LMFDB - Newform orbit 3248.1.ei.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3248,1,Mod(13,3248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3248.13"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3248, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([0, 21, 14, 18])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$N$	$=$	$3248 = 2^{4} \cdot 7 \cdot 29$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3248.ei (of order $28$ , degree $12$ , minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,2,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.62096316103$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{28})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{28}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{28} - \cdots)$

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

The $q$ -expansion and trace form are shown below.

$f(q)$	$=$	$q + \zeta_{28}^{3} q^{2} + \zeta_{28}^{6} q^{4} - \zeta_{28}^{13} q^{7} + \zeta_{28}^{9} q^{8} + \zeta_{28}^{5} q^{9} + (\zeta_{28}^{11} + 1) q^{11} + \zeta_{28}^{2} q^{14} + \zeta_{28}^{12} q^{16} + \zeta_{28}^{8} q^{18} + \cdots + (\zeta_{28}^{5} - \zeta_{28}^{2}) q^{99} +O(q^{100})$ q + z^3 * q^2 + z^6 * q^4 - z^13 * q^7 + z^9 * q^8 + z^5 * q^9 + (z^11 + 1) * q^11 + z^2 * q^14 + z^12 * q^16 + z^8 * q^18 + (z^3 - 1) * q^22 + (-z^4 - z^2) * q^23 + z * q^25 + z^5 * q^28 - z * q^29 - z * q^32 + z^11 * q^36 + (-z^9 + z^8) * q^37 + (-z^10 + z^3) * q^43 + (z^6 - z^3) * q^44 + (-z^7 - z^5) * q^46 - z^12 * q^49 + z^4 * q^50 + (z^9 - z^6) * q^53 + z^8 * q^56 - z^4 * q^58 + z^4 * q^63 - z^4 * q^64 + (z^13 - z^4) * q^67 + (-z^13 - z^5) * q^71 - q^72 + (-z^12 + z^11) * q^74 + (-z^13 + z^10) * q^77 + (z^7 + z^3) * q^79 + z^10 * q^81 + (-z^13 + z^6) * q^86 + (z^9 - z^6) * q^88 + (-z^10 - z^8) * q^92 + z * q^98 + (z^5 - z^2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 2 q^{4} + 12 q^{11} + 2 q^{14} - 2 q^{16} - 2 q^{18} - 12 q^{22} - 2 q^{37} - 2 q^{43} + 2 q^{44} + 2 q^{49} - 2 q^{50} - 2 q^{53} - 2 q^{56} + 2 q^{58} - 2 q^{63} + 2 q^{64} + 2 q^{67} - 12 q^{72}+ \cdots - 2 q^{99}+O(q^{100})$ 12 * q + 2 * q^4 + 12 * q^11 + 2 * q^14 - 2 * q^16 - 2 * q^18 - 12 * q^22 - 2 * q^37 - 2 * q^43 + 2 * q^44 + 2 * q^49 - 2 * q^50 - 2 * q^53 - 2 * q^56 + 2 * q^58 - 2 * q^63 + 2 * q^64 + 2 * q^67 - 12 * q^72 + 2 * q^74 + 2 * q^77 + 2 * q^81 + 2 * q^86 - 2 * q^88 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3248\mathbb{Z}\right)^\times$ .

$n$	$465$	$785$	$2031$	$2437$
$\chi(n)$	$-1$	$-\zeta_{28}^{8}$	$1$	$-\zeta_{28}^{7}$

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}

13.1

0.781831	−	0.623490i
−0.781831	−	0.623490i
0.974928	+	0.222521i
0.433884	+	0.900969i
−0.974928	+	0.222521i
0.433884	−	0.900969i
−0.781831	+	0.623490i
0.781831	+	0.623490i
−0.974928	−	0.222521i
−0.433884	−	0.900969i
0.974928	−	0.222521i
−0.433884	+	0.900969i

