LMFDB - Embedded newform 1388.3.d.a.693.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1388,3,Mod(693,1388)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1388, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1388.693");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1388 = 2^{2} \cdot 347 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1388.d (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:	$37.8202606932$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 30x^{8} + 315x^{6} - 25x^{5} - 1350x^{4} + 375x^{3} + 2025x^{2} - 1125x - 104 $ x^10 - 30x^8 + 315x^6 - 25x^5 - 1350x^4 + 375x^3 + 2025x^2 - 1125*x - 104
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			693.9
Root			$-3.31866$ of defining polynomial
Character	$\chi$	$=$	1388.693

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.01347 q^{3} +16.1349 q^{9} +O(q^{10})$	$q+5.01347 q^{3} +16.1349 q^{9} +1.74246 q^{11} +9.83551 q^{13} +25.0000 q^{25} +35.7707 q^{27} +48.7409 q^{29} -34.1059 q^{31} +8.73580 q^{33} +49.3101 q^{39} -9.73062 q^{43} +49.0000 q^{49} -105.941 q^{53} +117.914 q^{59} +121.214 q^{61} -129.630 q^{67} -114.010 q^{71} +37.0115 q^{73} +125.337 q^{75} +34.1214 q^{81} +156.686 q^{83} +244.361 q^{87} +132.897 q^{89} -170.989 q^{93} +28.1145 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 90 q^{9}+O(q^{10}) $ 10 * q + 90 * q^9	$ 10 q + 90 q^{9} + 250 q^{25} + 490 q^{49} + 810 q^{81} + 865 q^{87} + 845 q^{89} + 805 q^{93} + 745 q^{99}+O(q^{100}) $ 10 * q + 90 * q^9 + 250 * q^25 + 490 * q^49 + 810 * q^81 + 865 * q^87 + 845 * q^89 + 805 * q^93 + 745 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1388\mathbb{Z}\right)^\times$.

$n$	$349$	$695$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	5.01347	1.67116	0.835579	−	0.549370i	$-0.185132\pi$
$3$	5.01347	1.67116	0.835579	+	0.549370i	$0.185132\pi$
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	16.1349	1.79277
$10$	0	0
$11$	1.74246	0.158406	0.0792029	−	0.996859i	$-0.474762\pi$
$11$	1.74246	0.158406	0.0792029	+	0.996859i	$0.474762\pi$
$12$	0	0
$13$	9.83551	0.756577	0.378289	−	0.925688i	$-0.376513\pi$
$13$	9.83551	0.756577	0.378289	+	0.925688i	$0.376513\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	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
$27$	35.7707	1.32484
$28$	0	0
$29$	48.7409	1.68072	0.840360	−	0.542029i	$-0.182344\pi$
$29$	48.7409	1.68072	0.840360	+	0.542029i	$0.182344\pi$
$30$	0	0
$31$	−34.1059	−1.10019	−0.550095	−	0.835102i	$-0.685408\pi$
$31$	−34.1059	−1.10019	−0.550095	+	0.835102i	$0.685408\pi$
$32$	0	0
$33$	8.73580	0.264721
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	49.3101	1.26436
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−9.73062	−0.226294	−0.113147	−	0.993578i	$-0.536093\pi$
$43$	−9.73062	−0.226294	−0.113147	+	0.993578i	$0.536093\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	49.0000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−105.941	−1.99889	−0.999445	−	0.0333040i	$-0.989397\pi$
$53$	−105.941	−1.99889	−0.999445	+	0.0333040i	$0.989397\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	117.914	1.99854	0.999269	−	0.0382271i	$-0.0121710\pi$
$59$	117.914	1.99854	0.999269	+	0.0382271i	$0.0121710\pi$
$60$	0	0
$61$	121.214	1.98712	0.993558	−	0.113326i	$-0.0361504\pi$
$61$	121.214	1.98712	0.993558	+	0.113326i	$0.0361504\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−129.630	−1.93478	−0.967389	−	0.253295i	$-0.918486\pi$
$67$	−129.630	−1.93478	−0.967389	+	0.253295i	$0.918486\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−114.010	−1.60578	−0.802890	−	0.596128i	$-0.796705\pi$
$71$	−114.010	−1.60578	−0.802890	+	0.596128i	$0.796705\pi$
$72$	0	0
$73$	37.0115	0.507006	0.253503	−	0.967335i	$-0.418417\pi$
