LMFDB - Embedded newform 325.2.b.f.274.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("325.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	325.b (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:	$2.59513806569$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.2.b.f.274.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214i q^{2} +1.41421i q^{3} +1.82843 q^{4} +0.585786 q^{6} +0.828427i q^{7} -1.58579i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.414214i q^{2} +1.41421i q^{3} +1.82843 q^{4} +0.585786 q^{6} +0.828427i q^{7} -1.58579i q^{8} +1.00000 q^{9} +0.585786 q^{11} +2.58579i q^{12} -1.00000i q^{13} +0.343146 q^{14} +3.00000 q^{16} +4.82843i q^{17} -0.414214i q^{18} -3.41421 q^{19} -1.17157 q^{21} -0.242641i q^{22} -1.41421i q^{23} +2.24264 q^{24} -0.414214 q^{26} +5.65685i q^{27} +1.51472i q^{28} -5.65685 q^{29} +10.2426 q^{31} -4.41421i q^{32} +0.828427i q^{33} +2.00000 q^{34} +1.82843 q^{36} -8.48528i q^{37} +1.41421i q^{38} +1.41421 q^{39} -8.82843 q^{41} +0.485281i q^{42} +3.07107i q^{43} +1.07107 q^{44} -0.585786 q^{46} -0.828427i q^{47} +4.24264i q^{48} +6.31371 q^{49} -6.82843 q^{51} -1.82843i q^{52} -14.4853i q^{53} +2.34315 q^{54} +1.31371 q^{56} -4.82843i q^{57} +2.34315i q^{58} -10.2426 q^{59} -8.00000 q^{61} -4.24264i q^{62} +0.828427i q^{63} +4.17157 q^{64} +0.343146 q^{66} +2.00000i q^{67} +8.82843i q^{68} +2.00000 q^{69} -7.89949 q^{71} -1.58579i q^{72} -8.48528i q^{73} -3.51472 q^{74} -6.24264 q^{76} +0.485281i q^{77} -0.585786i q^{78} -8.48528 q^{79} -5.00000 q^{81} +3.65685i q^{82} -8.82843i q^{83} -2.14214 q^{84} +1.27208 q^{86} -8.00000i q^{87} -0.928932i q^{88} -6.00000 q^{89} +0.828427 q^{91} -2.58579i q^{92} +14.4853i q^{93} -0.343146 q^{94} +6.24264 q^{96} -3.65685i q^{97} -2.61522i q^{98} +0.585786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 8 * q^6 + 4 * q^9	$ 4 q - 4 q^{4} + 8 q^{6} + 4 q^{9} + 8 q^{11} + 24 q^{14} + 12 q^{16} - 8 q^{19} - 16 q^{21} - 8 q^{24} + 4 q^{26} + 24 q^{31} + 8 q^{34} - 4 q^{36} - 24 q^{41} - 24 q^{44} - 8 q^{46} - 20 q^{49} - 16 q^{51} + 32 q^{54} - 40 q^{56} - 24 q^{59} - 32 q^{61} + 28 q^{64} + 24 q^{66} + 8 q^{69} + 8 q^{71} - 48 q^{74} - 8 q^{76} - 20 q^{81} + 48 q^{84} + 56 q^{86} - 24 q^{89} - 8 q^{91} - 24 q^{94} + 8 q^{96} + 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^6 + 4 * q^9 + 8 * q^11 + 24 * q^14 + 12 * q^16 - 8 * q^19 - 16 * q^21 - 8 * q^24 + 4 * q^26 + 24 * q^31 + 8 * q^34 - 4 * q^36 - 24 * q^41 - 24 * q^44 - 8 * q^46 - 20 * q^49 - 16 * q^51 + 32 * q^54 - 40 * q^56 - 24 * q^59 - 32 * q^61 + 28 * q^64 + 24 * q^66 + 8 * q^69 + 8 * q^71 - 48 * q^74 - 8 * q^76 - 20 * q^81 + 48 * q^84 + 56 * q^86 - 24 * q^89 - 8 * q^91 - 24 * q^94 + 8 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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	$-0.953216\pi$
$2$	− 0.414214i	− 0.292893i	0.989219	−	0.146447i	$-0.0467837\pi$
$3$	1.41421i	0.816497i	0.912871	+	0.408248i	$0.133860\pi$
$3$	1.41421i	0.816497i	−0.912871	+	0.408248i	$0.866140\pi$
$4$	1.82843	0.914214
$5$	0	0
$5$	0	0
$6$	0.585786	0.239146
$7$	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
$9$	1.00000	0.333333
$10$	0	0
$11$	0.585786	0.176621	0.0883106	−	0.996093i	$-0.471853\pi$
$11$	0.585786	0.176621	0.0883106	+	0.996093i	$0.471853\pi$
$12$	2.58579i	0.746452i
$13$	− 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	$0.199115\pi$
$17$	4.82843i	1.17107i	−0.810649	+	0.585533i	$0.800885\pi$
$18$	− 0.414214i	− 0.0976311i
$19$	−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
$21$	−1.17157	−0.255658
$22$	− 0.242641i	− 0.0517312i
$23$	− 1.41421i	− 0.294884i	−0.989071	−	0.147442i	$-0.952896\pi$
$23$	− 1.41421i	− 0.294884i	0.989071	−	0.147442i	$-0.0471040\pi$
$24$	2.24264	0.457777
$25$	0	0
$26$	−0.414214	−0.0812340
$27$	5.65685i	1.08866i
$28$	1.51472i	0.286255i
$29$	−5.65685	−1.05045	−0.525226	−	0.850963i	$-0.676019\pi$
$29$	−5.65685	−1.05045	−0.525226	+	0.850963i	$0.676019\pi$
$30$	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
$33$	0.828427i	0.144211i
$34$	2.00000	0.342997
$35$	0	0
$36$	1.82843	0.304738
$37$	− 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
$39$	1.41421	0.226455
$40$	0	0
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	0.485281i	0.0748805i
$43$	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	$-0.980756\pi$
$47$	− 0.828427i	− 0.120839i	0.998173	−	0.0604193i	$-0.0192438\pi$
$48$	4.24264i	0.612372i
$49$	6.31371	0.901958
$50$	0	0
$51$	−6.82843	−0.956171
$52$	− 1.82843i	− 0.253557i
