LMFDB - Embedded newform 2432.2.c.j.1217.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2432,2,Mod(1217,2432)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [20,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.4196177716$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 170x^{16} + 6593x^{12} + 64168x^{8} + 95760x^{4} + 4096 $ x^20 + 170x^16 + 6593x^12 + 64168x^8 + 95760x^4 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.10
Root			$0.323980 - 0.323980i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.j.1217.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.458177i q^{3} +3.00397i q^{5} +2.21624 q^{7} +2.79007 q^{9} +5.02382i q^{11} +5.22021i q^{13} +1.37635 q^{15} +7.73025 q^{17} -1.00000i q^{19} -1.01543i q^{21} -6.51613 q^{23} -4.02382 q^{25} -2.65288i q^{27} -2.23562i q^{29} +3.33566 q^{31} +2.30180 q^{33} +6.65751i q^{35} +1.45208i q^{37} +2.39178 q^{39} -3.16461 q^{41} -2.16101i q^{43} +8.38129i q^{45} -9.95256 q^{47} -2.08828 q^{49} -3.54182i q^{51} -6.66810i q^{53} -15.0914 q^{55} -0.458177 q^{57} -3.89126i q^{59} +7.05259i q^{61} +6.18347 q^{63} -15.6813 q^{65} -13.2767i q^{67} +2.98554i q^{69} -5.72840 q^{71} +5.89754 q^{73} +1.84362i q^{75} +11.1340i q^{77} +15.9369 q^{79} +7.15473 q^{81} -8.71734i q^{83} +23.2214i q^{85} -1.02431 q^{87} +3.79938 q^{89} +11.5692i q^{91} -1.52833i q^{93} +3.00397 q^{95} -3.94645 q^{97} +14.0168i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 28 q^{9} + 8 q^{17} - 20 q^{25} - 16 q^{33} + 24 q^{41} + 52 q^{49} - 8 q^{57} - 48 q^{65} - 24 q^{73} + 68 q^{81} + 40 q^{89} - 56 q^{97}+O(q^{100}) $ 20 * q - 28 * q^9 + 8 * q^17 - 20 * q^25 - 16 * q^33 + 24 * q^41 + 52 * q^49 - 8 * q^57 - 48 * q^65 - 24 * q^73 + 68 * q^81 + 40 * q^89 - 56 * q^97

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
$3$	− 0.458177i	− 0.264529i	−0.991214	−	0.132264i	$-0.957775\pi$
$3$	− 0.458177i	− 0.264529i	0.991214	−	0.132264i	$-0.0422248\pi$
$4$	0	0
$5$	3.00397i	1.34341i	0.740817	+	0.671707i	$0.234439\pi$
$5$	3.00397i	1.34341i	−0.740817	+	0.671707i	$0.765561\pi$
$6$	0	0
$7$	2.21624	0.837660	0.418830	−	0.908065i	$-0.362440\pi$
$7$	2.21624	0.837660	0.418830	+	0.908065i	$0.362440\pi$
$8$	0	0
$9$	2.79007	0.930025
$10$	0	0
$11$	5.02382i	1.51474i	0.652987	+	0.757369i	$0.273516\pi$
$11$	5.02382i	1.51474i	−0.652987	+	0.757369i	$0.726484\pi$
$12$	0	0
$13$	5.22021i	1.44783i	0.689892	+	0.723913i	$0.257658\pi$
$13$	5.22021i	1.44783i	−0.689892	+	0.723913i	$0.742342\pi$
$14$	0	0
$15$	1.37635	0.355372
$16$	0	0
$17$	7.73025	1.87486	0.937430	−	0.348174i	$-0.113198\pi$
$17$	7.73025	1.87486	0.937430	+	0.348174i	$0.113198\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	− 1.01543i	− 0.221585i
$22$	0	0
$23$	−6.51613	−1.35871	−0.679353	−	0.733811i	$-0.737740\pi$
$23$	−6.51613	−1.35871	−0.679353	+	0.733811i	$0.737740\pi$
$24$	0	0
$25$	−4.02382	−0.804764
$26$	0	0
$27$	− 2.65288i	− 0.510547i
$28$	0	0
$29$	− 2.23562i	− 0.415145i	−0.978220	−	0.207572i	$-0.933444\pi$
$29$	− 2.23562i	− 0.415145i	0.978220	−	0.207572i	$-0.0665563\pi$
$30$	0	0
$31$	3.33566	0.599103	0.299551	−	0.954080i	$-0.403163\pi$
$31$	3.33566	0.599103	0.299551	+	0.954080i	$0.403163\pi$
$32$	0	0
$33$	2.30180	0.400692
$34$	0	0
$35$	6.65751i	1.12533i
$36$	0	0
$37$	1.45208i	0.238720i	0.992851	+	0.119360i	$0.0380843\pi$
$37$	1.45208i	0.238720i	−0.992851	+	0.119360i	$0.961916\pi$
$38$	0	0
$39$	2.39178	0.382991
$40$	0	0
$41$	−3.16461	−0.494228	−0.247114	−	0.968986i	$-0.579482\pi$
$41$	−3.16461	−0.494228	−0.247114	+	0.968986i	$0.579482\pi$
$42$	0	0
$43$	− 2.16101i	− 0.329551i	−0.986331	−	0.164776i	$-0.947310\pi$
$43$	− 2.16101i	− 0.329551i	0.986331	−	0.164776i	$-0.0526900\pi$
$44$	0	0
$45$	8.38129i	1.24941i
$46$	0	0
$47$	−9.95256	−1.45173	−0.725865	−	0.687837i	$-0.758560\pi$
$47$	−9.95256	−1.45173	−0.725865	+	0.687837i	$0.758560\pi$
$48$	0	0
$49$	−2.08828	−0.298325
$50$	0	0
$51$	− 3.54182i	− 0.495954i
$52$	0	0
$53$	− 6.66810i	− 0.915935i	−0.888969	−	0.457967i	$-0.848578\pi$
$53$	− 6.66810i	− 0.915935i	0.888969	−	0.457967i	$-0.151422\pi$
$54$	0	0
$55$	−15.0914	−2.03492
$56$	0	0
$57$	−0.458177	−0.0606871
$58$	0	0
$59$	− 3.89126i	− 0.506599i	−0.967388	−	0.253299i	$-0.918484\pi$
$59$	− 3.89126i	− 0.506599i	0.967388	−	0.253299i	$-0.0815157\pi$
$60$	0	0
$61$	7.05259i	0.902991i	0.892273	+	0.451496i	$0.149109\pi$
$61$	7.05259i	0.902991i	−0.892273	+	0.451496i	$0.850891\pi$
$62$	0	0
$63$	6.18347	0.779044
$64$	0	0
$65$	−15.6813	−1.94503
$66$	0	0
$67$	− 13.2767i	− 1.62201i	−0.585041	−	0.811004i	$-0.698922\pi$
$67$	− 13.2767i	− 1.62201i	0.585041	−	0.811004i	$-0.301078\pi$
$68$	0	0
$69$	2.98554i	0.359417i
$70$	0	0
$71$	−5.72840	−0.679836	−0.339918	−	0.940455i	$-0.610399\pi$
$71$	−5.72840	−0.679836	−0.339918	+	0.940455i	$0.610399\pi$
