LMFDB - Embedded newform 825.2.c.f.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.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:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.32001 - 1.32001i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.2.c.f.199.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.12489i q^{2} -1.00000i q^{3} -2.51514 q^{4} -2.12489 q^{6} -3.64002i q^{7} +1.09461i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.12489i q^{2} -1.00000i q^{3} -2.51514 q^{4} -2.12489 q^{6} -3.64002i q^{7} +1.09461i q^{8} -1.00000 q^{9} +1.00000 q^{11} +2.51514i q^{12} +1.51514i q^{13} -7.73463 q^{14} -2.70436 q^{16} -1.15516i q^{17} +2.12489i q^{18} -2.60975 q^{19} -3.64002 q^{21} -2.12489i q^{22} -5.73463i q^{23} +1.09461 q^{24} +3.21949 q^{26} +1.00000i q^{27} +9.15516i q^{28} -6.24977 q^{29} +5.51514 q^{31} +7.93567i q^{32} -1.00000i q^{33} -2.45459 q^{34} +2.51514 q^{36} -0.454586i q^{37} +5.54541i q^{38} +1.51514 q^{39} +4.12489 q^{41} +7.73463i q^{42} +11.7044i q^{43} -2.51514 q^{44} -12.1854 q^{46} +3.48486i q^{47} +2.70436i q^{48} -6.24977 q^{49} -1.15516 q^{51} -3.81078i q^{52} -12.5601i q^{53} +2.12489 q^{54} +3.98440 q^{56} +2.60975i q^{57} +13.2800i q^{58} +7.73463 q^{59} -12.0147 q^{61} -11.7190i q^{62} +3.64002i q^{63} +11.4537 q^{64} -2.12489 q^{66} +14.2645i q^{67} +2.90539i q^{68} -5.73463 q^{69} +8.51514 q^{71} -1.09461i q^{72} -9.21949i q^{73} -0.965943 q^{74} +6.56387 q^{76} -3.64002i q^{77} -3.21949i q^{78} -5.09461 q^{79} +1.00000 q^{81} -8.76491i q^{82} -14.7493i q^{83} +9.15516 q^{84} +24.8704 q^{86} +6.24977i q^{87} +1.09461i q^{88} +10.4995 q^{89} +5.51514 q^{91} +14.4234i q^{92} -5.51514i q^{93} +7.40493 q^{94} +7.93567 q^{96} +6.77959i q^{97} +13.2800i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 16 * q^4 + 4 * q^6 - 6 * q^9	$ 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9} + 6 q^{11} - 12 q^{14} + 20 q^{16} + 2 q^{19} - 6 q^{21} - 12 q^{24} - 16 q^{26} - 4 q^{29} + 34 q^{31} - 12 q^{34} + 16 q^{36} + 10 q^{39} + 8 q^{41} - 16 q^{44} - 60 q^{46} - 4 q^{49} + 8 q^{51} - 4 q^{54} - 44 q^{56} + 12 q^{59} - 6 q^{61} - 68 q^{64} + 4 q^{66} + 52 q^{71} - 28 q^{74} - 48 q^{76} - 12 q^{79} + 6 q^{81} + 40 q^{84} + 56 q^{86} - 4 q^{89} + 34 q^{91} - 4 q^{94} + 68 q^{96} - 6 q^{99}+O(q^{100}) $ 6 * q - 16 * q^4 + 4 * q^6 - 6 * q^9 + 6 * q^11 - 12 * q^14 + 20 * q^16 + 2 * q^19 - 6 * q^21 - 12 * q^24 - 16 * q^26 - 4 * q^29 + 34 * q^31 - 12 * q^34 + 16 * q^36 + 10 * q^39 + 8 * q^41 - 16 * q^44 - 60 * q^46 - 4 * q^49 + 8 * q^51 - 4 * q^54 - 44 * q^56 + 12 * q^59 - 6 * q^61 - 68 * q^64 + 4 * q^66 + 52 * q^71 - 28 * q^74 - 48 * q^76 - 12 * q^79 + 6 * q^81 + 40 * q^84 + 56 * q^86 - 4 * q^89 + 34 * q^91 - 4 * q^94 + 68 * q^96 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 2.12489i	− 1.50252i	−0.660006	−	0.751260i	$-0.729446\pi$
$2$	− 2.12489i	− 1.50252i	0.660006	−	0.751260i	$-0.270554\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−2.51514	−1.25757
$5$	0	0
$5$	0	0
$6$	−2.12489	−0.867481
$7$	− 3.64002i	− 1.37580i	−0.725806	−	0.687900i	$-0.758533\pi$
$7$	− 3.64002i	− 1.37580i	0.725806	−	0.687900i	$-0.241467\pi$
$8$	1.09461i	0.387003i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.51514i	0.726058i
$13$	1.51514i	0.420224i	0.977677	+	0.210112i	$0.0673828\pi$
$13$	1.51514i	0.420224i	−0.977677	+	0.210112i	$0.932617\pi$
$14$	−7.73463	−2.06717
$15$	0	0
$16$	−2.70436	−0.676089
$17$	− 1.15516i	− 0.280168i	−0.990140	−	0.140084i	$-0.955263\pi$
$17$	− 1.15516i	− 0.280168i	0.990140	−	0.140084i	$-0.0447372\pi$
$18$	2.12489i	0.500840i
$19$	−2.60975	−0.598717	−0.299359	−	0.954141i	$-0.596773\pi$
$19$	−2.60975	−0.598717	−0.299359	+	0.954141i	$0.596773\pi$
$20$	0	0
$21$	−3.64002	−0.794318
$22$	− 2.12489i	− 0.453027i
$23$	− 5.73463i	− 1.19575i	−0.801588	−	0.597877i	$-0.796011\pi$
$23$	− 5.73463i	− 1.19575i	0.801588	−	0.597877i	$-0.203989\pi$
$24$	1.09461	0.223436
$25$	0	0
$26$	3.21949	0.631395
$27$	1.00000i	0.192450i
$28$	9.15516i	1.73016i
$29$	−6.24977	−1.16055	−0.580277	−	0.814419i	$-0.697056\pi$
$29$	−6.24977	−1.16055	−0.580277	+	0.814419i	$0.697056\pi$
$30$	0	0
$31$	5.51514	0.990548	0.495274	−	0.868737i	$-0.335068\pi$
$31$	5.51514	0.990548	0.495274	+	0.868737i	$0.335068\pi$
$32$	7.93567i	1.40284i
$33$	− 1.00000i	− 0.174078i
$34$	−2.45459	−0.420958
$35$	0	0
$36$	2.51514	0.419190
$37$	− 0.454586i	− 0.0747335i	−0.999302	−	0.0373667i	$-0.988103\pi$
$37$	− 0.454586i	− 0.0747335i	0.999302	−	0.0373667i	$-0.0118970\pi$
$38$	5.54541i	0.899585i
$39$	1.51514	0.242616
$40$	0	0
$41$	4.12489	0.644199	0.322099	−	0.946706i	$-0.395611\pi$
$41$	4.12489	0.644199	0.322099	+	0.946706i	$0.395611\pi$
$42$	7.73463i	1.19348i
$43$	11.7044i	1.78490i	0.451149	+	0.892449i	$0.351014\pi$
$43$	11.7044i	1.78490i	−0.451149	+	0.892449i	$0.648986\pi$
$44$	−2.51514	−0.379171
$45$	0	0
$46$	−12.1854	−1.79664
$47$	3.48486i	0.508319i	0.967162	+	0.254160i	$0.0817989\pi$
$47$	3.48486i	0.508319i	−0.967162	+	0.254160i	$0.918201\pi$
$48$	2.70436i	0.390340i
$49$	−6.24977	−0.892824
$50$	0	0
$51$	−1.15516	−0.161755
$52$	− 3.81078i	− 0.528460i
$53$	− 12.5601i	− 1.72526i	−0.505834	−	0.862631i	$-0.668815\pi$
$53$	− 12.5601i	− 1.72526i	0.505834	−	0.862631i	$-0.331185\pi$
$54$	2.12489	0.289160
$55$	0	0
$56$	3.98440	0.532438
$57$	2.60975i	0.345669i
$58$	13.2800i	1.74376i
$59$	7.73463	1.00696	0.503482	−	0.864006i	$-0.332052\pi$
$59$	7.73463	1.00696	0.503482	+	0.864006i	$0.332052\pi$
$60$	0	0
$61$	−12.0147	−1.53832	−0.769161	−	0.639055i	$-0.779326\pi$
$61$	−12.0147	−1.53832	−0.769161	+	0.639055i	$0.779326\pi$
