LMFDB - Embedded newform 2432.2.c.f.1217.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2432,2,Mod(1217,2432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2432, 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("2432.1217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2432 = 2^{7} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2432.c (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:	$19.4196177716$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3182656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{3} + 25x^{2} - 10x + 2 $ x^6 - 2x^3 + 25x^2 - 10*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.2
Root			$-1.67298 - 1.67298i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.f.1217.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59774i q^{3} +1.59774i q^{5} -4.34596 q^{7} -3.74823 q^{9} +O(q^{10})$	$q-2.59774i q^{3} +1.59774i q^{5} -4.34596 q^{7} -3.74823 q^{9} +1.74823i q^{11} -0.748228i q^{13} +4.15049 q^{15} +3.00000 q^{17} +1.00000i q^{19} +11.2897i q^{21} +1.94370 q^{23} +2.44724 q^{25} +1.94370i q^{27} +7.28966i q^{29} +0.654037 q^{31} +4.54143 q^{33} -6.94370i q^{35} +3.84951i q^{37} -1.94370 q^{39} -4.54143 q^{41} -10.4402i q^{43} -5.98868i q^{45} +7.59774 q^{47} +11.8874 q^{49} -7.79321i q^{51} +6.59774i q^{53} -2.79321 q^{55} +2.59774 q^{57} -6.74823i q^{59} -9.09419i q^{61} +16.2897 q^{63} +1.19547 q^{65} -7.40226i q^{67} -5.04921i q^{69} +1.69901 q^{71} +16.6848 q^{73} -6.35729i q^{75} -7.59774i q^{77} +0.654037 q^{79} -6.19547 q^{81} +12.5793i q^{83} +4.79321i q^{85} +18.9366 q^{87} +14.5414 q^{89} +3.25177i q^{91} -1.69901i q^{93} -1.59774 q^{95} -10.5793 q^{97} -6.55276i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{7} - 16 q^{9}+O(q^{10}) $ 6 * q - 6 * q^7 - 16 * q^9	$ 6 q - 6 q^{7} - 16 q^{9} + 32 q^{15} + 18 q^{17} - 22 q^{23} - 6 q^{25} + 24 q^{31} - 20 q^{33} + 22 q^{39} + 20 q^{41} + 32 q^{47} + 4 q^{49} + 24 q^{55} + 2 q^{57} + 44 q^{63} - 20 q^{65} - 4 q^{71} + 34 q^{73} + 24 q^{79} - 10 q^{81} + 54 q^{87} + 40 q^{89} + 4 q^{95} + 44 q^{97}+O(q^{100}) $ 6 * q - 6 * q^7 - 16 * q^9 + 32 * q^15 + 18 * q^17 - 22 * q^23 - 6 * q^25 + 24 * q^31 - 20 * q^33 + 22 * q^39 + 20 * q^41 + 32 * q^47 + 4 * q^49 + 24 * q^55 + 2 * q^57 + 44 * q^63 - 20 * q^65 - 4 * q^71 + 34 * q^73 + 24 * q^79 - 10 * q^81 + 54 * q^87 + 40 * q^89 + 4 * q^95 + 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times$.

$n$	$1407$	$1921$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 2.59774i	− 1.49980i	−0.661549	−	0.749902i	$-0.730101\pi$
$3$	− 2.59774i	− 1.49980i	0.661549	−	0.749902i	$-0.269899\pi$
$4$	0	0
$5$	1.59774i	0.714529i	0.934003	+	0.357264i	$0.116291\pi$
$5$	1.59774i	0.714529i	−0.934003	+	0.357264i	$0.883709\pi$
$6$	0	0
$7$	−4.34596	−1.64262	−0.821310	−	0.570482i	$-0.806756\pi$
$7$	−4.34596	−1.64262	−0.821310	+	0.570482i	$0.806756\pi$
$8$	0	0
$9$	−3.74823	−1.24941
$10$	0	0
$11$	1.74823i	0.527111i	0.964644	+	0.263555i	$0.0848951\pi$
$11$	1.74823i	0.527111i	−0.964644	+	0.263555i	$0.915105\pi$
$12$	0	0
$13$	− 0.748228i	− 0.207521i	−0.994602	−	0.103761i	$-0.966912\pi$
$13$	− 0.748228i	− 0.207521i	0.994602	−	0.103761i	$-0.0330876\pi$
$14$	0	0
$15$	4.15049	1.07165
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	11.2897i	2.46361i
$22$	0	0
$23$	1.94370	0.405289	0.202645	−	0.979252i	$-0.435046\pi$
$23$	1.94370	0.405289	0.202645	+	0.979252i	$0.435046\pi$
$24$	0	0
$25$	2.44724	0.489448
$26$	0	0
$27$	1.94370i	0.374065i
$28$	0	0
$29$	7.28966i	1.35366i	0.736141	+	0.676828i	$0.236646\pi$
$29$	7.28966i	1.35366i	−0.736141	+	0.676828i	$0.763354\pi$
$30$	0	0
$31$	0.654037	0.117468	0.0587342	−	0.998274i	$-0.481294\pi$
$31$	0.654037	0.117468	0.0587342	+	0.998274i	$0.481294\pi$
$32$	0	0
$33$	4.54143	0.790562
$34$	0	0
$35$	− 6.94370i	− 1.17370i
$36$	0	0
$37$	3.84951i	0.632855i	0.948617	+	0.316428i	$0.102483\pi$
$37$	3.84951i	0.632855i	−0.948617	+	0.316428i	$0.897517\pi$
$38$	0	0
$39$	−1.94370	−0.311241
$40$	0	0
$41$	−4.54143	−0.709253	−0.354626	−	0.935008i	$-0.615392\pi$
$41$	−4.54143	−0.709253	−0.354626	+	0.935008i	$0.615392\pi$
$42$	0	0
$43$	− 10.4402i	− 1.59211i	−0.605225	−	0.796054i	$-0.706917\pi$
$43$	− 10.4402i	− 1.59211i	0.605225	−	0.796054i	$-0.293083\pi$
$44$	0	0
$45$	− 5.98868i	− 0.892739i
$46$	0	0
$47$	7.59774	1.10824	0.554122	−	0.832436i	$-0.313054\pi$
$47$	7.59774	1.10824	0.554122	+	0.832436i	$0.313054\pi$
$48$	0	0
$49$	11.8874	1.69820
$50$	0	0
$51$	− 7.79321i	− 1.09127i
$52$	0	0
$53$	6.59774i	0.906269i	0.891442	+	0.453134i	$0.149694\pi$
$53$	6.59774i	0.906269i	−0.891442	+	0.453134i	$0.850306\pi$
$54$	0	0
$55$	−2.79321	−0.376636
$56$	0	0
$57$	2.59774	0.344078
$58$	0	0
$59$	− 6.74823i	− 0.878544i	−0.898354	−	0.439272i	$-0.855236\pi$
$59$	− 6.74823i	− 0.878544i	0.898354	−	0.439272i	$-0.144764\pi$
$60$	0	0
$61$	− 9.09419i	− 1.16439i	−0.813049	−	0.582196i	$-0.802194\pi$