$72$	0	0
$73$	5.89754	0.690254	0.345127	−	0.938556i	$-0.387836\pi$
$73$	5.89754	0.690254	0.345127	+	0.938556i	$0.387836\pi$
$74$	0	0
$75$	1.84362i	0.212883i
$76$	0	0
$77$	11.1340i	1.26884i
$78$	0	0
$79$	15.9369	1.79304	0.896522	−	0.442998i	$-0.146085\pi$
$79$	15.9369	1.79304	0.896522	+	0.442998i	$0.146085\pi$
$80$	0	0
$81$	7.15473	0.794970
$82$	0	0
$83$	− 8.71734i	− 0.956852i	−0.878128	−	0.478426i	$-0.841207\pi$
$83$	− 8.71734i	− 0.956852i	0.878128	−	0.478426i	$-0.158793\pi$
$84$	0	0
$85$	23.2214i	2.51871i
$86$	0	0
$87$	−1.02431	−0.109818
$88$	0	0
$89$	3.79938	0.402734	0.201367	−	0.979516i	$-0.435462\pi$
$89$	3.79938	0.402734	0.201367	+	0.979516i	$0.435462\pi$
$90$	0	0
$91$	11.5692i	1.21279i
$92$	0	0
$93$	− 1.52833i	− 0.158480i
$94$	0	0
$95$	3.00397	0.308201
$96$	0	0
$97$	−3.94645	−0.400701	−0.200351	−	0.979724i	$-0.564208\pi$
$97$	−3.94645	−0.400701	−0.200351	+	0.979724i	$0.564208\pi$
$98$	0	0
$99$	14.0168i	1.40874i
$100$	0	0
$101$	6.32025i	0.628888i	0.949276	+	0.314444i	$0.101818\pi$
$101$	6.32025i	0.628888i	−0.949276	+	0.314444i	$0.898182\pi$
$102$	0	0
$103$	4.71201	0.464288	0.232144	−	0.972681i	$-0.425426\pi$
$103$	4.71201	0.464288	0.232144	+	0.972681i	$0.425426\pi$
$104$	0	0
$105$	3.05032	0.297681
$106$	0	0
$107$	18.5550i	1.79378i	0.442249	+	0.896892i	$0.354181\pi$
$107$	18.5550i	1.79378i	−0.442249	+	0.896892i	$0.645819\pi$
$108$	0	0
$109$	− 9.25995i	− 0.886942i	−0.896289	−	0.443471i	$-0.853747\pi$
$109$	− 9.25995i	− 0.886942i	0.896289	−	0.443471i	$-0.146253\pi$
$110$	0	0
$111$	0.665309	0.0631483
$112$	0	0
$113$	−7.97386	−0.750118	−0.375059	−	0.927001i	$-0.622378\pi$
$113$	−7.97386	−0.750118	−0.375059	+	0.927001i	$0.622378\pi$
$114$	0	0
$115$	− 19.5742i	− 1.82531i
$116$	0	0
$117$	14.5648i	1.34651i
$118$	0	0
$119$	17.1321	1.57050
$120$	0	0
$121$	−14.2387	−1.29443
$122$	0	0
$123$	1.44995i	0.130738i
$124$	0	0
$125$	2.93242i	0.262284i
$126$	0	0
$127$	8.32912	0.739090	0.369545	−	0.929213i	$-0.379514\pi$
$127$	8.32912	0.739090	0.369545	+	0.929213i	$0.379514\pi$
$128$	0	0
$129$	−0.990126	−0.0871758
$130$	0	0
$131$	− 16.3831i	− 1.43140i	−0.698408	−	0.715700i	$-0.746108\pi$
$131$	− 16.3831i	− 1.43140i	0.698408	−	0.715700i	$-0.253892\pi$
$132$	0	0
$133$	− 2.21624i	− 0.192172i
$134$	0	0
$135$	7.96916	0.685876
$136$	0	0
$137$	−16.2267	−1.38634	−0.693172	−	0.720772i	$-0.743788\pi$
$137$	−16.2267	−1.38634	−0.693172	+	0.720772i	$0.743788\pi$
$138$	0	0
$139$	4.38904i	0.372273i	0.982524	+	0.186137i	$0.0595967\pi$
$139$	4.38904i	0.372273i	−0.982524	+	0.186137i	$0.940403\pi$
$140$	0	0
$141$	4.56004i	0.384024i
$142$	0	0
$143$	−26.2254	−2.19308
$144$	0	0
$145$	6.71574	0.557712
$146$	0	0
$147$	0.956802i	0.0789157i
$148$	0	0
$149$	1.28542i	0.105306i	0.998613	+	0.0526528i	$0.0167677\pi$
$149$	1.28542i	0.105306i	−0.998613	+	0.0526528i	$0.983232\pi$
$150$	0	0
$151$	−9.07295	−0.738346	−0.369173	−	0.929361i	$-0.620359\pi$
$151$	−9.07295	−0.738346	−0.369173	+	0.929361i	$0.620359\pi$
$152$	0	0
$153$	21.5680	1.74367
$154$	0	0
$155$	10.0202i	0.804844i
$156$	0	0
$157$	− 3.13656i	− 0.250325i	−0.992136	−	0.125162i	$-0.960055\pi$
$157$	− 3.13656i	− 0.250325i	0.992136	−	0.125162i	$-0.0399452\pi$
$158$	0	0
$159$	−3.05517	−0.242291
$160$	0	0
$161$	−14.4413	−1.13813
$162$	0	0
$163$	− 14.9640i	− 1.17207i	−0.810286	−	0.586035i	$-0.800688\pi$
$163$	− 14.9640i	− 1.17207i	0.810286	−	0.586035i	$-0.199312\pi$
$164$	0	0
$165$	6.91453i	0.538295i
$166$	0	0
$167$	21.0799	1.63121	0.815607	−	0.578606i	$-0.196403\pi$
$167$	21.0799	1.63121	0.815607	+	0.578606i	$0.196403\pi$
$168$	0	0
$169$	−14.2506	−1.09620
$170$	0	0
$171$	− 2.79007i	− 0.213362i
$172$	0	0
$173$	23.7808i	1.80802i	0.427510	+	0.904011i	$0.359391\pi$
$173$	23.7808i	1.80802i	−0.427510	+	0.904011i	$0.640609\pi$
$174$	0	0
$175$	−8.91775	−0.674118
$176$	0	0
$177$	−1.78289	−0.134010
$178$	0	0
$179$	− 14.3018i	− 1.06897i	−0.845179	−	0.534483i	$-0.820506\pi$
$179$	− 14.3018i	− 1.06897i	0.845179	−	0.534483i	$-0.179494\pi$
$180$	0	0
$181$	− 13.9014i	− 1.03328i	−0.856203	−	0.516640i	$-0.827182\pi$
$181$	− 13.9014i	− 1.03328i	0.856203	−	0.516640i	$-0.172818\pi$
$182$	0	0
$183$	3.23133	0.238867
$184$	0	0
$185$	−4.36199	−0.320700
$186$	0	0
$187$	38.8353i	2.83992i
$188$	0	0
$189$	− 5.87942i	− 0.427665i
$190$	0	0
$191$	−2.17747	−0.157557	−0.0787783	−	0.996892i	$-0.525102\pi$
$191$	−2.17747	−0.157557	−0.0787783	+	0.996892i	$0.525102\pi$
$192$	0	0
$193$	26.6410	1.91766	0.958831	−	0.283978i	$-0.0916541\pi$
$193$	26.6410	1.91766	0.958831	+	0.283978i	$0.0916541\pi$
$194$	0	0
$195$	7.18483i	0.514516i
$196$	0	0
$197$	20.8500i	1.48550i	0.669570	+	0.742749i	$0.266479\pi$
$197$	20.8500i	1.48550i	−0.669570	+	0.742749i	$0.733521\pi$
$198$	0	0
$199$	−4.24710	−0.301069	−0.150535	−	0.988605i	$-0.548099\pi$
