LMFDB - Embedded newform 343.2.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [343,2,Mod(1,343)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(343, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("343.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 343 = 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	343.a (trivial)

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:	yes
Analytic conductor:	$2.73886878933$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	343.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.69202 q^{2} +5.24698 q^{4} +8.74094 q^{8} -3.00000 q^{9} -5.89977 q^{11} +13.0368 q^{16} -8.07606 q^{18} -15.8823 q^{22} +3.37867 q^{23} -5.00000 q^{25} +9.87263 q^{29} +17.6136 q^{32} -15.7409 q^{36} +0.814019 q^{37} -3.34481 q^{43} -30.9560 q^{44} +9.09544 q^{46} -13.4601 q^{50} +2.03923 q^{53} +26.5773 q^{58} +21.3424 q^{64} -16.3666 q^{67} +14.1129 q^{71} -26.2228 q^{72} +2.19136 q^{74} -7.42327 q^{79} +9.00000 q^{81} -9.00431 q^{86} -51.5695 q^{88} +17.7278 q^{92} +17.6993 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 11 q^{4} + 12 q^{8} - 9 q^{9} + 5 q^{11} + 11 q^{16} - 9 q^{18} - 9 q^{22} + 3 q^{23} - 15 q^{25} + 13 q^{29} + 22 q^{32} - 33 q^{36} + 17 q^{37} + 13 q^{43} + 2 q^{44} - 32 q^{46} - 15 q^{50}+ \cdots - 15 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 11 * q^4 + 12 * q^8 - 9 * q^9 + 5 * q^11 + 11 * q^16 - 9 * q^18 - 9 * q^22 + 3 * q^23 - 15 * q^25 + 13 * q^29 + 22 * q^32 - 33 * q^36 + 17 * q^37 + 13 * q^43 + 2 * q^44 - 32 * q^46 - 15 * q^50 + 19 * q^53 - 8 * q^58 - 23 * q^67 - q^71 - 36 * q^72 - 11 * q^74 - 25 * q^79 + 27 * q^81 - 8 * q^86 - 29 * q^88 + 4 * q^92 - 15 * q^99

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 $

$2$	2.69202	1.90355	0.951773	−	0.306802i	$-0.0992590\pi$
$2$	2.69202	1.90355	0.951773	+	0.306802i	$0.0992590\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	5.24698	2.62349
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.74094	3.09039
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−5.89977	−1.77885	−0.889424	−	0.457083i	$-0.848894\pi$
$11$	−5.89977	−1.77885	−0.889424	+	0.457083i	$0.848894\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	13.0368	3.25921
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−8.07606	−1.90355
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	−15.8823	−3.38612
$23$	3.37867	0.704501	0.352250	−	0.935906i	$-0.385417\pi$
$23$	3.37867	0.704501	0.352250	+	0.935906i	$0.385417\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.87263	1.83330	0.916650	−	0.399690i	$-0.130882\pi$
$29$	9.87263	1.83330	0.916650	+	0.399690i	$0.130882\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	17.6136	3.11367
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−15.7409	−2.62349
$37$	0.814019	0.133824	0.0669120	−	0.997759i	$-0.478685\pi$
$37$	0.814019	0.133824	0.0669120	+	0.997759i	$0.478685\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−3.34481	−0.510079	−0.255040	−	0.966931i	$-0.582089\pi$
$43$	−3.34481	−0.510079	−0.255040	+	0.966931i	$0.582089\pi$
$44$	−30.9560	−4.66679
$45$	0	0
$46$	9.09544	1.34105
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	−13.4601	−1.90355
$51$	0	0
$52$	0	0
$53$	2.03923	0.280110	0.140055	−	0.990144i	$-0.455272\pi$
$53$	2.03923	0.280110	0.140055	+	0.990144i	$0.455272\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	26.5773	3.48977
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	21.3424	2.66780
$65$	0	0
$66$	0	0
$67$	−16.3666	−1.99950	−0.999748	−	0.0224365i	$-0.992858\pi$
$67$	−16.3666	−1.99950	−0.999748	+	0.0224365i	$0.992858\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.1129	1.67489	0.837447	−	0.546519i	$-0.184047\pi$
$71$	14.1129	1.67489	0.837447	+	0.546519i	$0.184047\pi$
$72$	−26.2228	−3.09039
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	2.19136	0.254740
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.42327	−0.835183	−0.417592	−	0.908635i	$-0.637126\pi$
