LMFDB - Embedded newform 7605.2.a.cf.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.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:	$60.7262307372$
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			1.3
Root			$-0.219687$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.219687 q^{2} -1.95174 q^{4} +1.00000 q^{5} -0.332247 q^{7} -0.868145 q^{8} +O(q^{10})$	$q+0.219687 q^{2} -1.95174 q^{4} +1.00000 q^{5} -0.332247 q^{7} -0.868145 q^{8} +0.219687 q^{10} +5.37182 q^{11} -0.0729902 q^{14} +3.71276 q^{16} +5.06430 q^{17} +2.26795 q^{19} -1.95174 q^{20} +1.18012 q^{22} +2.83918 q^{23} +1.00000 q^{25} +0.648458 q^{28} +2.90348 q^{29} +5.46410 q^{31} +2.55193 q^{32} +1.11256 q^{34} -0.332247 q^{35} +5.97201 q^{37} +0.498239 q^{38} -0.868145 q^{40} -3.73205 q^{41} -5.06430 q^{43} -10.4844 q^{44} +0.623730 q^{46} -8.34285 q^{47} -6.88961 q^{49} +0.219687 q^{50} +1.56063 q^{53} +5.37182 q^{55} +0.288438 q^{56} +0.637855 q^{58} -2.70732 q^{59} +14.1039 q^{61} +1.20039 q^{62} -6.86488 q^{64} +10.3322 q^{67} -9.88418 q^{68} -0.0729902 q^{70} -12.7973 q^{71} -9.68922 q^{73} +1.31197 q^{74} -4.42644 q^{76} -1.78477 q^{77} +4.51851 q^{79} +3.71276 q^{80} -0.819883 q^{82} +4.26371 q^{83} +5.06430 q^{85} -1.11256 q^{86} -4.66351 q^{88} -3.22584 q^{89} -5.54133 q^{92} -1.83281 q^{94} +2.26795 q^{95} -2.50791 q^{97} -1.51356 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 10 q^{7} - 6 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 10 * q^7 - 6 * q^8	$ 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 10 q^{7} - 6 q^{8} - 2 q^{10} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 16 q^{19} + 2 q^{20} + 12 q^{22} + 10 q^{23} + 4 q^{25} + 8 q^{28} - 8 q^{29} + 8 q^{31} - 4 q^{32} - 4 q^{34} + 10 q^{35} - 2 q^{37} - 8 q^{38} - 6 q^{40} - 8 q^{41} - 2 q^{43} - 12 q^{44} + 16 q^{46} - 8 q^{47} + 12 q^{49} - 2 q^{50} + 12 q^{53} - 12 q^{56} + 22 q^{58} - 12 q^{59} + 28 q^{61} - 4 q^{62} + 4 q^{64} + 30 q^{67} - 14 q^{68} - 2 q^{70} - 4 q^{71} - 8 q^{73} + 10 q^{74} + 20 q^{76} - 18 q^{77} - 8 q^{79} + 2 q^{80} + 4 q^{82} + 12 q^{83} + 2 q^{85} + 4 q^{86} - 18 q^{88} + 12 q^{89} - 22 q^{92} - 32 q^{94} + 16 q^{95} + 2 q^{97} + 24 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 10 * q^7 - 6 * q^8 - 2 * q^10 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 16 * q^19 + 2 * q^20 + 12 * q^22 + 10 * q^23 + 4 * q^25 + 8 * q^28 - 8 * q^29 + 8 * q^31 - 4 * q^32 - 4 * q^34 + 10 * q^35 - 2 * q^37 - 8 * q^38 - 6 * q^40 - 8 * q^41 - 2 * q^43 - 12 * q^44 + 16 * q^46 - 8 * q^47 + 12 * q^49 - 2 * q^50 + 12 * q^53 - 12 * q^56 + 22 * q^58 - 12 * q^59 + 28 * q^61 - 4 * q^62 + 4 * q^64 + 30 * q^67 - 14 * q^68 - 2 * q^70 - 4 * q^71 - 8 * q^73 + 10 * q^74 + 20 * q^76 - 18 * q^77 - 8 * q^79 + 2 * q^80 + 4 * q^82 + 12 * q^83 + 2 * q^85 + 4 * q^86 - 18 * q^88 + 12 * q^89 - 22 * q^92 - 32 * q^94 + 16 * q^95 + 2 * q^97 + 24 * q^98

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 $

$2$	0.219687	0.155342	0.0776710	−	0.996979i	$-0.475252\pi$
$2$	0.219687	0.155342	0.0776710	+	0.996979i	$0.475252\pi$
$3$	0	0
$3$	0	0
$4$	−1.95174	−0.975869
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−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	0
$10$	0.219687	0.0694711
$11$	5.37182	1.61966	0.809832	−	0.586662i	$-0.199558\pi$
$11$	5.37182	1.61966	0.809832	+	0.586662i	$0.199558\pi$
$12$	0	0
$13$	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	0
$19$	2.26795	0.520303	0.260152	−	0.965568i	$-0.416227\pi$
$19$	2.26795	0.520303	0.260152	+	0.965568i	$0.416227\pi$
$20$	−1.95174	−0.436422
$21$	0	0
$22$	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	$0.404346\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0.648458	0.122547
$29$	2.90348	0.539162	0.269581	−	0.962978i	$-0.413115\pi$
$29$	2.90348	0.539162	0.269581	+	0.962978i	$0.413115\pi$
$30$	0	0
$31$	5.46410	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$31$	5.46410	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$32$	2.55193	0.451122
$33$	0	0
$34$	1.11256	0.190802
$35$	−0.332247	−0.0561599
$36$	0	0
$37$	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
$40$	−0.868145	−0.137266
$41$	−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	0
$43$	−5.06430	−0.772298	−0.386149	−	0.922436i	$-0.626195\pi$
$43$	−5.06430	−0.772298	−0.386149	+	0.922436i	$0.626195\pi$