$53$	− 14.4853i	− 1.98971i	−0.101327	−	0.994853i	$-0.532309\pi$
$53$	− 14.4853i	− 1.98971i	0.101327	−	0.994853i	$-0.467691\pi$
$54$	2.34315	0.318862
$55$	0	0
$56$	1.31371	0.175552
$57$	− 4.82843i	− 0.639541i
$58$	2.34315i	0.307670i
$59$	−10.2426	−1.33348	−0.666739	−	0.745291i	$-0.732310\pi$
$59$	−10.2426	−1.33348	−0.666739	+	0.745291i	$0.732310\pi$
$60$	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
$63$	0.828427i	0.104372i
$64$	4.17157	0.521447
$65$	0	0
$66$	0.343146	0.0422383
$67$	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
$69$	2.00000	0.240772
$70$	0	0
$71$	−7.89949	−0.937498	−0.468749	−	0.883332i	$-0.655295\pi$
$71$	−7.89949	−0.937498	−0.468749	+	0.883332i	$0.655295\pi$
$72$	− 1.58579i	− 0.186887i
$73$	− 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
$78$	− 0.585786i	− 0.0663273i
$79$	−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
$81$	−5.00000	−0.555556
$82$	3.65685i	0.403832i
$83$	− 8.82843i	− 0.969046i	−0.874779	−	0.484523i	$-0.838993\pi$
$83$	− 8.82843i	− 0.969046i	0.874779	−	0.484523i	$-0.161007\pi$
$84$	−2.14214	−0.233726
$85$	0	0
$86$	1.27208	0.137172
$87$	− 8.00000i	− 0.857690i
$88$	− 0.928932i	− 0.0990245i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0.828427	0.0868428
$92$	− 2.58579i	− 0.269587i
$93$	14.4853i	1.50205i
$94$	−0.343146	−0.0353928
$95$	0	0
$96$	6.24264	0.637137
$97$	− 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
$99$	0.585786	0.0588738
$100$	0	0
$101$	7.65685	0.761885	0.380943	−	0.924599i	$-0.375599\pi$
$101$	7.65685	0.761885	0.380943	+	0.924599i	$0.375599\pi$
$102$	2.82843i	0.280056i
$103$	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	$0.103124\pi$
$107$	6.58579i	0.636672i	−0.947978	+	0.318336i	$0.896876\pi$
$108$	10.3431i	0.995270i
$109$	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
$111$	12.0000	1.13899
$112$	2.48528i	0.234837i
$113$	− 3.17157i	− 0.298356i	−0.988810	−	0.149178i	$-0.952337\pi$
$113$	− 3.17157i	− 0.298356i	0.988810	−	0.149178i	$-0.0476628\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	−10.3431	−0.960337
$117$	− 1.00000i	− 0.0924500i
$118$	4.24264i	0.390567i
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−10.6569	−0.968805
$122$	3.31371i	0.300009i
$123$	− 12.4853i	− 1.12576i
$124$	18.7279	1.68182
$125$	0	0
$126$	0.343146	0.0305699
$127$	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
$129$	−4.34315	−0.382393
$130$	0	0
$131$	16.9706	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$132$	1.51472i	0.131839i
$133$	− 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	$-0.927111\pi$
$137$	− 5.31371i	− 0.453981i	0.973897	−	0.226990i	$-0.0728886\pi$
$138$	− 0.828427i	− 0.0705204i
$139$	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
$141$	1.17157	0.0986642
$142$	3.27208i	0.274587i
$143$	− 0.585786i	− 0.0489859i
$144$	3.00000	0.250000
$145$	0	0
$146$	−3.51472	−0.290880
$147$	8.92893i	0.736446i
$148$	− 15.5147i	− 1.27530i
$149$	0.343146	0.0281116	0.0140558	−	0.999901i	$-0.495526\pi$
$149$	0.343146	0.0281116	0.0140558	+	0.999901i	$0.495526\pi$
$150$	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
$153$	4.82843i	0.390355i
$154$	0.201010	0.0161979
$155$	0	0
$156$	2.58579	0.207029
$157$	− 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
$159$	20.4853	1.62459
$160$	0	0
$161$	1.17157	0.0923329
$162$	2.07107i	0.162718i
$163$	− 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	$0.110963\pi$
$167$	8.82843i	0.683164i	−0.939852	+	0.341582i	$0.889037\pi$
$168$	1.85786i	0.143337i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−3.41421	−0.261091
$172$	5.61522i	0.428157i
$173$	11.1716i	0.849359i	0.905344	+	0.424679i	$0.139613\pi$
$173$	11.1716i	0.849359i	−0.905344	+	0.424679i	$0.860387\pi$
$174$	−3.31371	−0.251212
$175$	0	0
$176$	1.75736	0.132466
$177$	− 14.4853i	− 1.08878i
$178$	2.48528i	0.186280i
$179$	5.65685	0.422813	0.211407	−	0.977398i	$-0.432196\pi$
$179$	5.65685	0.422813	0.211407	+	0.977398i	$0.432196\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	− 0.343146i	− 0.0254357i
$183$	− 11.3137i	− 0.836333i
$184$	−2.24264	−0.165330
$185$	0	0
$186$	6.00000	0.439941
$187$	2.82843i	0.206835i
$188$	− 1.51472i	− 0.110472i
$189$	−4.68629	−0.340878
$190$	0	0
$191$	13.6569	0.988175	0.494088	−	0.869412i	$-0.335502\pi$
$191$	13.6569	0.988175	0.494088	+	0.869412i	$0.335502\pi$
$192$	5.89949i	0.425759i
$193$	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	$0.305079\pi$
$197$	22.9706i	1.63658i	−0.574802	+	0.818292i	$0.694921\pi$
$198$	− 0.242641i	− 0.0172437i
$199$	−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
