LMFDB - Embedded newform 2925.2.c.r.2224.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(2224,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("2925.2224");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2224.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2925.2224
Dual form			2925.2.c.r.2224.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214i q^{2} +1.82843 q^{4} +0.828427i q^{7} +1.58579i q^{8} +O(q^{10})$	$q+0.414214i q^{2} +1.82843 q^{4} +0.828427i q^{7} +1.58579i q^{8} -0.585786 q^{11} -1.00000i q^{13} -0.343146 q^{14} +3.00000 q^{16} -4.82843i q^{17} -3.41421 q^{19} -0.242641i q^{22} +1.41421i q^{23} +0.414214 q^{26} +1.51472i q^{28} +5.65685 q^{29} +10.2426 q^{31} +4.41421i q^{32} +2.00000 q^{34} -8.48528i q^{37} -1.41421i q^{38} +8.82843 q^{41} +3.07107i q^{43} -1.07107 q^{44} -0.585786 q^{46} +0.828427i q^{47} +6.31371 q^{49} -1.82843i q^{52} +14.4853i q^{53} -1.31371 q^{56} +2.34315i q^{58} +10.2426 q^{59} -8.00000 q^{61} +4.24264i q^{62} +4.17157 q^{64} +2.00000i q^{67} -8.82843i q^{68} +7.89949 q^{71} -8.48528i q^{73} +3.51472 q^{74} -6.24264 q^{76} -0.485281i q^{77} -8.48528 q^{79} +3.65685i q^{82} +8.82843i q^{83} -1.27208 q^{86} -0.928932i q^{88} +6.00000 q^{89} +0.828427 q^{91} +2.58579i q^{92} -0.343146 q^{94} -3.65685i q^{97} +2.61522i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} - 8 q^{11} - 24 q^{14} + 12 q^{16} - 8 q^{19} - 4 q^{26} + 24 q^{31} + 8 q^{34} + 24 q^{41} + 24 q^{44} - 8 q^{46} - 20 q^{49} + 40 q^{56} + 24 q^{59} - 32 q^{61} + 28 q^{64} - 8 q^{71} + 48 q^{74} - 8 q^{76} - 56 q^{86} + 24 q^{89} - 8 q^{91} - 24 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 8 * q^11 - 24 * q^14 + 12 * q^16 - 8 * q^19 - 4 * q^26 + 24 * q^31 + 8 * q^34 + 24 * q^41 + 24 * q^44 - 8 * q^46 - 20 * q^49 + 40 * q^56 + 24 * q^59 - 32 * q^61 + 28 * q^64 - 8 * q^71 + 48 * q^74 - 8 * q^76 - 56 * q^86 + 24 * q^89 - 8 * q^91 - 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0.414214i	0.292893i	0.989219	+	0.146447i	$0.0467837\pi$
$2$	0.414214i	0.292893i	−0.989219	+	0.146447i	$0.953216\pi$
$3$	0	0
$3$	0	0
$4$	1.82843	0.914214
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.828427i	0.313116i	0.987669	+	0.156558i	$0.0500398\pi$
$7$	0.828427i	0.313116i	−0.987669	+	0.156558i	$0.949960\pi$
$8$	1.58579i	0.560660i
$9$	0	0
$10$	0	0
$11$	−0.585786	−0.176621	−0.0883106	−	0.996093i	$-0.528147\pi$
$11$	−0.585786	−0.176621	−0.0883106	+	0.996093i	$0.528147\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	−0.343146	−0.0917096
$15$	0	0
$16$	3.00000	0.750000
$17$	− 4.82843i	− 1.17107i	−0.810649	−	0.585533i	$-0.800885\pi$
$17$	− 4.82843i	− 1.17107i	0.810649	−	0.585533i	$-0.199115\pi$
$18$	0	0
$19$	−3.41421	−0.783274	−0.391637	−	0.920120i	$-0.628091\pi$
$19$	−3.41421	−0.783274	−0.391637	+	0.920120i	$0.628091\pi$
$20$	0	0
$21$	0	0
$22$	− 0.242641i	− 0.0517312i
$23$	1.41421i	0.294884i	0.989071	+	0.147442i	$0.0471040\pi$
$23$	1.41421i	0.294884i	−0.989071	+	0.147442i	$0.952896\pi$
$24$	0	0
$25$	0	0
$26$	0.414214	0.0812340
$27$	0	0
$28$	1.51472i	0.286255i
$29$	5.65685	1.05045	0.525226	−	0.850963i	$-0.323981\pi$
$29$	5.65685	1.05045	0.525226	+	0.850963i	$0.323981\pi$
$30$	0	0
$31$	10.2426	1.83963	0.919816	−	0.392349i	$-0.128338\pi$
$31$	10.2426	1.83963	0.919816	+	0.392349i	$0.128338\pi$
$32$	4.41421i	0.780330i
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	− 8.48528i	− 1.39497i	−0.716599	−	0.697486i	$-0.754302\pi$
$37$	− 8.48528i	− 1.39497i	0.716599	−	0.697486i	$-0.245698\pi$
$38$	− 1.41421i	− 0.229416i
$39$	0	0
$40$	0	0
$41$	8.82843	1.37877	0.689384	−	0.724396i	$-0.257881\pi$
$41$	8.82843	1.37877	0.689384	+	0.724396i	$0.257881\pi$
$42$	0	0
$43$	3.07107i	0.468333i	0.972196	+	0.234167i	$0.0752362\pi$
$43$	3.07107i	0.468333i	−0.972196	+	0.234167i	$0.924764\pi$
$44$	−1.07107	−0.161470
$45$	0	0
$46$	−0.585786	−0.0863695
$47$	0.828427i	0.120839i	0.998173	+	0.0604193i	$0.0192438\pi$
$47$	0.828427i	0.120839i	−0.998173	+	0.0604193i	$0.980756\pi$
$48$	0	0
$49$	6.31371	0.901958
$50$	0	0
$51$	0	0
$52$	− 1.82843i	− 0.253557i
$53$	14.4853i	1.98971i	0.101327	+	0.994853i	$0.467691\pi$
$53$	14.4853i	1.98971i	−0.101327	+	0.994853i	$0.532309\pi$
$54$	0	0
$55$	0	0
$56$	−1.31371	−0.175552
$57$	0	0
$58$	2.34315i	0.307670i
$59$	10.2426	1.33348	0.666739	−	0.745291i	$-0.267690\pi$
$59$	10.2426	1.33348	0.666739	+	0.745291i	$0.267690\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	4.24264i	0.538816i
$63$	0	0
$64$	4.17157	0.521447
$65$	0	0
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	− 8.82843i	− 1.07060i
$69$	0	0
$70$	0	0
$71$	7.89949	0.937498	0.468749	−	0.883332i	$-0.344705\pi$
$71$	7.89949	0.937498	0.468749	+	0.883332i	$0.344705\pi$
$72$	0	0
$73$	− 8.48528i	− 0.993127i	−0.868000	−	0.496564i	$-0.834595\pi$
$73$	− 8.48528i	− 0.993127i	0.868000	−	0.496564i	$-0.165405\pi$
$74$	3.51472	0.408578
$75$	0	0
$76$	−6.24264	−0.716080
$77$	− 0.485281i	− 0.0553029i
$78$	0	0