$79$	−7.42327	−0.835183	−0.417592	+	0.908635i	$0.637126\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	−9.00431	−0.970960
$87$	0	0
$88$	−51.5695	−5.49733
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	17.7278	1.84825
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	17.6993	1.77885
$100$	−26.2349	−2.62349
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	5.48965	0.533202
$107$	9.60925	0.928961	0.464481	−	0.885583i	$-0.346241\pi$
$107$	9.60925	0.928961	0.464481	+	0.885583i	$0.346241\pi$
$108$	0	0
$109$	19.4969	1.86747	0.933734	−	0.357967i	$-0.116530\pi$
$109$	19.4969	1.86747	0.933734	+	0.357967i	$0.116530\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.9855	−1.03343	−0.516716	−	0.856157i	$-0.672846\pi$
$113$	−10.9855	−1.03343	−0.516716	+	0.856157i	$0.672846\pi$
$114$	0	0
$115$	0	0
$116$	51.8015	4.80965
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	23.8073	2.16430
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.52781	−0.667985	−0.333993	−	0.942576i	$-0.608396\pi$
$127$	−7.52781	−0.667985	−0.333993	+	0.942576i	$0.608396\pi$
$128$	22.2271	1.96462
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	−44.0592	−3.80613
$135$	0	0
$136$	0	0
$137$	22.8605	1.95311	0.976554	−	0.215272i	$-0.0690640\pi$
$137$	22.8605	1.95311	0.976554	+	0.215272i	$0.0690640\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	37.9922	3.18824
$143$	0	0
$144$	−39.1105	−3.25921
$145$	0	0
$146$	0	0
$147$	0	0
$148$	4.27114	0.351086
$149$	−24.4131	−2.00000	−1.00000		0.000429442i	$-0.999863\pi$
$149$	−24.4131	−2.00000	−1.00000		0.000429442i	$0.999863\pi$
$150$	0	0
$151$	19.3274	1.57284	0.786419	−	0.617693i	$-0.211933\pi$
$151$	19.3274	1.57284	0.786419	+	0.617693i	$0.211933\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	−19.9836	−1.58981
$159$	0	0
$160$	0	0
$161$	0	0
$162$	24.2282	1.90355
$163$	−24.8810	−1.94883	−0.974415	−	0.224758i	$-0.927841\pi$
$163$	−24.8810	−1.94883	−0.974415	+	0.224758i	$0.927841\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	−17.5502	−1.33819
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	−76.9144	−5.79764
$177$	0	0
$178$	0	0
$179$	−18.1914	−1.35969	−0.679843	−	0.733358i	$-0.737952\pi$
$179$	−18.1914	−1.35969	−0.679843	+	0.733358i	$0.737952\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	29.5327	2.17718
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.27173	0.309092	0.154546	−	0.987986i	$-0.450609\pi$
$191$	4.27173	0.309092	0.154546	+	0.987986i	$0.450609\pi$
$192$	0	0
$193$	16.6300	1.19705	0.598525	−	0.801104i	$-0.295754\pi$
$193$	16.6300	1.19705	0.598525	+	0.801104i	$0.295754\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.1032	1.14731	0.573653	−	0.819098i	$-0.305526\pi$
$197$	16.1032	1.14731	0.573653	+	0.819098i	$0.305526\pi$
$198$	47.6469	3.38612
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−43.7047	−3.09039
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−10.1360	−0.704501
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−23.1239	−1.59192	−0.795958	−	0.605352i	$-0.793032\pi$
$211$	−23.1239	−1.59192	−0.795958	+	0.605352i	$0.793032\pi$
$212$	10.6998	0.734865
$213$	0	0
$214$	25.8683	1.76832
$215$	0	0
$216$	0	0
$217$	0	0
$218$	52.4862	3.55481
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	15.0000	1.00000
$226$	−29.5733	−1.96718
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	86.2960	5.66561
$233$	30.2650	1.98273	0.991364	−	0.131139i	$-0.0418634\pi$
$233$	30.2650	1.98273	0.991364	+	0.131139i	$0.0418634\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−29.3545	−1.89878	−0.949392	−	0.314092i	$-0.898300\pi$
$239$	−29.3545	−1.89878	−0.949392	+	0.314092i	$0.898300\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	64.0898	4.11985
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−19.9334	−1.25320
$254$	−20.2650	−1.27154
$255$	0	0
$256$	17.1511	1.07194
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−29.6179	−1.83330
$262$	0	0