$44$	−10.4844	−1.58058
$45$	0	0
$46$	0.623730	0.0919640
$47$	−8.34285	−1.21693	−0.608465	−	0.793581i	$-0.708214\pi$
$47$	−8.34285	−1.21693	−0.608465	+	0.793581i	$0.708214\pi$
$48$	0	0
$49$	−6.88961	−0.984230
$50$	0.219687	0.0310684
$51$	0	0
$52$	0	0
$53$	1.56063	0.214369	0.107184	−	0.994239i	$-0.465817\pi$
$53$	1.56063	0.214369	0.107184	+	0.994239i	$0.465817\pi$
$54$	0	0
$55$	5.37182	0.724336
$56$	0.288438	0.0385442
$57$	0	0
$58$	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	0
$64$	−6.86488	−0.858111
$65$	0	0
$66$	0	0
$67$	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
$69$	0	0
$70$	−0.0729902	−0.00872400
$71$	−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	0
$73$	−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	$0.418192\pi$
$80$	3.71276	0.415099
$81$	0	0
$82$	−0.819883	−0.0905409
$83$	4.26371	0.468003	0.234001	−	0.972236i	$-0.424818\pi$
$83$	4.26371	0.468003	0.234001	+	0.972236i	$0.424818\pi$
$84$	0	0
$85$	5.06430	0.549300
$86$	−1.11256	−0.119970
$87$	0	0
$88$	−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
$93$	0	0
$94$	−1.83281	−0.189040
$95$	2.26795	0.232687
$96$	0	0
$97$	−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
$99$	0	0
$100$	−1.95174	−0.195174
$101$	−12.4467	−1.23849	−0.619247	−	0.785196i	$-0.712562\pi$
$101$	−12.4467	−1.23849	−0.619247	+	0.785196i	$0.712562\pi$
$102$	0	0
$103$	−15.0247	−1.48043	−0.740215	−	0.672370i	$-0.765276\pi$
$103$	−15.0247	−1.48043	−0.740215	+	0.672370i	$0.765276\pi$
$104$	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	$0.282445\pi$
$108$	0	0
$109$	−11.2325	−1.07587	−0.537937	−	0.842985i	$-0.680796\pi$
$109$	−11.2325	−1.07587	−0.537937	+	0.842985i	$0.680796\pi$
$110$	1.18012	0.112520
$111$	0	0
$112$	−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	0
$115$	2.83918	0.264755
$116$	−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
$123$	0	0
$124$	−10.6645	−0.957700
$125$	1.00000	0.0894427
$126$	0	0
$127$	3.23996	0.287500	0.143750	−	0.989614i	$-0.454084\pi$
$127$	3.23996	0.287500	0.143750	+	0.989614i	$0.454084\pi$
$128$	−6.61199	−0.584423
$129$	0	0
$130$	0	0
$131$	−0.175664	−0.0153478	−0.00767390	−	0.999971i	$-0.502443\pi$
$131$	−0.175664	−0.0153478	−0.00767390	+	0.999971i	$0.502443\pi$
$132$	0	0
$133$	−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	$-0.221157\pi$
$137$	17.9829	1.53638	0.768190	+	0.640221i	$0.221157\pi$
$138$	0	0
$139$	11.9861	1.01665	0.508325	−	0.861165i	$-0.330265\pi$
$139$	11.9861	1.01665	0.508325	+	0.861165i	$0.330265\pi$
$140$	0.648458	0.0548047
$141$	0	0
$142$	−2.81140	−0.235928
$143$	0	0
$144$	0	0
$145$	2.90348	0.241121
$146$	−2.12859	−0.176164
$147$	0	0
$148$	−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	$-0.394991\pi$
$151$	7.96141	0.647890	0.323945	+	0.946076i	$0.394991\pi$
$152$	−1.96891	−0.159700
$153$	0	0
$154$	−0.392090	−0.0315955
$155$	5.46410	0.438887
$156$	0	0
$157$	−16.4329	−1.31148	−0.655742	−	0.754985i	$-0.727644\pi$
$157$	−16.4329	−1.31148	−0.655742	+	0.754985i	$0.727644\pi$
$158$	0.992658	0.0789716
$159$	0	0
$160$	2.55193	0.201748
$161$	−0.943307	−0.0743430
$162$	0	0
$163$	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	$-0.421702\pi$
$167$	6.29366	0.487018	0.243509	+	0.969899i	$0.421702\pi$
$168$	0	0
$169$	0	0
$170$	1.11256	0.0853294
$171$	0	0
$172$	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	$0.707740\pi$
$174$	0	0
$175$	−0.332247	−0.0251155
$176$	19.9442	1.50335
$177$	0	0
$178$	−0.708674	−0.0531173
$179$	−23.6174	−1.76525	−0.882625	−	0.470079i	$-0.844226\pi$
$179$	−23.6174	−1.76525	−0.882625	+	0.470079i	$0.844226\pi$
$180$	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
$183$	0	0
$184$	−2.46482	−0.181709
$185$	5.97201	0.439071
$186$	0	0
$187$	27.2045	1.98939
$188$	16.2831	1.18756
$189$	0	0
$190$	0.498239	0.0361460
$191$	2.01582	0.145860	0.0729298	−	0.997337i	$-0.476765\pi$
$191$	2.01582	0.145860	0.0729298	+	0.997337i	$0.476765\pi$
$192$	0	0
$193$	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	$-0.492708\pi$
$197$	0.643026	0.0458137	0.0229068	+	0.999738i	$0.492708\pi$
$198$	0	0
$199$	3.06684	0.217403	0.108701	−	0.994074i	$-0.465331\pi$
$199$	3.06684	0.217403	0.108701	+	0.994074i	$0.465331\pi$
$200$	−0.868145	−0.0613871
$201$	0	0
$202$	−2.73438	−0.192390
$203$	−0.964670	−0.0677065
$204$	0	0
$205$	−3.73205	−0.260658
$206$	−3.30074	−0.229973
$207$	0	0
