.748180	0.374090	−	0.927392i	$-0.377955\pi$
$283$	12.5863	0.748180	0.374090	+	0.927392i	$0.377955\pi$
$284$	24.9770	1.48211
$285$	0	0
$286$	0	0
$287$	1.23996	0.0731926
$288$	1.12125	0.0660704
$289$	8.64711	0.508653
$290$	0	0
$291$	−4.01315	−0.235255
$292$	18.9108	1.10667
$293$	16.9176	0.988337	0.494168	−	0.869366i	$-0.335473\pi$
$293$	16.9176	0.988337	0.494168	+	0.869366i	$0.335473\pi$
$294$	2.42199	0.141253
$295$	0	0
$296$	5.18457	0.301347
$297$	−29.5647	−1.71552
$298$	−0.749222	−0.0434012
$299$	0	0
$300$	0	0
$301$	−1.68260	−0.0969832
$302$	1.74902	0.100645
$303$	19.9172	1.14421
$304$	−8.42034	−0.482940
$305$	0	0
$306$	0.488829	0.0279445
$307$	−4.30426	−0.245657	−0.122828	−	0.992428i	$-0.539197\pi$
$307$	−4.30426	−0.245657	−0.122828	+	0.992428i	$0.539197\pi$
$308$	3.48340	0.198485
$309$	24.0425	1.36773
$310$	0	0
$311$	−2.22512	−0.126175	−0.0630875	−	0.998008i	$-0.520095\pi$
$311$	−2.22512	−0.126175	−0.0630875	+	0.998008i	$0.520095\pi$
$312$	0	0
$313$	−7.20887	−0.407469	−0.203735	−	0.979026i	$-0.565308\pi$
$313$	−7.20887	−0.407469	−0.203735	+	0.979026i	$0.565308\pi$
$314$	−3.61008	−0.203729
$315$	0	0
$316$	−8.81895	−0.496105
$317$	−0.321644	−0.0180653	−0.00903266	−	0.999959i	$-0.502875\pi$
$317$	−0.321644	−0.0180653	−0.00903266	+	0.999959i	$0.502875\pi$
$318$	−0.548626	−0.0307654
$319$	−15.5969	−0.873261
$320$	0	0
$321$	20.9054	1.16683
$322$	0.207232	0.0115486
$323$	−11.4856	−0.639074
$324$	14.6162	0.812013
$325$	0	0
$326$	−3.91201	−0.216666
$327$	17.9741	0.993972
$328$	−3.23996	−0.178897
$329$	−2.77188	−0.152819
$330$	0	0
$331$	−16.6320	−0.914178	−0.457089	−	0.889421i	$-0.651108\pi$
$331$	−16.6320	−0.914178	−0.457089	+	0.889421i	$0.651108\pi$
$332$	8.32164	0.456710
$333$	−2.62395	−0.143791
$334$	1.38263	0.0756543
$335$	0	0
$336$	−1.97392	−0.107686
$337$	−24.2186	−1.31927	−0.659636	−	0.751586i	$-0.729289\pi$
$337$	−24.2186	−1.31927	−0.659636	+	0.751586i	$0.729289\pi$
$338$	0	0
$339$	29.3537	1.59427
$340$	0	0
$341$	−29.3521	−1.58951
$342$	−0.218913	−0.0118375
$343$	4.61478	0.249174
$344$	4.39654	0.237046
$345$	0	0
$346$	3.50952	0.188673
$347$	6.27360	0.336784	0.168392	−	0.985720i	$-0.446143\pi$
$347$	6.27360	0.336784	0.168392	+	0.985720i	$0.446143\pi$
$348$	9.06802	0.486097
$349$	−7.06994	−0.378445	−0.189223	−	0.981934i	$-0.560597\pi$
$349$	−7.06994	−0.378445	−0.189223	+	0.981934i	$0.560597\pi$
$350$	0	0
$351$	0	0
$352$	−13.7085	−0.730666
$353$	−21.7898	−1.15976	−0.579878	−	0.814704i	$-0.696900\pi$
$353$	−21.7898	−1.15976	−0.579878	+	0.814704i	$0.696900\pi$
$354$	0.951738	0.0505843
$355$	0	0
$356$	6.29598	0.333687
$357$	−2.69248	−0.142501
$358$	−5.18844	−0.274217
$359$	−23.9737	−1.26528	−0.632642	−	0.774444i	$-0.718029\pi$
$359$	−23.9737	−1.26528	−0.632642	+	0.774444i	$0.718029\pi$
$360$	0	0
$361$	−13.8564	−0.729285
$362$	0.576876	0.0303199
$363$	28.5737	1.49973
$364$	0	0
$365$	0	0
$366$	−4.95810	−0.259164
$367$	−6.39133	−0.333625	−0.166812	−	0.985989i	$-0.553347\pi$
$367$	−6.39133	−0.333625	−0.166812	+	0.985989i	$0.553347\pi$
$368$	10.5412	0.549497
$369$	1.63977	0.0853628
$370$	0	0
$371$	−0.518513	−0.0269198
$372$	17.0653	0.884793
$373$	20.0801	1.03971	0.519855	−	0.854255i	$-0.325986\pi$
$373$	20.0801	1.03971	0.519855	+	0.854255i	$0.325986\pi$
$374$	−5.97647	−0.309036
$375$	0	0
$376$	7.24280	0.373519
$377$	0	0
$378$	−0.401714	−0.0206619
$379$	5.46182	0.280555	0.140277	−	0.990112i	$-0.455201\pi$
$379$	5.46182	0.280555	0.140277	+	0.990112i	$0.455201\pi$
$380$	0	0
$381$	−5.18457	−0.265614
$382$	0.442849	0.0226581
$383$	−5.66775	−0.289609	−0.144804	−	0.989460i	$-0.546255\pi$
$383$	−5.66775	−0.289609	−0.144804	+	0.989460i	$0.546255\pi$
$384$	10.5805	0.539933
$385$	0	0
$386$	−5.01349	−0.255180