$62$	− 11.7190i	− 1.48832i
$63$	3.64002i	0.458600i
$64$	11.4537	1.43171
$65$	0	0
$66$	−2.12489	−0.261555
$67$	14.2645i	1.74268i	0.490680	+	0.871340i	$0.336749\pi$
$67$	14.2645i	1.74268i	−0.490680	+	0.871340i	$0.663251\pi$
$68$	2.90539i	0.352330i
$69$	−5.73463	−0.690369
$70$	0	0
$71$	8.51514	1.01056	0.505280	−	0.862955i	$-0.331389\pi$
$71$	8.51514	1.01056	0.505280	+	0.862955i	$0.331389\pi$
$72$	− 1.09461i	− 0.129001i
$73$	− 9.21949i	− 1.07906i	−0.841966	−	0.539530i	$-0.818602\pi$
$73$	− 9.21949i	− 1.07906i	0.841966	−	0.539530i	$-0.181398\pi$
$74$	−0.965943	−0.112289
$75$	0	0
$76$	6.56387	0.752928
$77$	− 3.64002i	− 0.414819i
$78$	− 3.21949i	− 0.364536i
$79$	−5.09461	−0.573188	−0.286594	−	0.958052i	$-0.592523\pi$
$79$	−5.09461	−0.573188	−0.286594	+	0.958052i	$0.592523\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 8.76491i	− 0.967922i
$83$	− 14.7493i	− 1.61895i	−0.587156	−	0.809474i	$-0.699753\pi$
$83$	− 14.7493i	− 1.61895i	0.587156	−	0.809474i	$-0.300247\pi$
$84$	9.15516	0.998910
$85$	0	0
$86$	24.8704	2.68185
$87$	6.24977i	0.670046i
$88$	1.09461i	0.116686i
$89$	10.4995	1.11295	0.556475	−	0.830865i	$-0.312154\pi$
$89$	10.4995	1.11295	0.556475	+	0.830865i	$0.312154\pi$
$90$	0	0
$91$	5.51514	0.578144
$92$	14.4234i	1.50374i
$93$	− 5.51514i	− 0.571893i
$94$	7.40493	0.763760
$95$	0	0
$96$	7.93567	0.809931
$97$	6.77959i	0.688363i	0.938903	+	0.344181i	$0.111844\pi$
$97$	6.77959i	0.688363i	−0.938903	+	0.344181i	$0.888156\pi$
$98$	13.2800i	1.34149i
$99$	−1.00000	−0.100504
$100$	0	0
$101$	7.40493	0.736818	0.368409	−	0.929664i	$-0.379903\pi$
$101$	7.40493	0.736818	0.368409	+	0.929664i	$0.379903\pi$
$102$	2.45459i	0.243040i
$103$	− 16.4995i	− 1.62575i	−0.582439	−	0.812874i	$-0.697902\pi$
$103$	− 16.4995i	− 1.62575i	0.582439	−	0.812874i	$-0.302098\pi$
$104$	−1.65848	−0.162628
$105$	0	0
$106$	−26.6888	−2.59224
$107$	− 3.93945i	− 0.380841i	−0.981703	−	0.190420i	$-0.939015\pi$
$107$	− 3.93945i	− 0.380841i	0.981703	−	0.190420i	$-0.0609851\pi$
$108$	− 2.51514i	− 0.242019i
$109$	−6.73463	−0.645061	−0.322530	−	0.946559i	$-0.604533\pi$
$109$	−6.73463	−0.645061	−0.322530	+	0.946559i	$0.604533\pi$
$110$	0	0
$111$	−0.454586	−0.0431474
$112$	9.84392i	0.930163i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	5.54541	0.519376
$115$	0	0
$116$	15.7190	1.45948
$117$	− 1.51514i	− 0.140075i
$118$	− 16.4352i	− 1.51298i
$119$	−4.20482	−0.385455
$120$	0	0
$121$	1.00000	0.0909091
$122$	25.5298i	2.31136i
$123$	− 4.12489i	− 0.371928i
$124$	−13.8713	−1.24568
$125$	0	0
$126$	7.73463	0.689056
$127$	− 8.06433i	− 0.715594i	−0.933799	−	0.357797i	$-0.883528\pi$
$127$	− 8.06433i	− 0.715594i	0.933799	−	0.357797i	$-0.116472\pi$
$128$	− 8.46640i	− 0.748331i
$129$	11.7044	1.03051
$130$	0	0
$131$	−12.8099	−1.11920	−0.559602	−	0.828762i	$-0.689046\pi$
$131$	−12.8099	−1.11920	−0.559602	+	0.828762i	$0.689046\pi$
$132$	2.51514i	0.218915i
$133$	9.49954i	0.823715i
$134$	30.3103	2.61841
$135$	0	0
$136$	1.26445	0.108426
$137$	− 22.8099i	− 1.94878i	−0.224868	−	0.974389i	$-0.572195\pi$
$137$	− 22.8099i	− 1.94878i	0.224868	−	0.974389i	$-0.427805\pi$
$138$	12.1854i	1.03729i
$139$	−7.59037	−0.643807	−0.321903	−	0.946773i	$-0.604323\pi$
$139$	−7.59037	−0.643807	−0.321903	+	0.946773i	$0.604323\pi$
$140$	0	0
$141$	3.48486	0.293478
$142$	− 18.0937i	− 1.51839i
$143$	1.51514i	0.126702i
$144$	2.70436	0.225363
$145$	0	0
$146$	−19.5904	−1.62131
$147$	6.24977i	0.515472i
$148$	1.14335i	0.0939825i
$149$	1.81456	0.148655	0.0743274	−	0.997234i	$-0.476319\pi$
$149$	1.81456	0.148655	0.0743274	+	0.997234i	$0.476319\pi$
$150$	0	0
$151$	24.3250	1.97954	0.989770	−	0.142670i	$-0.0455687\pi$
$151$	24.3250	1.97954	0.989770	+	0.142670i	$0.0455687\pi$
$152$	− 2.85665i	− 0.231705i
$153$	1.15516i	0.0933893i
$154$	−7.73463	−0.623274
$155$	0	0
$156$	−3.81078	−0.305107
$157$	− 9.76491i	− 0.779325i	−0.920958	−	0.389662i	$-0.872592\pi$
$157$	− 9.76491i	− 0.779325i	0.920958	−	0.389662i	$-0.127408\pi$
$158$	10.8255i	0.861227i
$159$	−12.5601	−0.996080
$160$	0	0
$161$	−20.8742	−1.64512
$162$	− 2.12489i	− 0.166947i
$163$	− 6.98440i	− 0.547061i	−0.961863	−	0.273530i	$-0.911809\pi$
$163$	− 6.98440i	− 0.547061i	0.961863	−	0.273530i	$-0.0881914\pi$
$164$	−10.3747	−0.810125
$165$	0	0
$166$	−31.3406	−2.43250
$167$	6.31032i	0.488307i	0.969737	+	0.244154i	$0.0785102\pi$
$167$	6.31032i	0.488307i	−0.969737	+	0.244154i	$0.921490\pi$
$168$	− 3.98440i	− 0.307403i
$169$	10.7044	0.823412
$170$	0	0
$171$	2.60975	0.199572
$172$	− 29.4381i	− 2.24463i
$173$	− 12.8448i	− 0.976575i	−0.872683	−	0.488287i	$-0.837622\pi$
$173$	− 12.8448i	− 0.976575i	0.872683	−	0.488287i	$-0.162378\pi$
$174$	13.2800	1.00676
$175$	0	0
$176$	−2.70436	−0.203849
$177$	− 7.73463i	− 0.581371i
$178$	− 22.3103i	− 1.67223i
$179$	−13.4849	−1.00791	−0.503953	−	0.863731i	$-0.668122\pi$
$179$	−13.4849	−1.00791	−0.503953	+	0.863731i	$0.668122\pi$
$180$	0	0
$181$	−23.0899	−1.71626	−0.858130	−	0.513433i	$-0.828374\pi$
$181$	−23.0899	−1.71626	−0.858130	+	0.513433i	$0.828374\pi$
$182$	− 11.7190i	− 0.868673i
$183$	12.0147i	0.888151i
$184$	6.27718	0.462760
$185$	0	0
$186$	−11.7190	−0.859281
$187$	− 1.15516i	− 0.0844738i
$188$	− 8.76491i	− 0.639247i
$189$	3.64002	0.264773
$190$	0	0
$191$	−7.98440	−0.577731	−0.288866	−	0.957370i	$-0.593278\pi$
$191$	−7.98440	−0.577731	−0.288866	+	0.957370i	$0.593278\pi$
$192$	− 11.4537i	− 0.826597i
$193$	11.7649i	0.846857i	0.905930	+	0.423428i	$0.139173\pi$
$193$	11.7649i	0.846857i	−0.905930	+	0.423428i	$0.860827\pi$
$194$	14.4058	1.03428
$195$	0	0
$196$	15.7190	1.12279
$197$	3.81456i	0.271776i	0.990724	+	0.135888i	$0.0433888\pi$