$263$	−31.1269	−1.91937	−0.959683	−	0.281083i	$-0.909306\pi$
$263$	−31.1269	−1.91937	−0.959683	+	0.281083i	$0.909306\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−85.8751	−5.24566
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	61.5411	3.71783
$275$	29.4989	1.77885
$276$	0	0
$277$	22.7851	1.36902	0.684511	−	0.729002i	$-0.260016\pi$
$277$	22.7851	1.36902	0.684511	+	0.729002i	$0.260016\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.2416	0.849583	0.424791	−	0.905291i	$-0.360347\pi$
$281$	14.2416	0.849583	0.424791	+	0.905291i	$0.360347\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	74.0501	4.39406
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−52.8407	−3.11367
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	7.11529	0.413568
$297$	0	0
$298$	−65.7206	−3.80709
$299$	0	0
$300$	0	0
$301$	0	0
$302$	52.0297	2.99397
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	−38.9498	−2.19109
$317$	26.0411	1.46262	0.731308	−	0.682047i	$-0.238910\pi$
$317$	26.0411	1.46262	0.731308	+	0.682047i	$0.238910\pi$
$318$	0	0
$319$	−58.2462	−3.26116
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	47.2228	2.62349
$325$	0	0
$326$	−66.9801	−3.70969
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.85192	−0.156756	−0.0783779	−	0.996924i	$-0.524974\pi$
$331$	−2.85192	−0.156756	−0.0783779	+	0.996924i	$0.524974\pi$
$332$	0	0
$333$	−2.44206	−0.133824
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−36.2127	−1.97263	−0.986314	−	0.164875i	$-0.947278\pi$
$337$	−36.2127	−1.97263	−0.986314	+	0.164875i	$0.947278\pi$
$338$	−34.9963	−1.90355
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−29.2368	−1.57634
$345$	0	0
$346$	0	0
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	−103.916	−5.53874
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−48.9715	−2.58823
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	44.0471	2.29611
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−35.8485	−1.85616	−0.928082	−	0.372377i	$-0.878543\pi$
$373$	−35.8485	−1.85616	−0.928082	+	0.372377i	$0.878543\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	0	0
$382$	11.4996	0.588371
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	44.7682	2.27864
$387$	10.0344	0.510079
$388$	0	0
$389$	−15.4185	−0.781748	−0.390874	−	0.920444i	$-0.627827\pi$
$389$	−15.4185	−0.781748	−0.390874	+	0.920444i	$0.627827\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	43.3502	2.18395
$395$	0	0
$396$	92.8680	4.66679
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	−65.1842	−3.25921
$401$	−4.65040	−0.232230	−0.116115	−	0.993236i	$-0.537044\pi$
$401$	−4.65040	−0.232230	−0.116115	+	0.993236i	$0.537044\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.80253	−0.238053
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−27.2863	−1.34105
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−41.0331	−1.99983	−0.999916	−	0.0129745i	$-0.995870\pi$
$421$	−41.0331	−1.99983	−0.999916	+	0.0129745i	$0.995870\pi$
$422$	−62.2501	−3.03029
$423$	0	0
$424$	17.8248	0.865647
$425$	0	0
$426$	0	0
$427$	0	0
$428$	50.4196	2.43712
$429$	0	0
$430$	0	0
$431$	−0.733643	−0.0353383	−0.0176692	−	0.999844i	$-0.505625\pi$
$431$	−0.733643	−0.0353383	−0.0176692	+	0.999844i	$0.505625\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	102.300	4.89928
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	34.3435	1.62077	0.810385	−	0.585898i	$-0.199258\pi$
$449$	34.3435	1.62077	0.810385	+	0.585898i	$0.199258\pi$
$450$	40.3803	1.90355
$451$	0	0
$452$	−57.6408	−2.71120
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	42.6058	1.99302	0.996508	−	0.0835033i	$-0.0266109\pi$
$457$	42.6058	1.99302	0.996508	+	0.0835033i	$0.0266109\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	42.9264	1.99496	0.997481	−	0.0709404i	$-0.0226000\pi$
$463$	42.9264	1.99496	0.997481	+	0.0709404i	$0.0226000\pi$
$464$	128.708	5.97511
$465$	0	0
$466$	81.4741	3.77422
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	19.7336	0.907354
