LMFDB - Newform orbit 304.2.u.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,2,Mod(17,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 0, 10]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("304.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$304 = 2^{4} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	304.u (of order $9$ , degree $6$ , 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:	no
Analytic conductor:	$2.42745222145$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{3} + 1$ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a primitive root of unity $\zeta_{18}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\zeta_{18}^{5} - \zeta_{18}^{3} + \cdots - 1) q^{3} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \cdots - 1) q^{5} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \cdots - 2) q^{7} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + \cdots - 2) q^{9}+ \cdots + ( - 8 \zeta_{18}^{5} + \cdots - 5 \zeta_{18}) q^{99}+O(q^{100})$ q + (z^5 - z^3 - 2z^2 + z - 1) q^3 + (-z^5 - z^4 + z^2 - 1) * q^5 + (-z^5 - z^4 + 2z^3 + z^2 - 2) q^7 + (-2z^4 + z^3 + 4z^2 + z - 2) * q^9 + (z^4 - 2z^3 + z^2) q^11 + (3z^4 - z^3 - 2z - 2) * q^13 + (-z^5 + 3z^4 + 3z^3 - 4z + 1) q^15 + (3z^5 + 2z^4 + 3z^3 - 3z^2 - 2z - 3) q^17 + (z^5 - 2z^3 + z^2 - 2z - 2) * q^19 + (-4z^5 + 5z^4 - 5z + 4) q^21 + (2z^5 + 2z^4 + 2z^3 + 2z - 4) * q^23 + (-2z^5 + z^4 - 2z^3 + 2z^2 + z + 1) q^25 + (-6z^5 + 3z^4 + 4z^3 + 3z^2 - 6z) q^27 + (-z^4 + z^3 - 5z^2 + z - 1) q^29 + (-3z^5 - 3z^4 - z^3 + 3z^2 + 1) q^31 + (2z^5 - 5z^4 + 3z^3 + z^2 - 1) q^33 + (4z^5 - 4z^3 - 2z^2 + 3z + 2) * q^35 + (-z^5 - z^4 + 2z^2 + 2z - 4) * q^37 + (2z^5 - 5z^4 + 3z^2 + 3z + 6) * q^39 + (5z^5 - 5z^3 - 5z^2) q^41 + (3z^5 - 2z^4 - z^3 - 4z^2 + 4) q^43 + (3z^5 + 3z^4 - 7z^3 - 7z^2 + 4z + 7) q^45 + (-4z^4 + z^3 + z - 4) q^47 + (5z^5 - z^4 + z^3 - z^2 + 5z) * q^49 + (-7z^5 - 2z^4 - 5z^3 + 7z^2 + 7z + 7) q^51 + (-6z^5 - 3z^4 - 3z^3 + 6z - 3) * q^53 + (-z^5 + z^4 + 2z^3 - 2z^2 - z + 1) * q^55 + (-3z^5 - 3z^4 + 10z^3 + 4z^2 + 1) * q^57 + (2z^5 + 3z^4 + 6z^3 - 6z^2 - 3z - 2) q^59 + (5z^5 - 4z^4 - 4z^3 + 4) q^61 + (11z^5 + 3z^4 - 10z^3 - 11z^2 + 7z + 7) q^63 + (z^5 + 4z^3 + z) q^65 + (-8z^4 + 10z^3 + 2z^2 + 10z - 8) * q^67 + (-8z^5 - 8z^4 - 4z^3 + 10z^2 - 2z + 4) q^69 + (-8z^5 + 2z^4 - 10z^3 - 2z^2 + 2) * q^71 + (-4z^5 + 4z^3 + 8z^2 + 4) q^73 + (6z^5 - 3z^4 - 3z^2 - 3z - 1) * q^75 + (z^5 + 2z^4 - 3z^2 - 3z + 5) q^77 + (z^5 - z^3 - 5z^2 - 7z - 4) * q^79 + (5z^5 + z^4 - 8z^3 - 13z^2 + 13) q^81 + (4z^3 - z^2 + z - 4) q^83 + (z^4 + 2z^3 + 5z^2 + 2z + 1) q^85 + (2z^5 + 7z^4 - 5z^3 + 7z^2 + 2z) q^87 + (-5z^5 + 3z^4 - 2z^3 + 5z^2 - z - 1) * q^89 + (z^5 - z^4 - z^3 - 3z + 4) q^91 + (2z^5 + 8z^4 + 7z^3 - 7z^2 - 8z - 2) q^93 + (3z^5 + 5z^4 + z^3 - 5z^2 + z + 2) q^95 + (2z^5 + 7z^4 - 2z^3 + 2z^2 - 7z - 2) q^97 + (-8z^5 + 5z^4 + 5z^3 - 5z) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 9 q^{3} - 6 q^{5} - 6 q^{7} - 9 q^{9} - 6 q^{11} - 15 q^{13} + 15 q^{15} - 9 q^{17} - 18 q^{19} + 24 q^{21} - 18 q^{23} + 12 q^{27} - 3 q^{29} + 3 q^{31} + 3 q^{33} - 24 q^{37} + 36 q^{39} - 15 q^{41}+ \cdots + 15 q^{99}+O(q^{100})$ 6 * q - 9 * q^3 - 6 * q^5 - 6 * q^7 - 9 * q^9 - 6 * q^11 - 15 * q^13 + 15 * q^15 - 9 * q^17 - 18 * q^19 + 24 * q^21 - 18 * q^23 + 12 * q^27 - 3 * q^29 + 3 * q^31 + 3 * q^33 - 24 * q^37 + 36 * q^39 - 15 * q^41 + 21 * q^43 + 21 * q^45 - 21 * q^47 + 3 * q^49 + 27 * q^51 - 27 * q^53 + 12 * q^55 + 36 * q^57 + 6 * q^59 + 12 * q^61 + 12 * q^63 + 12 * q^65 - 18 * q^67 + 12 * q^69 - 18 * q^71 + 36 * q^73 - 6 * q^75 + 30 * q^77 - 27 * q^79 + 54 * q^81 - 12 * q^83 + 12 * q^85 - 15 * q^87 - 12 * q^89 + 21 * q^91 + 9 * q^93 + 15 * q^95 - 18 * q^97 + 15 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$ .