$73$	37.0115	0.507006	0.253503	+	0.967335i	$0.418417\pi$
$74$	0	0
$75$	125.337	1.67116
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	34.1214	0.421252
$82$	0	0
$83$	156.686	1.88779	0.943895	−	0.330247i	$-0.107132\pi$
$83$	156.686	1.88779	0.943895	+	0.330247i	$0.107132\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	244.361	2.80875
$88$	0	0
$89$	132.897	1.49322	0.746611	−	0.665261i	$-0.231680\pi$
$89$	132.897	1.49322	0.746611	+	0.665261i	$0.231680\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−170.989	−1.83859
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	28.1145	0.283985
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−78.6904	−0.735424	−0.367712	−	0.929940i	$-0.619859\pi$
$107$	−78.6904	−0.735424	−0.367712	+	0.929940i	$0.619859\pi$
$108$	0	0
$109$	198.098	1.81741	0.908707	−	0.417435i	$-0.137071\pi$
$109$	198.098	1.81741	0.908707	+	0.417435i	$0.137071\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−171.568	−1.51830	−0.759152	−	0.650913i	$-0.774386\pi$
$113$	−171.568	−1.51830	−0.759152	+	0.650913i	$0.774386\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	158.695	1.35637
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−117.964	−0.974908
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.6568	0.162652	0.0813258	−	0.996688i	$-0.474085\pi$
$127$	20.6568	0.162652	0.0813258	+	0.996688i	$0.474085\pi$
$128$	0	0
$129$	−48.7842	−0.378172
$130$	0	0
$131$	−33.1894	−0.253354	−0.126677	−	0.991944i	$-0.540431\pi$
$131$	−33.1894	−0.253354	−0.126677	+	0.991944i	$0.540431\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−48.4871	−0.353921	−0.176960	−	0.984218i	$-0.556626\pi$
$137$	−48.4871	−0.353921	−0.176960	+	0.984218i	$0.556626\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	17.1380	0.119846
$144$	0	0
$145$	0	0
$146$	0	0
$147$	245.660	1.67116
$148$	0	0
$149$	−274.040	−1.83920	−0.919599	−	0.392859i	$-0.871486\pi$
$149$	−274.040	−1.83920	−0.919599	+	0.392859i	$0.871486\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−196.069	−1.24885	−0.624425	−	0.781085i	$-0.714667\pi$
$157$	−196.069	−1.24885	−0.624425	+	0.781085i	$0.714667\pi$
$158$	0	0
$159$	−531.133	−3.34046
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−187.169	−1.12077	−0.560387	−	0.828231i	$-0.689348\pi$
$167$	−187.169	−1.12077	−0.560387	+	0.828231i	$0.689348\pi$
$168$	0	0
$169$	−72.2628	−0.427591
$170$	0	0
$171$	0	0
$172$	0	0
$173$	90.2472	0.521660	0.260830	−	0.965385i	$-0.416004\pi$
$173$	90.2472	0.521660	0.260830	+	0.965385i	$0.416004\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	591.158	3.33987
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	106.913	0.590679	0.295340	−	0.955392i	$-0.404567\pi$
$181$	106.913	0.590679	0.295340	+	0.955392i	$0.404567\pi$
$182$	0	0
$183$	607.704	3.32078
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−362.278	−1.83897	−0.919486	−	0.393122i	$-0.871395\pi$
$197$	−362.278	−1.83897	−0.919486	+	0.393122i	$0.871395\pi$
$198$	0	0
$199$	385.464	1.93700	0.968502	−	0.249005i	$-0.0801036\pi$
$199$	385.464	1.93700	0.968502	+	0.249005i	$0.0801036\pi$
$200$	0	0
$201$	−649.897	−3.23332
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	−571.588	−2.68351
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	185.556	0.847288
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	403.373	1.79277
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−442.490	−1.93227	−0.966135	−	0.258037i	$-0.916924\pi$
$229$	−442.490	−1.93227	−0.966135	+	0.258037i	$0.916924\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	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
$241$	−295.939	−1.22796	−0.613981	−	0.789321i	$-0.710433\pi$
$241$	−295.939	−1.22796	−0.613981	+	0.789321i	$0.710433\pi$
