LMFDB - Embedded newform 432.4.c.f.431.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,4,Mod(431,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("432.431");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$432 = 2^{4} \cdot 3^{3}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	432.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:	$25.4888251225$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{-17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 7x^{2} + 25$ x^4 + 7*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}\cdot 3^{4}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.3
Root			$0.866025 - 2.06155i$ of defining polynomial
Character	$\chi$	$=$	432.431
Dual form			432.4.c.f.431.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12.3693i q^{5} -21.4243i q^{7} +15.5885 q^{11} -2.00000 q^{13} +24.7386i q^{17} -85.6971i q^{19} +124.708 q^{23} -28.0000 q^{25} -272.125i q^{29} +278.516i q^{31} +265.004 q^{35} +128.000 q^{37} +296.864i q^{41} +42.8486i q^{43} +592.361 q^{47} -116.000 q^{49} +259.756i q^{53} +192.819i q^{55} +530.008 q^{59} -340.000 q^{61} -24.7386i q^{65} -899.820i q^{67} +966.484 q^{71} +817.000 q^{73} -333.972i q^{77} -214.243i q^{79} +358.535 q^{83} -306.000 q^{85} -915.329i q^{89} +42.8486i q^{91} +1060.02 q^{95} -965.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 8 q^{13} - 112 q^{25} + 512 q^{37} - 464 q^{49} - 1360 q^{61} + 3268 q^{73} - 1224 q^{85} - 3860 q^{97}+O(q^{100})$ 4 * q - 8 * q^13 - 112 * q^25 + 512 * q^37 - 464 * q^49 - 1360 * q^61 + 3268 * q^73 - 1224 * q^85 - 3860 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$ .

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$1$	$-1$

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	0
$3$	0	0
$4$	0	0
$5$	12.3693i	1.10635i	0.833067	+	0.553173i	$0.186583\pi$
$5$	12.3693i	1.10635i	−0.833067	+	0.553173i	$0.813417\pi$
$6$	0	0
$7$	− 21.4243i	− 1.15680i	−0.815752	−	0.578401i	$-0.803677\pi$
$7$	− 21.4243i	− 1.15680i	0.815752	−	0.578401i	$-0.196323\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	15.5885	0.427282	0.213641	−	0.976912i	$-0.431468\pi$
$11$	15.5885	0.427282	0.213641	+	0.976912i	$0.431468\pi$
$12$	0	0
$13$	−2.00000	−0.0426692	−0.0213346	−	0.999772i	$-0.506792\pi$
$13$	−2.00000	−0.0426692	−0.0213346	+	0.999772i	$0.506792\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	24.7386i	0.352941i	0.984306	+	0.176471i	$0.0564680\pi$
$17$	24.7386i	0.352941i	−0.984306	+	0.176471i	$0.943532\pi$
$18$	0	0
$19$	− 85.6971i	− 1.03475i	−0.855758	−	0.517376i	$-0.826909\pi$
$19$	− 85.6971i	− 1.03475i	0.855758	−	0.517376i	$-0.173091\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	124.708	1.13058	0.565290	−	0.824892i	$-0.308764\pi$
$23$	124.708	1.13058	0.565290	+	0.824892i	$0.308764\pi$
$24$	0	0
$25$	−28.0000	−0.224000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 272.125i	− 1.74249i	−0.490844	−	0.871247i	$-0.663312\pi$
$29$	− 272.125i	− 1.74249i	0.490844	−	0.871247i	$-0.336688\pi$
$30$	0	0
$31$	278.516i	1.61364i	0.590796	+	0.806821i	$0.298814\pi$
$31$	278.516i	1.61364i	−0.590796	+	0.806821i	$0.701186\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	265.004	1.27982
$36$	0	0
$37$	128.000	0.568732	0.284366	−	0.958716i	$-0.408217\pi$
$37$	128.000	0.568732	0.284366	+	0.958716i	$0.408217\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	296.864i	1.13079i	0.824821	+	0.565394i	$0.191276\pi$
$41$	296.864i	1.13079i	−0.824821	+	0.565394i	$0.808724\pi$
$42$	0	0
$43$	42.8486i	0.151962i	0.997109	+	0.0759808i	$0.0242088\pi$
$43$	42.8486i	0.151962i	−0.997109	+	0.0759808i	$0.975791\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	592.361	1.83840	0.919200	−	0.393791i	$-0.128837\pi$
$47$	592.361	1.83840	0.919200	+	0.393791i	$0.128837\pi$
$48$	0	0
$49$	−116.000	−0.338192
$50$	0	0
$51$	0	0
$52$	0	0
$53$	259.756i	0.673211i	0.941646	+	0.336606i	$0.109279\pi$
$53$	259.756i	0.673211i	−0.941646	+	0.336606i	$0.890721\pi$
$54$	0	0
$55$	192.819i	0.472721i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	530.008	1.16951	0.584755	−	0.811210i	$-0.301191\pi$
$59$	530.008	1.16951	0.584755	+	0.811210i	$0.301191\pi$
$60$	0	0
$61$	−340.000	−0.713648	−0.356824	−	0.934172i	$-0.616140\pi$
$61$	−340.000	−0.713648	−0.356824	+	0.934172i	$0.616140\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 24.7386i	− 0.0472069i
$66$	0	0
$67$	− 899.820i	− 1.64075i	−0.571823	−	0.820377i	$-0.693764\pi$
$67$	− 899.820i	− 1.64075i	0.571823	−	0.820377i	$-0.306236\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	966.484	1.61550	0.807751	−	0.589524i	$-0.200685\pi$