$197$	3.81456i	0.271776i	−0.990724	+	0.135888i	$0.956611\pi$
$198$	2.12489i	0.151009i
$199$	12.0752	0.855990	0.427995	−	0.903781i	$-0.359220\pi$
$199$	12.0752	0.855990	0.427995	+	0.903781i	$0.359220\pi$
$200$	0	0
$201$	14.2645	1.00614
$202$	− 15.7346i	− 1.10708i
$203$	22.7493i	1.59669i
$204$	2.90539	0.203418
$205$	0	0
$206$	−35.0596	−2.44272
$207$	5.73463i	0.398585i
$208$	− 4.09747i	− 0.284109i
$209$	−2.60975	−0.180520
$210$	0	0
$211$	10.2645	0.706634	0.353317	−	0.935504i	$-0.385054\pi$
$211$	10.2645	0.706634	0.353317	+	0.935504i	$0.385054\pi$
$212$	31.5904i	2.16964i
$213$	− 8.51514i	− 0.583448i
$214$	−8.37088	−0.572221
$215$	0	0
$216$	−1.09461	−0.0744787
$217$	− 20.0752i	− 1.36280i
$218$	14.3103i	0.969217i
$219$	−9.21949	−0.622996
$220$	0	0
$221$	1.75023	0.117733
$222$	0.965943i	0.0648298i
$223$	12.9239i	0.865445i	0.901527	+	0.432723i	$0.142447\pi$
$223$	12.9239i	0.865445i	−0.901527	+	0.432723i	$0.857553\pi$
$224$	28.8860	1.93003
$225$	0	0
$226$	−12.7493	−0.848072
$227$	− 22.8099i	− 1.51394i	−0.653447	−	0.756972i	$-0.726678\pi$
$227$	− 22.8099i	− 1.51394i	0.653447	−	0.756972i	$-0.273322\pi$
$228$	− 6.56387i	− 0.434703i
$229$	−14.7796	−0.976663	−0.488331	−	0.872658i	$-0.662394\pi$
$229$	−14.7796	−0.976663	−0.488331	+	0.872658i	$0.662394\pi$
$230$	0	0
$231$	−3.64002	−0.239496
$232$	− 6.84106i	− 0.449137i
$233$	4.96594i	0.325330i	0.986681	+	0.162665i	$0.0520089\pi$
$233$	4.96594i	0.325330i	−0.986681	+	0.162665i	$0.947991\pi$
$234$	−3.21949	−0.210465
$235$	0	0
$236$	−19.4537	−1.26633
$237$	5.09461i	0.330930i
$238$	8.93475i	0.579154i
$239$	14.9991	0.970210	0.485105	−	0.874456i	$-0.338781\pi$
$239$	14.9991	0.970210	0.485105	+	0.874456i	$0.338781\pi$
$240$	0	0
$241$	5.04496	0.324974	0.162487	−	0.986711i	$-0.448048\pi$
$241$	5.04496	0.324974	0.162487	+	0.986711i	$0.448048\pi$
$242$	− 2.12489i	− 0.136593i
$243$	− 1.00000i	− 0.0641500i
$244$	30.2186	1.93455
$245$	0	0
$246$	−8.76491	−0.558830
$247$	− 3.95413i	− 0.251595i
$248$	6.03692i	0.383345i
$249$	−14.7493	−0.934700
$250$	0	0
$251$	−3.03028	−0.191269	−0.0956347	−	0.995417i	$-0.530488\pi$
$251$	−3.03028	−0.191269	−0.0956347	+	0.995417i	$0.530488\pi$
$252$	− 9.15516i	− 0.576721i
$253$	− 5.73463i	− 0.360533i
$254$	−17.1358	−1.07519
$255$	0	0
$256$	4.91721	0.307325
$257$	13.6509i	0.851521i	0.904836	+	0.425761i	$0.139993\pi$
$257$	13.6509i	0.851521i	−0.904836	+	0.425761i	$0.860007\pi$
$258$	− 24.8704i	− 1.54836i
$259$	−1.65470	−0.102818
$260$	0	0
$261$	6.24977	0.386851
$262$	27.2195i	1.68163i
$263$	12.5601i	0.774489i	0.921977	+	0.387244i	$0.126573\pi$
$263$	12.5601i	0.774489i	−0.921977	+	0.387244i	$0.873427\pi$
$264$	1.09461	0.0673685
$265$	0	0
$266$	20.1854	1.23765
$267$	− 10.4995i	− 0.642562i
$268$	− 35.8771i	− 2.19154i
$269$	24.6888	1.50530	0.752650	−	0.658421i	$-0.228775\pi$
$269$	24.6888	1.50530	0.752650	+	0.658421i	$0.228775\pi$
$270$	0	0
$271$	7.56479	0.459528	0.229764	−	0.973246i	$-0.426205\pi$
$271$	7.56479	0.459528	0.229764	+	0.973246i	$0.426205\pi$
$272$	3.12397i	0.189418i
$273$	− 5.51514i	− 0.333791i
$274$	−48.4683	−2.92808
$275$	0	0
$276$	14.4234	0.868186
$277$	− 1.92477i	− 0.115648i	−0.998327	−	0.0578241i	$-0.981584\pi$
$277$	− 1.92477i	− 0.115648i	0.998327	−	0.0578241i	$-0.0184162\pi$
$278$	16.1287i	0.967333i
$279$	−5.51514	−0.330183
$280$	0	0
$281$	−1.87511	−0.111860	−0.0559300	−	0.998435i	$-0.517812\pi$
$281$	−1.87511	−0.111860	−0.0559300	+	0.998435i	$0.517812\pi$
$282$	− 7.40493i	− 0.440957i
$283$	30.1396i	1.79161i	0.444446	+	0.895806i	$0.353401\pi$
$283$	30.1396i	1.79161i	−0.444446	+	0.895806i	$0.646599\pi$
$284$	−21.4167	−1.27085
$285$	0	0
$286$	3.21949	0.190373
$287$	− 15.0147i	− 0.886289i
$288$	− 7.93567i	− 0.467614i
$289$	15.6656	0.921506
$290$	0	0
$291$	6.77959	0.397427
$292$	23.1883i	1.35699i
$293$	− 29.1552i	− 1.70326i	−0.524141	−	0.851631i	$-0.675614\pi$
$293$	− 29.1552i	− 1.70326i	0.524141	−	0.851631i	$-0.324386\pi$
$294$	13.2800	0.774508
$295$	0	0
$296$	0.497594	0.0289221
$297$	1.00000i	0.0580259i
$298$	− 3.85574i	− 0.223357i
$299$	8.68876	0.502484
$300$	0	0
$301$	42.6041	2.45566
$302$	− 51.6878i	− 2.97430i
$303$	− 7.40493i	− 0.425402i
$304$	7.05769	0.404786
$305$	0	0
$306$	2.45459	0.140319
$307$	− 27.8548i	− 1.58976i	−0.606768	−	0.794879i	$-0.707534\pi$
$307$	− 27.8548i	− 1.58976i	0.606768	−	0.794879i	$-0.292466\pi$
$308$	9.15516i	0.521664i
$309$	−16.4995	−0.938626
$310$	0	0
$311$	23.9083	1.35571	0.677856	−	0.735194i	$-0.262909\pi$
$311$	23.9083	1.35571	0.677856	+	0.735194i	$0.262909\pi$
$312$	1.65848i	0.0938932i
$313$	28.3094i	1.60014i	0.599905	+	0.800071i	$0.295205\pi$
$313$	28.3094i	1.60014i	−0.599905	+	0.800071i	$0.704795\pi$
$314$	−20.7493	−1.17095
$315$	0	0
$316$	12.8136	0.720824
$317$	8.80986i	0.494811i	0.968912	+	0.247406i	$0.0795780\pi$
$317$	8.80986i	0.494811i	−0.968912	+	0.247406i	$0.920422\pi$
$318$	26.6888i	1.49663i
$319$	−6.24977	−0.349920
$320$	0	0
$321$	−3.93945	−0.219879
$322$	44.3553i	2.47182i
$323$	3.01468i	0.167741i
$324$	−2.51514	−0.139730
$325$	0	0
$326$	−14.8411	−0.821970
$327$	6.73463i	0.372426i
$328$	4.51514i	0.249307i
$329$	12.6850	0.699346
$330$	0	0
$331$	32.2498	1.77261	0.886304	−	0.463104i	$-0.153264\pi$
$331$	32.2498	1.77261	0.886304	+	0.463104i	$0.153264\pi$
$332$	37.0966i	2.03594i
$333$	0.454586i	0.0249112i
$334$	13.4087	0.733692
$335$	0	0
$336$	9.84392	0.537030
$337$	− 28.9844i	− 1.57888i	−0.613827	−	0.789441i	$-0.710371\pi$
$337$	− 28.9844i	− 1.57888i	0.613827	−	0.789441i	$-0.289629\pi$
$338$	− 22.7455i	− 1.23719i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	5.51514	0.298661
$342$	− 5.54541i	− 0.299862i
$343$	− 2.73085i	− 0.147452i