$n$	$97$	$191$	$229$
$\chi(n)$	$-\zeta_{18}^{5}$	$1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

17.1

−0.766044	−	0.642788i
−0.173648	+	0.984808i
−0.766044	+	0.642788i
0.939693	+	0.342020i
0.939693	−	0.342020i
−0.173648	−	0.984808i

−1.67365

−

1.40436i

−0.826352

0.300767i

−0.826352

−

1.43128i

0.307934

1.74638i

81.1

−0.560307

3.17766i

−1.93969

−

1.62760i

−1.93969

−

3.35965i

−6.96451

−

2.53487i

161.1

−1.67365

1.40436i

−0.826352

−

0.300767i

−0.826352

1.43128i

0.307934

−

1.74638i

177.1

−2.26604

−

0.824773i

−0.233956

−

1.32683i

−0.233956

0.405223i

2.15657

1.80958i

225.1

−2.26604

0.824773i

−0.233956

1.32683i

−0.233956

−

0.405223i

2.15657

−

1.80958i

289.1

−0.560307

−

3.17766i

−1.93969

1.62760i

−1.93969

3.35965i

−6.96451

2.53487i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	304.2.u.a		6
4.b	odd	2	1		152.2.q.b	✓	6
19.e	even	9	1	inner	304.2.u.a		6
19.e	even	9	1		5776.2.a.bj		3
19.f	odd	18	1		5776.2.a.bs		3
76.k	even	18	1		2888.2.a.m		3
76.l	odd	18	1		152.2.q.b	✓	6
76.l	odd	18	1		2888.2.a.s		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
152.2.q.b	✓	6	4.b	odd	2	1
152.2.q.b	✓	6	76.l	odd	18	1
304.2.u.a		6	1.a	even	1	1	trivial
304.2.u.a		6	19.e	even	9	1	inner
2888.2.a.m		3	76.k	even	18	1
2888.2.a.s		3	76.l	odd	18	1
5776.2.a.bj		3	19.e	even	9	1
5776.2.a.bs		3	19.f	odd	18	1