$61$	− 9.09419i	− 1.16439i	0.813049	−	0.582196i	$-0.197806\pi$
$62$	0	0
$63$	16.2897	2.05230
$64$	0	0
$65$	1.19547	0.148280
$66$	0	0
$67$	− 7.40226i	− 0.904331i	−0.891934	−	0.452165i	$-0.850652\pi$
$67$	− 7.40226i	− 0.904331i	0.891934	−	0.452165i	$-0.149348\pi$
$68$	0	0
$69$	− 5.04921i	− 0.607854i
$70$	0	0
$71$	1.69901	0.201636	0.100818	−	0.994905i	$-0.467854\pi$
$71$	1.69901	0.201636	0.100818	+	0.994905i	$0.467854\pi$
$72$	0	0
$73$	16.6848	1.95281	0.976406	−	0.215941i	$-0.0692820\pi$
$73$	16.6848	1.95281	0.976406	+	0.215941i	$0.0692820\pi$
$74$	0	0
$75$	− 6.35729i	− 0.734076i
$76$	0	0
$77$	− 7.59774i	− 0.865842i
$78$	0	0
$79$	0.654037	0.0735849	0.0367924	−	0.999323i	$-0.488286\pi$
$79$	0.654037	0.0735849	0.0367924	+	0.999323i	$0.488286\pi$
$80$	0	0
$81$	−6.19547	−0.688386
$82$	0	0
$83$	12.5793i	1.38076i	0.723447	+	0.690380i	$0.242557\pi$
$83$	12.5793i	1.38076i	−0.723447	+	0.690380i	$0.757443\pi$
$84$	0	0
$85$	4.79321i	0.519896i
$86$	0	0
$87$	18.9366	2.03022
$88$	0	0
$89$	14.5414	1.54139	0.770694	−	0.637205i	$-0.219909\pi$
$89$	14.5414	1.54139	0.770694	+	0.637205i	$0.219909\pi$
$90$	0	0
$91$	3.25177i	0.340878i
$92$	0	0
$93$	− 1.69901i	− 0.176180i
$94$	0	0
$95$	−1.59774	−0.163924
$96$	0	0
$97$	−10.5793	−1.07417	−0.537084	−	0.843529i	$-0.680474\pi$
$97$	−10.5793	−1.07417	−0.537084	+	0.843529i	$0.680474\pi$
$98$	0	0
$99$	− 6.55276i	− 0.658577i
$100$	0	0
$101$	− 4.80453i	− 0.478069i	−0.971011	−	0.239034i	$-0.923169\pi$
$101$	− 4.80453i	− 0.478069i	0.971011	−	0.239034i	$-0.0768308\pi$
$102$	0	0
$103$	12.5793	1.23948	0.619739	−	0.784808i	$-0.287239\pi$
$103$	12.5793	1.23948	0.619739	+	0.784808i	$0.287239\pi$
$104$	0	0
$105$	−18.0379	−1.76032
$106$	0	0
$107$	5.44015i	0.525920i	0.964807	+	0.262960i	$0.0846987\pi$
$107$	5.44015i	0.525920i	−0.964807	+	0.262960i	$0.915301\pi$
$108$	0	0
$109$	− 14.7861i	− 1.41625i	−0.706085	−	0.708127i	$-0.749541\pi$
$109$	− 14.7861i	− 1.41625i	0.706085	−	0.708127i	$-0.250459\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	14.0379	1.32057	0.660287	−	0.751014i	$-0.270435\pi$
$113$	14.0379	1.32057	0.660287	+	0.751014i	$0.270435\pi$
$114$	0	0
$115$	3.10552i	0.289591i
$116$	0	0
$117$	2.80453i	0.259279i
$118$	0	0
$119$	−13.0379	−1.19518
$120$	0	0
$121$	7.94370	0.722154
$122$	0	0
$123$	11.7974i	1.06374i
$124$	0	0
$125$	11.8987i	1.06425i
$126$	0	0
$127$	7.54852	0.669823	0.334911	−	0.942250i	$-0.391294\pi$
$127$	7.54852	0.669823	0.334911	+	0.942250i	$0.391294\pi$
$128$	0	0
$129$	−27.1208	−2.38785
$130$	0	0
$131$	− 0.0492139i	− 0.00429984i	−0.999998	−	0.00214992i	$-0.999316\pi$
$131$	− 0.0492139i	− 0.00429984i	0.999998	−	0.00214992i	$-0.000684342\pi$
$132$	0	0
$133$	− 4.34596i	− 0.376843i
$134$	0	0
$135$	−3.10552	−0.267280
$136$	0	0
$137$	−17.3909	−1.48581	−0.742904	−	0.669398i	$-0.766552\pi$
$137$	−17.3909	−1.48581	−0.742904	+	0.669398i	$0.766552\pi$
$138$	0	0
$139$	− 8.25177i	− 0.699906i	−0.936767	−	0.349953i	$-0.886198\pi$
$139$	− 8.25177i	− 0.699906i	0.936767	−	0.349953i	$-0.113802\pi$
$140$	0	0
$141$	− 19.7369i	− 1.66215i
$142$	0	0
$143$	1.30807	0.109387
$144$	0	0
$145$	−11.6469	−0.967226
$146$	0	0
$147$	− 30.8803i	− 2.54696i
$148$	0	0
$149$	− 6.28966i	− 0.515269i	−0.966242	−	0.257635i	$-0.917057\pi$
$149$	− 6.28966i	− 0.515269i	0.966242	−	0.257635i	$-0.0829431\pi$
$150$	0	0
$151$	−22.1884	−1.80566	−0.902832	−	0.429992i	$-0.858516\pi$
$151$	−22.1884	−1.80566	−0.902832	+	0.429992i	$0.858516\pi$
$152$	0	0
$153$	−11.2447	−0.909079
$154$	0	0
$155$	1.04498i	0.0839346i
$156$	0	0
$157$	8.99291i	0.717713i	0.933393	+	0.358856i	$0.116833\pi$
$157$	8.99291i	0.717713i	−0.933393	+	0.358856i	$0.883167\pi$
$158$	0	0
$159$	17.1392	1.35922
$160$	0	0
$161$	−8.44724	−0.665736
$162$	0	0
$163$	16.9929i	1.33099i	0.746403	+	0.665494i	$0.231779\pi$
$163$	16.9929i	1.33099i	−0.746403	+	0.665494i	$0.768221\pi$
$164$	0	0
$165$	7.25601i	0.564879i
$166$	0	0
$167$	16.3389	1.26434	0.632170	−	0.774830i	$-0.282164\pi$
$167$	16.3389	1.26434	0.632170	+	0.774830i	$0.282164\pi$
$168$	0	0
$169$	12.4402	0.956935
$170$	0	0
$171$	− 3.74823i	− 0.286634i
$172$	0	0
$173$	23.3839i	1.77784i	0.458061	+	0.888921i	$0.348544\pi$
$173$	23.3839i	1.77784i	−0.458061	+	0.888921i	$0.651456\pi$
$174$	0	0
$175$	−10.6356	−0.803978
$176$	0	0
$177$	−17.5301	−1.31764
$178$	0	0
$179$	4.54143i	0.339443i	0.985492	+	0.169721i	$0.0542868\pi$
$179$	4.54143i	0.339443i	−0.985492	+	0.169721i	$0.945713\pi$
$180$	0	0
$181$	− 12.0900i	− 0.898639i	−0.893371	−	0.449320i	$-0.851666\pi$
$181$	− 12.0900i	− 0.898639i	0.893371	−	0.449320i	$-0.148334\pi$
$182$	0	0
$183$	−23.6243	−1.74636
$184$	0	0
$185$	−6.15049	−0.452193
$186$	0	0
$187$	5.24468i	0.383529i
$188$	0	0
$189$	− 8.44724i	− 0.614446i
$190$	0	0
$191$	12.5485	0.907979	0.453990	−	0.891007i	$-0.350000\pi$
$191$	12.5485	0.907979	0.453990	+	0.891007i	$0.350000\pi$
$192$	0	0