$242$	0	0
$243$	−150.870	−0.620864
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	785.544	3.15479
$250$	0	0
$251$	436.021	1.73714	0.868568	−	0.495570i	$-0.165041\pi$
$251$	436.021	1.73714	0.868568	+	0.495570i	$0.165041\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	786.430	3.01314
$262$	0	0
$263$	−304.749	−1.15874	−0.579371	−	0.815064i	$-0.696702\pi$
$263$	−304.749	−1.15874	−0.579371	+	0.815064i	$0.696702\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	666.275	2.49541
$268$	0	0
$269$	340.071	1.26421	0.632103	−	0.774885i	$-0.282192\pi$
$269$	340.071	1.26421	0.632103	+	0.774885i	$0.282192\pi$
$270$	0	0
$271$	−476.818	−1.75948	−0.879738	−	0.475458i	$-0.842282\pi$
$271$	−476.818	−1.75948	−0.879738	+	0.475458i	$0.842282\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	43.5616	0.158406
$276$	0	0
$277$	−223.870	−0.808196	−0.404098	−	0.914716i	$-0.632415\pi$
$277$	−223.870	−0.808196	−0.404098	+	0.914716i	$0.632415\pi$
$278$	0	0
$279$	−550.296	−1.97239
$280$	0	0
$281$	366.816	1.30539	0.652697	−	0.757619i	$-0.273637\pi$
$281$	366.816	1.30539	0.652697	+	0.757619i	$0.273637\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	159.043	0.542808	0.271404	−	0.962466i	$-0.412512\pi$
$293$	159.043	0.542808	0.271404	+	0.962466i	$0.412512\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	62.3292	0.209863
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	84.9292	0.266236
$320$	0	0
$321$	−394.512	−1.22901
$322$	0	0
$323$	0	0
$324$	0	0
$325$	245.888	0.756577
$326$	0	0
$327$	993.159	3.03719
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	−860.154	−2.53733
$340$	0	0
$341$	−59.4283	−0.174276
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−347.000	−1.00000
$347$	−347.000	−1.00000
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	351.823	1.00235
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−717.865	−1.99962	−0.999812	−	0.0194074i	$-0.993822\pi$
$359$	−717.865	−1.99962	−0.999812	+	0.0194074i	$0.993822\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	−591.409	−1.62922
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	479.391	1.27159
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	103.562	0.271817
$382$	0	0
$383$	−622.000	−1.62402	−0.812010	−	0.583643i	$-0.801627\pi$
$383$	−622.000	−1.62402	−0.812010	+	0.583643i	$0.801627\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−157.003	−0.405692
$388$	0	0
$389$	−717.082	−1.84340	−0.921700	−	0.387904i	$-0.873199\pi$
$389$	−717.082	−1.84340	−0.921700	+	0.387904i	$0.873199\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−166.394	−0.423395
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−335.449	−0.832379
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	−243.089	−0.591457
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	834.119	1.98128	0.990640	−	0.136503i	$-0.0435863\pi$
$421$	834.119	1.98128	0.990640	+	0.136503i	$0.0435863\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	85.9210	0.200282
$430$	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
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	790.611	1.79277
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1373.89	−3.07359
$448$	0	0
$449$	−501.137	−1.11612	−0.558059	−	0.829801i	$-0.688454\pi$
$449$	−501.137	−1.11612	−0.558059	+	0.829801i	$0.688454\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	813.760	1.78066	0.890328	−	0.455320i	$-0.150475\pi$
$457$	813.760	1.78066	0.890328	+	0.455320i	$0.150475\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	148.231	0.321542	0.160771	−	0.986992i	$-0.448602\pi$
$461$	148.231	0.321542	0.160771	+	0.986992i	$0.448602\pi$
$462$	0	0
$463$	−745.342	−1.60981	−0.804905	−	0.593403i	$-0.797784\pi$