$387$	−2.22512	−0.113109
$388$	4.89478	0.248495
$389$	10.6174	0.538325	0.269162	−	0.963095i	$-0.413253\pi$
$389$	10.6174	0.538325	0.269162	+	0.963095i	$0.413253\pi$
$390$	0	0
$391$	14.3784	0.727149
$392$	−5.98118	−0.302095
$393$	0.281096	0.0141794
$394$	0.141264	0.00711679
$395$	0	0
$396$	4.60656	0.231488
$397$	28.0338	1.40697	0.703487	−	0.710708i	$-0.251625\pi$
$397$	28.0338	1.40697	0.703487	+	0.710708i	$0.251625\pi$
$398$	−0.673745	−0.0337718
$399$	1.20578	0.0603643
$400$	0	0
$401$	22.5143	1.12431	0.562155	−	0.827032i	$-0.309973\pi$
$401$	22.5143	1.12431	0.562155	+	0.827032i	$0.309973\pi$
$402$	−3.63222	−0.181159
$403$	0	0
$404$	−24.2927	−1.20861
$405$	0	0
$406$	−0.211925	−0.0105177
$407$	32.0805	1.59017
$408$	7.03533	0.348301
$409$	−4.28772	−0.212014	−0.106007	−	0.994365i	$-0.533807\pi$
$409$	−4.28772	−0.212014	−0.106007	+	0.994365i	$0.533807\pi$
$410$	0	0
$411$	−28.7761	−1.41942
$412$	−29.3243	−1.44471
$413$	0.899499	0.0442614
$414$	0.274051	0.0134689
$415$	0	0
$416$	0	0
$417$	19.1802	0.939257
$418$	2.67645	0.130909
$419$	17.7116	0.865266	0.432633	−	0.901570i	$-0.357585\pi$
$419$	17.7116	0.865266	0.432633	+	0.901570i	$0.357585\pi$
$420$	0	0
$421$	−12.8787	−0.627672	−0.313836	−	0.949477i	$-0.601614\pi$
$421$	−12.8787	−0.627672	−0.313836	+	0.949477i	$0.601614\pi$
$422$	1.80152	0.0876965
$423$	−3.66563	−0.178229
$424$	1.35485	0.0657973
$425$	0	0
$426$	4.49880	0.217967
$427$	−4.68596	−0.226770
$428$	−25.4981	−1.23250
$429$	0	0
$430$	0	0
$431$	−9.49845	−0.457524	−0.228762	−	0.973482i	$-0.573468\pi$
$431$	−9.49845	−0.457524	−0.228762	+	0.973482i	$0.573468\pi$
$432$	−20.4338	−0.983121
$433$	1.39628	0.0671010	0.0335505	−	0.999437i	$-0.489319\pi$
$433$	1.39628	0.0671010	0.0335505	+	0.999437i	$0.489319\pi$
$434$	−0.398826	−0.0191443
$435$	0	0
$436$	−21.9228	−1.04991
$437$	−6.43911	−0.308024
$438$	3.40617	0.162753
$439$	4.16180	0.198632	0.0993159	−	0.995056i	$-0.468335\pi$
$439$	4.16180	0.198632	0.0993159	+	0.995056i	$0.468335\pi$
$440$	0	0
$441$	3.02711	0.144148
$442$	0	0
$443$	−9.54563	−0.453526	−0.226763	−	0.973950i	$-0.572814\pi$
$443$	−9.54563	−0.453526	−0.226763	+	0.973950i	$0.572814\pi$
$444$	−18.6516	−0.885164
$445$	0	0
$446$	−2.25052	−0.106565
$447$	5.45732	0.258122
$448$	2.28083	0.107759
$449$	−21.7171	−1.02489	−0.512446	−	0.858720i	$-0.671260\pi$
$449$	−21.7171	−1.02489	−0.512446	+	0.858720i	$0.671260\pi$
$450$	0	0
$451$	−20.0479	−0.944018
$452$	−35.8023	−1.68400
$453$	−12.7398	−0.598569
$454$	−1.54743	−0.0726246
$455$	0	0
$456$	−3.15064	−0.147542
$457$	4.72259	0.220914	0.110457	−	0.993881i	$-0.464769\pi$
$457$	4.72259	0.220914	0.110457	+	0.993881i	$0.464769\pi$
$458$	0.291961	0.0136424
$459$	−27.8722	−1.30096
$460$	0	0
$461$	1.78151	0.0829730	0.0414865	−	0.999139i	$-0.486791\pi$
$461$	1.78151	0.0829730	0.0414865	+	0.999139i	$0.486791\pi$
$462$	0.627421	0.0291902
$463$	6.80200	0.316116	0.158058	−	0.987430i	$-0.449477\pi$
$463$	6.80200	0.316116	0.158058	+	0.987430i	$0.449477\pi$
$464$	−10.7799	−0.500444
$465$	0	0
$466$	−0.274051	−0.0126952
$467$	−18.2374	−0.843927	−0.421963	−	0.906613i	$-0.638659\pi$
$467$	−18.2374	−0.843927	−0.421963	+	0.906613i	$0.638659\pi$
$468$	0	0
$469$	−3.43285	−0.158514
$470$	0	0
$471$	26.2958	1.21165
$472$	−2.35035	−0.108184
$473$	27.2045	1.25086
$474$	−1.58845	−0.0729598
$475$	0	0
$476$	3.28398	0.150521
$477$	−0.685698	−0.0313960
$478$	−2.18414	−0.0999003
$479$	35.1807	1.60745	0.803724	−	0.595002i	$-0.202849\pi$
$479$	35.1807	1.60745	0.803724	+	0.595002i	$0.202849\pi$
$480$	0	0
$481$	0	0
$482$	−4.96204	−0.226015
$483$	−1.50948	−0.0686835
$484$	−34.8510	−1.58414
$485$	0	0
$486$	−0.994615	−0.0451166
$487$	−10.3040	−0.466919	−0.233459	−	0.972367i	$-0.575005\pi$
