LMFDB - Embedded newform 307.3.b.a.306.3

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			306.3
Root			$4.84527$ 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)

$f(q)$	$=$	$q+4.00000 q^{4} +9.47664 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+4.00000 q^{4} +9.47664 q^{7} +9.00000 q^{9} -16.6376 q^{11} +16.0000 q^{16} +17.2793 q^{17} -31.6012 q^{19} +25.0000 q^{25} +37.9066 q^{28} +36.0000 q^{36} +70.1495 q^{37} -27.6115 q^{41} -66.5504 q^{44} +40.8067 q^{49} -105.954 q^{53} +85.2898 q^{63} +64.0000 q^{64} +69.1172 q^{68} +141.973 q^{71} -126.405 q^{76} -157.669 q^{77} -149.000 q^{79} +81.0000 q^{81} -141.000 q^{83} -129.000 q^{89} -113.000 q^{97} -149.738 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	9.47664	1.35381	0.676903	−	0.736072i	$-0.263322\pi$
$7$	9.47664	1.35381	0.676903	+	0.736072i	$0.263322\pi$
$8$	0	0
$9$	9.00000	1.00000
$10$	0	0
$11$	−16.6376	−1.51251	−0.756254	−	0.654278i	$-0.772973\pi$
$11$	−16.6376	−1.51251	−0.756254	+	0.654278i	$0.772973\pi$
$12$	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
$17$	17.2793	1.01643	0.508214	−	0.861231i	$-0.330306\pi$
$17$	17.2793	1.01643	0.508214	+	0.861231i	$0.330306\pi$
$18$	0	0
$19$	−31.6012	−1.66322	−0.831611	−	0.555359i	$-0.812581\pi$
$19$	−31.6012	−1.66322	−0.831611	+	0.555359i	$0.812581\pi$
$20$	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
$28$	37.9066	1.35381
$29$	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
$37$	70.1495	1.89593	0.947966	−	0.318372i	$-0.103136\pi$
$37$	70.1495	1.89593	0.947966	+	0.318372i	$0.103136\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−27.6115	−0.673452	−0.336726	−	0.941603i	$-0.609320\pi$
$41$	−27.6115	−0.673452	−0.336726	+	0.941603i	$0.609320\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−66.5504	−1.51251
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	40.8067	0.832791
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−105.954	−1.99914	−0.999568	−	0.0293780i	$-0.990647\pi$
$53$	−105.954	−1.99914	−0.999568	+	0.0293780i	$0.990647\pi$
$54$	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
$63$	85.2898	1.35381
$64$	64.0000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	69.1172	1.01643
$69$	0	0
$70$	0	0
$71$	141.973	1.99962	0.999809	−	0.0195265i	$-0.00621588\pi$
$71$	141.973	1.99962	0.999809	+	0.0195265i	$0.00621588\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	−126.405	−1.66322
$77$	−157.669	−2.04764
$78$	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
$99$	−149.738	−1.51251
$100$	100.000	1.00000
$101$	165.381	1.63743	0.818717	−	0.574197i	$-0.194686\pi$
$101$	165.381	1.63743	0.818717	+	0.574197i	$0.194686\pi$
$102$	0	0
$103$	−66.9410	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$103$	−66.9410	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$104$	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
$109$	−213.582	−1.95947	−0.979736	−	0.200292i	$-0.935811\pi$
$109$	−213.582	−1.95947	−0.979736	+	0.200292i	$0.935811\pi$
$110$	0	0
$111$	0	0
$112$	151.626	1.35381
$113$	−127.902	−1.13188	−0.565939	−	0.824447i	$-0.691486\pi$
$113$	−127.902	−1.13188	−0.565939	+	0.824447i	$0.691486\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	163.750	1.37605
$120$	0	0
$121$	155.810	1.28768
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	37.2277	0.293131	0.146566	−	0.989201i	$-0.453178\pi$
$127$	37.2277	0.293131	0.146566	+	0.989201i	$0.453178\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	−299.473	−2.25168
$134$	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