$463$	−745.342	−1.60981	−0.804905	+	0.593403i	$0.797784\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	335.658	0.718755	0.359377	−	0.933192i	$-0.382989\pi$
$467$	335.658	0.718755	0.359377	+	0.933192i	$0.382989\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−982.989	−2.08703
$472$	0	0
$473$	−16.9553	−0.0358462
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1709.35	−3.58355
$478$	0	0
$479$	−554.866	−1.15838	−0.579192	−	0.815191i	$-0.696632\pi$
$479$	−554.866	−1.15838	−0.579192	+	0.815191i	$0.696632\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−813.649	−1.67074	−0.835369	−	0.549690i	$-0.814746\pi$
$487$	−813.649	−1.67074	−0.835369	+	0.549690i	$0.814746\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−406.000	−0.826884	−0.413442	−	0.910530i	$-0.635674\pi$
$491$	−406.000	−0.826884	−0.413442	+	0.910530i	$0.635674\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	937.224	1.87821	0.939103	−	0.343637i	$-0.111659\pi$
$499$	937.224	1.87821	0.939103	+	0.343637i	$0.111659\pi$
$500$	0	0
$501$	−938.369	−1.87299
$502$	0	0
$503$	−1005.97	−1.99995	−0.999973	−	0.00741370i	$-0.997640\pi$
$503$	−1005.97	−1.99995	−0.999973	+	0.00741370i	$0.997640\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−362.288	−0.714572
$508$	0	0
$509$	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
$519$	452.452	0.871776
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	−423.373	−0.809508	−0.404754	−	0.914426i	$-0.632643\pi$
$523$	−423.373	−0.809508	−0.404754	+	0.914426i	$0.632643\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	1902.53	3.58292
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	85.3807	0.158406
$540$	0	0
$541$	−712.543	−1.31709	−0.658543	−	0.752543i	$-0.728827\pi$
$541$	−712.543	−1.31709	−0.658543	+	0.752543i	$0.728827\pi$
$542$	0	0
$543$	536.005	0.987118
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	1955.78	3.56244
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	−95.7056	−0.171209
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1134.24	−1.99338	−0.996692	−	0.0812732i	$-0.974101\pi$
$569$	−1134.24	−1.99338	−0.996692	+	0.0812732i	$0.974101\pi$
$570$	0	0
$571$	−798.725	−1.39882	−0.699409	−	0.714721i	$-0.746553\pi$
$571$	−798.725	−1.39882	−0.699409	+	0.714721i	$0.746553\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−184.599	−0.316636
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−1816.27	−3.07321
$592$	0	0
$593$	−1145.45	−1.93162	−0.965811	−	0.259247i	$-0.916526\pi$
$593$	−1145.45	−1.93162	−0.965811	+	0.259247i	$0.916526\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1932.51	3.23704
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	−2091.57	−3.46861
$604$	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
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−519.366	−0.841760	−0.420880	−	0.907116i	$-0.638279\pi$
$617$	−519.366	−0.841760	−0.420880	+	0.907116i	$0.638279\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1261.27	−1.99884	−0.999419	−	0.0340975i	$-0.989144\pi$
$631$	−1261.27	−1.99884	−0.999419	+	0.0340975i	$0.989144\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	481.940	0.756577
$638$	0	0
$639$	−1839.55	−2.87879
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	995.344	1.54797	0.773984	−	0.633205i	$-0.218261\pi$
$643$	995.344	1.54797	0.773984	+	0.633205i	$0.218261\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1014.26	1.56763	0.783815	−	0.620994i	$-0.213271\pi$
$647$	1014.26	1.56763	0.783815	+	0.620994i	$0.213271\pi$
$648$	0	0
$649$	205.460	0.316580
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1291.67	1.97806	0.989030	−	0.147714i	$-0.0471915\pi$
$653$	1291.67	1.97806	0.989030	+	0.147714i	$0.0471915\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	597.177	0.908945