$487$	−10.3040	−0.466919	−0.233459	+	0.972367i	$0.575005\pi$
$488$	12.2442	0.554269
$489$	28.4950	1.28859
$490$	0	0
$491$	9.33198	0.421147	0.210573	−	0.977578i	$-0.432467\pi$
$491$	9.33198	0.421147	0.210573	+	0.977578i	$0.432467\pi$
$492$	11.6558	0.525484
$493$	−14.7041	−0.662238
$494$	0	0
$495$	0	0
$496$	−20.2869	−0.910907
$497$	4.25187	0.190722
$498$	1.49887	0.0671661
$499$	−23.9421	−1.07179	−0.535897	−	0.844283i	$-0.680026\pi$
$499$	−23.9421	−1.07179	−0.535897	+	0.844283i	$0.680026\pi$
$500$	0	0
$501$	−10.0711	−0.449943
$502$	1.48692	0.0663645
$503$	−42.1443	−1.87912	−0.939560	−	0.342385i	$-0.888765\pi$
$503$	−42.1443	−1.87912	−0.939560	+	0.342385i	$0.888765\pi$
$504$	0.126732	0.00564510
$505$	0	0
$506$	−3.35056	−0.148951
$507$	0	0
$508$	6.32355	0.280562
$509$	33.5602	1.48753	0.743765	−	0.668441i	$-0.233038\pi$
$509$	33.5602	1.48753	0.743765	+	0.668441i	$0.233038\pi$
$510$	0	0
$511$	3.21921	0.142409
$512$	−15.9211	−0.703621
$513$	12.4820	0.551096
$514$	−2.25263	−0.0993593
$515$	0	0
$516$	−15.8166	−0.696288
$517$	44.8162	1.97102
$518$	0.435898	0.0191523
$519$	−25.5633	−1.12210
$520$	0	0
$521$	12.4649	0.546098	0.273049	−	0.962000i	$-0.411968\pi$
$521$	12.4649	0.546098	0.273049	+	0.962000i	$0.411968\pi$
$522$	−0.280257	−0.0122665
$523$	5.65956	0.247475	0.123738	−	0.992315i	$-0.460512\pi$
$523$	5.65956	0.247475	0.123738	+	0.992315i	$0.460512\pi$
$524$	−0.342849	−0.0149774
$525$	0	0
$526$	4.09870	0.178712
$527$	−27.6718	−1.20540
$528$	31.9147	1.38891
$529$	−14.9391	−0.649525
$530$	0	0
$531$	1.18953	0.0516211
$532$	−1.47067	−0.0637616
$533$	0	0
$534$	1.13402	0.0490737
$535$	0	0
$536$	8.96989	0.387440
$537$	37.7925	1.63087
$538$	−3.94511	−0.170086
$539$	−37.0097	−1.59412
$540$	0	0
$541$	15.4750	0.665321	0.332660	−	0.943047i	$-0.392054\pi$
$541$	15.4750	0.665321	0.332660	+	0.943047i	$0.392054\pi$
$542$	6.78781	0.291562
$543$	−4.20196	−0.180323
$544$	−12.9237	−0.554101
$545$	0	0
$546$	0	0
$547$	−25.1765	−1.07647	−0.538234	−	0.842795i	$-0.680908\pi$
$547$	−25.1765	−1.07647	−0.538234	+	0.842795i	$0.680908\pi$
$548$	35.0979	1.49931
$549$	−6.19687	−0.264476
$550$	0	0
$551$	6.58493	0.280528
$552$	3.94419	0.167876
$553$	−1.50126	−0.0638401
$554$	5.82269	0.247382
$555$	0	0
$556$	−23.3938	−0.992118
$557$	42.3489	1.79438	0.897190	−	0.441645i	$-0.145605\pi$
$557$	42.3489	1.79438	0.897190	+	0.441645i	$0.145605\pi$
$558$	−0.527420	−0.0223275
$559$	0	0
$560$	0	0
$561$	43.5325	1.83794
$562$	1.09353	0.0461276
$563$	−23.7905	−1.00265	−0.501326	−	0.865259i	$-0.667154\pi$
$563$	−23.7905	−1.00265	−0.501326	+	0.865259i	$0.667154\pi$
$564$	−26.0561	−1.09716
$565$	0	0
$566$	−2.76505	−0.116224
$567$	2.48814	0.104492
$568$	−11.1099	−0.466162
$569$	−26.7421	−1.12109	−0.560543	−	0.828125i	$-0.689407\pi$
$569$	−26.7421	−1.12109	−0.560543	+	0.828125i	$0.689407\pi$
$570$	0	0
$571$	−16.7159	−0.699539	−0.349769	−	0.936836i	$-0.613740\pi$
$571$	−16.7159	−0.699539	−0.349769	+	0.936836i	$0.613740\pi$
$572$	0	0
$573$	−3.22571	−0.134756
$574$	−0.272403	−0.0113699
$575$	0	0
$576$	3.01625	0.125677
$577$	−20.6768	−0.860786	−0.430393	−	0.902642i	$-0.641625\pi$
$577$	−20.6768	−0.860786	−0.430393	+	0.902642i	$0.641625\pi$
$578$	−1.89966	−0.0790153
$579$	36.5182	1.51764
$580$	0	0
$581$	1.41660	0.0587706
$582$	0.881636	0.0365450
$583$	8.38340	0.347205
$584$	−8.41165	−0.348076
$585$	0	0
$586$	−3.71657	−0.153530
$587$	20.7972	0.858391	0.429196	−	0.903212i	$-0.358797\pi$
$587$	20.7972	0.858391	0.429196	+	0.903212i	$0.358797\pi$
$588$	21.5174	0.887362
$589$	12.3923	0.510616
$590$	0	0
$591$	−1.02897	−0.0423260
$592$	22.1726	0.911289
$593$	−21.8475	−0.897169	−0.448585	−	0.893740i	$-0.648072\pi$
$593$	−21.8475	−0.897169	−0.448585	+	0.893740i	$0.648072\pi$
$594$	6.49498	0.266492
