LMFDB - Embedded newform 307.3.b.a.306.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [307,3,Mod(306,307)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(307, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("307.306");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 307 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	307.b (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:	yes
Analytic conductor:	$8.36514411586$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.8289.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 21x - 12 $ x^3 - 21*x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			306.2
Root			$-4.26451$ of defining polynomial
Character	$\chi$	$=$	307.306

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{4} +4.18608 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+4.00000 q^{4} +4.18608 q^{7} +9.00000 q^{9} -4.14686 q^{11} +16.0000 q^{16} -33.9985 q^{17} +34.0769 q^{19} +25.0000 q^{25} +16.7443 q^{28} +36.0000 q^{36} -55.4779 q^{37} +80.6728 q^{41} -16.5875 q^{44} -31.4767 q^{49} +55.6740 q^{53} +37.6747 q^{63} +64.0000 q^{64} -135.994 q^{68} -68.5852 q^{71} +136.308 q^{76} -17.3591 q^{77} -149.000 q^{79} +81.0000 q^{81} -141.000 q^{83} -129.000 q^{89} -113.000 q^{97} -37.3218 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 12 q^{4} + 27 q^{9}+O(q^{10}) $ 3 * q + 12 * q^4 + 27 * q^9	$ 3 q + 12 q^{4} + 27 q^{9} + 48 q^{16} + 75 q^{25} + 108 q^{36} + 147 q^{49} + 192 q^{64} - 459 q^{77} - 447 q^{79} + 243 q^{81} - 423 q^{83} - 387 q^{89} - 339 q^{97}+O(q^{100}) $ 3 * q + 12 * q^4 + 27 * q^9 + 48 * q^16 + 75 * q^25 + 108 * q^36 + 147 * q^49 + 192 * q^64 - 459 * q^77 - 447 * q^79 + 243 * q^81 - 423 * q^83 - 387 * q^89 - 339 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/307\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

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	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	4.00000	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	4.18608	0.598011	0.299006	−	0.954251i	$-0.403345\pi$
$7$	4.18608	0.598011	0.299006	+	0.954251i	$0.403345\pi$
$8$	0	0
$9$	9.00000	1.00000
$10$	0	0
$11$	−4.14686	−0.376988	−0.188494	−	0.982074i	$-0.560361\pi$
$11$	−4.14686	−0.376988	−0.188494	+	0.982074i	$0.560361\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	−33.9985	−1.99991	−0.999955	−	0.00951138i	$-0.996972\pi$
$17$	−33.9985	−1.99991	−0.999955	+	0.00951138i	$0.996972\pi$
$18$	0	0
$19$	34.0769	1.79352	0.896760	−	0.442516i	$-0.145914\pi$
$19$	34.0769	1.79352	0.896760	+	0.442516i	$0.145914\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	25.0000	1.00000
$26$	0	0
$27$	0	0
$28$	16.7443	0.598011
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	36.0000	1.00000
$37$	−55.4779	−1.49940	−0.749701	−	0.661776i	$-0.769803\pi$
$37$	−55.4779	−1.49940	−0.749701	+	0.661776i	$0.769803\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	80.6728	1.96763	0.983815	−	0.179188i	$-0.0573471\pi$
$41$	80.6728	1.96763	0.983815	+	0.179188i	$0.0573471\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−16.5875	−0.376988
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−31.4767	−0.642382
$50$	0	0
$51$	0	0
$52$	0	0
$53$	55.6740	1.05045	0.525226	−	0.850963i	$-0.323981\pi$
$53$	55.6740	1.05045	0.525226	+	0.850963i	$0.323981\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	37.6747	0.598011
$64$	64.0000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−135.994	−1.99991
$69$	0	0
$70$	0	0
$71$	−68.5852	−0.965988	−0.482994	−	0.875624i	$-0.660451\pi$
$71$	−68.5852	−0.965988	−0.482994	+	0.875624i	$0.660451\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	136.308	1.79352
$77$	−17.3591	−0.225443
$78$	0	0
$79$	−149.000	−1.88608	−0.943038	−	0.332685i	$-0.892045\pi$
$79$	−149.000	−1.88608	−0.943038	+	0.332685i	$0.892045\pi$
$80$	0	0
$81$	81.0000	1.00000
$82$	0	0
$83$	−141.000	−1.69880	−0.849398	−	0.527753i	$-0.823034\pi$