$199$	−4.24710	−0.301069	−0.150535	+	0.988605i	$0.548099\pi$
$200$	0	0
$201$	−6.08308	−0.429068
$202$	0	0
$203$	− 4.95468i	− 0.347750i
$204$	0	0
$205$	− 9.50637i	− 0.663954i
$206$	0	0
$207$	−18.1805	−1.26363
$208$	0	0
$209$	5.02382	0.347505
$210$	0	0
$211$	18.1134i	1.24698i	0.781833	+	0.623488i	$0.214285\pi$
$211$	18.1134i	1.24698i	−0.781833	+	0.623488i	$0.785715\pi$
$212$	0	0
$213$	2.62462i	0.179836i
$214$	0	0
$215$	6.49161	0.442724
$216$	0	0
$217$	7.39263	0.501845
$218$	0	0
$219$	− 2.70212i	− 0.182592i
$220$	0	0
$221$	40.3535i	2.71447i
$222$	0	0
$223$	3.71952	0.249078	0.124539	−	0.992215i	$-0.460255\pi$
$223$	3.71952	0.249078	0.124539	+	0.992215i	$0.460255\pi$
$224$	0	0
$225$	−11.2267	−0.748450
$226$	0	0
$227$	17.8318i	1.18354i	0.806108	+	0.591769i	$0.201570\pi$
$227$	17.8318i	1.18354i	−0.806108	+	0.591769i	$0.798430\pi$
$228$	0	0
$229$	13.5159i	0.893158i	0.894744	+	0.446579i	$0.147358\pi$
$229$	13.5159i	0.893158i	−0.894744	+	0.446579i	$0.852642\pi$
$230$	0	0
$231$	5.10134	0.335644
$232$	0	0
$233$	18.5061	1.21238	0.606189	−	0.795321i	$-0.292698\pi$
$233$	18.5061	1.21238	0.606189	+	0.795321i	$0.292698\pi$
$234$	0	0
$235$	− 29.8972i	− 1.95028i
$236$	0	0
$237$	− 7.30194i	− 0.474312i
$238$	0	0
$239$	−11.6874	−0.755995	−0.377997	−	0.925807i	$-0.623387\pi$
$239$	−11.6874	−0.755995	−0.377997	+	0.925807i	$0.623387\pi$
$240$	0	0
$241$	10.2701	0.661554	0.330777	−	0.943709i	$-0.392689\pi$
$241$	10.2701	0.661554	0.330777	+	0.943709i	$0.392689\pi$
$242$	0	0
$243$	− 11.2368i	− 0.720839i
$244$	0	0
$245$	− 6.27312i	− 0.400775i
$246$	0	0
$247$	5.22021	0.332154
$248$	0	0
$249$	−3.99409	−0.253115
$250$	0	0
$251$	25.1064i	1.58470i	0.610066	+	0.792350i	$0.291143\pi$
$251$	25.1064i	1.58470i	−0.610066	+	0.792350i	$0.708857\pi$
$252$	0	0
$253$	− 32.7358i	− 2.05808i
$254$	0	0
$255$	10.6395	0.666273
$256$	0	0
$257$	12.5905	0.785374	0.392687	−	0.919672i	$-0.371546\pi$
$257$	12.5905	0.785374	0.392687	+	0.919672i	$0.371546\pi$
$258$	0	0
$259$	3.21815i	0.199966i
$260$	0	0
$261$	− 6.23755i	− 0.386095i
$262$	0	0
$263$	−22.7749	−1.40436	−0.702181	−	0.711999i	$-0.747790\pi$
$263$	−22.7749	−1.40436	−0.702181	+	0.711999i	$0.747790\pi$
$264$	0	0
$265$	20.0308	1.23048
$266$	0	0
$267$	− 1.74079i	− 0.106535i
$268$	0	0
$269$	− 17.0271i	− 1.03816i	−0.854725	−	0.519081i	$-0.826274\pi$
$269$	− 17.0271i	− 1.03816i	0.854725	−	0.519081i	$-0.173726\pi$
$270$	0	0
$271$	−8.19591	−0.497866	−0.248933	−	0.968521i	$-0.580080\pi$
$271$	−8.19591	−0.497866	−0.248933	+	0.968521i	$0.580080\pi$
$272$	0	0
$273$	5.30076	0.320817
$274$	0	0
$275$	− 20.2149i	− 1.21901i
$276$	0	0
$277$	− 18.7174i	− 1.12462i	−0.826927	−	0.562309i	$-0.809913\pi$
$277$	− 18.7174i	− 1.12462i	0.826927	−	0.562309i	$-0.190087\pi$
$278$	0	0
$279$	9.30675	0.557180
$280$	0	0
$281$	−0.189116	−0.0112817	−0.00564087	−	0.999984i	$-0.501796\pi$
$281$	−0.189116	−0.0112817	−0.00564087	+	0.999984i	$0.501796\pi$
$282$	0	0
$283$	10.6357i	0.632226i	0.948722	+	0.316113i	$0.102378\pi$
$283$	10.6357i	0.632226i	−0.948722	+	0.316113i	$0.897622\pi$
$284$	0	0
$285$	− 1.37635i	− 0.0815279i
$286$	0	0
$287$	−7.01353	−0.413995
$288$	0	0
$289$	42.7567	2.51510
$290$	0	0
$291$	1.80817i	0.105997i
$292$	0	0
$293$	− 27.6765i	− 1.61688i	−0.588580	−	0.808439i	$-0.700313\pi$
$293$	− 27.6765i	− 1.61688i	0.588580	−	0.808439i	$-0.299687\pi$
$294$	0	0
$295$	11.6892	0.680572
$296$	0	0
$297$	13.3276	0.773345
$298$	0	0
$299$	− 34.0155i	− 1.96717i
$300$	0	0
$301$	− 4.78932i	− 0.276052i
$302$	0	0
$303$	2.89579	0.166359
$304$	0	0
$305$	−21.1857	−1.21309
$306$	0	0
$307$	− 8.14707i	− 0.464978i	−0.972599	−	0.232489i	$-0.925313\pi$
$307$	− 8.14707i	− 0.464978i	0.972599	−	0.232489i	$-0.0746869\pi$
$308$	0	0
$309$	− 2.15894i	− 0.122818i
$310$	0	0
$311$	11.1283	0.631030	0.315515	−	0.948921i	$-0.397823\pi$
$311$	11.1283	0.631030	0.315515	+	0.948921i	$0.397823\pi$
$312$	0	0
$313$	10.2647	0.580198	0.290099	−	0.956997i	$-0.406312\pi$
$313$	10.2647	0.580198	0.290099	+	0.956997i	$0.406312\pi$
$314$	0	0
$315$	18.5750i	1.04658i
$316$	0	0
$317$	− 7.14866i	− 0.401509i	−0.979642	−	0.200754i	$-0.935661\pi$
$317$	− 7.14866i	− 0.401509i	0.979642	−	0.200754i	$-0.0643393\pi$
$318$	0	0
$319$	11.2314	0.628836
$320$	0	0
$321$	8.50150	0.474508
$322$	0	0
$323$	− 7.73025i	− 0.430122i
$324$	0	0
$325$	− 21.0052i	− 1.16516i
$326$	0	0
$327$	−4.24270	−0.234622
$328$	0	0
$329$	−22.0573	−1.21606
$330$	0	0
$331$	− 28.4370i	− 1.56304i	−0.623881	−	0.781519i	$-0.714445\pi$
$331$	− 28.4370i	− 1.56304i	0.623881	−	0.781519i	$-0.285555\pi$
$332$	0	0
$333$	4.05140i	0.222016i
$334$	0	0
$335$	39.8828	2.17903
$336$	0	0
$337$	−18.9197	−1.03062	−0.515311	−	0.857003i	$-0.672324\pi$
$337$	−18.9197	−1.03062	−0.515311	+	0.857003i	$0.672324\pi$
