LMFDB - Embedded newform 2432.2.c.j.1217.1

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			1217.1
Root			$2.33603 - 2.33603i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.j.1217.20

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.30364i q^{3} -2.55295i q^{5} +1.32515 q^{7} -7.91406 q^{9} +2.51756i q^{11} -1.22781i q^{13} -8.43404 q^{15} +0.210784 q^{17} -1.00000i q^{19} -4.37781i q^{21} -8.11631 q^{23} -1.51756 q^{25} +16.2343i q^{27} +5.97458i q^{29} -6.01598 q^{31} +8.31712 q^{33} -3.38303i q^{35} +11.2625i q^{37} -4.05623 q^{39} -0.996870 q^{41} -7.83781i q^{43} +20.2042i q^{45} +0.910938 q^{47} -5.24399 q^{49} -0.696356i q^{51} +3.32429i q^{53} +6.42720 q^{55} -3.30364 q^{57} -2.04859i q^{59} -10.0769i q^{61} -10.4873 q^{63} -3.13453 q^{65} -11.7584i q^{67} +26.8134i q^{69} -11.9944 q^{71} -13.0038 q^{73} +5.01347i q^{75} +3.33613i q^{77} -12.0761 q^{79} +29.8902 q^{81} +4.50788i q^{83} -0.538122i q^{85} +19.7379 q^{87} +6.64553 q^{89} -1.62702i q^{91} +19.8746i q^{93} -2.55295 q^{95} +9.92753 q^{97} -19.9241i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 3.30364i	− 1.90736i	−0.300824	−	0.953680i	$-0.597262\pi$
$3$	− 3.30364i	− 1.90736i	0.300824	−	0.953680i	$-0.402738\pi$
$4$	0	0
$5$	− 2.55295i	− 1.14171i	−0.821049	−	0.570857i	$-0.806611\pi$
$5$	− 2.55295i	− 1.14171i	0.821049	−	0.570857i	$-0.193389\pi$
$6$	0	0
$7$	1.32515	0.500858	0.250429	−	0.968135i	$-0.419428\pi$
$7$	1.32515	0.500858	0.250429	+	0.968135i	$0.419428\pi$
$8$	0	0
$9$	−7.91406	−2.63802
$10$	0	0
$11$	2.51756i	0.759072i	0.925177	+	0.379536i	$0.123916\pi$
$11$	2.51756i	0.759072i	−0.925177	+	0.379536i	$0.876084\pi$
$12$	0	0
$13$	− 1.22781i	− 0.340532i	−0.985398	−	0.170266i	$-0.945537\pi$
$13$	− 1.22781i	− 0.340532i	0.985398	−	0.170266i	$-0.0544627\pi$
$14$	0	0
$15$	−8.43404	−2.17766
$16$	0	0
$17$	0.210784	0.0511227	0.0255614	−	0.999673i	$-0.491863\pi$
$17$	0.210784	0.0511227	0.0255614	+	0.999673i	$0.491863\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	− 4.37781i	− 0.955316i
$22$	0	0
$23$	−8.11631	−1.69237	−0.846184	−	0.532891i	$-0.821105\pi$
$23$	−8.11631	−1.69237	−0.846184	+	0.532891i	$0.821105\pi$
$24$	0	0
$25$	−1.51756	−0.303512
$26$	0	0
$27$	16.2343i	3.12429i
$28$	0	0
$29$	5.97458i	1.10945i	0.832033	+	0.554726i	$0.187177\pi$
$29$	5.97458i	1.10945i	−0.832033	+	0.554726i	$0.812823\pi$
$30$	0	0
$31$	−6.01598	−1.08050	−0.540251	−	0.841504i	$-0.681671\pi$
$31$	−6.01598	−1.08050	−0.540251	+	0.841504i	$0.681671\pi$
$32$	0	0
$33$	8.31712	1.44782
$34$	0	0
$35$	− 3.38303i	− 0.571837i
$36$	0	0
$37$	11.2625i	1.85154i	0.378089	+	0.925769i	$0.376581\pi$
$37$	11.2625i	1.85154i	−0.378089	+	0.925769i	$0.623419\pi$
$38$	0	0
$39$	−4.05623	−0.649517
$40$	0	0
$41$	−0.996870	−0.155685	−0.0778424	−	0.996966i	$-0.524803\pi$
$41$	−0.996870	−0.155685	−0.0778424	+	0.996966i	$0.524803\pi$
$42$	0	0
$43$	− 7.83781i	− 1.19525i	−0.801774	−	0.597627i	$-0.796110\pi$
$43$	− 7.83781i	− 1.19525i	0.801774	−	0.597627i	$-0.203890\pi$
$44$	0	0
$45$	20.2042i	3.01187i
$46$	0	0
$47$	0.910938	0.132874	0.0664369	−	0.997791i	$-0.478837\pi$
$47$	0.910938	0.132874	0.0664369	+	0.997791i	$0.478837\pi$
$48$	0	0
$49$	−5.24399	−0.749141
$50$	0	0
$51$	− 0.696356i	− 0.0975094i
$52$	0	0
$53$	3.32429i	0.456627i	0.973588	+	0.228313i	$0.0733211\pi$
$53$	3.32429i	0.456627i	−0.973588	+	0.228313i	$0.926679\pi$
$54$	0	0
$55$	6.42720	0.866644
$56$	0	0
$57$	−3.30364	−0.437578
$58$	0	0
$59$	− 2.04859i	− 0.266704i	−0.991069	−	0.133352i	$-0.957426\pi$
$59$	− 2.04859i	− 0.266704i	0.991069	−	0.133352i	$-0.0425740\pi$
$60$	0	0
$61$	− 10.0769i	− 1.29022i	−0.764091	−	0.645108i	$-0.776812\pi$
$61$	− 10.0769i	− 1.29022i	0.764091	−	0.645108i	$-0.223188\pi$
$62$	0	0
$63$	−10.4873	−1.32127
$64$	0	0
$65$	−3.13453	−0.388790
$66$	0	0
$67$	− 11.7584i	− 1.43652i	−0.695775	−	0.718260i	$-0.744939\pi$
$67$	− 11.7584i	− 1.43652i	0.695775	−	0.718260i	$-0.255061\pi$
$68$	0	0
$69$	26.8134i	3.22795i
$70$	0	0
$71$	−11.9944	−1.42347	−0.711737	−	0.702446i	$-0.752091\pi$
$71$	−11.9944	−1.42347	−0.711737	+	0.702446i	$0.752091\pi$
$72$	0	0
$73$	−13.0038	−1.52198	−0.760989	−	0.648764i	$-0.775286\pi$
$73$	−13.0038	−1.52198	−0.760989	+	0.648764i	$0.775286\pi$
$74$	0	0
$75$	5.01347i	0.578906i
$76$	0	0
$77$	3.33613i	0.380187i
$78$	0	0
$79$	−12.0761	−1.35866	−0.679332	−	0.733831i	$-0.737730\pi$
$79$	−12.0761	−1.35866	−0.679332	+	0.733831i	$0.737730\pi$
$80$	0	0
$81$	29.8902	3.32113
$82$	0	0
$83$	4.50788i	0.494804i	0.968913	+	0.247402i	$0.0795768\pi$
$83$	4.50788i	0.494804i	−0.968913	+	0.247402i	$0.920423\pi$
$84$	0	0
$85$	− 0.538122i	− 0.0583675i
$86$	0	0
$87$	19.7379	2.11612
$88$	0	0