$201$	−2.82843	−0.199502
$202$	− 3.17157i	− 0.223151i
$203$	− 4.68629i	− 0.328913i
$204$	−12.4853	−0.874145
$205$	0	0
$206$	7.21320	0.502568
$207$	− 1.41421i	− 0.0982946i
$208$	− 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
$213$	− 11.1716i	− 0.765464i
$214$	2.72792	0.186477
$215$	0	0
$216$	8.97056	0.610369
$217$	8.48528i	0.576018i
$218$	− 0.828427i	− 0.0561082i
$219$	12.0000	0.810885
$220$	0	0
$221$	4.82843	0.324795
$222$	− 4.97056i	− 0.333602i
$223$	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	$0.370056\pi$
$227$	27.6569i	1.83565i	−0.396985	+	0.917825i	$0.629944\pi$
$228$	− 8.82843i	− 0.584677i
$229$	−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
$231$	−0.686292	−0.0451547
$232$	8.97056i	0.588946i
$233$	24.6274i	1.61340i	0.590964	+	0.806698i	$0.298747\pi$
$233$	24.6274i	1.61340i	−0.590964	+	0.806698i	$0.701253\pi$
$234$	−0.414214	−0.0270780
$235$	0	0
$236$	−18.7279	−1.21908
$237$	− 12.0000i	− 0.779484i
$238$	1.65685i	0.107398i
$239$	0.585786	0.0378914	0.0189457	−	0.999821i	$-0.493969\pi$
$239$	0.585786	0.0378914	0.0189457	+	0.999821i	$0.493969\pi$
$240$	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
$243$	9.89949i	0.635053i
$244$	−14.6274	−0.936424
$245$	0	0
$246$	−5.17157	−0.329727
$247$	3.41421i	0.217241i
$248$	− 16.2426i	− 1.03141i
$249$	12.4853	0.791223
$250$	0	0
$251$	−19.7990	−1.24970	−0.624851	−	0.780744i	$-0.714840\pi$
$251$	−19.7990	−1.24970	−0.624851	+	0.780744i	$0.714840\pi$
$252$	1.51472i	0.0954183i
$253$	− 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	$-0.829746\pi$
$257$	− 16.3431i	− 1.01946i	0.860335	−	0.509729i	$-0.170254\pi$
$258$	1.79899i	0.112000i
$259$	7.02944	0.436788
$260$	0	0
$261$	−5.65685	−0.350150
$262$	− 7.02944i	− 0.434280i
$263$	− 13.4142i	− 0.827156i	−0.910469	−	0.413578i	$-0.864279\pi$
$263$	− 13.4142i	− 0.827156i	0.910469	−	0.413578i	$-0.135721\pi$
$264$	1.31371	0.0808532
$265$	0	0
$266$	−1.17157	−0.0718337
$267$	− 8.48528i	− 0.519291i
$268$	3.65685i	0.223378i
$269$	2.68629	0.163786	0.0818930	−	0.996641i	$-0.473903\pi$
$269$	2.68629	0.163786	0.0818930	+	0.996641i	$0.473903\pi$
$270$	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
$273$	1.17157i	0.0709068i
$274$	−2.20101	−0.132968
$275$	0	0
$276$	3.65685	0.220117
$277$	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
$279$	10.2426	0.613211
$280$	0	0
$281$	−17.7990	−1.06180	−0.530899	−	0.847435i	$-0.678146\pi$
$281$	−17.7990	−1.06180	−0.530899	+	0.847435i	$0.678146\pi$
$282$	− 0.485281i	− 0.0288981i
$283$	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
$288$	− 4.41421i	− 0.260110i
$289$	−6.31371	−0.371395
$290$	0	0
$291$	5.17157	0.303163
$292$	− 15.5147i	− 0.907930i
$293$	− 2.14214i	− 0.125145i	−0.998040	−	0.0625724i	$-0.980070\pi$
$293$	− 2.14214i	− 0.125145i	0.998040	−	0.0625724i	$-0.0199304\pi$
$294$	3.69848	0.215700
$295$	0	0
$296$	−13.4558	−0.782105
$297$	3.31371i	0.192281i
$298$	− 0.142136i	− 0.00823370i
$299$	−1.41421	−0.0817861
$300$	0	0
$301$	−2.54416	−0.146643
$302$	− 7.55635i	− 0.434819i
$303$	10.8284i	0.622077i
$304$	−10.2426	−0.587456
$305$	0	0
$306$	2.00000	0.114332
$307$	− 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
$309$	−24.6274	−1.40100
$310$	0	0
$311$	−8.48528	−0.481156	−0.240578	−	0.970630i	$-0.577337\pi$
$311$	−8.48528	−0.481156	−0.240578	+	0.970630i	$0.577337\pi$
$312$	− 2.24264i	− 0.126965i
$313$	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	$-0.737584\pi$
$317$	− 26.1421i	− 1.46829i	0.678993	−	0.734144i	$-0.262416\pi$
$318$	− 8.48528i	− 0.475831i
$319$	−3.31371	−0.185532
$320$	0	0
$321$	−9.31371	−0.519841
$322$	− 0.485281i	− 0.0270437i
$323$	− 16.4853i	− 0.917266i
$324$	−9.14214	−0.507896
$325$	0	0
$326$	−6.20101	−0.343442
$327$	2.82843i	0.156412i
$328$	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
$333$	− 8.48528i	− 0.464991i
$334$	3.65685	0.200094
$335$	0	0
$336$	−3.51472	−0.191744
$337$	− 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
$339$	4.48528	0.243607
$340$	0	0
$341$	6.00000	0.324918
$342$	1.41421i	0.0764719i
$343$	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	$-0.963673\pi$
$347$	− 4.24264i	− 0.227757i	0.993495	−	0.113878i	$-0.0363274\pi$
$348$	− 14.6274i	− 0.784112i
$349$	−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
$351$	5.65685	0.301941
$352$	− 2.58579i	− 0.137823i
$353$	9.17157i	0.488154i	0.969756	+	0.244077i	$0.0784849\pi$
$353$	9.17157i	0.488154i	−0.969756	+	0.244077i	$0.921515\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	−10.9706	−0.581439
$357$	− 5.65685i	− 0.299392i
