LMFDB - Embedded newform 48.13.g.a.31.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,13,Mod(31,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 13, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 13);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 13 $
Character orbit:	$[\chi]$	$=$	48.g (of order $2$, degree $1$, 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:	$43.8717032293$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 4333x^{2} + 4332x + 18766224 $ x^4 - x^3 + 4333x^2 + 4332x + 18766224
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.3
Root			$-32.6599 + 56.5686i$ of defining polynomial
Character	$\chi$	$=$	48.31
Dual form			48.13.g.a.31.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+420.888i q^{3} -21286.8 q^{5} -7111.90i q^{7} -177147. q^{9} +O(q^{10})$	$q+420.888i q^{3} -21286.8 q^{5} -7111.90i q^{7} -177147. q^{9} +1.88540e6i q^{11} -8.56106e6 q^{13} -8.95935e6i q^{15} +6.77647e6 q^{17} -645743. i q^{19} +2.99331e6 q^{21} -2.15694e8i q^{23} +2.08985e8 q^{25} -7.45591e7i q^{27} -3.66716e8 q^{29} +1.03858e9i q^{31} -7.93544e8 q^{33} +1.51389e8i q^{35} +3.52823e9 q^{37} -3.60325e9i q^{39} +5.66160e9 q^{41} +4.86430e9i q^{43} +3.77089e9 q^{45} -1.89030e10i q^{47} +1.37907e10 q^{49} +2.85214e9i q^{51} +1.61412e9 q^{53} -4.01341e10i q^{55} +2.71786e8 q^{57} +1.28375e10i q^{59} +3.86920e10 q^{61} +1.25985e9i q^{63} +1.82237e11 q^{65} -1.56544e11i q^{67} +9.07829e10 q^{69} -4.16779e10i q^{71} +1.00107e11 q^{73} +8.79596e10i q^{75} +1.34088e10 q^{77} +1.99076e11i q^{79} +3.13811e10 q^{81} -2.30690e11i q^{83} -1.44249e11 q^{85} -1.54347e11i q^{87} -4.73312e11 q^{89} +6.08854e10i q^{91} -4.37126e11 q^{93} +1.37458e10i q^{95} +9.29103e11 q^{97} -3.33993e11i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 21960 q^{5} - 708588 q^{9}+O(q^{10}) $ 4 * q - 21960 * q^5 - 708588 * q^9	$ 4 q - 21960 q^{5} - 708588 q^{9} - 5810072 q^{13} - 33882264 q^{17} - 15664752 q^{21} + 142148300 q^{25} + 42583896 q^{29} + 271362960 q^{33} + 9402865736 q^{37} + 17968882536 q^{41} + 3890148120 q^{45} + 53940845764 q^{49} + 60546956760 q^{53} + 46910963376 q^{57} + 168287201672 q^{61} + 481064975280 q^{65} + 163198209024 q^{69} + 789026629000 q^{73} + 140389989696 q^{77} + 125524238436 q^{81} - 777401136720 q^{85} - 638670460536 q^{89} - 607412656080 q^{93} - 563542043000 q^{97}+O(q^{100}) $ 4 * q - 21960 * q^5 - 708588 * q^9 - 5810072 * q^13 - 33882264 * q^17 - 15664752 * q^21 + 142148300 * q^25 + 42583896 * q^29 + 271362960 * q^33 + 9402865736 * q^37 + 17968882536 * q^41 + 3890148120 * q^45 + 53940845764 * q^49 + 60546956760 * q^53 + 46910963376 * q^57 + 168287201672 * q^61 + 481064975280 * q^65 + 163198209024 * q^69 + 789026629000 * q^73 + 140389989696 * q^77 + 125524238436 * q^81 - 777401136720 * q^85 - 638670460536 * q^89 - 607412656080 * q^93 - 563542043000 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	420.888i	0.577350i
$3$	420.888i	0.577350i
$4$	0	0
$5$	−21286.8	−1.36235	−0.681176	−	0.732119i	$-0.738531\pi$
$5$	−21286.8	−1.36235	−0.681176	+	0.732119i	$0.738531\pi$
$6$	0	0
$7$	− 7111.90i	− 0.0604501i	−0.999543	−	0.0302251i	$-0.990378\pi$
$7$	− 7111.90i	− 0.0604501i	0.999543	−	0.0302251i	$-0.00962240\pi$
$8$	0	0
$9$	−177147.	−0.333333
$10$	0	0
$11$	1.88540e6i	1.06426i	0.846663	+	0.532130i	$0.178608\pi$
$11$	1.88540e6i	1.06426i	−0.846663	+	0.532130i	$0.821392\pi$
$12$	0	0
$13$	−8.56106e6	−1.77365	−0.886824	−	0.462107i	$-0.847093\pi$
$13$	−8.56106e6	−1.77365	−0.886824	+	0.462107i	$0.847093\pi$
$14$	0	0
$15$	− 8.95935e6i	− 0.786555i
$16$	0	0
$17$	6.77647e6	0.280744	0.140372	−	0.990099i	$-0.455170\pi$
$17$	6.77647e6	0.280744	0.140372	+	0.990099i	$0.455170\pi$
$18$	0	0
$19$	− 645743.i	− 0.0137258i	−0.999976	−	0.00686291i	$-0.997815\pi$
$19$	− 645743.i	− 0.0137258i	0.999976	−	0.00686291i	$-0.00218455\pi$
$20$	0	0
$21$	2.99331e6	0.0349009
$22$	0	0
$23$	− 2.15694e8i	− 1.45704i	−0.685027	−	0.728518i	$-0.740210\pi$
$23$	− 2.15694e8i	− 1.45704i	0.685027	−	0.728518i	$-0.259790\pi$
$24$	0	0
$25$	2.08985e8	0.856005
$26$	0	0
$27$	− 7.45591e7i	− 0.192450i
$28$	0	0
$29$	−3.66716e8	−0.616513	−0.308257	−	0.951303i	$-0.599746\pi$
$29$	−3.66716e8	−0.616513	−0.308257	+	0.951303i	$0.599746\pi$
$30$	0	0
$31$	1.03858e9i	1.17022i	0.810952	+	0.585112i	$0.198950\pi$
$31$	1.03858e9i	1.17022i	−0.810952	+	0.585112i	$0.801050\pi$
$32$	0	0
$33$	−7.93544e8	−0.614451
$34$	0	0
$35$	1.51389e8i	0.0823544i
$36$	0	0
$37$	3.52823e9	1.37514	0.687570	−	0.726118i	$-0.258678\pi$
$37$	3.52823e9	1.37514	0.687570	+	0.726118i	$0.258678\pi$
$38$	0	0
$39$	− 3.60325e9i	− 1.02402i
$40$	0	0
$41$	5.66160e9	1.19189	0.595945	−	0.803025i	$-0.296778\pi$
$41$	5.66160e9	1.19189	0.595945	+	0.803025i	$0.296778\pi$
$42$	0	0
$43$	4.86430e9i	0.769502i	0.923020	+	0.384751i	$0.125713\pi$
$43$	4.86430e9i	0.769502i	−0.923020	+	0.384751i	$0.874287\pi$
$44$	0	0
$45$	3.77089e9	0.454118
$46$	0	0
$47$	− 1.89030e10i	− 1.75365i	−0.480811	−	0.876824i	$-0.659658\pi$
$47$	− 1.89030e10i	− 1.75365i	0.480811	−	0.876824i	$-0.340342\pi$
$48$	0	0
$49$	1.37907e10	0.996346
$50$	0	0
$51$	2.85214e9i	0.162087i
$52$	0	0
$53$	1.61412e9	0.0728251	0.0364126	−	0.999337i	$-0.488407\pi$
$53$	1.61412e9	0.0728251	0.0364126	+	0.999337i	$0.488407\pi$
