LMFDB - Embedded newform 343.4.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [343,4,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, 4, names="a")

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

chi := DirichletCharacter("343.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 343 = 7^{3} $
Weight:	$ k $	$=$	$ 4 $
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:	$20.2376551320$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.3
Root			$1.80194$ 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)

$f(q)$	$=$	$q+5.65279 q^{2} +23.9541 q^{4} +90.1850 q^{8} -27.0000 q^{9} +40.9256 q^{11} +318.165 q^{16} -152.625 q^{18} +231.344 q^{22} +144.680 q^{23} -125.000 q^{25} -310.352 q^{29} +1077.04 q^{32} -646.760 q^{36} -110.452 q^{37} -69.7112 q^{43} +980.334 q^{44} +817.846 q^{46} -706.599 q^{50} -747.386 q^{53} -1754.36 q^{58} +3542.96 q^{64} +171.588 q^{67} -1044.61 q^{71} -2435.00 q^{72} -624.363 q^{74} +1143.63 q^{79} +729.000 q^{81} -394.063 q^{86} +3690.88 q^{88} +3465.67 q^{92} -1104.99 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 37 q^{4} + 54 q^{8} - 81 q^{9} + 69 q^{11} + 477 q^{16} + 27 q^{18} + 5 q^{22} + 307 q^{23} - 375 q^{25} - 267 q^{29} + 606 q^{32} - 999 q^{36} - 239 q^{37} + 797 q^{43} + 2048 q^{44} - 240 q^{46}+ \cdots - 1863 q^{99}+O(q^{100}) $ 3 * q - q^2 + 37 * q^4 + 54 * q^8 - 81 * q^9 + 69 * q^11 + 477 * q^16 + 27 * q^18 + 5 * q^22 + 307 * q^23 - 375 * q^25 - 267 * q^29 + 606 * q^32 - 999 * q^36 - 239 * q^37 + 797 * q^43 + 2048 * q^44 - 240 * q^46 + 125 * q^50 + 363 * q^53 - 996 * q^58 + 4018 * q^64 + 1441 * q^67 + 951 * q^71 - 1458 * q^72 + 1055 * q^74 + 1007 * q^79 + 2187 * q^81 - 2802 * q^86 + 437 * q^88 + 7970 * q^92 - 1863 * 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 $

$2$	5.65279	1.99856	0.999282	−	0.0378869i	$-0.0120627\pi$
$2$	5.65279	1.99856	0.999282	+	0.0378869i	$0.0120627\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	23.9541	2.99426
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	90.1850	3.98565
$9$	−27.0000	−1.00000
$10$	0	0
$11$	40.9256	1.12178	0.560888	−	0.827892i	$-0.310460\pi$
$11$	40.9256	1.12178	0.560888	+	0.827892i	$0.310460\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	318.165	4.97132
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−152.625	−1.99856
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	231.344	2.24194
$23$	144.680	1.31165	0.655823	−	0.754915i	$-0.272322\pi$
$23$	144.680	1.31165	0.655823	+	0.754915i	$0.272322\pi$
$24$	0	0
$25$	−125.000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−310.352	−1.98728	−0.993638	−	0.112621i	$-0.964076\pi$
$29$	−310.352	−1.98728	−0.993638	+	0.112621i	$0.964076\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1077.04	5.94986
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−646.760	−2.99426
$37$	−110.452	−0.490762	−0.245381	−	0.969427i	$-0.578913\pi$
$37$	−110.452	−0.490762	−0.245381	+	0.969427i	$0.578913\pi$
$38$	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
$43$	−69.7112	−0.247229	−0.123615	−	0.992330i	$-0.539449\pi$
$43$	−69.7112	−0.247229	−0.123615	+	0.992330i	$0.539449\pi$
$44$	980.334	3.35889
$45$	0	0
$46$	817.846	2.62141
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	−706.599	−1.99856
$51$	0	0
$52$	0	0
$53$	−747.386	−1.93701	−0.968503	−	0.249001i	$-0.919898\pi$
$53$	−747.386	−1.93701	−0.968503	+	0.249001i	$0.919898\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−1754.36	−3.97170
$59$	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
$64$	3542.96	6.91985
$65$	0	0
$66$	0	0
$67$	171.588	0.312878	0.156439	−	0.987688i	$-0.449999\pi$