$71$	966.484	1.61550	0.807751	+	0.589524i	$0.200685\pi$
$72$	0	0
$73$	817.000	1.30990	0.654949	−	0.755673i	$-0.272690\pi$
$73$	817.000	1.30990	0.654949	+	0.755673i	$0.272690\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 333.972i	− 0.494280i
$78$	0	0
$79$	− 214.243i	− 0.305117i	−0.988294	−	0.152558i	$-0.951249\pi$
$79$	− 214.243i	− 0.305117i	0.988294	−	0.152558i	$-0.0487512\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	358.535	0.474148	0.237074	−	0.971492i	$-0.423812\pi$
$83$	358.535	0.474148	0.237074	+	0.971492i	$0.423812\pi$
$84$	0	0
$85$	−306.000	−0.390475
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 915.329i	− 1.09017i	−0.838382	−	0.545083i	$-0.816498\pi$
$89$	− 915.329i	− 1.09017i	0.838382	−	0.545083i	$-0.183502\pi$
$90$	0	0
$91$	42.8486i	0.0493599i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1060.02	1.14479
$96$	0	0
$97$	−965.000	−1.01011	−0.505056	−	0.863086i	$-0.668528\pi$
$97$	−965.000	−1.01011	−0.505056	+	0.863086i	$0.668528\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	853.483i	0.840839i	0.907330	+	0.420419i	$0.138117\pi$
$101$	853.483i	0.840839i	−0.907330	+	0.420419i	$0.861883\pi$
$102$	0	0
$103$	− 128.546i	− 0.122971i	−0.998108	−	0.0614854i	$-0.980416\pi$
$103$	− 128.546i	− 0.122971i	0.998108	−	0.0614854i	$-0.0195838\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−607.950	−0.549278	−0.274639	−	0.961547i	$-0.588558\pi$
$107$	−607.950	−0.549278	−0.274639	+	0.961547i	$0.588558\pi$
$108$	0	0
$109$	1816.00	1.59579	0.797896	−	0.602796i	$-0.205947\pi$
$109$	1816.00	1.59579	0.797896	+	0.602796i	$0.205947\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 940.068i	− 0.782603i	−0.920262	−	0.391302i	$-0.872025\pi$
$113$	− 940.068i	− 0.782603i	0.920262	−	0.391302i	$-0.127975\pi$
$114$	0	0
$115$	1542.55i	1.25081i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	530.008	0.408283
$120$	0	0
$121$	−1088.00	−0.817431
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1199.82i	0.858524i
$126$	0	0
$127$	792.699i	0.553863i	0.960890	+	0.276932i	$0.0893175\pi$
$127$	792.699i	0.553863i	−0.960890	+	0.276932i	$0.910682\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1761.50	−1.17483	−0.587414	−	0.809286i	$-0.699854\pi$
$131$	−1761.50	−1.17483	−0.587414	+	0.809286i	$0.699854\pi$
$132$	0	0
$133$	−1836.00	−1.19700
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2647.03i	1.65074i	0.564592	+	0.825370i	$0.309033\pi$
$137$	2647.03i	1.65074i	−0.564592	+	0.825370i	$0.690967\pi$
$138$	0	0
$139$	1971.03i	1.20274i	0.798970	+	0.601370i	$0.205378\pi$
$139$	1971.03i	1.20274i	−0.798970	+	0.601370i	$0.794622\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−31.1769	−0.0182318
$144$	0	0
$145$	3366.00	1.92780
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 235.017i	− 0.129217i	−0.997911	−	0.0646085i	$-0.979420\pi$
$149$	− 235.017i	− 0.129217i	0.997911	−	0.0646085i	$-0.0205799\pi$
$150$	0	0
$151$	− 3020.82i	− 1.62802i	−0.580850	−	0.814011i	$-0.697280\pi$
$151$	− 3020.82i	− 1.62802i	0.580850	−	0.814011i	$-0.302720\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3445.05	−1.78525
$156$	0	0
$157$	−484.000	−0.246034	−0.123017	−	0.992405i	$-0.539257\pi$
$157$	−484.000	−0.246034	−0.123017	+	0.992405i	$0.539257\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2671.77i	− 1.30786i
$162$	0	0
$163$	− 2013.88i	− 0.967727i	−0.875144	−	0.483863i	$-0.839233\pi$
$163$	− 2013.88i	− 0.967727i	0.875144	−	0.483863i	$-0.160767\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−841.777	−0.390052	−0.195026	−	0.980798i	$-0.562479\pi$
$167$	−841.777	−0.390052	−0.195026	+	0.980798i	$0.562479\pi$
$168$	0	0
$169$	−2193.00	−0.998179
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3871.60i	− 1.70146i	−0.525606	−	0.850728i	$-0.676161\pi$
$173$	− 3871.60i	− 1.70146i	0.525606	−	0.850728i	$-0.323839\pi$
$174$	0	0
$175$	599.880i	0.259124i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3756.82	−1.56870	−0.784351	−	0.620317i	$-0.787004\pi$
$179$	−3756.82	−1.56870	−0.784351	+	0.620317i	$0.787004\pi$
$180$	0	0
$181$	3400.00	1.39624	0.698122	−	0.715979i	$-0.254019\pi$
$181$	3400.00	1.39624	0.698122	+	0.715979i	$0.254019\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1583.27i	0.629214i
$186$	0	0
$187$	385.637i	0.150805i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−280.592	−0.106298	−0.0531491	−	0.998587i	$-0.516926\pi$
$191$	−280.592	−0.106298	−0.0531491	+	0.998587i	$0.516926\pi$
$192$	0	0
$193$	−2941.00	−1.09688	−0.548440	−	0.836190i	$-0.684778\pi$