$79$	−8.48528	−0.954669	−0.477334	−	0.878722i	$-0.658397\pi$
$79$	−8.48528	−0.954669	−0.477334	+	0.878722i	$0.658397\pi$
$80$	0	0
$81$	0	0
$82$	3.65685i	0.403832i
$83$	8.82843i	0.969046i	0.874779	+	0.484523i	$0.161007\pi$
$83$	8.82843i	0.969046i	−0.874779	+	0.484523i	$0.838993\pi$
$84$	0	0
$85$	0	0
$86$	−1.27208	−0.137172
$87$	0	0
$88$	− 0.928932i	− 0.0990245i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0.828427	0.0868428
$92$	2.58579i	0.269587i
$93$	0	0
$94$	−0.343146	−0.0353928
$95$	0	0
$96$	0	0
$97$	− 3.65685i	− 0.371297i	−0.982616	−	0.185649i	$-0.940561\pi$
$97$	− 3.65685i	− 0.371297i	0.982616	−	0.185649i	$-0.0594386\pi$
$98$	2.61522i	0.264177i
$99$	0	0
$100$	0	0
$101$	−7.65685	−0.761885	−0.380943	−	0.924599i	$-0.624401\pi$
$101$	−7.65685	−0.761885	−0.380943	+	0.924599i	$0.624401\pi$
$102$	0	0
$103$	17.4142i	1.71587i	0.513755	+	0.857937i	$0.328254\pi$
$103$	17.4142i	1.71587i	−0.513755	+	0.857937i	$0.671746\pi$
$104$	1.58579	0.155499
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 6.58579i	− 0.636672i	−0.947978	−	0.318336i	$-0.896876\pi$
$107$	− 6.58579i	− 0.636672i	0.947978	−	0.318336i	$-0.103124\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	2.48528i	0.234837i
$113$	3.17157i	0.298356i	0.988810	+	0.149178i	$0.0476628\pi$
$113$	3.17157i	0.298356i	−0.988810	+	0.149178i	$0.952337\pi$
$114$	0	0
$115$	0	0
$116$	10.3431	0.960337
$117$	0	0
$118$	4.24264i	0.390567i
$119$	4.00000	0.366679
$120$	0	0
$121$	−10.6569	−0.968805
$122$	− 3.31371i	− 0.300009i
$123$	0	0
$124$	18.7279	1.68182
$125$	0	0
$126$	0	0
$127$	9.41421i	0.835376i	0.908590	+	0.417688i	$0.137160\pi$
$127$	9.41421i	0.835376i	−0.908590	+	0.417688i	$0.862840\pi$
$128$	10.5563i	0.933058i
$129$	0	0
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0	0
$133$	− 2.82843i	− 0.245256i
$134$	−0.828427	−0.0715652
$135$	0	0
$136$	7.65685	0.656570
$137$	5.31371i	0.453981i	0.973897	+	0.226990i	$0.0728886\pi$
$137$	5.31371i	0.453981i	−0.973897	+	0.226990i	$0.927111\pi$
$138$	0	0
$139$	12.4853	1.05899	0.529494	−	0.848314i	$-0.322382\pi$
$139$	12.4853	1.05899	0.529494	+	0.848314i	$0.322382\pi$
$140$	0	0
$141$	0	0
$142$	3.27208i	0.274587i
$143$	0.585786i	0.0489859i
$144$	0	0
$145$	0	0
$146$	3.51472	0.290880
$147$	0	0
$148$	− 15.5147i	− 1.27530i
$149$	−0.343146	−0.0281116	−0.0140558	−	0.999901i	$-0.504474\pi$
$149$	−0.343146	−0.0281116	−0.0140558	+	0.999901i	$0.504474\pi$
$150$	0	0
$151$	18.2426	1.48457	0.742283	−	0.670087i	$-0.233743\pi$
$151$	18.2426	1.48457	0.742283	+	0.670087i	$0.233743\pi$
$152$	− 5.41421i	− 0.439151i
$153$	0	0
$154$	0.201010	0.0161979
$155$	0	0
$156$	0	0
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	− 3.51472i	− 0.279616i
$159$	0	0
$160$	0	0
$161$	−1.17157	−0.0923329
$162$	0	0
$163$	− 14.9706i	− 1.17258i	−0.810099	−	0.586292i	$-0.800587\pi$
$163$	− 14.9706i	− 1.17258i	0.810099	−	0.586292i	$-0.199413\pi$
$164$	16.1421	1.26049
$165$	0	0
$166$	−3.65685	−0.283827
$167$	− 8.82843i	− 0.683164i	−0.939852	−	0.341582i	$-0.889037\pi$
$167$	− 8.82843i	− 0.683164i	0.939852	−	0.341582i	$-0.110963\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	5.61522i	0.428157i
$173$	− 11.1716i	− 0.849359i	−0.905344	−	0.424679i	$-0.860387\pi$
$173$	− 11.1716i	− 0.849359i	0.905344	−	0.424679i	$-0.139613\pi$
$174$	0	0
$175$	0	0
$176$	−1.75736	−0.132466
$177$	0	0
$178$	2.48528i	0.186280i
$179$	−5.65685	−0.422813	−0.211407	−	0.977398i	$-0.567804\pi$
$179$	−5.65685	−0.422813	−0.211407	+	0.977398i	$0.567804\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0.343146i	0.0254357i
$183$	0	0
$184$	−2.24264	−0.165330
$185$	0	0
$186$	0	0
$187$	2.82843i	0.206835i
$188$	1.51472i	0.110472i
$189$	0	0
$190$	0	0
$191$	−13.6569	−0.988175	−0.494088	−	0.869412i	$-0.664498\pi$
$191$	−13.6569	−0.988175	−0.494088	+	0.869412i	$0.664498\pi$
$192$	0	0
$193$	15.6569i	1.12701i	0.826114	+	0.563503i	$0.190546\pi$
$193$	15.6569i	1.12701i	−0.826114	+	0.563503i	$0.809454\pi$
$194$	1.51472	0.108750
$195$	0	0
$196$	11.5442	0.824583
$197$	− 22.9706i	− 1.63658i	−0.574802	−	0.818292i	$-0.694921\pi$
$197$	− 22.9706i	− 1.63658i	0.574802	−	0.818292i	$-0.305079\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	0	0
$202$	− 3.17157i	− 0.223151i
$203$	4.68629i	0.328913i
$204$	0	0
$205$	0	0
$206$	−7.21320	−0.502568
$207$	0	0
$208$	− 3.00000i	− 0.208013i
$209$	2.00000	0.138343
$210$	0	0
$211$	−19.3137	−1.32961	−0.664805	−	0.747017i	$-0.731485\pi$
$211$	−19.3137	−1.32961	−0.664805	+	0.747017i	$0.731485\pi$
$212$	26.4853i	1.81902i
$213$	0	0
$214$	2.72792	0.186477
$215$	0	0
$216$	0	0
$217$	8.48528i	0.576018i
$218$	0.828427i	0.0561082i
$219$	0	0
$220$	0	0
$221$	−4.82843	−0.324795
$222$	0	0
$223$	26.4853i	1.77359i	0.462167	+	0.886793i	$0.347072\pi$
$223$	26.4853i	1.77359i	−0.462167	+	0.886793i	$0.652928\pi$
$224$	−3.65685	−0.244334
$225$	0	0
$226$	−1.31371	−0.0873866
$227$	− 27.6569i	− 1.83565i	−0.396985	−	0.917825i	$-0.629944\pi$
$227$	− 27.6569i	− 1.83565i	0.396985	−	0.917825i	$-0.370056\pi$