$89$	6.64553	0.704425	0.352213	−	0.935920i	$-0.385429\pi$
$89$	6.64553	0.704425	0.352213	+	0.935920i	$0.385429\pi$
$90$	0	0
$91$	− 1.62702i	− 0.170558i
$92$	0	0
$93$	19.8746i	2.06090i
$94$	0	0
$95$	−2.55295	−0.261927
$96$	0	0
$97$	9.92753	1.00799	0.503994	−	0.863707i	$-0.331863\pi$
$97$	9.92753	1.00799	0.503994	+	0.863707i	$0.331863\pi$
$98$	0	0
$99$	− 19.9241i	− 2.00245i
$100$	0	0
$101$	− 1.26920i	− 0.126290i	−0.998004	−	0.0631450i	$-0.979887\pi$
$101$	− 1.26920i	− 0.126290i	0.998004	−	0.0631450i	$-0.0201130\pi$
$102$	0	0
$103$	−14.4500	−1.42380	−0.711901	−	0.702280i	$-0.752166\pi$
$103$	−14.4500	−1.42380	−0.711901	+	0.702280i	$0.752166\pi$
$104$	0	0
$105$	−11.1763	−1.09070
$106$	0	0
$107$	− 10.3868i	− 1.00413i	−0.864829	−	0.502066i	$-0.832573\pi$
$107$	− 10.3868i	− 1.00413i	0.864829	−	0.502066i	$-0.167427\pi$
$108$	0	0
$109$	− 15.3639i	− 1.47160i	−0.677200	−	0.735799i	$-0.736807\pi$
$109$	− 15.3639i	− 1.47160i	0.677200	−	0.735799i	$-0.263193\pi$
$110$	0	0
$111$	37.2072	3.53155
$112$	0	0
$113$	16.2509	1.52876	0.764379	−	0.644768i	$-0.223046\pi$
$113$	16.2509	1.52876	0.764379	+	0.644768i	$0.223046\pi$
$114$	0	0
$115$	20.7205i	1.93220i
$116$	0	0
$117$	9.71693i	0.898331i
$118$	0	0
$119$	0.279320	0.0256052
$120$	0	0
$121$	4.66190	0.423809
$122$	0	0
$123$	3.29330i	0.296947i
$124$	0	0
$125$	− 8.89050i	− 0.795191i
$126$	0	0
$127$	6.56693	0.582721	0.291360	−	0.956613i	$-0.405892\pi$
$127$	6.56693	0.582721	0.291360	+	0.956613i	$0.405892\pi$
$128$	0	0
$129$	−25.8933	−2.27978
$130$	0	0
$131$	10.0235i	0.875759i	0.899034	+	0.437880i	$0.144270\pi$
$131$	10.0235i	0.875759i	−0.899034	+	0.437880i	$0.855730\pi$
$132$	0	0
$133$	− 1.32515i	− 0.114905i
$134$	0	0
$135$	41.4454	3.56705
$136$	0	0
$137$	7.01005	0.598909	0.299455	−	0.954111i	$-0.403195\pi$
$137$	7.01005	0.598909	0.299455	+	0.954111i	$0.403195\pi$
$138$	0	0
$139$	− 3.13111i	− 0.265577i	−0.991144	−	0.132789i	$-0.957607\pi$
$139$	− 3.13111i	− 0.265577i	0.991144	−	0.132789i	$-0.0423931\pi$
$140$	0	0
$141$	− 3.00941i	− 0.253438i
$142$	0	0
$143$	3.09107	0.258489
$144$	0	0
$145$	15.2528	1.26668
$146$	0	0
$147$	17.3243i	1.42888i
$148$	0	0
$149$	− 7.47186i	− 0.612119i	−0.952012	−	0.306059i	$-0.900989\pi$
$149$	− 7.47186i	− 0.612119i	0.952012	−	0.306059i	$-0.0990106\pi$
$150$	0	0
$151$	18.1373	1.47599	0.737995	−	0.674806i	$-0.235773\pi$
$151$	18.1373	1.47599	0.737995	+	0.674806i	$0.235773\pi$
$152$	0	0
$153$	−1.66816	−0.134863
$154$	0	0
$155$	15.3585i	1.23362i
$156$	0	0
$157$	6.69383i	0.534225i	0.963665	+	0.267113i	$0.0860696\pi$
$157$	6.69383i	0.534225i	−0.963665	+	0.267113i	$0.913930\pi$
$158$	0	0
$159$	10.9823	0.870952
$160$	0	0
$161$	−10.7553	−0.847635
$162$	0	0
$163$	− 15.6424i	− 1.22521i	−0.790390	−	0.612604i	$-0.790122\pi$
$163$	− 15.6424i	− 1.22521i	0.790390	−	0.612604i	$-0.209878\pi$
$164$	0	0
$165$	− 21.2332i	− 1.65300i
$166$	0	0
$167$	−4.23338	−0.327588	−0.163794	−	0.986495i	$-0.552373\pi$
$167$	−4.23338	−0.327588	−0.163794	+	0.986495i	$0.552373\pi$
$168$	0	0
$169$	11.4925	0.884038
$170$	0	0
$171$	7.91406i	0.605203i
$172$	0	0
$173$	4.25476i	0.323483i	0.986833	+	0.161742i	$0.0517111\pi$
$173$	4.25476i	0.323483i	−0.986833	+	0.161742i	$0.948289\pi$
$174$	0	0
$175$	−2.01099	−0.152016
$176$	0	0
$177$	−6.76781	−0.508700
$178$	0	0
$179$	− 20.3171i	− 1.51857i	−0.650757	−	0.759286i	$-0.725548\pi$
$179$	− 20.3171i	− 1.51857i	0.650757	−	0.759286i	$-0.274452\pi$
$180$	0	0
$181$	− 16.6430i	− 1.23706i	−0.785760	−	0.618532i	$-0.787728\pi$
$181$	− 16.6430i	− 1.23706i	0.785760	−	0.618532i	$-0.212272\pi$
$182$	0	0
$183$	−33.2905	−2.46091
$184$	0	0
$185$	28.7525	2.11393
$186$	0	0
$187$	0.530662i	0.0388058i
$188$	0	0
$189$	21.5128i	1.56483i
$190$	0	0
$191$	−15.9246	−1.15226	−0.576132	−	0.817357i	$-0.695439\pi$
$191$	−15.9246	−1.15226	−0.576132	+	0.817357i	$0.695439\pi$
$192$	0	0
$193$	−25.8762	−1.86261	−0.931304	−	0.364242i	$-0.881328\pi$
$193$	−25.8762	−1.86261	−0.931304	+	0.364242i	$0.881328\pi$
$194$	0	0
$195$	10.3554i	0.741563i
$196$	0	0
$197$	18.2850i	1.30275i	0.758755	+	0.651376i	$0.225808\pi$
$197$	18.2850i	1.30275i	−0.758755	+	0.651376i	$0.774192\pi$
$198$	0	0
$199$	−10.0808	−0.714606	−0.357303	−	0.933989i	$-0.616304\pi$
$199$	−10.0808	−0.714606	−0.357303	+	0.933989i	$0.616304\pi$
$200$	0	0
$201$	−38.8456	−2.73996
$202$	0	0
$203$	7.91719i	0.555678i
$204$	0	0
$205$	2.54496i	0.177748i
$206$	0	0
$207$	64.2330	4.46450
$208$	0	0
$209$	2.51756	0.174143
$210$	0	0
$211$	− 15.8127i	− 1.08859i	−0.838893	−	0.544297i	$-0.816796\pi$
$211$	− 15.8127i	− 1.08859i	0.838893	−	0.544297i	$-0.183204\pi$
$212$	0	0
$213$	39.6252i	2.71508i
$214$	0	0
$215$	−20.0095	−1.36464