$67$	171.588	0.312878	0.156439	+	0.987688i	$0.449999\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1044.61	−1.74609	−0.873043	−	0.487643i	$-0.837857\pi$
$71$	−1044.61	−1.74609	−0.873043	+	0.487643i	$0.837857\pi$
$72$	−2435.00	−3.98565
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−624.363	−0.980820
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1143.63	1.62871	0.814354	−	0.580369i	$-0.197091\pi$
$79$	1143.63	1.62871	0.814354	+	0.580369i	$0.197091\pi$
$80$	0	0
$81$	729.000	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
$86$	−394.063	−0.494103
$87$	0	0
$88$	3690.88	4.47101
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	3465.67	3.92741
$93$	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
$99$	−1104.99	−1.12178
$100$	−2994.26	−2.99426
$101$	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
$106$	−4224.82	−3.87123
$107$	227.046	0.205134	0.102567	−	0.994726i	$-0.467294\pi$
$107$	227.046	0.205134	0.102567	+	0.994726i	$0.467294\pi$
$108$	0	0
$109$	−938.583	−0.824770	−0.412385	−	0.911010i	$-0.635304\pi$
$109$	−938.583	−0.824770	−0.412385	+	0.911010i	$0.635304\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2100.16	−1.74838	−0.874189	−	0.485586i	$-0.838606\pi$
$113$	−2100.16	−1.74838	−0.874189	+	0.485586i	$0.838606\pi$
$114$	0	0
$115$	0	0
$116$	−7434.20	−5.95042
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	343.904	0.258380
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1551.43	−1.08399	−0.541996	−	0.840381i	$-0.682331\pi$
$127$	−1551.43	−1.08399	−0.541996	+	0.840381i	$0.682331\pi$
$128$	11411.3	7.87990
$129$	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
$134$	969.953	0.625307
$135$	0	0
$136$	0	0
$137$	1326.77	0.827397	0.413699	−	0.910414i	$-0.364237\pi$
$137$	1326.77	0.827397	0.413699	+	0.910414i	$0.364237\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	−5904.95	−3.48966
$143$	0	0
$144$	−8590.45	−4.97132
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−2645.78	−1.46947
$149$	3275.29	1.80082	0.900409	−	0.435044i	$-0.143267\pi$
$149$	3275.29	1.80082	0.900409	+	0.435044i	$0.143267\pi$
$150$	0	0
$151$	2849.39	1.53563	0.767814	−	0.640673i	$-0.221345\pi$
$151$	2849.39	1.53563	0.767814	+	0.640673i	$0.221345\pi$
$152$	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	$0.5\pi$
$158$	6464.68	3.25508
$159$	0	0
$160$	0	0
$161$	0	0
$162$	4120.89	1.99856
$163$	28.6581	0.0137710	0.00688552	−	0.999976i	$-0.497808\pi$
$163$	28.6581	0.0137710	0.00688552	+	0.999976i	$0.497808\pi$
$164$	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
$169$	−2197.00	−1.00000
$170$	0	0
$171$	0	0
$172$	−1669.87	−0.740268
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	13021.1	5.57671
$177$	0	0
$178$	0	0
$179$	6.46311	0.00269875	0.00134937	−	0.999999i	$-0.499570\pi$
$179$	6.46311	0.00269875	0.00134937	+	0.999999i	$0.499570\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	13048.0	5.22777
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4181.96	1.58427	0.792137	−	0.610343i	$-0.208969\pi$
$191$	4181.96	1.58427	0.792137	+	0.610343i	$0.208969\pi$
$192$	0	0
$193$	−693.997	−0.258834	−0.129417	−	0.991590i	$-0.541311\pi$
$193$	−693.997	−0.258834	−0.129417	+	0.991590i	$0.541311\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2585.39	0.935034	0.467517	−	0.883984i	$-0.345149\pi$
$197$	2585.39	0.935034	0.467517	+	0.883984i	$0.345149\pi$
$198$	−6246.29	−2.24194
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−11273.1	−3.98565
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3906.36	−1.31165
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5044.55	1.64588	0.822942	−	0.568126i	$-0.192331\pi$