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{6} + 9T_{3}^{5} + 45T_{3}^{4} + 152T_{3}^{3} + 342T_{3}^{2} + 459T_{3} + 289$ T3^6 + 9*T3^5 + 45*T3^4 + 152*T3^3 + 342*T3^2 + 459*T3 + 289 acting on $S_{2}^{\mathrm{new}}(304, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6} + 9 T^{5} + \cdots + 289$ T^6 + 9T^5 + 45T^4 + 152T^3 + 342T^2 + 459*T + 289
$5$	$T^{6} + 6 T^{5} + \cdots + 9$ T^6 + 6T^5 + 18T^4 + 30T^3 + 36T^2 + 27*T + 9
$7$	$T^{6} + 6 T^{5} + \cdots + 9$ T^6 + 6T^5 + 27T^4 + 48T^3 + 63T^2 + 27*T + 9
$11$	$T^{6} + 6 T^{5} + \cdots + 1$ T^6 + 6T^5 + 27T^4 + 52T^3 + 75T^2 + 9*T + 1
$13$	$T^{6} + 15 T^{5} + \cdots + 289$ T^6 + 15T^5 + 96T^4 + 298T^3 + 390T^2 + 102*T + 289
$17$	$T^{6} + 9 T^{5} + \cdots + 361$ T^6 + 9T^5 + 36T^4 - 62T^3 + 396T^2 - 684*T + 361
$19$	$T^{6} + 18 T^{5} + \cdots + 6859$ T^6 + 18T^5 + 144T^4 + 737T^3 + 2736T^2 + 6498*T + 6859
$23$	$T^{6} + 18 T^{5} + \cdots + 18496$ T^6 + 18T^5 + 180T^4 + 1216T^3 + 5472T^2 + 14688*T + 18496
$29$	$T^{6} + 3 T^{5} + \cdots + 12321$ T^6 + 3T^5 + 36T^4 - 57T^3 - 477T^2 - 1998*T + 12321
$31$	$T^{6} - 3 T^{5} + \cdots + 2809$ T^6 - 3T^5 + 33T^4 - 34T^3 + 735T^2 - 1272*T + 2809
$37$	$(T^{3} + 12 T^{2} + \cdots + 19)^{2}$ (T^3 + 12T^2 + 39T + 19)^2
$41$	$T^{6} + 15 T^{5} + \cdots + 15625$ T^6 + 15T^5 + 150T^4 + 1000T^3 + 1875T^2 - 9375*T + 15625
$43$	$T^{6} - 21 T^{5} + \cdots + 7921$ T^6 - 21T^5 + 174T^4 - 712T^3 + 1797T^2 - 3471*T + 7921
$47$	$T^{6} + 21 T^{5} + \cdots + 7921$ T^6 + 21T^5 + 186T^4 + 908T^3 + 2856T^2 + 5874*T + 7921
$53$	$T^{6} + 27 T^{5} + \cdots + 263169$ T^6 + 27T^5 + 324T^4 + 3132T^3 + 28431T^2 + 124659*T + 263169
$59$	$T^{6} - 6 T^{5} + \cdots + 271441$ T^6 - 6T^5 + 6T^4 - 199T^3 + 1686T^2 - 21882*T + 271441
$61$	$T^{6} - 12 T^{5} + \cdots + 32041$ T^6 - 12T^5 - 24T^4 + 323T^3 + 4488T^2 - 2148*T + 32041
$67$	$T^{6} + 18 T^{5} + \cdots + 3474496$ T^6 + 18T^5 + 252T^4 + 2240T^3 - 9504T^2 - 67104*T + 3474496
$71$	$T^{6} + 18 T^{5} + \cdots + 3474496$ T^6 + 18T^5 + 252T^4 + 2240T^3 - 9504T^2 - 67104*T + 3474496
$73$	$T^{6} - 36 T^{5} + \cdots + 331776$ T^6 - 36T^5 + 576T^4 - 5760T^3 + 41472T^2 - 165888*T + 331776
$79$	$T^{6} + 27 T^{5} + \cdots + 5329$ T^6 + 27T^5 + 180T^4 - 305T^3 + 3771T^2 - 6570*T + 5329
$83$	$T^{6} + 12 T^{5} + \cdots + 2809$ T^6 + 12T^5 + 99T^4 + 434T^3 + 1389T^2 + 2385*T + 2809
$89$	$T^{6} + 12 T^{5} + \cdots + 3249$ T^6 + 12T^5 + 54T^4 - 300T^3 + 522T^2 - 1539*T + 3249
$97$	$T^{6} + 18 T^{5} + \cdots + 361$ T^6 + 18T^5 + 270T^4 + 2141T^3 + 10782T^2 - 3762*T + 361
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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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 17.1
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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	304.2.u.a

Level	$304$
Weight	$2$
Character orbit	304.u
Analytic conductor	$2.427$
Analytic rank	$1$
Dimension	$6$
Inner twists	$2$