$148$	280.598	1.89593
$149$	−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
$153$	155.514	1.01643
$154$	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			$\pi$
$164$	−110.446	−0.673452
$165$	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
$171$	−284.411	−1.66322
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	236.916	1.35381
$176$	−266.202	−1.51251
$177$	0	0
$178$	0	0
$179$	−158.964	−0.888067	−0.444033	−	0.896010i	$-0.646453\pi$
$179$	−158.964	−0.888067	−0.444033	+	0.896010i	$0.646453\pi$
$180$	0	0
$181$	342.480	1.89215	0.946077	−	0.323943i	$-0.105009\pi$
$181$	342.480	1.89215	0.946077	+	0.323943i	$0.105009\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−287.486	−1.53736
$188$	0	0
$189$	0	0
$190$	0	0
$191$	98.0772	0.513493	0.256747	−	0.966479i	$-0.417349\pi$
$191$	98.0772	0.513493	0.256747	+	0.966479i	$0.417349\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	163.227	0.832791
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	295.171	1.48327	0.741635	−	0.670803i	$-0.234051\pi$
$199$	295.171	1.48327	0.741635	+	0.670803i	$0.234051\pi$
$200$	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
$209$	525.768	2.51564
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−423.817	−1.99914
$213$	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
$223$	−291.395	−1.30670	−0.653352	−	0.757054i	$-0.726638\pi$
$223$	−291.395	−1.30670	−0.653352	+	0.757054i	$0.726638\pi$
$224$	0	0
$225$	225.000	1.00000
$226$	0	0
$227$	112.371	0.495027	0.247514	−	0.968884i	$-0.420386\pi$
$227$	112.371	0.495027	0.247514	+	0.968884i	$0.420386\pi$
$228$	0	0
$229$	−74.0368	−0.323305	−0.161652	−	0.986848i	$-0.551682\pi$
$229$	−74.0368	−0.323305	−0.161652	+	0.986848i	$0.551682\pi$
$230$	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
$251$	−474.195	−1.88922	−0.944611	−	0.328192i	$-0.893561\pi$
$251$	−474.195	−1.88922	−0.944611	+	0.328192i	$0.893561\pi$
$252$	341.159	1.35381
$253$	0	0
$254$	0	0
$255$	0	0
$256$	256.000	1.00000
$257$	471.163	1.83332	0.916660	−	0.399669i	$-0.130875\pi$
$257$	471.163	1.83332	0.916660	+	0.399669i	$0.130875\pi$
$258$	0	0
$259$	664.781	2.56672
$260$	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
$269$	177.880	0.661264	0.330632	−	0.943760i	$-0.392738\pi$
$269$	177.880	0.661264	0.330632	+	0.943760i	$0.392738\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	276.469	1.01643
$273$	0	0
$274$	0	0
$275$	−415.940	−1.51251
$276$	0	0
$277$	−284.262	−1.02622	−0.513108	−	0.858324i	$-0.671506\pi$
$277$	−284.262	−1.02622	−0.513108	+	0.858324i	$0.671506\pi$
$278$	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			$\pi$
$284$	567.892	1.99962
$285$	0	0
$286$	0	0
$287$	−261.665	−0.911723
$288$	0	0
$289$	9.57394	0.0331278
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−133.631	−0.456078	−0.228039	−	0.973652i	$-0.573231\pi$
$293$	−133.631	−0.456078	−0.228039	+	0.973652i	$0.573231\pi$
$294$	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
$304$	−505.619	−1.66322
$305$	0	0
$306$	0	0
$307$	−307.000	−1.00000
$307$	−307.000	−1.00000
$308$	−630.674	−2.04764
$309$	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
$313$	−622.436	−1.98861	−0.994307	−	0.106558i	$-0.966017\pi$
$313$	−622.436	−1.98861	−0.994307	+	0.106558i	$0.966017\pi$
$314$	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
$323$	−546.046	−1.69055
$324$	324.000	1.00000
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	459.519	1.38828	0.694138	−	0.719842i	$-0.255786\pi$
$331$	459.519	1.38828	0.694138	+	0.719842i	$0.255786\pi$