$193$	−2941.00	−1.09688	−0.548440	+	0.836190i	$0.684778\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2436.76i	0.881277i	0.897685	+	0.440639i	$0.145248\pi$
$197$	2436.76i	0.881277i	−0.897685	+	0.440639i	$0.854752\pi$
$198$	0	0
$199$	− 3706.40i	− 1.32030i	−0.751134	−	0.660150i	$-0.770493\pi$
$199$	− 3706.40i	− 1.32030i	0.751134	−	0.660150i	$-0.229507\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5830.08	−2.01572
$204$	0	0
$205$	−3672.00	−1.25104
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1335.89i	− 0.442130i
$210$	0	0
$211$	− 1713.94i	− 0.559207i	−0.960116	−	0.279604i	$-0.909797\pi$
$211$	− 1713.94i	− 0.559207i	0.960116	−	0.279604i	$-0.0902030\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−530.008	−0.168122
$216$	0	0
$217$	5967.00	1.86667
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 49.4773i	− 0.0150597i
$222$	0	0
$223$	3042.25i	0.913561i	0.889580	+	0.456780i	$0.150997\pi$
$223$	3042.25i	0.913561i	−0.889580	+	0.456780i	$0.849003\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3772.41	−1.10301	−0.551505	−	0.834171i	$-0.685946\pi$
$227$	−3772.41	−1.10301	−0.551505	+	0.834171i	$0.685946\pi$
$228$	0	0
$229$	538.000	0.155249	0.0776245	−	0.996983i	$-0.475266\pi$
$229$	538.000	0.155249	0.0776245	+	0.996983i	$0.475266\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3834.49i	1.07814i	0.842262	+	0.539068i	$0.181223\pi$
$233$	3834.49i	1.07814i	−0.842262	+	0.539068i	$0.818777\pi$
$234$	0	0
$235$	7327.11i	2.03390i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3180.05	0.860670	0.430335	−	0.902669i	$-0.358396\pi$
$239$	3180.05	0.860670	0.430335	+	0.902669i	$0.358396\pi$
$240$	0	0
$241$	−1658.00	−0.443158	−0.221579	−	0.975142i	$-0.571121\pi$
$241$	−1658.00	−0.443158	−0.221579	+	0.975142i	$0.571121\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1434.84i	− 0.374158i
$246$	0	0
$247$	171.394i	0.0441521i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4707.71	−1.18386	−0.591929	−	0.805990i	$-0.701633\pi$
$251$	−4707.71	−1.18386	−0.591929	+	0.805990i	$0.701633\pi$
$252$	0	0
$253$	1944.00	0.483076
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6506.26i	1.57918i	0.613634	+	0.789590i	$0.289707\pi$
$257$	6506.26i	1.57918i	−0.613634	+	0.789590i	$0.710293\pi$
$258$	0	0
$259$	− 2742.31i	− 0.657910i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	872.954	0.204672	0.102336	−	0.994750i	$-0.467368\pi$
$263$	872.954	0.204672	0.102336	+	0.994750i	$0.467368\pi$
$264$	0	0
$265$	−3213.00	−0.744804
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 3141.81i	− 0.712116i	−0.934464	−	0.356058i	$-0.884120\pi$
$269$	− 3141.81i	− 0.712116i	0.934464	−	0.356058i	$-0.115880\pi$
$270$	0	0
$271$	5848.83i	1.31104i	0.755179	+	0.655519i	$0.227550\pi$
$271$	5848.83i	1.31104i	−0.755179	+	0.655519i	$0.772450\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−436.477	−0.0957111
$276$	0	0
$277$	−3044.00	−0.660275	−0.330138	−	0.943933i	$-0.607095\pi$
$277$	−3044.00	−0.660275	−0.330138	+	0.943933i	$0.607095\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 4131.35i	− 0.877067i	−0.898715	−	0.438533i	$-0.855498\pi$
$281$	− 4131.35i	− 0.877067i	0.898715	−	0.438533i	$-0.144502\pi$
$282$	0	0
$283$	− 4713.34i	− 0.990032i	−0.868884	−	0.495016i	$-0.835162\pi$
$283$	− 4713.34i	− 0.990032i	0.868884	−	0.495016i	$-0.164838\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6360.09	1.30810
$288$	0	0
$289$	4301.00	0.875433
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 3290.24i	− 0.656033i	−0.944672	−	0.328017i	$-0.893620\pi$
$293$	− 3290.24i	− 0.656033i	0.944672	−	0.328017i	$-0.106380\pi$
$294$	0	0
$295$	6555.83i	1.29388i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−249.415	−0.0482410
$300$	0	0
$301$	918.000	0.175790
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4205.57i	− 0.789541i
$306$	0	0
$307$	− 8612.56i	− 1.60112i	−0.599250	−	0.800562i	$-0.704535\pi$
$307$	− 8612.56i	− 1.60112i	0.599250	−	0.800562i	$-0.295465\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4427.12	0.807200	0.403600	−	0.914936i	$-0.367759\pi$
$311$	4427.12	0.807200	0.403600	+	0.914936i	$0.367759\pi$
$312$	0	0
$313$	−799.000	−0.144288	−0.0721440	−	0.997394i	$-0.522984\pi$
$313$	−799.000	−0.144288	−0.0721440	+	0.997394i	$0.522984\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7062.88i	1.25139i	0.780068	+	0.625695i	$0.215185\pi$
$317$	7062.88i	1.25139i	−0.780068	+	0.625695i	$0.784815\pi$
$318$	0	0
$319$	− 4242.01i	− 0.744536i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2120.03	0.365206
$324$	0	0
$325$	56.0000	0.00955791