$216$	0	0
$217$	−7.97204	−0.541178
$218$	0	0
$219$	42.9599i	2.90296i
$220$	0	0
$221$	− 0.258802i	− 0.0174089i
$222$	0	0
$223$	4.15828	0.278459	0.139229	−	0.990260i	$-0.455537\pi$
$223$	4.15828	0.278459	0.139229	+	0.990260i	$0.455537\pi$
$224$	0	0
$225$	12.0101	0.800670
$226$	0	0
$227$	− 16.7714i	− 1.11315i	−0.830796	−	0.556577i	$-0.812114\pi$
$227$	− 16.7714i	− 1.11315i	0.830796	−	0.556577i	$-0.187886\pi$
$228$	0	0
$229$	1.32899i	0.0878220i	0.999035	+	0.0439110i	$0.0139818\pi$
$229$	1.32899i	0.0878220i	−0.999035	+	0.0439110i	$0.986018\pi$
$230$	0	0
$231$	11.0214	0.725154
$232$	0	0
$233$	−15.4631	−1.01302	−0.506510	−	0.862234i	$-0.669065\pi$
$233$	−15.4631	−1.01302	−0.506510	+	0.862234i	$0.669065\pi$
$234$	0	0
$235$	− 2.32558i	− 0.151704i
$236$	0	0
$237$	39.8950i	2.59146i
$238$	0	0
$239$	15.4564	0.999794	0.499897	−	0.866085i	$-0.333371\pi$
$239$	15.4564	0.999794	0.499897	+	0.866085i	$0.333371\pi$
$240$	0	0
$241$	−14.0126	−0.902632	−0.451316	−	0.892364i	$-0.649045\pi$
$241$	−14.0126	−0.902632	−0.451316	+	0.892364i	$0.649045\pi$
$242$	0	0
$243$	− 50.0436i	− 3.21030i
$244$	0	0
$245$	13.3877i	0.855306i
$246$	0	0
$247$	−1.22781	−0.0781234
$248$	0	0
$249$	14.8924	0.943769
$250$	0	0
$251$	4.36102i	0.275265i	0.990483	+	0.137632i	$0.0439493\pi$
$251$	4.36102i	0.275265i	−0.990483	+	0.137632i	$0.956051\pi$
$252$	0	0
$253$	− 20.4333i	− 1.28463i
$254$	0	0
$255$	−1.77776	−0.111328
$256$	0	0
$257$	21.4237	1.33637	0.668186	−	0.743994i	$-0.267071\pi$
$257$	21.4237	1.33637	0.668186	+	0.743994i	$0.267071\pi$
$258$	0	0
$259$	14.9244i	0.927357i
$260$	0	0
$261$	− 47.2832i	− 2.92676i
$262$	0	0
$263$	5.37369	0.331356	0.165678	−	0.986180i	$-0.447019\pi$
$263$	5.37369	0.331356	0.165678	+	0.986180i	$0.447019\pi$
$264$	0	0
$265$	8.48676	0.521338
$266$	0	0
$267$	− 21.9545i	− 1.34359i
$268$	0	0
$269$	10.0684i	0.613879i	0.951729	+	0.306940i	$0.0993049\pi$
$269$	10.0684i	0.613879i	−0.951729	+	0.306940i	$0.900695\pi$
$270$	0	0
$271$	−27.6347	−1.67869	−0.839344	−	0.543601i	$-0.817060\pi$
$271$	−27.6347	−1.67869	−0.839344	+	0.543601i	$0.817060\pi$
$272$	0	0
$273$	−5.37510	−0.325316
$274$	0	0
$275$	− 3.82054i	− 0.230387i
$276$	0	0
$277$	9.52596i	0.572360i	0.958176	+	0.286180i	$0.0923855\pi$
$277$	9.52596i	0.572360i	−0.958176	+	0.286180i	$0.907615\pi$
$278$	0	0
$279$	47.6108	2.85038
$280$	0	0
$281$	27.6168	1.64748	0.823739	−	0.566969i	$-0.191884\pi$
$281$	27.6168	1.64748	0.823739	+	0.566969i	$0.191884\pi$
$282$	0	0
$283$	17.0192i	1.01168i	0.862626	+	0.505842i	$0.168818\pi$
$283$	17.0192i	1.01168i	−0.862626	+	0.505842i	$0.831182\pi$
$284$	0	0
$285$	8.43404i	0.499589i
$286$	0	0
$287$	−1.32100	−0.0779760
$288$	0	0
$289$	−16.9556	−0.997386
$290$	0	0
$291$	− 32.7970i	− 1.92260i
$292$	0	0
$293$	13.8952i	0.811767i	0.913925	+	0.405884i	$0.133036\pi$
$293$	13.8952i	0.811767i	−0.913925	+	0.405884i	$0.866964\pi$
$294$	0	0
$295$	−5.22995	−0.304499
$296$	0	0
$297$	−40.8708	−2.37157
$298$	0	0
$299$	9.96525i	0.576305i
$300$	0	0
$301$	− 10.3862i	− 0.598652i
$302$	0	0
$303$	−4.19298	−0.240880
$304$	0	0
$305$	−25.7259	−1.47306
$306$	0	0
$307$	8.57307i	0.489291i	0.969613	+	0.244645i	$0.0786716\pi$
$307$	8.57307i	0.489291i	−0.969613	+	0.244645i	$0.921328\pi$
$308$	0	0
$309$	47.7377i	2.71570i
$310$	0	0
$311$	18.7442	1.06288	0.531442	−	0.847094i	$-0.321650\pi$
$311$	18.7442	1.06288	0.531442	+	0.847094i	$0.321650\pi$
$312$	0	0
$313$	0.267307	0.0151091	0.00755454	−	0.999971i	$-0.497595\pi$
$313$	0.267307	0.0151091	0.00755454	+	0.999971i	$0.497595\pi$
$314$	0	0
$315$	26.7735i	1.50852i
$316$	0	0
$317$	− 24.3865i	− 1.36968i	−0.728693	−	0.684841i	$-0.759872\pi$
$317$	− 24.3865i	− 1.36968i	0.728693	−	0.684841i	$-0.240128\pi$
$318$	0	0
$319$	−15.0414	−0.842155
$320$	0	0
$321$	−34.3144	−1.91524
$322$	0	0
$323$	− 0.210784i	− 0.0117284i
$324$	0	0
$325$	1.86327i	0.103355i
$326$	0	0
$327$	−50.7570	−2.80687
$328$	0	0
$329$	1.20712	0.0665509
$330$	0	0
$331$	15.8978i	0.873824i	0.899504	+	0.436912i	$0.143928\pi$
$331$	15.8978i	0.873824i	−0.899504	+	0.436912i	$0.856072\pi$
$332$	0	0
$333$	− 89.1319i	− 4.88440i
$334$	0	0
$335$	−30.0187	−1.64009
$336$	0	0
$337$	−23.4174	−1.27563	−0.637814	−	0.770190i	$-0.720161\pi$
$337$	−23.4174	−1.27563	−0.637814	+	0.770190i	$0.720161\pi$
$338$	0	0
$339$	− 53.6872i	− 2.91589i
$340$	0	0
$341$	− 15.1456i	− 0.820179i
$342$	0	0
$343$	−16.2251	−0.876071
$344$	0	0
$345$	68.4533	3.68540
$346$	0	0
$347$	− 0.166224i	− 0.00892336i	−0.999990	−	0.00446168i	$-0.998580\pi$
$347$	− 0.166224i	− 0.00892336i	0.999990	−	0.00446168i	$-0.00142020\pi$
$348$	0	0
$349$	− 3.92636i	− 0.210173i	−0.994463	−	0.105087i	$-0.966488\pi$