$211$	5044.55	1.64588	0.822942	+	0.568126i	$0.192331\pi$
$212$	−17902.9	−5.79990
$213$	0	0
$214$	1283.44	0.409973
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−5305.62	−1.64836
$219$	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
$225$	3375.00	1.00000
$226$	−11871.8	−3.49424
$227$	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
$232$	−27989.1	−7.92059
$233$	1956.50	0.550106	0.275053	−	0.961429i	$-0.411305\pi$
$233$	1956.50	0.550106	0.275053	+	0.961429i	$0.411305\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4950.51	−1.33984	−0.669920	−	0.742433i	$-0.733672\pi$
$239$	−4950.51	−1.33984	−0.669920	+	0.742433i	$0.733672\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	1944.02	0.516389
$243$	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
$253$	5921.11	1.47137
$254$	−8769.89	−2.16643
$255$	0	0
$256$	36162.1	8.82864
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8379.52	1.98728
$262$	0	0
$263$	2252.51	0.528122	0.264061	−	0.964506i	$-0.414938\pi$
$263$	2252.51	0.528122	0.264061	+	0.964506i	$0.414938\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	4110.24	0.936838
$269$	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
$274$	7499.94	1.65361
$275$	−5115.70	−1.12178
$276$	0	0
$277$	3852.05	0.835550	0.417775	−	0.908550i	$-0.362810\pi$
$277$	3852.05	0.835550	0.417775	+	0.908550i	$0.362810\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7184.77	1.52529	0.762647	−	0.646815i	$-0.223899\pi$
$281$	7184.77	1.52529	0.762647	+	0.646815i	$0.223899\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−25022.6	−5.22823
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−29080.1	−5.94986
$289$	−4913.00	−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
$296$	−9961.13	−1.95601
$297$	0	0
$298$	18514.5	3.59905
$299$	0	0
$300$	0	0
$301$	0	0
$302$	16107.0	3.06905
$303$	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
$316$	27394.5	4.87677
$317$	10207.5	1.80855	0.904274	−	0.426952i	$-0.140413\pi$
$317$	10207.5	1.80855	0.904274	+	0.426952i	$0.140413\pi$
$318$	0	0
$319$	−12701.4	−2.22928
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	17462.5	2.99426
$325$	0	0
$326$	161.999	0.0275223
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10793.3	1.79231	0.896153	−	0.443746i	$-0.146351\pi$
$331$	10793.3	1.79231	0.896153	+	0.443746i	$0.146351\pi$
$332$	0	0
$333$	2982.21	0.490762
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12358.7	1.99769	0.998844	−	0.0480763i	$-0.0153090\pi$
$337$	12358.7	1.99769	0.998844	+	0.0480763i	$0.0153090\pi$
$338$	−12419.2	−1.99856
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−6286.91	−0.985370
$345$	0	0
$346$	0	0
$347$	−4100.00	−0.634293	−0.317146	−	0.948377i	$-0.602725\pi$
$347$	−4100.00	−0.634293	−0.317146	+	0.948377i	$0.602725\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	44078.5	6.67440
$353$	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
$358$	36.5346	0.00539362
$359$	−8104.00	−1.19140	−0.595700	−	0.803207i	$-0.703125\pi$
$359$	−8104.00	−1.19140	−0.595700	+	0.803207i	$0.703125\pi$
$360$	0	0
$361$	−6859.00	−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	$0.5\pi$
$368$	46032.1	6.52062
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−11465.4	−1.59158	−0.795788	−	0.605575i	$-0.792943\pi$
$373$	−11465.4	−1.59158	−0.795788	+	0.605575i	$0.792943\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	11916.0	1.61500	0.807498	−	0.589870i	$-0.200821\pi$
$379$	11916.0	1.61500	0.807498	+	0.589870i	$0.200821\pi$
$380$	0	0
$381$	0	0
$382$	23639.8	3.16627