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 12690.9i	− 2.12667i
$330$	0	0
$331$	− 6984.32i	− 1.15980i	−0.814689	−	0.579898i	$-0.803092\pi$
$331$	− 6984.32i	− 1.15980i	0.814689	−	0.579898i	$-0.196908\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11130.2	1.81524
$336$	0	0
$337$	−3602.00	−0.582236	−0.291118	−	0.956687i	$-0.594027\pi$
$337$	−3602.00	−0.582236	−0.291118	+	0.956687i	$0.594027\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4341.63i	0.689479i
$342$	0	0
$343$	− 4863.31i	− 0.765581i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−296.181	−0.0458208	−0.0229104	−	0.999738i	$-0.507293\pi$
$347$	−296.181	−0.0458208	−0.0229104	+	0.999738i	$0.507293\pi$
$348$	0	0
$349$	1316.00	0.201845	0.100922	−	0.994894i	$-0.467821\pi$
$349$	1316.00	0.201845	0.100922	+	0.994894i	$0.467821\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7866.89i	1.18615i	0.805146	+	0.593077i	$0.202087\pi$
$353$	7866.89i	1.18615i	−0.805146	+	0.593077i	$0.797913\pi$
$354$	0	0
$355$	11954.8i	1.78730i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8636.01	1.26961	0.634806	−	0.772671i	$-0.281080\pi$
$359$	8636.01	1.26961	0.634806	+	0.772671i	$0.281080\pi$
$360$	0	0
$361$	−485.000	−0.0707100
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10105.7i	1.44920i
$366$	0	0
$367$	− 2635.19i	− 0.374811i	−0.982283	−	0.187406i	$-0.939992\pi$
$367$	− 2635.19i	− 0.374811i	0.982283	−	0.187406i	$-0.0600078\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5565.08	0.778772
$372$	0	0
$373$	9218.00	1.27960	0.639799	−	0.768542i	$-0.279018\pi$
$373$	9218.00	1.27960	0.639799	+	0.768542i	$0.279018\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	544.250i	0.0743509i
$378$	0	0
$379$	342.789i	0.0464587i	0.999730	+	0.0232294i	$0.00739480\pi$
$379$	342.789i	0.0464587i	−0.999730	+	0.0232294i	$0.992605\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7638.34	1.01906	0.509532	−	0.860452i	$-0.329819\pi$
$383$	7638.34	1.01906	0.509532	+	0.860452i	$0.329819\pi$
$384$	0	0
$385$	4131.00	0.546845
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3228.39i	0.420787i	0.977617	+	0.210393i	$0.0674744\pi$
$389$	3228.39i	0.420787i	−0.977617	+	0.210393i	$0.932526\pi$
$390$	0	0
$391$	3085.10i	0.399028i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2650.04	0.337564
$396$	0	0
$397$	−10942.0	−1.38328	−0.691641	−	0.722241i	$-0.743112\pi$
$397$	−10942.0	−1.38328	−0.691641	+	0.722241i	$0.743112\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 2473.86i	− 0.308077i	−0.988065	−	0.154038i	$-0.950772\pi$
$401$	− 2473.86i	− 0.308077i	0.988065	−	0.154038i	$-0.0492280\pi$
$402$	0	0
$403$	− 557.031i	− 0.0688529i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1995.32	0.243009
$408$	0	0
$409$	−8399.00	−1.01541	−0.507706	−	0.861530i	$-0.669507\pi$
$409$	−8399.00	−1.01541	−0.507706	+	0.861530i	$0.669507\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 11355.0i	− 1.35289i
$414$	0	0
$415$	4434.83i	0.524571i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11940.8	1.39223	0.696115	−	0.717930i	$-0.254910\pi$
$419$	11940.8	1.39223	0.696115	+	0.717930i	$0.254910\pi$
$420$	0	0
$421$	−2684.00	−0.310713	−0.155356	−	0.987858i	$-0.549653\pi$
$421$	−2684.00	−0.310713	−0.155356	+	0.987858i	$0.549653\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 692.682i	− 0.0790588i
$426$	0	0
$427$	7284.26i	0.825550i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3772.41	0.421602	0.210801	−	0.977529i	$-0.432393\pi$
$431$	3772.41	0.421602	0.210801	+	0.977529i	$0.432393\pi$
$432$	0	0
$433$	1451.00	0.161041	0.0805203	−	0.996753i	$-0.474342\pi$
$433$	1451.00	0.161041	0.0805203	+	0.996753i	$0.474342\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 10687.1i	− 1.16987i
$438$	0	0
$439$	8034.11i	0.873456i	0.899594	+	0.436728i	$0.143863\pi$
$439$	8034.11i	0.873456i	−0.899594	+	0.436728i	$0.856137\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6328.91	−0.678771	−0.339386	−	0.940647i	$-0.610219\pi$
$443$	−6328.91	−0.678771	−0.339386	+	0.940647i	$0.610219\pi$
$444$	0	0
$445$	11322.0	1.20610
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11132.4i	1.17009i	0.811001	+	0.585044i	$0.198923\pi$
$449$	11132.4i	1.17009i	−0.811001	+	0.585044i	$0.801077\pi$
$450$	0	0
$451$	4627.65i	0.483165i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−530.008	−0.0546091
$456$	0	0
$457$	16297.0	1.66814	0.834072	−	0.551656i	$-0.186004\pi$
$457$	16297.0	1.66814	0.834072	+	0.551656i	$0.186004\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1818.29i	0.183701i	0.995773	+	0.0918505i	$0.0292782\pi$