$349$	− 3.92636i	− 0.210173i	0.994463	−	0.105087i	$-0.0335119\pi$
$350$	0	0
$351$	19.9326	1.06392
$352$	0	0
$353$	9.45111	0.503032	0.251516	−	0.967853i	$-0.419071\pi$
$353$	9.45111	0.503032	0.251516	+	0.967853i	$0.419071\pi$
$354$	0	0
$355$	30.6211i	1.62520i
$356$	0	0
$357$	− 0.922773i	− 0.0488383i
$358$	0	0
$359$	7.81041	0.412218	0.206109	−	0.978529i	$-0.433920\pi$
$359$	7.81041	0.412218	0.206109	+	0.978529i	$0.433920\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 15.4013i	− 0.808356i
$364$	0	0
$365$	33.1980i	1.73766i
$366$	0	0
$367$	−34.3412	−1.79260	−0.896298	−	0.443452i	$-0.853754\pi$
$367$	−34.3412	−1.79260	−0.896298	+	0.443452i	$0.853754\pi$
$368$	0	0
$369$	7.88929	0.410700
$370$	0	0
$371$	4.40517i	0.228705i
$372$	0	0
$373$	− 33.9919i	− 1.76004i	−0.474941	−	0.880018i	$-0.657531\pi$
$373$	− 33.9919i	− 1.76004i	0.474941	−	0.880018i	$-0.342469\pi$
$374$	0	0
$375$	−29.3711	−1.51671
$376$	0	0
$377$	7.33563	0.377804
$378$	0	0
$379$	− 4.11882i	− 0.211570i	−0.994389	−	0.105785i	$-0.966265\pi$
$379$	− 4.11882i	− 0.211570i	0.994389	−	0.105785i	$-0.0337355\pi$
$380$	0	0
$381$	− 21.6948i	− 1.11146i
$382$	0	0
$383$	−17.5442	−0.896467	−0.448233	−	0.893917i	$-0.647947\pi$
$383$	−17.5442	−0.896467	−0.448233	+	0.893917i	$0.647947\pi$
$384$	0	0
$385$	8.51698	0.434065
$386$	0	0
$387$	62.0289i	3.15310i
$388$	0	0
$389$	− 2.23423i	− 0.113280i	−0.998395	−	0.0566399i	$-0.981961\pi$
$389$	− 2.23423i	− 0.113280i	0.998395	−	0.0566399i	$-0.0180387\pi$
$390$	0	0
$391$	−1.71079	−0.0865184
$392$	0	0
$393$	33.1142	1.67039
$394$	0	0
$395$	30.8296i	1.55121i
$396$	0	0
$397$	15.4349i	0.774655i	0.921942	+	0.387327i	$0.126602\pi$
$397$	15.4349i	0.774655i	−0.921942	+	0.387327i	$0.873398\pi$
$398$	0	0
$399$	−4.37781	−0.219164
$400$	0	0
$401$	−22.3503	−1.11612	−0.558061	−	0.829800i	$-0.688454\pi$
$401$	−22.3503	−1.11612	−0.558061	+	0.829800i	$0.688454\pi$
$402$	0	0
$403$	7.38645i	0.367945i
$404$	0	0
$405$	− 76.3082i	− 3.79178i
$406$	0	0
$407$	−28.3539	−1.40545
$408$	0	0
$409$	8.58617	0.424559	0.212279	−	0.977209i	$-0.431911\pi$
$409$	8.58617	0.424559	0.212279	+	0.977209i	$0.431911\pi$
$410$	0	0
$411$	− 23.1587i	− 1.14234i
$412$	0	0
$413$	− 2.71468i	− 0.133581i
$414$	0	0
$415$	11.5084	0.564925
$416$	0	0
$417$	−10.3441	−0.506551
$418$	0	0
$419$	− 21.9670i	− 1.07316i	−0.843850	−	0.536579i	$-0.819716\pi$
$419$	− 21.9670i	− 1.07316i	0.843850	−	0.536579i	$-0.180284\pi$
$420$	0	0
$421$	− 22.3942i	− 1.09143i	−0.837972	−	0.545713i	$-0.816259\pi$
$421$	− 22.3942i	− 1.09143i	0.837972	−	0.545713i	$-0.183741\pi$
$422$	0	0
$423$	−7.20922	−0.350524
$424$	0	0
$425$	−0.319878	−0.0155163
$426$	0	0
$427$	− 13.3534i	− 0.646215i
$428$	0	0
$429$	− 10.2118i	− 0.493031i
$430$	0	0
$431$	7.39422	0.356167	0.178084	−	0.984015i	$-0.443010\pi$
$431$	7.39422	0.356167	0.178084	+	0.984015i	$0.443010\pi$
$432$	0	0
$433$	−30.0766	−1.44539	−0.722694	−	0.691168i	$-0.757097\pi$
$433$	−30.0766	−1.44539	−0.722694	+	0.691168i	$0.757097\pi$
$434$	0	0
$435$	− 50.3899i	− 2.41601i
$436$	0	0
$437$	8.11631i	0.388256i
$438$	0	0
$439$	16.3397	0.779850	0.389925	−	0.920847i	$-0.372501\pi$
$439$	16.3397	0.779850	0.389925	+	0.920847i	$0.372501\pi$
$440$	0	0
$441$	41.5013	1.97625
$442$	0	0
$443$	4.07589i	0.193651i	0.995301	+	0.0968256i	$0.0308689\pi$
$443$	4.07589i	0.193651i	−0.995301	+	0.0968256i	$0.969131\pi$
$444$	0	0
$445$	− 16.9657i	− 0.804252i
$446$	0	0
$447$	−24.6844	−1.16753
$448$	0	0
$449$	4.21892	0.199103	0.0995517	−	0.995032i	$-0.468259\pi$
$449$	4.21892	0.199103	0.0995517	+	0.995032i	$0.468259\pi$
$450$	0	0
$451$	− 2.50968i	− 0.118176i
$452$	0	0
$453$	− 59.9191i	− 2.81525i
$454$	0	0
$455$	−4.15370	−0.194729
$456$	0	0
$457$	28.8382	1.34899	0.674496	−	0.738278i	$-0.264361\pi$
$457$	28.8382	1.34899	0.674496	+	0.738278i	$0.264361\pi$
$458$	0	0
$459$	3.42194i	0.159722i
$460$	0	0
$461$	31.0261i	1.44503i	0.691356	+	0.722514i	$0.257014\pi$
$461$	31.0261i	1.44503i	−0.691356	+	0.722514i	$0.742986\pi$
$462$	0	0
$463$	−12.5849	−0.584870	−0.292435	−	0.956285i	$-0.594465\pi$
$463$	−12.5849	−0.584870	−0.292435	+	0.956285i	$0.594465\pi$
$464$	0	0
$465$	50.7390	2.35296
$466$	0	0
$467$	27.8849i	1.29036i	0.764032	+	0.645179i	$0.223217\pi$
$467$	27.8849i	1.29036i	−0.764032	+	0.645179i	$0.776783\pi$
$468$	0	0
$469$	− 15.5816i	− 0.719492i
$470$	0	0
$471$	22.1140	1.01896
$472$	0	0
$473$	19.7321	0.907284
$474$	0	0
$475$	1.51756i	0.0696304i
$476$	0	0
$477$	− 26.3087i	− 1.20459i
$478$	0	0
$479$	11.7138	0.535217	0.267608	−	0.963528i	$-0.413767\pi$
$479$	11.7138	0.535217	0.267608	+	0.963528i	$0.413767\pi$
$480$	0	0
$481$	13.8281	0.630508
$482$	0	0
$483$	35.5316i	1.61675i
$484$	0	0
$485$	− 25.3445i	− 1.15083i