$83$	−141.000	−1.69880	−0.849398	+	0.527753i	$0.823034\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−129.000	−1.44944	−0.724719	−	0.689044i	$-0.758030\pi$
$89$	−129.000	−1.44944	−0.724719	+	0.689044i	$0.758030\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−113.000	−1.16495	−0.582474	−	0.812849i	$-0.697915\pi$
$97$	−113.000	−1.16495	−0.582474	+	0.812849i	$0.697915\pi$
$98$	0	0
$99$	−37.3218	−0.376988
$100$	100.000	1.00000
$101$	−183.139	−1.81326	−0.906628	−	0.421931i	$-0.861352\pi$
$101$	−183.139	−1.81326	−0.906628	+	0.421931i	$0.861352\pi$
$102$	0	0
$103$	−135.249	−1.31309	−0.656547	−	0.754285i	$-0.727984\pi$
$103$	−135.249	−1.31309	−0.656547	+	0.754285i	$0.727984\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−93.0000	−0.869159	−0.434579	−	0.900634i	$-0.643103\pi$
$107$	−93.0000	−0.869159	−0.434579	+	0.900634i	$0.643103\pi$
$108$	0	0
$109$	68.9773	0.632820	0.316410	−	0.948623i	$-0.397523\pi$
$109$	68.9773	0.632820	0.316410	+	0.948623i	$0.397523\pi$
$110$	0	0
$111$	0	0
$112$	66.9773	0.598011
$113$	225.313	1.99392	0.996962	−	0.0778934i	$-0.0248194\pi$
$113$	225.313	1.99392	0.996962	+	0.0778934i	$0.0248194\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−142.320	−1.19597
$120$	0	0
$121$	−103.804	−0.857880
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	198.981	1.56678	0.783390	−	0.621530i	$-0.213489\pi$
$127$	198.981	1.56678	0.783390	+	0.621530i	$0.213489\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	142.649	1.07255
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	144.000	1.00000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−221.912	−1.49940
$149$	−9.00000	−0.0604027	−0.0302013	−	0.999544i	$-0.509615\pi$
$149$	−9.00000	−0.0604027	−0.0302013	+	0.999544i	$0.509615\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−305.986	−1.99991
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	322.691	1.96763
$165$	0	0
$166$	0	0
$167$	27.0000	0.161677	0.0808383	−	0.996727i	$-0.474240\pi$
$167$	27.0000	0.161677	0.0808383	+	0.996727i	$0.474240\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	306.692	1.79352
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	104.652	0.598011
$176$	−66.3498	−0.376988
$177$	0	0
$178$	0	0
$179$	−198.314	−1.10790	−0.553951	−	0.832549i	$-0.686881\pi$
$179$	−198.314	−1.10790	−0.553951	+	0.832549i	$0.686881\pi$
$180$	0	0
$181$	−69.6832	−0.384990	−0.192495	−	0.981298i	$-0.561658\pi$
$181$	−69.6832	−0.384990	−0.192495	+	0.981298i	$0.561658\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	140.987	0.753941
$188$	0	0
$189$	0	0
$190$	0	0
$191$	270.694	1.41724	0.708622	−	0.705588i	$-0.249317\pi$
$191$	270.694	1.41724	0.708622	+	0.705588i	$0.249317\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−125.907	−0.642382
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−378.797	−1.90350	−0.951750	−	0.306873i	$-0.900717\pi$
$199$	−378.797	−1.90350	−0.951750	+	0.306873i	$0.900717\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−141.312	−0.676135
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	222.696	1.05045
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	438.108	1.96461	0.982304	−	0.187293i	$-0.0599712\pi$
$223$	438.108	1.96461	0.982304	+	0.187293i	$0.0599712\pi$
$224$	0	0
$225$	225.000	1.00000
$226$	0	0
$227$	−437.127	−1.92567	−0.962835	−	0.270089i	$-0.912947\pi$
$227$	−437.127	−1.92567	−0.962835	+	0.270089i	$0.912947\pi$
$228$	0	0
$229$	428.441	1.87092	0.935461	−	0.353429i	$-0.114984\pi$
$229$	428.441	1.87092	0.935461	+	0.353429i	$0.114984\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	159.000	0.682403	0.341202	−	0.939990i	$-0.389166\pi$
$233$	159.000	0.682403	0.341202	+	0.939990i	$0.389166\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	379.777	1.51306	0.756528	−	0.653961i	$-0.226894\pi$
$251$	379.777	1.51306	0.756528	+	0.653961i	$0.226894\pi$
$252$	150.699	0.598011
$253$	0	0
$254$	0	0
$255$	0	0