$193$	10.7298	0.772349	0.386175	−	0.922426i	$-0.373796\pi$
$193$	10.7298	0.772349	0.386175	+	0.922426i	$0.373796\pi$
$194$	0	0
$195$	− 3.10552i	− 0.222391i
$196$	0	0
$197$	9.38385i	0.668572i	0.942472	+	0.334286i	$0.108495\pi$
$197$	9.38385i	0.668572i	−0.942472	+	0.334286i	$0.891505\pi$
$198$	0	0
$199$	−11.2404	−0.796814	−0.398407	−	0.917209i	$-0.630437\pi$
$199$	−11.2404	−0.796814	−0.398407	+	0.917209i	$0.630437\pi$
$200$	0	0
$201$	−19.2291	−1.35632
$202$	0	0
$203$	− 31.6806i	− 2.22354i
$204$	0	0
$205$	− 7.25601i	− 0.506782i
$206$	0	0
$207$	−7.28543	−0.506372
$208$	0	0
$209$	−1.74823	−0.120927
$210$	0	0
$211$	5.44015i	0.374516i	0.982311	+	0.187258i	$0.0599600\pi$
$211$	5.44015i	0.374516i	−0.982311	+	0.187258i	$0.940040\pi$
$212$	0	0
$213$	− 4.41359i	− 0.302414i
$214$	0	0
$215$	16.6806	1.13761
$216$	0	0
$217$	−2.84242	−0.192956
$218$	0	0
$219$	− 43.3428i	− 2.92883i
$220$	0	0
$221$	− 2.24468i	− 0.150994i
$222$	0	0
$223$	25.4217	1.70237	0.851183	−	0.524869i	$-0.175886\pi$
$223$	25.4217	1.70237	0.851183	+	0.524869i	$0.175886\pi$
$224$	0	0
$225$	−9.17282	−0.611521
$226$	0	0
$227$	− 3.74114i	− 0.248308i	−0.992263	−	0.124154i	$-0.960378\pi$
$227$	− 3.74114i	− 0.248308i	0.992263	−	0.124154i	$-0.0396217\pi$
$228$	0	0
$229$	− 5.29675i	− 0.350019i	−0.984567	−	0.175010i	$-0.944004\pi$
$229$	− 5.29675i	− 0.350019i	0.984567	−	0.175010i	$-0.0559956\pi$
$230$	0	0
$231$	−19.7369	−1.29859
$232$	0	0
$233$	−9.93661	−0.650969	−0.325485	−	0.945547i	$-0.605527\pi$
$233$	−9.93661	−0.650969	−0.325485	+	0.945547i	$0.605527\pi$
$234$	0	0
$235$	12.1392i	0.791872i
$236$	0	0
$237$	− 1.69901i	− 0.110363i
$238$	0	0
$239$	27.8424	1.80098	0.900488	−	0.434880i	$-0.143209\pi$
$239$	27.8424	1.80098	0.900488	+	0.434880i	$0.143209\pi$
$240$	0	0
$241$	5.84951	0.376800	0.188400	−	0.982092i	$-0.439670\pi$
$241$	5.84951	0.376800	0.188400	+	0.982092i	$0.439670\pi$
$242$	0	0
$243$	21.9253i	1.40651i
$244$	0	0
$245$	18.9929i	1.21341i
$246$	0	0
$247$	0.748228	0.0476086
$248$	0	0
$249$	32.6778	2.07087
$250$	0	0
$251$	15.2447i	0.962236i	0.876656	+	0.481118i	$0.159769\pi$
$251$	15.2447i	0.962236i	−0.876656	+	0.481118i	$0.840231\pi$
$252$	0	0
$253$	3.39803i	0.213632i
$254$	0	0
$255$	12.4515	0.779742
$256$	0	0
$257$	7.58641	0.473227	0.236614	−	0.971604i	$-0.423962\pi$
$257$	7.58641	0.473227	0.236614	+	0.971604i	$0.423962\pi$
$258$	0	0
$259$	− 16.7298i	− 1.03954i
$260$	0	0
$261$	− 27.3233i	− 1.69127i
$262$	0	0
$263$	2.79321	0.172236	0.0861182	−	0.996285i	$-0.472554\pi$
$263$	2.79321	0.172236	0.0861182	+	0.996285i	$0.472554\pi$
$264$	0	0
$265$	−10.5414	−0.647555
$266$	0	0
$267$	− 37.7748i	− 2.31178i
$268$	0	0
$269$	− 25.0308i	− 1.52615i	−0.646307	−	0.763077i	$-0.723687\pi$
$269$	− 25.0308i	− 1.52615i	0.646307	−	0.763077i	$-0.276313\pi$
$270$	0	0
$271$	−27.5301	−1.67234	−0.836168	−	0.548474i	$-0.815209\pi$
$271$	−27.5301	−1.67234	−0.836168	+	0.548474i	$0.815209\pi$
$272$	0	0
$273$	8.44724	0.511250
$274$	0	0
$275$	4.27834i	0.257993i
$276$	0	0
$277$	− 10.4023i	− 0.625012i	−0.949916	−	0.312506i	$-0.898832\pi$
$277$	− 10.4023i	− 0.625012i	0.949916	−	0.312506i	$-0.101168\pi$
$278$	0	0
$279$	−2.45148	−0.146766
$280$	0	0
$281$	13.8874	0.828453	0.414226	−	0.910174i	$-0.364052\pi$
$281$	13.8874	0.828453	0.414226	+	0.910174i	$0.364052\pi$
$282$	0	0
$283$	27.3346i	1.62488i	0.583048	+	0.812438i	$0.301860\pi$
$283$	27.3346i	1.62488i	−0.583048	+	0.812438i	$0.698140\pi$
$284$	0	0
$285$	4.15049i	0.245854i
$286$	0	0
$287$	19.7369	1.16503
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	27.4823i	1.61104i
$292$	0	0
$293$	− 10.1321i	− 0.591923i	−0.955200	−	0.295961i	$-0.904360\pi$
$293$	− 10.1321i	− 0.591923i	0.955200	−	0.295961i	$-0.0956399\pi$
$294$	0	0
$295$	10.7819	0.627745
$296$	0	0
$297$	−3.39803	−0.197174
$298$	0	0
$299$	− 1.45433i	− 0.0841061i
$300$	0	0
$301$	45.3725i	2.61523i
$302$	0	0
$303$	−12.4809	−0.717009
$304$	0	0
$305$	14.5301	0.831992
$306$	0	0
$307$	− 17.1208i	− 0.977133i	−0.872527	−	0.488567i	$-0.837520\pi$
$307$	− 17.1208i	− 0.977133i	0.872527	−	0.488567i	$-0.162480\pi$
$308$	0	0
$309$	− 32.6778i	− 1.85897i
$310$	0	0
$311$	−31.2404	−1.77148	−0.885742	−	0.464179i	$-0.846350\pi$
$311$	−31.2404	−1.77148	−0.885742	+	0.464179i	$0.846350\pi$
$312$	0	0
$313$	25.2291	1.42603	0.713017	−	0.701147i	$-0.247328\pi$
$313$	25.2291	1.42603	0.713017	+	0.701147i	$0.247328\pi$
$314$	0	0
$315$	26.0266i	1.46643i
$316$	0	0
$317$	11.6427i	0.653920i	0.945038	+	0.326960i	$0.106024\pi$
$317$	11.6427i	0.653920i	−0.945038	+	0.326960i	$0.893976\pi$
$318$	0	0
$319$	−12.7440	−0.713527
$320$	0	0
$321$	14.1321	0.788776
$322$	0	0
$323$	3.00000i	0.166924i
$324$	0	0
$325$	− 1.83110i	− 0.101571i
$326$	0	0
$327$	−38.4104	−2.12410
$328$	0	0
$329$	−33.0195	−1.82042
$330$	0	0
$331$	12.5977i	0.692434i	0.938154	+	0.346217i	$0.112534\pi$
$331$	12.5977i	0.692434i	−0.938154	+	0.346217i	$0.887466\pi$