$228$	0	0
$229$	−0.828427	−0.0547440	−0.0273720	−	0.999625i	$-0.508714\pi$
$229$	−0.828427	−0.0547440	−0.0273720	+	0.999625i	$0.508714\pi$
$230$	0	0
$231$	0	0
$232$	8.97056i	0.588946i
$233$	− 24.6274i	− 1.61340i	−0.590964	−	0.806698i	$-0.701253\pi$
$233$	− 24.6274i	− 1.61340i	0.590964	−	0.806698i	$-0.298747\pi$
$234$	0	0
$235$	0	0
$236$	18.7279	1.21908
$237$	0	0
$238$	1.65685i	0.107398i
$239$	−0.585786	−0.0378914	−0.0189457	−	0.999821i	$-0.506031\pi$
$239$	−0.585786	−0.0378914	−0.0189457	+	0.999821i	$0.506031\pi$
$240$	0	0
$241$	2.48528	0.160091	0.0800455	−	0.996791i	$-0.474493\pi$
$241$	2.48528	0.160091	0.0800455	+	0.996791i	$0.474493\pi$
$242$	− 4.41421i	− 0.283756i
$243$	0	0
$244$	−14.6274	−0.936424
$245$	0	0
$246$	0	0
$247$	3.41421i	0.217241i
$248$	16.2426i	1.03141i
$249$	0	0
$250$	0	0
$251$	19.7990	1.24970	0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990	1.24970	0.624851	+	0.780744i	$0.285160\pi$
$252$	0	0
$253$	− 0.828427i	− 0.0520828i
$254$	−3.89949	−0.244676
$255$	0	0
$256$	3.97056	0.248160
$257$	16.3431i	1.01946i	0.860335	+	0.509729i	$0.170254\pi$
$257$	16.3431i	1.01946i	−0.860335	+	0.509729i	$0.829746\pi$
$258$	0	0
$259$	7.02944	0.436788
$260$	0	0
$261$	0	0
$262$	− 7.02944i	− 0.434280i
$263$	13.4142i	0.827156i	0.910469	+	0.413578i	$0.135721\pi$
$263$	13.4142i	0.827156i	−0.910469	+	0.413578i	$0.864279\pi$
$264$	0	0
$265$	0	0
$266$	1.17157	0.0718337
$267$	0	0
$268$	3.65685i	0.223378i
$269$	−2.68629	−0.163786	−0.0818930	−	0.996641i	$-0.526097\pi$
$269$	−2.68629	−0.163786	−0.0818930	+	0.996641i	$0.526097\pi$
$270$	0	0
$271$	1.27208	0.0772732	0.0386366	−	0.999253i	$-0.487699\pi$
$271$	1.27208	0.0772732	0.0386366	+	0.999253i	$0.487699\pi$
$272$	− 14.4853i	− 0.878299i
$273$	0	0
$274$	−2.20101	−0.132968
$275$	0	0
$276$	0	0
$277$	7.17157i	0.430898i	0.976515	+	0.215449i	$0.0691215\pi$
$277$	7.17157i	0.430898i	−0.976515	+	0.215449i	$0.930878\pi$
$278$	5.17157i	0.310170i
$279$	0	0
$280$	0	0
$281$	17.7990	1.06180	0.530899	−	0.847435i	$-0.321854\pi$
$281$	17.7990	1.06180	0.530899	+	0.847435i	$0.321854\pi$
$282$	0	0
$283$	8.72792i	0.518821i	0.965767	+	0.259411i	$0.0835283\pi$
$283$	8.72792i	0.518821i	−0.965767	+	0.259411i	$0.916472\pi$
$284$	14.4437	0.857073
$285$	0	0
$286$	−0.242641	−0.0143476
$287$	7.31371i	0.431715i
$288$	0	0
$289$	−6.31371	−0.371395
$290$	0	0
$291$	0	0
$292$	− 15.5147i	− 0.907930i
$293$	2.14214i	0.125145i	0.998040	+	0.0625724i	$0.0199304\pi$
$293$	2.14214i	0.125145i	−0.998040	+	0.0625724i	$0.980070\pi$
$294$	0	0
$295$	0	0
$296$	13.4558	0.782105
$297$	0	0
$298$	− 0.142136i	− 0.00823370i
$299$	1.41421	0.0817861
$300$	0	0
$301$	−2.54416	−0.146643
$302$	7.55635i	0.434819i
$303$	0	0
$304$	−10.2426	−0.587456
$305$	0	0
$306$	0	0
$307$	− 19.1716i	− 1.09418i	−0.837074	−	0.547090i	$-0.815736\pi$
$307$	− 19.1716i	− 1.09418i	0.837074	−	0.547090i	$-0.184264\pi$
$308$	− 0.887302i	− 0.0505587i
$309$	0	0
$310$	0	0
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	0.828427i	0.0468255i	0.999726	+	0.0234127i	$0.00745319\pi$
$313$	0.828427i	0.0468255i	−0.999726	+	0.0234127i	$0.992547\pi$
$314$	7.45584	0.420758
$315$	0	0
$316$	−15.5147	−0.872771
$317$	26.1421i	1.46829i	0.678993	+	0.734144i	$0.262416\pi$
$317$	26.1421i	1.46829i	−0.678993	+	0.734144i	$0.737584\pi$
$318$	0	0
$319$	−3.31371	−0.185532
$320$	0	0
$321$	0	0
$322$	− 0.485281i	− 0.0270437i
$323$	16.4853i	0.917266i
$324$	0	0
$325$	0	0
$326$	6.20101	0.343442
$327$	0	0
$328$	14.0000i	0.773021i
$329$	−0.686292	−0.0378365
$330$	0	0
$331$	22.0416	1.21152	0.605759	−	0.795648i	$-0.292870\pi$
$331$	22.0416	1.21152	0.605759	+	0.795648i	$0.292870\pi$
$332$	16.1421i	0.885915i
$333$	0	0
$334$	3.65685	0.200094
$335$	0	0
$336$	0	0
$337$	− 7.17157i	− 0.390660i	−0.980738	−	0.195330i	$-0.937422\pi$
$337$	− 7.17157i	− 0.390660i	0.980738	−	0.195330i	$-0.0625779\pi$
$338$	− 0.414214i	− 0.0225302i
$339$	0	0
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	11.0294i	0.595534i
$344$	−4.87006	−0.262576
$345$	0	0
$346$	4.62742	0.248771
$347$	4.24264i	0.227757i	0.993495	+	0.113878i	$0.0363274\pi$
$347$	4.24264i	0.227757i	−0.993495	+	0.113878i	$0.963673\pi$
$348$	0	0
$349$	−1.51472	−0.0810810	−0.0405405	−	0.999178i	$-0.512908\pi$
$349$	−1.51472	−0.0810810	−0.0405405	+	0.999178i	$0.512908\pi$
$350$	0	0
$351$	0	0
$352$	− 2.58579i	− 0.137823i
$353$	− 9.17157i	− 0.488154i	−0.969756	−	0.244077i	$-0.921515\pi$
$353$	− 9.17157i	− 0.488154i	0.969756	−	0.244077i	$-0.0784849\pi$
$354$	0	0
$355$	0	0
$356$	10.9706	0.581439
$357$	0	0
$358$	− 2.34315i	− 0.123839i
$359$	−27.8995	−1.47248	−0.736240	−	0.676721i	$-0.763400\pi$
$359$	−27.8995	−1.47248	−0.736240	+	0.676721i	$0.763400\pi$
$360$	0	0
$361$	−7.34315	−0.386481
$362$	0	0
$363$	0	0
$364$	1.51472	0.0793928
$365$	0	0
$366$	0	0
$367$	− 4.44365i	− 0.231957i	−0.993252	−	0.115978i	$-0.963000\pi$
$367$	− 4.44365i	− 0.231957i	0.993252	−	0.115978i	$-0.0370003\pi$
$368$	4.24264i	0.221163i