$256$	256.000	1.00000
$257$	−57.6740	−0.224413	−0.112206	−	0.993685i	$-0.535792\pi$
$257$	−57.6740	−0.224413	−0.112206	+	0.993685i	$0.535792\pi$
$258$	0	0
$259$	−232.235	−0.896660
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	350.778	1.30401	0.652004	−	0.758216i	$-0.273929\pi$
$269$	350.778	1.30401	0.652004	+	0.758216i	$0.273929\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−543.975	−1.99991
$273$	0	0
$274$	0	0
$275$	−103.672	−0.376988
$276$	0	0
$277$	−269.674	−0.973552	−0.486776	−	0.873527i	$-0.661827\pi$
$277$	−269.674	−0.973552	−0.486776	+	0.873527i	$0.661827\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	−274.341	−0.965988
$285$	0	0
$286$	0	0
$287$	337.703	1.17667
$288$	0	0
$289$	866.895	2.99964
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−427.304	−1.45838	−0.729188	−	0.684314i	$-0.760102\pi$
$293$	−427.304	−1.45838	−0.729188	+	0.684314i	$0.760102\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	545.230	1.79352
$305$	0	0
$306$	0	0
$307$	−307.000	−1.00000
$307$	−307.000	−1.00000
$308$	−69.4364	−0.225443
$309$	0	0
$310$	0	0
$311$	−606.000	−1.94855	−0.974277	−	0.225356i	$-0.927646\pi$
$311$	−606.000	−1.94855	−0.974277	+	0.225356i	$0.927646\pi$
$312$	0	0
$313$	253.450	0.809743	0.404871	−	0.914374i	$-0.367316\pi$
$313$	253.450	0.809743	0.404871	+	0.914374i	$0.367316\pi$
$314$	0	0
$315$	0	0
$316$	−596.000	−1.88608
$317$	327.000	1.03155	0.515773	−	0.856725i	$-0.327505\pi$
$317$	327.000	1.03155	0.515773	+	0.856725i	$0.327505\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1158.56	−3.58688
$324$	324.000	1.00000
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−642.451	−1.94094	−0.970470	−	0.241220i	$-0.922452\pi$
$331$	−642.451	−1.94094	−0.970470	+	0.241220i	$0.922452\pi$
$332$	−564.000	−1.69880
$333$	−499.301	−1.49940
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−336.882	−0.982163
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−37.2927	−0.107472	−0.0537358	−	0.998555i	$-0.517113\pi$
$347$	−37.2927	−0.107472	−0.0537358	+	0.998555i	$0.517113\pi$
$348$	0	0
$349$	391.000	1.12034	0.560172	−	0.828376i	$-0.310735\pi$
$349$	391.000	1.12034	0.560172	+	0.828376i	$0.310735\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	444.568	1.25940	0.629700	−	0.776838i	$-0.283178\pi$
$353$	444.568	1.25940	0.629700	+	0.776838i	$0.283178\pi$
$354$	0	0
$355$	0	0
$356$	−516.000	−1.44944
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	800.235	2.21672
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−717.448	−1.95490	−0.977449	−	0.211170i	$-0.932273\pi$
$367$	−717.448	−1.95490	−0.977449	+	0.211170i	$0.932273\pi$
$368$	0	0
$369$	726.055	1.96763
$370$	0	0
$371$	233.056	0.628183
$372$	0	0
$373$	−287.997	−0.772110	−0.386055	−	0.922476i	$-0.626163\pi$
$373$	−287.997	−0.772110	−0.386055	+	0.922476i	$0.626163\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	459.000	1.19843	0.599217	−	0.800587i	$-0.295479\pi$
$383$	459.000	1.19843	0.599217	+	0.800587i	$0.295479\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	−452.000	−1.16495
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	−149.287	−0.376988
$397$	−349.327	−0.879917	−0.439959	−	0.898018i	$-0.645007\pi$
$397$	−349.327	−0.879917	−0.439959	+	0.898018i	$0.645007\pi$
$398$	0	0
$399$	0	0
$400$	400.000	1.00000
$401$	665.058	1.65850	0.829249	−	0.558879i	$-0.188768\pi$
$401$	665.058	1.65850	0.829249	+	0.558879i	$0.188768\pi$
$402$	0	0
$403$	0	0
$404$	−732.555	−1.81326
$405$	0	0
$406$	0	0
$407$	230.059	0.565256
$408$	0	0
$409$	59.1251	0.144560	0.0722800	−	0.997384i	$-0.476972\pi$
$409$	59.1251	0.144560	0.0722800	+	0.997384i	$0.476972\pi$
$410$	0	0
$411$	0	0
$412$	−540.995	−1.31309
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	531.000	1.26730	0.633652	−	0.773619i	$-0.281555\pi$
$419$	531.000	1.26730	0.633652	+	0.773619i	$0.281555\pi$
$420$	0	0
$421$	−546.995	−1.29928	−0.649638	−	0.760244i	$-0.725079\pi$