$338$	0	0
$339$	3.65344i	0.198428i
$340$	0	0
$341$	16.7578i	0.907484i
$342$	0	0
$343$	−20.1418	−1.08756
$344$	0	0
$345$	−8.96847	−0.482846
$346$	0	0
$347$	2.34140i	0.125693i	0.998023	+	0.0628466i	$0.0200179\pi$
$347$	2.34140i	0.125693i	−0.998023	+	0.0628466i	$0.979982\pi$
$348$	0	0
$349$	14.8506i	0.794935i	0.917616	+	0.397468i	$0.130111\pi$
$349$	14.8506i	0.794935i	−0.917616	+	0.397468i	$0.869889\pi$
$350$	0	0
$351$	13.8486	0.739183
$352$	0	0
$353$	−16.9690	−0.903168	−0.451584	−	0.892229i	$-0.649141\pi$
$353$	−16.9690	−0.903168	−0.451584	+	0.892229i	$0.649141\pi$
$354$	0	0
$355$	− 17.2079i	− 0.913302i
$356$	0	0
$357$	− 7.84953i	− 0.415441i
$358$	0	0
$359$	32.2107	1.70002	0.850008	−	0.526770i	$-0.176597\pi$
$359$	32.2107	1.70002	0.850008	+	0.526770i	$0.176597\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	6.52387i	0.342414i
$364$	0	0
$365$	17.7160i	0.927298i
$366$	0	0
$367$	−10.2366	−0.534347	−0.267173	−	0.963648i	$-0.586090\pi$
$367$	−10.2366	−0.534347	−0.267173	+	0.963648i	$0.586090\pi$
$368$	0	0
$369$	−8.82948	−0.459644
$370$	0	0
$371$	− 14.7781i	− 0.767242i
$372$	0	0
$373$	− 30.1908i	− 1.56322i	−0.623767	−	0.781610i	$-0.714399\pi$
$373$	− 30.1908i	− 1.56322i	0.623767	−	0.781610i	$-0.285601\pi$
$374$	0	0
$375$	1.34357	0.0693815
$376$	0	0
$377$	11.6704	0.601057
$378$	0	0
$379$	− 15.9865i	− 0.821173i	−0.911822	−	0.410586i	$-0.865324\pi$
$379$	− 15.9865i	− 0.821173i	0.911822	−	0.410586i	$-0.134676\pi$
$380$	0	0
$381$	− 3.81622i	− 0.195511i
$382$	0	0
$383$	−22.6961	−1.15971	−0.579857	−	0.814718i	$-0.696892\pi$
$383$	−22.6961	−1.15971	−0.579857	+	0.814718i	$0.696892\pi$
$384$	0	0
$385$	−33.4461	−1.70457
$386$	0	0
$387$	− 6.02938i	− 0.306491i
$388$	0	0
$389$	12.1956i	0.618341i	0.951007	+	0.309170i	$0.100051\pi$
$389$	12.1956i	0.618341i	−0.951007	+	0.309170i	$0.899949\pi$
$390$	0	0
$391$	−50.3713	−2.54738
$392$	0	0
$393$	−7.50638	−0.378647
$394$	0	0
$395$	47.8740i	2.40880i
$396$	0	0
$397$	− 10.7422i	− 0.539137i	−0.962981	−	0.269568i	$-0.913119\pi$
$397$	− 10.7422i	− 0.539137i	0.962981	−	0.269568i	$-0.0868810\pi$
$398$	0	0
$399$	−1.01543	−0.0508351
$400$	0	0
$401$	−5.65983	−0.282639	−0.141319	−	0.989964i	$-0.545134\pi$
$401$	−5.65983	−0.282639	−0.141319	+	0.989964i	$0.545134\pi$
$402$	0	0
$403$	17.4129i	0.867396i
$404$	0	0
$405$	21.4926i	1.06797i
$406$	0	0
$407$	−7.29497	−0.361598
$408$	0	0
$409$	27.6645	1.36792	0.683960	−	0.729520i	$-0.260256\pi$
$409$	27.6645	1.36792	0.683960	+	0.729520i	$0.260256\pi$
$410$	0	0
$411$	7.43473i	0.366728i
$412$	0	0
$413$	− 8.62396i	− 0.424357i
$414$	0	0
$415$	26.1866	1.28545
$416$	0	0
$417$	2.01096	0.0984770
$418$	0	0
$419$	6.91763i	0.337948i	0.985620	+	0.168974i	$0.0540454\pi$
$419$	6.91763i	0.337948i	−0.985620	+	0.168974i	$0.945955\pi$
$420$	0	0
$421$	− 3.07015i	− 0.149630i	−0.997197	−	0.0748150i	$-0.976163\pi$
$421$	− 3.07015i	− 0.149630i	0.997197	−	0.0748150i	$-0.0238366\pi$
$422$	0	0
$423$	−27.7684	−1.35014
$424$	0	0
$425$	−31.1051	−1.50882
$426$	0	0
$427$	15.6302i	0.756400i
$428$	0	0
$429$	12.0159i	0.580132i
$430$	0	0
$431$	13.2015	0.635893	0.317946	−	0.948109i	$-0.397007\pi$
$431$	13.2015	0.635893	0.317946	+	0.948109i	$0.397007\pi$
$432$	0	0
$433$	−20.1544	−0.968559	−0.484279	−	0.874913i	$-0.660918\pi$
$433$	−20.1544	−0.968559	−0.484279	+	0.874913i	$0.660918\pi$
$434$	0	0
$435$	− 3.07700i	− 0.147531i
$436$	0	0
$437$	6.51613i	0.311709i
$438$	0	0
$439$	1.72144	0.0821601	0.0410800	−	0.999156i	$-0.486920\pi$
$439$	1.72144	0.0821601	0.0410800	+	0.999156i	$0.486920\pi$
$440$	0	0
$441$	−5.82645	−0.277450
$442$	0	0
$443$	38.0168i	1.80623i	0.429395	+	0.903117i	$0.358727\pi$
$443$	38.0168i	1.80623i	−0.429395	+	0.903117i	$0.641273\pi$
$444$	0	0
$445$	11.4132i	0.541039i
$446$	0	0
$447$	0.588950	0.0278564
$448$	0	0
$449$	−27.1861	−1.28299	−0.641496	−	0.767127i	$-0.721686\pi$
$449$	−27.1861	−1.28299	−0.641496	+	0.767127i	$0.721686\pi$
$450$	0	0
$451$	− 15.8984i	− 0.748626i
$452$	0	0
$453$	4.15702i	0.195314i
$454$	0	0
$455$	−34.7536	−1.62927
$456$	0	0
$457$	−15.8069	−0.739415	−0.369708	−	0.929148i	$-0.620542\pi$
$457$	−15.8069	−0.739415	−0.369708	+	0.929148i	$0.620542\pi$
$458$	0	0
$459$	− 20.5074i	− 0.957204i
$460$	0	0
$461$	− 29.7741i	− 1.38672i	−0.720593	−	0.693358i	$-0.756130\pi$
$461$	− 29.7741i	− 1.38672i	0.720593	−	0.693358i	$-0.243870\pi$
$462$	0	0
$463$	−11.8339	−0.549967	−0.274984	−	0.961449i	$-0.588672\pi$
$463$	−11.8339	−0.549967	−0.274984	+	0.961449i	$0.588672\pi$
$464$	0	0
$465$	4.59104	0.212904
$466$	0	0
$467$	− 30.6052i	− 1.41624i	−0.706091	−	0.708121i	$-0.749543\pi$
$467$	− 30.6052i	− 1.41624i	0.706091	−	0.708121i	$-0.250457\pi$
$468$	0	0
$469$	− 29.4244i	− 1.35869i
$470$	0	0
$471$	−1.43710	−0.0662181
$472$	0	0
$473$	10.8565	0.499184
$474$	0	0