$461$	1818.29i	0.183701i	−0.995773	+	0.0918505i	$0.970722\pi$
$462$	0	0
$463$	− 5891.68i	− 0.591381i	−0.955284	−	0.295691i	$-0.904450\pi$
$463$	− 5891.68i	− 0.591381i	0.955284	−	0.295691i	$-0.0955497\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3788.00	0.375348	0.187674	−	0.982231i	$-0.439905\pi$
$467$	3788.00	0.375348	0.187674	+	0.982231i	$0.439905\pi$
$468$	0	0
$469$	−19278.0	−1.89803
$470$	0	0
$471$	0	0
$472$	0	0
$473$	667.943i	0.0649304i
$474$	0	0
$475$	2399.52i	0.231784i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9633.67	0.918943	0.459471	−	0.888193i	$-0.348039\pi$
$479$	9633.67	0.918943	0.459471	+	0.888193i	$0.348039\pi$
$480$	0	0
$481$	−256.000	−0.0242673
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 11936.4i	− 1.11753i
$486$	0	0
$487$	− 12897.4i	− 1.20008i	−0.799971	−	0.600039i	$-0.795152\pi$
$487$	− 12897.4i	− 1.20008i	0.799971	−	0.600039i	$-0.204848\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8838.66	−0.812389	−0.406194	−	0.913787i	$-0.633144\pi$
$491$	−8838.66	−0.812389	−0.406194	+	0.913787i	$0.633144\pi$
$492$	0	0
$493$	6732.00	0.614998
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 20706.2i	− 1.86882i
$498$	0	0
$499$	− 9383.84i	− 0.841840i	−0.907098	−	0.420920i	$-0.861707\pi$
$499$	− 9383.84i	− 0.841840i	0.907098	−	0.420920i	$-0.138293\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17209.7	−1.52553	−0.762764	−	0.646677i	$-0.776158\pi$
$503$	−17209.7	−1.52553	−0.762764	+	0.646677i	$0.776158\pi$
$504$	0	0
$505$	−10557.0	−0.930258
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14583.4i	1.26994i	0.772537	+	0.634969i	$0.218987\pi$
$509$	14583.4i	1.26994i	−0.772537	+	0.634969i	$0.781013\pi$
$510$	0	0
$511$	− 17503.6i	− 1.51529i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1590.02	0.136048
$516$	0	0
$517$	9234.00	0.785514
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17292.3i	1.45411i	0.686581	+	0.727053i	$0.259111\pi$
$521$	17292.3i	1.45411i	−0.686581	+	0.727053i	$0.740889\pi$
$522$	0	0
$523$	9126.75i	0.763068i	0.924355	+	0.381534i	$0.124604\pi$
$523$	9126.75i	0.763068i	−0.924355	+	0.381534i	$0.875396\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6890.10	−0.569521
$528$	0	0
$529$	3385.00	0.278212
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 593.727i	− 0.0482499i
$534$	0	0
$535$	− 7519.92i	− 0.607691i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1808.26	−0.144503
$540$	0	0
$541$	−15572.0	−1.23751	−0.618755	−	0.785584i	$-0.712363\pi$
$541$	−15572.0	−1.23751	−0.618755	+	0.785584i	$0.712363\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22462.7i	1.76550i
$546$	0	0
$547$	15039.8i	1.17561i	0.809003	+	0.587804i	$0.200007\pi$
$547$	15039.8i	1.17561i	−0.809003	+	0.587804i	$0.799993\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−23320.3	−1.80305
$552$	0	0
$553$	−4590.00	−0.352960
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3401.56i	0.258759i	0.991595	+	0.129380i	$0.0412986\pi$
$557$	3401.56i	0.258759i	−0.991595	+	0.129380i	$0.958701\pi$
$558$	0	0
$559$	− 85.6971i	− 0.00648408i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3070.93	−0.229883	−0.114941	−	0.993372i	$-0.536668\pi$
$563$	−3070.93	−0.229883	−0.114941	+	0.993372i	$0.536668\pi$
$564$	0	0
$565$	11628.0	0.865829
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1682.23i	− 0.123941i	−0.998078	−	0.0619707i	$-0.980261\pi$
$569$	− 1682.23i	− 0.123941i	0.998078	−	0.0619707i	$-0.0197385\pi$
$570$	0	0
$571$	− 24980.7i	− 1.83084i	−0.402498	−	0.915421i	$-0.631858\pi$
$571$	− 24980.7i	− 1.83084i	0.402498	−	0.915421i	$-0.368142\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3491.81	−0.253250
$576$	0	0
$577$	−12346.0	−0.890764	−0.445382	−	0.895341i	$-0.646932\pi$
$577$	−12346.0	−0.890764	−0.445382	+	0.895341i	$0.646932\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 7681.35i	− 0.548496i
$582$	0	0
$583$	4049.19i	0.287651i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10335.1	−0.726707	−0.363354	−	0.931651i	$-0.618368\pi$
$587$	−10335.1	−0.726707	−0.363354	+	0.931651i	$0.618368\pi$
$588$	0	0
$589$	23868.0	1.66972
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 321.602i	− 0.0222709i	−0.999938	−	0.0111354i	$-0.996455\pi$
$593$	− 321.602i	− 0.0222709i	0.999938	−	0.0111354i	$-0.00354459\pi$
$594$	0	0
$595$	6555.83i	0.451702i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21480.9	−1.46525	−0.732626	−	0.680632i	$-0.761705\pi$
$599$	−21480.9	−1.46525	−0.732626	+	0.680632i	$0.761705\pi$
$600$	0	0
$601$	17453.0	1.18456	0.592282	−	0.805731i	$-0.298227\pi$