$332$	−564.000	−1.69880
$333$	631.345	1.89593
$334$	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
$343$	−77.6445	−0.226369
$344$	0	0
$345$	0	0
$346$	0	0
$347$	618.800	1.78328	0.891642	−	0.452741i	$-0.149554\pi$
$347$	618.800	1.78328	0.891642	+	0.452741i	$0.149554\pi$
$348$	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
$353$	−697.254	−1.97522	−0.987612	−	0.156917i	$-0.949844\pi$
$353$	−697.254	−1.97522	−0.987612	+	0.156917i	$0.949844\pi$
$354$	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
$361$	637.636	1.76631
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	224.491	0.611693	0.305846	−	0.952081i	$-0.401061\pi$
$367$	224.491	0.611693	0.305846	+	0.952081i	$0.401061\pi$
$368$	0	0
$369$	−248.504	−0.673452
$370$	0	0
$371$	−1004.09	−2.70644
$372$	0	0
$373$	739.969	1.98383	0.991915	−	0.126904i	$-0.0405042\pi$
$373$	739.969	1.98383	0.991915	+	0.126904i	$0.0405042\pi$
$374$	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
$396$	−598.954	−1.51251
$397$	−442.835	−1.11545	−0.557727	−	0.830024i	$-0.688326\pi$
$397$	−442.835	−1.11545	−0.557727	+	0.830024i	$0.688326\pi$
$398$	0	0
$399$	0	0
$400$	400.000	1.00000
$401$	55.6416	0.138757	0.0693785	−	0.997590i	$-0.477898\pi$
$401$	55.6416	0.138757	0.0693785	+	0.997590i	$0.477898\pi$
$402$	0	0
$403$	0	0
$404$	661.524	1.63743
$405$	0	0
$406$	0	0
$407$	−1167.12	−2.86761
$408$	0	0
$409$	−736.118	−1.79980	−0.899900	−	0.436096i	$-0.856361\pi$
$409$	−736.118	−1.79980	−0.899900	+	0.436096i	$0.856361\pi$
$410$	0	0
$411$	0	0
$412$	−267.764	−0.649913
$413$	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
$421$	827.862	1.96642	0.983209	−	0.182481i	$-0.0584128\pi$
$421$	827.862	1.96642	0.983209	+	0.182481i	$0.0584128\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	431.982	1.01643
$426$	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
$436$	−854.330	−1.95947
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	367.261	0.832791
$442$	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
$448$	606.505	1.35381
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	459.389	1.01860
$452$	−511.608	−1.13188
$453$	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
$463$	−198.628	−0.429002	−0.214501	−	0.976724i	$-0.568813\pi$
$463$	−198.628	−0.429002	−0.214501	+	0.976724i	$0.568813\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	394.346	0.844423	0.422211	−	0.906497i	$-0.361254\pi$
$467$	394.346	0.844423	0.422211	+	0.906497i	$0.361254\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−790.030	−1.66322
$476$	654.999	1.37605
$477$	−953.588	−1.99914
$478$	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
$484$	623.239	1.28768
$485$	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
$491$	882.481	1.79731	0.898657	−	0.438653i	$-0.144544\pi$
$491$	882.481	1.79731	0.898657	+	0.438653i	$0.144544\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1345.43	2.70710
$498$	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
$508$	148.911	0.293131
$509$	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
$523$	1043.33	1.99490	0.997450	−	0.0713691i	$-0.0227368\pi$
$523$	1043.33	1.99490	0.997450	+	0.0713691i	$0.0227368\pi$
$524$	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
$532$	−1197.89	−2.25168
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−678.926	−1.25960
$540$	0	0
$541$	371.170	0.686081	0.343041	−	0.939321i	$-0.388543\pi$
$541$	371.170	0.686081	0.343041	+	0.939321i	$0.388543\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−486.173	−0.888799	−0.444400	−	0.895829i	$-0.646583\pi$