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.11769	−0.280110
$478$	−79.0230	−3.61443
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	124.916	5.67802
$485$	0	0
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.5706	1.42476	0.712381	−	0.701793i	$-0.247617\pi$
$491$	31.5706	1.42476	0.712381	+	0.701793i	$0.247617\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20.9554	−0.938092	−0.469046	−	0.883174i	$-0.655402\pi$
$499$	−20.9554	−0.938092	−0.469046	+	0.883174i	$0.655402\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	−53.6610	−2.38552
$507$	0	0
$508$	−39.4983	−1.75245
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	1.71678	0.0758715
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	−79.7320	−3.48977
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	−83.7943	−3.65360
$527$	0	0
$528$	0	0
$529$	−11.5846	−0.503679
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−143.059	−6.17922
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−23.3873	−1.00550	−0.502749	−	0.864432i	$-0.667678\pi$
$541$	−23.3873	−1.00550	−0.502749	+	0.864432i	$0.667678\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−32.7549	−1.40050	−0.700250	−	0.713898i	$-0.746928\pi$
$547$	−32.7549	−1.40050	−0.700250	+	0.713898i	$0.746928\pi$
$548$	119.949	5.12396
$549$	0	0
$550$	79.4116	3.38612
$551$	0	0
$552$	0	0
$553$	0	0
$554$	61.3379	2.60600
$555$	0	0
$556$	0	0
$557$	36.9547	1.56582	0.782910	−	0.622136i	$-0.213735\pi$
$557$	36.9547	1.56582	0.782910	+	0.622136i	$0.213735\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	38.3387	1.61722
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	123.360	5.17607
$569$	36.3752	1.52493	0.762464	−	0.647031i	$-0.223989\pi$
$569$	36.3752	1.52493	0.762464	+	0.647031i	$0.223989\pi$
$570$	0	0
$571$	39.7275	1.66255	0.831273	−	0.555865i	$-0.187613\pi$
$571$	39.7275	1.66255	0.831273	+	0.555865i	$0.187613\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−16.8933	−0.704501
$576$	−64.0273	−2.66780
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−45.7644	−1.90355
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−12.0310	−0.498272
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	10.6122	0.436160
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	−128.095	−5.24698
$597$	0	0
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	49.0998	1.99950
$604$	101.410	4.12632
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	48.0122	1.93919	0.969597	−	0.244706i	$-0.0786916\pi$
$613$	48.0122	1.93919	0.969597	+	0.244706i	$0.0786916\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.8858	0.679796	0.339898	−	0.940462i	$-0.389608\pi$
$617$	16.8858	0.679796	0.339898	+	0.940462i	$0.389608\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	42.8692	1.70659	0.853297	−	0.521425i	$-0.174599\pi$
$631$	42.8692	1.70659	0.853297	+	0.521425i	$0.174599\pi$
$632$	−64.8864	−2.58104
$633$	0	0
$634$	70.1033	2.78416
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−156.800	−6.20778
$639$	−42.3387	−1.67489
$640$	0	0
$641$	10.3994	0.410750	0.205375	−	0.978683i	$-0.434159\pi$
$641$	10.3994	0.410750	0.205375	+	0.978683i	$0.434159\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	78.6684	3.09039
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−130.550	−5.11273
$653$	−49.6402	−1.94257	−0.971286	−	0.237914i	$-0.923536\pi$
$653$	−49.6402	−1.94257	−0.971286	+	0.237914i	$0.923536\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−35.5851	−1.38620	−0.693099	−	0.720842i	$-0.743755\pi$
$659$	−35.5851	−1.38620	−0.693099	+	0.720842i	$0.743755\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	−7.67743	−0.298392
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−6.57407	−0.254740
$667$	33.3563	1.29156
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	51.8012	1.99679	0.998395	−	0.0566381i	$-0.0180381\pi$
$673$	51.8012	1.99679	0.998395	+	0.0566381i	$0.0180381\pi$
$674$	−97.4852	−3.75499
$675$	0	0