$344$	−12.8117	−0.690760
$345$	0	0
$346$	−27.2938	−1.46732
$347$	35.7190i	1.91750i	0.284253	+	0.958749i	$0.408254\pi$
$347$	35.7190i	1.91750i	−0.284253	+	0.958749i	$0.591746\pi$
$348$	− 15.7190i	− 0.842629i
$349$	23.2800	1.24615	0.623076	−	0.782161i	$-0.285883\pi$
$349$	23.2800	1.24615	0.623076	+	0.782161i	$0.285883\pi$
$350$	0	0
$351$	−1.51514	−0.0808721
$352$	7.93567i	0.422972i
$353$	− 9.75023i	− 0.518952i	−0.965750	−	0.259476i	$-0.916450\pi$
$353$	− 9.75023i	− 0.518952i	0.965750	−	0.259476i	$-0.0835499\pi$
$354$	−16.4352	−0.873521
$355$	0	0
$356$	−26.4078	−1.39961
$357$	4.20482i	0.222542i
$358$	28.6538i	1.51440i
$359$	−33.9007	−1.78921	−0.894605	−	0.446858i	$-0.852543\pi$
$359$	−33.9007	−1.78921	−0.894605	+	0.446858i	$0.852543\pi$
$360$	0	0
$361$	−12.1892	−0.641538
$362$	49.0634i	2.57872i
$363$	− 1.00000i	− 0.0524864i
$364$	−13.8713	−0.727055
$365$	0	0
$366$	25.5298	1.33446
$367$	− 1.88601i	− 0.0984491i	−0.998788	−	0.0492245i	$-0.984325\pi$
$367$	− 1.88601i	− 0.0984491i	0.998788	−	0.0492245i	$-0.0156750\pi$
$368$	15.5085i	0.808436i
$369$	−4.12489	−0.214733
$370$	0	0
$371$	−45.7190	−2.37361
$372$	13.8713i	0.719195i
$373$	− 16.3250i	− 0.845277i	−0.906298	−	0.422638i	$-0.861104\pi$
$373$	− 16.3250i	− 0.845277i	0.906298	−	0.422638i	$-0.138896\pi$
$374$	−2.45459	−0.126924
$375$	0	0
$376$	−3.81456	−0.196721
$377$	− 9.46927i	− 0.487692i
$378$	− 7.73463i	− 0.397827i
$379$	26.0440	1.33779	0.668896	−	0.743356i	$-0.266767\pi$
$379$	26.0440	1.33779	0.668896	+	0.743356i	$0.266767\pi$
$380$	0	0
$381$	−8.06433	−0.413148
$382$	16.9659i	0.868053i
$383$	12.4702i	0.637197i	0.947890	+	0.318598i	$0.103212\pi$
$383$	12.4702i	0.637197i	−0.947890	+	0.318598i	$0.896788\pi$
$384$	−8.46640	−0.432049
$385$	0	0
$386$	24.9991	1.27242
$387$	− 11.7044i	− 0.594966i
$388$	− 17.0516i	− 0.865664i
$389$	18.0899	0.917195	0.458597	−	0.888644i	$-0.348352\pi$
$389$	18.0899	0.917195	0.458597	+	0.888644i	$0.348352\pi$
$390$	0	0
$391$	−6.62443	−0.335012
$392$	− 6.84106i	− 0.345526i
$393$	12.8099i	0.646172i
$394$	8.10551	0.408350
$395$	0	0
$396$	2.51514	0.126390
$397$	15.2342i	0.764581i	0.924042	+	0.382291i	$0.124865\pi$
$397$	15.2342i	0.764581i	−0.924042	+	0.382291i	$0.875135\pi$
$398$	− 25.6585i	− 1.28614i
$399$	9.49954	0.475572
$400$	0	0
$401$	2.74931	0.137294	0.0686471	−	0.997641i	$-0.478132\pi$
$401$	2.74931	0.137294	0.0686471	+	0.997641i	$0.478132\pi$
$402$	− 30.3103i	− 1.51174i
$403$	8.35620i	0.416252i
$404$	−18.6244	−0.926600
$405$	0	0
$406$	48.3397	2.39906
$407$	− 0.454586i	− 0.0225330i
$408$	− 1.26445i	− 0.0625996i
$409$	3.98532	0.197061	0.0985307	−	0.995134i	$-0.468586\pi$
$409$	3.98532	0.197061	0.0985307	+	0.995134i	$0.468586\pi$
$410$	0	0
$411$	−22.8099	−1.12513
$412$	41.4986i	2.04449i
$413$	− 28.1542i	− 1.38538i
$414$	12.1854	0.598882
$415$	0	0
$416$	−12.0236	−0.589507
$417$	7.59037i	0.371702i
$418$	5.54541i	0.271235i
$419$	5.13578	0.250899	0.125450	−	0.992100i	$-0.459963\pi$
$419$	5.13578	0.250899	0.125450	+	0.992100i	$0.459963\pi$
$420$	0	0
$421$	−8.94657	−0.436029	−0.218014	−	0.975946i	$-0.569958\pi$
$421$	−8.94657	−0.436029	−0.218014	+	0.975946i	$0.569958\pi$
$422$	− 21.8108i	− 1.06173i
$423$	− 3.48486i	− 0.169440i
$424$	13.7484	0.667681
$425$	0	0
$426$	−18.0937	−0.876642
$427$	43.7337i	2.11642i
$428$	9.90826i	0.478934i
$429$	1.51514	0.0731516
$430$	0	0
$431$	22.7493	1.09580	0.547898	−	0.836545i	$-0.315428\pi$
$431$	22.7493	1.09580	0.547898	+	0.836545i	$0.315428\pi$
$432$	− 2.70436i	− 0.130113i
$433$	7.58325i	0.364428i	0.983259	+	0.182214i	$0.0583263\pi$
$433$	7.58325i	0.364428i	−0.983259	+	0.182214i	$0.941674\pi$
$434$	−42.6576	−2.04763
$435$	0	0
$436$	16.9385	0.811209
$437$	14.9659i	0.715918i
$438$	19.5904i	0.936064i
$439$	−17.3903	−0.829991	−0.414996	−	0.909823i	$-0.636217\pi$
$439$	−17.3903	−0.829991	−0.414996	+	0.909823i	$0.636217\pi$
$440$	0	0
$441$	6.24977	0.297608
$442$	− 3.71904i	− 0.176897i
$443$	10.6438i	0.505702i	0.967505	+	0.252851i	$0.0813683\pi$
$443$	10.6438i	0.505702i	−0.967505	+	0.252851i	$0.918632\pi$
$444$	1.14335	0.0542608
$445$	0	0
$446$	27.4617	1.30035
$447$	− 1.81456i	− 0.0858259i
$448$	− 41.6916i	− 1.96974i
$449$	26.9310	1.27095	0.635476	−	0.772121i	$-0.280804\pi$
$449$	26.9310	1.27095	0.635476	+	0.772121i	$0.280804\pi$
$450$	0	0
$451$	4.12489	0.194233
$452$	15.0908i	0.709813i
$453$	− 24.3250i	− 1.14289i
$454$	−48.4683	−2.27473
$455$	0	0
$456$	−2.85665	−0.133775
$457$	15.7796i	0.738138i	0.929402	+	0.369069i	$0.120323\pi$
$457$	15.7796i	0.738138i	−0.929402	+	0.369069i	$0.879677\pi$
$458$	31.4049i	1.46746i
$459$	1.15516	0.0539183
$460$	0	0
$461$	8.18922	0.381410	0.190705	−	0.981647i	$-0.438923\pi$
$461$	8.18922	0.381410	0.190705	+	0.981647i	$0.438923\pi$
$462$	7.73463i	0.359848i
$463$	16.0899i	0.747762i	0.927477	+	0.373881i	$0.121973\pi$
$463$	16.0899i	0.747762i	−0.927477	+	0.373881i	$0.878027\pi$
$464$	16.9016	0.784638
$465$	0	0
$466$	10.5521	0.488815
$467$	29.4693i	1.36367i	0.731504	+	0.681837i	$0.238819\pi$
$467$	29.4693i	1.36367i	−0.731504	+	0.681837i	$0.761181\pi$
$468$	3.81078i	0.176153i
$469$	51.9229	2.39758
$470$	0	0
$471$	−9.76491	−0.449943
$472$	8.46640i	0.389698i
$473$	11.7044i	0.538167i
$474$	10.8255	0.497230
$475$	0	0
$476$	10.5757	0.484736
$477$	12.5601i	0.575087i
$478$	− 31.8713i	− 1.45776i
$479$	−32.2186	−1.47210	−0.736052	−	0.676925i	$-0.763312\pi$
$479$	−32.2186	−1.47210	−0.736052	+	0.676925i	$0.763312\pi$
$480$	0	0
$481$	0.688760	0.0314048
$482$	− 10.7200i	− 0.488280i
$483$	20.8742i	0.949809i
$484$	−2.51514	−0.114324
$485$	0	0
$486$	−2.12489	−0.0963868
$487$	35.8936i	1.62649i	0.581919	+	0.813247i	$0.302302\pi$
