LMFDB - Embedded newform 3375.1.d.a.3374.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3375,1,Mod(3374,3375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3375, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3375.3374");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3375 = 3^{3} \cdot 5^{3}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3375.d (of order $2$ , degree $1$ , not 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:	$1.68434441764$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 4x^{2} + 4x + 1$ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{15}$
Projective field:	Galois closure of 15.1.2463153133392333984375.1

Embedding invariants

Embedding label			3374.3
Root			$-0.209057$ of defining polynomial
Character	$\chi$	$=$	3375.3374

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.209057 q^{2} -0.956295 q^{4} -0.408977 q^{8} +0.870796 q^{16} -1.33826 q^{17} -0.209057 q^{19} +1.95630 q^{23} +1.33826 q^{31} +0.591023 q^{32} -0.279773 q^{34} -0.0437048 q^{38} +0.408977 q^{46} +1.61803 q^{47} +1.00000 q^{49} -1.82709 q^{53} +1.82709 q^{61} +0.279773 q^{62} -0.747238 q^{64} +1.27977 q^{68} +0.199920 q^{76} +1.82709 q^{79} -1.82709 q^{83} -1.87080 q^{92} +0.338261 q^{94} +0.209057 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - q^{2} + 5 q^{4} + q^{8} + 6 q^{16} - q^{17} + q^{19} - q^{23} + q^{31} + 5 q^{32} - q^{34} - 9 q^{38} - q^{46} + 2 q^{47} + 4 q^{49} - q^{53} + q^{61} + q^{62} + 4 q^{64} + 5 q^{68} + q^{79}+ \cdots - q^{98}+O(q^{100})$ 4 * q - q^2 + 5 * q^4 + q^8 + 6 * q^16 - q^17 + q^19 - q^23 + q^31 + 5 * q^32 - q^34 - 9 * q^38 - q^46 + 2 * q^47 + 4 * q^49 - q^53 + q^61 + q^62 + 4 * q^64 + 5 * q^68 + q^79 - q^83 - 10 * q^92 - 3 * q^94 - q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3375\mathbb{Z}\right)^\times$ .

$n$	$1001$	$2377$
$\chi(n)$	$-1$	$-1$

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

)