$676$	−68.2107	−2.62349
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.5545	1.70483	0.852415	−	0.522866i	$-0.175137\pi$
$683$	44.5545	1.70483	0.852415	+	0.522866i	$0.175137\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−43.6058	−1.66246
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	10.7681	0.408751
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.1570	0.799090	0.399545	−	0.916714i	$-0.369168\pi$
$701$	21.1570	0.799090	0.399545	+	0.916714i	$0.369168\pi$
$702$	0	0
$703$	0	0
$704$	−125.915	−4.74562
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−45.1116	−1.69420	−0.847100	−	0.531433i	$-0.821654\pi$
$709$	−45.1116	−1.69420	−0.847100	+	0.531433i	$0.821654\pi$
$710$	0	0
$711$	22.2698	0.835183
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−95.4497	−3.56712
$717$	0	0
$718$	21.5362	0.803723
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	−51.1484	−1.90355
$723$	0	0
$724$	0	0
$725$	−49.3631	−1.83330
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	59.5104	2.19358
$737$	96.5591	3.55680
$738$	0	0
$739$	−3.90541	−0.143663	−0.0718315	−	0.997417i	$-0.522884\pi$
$739$	−3.90541	−0.143663	−0.0718315	+	0.997417i	$0.522884\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	0	0
$745$	0	0
$746$	−96.5048	−3.53329
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	54.7260	1.99698	0.998490	−	0.0549360i	$-0.0174955\pi$
$751$	54.7260	1.99698	0.998490	+	0.0549360i	$0.0174955\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.3338	0.811736	0.405868	−	0.913932i	$-0.366969\pi$
$757$	22.3338	0.811736	0.405868	+	0.913932i	$0.366969\pi$
$758$	−32.3043	−1.17334
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	22.4137	0.810899
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	87.2570	3.14045
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	27.0129	0.970960
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−41.5069	−1.48809
$779$	0	0
$780$	0	0
$781$	−83.2629	−2.97938
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	84.4932	3.00995
$789$	0	0
$790$	0	0
$791$	0	0
$792$	154.709	5.49733
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	−88.0678	−3.11367
$801$	0	0
$802$	−12.5190	−0.442060
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15.8696	0.557947	0.278973	−	0.960299i	$-0.410006\pi$
$809$	15.8696	0.557947	0.278973	+	0.960299i	$0.410006\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	−12.9285	−0.453144
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.6539	−0.965126	−0.482563	−	0.875861i	$-0.660294\pi$
$821$	−27.6539	−0.965126	−0.482563	+	0.875861i	$0.660294\pi$
$822$	0	0
$823$	57.1852	1.99335	0.996676	−	0.0814657i	$-0.0259601\pi$
$823$	57.1852	1.99335	0.996676	+	0.0814657i	$0.0259601\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	−53.1834	−1.84825
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	68.4687	2.36099
$842$	−110.462	−3.80677
$843$	0	0
$844$	−121.331	−4.17638
$845$	0	0
$846$	0	0
$847$	0	0
$848$	26.5851	0.912936
$849$	0	0
$850$	0	0
$851$	2.75030	0.0942791
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	83.9939	2.87085
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	−1.97498	−0.0672682
$863$	50.4956	1.71889	0.859445	−	0.511228i	$-0.170809\pi$
$863$	50.4956	1.71889	0.859445	+	0.511228i	$0.170809\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	43.7956	1.48566
$870$	0	0
$871$	0	0
$872$	170.422	5.77120
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−42.0790	−1.42091	−0.710454	−	0.703743i	$-0.751510\pi$
$877$	−42.0790	−1.42091	−0.710454	+	0.703743i	$0.751510\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−14.4432	−0.486054	−0.243027	−	0.970020i	$-0.578140\pi$
$883$	−14.4432	−0.486054	−0.243027	+	0.970020i	$0.578140\pi$
$884$	0	0
$885$	0	0
$886$	−53.8404	−1.80880
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−53.0980	−1.77885
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	92.4534	3.08521
$899$	0	0
$900$	78.7047	2.62349