$486$	0	0
$487$	8.07492	0.365909	0.182955	−	0.983121i	$-0.441434\pi$
$487$	8.07492	0.365909	0.182955	+	0.983121i	$0.441434\pi$
$488$	0	0
$489$	−51.6769	−2.33691
$490$	0	0
$491$	− 15.0953i	− 0.681240i	−0.940201	−	0.340620i	$-0.889363\pi$
$491$	− 15.0953i	− 0.681240i	0.940201	−	0.340620i	$-0.110637\pi$
$492$	0	0
$493$	1.25935i	0.0567182i
$494$	0	0
$495$	−50.8653	−2.28622
$496$	0	0
$497$	−15.8943	−0.712958
$498$	0	0
$499$	− 31.2381i	− 1.39841i	−0.714922	−	0.699205i	$-0.753538\pi$
$499$	− 31.2381i	− 1.39841i	0.714922	−	0.699205i	$-0.246462\pi$
$500$	0	0
$501$	13.9856i	0.624829i
$502$	0	0
$503$	−21.7755	−0.970920	−0.485460	−	0.874259i	$-0.661348\pi$
$503$	−21.7755	−0.970920	−0.485460	+	0.874259i	$0.661348\pi$
$504$	0	0
$505$	−3.24020	−0.144187
$506$	0	0
$507$	− 37.9671i	− 1.68618i
$508$	0	0
$509$	4.88080i	0.216338i	0.994133	+	0.108169i	$0.0344987\pi$
$509$	4.88080i	0.216338i	−0.994133	+	0.108169i	$0.965501\pi$
$510$	0	0
$511$	−17.2319	−0.762295
$512$	0	0
$513$	16.2343	0.716762
$514$	0	0
$515$	36.8902i	1.62558i
$516$	0	0
$517$	2.29334i	0.100861i
$518$	0	0
$519$	14.0562	0.616998
$520$	0	0
$521$	10.8112	0.473645	0.236823	−	0.971553i	$-0.423894\pi$
$521$	10.8112	0.473645	0.236823	+	0.971553i	$0.423894\pi$
$522$	0	0
$523$	− 21.1722i	− 0.925798i	−0.886411	−	0.462899i	$-0.846809\pi$
$523$	− 21.1722i	− 0.925798i	0.886411	−	0.462899i	$-0.153191\pi$
$524$	0	0
$525$	6.64358i	0.289950i
$526$	0	0
$527$	−1.26807	−0.0552382
$528$	0	0
$529$	42.8745	1.86411
$530$	0	0
$531$	16.2127i	0.703570i
$532$	0	0
$533$	1.22396i	0.0530157i
$534$	0	0
$535$	−26.5170	−1.14643
$536$	0	0
$537$	−67.1205	−2.89646
$538$	0	0
$539$	− 13.2021i	− 0.568653i
$540$	0	0
$541$	15.1179i	0.649969i	0.945719	+	0.324984i	$0.105359\pi$
$541$	15.1179i	0.649969i	−0.945719	+	0.324984i	$0.894641\pi$
$542$	0	0
$543$	−54.9825	−2.35952
$544$	0	0
$545$	−39.2234	−1.68014
$546$	0	0
$547$	− 5.51960i	− 0.236001i	−0.993014	−	0.118001i	$-0.962352\pi$
$547$	− 5.51960i	− 0.236001i	0.993014	−	0.118001i	$-0.0376485\pi$
$548$	0	0
$549$	79.7493i	3.40362i
$550$	0	0
$551$	5.97458	0.254526
$552$	0	0
$553$	−16.0025	−0.680497
$554$	0	0
$555$	− 94.9881i	− 4.03202i
$556$	0	0
$557$	− 18.9708i	− 0.803820i	−0.915679	−	0.401910i	$-0.868346\pi$
$557$	− 18.9708i	− 0.803820i	0.915679	−	0.401910i	$-0.131654\pi$
$558$	0	0
$559$	−9.62330	−0.407022
$560$	0	0
$561$	1.75312	0.0740167
$562$	0	0
$563$	− 15.5505i	− 0.655374i	−0.944786	−	0.327687i	$-0.893731\pi$
$563$	− 15.5505i	− 0.655374i	0.944786	−	0.327687i	$-0.106269\pi$
$564$	0	0
$565$	− 41.4878i	− 1.74540i
$566$	0	0
$567$	39.6088	1.66341
$568$	0	0
$569$	−3.32025	−0.139192	−0.0695960	−	0.997575i	$-0.522171\pi$
$569$	−3.32025	−0.139192	−0.0695960	+	0.997575i	$0.522171\pi$
$570$	0	0
$571$	8.41420i	0.352123i	0.984379	+	0.176062i	$0.0563358\pi$
$571$	8.41420i	0.352123i	−0.984379	+	0.176062i	$0.943664\pi$
$572$	0	0
$573$	52.6092i	2.19778i
$574$	0	0
$575$	12.3170	0.513653
$576$	0	0
$577$	5.94857	0.247642	0.123821	−	0.992305i	$-0.460485\pi$
$577$	5.94857	0.247642	0.123821	+	0.992305i	$0.460485\pi$
$578$	0	0
$579$	85.4857i	3.55266i
$580$	0	0
$581$	5.97359i	0.247826i
$582$	0	0
$583$	−8.36911	−0.346613
$584$	0	0
$585$	24.8068	1.02564
$586$	0	0
$587$	27.1449i	1.12039i	0.828360	+	0.560196i	$0.189274\pi$
$587$	27.1449i	1.12039i	−0.828360	+	0.560196i	$0.810726\pi$
$588$	0	0
$589$	6.01598i	0.247884i
$590$	0	0
$591$	60.4071	2.48482
$592$	0	0
$593$	−24.3801	−1.00117	−0.500585	−	0.865687i	$-0.666882\pi$
$593$	−24.3801	−1.00117	−0.500585	+	0.865687i	$0.666882\pi$
$594$	0	0
$595$	− 0.713090i	− 0.0292338i
$596$	0	0
$597$	33.3032i	1.36301i
$598$	0	0
$599$	−18.7108	−0.764502	−0.382251	−	0.924059i	$-0.624851\pi$
$599$	−18.7108	−0.764502	−0.382251	+	0.924059i	$0.624851\pi$
$600$	0	0
$601$	44.2967	1.80690	0.903451	−	0.428692i	$-0.141025\pi$
$601$	44.2967	1.80690	0.903451	+	0.428692i	$0.141025\pi$
$602$	0	0
$603$	93.0568i	3.78957i
$604$	0	0
$605$	− 11.9016i	− 0.483869i
$606$	0	0
$607$	−30.5153	−1.23858	−0.619290	−	0.785163i	$-0.712579\pi$
$607$	−30.5153	−1.23858	−0.619290	+	0.785163i	$0.712579\pi$
$608$	0	0
$609$	26.1556	1.05988
$610$	0	0
$611$	− 1.11845i	− 0.0452478i
$612$	0	0
$613$	− 21.3377i	− 0.861822i	−0.902395	−	0.430911i	$-0.858192\pi$
$613$	− 21.3377i	− 0.861822i	0.902395	−	0.430911i	$-0.141808\pi$
$614$	0	0
$615$	8.40764	0.339029
$616$	0	0
$617$	−23.5271	−0.947163	−0.473582	−	0.880750i	$-0.657039\pi$
$617$	−23.5271	−0.947163	−0.473582	+	0.880750i	$0.657039\pi$
$618$	0	0
$619$	18.0716i	0.726357i	0.931720	+	0.363179i	$0.118308\pi$
$619$	18.0716i	0.726357i	−0.931720	+	0.363179i	$0.881692\pi$
$620$	0	0
$621$	− 131.763i	− 5.28745i
$622$	0	0