$358$	− 2.34315i	− 0.123839i
$359$	27.8995	1.47248	0.736240	−	0.676721i	$-0.236600\pi$
$359$	27.8995	1.47248	0.736240	+	0.676721i	$0.236600\pi$
$360$	0	0
$361$	−7.34315	−0.386481
$362$	0	0
$363$	− 15.0711i	− 0.791026i
$364$	1.51472	0.0793928
$365$	0	0
$366$	−4.68629	−0.244956
$367$	− 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
$369$	−8.82843	−0.459590
$370$	0	0
$371$	12.0000	0.623009
$372$	26.4853i	1.37320i
$373$	− 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
$378$	1.94113i	0.0998407i
$379$	−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
$381$	−13.3137	−0.682082
$382$	− 5.65685i	− 0.289430i
$383$	33.1127i	1.69198i	0.533199	+	0.845990i	$0.320990\pi$
$383$	33.1127i	1.69198i	−0.533199	+	0.845990i	$0.679010\pi$
$384$	14.9289	0.761839
$385$	0	0
$386$	6.48528	0.330092
$387$	3.07107i	0.156111i
$388$	− 6.68629i	− 0.339445i
$389$	16.6274	0.843044	0.421522	−	0.906818i	$-0.361496\pi$
$389$	16.6274	0.843044	0.421522	+	0.906818i	$0.361496\pi$
$390$	0	0
$391$	6.82843	0.345328
$392$	− 10.0122i	− 0.505692i
$393$	24.0000i	1.21064i
$394$	9.51472	0.479345
$395$	0	0
$396$	1.07107	0.0538232
$397$	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
$399$	4.00000	0.200250
$400$	0	0
$401$	17.3137	0.864605	0.432303	−	0.901729i	$-0.357701\pi$
$401$	17.3137	0.864605	0.432303	+	0.901729i	$0.357701\pi$
$402$	1.17157i	0.0584327i
$403$	− 10.2426i	− 0.510222i
$404$	14.0000	0.696526
$405$	0	0
$406$	−1.94113	−0.0963364
$407$	− 4.97056i	− 0.246382i
$408$	10.8284i	0.536087i
$409$	−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
$411$	7.51472	0.370674
$412$	31.8406i	1.56867i
$413$	− 8.48528i	− 0.417533i
$414$	−0.585786	−0.0287898
$415$	0	0
$416$	−4.41421	−0.216425
$417$	17.6569i	0.864660i
$418$	0.828427i	0.0405197i
$419$	−5.17157	−0.252648	−0.126324	−	0.991989i	$-0.540318\pi$
$419$	−5.17157	−0.252648	−0.126324	+	0.991989i	$0.540318\pi$
$420$	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
$423$	− 0.828427i	− 0.0402795i
$424$	−22.9706	−1.11555
$425$	0	0
$426$	−4.62742	−0.224199
$427$	− 6.62742i	− 0.320723i
$428$	12.0416i	0.582054i
$429$	0.828427	0.0399968
$430$	0	0
$431$	3.61522	0.174139	0.0870696	−	0.996202i	$-0.472250\pi$
$431$	3.61522	0.174139	0.0870696	+	0.996202i	$0.472250\pi$
$432$	16.9706i	0.816497i
$433$	− 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
$438$	− 4.97056i	− 0.237503i
$439$	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
$441$	6.31371	0.300653
$442$	− 2.00000i	− 0.0951303i
$443$	− 6.58579i	− 0.312900i	−0.987686	−	0.156450i	$-0.949995\pi$
$443$	− 6.58579i	− 0.312900i	0.987686	−	0.156450i	$-0.0500050\pi$
$444$	21.9411	1.04128
$445$	0	0
$446$	10.9706	0.519471
$447$	0.485281i	0.0229530i
$448$	3.45584i	0.163273i
$449$	−29.1127	−1.37391	−0.686957	−	0.726698i	$-0.741054\pi$
$449$	−29.1127	−1.37391	−0.686957	+	0.726698i	$0.741054\pi$
$450$	0	0
$451$	−5.17157	−0.243520
$452$	− 5.79899i	− 0.272762i
$453$	25.7990i	1.21214i
$454$	11.4558	0.537649
$455$	0	0
$456$	−7.65685	−0.358565
$457$	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
$459$	−27.3137	−1.27489
$460$	0	0
$461$	26.4853	1.23354	0.616771	−	0.787142i	$-0.288440\pi$
$461$	26.4853	1.23354	0.616771	+	0.787142i	$0.288440\pi$
$462$	0.284271i	0.0132255i
$463$	− 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	$0.0787636\pi$
$467$	10.5858i	0.489852i	−0.969542	+	0.244926i	$0.921236\pi$
$468$	− 1.82843i	− 0.0845191i
$469$	−1.65685	−0.0765064
$470$	0	0
$471$	25.4558	1.17294
$472$	16.2426i	0.747628i
$473$	1.79899i	0.0827176i
$474$	−4.97056	−0.228306
$475$	0	0
$476$	−7.31371	−0.335223
$477$	− 14.4853i	− 0.663235i
$478$	− 0.242641i	− 0.0110981i
$479$	5.27208	0.240887	0.120444	−	0.992720i	$-0.461568\pi$
$479$	5.27208	0.240887	0.120444	+	0.992720i	$0.461568\pi$
$480$	0	0
$481$	−8.48528	−0.386896
$482$	− 1.02944i	− 0.0468896i
$483$	1.65685i	0.0753895i
$484$	−19.4853	−0.885695
$485$	0	0
$486$	4.10051	0.186003
$487$	− 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
$489$	21.1716	0.957412
$490$	0	0
$491$	10.8284	0.488680	0.244340	−	0.969690i	$-0.421429\pi$
$491$	10.8284	0.488680	0.244340	+	0.969690i	$0.421429\pi$
$492$	− 22.8284i	− 1.02918i
$493$	− 27.3137i	− 1.23015i
$494$	1.41421	0.0636285
$495$	0	0
$496$	30.7279	1.37972
$497$	− 6.54416i	− 0.293546i
$498$	− 5.17157i	− 0.231744i
$499$	−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
$501$	−12.4853	−0.557801
$502$	8.20101i	0.366029i
$503$	18.1005i	0.807062i	0.914966	+	0.403531i	$0.132217\pi$
$503$	18.1005i	0.807062i	−0.914966	+	0.403531i	$0.867783\pi$
$504$	1.31371	0.0585172
$505$	0	0