$487$	35.8936i	1.62649i	−0.581919	+	0.813247i	$0.697698\pi$
$488$	− 13.1514i	− 0.595335i
$489$	−6.98440	−0.315846
$490$	0	0
$491$	−7.15894	−0.323079	−0.161539	−	0.986866i	$-0.551646\pi$
$491$	−7.15894	−0.323079	−0.161539	+	0.986866i	$0.551646\pi$
$492$	10.3747i	0.467726i
$493$	7.21949i	0.325150i
$494$	−8.40207	−0.378027
$495$	0	0
$496$	−14.9149	−0.669699
$497$	− 30.9953i	− 1.39033i
$498$	31.3406i	1.40441i
$499$	−27.0743	−1.21201	−0.606006	−	0.795460i	$-0.707229\pi$
$499$	−27.0743	−1.21201	−0.606006	+	0.795460i	$0.707229\pi$
$500$	0	0
$501$	6.31032	0.281924
$502$	6.43899i	0.287386i
$503$	26.9991i	1.20383i	0.798560	+	0.601915i	$0.205595\pi$
$503$	26.9991i	1.20383i	−0.798560	+	0.601915i	$0.794405\pi$
$504$	−3.98440	−0.177479
$505$	0	0
$506$	−12.1854	−0.541709
$507$	− 10.7044i	− 0.475397i
$508$	20.2829i	0.899909i
$509$	−15.5904	−0.691031	−0.345515	−	0.938413i	$-0.612296\pi$
$509$	−15.5904	−0.691031	−0.345515	+	0.938413i	$0.612296\pi$
$510$	0	0
$511$	−33.5592	−1.48457
$512$	− 27.3813i	− 1.21009i
$513$	− 2.60975i	− 0.115223i
$514$	29.0066	1.27943
$515$	0	0
$516$	−29.4381	−1.29594
$517$	3.48486i	0.153264i
$518$	3.51605i	0.154487i
$519$	−12.8448	−0.563826
$520$	0	0
$521$	11.1589	0.488882	0.244441	−	0.969664i	$-0.421396\pi$
$521$	11.1589	0.488882	0.244441	+	0.969664i	$0.421396\pi$
$522$	− 13.2800i	− 0.581252i
$523$	− 10.5786i	− 0.462568i	−0.972886	−	0.231284i	$-0.925707\pi$
$523$	− 10.5786i	− 0.462568i	0.972886	−	0.231284i	$-0.0742926\pi$
$524$	32.2186	1.40748
$525$	0	0
$526$	26.6888	1.16369
$527$	− 6.37088i	− 0.277520i
$528$	2.70436i	0.117692i
$529$	−9.88601	−0.429827
$530$	0	0
$531$	−7.73463	−0.335654
$532$	− 23.8927i	− 1.03588i
$533$	6.24977i	0.270708i
$534$	−22.3103	−0.965462
$535$	0	0
$536$	−15.6140	−0.674422
$537$	13.4849i	0.581915i
$538$	− 52.4608i	− 2.26175i
$539$	−6.24977	−0.269197
$540$	0	0
$541$	−11.2947	−0.485598	−0.242799	−	0.970077i	$-0.578066\pi$
$541$	−11.2947	−0.485598	−0.242799	+	0.970077i	$0.578066\pi$
$542$	− 16.0743i	− 0.690451i
$543$	23.0899i	0.990883i
$544$	9.16698	0.393031
$545$	0	0
$546$	−11.7190	−0.501528
$547$	− 6.09369i	− 0.260547i	−0.991478	−	0.130274i	$-0.958414\pi$
$547$	− 6.09369i	− 0.260547i	0.991478	−	0.130274i	$-0.0415856\pi$
$548$	57.3700i	2.45072i
$549$	12.0147	0.512774
$550$	0	0
$551$	16.3103	0.694843
$552$	− 6.27718i	− 0.267175i
$553$	18.5445i	0.788592i
$554$	−4.08991	−0.173764
$555$	0	0
$556$	19.0908	0.809631
$557$	5.90069i	0.250020i	0.992155	+	0.125010i	$0.0398964\pi$
$557$	5.90069i	0.250020i	−0.992155	+	0.125010i	$0.960104\pi$
$558$	11.7190i	0.496106i
$559$	−17.7337	−0.750056
$560$	0	0
$561$	−1.15516	−0.0487710
$562$	3.98440i	0.168072i
$563$	3.03028i	0.127711i	0.997959	+	0.0638555i	$0.0203397\pi$
$563$	3.03028i	0.127711i	−0.997959	+	0.0638555i	$0.979660\pi$
$564$	−8.76491	−0.369069
$565$	0	0
$566$	64.0431	2.69193
$567$	− 3.64002i	− 0.152867i
$568$	9.32075i	0.391090i
$569$	−13.4049	−0.561964	−0.280982	−	0.959713i	$-0.590660\pi$
$569$	−13.4049	−0.561964	−0.280982	+	0.959713i	$0.590660\pi$
$570$	0	0
$571$	26.8851	1.12511	0.562553	−	0.826761i	$-0.309819\pi$
$571$	26.8851	1.12511	0.562553	+	0.826761i	$0.309819\pi$
$572$	− 3.81078i	− 0.159337i
$573$	7.98440i	0.333553i
$574$	−31.9045	−1.33167
$575$	0	0
$576$	−11.4537	−0.477236
$577$	− 2.03028i	− 0.0845215i	−0.999107	−	0.0422607i	$-0.986544\pi$
$577$	− 2.03028i	− 0.0845215i	0.999107	−	0.0422607i	$-0.0134560\pi$
$578$	− 33.2876i	− 1.38458i
$579$	11.7649	0.488933
$580$	0	0
$581$	−53.6878	−2.22735
$582$	− 14.4058i	− 0.597142i
$583$	− 12.5601i	− 0.520186i
$584$	10.0917	0.417599
$585$	0	0
$586$	−61.9514	−2.55919
$587$	− 21.8245i	− 0.900795i	−0.892828	−	0.450398i	$-0.851282\pi$
$587$	− 21.8245i	− 0.900795i	0.892828	−	0.450398i	$-0.148718\pi$
$588$	− 15.7190i	− 0.648242i
$589$	−14.3931	−0.593058
$590$	0	0
$591$	3.81456	0.156910
$592$	1.22936i	0.0505265i
$593$	8.06811i	0.331318i	0.986183	+	0.165659i	$0.0529751\pi$
$593$	8.06811i	0.331318i	−0.986183	+	0.165659i	$0.947025\pi$
$594$	2.12489	0.0871851
$595$	0	0
$596$	−4.56387	−0.186944
$597$	− 12.0752i	− 0.494206i
$598$	− 18.4626i	− 0.754993i
$599$	−7.61353	−0.311080	−0.155540	−	0.987830i	$-0.549712\pi$
$599$	−7.61353	−0.311080	−0.155540	+	0.987830i	$0.549712\pi$
$600$	0	0
$601$	3.57569	0.145855	0.0729277	−	0.997337i	$-0.476766\pi$
$601$	3.57569	0.145855	0.0729277	+	0.997337i	$0.476766\pi$
$602$	− 90.5289i	− 3.68968i
$603$	− 14.2645i	− 0.580893i
$604$	−61.1807	−2.48941
$605$	0	0
$606$	−15.7346	−0.639176
$607$	− 17.5298i	− 0.711513i	−0.934579	−	0.355757i	$-0.884223\pi$
$607$	− 17.5298i	− 0.711513i	0.934579	−	0.355757i	$-0.115777\pi$
$608$	− 20.7101i	− 0.839905i
$609$	22.7493	0.921849
$610$	0	0
$611$	−5.28005	−0.213608
$612$	− 2.90539i	− 0.117443i
$613$	12.5601i	0.507297i	0.967296	+	0.253649i	$0.0816307\pi$
$613$	12.5601i	0.507297i	−0.967296	+	0.253649i	$0.918369\pi$
$614$	−59.1883	−2.38865
$615$	0	0
$616$	3.98440	0.160536
$617$	− 15.9612i	− 0.642576i	−0.946982	−	0.321288i	$-0.895884\pi$
$617$	− 15.9612i	− 0.642576i	0.946982	−	0.321288i	$-0.104116\pi$
$618$	35.0596i	1.41031i
$619$	9.23417	0.371153	0.185576	−	0.982630i	$-0.440585\pi$
$619$	9.23417	0.371153	0.185576	+	0.982630i	$0.440585\pi$
$620$	0	0
$621$	5.73463	0.230123
$622$	− 50.8023i	− 2.03699i
$623$	− 38.2186i	− 1.53119i
$624$	−4.09747	−0.164030
$625$	0	0
$626$	60.1542	2.40425
$627$	2.60975i	0.104223i
$628$	24.5601i	0.980054i
$629$	−0.525120	−0.0209379
$630$	0	0
$631$	29.2342	1.16379	0.581897	−	0.813262i	$-0.302311\pi$
$631$	29.2342	1.16379	0.581897	+	0.813262i	$0.302311\pi$
$632$	− 5.57661i	− 0.221826i
$633$	− 10.2645i	− 0.407975i