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	− 25.3137i	− 1.31069i	−0.755328	−	0.655347i	$-0.772522\pi$
$373$	− 25.3137i	− 1.31069i	0.755328	−	0.655347i	$-0.227478\pi$
$374$	−1.17157	−0.0605806
$375$	0	0
$376$	−1.31371	−0.0677493
$377$	− 5.65685i	− 0.291343i
$378$	0	0
$379$	−14.9289	−0.766848	−0.383424	−	0.923572i	$-0.625255\pi$
$379$	−14.9289	−0.766848	−0.383424	+	0.923572i	$0.625255\pi$
$380$	0	0
$381$	0	0
$382$	− 5.65685i	− 0.289430i
$383$	− 33.1127i	− 1.69198i	−0.533199	−	0.845990i	$-0.679010\pi$
$383$	− 33.1127i	− 1.69198i	0.533199	−	0.845990i	$-0.320990\pi$
$384$	0	0
$385$	0	0
$386$	−6.48528	−0.330092
$387$	0	0
$388$	− 6.68629i	− 0.339445i
$389$	−16.6274	−0.843044	−0.421522	−	0.906818i	$-0.638504\pi$
$389$	−16.6274	−0.843044	−0.421522	+	0.906818i	$0.638504\pi$
$390$	0	0
$391$	6.82843	0.345328
$392$	10.0122i	0.505692i
$393$	0	0
$394$	9.51472	0.479345
$395$	0	0
$396$	0	0
$397$	27.7990i	1.39519i	0.716492	+	0.697596i	$0.245747\pi$
$397$	27.7990i	1.39519i	−0.716492	+	0.697596i	$0.754253\pi$
$398$	− 1.65685i	− 0.0830506i
$399$	0	0
$400$	0	0
$401$	−17.3137	−0.864605	−0.432303	−	0.901729i	$-0.642299\pi$
$401$	−17.3137	−0.864605	−0.432303	+	0.901729i	$0.642299\pi$
$402$	0	0
$403$	− 10.2426i	− 0.510222i
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−1.94113	−0.0963364
$407$	4.97056i	0.246382i
$408$	0	0
$409$	−12.8284	−0.634325	−0.317162	−	0.948371i	$-0.602730\pi$
$409$	−12.8284	−0.634325	−0.317162	+	0.948371i	$0.602730\pi$
$410$	0	0
$411$	0	0
$412$	31.8406i	1.56867i
$413$	8.48528i	0.417533i
$414$	0	0
$415$	0	0
$416$	4.41421	0.216425
$417$	0	0
$418$	0.828427i	0.0405197i
$419$	5.17157	0.252648	0.126324	−	0.991989i	$-0.459682\pi$
$419$	5.17157	0.252648	0.126324	+	0.991989i	$0.459682\pi$
$420$	0	0
$421$	−1.02944	−0.0501717	−0.0250859	−	0.999685i	$-0.507986\pi$
$421$	−1.02944	−0.0501717	−0.0250859	+	0.999685i	$0.507986\pi$
$422$	− 8.00000i	− 0.389434i
$423$	0	0
$424$	−22.9706	−1.11555
$425$	0	0
$426$	0	0
$427$	− 6.62742i	− 0.320723i
$428$	− 12.0416i	− 0.582054i
$429$	0	0
$430$	0	0
$431$	−3.61522	−0.174139	−0.0870696	−	0.996202i	$-0.527750\pi$
$431$	−3.61522	−0.174139	−0.0870696	+	0.996202i	$0.527750\pi$
$432$	0	0
$433$	− 3.65685i	− 0.175737i	−0.996132	−	0.0878686i	$-0.971994\pi$
$433$	− 3.65685i	− 0.175737i	0.996132	−	0.0878686i	$-0.0280056\pi$
$434$	−3.51472	−0.168712
$435$	0	0
$436$	3.65685	0.175132
$437$	− 4.82843i	− 0.230975i
$438$	0	0
$439$	32.9706	1.57360	0.786800	−	0.617209i	$-0.211737\pi$
$439$	32.9706	1.57360	0.786800	+	0.617209i	$0.211737\pi$
$440$	0	0
$441$	0	0
$442$	− 2.00000i	− 0.0951303i
$443$	6.58579i	0.312900i	0.987686	+	0.156450i	$0.0500050\pi$
$443$	6.58579i	0.312900i	−0.987686	+	0.156450i	$0.949995\pi$
$444$	0	0
$445$	0	0
$446$	−10.9706	−0.519471
$447$	0	0
$448$	3.45584i	0.163273i
$449$	29.1127	1.37391	0.686957	−	0.726698i	$-0.258946\pi$
$449$	29.1127	1.37391	0.686957	+	0.726698i	$0.258946\pi$
$450$	0	0
$451$	−5.17157	−0.243520
$452$	5.79899i	0.272762i
$453$	0	0
$454$	11.4558	0.537649
$455$	0	0
$456$	0	0
$457$	18.0000i	0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$	18.0000i	0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	− 0.343146i	− 0.0160341i
$459$	0	0
$460$	0	0
$461$	−26.4853	−1.23354	−0.616771	−	0.787142i	$-0.711560\pi$
$461$	−26.4853	−1.23354	−0.616771	+	0.787142i	$0.711560\pi$
$462$	0	0
$463$	− 15.6569i	− 0.727636i	−0.931470	−	0.363818i	$-0.881473\pi$
$463$	− 15.6569i	− 0.727636i	0.931470	−	0.363818i	$-0.118527\pi$
$464$	16.9706	0.787839
$465$	0	0
$466$	10.2010	0.472553
$467$	− 10.5858i	− 0.489852i	−0.969542	−	0.244926i	$-0.921236\pi$
$467$	− 10.5858i	− 0.489852i	0.969542	−	0.244926i	$-0.0787636\pi$
$468$	0	0
$469$	−1.65685	−0.0765064
$470$	0	0
$471$	0	0
$472$	16.2426i	0.747628i
$473$	− 1.79899i	− 0.0827176i
$474$	0	0
$475$	0	0
$476$	7.31371	0.335223
$477$	0	0
$478$	− 0.242641i	− 0.0110981i
$479$	−5.27208	−0.240887	−0.120444	−	0.992720i	$-0.538432\pi$
$479$	−5.27208	−0.240887	−0.120444	+	0.992720i	$0.538432\pi$
$480$	0	0
$481$	−8.48528	−0.386896
$482$	1.02944i	0.0468896i
$483$	0	0
$484$	−19.4853	−0.885695
$485$	0	0
$486$	0	0
$487$	− 22.9706i	− 1.04090i	−0.853894	−	0.520448i	$-0.825765\pi$
$487$	− 22.9706i	− 1.04090i	0.853894	−	0.520448i	$-0.174235\pi$
$488$	− 12.6863i	− 0.574281i
$489$	0	0
$490$	0	0
$491$	−10.8284	−0.488680	−0.244340	−	0.969690i	$-0.578571\pi$
$491$	−10.8284	−0.488680	−0.244340	+	0.969690i	$0.578571\pi$
$492$	0	0
$493$	− 27.3137i	− 1.23015i
$494$	−1.41421	−0.0636285
$495$	0	0
$496$	30.7279	1.37972
$497$	6.54416i	0.293546i
$498$	0	0
$499$	−10.4437	−0.467522	−0.233761	−	0.972294i	$-0.575103\pi$
$499$	−10.4437	−0.467522	−0.233761	+	0.972294i	$0.575103\pi$
$500$	0	0
$501$	0	0
$502$	8.20101i	0.366029i
$503$	− 18.1005i	− 0.807062i	−0.914966	−	0.403531i	$-0.867783\pi$
$503$	− 18.1005i	− 0.807062i	0.914966	−	0.403531i	$-0.132217\pi$
$504$	0	0
$505$	0	0
$506$	0.343146	0.0152547
$507$	0	0
$508$	17.2132i	0.763712i