$54$	0	0
$55$	− 4.01341e10i	− 1.44990i
$56$	0	0
$57$	2.71786e8	0.00792460
$58$	0	0
$59$	1.28375e10i	0.304346i	0.988354	+	0.152173i	$0.0486271\pi$
$59$	1.28375e10i	0.304346i	−0.988354	+	0.152173i	$0.951373\pi$
$60$	0	0
$61$	3.86920e10	0.751004	0.375502	−	0.926822i	$-0.377470\pi$
$61$	3.86920e10	0.751004	0.375502	+	0.926822i	$0.377470\pi$
$62$	0	0
$63$	1.25985e9i	0.0201500i
$64$	0	0
$65$	1.82237e11	2.41633
$66$	0	0
$67$	− 1.56544e11i	− 1.73056i	−0.501289	−	0.865280i	$-0.667141\pi$
$67$	− 1.56544e11i	− 1.73056i	0.501289	−	0.865280i	$-0.332859\pi$
$68$	0	0
$69$	9.07829e10	0.841220
$70$	0	0
$71$	− 4.16779e10i	− 0.325354i	−0.986679	−	0.162677i	$-0.947987\pi$
$71$	− 4.16779e10i	− 0.325354i	0.986679	−	0.162677i	$-0.0520128\pi$
$72$	0	0
$73$	1.00107e11	0.661497	0.330748	−	0.943719i	$-0.392699\pi$
$73$	1.00107e11	0.661497	0.330748	+	0.943719i	$0.392699\pi$
$74$	0	0
$75$	8.79596e10i	0.494214i
$76$	0	0
$77$	1.34088e10	0.0643347
$78$	0	0
$79$	1.99076e11i	0.818947i	0.912322	+	0.409473i	$0.134288\pi$
$79$	1.99076e11i	0.818947i	−0.912322	+	0.409473i	$0.865712\pi$
$80$	0	0
$81$	3.13811e10	0.111111
$82$	0	0
$83$	− 2.30690e11i	− 0.705604i	−0.935698	−	0.352802i	$-0.885229\pi$
$83$	− 2.30690e11i	− 0.705604i	0.935698	−	0.352802i	$-0.114771\pi$
$84$	0	0
$85$	−1.44249e11	−0.382472
$86$	0	0
$87$	− 1.54347e11i	− 0.355944i
$88$	0	0
$89$	−4.73312e11	−0.952375	−0.476187	−	0.879344i	$-0.657982\pi$
$89$	−4.73312e11	−0.952375	−0.476187	+	0.879344i	$0.657982\pi$
$90$	0	0
$91$	6.08854e10i	0.107217i
$92$	0	0
$93$	−4.37126e11	−0.675630
$94$	0	0
$95$	1.37458e10i	0.0186994i
$96$	0	0
$97$	9.29103e11	1.11541	0.557704	−	0.830040i	$-0.311682\pi$
$97$	9.29103e11	1.11541	0.557704	+	0.830040i	$0.311682\pi$
$98$	0	0
$99$	− 3.33993e11i	− 0.354754i
$100$	0	0
$101$	−4.30494e11	−0.405545	−0.202773	−	0.979226i	$-0.564995\pi$
$101$	−4.30494e11	−0.405545	−0.202773	+	0.979226i	$0.564995\pi$
$102$	0	0
$103$	1.24450e12i	1.04225i	0.853482	+	0.521123i	$0.174487\pi$
$103$	1.24450e12i	1.04225i	−0.853482	+	0.521123i	$0.825513\pi$
$104$	0	0
$105$	−6.37180e10	−0.0475473
$106$	0	0
$107$	− 3.40478e11i	− 0.226875i	−0.993545	−	0.113437i	$-0.963814\pi$
$107$	− 3.40478e11i	− 0.226875i	0.993545	−	0.113437i	$-0.0361861\pi$
$108$	0	0
$109$	−3.31888e12	−1.97894	−0.989470	−	0.144738i	$-0.953766\pi$
$109$	−3.31888e12	−1.97894	−0.989470	+	0.144738i	$0.953766\pi$
$110$	0	0
$111$	1.48499e12i	0.793937i
$112$	0	0
$113$	−3.87063e12	−1.85913	−0.929567	−	0.368654i	$-0.879819\pi$
$113$	−3.87063e12	−1.85913	−0.929567	+	0.368654i	$0.879819\pi$
$114$	0	0
$115$	4.59142e12i	1.98500i
$116$	0	0
$117$	1.51657e12	0.591216
$118$	0	0
$119$	− 4.81935e10i	− 0.0169710i
$120$	0	0
$121$	−4.16314e11	−0.132650
$122$	0	0
$123$	2.38290e12i	0.688138i
$124$	0	0
$125$	7.48339e11	0.196173
$126$	0	0
$127$	− 8.23613e12i	− 1.96291i	−0.191688	−	0.981456i	$-0.561396\pi$
$127$	− 8.23613e12i	− 1.96291i	0.191688	−	0.981456i	$-0.438604\pi$
$128$	0	0
$129$	−2.04733e12	−0.444272
$130$	0	0
$131$	1.61593e12i	0.319738i	0.987138	+	0.159869i	$0.0511072\pi$
$131$	1.61593e12i	0.319738i	−0.987138	+	0.159869i	$0.948893\pi$
$132$	0	0
$133$	−4.59246e9	−0.000829727 0
$134$	0	0
$135$	1.58712e12i	0.262185i
$136$	0	0
$137$	−4.29703e11	−0.0649897	−0.0324948	−	0.999472i	$-0.510345\pi$
$137$	−4.29703e11	−0.0649897	−0.0324948	+	0.999472i	$0.510345\pi$
$138$	0	0
$139$	− 1.70332e12i	− 0.236160i	−0.993004	−	0.118080i	$-0.962326\pi$
$139$	− 1.70332e12i	− 0.236160i	0.993004	−	0.118080i	$-0.0376739\pi$
$140$	0	0
$141$	7.95603e12	1.01247
$142$	0	0
$143$	− 1.61410e13i	− 1.88762i
$144$	0	0
$145$	7.80621e12	0.839908
$146$	0	0
$147$	5.80435e12i	0.575241i
$148$	0	0
$149$	−5.75418e12	−0.525854	−0.262927	−	0.964816i	$-0.584688\pi$
$149$	−5.75418e12	−0.525854	−0.262927	+	0.964816i	$0.584688\pi$
$150$	0	0
$151$	1.73435e13i	1.46311i	0.681785	+	0.731553i	$0.261204\pi$
$151$	1.73435e13i	1.46311i	−0.681785	+	0.731553i	$0.738796\pi$
$152$	0	0
$153$	−1.20043e12	−0.0935812
$154$	0	0
$155$	− 2.21080e13i	− 1.59426i
$156$	0	0
$157$	1.18262e13	0.789672	0.394836	−	0.918752i	$-0.370801\pi$
$157$	1.18262e13	0.789672	0.394836	+	0.918752i	$0.370801\pi$
$158$	0	0
$159$	6.79365e11i	0.0420456i
$160$	0	0
$161$	−1.53399e12	−0.0880780
$162$	0	0
$163$	1.72346e13i	0.918917i	0.888199	+	0.459458i	$0.151956\pi$
$163$	1.72346e13i	0.918917i	−0.888199	+	0.459458i	$0.848044\pi$
$164$	0	0
$165$	1.68920e13	0.837099
$166$	0	0
$167$	− 3.25246e13i	− 1.49939i	−0.661785	−	0.749694i	$-0.730201\pi$
$167$	− 3.25246e13i	− 1.49939i	0.661785	−	0.749694i	$-0.269799\pi$
$168$	0	0
$169$	4.99937e13	2.14583
$170$	0	0
$171$	1.14391e11i	0.00457527i
$172$	0	0
$173$	2.94963e13	1.10025	0.550125	−	0.835082i	$-0.314580\pi$
$173$	2.94963e13	1.10025	0.550125	+	0.835082i	$0.314580\pi$
$174$	0	0
$175$	− 1.48628e12i	− 0.0517456i
$176$	0	0
$177$	−5.40314e12	−0.175714
$178$	0	0
$179$	− 2.17325e13i	− 0.660682i	−0.943862	−	0.330341i	$-0.892836\pi$
$179$	− 2.17325e13i	− 0.660682i	0.943862	−	0.330341i	$-0.107164\pi$
$180$	0	0
$181$	−1.24786e12	−0.0354891	−0.0177446	−	0.999843i	$-0.505649\pi$
$181$	−1.24786e12	−0.0354891	−0.0177446	+	0.999843i	$0.505649\pi$
$182$	0	0
$183$	1.62850e13i	0.433592i
$184$	0	0
$185$	−7.51046e13	−1.87343
$186$	0	0
$187$	1.27764e13i	0.298784i
$188$	0	0
$189$	−5.30257e11	−0.0116336