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	−3923.02	−0.517297
$387$	1882.20	0.247229
$388$	0	0
$389$	4639.26	0.604678	0.302339	−	0.953201i	$-0.402233\pi$
$389$	4639.26	0.604678	0.302339	+	0.953201i	$0.402233\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	14614.7	1.86873
$395$	0	0
$396$	−26469.0	−3.35889
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	−39770.6	−4.97132
$401$	−8373.36	−1.04276	−0.521378	−	0.853326i	$-0.674582\pi$
$401$	−8373.36	−1.04276	−0.521378	+	0.853326i	$0.674582\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4520.32	−0.550525
$408$	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
$414$	−22081.8	−2.62141
$415$	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
$421$	−10237.9	−1.18519	−0.592593	−	0.805502i	$-0.701896\pi$
$421$	−10237.9	−1.18519	−0.592593	+	0.805502i	$0.701896\pi$
$422$	28515.8	3.28940
$423$	0	0
$424$	−67403.0	−7.72024
$425$	0	0
$426$	0	0
$427$	0	0
$428$	5438.67	0.614224
$429$	0	0
$430$	0	0
$431$	14562.9	1.62754	0.813769	−	0.581188i	$-0.197412\pi$
$431$	14562.9	1.62754	0.813769	+	0.581188i	$0.197412\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−22482.9	−2.46958
$437$	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
$443$	18580.0	1.99269	0.996346	−	0.0854102i	$-0.0272201\pi$
$443$	18580.0	1.99269	0.996346	+	0.0854102i	$0.0272201\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10593.4	1.11344	0.556719	−	0.830701i	$-0.312060\pi$
$449$	10593.4	1.11344	0.556719	+	0.830701i	$0.312060\pi$
$450$	19078.2	1.99856
$451$	0	0
$452$	−50307.4	−5.23509
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8939.48	−0.915035	−0.457518	−	0.889201i	$-0.651261\pi$
$457$	−8939.48	−0.915035	−0.457518	+	0.889201i	$0.651261\pi$
$458$	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
$463$	−15718.7	−1.57778	−0.788888	−	0.614537i	$-0.789343\pi$
$463$	−15718.7	−1.57778	−0.788888	+	0.614537i	$0.789343\pi$
$464$	−98743.2	−9.87939
$465$	0	0
$466$	11059.7	1.09942
$467$	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
$473$	−2852.97	−0.277336
$474$	0	0
$475$	0	0
$476$	0	0
$477$	20179.4	1.93701
$478$	−27984.2	−2.67776
$479$	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
$484$	8237.90	0.773657
$485$	0	0
$486$	0	0
$487$	21240.0	1.97634	0.988169	−	0.153371i	$-0.0490130\pi$
$487$	21240.0	1.97634	0.988169	+	0.153371i	$0.0490130\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−21672.1	−1.99195	−0.995976	−	0.0896187i	$-0.971435\pi$
$491$	−21672.1	−1.99195	−0.995976	+	0.0896187i	$0.971435\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−18947.8	−1.69984	−0.849919	−	0.526913i	$-0.823349\pi$
$499$	−18947.8	−1.69984	−0.849919	+	0.526913i	$0.823349\pi$
$500$	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
$506$	33470.8	2.94063
$507$	0	0
$508$	−37163.0	−3.24575
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	113126.	9.76470
$513$	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	$0.5\pi$
$522$	47367.7	3.97170
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	12733.0	1.05549
$527$	0	0
$528$	0	0
$529$	8765.30	0.720416
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	15474.7	1.24702
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5365.99	−0.426436	−0.213218	−	0.977005i	$-0.568395\pi$
$541$	−5365.99	−0.426436	−0.213218	+	0.977005i	$0.568395\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−24385.2	−1.90610	−0.953049	−	0.302818i	$-0.902073\pi$
$547$	−24385.2	−1.90610	−0.953049	+	0.302818i	$0.902073\pi$
$548$	31781.5	2.47744
$549$	0	0
$550$	−28918.0	−2.24194
$551$	0	0
$552$	0	0
$553$	0	0
$554$	21774.9	1.66990
$555$	0	0
$556$	0	0