$506$	−0.343146	−0.0152547
$507$	− 1.41421i	− 0.0628074i
$508$	17.2132i	0.763712i
$509$	21.1127	0.935804	0.467902	−	0.883780i	$-0.345010\pi$
$509$	21.1127	0.935804	0.467902	+	0.883780i	$0.345010\pi$
$510$	0	0
$511$	7.02944	0.310964
$512$	− 22.7574i	− 1.00574i
$513$	− 19.3137i	− 0.852721i
$514$	−6.76955	−0.298592
$515$	0	0
$516$	−7.94113	−0.349589
$517$	− 0.485281i	− 0.0213427i
$518$	− 2.91169i	− 0.127932i
$519$	−15.7990	−0.693499
$520$	0	0
$521$	−6.34315	−0.277898	−0.138949	−	0.990300i	$-0.544372\pi$
$521$	−6.34315	−0.277898	−0.138949	+	0.990300i	$0.544372\pi$
$522$	2.34315i	0.102557i
$523$	− 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
$528$	2.48528i	0.108158i
$529$	21.0000	0.913043
$530$	0	0
$531$	−10.2426	−0.444493
$532$	− 5.17157i	− 0.224216i
$533$	8.82843i	0.382402i
$534$	−3.51472	−0.152097
$535$	0	0
$536$	3.17157	0.136991
$537$	8.00000i	0.345225i
$538$	− 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
$546$	0.485281	0.0207681
$547$	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
$549$	−8.00000	−0.341432
$550$	0	0
$551$	19.3137	0.822792
$552$	− 3.17157i	− 0.134991i
$553$	− 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	$-0.974353\pi$
$557$	− 3.79899i	− 0.160968i	0.996756	−	0.0804842i	$-0.0256467\pi$
$558$	− 4.24264i	− 0.179605i
$559$	3.07107	0.129892
$560$	0	0
$561$	−4.00000	−0.168880
$562$	7.37258i	0.310994i
$563$	− 16.2426i	− 0.684546i	−0.939601	−	0.342273i	$-0.888803\pi$
$563$	− 16.2426i	− 0.684546i	0.939601	−	0.342273i	$-0.111197\pi$
$564$	2.14214	0.0902002
$565$	0	0
$566$	3.61522	0.151959
$567$	− 4.14214i	− 0.173953i
$568$	12.5269i	0.525618i
$569$	21.6569	0.907903	0.453951	−	0.891027i	$-0.350014\pi$
$569$	21.6569	0.907903	0.453951	+	0.891027i	$0.350014\pi$
$570$	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
$573$	19.3137i	0.806842i
$574$	−3.02944	−0.126446
$575$	0	0
$576$	4.17157	0.173816
$577$	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
$579$	−22.1421	−0.920196
$580$	0	0
$581$	7.31371	0.303424
$582$	− 2.14214i	− 0.0887944i
$583$	− 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	$-0.773380\pi$
$587$	− 31.6569i	− 1.30662i	0.757091	−	0.653309i	$-0.226620\pi$
$588$	16.3259i	0.673269i
$589$	−34.9706	−1.44094
$590$	0	0
$591$	−32.4853	−1.33627
$592$	− 25.4558i	− 1.04623i
$593$	20.6274i	0.847066i	0.905881	+	0.423533i	$0.139210\pi$
$593$	20.6274i	0.847066i	−0.905881	+	0.423533i	$0.860790\pi$
$594$	1.37258	0.0563178
$595$	0	0
$596$	0.627417	0.0257000
$597$	− 5.65685i	− 0.231520i
$598$	0.585786i	0.0239546i
$599$	25.4558	1.04010	0.520049	−	0.854137i	$-0.325914\pi$
$599$	25.4558	1.04010	0.520049	+	0.854137i	$0.325914\pi$
$600$	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
$603$	2.00000i	0.0814463i
$604$	33.3553	1.35721
$605$	0	0
$606$	4.48528	0.182202
$607$	− 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
$609$	6.62742	0.268556
$610$	0	0
$611$	−0.828427	−0.0335146
$612$	8.82843i	0.356868i
$613$	− 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	$0.153004\pi$
$617$	22.9706i	0.924760i	−0.886682	+	0.462380i	$0.846996\pi$
$618$	10.2010i	0.410345i
$619$	−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
$621$	8.00000	0.321029
$622$	3.51472i	0.140927i
$623$	− 4.97056i	− 0.199141i
$624$	4.24264	0.169842
$625$	0	0
$626$	0.343146	0.0137149
$627$	− 2.82843i	− 0.112956i
$628$	− 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
$633$	− 27.3137i	− 1.08562i
$634$	−10.8284	−0.430052
$635$	0	0
$636$	37.4558	1.48522
$637$	− 6.31371i	− 0.250158i
$638$	1.37258i	0.0543411i
$639$	−7.89949	−0.312499
$640$	0	0
$641$	36.3431	1.43547	0.717734	−	0.696317i	$-0.245179\pi$
$641$	36.3431	1.43547	0.717734	+	0.696317i	$0.245179\pi$
$642$	3.85786i	0.152258i
$643$	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	$-0.958677\pi$
$647$	− 6.58579i	− 0.258914i	0.991585	−	0.129457i	$-0.0413234\pi$
$648$	7.92893i	0.311478i
$649$	−6.00000	−0.235521
$650$	0	0
$651$	−12.0000	−0.470317
$652$	− 27.3726i	− 1.07199i
$653$	13.0294i	0.509881i	0.966957	+	0.254941i	$0.0820559\pi$
$653$	13.0294i	0.509881i	−0.966957	+	0.254941i	$0.917944\pi$
$654$	1.17157	0.0458121
$655$	0	0
$656$	−26.4853	−1.03408
$657$	− 8.48528i	− 0.331042i
$658$	− 0.284271i	− 0.0110820i
$659$	−46.1421	−1.79744	−0.898721	−	0.438520i	$-0.855503\pi$
$659$	−46.1421	−1.79744	−0.898721	+	0.438520i	$0.855503\pi$
$660$	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
$663$	6.82843i	0.265194i
$664$	−14.0000	−0.543305
$665$	0	0
$666$	−3.51472	−0.136193
$667$	8.00000i	0.309761i
$668$	16.1421i	0.624558i
$669$	−37.4558	−1.44813