$595$	0	0
$596$	−6.65622	−0.272649
$597$	4.90755	0.200853
$598$	0	0
$599$	−3.58040	−0.146291	−0.0731456	−	0.997321i	$-0.523304\pi$
$599$	−3.58040	−0.146291	−0.0731456	+	0.997321i	$0.523304\pi$
$600$	0	0
$601$	21.3486	0.870829	0.435414	−	0.900230i	$-0.356602\pi$
$601$	21.3486	0.870829	0.435414	+	0.900230i	$0.356602\pi$
$602$	0.369644	0.0150656
$603$	−4.53972	−0.184872
$604$	15.5386	0.632256
$605$	0	0
$606$	−4.37554	−0.177744
$607$	3.29976	0.133933	0.0669665	−	0.997755i	$-0.478668\pi$
$607$	3.29976	0.133933	0.0669665	+	0.997755i	$0.478668\pi$
$608$	5.78766	0.234720
$609$	1.54366	0.0625523
$610$	0	0
$611$	0	0
$612$	4.34285	0.175549
$613$	9.88635	0.399306	0.199653	−	0.979867i	$-0.436019\pi$
$613$	9.88635	0.399306	0.199653	+	0.979867i	$0.436019\pi$
$614$	0.945589	0.0381609
$615$	0	0
$616$	−1.54944	−0.0624286
$617$	−45.7169	−1.84049	−0.920246	−	0.391339i	$-0.872012\pi$
$617$	−45.7169	−1.84049	−0.920246	+	0.391339i	$0.872012\pi$
$618$	−5.28182	−0.212466
$619$	−19.9143	−0.800425	−0.400212	−	0.916422i	$-0.631064\pi$
$619$	−19.9143	−0.800425	−0.400212	+	0.916422i	$0.631064\pi$
$620$	0	0
$621$	−15.6259	−0.627046
$622$	0.488829	0.0196003
$623$	1.07177	0.0429397
$624$	0	0
$625$	0	0
$626$	1.58369	0.0632971
$627$	−19.4952	−0.778563
$628$	−32.0726	−1.27984
$629$	30.2440	1.20591
$630$	0	0
$631$	−14.5958	−0.581050	−0.290525	−	0.956867i	$-0.593830\pi$
$631$	−14.5958	−0.581050	−0.290525	+	0.956867i	$0.593830\pi$
$632$	3.92272	0.156038
$633$	−13.1222	−0.521562
$634$	0.0706609	0.00280630
$635$	0	0
$636$	−4.87409	−0.193270
$637$	0	0
$638$	3.42644	0.135654
$639$	5.62281	0.222435
$640$	0	0
$641$	−14.1637	−0.559431	−0.279716	−	0.960083i	$-0.590240\pi$
$641$	−14.1637	−0.559431	−0.279716	+	0.960083i	$0.590240\pi$
$642$	−4.59265	−0.181257
$643$	16.7716	0.661408	0.330704	−	0.943735i	$-0.392714\pi$
$643$	16.7716	0.661408	0.330704	+	0.943735i	$0.392714\pi$
$644$	1.84109	0.0725490
$645$	0	0
$646$	2.52323	0.0992751
$647$	2.99168	0.117615	0.0588075	−	0.998269i	$-0.481270\pi$
$647$	2.99168	0.117615	0.0588075	+	0.998269i	$0.481270\pi$
$648$	−6.50139	−0.255399
$649$	−14.5432	−0.570872
$650$	0	0
$651$	2.90504	0.113858
$652$	−34.7550	−1.36111
$653$	11.6643	0.456461	0.228230	−	0.973607i	$-0.426706\pi$
$653$	11.6643	0.456461	0.228230	+	0.973607i	$0.426706\pi$
$654$	−3.94868	−0.154406
$655$	0	0
$656$	−13.8562	−0.540993
$657$	4.25719	0.166089
$658$	0.608946	0.0237392
$659$	−1.81047	−0.0705260	−0.0352630	−	0.999378i	$-0.511227\pi$
$659$	−1.81047	−0.0705260	−0.0352630	+	0.999378i	$0.511227\pi$
$660$	0	0
$661$	12.3406	0.479992	0.239996	−	0.970774i	$-0.422854\pi$
$661$	12.3406	0.479992	0.239996	+	0.970774i	$0.422854\pi$
$662$	3.65383	0.142010
$663$	0	0
$664$	−3.70152	−0.143647
$665$	0	0
$666$	0.576446	0.0223368
$667$	−8.24348	−0.319189
$668$	12.2836	0.475265
$669$	16.3927	0.633779
$670$	0	0
$671$	75.7634	2.92481
$672$	1.35676	0.0523381
$673$	−9.26625	−0.357188	−0.178594	−	0.983923i	$-0.557155\pi$
$673$	−9.26625	−0.357188	−0.178594	+	0.983923i	$0.557155\pi$
$674$	5.32051	0.204938
$675$	0	0
$676$	0	0
$677$	−13.8984	−0.534158	−0.267079	−	0.963675i	$-0.586059\pi$
$677$	−13.8984	−0.534158	−0.267079	+	0.963675i	$0.586059\pi$
$678$	−6.44863	−0.247658
$679$	0.833244	0.0319770
$680$	0	0
$681$	11.2715	0.431924
$682$	6.44828	0.246917
$683$	−37.7512	−1.44451	−0.722255	−	0.691626i	$-0.756895\pi$
$683$	−37.7512	−1.44451	−0.722255	+	0.691626i	$0.756895\pi$
$684$	−1.94486	−0.0743637
$685$	0	0
$686$	−1.01381	−0.0387073
$687$	−2.12664	−0.0811363
$688$	18.8025	0.716838
$689$	0	0
$690$	0	0
$691$	1.65291	0.0628797	0.0314399	−	0.999506i	$-0.489991\pi$
$691$	1.65291	0.0628797	0.0314399	+	0.999506i	$0.489991\pi$
$692$	31.1792	1.18526
$693$	0.784180	0.0297885