$332$	0	0
$333$	− 14.4288i	− 0.790695i
$334$	0	0
$335$	11.8269	0.646170
$336$	0	0
$337$	−5.49646	−0.299411	−0.149706	−	0.988731i	$-0.547833\pi$
$337$	−5.49646	−0.299411	−0.149706	+	0.988731i	$0.547833\pi$
$338$	0	0
$339$	− 36.4667i	− 1.98060i
$340$	0	0
$341$	1.14341i	0.0619189i
$342$	0	0
$343$	−21.2404	−1.14688
$344$	0	0
$345$	8.06731	0.434329
$346$	0	0
$347$	10.3417i	0.555173i	0.960701	+	0.277586i	$0.0895345\pi$
$347$	10.3417i	0.555173i	−0.960701	+	0.277586i	$0.910465\pi$
$348$	0	0
$349$	30.6806i	1.64230i	0.570716	+	0.821148i	$0.306666\pi$
$349$	30.6806i	1.64230i	−0.570716	+	0.821148i	$0.693334\pi$
$350$	0	0
$351$	1.45433	0.0776264
$352$	0	0
$353$	23.3276	1.24160	0.620800	−	0.783969i	$-0.286808\pi$
$353$	23.3276	1.24160	0.620800	+	0.783969i	$0.286808\pi$
$354$	0	0
$355$	2.71457i	0.144075i
$356$	0	0
$357$	33.8690i	1.79254i
$358$	0	0
$359$	−12.1576	−0.641653	−0.320826	−	0.947138i	$-0.603961\pi$
$359$	−12.1576	−0.641653	−0.320826	+	0.947138i	$0.603961\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 20.6356i	− 1.08309i
$364$	0	0
$365$	26.6580i	1.39534i
$366$	0	0
$367$	9.08287	0.474122	0.237061	−	0.971495i	$-0.423816\pi$
$367$	9.08287	0.474122	0.237061	+	0.971495i	$0.423816\pi$
$368$	0	0
$369$	17.0223	0.886147
$370$	0	0
$371$	− 28.6735i	− 1.48865i
$372$	0	0
$373$	− 18.1321i	− 0.938844i	−0.882974	−	0.469422i	$-0.844462\pi$
$373$	− 18.1321i	− 0.938844i	0.882974	−	0.469422i	$-0.155538\pi$
$374$	0	0
$375$	30.9097	1.59617
$376$	0	0
$377$	5.45433	0.280912
$378$	0	0
$379$	23.6427i	1.21444i	0.794532	+	0.607222i	$0.207716\pi$
$379$	23.6427i	1.21444i	−0.794532	+	0.607222i	$0.792284\pi$
$380$	0	0
$381$	− 19.6091i	− 1.00460i
$382$	0	0
$383$	−21.9253	−1.12033	−0.560165	−	0.828381i	$-0.689262\pi$
$383$	−21.9253	−1.12033	−0.560165	+	0.828381i	$0.689262\pi$
$384$	0	0
$385$	12.1392	0.618669
$386$	0	0
$387$	39.1321i	1.98920i
$388$	0	0
$389$	− 29.2599i	− 1.48354i	−0.670656	−	0.741769i	$-0.733987\pi$
$389$	− 29.2599i	− 1.48354i	0.670656	−	0.741769i	$-0.266013\pi$
$390$	0	0
$391$	5.83110	0.294891
$392$	0	0
$393$	−0.127845	−0.00644892
$394$	0	0
$395$	1.04498i	0.0525785i
$396$	0	0
$397$	6.10128i	0.306214i	0.988210	+	0.153107i	$0.0489280\pi$
$397$	6.10128i	0.306214i	−0.988210	+	0.153107i	$0.951072\pi$
$398$	0	0
$399$	−11.2897	−0.565190
$400$	0	0
$401$	16.0379	0.800894	0.400447	−	0.916320i	$-0.368855\pi$
$401$	16.0379	0.800894	0.400447	+	0.916320i	$0.368855\pi$
$402$	0	0
$403$	− 0.489369i	− 0.0243772i
$404$	0	0
$405$	− 9.89872i	− 0.491871i
$406$	0	0
$407$	−6.72982	−0.333585
$408$	0	0
$409$	25.5343	1.26259	0.631296	−	0.775542i	$-0.282523\pi$
$409$	25.5343	1.26259	0.631296	+	0.775542i	$0.282523\pi$
$410$	0	0
$411$	45.1771i	2.22842i
$412$	0	0
$413$	29.3276i	1.44311i
$414$	0	0
$415$	−20.0984	−0.986593
$416$	0	0
$417$	−21.4359	−1.04972
$418$	0	0
$419$	35.1586i	1.71761i	0.512301	+	0.858806i	$0.328793\pi$
$419$	35.1586i	1.71761i	−0.512301	+	0.858806i	$0.671207\pi$
$420$	0	0
$421$	2.89872i	0.141275i	0.997502	+	0.0706375i	$0.0225034\pi$
$421$	2.89872i	0.141275i	−0.997502	+	0.0706375i	$0.977497\pi$
$422$	0	0
$423$	−28.4780	−1.38465
$424$	0	0
$425$	7.34173	0.356126
$426$	0	0
$427$	39.5230i	1.91265i
$428$	0	0
$429$	− 3.39803i	− 0.164058i
$430$	0	0
$431$	25.9774	1.25129	0.625643	−	0.780110i	$-0.284837\pi$
$431$	25.9774	1.25129	0.625643	+	0.780110i	$0.284837\pi$
$432$	0	0
$433$	30.9324	1.48652	0.743258	−	0.669005i	$-0.233280\pi$
$433$	30.9324	1.48652	0.743258	+	0.669005i	$0.233280\pi$
$434$	0	0
$435$	30.2557i	1.45065i
$436$	0	0
$437$	1.94370i	0.0929797i
$438$	0	0
$439$	−10.1278	−0.483376	−0.241688	−	0.970354i	$-0.577701\pi$
$439$	−10.1278	−0.483376	−0.241688	+	0.970354i	$0.577701\pi$
$440$	0	0
$441$	−44.5567	−2.12175
$442$	0	0
$443$	− 14.3276i	− 0.680723i	−0.940295	−	0.340361i	$-0.889451\pi$
$443$	− 14.3276i	− 0.680723i	0.940295	−	0.340361i	$-0.110549\pi$
$444$	0	0
$445$	23.2334i	1.10137i
$446$	0	0
$447$	−16.3389	−0.772802
$448$	0	0
$449$	6.61615	0.312235	0.156118	−	0.987738i	$-0.450102\pi$
$449$	6.61615	0.312235	0.156118	+	0.987738i	$0.450102\pi$
$450$	0	0
$451$	− 7.93946i	− 0.373855i
$452$	0	0
$453$	57.6395i	2.70814i
$454$	0	0
$455$	−5.19547	−0.243567
$456$	0	0
$457$	−38.5864	−1.80500	−0.902498	−	0.430694i	$-0.858269\pi$
$457$	−38.5864	−1.80500	−0.902498	+	0.430694i	$0.858269\pi$
$458$	0	0
$459$	5.83110i	0.272172i
$460$	0	0
$461$	33.1841i	1.54554i	0.634686	+	0.772770i	$0.281129\pi$
$461$	33.1841i	1.54554i	−0.634686	+	0.772770i	$0.718871\pi$
$462$	0	0
$463$	13.2826	0.617294	0.308647	−	0.951177i	$-0.400124\pi$
$463$	13.2826	0.617294	0.308647	+	0.951177i	$0.400124\pi$
$464$	0	0
$465$	2.71457	0.125885
$466$	0	0
$467$	− 20.8311i	− 0.963948i	−0.876185	−	0.481974i	$-0.839920\pi$
$467$	− 20.8311i	− 0.963948i	0.876185	−	0.481974i	$-0.160080\pi$
$468$	0	0
$469$	32.1700i	1.48547i
$470$	0	0
$471$	23.3612	1.07643
$472$	0	0
$473$	18.2518	0.839217