$670$	0	0
$671$	−4.68629	−0.180912
$672$	5.17157i	0.199498i
$673$	− 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	$-0.949991\pi$
$677$	− 8.14214i	− 0.312928i	0.987684	−	0.156464i	$-0.0500095\pi$
$678$	− 1.85786i	− 0.0713509i
$679$	3.02944	0.116259
$680$	0	0
$681$	−39.1127	−1.49880
$682$	− 2.48528i	− 0.0951663i
$683$	33.3137i	1.27471i	0.770569	+	0.637357i	$0.219972\pi$
$683$	33.3137i	1.27471i	−0.770569	+	0.637357i	$0.780028\pi$
$684$	−6.24264	−0.238693
$685$	0	0
$686$	4.56854	0.174428
$687$	− 1.17157i	− 0.0446983i
$688$	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
$693$	0.485281i	0.0184343i
$694$	−1.75736	−0.0667084
$695$	0	0
$696$	−12.6863	−0.480873
$697$	− 42.6274i	− 1.61463i
$698$	0.627417i	0.0237481i
$699$	−34.8284	−1.31733
$700$	0	0
$701$	37.3137	1.40932	0.704660	−	0.709545i	$-0.251100\pi$
$701$	37.3137	1.40932	0.704660	+	0.709545i	$0.251100\pi$
$702$	− 2.34315i	− 0.0884363i
$703$	28.9706i	1.09265i
$704$	2.44365	0.0920986
$705$	0	0
$706$	3.79899	0.142977
$707$	6.34315i	0.238559i
$708$	− 26.4853i	− 0.995378i
$709$	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
$711$	−8.48528	−0.318223
$712$	9.51472i	0.356579i
$713$	− 14.4853i	− 0.542478i
$714$	−2.34315	−0.0876900
$715$	0	0
$716$	10.3431	0.386542
$717$	0.828427i	0.0309382i
$718$	− 11.5563i	− 0.431279i
$719$	4.97056	0.185371	0.0926854	−	0.995695i	$-0.470455\pi$
$719$	4.97056	0.185371	0.0926854	+	0.995695i	$0.470455\pi$
$720$	0	0
$721$	−14.4264	−0.537267
$722$	3.04163i	0.113198i
$723$	3.51472i	0.130714i
$724$	0	0
$725$	0	0
$726$	−6.24264	−0.231686
$727$	− 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
$729$	−29.0000	−1.07407
$730$	0	0
$731$	−14.8284	−0.548449
$732$	− 20.6863i	− 0.764587i
$733$	− 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
$738$	3.65685i	0.134611i
$739$	−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
$741$	−4.82843	−0.177377
$742$	− 4.97056i	− 0.182475i
$743$	− 38.4853i	− 1.41189i	−0.708268	−	0.705944i	$-0.750523\pi$
$743$	− 38.4853i	− 1.41189i	0.708268	−	0.705944i	$-0.249477\pi$
$744$	22.9706	0.842142
$745$	0	0
$746$	−10.4853	−0.383893
$747$	− 8.82843i	− 0.323015i
$748$	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
$753$	− 28.0000i	− 1.02038i
$754$	2.34315	0.0853323
$755$	0	0
$756$	−8.56854	−0.311635
$757$	− 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
$759$	1.17157	0.0425254
$760$	0	0
$761$	−36.6274	−1.32774	−0.663871	−	0.747847i	$-0.731088\pi$
$761$	−36.6274	−1.32774	−0.663871	+	0.747847i	$0.731088\pi$
$762$	5.51472i	0.199777i
$763$	1.65685i	0.0599822i
$764$	24.9706	0.903403
$765$	0	0
$766$	13.7157	0.495569
$767$	10.2426i	0.369840i
$768$	5.61522i	0.202622i
$769$	−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
$771$	23.1127	0.832383
$772$	28.6274i	1.03032i
$773$	6.14214i	0.220917i	0.993881	+	0.110459i	$0.0352320\pi$
$773$	6.14214i	0.220917i	−0.993881	+	0.110459i	$0.964768\pi$
$774$	1.27208	0.0457239
$775$	0	0
$776$	−5.79899	−0.208172
$777$	9.94113i	0.356636i
$778$	− 6.88730i	− 0.246922i
$779$	30.1421	1.07995
$780$	0	0
$781$	−4.62742	−0.165582
$782$	− 2.82843i	− 0.101144i
$783$	− 32.0000i	− 1.14359i
$784$	18.9411	0.676469
$785$	0	0
$786$	9.94113	0.354588
$787$	− 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
$789$	18.9706	0.675370
$790$	0	0
$791$	2.62742	0.0934202
$792$	− 0.928932i	− 0.0330082i
$793$	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	$-0.937757\pi$
$797$	− 10.9706i	− 0.388597i	0.980942	−	0.194299i	$-0.0622431\pi$
$798$	− 1.65685i	− 0.0586520i
$799$	4.00000	0.141510
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 7.17157i	− 0.253237i
$803$	− 4.97056i	− 0.175407i
$804$	−5.17157	−0.182387
$805$	0	0
$806$	−4.24264	−0.149441
$807$	3.79899i	0.133731i
$808$	− 12.1421i	− 0.427159i
$809$	45.2548	1.59108	0.795538	−	0.605904i	$-0.207189\pi$
$809$	45.2548	1.59108	0.795538	+	0.605904i	$0.207189\pi$
$810$	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
$813$	1.79899i	0.0630933i
$814$	−2.05887	−0.0721635
$815$	0	0
$816$	−20.4853	−0.717128
$817$	− 10.4853i	− 0.366834i
$818$	5.31371i	0.185789i
$819$	0.828427	0.0289476
$820$	0	0
$821$	39.2548	1.37000	0.685002	−	0.728542i	$-0.259801\pi$
$821$	39.2548	1.37000	0.685002	+	0.728542i	$0.259801\pi$
$822$	− 3.11270i	− 0.108568i
$823$	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	$-0.839054\pi$
$827$	− 27.8579i	− 0.968713i	0.874871	−	0.484356i	$-0.160946\pi$
$828$	− 2.58579i	− 0.0898623i
$829$	−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
$831$	−10.1421	−0.351827
$832$	− 4.17157i	− 0.144623i
$833$	30.4853i	1.05625i