$557$	−25601.4	−1.94752	−0.973759	−	0.227581i	$-0.926918\pi$
$557$	−25601.4	−1.94752	−0.973759	+	0.227581i	$0.926918\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	40614.0	3.04840
$563$	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
$568$	−94208.0	−6.95929
$569$	−19588.8	−1.44324	−0.721622	−	0.692287i	$-0.756603\pi$
$569$	−19588.8	−1.44324	−0.721622	+	0.692287i	$0.756603\pi$
$570$	0	0
$571$	17583.8	1.28872	0.644359	−	0.764723i	$-0.277124\pi$
$571$	17583.8	1.28872	0.644359	+	0.764723i	$0.277124\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−18085.0	−1.31165
$576$	−95660.0	−6.91985
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−27772.2	−1.99856
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−30587.2	−2.17289
$584$	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
$592$	−35142.0	−2.43974
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	78456.4	5.39212
$597$	0	0
$598$	0	0
$599$	−24736.0	−1.68729	−0.843644	−	0.536903i	$-0.819594\pi$
$599$	−24736.0	−1.68729	−0.843644	+	0.536903i	$0.819594\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	−4632.88	−0.312878
$604$	68254.4	4.59807
$605$	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
$613$	−29062.0	−1.91485	−0.957425	−	0.288684i	$-0.906782\pi$
$613$	−29062.0	−1.91485	−0.957425	+	0.288684i	$0.906782\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28608.3	−1.86665	−0.933327	−	0.359028i	$-0.883108\pi$
$617$	−28608.3	−1.86665	−0.933327	+	0.359028i	$0.883108\pi$
$618$	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
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−30293.0	−1.91117	−0.955584	−	0.294720i	$-0.904774\pi$
$631$	−30293.0	−1.91117	−0.955584	+	0.294720i	$0.904774\pi$
$632$	103138.	6.49146
$633$	0	0
$634$	57700.8	3.61450
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−71798.1	−4.45535
$639$	28204.4	1.74609
$640$	0	0
$641$	29944.0	1.84511	0.922557	−	0.385862i	$-0.126096\pi$
$641$	29944.0	1.84511	0.922557	+	0.385862i	$0.126096\pi$
$642$	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	$0.5\pi$
$648$	65744.9	3.98565
$649$	0	0
$650$	0	0
$651$	0	0
$652$	686.479	0.0412340
$653$	13037.5	0.781315	0.390657	−	0.920536i	$-0.372248\pi$
$653$	13037.5	0.781315	0.390657	+	0.920536i	$0.372248\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−27539.9	−1.62793	−0.813963	−	0.580917i	$-0.802694\pi$
$659$	−27539.9	−1.62793	−0.813963	+	0.580917i	$0.802694\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	61012.2	3.58204
$663$	0	0
$664$	0	0
$665$	0	0
$666$	16857.8	0.980820
$667$	−44901.8	−2.60660
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	26076.2	1.49356	0.746778	−	0.665074i	$-0.231600\pi$
$673$	26076.2	1.49356	0.746778	+	0.665074i	$0.231600\pi$
$674$	69861.1	3.99251
$675$	0	0
$676$	−52627.1	−2.99426
$677$	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
$683$	18005.1	1.00870	0.504352	−	0.863498i	$-0.331731\pi$
$683$	18005.1	1.00870	0.504352	+	0.863498i	$0.331731\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−22179.6	−1.22906
$689$	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
$694$	−23176.5	−1.26767
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	36891.2	1.98768	0.993839	−	0.110836i	$-0.0353528\pi$
$701$	36891.2	1.98768	0.993839	+	0.110836i	$0.0353528\pi$
$702$	0	0
$703$	0	0
$704$	144998.	7.76251
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−26754.9	−1.41721	−0.708606	−	0.705605i	$-0.750675\pi$
$709$	−26754.9	−1.41721	−0.708606	+	0.705605i	$0.750675\pi$
$710$	0	0
$711$	−30877.9	−1.62871
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	154.818	0.00808074
$717$	0	0