$208$	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
$213$	0	0
$214$	2.87005	0.196193
$215$	−5.06430	−0.345382
$216$	0	0
$217$	−1.81543	−0.123239
$218$	−2.46762	−0.167129
$219$	0	0
$220$	−10.4844	−0.706856
$221$	0	0
$222$	0	0
$223$	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	$-0.575102\pi$
$227$	−7.04381	−0.467514	−0.233757	+	0.972295i	$0.575102\pi$
$228$	0	0
$229$	1.32899	0.0878219	0.0439109	−	0.999035i	$-0.486018\pi$
$229$	1.32899	0.0878219	0.0439109	+	0.999035i	$0.486018\pi$
$230$	0.623730	0.0411275
$231$	0	0
$232$	−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
$235$	−8.34285	−0.544227
$236$	5.28398	0.343958
$237$	0	0
$238$	−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	$-0.759306\pi$
$241$	−22.5869	−1.45495	−0.727475	+	0.686134i	$0.759306\pi$
$242$	3.92282	0.252168
$243$	0	0
$244$	−27.5270	−1.76224
$245$	−6.88961	−0.440161
$246$	0	0
$247$	0	0
$248$	−4.74363	−0.301221
$249$	0	0
$250$	0.219687	0.0138942
$251$	6.76836	0.427215	0.213608	−	0.976920i	$-0.431479\pi$
$251$	6.76836	0.427215	0.213608	+	0.976920i	$0.431479\pi$
$252$	0	0
$253$	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	$0.396382\pi$
$258$	0	0
$259$	−1.98418	−0.123291
$260$	0	0
$261$	0	0
$262$	−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	$0.695083\pi$
$264$	0	0
$265$	1.56063	0.0958685
$266$	−0.165538	−0.0101498
$267$	0	0
$268$	−20.1658	−1.23182
$269$	−17.9579	−1.09491	−0.547456	−	0.836835i	$-0.684404\pi$
$269$	−17.9579	−1.09491	−0.547456	+	0.836835i	$0.684404\pi$
$270$	0	0
$271$	30.8977	1.87690	0.938450	−	0.345415i	$-0.112262\pi$
$271$	30.8977	1.87690	0.938450	+	0.345415i	$0.112262\pi$
$272$	18.8025	1.14007
$273$	0	0
$274$	3.95060	0.238665
$275$	5.37182	0.323933
$276$	0	0
$277$	26.5045	1.59250	0.796250	−	0.604967i	$-0.206814\pi$
$277$	26.5045	1.59250	0.796250	+	0.604967i	$0.206814\pi$
$278$	2.63320	0.157929
$279$	0	0
$280$	0.288438	0.0172375
$281$	−4.97766	−0.296942	−0.148471	−	0.988917i	$-0.547435\pi$
$281$	−4.97766	−0.296942	−0.148471	+	0.988917i	$0.547435\pi$
$282$	0	0
$283$	−12.5863	−0.748180	−0.374090	−	0.927392i	$-0.622045\pi$
$283$	−12.5863	−0.748180	−0.374090	+	0.927392i	$0.622045\pi$
$284$	24.9770	1.48211
$285$	0	0
$286$	0	0
$287$	1.23996	0.0731926
$288$	0	0
$289$	8.64711	0.508653
$290$	0.637855	0.0374562
$291$	0	0
$292$	18.9108	1.10667
$293$	−16.9176	−0.988337	−0.494168	−	0.869366i	$-0.664527\pi$
$293$	−16.9176	−0.988337	−0.494168	+	0.869366i	$0.664527\pi$
$294$	0	0
$295$	−2.70732	−0.157626
$296$	−5.18457	−0.301347
$297$	0	0
$298$	0.749222	0.0434012
$299$	0	0
$300$	0	0
$301$	1.68260	0.0969832
$302$	1.74902	0.100645
$303$	0	0
$304$	8.42034	0.482940
$305$	14.1039	0.807585
$306$	0	0
$307$	−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
$309$	0	0
$310$	1.20039	0.0681776
$311$	2.22512	0.126175	0.0630875	−	0.998008i	$-0.479905\pi$
$311$	2.22512	0.126175	0.0630875	+	0.998008i	$0.479905\pi$
$312$	0	0
$313$	7.20887	0.407469	0.203735	−	0.979026i	$-0.434692\pi$
$313$	7.20887	0.407469	0.203735	+	0.979026i	$0.434692\pi$
$314$	−3.61008	−0.203729
$315$	0	0
$316$	−8.81895	−0.496105
$317$	0.321644	0.0180653	0.00903266	−	0.999959i	$-0.497125\pi$
$317$	0.321644	0.0180653	0.00903266	+	0.999959i	$0.497125\pi$
$318$	0	0
$319$	15.5969	0.873261
$320$	−6.86488	−0.383759
$321$	0	0
$322$	−0.207232	−0.0115486
$323$	11.4856	0.639074
$324$	0	0
$325$	0	0
$326$	3.91201	0.216666
$327$	0	0
$328$	3.23996	0.178897
$329$	2.77188	0.152819
$330$	0	0
$331$	16.6320	0.914178	0.457089	−	0.889421i	$-0.348892\pi$
$331$	16.6320	0.914178	0.457089	+	0.889421i	$0.348892\pi$
$332$	−8.32164	−0.456710
$333$	0	0
$334$	1.38263	0.0756543
$335$	10.3322	0.564511
$336$	0	0
$337$	24.2186	1.31927	0.659636	−	0.751586i	$-0.270711\pi$
$337$	24.2186	1.31927	0.659636	+	0.751586i	$0.270711\pi$
$338$	0	0
$339$	0	0
$340$	−9.88418	−0.536045
$341$	29.3521	1.58951
$342$	0	0
$343$	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	$0.446143\pi$
$348$	0	0
$349$	7.06994	0.378445	0.189223	−	0.981934i	$-0.439403\pi$
$349$	7.06994	0.378445	0.189223	+	0.981934i	$0.439403\pi$
$350$	−0.0729902	−0.00390149
$351$	0	0
$352$	13.7085	0.730666
$353$	21.7898	1.15976	0.579878	−	0.814704i	$-0.303100\pi$
$353$	21.7898	1.15976	0.579878	+	0.814704i	$0.303100\pi$
$354$	0	0
$355$	−12.7973	−0.679212
$356$	6.29598	0.333687
$357$	0	0
$358$	−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
$363$	0	0