$623$	8.80630	0.352817
$624$	0	0
$625$	−30.2848	−1.21139
$626$	0	0
$627$	− 8.31712i	− 0.332154i
$628$	0	0
$629$	2.37395i	0.0946557i
$630$	0	0
$631$	13.2280	0.526600	0.263300	−	0.964714i	$-0.415189\pi$
$631$	13.2280	0.526600	0.263300	+	0.964714i	$0.415189\pi$
$632$	0	0
$633$	−52.2397	−2.07634
$634$	0	0
$635$	− 16.7650i	− 0.665301i
$636$	0	0
$637$	6.43860i	0.255107i
$638$	0	0
$639$	94.9245	3.75515
$640$	0	0
$641$	−30.1208	−1.18970	−0.594851	−	0.803836i	$-0.702789\pi$
$641$	−30.1208	−1.18970	−0.594851	+	0.803836i	$0.702789\pi$
$642$	0	0
$643$	24.7964i	0.977876i	0.872319	+	0.488938i	$0.162616\pi$
$643$	24.7964i	0.977876i	−0.872319	+	0.488938i	$0.837384\pi$
$644$	0	0
$645$	66.1044i	2.60286i
$646$	0	0
$647$	22.8386	0.897880	0.448940	−	0.893562i	$-0.351802\pi$
$647$	22.8386	0.897880	0.448940	+	0.893562i	$0.351802\pi$
$648$	0	0
$649$	5.15744	0.202447
$650$	0	0
$651$	26.3368i	1.03222i
$652$	0	0
$653$	32.4823i	1.27113i	0.772048	+	0.635565i	$0.219233\pi$
$653$	32.4823i	1.27113i	−0.772048	+	0.635565i	$0.780767\pi$
$654$	0	0
$655$	25.5896	0.999867
$656$	0	0
$657$	102.913	4.01501
$658$	0	0
$659$	− 17.2055i	− 0.670229i	−0.942177	−	0.335115i	$-0.891225\pi$
$659$	− 17.2055i	− 0.670229i	0.942177	−	0.335115i	$-0.108775\pi$
$660$	0	0
$661$	− 13.0942i	− 0.509305i	−0.967033	−	0.254653i	$-0.918039\pi$
$661$	− 13.0942i	− 0.509305i	0.967033	−	0.254653i	$-0.0819611\pi$
$662$	0	0
$663$	−0.854990	−0.0332051
$664$	0	0
$665$	−3.38303	−0.131188
$666$	0	0
$667$	− 48.4916i	− 1.87760i
$668$	0	0
$669$	− 13.7375i	− 0.531121i
$670$	0	0
$671$	25.3692	0.979368
$672$	0	0
$673$	−11.3396	−0.437110	−0.218555	−	0.975825i	$-0.570134\pi$
$673$	−11.3396	−0.437110	−0.218555	+	0.975825i	$0.570134\pi$
$674$	0	0
$675$	− 24.6365i	− 0.948260i
$676$	0	0
$677$	43.2663i	1.66286i	0.555631	+	0.831429i	$0.312477\pi$
$677$	43.2663i	1.66286i	−0.555631	+	0.831429i	$0.687523\pi$
$678$	0	0
$679$	13.1554	0.504859
$680$	0	0
$681$	−55.4066	−2.12319
$682$	0	0
$683$	− 7.63100i	− 0.291992i	−0.989285	−	0.145996i	$-0.953361\pi$
$683$	− 7.63100i	− 0.291992i	0.989285	−	0.145996i	$-0.0466387\pi$
$684$	0	0
$685$	− 17.8963i	− 0.683783i
$686$	0	0
$687$	4.39051	0.167508
$688$	0	0
$689$	4.08159	0.155496
$690$	0	0
$691$	20.2177i	0.769117i	0.923101	+	0.384558i	$0.125646\pi$
$691$	20.2177i	0.769117i	−0.923101	+	0.384558i	$0.874354\pi$
$692$	0	0
$693$	− 26.4023i	− 1.00294i
$694$	0	0
$695$	−7.99356	−0.303213
$696$	0	0
$697$	−0.210124	−0.00795903
$698$	0	0
$699$	51.0845i	1.93219i
$700$	0	0
$701$	13.3108i	0.502743i	0.967891	+	0.251372i	$0.0808816\pi$
$701$	13.3108i	0.502743i	−0.967891	+	0.251372i	$0.919118\pi$
$702$	0	0
$703$	11.2625	0.424772
$704$	0	0
$705$	−7.68288	−0.289354
$706$	0	0
$707$	− 1.68187i	− 0.0632533i
$708$	0	0
$709$	− 23.7864i	− 0.893319i	−0.894704	−	0.446659i	$-0.852614\pi$
$709$	− 23.7864i	− 0.893319i	0.894704	−	0.446659i	$-0.147386\pi$
$710$	0	0
$711$	95.5707	3.58418
$712$	0	0
$713$	48.8275	1.82861
$714$	0	0
$715$	− 7.89136i	− 0.295120i
$716$	0	0
$717$	− 51.0626i	− 1.90697i
$718$	0	0
$719$	−8.77476	−0.327243	−0.163622	−	0.986523i	$-0.552318\pi$
$719$	−8.77476	−0.327243	−0.163622	+	0.986523i	$0.552318\pi$
$720$	0	0
$721$	−19.1484	−0.713123
$722$	0	0
$723$	46.2927i	1.72164i
$724$	0	0
$725$	− 9.06678i	− 0.336732i
$726$	0	0
$727$	41.1826	1.52738	0.763689	−	0.645584i	$-0.223386\pi$
$727$	41.1826	1.52738	0.763689	+	0.645584i	$0.223386\pi$
$728$	0	0
$729$	−75.6557	−2.80206
$730$	0	0
$731$	− 1.65209i	− 0.0611046i
$732$	0	0
$733$	− 48.2810i	− 1.78330i	−0.452727	−	0.891649i	$-0.649549\pi$
$733$	− 48.2810i	− 1.78330i	0.452727	−	0.891649i	$-0.350451\pi$
$734$	0	0
$735$	44.2280	1.63138
$736$	0	0
$737$	29.6025	1.09042
$738$	0	0
$739$	− 24.7479i	− 0.910366i	−0.890398	−	0.455183i	$-0.849574\pi$
$739$	− 24.7479i	− 0.910366i	0.890398	−	0.455183i	$-0.150426\pi$
$740$	0	0
$741$	4.05623i	0.149009i
$742$	0	0
$743$	−40.1141	−1.47164	−0.735822	−	0.677175i	$-0.763204\pi$
$743$	−40.1141	−1.47164	−0.735822	+	0.677175i	$0.763204\pi$
$744$	0	0
$745$	−19.0753	−0.698865
$746$	0	0
$747$	− 35.6756i	− 1.30530i
$748$	0	0
$749$	− 13.7640i	− 0.502927i
$750$	0	0
$751$	31.9595	1.16622	0.583110	−	0.812393i	$-0.301836\pi$
$751$	31.9595	1.16622	0.583110	+	0.812393i	$0.301836\pi$
$752$	0	0
$753$	14.4072	0.525029
$754$	0	0
$755$	− 46.3036i	− 1.68516i
$756$	0	0
$757$	− 25.7492i	− 0.935870i	−0.883763	−	0.467935i	$-0.844998\pi$
$757$	− 25.7492i	− 0.935870i	0.883763	−	0.467935i	$-0.155002\pi$
$758$	0	0
$759$	−67.5043	−2.45025
$760$	0	0
$761$	−21.4825	−0.778740	−0.389370	−	0.921082i	$-0.627307\pi$
$761$	−21.4825	−0.778740	−0.389370	+	0.921082i	$0.627307\pi$
$762$	0	0
$763$	− 20.3594i	− 0.737061i
$764$	0	0