$474$	0	0
$475$	2.44724i	0.112287i
$476$	0	0
$477$	− 24.7298i	− 1.13230i
$478$	0	0
$479$	25.6848	1.17357	0.586785	−	0.809743i	$-0.300393\pi$
$479$	25.6848	1.17357	0.586785	+	0.809743i	$0.300393\pi$
$480$	0	0
$481$	2.88031	0.131331
$482$	0	0
$483$	21.9437i	0.998473i
$484$	0	0
$485$	− 16.9030i	− 0.767524i
$486$	0	0
$487$	−9.51063	−0.430968	−0.215484	−	0.976507i	$-0.569133\pi$
$487$	−9.51063	−0.430968	−0.215484	+	0.976507i	$0.569133\pi$
$488$	0	0
$489$	44.1431	1.99622
$490$	0	0
$491$	− 6.89448i	− 0.311144i	−0.987825	−	0.155572i	$-0.950278\pi$
$491$	− 6.89448i	− 0.311144i	0.987825	−	0.155572i	$-0.0497220\pi$
$492$	0	0
$493$	21.8690i	0.984930i
$494$	0	0
$495$	10.4696	0.470572
$496$	0	0
$497$	−7.38385	−0.331211
$498$	0	0
$499$	29.1321i	1.30413i	0.758163	+	0.652066i	$0.226097\pi$
$499$	29.1321i	1.30413i	−0.758163	+	0.652066i	$0.773903\pi$
$500$	0	0
$501$	− 42.4441i	− 1.89626i
$502$	0	0
$503$	−13.1533	−0.586479	−0.293239	−	0.956039i	$-0.594733\pi$
$503$	−13.1533	−0.586479	−0.293239	+	0.956039i	$0.594733\pi$
$504$	0	0
$505$	7.67637	0.341594
$506$	0	0
$507$	− 32.3162i	− 1.43521i
$508$	0	0
$509$	− 32.5651i	− 1.44342i	−0.692193	−	0.721712i	$-0.743355\pi$
$509$	− 32.5651i	− 1.44342i	0.692193	−	0.721712i	$-0.256645\pi$
$510$	0	0
$511$	−72.5117	−3.20773
$512$	0	0
$513$	−1.94370	−0.0858164
$514$	0	0
$515$	20.0984i	0.885643i
$516$	0	0
$517$	13.2826i	0.584167i
$518$	0	0
$519$	60.7451	2.66641
$520$	0	0
$521$	−13.0450	−0.571511	−0.285755	−	0.958303i	$-0.592244\pi$
$521$	−13.0450	−0.571511	−0.285755	+	0.958303i	$0.592244\pi$
$522$	0	0
$523$	− 11.9437i	− 0.522261i	−0.965303	−	0.261131i	$-0.915905\pi$
$523$	− 11.9437i	− 0.522261i	0.965303	−	0.261131i	$-0.0840954\pi$
$524$	0	0
$525$	27.6285i	1.20581i
$526$	0	0
$527$	1.96211	0.0854709
$528$	0	0
$529$	−19.2220	−0.835741
$530$	0	0
$531$	25.2939i	1.09766i
$532$	0	0
$533$	3.39803i	0.147185i
$534$	0	0
$535$	−8.69193	−0.375785
$536$	0	0
$537$	11.7974	0.509097
$538$	0	0
$539$	20.7819i	0.895139i
$540$	0	0
$541$	− 1.48513i	− 0.0638508i	−0.999490	−	0.0319254i	$-0.989836\pi$
$541$	− 1.48513i	− 0.0638508i	0.999490	−	0.0319254i	$-0.0101639\pi$
$542$	0	0
$543$	−31.4065	−1.34778
$544$	0	0
$545$	23.6243	1.01195
$546$	0	0
$547$	6.01418i	0.257148i	0.991700	+	0.128574i	$0.0410399\pi$
$547$	6.01418i	0.257148i	−0.991700	+	0.128574i	$0.958960\pi$
$548$	0	0
$549$	34.0871i	1.45480i
$550$	0	0
$551$	−7.28966	−0.310550
$552$	0	0
$553$	−2.84242	−0.120872
$554$	0	0
$555$	15.9774i	0.678201i
$556$	0	0
$557$	27.3584i	1.15921i	0.814897	+	0.579605i	$0.196793\pi$
$557$	27.3584i	1.15921i	−0.814897	+	0.579605i	$0.803207\pi$
$558$	0	0
$559$	−7.81162	−0.330396
$560$	0	0
$561$	13.6243	0.575218
$562$	0	0
$563$	− 21.1434i	− 0.891088i	−0.895260	−	0.445544i	$-0.853010\pi$
$563$	− 21.1434i	− 0.891088i	0.895260	−	0.445544i	$-0.146990\pi$
$564$	0	0
$565$	22.4288i	0.943588i
$566$	0	0
$567$	26.9253	1.13076
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	− 9.18130i	− 0.384225i	−0.981373	−	0.192113i	$-0.938466\pi$
$571$	− 9.18130i	− 0.384225i	0.981373	−	0.192113i	$-0.0615339\pi$
$572$	0	0
$573$	− 32.5977i	− 1.36179i
$574$	0	0
$575$	4.75670	0.198368
$576$	0	0
$577$	−4.79744	−0.199720	−0.0998601	−	0.995001i	$-0.531840\pi$
$577$	−4.79744	−0.199720	−0.0998601	+	0.995001i	$0.531840\pi$
$578$	0	0
$579$	− 27.8732i	− 1.15837i
$580$	0	0
$581$	− 54.6693i	− 2.26806i
$582$	0	0
$583$	−11.5343	−0.477704
$584$	0	0
$585$	−4.48090	−0.185262
$586$	0	0
$587$	1.64980i	0.0680945i	0.999420	+	0.0340473i	$0.0108397\pi$
$587$	1.64980i	0.0680945i	−0.999420	+	0.0340473i	$0.989160\pi$
$588$	0	0
$589$	0.654037i	0.0269491i
$590$	0	0
$591$	24.3768	1.00273
$592$	0	0
$593$	−19.8874	−0.816678	−0.408339	−	0.912830i	$-0.633892\pi$
$593$	−19.8874	−0.816678	−0.408339	+	0.912830i	$0.633892\pi$
$594$	0	0
$595$	− 20.8311i	− 0.853992i
$596$	0	0
$597$	29.1997i	1.19506i
$598$	0	0
$599$	2.25462	0.0921214	0.0460607	−	0.998939i	$-0.485333\pi$
$599$	2.25462	0.0921214	0.0460607	+	0.998939i	$0.485333\pi$
$600$	0	0
$601$	−24.7677	−1.01030	−0.505148	−	0.863033i	$-0.668562\pi$
$601$	−24.7677	−1.01030	−0.505148	+	0.863033i	$0.668562\pi$
$602$	0	0
$603$	27.7454i	1.12988i
$604$	0	0
$605$	12.6919i	0.516000i
$606$	0	0
$607$	−45.9774	−1.86616	−0.933082	−	0.359665i	$-0.882891\pi$
$607$	−45.9774	−1.86616	−0.933082	+	0.359665i	$0.882891\pi$
$608$	0	0
$609$	−82.2978	−3.33488
$610$	0	0
$611$	− 5.68484i	− 0.229984i
$612$	0	0
$613$	− 0.326485i	− 0.0131866i	−0.999978	−	0.00659330i	$-0.997901\pi$
$613$	− 0.326485i	− 0.0131866i	0.999978	−	0.00659330i	$-0.00209873\pi$
$614$	0	0
$615$	−18.8492	−0.760073
$616$	0	0
$617$	37.9140	1.52636	0.763179	−	0.646187i	$-0.223637\pi$
$617$	37.9140	1.52636	0.763179	+	0.646187i	$0.223637\pi$
$618$	0	0
$619$	4.76771i	0.191630i	0.995399	+	0.0958151i	$0.0305458\pi$
$619$	4.76771i	0.191630i	−0.995399	+	0.0958151i	$0.969454\pi$
$620$	0	0