$694$	−1.37823	−0.0523168
$695$	0	0
$696$	−4.03351	−0.152890
$697$	−18.9002	−0.715897
$698$	1.55317	0.0587885
$699$	1.99618	0.0755025
$700$	0	0
$701$	20.4819	0.773590	0.386795	−	0.922166i	$-0.373582\pi$
$701$	20.4819	0.773590	0.386795	+	0.922166i	$0.373582\pi$
$702$	0	0
$703$	−13.5442	−0.510830
$704$	−36.8769	−1.38985
$705$	0	0
$706$	4.78694	0.180159
$707$	−4.13538	−0.155527
$708$	8.45541	0.317774
$709$	−21.9417	−0.824039	−0.412020	−	0.911175i	$-0.635177\pi$
$709$	−21.9417	−0.824039	−0.412020	+	0.911175i	$0.635177\pi$
$710$	0	0
$711$	−1.98532	−0.0744552
$712$	−2.80049	−0.104953
$713$	−15.5136	−0.580987
$714$	0.591503	0.0221364
$715$	0	0
$716$	−46.0950	−1.72265
$717$	15.9093	0.594142
$718$	5.26671	0.196552
$719$	38.8475	1.44877	0.724384	−	0.689397i	$-0.242124\pi$
$719$	38.8475	1.44877	0.724384	+	0.689397i	$0.242124\pi$
$720$	0	0
$721$	−4.99191	−0.185909
$722$	3.04407	0.113289
$723$	36.1434	1.34419
$724$	5.12507	0.190472
$725$	0	0
$726$	−6.27728	−0.232972
$727$	30.6598	1.13711	0.568555	−	0.822645i	$-0.307503\pi$
$727$	30.6598	1.13711	0.568555	+	0.822645i	$0.307503\pi$
$728$	0	0
$729$	29.7112	1.10042
$730$	0	0
$731$	25.6471	0.948593
$732$	−44.0487	−1.62809
$733$	24.3858	0.900709	0.450355	−	0.892850i	$-0.351298\pi$
$733$	24.3858	0.900709	0.450355	+	0.892850i	$0.351298\pi$
$734$	1.40409	0.0518259
$735$	0	0
$736$	−7.24539	−0.267069
$737$	55.5029	2.04448
$738$	−0.360235	−0.0132604
$739$	−38.2788	−1.40811	−0.704054	−	0.710146i	$-0.748629\pi$
$739$	−38.2788	−1.40811	−0.704054	+	0.710146i	$0.748629\pi$
$740$	0	0
$741$	0	0
$742$	0.113910	0.00418178
$743$	−40.0079	−1.46775	−0.733874	−	0.679286i	$-0.762290\pi$
$743$	−40.0079	−1.46775	−0.733874	+	0.679286i	$0.762290\pi$
$744$	−7.59074	−0.278290
$745$	0	0
$746$	−4.41134	−0.161511
$747$	1.87336	0.0685427
$748$	−53.0960	−1.94138
$749$	−4.34057	−0.158601
$750$	0	0
$751$	25.6020	0.934230	0.467115	−	0.884197i	$-0.345293\pi$
$751$	25.6020	0.934230	0.467115	+	0.884197i	$0.345293\pi$
$752$	30.9750	1.12954
$753$	−10.8307	−0.394693
$754$	0	0
$755$	0	0
$756$	−3.56890	−0.129800
$757$	1.84848	0.0671840	0.0335920	−	0.999436i	$-0.489305\pi$
$757$	1.84848	0.0671840	0.0335920	+	0.999436i	$0.489305\pi$
$758$	−1.19989	−0.0435820
$759$	24.4055	0.885862
$760$	0	0
$761$	−26.2124	−0.950199	−0.475099	−	0.879932i	$-0.657588\pi$
$761$	−26.2124	−0.950199	−0.475099	+	0.879932i	$0.657588\pi$
$762$	1.13898	0.0412610
$763$	−3.73195	−0.135106
$764$	3.93435	0.142340
$765$	0	0
$766$	1.24513	0.0449884
$767$	0	0
$768$	19.6459	0.708911
$769$	44.3495	1.59928	0.799641	−	0.600478i	$-0.205023\pi$
$769$	44.3495	1.59928	0.799641	+	0.600478i	$0.205023\pi$
$770$	0	0
$771$	16.4081	0.590924
$772$	−44.5408	−1.60306
$773$	−23.2638	−0.836742	−0.418371	−	0.908276i	$-0.637399\pi$
$773$	−23.2638	−0.836742	−0.418371	+	0.908276i	$0.637399\pi$
$774$	0.488829	0.0175706
$775$	0	0
$776$	−2.17723	−0.0781580
$777$	−3.17508	−0.113905
$778$	−2.33251	−0.0836245
$779$	8.46410	0.303258
$780$	0	0
$781$	−68.7449	−2.45989
$782$	−3.15875	−0.112957
$783$	15.9798	0.571071
$784$	−25.5794	−0.913552
$785$	0	0
$786$	−0.0617531	−0.00220266
$787$	−47.9133	−1.70793	−0.853963	−	0.520334i	$-0.825808\pi$
$787$	−47.9133	−1.70793	−0.853963	+	0.520334i	$0.825808\pi$
$788$	1.25502	0.0447081
$789$	−29.8548	−1.06286
$790$	0	0
$791$	−6.09467	−0.216702
$792$	−2.04903	−0.0728090
$793$	0	0
$794$	−6.15865	−0.218562
$795$	0	0
$796$	−5.98567	−0.212156
$797$	20.6952	0.733060	0.366530	−	0.930406i	$-0.380546\pi$
$797$	20.6952	0.733060	0.366530	+	0.930406i	$0.380546\pi$
$798$	−0.264893	−0.00937712
$799$	42.2507	1.49472
$800$	0	0
$801$	1.41735	0.0500795
$802$	−4.94609	−0.174653
$803$	−52.0487	−1.83676
$804$	−32.2693	−1.13805
$805$	0	0
$806$	0	0
$807$	28.7361	1.01156