$658$	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
$668$	0	0
$669$	0	0
$670$	0	0
$671$	211.211	0.314771
$672$	0	0
$673$	663.824	0.986366	0.493183	−	0.869926i	$-0.335833\pi$
$673$	663.824	0.986366	0.493183	+	0.869926i	$0.335833\pi$
$674$	0	0
$675$	894.269	1.32484
$676$	0	0
$677$	−1343.74	−1.98484	−0.992422	−	0.122878i	$-0.960787\pi$
$677$	−1343.74	−1.98484	−0.992422	+	0.122878i	$0.960787\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2218.41	−3.22913
$688$	0	0
$689$	−1041.99	−1.51232
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	640.765	0.891189	0.445594	−	0.895235i	$-0.352992\pi$
$719$	640.765	0.891189	0.445594	+	0.895235i	$0.352992\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1483.68	−2.05212
$724$	0	0
$725$	1218.52	1.68072
$726$	0	0
$727$	1382.21	1.90125	0.950624	−	0.310346i	$-0.100445\pi$
$727$	1382.21	1.90125	0.950624	+	0.310346i	$0.100445\pi$
$728$	0	0
$729$	−1063.48	−1.45881
$730$	0	0
$731$	0	0
$732$	0	0
$733$	258.279	0.352358	0.176179	−	0.984358i	$-0.443626\pi$
$733$	258.279	0.352358	0.176179	+	0.984358i	$0.443626\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−225.876	−0.306480
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1096.11	−1.47525	−0.737624	−	0.675212i	$-0.764052\pi$
$743$	−1096.11	−1.47525	−0.737624	+	0.675212i	$0.764052\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2528.12	3.38437
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	2185.98	2.90303
$754$	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
$761$	1485.53	1.95208	0.976038	−	0.217600i	$-0.0698228\pi$
$761$	1485.53	1.95208	0.976038	+	0.217600i	$0.0698228\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1159.74	1.51205
$768$	0	0
$769$	−1493.39	−1.94200	−0.970998	−	0.239089i	$-0.923151\pi$
$769$	−1493.39	−1.94200	−0.970998	+	0.239089i	$0.923151\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−852.647	−1.10019
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−198.659	−0.254365
$782$	0	0
$783$	1743.50	2.22669
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1016.48	1.29159	0.645797	−	0.763509i	$-0.276525\pi$
$787$	1016.48	1.29159	0.645797	+	0.763509i	$0.276525\pi$
$788$	0	0
$789$	−1527.85	−1.93644
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1192.20	1.50341
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	2144.28	2.67700
$802$	0	0
$803$	64.4912	0.0803128
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1704.94	2.11269
$808$	0	0
$809$	1577.78	1.95028	0.975141	−	0.221585i	$-0.0711232\pi$
$809$	1577.78	1.95028	0.975141	+	0.221585i	$0.0711232\pi$
$810$	0	0
$811$	218.135	0.268971	0.134485	−	0.990916i	$-0.457062\pi$
$811$	218.135	0.268971	0.134485	+	0.990916i	$0.457062\pi$
$812$	0	0
$813$	−2390.52	−2.94036
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	357.763	0.435765	0.217883	−	0.975975i	$-0.430085\pi$
$821$	357.763	0.435765	0.217883	+	0.975975i	$0.430085\pi$
$822$	0	0
$823$	1367.66	1.66180	0.830898	−	0.556425i	$-0.187827\pi$
$823$	1367.66	1.66180	0.830898	+	0.556425i	$0.187827\pi$
$824$	0	0
$825$	218.395	0.264721
$826$	0	0
$827$	44.1145	0.0533428	0.0266714	−	0.999644i	$-0.491509\pi$
$827$	44.1145	0.0533428	0.0266714	+	0.999644i	$0.491509\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	−1122.37	−1.35062
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1219.99	−1.45758
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1534.67	1.82482
$842$	0	0
$843$	1839.02	2.18152
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1531.25	1.79513	0.897565	−	0.440883i	$-0.145334\pi$
$853$	1531.25	1.79513	0.897565	+	0.440883i	$0.145334\pi$
$854$	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
$863$	470.122	0.544753	0.272376	−	0.962191i	$-0.412190\pi$