$364$	0	0
$365$	−9.68922	−0.507157
$366$	0	0
$367$	6.39133	0.333625	0.166812	−	0.985989i	$-0.446653\pi$
$367$	6.39133	0.333625	0.166812	+	0.985989i	$0.446653\pi$
$368$	10.5412	0.549497
$369$	0	0
$370$	1.31197	0.0682062
$371$	−0.518513	−0.0269198
$372$	0	0
$373$	−20.0801	−1.03971	−0.519855	−	0.854255i	$-0.674014\pi$
$373$	−20.0801	−1.03971	−0.519855	+	0.854255i	$0.674014\pi$
$374$	5.97647	0.309036
$375$	0	0
$376$	7.24280	0.373519
$377$	0	0
$378$	0	0
$379$	−5.46182	−0.280555	−0.140277	−	0.990112i	$-0.544799\pi$
$379$	−5.46182	−0.280555	−0.140277	+	0.990112i	$0.544799\pi$
$380$	−4.42644	−0.227072
$381$	0	0
$382$	0.442849	0.0226581
$383$	5.66775	0.289609	0.144804	−	0.989460i	$-0.453745\pi$
$383$	5.66775	0.289609	0.144804	+	0.989460i	$0.453745\pi$
$384$	0	0
$385$	−1.78477	−0.0909602
$386$	5.01349	0.255180
$387$	0	0
$388$	4.89478	0.248495
$389$	−10.6174	−0.538325	−0.269162	−	0.963095i	$-0.586747\pi$
$389$	−10.6174	−0.538325	−0.269162	+	0.963095i	$0.586747\pi$
$390$	0	0
$391$	14.3784	0.727149
$392$	5.98118	0.302095
$393$	0	0
$394$	0.141264	0.00711679
$395$	4.51851	0.227351
$396$	0	0
$397$	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
$399$	0	0
$400$	3.71276	0.185638
$401$	22.5143	1.12431	0.562155	−	0.827032i	$-0.309973\pi$
$401$	22.5143	1.12431	0.562155	+	0.827032i	$0.309973\pi$
$402$	0	0
$403$	0	0
$404$	24.2927	1.20861
$405$	0	0
$406$	−0.211925	−0.0105177
$407$	32.0805	1.59017
$408$	0	0
$409$	4.28772	0.212014	0.106007	−	0.994365i	$-0.466193\pi$
$409$	4.28772	0.212014	0.106007	+	0.994365i	$0.466193\pi$
$410$	−0.819883	−0.0404911
$411$	0	0
$412$	29.3243	1.44471
$413$	0.899499	0.0442614
$414$	0	0
$415$	4.26371	0.209297
$416$	0	0
$417$	0	0
$418$	2.67645	0.130909
$419$	−17.7116	−0.865266	−0.432633	−	0.901570i	$-0.642415\pi$
$419$	−17.7116	−0.865266	−0.432633	+	0.901570i	$0.642415\pi$
$420$	0	0
$421$	12.8787	0.627672	0.313836	−	0.949477i	$-0.398386\pi$
$421$	12.8787	0.627672	0.313836	+	0.949477i	$0.398386\pi$
$422$	−1.80152	−0.0876965
$423$	0	0
$424$	−1.35485	−0.0657973
$425$	5.06430	0.245655
$426$	0	0
$427$	−4.68596	−0.226770
$428$	−25.4981	−1.23250
$429$	0	0
$430$	−1.11256	−0.0536524
$431$	−9.49845	−0.457524	−0.228762	−	0.973482i	$-0.573468\pi$
$431$	−9.49845	−0.457524	−0.228762	+	0.973482i	$0.573468\pi$
$432$	0	0
$433$	−1.39628	−0.0671010	−0.0335505	−	0.999437i	$-0.510681\pi$
$433$	−1.39628	−0.0671010	−0.0335505	+	0.999437i	$0.510681\pi$
$434$	−0.398826	−0.0191443
$435$	0	0
$436$	21.9228	1.04991
$437$	6.43911	0.308024
$438$	0	0
$439$	4.16180	0.198632	0.0993159	−	0.995056i	$-0.468335\pi$
$439$	4.16180	0.198632	0.0993159	+	0.995056i	$0.468335\pi$
$440$	−4.66351	−0.222324
$441$	0	0
$442$	0	0
$443$	−9.54563	−0.453526	−0.226763	−	0.973950i	$-0.572814\pi$
$443$	−9.54563	−0.453526	−0.226763	+	0.973950i	$0.572814\pi$
$444$	0	0
$445$	−3.22584	−0.152919
$446$	2.25052	0.106565
$447$	0	0
$448$	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
$453$	0	0
$454$	−1.54743	−0.0726246
$455$	0	0
$456$	0	0
$457$	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
$459$	0	0
$460$	−5.54133	−0.258366
$461$	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	0
$463$	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
$470$	−1.83281	−0.0845414
$471$	0	0
$472$	2.35035	0.108184
$473$	−27.2045	−1.25086
$474$	0	0
$475$	2.26795	0.104061
$476$	3.28398	0.150521
$477$	0	0
$478$	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
$483$	0	0
$484$	−34.8510	−1.58414
$485$	−2.50791	−0.113878
$486$	0	0
$487$	−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
$489$	0	0
$490$	−1.51356	−0.0683756
$491$	−9.33198	−0.421147	−0.210573	−	0.977578i	$-0.567533\pi$
$491$	−9.33198	−0.421147	−0.210573	+	0.977578i	$0.567533\pi$
$492$	0	0
$493$	14.7041	0.662238
$494$	0	0
$495$	0	0
$496$	20.2869	0.910907
$497$	4.25187	0.190722
$498$	0	0
$499$	23.9421	1.07179	0.535897	−	0.844283i	$-0.319974\pi$
$499$	23.9421	1.07179	0.535897	+	0.844283i	$0.319974\pi$
$500$	−1.95174	−0.0872844
$501$	0	0
$502$	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	0
$505$	−12.4467	−0.553872
$506$	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
$513$	0	0
$514$	2.25263	0.0993593
$515$	−15.0247	−0.662069
$516$	0	0
$517$	−44.8162	−1.97102
$518$	−0.435898	−0.0191523
$519$	0	0
$520$	0	0
$521$	−12.4649	−0.546098	−0.273049	−	0.962000i	$-0.588032\pi$
$521$	−12.4649	−0.546098	−0.273049	+	0.962000i	$0.588032\pi$