$601$	17453.0	1.18456	0.592282	+	0.805731i	$0.298227\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 13457.8i	− 0.904360i
$606$	0	0
$607$	18553.4i	1.24063i	0.784354	+	0.620313i	$0.212995\pi$
$607$	18553.4i	1.24063i	−0.784354	+	0.620313i	$0.787005\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1184.72	−0.0784431
$612$	0	0
$613$	−26674.0	−1.75751	−0.878754	−	0.477275i	$-0.841625\pi$
$613$	−26674.0	−1.75751	−0.878754	+	0.477275i	$0.841625\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 19246.7i	− 1.25582i	−0.778286	−	0.627910i	$-0.783910\pi$
$617$	− 19246.7i	− 1.25582i	0.778286	−	0.627910i	$-0.216090\pi$
$618$	0	0
$619$	3470.73i	0.225364i	0.993631	+	0.112682i	$0.0359442\pi$
$619$	3470.73i	0.225364i	−0.993631	+	0.112682i	$0.964056\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−19610.3	−1.26111
$624$	0	0
$625$	−18341.0	−1.17382
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3166.55i	0.200729i
$630$	0	0
$631$	27915.8i	1.76119i	0.473868	+	0.880596i	$0.342857\pi$
$631$	27915.8i	1.76119i	−0.473868	+	0.880596i	$0.657143\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9805.14	−0.612764
$636$	0	0
$637$	232.000	0.0144304
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 989.545i	− 0.0609746i	−0.999535	−	0.0304873i	$-0.990294\pi$
$641$	− 989.545i	− 0.0609746i	0.999535	−	0.0304873i	$-0.00970591\pi$
$642$	0	0
$643$	19538.9i	1.19835i	0.800617	+	0.599176i	$0.204505\pi$
$643$	19538.9i	1.19835i	−0.800617	+	0.599176i	$0.795495\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22759.1	−1.38293	−0.691464	−	0.722411i	$-0.743034\pi$
$647$	−22759.1	−1.38293	−0.691464	+	0.722411i	$0.743034\pi$
$648$	0	0
$649$	8262.00	0.499710
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3302.61i	0.197919i	0.995091	+	0.0989594i	$0.0315514\pi$
$653$	3302.61i	0.197919i	−0.995091	+	0.0989594i	$0.968449\pi$
$654$	0	0
$655$	− 21788.5i	− 1.29977i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26297.7	1.55450	0.777249	−	0.629193i	$-0.216615\pi$
$659$	26297.7	1.55450	0.777249	+	0.629193i	$0.216615\pi$
$660$	0	0
$661$	25868.0	1.52216	0.761081	−	0.648657i	$-0.224669\pi$
$661$	25868.0	1.52216	0.761081	+	0.648657i	$0.224669\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 22710.1i	− 1.32430i
$666$	0	0
$667$	− 33936.1i	− 1.97003i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5300.08	−0.304929
$672$	0	0
$673$	18263.0	1.04604	0.523021	−	0.852319i	$-0.324805\pi$
$673$	18263.0	1.04604	0.523021	+	0.852319i	$0.324805\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10464.4i	0.594064i	0.954868	+	0.297032i	$0.0959968\pi$
$677$	10464.4i	0.594064i	−0.954868	+	0.297032i	$0.904003\pi$
$678$	0	0
$679$	20674.4i	1.16850i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24723.3	−1.38508	−0.692541	−	0.721379i	$-0.743509\pi$
$683$	−24723.3	−1.38508	−0.692541	+	0.721379i	$0.743509\pi$
$684$	0	0
$685$	−32742.0	−1.82629
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 519.511i	− 0.0287254i
$690$	0	0
$691$	17567.9i	0.967171i	0.875297	+	0.483585i	$0.160666\pi$
$691$	17567.9i	0.967171i	−0.875297	+	0.483585i	$0.839334\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24380.3	−1.33065
$696$	0	0
$697$	−7344.00	−0.399102
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5726.99i	0.308567i	0.988027	+	0.154284i	$0.0493069\pi$
$701$	5726.99i	0.308567i	−0.988027	+	0.154284i	$0.950693\pi$
$702$	0	0
$703$	− 10969.2i	− 0.588496i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18285.3	0.972685
$708$	0	0
$709$	−6464.00	−0.342399	−0.171199	−	0.985236i	$-0.554764\pi$
$709$	−6464.00	−0.342399	−0.171199	+	0.985236i	$0.554764\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	34733.0i	1.82435i
$714$	0	0
$715$	− 385.637i	− 0.0201706i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10787.2	0.559520	0.279760	−	0.960070i	$-0.409745\pi$
$719$	10787.2	0.559520	0.279760	+	0.960070i	$0.409745\pi$
$720$	0	0
$721$	−2754.00	−0.142253
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7619.50i	0.390319i
$726$	0	0
$727$	5720.28i	0.291821i	0.989298	+	0.145910i	$0.0466111\pi$
$727$	5720.28i	0.291821i	−0.989298	+	0.145910i	$0.953389\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1060.02	−0.0536335
$732$	0	0
$733$	556.000	0.0280168	0.0140084	−	0.999902i	$-0.495541\pi$
$733$	556.000	0.0280168	0.0140084	+	0.999902i	$0.495541\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 14026.8i	− 0.701064i
$738$	0	0
$739$	− 24209.4i	− 1.20509i	−0.798086	−	0.602543i	$-0.794154\pi$