$808$	10.8056	0.380138
$809$	15.8915	0.558714	0.279357	−	0.960187i	$-0.409879\pi$
$809$	15.8915	0.558714	0.279357	+	0.960187i	$0.409879\pi$
$810$	0	0
$811$	23.8796	0.838525	0.419263	−	0.907865i	$-0.362289\pi$
$811$	23.8796	0.838525	0.419263	+	0.907865i	$0.362289\pi$
$812$	−1.88278	−0.0660727
$813$	−49.4423	−1.73402
$814$	−7.04768	−0.247021
$815$	0	0
$816$	30.0877	1.05328
$817$	−11.4856	−0.401829
$818$	0.941956	0.0329347
$819$	0	0
$820$	0	0
$821$	−15.9097	−0.555251	−0.277626	−	0.960689i	$-0.589547\pi$
$821$	−15.9097	−0.555251	−0.277626	+	0.960689i	$0.589547\pi$
$822$	6.32174	0.220496
$823$	14.8115	0.516295	0.258147	−	0.966106i	$-0.416888\pi$
$823$	14.8115	0.516295	0.258147	+	0.966106i	$0.416888\pi$
$824$	13.0436	0.454397
$825$	0	0
$826$	−0.197608	−0.00687566
$827$	33.9498	1.18055	0.590275	−	0.807202i	$-0.299019\pi$
$827$	33.9498	1.18055	0.590275	+	0.807202i	$0.299019\pi$
$828$	2.43472	0.0846122
$829$	23.3146	0.809749	0.404875	−	0.914372i	$-0.367315\pi$
$829$	23.3146	0.809749	0.404875	+	0.914372i	$0.367315\pi$
$830$	0	0
$831$	−42.4124	−1.47127
$832$	0	0
$833$	−34.8910	−1.20890
$834$	−4.21363	−0.145906
$835$	0	0
$836$	23.7780	0.822380
$837$	30.0726	1.03946
$838$	−3.89100	−0.134412
$839$	14.7930	0.510710	0.255355	−	0.966847i	$-0.417808\pi$
$839$	14.7930	0.510710	0.255355	+	0.966847i	$0.417808\pi$
$840$	0	0
$841$	−20.5698	−0.709305
$842$	2.82929	0.0975038
$843$	−7.96523	−0.274337
$844$	16.0050	0.550915
$845$	0	0
$846$	0.805291	0.0276865
$847$	−5.93273	−0.203851
$848$	5.79422	0.198974
$849$	20.1406	0.691223
$850$	0	0
$851$	16.9556	0.581231
$852$	39.9681	1.36929
$853$	−16.3452	−0.559650	−0.279825	−	0.960051i	$-0.590276\pi$
$853$	−16.3452	−0.559650	−0.279825	+	0.960051i	$0.590276\pi$
$854$	1.02944	0.0352268
$855$	0	0
$856$	11.3417	0.387651
$857$	34.1418	1.16626	0.583132	−	0.812378i	$-0.301827\pi$
$857$	34.1418	1.16626	0.583132	+	0.812378i	$0.301827\pi$
$858$	0	0
$859$	−45.1996	−1.54219	−0.771096	−	0.636719i	$-0.780291\pi$
$859$	−45.1996	−1.54219	−0.771096	+	0.636719i	$0.780291\pi$
$860$	0	0
$861$	1.98418	0.0676207
$862$	2.08669	0.0710728
$863$	4.75058	0.161712	0.0808559	−	0.996726i	$-0.474235\pi$
$863$	4.75058	0.161712	0.0808559	+	0.996726i	$0.474235\pi$
$864$	14.0450	0.477821
$865$	0	0
$866$	−0.306745	−0.0104236
$867$	13.8371	0.469931
$868$	−3.54324	−0.120265
$869$	24.2726	0.823392
$870$	0	0
$871$	0	0
$872$	9.75140	0.330224
$873$	1.10191	0.0372940
$874$	1.41459	0.0478492
$875$	0	0
$876$	30.2610	1.02242
$877$	2.25506	0.0761481	0.0380741	−	0.999275i	$-0.487878\pi$
$877$	2.25506	0.0761481	0.0380741	+	0.999275i	$0.487878\pi$
$878$	−0.914293	−0.0308559
$879$	27.0715	0.913098
$880$	0	0
$881$	−2.98304	−0.100501	−0.0502507	−	0.998737i	$-0.516002\pi$
$881$	−2.98304	−0.100501	−0.0502507	+	0.998737i	$0.516002\pi$
$882$	−0.665017	−0.0223923
$883$	−28.2874	−0.951947	−0.475973	−	0.879460i	$-0.657904\pi$
$883$	−28.2874	−0.951947	−0.475973	+	0.879460i	$0.657904\pi$
$884$	0	0
$885$	0	0
$886$	2.09705	0.0704517
$887$	27.9816	0.939531	0.469766	−	0.882791i	$-0.344338\pi$
$887$	27.9816	0.939531	0.469766	+	0.882791i	$0.344338\pi$
$888$	8.29633	0.278407
$889$	1.07647	0.0361035
$890$	0	0
$891$	−40.2286	−1.34771
$892$	−19.9940	−0.669448
$893$	−18.9212	−0.633172
$894$	−1.19890	−0.0400973
$895$	0	0
$896$	−2.19681	−0.0733903
$897$	0	0
$898$	4.77095	0.159209
$899$	15.8649	0.529124
$900$	0	0
$901$	7.90348	0.263303
$902$	4.40426	0.146646
$903$	−2.69248	−0.0896002
$904$	15.9251	0.529661
$905$	0	0
$906$	2.79877	0.0929829
$907$	−16.5520	−0.549600	−0.274800	−	0.961501i	$-0.588612\pi$
$907$	−16.5520	−0.549600	−0.274800	+	0.961501i	$0.588612\pi$
$908$	−13.7477	−0.456232
$909$	−5.46876	−0.181387
$910$	0	0