$509$	−21.1127	−0.935804	−0.467902	−	0.883780i	$-0.654990\pi$
$509$	−21.1127	−0.935804	−0.467902	+	0.883780i	$0.654990\pi$
$510$	0	0
$511$	7.02944	0.310964
$512$	22.7574i	1.00574i
$513$	0	0
$514$	−6.76955	−0.298592
$515$	0	0
$516$	0	0
$517$	− 0.485281i	− 0.0213427i
$518$	2.91169i	0.127932i
$519$	0	0
$520$	0	0
$521$	6.34315	0.277898	0.138949	−	0.990300i	$-0.455628\pi$
$521$	6.34315	0.277898	0.138949	+	0.990300i	$0.455628\pi$
$522$	0	0
$523$	− 28.2426i	− 1.23496i	−0.786585	−	0.617482i	$-0.788153\pi$
$523$	− 28.2426i	− 1.23496i	0.786585	−	0.617482i	$-0.211847\pi$
$524$	−31.0294	−1.35553
$525$	0	0
$526$	−5.55635	−0.242268
$527$	− 49.4558i	− 2.15433i
$528$	0	0
$529$	21.0000	0.913043
$530$	0	0
$531$	0	0
$532$	− 5.17157i	− 0.224216i
$533$	− 8.82843i	− 0.382402i
$534$	0	0
$535$	0	0
$536$	−3.17157	−0.136991
$537$	0	0
$538$	− 1.11270i	− 0.0479718i
$539$	−3.69848	−0.159305
$540$	0	0
$541$	−12.8284	−0.551537	−0.275769	−	0.961224i	$-0.588932\pi$
$541$	−12.8284	−0.551537	−0.275769	+	0.961224i	$0.588932\pi$
$542$	0.526912i	0.0226328i
$543$	0	0
$544$	21.3137	0.913818
$545$	0	0
$546$	0	0
$547$	29.2132i	1.24907i	0.780998	+	0.624533i	$0.214711\pi$
$547$	29.2132i	1.24907i	−0.780998	+	0.624533i	$0.785289\pi$
$548$	9.71573i	0.415035i
$549$	0	0
$550$	0	0
$551$	−19.3137	−0.822792
$552$	0	0
$553$	− 7.02944i	− 0.298922i
$554$	−2.97056	−0.126207
$555$	0	0
$556$	22.8284	0.968141
$557$	3.79899i	0.160968i	0.996756	+	0.0804842i	$0.0256467\pi$
$557$	3.79899i	0.160968i	−0.996756	+	0.0804842i	$0.974353\pi$
$558$	0	0
$559$	3.07107	0.129892
$560$	0	0
$561$	0	0
$562$	7.37258i	0.310994i
$563$	16.2426i	0.684546i	0.939601	+	0.342273i	$0.111197\pi$
$563$	16.2426i	0.684546i	−0.939601	+	0.342273i	$0.888803\pi$
$564$	0	0
$565$	0	0
$566$	−3.61522	−0.151959
$567$	0	0
$568$	12.5269i	0.525618i
$569$	−21.6569	−0.907903	−0.453951	−	0.891027i	$-0.649986\pi$
$569$	−21.6569	−0.907903	−0.453951	+	0.891027i	$0.649986\pi$
$570$	0	0
$571$	−28.4853	−1.19207	−0.596036	−	0.802958i	$-0.703258\pi$
$571$	−28.4853	−1.19207	−0.596036	+	0.802958i	$0.703258\pi$
$572$	1.07107i	0.0447836i
$573$	0	0
$574$	−3.02944	−0.126446
$575$	0	0
$576$	0	0
$577$	29.1716i	1.21443i	0.794538	+	0.607214i	$0.207713\pi$
$577$	29.1716i	1.21443i	−0.794538	+	0.607214i	$0.792287\pi$
$578$	− 2.61522i	− 0.108779i
$579$	0	0
$580$	0	0
$581$	−7.31371	−0.303424
$582$	0	0
$583$	− 8.48528i	− 0.351424i
$584$	13.4558	0.556807
$585$	0	0
$586$	−0.887302	−0.0366541
$587$	31.6569i	1.30662i	0.757091	+	0.653309i	$0.226620\pi$
$587$	31.6569i	1.30662i	−0.757091	+	0.653309i	$0.773380\pi$
$588$	0	0
$589$	−34.9706	−1.44094
$590$	0	0
$591$	0	0
$592$	− 25.4558i	− 1.04623i
$593$	− 20.6274i	− 0.847066i	−0.905881	−	0.423533i	$-0.860790\pi$
$593$	− 20.6274i	− 0.847066i	0.905881	−	0.423533i	$-0.139210\pi$
$594$	0	0
$595$	0	0
$596$	−0.627417	−0.0257000
$597$	0	0
$598$	0.585786i	0.0239546i
$599$	−25.4558	−1.04010	−0.520049	−	0.854137i	$-0.674086\pi$
$599$	−25.4558	−1.04010	−0.520049	+	0.854137i	$0.674086\pi$
$600$	0	0
$601$	0.627417	0.0255929	0.0127964	−	0.999918i	$-0.495927\pi$
$601$	0.627417	0.0255929	0.0127964	+	0.999918i	$0.495927\pi$
$602$	− 1.05382i	− 0.0429507i
$603$	0	0
$604$	33.3553	1.35721
$605$	0	0
$606$	0	0
$607$	− 40.2426i	− 1.63340i	−0.577064	−	0.816699i	$-0.695802\pi$
$607$	− 40.2426i	− 1.63340i	0.577064	−	0.816699i	$-0.304198\pi$
$608$	− 15.0711i	− 0.611213i
$609$	0	0
$610$	0	0
$611$	0.828427	0.0335146
$612$	0	0
$613$	− 37.3137i	− 1.50709i	−0.657398	−	0.753543i	$-0.728343\pi$
$613$	− 37.3137i	− 1.50709i	0.657398	−	0.753543i	$-0.271657\pi$
$614$	7.94113	0.320478
$615$	0	0
$616$	0.769553	0.0310062
$617$	− 22.9706i	− 0.924760i	−0.886682	−	0.462380i	$-0.846996\pi$
$617$	− 22.9706i	− 0.924760i	0.886682	−	0.462380i	$-0.153004\pi$
$618$	0	0
$619$	−10.2426	−0.411686	−0.205843	−	0.978585i	$-0.565994\pi$
$619$	−10.2426	−0.411686	−0.205843	+	0.978585i	$0.565994\pi$
$620$	0	0
$621$	0	0
$622$	3.51472i	0.140927i
$623$	4.97056i	0.199141i
$624$	0	0
$625$	0	0
$626$	−0.343146	−0.0137149
$627$	0	0
$628$	− 32.9117i	− 1.31332i
$629$	−40.9706	−1.63360
$630$	0	0
$631$	−18.2426	−0.726228	−0.363114	−	0.931745i	$-0.618286\pi$
$631$	−18.2426	−0.726228	−0.363114	+	0.931745i	$0.618286\pi$
$632$	− 13.4558i	− 0.535245i
$633$	0	0
$634$	−10.8284	−0.430052
$635$	0	0
$636$	0	0
$637$	− 6.31371i	− 0.250158i
$638$	− 1.37258i	− 0.0543411i
$639$	0	0
$640$	0	0
$641$	−36.3431	−1.43547	−0.717734	−	0.696317i	$-0.754821\pi$
$641$	−36.3431	−1.43547	−0.717734	+	0.696317i	$0.754821\pi$
$642$	0	0
$643$	26.4853i	1.04448i	0.852799	+	0.522239i	$0.174903\pi$
$643$	26.4853i	1.04448i	−0.852799	+	0.522239i	$0.825097\pi$
$644$	−2.14214	−0.0844120
$645$	0	0
$646$	−6.82843	−0.268661
$647$	6.58579i	0.258914i	0.991585	+	0.129457i	$0.0413234\pi$
$647$	6.58579i	0.258914i	−0.991585	+	0.129457i	$0.958677\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	0	0
$651$	0	0
$652$	− 27.3726i	− 1.07199i
$653$	− 13.0294i	− 0.509881i	−0.966957	−	0.254941i	$-0.917944\pi$