$190$	0	0
$191$	− 6.52191e13i	− 1.34330i	−0.740866	−	0.671652i	$-0.765585\pi$
$191$	− 6.52191e13i	− 1.34330i	0.740866	−	0.671652i	$-0.234415\pi$
$192$	0	0
$193$	−1.62613e13	−0.314638	−0.157319	−	0.987548i	$-0.550285\pi$
$193$	−1.62613e13	−0.314638	−0.157319	+	0.987548i	$0.550285\pi$
$194$	0	0
$195$	7.67015e13i	1.39507i
$196$	0	0
$197$	−8.69576e13	−1.48768	−0.743841	−	0.668356i	$-0.766998\pi$
$197$	−8.69576e13	−1.48768	−0.743841	+	0.668356i	$0.766998\pi$
$198$	0	0
$199$	1.03084e13i	0.165986i	0.996550	+	0.0829932i	$0.0264480\pi$
$199$	1.03084e13i	0.165986i	−0.996550	+	0.0829932i	$0.973552\pi$
$200$	0	0
$201$	6.58874e13	0.999139
$202$	0	0
$203$	2.60805e12i	0.0372683i
$204$	0	0
$205$	−1.20517e14	−1.62378
$206$	0	0
$207$	3.82095e13i	0.485678i
$208$	0	0
$209$	1.21749e12	0.0146078
$210$	0	0
$211$	1.21337e14i	1.37499i	0.726189	+	0.687495i	$0.241290\pi$
$211$	1.21337e14i	1.37499i	−0.726189	+	0.687495i	$0.758710\pi$
$212$	0	0
$213$	1.75417e13	0.187843
$214$	0	0
$215$	− 1.03545e14i	− 1.04833i
$216$	0	0
$217$	7.38627e12	0.0707402
$218$	0	0
$219$	4.21339e13i	0.381915i
$220$	0	0
$221$	−5.80137e13	−0.497940
$222$	0	0
$223$	− 1.52486e14i	− 1.23994i	−0.784624	−	0.619972i	$-0.787144\pi$
$223$	− 1.52486e14i	− 1.23994i	0.784624	−	0.619972i	$-0.212856\pi$
$224$	0	0
$225$	−3.70212e13	−0.285335
$226$	0	0
$227$	2.60931e14i	1.90709i	0.301249	+	0.953545i	$0.402596\pi$
$227$	2.60931e14i	1.90709i	−0.301249	+	0.953545i	$0.597404\pi$
$228$	0	0
$229$	1.11017e14	0.769798	0.384899	−	0.922959i	$-0.374236\pi$
$229$	1.11017e14	0.769798	0.384899	+	0.922959i	$0.374236\pi$
$230$	0	0
$231$	5.64360e12i	0.0371436i
$232$	0	0
$233$	−1.48696e14	−0.929318	−0.464659	−	0.885490i	$-0.653823\pi$
$233$	−1.48696e14	−0.929318	−0.464659	+	0.885490i	$0.653823\pi$
$234$	0	0
$235$	4.02383e14i	2.38909i
$236$	0	0
$237$	−8.37887e13	−0.472819
$238$	0	0
$239$	− 3.25593e14i	− 1.74698i	−0.486841	−	0.873491i	$-0.661851\pi$
$239$	− 3.25593e14i	− 1.74698i	0.486841	−	0.873491i	$-0.338149\pi$
$240$	0	0
$241$	−1.46403e14	−0.747220	−0.373610	−	0.927586i	$-0.621880\pi$
$241$	−1.46403e14	−0.747220	−0.373610	+	0.927586i	$0.621880\pi$
$242$	0	0
$243$	1.32079e13i	0.0641500i
$244$	0	0
$245$	−2.93559e14	−1.35737
$246$	0	0
$247$	5.52825e12i	0.0243448i
$248$	0	0
$249$	9.70949e13	0.407381
$250$	0	0
$251$	− 4.56243e14i	− 1.82454i	−0.409589	−	0.912270i	$-0.634328\pi$
$251$	− 4.56243e14i	− 1.82454i	0.409589	−	0.912270i	$-0.365672\pi$
$252$	0	0
$253$	4.06669e14	1.55067
$254$	0	0
$255$	− 6.07127e13i	− 0.220820i
$256$	0	0
$257$	4.70340e14	1.63235	0.816175	−	0.577804i	$-0.196090\pi$
$257$	4.70340e14	1.63235	0.816175	+	0.577804i	$0.196090\pi$
$258$	0	0
$259$	− 2.50924e13i	− 0.0831274i
$260$	0	0
$261$	6.49627e13	0.205504
$262$	0	0
$263$	1.42061e14i	0.429280i	0.976693	+	0.214640i	$0.0688578\pi$
$263$	1.42061e14i	0.429280i	−0.976693	+	0.214640i	$0.931142\pi$
$264$	0	0
$265$	−3.43594e13	−0.0992135
$266$	0	0
$267$	− 1.99212e14i	− 0.549854i
$268$	0	0
$269$	2.94994e14	0.778573	0.389286	−	0.921117i	$-0.372722\pi$
$269$	2.94994e14	0.778573	0.389286	+	0.921117i	$0.372722\pi$
$270$	0	0
$271$	− 4.70773e14i	− 1.18849i	−0.804284	−	0.594246i	$-0.797451\pi$
$271$	− 4.70773e14i	− 1.18849i	0.804284	−	0.594246i	$-0.202549\pi$
$272$	0	0
$273$	−2.56259e13	−0.0619019
$274$	0	0
$275$	3.94022e14i	0.911012i
$276$	0	0
$277$	4.15318e14	0.919395	0.459698	−	0.888075i	$-0.347958\pi$
$277$	4.15318e14	0.919395	0.459698	+	0.888075i	$0.347958\pi$
$278$	0	0
$279$	− 1.83981e14i	− 0.390075i
$280$	0	0
$281$	6.73345e14	1.36773	0.683864	−	0.729609i	$-0.260298\pi$
$281$	6.73345e14	1.36773	0.683864	+	0.729609i	$0.260298\pi$
$282$	0	0
$283$	− 3.70518e14i	− 0.721259i	−0.932709	−	0.360629i	$-0.882562\pi$
$283$	− 3.70518e14i	− 0.721259i	0.932709	−	0.360629i	$-0.117438\pi$
$284$	0	0
$285$	−5.78544e12	−0.0107961
$286$	0	0
$287$	− 4.02647e13i	− 0.0720499i
$288$	0	0
$289$	−5.36702e14	−0.921183
$290$	0	0
$291$	3.91049e14i	0.643981i
$292$	0	0
$293$	2.56469e14	0.405350	0.202675	−	0.979246i	$-0.435037\pi$
$293$	2.56469e14	0.405350	0.202675	+	0.979246i	$0.435037\pi$
$294$	0	0
$295$	− 2.73268e14i	− 0.414626i
$296$	0	0
$297$	1.40574e14	0.204817
$298$	0	0
$299$	1.84657e15i	2.58427i
$300$	0	0
$301$	3.45944e13	0.0465165
$302$	0	0
$303$	− 1.81190e14i	− 0.234142i
$304$	0	0
$305$	−8.23627e14	−1.02313
$306$	0	0
$307$	− 8.24545e14i	− 0.984882i	−0.870346	−	0.492441i	$-0.836105\pi$
$307$	− 8.24545e14i	− 0.984882i	0.870346	−	0.492441i	$-0.163895\pi$
$308$	0	0
$309$	−5.23793e14	−0.601740
$310$	0	0
$311$	− 7.97009e14i	− 0.880848i	−0.897790	−	0.440424i	$-0.854828\pi$
$311$	− 7.97009e14i	− 0.880848i	0.897790	−	0.440424i	$-0.145172\pi$
$312$	0	0
$313$	1.79978e14	0.191406	0.0957028	−	0.995410i	$-0.469490\pi$
$313$	1.79978e14	0.191406	0.0957028	+	0.995410i	$0.469490\pi$
$314$	0	0
$315$	− 2.68181e13i	− 0.0274515i
$316$	0	0
$317$	1.63990e15	1.61608	0.808039	−	0.589130i	$-0.200529\pi$
$317$	1.63990e15	1.61608	0.808039	+	0.589130i	$0.200529\pi$
$318$	0	0
$319$	− 6.91408e14i	− 0.656131i
$320$	0	0
$321$	1.43303e14	0.130986
$322$	0	0
$323$	− 4.37586e12i	− 0.00385343i
$324$	0	0
$325$	−1.78914e15	−1.51825
$326$	0	0
$327$	− 1.39688e15i	− 1.14254i
$328$	0	0
$329$	−1.34436e14	−0.106008
$330$	0	0
$331$	2.09770e15i	1.59505i	0.603283	+	0.797527i	$0.293859\pi$