$547$	−486.173	−0.888799	−0.444400	+	0.895829i	$0.646583\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−1412.02	−2.55338
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−807.988	−1.45061	−0.725304	−	0.688429i	$-0.758301\pi$
$557$	−807.988	−1.45061	−0.725304	+	0.688429i	$0.758301\pi$
$558$	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
$567$	767.608	1.35381
$568$	0	0
$569$	970.933	1.70638	0.853192	−	0.521597i	$-0.174663\pi$
$569$	970.933	1.70638	0.853192	+	0.521597i	$0.174663\pi$
$570$	0	0
$571$	−129.799	−0.227319	−0.113660	−	0.993520i	$-0.536257\pi$
$571$	−129.799	−0.227319	−0.113660	+	0.993520i	$0.536257\pi$
$572$	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
$581$	−1336.21	−2.29984
$582$	0	0
$583$	1762.82	3.02371
$584$	0	0
$585$	0	0
$586$	0	0
$587$	531.008	0.904614	0.452307	−	0.891862i	$-0.350601\pi$
$587$	531.008	0.904614	0.452307	+	0.891862i	$0.350601\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	1122.39	1.89593
$593$	1140.98	1.92408	0.962042	−	0.272901i	$-0.0879830\pi$
$593$	1140.98	1.92408	0.962042	+	0.272901i	$0.0879830\pi$
$594$	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
$601$	−1095.24	−1.82237	−0.911185	−	0.411996i	$-0.864832\pi$
$601$	−1095.24	−1.82237	−0.911185	+	0.411996i	$0.864832\pi$
$602$	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
$612$	622.055	1.01643
$613$	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
$623$	−1222.49	−1.96226
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1212.13	1.92708
$630$	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
$639$	1277.76	1.99962
$640$	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
$653$	−806.993	−1.23582	−0.617912	−	0.786247i	$-0.712021\pi$
$653$	−806.993	−1.23582	−0.617912	+	0.786247i	$0.712021\pi$
$654$	0	0
$655$	0	0
$656$	−441.784	−0.673452
$657$	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
$679$	−1070.86	−1.57711
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1168.18	1.71036	0.855179	−	0.518332i	$-0.173447\pi$
$683$	1168.18	1.71036	0.855179	+	0.518332i	$0.173447\pi$
$684$	−1137.64	−1.66322
$685$	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
$693$	−1419.02	−2.04764
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−477.108	−0.684516
$698$	0	0
$699$	0	0
$700$	947.664	1.35381
$701$	1175.43	1.67679	0.838395	−	0.545064i	$-0.183495\pi$
$701$	1175.43	1.67679	0.838395	+	0.545064i	$0.183495\pi$
$702$	0	0
$703$	−2216.81	−3.15335
$704$	−1064.81	−1.51251
$705$	0	0
$706$	0	0
$707$	1567.26	2.21677
$708$	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
$716$	−635.856	−0.888067
$717$	0	0
$718$	0	0
$719$	−1250.85	−1.73971	−0.869855	−	0.493307i	$-0.835788\pi$
$719$	−1250.85	−1.73971	−0.869855	+	0.493307i	$0.835788\pi$
$720$	0	0
$721$	−634.376	−0.879856
$722$	0	0
$723$	0	0
$724$	1369.92	1.89215
$725$	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
$743$	−392.951	−0.528870	−0.264435	−	0.964403i	$-0.585185\pi$
$743$	−392.951	−0.528870	−0.264435	+	0.964403i	$0.585185\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1269.00	−1.69880
$748$	−1149.94	−1.53736
$749$	−881.328	−1.17667
$750$	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
$763$	−2024.04	−2.65275
$764$	392.309	0.513493
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1531.13	1.99107	0.995535	−	0.0943900i	$-0.0300901\pi$
$769$	1531.13	1.99107	0.995535	+	0.0943900i	$0.0300901\pi$
$770$	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
$779$	872.557	1.12010
$780$	0	0
$781$	−2362.09	−3.02444
$782$	0	0
$783$	0	0
$784$	652.908	0.832791