$475$	4.02382i	0.184625i
$476$	0	0
$477$	− 18.6045i	− 0.851842i
$478$	0	0
$479$	−12.3908	−0.566150	−0.283075	−	0.959098i	$-0.591355\pi$
$479$	−12.3908	−0.566150	−0.283075	+	0.959098i	$0.591355\pi$
$480$	0	0
$481$	−7.58015	−0.345625
$482$	0	0
$483$	6.61668i	0.301069i
$484$	0	0
$485$	− 11.8550i	− 0.538308i
$486$	0	0
$487$	7.61617	0.345122	0.172561	−	0.984999i	$-0.444796\pi$
$487$	7.61617	0.345122	0.172561	+	0.984999i	$0.444796\pi$
$488$	0	0
$489$	−6.85616	−0.310046
$490$	0	0
$491$	− 3.09291i	− 0.139581i	−0.997562	−	0.0697906i	$-0.977767\pi$
$491$	− 3.09291i	− 0.139581i	0.997562	−	0.0697906i	$-0.0222331\pi$
$492$	0	0
$493$	− 17.2819i	− 0.778338i
$494$	0	0
$495$	−42.1061	−1.89253
$496$	0	0
$497$	−12.6955	−0.569472
$498$	0	0
$499$	6.55042i	0.293237i	0.989193	+	0.146618i	$0.0468390\pi$
$499$	6.55042i	0.293237i	−0.989193	+	0.146618i	$0.953161\pi$
$500$	0	0
$501$	− 9.65835i	− 0.431503i
$502$	0	0
$503$	34.9332	1.55760	0.778798	−	0.627275i	$-0.215830\pi$
$503$	34.9332	1.55760	0.778798	+	0.627275i	$0.215830\pi$
$504$	0	0
$505$	−18.9858	−0.844858
$506$	0	0
$507$	6.52928i	0.289976i
$508$	0	0
$509$	10.6461i	0.471882i	0.971767	+	0.235941i	$0.0758171\pi$
$509$	10.6461i	0.471882i	−0.971767	+	0.235941i	$0.924183\pi$
$510$	0	0
$511$	13.0704	0.578199
$512$	0	0
$513$	−2.65288	−0.117128
$514$	0	0
$515$	14.1547i	0.623732i
$516$	0	0
$517$	− 49.9999i	− 2.19899i
$518$	0	0
$519$	10.8958	0.478274
$520$	0	0
$521$	33.7087	1.47681	0.738403	−	0.674359i	$-0.235580\pi$
$521$	33.7087	1.47681	0.738403	+	0.674359i	$0.235580\pi$
$522$	0	0
$523$	− 3.61224i	− 0.157952i	−0.996877	−	0.0789761i	$-0.974835\pi$
$523$	− 3.61224i	− 0.157952i	0.996877	−	0.0789761i	$-0.0251651\pi$
$524$	0	0
$525$	4.08591i	0.178324i
$526$	0	0
$527$	25.7855	1.12323
$528$	0	0
$529$	19.4599	0.846084
$530$	0	0
$531$	− 10.8569i	− 0.471149i
$532$	0	0
$533$	− 16.5199i	− 0.715556i
$534$	0	0
$535$	−55.7388	−2.40980
$536$	0	0
$537$	−6.55276	−0.282772
$538$	0	0
$539$	− 10.4911i	− 0.451885i
$540$	0	0
$541$	− 45.1835i	− 1.94259i	−0.237876	−	0.971295i	$-0.576451\pi$
$541$	− 45.1835i	− 1.94259i	0.237876	−	0.971295i	$-0.423549\pi$
$542$	0	0
$543$	−6.36929	−0.273332
$544$	0	0
$545$	27.8166	1.19153
$546$	0	0
$547$	25.9716i	1.11047i	0.831695	+	0.555233i	$0.187371\pi$
$547$	25.9716i	1.11047i	−0.831695	+	0.555233i	$0.812629\pi$
$548$	0	0
$549$	19.6772i	0.839804i
$550$	0	0
$551$	−2.23562	−0.0952408
$552$	0	0
$553$	35.3201	1.50196
$554$	0	0
$555$	1.99857i	0.0848344i
$556$	0	0
$557$	− 27.5515i	− 1.16739i	−0.811972	−	0.583697i	$-0.801606\pi$
$557$	− 27.5515i	− 1.16739i	0.811972	−	0.583697i	$-0.198394\pi$
$558$	0	0
$559$	11.2809	0.477132
$560$	0	0
$561$	17.7935	0.751241
$562$	0	0
$563$	14.8531i	0.625985i	0.949756	+	0.312992i	$0.101331\pi$
$563$	14.8531i	0.625985i	−0.949756	+	0.312992i	$0.898669\pi$
$564$	0	0
$565$	− 23.9532i	− 1.00772i
$566$	0	0
$567$	15.8566	0.665915
$568$	0	0
$569$	4.86281	0.203859	0.101930	−	0.994792i	$-0.467498\pi$
$569$	4.86281	0.203859	0.101930	+	0.994792i	$0.467498\pi$
$570$	0	0
$571$	21.6960i	0.907948i	0.891015	+	0.453974i	$0.149994\pi$
$571$	21.6960i	0.907948i	−0.891015	+	0.453974i	$0.850006\pi$
$572$	0	0
$573$	0.997669i	0.0416782i
$574$	0	0
$575$	26.2197	1.09344
$576$	0	0
$577$	28.5083	1.18682	0.593409	−	0.804901i	$-0.297782\pi$
$577$	28.5083	1.18682	0.593409	+	0.804901i	$0.297782\pi$
$578$	0	0
$579$	− 12.2063i	− 0.507277i
$580$	0	0
$581$	− 19.3197i	− 0.801517i
$582$	0	0
$583$	33.4993	1.38740
$584$	0	0
$585$	−43.7521	−1.80893
$586$	0	0
$587$	− 22.5133i	− 0.929224i	−0.885514	−	0.464612i	$-0.846194\pi$
$587$	− 22.5133i	− 0.929224i	0.885514	−	0.464612i	$-0.153806\pi$
$588$	0	0
$589$	− 3.33566i	− 0.137444i
$590$	0	0
$591$	9.55298	0.392957
$592$	0	0
$593$	−11.0209	−0.452574	−0.226287	−	0.974061i	$-0.572659\pi$
$593$	−11.0209	−0.452574	−0.226287	+	0.974061i	$0.572659\pi$
$594$	0	0
$595$	51.4642i	2.10983i
$596$	0	0
$597$	1.94593i	0.0796414i
$598$	0	0
$599$	−18.7308	−0.765319	−0.382659	−	0.923890i	$-0.624992\pi$
$599$	−18.7308	−0.765319	−0.382659	+	0.923890i	$0.624992\pi$
$600$	0	0
$601$	−14.8859	−0.607209	−0.303604	−	0.952798i	$-0.598190\pi$
$601$	−14.8859	−0.607209	−0.303604	+	0.952798i	$0.598190\pi$
$602$	0	0
$603$	− 37.0430i	− 1.50851i
$604$	0	0
$605$	− 42.7727i	− 1.73896i
$606$	0	0
$607$	−37.9701	−1.54116	−0.770579	−	0.637345i	$-0.780033\pi$
$607$	−37.9701	−1.54116	−0.770579	+	0.637345i	$0.780033\pi$
$608$	0	0
$609$	−2.27012	−0.0919900
$610$	0	0
$611$	− 51.9544i	− 2.10185i
$612$	0	0
$613$	28.5379i	1.15264i	0.817226	+	0.576318i	$0.195511\pi$
$613$	28.5379i	1.15264i	−0.817226	+	0.576318i	$0.804489\pi$
$614$	0	0
$615$	−4.35560	−0.175635
$616$	0	0
$617$	−3.91835	−0.157747	−0.0788733	−	0.996885i	$-0.525132\pi$
$617$	−3.91835	−0.157747	−0.0788733	+	0.996885i	$0.525132\pi$