$331$	2.09770e15i	1.59505i	−0.603283	+	0.797527i	$0.706141\pi$
$332$	0	0
$333$	−6.25016e14	−0.458380
$334$	0	0
$335$	3.33231e15i	2.35763i
$336$	0	0
$337$	−1.36274e15	−0.930321	−0.465160	−	0.885226i	$-0.654003\pi$
$337$	−1.36274e15	−0.930321	−0.465160	+	0.885226i	$0.654003\pi$
$338$	0	0
$339$	− 1.62910e15i	− 1.07337i
$340$	0	0
$341$	−1.95814e15	−1.24542
$342$	0	0
$343$	− 1.96516e14i	− 0.120679i
$344$	0	0
$345$	−1.93247e15	−1.14604
$346$	0	0
$347$	2.21109e15i	1.26657i	0.773917	+	0.633287i	$0.218295\pi$
$347$	2.21109e15i	1.26657i	−0.773917	+	0.633287i	$0.781705\pi$
$348$	0	0
$349$	−8.25554e14	−0.456870	−0.228435	−	0.973559i	$-0.573361\pi$
$349$	−8.25554e14	−0.456870	−0.228435	+	0.973559i	$0.573361\pi$
$350$	0	0
$351$	6.38305e14i	0.341339i
$352$	0	0
$353$	9.06854e14	0.468694	0.234347	−	0.972153i	$-0.424705\pi$
$353$	9.06854e14	0.468694	0.234347	+	0.972153i	$0.424705\pi$
$354$	0	0
$355$	8.87187e14i	0.443246i
$356$	0	0
$357$	2.02841e13	0.00979820
$358$	0	0
$359$	− 5.13417e14i	− 0.239830i	−0.992784	−	0.119915i	$-0.961738\pi$
$359$	− 5.13417e14i	− 0.239830i	0.992784	−	0.119915i	$-0.0382622\pi$
$360$	0	0
$361$	2.21290e15	0.999812
$362$	0	0
$363$	− 1.75222e14i	− 0.0765858i
$364$	0	0
$365$	−2.13096e15	−0.901192
$366$	0	0
$367$	6.68311e14i	0.273516i	0.990605	+	0.136758i	$0.0436682\pi$
$367$	6.68311e14i	0.273516i	−0.990605	+	0.136758i	$0.956332\pi$
$368$	0	0
$369$	−1.00294e15	−0.397297
$370$	0	0
$371$	− 1.14795e13i	− 0.00440229i
$372$	0	0
$373$	9.11568e13	0.0338483	0.0169241	−	0.999857i	$-0.494613\pi$
$373$	9.11568e13	0.0338483	0.0169241	+	0.999857i	$0.494613\pi$
$374$	0	0
$375$	3.14967e14i	0.113260i
$376$	0	0
$377$	3.13948e15	1.09348
$378$	0	0
$379$	− 3.53114e15i	− 1.19146i	−0.803184	−	0.595731i	$-0.796863\pi$
$379$	− 3.53114e15i	− 1.19146i	0.803184	−	0.595731i	$-0.203137\pi$
$380$	0	0
$381$	3.46649e15	1.13329
$382$	0	0
$383$	2.41943e15i	0.766515i	0.923642	+	0.383257i	$0.125198\pi$
$383$	2.41943e15i	0.766515i	−0.923642	+	0.383257i	$0.874802\pi$
$384$	0	0
$385$	−2.85430e14	−0.0876465
$386$	0	0
$387$	− 8.61697e14i	− 0.256501i
$388$	0	0
$389$	4.88362e15	1.40943	0.704717	−	0.709489i	$-0.251074\pi$
$389$	4.88362e15	1.40943	0.704717	+	0.709489i	$0.251074\pi$
$390$	0	0
$391$	− 1.46164e15i	− 0.409053i
$392$	0	0
$393$	−6.80125e14	−0.184601
$394$	0	0
$395$	− 4.23768e15i	− 1.11569i
$396$	0	0
$397$	−3.63632e15	−0.928792	−0.464396	−	0.885628i	$-0.653729\pi$
$397$	−3.63632e15	−0.928792	−0.464396	+	0.885628i	$0.653729\pi$
$398$	0	0
$399$	− 1.93291e12i	0 0.000479043i
$400$	0	0
$401$	2.72304e14	0.0654919	0.0327459	−	0.999464i	$-0.489575\pi$
$401$	2.72304e14	0.0654919	0.0327459	+	0.999464i	$0.489575\pi$
$402$	0	0
$403$	− 8.89134e15i	− 2.07557i
$404$	0	0
$405$	−6.68001e14	−0.151373
$406$	0	0
$407$	6.65214e15i	1.46351i
$408$	0	0
$409$	1.47132e15	0.314317	0.157158	−	0.987573i	$-0.449767\pi$
$409$	1.47132e15	0.314317	0.157158	+	0.987573i	$0.449767\pi$
$410$	0	0
$411$	− 1.80857e14i	− 0.0375218i
$412$	0	0
$413$	9.12988e13	0.0183977
$414$	0	0
$415$	4.91065e15i	0.961281i
$416$	0	0
$417$	7.16906e14	0.136347
$418$	0	0
$419$	− 6.30146e15i	− 1.16455i	−0.812993	−	0.582273i	$-0.802163\pi$
$419$	− 6.30146e15i	− 1.16455i	0.812993	−	0.582273i	$-0.197837\pi$
$420$	0	0
$421$	3.25314e15	0.584266	0.292133	−	0.956378i	$-0.405635\pi$
$421$	3.25314e15	0.584266	0.292133	+	0.956378i	$0.405635\pi$
$422$	0	0
$423$	3.34860e15i	0.584550i
$424$	0	0
$425$	1.41618e15	0.240318
$426$	0	0
$427$	− 2.75174e14i	− 0.0453983i
$428$	0	0
$429$	6.79358e15	1.08982
$430$	0	0
$431$	− 1.03071e16i	− 1.60795i	−0.594660	−	0.803977i	$-0.702713\pi$
$431$	− 1.03071e16i	− 1.60795i	0.594660	−	0.803977i	$-0.297287\pi$
$432$	0	0
$433$	9.64406e15	1.46330	0.731648	−	0.681682i	$-0.238751\pi$
$433$	9.64406e15	1.46330	0.731648	+	0.681682i	$0.238751\pi$
$434$	0	0
$435$	3.28554e15i	0.484921i
$436$	0	0
$437$	−1.39283e14	−0.0199990
$438$	0	0
$439$	3.83012e15i	0.535087i	0.963546	+	0.267544i	$0.0862120\pi$
$439$	3.83012e15i	0.535087i	−0.963546	+	0.267544i	$0.913788\pi$
$440$	0	0
$441$	−2.44298e15	−0.332115
$442$	0	0
$443$	− 6.98519e15i	− 0.924179i	−0.886833	−	0.462089i	$-0.847100\pi$
$443$	− 6.98519e15i	− 0.924179i	0.886833	−	0.462089i	$-0.152900\pi$
$444$	0	0
$445$	1.00753e16	1.29747
$446$	0	0
$447$	− 2.42187e15i	− 0.303602i
$448$	0	0
$449$	−6.54024e15	−0.798207	−0.399103	−	0.916906i	$-0.630679\pi$
$449$	−6.54024e15	−0.798207	−0.399103	+	0.916906i	$0.630679\pi$
$450$	0	0
$451$	1.06744e16i	1.26848i
$452$	0	0
$453$	−7.29969e15	−0.844725
$454$	0	0
$455$	− 1.29605e15i	− 0.146068i
$456$	0	0
$457$	−3.93711e15	−0.432196	−0.216098	−	0.976372i	$-0.569333\pi$
$457$	−3.93711e15	−0.432196	−0.216098	+	0.976372i	$0.569333\pi$
$458$	0	0
$459$	− 5.05247e14i	− 0.0540291i
$460$	0	0
$461$	−6.78296e15	−0.706665	−0.353333	−	0.935498i	$-0.614952\pi$
$461$	−6.78296e15	−0.706665	−0.353333	+	0.935498i	$0.614952\pi$
$462$	0	0
$463$	1.21932e16i	1.23774i	0.785492	+	0.618872i	$0.212410\pi$
$463$	1.21932e16i	1.23774i	−0.785492	+	0.618872i	$0.787590\pi$
$464$	0	0
$465$	9.30499e15	0.920446
$466$	0	0
$467$	9.69994e14i	0.0935121i	0.998906	+	0.0467560i	$0.0148883\pi$
$467$	9.69994e14i	0.0935121i	−0.998906	+	0.0467560i	$0.985112\pi$
$468$	0	0
$469$	−1.11332e15	−0.104613
$470$	0	0
$471$	4.97750e15i	0.455917i
$472$	0	0