$634$	18.7200	0.743464
$635$	0	0
$636$	31.5904	1.25264
$637$	− 9.46927i	− 0.375186i
$638$	13.2800i	0.525762i
$639$	−8.51514	−0.336854
$640$	0	0
$641$	13.9612	0.551436	0.275718	−	0.961239i	$-0.411084\pi$
$641$	13.9612	0.551436	0.275718	+	0.961239i	$0.411084\pi$
$642$	8.37088i	0.330372i
$643$	12.6206i	0.497710i	0.968541	+	0.248855i	$0.0800542\pi$
$643$	12.6206i	0.497710i	−0.968541	+	0.248855i	$0.919946\pi$
$644$	52.5015	2.06885
$645$	0	0
$646$	6.40585	0.252035
$647$	− 29.6429i	− 1.16538i	−0.812694	−	0.582691i	$-0.802000\pi$
$647$	− 29.6429i	− 1.16538i	0.812694	−	0.582691i	$-0.198000\pi$
$648$	1.09461i	0.0430003i
$649$	7.73463	0.303611
$650$	0	0
$651$	−20.0752	−0.786810
$652$	17.5667i	0.687967i
$653$	9.90069i	0.387444i	0.981056	+	0.193722i	$0.0620560\pi$
$653$	9.90069i	0.387444i	−0.981056	+	0.193722i	$0.937944\pi$
$654$	14.3103	0.559578
$655$	0	0
$656$	−11.1552	−0.435536
$657$	9.21949i	0.359687i
$658$	− 26.9541i	− 1.05078i
$659$	5.28005	0.205681	0.102841	−	0.994698i	$-0.467207\pi$
$659$	5.28005	0.205681	0.102841	+	0.994698i	$0.467207\pi$
$660$	0	0
$661$	−26.8548	−1.04453	−0.522266	−	0.852783i	$-0.674913\pi$
$661$	−26.8548	−1.04453	−0.522266	+	0.852783i	$0.674913\pi$
$662$	− 68.5271i	− 2.66338i
$663$	− 1.75023i	− 0.0679733i
$664$	16.1447	0.626537
$665$	0	0
$666$	0.965943	0.0374295
$667$	35.8401i	1.38774i
$668$	− 15.8713i	− 0.614080i
$669$	12.9239	0.499665
$670$	0	0
$671$	−12.0147	−0.463822
$672$	− 28.8860i	− 1.11430i
$673$	− 3.81834i	− 0.147186i	−0.997288	−	0.0735932i	$-0.976553\pi$
$673$	− 3.81834i	− 0.147186i	0.997288	−	0.0735932i	$-0.0234466\pi$
$674$	−61.5885	−2.37230
$675$	0	0
$676$	−26.9229	−1.03550
$677$	15.6897i	0.603003i	0.953466	+	0.301502i	$0.0974879\pi$
$677$	15.6897i	0.603003i	−0.953466	+	0.301502i	$0.902512\pi$
$678$	12.7493i	0.489634i
$679$	24.6779	0.947049
$680$	0	0
$681$	−22.8099	−0.874076
$682$	− 11.7190i	− 0.448745i
$683$	− 15.6353i	− 0.598269i	−0.954211	−	0.299135i	$-0.903302\pi$
$683$	− 15.6353i	− 0.598269i	0.954211	−	0.299135i	$-0.0966979\pi$
$684$	−6.56387	−0.250976
$685$	0	0
$686$	−5.80275	−0.221550
$687$	14.7796i	0.563876i
$688$	− 31.6528i	− 1.20675i
$689$	19.0303	0.724996
$690$	0	0
$691$	−31.4305	−1.19567	−0.597836	−	0.801618i	$-0.703973\pi$
$691$	−31.4305	−1.19567	−0.597836	+	0.801618i	$0.703973\pi$
$692$	32.3065i	1.22811i
$693$	3.64002i	0.138273i
$694$	75.8989	2.88108
$695$	0	0
$696$	−6.84106	−0.259310
$697$	− 4.76491i	− 0.180484i
$698$	− 49.4674i	− 1.87237i
$699$	4.96594	0.187829
$700$	0	0
$701$	3.24507	0.122565	0.0612824	−	0.998120i	$-0.480481\pi$
$701$	3.24507	0.122565	0.0612824	+	0.998120i	$0.480481\pi$
$702$	3.21949i	0.121512i
$703$	1.18635i	0.0447442i
$704$	11.4537	0.431676
$705$	0	0
$706$	−20.7181	−0.779737
$707$	− 26.9541i	− 1.01371i
$708$	19.4537i	0.731114i
$709$	26.7190	1.00345	0.501727	−	0.865026i	$-0.332698\pi$
$709$	26.7190	1.00345	0.501727	+	0.865026i	$0.332698\pi$
$710$	0	0
$711$	5.09461	0.191063
$712$	11.4929i	0.430714i
$713$	− 31.6273i	− 1.18445i
$714$	8.93475	0.334375
$715$	0	0
$716$	33.9163	1.26751
$717$	− 14.9991i	− 0.560151i
$718$	72.0351i	2.68833i
$719$	6.78807	0.253152	0.126576	−	0.991957i	$-0.459601\pi$
$719$	6.78807	0.253152	0.126576	+	0.991957i	$0.459601\pi$
$720$	0	0
$721$	−60.0587	−2.23670
$722$	25.9007i	0.963924i
$723$	− 5.04496i	− 0.187624i
$724$	58.0743	2.15831
$725$	0	0
$726$	−2.12489	−0.0788619
$727$	19.9154i	0.738620i	0.929306	+	0.369310i	$0.120406\pi$
$727$	19.9154i	0.738620i	−0.929306	+	0.369310i	$0.879594\pi$
$728$	6.03692i	0.223743i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	13.5204	0.500071
$732$	− 30.2186i	− 1.11691i
$733$	− 31.3388i	− 1.15752i	−0.815497	−	0.578762i	$-0.803536\pi$
$733$	− 31.3388i	− 1.15752i	0.815497	−	0.578762i	$-0.196464\pi$
$734$	−4.00756	−0.147922
$735$	0	0
$736$	45.5081	1.67745
$737$	14.2645i	0.525438i
$738$	8.76491i	0.322641i
$739$	2.25355	0.0828982	0.0414491	−	0.999141i	$-0.486803\pi$
$739$	2.25355	0.0828982	0.0414491	+	0.999141i	$0.486803\pi$
$740$	0	0
$741$	−3.95413	−0.145259
$742$	97.1477i	3.56640i
$743$	− 6.74931i	− 0.247608i	−0.992307	−	0.123804i	$-0.960491\pi$
$743$	− 6.74931i	− 0.247608i	0.992307	−	0.123804i	$-0.0395095\pi$
$744$	6.03692	0.221324
$745$	0	0
$746$	−34.6888	−1.27005
$747$	14.7493i	0.539649i
$748$	2.90539i	0.106232i
$749$	−14.3397	−0.523961
$750$	0	0
$751$	22.4390	0.818810	0.409405	−	0.912353i	$-0.365736\pi$
$751$	22.4390	0.818810	0.409405	+	0.912353i	$0.365736\pi$
$752$	− 9.42431i	− 0.343669i
$753$	3.03028i	0.110429i
$754$	−20.1211	−0.732767
$755$	0	0
$756$	−9.15516	−0.332970
$757$	− 25.4158i	− 0.923754i	−0.886944	−	0.461877i	$-0.847176\pi$
$757$	− 25.4158i	− 0.923754i	0.886944	−	0.461877i	$-0.152824\pi$
$758$	− 55.3406i	− 2.01006i
$759$	−5.73463	−0.208154
$760$	0	0
$761$	−30.7493	−1.11466	−0.557331	−	0.830291i	$-0.688174\pi$
$761$	−30.7493	−1.11466	−0.557331	+	0.830291i	$0.688174\pi$
$762$	17.1358i	0.620764i
$763$	24.5142i	0.887474i
$764$	20.0819	0.726537
$765$	0	0
$766$	26.4977	0.957401
$767$	11.7190i	0.423150i
$768$	− 4.91721i	− 0.177434i
$769$	16.2956	0.587636	0.293818	−	0.955861i	$-0.405074\pi$
$769$	16.2956	0.587636	0.293818	+	0.955861i	$0.405074\pi$
$770$	0	0
$771$	13.6509	0.491626
$772$	− 29.5904i	− 1.06498i
$773$	48.7787i	1.75445i	0.480082	+	0.877223i	$0.340607\pi$
$773$	48.7787i	1.75445i	−0.480082	+	0.877223i	$0.659393\pi$
$774$	−24.8704	−0.893949
$775$	0	0
$776$	−7.42100	−0.266398
$777$	1.65470i	0.0593621i
$778$	− 38.4390i	− 1.37810i
$779$	−10.7649	−0.385693
$780$	0	0
$781$	8.51514	0.304696
$782$	14.0761i	0.503362i
$783$	− 6.24977i	− 0.223349i
$784$	16.9016	0.603629
$785$	0	0
$786$	27.2195	0.970887