$653$	− 13.0294i	− 0.509881i	0.966957	−	0.254941i	$-0.0820559\pi$
$654$	0	0
$655$	0	0
$656$	26.4853	1.03408
$657$	0	0
$658$	− 0.284271i	− 0.0110820i
$659$	46.1421	1.79744	0.898721	−	0.438520i	$-0.144497\pi$
$659$	46.1421	1.79744	0.898721	+	0.438520i	$0.144497\pi$
$660$	0	0
$661$	−49.5980	−1.92914	−0.964569	−	0.263831i	$-0.915014\pi$
$661$	−49.5980	−1.92914	−0.964569	+	0.263831i	$0.915014\pi$
$662$	9.12994i	0.354845i
$663$	0	0
$664$	−14.0000	−0.543305
$665$	0	0
$666$	0	0
$667$	8.00000i	0.309761i
$668$	− 16.1421i	− 0.624558i
$669$	0	0
$670$	0	0
$671$	4.68629	0.180912
$672$	0	0
$673$	− 10.4853i	− 0.404178i	−0.979367	−	0.202089i	$-0.935227\pi$
$673$	− 10.4853i	− 0.404178i	0.979367	−	0.202089i	$-0.0647730\pi$
$674$	2.97056	0.114422
$675$	0	0
$676$	−1.82843	−0.0703241
$677$	8.14214i	0.312928i	0.987684	+	0.156464i	$0.0500095\pi$
$677$	8.14214i	0.312928i	−0.987684	+	0.156464i	$0.949991\pi$
$678$	0	0
$679$	3.02944	0.116259
$680$	0	0
$681$	0	0
$682$	− 2.48528i	− 0.0951663i
$683$	− 33.3137i	− 1.27471i	−0.770569	−	0.637357i	$-0.780028\pi$
$683$	− 33.3137i	− 1.27471i	0.770569	−	0.637357i	$-0.219972\pi$
$684$	0	0
$685$	0	0
$686$	−4.56854	−0.174428
$687$	0	0
$688$	9.21320i	0.351250i
$689$	14.4853	0.551845
$690$	0	0
$691$	21.0711	0.801581	0.400791	−	0.916170i	$-0.368735\pi$
$691$	21.0711	0.801581	0.400791	+	0.916170i	$0.368735\pi$
$692$	− 20.4264i	− 0.776495i
$693$	0	0
$694$	−1.75736	−0.0667084
$695$	0	0
$696$	0	0
$697$	− 42.6274i	− 1.61463i
$698$	− 0.627417i	− 0.0237481i
$699$	0	0
$700$	0	0
$701$	−37.3137	−1.40932	−0.704660	−	0.709545i	$-0.748900\pi$
$701$	−37.3137	−1.40932	−0.704660	+	0.709545i	$0.748900\pi$
$702$	0	0
$703$	28.9706i	1.09265i
$704$	−2.44365	−0.0920986
$705$	0	0
$706$	3.79899	0.142977
$707$	− 6.34315i	− 0.238559i
$708$	0	0
$709$	17.1127	0.642681	0.321340	−	0.946964i	$-0.395867\pi$
$709$	17.1127	0.642681	0.321340	+	0.946964i	$0.395867\pi$
$710$	0	0
$711$	0	0
$712$	9.51472i	0.356579i
$713$	14.4853i	0.542478i
$714$	0	0
$715$	0	0
$716$	−10.3431	−0.386542
$717$	0	0
$718$	− 11.5563i	− 0.431279i
$719$	−4.97056	−0.185371	−0.0926854	−	0.995695i	$-0.529545\pi$
$719$	−4.97056	−0.185371	−0.0926854	+	0.995695i	$0.529545\pi$
$720$	0	0
$721$	−14.4264	−0.537267
$722$	− 3.04163i	− 0.113198i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 19.3553i	− 0.717850i	−0.933366	−	0.358925i	$-0.883143\pi$
$727$	− 19.3553i	− 0.717850i	0.933366	−	0.358925i	$-0.116857\pi$
$728$	1.31371i	0.0486893i
$729$	0	0
$730$	0	0
$731$	14.8284	0.548449
$732$	0	0
$733$	− 1.31371i	− 0.0485229i	−0.999706	−	0.0242615i	$-0.992277\pi$
$733$	− 1.31371i	− 0.0485229i	0.999706	−	0.0242615i	$-0.00772342\pi$
$734$	1.84062	0.0679385
$735$	0	0
$736$	−6.24264	−0.230107
$737$	− 1.17157i	− 0.0431554i
$738$	0	0
$739$	−30.7279	−1.13034	−0.565172	−	0.824973i	$-0.691190\pi$
$739$	−30.7279	−1.13034	−0.565172	+	0.824973i	$0.691190\pi$
$740$	0	0
$741$	0	0
$742$	− 4.97056i	− 0.182475i
$743$	38.4853i	1.41189i	0.708268	+	0.705944i	$0.249477\pi$
$743$	38.4853i	1.41189i	−0.708268	+	0.705944i	$0.750523\pi$
$744$	0	0
$745$	0	0
$746$	10.4853	0.383893
$747$	0	0
$748$	5.17157i	0.189091i
$749$	5.45584	0.199352
$750$	0	0
$751$	−44.4853	−1.62329	−0.811645	−	0.584150i	$-0.801428\pi$
$751$	−44.4853	−1.62329	−0.811645	+	0.584150i	$0.801428\pi$
$752$	2.48528i	0.0906289i
$753$	0	0
$754$	2.34315	0.0853323
$755$	0	0
$756$	0	0
$757$	− 4.14214i	− 0.150548i	−0.997163	−	0.0752742i	$-0.976017\pi$
$757$	− 4.14214i	− 0.150548i	0.997163	−	0.0752742i	$-0.0239832\pi$
$758$	− 6.18377i	− 0.224605i
$759$	0	0
$760$	0	0
$761$	36.6274	1.32774	0.663871	−	0.747847i	$-0.268912\pi$
$761$	36.6274	1.32774	0.663871	+	0.747847i	$0.268912\pi$
$762$	0	0
$763$	1.65685i	0.0599822i
$764$	−24.9706	−0.903403
$765$	0	0
$766$	13.7157	0.495569
$767$	− 10.2426i	− 0.369840i
$768$	0	0
$769$	−10.9706	−0.395609	−0.197804	−	0.980242i	$-0.563381\pi$
$769$	−10.9706	−0.395609	−0.197804	+	0.980242i	$0.563381\pi$
$770$	0	0
$771$	0	0
$772$	28.6274i	1.03032i
$773$	− 6.14214i	− 0.220917i	−0.993881	−	0.110459i	$-0.964768\pi$
$773$	− 6.14214i	− 0.220917i	0.993881	−	0.110459i	$-0.0352320\pi$
$774$	0	0
$775$	0	0
$776$	5.79899	0.208172
$777$	0	0
$778$	− 6.88730i	− 0.246922i
$779$	−30.1421	−1.07995
$780$	0	0
$781$	−4.62742	−0.165582
$782$	2.82843i	0.101144i
$783$	0	0
$784$	18.9411	0.676469
$785$	0	0
$786$	0	0
$787$	− 5.51472i	− 0.196578i	−0.995158	−	0.0982892i	$-0.968663\pi$
$787$	− 5.51472i	− 0.196578i	0.995158	−	0.0982892i	$-0.0313370\pi$
$788$	− 42.0000i	− 1.49619i
$789$	0	0
$790$	0	0
$791$	−2.62742	−0.0934202
$792$	0	0
$793$	8.00000i	0.284088i
$794$	−11.5147	−0.408642
$795$	0	0
$796$	−7.31371	−0.259228
$797$	10.9706i	0.388597i	0.980942	+	0.194299i	$0.0622431\pi$
$797$	10.9706i	0.388597i	−0.980942	+	0.194299i	$0.937757\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	0	0
$802$	− 7.17157i	− 0.253237i
$803$	4.97056i	0.175407i
$804$	0	0
$805$	0	0
$806$	4.24264	0.149441
$807$	0	0
$808$	− 12.1421i	− 0.427159i
$809$	−45.2548	−1.59108	−0.795538	−	0.605904i	$-0.792811\pi$