$739$	− 24209.4i	− 1.20509i	0.798086	−	0.602543i	$-0.205846\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23538.6	−1.16224	−0.581121	−	0.813817i	$-0.697386\pi$
$743$	−23538.6	−1.16224	−0.581121	+	0.813817i	$0.697386\pi$
$744$	0	0
$745$	2907.00	0.142959
$746$	0	0
$747$	0	0
$748$	0	0
$749$	13024.9i	0.635406i
$750$	0	0
$751$	− 5891.68i	− 0.286272i	−0.989703	−	0.143136i	$-0.954281\pi$
$751$	− 5891.68i	− 0.286272i	0.989703	−	0.143136i	$-0.0457187\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	37365.5	1.80115
$756$	0	0
$757$	3886.00	0.186577	0.0932887	−	0.995639i	$-0.470262\pi$
$757$	3886.00	0.186577	0.0932887	+	0.995639i	$0.470262\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 15535.9i	− 0.740045i	−0.929023	−	0.370023i	$-0.879350\pi$
$761$	− 15535.9i	− 0.740045i	0.929023	−	0.370023i	$-0.120650\pi$
$762$	0	0
$763$	− 38906.5i	− 1.84602i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1060.02	−0.0499021
$768$	0	0
$769$	−3697.00	−0.173364	−0.0866822	−	0.996236i	$-0.527626\pi$
$769$	−3697.00	−0.173364	−0.0866822	+	0.996236i	$0.527626\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	21695.8i	1.00950i	0.863266	+	0.504750i	$0.168415\pi$
$773$	21695.8i	1.00950i	−0.863266	+	0.504750i	$0.831585\pi$
$774$	0	0
$775$	− 7798.44i	− 0.361456i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25440.4	1.17008
$780$	0	0
$781$	15066.0	0.690274
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 5986.75i	− 0.272199i
$786$	0	0
$787$	− 22367.0i	− 1.01308i	−0.862216	−	0.506541i	$-0.830924\pi$
$787$	− 22367.0i	− 1.01308i	0.862216	−	0.506541i	$-0.169076\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−20140.3	−0.905318
$792$	0	0
$793$	680.000	0.0304508
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 26878.5i	− 1.19459i	−0.802023	−	0.597294i	$-0.796243\pi$
$797$	− 26878.5i	− 1.19459i	0.802023	−	0.597294i	$-0.203757\pi$
$798$	0	0
$799$	14654.2i	0.648847i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12735.8	0.559696
$804$	0	0
$805$	33048.0	1.44694
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14521.6i	0.631090i	0.948911	+	0.315545i	$0.102187\pi$
$809$	14521.6i	0.631090i	−0.948911	+	0.315545i	$0.897813\pi$
$810$	0	0
$811$	33336.2i	1.44339i	0.692210	+	0.721697i	$0.256637\pi$
$811$	33336.2i	1.44339i	−0.692210	+	0.721697i	$0.743363\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	24910.4	1.07064
$816$	0	0
$817$	3672.00	0.157242
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19123.0i	0.812906i	0.913672	+	0.406453i	$0.133235\pi$
$821$	19123.0i	0.812906i	−0.913672	+	0.406453i	$0.866765\pi$
$822$	0	0
$823$	10519.3i	0.445541i	0.974871	+	0.222771i	$0.0715101\pi$
$823$	10519.3i	0.445541i	−0.974871	+	0.222771i	$0.928490\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15619.6	0.656769	0.328384	−	0.944544i	$-0.393496\pi$
$827$	15619.6	0.656769	0.328384	+	0.944544i	$0.393496\pi$
$828$	0	0
$829$	−11666.0	−0.488754	−0.244377	−	0.969680i	$-0.578583\pi$
$829$	−11666.0	−0.488754	−0.244377	+	0.969680i	$0.578583\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2869.68i	− 0.119362i
$834$	0	0
$835$	− 10412.2i	− 0.431532i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	27092.7	1.11483	0.557417	−	0.830233i	$-0.311793\pi$
$839$	27092.7	1.11483	0.557417	+	0.830233i	$0.311793\pi$
$840$	0	0
$841$	−49663.0	−2.03629
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 27125.9i	− 1.10433i
$846$	0	0
$847$	23309.6i	0.945606i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15962.6	0.642997
$852$	0	0
$853$	−47014.0	−1.88714	−0.943569	−	0.331176i	$-0.892555\pi$
$853$	−47014.0	−1.88714	−0.943569	+	0.331176i	$0.892555\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 10563.4i	− 0.421049i	−0.977589	−	0.210524i	$-0.932483\pi$
$857$	− 10563.4i	− 0.421049i	0.977589	−	0.210524i	$-0.0675171\pi$
$858$	0	0
$859$	12254.7i	0.486757i	0.969931	+	0.243379i	$0.0782558\pi$
$859$	12254.7i	0.486757i	−0.969931	+	0.243379i	$0.921744\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2182.38	0.0860825	0.0430413	−	0.999073i	$-0.486295\pi$
$863$	2182.38	0.0860825	0.0430413	+	0.999073i	$0.486295\pi$
$864$	0	0
$865$	47889.0	1.88240
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 3339.72i	− 0.130371i
$870$	0	0
$871$	1799.64i	0.0700097i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	25705.4	0.993143
$876$	0	0
$877$	8318.00	0.320272	0.160136	−	0.987095i	$-0.448807\pi$
$877$	8318.00	0.320272	0.160136	+	0.987095i	$0.448807\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 49502.0i	− 1.89304i	−0.322651	−	0.946518i	$-0.604574\pi$
$881$	− 49502.0i	− 1.89304i	0.322651	−	0.946518i	$-0.395426\pi$
$882$	0	0