$522$	0	0
$523$	−5.65956	−0.247475	−0.123738	−	0.992315i	$-0.539488\pi$
$523$	−5.65956	−0.247475	−0.123738	+	0.992315i	$0.539488\pi$
$524$	0.342849	0.0149774
$525$	0	0
$526$	−4.09870	−0.178712
$527$	27.6718	1.20540
$528$	0	0
$529$	−14.9391	−0.649525
$530$	0.342849	0.0148924
$531$	0	0
$532$	1.47067	0.0637616
$533$	0	0
$534$	0	0
$535$	13.0643	0.564819
$536$	−8.96989	−0.387440
$537$	0	0
$538$	−3.94511	−0.170086
$539$	−37.0097	−1.59412
$540$	0	0
$541$	−15.4750	−0.665321	−0.332660	−	0.943047i	$-0.607946\pi$
$541$	−15.4750	−0.665321	−0.332660	+	0.943047i	$0.607946\pi$
$542$	6.78781	0.291562
$543$	0	0
$544$	12.9237	0.554101
$545$	−11.2325	−0.481146
$546$	0	0
$547$	25.1765	1.07647	0.538234	−	0.842795i	$-0.319092\pi$
$547$	25.1765	1.07647	0.538234	+	0.842795i	$0.319092\pi$
$548$	−35.0979	−1.49931
$549$	0	0
$550$	1.18012	0.0503204
$551$	6.58493	0.280528
$552$	0	0
$553$	−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	$-0.854395\pi$
$557$	−42.3489	−1.79438	−0.897190	+	0.441645i	$0.854395\pi$
$558$	0	0
$559$	0	0
$560$	−1.23355	−0.0521270
$561$	0	0
$562$	−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	$0.667154\pi$
$564$	0	0
$565$	18.3438	0.771731
$566$	−2.76505	−0.116224
$567$	0	0
$568$	11.1099	0.466162
$569$	26.7421	1.12109	0.560543	−	0.828125i	$-0.310593\pi$
$569$	26.7421	1.12109	0.560543	+	0.828125i	$0.310593\pi$
$570$	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
$573$	0	0
$574$	0.272403	0.0113699
$575$	2.83918	0.118402
$576$	0	0
$577$	−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
$579$	0	0
$580$	−5.66682	−0.235302
$581$	−1.41660	−0.0587706
$582$	0	0
$583$	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	$-0.641203\pi$
$587$	−20.7972	−0.858391	−0.429196	+	0.903212i	$0.641203\pi$
$588$	0	0
$589$	12.3923	0.510616
$590$	−0.594763	−0.0244860
$591$	0	0
$592$	22.1726	0.911289
$593$	21.8475	0.897169	0.448585	−	0.893740i	$-0.351928\pi$
$593$	21.8475	0.897169	0.448585	+	0.893740i	$0.351928\pi$
$594$	0	0
$595$	−1.68260	−0.0689797
$596$	−6.65622	−0.272649
$597$	0	0
$598$	0	0
$599$	3.58040	0.146291	0.0731456	−	0.997321i	$-0.476696\pi$
$599$	3.58040	0.146291	0.0731456	+	0.997321i	$0.476696\pi$
$600$	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
$603$	0	0
$604$	−15.5386	−0.632256
$605$	17.8564	0.725966
$606$	0	0
$607$	−3.29976	−0.133933	−0.0669665	−	0.997755i	$-0.521332\pi$
$607$	−3.29976	−0.133933	−0.0669665	+	0.997755i	$0.521332\pi$
$608$	5.78766	0.234720
$609$	0	0
$610$	3.09843	0.125452
$611$	0	0
$612$	0	0
$613$	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	$-0.127988\pi$
$617$	45.7169	1.84049	0.920246	+	0.391339i	$0.127988\pi$
$618$	0	0
$619$	19.9143	0.800425	0.400212	−	0.916422i	$-0.368936\pi$
$619$	19.9143	0.800425	0.400212	+	0.916422i	$0.368936\pi$
$620$	−10.6645	−0.428296
$621$	0	0
$622$	0.488829	0.0196003
$623$	1.07177	0.0429397
$624$	0	0
$625$	1.00000	0.0400000
$626$	1.58369	0.0632971
$627$	0	0
$628$	32.0726	1.27984
$629$	30.2440	1.20591
$630$	0	0
$631$	14.5958	0.581050	0.290525	−	0.956867i	$-0.406170\pi$
$631$	14.5958	0.581050	0.290525	+	0.956867i	$0.406170\pi$
$632$	−3.92272	−0.156038
$633$	0	0
$634$	0.0706609	0.00280630
$635$	3.23996	0.128574
$636$	0	0
$637$	0	0
$638$	3.42644	0.135654
$639$	0	0
$640$	−6.61199	−0.261362
$641$	14.1637	0.559431	0.279716	−	0.960083i	$-0.409760\pi$
$641$	14.1637	0.559431	0.279716	+	0.960083i	$0.409760\pi$
$642$	0	0
$643$	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	$0.481270\pi$
$648$	0	0
$649$	−14.5432	−0.570872
$650$	0	0
$651$	0	0
$652$	−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	$0.426706\pi$
$654$	0	0
$655$	−0.175664	−0.00686374
$656$	−13.8562	−0.540993
$657$	0	0
$658$	0.608946	0.0237392
$659$	1.81047	0.0705260	0.0352630	−	0.999378i	$-0.488773\pi$
$659$	1.81047	0.0705260	0.0352630	+	0.999378i	$0.488773\pi$
$660$	0	0
$661$	−12.3406	−0.479992	−0.239996	−	0.970774i	$-0.577146\pi$
$661$	−12.3406	−0.479992	−0.239996	+	0.970774i	$0.577146\pi$
$662$	3.65383	0.142010
$663$	0	0
$664$	−3.70152	−0.143647
$665$	−0.753518	−0.0292202
$666$	0	0
$667$	8.24348	0.319189
$668$	−12.2836	−0.475265
$669$	0	0
$670$	2.26986	0.0876923
$671$	75.7634	2.92481
$672$	0	0
$673$	9.26625	0.357188	0.178594	−	0.983923i	$-0.442845\pi$
$673$	9.26625	0.357188	0.178594	+	0.983923i	$0.442845\pi$
$674$	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	$0.586059\pi$