$765$	4.25873i	0.153975i
$766$	0	0
$767$	−2.51527	−0.0908211
$768$	0	0
$769$	13.9749	0.503949	0.251975	−	0.967734i	$-0.418920\pi$
$769$	13.9749	0.503949	0.251975	+	0.967734i	$0.418920\pi$
$770$	0	0
$771$	− 70.7762i	− 2.54894i
$772$	0	0
$773$	− 15.8238i	− 0.569143i	−0.958655	−	0.284572i	$-0.908149\pi$
$773$	− 15.8238i	− 0.569143i	0.958655	−	0.284572i	$-0.0918513\pi$
$774$	0	0
$775$	9.12960	0.327945
$776$	0	0
$777$	49.3049	1.76880
$778$	0	0
$779$	0.996870i	0.0357166i
$780$	0	0
$781$	− 30.1966i	− 1.08052i
$782$	0	0
$783$	−96.9933	−3.46626
$784$	0	0
$785$	17.0890	0.609933
$786$	0	0
$787$	24.5418i	0.874820i	0.899262	+	0.437410i	$0.144104\pi$
$787$	24.5418i	0.874820i	−0.899262	+	0.437410i	$0.855896\pi$
$788$	0	0
$789$	− 17.7528i	− 0.632015i
$790$	0	0
$791$	21.5348	0.765690
$792$	0	0
$793$	−12.3725	−0.439360
$794$	0	0
$795$	− 28.0372i	− 0.994378i
$796$	0	0
$797$	48.3819i	1.71377i	0.515505	+	0.856887i	$0.327604\pi$
$797$	48.3819i	1.71377i	−0.515505	+	0.856887i	$0.672396\pi$
$798$	0	0
$799$	0.192011	0.00679287
$800$	0	0
$801$	−52.5932	−1.85829
$802$	0	0
$803$	− 32.7378i	− 1.15529i
$804$	0	0
$805$	27.4577i	0.967758i
$806$	0	0
$807$	33.2623	1.17089
$808$	0	0
$809$	−13.1294	−0.461604	−0.230802	−	0.973001i	$-0.574135\pi$
$809$	−13.1294	−0.461604	−0.230802	+	0.973001i	$0.574135\pi$
$810$	0	0
$811$	− 3.76569i	− 0.132231i	−0.997812	−	0.0661157i	$-0.978939\pi$
$811$	− 3.76569i	− 0.132231i	0.997812	−	0.0661157i	$-0.0210606\pi$
$812$	0	0
$813$	91.2951i	3.20186i
$814$	0	0
$815$	−39.9343	−1.39884
$816$	0	0
$817$	−7.83781	−0.274210
$818$	0	0
$819$	12.8763i	0.449936i
$820$	0	0
$821$	− 40.5291i	− 1.41447i	−0.706977	−	0.707237i	$-0.749941\pi$
$821$	− 40.5291i	− 1.41447i	0.706977	−	0.707237i	$-0.250059\pi$
$822$	0	0
$823$	13.7979	0.480964	0.240482	−	0.970654i	$-0.422695\pi$
$823$	13.7979	0.480964	0.240482	+	0.970654i	$0.422695\pi$
$824$	0	0
$825$	−12.6217	−0.439432
$826$	0	0
$827$	47.3483i	1.64646i	0.567708	+	0.823230i	$0.307830\pi$
$827$	47.3483i	1.64646i	−0.567708	+	0.823230i	$0.692170\pi$
$828$	0	0
$829$	2.49303i	0.0865865i	0.999062	+	0.0432933i	$0.0137850\pi$
$829$	2.49303i	0.0865865i	−0.999062	+	0.0432933i	$0.986215\pi$
$830$	0	0
$831$	31.4704	1.09170
$832$	0	0
$833$	−1.10535	−0.0382981
$834$	0	0
$835$	10.8076i	0.374012i
$836$	0	0
$837$	− 97.6652i	− 3.37580i
$838$	0	0
$839$	3.94935	0.136347	0.0681733	−	0.997673i	$-0.478283\pi$
$839$	3.94935	0.136347	0.0681733	+	0.997673i	$0.478283\pi$
$840$	0	0
$841$	−6.69567	−0.230885
$842$	0	0
$843$	− 91.2360i	− 3.14233i
$844$	0	0
$845$	− 29.3398i	− 1.00932i
$846$	0	0
$847$	6.17769	0.212268
$848$	0	0
$849$	56.2253	1.92965
$850$	0	0
$851$	− 91.4097i	− 3.13348i
$852$	0	0
$853$	47.6335i	1.63094i	0.578799	+	0.815470i	$0.303521\pi$
$853$	47.6335i	1.63094i	−0.578799	+	0.815470i	$0.696479\pi$
$854$	0	0
$855$	20.2042	0.690969
$856$	0	0
$857$	−50.1888	−1.71442	−0.857209	−	0.514969i	$-0.827804\pi$
$857$	−50.1888	−1.71442	−0.857209	+	0.514969i	$0.827804\pi$
$858$	0	0
$859$	11.0894i	0.378366i	0.981942	+	0.189183i	$0.0605839\pi$
$859$	11.0894i	0.378366i	−0.981942	+	0.189183i	$0.939416\pi$
$860$	0	0
$861$	4.36410i	0.148728i
$862$	0	0
$863$	37.7550	1.28520	0.642598	−	0.766204i	$-0.277857\pi$
$863$	37.7550	1.28520	0.642598	+	0.766204i	$0.277857\pi$
$864$	0	0
$865$	10.8622	0.369325
$866$	0	0
$867$	56.0152i	1.90237i
$868$	0	0
$869$	− 30.4022i	− 1.03132i
$870$	0	0
$871$	−14.4371	−0.489181
$872$	0	0
$873$	−78.5671	−2.65909
$874$	0	0
$875$	− 11.7812i	− 0.398277i
$876$	0	0
$877$	− 43.6274i	− 1.47319i	−0.676333	−	0.736596i	$-0.736432\pi$
$877$	− 43.6274i	− 1.47319i	0.676333	−	0.736596i	$-0.263568\pi$
$878$	0	0
$879$	45.9049	1.54833
$880$	0	0
$881$	−15.5145	−0.522698	−0.261349	−	0.965244i	$-0.584167\pi$
$881$	−15.5145	−0.522698	−0.261349	+	0.965244i	$0.584167\pi$
$882$	0	0
$883$	− 0.166224i	− 0.00559388i	−0.999996	−	0.00279694i	$-0.999110\pi$
$883$	− 0.166224i	− 0.00559388i	0.999996	−	0.00279694i	$-0.000890295\pi$
$884$	0	0
$885$	17.2779i	0.580790i
$886$	0	0
$887$	9.04708	0.303771	0.151886	−	0.988398i	$-0.451465\pi$
$887$	9.04708	0.303771	0.151886	+	0.988398i	$0.451465\pi$
$888$	0	0
$889$	8.70213	0.291860
$890$	0	0
$891$	75.2503i	2.52098i
$892$	0	0
$893$	− 0.910938i	− 0.0304834i
$894$	0	0
$895$	−51.8686	−1.73378
$896$	0	0
$897$	32.9216	1.09922
$898$	0	0
$899$	− 35.9430i	− 1.19877i
$900$	0	0
$901$	0.700709i	0.0233440i
$902$	0	0
$903$	−34.3124	−1.14185
$904$	0	0
$905$	−42.4887	−1.41237
$906$	0	0
$907$	− 16.2748i	− 0.540395i	−0.962805	−	0.270198i	$-0.912911\pi$
$907$	− 16.2748i	− 0.540395i	0.962805	−	0.270198i	$-0.0870891\pi$
$908$	0	0
$909$	10.0445i	0.333155i
$910$	0	0
$911$	11.6990	0.387604	0.193802	−	0.981041i	$-0.437918\pi$