$621$	3.77796i	0.151604i
$622$	0	0
$623$	−63.1965	−2.53192
$624$	0	0
$625$	−6.77479	−0.270992
$626$	0	0
$627$	4.54143i	0.181367i
$628$	0	0
$629$	11.5485i	0.460470i
$630$	0	0
$631$	−10.9958	−0.437734	−0.218867	−	0.975755i	$-0.570236\pi$
$631$	−10.9958	−0.437734	−0.218867	+	0.975755i	$0.570236\pi$
$632$	0	0
$633$	14.1321	0.561700
$634$	0	0
$635$	12.0605i	0.478608i
$636$	0	0
$637$	− 8.89448i	− 0.352412i
$638$	0	0
$639$	−6.36829	−0.251926
$640$	0	0
$641$	−25.1586	−0.993707	−0.496853	−	0.867834i	$-0.665511\pi$
$641$	−25.1586	−0.993707	−0.496853	+	0.867834i	$0.665511\pi$
$642$	0	0
$643$	− 36.9069i	− 1.45547i	−0.685861	−	0.727733i	$-0.740574\pi$
$643$	− 36.9069i	− 1.45547i	0.685861	−	0.727733i	$-0.259426\pi$
$644$	0	0
$645$	− 43.3318i	− 1.70619i
$646$	0	0
$647$	−17.3531	−0.682219	−0.341109	−	0.940024i	$-0.610803\pi$
$647$	−17.3531	−0.682219	−0.341109	+	0.940024i	$0.610803\pi$
$648$	0	0
$649$	11.7974	0.463090
$650$	0	0
$651$	7.38385i	0.289396i
$652$	0	0
$653$	− 14.4780i	− 0.566570i	−0.959036	−	0.283285i	$-0.908576\pi$
$653$	− 14.4780i	− 0.566570i	0.959036	−	0.283285i	$-0.0914242\pi$
$654$	0	0
$655$	0.0786309	0.00307236
$656$	0	0
$657$	−62.5386	−2.43986
$658$	0	0
$659$	− 47.5301i	− 1.85151i	−0.378124	−	0.925755i	$-0.623431\pi$
$659$	− 47.5301i	− 1.85151i	0.378124	−	0.925755i	$-0.376569\pi$
$660$	0	0
$661$	− 8.22204i	− 0.319800i	−0.987133	−	0.159900i	$-0.948883\pi$
$661$	− 8.22204i	− 0.319800i	0.987133	−	0.159900i	$-0.0511172\pi$
$662$	0	0
$663$	−5.83110	−0.226461
$664$	0	0
$665$	6.94370	0.269265
$666$	0	0
$667$	14.1689i	0.548622i
$668$	0	0
$669$	− 66.0390i	− 2.55321i
$670$	0	0
$671$	15.8987	0.613763
$672$	0	0
$673$	15.9016	0.612961	0.306480	−	0.951877i	$-0.400849\pi$
$673$	15.9016	0.612961	0.306480	+	0.951877i	$0.400849\pi$
$674$	0	0
$675$	4.75670i	0.183086i
$676$	0	0
$677$	21.9816i	0.844821i	0.906405	+	0.422411i	$0.138816\pi$
$677$	21.9816i	0.844821i	−0.906405	+	0.422411i	$0.861184\pi$
$678$	0	0
$679$	45.9774	1.76445
$680$	0	0
$681$	−9.71849	−0.372413
$682$	0	0
$683$	1.30807i	0.0500520i	0.999687	+	0.0250260i	$0.00796686\pi$
$683$	1.30807i	0.0500520i	−0.999687	+	0.0250260i	$0.992033\pi$
$684$	0	0
$685$	− 27.7861i	− 1.06165i
$686$	0	0
$687$	−13.7596	−0.524960
$688$	0	0
$689$	4.93661	0.188070
$690$	0	0
$691$	44.2291i	1.68256i	0.540603	+	0.841278i	$0.318196\pi$
$691$	44.2291i	1.68256i	−0.540603	+	0.841278i	$0.681804\pi$
$692$	0	0
$693$	28.4780i	1.08179i
$694$	0	0
$695$	13.1841	0.500103
$696$	0	0
$697$	−13.6243	−0.516057
$698$	0	0
$699$	25.8127i	0.976325i
$700$	0	0
$701$	− 24.5793i	− 0.928348i	−0.885744	−	0.464174i	$-0.846351\pi$
$701$	− 24.5793i	− 0.928348i	0.885744	−	0.464174i	$-0.153649\pi$
$702$	0	0
$703$	−3.84951	−0.145187
$704$	0	0
$705$	31.5343	1.18765
$706$	0	0
$707$	20.8803i	0.785285i
$708$	0	0
$709$	9.39803i	0.352950i	0.984305	+	0.176475i	$0.0564695\pi$
$709$	9.39803i	0.352950i	−0.984305	+	0.176475i	$0.943530\pi$
$710$	0	0
$711$	−2.45148	−0.0919376
$712$	0	0
$713$	1.27125	0.0476087
$714$	0	0
$715$	2.08995i	0.0781599i
$716$	0	0
$717$	− 72.3272i	− 2.70111i
$718$	0	0
$719$	24.3460	0.907951	0.453976	−	0.891014i	$-0.350005\pi$
$719$	24.3460	0.907951	0.453976	+	0.891014i	$0.350005\pi$
$720$	0	0
$721$	−54.6693	−2.03599
$722$	0	0
$723$	− 15.1955i	− 0.565126i
$724$	0	0
$725$	17.8396i	0.662545i
$726$	0	0
$727$	29.1137	1.07977	0.539883	−	0.841740i	$-0.318468\pi$
$727$	29.1137	1.07977	0.539883	+	0.841740i	$0.318468\pi$
$728$	0	0
$729$	38.3697	1.42110
$730$	0	0
$731$	− 31.3205i	− 1.15843i
$732$	0	0
$733$	− 4.91713i	− 0.181618i	−0.995868	−	0.0908092i	$-0.971055\pi$
$733$	− 4.91713i	− 0.181618i	0.995868	−	0.0908092i	$-0.0289453\pi$
$734$	0	0
$735$	49.3386	1.81988
$736$	0	0
$737$	12.9408	0.476682
$738$	0	0
$739$	51.7114i	1.90223i	0.308830	+	0.951117i	$0.400063\pi$
$739$	51.7114i	1.90223i	−0.308830	+	0.951117i	$0.599937\pi$
$740$	0	0
$741$	− 1.94370i	− 0.0714035i
$742$	0	0
$743$	−18.0379	−0.661746	−0.330873	−	0.943675i	$-0.607343\pi$
$743$	−18.0379	−0.661746	−0.330873	+	0.943675i	$0.607343\pi$
$744$	0	0
$745$	10.0492	0.368175
$746$	0	0
$747$	− 47.1502i	− 1.72513i
$748$	0	0
$749$	− 23.6427i	− 0.863886i
$750$	0	0
$751$	−10.9171	−0.398372	−0.199186	−	0.979962i	$-0.563830\pi$
$751$	−10.9171	−0.398372	−0.199186	+	0.979962i	$0.563830\pi$
$752$	0	0
$753$	39.6017	1.44316
$754$	0	0
$755$	− 35.4512i	− 1.29020i
$756$	0	0
$757$	− 33.4710i	− 1.21652i	−0.793737	−	0.608261i	$-0.791867\pi$
$757$	− 33.4710i	− 1.21652i	0.793737	−	0.608261i	$-0.208133\pi$
$758$	0	0
$759$	8.82718	0.320406
$760$	0	0
$761$	51.3399	1.86107	0.930536	−	0.366201i	$-0.119342\pi$
$761$	51.3399	1.86107	0.930536	+	0.366201i	$0.119342\pi$
$762$	0	0
$763$	64.2599i	2.32637i
$764$	0	0
$765$	− 17.9660i	− 0.649563i
$766$	0	0
$767$	−5.04921	−0.182317
$768$	0	0
$769$	16.7974	0.605731	0.302866	−	0.953033i	$-0.402057\pi$
$769$	16.7974	0.605731	0.302866	+	0.953033i	$0.402057\pi$
$770$	0	0