$618$	0	0
$619$	− 5.37059i	− 0.215862i	−0.994158	−	0.107931i	$-0.965577\pi$
$619$	− 5.37059i	− 0.215862i	0.994158	−	0.107931i	$-0.0344226\pi$
$620$	0	0
$621$	17.2865i	0.693684i
$622$	0	0
$623$	8.42035	0.337354
$624$	0	0
$625$	−28.9280	−1.15712
$626$	0	0
$627$	− 2.30180i	− 0.0919250i
$628$	0	0
$629$	11.2249i	0.447567i
$630$	0	0
$631$	18.6483	0.742377	0.371189	−	0.928557i	$-0.378950\pi$
$631$	18.6483	0.742377	0.371189	+	0.928557i	$0.378950\pi$
$632$	0	0
$633$	8.29913	0.329861
$634$	0	0
$635$	25.0204i	0.992905i
$636$	0	0
$637$	− 10.9012i	− 0.431923i
$638$	0	0
$639$	−15.9827	−0.632264
$640$	0	0
$641$	42.2857	1.67018	0.835092	−	0.550111i	$-0.185415\pi$
$641$	42.2857	1.67018	0.835092	+	0.550111i	$0.185415\pi$
$642$	0	0
$643$	18.4426i	0.727305i	0.931535	+	0.363652i	$0.118470\pi$
$643$	18.4426i	0.727305i	−0.931535	+	0.363652i	$0.881530\pi$
$644$	0	0
$645$	− 2.97431i	− 0.117113i
$646$	0	0
$647$	2.19012	0.0861024	0.0430512	−	0.999073i	$-0.486292\pi$
$647$	2.19012	0.0861024	0.0430512	+	0.999073i	$0.486292\pi$
$648$	0	0
$649$	19.5490	0.767364
$650$	0	0
$651$	− 3.38714i	− 0.132752i
$652$	0	0
$653$	− 43.8208i	− 1.71484i	−0.514617	−	0.857420i	$-0.672066\pi$
$653$	− 43.8208i	− 1.71484i	0.514617	−	0.857420i	$-0.327934\pi$
$654$	0	0
$655$	49.2144	1.92296
$656$	0	0
$657$	16.4546	0.641954
$658$	0	0
$659$	11.0297i	0.429657i	0.976652	+	0.214829i	$0.0689193\pi$
$659$	11.0297i	0.429657i	−0.976652	+	0.214829i	$0.931081\pi$
$660$	0	0
$661$	7.49137i	0.291381i	0.989330	+	0.145690i	$0.0465403\pi$
$661$	7.49137i	0.291381i	−0.989330	+	0.145690i	$0.953460\pi$
$662$	0	0
$663$	18.4890	0.718055
$664$	0	0
$665$	6.65751	0.258167
$666$	0	0
$667$	14.5676i	0.564060i
$668$	0	0
$669$	− 1.70420i	− 0.0658882i
$670$	0	0
$671$	−35.4309	−1.36780
$672$	0	0
$673$	−34.6195	−1.33448	−0.667242	−	0.744841i	$-0.732525\pi$
$673$	−34.6195	−1.33448	−0.667242	+	0.744841i	$0.732525\pi$
$674$	0	0
$675$	10.6747i	0.410870i
$676$	0	0
$677$	− 29.0201i	− 1.11533i	−0.830066	−	0.557666i	$-0.811697\pi$
$677$	− 29.0201i	− 1.11533i	0.830066	−	0.557666i	$-0.188303\pi$
$678$	0	0
$679$	−8.74629	−0.335652
$680$	0	0
$681$	8.17012	0.313080
$682$	0	0
$683$	− 13.3799i	− 0.511966i	−0.966681	−	0.255983i	$-0.917601\pi$
$683$	− 13.3799i	− 0.511966i	0.966681	−	0.255983i	$-0.0823991\pi$
$684$	0	0
$685$	− 48.7446i	− 1.86244i
$686$	0	0
$687$	6.19269	0.236266
$688$	0	0
$689$	34.8089	1.32611
$690$	0	0
$691$	40.7415i	1.54988i	0.632035	+	0.774940i	$0.282220\pi$
$691$	40.7415i	1.54988i	−0.632035	+	0.774940i	$0.717780\pi$
$692$	0	0
$693$	31.0646i	1.18005i
$694$	0	0
$695$	−13.1845	−0.500118
$696$	0	0
$697$	−24.4632	−0.926609
$698$	0	0
$699$	− 8.47909i	− 0.320709i
$700$	0	0
$701$	43.7760i	1.65340i	0.562645	+	0.826698i	$0.309784\pi$
$701$	43.7760i	1.65340i	−0.562645	+	0.826698i	$0.690216\pi$
$702$	0	0
$703$	1.45208	0.0547662
$704$	0	0
$705$	−13.6982	−0.515904
$706$	0	0
$707$	14.0072i	0.526795i
$708$	0	0
$709$	26.0155i	0.977031i	0.872555	+	0.488516i	$0.162462\pi$
$709$	26.0155i	0.977031i	−0.872555	+	0.488516i	$0.837538\pi$
$710$	0	0
$711$	44.4652	1.66758
$712$	0	0
$713$	−21.7356	−0.814005
$714$	0	0
$715$	− 78.7801i	− 2.94621i
$716$	0	0
$717$	5.35490i	0.199982i
$718$	0	0
$719$	31.3051	1.16748	0.583741	−	0.811940i	$-0.301588\pi$
$719$	31.3051	1.16748	0.583741	+	0.811940i	$0.301588\pi$
$720$	0	0
$721$	10.4430	0.388916
$722$	0	0
$723$	− 4.70552i	− 0.175000i
$724$	0	0
$725$	8.99574i	0.334093i
$726$	0	0
$727$	7.31403	0.271262	0.135631	−	0.990759i	$-0.456694\pi$
$727$	7.31403	0.271262	0.135631	+	0.990759i	$0.456694\pi$
$728$	0	0
$729$	16.3158	0.604287
$730$	0	0
$731$	− 16.7051i	− 0.617862i
$732$	0	0
$733$	13.0935i	0.483620i	0.970324	+	0.241810i	$0.0777411\pi$
$733$	13.0935i	0.483620i	−0.970324	+	0.241810i	$0.922259\pi$
$734$	0	0
$735$	−2.87420	−0.106016
$736$	0	0
$737$	66.6997	2.45692
$738$	0	0
$739$	10.5782i	0.389124i	0.980890	+	0.194562i	$0.0623285\pi$
$739$	10.5782i	0.389124i	−0.980890	+	0.194562i	$0.937672\pi$
$740$	0	0
$741$	− 2.39178i	− 0.0878642i
$742$	0	0
$743$	23.6180	0.866460	0.433230	−	0.901283i	$-0.357374\pi$
$743$	23.6180	0.866460	0.433230	+	0.901283i	$0.357374\pi$
$744$	0	0
$745$	−3.86136	−0.141469
$746$	0	0
$747$	− 24.3220i	− 0.889896i
$748$	0	0
$749$	41.1225i	1.50258i
$750$	0	0
$751$	24.5994	0.897644	0.448822	−	0.893621i	$-0.351844\pi$
$751$	24.5994	0.897644	0.448822	+	0.893621i	$0.351844\pi$
$752$	0	0
$753$	11.5032	0.419199
$754$	0	0
$755$	− 27.2548i	− 0.991905i
$756$	0	0
$757$	3.03483i	0.110303i	0.998478	+	0.0551514i	$0.0175642\pi$
$757$	3.03483i	0.110303i	−0.998478	+	0.0551514i	$0.982436\pi$
$758$	0	0
$759$	−14.9988	−0.544423
$760$	0	0
$761$	8.15330	0.295557	0.147778	−	0.989020i	$-0.452788\pi$
$761$	8.15330	0.295557	0.147778	+	0.989020i	$0.452788\pi$
$762$	0	0