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

$2$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$2$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$3$	0	0
$3$	0	0
$4$	−0.956295	−0.956295
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−0.408977	−0.408977
$9$	0	0
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0.870796	0.870796
$17$	−1.33826	−1.33826	−0.669131	−	0.743145i	$-0.733333\pi$
$17$	−1.33826	−1.33826	−0.669131	+	0.743145i	$0.733333\pi$
$18$	0	0
$19$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$19$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.95630	1.95630	0.978148	−	0.207912i	$-0.0666667\pi$
$23$	1.95630	1.95630	0.978148	+	0.207912i	$0.0666667\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$31$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$32$	0.591023	0.591023
$33$	0	0
$34$	−0.279773	−0.279773
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−0.0437048	−0.0437048
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0.408977	0.408977
$47$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$47$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$48$	0	0
$49$	1.00000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.82709	−1.82709	−0.913545	−	0.406737i	$-0.866667\pi$
$53$	−1.82709	−1.82709	−0.913545	+	0.406737i	$0.866667\pi$
$54$	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
$61$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$61$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$62$	0.279773	0.279773
$63$	0	0
$64$	−0.747238	−0.747238
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	1.27977	1.27977
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0.199920	0.199920
$77$	0	0
$78$	0	0
$79$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$79$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.82709	−1.82709	−0.913545	−	0.406737i	$-0.866667\pi$
$83$	−1.82709	−1.82709	−0.913545	+	0.406737i	$0.866667\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	−1.87080	−1.87080
$93$	0	0
$94$	0.338261	0.338261
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.209057	0.209057
$99$	0	0
$100$	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
$106$	−0.381966	−0.381966
$107$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$107$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$108$	0	0
$109$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$109$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$113$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	0.381966	0.381966
$123$	0	0
$124$	−1.27977	−1.27977
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−0.747238	−0.747238
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0.547318	0.547318
$137$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$137$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$138$	0	0
$139$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$139$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$151$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$152$	0.0854995	0.0854995
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0.381966	0.381966
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	−0.381966	−0.381966
$167$	−1.82709	−1.82709	−0.913545	−	0.406737i	$-0.866667\pi$
$167$	−1.82709	−1.82709	−0.913545	+	0.406737i	$0.866667\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$173$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$181$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$182$	0	0
$183$	0	0
$184$	−0.800080	−0.800080
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−1.54732	−1.54732
$189$	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
$196$	−0.956295	−0.956295
$197$	1.95630	1.95630	0.978148	−	0.207912i	$-0.0666667\pi$
$197$	1.95630	1.95630	0.978148	+	0.207912i	$0.0666667\pi$
$198$	0	0
$199$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$199$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$200$	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
$209$	0	0
$210$	0	0
$211$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$211$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$212$	1.74724	1.74724
$213$	0	0
$214$	−0.129204	−0.129204
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0.279773	0.279773
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0.338261	0.338261
$227$	−1.33826	−1.33826	−0.669131	−	0.743145i	$-0.733333\pi$
$227$	−1.33826	−1.33826	−0.669131	+	0.743145i	$0.733333\pi$
$228$	0	0
$229$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$229$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.00000	−2.00000	−1.00000			$\pi$
$233$	−2.00000	−2.00000	−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
$241$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$241$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$242$	0.209057	0.209057
$243$	0	0
$244$	−1.74724	−1.74724
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−0.547318	−0.547318
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0.591023	0.591023
$257$	−1.82709	−1.82709	−0.913545	−	0.406737i	$-0.866667\pi$
$257$	−1.82709	−1.82709	−0.913545	+	0.406737i	$0.866667\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$263$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$271$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$272$	−1.16535	−1.16535
$273$	0	0
$274$	0.209057	0.209057
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0.129204	0.129204
$279$	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			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	0.790943	0.790943
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.95630	1.95630	0.978148	−	0.207912i	$-0.0666667\pi$
$293$	1.95630	1.95630	0.978148	+	0.207912i	$0.0666667\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−0.338261	−0.338261
$303$	0	0
$304$	−0.182046	−0.182046
$305$	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
$316$	−1.74724	−1.74724
$317$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$317$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.279773	0.279773
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$331$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$332$	1.74724	1.74724
$333$	0	0
$334$	−0.381966	−0.381966
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0.209057	0.209057
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0.209057	0.209057
$347$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$347$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$348$	0	0
$349$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$349$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$353$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−0.956295	−0.956295
$362$	−0.0437048	−0.0437048
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	1.70353	1.70353
$369$	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
$376$	−0.661739	−0.661739
$377$	0	0
$378$	0	0
$379$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.33826	−1.33826	−0.669131	−	0.743145i	$-0.733333\pi$
$383$	−1.33826	−1.33826	−0.669131	+	0.743145i	$0.733333\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	−2.61803	−2.61803
$392$	−0.408977	−0.408977
$393$	0	0
$394$	0.408977	0.408977
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0.129204	0.129204
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$409$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$410$	0	0
$411$	0	0
$412$	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
$421$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$421$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$422$	−0.408977	−0.408977
$423$	0	0
$424$	0.747238	0.747238
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0.591023	0.591023
$429$	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
$436$	−1.27977	−1.27977
$437$	−0.408977	−0.408977
$438$	0	0
$439$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$439$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$443$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	−1.54732	−1.54732
$453$	0	0
$454$	−0.279773	−0.279773
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−0.408977	−0.408977
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	−0.418114	−0.418114
$467$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$467$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0.279773	0.279773