$473$	−9.17117e15	−0.818951
$474$	0	0
$475$	− 1.34951e14i	− 0.0117494i
$476$	0	0
$477$	−2.85937e14	−0.0242750
$478$	0	0
$479$	− 6.76625e15i	− 0.560190i	−0.959972	−	0.280095i	$-0.909634\pi$
$479$	− 6.76625e15i	− 0.560190i	0.959972	−	0.280095i	$-0.0903659\pi$
$480$	0	0
$481$	−3.02054e16	−2.43901
$482$	0	0
$483$	− 6.45639e14i	− 0.0508518i
$484$	0	0
$485$	−1.97776e16	−1.51958
$486$	0	0
$487$	1.15311e16i	0.864367i	0.901786	+	0.432184i	$0.142257\pi$
$487$	1.15311e16i	0.864367i	−0.901786	+	0.432184i	$0.857743\pi$
$488$	0	0
$489$	−7.25385e15	−0.530537
$490$	0	0
$491$	− 1.32213e16i	− 0.943596i	−0.881707	−	0.471798i	$-0.843605\pi$
$491$	− 1.32213e16i	− 0.943596i	0.881707	−	0.471798i	$-0.156395\pi$
$492$	0	0
$493$	−2.48504e15	−0.173082
$494$	0	0
$495$	7.10964e15i	0.483299i
$496$	0	0
$497$	−2.96409e14	−0.0196677
$498$	0	0
$499$	− 4.97689e15i	− 0.322370i	−0.986924	−	0.161185i	$-0.948468\pi$
$499$	− 4.97689e15i	− 0.322370i	0.986924	−	0.161185i	$-0.0515316\pi$
$500$	0	0
$501$	1.36892e16	0.865672
$502$	0	0
$503$	1.01640e16i	0.627562i	0.949495	+	0.313781i	$0.101596\pi$
$503$	1.01640e16i	0.627562i	−0.949495	+	0.313781i	$0.898404\pi$
$504$	0	0
$505$	9.16383e15	0.552496
$506$	0	0
$507$	2.10417e16i	1.23889i
$508$	0	0
$509$	−1.39231e16	−0.800626	−0.400313	−	0.916379i	$-0.631099\pi$
$509$	−1.39231e16	−0.800626	−0.400313	+	0.916379i	$0.631099\pi$
$510$	0	0
$511$	− 7.11951e14i	− 0.0399876i
$512$	0	0
$513$	−4.81460e13	−0.00264153
$514$	0	0
$515$	− 2.64913e16i	− 1.41991i
$516$	0	0
$517$	3.56397e16	1.86634
$518$	0	0
$519$	1.24147e16i	0.635230i
$520$	0	0
$521$	1.75513e16	0.877572	0.438786	−	0.898592i	$-0.355409\pi$
$521$	1.75513e16	0.877572	0.438786	+	0.898592i	$0.355409\pi$
$522$	0	0
$523$	− 1.65340e16i	− 0.807919i	−0.914777	−	0.403959i	$-0.867634\pi$
$523$	− 1.65340e16i	− 0.807919i	0.914777	−	0.403959i	$-0.132366\pi$
$524$	0	0
$525$	6.25559e14	0.0298753
$526$	0	0
$527$	7.03790e15i	0.328533i
$528$	0	0
$529$	−2.46091e16	−1.12295
$530$	0	0
$531$	− 2.27412e15i	− 0.101449i
$532$	0	0
$533$	−4.84693e16	−2.11399
$534$	0	0
$535$	7.24767e15i	0.309083i
$536$	0	0
$537$	9.14697e15	0.381445
$538$	0	0
$539$	2.60010e16i	1.06037i
$540$	0	0
$541$	−3.40764e16	−1.35916	−0.679580	−	0.733602i	$-0.737838\pi$
$541$	−3.40764e16	−1.35916	−0.679580	+	0.733602i	$0.737838\pi$
$542$	0	0
$543$	− 5.25211e14i	− 0.0204897i
$544$	0	0
$545$	7.06482e16	2.69601
$546$	0	0
$547$	3.02110e16i	1.12782i	0.825835	+	0.563912i	$0.190704\pi$
$547$	3.02110e16i	1.12782i	−0.825835	+	0.563912i	$0.809296\pi$
$548$	0	0
$549$	−6.85417e15	−0.250335
$550$	0	0
$551$	2.36805e14i	0.00846215i
$552$	0	0
$553$	1.41581e15	0.0495054
$554$	0	0
$555$	− 3.16107e16i	− 1.08162i
$556$	0	0
$557$	1.86821e16	0.625596	0.312798	−	0.949820i	$-0.398734\pi$
$557$	1.86821e16	0.625596	0.312798	+	0.949820i	$0.398734\pi$
$558$	0	0
$559$	− 4.16436e16i	− 1.36483i
$560$	0	0
$561$	−5.37742e15	−0.172503
$562$	0	0
$563$	− 1.77453e16i	− 0.557228i	−0.960403	−	0.278614i	$-0.910125\pi$
$563$	− 1.77453e16i	− 0.557228i	0.960403	−	0.278614i	$-0.0898750\pi$
$564$	0	0
$565$	8.23931e16	2.53280
$566$	0	0
$567$	− 2.23179e14i	− 0.00671668i
$568$	0	0
$569$	−2.04279e16	−0.601935	−0.300968	−	0.953634i	$-0.597310\pi$
$569$	−2.04279e16	−0.601935	−0.300968	+	0.953634i	$0.597310\pi$
$570$	0	0
$571$	5.67402e15i	0.163709i	0.996644	+	0.0818547i	$0.0260844\pi$
$571$	5.67402e15i	0.163709i	−0.996644	+	0.0818547i	$0.973916\pi$
$572$	0	0
$573$	2.74500e16	0.775557
$574$	0	0
$575$	− 4.50768e16i	− 1.24723i
$576$	0	0
$577$	1.70028e16	0.460750	0.230375	−	0.973102i	$-0.426005\pi$
$577$	1.70028e16	0.460750	0.230375	+	0.973102i	$0.426005\pi$
$578$	0	0
$579$	− 6.84418e15i	− 0.181656i
$580$	0	0
$581$	−1.64065e15	−0.0426538
$582$	0	0
$583$	3.04327e15i	0.0775049i
$584$	0	0
$585$	−3.22828e16	−0.805445
$586$	0	0
$587$	− 3.27992e16i	− 0.801742i	−0.916134	−	0.400871i	$-0.868708\pi$
$587$	− 3.27992e16i	− 0.801742i	0.916134	−	0.400871i	$-0.131292\pi$
$588$	0	0
$589$	6.70655e14	0.0160623
$590$	0	0
$591$	− 3.65995e16i	− 0.858914i
$592$	0	0
$593$	6.54880e16	1.50603	0.753015	−	0.658003i	$-0.228599\pi$
$593$	6.54880e16	1.50603	0.753015	+	0.658003i	$0.228599\pi$
$594$	0	0
$595$	1.02588e15i	0.0231205i
$596$	0	0
$597$	−4.33868e15	−0.0958322
$598$	0	0
$599$	− 1.46157e16i	− 0.316417i	−0.987406	−	0.158209i	$-0.949428\pi$
$599$	− 1.46157e16i	− 0.316417i	0.987406	−	0.158209i	$-0.0505718\pi$
$600$	0	0
$601$	3.63481e16	0.771320	0.385660	−	0.922641i	$-0.373974\pi$
$601$	3.63481e16	0.771320	0.385660	+	0.922641i	$0.373974\pi$
$602$	0	0
$603$	2.77312e16i	0.576853i
$604$	0	0
$605$	8.86197e15	0.180717
$606$	0	0
$607$	− 4.19339e16i	− 0.838364i	−0.907902	−	0.419182i	$-0.862317\pi$
$607$	− 4.19339e16i	− 0.838364i	0.907902	−	0.419182i	$-0.137683\pi$
$608$	0	0
$609$	−1.09770e15	−0.0215169
$610$	0	0
$611$	1.61829e17i	3.11036i
$612$	0	0
$613$	−2.03819e16	−0.384134	−0.192067	−	0.981382i	$-0.561519\pi$
$613$	−2.03819e16	−0.384134	−0.192067	+	0.981382i	$0.561519\pi$
$614$	0	0
$615$	− 5.07243e16i	− 0.937487i
$616$	0	0
$617$	−3.74912e16	−0.679545	−0.339772	−	0.940508i	$-0.610350\pi$
$617$	−3.74912e16	−0.679545	−0.339772	+	0.940508i	$0.610350\pi$
$618$	0	0
$619$	2.18233e16i	0.387951i	0.981006	+	0.193975i	$0.0621382\pi$
$619$	2.18233e16i	0.387951i	−0.981006	+	0.193975i	$0.937862\pi$