$718$	−45810.2	−2.38109
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	−38772.5	−1.99856
$723$	0	0
$724$	0	0
$725$	38794.1	1.98728
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−19683.0	−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
$736$	155826.	7.80411
$737$	7022.35	0.350979
$738$	0	0
$739$	−40177.2	−1.99992	−0.999962	−	0.00871925i	$-0.997225\pi$
$739$	−40177.2	−1.99992	−0.999962	+	0.00871925i	$0.997225\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25160.0	1.24230	0.621151	−	0.783691i	$-0.286665\pi$
$743$	25160.0	1.24230	0.621151	+	0.783691i	$0.286665\pi$
$744$	0	0
$745$	0	0
$746$	−64811.7	−3.18087
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−39513.6	−1.91994	−0.959968	−	0.280109i	$-0.909629\pi$
$751$	−39513.6	−1.91994	−0.959968	+	0.280109i	$0.909629\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−3852.31	−0.184960	−0.0924799	−	0.995715i	$-0.529479\pi$
$757$	−3852.31	−0.184960	−0.0924799	+	0.995715i	$0.529479\pi$
$758$	67358.7	3.22767
$759$	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
$764$	100175.	4.74373
$765$	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
$772$	−16624.1	−0.775016
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	10639.7	0.494103
$775$	0	0
$776$	0	0
$777$	0	0
$778$	26224.8	1.20849
$779$	0	0
$780$	0	0
$781$	−42751.2	−1.95872
$782$	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	$0.5\pi$
$788$	61930.7	2.79973
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−99653.6	−4.47101
$793$	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
$800$	−134630.	−5.94986
$801$	0	0
$802$	−47332.9	−2.08402
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	17894.8	0.777687	0.388843	−	0.921304i	$-0.372875\pi$
$809$	17894.8	0.777687	0.388843	+	0.921304i	$0.372875\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	−25552.4	−1.10026
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31489.2	1.33859	0.669294	−	0.742998i	$-0.266597\pi$
$821$	31489.2	1.33859	0.669294	+	0.742998i	$0.266597\pi$
$822$	0	0
$823$	37507.5	1.58861	0.794307	−	0.607516i	$-0.207834\pi$
$823$	37507.5	1.58861	0.794307	+	0.607516i	$0.207834\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23980.0	−1.00830	−0.504151	−	0.863615i	$-0.668195\pi$
$827$	−23980.0	−1.00830	−0.504151	+	0.863615i	$0.668195\pi$
$828$	−93573.2	−3.92741
$829$	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
$841$	71929.7	2.94927
$842$	−57872.5	−2.36867
$843$	0	0
$844$	120838.	4.92820
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−237792.	−9.62949
$849$	0	0
$850$	0	0
$851$	−15980.2	−0.643707
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	20476.1	0.817593
$857$	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
$862$	82320.9	3.25274
$863$	38378.2	1.51380	0.756900	−	0.653531i	$-0.226713\pi$
$863$	38378.2	1.51380	0.756900	+	0.653531i	$0.226713\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	46803.6	1.82704
$870$	0	0
$871$	0	0
$872$	−84646.2	−3.28725
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−44370.6	−1.70842	−0.854212	−	0.519925i	$-0.825960\pi$
$877$	−44370.6	−1.70842	−0.854212	+	0.519925i	$0.825960\pi$
$878$	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
$883$	−48625.1	−1.85319	−0.926595	−	0.376062i	$-0.877278\pi$
$883$	−48625.1	−1.85319	−0.926595	+	0.376062i	$0.877278\pi$
$884$	0	0
$885$	0	0
$886$	105029.	3.98252
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	29834.8	1.12178
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	59882.3	2.22528
$899$	0	0
$900$	80845.0	2.99426
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−189403.	−6.96843
$905$	0	0