$421$	−546.995	−1.29928	−0.649638	+	0.760244i	$0.725079\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−849.962	−1.99991
$426$	0	0
$427$	0	0
$428$	−372.000	−0.869159
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	275.909	0.632820
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−283.291	−0.642382
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	267.909	0.598011
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	−334.539	−0.741772
$452$	901.253	1.99392
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	607.000	1.32823	0.664114	−	0.747632i	$-0.268809\pi$
$457$	607.000	1.32823	0.664114	+	0.747632i	$0.268809\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	882.587	1.90624	0.953118	−	0.302598i	$-0.0978541\pi$
$463$	882.587	1.90624	0.953118	+	0.302598i	$0.0978541\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	536.064	1.14789	0.573944	−	0.818895i	$-0.305413\pi$
$467$	536.064	1.14789	0.573944	+	0.818895i	$0.305413\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	851.922	1.79352
$476$	−569.281	−1.19597
$477$	501.066	1.05045
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−415.214	−0.857880
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−814.287	−1.65843	−0.829213	−	0.558933i	$-0.811211\pi$
$491$	−814.287	−1.65843	−0.829213	+	0.558933i	$0.811211\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−287.103	−0.577672
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−222.000	−0.441352	−0.220676	−	0.975347i	$-0.570826\pi$
$503$	−222.000	−0.441352	−0.220676	+	0.975347i	$0.570826\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	795.925	1.56678
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	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	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	−586.317	−1.12106	−0.560532	−	0.828132i	$-0.689403\pi$
$523$	−586.317	−1.12106	−0.560532	+	0.828132i	$0.689403\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	0	0
$532$	570.594	1.07255
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	130.530	0.242170
$540$	0	0
$541$	−1065.77	−1.96999	−0.984996	−	0.172578i	$-0.944790\pi$
$541$	−1065.77	−1.96999	−0.984996	+	0.172578i	$0.944790\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1091.82	1.99602	0.998010	−	0.0630531i	$-0.0200837\pi$
$547$	1091.82	1.99602	0.998010	+	0.0630531i	$0.0200837\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−623.726	−1.12790
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1068.16	1.91770	0.958849	−	0.283917i	$-0.0916340\pi$
$557$	1068.16	1.91770	0.958849	+	0.283917i	$0.0916340\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−102.000	−0.181172	−0.0905861	−	0.995889i	$-0.528874\pi$
$563$	−102.000	−0.181172	−0.0905861	+	0.995889i	$0.528874\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	339.073	0.598011
$568$	0	0
$569$	28.5865	0.0502400	0.0251200	−	0.999684i	$-0.492003\pi$
$569$	28.5865	0.0502400	0.0251200	+	0.999684i	$0.492003\pi$
$570$	0	0
$571$	1047.49	1.83449	0.917243	−	0.398328i	$-0.130409\pi$
$571$	1047.49	1.83449	0.917243	+	0.398328i	$0.130409\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	576.000	1.00000
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−590.237	−1.01590
$582$	0	0
$583$	−230.872	−0.396008
$584$	0	0
$585$	0	0
$586$	0	0
$587$	641.265	1.09244	0.546222	−	0.837640i	$-0.316065\pi$
$587$	641.265	1.09244	0.546222	+	0.837640i	$0.316065\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−887.646	−1.49940
$593$	−290.193	−0.489365	−0.244682	−	0.969603i	$-0.578684\pi$
$593$	−290.193	−0.489365	−0.244682	+	0.969603i	$0.578684\pi$
$594$	0	0
$595$	0	0
$596$	−36.0000	−0.0604027
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	976.495	1.62478	0.812392	−	0.583112i	$-0.198165\pi$
$601$	976.495	1.62478	0.812392	+	0.583112i	$0.198165\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−1223.94	−1.99991
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−540.004	−0.866781