$911$	11.6990	0.387604	0.193802	+	0.981041i	$0.437918\pi$
$912$	0	0
$913$	−11.3488	−0.375592
$914$	0	0
$915$	84.9891i	2.80965i
$916$	0	0
$917$	13.2826i	0.438631i
$918$	0	0
$919$	24.0697	0.793987	0.396994	−	0.917821i	$-0.370053\pi$
$919$	24.0697	0.793987	0.396994	+	0.917821i	$0.370053\pi$
$920$	0	0
$921$	28.3224	0.933254
$922$	0	0
$923$	14.7268i	0.484739i
$924$	0	0
$925$	− 17.0915i	− 0.561964i
$926$	0	0
$927$	114.358	3.75602
$928$	0	0
$929$	44.5910	1.46298	0.731491	−	0.681851i	$-0.238825\pi$
$929$	44.5910	1.46298	0.731491	+	0.681851i	$0.238825\pi$
$930$	0	0
$931$	5.24399i	0.171865i
$932$	0	0
$933$	− 61.9241i	− 2.02730i
$934$	0	0
$935$	1.35475	0.0443052
$936$	0	0
$937$	46.6674	1.52456	0.762278	−	0.647250i	$-0.224081\pi$
$937$	46.6674	1.52456	0.762278	+	0.647250i	$0.224081\pi$
$938$	0	0
$939$	− 0.883087i	− 0.0288185i
$940$	0	0
$941$	38.4624i	1.25384i	0.779085	+	0.626919i	$0.215684\pi$
$941$	38.4624i	1.25384i	−0.779085	+	0.626919i	$0.784316\pi$
$942$	0	0
$943$	8.09090	0.263476
$944$	0	0
$945$	54.9212	1.78659
$946$	0	0
$947$	28.6521i	0.931068i	0.885030	+	0.465534i	$0.154138\pi$
$947$	28.6521i	0.931068i	−0.885030	+	0.465534i	$0.845862\pi$
$948$	0	0
$949$	15.9661i	0.518282i
$950$	0	0
$951$	−80.5642	−2.61247
$952$	0	0
$953$	55.7585	1.80619	0.903097	−	0.429437i	$-0.141288\pi$
$953$	55.7585	1.80619	0.903097	+	0.429437i	$0.141288\pi$
$954$	0	0
$955$	40.6547i	1.31556i
$956$	0	0
$957$	49.6913i	1.60629i
$958$	0	0
$959$	9.28934	0.299968
$960$	0	0
$961$	5.19198	0.167483
$962$	0	0
$963$	82.2019i	2.64892i
$964$	0	0
$965$	66.0606i	2.12657i
$966$	0	0
$967$	−1.84686	−0.0593909	−0.0296954	−	0.999559i	$-0.509454\pi$
$967$	−1.84686	−0.0593909	−0.0296954	+	0.999559i	$0.509454\pi$
$968$	0	0
$969$	−0.696356	−0.0223702
$970$	0	0
$971$	6.95804i	0.223294i	0.993748	+	0.111647i	$0.0356126\pi$
$971$	6.95804i	0.223294i	−0.993748	+	0.111647i	$0.964387\pi$
$972$	0	0
$973$	− 4.14917i	− 0.133016i
$974$	0	0
$975$	6.15557	0.197136
$976$	0	0
$977$	−2.75699	−0.0882039	−0.0441020	−	0.999027i	$-0.514043\pi$
$977$	−2.75699	−0.0882039	−0.0441020	+	0.999027i	$0.514043\pi$
$978$	0	0
$979$	16.7305i	0.534710i
$980$	0	0
$981$	121.591i	3.88211i
$982$	0	0
$983$	42.9562	1.37009	0.685045	−	0.728500i	$-0.259782\pi$
$983$	42.9562	1.37009	0.685045	+	0.728500i	$0.259782\pi$
$984$	0	0
$985$	46.6807	1.48737
$986$	0	0
$987$	− 3.98791i	− 0.126937i
$988$	0	0
$989$	63.6141i	2.02281i
$990$	0	0
$991$	45.1884	1.43546	0.717729	−	0.696323i	$-0.245182\pi$
$991$	45.1884	1.43546	0.717729	+	0.696323i	$0.245182\pi$
$992$	0	0
$993$	52.5208	1.66670
$994$	0	0
$995$	25.7357i	0.815876i
$996$	0	0
$997$	− 33.7095i	− 1.06759i	−0.845614	−	0.533795i	$-0.820766\pi$
$997$	− 33.7095i	− 1.06759i	0.845614	−	0.533795i	$-0.179234\pi$
$998$	0	0
$999$	−182.838	−5.78475

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.j.1217.1	✓	20
4.3	odd	2	inner	2432.2.c.j.1217.19	yes	20
8.3	odd	2	inner	2432.2.c.j.1217.2	yes	20
8.5	even	2	inner	2432.2.c.j.1217.20	yes	20
16.3	odd	4		4864.2.a.bs.1.1		10
16.5	even	4		4864.2.a.bs.1.2		10
16.11	odd	4		4864.2.a.bt.1.10		10
16.13	even	4		4864.2.a.bt.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.j.1217.1	✓	20	1.1	even	1	trivial
2432.2.c.j.1217.2	yes	20	8.3	odd	2	inner
2432.2.c.j.1217.19	yes	20	4.3	odd	2	inner
2432.2.c.j.1217.20	yes	20	8.5	even	2	inner
4864.2.a.bs.1.1		10	16.3	odd	4
4864.2.a.bs.1.2		10	16.5	even	4
4864.2.a.bt.1.9		10	16.13	even	4
4864.2.a.bt.1.10		10	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-3.30364i q^{3} -2.55295i q^{5} +1.32515 q^{7} -7.91406 q^{9} +2.51756i q^{11} -1.22781i q^{13} -8.43404 q^{15} +0.210784 q^{17} -1.00000i q^{19} -4.37781i q^{21} -8.11631 q^{23} -1.51756 q^{25} +16.2343i q^{27} +5.97458i q^{29} -6.01598 q^{31} +8.31712 q^{33} -3.38303i q^{35} +11.2625i q^{37} -4.05623 q^{39} -0.996870 q^{41} -7.83781i q^{43} +20.2042i q^{45} +0.910938 q^{47} -5.24399 q^{49} -0.696356i q^{51} +3.32429i q^{53} +6.42720 q^{55} -3.30364 q^{57} -2.04859i q^{59} -10.0769i q^{61} -10.4873 q^{63} -3.13453 q^{65} -11.7584i q^{67} +26.8134i q^{69} -11.9944 q^{71} -13.0038 q^{73} +5.01347i q^{75} +3.33613i q^{77} -12.0761 q^{79} +29.8902 q^{81} +4.50788i q^{83} -0.538122i q^{85} +19.7379 q^{87} +6.64553 q^{89} -1.62702i q^{91} +19.8746i q^{93} -2.55295 q^{95} +9.92753 q^{97} -19.9241i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 28 q^{9} + 8 q^{17} - 20 q^{25} - 16 q^{33} + 24 q^{41} + 52 q^{49} - 8 q^{57} - 48 q^{65} - 24 q^{73} + 68 q^{81} + 40 q^{89} - 56 q^{97}+O(q^{100}) \) 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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