$678$	0	0
$679$	0.833244	0.0319770
$680$	−4.39654	−0.168600
$681$	0	0
$682$	6.44828	0.246917
$683$	37.7512	1.44451	0.722255	−	0.691626i	$-0.243105\pi$
$683$	37.7512	1.44451	0.722255	+	0.691626i	$0.243105\pi$
$684$	0	0
$685$	17.9829	0.687090
$686$	1.01381	0.0387073
$687$	0	0
$688$	−18.8025	−0.716838
$689$	0	0
$690$	0	0
$691$	−1.65291	−0.0628797	−0.0314399	−	0.999506i	$-0.510009\pi$
$691$	−1.65291	−0.0628797	−0.0314399	+	0.999506i	$0.510009\pi$
$692$	31.1792	1.18526
$693$	0	0
$694$	1.37823	0.0523168
$695$	11.9861	0.454660
$696$	0	0
$697$	−18.9002	−0.715897
$698$	1.55317	0.0587885
$699$	0	0
$700$	0.648458	0.0245094
$701$	−20.4819	−0.773590	−0.386795	−	0.922166i	$-0.626418\pi$
$701$	−20.4819	−0.773590	−0.386795	+	0.922166i	$0.626418\pi$
$702$	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
$708$	0	0
$709$	21.9417	0.824039	0.412020	−	0.911175i	$-0.364823\pi$
$709$	21.9417	0.824039	0.412020	+	0.911175i	$0.364823\pi$
$710$	−2.81140	−0.105510
$711$	0	0
$712$	2.80049	0.104953
$713$	15.5136	0.580987
$714$	0	0
$715$	0	0
$716$	46.0950	1.72265
$717$	0	0
$718$	−5.26671	−0.196552
$719$	−38.8475	−1.44877	−0.724384	−	0.689397i	$-0.757876\pi$
$719$	−38.8475	−1.44877	−0.724384	+	0.689397i	$0.757876\pi$
$720$	0	0
$721$	4.99191	0.185909
$722$	−3.04407	−0.113289
$723$	0	0
$724$	5.12507	0.190472
$725$	2.90348	0.107832
$726$	0	0
$727$	−30.6598	−1.13711	−0.568555	−	0.822645i	$-0.692497\pi$
$727$	−30.6598	−1.13711	−0.568555	+	0.822645i	$0.692497\pi$
$728$	0	0
$729$	0	0
$730$	−2.12859	−0.0787828
$731$	−25.6471	−0.948593
$732$	0	0
$733$	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	0
$739$	38.2788	1.40811	0.704054	−	0.710146i	$-0.251371\pi$
$739$	38.2788	1.40811	0.704054	+	0.710146i	$0.251371\pi$
$740$	−11.6558	−0.428476
$741$	0	0
$742$	−0.113910	−0.00418178
$743$	40.0079	1.46775	0.733874	−	0.679286i	$-0.237710\pi$
$743$	40.0079	1.46775	0.733874	+	0.679286i	$0.237710\pi$
$744$	0	0
$745$	3.41041	0.124948
$746$	−4.41134	−0.161511
$747$	0	0
$748$	−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
$753$	0	0
$754$	0	0
$755$	7.96141	0.289745
$756$	0	0
$757$	−1.84848	−0.0671840	−0.0335920	−	0.999436i	$-0.510695\pi$
$757$	−1.84848	−0.0671840	−0.0335920	+	0.999436i	$0.510695\pi$
$758$	−1.19989	−0.0435820
$759$	0	0
$760$	−1.96891	−0.0714198
$761$	−26.2124	−0.950199	−0.475099	−	0.879932i	$-0.657588\pi$
$761$	−26.2124	−0.950199	−0.475099	+	0.879932i	$0.657588\pi$
$762$	0	0
$763$	3.73195	0.135106
$764$	−3.93435	−0.142340
$765$	0	0
$766$	1.24513	0.0449884
$767$	0	0
$768$	0	0
$769$	−44.3495	−1.59928	−0.799641	−	0.600478i	$-0.794977\pi$
$769$	−44.3495	−1.59928	−0.799641	+	0.600478i	$0.794977\pi$
$770$	−0.392090	−0.0141299
$771$	0	0
$772$	−44.5408	−1.60306
$773$	23.2638	0.836742	0.418371	−	0.908276i	$-0.362601\pi$
$773$	23.2638	0.836742	0.418371	+	0.908276i	$0.362601\pi$
$774$	0	0
$775$	5.46410	0.196276
$776$	2.17723	0.0781580
$777$	0	0
$778$	−2.33251	−0.0836245
$779$	−8.46410	−0.303258
$780$	0	0
$781$	−68.7449	−2.45989
$782$	3.15875	0.112957
$783$	0	0
$784$	−25.5794	−0.913552
$785$	−16.4329	−0.586514
$786$	0	0
$787$	−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
$789$	0	0
$790$	0.992658	0.0353172
$791$	−6.09467	−0.216702
$792$	0	0
$793$	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	0
$799$	−42.2507	−1.49472
$800$	2.55193	0.0902245
$801$	0	0
$802$	4.94609	0.174653
$803$	−52.0487	−1.83676
$804$	0	0
$805$	−0.943307	−0.0332472
$806$	0	0
$807$	0	0
$808$	10.8056	0.380138
$809$	−15.8915	−0.558714	−0.279357	−	0.960187i	$-0.590121\pi$
$809$	−15.8915	−0.558714	−0.279357	+	0.960187i	$0.590121\pi$
$810$	0	0
$811$	−23.8796	−0.838525	−0.419263	−	0.907865i	$-0.637711\pi$
$811$	−23.8796	−0.838525	−0.419263	+	0.907865i	$0.637711\pi$
$812$	1.88278	0.0660727
$813$	0	0
$814$	7.04768	0.247021
$815$	17.8072	0.623759
$816$	0	0
$817$	−11.4856	−0.401829
$818$	0.941956	0.0329347
$819$	0	0
$820$	7.28398	0.254368
$821$	−15.9097	−0.555251	−0.277626	−	0.960689i	$-0.589547\pi$
$821$	−15.9097	−0.555251	−0.277626	+	0.960689i	$0.589547\pi$
$822$	0	0
$823$	−14.8115	−0.516295	−0.258147	−	0.966106i	$-0.583112\pi$
$823$	−14.8115	−0.516295	−0.258147	+	0.966106i	$0.583112\pi$
$824$	13.0436	0.454397
$825$	0	0
$826$	0.197608	0.00687566
$827$	−33.9498	−1.18055	−0.590275	−	0.807202i	$-0.700981\pi$
$827$	−33.9498	−1.18055	−0.590275	+	0.807202i	$0.700981\pi$
$828$	0	0