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1886.16	2.99867
$630$	0	0
$631$	34.0000	0.0538827	0.0269414	−	0.999637i	$-0.491423\pi$
$631$	34.0000	0.0538827	0.0269414	+	0.999637i	$0.491423\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−617.267	−0.965988
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1292.77	1.97973	0.989866	−	0.142004i	$-0.0453546\pi$
$653$	1292.77	1.97973	0.989866	+	0.142004i	$0.0453546\pi$
$654$	0	0
$655$	0	0
$656$	1290.77	1.96763
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	108.000	0.161677
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	676.000	1.00000
$677$	1047.00	1.54653	0.773264	−	0.634084i	$-0.218623\pi$
$677$	1047.00	1.54653	0.773264	+	0.634084i	$0.218623\pi$
$678$	0	0
$679$	−473.027	−0.696653
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1197.27	−1.75296	−0.876478	−	0.481441i	$-0.840114\pi$
$683$	−1197.27	−1.75296	−0.876478	+	0.481441i	$0.840114\pi$
$684$	1226.77	1.79352
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1381.00	−1.99855	−0.999276	−	0.0380349i	$-0.987890\pi$
$691$	−1381.00	−1.99855	−0.999276	+	0.0380349i	$0.987890\pi$
$692$	0	0
$693$	−156.232	−0.225443
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2742.75	−3.93508
$698$	0	0
$699$	0	0
$700$	418.608	0.598011
$701$	74.0844	0.105684	0.0528419	−	0.998603i	$-0.483172\pi$
$701$	74.0844	0.105684	0.0528419	+	0.998603i	$0.483172\pi$
$702$	0	0
$703$	−1890.51	−2.68921
$704$	−265.399	−0.376988
$705$	0	0
$706$	0	0
$707$	−766.634	−1.08435
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	−1341.00	−1.88608
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−793.258	−1.10790
$717$	0	0
$718$	0	0
$719$	11.0882	0.0154218	0.00771088	−	0.999970i	$-0.497546\pi$
$719$	11.0882	0.0154218	0.00771088	+	0.999970i	$0.497546\pi$
$720$	0	0
$721$	−566.162	−0.785246
$722$	0	0
$723$	0	0
$724$	−278.733	−0.384990
$725$	0	0
$726$	0	0
$727$	1147.00	1.57772	0.788858	−	0.614575i	$-0.210673\pi$
$727$	1147.00	1.57772	0.788858	+	0.614575i	$0.210673\pi$
$728$	0	0
$729$	729.000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1297.00	−1.76944	−0.884720	−	0.466122i	$-0.845651\pi$
$733$	−1297.00	−1.76944	−0.884720	+	0.466122i	$0.845651\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1044.63	−1.40596	−0.702980	−	0.711209i	$-0.748148\pi$
$743$	−1044.63	−1.40596	−0.702980	+	0.711209i	$0.748148\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1269.00	−1.69880
$748$	563.948	0.753941
$749$	−389.305	−0.519767
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	288.745	0.378433
$764$	1082.77	1.41724
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−639.844	−0.832047	−0.416023	−	0.909354i	$-0.636577\pi$
$769$	−639.844	−0.832047	−0.416023	+	0.909354i	$0.636577\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2749.08	3.52898
$780$	0	0
$781$	284.413	0.364166
$782$	0	0
$783$	0	0
$784$	−503.628	−0.642382
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	943.180	1.19239
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−1515.19	−1.90350
$797$	−1296.29	−1.62647	−0.813234	−	0.581937i	$-0.802295\pi$
$797$	−1296.29	−1.62647	−0.813234	+	0.581937i	$0.802295\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−1161.00	−1.44944
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−1117.00	−1.35723	−0.678615	−	0.734494i	$-0.737419\pi$
$823$	−1117.00	−1.35723	−0.678615	+	0.734494i	$0.737419\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	1402.42	1.69170	0.845848	−	0.533423i	$-0.179095\pi$
$829$	1402.42	1.69170	0.845848	+	0.533423i	$0.179095\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1070.16	1.28471
$834$	0	0
$835$	0	0
$836$	−565.249	−0.676135
$837$	0	0
$838$	0	0
$839$	−1522.25	−1.81437	−0.907184	−	0.420735i	$-0.861772\pi$
$839$	−1522.25	−1.81437	−0.907184	+	0.420735i	$0.861772\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−434.530	−0.513022
$848$	890.784	1.05045