$911$	7.04863	0.233532	0.116766	−	0.993159i	$-0.462747\pi$
$911$	7.04863	0.233532	0.116766	+	0.993159i	$0.462747\pi$
$912$	−13.4742	−0.446175
$913$	−22.9039	−0.758007
$914$	−1.03749	−0.0343172
$915$	0	0
$916$	2.59383	0.0857026
$917$	−0.0583636	−0.00192734
$918$	6.12316	0.202094
$919$	16.5438	0.545728	0.272864	−	0.962053i	$-0.412029\pi$
$919$	16.5438	0.545728	0.272864	+	0.962053i	$0.412029\pi$
$920$	0	0
$921$	−6.88766	−0.226956
$922$	−0.391374	−0.0128892
$923$	0	0
$924$	5.57412	0.183375
$925$	0	0
$926$	−1.49431	−0.0491060
$927$	−6.60147	−0.216821
$928$	7.40948	0.243228
$929$	−33.8367	−1.11015	−0.555074	−	0.831801i	$-0.687310\pi$
$929$	−33.8367	−1.11015	−0.555074	+	0.831801i	$0.687310\pi$
$930$	0	0
$931$	15.6253	0.512098
$932$	−2.43472	−0.0797517
$933$	−3.56063	−0.116570
$934$	4.00652	0.131097
$935$	0	0
$936$	0	0
$937$	30.4606	0.995104	0.497552	−	0.867434i	$-0.334232\pi$
$937$	30.4606	0.995104	0.497552	+	0.867434i	$0.334232\pi$
$938$	0.754153	0.0246240
$939$	−11.5356	−0.376450
$940$	0	0
$941$	38.2101	1.24561	0.622807	−	0.782375i	$-0.285992\pi$
$941$	38.2101	1.24561	0.622807	+	0.782375i	$0.285992\pi$
$942$	−5.77684	−0.188220
$943$	−10.5960	−0.345052
$944$	−10.0516	−0.327153
$945$	0	0
$946$	−5.97647	−0.194312
$947$	52.4482	1.70434	0.852169	−	0.523266i	$-0.175287\pi$
$947$	52.4482	1.70434	0.852169	+	0.523266i	$0.175287\pi$
$948$	−14.1120	−0.458338
$949$	0	0
$950$	0	0
$951$	−0.514693	−0.0166901
$952$	−1.46074	−0.0473427
$953$	−39.7500	−1.28763	−0.643814	−	0.765182i	$-0.722649\pi$
$953$	−39.7500	−1.28763	−0.643814	+	0.765182i	$0.722649\pi$
$954$	0.150639	0.00487711
$955$	0	0
$956$	−19.4043	−0.627580
$957$	−24.9581	−0.806782
$958$	−7.72874	−0.249704
$959$	5.97475	0.192935
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	−5.74011	−0.184972
$964$	−44.0837	−1.41984
$965$	0	0
$966$	0.331612	0.0106694
$967$	25.7857	0.829214	0.414607	−	0.910001i	$-0.363919\pi$
$967$	25.7857	0.829214	0.414607	+	0.910001i	$0.363919\pi$
$968$	15.5019	0.498251
$969$	−18.3792	−0.590424
$970$	0	0
$971$	−55.5252	−1.78189	−0.890945	−	0.454111i	$-0.849957\pi$
$971$	−55.5252	−1.78189	−0.890945	+	0.454111i	$0.849957\pi$
$972$	−8.83634	−0.283426
$973$	−3.98235	−0.127668
$974$	2.26365	0.0725321
$975$	0	0
$976$	52.3642	1.67614
$977$	−40.5161	−1.29622	−0.648112	−	0.761545i	$-0.724441\pi$
$977$	−40.5161	−1.29622	−0.648112	+	0.761545i	$0.724441\pi$
$978$	−6.25998	−0.200172
$979$	−17.3286	−0.553824
$980$	0	0
$981$	−4.93525	−0.157570
$982$	−2.05011	−0.0654218
$983$	34.8059	1.11014	0.555068	−	0.831805i	$-0.312692\pi$
$983$	34.8059	1.11014	0.555068	+	0.831805i	$0.312692\pi$
$984$	−5.18457	−0.165278
$985$	0	0
$986$	3.23029	0.102873
$987$	−4.43555	−0.141185
$988$	0	0
$989$	14.3784	0.457208
$990$	0	0
$991$	43.8855	1.39407	0.697034	−	0.717038i	$-0.254503\pi$
$991$	43.8855	1.39407	0.697034	+	0.717038i	$0.254503\pi$
$992$	13.9440	0.442723
$993$	−26.6145	−0.844584
$994$	−0.934079	−0.0296272
$995$	0	0
$996$	13.3163	0.421942
$997$	5.49137	0.173914	0.0869568	−	0.996212i	$-0.472286\pi$
$997$	5.49137	0.173914	0.0869568	+	0.996212i	$0.472286\pi$
$998$	5.25976	0.166495
$999$	−32.8680	−1.03990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bl.1.2		4
5.4	even	2		845.2.a.l.1.3		4
13.6	odd	12		325.2.n.d.101.3		8
13.11	odd	12		325.2.n.d.251.3		8
13.12	even	2		4225.2.a.bi.1.3		4
15.14	odd	2		7605.2.a.cj.1.2		4
65.4	even	6		845.2.e.m.146.3		8
65.9	even	6		845.2.e.n.146.2		8
65.19	odd	12		65.2.m.a.36.2	✓	8
65.24	odd	12		65.2.m.a.56.2	yes	8
65.29	even	6		845.2.e.n.191.2		8
65.32	even	12		325.2.m.b.49.3		8
65.34	odd	4		845.2.c.g.506.5		8
65.37	even	12		325.2.m.c.199.2		8
65.44	odd	4		845.2.c.g.506.4		8
65.49	even	6		845.2.e.m.191.3		8
65.54	odd	12		845.2.m.g.316.3		8
65.58	even	12		325.2.m.c.49.2		8