$771$	− 19.7075i	− 0.709748i
$772$	0	0
$773$	− 8.62145i	− 0.310092i	−0.987907	−	0.155046i	$-0.950447\pi$
$773$	− 8.62145i	− 0.310092i	0.987907	−	0.155046i	$-0.0495526\pi$
$774$	0	0
$775$	1.60059	0.0574948
$776$	0	0
$777$	−43.4596	−1.55911
$778$	0	0
$779$	− 4.54143i	− 0.162714i
$780$	0	0
$781$	2.97026i	0.106284i
$782$	0	0
$783$	−14.1689	−0.506355
$784$	0	0
$785$	−14.3683	−0.512826
$786$	0	0
$787$	25.0124i	0.891595i	0.895134	+	0.445798i	$0.147080\pi$
$787$	25.0124i	0.891595i	−0.895134	+	0.445798i	$0.852920\pi$
$788$	0	0
$789$	− 7.25601i	− 0.258321i
$790$	0	0
$791$	−61.0082	−2.16920
$792$	0	0
$793$	−6.80453	−0.241636
$794$	0	0
$795$	27.3839i	0.971205i
$796$	0	0
$797$	− 48.0574i	− 1.70228i	−0.524939	−	0.851140i	$-0.675912\pi$
$797$	− 48.0574i	− 1.70228i	0.524939	−	0.851140i	$-0.324088\pi$
$798$	0	0
$799$	22.7932	0.806366
$800$	0	0
$801$	−54.5046	−1.92583
$802$	0	0
$803$	29.1689i	1.02935i
$804$	0	0
$805$	− 13.4965i	− 0.475688i
$806$	0	0
$807$	−65.0234	−2.28893
$808$	0	0
$809$	−6.20965	−0.218320	−0.109160	−	0.994024i	$-0.534816\pi$
$809$	−6.20965	−0.218320	−0.109160	+	0.994024i	$0.534816\pi$
$810$	0	0
$811$	− 26.6130i	− 0.934508i	−0.884123	−	0.467254i	$-0.845243\pi$
$811$	− 26.6130i	− 0.934508i	0.884123	−	0.467254i	$-0.154757\pi$
$812$	0	0
$813$	71.5159i	2.50817i
$814$	0	0
$815$	−27.1502	−0.951029
$816$	0	0
$817$	10.4402	0.365255
$818$	0	0
$819$	− 12.1884i	− 0.425897i
$820$	0	0
$821$	− 25.4993i	− 0.889932i	−0.895547	−	0.444966i	$-0.853216\pi$
$821$	− 25.4993i	− 0.889932i	0.895547	−	0.444966i	$-0.146784\pi$
$822$	0	0
$823$	−34.0166	−1.18575	−0.592873	−	0.805296i	$-0.702006\pi$
$823$	−34.0166	−1.18575	−0.592873	+	0.805296i	$0.702006\pi$
$824$	0	0
$825$	11.1140	0.386939
$826$	0	0
$827$	− 25.6048i	− 0.890367i	−0.895439	−	0.445183i	$-0.853138\pi$
$827$	− 25.6048i	− 0.890367i	0.895439	−	0.445183i	$-0.146862\pi$
$828$	0	0
$829$	19.0350i	0.661114i	0.943786	+	0.330557i	$0.107237\pi$
$829$	19.0350i	0.661114i	−0.943786	+	0.330557i	$0.892763\pi$
$830$	0	0
$831$	−27.0223	−0.937394
$832$	0	0
$833$	35.6622	1.23562
$834$	0	0
$835$	26.1052i	0.903408i
$836$	0	0
$837$	1.27125i	0.0439408i
$838$	0	0
$839$	15.6848	0.541501	0.270750	−	0.962650i	$-0.412728\pi$
$839$	15.6848	0.541501	0.270750	+	0.962650i	$0.412728\pi$
$840$	0	0
$841$	−24.1392	−0.832385
$842$	0	0
$843$	− 36.0758i	− 1.24252i
$844$	0	0
$845$	19.8761i	0.683758i
$846$	0	0
$847$	−34.5230	−1.18623
$848$	0	0
$849$	71.0082	2.43699
$850$	0	0
$851$	7.48228i	0.256489i
$852$	0	0
$853$	− 3.57224i	− 0.122311i	−0.998128	−	0.0611555i	$-0.980521\pi$
$853$	− 3.57224i	− 0.122311i	0.998128	−	0.0611555i	$-0.0194786\pi$
$854$	0	0
$855$	5.98868	0.204808
$856$	0	0
$857$	−56.7309	−1.93789	−0.968945	−	0.247276i	$-0.920464\pi$
$857$	−56.7309	−1.93789	−0.968945	+	0.247276i	$0.920464\pi$
$858$	0	0
$859$	− 18.6059i	− 0.634825i	−0.948288	−	0.317412i	$-0.897186\pi$
$859$	− 18.6059i	− 0.634825i	0.948288	−	0.317412i	$-0.102814\pi$
$860$	0	0
$861$	− 51.2713i	− 1.74732i
$862$	0	0
$863$	−17.9395	−0.610666	−0.305333	−	0.952246i	$-0.598768\pi$
$863$	−17.9395	−0.610666	−0.305333	+	0.952246i	$0.598768\pi$
$864$	0	0
$865$	−37.3612	−1.27032
$866$	0	0
$867$	20.7819i	0.705790i
$868$	0	0
$869$	1.14341i	0.0387874i
$870$	0	0
$871$	−5.53858	−0.187668
$872$	0	0
$873$	39.6537	1.34208
$874$	0	0
$875$	− 51.7114i	− 1.74816i
$876$	0	0
$877$	− 57.6201i	− 1.94569i	−0.231455	−	0.972846i	$-0.574349\pi$
$877$	− 57.6201i	− 1.94569i	0.231455	−	0.972846i	$-0.425651\pi$
$878$	0	0
$879$	−26.3205	−0.887767
$880$	0	0
$881$	15.3431	0.516923	0.258461	−	0.966022i	$-0.416785\pi$
$881$	15.3431	0.516923	0.258461	+	0.966022i	$0.416785\pi$
$882$	0	0
$883$	17.8608i	0.601065i	0.953772	+	0.300532i	$0.0971644\pi$
$883$	17.8608i	0.601065i	−0.953772	+	0.300532i	$0.902836\pi$
$884$	0	0
$885$	− 28.0085i	− 0.941495i
$886$	0	0
$887$	49.1445	1.65011	0.825055	−	0.565053i	$-0.191144\pi$
$887$	49.1445	1.65011	0.825055	+	0.565053i	$0.191144\pi$
$888$	0	0
$889$	−32.8056	−1.10026
$890$	0	0
$891$	− 10.8311i	− 0.362855i
$892$	0	0
$893$	7.59774i	0.254249i
$894$	0	0
$895$	−7.25601	−0.242542
$896$	0	0
$897$	−3.77796	−0.126143
$898$	0	0
$899$	4.76771i	0.159012i
$900$	0	0
$901$	19.7932i	0.659407i
$902$	0	0
$903$	117.866	3.92233
$904$	0	0
$905$	19.3165	0.642104
$906$	0	0
$907$	9.19971i	0.305471i	0.988267	+	0.152736i	$0.0488083\pi$
$907$	9.19971i	0.305471i	−0.988267	+	0.152736i	$0.951192\pi$
$908$	0	0
$909$	18.0085i	0.597303i
$910$	0	0
$911$	25.2939	0.838024	0.419012	−	0.907981i	$-0.362376\pi$
$911$	25.2939	0.838024	0.419012	+	0.907981i	$0.362376\pi$
$912$	0	0
$913$	−21.9915	−0.727813
$914$	0	0
$915$	− 37.7454i	− 1.24782i
$916$	0	0
$917$	0.213882i	0.00706301i
$918$	0	0
$919$	−44.8155	−1.47833	−0.739164	−	0.673525i	$-0.764779\pi$
$919$	−44.8155	−1.47833	−0.739164	+	0.673525i	$0.764779\pi$
$920$	0	0
$921$	−44.4752	−1.46551
$922$	0	0
$923$	− 1.27125i	− 0.0418437i