$620$	0	0
$621$	−1.60819e16	−0.280407
$622$	0	0
$623$	3.36615e15i	0.0575712i
$624$	0	0
$625$	−6.69516e16	−1.12326
$626$	0	0
$627$	5.12426e14i	0.00843384i
$628$	0	0
$629$	2.39089e16	0.386062
$630$	0	0
$631$	− 4.54812e16i	− 0.720537i	−0.932849	−	0.360268i	$-0.882685\pi$
$631$	− 4.54812e16i	− 0.720537i	0.932849	−	0.360268i	$-0.117315\pi$
$632$	0	0
$633$	−5.10695e16	−0.793851
$634$	0	0
$635$	1.75320e17i	2.67418i
$636$	0	0
$637$	−1.18063e17	−1.76717
$638$	0	0
$639$	7.38311e15i	0.108451i
$640$	0	0
$641$	7.57002e16	1.09131	0.545656	−	0.838009i	$-0.316281\pi$
$641$	7.57002e16	1.09131	0.545656	+	0.838009i	$0.316281\pi$
$642$	0	0
$643$	− 5.74737e16i	− 0.813212i	−0.913604	−	0.406606i	$-0.866712\pi$
$643$	− 5.74737e16i	− 0.813212i	0.913604	−	0.406606i	$-0.133288\pi$
$644$	0	0
$645$	4.35810e16	0.605255
$646$	0	0
$647$	5.03250e16i	0.686053i	0.939326	+	0.343027i	$0.111452\pi$
$647$	5.03250e16i	0.686053i	−0.939326	+	0.343027i	$0.888548\pi$
$648$	0	0
$649$	−2.42038e16	−0.323903
$650$	0	0
$651$	3.10879e15i	0.0408419i
$652$	0	0
$653$	1.84738e15	0.0238275	0.0119137	−	0.999929i	$-0.496208\pi$
$653$	1.84738e15	0.0238275	0.0119137	+	0.999929i	$0.496208\pi$
$654$	0	0
$655$	− 3.43979e16i	− 0.435596i
$656$	0	0
$657$	−1.77337e16	−0.220499
$658$	0	0
$659$	− 8.93078e16i	− 1.09038i	−0.838313	−	0.545189i	$-0.816458\pi$
$659$	− 8.93078e16i	− 1.09038i	0.838313	−	0.545189i	$-0.183542\pi$
$660$	0	0
$661$	1.13618e16	0.136219	0.0681095	−	0.997678i	$-0.478303\pi$
$661$	1.13618e16	0.136219	0.0681095	+	0.997678i	$0.478303\pi$
$662$	0	0
$663$	− 2.44173e16i	− 0.287486i
$664$	0	0
$665$	9.77586e13	0.00113038
$666$	0	0
$667$	7.90984e16i	0.898282i
$668$	0	0
$669$	6.41797e16	0.715881
$670$	0	0
$671$	7.29500e16i	0.799264i
$672$	0	0
$673$	−1.40873e17	−1.51614	−0.758069	−	0.652174i	$-0.773857\pi$
$673$	−1.40873e17	−1.51614	−0.758069	+	0.652174i	$0.773857\pi$
$674$	0	0
$675$	− 1.55818e16i	− 0.164738i
$676$	0	0
$677$	−1.06344e17	−1.10454	−0.552268	−	0.833667i	$-0.686238\pi$
$677$	−1.06344e17	−1.10454	−0.552268	+	0.833667i	$0.686238\pi$
$678$	0	0
$679$	− 6.60769e15i	− 0.0674265i
$680$	0	0
$681$	−1.09823e17	−1.10106
$682$	0	0
$683$	7.82367e16i	0.770702i	0.922770	+	0.385351i	$0.125920\pi$
$683$	7.82367e16i	0.770702i	−0.922770	+	0.385351i	$0.874080\pi$
$684$	0	0
$685$	9.14697e15	0.0885389
$686$	0	0
$687$	4.67258e16i	0.444443i
$688$	0	0
$689$	−1.38186e16	−0.129166
$690$	0	0
$691$	− 1.63249e17i	− 1.49963i	−0.661650	−	0.749813i	$-0.730143\pi$
$691$	− 1.63249e17i	− 1.49963i	0.661650	−	0.749813i	$-0.269857\pi$
$692$	0	0
$693$	−2.37533e15	−0.0214449
$694$	0	0
$695$	3.62581e16i	0.321733i
$696$	0	0
$697$	3.83657e16	0.334616
$698$	0	0
$699$	− 6.25845e16i	− 0.536542i
$700$	0	0
$701$	−1.46995e17	−1.23878	−0.619391	−	0.785083i	$-0.712620\pi$
$701$	−1.46995e17	−1.23878	−0.619391	+	0.785083i	$0.712620\pi$
$702$	0	0
$703$	− 2.27833e15i	− 0.0188749i
$704$	0	0
$705$	−1.69358e17	−1.37934
$706$	0	0
$707$	3.06163e15i	0.0245153i
$708$	0	0
$709$	5.55937e16	0.437672	0.218836	−	0.975762i	$-0.429774\pi$
$709$	5.55937e16	0.437672	0.218836	+	0.975762i	$0.429774\pi$
$710$	0	0
$711$	− 3.52657e16i	− 0.272982i
$712$	0	0
$713$	2.24015e17	1.70506
$714$	0	0
$715$	3.43590e17i	2.57161i
$716$	0	0
$717$	1.37038e17	1.00862
$718$	0	0
$719$	− 1.11386e16i	− 0.0806229i	−0.999187	−	0.0403115i	$-0.987165\pi$
$719$	− 1.11386e16i	− 0.0806229i	0.999187	−	0.0403115i	$-0.0128350\pi$
$720$	0	0
$721$	8.85072e15	0.0630038
$722$	0	0
$723$	− 6.16194e16i	− 0.431408i
$724$	0	0
$725$	−7.66384e16	−0.527738
$726$	0	0
$727$	− 4.35822e16i	− 0.295191i	−0.989048	−	0.147595i	$-0.952847\pi$
$727$	− 4.35822e16i	− 0.295191i	0.989048	−	0.147595i	$-0.0471533\pi$
$728$	0	0
$729$	−5.55906e15	−0.0370370
$730$	0	0
$731$	3.29628e16i	0.216033i
$732$	0	0
$733$	1.02849e17	0.663095	0.331547	−	0.943439i	$-0.392429\pi$
$733$	1.02849e17	0.663095	0.331547	+	0.943439i	$0.392429\pi$
$734$	0	0
$735$	− 1.23556e17i	− 0.783680i
$736$	0	0
$737$	2.95148e17	1.84177
$738$	0	0
$739$	− 2.89187e16i	− 0.177546i	−0.996052	−	0.0887732i	$-0.971705\pi$
$739$	− 2.89187e16i	− 0.177546i	0.996052	−	0.0887732i	$-0.0282946\pi$
$740$	0	0
$741$	−2.32677e15	−0.0140555
$742$	0	0
$743$	− 1.84464e17i	− 1.09642i	−0.836339	−	0.548212i	$-0.815309\pi$
$743$	− 1.84464e17i	− 1.09642i	0.836339	−	0.548212i	$-0.184691\pi$
$744$	0	0
$745$	1.22488e17	0.716399
$746$	0	0
$747$	4.08661e16i	0.235201i
$748$	0	0
$749$	−2.42144e15	−0.0137146
$750$	0	0
$751$	− 1.07917e17i	− 0.601522i	−0.953700	−	0.300761i	$-0.902759\pi$
$751$	− 1.07917e17i	− 0.601522i	0.953700	−	0.300761i	$-0.0972406\pi$
$752$	0	0
$753$	1.92027e17	1.05340
$754$	0	0
$755$	− 3.69188e17i	− 1.99327i
$756$	0	0
$757$	2.20280e17	1.17058	0.585288	−	0.810825i	$-0.300981\pi$
$757$	2.20280e17	1.17058	0.585288	+	0.810825i	$0.300981\pi$
$758$	0	0
$759$	1.71162e17i	0.895277i
$760$	0	0
$761$	1.15648e17	0.595430	0.297715	−	0.954655i	$-0.403775\pi$
$761$	1.15648e17	0.595430	0.297715	+	0.954655i	$0.403775\pi$
$762$	0	0
$763$	2.36035e16i	0.119627i
$764$	0	0
$765$	2.55533e16	0.127491
$766$	0	0
$767$	− 1.09902e17i	− 0.539803i
$768$	0	0
$769$	−9.40988e14	−0.00455016	−0.00227508	−	0.999997i	$-0.500724\pi$
$769$	−9.40988e14	−0.00455016	−0.00227508	+	0.999997i	$0.500724\pi$
$770$	0	0
$771$	1.97961e17i	0.942438i
$772$	0	0