$787$	46.2001i	1.64686i	0.567421	+	0.823428i	$0.307941\pi$
$787$	46.2001i	1.64686i	−0.567421	+	0.823428i	$0.692059\pi$
$788$	− 9.59415i	− 0.341777i
$789$	12.5601	0.447151
$790$	0	0
$791$	−21.8401	−0.776546
$792$	− 1.09461i	− 0.0388952i
$793$	− 18.2039i	− 0.646439i
$794$	32.3709	1.14880
$795$	0	0
$796$	−30.3709	−1.07647
$797$	− 36.3784i	− 1.28859i	−0.764777	−	0.644295i	$-0.777151\pi$
$797$	− 36.3784i	− 1.28859i	0.764777	−	0.644295i	$-0.222849\pi$
$798$	− 20.1854i	− 0.714557i
$799$	4.02558	0.142415
$800$	0	0
$801$	−10.4995	−0.370983
$802$	− 5.84197i	− 0.206287i
$803$	− 9.21949i	− 0.325349i
$804$	−35.8771	−1.26529
$805$	0	0
$806$	17.7560	0.625427
$807$	− 24.6888i	− 0.869086i
$808$	8.10551i	0.285151i
$809$	−11.3737	−0.399879	−0.199940	−	0.979808i	$-0.564075\pi$
$809$	−11.3737	−0.399879	−0.199940	+	0.979808i	$0.564075\pi$
$810$	0	0
$811$	−13.3903	−0.470195	−0.235098	−	0.971972i	$-0.575541\pi$
$811$	−13.3903	−0.470195	−0.235098	+	0.971972i	$0.575541\pi$
$812$	− 57.2177i	− 2.00795i
$813$	− 7.56479i	− 0.265309i
$814$	−0.965943	−0.0338563
$815$	0	0
$816$	3.12397	0.109361
$817$	− 30.5454i	− 1.06865i
$818$	− 8.46835i	− 0.296089i
$819$	−5.51514	−0.192715
$820$	0	0
$821$	32.0975	1.12021	0.560105	−	0.828422i	$-0.310761\pi$
$821$	32.0975	1.12021	0.560105	+	0.828422i	$0.310761\pi$
$822$	48.4683i	1.69053i
$823$	16.7952i	0.585443i	0.956198	+	0.292722i	$0.0945609\pi$
$823$	16.7952i	0.585443i	−0.956198	+	0.292722i	$0.905439\pi$
$824$	18.0606	0.629169
$825$	0	0
$826$	−59.8245	−2.08156
$827$	45.5904i	1.58533i	0.609656	+	0.792666i	$0.291308\pi$
$827$	45.5904i	1.58533i	−0.609656	+	0.792666i	$0.708692\pi$
$828$	− 14.4234i	− 0.501248i
$829$	−12.9385	−0.449374	−0.224687	−	0.974431i	$-0.572136\pi$
$829$	−12.9385	−0.449374	−0.224687	+	0.974431i	$0.572136\pi$
$830$	0	0
$831$	−1.92477	−0.0667695
$832$	17.3539i	0.601638i
$833$	7.21949i	0.250141i
$834$	16.1287	0.558490
$835$	0	0
$836$	6.56387	0.227016
$837$	5.51514i	0.190631i
$838$	− 10.9130i	− 0.376982i
$839$	1.59037	0.0549057	0.0274528	−	0.999623i	$-0.491260\pi$
$839$	1.59037	0.0549057	0.0274528	+	0.999623i	$0.491260\pi$
$840$	0	0
$841$	10.0596	0.346884
$842$	19.0104i	0.655143i
$843$	1.87511i	0.0645824i
$844$	−25.8165	−0.888641
$845$	0	0
$846$	−7.40493	−0.254587
$847$	− 3.64002i	− 0.125073i
$848$	33.9670i	1.16643i
$849$	30.1396	1.03439
$850$	0	0
$851$	−2.60688	−0.0893628
$852$	21.4167i	0.733726i
$853$	10.5161i	0.360063i	0.983661	+	0.180031i	$0.0576199\pi$
$853$	10.5161i	0.360063i	−0.983661	+	0.180031i	$0.942380\pi$
$854$	92.9291	3.17997
$855$	0	0
$856$	4.31216	0.147386
$857$	57.4637i	1.96292i	0.191665	+	0.981460i	$0.438611\pi$
$857$	57.4637i	1.96292i	−0.191665	+	0.981460i	$0.561389\pi$
$858$	− 3.21949i	− 0.109912i
$859$	−32.7181	−1.11633	−0.558164	−	0.829731i	$-0.688494\pi$
$859$	−32.7181	−1.11633	−0.558164	+	0.829731i	$0.688494\pi$
$860$	0	0
$861$	−15.0147	−0.511699
$862$	− 48.3397i	− 1.64646i
$863$	− 43.1807i	− 1.46989i	−0.678127	−	0.734945i	$-0.737208\pi$
$863$	− 43.1807i	− 1.46989i	0.678127	−	0.734945i	$-0.262792\pi$
$864$	−7.93567	−0.269977
$865$	0	0
$866$	16.1135	0.547560
$867$	− 15.6656i	− 0.532032i
$868$	50.4920i	1.71381i
$869$	−5.09461	−0.172823
$870$	0	0
$871$	−21.6126	−0.732315
$872$	− 7.37179i	− 0.249640i
$873$	− 6.77959i	− 0.229454i
$874$	31.8009	1.07568
$875$	0	0
$876$	23.1883	0.783460
$877$	16.0752i	0.542822i	0.962464	+	0.271411i	$0.0874903\pi$
$877$	16.0752i	0.542822i	−0.962464	+	0.271411i	$0.912510\pi$
$878$	36.9523i	1.24708i
$879$	−29.1552	−0.983379
$880$	0	0
$881$	31.2876	1.05411	0.527053	−	0.849832i	$-0.323297\pi$
$881$	31.2876	1.05411	0.527053	+	0.849832i	$0.323297\pi$
$882$	− 13.2800i	− 0.447162i
$883$	24.7640i	0.833375i	0.909050	+	0.416687i	$0.136809\pi$
$883$	24.7640i	0.833375i	−0.909050	+	0.416687i	$0.863191\pi$
$884$	−4.40207	−0.148058
$885$	0	0
$886$	22.6169	0.759828
$887$	26.3085i	0.883353i	0.897174	+	0.441676i	$0.145616\pi$
$887$	26.3085i	0.883353i	−0.897174	+	0.441676i	$0.854384\pi$
$888$	− 0.497594i	− 0.0166982i
$889$	−29.3544	−0.984514
$890$	0	0
$891$	1.00000	0.0335013
$892$	− 32.5053i	− 1.08836i
$893$	− 9.09461i	− 0.304339i
$894$	−3.85574	−0.128955
$895$	0	0
$896$	−30.8179	−1.02955
$897$	− 8.68876i	− 0.290109i
$898$	− 57.2252i	− 1.90963i
$899$	−34.4683	−1.14958
$900$	0	0
$901$	−14.5089	−0.483363
$902$	− 8.76491i	− 0.291840i
$903$	− 42.6041i	− 1.41778i
$904$	6.56766	0.218437
$905$	0	0
$906$	−51.6878	−1.71721
$907$	55.9301i	1.85713i	0.371174	+	0.928563i	$0.378955\pi$
$907$	55.9301i	1.85713i	−0.371174	+	0.928563i	$0.621045\pi$
$908$	57.3700i	1.90389i
$909$	−7.40493	−0.245606
$910$	0	0
$911$	−15.3931	−0.509997	−0.254998	−	0.966941i	$-0.582075\pi$
$911$	−15.3931	−0.509997	−0.254998	+	0.966941i	$0.582075\pi$
$912$	− 7.05769i	− 0.233703i
$913$	− 14.7493i	− 0.488131i
$914$	33.5298	1.10907
$915$	0	0
$916$	37.1727	1.22822
$917$	46.6282i	1.53980i
$918$	− 2.45459i	− 0.0810134i
$919$	51.2598	1.69090	0.845452	−	0.534052i	$-0.179331\pi$
$919$	51.2598	1.69090	0.845452	+	0.534052i	$0.179331\pi$
$920$	0	0
$921$	−27.8548	−0.917848
$922$	− 17.4012i	− 0.573076i
$923$	12.9016i	0.424662i
$924$	9.15516	0.301183
$925$	0	0
$926$	34.1892	1.12353
$927$	16.4995i	0.541916i
$928$	− 49.5961i	− 1.62807i
$929$	−14.8099	−0.485896	−0.242948	−	0.970039i	$-0.578114\pi$
$929$	−14.8099	−0.485896	−0.242948	+	0.970039i	$0.578114\pi$
$930$	0	0
$931$	16.3103	0.534549
$932$	− 12.4900i	− 0.409125i
$933$	− 23.9083i	− 0.782721i
$934$	62.6188	2.04895
$935$	0	0
$936$	1.65848	0.0542093
$937$	− 27.2654i	− 0.890721i	−0.895351	−	0.445360i	$-0.853076\pi$
$937$	− 27.2654i	− 0.890721i	0.895351	−	0.445360i	$-0.146924\pi$