$924$	0	0
$925$	9.42068i	0.309750i
$926$	0	0
$927$	−47.1502	−1.54861
$928$	0	0
$929$	−5.65119	−0.185409	−0.0927047	−	0.995694i	$-0.529551\pi$
$929$	−5.65119	−0.185409	−0.0927047	+	0.995694i	$0.529551\pi$
$930$	0	0
$931$	11.8874i	0.389594i
$932$	0	0
$933$	81.1544i	2.65688i
$934$	0	0
$935$	−8.37962	−0.274043
$936$	0	0
$937$	8.58641	0.280506	0.140253	−	0.990116i	$-0.455208\pi$
$937$	8.58641	0.280506	0.140253	+	0.990116i	$0.455208\pi$
$938$	0	0
$939$	− 65.5386i	− 2.13877i
$940$	0	0
$941$	− 34.8155i	− 1.13495i	−0.823389	−	0.567477i	$-0.807920\pi$
$941$	− 34.8155i	− 1.13495i	0.823389	−	0.567477i	$-0.192080\pi$
$942$	0	0
$943$	−8.82718	−0.287452
$944$	0	0
$945$	13.4965	0.439040
$946$	0	0
$947$	18.6551i	0.606209i	0.952957	+	0.303105i	$0.0980232\pi$
$947$	18.6551i	0.606209i	−0.952957	+	0.303105i	$0.901977\pi$
$948$	0	0
$949$	− 12.4841i	− 0.405250i
$950$	0	0
$951$	30.2447	0.980751
$952$	0	0
$953$	−37.7227	−1.22196	−0.610980	−	0.791646i	$-0.709224\pi$
$953$	−37.7227	−1.22196	−0.610980	+	0.791646i	$0.709224\pi$
$954$	0	0
$955$	20.0492i	0.648777i
$956$	0	0
$957$	33.1055i	1.07015i
$958$	0	0
$959$	75.5804	2.44062
$960$	0	0
$961$	−30.5722	−0.986201
$962$	0	0
$963$	− 20.3909i	− 0.657089i
$964$	0	0
$965$	17.1434i	0.551866i
$966$	0	0
$967$	26.3683	0.847947	0.423974	−	0.905675i	$-0.360635\pi$
$967$	26.3683	0.847947	0.423974	+	0.905675i	$0.360635\pi$
$968$	0	0
$969$	7.79321	0.250354
$970$	0	0
$971$	0.978737i	0.0314092i	0.999877	+	0.0157046i	$0.00499913\pi$
$971$	0.978737i	0.0314092i	−0.999877	+	0.0157046i	$0.995001\pi$
$972$	0	0
$973$	35.8619i	1.14968i
$974$	0	0
$975$	−4.75670	−0.152336
$976$	0	0
$977$	−0.254623	−0.00814611	−0.00407306	−	0.999992i	$-0.501296\pi$
$977$	−0.254623	−0.00814611	−0.00407306	+	0.999992i	$0.501296\pi$
$978$	0	0
$979$	25.4217i	0.812482i
$980$	0	0
$981$	55.4217i	1.76948i
$982$	0	0
$983$	49.3754	1.57483	0.787415	−	0.616423i	$-0.211419\pi$
$983$	49.3754	1.57483	0.787415	+	0.616423i	$0.211419\pi$
$984$	0	0
$985$	−14.9929	−0.477714
$986$	0	0
$987$	85.7759i	2.73028i
$988$	0	0
$989$	− 20.2925i	− 0.645264i
$990$	0	0
$991$	37.6101	1.19473	0.597363	−	0.801971i	$-0.296215\pi$
$991$	37.6101	1.19473	0.597363	+	0.801971i	$0.296215\pi$
$992$	0	0
$993$	32.7256	1.03851
$994$	0	0
$995$	− 17.9593i	− 0.569347i
$996$	0	0
$997$	19.3867i	0.613983i	0.951712	+	0.306992i	$0.0993223\pi$
$997$	19.3867i	0.613983i	−0.951712	+	0.306992i	$0.900678\pi$
$998$	0	0
$999$	−7.48228	−0.236729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.f.1217.2	✓	6
4.3	odd	2		2432.2.c.g.1217.5	yes	6
8.3	odd	2		2432.2.c.g.1217.2	yes	6
8.5	even	2	inner	2432.2.c.f.1217.5	yes	6
16.3	odd	4		4864.2.a.bc.1.1		3
16.5	even	4		4864.2.a.bd.1.1		3
16.11	odd	4		4864.2.a.bf.1.3		3
16.13	even	4		4864.2.a.be.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.f.1217.2	✓	6	1.1	even	1	trivial
2432.2.c.f.1217.5	yes	6	8.5	even	2	inner
2432.2.c.g.1217.2	yes	6	8.3	odd	2
2432.2.c.g.1217.5	yes	6	4.3	odd	2
4864.2.a.bc.1.1		3	16.3	odd	4
4864.2.a.bd.1.1		3	16.5	even	4
4864.2.a.be.1.3		3	16.13	even	4
4864.2.a.bf.1.3		3	16.11	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:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.59774i q^{3} +1.59774i q^{5} -4.34596 q^{7} -3.74823 q^{9} +O(q^{10})\)	\(q-2.59774i q^{3} +1.59774i q^{5} -4.34596 q^{7} -3.74823 q^{9} +1.74823i q^{11} -0.748228i q^{13} +4.15049 q^{15} +3.00000 q^{17} +1.00000i q^{19} +11.2897i q^{21} +1.94370 q^{23} +2.44724 q^{25} +1.94370i q^{27} +7.28966i q^{29} +0.654037 q^{31} +4.54143 q^{33} -6.94370i q^{35} +3.84951i q^{37} -1.94370 q^{39} -4.54143 q^{41} -10.4402i q^{43} -5.98868i q^{45} +7.59774 q^{47} +11.8874 q^{49} -7.79321i q^{51} +6.59774i q^{53} -2.79321 q^{55} +2.59774 q^{57} -6.74823i q^{59} -9.09419i q^{61} +16.2897 q^{63} +1.19547 q^{65} -7.40226i q^{67} -5.04921i q^{69} +1.69901 q^{71} +16.6848 q^{73} -6.35729i q^{75} -7.59774i q^{77} +0.654037 q^{79} -6.19547 q^{81} +12.5793i q^{83} +4.79321i q^{85} +18.9366 q^{87} +14.5414 q^{89} +3.25177i q^{91} -1.69901i q^{93} -1.59774 q^{95} -10.5793 q^{97} -6.55276i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{7} - 16 q^{9}+O(q^{10}) \) 6 * q - 6 * q^7 - 16 * q^9	\( 6 q - 6 q^{7} - 16 q^{9} + 32 q^{15} + 18 q^{17} - 22 q^{23} - 6 q^{25} + 24 q^{31} - 20 q^{33} + 22 q^{39} + 20 q^{41} + 32 q^{47} + 4 q^{49} + 24 q^{55} + 2 q^{57} + 44 q^{63} - 20 q^{65} - 4 q^{71} + 34 q^{73} + 24 q^{79} - 10 q^{81} + 54 q^{87} + 40 q^{89} + 4 q^{95} + 44 q^{97}+O(q^{100}) \) 6 * q - 6 * q^7 - 16 * q^9 + 32 * q^15 + 18 * q^17 - 22 * q^23 - 6 * q^25 + 24 * q^31 - 20 * q^33 + 22 * q^39 + 20 * q^41 + 32 * q^47 + 4 * q^49 + 24 * q^55 + 2 * q^57 + 44 * q^63 - 20 * q^65 - 4 * q^71 + 34 * q^73 + 24 * q^79 - 10 * q^81 + 54 * q^87 + 40 * q^89 + 4 * q^95 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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