−0.433884

−

0.900969i

−0.623490

0.781831i

0.781831

0.623490i

0.974928

0.222521i

−0.974928

0.222521i

125.1

0.433884

−

0.900969i

−0.623490

−

0.781831i

−0.781831

0.623490i

−0.974928

0.222521i

0.974928

0.222521i

237.1

0.781831

0.623490i

0.222521

0.974928i

0.974928

−

0.222521i

−0.433884

0.900969i

0.433884

0.900969i

573.1

−0.974928

−

0.222521i

0.900969

0.433884i

0.433884

−

0.900969i

−0.781831

−

0.623490i

0.781831

−

0.623490i

1021.1

−0.781831

0.623490i

0.222521

−

0.974928i

−0.974928

−

0.222521i

0.433884

0.900969i

−0.433884

0.900969i

1077.1

−0.974928

0.222521i

0.900969

−

0.433884i

0.433884

0.900969i

−0.781831

0.623490i

0.781831

0.623490i

1637.1

0.433884

0.900969i

−0.623490

0.781831i

−0.781831

−

0.623490i

−0.974928

−

0.222521i

0.974928

−

0.222521i

1749.1

−0.433884

0.900969i

−0.623490

−

0.781831i

0.781831

−

0.623490i

0.974928

−

0.222521i

−0.974928

−

0.222521i

1861.1

−0.781831

−

0.623490i

0.222521

0.974928i

−0.974928

0.222521i

0.433884

−

0.900969i

−0.433884

−

0.900969i

2197.1

0.974928

0.222521i

0.900969

0.433884i

−0.433884

0.900969i

0.781831

0.623490i

−0.781831

0.623490i

2645.1

0.781831

−

0.623490i

0.222521

−

0.974928i

0.974928

0.222521i

−0.433884

−

0.900969i

0.433884

−

0.900969i

2701.1

0.974928

−

0.222521i

0.900969

−

0.433884i

−0.433884

−

0.900969i

0.781831

−

0.623490i

−0.781831

−

0.623490i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7})$
464.bj	even	28	1	inner
3248.ei	odd	28	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3248.1.ei.b	yes	12
7.b	odd	2	1	CM	3248.1.ei.b	yes	12
16.e	even	4	1		3248.1.ei.a	✓	12
29.e	even	14	1		3248.1.ei.a	✓	12
112.l	odd	4	1		3248.1.ei.a	✓	12
203.o	odd	14	1		3248.1.ei.a	✓	12
464.bj	even	28	1	inner	3248.1.ei.b	yes	12
3248.ei	odd	28	1	inner	3248.1.ei.b	yes	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3248.1.ei.a	✓	12	16.e	even	4	1
3248.1.ei.a	✓	12	29.e	even	14	1
3248.1.ei.a	✓	12	112.l	odd	4	1
3248.1.ei.a	✓	12	203.o	odd	14	1
3248.1.ei.b	yes	12	1.a	even	1	1	trivial
3248.1.ei.b	yes	12	7.b	odd	2	1	CM
3248.1.ei.b	yes	12	464.bj	even	28	1	inner
3248.1.ei.b	yes	12	3248.ei	odd	28	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{11}^{12} - 12 T_{11}^{11} + 65 T_{11}^{10} - 210 T_{11}^{9} + 451 T_{11}^{8} - 680 T_{11}^{7} + \cdots + 1$ T11^12 - 12*T11^11 + 65*T11^10 - 210*T11^9 + 451*T11^8 - 680*T11^7 + 741*T11^6 - 590*T11^5 + 341*T11^4 - 140*T11^3 + 39*T11^2 - 6*T11 + 1 acting on $S_{1}^{\mathrm{new}}(3248, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12} - T^{10} + \cdots + 1$ T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$3$	$T^{12}$ T^12
$5$	$T^{12}$ T^12
$7$	$T^{12} - T^{10} + \cdots + 1$ T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$11$	$T^{12} - 12 T^{11} + \cdots + 1$ T^12 - 12T^11 + 65T^10 - 210T^9 + 451T^8 - 680T^7 + 741T^6 - 590T^5 + 341T^4 - 140T^3 + 39T^2 - 6*T + 1
$13$	$T^{12}$ T^12
$17$	$T^{12}$ T^12
$19$	$T^{12}$ T^12
$23$	$(T^{6} + 7 T^{2} - 14 T + 7)^{2}$ (T^6 + 7T^2 - 14T + 7)^2
$29$	$T^{12} - T^{10} + \cdots + 1$ T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$31$	$T^{12}$ T^12
$37$	$T^{12} + 2 T^{11} + \cdots + 1$ T^12 + 2T^11 + 2T^10 - 4T^8 + 20T^7 + 48T^6 + 54T^5 + 61T^4 + 14T^3 + 4T^2 - 6T + 1
$41$	$T^{12}$ T^12
$43$	$T^{12} + 2 T^{11} + \cdots + 64$ T^12 + 2T^11 + 2T^10 - 4T^8 - 8T^7 - 8T^6 - 16T^5 - 16T^4 + 32T^2 + 64*T + 64
$47$	$T^{12}$ T^12
$53$	$T^{12} + 2 T^{11} + \cdots + 1$ T^12 + 2T^11 + 2T^10 - 4T^8 - 22T^7 - T^6 + 40T^5 + 82T^4 + 84T^3 + 39T^2 + 8*T + 1
$59$	$T^{12}$ T^12
$61$	$T^{12}$ T^12
$67$	$T^{12} - 2 T^{11} + \cdots + 1$ T^12 - 2T^11 + 2T^10 - 4T^8 + 22T^7 - T^6 - 40T^5 + 82T^4 - 84T^3 + 39T^2 - 8*T + 1
$71$	$T^{12} - 4 T^{10} + \cdots + 1$ T^12 - 4T^10 + 2T^8 + 6T^6 + 25T^4 - 2*T^2 + 1
$73$	$T^{12}$ T^12
$79$	$T^{12} + 3 T^{10} + \cdots + 1$ T^12 + 3T^10 + 9T^8 - T^6 + 25T^4 - 9T^2 + 1
$83$	$T^{12}$ T^12
$89$	$T^{12}$ T^12
$97$	$T^{12}$ T^12
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 13.1
Significant digits:
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Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	3248.1.ei.b

Level	$3248$
Weight	$1$
Character orbit	3248.ei
Analytic conductor	$1.621$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{28}$
CM discriminant	-7
Inner twists	$4$