$483$	0	0
$484$	−0.956295	−0.956295
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	−0.747238	−0.747238
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	1.16535	1.16535
$497$	0	0
$498$	0	0
$499$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$499$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$503$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0.870796	0.870796
$513$	0	0
$514$	−0.381966	−0.381966
$515$	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
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	−0.129204	−0.129204
$527$	−1.79094	−1.79094
$528$	0	0
$529$	2.82709	2.82709
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$541$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$542$	−0.408977	−0.408977
$543$	0	0
$544$	−0.790943	−0.790943
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−0.956295	−0.956295
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−0.591023	−0.591023
$557$	−2.00000	−2.00000	−1.00000			$\pi$
$557$	−2.00000	−2.00000	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$563$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$572$	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			$\pi$
$578$	0.165352	0.165352
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0.408977	0.408977
$587$	1.95630	1.95630	0.978148	−	0.207912i	$-0.0666667\pi$
$587$	1.95630	1.95630	0.978148	+	0.207912i	$0.0666667\pi$
$588$	0	0
$589$	−0.279773	−0.279773
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$593$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$601$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$602$	0	0
$603$	0	0
$604$	1.54732	1.54732
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−0.123557	−0.123557
$609$	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
$617$	1.95630	1.95630	0.978148	−	0.207912i	$-0.0666667\pi$
$617$	1.95630	1.95630	0.978148	+	0.207912i	$0.0666667\pi$
$618$	0	0
$619$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$619$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$632$	−0.747238	−0.747238
$633$	0	0
$634$	0.209057	0.209057
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	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
$646$	0.0584884	0.0584884
$647$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$647$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$653$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$661$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$662$	−0.338261	−0.338261
$663$	0	0
$664$	0.747238	0.747238
$665$	0	0
$666$	0	0
$667$	0	0
$668$	1.74724	1.74724
$669$	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
$676$	−0.956295	−0.956295
$677$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$677$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.95630	1.95630	0.978148	−	0.207912i	$-0.0666667\pi$
$683$	1.95630	1.95630	0.978148	+	0.207912i	$0.0666667\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$691$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$692$	−0.956295	−0.956295
$693$	0	0
$694$	0.338261	0.338261
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−0.408977	−0.408977
$699$	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
$706$	−0.129204	−0.129204
$707$	0	0
$708$	0	0
$709$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$709$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.61803	2.61803
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	−0.199920	−0.199920
$723$	0	0
$724$	0.199920	0.199920
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	1.15622	1.15622
$737$	0	0
$738$	0	0
$739$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$739$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.00000	−2.00000	−1.00000			$\pi$
$743$	−2.00000	−2.00000	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$751$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$752$	1.40898	1.40898
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−0.209057	−0.209057
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−0.279773	−0.279773
$767$	0	0
$768$	0	0
$769$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$769$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$773$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−0.547318	−0.547318
$783$	0	0
$784$	0.870796	0.870796
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−1.87080	−1.87080
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−0.591023	−0.591023
$797$	−1.33826	−1.33826	−0.669131	−	0.743145i	$-0.733333\pi$
$797$	−1.33826	−1.33826	−0.669131	+	0.743145i	$0.733333\pi$
$798$	0	0
$799$	−2.16535	−2.16535
$800$	0	0
$801$	0	0
$802$	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
$811$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$811$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0.381966	0.381966
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.82709	−1.82709	−0.913545	−	0.406737i	$-0.866667\pi$
$827$	−1.82709	−1.82709	−0.913545	+	0.406737i	$0.866667\pi$
$828$	0	0
$829$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$829$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.33826	−1.33826
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	−0.0437048	−0.0437048
$843$	0	0
$844$	1.87080	1.87080
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−1.59102	−1.59102
$849$	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
$856$	0.252762	0.252762
$857$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$857$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$858$	0	0
$859$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$859$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$863$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−0.547318	−0.547318
$873$	0	0
$874$	−0.0854995	−0.0854995
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−0.209057	−0.209057
$879$	0	0
$880$	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
$886$	0.0437048	0.0437048
$887$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$887$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.338261	−0.338261
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	2.44512	2.44512
$902$	0	0
$903$	0	0
$904$	−0.661739	−0.661739
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	1.27977	1.27977
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	1.87080	1.87080
$917$	0	0
$918$	0	0
$919$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$919$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−0.209057	−0.209057
$932$	1.91259	1.91259
$933$	0	0
$934$	0.0437048	0.0437048
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	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
$947$	1.95630	1.95630	0.978148	−	0.207912i	$-0.0666667\pi$
$947$	1.95630	1.95630	0.978148	+	0.207912i	$0.0666667\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$953$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	0.790943	0.790943
$962$	0	0
$963$	0	0
$964$	−1.27977	−1.27977
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−0.408977	−0.408977
$969$	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
$976$	1.59102	1.59102
$977$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$977$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$983$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$991$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$992$	0.790943	0.790943
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0.381966	0.381966
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3375.1.d.a.3374.3		4
3.2	odd	2		3375.1.d.b.3374.2		4
5.2	odd	4		3375.1.c.a.1376.5	yes	8
5.3	odd	4		3375.1.c.a.1376.4	✓	8
5.4	even	2		3375.1.d.b.3374.2		4
15.2	even	4		3375.1.c.a.1376.4	✓	8
15.8	even	4		3375.1.c.a.1376.5	yes	8
15.14	odd	2	CM	3375.1.d.a.3374.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3375.1.c.a.1376.4	✓	8	5.3	odd	4
3375.1.c.a.1376.4	✓	8	15.2	even	4
3375.1.c.a.1376.5	yes	8	5.2	odd	4
3375.1.c.a.1376.5	yes	8	15.8	even	4
3375.1.d.a.3374.3		4	1.1	even	1	trivial
3375.1.d.a.3374.3		4	15.14	odd	2	CM
3375.1.d.b.3374.2		4	3.2	odd	2
3375.1.d.b.3374.2		4	5.4	even	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3375.1.d.a.3374.3

Level	$3375$
Weight	$1$
Character	3375.3374
Self dual	yes
Analytic conductor	$1.684$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{15}$
CM discriminant	-15
Inner twists	$2$