$773$	−2.12581e17	−0.996434	−0.498217	−	0.867052i	$-0.666012\pi$
$773$	−2.12581e17	−0.996434	−0.498217	+	0.867052i	$0.666012\pi$
$774$	0	0
$775$	2.17048e17i	1.00172i
$776$	0	0
$777$	1.05611e16	0.0479936
$778$	0	0
$779$	− 3.65594e15i	− 0.0163597i
$780$	0	0
$781$	7.85796e16	0.346261
$782$	0	0
$783$	2.73421e16i	0.118648i
$784$	0	0
$785$	−2.51741e17	−1.07581
$786$	0	0
$787$	− 3.88757e17i	− 1.63617i	−0.575095	−	0.818087i	$-0.695035\pi$
$787$	− 3.88757e17i	− 1.63617i	0.575095	−	0.818087i	$-0.304965\pi$
$788$	0	0
$789$	−5.97919e16	−0.247845
$790$	0	0
$791$	2.75275e16i	0.112385i
$792$	0	0
$793$	−3.31245e17	−1.33202
$794$	0	0
$795$	− 1.44615e16i	− 0.0572809i
$796$	0	0
$797$	−2.11775e16	−0.0826276	−0.0413138	−	0.999146i	$-0.513154\pi$
$797$	−2.11775e16	−0.0826276	−0.0413138	+	0.999146i	$0.513154\pi$
$798$	0	0
$799$	− 1.28095e17i	− 0.492326i
$800$	0	0
$801$	8.38459e16	0.317458
$802$	0	0
$803$	1.88742e17i	0.704005i
$804$	0	0
$805$	3.26537e16	0.119993
$806$	0	0
$807$	1.24159e17i	0.449509i
$808$	0	0
$809$	2.74837e17	0.980357	0.490179	−	0.871622i	$-0.336932\pi$
$809$	2.74837e17	0.980357	0.490179	+	0.871622i	$0.336932\pi$
$810$	0	0
$811$	− 3.56743e17i	− 1.25381i	−0.779098	−	0.626903i	$-0.784322\pi$
$811$	− 3.56743e17i	− 1.25381i	0.779098	−	0.626903i	$-0.215678\pi$
$812$	0	0
$813$	1.98143e17	0.686176
$814$	0	0
$815$	− 3.66869e17i	− 1.25189i
$816$	0	0
$817$	3.14109e15	0.0105620
$818$	0	0
$819$	− 1.07857e16i	− 0.0357391i
$820$	0	0
$821$	−1.70191e17	−0.555748	−0.277874	−	0.960617i	$-0.589630\pi$
$821$	−1.70191e17	−0.555748	−0.277874	+	0.960617i	$0.589630\pi$
$822$	0	0
$823$	4.53033e17i	1.45791i	0.684562	+	0.728955i	$0.259994\pi$
$823$	4.53033e17i	1.45791i	−0.684562	+	0.728955i	$0.740006\pi$
$824$	0	0
$825$	−1.65839e17	−0.525973
$826$	0	0
$827$	− 4.07186e17i	− 1.27280i	−0.771360	−	0.636399i	$-0.780423\pi$
$827$	− 4.07186e17i	− 1.27280i	0.771360	−	0.636399i	$-0.219577\pi$
$828$	0	0
$829$	2.99377e17	0.922342	0.461171	−	0.887311i	$-0.347429\pi$
$829$	2.99377e17	0.922342	0.461171	+	0.887311i	$0.347429\pi$
$830$	0	0
$831$	1.74803e17i	0.530813i
$832$	0	0
$833$	9.34523e16	0.279718
$834$	0	0
$835$	6.92344e17i	2.04269i
$836$	0	0
$837$	7.74355e16	0.225210
$838$	0	0
$839$	6.54448e17i	1.87631i	0.346222	+	0.938153i	$0.387464\pi$
$839$	6.54448e17i	1.87631i	−0.346222	+	0.938153i	$0.612536\pi$
$840$	0	0
$841$	−2.19334e17	−0.619911
$842$	0	0
$843$	2.83403e17i	0.789659i
$844$	0	0
$845$	−1.06420e18	−2.92337
$846$	0	0
$847$	2.96078e15i	0.00801873i
$848$	0	0
$849$	1.55947e17	0.416419
$850$	0	0
$851$	− 7.61017e17i	− 2.00363i
$852$	0	0
$853$	9.94589e16	0.258196	0.129098	−	0.991632i	$-0.458792\pi$
$853$	9.94589e16	0.258196	0.129098	+	0.991632i	$0.458792\pi$
$854$	0	0
$855$	− 2.43502e15i	− 0.00623313i
$856$	0	0
$857$	1.81107e17	0.457142	0.228571	−	0.973527i	$-0.426595\pi$
$857$	1.81107e17	0.457142	0.228571	+	0.973527i	$0.426595\pi$
$858$	0	0
$859$	2.11062e17i	0.525353i	0.964884	+	0.262677i	$0.0846052\pi$
$859$	2.11062e17i	0.525353i	−0.964884	+	0.262677i	$0.915395\pi$
$860$	0	0
$861$	1.69470e16	0.0415980
$862$	0	0
$863$	− 3.64870e17i	− 0.883229i	−0.897205	−	0.441614i	$-0.854406\pi$
$863$	− 3.64870e17i	− 0.883229i	0.897205	−	0.441614i	$-0.145594\pi$
$864$	0	0
$865$	−6.27881e17	−1.49893
$866$	0	0
$867$	− 2.25892e17i	− 0.531845i
$868$	0	0
$869$	−3.75338e17	−0.871573
$870$	0	0
$871$	1.34018e18i	3.06940i
$872$	0	0
$873$	−1.64588e17	−0.371803
$874$	0	0
$875$	− 5.32211e15i	− 0.0118587i
$876$	0	0
$877$	2.20060e17	0.483663	0.241832	−	0.970318i	$-0.422252\pi$
$877$	2.20060e17	0.483663	0.241832	+	0.970318i	$0.422252\pi$
$878$	0	0
$879$	1.07945e17i	0.234029i
$880$	0	0
$881$	−5.23854e17	−1.12035	−0.560176	−	0.828373i	$-0.689267\pi$
$881$	−5.23854e17	−1.12035	−0.560176	+	0.828373i	$0.689267\pi$
$882$	0	0
$883$	− 5.19490e17i	− 1.09601i	−0.836476	−	0.548003i	$-0.815388\pi$
$883$	− 5.19490e17i	− 1.09601i	0.836476	−	0.548003i	$-0.184612\pi$
$884$	0	0
$885$	1.15015e17	0.239385
$886$	0	0
$887$	6.43189e17i	1.32068i	0.750968	+	0.660339i	$0.229587\pi$
$887$	6.43189e17i	1.32068i	−0.750968	+	0.660339i	$0.770413\pi$
$888$	0	0
$889$	−5.85745e16	−0.118658
$890$	0	0
$891$	5.91659e16i	0.118251i
$892$	0	0
$893$	−1.22065e16	−0.0240703
$894$	0	0
$895$	4.62615e17i	0.900081i
$896$	0	0
$897$	−7.77198e17	−1.49203
$898$	0	0
$899$	− 3.80864e17i	− 0.721459i
$900$	0	0
$901$	1.09380e16	0.0204452
$902$	0	0
$903$	1.45604e16i	0.0268563i
$904$	0	0
$905$	2.65630e16	0.0483487
$906$	0	0
$907$	− 8.20905e17i	− 1.47452i	−0.675612	−	0.737258i	$-0.736120\pi$
$907$	− 8.20905e17i	− 1.47452i	0.675612	−	0.737258i	$-0.263880\pi$
$908$	0	0
$909$	7.62608e16	0.135182
$910$	0	0
$911$	− 7.92986e16i	− 0.138725i	−0.997592	−	0.0693625i	$-0.977903\pi$
$911$	− 7.92986e16i	− 0.138725i	0.997592	−	0.0693625i	$-0.0220965\pi$
$912$	0	0
$913$	4.34944e17	0.750946
$914$	0	0
$915$	− 3.46655e17i	− 0.590706i
$916$	0	0
$917$	1.14923e16	0.0193282
$918$	0	0
$919$	− 7.38065e17i	− 1.22518i	−0.790399	−	0.612592i	$-0.790127\pi$
$919$	− 7.38065e17i	− 1.22518i	0.790399	−	0.612592i	$-0.209873\pi$
$920$	0	0
$921$	3.47041e17	0.568622
$922$	0	0
$923$	3.56807e17i	0.577063i
$924$	0	0
$925$	7.37349e17	1.17713
$926$	0	0
$927$	− 2.20459e17i	− 0.347415i
$928$	0	0
$929$	1.11243e18	1.73052	0.865261	−	0.501322i	$-0.167153\pi$