$809$	−45.2548	−1.59108	−0.795538	+	0.605904i	$0.792811\pi$
$810$	0	0
$811$	8.38478	0.294429	0.147215	−	0.989105i	$-0.452969\pi$
$811$	8.38478	0.294429	0.147215	+	0.989105i	$0.452969\pi$
$812$	8.56854i	0.300697i
$813$	0	0
$814$	−2.05887	−0.0721635
$815$	0	0
$816$	0	0
$817$	− 10.4853i	− 0.366834i
$818$	− 5.31371i	− 0.185789i
$819$	0	0
$820$	0	0
$821$	−39.2548	−1.37000	−0.685002	−	0.728542i	$-0.740199\pi$
$821$	−39.2548	−1.37000	−0.685002	+	0.728542i	$0.740199\pi$
$822$	0	0
$823$	34.3848i	1.19858i	0.800533	+	0.599289i	$0.204550\pi$
$823$	34.3848i	1.19858i	−0.800533	+	0.599289i	$0.795450\pi$
$824$	−27.6152	−0.962022
$825$	0	0
$826$	−3.51472	−0.122293
$827$	27.8579i	0.968713i	0.874871	+	0.484356i	$0.160946\pi$
$827$	27.8579i	0.968713i	−0.874871	+	0.484356i	$0.839054\pi$
$828$	0	0
$829$	−7.02944	−0.244142	−0.122071	−	0.992521i	$-0.538954\pi$
$829$	−7.02944	−0.244142	−0.122071	+	0.992521i	$0.538954\pi$
$830$	0	0
$831$	0	0
$832$	− 4.17157i	− 0.144623i
$833$	− 30.4853i	− 1.05625i
$834$	0	0
$835$	0	0
$836$	3.65685	0.126475
$837$	0	0
$838$	2.14214i	0.0739988i
$839$	−18.7279	−0.646560	−0.323280	−	0.946303i	$-0.604786\pi$
$839$	−18.7279	−0.646560	−0.323280	+	0.946303i	$0.604786\pi$
$840$	0	0
$841$	3.00000	0.103448
$842$	− 0.426407i	− 0.0146950i
$843$	0	0
$844$	−35.3137	−1.21555
$845$	0	0
$846$	0	0
$847$	− 8.82843i	− 0.303348i
$848$	43.4558i	1.49228i
$849$	0	0
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	37.4558i	1.28246i	0.767347	+	0.641232i	$0.221576\pi$
$853$	37.4558i	1.28246i	−0.767347	+	0.641232i	$0.778424\pi$
$854$	2.74517	0.0939376
$855$	0	0
$856$	10.4437	0.356957
$857$	0.343146i	0.0117216i	0.999983	+	0.00586082i	$0.00186557\pi$
$857$	0.343146i	0.0117216i	−0.999983	+	0.00586082i	$0.998134\pi$
$858$	0	0
$859$	−11.7990	−0.402576	−0.201288	−	0.979532i	$-0.564513\pi$
$859$	−11.7990	−0.402576	−0.201288	+	0.979532i	$0.564513\pi$
$860$	0	0
$861$	0	0
$862$	− 1.49747i	− 0.0510042i
$863$	− 19.4558i	− 0.662285i	−0.943581	−	0.331142i	$-0.892566\pi$
$863$	− 19.4558i	− 0.662285i	0.943581	−	0.331142i	$-0.107434\pi$
$864$	0	0
$865$	0	0
$866$	1.51472	0.0514722
$867$	0	0
$868$	15.5147i	0.526604i
$869$	4.97056	0.168615
$870$	0	0
$871$	2.00000	0.0677674
$872$	3.17157i	0.107403i
$873$	0	0
$874$	2.00000	0.0676510
$875$	0	0
$876$	0	0
$877$	− 2.68629i	− 0.0907096i	−0.998971	−	0.0453548i	$-0.985558\pi$
$877$	− 2.68629i	− 0.0907096i	0.998971	−	0.0453548i	$-0.0144418\pi$
$878$	13.6569i	0.460897i
$879$	0	0
$880$	0	0
$881$	−52.9706	−1.78462	−0.892312	−	0.451420i	$-0.850918\pi$
$881$	−52.9706	−1.78462	−0.892312	+	0.451420i	$0.850918\pi$
$882$	0	0
$883$	− 32.2426i	− 1.08505i	−0.840039	−	0.542526i	$-0.817468\pi$
$883$	− 32.2426i	− 1.08505i	0.840039	−	0.542526i	$-0.182532\pi$
$884$	−8.82843	−0.296932
$885$	0	0
$886$	−2.72792	−0.0916463
$887$	14.3848i	0.482994i	0.970402	+	0.241497i	$0.0776383\pi$
$887$	14.3848i	0.482994i	−0.970402	+	0.241497i	$0.922362\pi$
$888$	0	0
$889$	−7.79899	−0.261570
$890$	0	0
$891$	0	0
$892$	48.4264i	1.62144i
$893$	− 2.82843i	− 0.0946497i
$894$	0	0
$895$	0	0
$896$	−8.74517	−0.292155
$897$	0	0
$898$	12.0589i	0.402410i
$899$	57.9411	1.93244
$900$	0	0
$901$	69.9411	2.33008
$902$	− 2.14214i	− 0.0713253i
$903$	0	0
$904$	−5.02944	−0.167277
$905$	0	0
$906$	0	0
$907$	33.2132i	1.10283i	0.834232	+	0.551413i	$0.185911\pi$
$907$	33.2132i	1.10283i	−0.834232	+	0.551413i	$0.814089\pi$
$908$	− 50.5685i	− 1.67818i
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	− 5.17157i	− 0.171154i
$914$	−7.45584	−0.246617
$915$	0	0
$916$	−1.51472	−0.0500477
$917$	− 14.0589i	− 0.464265i
$918$	0	0
$919$	16.4853	0.543799	0.271900	−	0.962326i	$-0.412348\pi$
$919$	16.4853	0.543799	0.271900	+	0.962326i	$0.412348\pi$
$920$	0	0
$921$	0	0
$922$	− 10.9706i	− 0.361296i
$923$	− 7.89949i	− 0.260015i
$924$	0	0
$925$	0	0
$926$	6.48528	0.213120
$927$	0	0
$928$	24.9706i	0.819699i
$929$	11.1716	0.366527	0.183264	−	0.983064i	$-0.441334\pi$
$929$	11.1716	0.366527	0.183264	+	0.983064i	$0.441334\pi$
$930$	0	0
$931$	−21.5563	−0.706481
$932$	− 45.0294i	− 1.47499i
$933$	0	0
$934$	4.38478	0.143474
$935$	0	0
$936$	0	0
$937$	− 10.9706i	− 0.358393i	−0.983813	−	0.179196i	$-0.942650\pi$
$937$	− 10.9706i	− 0.358393i	0.983813	−	0.179196i	$-0.0573497\pi$
$938$	− 0.686292i	− 0.0224082i
$939$	0	0
$940$	0	0
$941$	54.7696	1.78544	0.892718	−	0.450615i	$-0.148795\pi$
$941$	54.7696	1.78544	0.892718	+	0.450615i	$0.148795\pi$
$942$	0	0
$943$	12.4853i	0.406577i
$944$	30.7279	1.00011
$945$	0	0
$946$	0.745166	0.0242274
$947$	− 45.1127i	− 1.46597i	−0.680247	−	0.732983i	$-0.738128\pi$
$947$	− 45.1127i	− 1.46597i	0.680247	−	0.732983i	$-0.261872\pi$
$948$	0	0
$949$	−8.48528	−0.275444
$950$	0	0
$951$	0	0
$952$	6.34315i	0.205583i
$953$	55.2548i	1.78988i	0.446187	+	0.894940i	$0.352782\pi$
$953$	55.2548i	1.78988i	−0.446187	+	0.894940i	$0.647218\pi$
$954$	0	0
$955$	0	0
$956$	−1.07107	−0.0346408
$957$	0	0
$958$	− 2.18377i	− 0.0705543i
$959$	−4.40202	−0.142149