$938$	− 110.330i	− 3.60241i
$939$	28.3094	0.923843
$940$	0	0
$941$	−49.5630	−1.61571	−0.807853	−	0.589384i	$-0.799371\pi$
$941$	−49.5630	−1.61571	−0.807853	+	0.589384i	$0.799371\pi$
$942$	20.7493i	0.676049i
$943$	− 23.6547i	− 0.770303i
$944$	−20.9172	−0.680797
$945$	0	0
$946$	24.8704	0.808607
$947$	13.9844i	0.454432i	0.973844	+	0.227216i	$0.0729623\pi$
$947$	13.9844i	0.454432i	−0.973844	+	0.227216i	$0.927038\pi$
$948$	− 12.8136i	− 0.416168i
$949$	13.9688	0.453447
$950$	0	0
$951$	8.80986	0.285679
$952$	− 4.60263i	− 0.149172i
$953$	17.1240i	0.554700i	0.960769	+	0.277350i	$0.0894561\pi$
$953$	17.1240i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$954$	26.6888	0.864081
$955$	0	0
$956$	−37.7248	−1.22011
$957$	6.24977i	0.202026i
$958$	68.4608i	2.21187i
$959$	−83.0284	−2.68113
$960$	0	0
$961$	−0.583252	−0.0188146
$962$	− 1.46354i	− 0.0471863i
$963$	3.93945i	0.126947i
$964$	−12.6888	−0.408677
$965$	0	0
$966$	44.3553	1.42711
$967$	− 1.90826i	− 0.0613653i	−0.999529	−	0.0306827i	$-0.990232\pi$
$967$	− 1.90826i	− 0.0613653i	0.999529	−	0.0306827i	$-0.00976813\pi$
$968$	1.09461i	0.0351821i
$969$	3.01468	0.0968455
$970$	0	0
$971$	31.3856	1.00721	0.503605	−	0.863934i	$-0.332007\pi$
$971$	31.3856	1.00721	0.503605	+	0.863934i	$0.332007\pi$
$972$	2.51514i	0.0806731i
$973$	27.6291i	0.885749i
$974$	76.2697	2.44384
$975$	0	0
$976$	32.4920	1.04004
$977$	− 41.4693i	− 1.32672i	−0.748301	−	0.663360i	$-0.769130\pi$
$977$	− 41.4693i	− 1.32672i	0.748301	−	0.663360i	$-0.230870\pi$
$978$	14.8411i	0.474565i
$979$	10.4995	0.335567
$980$	0	0
$981$	6.73463	0.215020
$982$	15.2119i	0.485432i
$983$	− 6.23601i	− 0.198898i	−0.995043	−	0.0994489i	$-0.968292\pi$
$983$	− 6.23601i	− 0.198898i	0.995043	−	0.0994489i	$-0.0317080\pi$
$984$	4.51514	0.143937
$985$	0	0
$986$	15.3406	0.488544
$987$	− 12.6850i	− 0.403767i
$988$	9.94518i	0.316398i
$989$	67.1202	2.13430
$990$	0	0
$991$	27.2048	0.864189	0.432095	−	0.901828i	$-0.357775\pi$
$991$	27.2048	0.864189	0.432095	+	0.901828i	$0.357775\pi$
$992$	43.7663i	1.38958i
$993$	− 32.2498i	− 1.02342i
$994$	−65.8615	−2.08900
$995$	0	0
$996$	37.0966	1.17545
$997$	− 28.0606i	− 0.888687i	−0.895857	−	0.444343i	$-0.853437\pi$
$997$	− 28.0606i	− 0.888687i	0.895857	−	0.444343i	$-0.146563\pi$
$998$	57.5298i	1.82107i
$999$	0.454586	0.0143825

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.c.f.199.2		6
3.2	odd	2		2475.2.c.q.199.5		6
5.2	odd	4		825.2.a.i.1.3	✓	3
5.3	odd	4		825.2.a.m.1.1	yes	3
5.4	even	2	inner	825.2.c.f.199.5		6
15.2	even	4		2475.2.a.bd.1.1		3
15.8	even	4		2475.2.a.z.1.3		3
15.14	odd	2		2475.2.c.q.199.2		6
55.32	even	4		9075.2.a.cj.1.1		3
55.43	even	4		9075.2.a.cd.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.i.1.3	✓	3	5.2	odd	4
825.2.a.m.1.1	yes	3	5.3	odd	4
825.2.c.f.199.2		6	1.1	even	1	trivial
825.2.c.f.199.5		6	5.4	even	2	inner
2475.2.a.z.1.3		3	15.8	even	4
2475.2.a.bd.1.1		3	15.2	even	4
2475.2.c.q.199.2		6	15.14	odd	2
2475.2.c.q.199.5		6	3.2	odd	2
9075.2.a.cd.1.3		3	55.43	even	4
9075.2.a.cj.1.1		3	55.32	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.12489i q^{2} -1.00000i q^{3} -2.51514 q^{4} -2.12489 q^{6} -3.64002i q^{7} +1.09461i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.12489i q^{2} -1.00000i q^{3} -2.51514 q^{4} -2.12489 q^{6} -3.64002i q^{7} +1.09461i q^{8} -1.00000 q^{9} +1.00000 q^{11} +2.51514i q^{12} +1.51514i q^{13} -7.73463 q^{14} -2.70436 q^{16} -1.15516i q^{17} +2.12489i q^{18} -2.60975 q^{19} -3.64002 q^{21} -2.12489i q^{22} -5.73463i q^{23} +1.09461 q^{24} +3.21949 q^{26} +1.00000i q^{27} +9.15516i q^{28} -6.24977 q^{29} +5.51514 q^{31} +7.93567i q^{32} -1.00000i q^{33} -2.45459 q^{34} +2.51514 q^{36} -0.454586i q^{37} +5.54541i q^{38} +1.51514 q^{39} +4.12489 q^{41} +7.73463i q^{42} +11.7044i q^{43} -2.51514 q^{44} -12.1854 q^{46} +3.48486i q^{47} +2.70436i q^{48} -6.24977 q^{49} -1.15516 q^{51} -3.81078i q^{52} -12.5601i q^{53} +2.12489 q^{54} +3.98440 q^{56} +2.60975i q^{57} +13.2800i q^{58} +7.73463 q^{59} -12.0147 q^{61} -11.7190i q^{62} +3.64002i q^{63} +11.4537 q^{64} -2.12489 q^{66} +14.2645i q^{67} +2.90539i q^{68} -5.73463 q^{69} +8.51514 q^{71} -1.09461i q^{72} -9.21949i q^{73} -0.965943 q^{74} +6.56387 q^{76} -3.64002i q^{77} -3.21949i q^{78} -5.09461 q^{79} +1.00000 q^{81} -8.76491i q^{82} -14.7493i q^{83} +9.15516 q^{84} +24.8704 q^{86} +6.24977i q^{87} +1.09461i q^{88} +10.4995 q^{89} +5.51514 q^{91} +14.4234i q^{92} -5.51514i q^{93} +7.40493 q^{94} +7.93567 q^{96} +6.77959i q^{97} +13.2800i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 16 * q^4 + 4 * q^6 - 6 * q^9	\( 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9} + 6 q^{11} - 12 q^{14} + 20 q^{16} + 2 q^{19} - 6 q^{21} - 12 q^{24} - 16 q^{26} - 4 q^{29} + 34 q^{31} - 12 q^{34} + 16 q^{36} + 10 q^{39} + 8 q^{41} - 16 q^{44} - 60 q^{46} - 4 q^{49} + 8 q^{51} - 4 q^{54} - 44 q^{56} + 12 q^{59} - 6 q^{61} - 68 q^{64} + 4 q^{66} + 52 q^{71} - 28 q^{74} - 48 q^{76} - 12 q^{79} + 6 q^{81} + 40 q^{84} + 56 q^{86} - 4 q^{89} + 34 q^{91} - 4 q^{94} + 68 q^{96} - 6 q^{99}+O(q^{100}) \) 6 * q - 16 * q^4 + 4 * q^6 - 6 * q^9 + 6 * q^11 - 12 * q^14 + 20 * q^16 + 2 * q^19 - 6 * q^21 - 12 * q^24 - 16 * q^26 - 4 * q^29 + 34 * q^31 - 12 * q^34 + 16 * q^36 + 10 * q^39 + 8 * q^41 - 16 * q^44 - 60 * q^46 - 4 * q^49 + 8 * q^51 - 4 * q^54 - 44 * q^56 + 12 * q^59 - 6 * q^61 - 68 * q^64 + 4 * q^66 + 52 * q^71 - 28 * q^74 - 48 * q^76 - 12 * q^79 + 6 * q^81 + 40 * q^84 + 56 * q^86 - 4 * q^89 + 34 * q^91 - 4 * q^94 + 68 * q^96 - 6 * q^99

Introduction

L-functions

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