$906$	0	0
$907$	2156.98	0.0789653	0.0394826	−	0.999220i	$-0.487429\pi$
$907$	2156.98	0.0789653	0.0394826	+	0.999220i	$0.487429\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−28350.4	−1.03106	−0.515528	−	0.856873i	$-0.672404\pi$
$911$	−28350.4	−1.03106	−0.515528	+	0.856873i	$0.672404\pi$
$912$	0	0
$913$	0	0
$914$	−50533.0	−1.82876
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	21744.0	0.780488	0.390244	−	0.920711i	$-0.372391\pi$
$919$	21744.0	0.780488	0.390244	+	0.920711i	$0.372391\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	13806.5	0.490762
$926$	−88854.6	−3.15329
$927$	0	0
$928$	−334262.	−11.8240
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	46866.2	1.64716
$933$	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
$946$	−16127.3	−0.554273
$947$	−30170.9	−1.03529	−0.517646	−	0.855595i	$-0.673192\pi$
$947$	−30170.9	−1.03529	−0.517646	+	0.855595i	$0.673192\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	43234.2	1.46956	0.734780	−	0.678305i	$-0.237285\pi$
$953$	43234.2	1.46956	0.734780	+	0.678305i	$0.237285\pi$
$954$	114070.	3.87123
$955$	0	0
$956$	−118585.	−4.01183
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29791.0	−1.00000
$962$	0	0
$963$	−6130.23	−0.205134
$964$	0	0
$965$	0	0
$966$	0	0
$967$	56015.3	1.86280	0.931401	−	0.363995i	$-0.118588\pi$
$967$	56015.3	1.86280	0.931401	+	0.363995i	$0.118588\pi$
$968$	31015.0	1.02981
$969$	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
$974$	120065.	3.94984
$975$	0	0
$976$	0	0
$977$	55349.6	1.81248	0.906239	−	0.422766i	$-0.138941\pi$
$977$	55349.6	1.81248	0.906239	+	0.422766i	$0.138941\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	25341.8	0.824770
$982$	−122508.	−3.98104
$983$	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
$989$	−10085.8	−0.324277
$990$	0	0
$991$	16982.4	0.544364	0.272182	−	0.962246i	$-0.412255\pi$
$991$	16982.4	0.544364	0.272182	+	0.962246i	$0.412255\pi$
$992$	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	$0.5\pi$
$998$	−107108.	−3.39724
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	343.4.a.a.1.3	✓	3
7.6	odd	2	CM	343.4.a.a.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
343.4.a.a.1.3	✓	3	1.1	even	1	trivial
343.4.a.a.1.3	✓	3	7.6	odd	2	CM

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	343.a (trivial)

\(f(q)\)	\(=\)	\(q+5.65279 q^{2} +23.9541 q^{4} +90.1850 q^{8} -27.0000 q^{9} +40.9256 q^{11} +318.165 q^{16} -152.625 q^{18} +231.344 q^{22} +144.680 q^{23} -125.000 q^{25} -310.352 q^{29} +1077.04 q^{32} -646.760 q^{36} -110.452 q^{37} -69.7112 q^{43} +980.334 q^{44} +817.846 q^{46} -706.599 q^{50} -747.386 q^{53} -1754.36 q^{58} +3542.96 q^{64} +171.588 q^{67} -1044.61 q^{71} -2435.00 q^{72} -624.363 q^{74} +1143.63 q^{79} +729.000 q^{81} -394.063 q^{86} +3690.88 q^{88} +3465.67 q^{92} -1104.99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 37 q^{4} + 54 q^{8} - 81 q^{9} + 69 q^{11} + 477 q^{16} + 27 q^{18} + 5 q^{22} + 307 q^{23} - 375 q^{25} - 267 q^{29} + 606 q^{32} - 999 q^{36} - 239 q^{37} + 797 q^{43} + 2048 q^{44} - 240 q^{46}+ \cdots - 1863 q^{99}+O(q^{100}) \) 3 * q - q^2 + 37 * q^4 + 54 * q^8 - 81 * q^9 + 69 * q^11 + 477 * q^16 + 27 * q^18 + 5 * q^22 + 307 * q^23 - 375 * q^25 - 267 * q^29 + 606 * q^32 - 999 * q^36 - 239 * q^37 + 797 * q^43 + 2048 * q^44 - 240 * q^46 + 125 * q^50 + 363 * q^53 - 996 * q^58 + 4018 * q^64 + 1441 * q^67 + 951 * q^71 - 1458 * q^72 + 1055 * q^74 + 1007 * q^79 + 2187 * q^81 - 2802 * q^86 + 437 * q^88 + 7970 * q^92 - 1863 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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