65.59	odd	12		845.2.m.g.361.3		8
65.63	even	12		325.2.m.b.199.3		8
65.64	even	2		845.2.a.m.1.2		4
195.89	even	12		585.2.bu.c.316.3		8
195.149	even	12		585.2.bu.c.361.3		8
195.194	odd	2		7605.2.a.cf.1.3		4
260.19	even	12		1040.2.da.b.881.2		8
260.219	even	12		1040.2.da.b.641.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.m.a.36.2	✓	8	65.19	odd	12
65.2.m.a.56.2	yes	8	65.24	odd	12
325.2.m.b.49.3		8	65.32	even	12
325.2.m.b.199.3		8	65.63	even	12
325.2.m.c.49.2		8	65.58	even	12
325.2.m.c.199.2		8	65.37	even	12
325.2.n.d.101.3		8	13.6	odd	12
325.2.n.d.251.3		8	13.11	odd	12
585.2.bu.c.316.3		8	195.89	even	12
585.2.bu.c.361.3		8	195.149	even	12
845.2.a.l.1.3		4	5.4	even	2
845.2.a.m.1.2		4	65.64	even	2
845.2.c.g.506.4		8	65.44	odd	4
845.2.c.g.506.5		8	65.34	odd	4
845.2.e.m.146.3		8	65.4	even	6
845.2.e.m.191.3		8	65.49	even	6
845.2.e.n.146.2		8	65.9	even	6
845.2.e.n.191.2		8	65.29	even	6
845.2.m.g.316.3		8	65.54	odd	12
845.2.m.g.361.3		8	65.59	odd	12
1040.2.da.b.641.2		8	260.219	even	12
1040.2.da.b.881.2		8	260.19	even	12
4225.2.a.bi.1.3		4	13.12	even	2
4225.2.a.bl.1.2		4	1.1	even	1	trivial
7605.2.a.cf.1.3		4	195.194	odd	2
7605.2.a.cj.1.2		4	15.14	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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.219687 q^{2} +1.60020 q^{3} -1.95174 q^{4} -0.351542 q^{6} -0.332247 q^{7} +0.868145 q^{8} -0.439374 q^{9} +5.37182 q^{11} -3.12316 q^{12} +0.0729902 q^{14} +3.71276 q^{16} +5.06430 q^{17} +0.0965246 q^{18} -2.26795 q^{19} -0.531659 q^{21} -1.18012 q^{22} +2.83918 q^{23} +1.38920 q^{24} -5.50367 q^{27} +0.648458 q^{28} -2.90348 q^{29} -5.46410 q^{31} -2.55193 q^{32} +8.59596 q^{33} -1.11256 q^{34} +0.857542 q^{36} +5.97201 q^{37} +0.498239 q^{38} -3.73205 q^{41} +0.116799 q^{42} +5.06430 q^{43} -10.4844 q^{44} -0.623730 q^{46} +8.34285 q^{47} +5.94114 q^{48} -6.88961 q^{49} +8.10387 q^{51} +1.56063 q^{53} +1.20908 q^{54} -0.288438 q^{56} -3.62916 q^{57} +0.637855 q^{58} -2.70732 q^{59} +14.1039 q^{61} +1.20039 q^{62} +0.145980 q^{63} -6.86488 q^{64} -1.88842 q^{66} +10.3322 q^{67} -9.88418 q^{68} +4.54324 q^{69} -12.7973 q^{71} -0.381440 q^{72} -9.68922 q^{73} -1.31197 q^{74} +4.42644 q^{76} -1.78477 q^{77} +4.51851 q^{79} -7.48883 q^{81} +0.819883 q^{82} -4.26371 q^{83} +1.03766 q^{84} -1.11256 q^{86} -4.64613 q^{87} +4.66351 q^{88} -3.22584 q^{89} -5.54133 q^{92} -8.74363 q^{93} -1.83281 q^{94} -4.08359 q^{96} -2.50791 q^{97} +1.51356 q^{98} -2.36023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{6} + 10 q^{7} + 6 q^{8} + 4 q^{9} + 10 q^{12} + 2 q^{14} + 2 q^{16} + 2 q^{17} + 20 q^{18} - 16 q^{19} - 4 q^{21} - 12 q^{22} + 10 q^{23} + 24 q^{24} + 2 q^{27}+ \cdots - 24 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^6 + 10 * q^7 + 6 * q^8 + 4 * q^9 + 10 * q^12 + 2 * q^14 + 2 * q^16 + 2 * q^17 + 20 * q^18 - 16 * q^19 - 4 * q^21 - 12 * q^22 + 10 * q^23 + 24 * q^24 + 2 * q^27 + 8 * q^28 + 8 * q^29 - 8 * q^31 + 4 * q^32 + 18 * q^33 + 4 * q^34 + 20 * q^36 - 2 * q^37 - 8 * q^38 - 8 * q^41 + 4 * q^42 + 2 * q^43 - 12 * q^44 - 16 * q^46 + 8 * q^47 + 28 * q^48 + 12 * q^49 + 4 * q^51 + 12 * q^53 + 16 * q^54 + 12 * q^56 - 14 * q^57 + 22 * q^58 - 12 * q^59 + 28 * q^61 - 4 * q^62 + 4 * q^63 + 4 * q^64 + 6 * q^66 + 30 * q^67 - 14 * q^68 - 16 * q^69 - 4 * q^71 + 12 * q^72 - 8 * q^73 - 10 * q^74 - 20 * q^76 - 18 * q^77 - 8 * q^79 - 8 * q^81 - 4 * q^82 - 12 * q^83 + 28 * q^84 + 4 * q^86 + 22 * q^87 + 18 * q^88 + 12 * q^89 - 22 * q^92 + 8 * q^93 - 32 * q^94 - 4 * q^96 + 2 * q^97 - 24 * q^98 - 24 * q^99

Introduction

L-functions

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Database

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