$929$	1.11243e18	1.73052	0.865261	+	0.501322i	$0.167153\pi$
$930$	0	0
$931$	− 8.90526e15i	− 0.0136757i
$932$	0	0
$933$	3.35452e17	0.508558
$934$	0	0
$935$	− 2.71967e17i	− 0.407049i
$936$	0	0
$937$	−3.87919e17	−0.573197	−0.286598	−	0.958051i	$-0.592525\pi$
$937$	−3.87919e17	−0.573197	−0.286598	+	0.958051i	$0.592525\pi$
$938$	0	0
$939$	7.57508e16i	0.110508i
$940$	0	0
$941$	3.15087e17	0.453829	0.226914	−	0.973915i	$-0.427136\pi$
$941$	3.15087e17	0.453829	0.226914	+	0.973915i	$0.427136\pi$
$942$	0	0
$943$	− 1.22117e18i	− 1.73663i
$944$	0	0
$945$	1.12874e16	0.0158491
$946$	0	0
$947$	3.79195e16i	0.0525730i	0.999654	+	0.0262865i	$0.00836821\pi$
$947$	3.79195e16i	0.0525730i	−0.999654	+	0.0262865i	$0.991632\pi$
$948$	0	0
$949$	−8.57023e17	−1.17326
$950$	0	0
$951$	6.90215e17i	0.933043i
$952$	0	0
$953$	6.56690e17	0.876603	0.438302	−	0.898828i	$-0.355580\pi$
$953$	6.56690e17	0.876603	0.438302	+	0.898828i	$0.355580\pi$
$954$	0	0
$955$	1.38830e18i	1.83005i
$956$	0	0
$957$	2.91006e17	0.378817
$958$	0	0
$959$	3.05600e15i	0.00392864i
$960$	0	0
$961$	−2.90984e17	−0.369427
$962$	0	0
$963$	6.03146e16i	0.0756249i
$964$	0	0
$965$	3.46150e17	0.428647
$966$	0	0
$967$	− 6.24136e17i	− 0.763345i	−0.924298	−	0.381672i	$-0.875348\pi$
$967$	− 6.24136e17i	− 0.763345i	0.924298	−	0.381672i	$-0.124652\pi$
$968$	0	0
$969$	1.84175e15	0.00222478
$970$	0	0
$971$	9.04996e17i	1.07977i	0.841739	+	0.539885i	$0.181532\pi$
$971$	9.04996e17i	1.07977i	−0.841739	+	0.539885i	$0.818468\pi$
$972$	0	0
$973$	−1.21138e16	−0.0142759
$974$	0	0
$975$	− 7.53027e17i	− 0.876562i
$976$	0	0
$977$	−1.37695e17	−0.158326	−0.0791628	−	0.996862i	$-0.525225\pi$
$977$	−1.37695e17	−0.158326	−0.0791628	+	0.996862i	$0.525225\pi$
$978$	0	0
$979$	− 8.92384e17i	− 1.01357i
$980$	0	0
$981$	5.87930e17	0.659647
$982$	0	0
$983$	1.06312e18i	1.17831i	0.808018	+	0.589157i	$0.200540\pi$
$983$	1.06312e18i	1.17831i	−0.808018	+	0.589157i	$0.799460\pi$
$984$	0	0
$985$	1.85105e18	2.02675
$986$	0	0
$987$	− 5.65825e16i	− 0.0612039i
$988$	0	0
$989$	1.04920e18	1.12119
$990$	0	0
$991$	− 9.28788e17i	− 0.980561i	−0.871565	−	0.490281i	$-0.836894\pi$
$991$	− 9.28788e17i	− 0.980561i	0.871565	−	0.490281i	$-0.163106\pi$
$992$	0	0
$993$	−8.82898e17	−0.920905
$994$	0	0
$995$	− 2.19432e17i	− 0.226132i
$996$	0	0
$997$	−9.52810e17	−0.970142	−0.485071	−	0.874475i	$-0.661206\pi$
$997$	−9.52810e17	−0.970142	−0.485071	+	0.874475i	$0.661206\pi$
$998$	0	0
$999$	− 2.63062e17i	− 0.264646i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.13.g.a.31.3	yes	4
3.2	odd	2		144.13.g.h.127.3		4
4.3	odd	2	inner	48.13.g.a.31.1	✓	4
8.3	odd	2		192.13.g.c.127.4		4
8.5	even	2		192.13.g.c.127.2		4
12.11	even	2		144.13.g.h.127.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.13.g.a.31.1	✓	4	4.3	odd	2	inner
48.13.g.a.31.3	yes	4	1.1	even	1	trivial
144.13.g.h.127.3		4	3.2	odd	2
144.13.g.h.127.4		4	12.11	even	2
192.13.g.c.127.2		4	8.5	even	2
192.13.g.c.127.4		4	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	48.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+420.888i q^{3} -21286.8 q^{5} -7111.90i q^{7} -177147. q^{9} +O(q^{10})\)	\(q+420.888i q^{3} -21286.8 q^{5} -7111.90i q^{7} -177147. q^{9} +1.88540e6i q^{11} -8.56106e6 q^{13} -8.95935e6i q^{15} +6.77647e6 q^{17} -645743. i q^{19} +2.99331e6 q^{21} -2.15694e8i q^{23} +2.08985e8 q^{25} -7.45591e7i q^{27} -3.66716e8 q^{29} +1.03858e9i q^{31} -7.93544e8 q^{33} +1.51389e8i q^{35} +3.52823e9 q^{37} -3.60325e9i q^{39} +5.66160e9 q^{41} +4.86430e9i q^{43} +3.77089e9 q^{45} -1.89030e10i q^{47} +1.37907e10 q^{49} +2.85214e9i q^{51} +1.61412e9 q^{53} -4.01341e10i q^{55} +2.71786e8 q^{57} +1.28375e10i q^{59} +3.86920e10 q^{61} +1.25985e9i q^{63} +1.82237e11 q^{65} -1.56544e11i q^{67} +9.07829e10 q^{69} -4.16779e10i q^{71} +1.00107e11 q^{73} +8.79596e10i q^{75} +1.34088e10 q^{77} +1.99076e11i q^{79} +3.13811e10 q^{81} -2.30690e11i q^{83} -1.44249e11 q^{85} -1.54347e11i q^{87} -4.73312e11 q^{89} +6.08854e10i q^{91} -4.37126e11 q^{93} +1.37458e10i q^{95} +9.29103e11 q^{97} -3.33993e11i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 21960 q^{5} - 708588 q^{9}+O(q^{10}) \) 4 * q - 21960 * q^5 - 708588 * q^9	\( 4 q - 21960 q^{5} - 708588 q^{9} - 5810072 q^{13} - 33882264 q^{17} - 15664752 q^{21} + 142148300 q^{25} + 42583896 q^{29} + 271362960 q^{33} + 9402865736 q^{37} + 17968882536 q^{41} + 3890148120 q^{45} + 53940845764 q^{49} + 60546956760 q^{53} + 46910963376 q^{57} + 168287201672 q^{61} + 481064975280 q^{65} + 163198209024 q^{69} + 789026629000 q^{73} + 140389989696 q^{77} + 125524238436 q^{81} - 777401136720 q^{85} - 638670460536 q^{89} - 607412656080 q^{93} - 563542043000 q^{97}+O(q^{100}) \) 4 * q - 21960 * q^5 - 708588 * q^9 - 5810072 * q^13 - 33882264 * q^17 - 15664752 * q^21 + 142148300 * q^25 + 42583896 * q^29 + 271362960 * q^33 + 9402865736 * q^37 + 17968882536 * q^41 + 3890148120 * q^45 + 53940845764 * q^49 + 60546956760 * q^53 + 46910963376 * q^57 + 168287201672 * q^61 + 481064975280 * q^65 + 163198209024 * q^69 + 789026629000 * q^73 + 140389989696 * q^77 + 125524238436 * q^81 - 777401136720 * q^85 - 638670460536 * q^89 - 607412656080 * q^93 - 563542043000 * q^97

Introduction

L-functions

Modular forms

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