$834$	7.31371	0.253253
$835$	0	0
$836$	−3.65685	−0.126475
$837$	57.9411i	2.00274i
$838$	2.14214i	0.0739988i
$839$	18.7279	0.646560	0.323280	−	0.946303i	$-0.395214\pi$
$839$	18.7279	0.646560	0.323280	+	0.946303i	$0.395214\pi$
$840$	0	0
$841$	3.00000	0.103448
$842$	0.426407i	0.0146950i
$843$	− 25.1716i	− 0.866955i
$844$	−35.3137	−1.21555
$845$	0	0
$846$	−0.343146	−0.0117976
$847$	− 8.82843i	− 0.303348i
$848$	− 43.4558i	− 1.49228i
$849$	−12.3431	−0.423616
$850$	0	0
$851$	−12.0000	−0.411355
$852$	− 20.4264i	− 0.699797i
$853$	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	$-0.998134\pi$
$857$	− 0.343146i	− 0.0117216i	0.999983	−	0.00586082i	$-0.00186557\pi$
$858$	− 0.343146i	− 0.0117148i
$859$	−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
$861$	10.3431	0.352493
$862$	− 1.49747i	− 0.0510042i
$863$	19.4558i	0.662285i	0.943581	+	0.331142i	$0.107434\pi$
$863$	19.4558i	0.662285i	−0.943581	+	0.331142i	$0.892566\pi$
$864$	24.9706	0.849516
$865$	0	0
$866$	−1.51472	−0.0514722
$867$	− 8.92893i	− 0.303242i
$868$	15.5147i	0.526604i
$869$	−4.97056	−0.168615
$870$	0	0
$871$	2.00000	0.0677674
$872$	− 3.17157i	− 0.107403i
$873$	− 3.65685i	− 0.123766i
$874$	2.00000	0.0676510
$875$	0	0
$876$	21.9411	0.741322
$877$	− 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
$879$	3.02944	0.102180
$880$	0	0
$881$	52.9706	1.78462	0.892312	−	0.451420i	$-0.149082\pi$
$881$	52.9706	1.78462	0.892312	+	0.451420i	$0.149082\pi$
$882$	− 2.61522i	− 0.0880592i
$883$	− 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	$-0.922362\pi$
$887$	− 14.3848i	− 0.482994i	0.970402	−	0.241497i	$-0.0776383\pi$
$888$	− 19.0294i	− 0.638586i
$889$	−7.79899	−0.261570
$890$	0	0
$891$	−2.92893	−0.0981229
$892$	48.4264i	1.62144i
$893$	2.82843i	0.0946497i
$894$	0.201010	0.00672278
$895$	0	0
$896$	8.74517	0.292155
$897$	− 2.00000i	− 0.0667781i
$898$	12.0589i	0.402410i
$899$	−57.9411	−1.93244
$900$	0	0
$901$	69.9411	2.33008
$902$	2.14214i	0.0713253i
$903$	− 3.59798i	− 0.119733i
$904$	−5.02944	−0.167277
$905$	0	0
$906$	10.6863	0.355028
$907$	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
$909$	7.65685	0.253962
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	− 14.4853i	− 0.479656i
$913$	− 5.17157i	− 0.171154i
$914$	7.45584	0.246617
$915$	0	0
$916$	−1.51472	−0.0500477
$917$	14.0589i	0.464265i
$918$	11.3137i	0.373408i
$919$	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
$921$	27.1127	0.893394
$922$	− 10.9706i	− 0.361296i
$923$	7.89949i	0.260015i
$924$	−1.25483	−0.0412810
$925$	0	0
$926$	−6.48528	−0.213120
$927$	17.4142i	0.571958i
$928$	24.9706i	0.819699i
$929$	−11.1716	−0.366527	−0.183264	−	0.983064i	$-0.558666\pi$
$929$	−11.1716	−0.366527	−0.183264	+	0.983064i	$0.558666\pi$
$930$	0	0
$931$	−21.5563	−0.706481
$932$	45.0294i	1.47499i
$933$	− 12.0000i	− 0.392862i
$934$	4.38478	0.143474
$935$	0	0
$936$	−1.58579	−0.0518331
$937$	− 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
$939$	−1.17157	−0.0382328
$940$	0	0
$941$	−54.7696	−1.78544	−0.892718	−	0.450615i	$-0.851205\pi$
$941$	−54.7696	−1.78544	−0.892718	+	0.450615i	$0.851205\pi$
$942$	− 10.5442i	− 0.343547i
$943$	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	$0.261872\pi$
$947$	45.1127i	1.46597i	−0.680247	+	0.732983i	$0.738128\pi$
$948$	− 21.9411i	− 0.712615i
$949$	−8.48528	−0.275444
$950$	0	0
$951$	36.9706	1.19885
$952$	6.34315i	0.205583i
$953$	− 55.2548i	− 1.78988i	−0.446187	−	0.894940i	$-0.647218\pi$
$953$	− 55.2548i	− 1.78988i	0.446187	−	0.894940i	$-0.352782\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	1.07107	0.0346408
$957$	− 4.68629i	− 0.151486i
$958$	− 2.18377i	− 0.0705543i
$959$	4.40202	0.142149
$960$	0	0
$961$	73.9117	2.38425
$962$	3.51472i	0.113319i
$963$	6.58579i	0.212224i
$964$	4.54416	0.146357
$965$	0	0
$966$	0.686292	0.0220811
$967$	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
$969$	23.3137	0.748944
$970$	0	0
$971$	−12.2843	−0.394221	−0.197111	−	0.980381i	$-0.563156\pi$
$971$	−12.2843	−0.394221	−0.197111	+	0.980381i	$0.563156\pi$
$972$	18.1005i	0.580574i
$973$	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	$0.359057\pi$
$977$	56.4853i	1.80712i	−0.428457	+	0.903562i	$0.640943\pi$
$978$	− 8.76955i	− 0.280419i
$979$	−3.51472	−0.112331
$980$	0	0
$981$	2.00000	0.0638551
$982$	− 4.48528i	− 0.143131i
$983$	− 34.9706i	− 1.11539i	−0.830047	−	0.557694i	$-0.811686\pi$
$983$	− 34.9706i	− 1.11539i	0.830047	−	0.557694i	$-0.188314\pi$
$984$	−19.7990	−0.631169
$985$	0	0
$986$	−11.3137	−0.360302
$987$	0.970563i	0.0308934i
$988$	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