$763$	− 20.5223i	− 0.742956i
$764$	0	0
$765$	64.7894i	2.34247i
$766$	0	0
$767$	20.3132	0.733466
$768$	0	0
$769$	−14.2744	−0.514747	−0.257374	−	0.966312i	$-0.582857\pi$
$769$	−14.2744	−0.514747	−0.257374	+	0.966312i	$0.582857\pi$
$770$	0	0
$771$	− 5.76868i	− 0.207754i
$772$	0	0
$773$	− 45.2404i	− 1.62718i	−0.581437	−	0.813591i	$-0.697509\pi$
$773$	− 45.2404i	− 1.62718i	0.581437	−	0.813591i	$-0.302491\pi$
$774$	0	0
$775$	−13.4221	−0.482136
$776$	0	0
$777$	1.47448	0.0528968
$778$	0	0
$779$	3.16461i	0.113384i
$780$	0	0
$781$	− 28.7784i	− 1.02977i
$782$	0	0
$783$	−5.93084	−0.211951
$784$	0	0
$785$	9.42212	0.336290
$786$	0	0
$787$	− 50.1656i	− 1.78821i	−0.447856	−	0.894106i	$-0.647812\pi$
$787$	− 50.1656i	− 1.78821i	0.447856	−	0.894106i	$-0.352188\pi$
$788$	0	0
$789$	10.4349i	0.371494i
$790$	0	0
$791$	−17.6720	−0.628344
$792$	0	0
$793$	−36.8160	−1.30737
$794$	0	0
$795$	− 9.17764i	− 0.325497i
$796$	0	0
$797$	21.7396i	0.770056i	0.922905	+	0.385028i	$0.125808\pi$
$797$	21.7396i	0.770056i	−0.922905	+	0.385028i	$0.874192\pi$
$798$	0	0
$799$	−76.9357	−2.72179
$800$	0	0
$801$	10.6006	0.374552
$802$	0	0
$803$	29.6281i	1.04555i
$804$	0	0
$805$	− 43.3812i	− 1.52899i
$806$	0	0
$807$	−7.80144	−0.274624
$808$	0	0
$809$	9.48695	0.333543	0.166772	−	0.985996i	$-0.446666\pi$
$809$	9.48695	0.333543	0.166772	+	0.985996i	$0.446666\pi$
$810$	0	0
$811$	− 22.6529i	− 0.795450i	−0.917505	−	0.397725i	$-0.869800\pi$
$811$	− 22.6529i	− 0.795450i	0.917505	−	0.397725i	$-0.130200\pi$
$812$	0	0
$813$	3.75518i	0.131700i
$814$	0	0
$815$	44.9513	1.57458
$816$	0	0
$817$	−2.16101	−0.0756042
$818$	0	0
$819$	32.2790i	1.12792i
$820$	0	0
$821$	26.4552i	0.923292i	0.887064	+	0.461646i	$0.152741\pi$
$821$	26.4552i	0.923292i	−0.887064	+	0.461646i	$0.847259\pi$
$822$	0	0
$823$	−35.1129	−1.22396	−0.611980	−	0.790873i	$-0.709627\pi$
$823$	−35.1129	−1.22396	−0.611980	+	0.790873i	$0.709627\pi$
$824$	0	0
$825$	−9.26202	−0.322462
$826$	0	0
$827$	27.4003i	0.952803i	0.879228	+	0.476402i	$0.158059\pi$
$827$	27.4003i	0.952803i	−0.879228	+	0.476402i	$0.841941\pi$
$828$	0	0
$829$	− 25.2213i	− 0.875972i	−0.898982	−	0.437986i	$-0.855692\pi$
$829$	− 25.2213i	− 0.875972i	0.898982	−	0.437986i	$-0.144308\pi$
$830$	0	0
$831$	−8.57588	−0.297494
$832$	0	0
$833$	−16.1429	−0.559319
$834$	0	0
$835$	63.3234i	2.19140i
$836$	0	0
$837$	− 8.84912i	− 0.305870i
$838$	0	0
$839$	−36.8869	−1.27348	−0.636738	−	0.771080i	$-0.719717\pi$
$839$	−36.8869	−1.27348	−0.636738	+	0.771080i	$0.719717\pi$
$840$	0	0
$841$	24.0020	0.827655
$842$	0	0
$843$	0.0866488i	0.00298434i
$844$	0	0
$845$	− 42.8082i	− 1.47265i
$846$	0	0
$847$	−31.5565	−1.08429
$848$	0	0
$849$	4.87303	0.167242
$850$	0	0
$851$	− 9.46193i	− 0.324351i
$852$	0	0
$853$	3.36746i	0.115300i	0.998337	+	0.0576499i	$0.0183607\pi$
$853$	3.36746i	0.115300i	−0.998337	+	0.0576499i	$0.981639\pi$
$854$	0	0
$855$	8.38129	0.286634
$856$	0	0
$857$	16.0416	0.547970	0.273985	−	0.961734i	$-0.411658\pi$
$857$	16.0416	0.547970	0.273985	+	0.961734i	$0.411658\pi$
$858$	0	0
$859$	14.7310i	0.502614i	0.967907	+	0.251307i	$0.0808603\pi$
$859$	14.7310i	0.502614i	−0.967907	+	0.251307i	$0.919140\pi$
$860$	0	0
$861$	3.21344i	0.109514i
$862$	0	0
$863$	−41.3034	−1.40598	−0.702991	−	0.711198i	$-0.748153\pi$
$863$	−41.3034	−1.40598	−0.702991	+	0.711198i	$0.748153\pi$
$864$	0	0
$865$	−71.4368	−2.42892
$866$	0	0
$867$	− 19.5901i	− 0.665316i
$868$	0	0
$869$	80.0642i	2.71599i
$870$	0	0
$871$	69.3071	2.34838
$872$	0	0
$873$	−11.0109	−0.372662
$874$	0	0
$875$	6.49895i	0.219704i
$876$	0	0
$877$	3.84438i	0.129815i	0.997891	+	0.0649077i	$0.0206753\pi$
$877$	3.84438i	0.129815i	−0.997891	+	0.0649077i	$0.979325\pi$
$878$	0	0
$879$	−12.6807	−0.427711
$880$	0	0
$881$	−4.57677	−0.154195	−0.0770976	−	0.997024i	$-0.524565\pi$
$881$	−4.57677	−0.154195	−0.0770976	+	0.997024i	$0.524565\pi$
$882$	0	0
$883$	2.34140i	0.0787945i	0.999224	+	0.0393972i	$0.0125438\pi$
$883$	2.34140i	0.0787945i	−0.999224	+	0.0393972i	$0.987456\pi$
$884$	0	0
$885$	− 5.35573i	− 0.180031i
$886$	0	0
$887$	48.1958	1.61826	0.809129	−	0.587631i	$-0.199939\pi$
$887$	48.1958	1.61826	0.809129	+	0.587631i	$0.199939\pi$
$888$	0	0
$889$	18.4593	0.619106
$890$	0	0
$891$	35.9441i	1.20417i
$892$	0	0
$893$	9.95256i	0.333050i
$894$	0	0
$895$	42.9621	1.43607
$896$	0	0
$897$	−15.5851	−0.520373
$898$	0	0
$899$	− 7.45729i	− 0.248714i
$900$	0	0
$901$	− 51.5461i	− 1.71725i
$902$	0	0
$903$	−2.19436	−0.0730237
$904$	0	0
$905$	41.7593	1.38812
$906$	0	0
$907$	− 4.08133i	− 0.135518i	−0.997702	−	0.0677592i	$-0.978415\pi$
$907$	− 4.08133i	− 0.135518i	0.997702	−	0.0677592i	$-0.0215850\pi$
$908$	0	0
$909$	17.6340i	0.584881i
$910$	0	0
$911$	−54.1174	−1.79299	−0.896495	−	0.443054i	$-0.853895\pi$
$911$	−54.1174	−1.79299	−0.896495	+	0.443054i	$0.853895\pi$