$883$	20653.0i	0.787122i	0.919298	+	0.393561i	$0.128757\pi$
$883$	20653.0i	0.787122i	−0.919298	+	0.393561i	$0.871243\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24473.9	−0.926440	−0.463220	−	0.886243i	$-0.653306\pi$
$887$	−24473.9	−0.926440	−0.463220	+	0.886243i	$0.653306\pi$
$888$	0	0
$889$	16983.0	0.640710
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 50763.7i	− 1.90229i
$894$	0	0
$895$	− 46469.3i	− 1.73553i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	75791.1	2.81176
$900$	0	0
$901$	−6426.00	−0.237604
$902$	0	0
$903$	0	0
$904$	0	0
$905$	42055.7i	1.54473i
$906$	0	0
$907$	50561.3i	1.85100i	0.378742	+	0.925502i	$0.376357\pi$
$907$	50561.3i	1.85100i	−0.378742	+	0.925502i	$0.623643\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20795.0	0.756278	0.378139	−	0.925749i	$-0.376564\pi$
$911$	20795.0	0.756278	0.378139	+	0.925749i	$0.376564\pi$
$912$	0	0
$913$	5589.00	0.202595
$914$	0	0
$915$	0	0
$916$	0	0
$917$	37738.8i	1.35905i
$918$	0	0
$919$	− 8933.93i	− 0.320678i	−0.987062	−	0.160339i	$-0.948741\pi$
$919$	− 8933.93i	− 0.320678i	0.987062	−	0.160339i	$-0.0512587\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1932.97	−0.0689322
$924$	0	0
$925$	−3584.00	−0.127396
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30230.6i	1.06764i	0.845599	+	0.533818i	$0.179243\pi$
$929$	30230.6i	1.06764i	−0.845599	+	0.533818i	$0.820757\pi$
$930$	0	0
$931$	9940.87i	0.349945i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4770.07	−0.166843
$936$	0	0
$937$	14681.0	0.511854	0.255927	−	0.966696i	$-0.417619\pi$
$937$	14681.0	0.511854	0.255927	+	0.966696i	$0.417619\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	15424.5i	0.534352i	0.963648	+	0.267176i	$0.0860906\pi$
$941$	15424.5i	0.534352i	−0.963648	+	0.267176i	$0.913909\pi$
$942$	0	0
$943$	37021.2i	1.27845i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25798.9	0.885271	0.442635	−	0.896702i	$-0.354044\pi$
$947$	25798.9	0.885271	0.442635	+	0.896702i	$0.354044\pi$
$948$	0	0
$949$	−1634.00	−0.0558924
$950$	0	0
$951$	0	0
$952$	0	0
$953$	47992.9i	1.63132i	0.578534	+	0.815658i	$0.303625\pi$
$953$	47992.9i	1.63132i	−0.578534	+	0.815658i	$0.696375\pi$
$954$	0	0
$955$	− 3470.73i	− 0.117602i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	56710.8	1.90958
$960$	0	0
$961$	−47780.0	−1.60384
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 36378.2i	− 1.21353i
$966$	0	0
$967$	− 21017.2i	− 0.698933i	−0.936949	−	0.349467i	$-0.886363\pi$
$967$	− 21017.2i	− 0.698933i	0.936949	−	0.349467i	$-0.113637\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−25861.3	−0.854714	−0.427357	−	0.904083i	$-0.640555\pi$
$971$	−25861.3	−0.854714	−0.427357	+	0.904083i	$0.640555\pi$
$972$	0	0
$973$	42228.0	1.39133
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34510.4i	1.13008i	0.825065	+	0.565039i	$0.191139\pi$
$977$	34510.4i	1.13008i	−0.825065	+	0.565039i	$0.808861\pi$
$978$	0	0
$979$	− 14268.6i	− 0.465808i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−37256.4	−1.20885	−0.604423	−	0.796664i	$-0.706596\pi$
$983$	−37256.4	−1.20885	−0.604423	+	0.796664i	$0.706596\pi$
$984$	0	0
$985$	−30141.0	−0.974997
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5343.54i	0.171805i
$990$	0	0
$991$	− 18060.7i	− 0.578927i	−0.957189	−	0.289463i	$-0.906523\pi$
$991$	− 18060.7i	− 0.578927i	0.957189	−	0.289463i	$-0.0934768\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	45845.7	1.46071
$996$	0	0
$997$	4250.00	0.135004	0.0675019	−	0.997719i	$-0.478497\pi$
$997$	4250.00	0.135004	0.0675019	+	0.997719i	$0.478497\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.4.c.f.431.3	yes	4
3.2	odd	2	inner	432.4.c.f.431.1	✓	4
4.3	odd	2	inner	432.4.c.f.431.4	yes	4
8.3	odd	2		1728.4.c.g.1727.2		4
8.5	even	2		1728.4.c.g.1727.1		4
12.11	even	2	inner	432.4.c.f.431.2	yes	4
24.5	odd	2		1728.4.c.g.1727.3		4
24.11	even	2		1728.4.c.g.1727.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.4.c.f.431.1	✓	4	3.2	odd	2	inner
432.4.c.f.431.2	yes	4	12.11	even	2	inner
432.4.c.f.431.3	yes	4	1.1	even	1	trivial
432.4.c.f.431.4	yes	4	4.3	odd	2	inner
1728.4.c.g.1727.1		4	8.5	even	2
1728.4.c.g.1727.2		4	8.3	odd	2
1728.4.c.g.1727.3		4	24.5	odd	2
1728.4.c.g.1727.4		4	24.11	even	2

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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$
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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	432.4.c.f.431.3

Level	$432$
Weight	$4$
Character	432.431
Analytic conductor	$25.489$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$