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−96.0238	−3.19370
$905$	0	0
$906$	0	0
$907$	−56.3540	−1.87120	−0.935602	−	0.353055i	$-0.885143\pi$
$907$	−56.3540	−1.87120	−0.935602	+	0.353055i	$0.885143\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−39.6704	−1.31434	−0.657169	−	0.753743i	$-0.728246\pi$
$911$	−39.6704	−1.31434	−0.657169	+	0.753743i	$0.728246\pi$
$912$	0	0
$913$	0	0
$914$	114.696	3.79380
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−4.07010	−0.133824
$926$	115.559	3.79750
$927$	0	0
$928$	173.892	5.70829
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	158.800	5.20167
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	53.1234	1.72719
$947$	33.0379	1.07359	0.536794	−	0.843713i	$-0.319635\pi$
$947$	33.0379	1.07359	0.536794	+	0.843713i	$0.319635\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−61.4398	−1.99023	−0.995115	−	0.0987237i	$-0.968524\pi$
$953$	−61.4398	−1.99023	−0.995115	+	0.0987237i	$0.968524\pi$
$954$	−16.4689	−0.533202
$955$	0	0
$956$	−154.022	−4.98144
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−28.8278	−0.928961
$964$	0	0
$965$	0	0
$966$	0	0
$967$	55.3303	1.77930	0.889652	−	0.456639i	$-0.150947\pi$
$967$	55.3303	1.77930	0.889652	+	0.456639i	$0.150947\pi$
$968$	208.098	6.68853
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	−64.6085	−2.07019
$975$	0	0
$976$	0	0
$977$	−59.8117	−1.91355	−0.956774	−	0.290834i	$-0.906067\pi$
$977$	−59.8117	−1.91355	−0.956774	+	0.290834i	$0.906067\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−58.4908	−1.86747
$982$	84.9888	2.71210
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.3010	−0.359351
$990$	0	0
$991$	62.0877	1.97228	0.986140	−	0.165916i	$-0.0530579\pi$
$991$	62.0877	1.97228	0.986140	+	0.165916i	$0.0530579\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	−56.4124	−1.78570
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	343.2.a.b.1.3	✓	3
3.2	odd	2		3087.2.a.a.1.1		3
4.3	odd	2		5488.2.a.c.1.3		3
5.4	even	2		8575.2.a.a.1.1		3
7.2	even	3		343.2.c.a.18.1		6
7.3	odd	6		343.2.c.a.324.1		6
7.4	even	3		343.2.c.a.324.1		6
7.5	odd	6		343.2.c.a.18.1		6
7.6	odd	2	CM	343.2.a.b.1.3	✓	3
21.20	even	2		3087.2.a.a.1.1		3
28.27	even	2		5488.2.a.c.1.3		3
35.34	odd	2		8575.2.a.a.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
343.2.a.b.1.3	✓	3	1.1	even	1	trivial
343.2.a.b.1.3	✓	3	7.6	odd	2	CM
343.2.c.a.18.1		6	7.2	even	3
343.2.c.a.18.1		6	7.5	odd	6
343.2.c.a.324.1		6	7.3	odd	6
343.2.c.a.324.1		6	7.4	even	3
3087.2.a.a.1.1		3	3.2	odd	2
3087.2.a.a.1.1		3	21.20	even	2
5488.2.a.c.1.3		3	4.3	odd	2
5488.2.a.c.1.3		3	28.27	even	2
8575.2.a.a.1.1		3	5.4	even	2
8575.2.a.a.1.1		3	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 343 = 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	343.a (trivial)

\(f(q)\)	\(=\)	\(q+2.69202 q^{2} +5.24698 q^{4} +8.74094 q^{8} -3.00000 q^{9} -5.89977 q^{11} +13.0368 q^{16} -8.07606 q^{18} -15.8823 q^{22} +3.37867 q^{23} -5.00000 q^{25} +9.87263 q^{29} +17.6136 q^{32} -15.7409 q^{36} +0.814019 q^{37} -3.34481 q^{43} -30.9560 q^{44} +9.09544 q^{46} -13.4601 q^{50} +2.03923 q^{53} +26.5773 q^{58} +21.3424 q^{64} -16.3666 q^{67} +14.1129 q^{71} -26.2228 q^{72} +2.19136 q^{74} -7.42327 q^{79} +9.00000 q^{81} -9.00431 q^{86} -51.5695 q^{88} +17.7278 q^{92} +17.6993 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 11 q^{4} + 12 q^{8} - 9 q^{9} + 5 q^{11} + 11 q^{16} - 9 q^{18} - 9 q^{22} + 3 q^{23} - 15 q^{25} + 13 q^{29} + 22 q^{32} - 33 q^{36} + 17 q^{37} + 13 q^{43} + 2 q^{44} - 32 q^{46} - 15 q^{50}+ \cdots - 15 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 11 * q^4 + 12 * q^8 - 9 * q^9 + 5 * q^11 + 11 * q^16 - 9 * q^18 - 9 * q^22 + 3 * q^23 - 15 * q^25 + 13 * q^29 + 22 * q^32 - 33 * q^36 + 17 * q^37 + 13 * q^43 + 2 * q^44 - 32 * q^46 - 15 * q^50 + 19 * q^53 - 8 * q^58 - 23 * q^67 - q^71 - 36 * q^72 - 11 * q^74 - 25 * q^79 + 27 * q^81 - 8 * q^86 - 29 * q^88 + 4 * q^92 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more