$829$	23.3146	0.809749	0.404875	−	0.914372i	$-0.367315\pi$
$829$	23.3146	0.809749	0.404875	+	0.914372i	$0.367315\pi$
$830$	0.936681	0.0325127
$831$	0	0
$832$	0	0
$833$	−34.8910	−1.20890
$834$	0	0
$835$	6.29366	0.217801
$836$	−23.7780	−0.822380
$837$	0	0
$838$	−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
$843$	0	0
$844$	16.0050	0.550915
$845$	0	0
$846$	0	0
$847$	−5.93273	−0.203851
$848$	5.79422	0.198974
$849$	0	0
$850$	1.11256	0.0381605
$851$	16.9556	0.581231
$852$	0	0
$853$	−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	$0.780291\pi$
$860$	9.88418	0.337048
$861$	0	0
$862$	−2.08669	−0.0710728
$863$	−4.75058	−0.161712	−0.0808559	−	0.996726i	$-0.525765\pi$
$863$	−4.75058	−0.161712	−0.0808559	+	0.996726i	$0.525765\pi$
$864$	0	0
$865$	−15.9751	−0.543170
$866$	−0.306745	−0.0104236
$867$	0	0
$868$	3.54324	0.120265
$869$	24.2726	0.823392
$870$	0	0
$871$	0	0
$872$	9.75140	0.330224
$873$	0	0
$874$	1.41459	0.0478492
$875$	−0.332247	−0.0112320
$876$	0	0
$877$	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
$879$	0	0
$880$	19.9442	0.672320
$881$	2.98304	0.100501	0.0502507	−	0.998737i	$-0.483998\pi$
$881$	2.98304	0.100501	0.0502507	+	0.998737i	$0.483998\pi$
$882$	0	0
$883$	28.2874	0.951947	0.475973	−	0.879460i	$-0.342096\pi$
$883$	28.2874	0.951947	0.475973	+	0.879460i	$0.342096\pi$
$884$	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	$0.344338\pi$
$888$	0	0
$889$	−1.07647	−0.0361035
$890$	−0.708674	−0.0237548
$891$	0	0
$892$	−19.9940	−0.669448
$893$	−18.9212	−0.633172
$894$	0	0
$895$	−23.6174	−0.789443
$896$	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
$903$	0	0
$904$	−15.9251	−0.529661
$905$	−2.62590	−0.0872879
$906$	0	0
$907$	16.5520	0.549600	0.274800	−	0.961501i	$-0.411388\pi$
$907$	16.5520	0.549600	0.274800	+	0.961501i	$0.411388\pi$
$908$	13.7477	0.456232
$909$	0	0
$910$	0	0
$911$	−7.04863	−0.233532	−0.116766	−	0.993159i	$-0.537253\pi$
$911$	−7.04863	−0.233532	−0.116766	+	0.993159i	$0.537253\pi$
$912$	0	0
$913$	22.9039	0.758007
$914$	1.03749	0.0343172
$915$	0	0
$916$	−2.59383	−0.0857026
$917$	0.0583636	0.00192734
$918$	0	0
$919$	16.5438	0.545728	0.272864	−	0.962053i	$-0.412029\pi$
$919$	16.5438	0.545728	0.272864	+	0.962053i	$0.412029\pi$
$920$	−2.46482	−0.0812626
$921$	0	0
$922$	0.391374	0.0128892
$923$	0	0
$924$	0	0
$925$	5.97201	0.196359
$926$	1.49431	0.0491060
$927$	0	0
$928$	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
$933$	0	0
$934$	−4.00652	−0.131097
$935$	27.2045	0.889681
$936$	0	0
$937$	−30.4606	−0.995104	−0.497552	−	0.867434i	$-0.665768\pi$
$937$	−30.4606	−0.995104	−0.497552	+	0.867434i	$0.665768\pi$
$938$	−0.754153	−0.0246240
$939$	0	0
$940$	16.2831	0.531095
$941$	38.2101	1.24561	0.622807	−	0.782375i	$-0.285992\pi$
$941$	38.2101	1.24561	0.622807	+	0.782375i	$0.285992\pi$
$942$	0	0
$943$	−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	$-0.824713\pi$
$947$	−52.4482	−1.70434	−0.852169	+	0.523266i	$0.824713\pi$
$948$	0	0
$949$	0	0
$950$	0.498239	0.0161650
$951$	0	0
$952$	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	0
$955$	2.01582	0.0652304
$956$	−19.4043	−0.627580
$957$	0	0
$958$	7.72874	0.249704
$959$	−5.97475	−0.192935
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	0	0
$964$	44.0837	1.41984
$965$	22.8211	0.734636
$966$	0	0
$967$	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
$969$	0	0
$970$	−0.550955	−0.0176901
$971$	55.5252	1.78189	0.890945	−	0.454111i	$-0.150043\pi$
$971$	55.5252	1.78189	0.890945	+	0.454111i	$0.150043\pi$
$972$	0	0
$973$	−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	$-0.275559\pi$
$977$	40.5161	1.29622	0.648112	+	0.761545i	$0.275559\pi$
$978$	0	0
$979$	−17.3286	−0.553824
$980$	13.4467	0.429540
$981$	0	0
$982$	−2.05011	−0.0654218
$983$	−34.8059	−1.11014	−0.555068	−	0.831805i	$-0.687308\pi$
$983$	−34.8059	−1.11014	−0.555068	+	0.831805i	$0.687308\pi$
$984$	0	0
$985$	0.643026	0.0204885
$986$	3.23029	0.102873
$987$	0	0
$988$	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
$993$	0	0
$994$	0.934079	0.0296272
$995$	3.06684	0.0972254
$996$	0	0
$997$	−5.49137	−0.173914	−0.0869568	−	0.996212i	$-0.527714\pi$
$997$	−5.49137	−0.173914	−0.0869568	+	0.996212i	$0.527714\pi$
$998$	5.25976	0.166495
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.cf.1.3		4
3.2	odd	2		845.2.a.m.1.2		4
13.2	odd	12		585.2.bu.c.316.3		8
13.7	odd	12		585.2.bu.c.361.3		8