$863$	470.122	0.544753	0.272376	+	0.962191i	$0.412190\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1448.89	1.67116
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−1274.98	−1.46381
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1615.92	1.84255	0.921275	−	0.388911i	$-0.127149\pi$
$877$	1615.92	1.84255	0.921275	+	0.388911i	$0.127149\pi$
$878$	0	0
$879$	797.356	0.907117
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	59.4553	0.0667287
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1662.35	−1.84911
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1394.25	1.53721	0.768606	−	0.639722i	$-0.220951\pi$
$907$	1394.25	1.53721	0.768606	+	0.639722i	$0.220951\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	273.021	0.299037
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	999.135	1.08720	0.543599	−	0.839345i	$-0.317061\pi$
$919$	999.135	1.08720	0.543599	+	0.839345i	$0.317061\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1121.35	−1.21490
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1369.77	1.47446	0.737229	−	0.675643i	$-0.236134\pi$
$929$	1369.77	1.47446	0.737229	+	0.675643i	$0.236134\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1834.42	1.95776	0.978879	−	0.204439i	$-0.0655369\pi$
$937$	1834.42	1.95776	0.978879	+	0.204439i	$0.0655369\pi$
$938$	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
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	364.026	0.383590
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1894.79	−1.98824	−0.994118	−	0.108299i	$-0.965460\pi$
$953$	−1894.79	−1.98824	−0.994118	+	0.108299i	$0.965460\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	425.790	0.444922
$958$	0	0
$959$	0	0
$960$	0	0
$961$	202.211	0.210417
$962$	0	0
$963$	−1269.66	−1.31845
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1062.54	−1.09427	−0.547137	−	0.837043i	$-0.684282\pi$
$971$	−1062.54	−1.09427	−0.547137	+	0.837043i	$0.684282\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1232.75	1.26436
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	231.568	0.236535
$980$	0	0
$981$	3196.30	3.25820
$982$	0	0
$983$	−1355.10	−1.37854	−0.689268	−	0.724506i	$-0.742068\pi$
$983$	−1355.10	−1.37854	−0.689268	+	0.724506i	$0.742068\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1974.01	1.99193	0.995967	−	0.0897173i	$-0.0285964\pi$
$991$	1974.01	1.99193	0.995967	+	0.0897173i	$0.0285964\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1388.3.d.a.693.9	✓	10
347.346	odd	2	CM	1388.3.d.a.693.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1388.3.d.a.693.9	✓	10	1.1	even	1	trivial
1388.3.d.a.693.9	✓	10	347.346	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:	\( N \)	\(=\)	\( 1388 = 2^{2} \cdot 347 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1388.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.01347 q^{3} +16.1349 q^{9} +O(q^{10})\)	\(q+5.01347 q^{3} +16.1349 q^{9} +1.74246 q^{11} +9.83551 q^{13} +25.0000 q^{25} +35.7707 q^{27} +48.7409 q^{29} -34.1059 q^{31} +8.73580 q^{33} +49.3101 q^{39} -9.73062 q^{43} +49.0000 q^{49} -105.941 q^{53} +117.914 q^{59} +121.214 q^{61} -129.630 q^{67} -114.010 q^{71} +37.0115 q^{73} +125.337 q^{75} +34.1214 q^{81} +156.686 q^{83} +244.361 q^{87} +132.897 q^{89} -170.989 q^{93} +28.1145 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 90 q^{9}+O(q^{10}) \) 10 * q + 90 * q^9	\( 10 q + 90 q^{9} + 250 q^{25} + 490 q^{49} + 810 q^{81} + 865 q^{87} + 845 q^{89} + 805 q^{93} + 745 q^{99}+O(q^{100}) \) 10 * q + 90 * q^9 + 250 * q^25 + 490 * q^49 + 810 * q^81 + 865 * q^87 + 845 * q^89 + 805 * q^93 + 745 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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