$785$	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
$791$	−1212.08	−1.53234
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	1180.68	1.48327
$797$	1451.48	1.82118	0.910589	−	0.413313i	$-0.135628\pi$
$797$	1451.48	1.82118	0.910589	+	0.413313i	$0.135628\pi$
$798$	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
$829$	64.7184	0.0780680	0.0390340	−	0.999238i	$-0.487572\pi$
$829$	64.7184	0.0780680	0.0390340	+	0.999238i	$0.487572\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	705.112	0.846473
$834$	0	0
$835$	0	0
$836$	2103.07	2.51564
$837$	0	0
$838$	0	0
$839$	149.720	0.178450	0.0892251	−	0.996011i	$-0.471561\pi$
$839$	149.720	0.178450	0.0892251	+	0.996011i	$0.471561\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1476.55	1.74327
$848$	−1695.27	−1.99914
$849$	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
$869$	2479.00	2.85271
$870$	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
$887$	−748.143	−0.843453	−0.421727	−	0.906723i	$-0.638576\pi$
$887$	−748.143	−0.843453	−0.421727	+	0.906723i	$0.638576\pi$
$888$	0	0
$889$	352.793	0.396843
$890$	0	0
$891$	−1347.65	−1.51251
$892$	−1165.58	−1.30670
$893$	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
$901$	−1830.81	−2.03198
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1558.13	−1.71790	−0.858948	−	0.512063i	$-0.828881\pi$
$907$	−1558.13	−1.71790	−0.858948	+	0.512063i	$0.828881\pi$
$908$	449.485	0.495027
$909$	1488.43	1.63743
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	2345.90	2.56944
$914$	0	0
$915$	0	0
$916$	−296.147	−0.323305
$917$	0	0
$918$	0	0
$919$	−1805.50	−1.96464	−0.982318	−	0.187221i	$-0.940052\pi$
$919$	−1805.50	−1.96464	−0.982318	+	0.187221i	$0.940052\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1753.74	1.89593
$926$	0	0
$927$	−602.469	−0.649913
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−1289.54	−1.38512
$932$	636.000	0.682403
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1749.66	−1.86730	−0.933652	−	0.358182i	$-0.883397\pi$
$937$	−1749.66	−1.86730	−0.933652	+	0.358182i	$0.883397\pi$
$938$	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
$967$	−316.170	−0.326960	−0.163480	−	0.986547i	$-0.552272\pi$
$967$	−316.170	−0.326960	−0.163480	+	0.986547i	$0.552272\pi$
$968$	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
$979$	2146.25	2.19229
$980$	0	0
$981$	−1922.24	−1.95947
$982$	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
$991$	768.203	0.775180	0.387590	−	0.921832i	$-0.373308\pi$
$991$	768.203	0.775180	0.387590	+	0.921832i	$0.373308\pi$
$992$	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.306.3	✓	3
307.306	odd	2	CM	307.3.b.a.306.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
307.3.b.a.306.3	✓	3	1.1	even	1	trivial
307.3.b.a.306.3	✓	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)

\(f(q)\)	\(=\)	\(q+4.00000 q^{4} +9.47664 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+4.00000 q^{4} +9.47664 q^{7} +9.00000 q^{9} -16.6376 q^{11} +16.0000 q^{16} +17.2793 q^{17} -31.6012 q^{19} +25.0000 q^{25} +37.9066 q^{28} +36.0000 q^{36} +70.1495 q^{37} -27.6115 q^{41} -66.5504 q^{44} +40.8067 q^{49} -105.954 q^{53} +85.2898 q^{63} +64.0000 q^{64} +69.1172 q^{68} +141.973 q^{71} -126.405 q^{76} -157.669 q^{77} -149.000 q^{79} +81.0000 q^{81} -141.000 q^{83} -129.000 q^{89} -113.000 q^{97} -149.738 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{4} + 27 q^{9}+O(q^{10}) \) 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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