13.12	even	2		7605.2.a.cj.1.2		4
15.14	odd	2		4225.2.a.bi.1.3		4
39.2	even	12		65.2.m.a.56.2	yes	8
39.5	even	4		845.2.c.g.506.5		8
39.8	even	4		845.2.c.g.506.4		8
39.11	even	12		845.2.m.g.316.3		8
39.17	odd	6		845.2.e.n.146.2		8
39.20	even	12		65.2.m.a.36.2	✓	8
39.23	odd	6		845.2.e.n.191.2		8
39.29	odd	6		845.2.e.m.191.3		8
39.32	even	12		845.2.m.g.361.3		8
39.35	odd	6		845.2.e.m.146.3		8
39.38	odd	2		845.2.a.l.1.3		4
156.59	odd	12		1040.2.da.b.881.2		8
156.119	odd	12		1040.2.da.b.641.2		8
195.2	odd	12		325.2.m.b.199.3		8
195.59	even	12		325.2.n.d.101.3		8
195.98	odd	12		325.2.m.b.49.3		8
195.119	even	12		325.2.n.d.251.3		8
195.137	odd	12		325.2.m.c.49.2		8
195.158	odd	12		325.2.m.c.199.2		8
195.194	odd	2		4225.2.a.bl.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.m.a.36.2	✓	8	39.20	even	12
65.2.m.a.56.2	yes	8	39.2	even	12
325.2.m.b.49.3		8	195.98	odd	12
325.2.m.b.199.3		8	195.2	odd	12
325.2.m.c.49.2		8	195.137	odd	12
325.2.m.c.199.2		8	195.158	odd	12
325.2.n.d.101.3		8	195.59	even	12
325.2.n.d.251.3		8	195.119	even	12
585.2.bu.c.316.3		8	13.2	odd	12
585.2.bu.c.361.3		8	13.7	odd	12
845.2.a.l.1.3		4	39.38	odd	2
845.2.a.m.1.2		4	3.2	odd	2
845.2.c.g.506.4		8	39.8	even	4
845.2.c.g.506.5		8	39.5	even	4
845.2.e.m.146.3		8	39.35	odd	6
845.2.e.m.191.3		8	39.29	odd	6
845.2.e.n.146.2		8	39.17	odd	6
845.2.e.n.191.2		8	39.23	odd	6
845.2.m.g.316.3		8	39.11	even	12
845.2.m.g.361.3		8	39.32	even	12
1040.2.da.b.641.2		8	156.119	odd	12
1040.2.da.b.881.2		8	156.59	odd	12
4225.2.a.bi.1.3		4	15.14	odd	2
4225.2.a.bl.1.2		4	195.194	odd	2
7605.2.a.cf.1.3		4	1.1	even	1	trivial
7605.2.a.cj.1.2		4	13.12	even	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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q+0.219687 q^{2} -1.95174 q^{4} +1.00000 q^{5} -0.332247 q^{7} -0.868145 q^{8} +O(q^{10})\)	\(q+0.219687 q^{2} -1.95174 q^{4} +1.00000 q^{5} -0.332247 q^{7} -0.868145 q^{8} +0.219687 q^{10} +5.37182 q^{11} -0.0729902 q^{14} +3.71276 q^{16} +5.06430 q^{17} +2.26795 q^{19} -1.95174 q^{20} +1.18012 q^{22} +2.83918 q^{23} +1.00000 q^{25} +0.648458 q^{28} +2.90348 q^{29} +5.46410 q^{31} +2.55193 q^{32} +1.11256 q^{34} -0.332247 q^{35} +5.97201 q^{37} +0.498239 q^{38} -0.868145 q^{40} -3.73205 q^{41} -5.06430 q^{43} -10.4844 q^{44} +0.623730 q^{46} -8.34285 q^{47} -6.88961 q^{49} +0.219687 q^{50} +1.56063 q^{53} +5.37182 q^{55} +0.288438 q^{56} +0.637855 q^{58} -2.70732 q^{59} +14.1039 q^{61} +1.20039 q^{62} -6.86488 q^{64} +10.3322 q^{67} -9.88418 q^{68} -0.0729902 q^{70} -12.7973 q^{71} -9.68922 q^{73} +1.31197 q^{74} -4.42644 q^{76} -1.78477 q^{77} +4.51851 q^{79} +3.71276 q^{80} -0.819883 q^{82} +4.26371 q^{83} +5.06430 q^{85} -1.11256 q^{86} -4.66351 q^{88} -3.22584 q^{89} -5.54133 q^{92} -1.83281 q^{94} +2.26795 q^{95} -2.50791 q^{97} -1.51356 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 10 * q^7 - 6 * q^8	\( 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 10 q^{7} - 6 q^{8} - 2 q^{10} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 16 q^{19} + 2 q^{20} + 12 q^{22} + 10 q^{23} + 4 q^{25} + 8 q^{28} - 8 q^{29} + 8 q^{31} - 4 q^{32} - 4 q^{34} + 10 q^{35} - 2 q^{37} - 8 q^{38} - 6 q^{40} - 8 q^{41} - 2 q^{43} - 12 q^{44} + 16 q^{46} - 8 q^{47} + 12 q^{49} - 2 q^{50} + 12 q^{53} - 12 q^{56} + 22 q^{58} - 12 q^{59} + 28 q^{61} - 4 q^{62} + 4 q^{64} + 30 q^{67} - 14 q^{68} - 2 q^{70} - 4 q^{71} - 8 q^{73} + 10 q^{74} + 20 q^{76} - 18 q^{77} - 8 q^{79} + 2 q^{80} + 4 q^{82} + 12 q^{83} + 2 q^{85} + 4 q^{86} - 18 q^{88} + 12 q^{89} - 22 q^{92} - 32 q^{94} + 16 q^{95} + 2 q^{97} + 24 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 10 * q^7 - 6 * q^8 - 2 * q^10 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 16 * q^19 + 2 * q^20 + 12 * q^22 + 10 * q^23 + 4 * q^25 + 8 * q^28 - 8 * q^29 + 8 * q^31 - 4 * q^32 - 4 * q^34 + 10 * q^35 - 2 * q^37 - 8 * q^38 - 6 * q^40 - 8 * q^41 - 2 * q^43 - 12 * q^44 + 16 * q^46 - 8 * q^47 + 12 * q^49 - 2 * q^50 + 12 * q^53 - 12 * q^56 + 22 * q^58 - 12 * q^59 + 28 * q^61 - 4 * q^62 + 4 * q^64 + 30 * q^67 - 14 * q^68 - 2 * q^70 - 4 * q^71 - 8 * q^73 + 10 * q^74 + 20 * q^76 - 18 * q^77 - 8 * q^79 + 2 * q^80 + 4 * q^82 + 12 * q^83 + 2 * q^85 + 4 * q^86 - 18 * q^88 + 12 * q^89 - 22 * q^92 - 32 * q^94 + 16 * q^95 + 2 * q^97 + 24 * q^98

Introduction

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