$912$	0	0
$913$	43.7943	1.44938
$914$	0	0
$915$	9.70682i	0.320898i
$916$	0	0
$917$	− 36.3089i	− 1.19903i
$918$	0	0
$919$	32.1252	1.05971	0.529857	−	0.848087i	$-0.322246\pi$
$919$	32.1252	1.05971	0.529857	+	0.848087i	$0.322246\pi$
$920$	0	0
$921$	−3.73280	−0.123000
$922$	0	0
$923$	− 29.9034i	− 0.984284i
$924$	0	0
$925$	− 5.84290i	− 0.192113i
$926$	0	0
$927$	13.1469	0.431800
$928$	0	0
$929$	−10.9339	−0.358730	−0.179365	−	0.983783i	$-0.557404\pi$
$929$	−10.9339	−0.358730	−0.179365	+	0.983783i	$0.557404\pi$
$930$	0	0
$931$	2.08828i	0.0684406i
$932$	0	0
$933$	− 5.09875i	− 0.166926i
$934$	0	0
$935$	−116.660	−3.81519
$936$	0	0
$937$	46.5172	1.51965	0.759825	−	0.650127i	$-0.225284\pi$
$937$	46.5172	1.51965	0.759825	+	0.650127i	$0.225284\pi$
$938$	0	0
$939$	− 4.70307i	− 0.153479i
$940$	0	0
$941$	39.9678i	1.30291i	0.758686	+	0.651457i	$0.225842\pi$
$941$	39.9678i	1.30291i	−0.758686	+	0.651457i	$0.774158\pi$
$942$	0	0
$943$	20.6210	0.671511
$944$	0	0
$945$	17.6616	0.574531
$946$	0	0
$947$	− 33.7011i	− 1.09514i	−0.836761	−	0.547569i	$-0.815553\pi$
$947$	− 33.7011i	− 1.09514i	0.836761	−	0.547569i	$-0.184447\pi$
$948$	0	0
$949$	30.7864i	0.999368i
$950$	0	0
$951$	−3.27535	−0.106211
$952$	0	0
$953$	30.1472	0.976564	0.488282	−	0.872686i	$-0.337624\pi$
$953$	30.1472	0.976564	0.488282	+	0.872686i	$0.337624\pi$
$954$	0	0
$955$	− 6.54106i	− 0.211664i
$956$	0	0
$957$	− 5.14596i	− 0.166345i
$958$	0	0
$959$	−35.9624	−1.16129
$960$	0	0
$961$	−19.8733	−0.641076
$962$	0	0
$963$	51.7700i	1.66826i
$964$	0	0
$965$	80.0287i	2.57622i
$966$	0	0
$967$	35.1010	1.12877	0.564385	−	0.825511i	$-0.309113\pi$
$967$	35.1010	1.12877	0.564385	+	0.825511i	$0.309113\pi$
$968$	0	0
$969$	−3.54182	−0.113780
$970$	0	0
$971$	− 38.1884i	− 1.22552i	−0.790268	−	0.612762i	$-0.790059\pi$
$971$	− 38.1884i	− 1.22552i	0.790268	−	0.612762i	$-0.209941\pi$
$972$	0	0
$973$	9.72716i	0.311839i
$974$	0	0
$975$	−9.62409	−0.308217
$976$	0	0
$977$	22.6596	0.724946	0.362473	−	0.931994i	$-0.381933\pi$
$977$	22.6596	0.724946	0.362473	+	0.931994i	$0.381933\pi$
$978$	0	0
$979$	19.0874i	0.610036i
$980$	0	0
$981$	− 25.8359i	− 0.824878i
$982$	0	0
$983$	1.47180	0.0469430	0.0234715	−	0.999725i	$-0.492528\pi$
$983$	1.47180	0.0469430	0.0234715	+	0.999725i	$0.492528\pi$
$984$	0	0
$985$	−62.6326	−1.99564
$986$	0	0
$987$	10.1061i	0.321682i
$988$	0	0
$989$	14.0814i	0.447763i
$990$	0	0
$991$	30.6228	0.972767	0.486383	−	0.873746i	$-0.338316\pi$
$991$	30.6228	0.972767	0.486383	+	0.873746i	$0.338316\pi$
$992$	0	0
$993$	−13.0292	−0.413469
$994$	0	0
$995$	− 12.7582i	− 0.404461i
$996$	0	0
$997$	− 7.61232i	− 0.241085i	−0.992708	−	0.120542i	$-0.961537\pi$
$997$	− 7.61232i	− 0.241085i	0.992708	−	0.120542i	$-0.0384634\pi$
$998$	0	0
$999$	3.85219	0.121878

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.j.1217.10	yes	20
4.3	odd	2	inner	2432.2.c.j.1217.12	yes	20
8.3	odd	2	inner	2432.2.c.j.1217.9	✓	20
8.5	even	2	inner	2432.2.c.j.1217.11	yes	20
16.3	odd	4		4864.2.a.bs.1.6		10
16.5	even	4		4864.2.a.bs.1.5		10
16.11	odd	4		4864.2.a.bt.1.5		10
16.13	even	4		4864.2.a.bt.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.j.1217.9	✓	20	8.3	odd	2	inner
2432.2.c.j.1217.10	yes	20	1.1	even	1	trivial
2432.2.c.j.1217.11	yes	20	8.5	even	2	inner
2432.2.c.j.1217.12	yes	20	4.3	odd	2	inner
4864.2.a.bs.1.5		10	16.5	even	4
4864.2.a.bs.1.6		10	16.3	odd	4
4864.2.a.bt.1.5		10	16.11	odd	4
4864.2.a.bt.1.6		10	16.13	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.458177i q^{3} +3.00397i q^{5} +2.21624 q^{7} +2.79007 q^{9} +5.02382i q^{11} +5.22021i q^{13} +1.37635 q^{15} +7.73025 q^{17} -1.00000i q^{19} -1.01543i q^{21} -6.51613 q^{23} -4.02382 q^{25} -2.65288i q^{27} -2.23562i q^{29} +3.33566 q^{31} +2.30180 q^{33} +6.65751i q^{35} +1.45208i q^{37} +2.39178 q^{39} -3.16461 q^{41} -2.16101i q^{43} +8.38129i q^{45} -9.95256 q^{47} -2.08828 q^{49} -3.54182i q^{51} -6.66810i q^{53} -15.0914 q^{55} -0.458177 q^{57} -3.89126i q^{59} +7.05259i q^{61} +6.18347 q^{63} -15.6813 q^{65} -13.2767i q^{67} +2.98554i q^{69} -5.72840 q^{71} +5.89754 q^{73} +1.84362i q^{75} +11.1340i q^{77} +15.9369 q^{79} +7.15473 q^{81} -8.71734i q^{83} +23.2214i q^{85} -1.02431 q^{87} +3.79938 q^{89} +11.5692i q^{91} -1.52833i q^{93} +3.00397 q^{95} -3.94645 q^{97} +14.0168i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 28 q^{9} + 8 q^{17} - 20 q^{25} - 16 q^{33} + 24 q^{41} + 52 q^{49} - 8 q^{57} - 48 q^{65} - 24 q^{73} + 68 q^{81} + 40 q^{89} - 56 q^{97}+O(q^{100}) \) 20 * q - 28 * q^9 + 8 * q^17 - 20 * q^25 - 16 * q^33 + 24 * q^41 + 52 * q^49 - 8 * q^57 - 48 * q^65 - 24 * q^73 + 68 * q^81 + 40 * q^89 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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