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	617.883	0.711027
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−1017.00	−1.16495
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	538.000	0.609287	0.304643	−	0.952467i	$-0.401463\pi$
$883$	538.000	0.609287	0.304643	+	0.952467i	$0.401463\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1767.10	1.99222	0.996108	−	0.0881355i	$-0.0280908\pi$
$887$	1767.10	1.99222	0.996108	+	0.0881355i	$0.0280908\pi$
$888$	0	0
$889$	832.951	0.936953
$890$	0	0
$891$	−335.896	−0.376988
$892$	1752.43	1.96461
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	900.000	1.00000
$901$	−1892.83	−2.10081
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−25.3708	−0.0279722	−0.0139861	−	0.999902i	$-0.504452\pi$
$907$	−25.3708	−0.0279722	−0.0139861	+	0.999902i	$0.504452\pi$
$908$	−1748.51	−1.92567
$909$	−1648.25	−1.81326
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	584.708	0.640425
$914$	0	0
$915$	0	0
$916$	1713.77	1.87092
$917$	0	0
$918$	0	0
$919$	1200.76	1.30659	0.653297	−	0.757102i	$-0.273385\pi$
$919$	1200.76	1.30659	0.653297	+	0.757102i	$0.273385\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−1386.95	−1.49940
$926$	0	0
$927$	−1217.24	−1.31309
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−1072.63	−1.15213
$932$	636.000	0.682403
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	293.527	0.313262	0.156631	−	0.987657i	$-0.449937\pi$
$937$	293.527	0.313262	0.156631	+	0.987657i	$0.449937\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1587.00	1.67582	0.837909	−	0.545810i	$-0.183778\pi$
$947$	1587.00	1.67582	0.837909	+	0.545810i	$0.183778\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	−837.000	−0.869159
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1494.28	−1.54527	−0.772634	−	0.634851i	$-0.781061\pi$
$967$	−1494.28	−1.54527	−0.772634	+	0.634851i	$0.781061\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	534.945	0.546420
$980$	0	0
$981$	620.796	0.632820
$982$	0	0
$983$	738.000	0.750763	0.375381	−	0.926870i	$-0.377512\pi$
$983$	738.000	0.750763	0.375381	+	0.926870i	$0.377512\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1966.39	−1.98425	−0.992125	−	0.125253i	$-0.960026\pi$
$991$	−1966.39	−1.98425	−0.992125	+	0.125253i	$0.960026\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−769.000	−0.771314	−0.385657	−	0.922642i	$-0.626025\pi$
$997$	−769.000	−0.771314	−0.385657	+	0.922642i	$0.626025\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	307.3.b.a.306.2	✓	3
307.306	odd	2	CM	307.3.b.a.306.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
307.3.b.a.306.2	✓	3	1.1	even	1	trivial
307.3.b.a.306.2	✓	3	307.306	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:	\( N \)	\(=\)	\( 307 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	307.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.00000 q^{4} +4.18608 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+4.00000 q^{4} +4.18608 q^{7} +9.00000 q^{9} -4.14686 q^{11} +16.0000 q^{16} -33.9985 q^{17} +34.0769 q^{19} +25.0000 q^{25} +16.7443 q^{28} +36.0000 q^{36} -55.4779 q^{37} +80.6728 q^{41} -16.5875 q^{44} -31.4767 q^{49} +55.6740 q^{53} +37.6747 q^{63} +64.0000 q^{64} -135.994 q^{68} -68.5852 q^{71} +136.308 q^{76} -17.3591 q^{77} -149.000 q^{79} +81.0000 q^{81} -141.000 q^{83} -129.000 q^{89} -113.000 q^{97} -37.3218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{4} + 27 q^{9}+O(q^{10}) \) 3 * q + 12 * q^4 + 27 * q^9	\( 3 q + 12 q^{4} + 27 q^{9} + 48 q^{16} + 75 q^{25} + 108 q^{36} + 147 q^{49} + 192 q^{64} - 459 q^{77} - 447 q^{79} + 243 q^{81} - 423 q^{83} - 387 q^{89} - 339 q^{97}+O(q^{100}) \) 3 * q + 12 * q^4 + 27 * q^9 + 48 * q^16 + 75 * q^25 + 108 * q^36 + 147 * q^49 + 192 * q^64 - 459 * q^77 - 447 * q^79 + 243 * q^81 - 423 * q^83 - 387 * q^89 - 339 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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