$960$	0	0
$961$	73.9117	2.38425
$962$	− 3.51472i	− 0.113319i
$963$	0	0
$964$	4.54416	0.146357
$965$	0	0
$966$	0	0
$967$	19.9411i	0.641263i	0.947204	+	0.320632i	$0.103895\pi$
$967$	19.9411i	0.641263i	−0.947204	+	0.320632i	$0.896105\pi$
$968$	− 16.8995i	− 0.543170i
$969$	0	0
$970$	0	0
$971$	12.2843	0.394221	0.197111	−	0.980381i	$-0.436844\pi$
$971$	12.2843	0.394221	0.197111	+	0.980381i	$0.436844\pi$
$972$	0	0
$973$	10.3431i	0.331586i
$974$	9.51472	0.304871
$975$	0	0
$976$	−24.0000	−0.768221
$977$	− 56.4853i	− 1.80712i	−0.428457	−	0.903562i	$-0.640943\pi$
$977$	− 56.4853i	− 1.80712i	0.428457	−	0.903562i	$-0.359057\pi$
$978$	0	0
$979$	−3.51472	−0.112331
$980$	0	0
$981$	0	0
$982$	− 4.48528i	− 0.143131i
$983$	34.9706i	1.11539i	0.830047	+	0.557694i	$0.188314\pi$
$983$	34.9706i	1.11539i	−0.830047	+	0.557694i	$0.811686\pi$
$984$	0	0
$985$	0	0
$986$	11.3137	0.360302
$987$	0	0
$988$	6.24264i	0.198605i
$989$	−4.34315	−0.138104
$990$	0	0
$991$	15.0294	0.477426	0.238713	−	0.971090i	$-0.423275\pi$
$991$	15.0294	0.477426	0.238713	+	0.971090i	$0.423275\pi$
$992$	45.2132i	1.43552i
$993$	0	0
$994$	−2.71068	−0.0859775
$995$	0	0
$996$	0	0
$997$	− 23.1716i	− 0.733851i	−0.930250	−	0.366926i	$-0.880410\pi$
$997$	− 23.1716i	− 0.733851i	0.930250	−	0.366926i	$-0.119590\pi$
$998$	− 4.32590i	− 0.136934i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.r.2224.3		4
3.2	odd	2		325.2.b.f.274.2		4
5.2	odd	4		585.2.a.m.1.1		2
5.3	odd	4		2925.2.a.u.1.2		2
5.4	even	2	inner	2925.2.c.r.2224.2		4
15.2	even	4		65.2.a.b.1.2	✓	2
15.8	even	4		325.2.a.i.1.1		2
15.14	odd	2		325.2.b.f.274.3		4
20.7	even	4		9360.2.a.cd.1.2		2
60.23	odd	4		5200.2.a.bu.1.2		2
60.47	odd	4		1040.2.a.j.1.1		2
65.12	odd	4		7605.2.a.x.1.2		2
105.62	odd	4		3185.2.a.j.1.2		2
120.77	even	4		4160.2.a.bf.1.1		2
120.107	odd	4		4160.2.a.z.1.2		2
165.32	odd	4		7865.2.a.j.1.1		2
195.2	odd	12		845.2.m.f.316.3		8
195.17	even	12		845.2.e.c.146.2		4
195.32	odd	12		845.2.m.f.361.2		8
195.38	even	4		4225.2.a.r.1.2		2
195.47	odd	4		845.2.c.b.506.3		4
195.62	even	12		845.2.e.c.191.2		4
195.77	even	4		845.2.a.g.1.1		2
195.107	even	12		845.2.e.h.191.1		4
195.122	odd	4		845.2.c.b.506.2		4
195.137	odd	12		845.2.m.f.361.3		8
195.152	even	12		845.2.e.h.146.1		4
195.167	odd	12		845.2.m.f.316.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.b.1.2	✓	2	15.2	even	4
325.2.a.i.1.1		2	15.8	even	4
325.2.b.f.274.2		4	3.2	odd	2
325.2.b.f.274.3		4	15.14	odd	2
585.2.a.m.1.1		2	5.2	odd	4
845.2.a.g.1.1		2	195.77	even	4
845.2.c.b.506.2		4	195.122	odd	4
845.2.c.b.506.3		4	195.47	odd	4
845.2.e.c.146.2		4	195.17	even	12
845.2.e.c.191.2		4	195.62	even	12
845.2.e.h.146.1		4	195.152	even	12
845.2.e.h.191.1		4	195.107	even	12
845.2.m.f.316.2		8	195.167	odd	12
845.2.m.f.316.3		8	195.2	odd	12
845.2.m.f.361.2		8	195.32	odd	12
845.2.m.f.361.3		8	195.137	odd	12
1040.2.a.j.1.1		2	60.47	odd	4
2925.2.a.u.1.2		2	5.3	odd	4
2925.2.c.r.2224.2		4	5.4	even	2	inner
2925.2.c.r.2224.3		4	1.1	even	1	trivial
3185.2.a.j.1.2		2	105.62	odd	4
4160.2.a.z.1.2		2	120.107	odd	4
4160.2.a.bf.1.1		2	120.77	even	4
4225.2.a.r.1.2		2	195.38	even	4
5200.2.a.bu.1.2		2	60.23	odd	4
7605.2.a.x.1.2		2	65.12	odd	4
7865.2.a.j.1.1		2	165.32	odd	4
9360.2.a.cd.1.2		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.414214i q^{2} +1.82843 q^{4} +0.828427i q^{7} +1.58579i q^{8} +O(q^{10})\)	\(q+0.414214i q^{2} +1.82843 q^{4} +0.828427i q^{7} +1.58579i q^{8} -0.585786 q^{11} -1.00000i q^{13} -0.343146 q^{14} +3.00000 q^{16} -4.82843i q^{17} -3.41421 q^{19} -0.242641i q^{22} +1.41421i q^{23} +0.414214 q^{26} +1.51472i q^{28} +5.65685 q^{29} +10.2426 q^{31} +4.41421i q^{32} +2.00000 q^{34} -8.48528i q^{37} -1.41421i q^{38} +8.82843 q^{41} +3.07107i q^{43} -1.07107 q^{44} -0.585786 q^{46} +0.828427i q^{47} +6.31371 q^{49} -1.82843i q^{52} +14.4853i q^{53} -1.31371 q^{56} +2.34315i q^{58} +10.2426 q^{59} -8.00000 q^{61} +4.24264i q^{62} +4.17157 q^{64} +2.00000i q^{67} -8.82843i q^{68} +7.89949 q^{71} -8.48528i q^{73} +3.51472 q^{74} -6.24264 q^{76} -0.485281i q^{77} -8.48528 q^{79} +3.65685i q^{82} +8.82843i q^{83} -1.27208 q^{86} -0.928932i q^{88} +6.00000 q^{89} +0.828427 q^{91} +2.58579i q^{92} -0.343146 q^{94} -3.65685i q^{97} +2.61522i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} - 8 q^{11} - 24 q^{14} + 12 q^{16} - 8 q^{19} - 4 q^{26} + 24 q^{31} + 8 q^{34} + 24 q^{41} + 24 q^{44} - 8 q^{46} - 20 q^{49} + 40 q^{56} + 24 q^{59} - 32 q^{61} + 28 q^{64} - 8 q^{71} + 48 q^{74} - 8 q^{76} - 56 q^{86} + 24 q^{89} - 8 q^{91} - 24 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 8 * q^11 - 24 * q^14 + 12 * q^16 - 8 * q^19 - 4 * q^26 + 24 * q^31 + 8 * q^34 + 24 * q^41 + 24 * q^44 - 8 * q^46 - 20 * q^49 + 40 * q^56 + 24 * q^59 - 32 * q^61 + 28 * q^64 - 8 * q^71 + 48 * q^74 - 8 * q^76 - 56 * q^86 + 24 * q^89 - 8 * q^91 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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