$993$	31.1716i	0.989200i
$994$	−2.71068	−0.0859775
$995$	0	0
$996$	22.8284	0.723346
$997$	− 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
$999$	48.0000	1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.2.b.f.274.2		4
3.2	odd	2		2925.2.c.r.2224.3		4
5.2	odd	4		65.2.a.b.1.2	✓	2
5.3	odd	4		325.2.a.i.1.1		2
5.4	even	2	inner	325.2.b.f.274.3		4
15.2	even	4		585.2.a.m.1.1		2
15.8	even	4		2925.2.a.u.1.2		2
15.14	odd	2		2925.2.c.r.2224.2		4
20.3	even	4		5200.2.a.bu.1.2		2
20.7	even	4		1040.2.a.j.1.1		2
35.27	even	4		3185.2.a.j.1.2		2
40.27	even	4		4160.2.a.z.1.2		2
40.37	odd	4		4160.2.a.bf.1.1		2
55.32	even	4		7865.2.a.j.1.1		2
60.47	odd	4		9360.2.a.cd.1.2		2
65.2	even	12		845.2.m.f.316.3		8
65.7	even	12		845.2.m.f.361.3		8
65.12	odd	4		845.2.a.g.1.1		2
65.17	odd	12		845.2.e.c.146.2		4
65.22	odd	12		845.2.e.h.146.1		4
65.32	even	12		845.2.m.f.361.2		8
65.37	even	12		845.2.m.f.316.2		8
65.38	odd	4		4225.2.a.r.1.2		2
65.42	odd	12		845.2.e.h.191.1		4
65.47	even	4		845.2.c.b.506.3		4
65.57	even	4		845.2.c.b.506.2		4
65.62	odd	12		845.2.e.c.191.2		4
195.77	even	4		7605.2.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.b.1.2	✓	2	5.2	odd	4
325.2.a.i.1.1		2	5.3	odd	4
325.2.b.f.274.2		4	1.1	even	1	trivial
325.2.b.f.274.3		4	5.4	even	2	inner
585.2.a.m.1.1		2	15.2	even	4
845.2.a.g.1.1		2	65.12	odd	4
845.2.c.b.506.2		4	65.57	even	4
845.2.c.b.506.3		4	65.47	even	4
845.2.e.c.146.2		4	65.17	odd	12
845.2.e.c.191.2		4	65.62	odd	12
845.2.e.h.146.1		4	65.22	odd	12
845.2.e.h.191.1		4	65.42	odd	12
845.2.m.f.316.2		8	65.37	even	12
845.2.m.f.316.3		8	65.2	even	12
845.2.m.f.361.2		8	65.32	even	12
845.2.m.f.361.3		8	65.7	even	12
1040.2.a.j.1.1		2	20.7	even	4
2925.2.a.u.1.2		2	15.8	even	4
2925.2.c.r.2224.2		4	15.14	odd	2
2925.2.c.r.2224.3		4	3.2	odd	2
3185.2.a.j.1.2		2	35.27	even	4
4160.2.a.z.1.2		2	40.27	even	4
4160.2.a.bf.1.1		2	40.37	odd	4
4225.2.a.r.1.2		2	65.38	odd	4
5200.2.a.bu.1.2		2	20.3	even	4
7605.2.a.x.1.2		2	195.77	even	4
7865.2.a.j.1.1		2	55.32	even	4
9360.2.a.cd.1.2		2	60.47	odd	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.414214i q^{2} +1.41421i q^{3} +1.82843 q^{4} +0.585786 q^{6} +0.828427i q^{7} -1.58579i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.414214i q^{2} +1.41421i q^{3} +1.82843 q^{4} +0.585786 q^{6} +0.828427i q^{7} -1.58579i q^{8} +1.00000 q^{9} +0.585786 q^{11} +2.58579i q^{12} -1.00000i q^{13} +0.343146 q^{14} +3.00000 q^{16} +4.82843i q^{17} -0.414214i q^{18} -3.41421 q^{19} -1.17157 q^{21} -0.242641i q^{22} -1.41421i q^{23} +2.24264 q^{24} -0.414214 q^{26} +5.65685i q^{27} +1.51472i q^{28} -5.65685 q^{29} +10.2426 q^{31} -4.41421i q^{32} +0.828427i q^{33} +2.00000 q^{34} +1.82843 q^{36} -8.48528i q^{37} +1.41421i q^{38} +1.41421 q^{39} -8.82843 q^{41} +0.485281i q^{42} +3.07107i q^{43} +1.07107 q^{44} -0.585786 q^{46} -0.828427i q^{47} +4.24264i q^{48} +6.31371 q^{49} -6.82843 q^{51} -1.82843i q^{52} -14.4853i q^{53} +2.34315 q^{54} +1.31371 q^{56} -4.82843i q^{57} +2.34315i q^{58} -10.2426 q^{59} -8.00000 q^{61} -4.24264i q^{62} +0.828427i q^{63} +4.17157 q^{64} +0.343146 q^{66} +2.00000i q^{67} +8.82843i q^{68} +2.00000 q^{69} -7.89949 q^{71} -1.58579i q^{72} -8.48528i q^{73} -3.51472 q^{74} -6.24264 q^{76} +0.485281i q^{77} -0.585786i q^{78} -8.48528 q^{79} -5.00000 q^{81} +3.65685i q^{82} -8.82843i q^{83} -2.14214 q^{84} +1.27208 q^{86} -8.00000i q^{87} -0.928932i q^{88} -6.00000 q^{89} +0.828427 q^{91} -2.58579i q^{92} +14.4853i q^{93} -0.343146 q^{94} +6.24264 q^{96} -3.65685i q^{97} -2.61522i q^{98} +0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 8 * q^6 + 4 * q^9	\( 4 q - 4 q^{4} + 8 q^{6} + 4 q^{9} + 8 q^{11} + 24 q^{14} + 12 q^{16} - 8 q^{19} - 16 q^{21} - 8 q^{24} + 4 q^{26} + 24 q^{31} + 8 q^{34} - 4 q^{36} - 24 q^{41} - 24 q^{44} - 8 q^{46} - 20 q^{49} - 16 q^{51} + 32 q^{54} - 40 q^{56} - 24 q^{59} - 32 q^{61} + 28 q^{64} + 24 q^{66} + 8 q^{69} + 8 q^{71} - 48 q^{74} - 8 q^{76} - 20 q^{81} + 48 q^{84} + 56 q^{86} - 24 q^{89} - 8 q^{91} - 24 q^{94} + 8 q^{96} + 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^6 + 4 * q^9 + 8 * q^11 + 24 * q^14 + 12 * q^16 - 8 * q^19 - 16 * q^21 - 8 * q^24 + 4 * q^26 + 24 * q^31 + 8 * q^34 - 4 * q^36 - 24 * q^41 - 24 * q^44 - 8 * q^46 - 20 * q^49 - 16 * q^51 + 32 * q^54 - 40 * q^56 - 24 * q^59 - 32 * q^61 + 28 * q^64 + 24 * q^66 + 8 * q^69 + 8 * q^71 - 48 * q^74 - 8 * q^76 - 20 * q^81 + 48 * q^84 + 56 * q^86 - 24 * q^89 - 8 * q^91 - 24 * q^94 + 8 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more