LMFDB - Newform orbit 2312.1.e.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,1,Mod(1155,2312)] 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("2312.1155"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2312.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$1.15383830921$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2 + \sqrt{2}})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 136)
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.0.1680747204608.3

$q$-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 basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + q^{8} + (\beta_{2} - 1) q^{9} + \beta_{3} q^{11} - \beta_1 q^{12} + q^{16} + (\beta_{2} - 1) q^{18} + \beta_{3} q^{22} - \beta_1 q^{24} - q^{25}+ \cdots + \beta_1 q^{99}+O(q^{100}) $ q + q^2 - b1 * q^3 + q^4 - b1 * q^6 + q^8 + (b2 - 1) * q^9 + b3 * q^11 - b1 * q^12 + q^16 + (b2 - 1) * q^18 + b3 * q^22 - b1 * q^24 - q^25 + (-b3 + b1) * q^27 + q^32 + b2 * q^33 + (b2 - 1) * q^36 - b3 * q^41 - b2 * q^43 + b3 * q^44 - b1 * q^48 - q^49 - q^50 + (-b3 + b1) * q^54 - b2 * q^59 + q^64 + b2 * q^66 - b2 * q^67 + (b2 - 1) * q^72 - b3 * q^73 + b1 * q^75 + (-b2 + 1) * q^81 - b3 * q^82 + b2 * q^83 - b2 * q^86 + b3 * q^88 + b2 * q^89 - b1 * q^96 + b1 * q^97 - q^98 + b1 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 4 q^{9} + 4 q^{16} - 4 q^{18} - 4 q^{25} + 4 q^{32} - 4 q^{36} - 4 q^{49} - 4 q^{50} + 4 q^{64} - 4 q^{72} + 4 q^{81} - 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 4 * q^9 + 4 * q^16 - 4 * q^18 - 4 * q^25 + 4 * q^32 - 4 * q^36 - 4 * q^49 - 4 * q^50 + 4 * q^64 - 4 * q^72 + 4 * q^81 - 4 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 2 $ v^2 + 2
$\beta_{3}$	$=$	$ \nu^{3} + 3\nu $ v^3 + 3*v

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 2 $ b2 - 2
$\nu^{3}$	$=$	$ \beta_{3} - 3\beta_1 $ b3 - 3*b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

1155.1

		1.84776i
		0.765367i
	−	0.765367i
	−	1.84776i

1.00000

−

1.84776i

1.00000

−

1.84776i

1.00000

−2.41421

1155.2

1.00000

−

0.765367i

1.00000

−

0.765367i

1.00000

0.414214

1155.3

1.00000

0.765367i

1.00000

0.765367i

1.00000

0.414214

1155.4

1.00000

1.84776i

1.00000

1.84776i

1.00000

−2.41421

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
17.b	even	2	1	inner
136.e	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2312.1.e.b		4
8.d	odd	2	1	CM	2312.1.e.b		4
17.b	even	2	1	inner	2312.1.e.b		4
17.c	even	4	2		2312.1.f.c		4
17.d	even	8	4		2312.1.j.c		8
17.e	odd	16	2		136.1.p.a	✓	4
17.e	odd	16	2		2312.1.p.a		4
17.e	odd	16	2		2312.1.p.b		4
17.e	odd	16	2		2312.1.p.d		4
51.i	even	16	2		1224.1.bv.a		4
68.i	even	16	2		544.1.bl.a		4
85.o	even	16	2		3400.1.br.a		4
85.p	odd	16	2		3400.1.ce.a		4
85.r	even	16	2		3400.1.br.b		4
136.e	odd	2	1	inner	2312.1.e.b		4
136.j	odd	4	2		2312.1.f.c		4
136.p	odd	8	4		2312.1.j.c		8
136.q	odd	16	2		544.1.bl.a		4
136.s	even	16	2		136.1.p.a	✓	4
136.s	even	16	2		2312.1.p.a		4
136.s	even	16	2		2312.1.p.b		4
136.s	even	16	2		2312.1.p.d		4
408.bg	odd	16	2		1224.1.bv.a		4
680.ch	odd	16	2		3400.1.br.b		4
680.co	even	16	2		3400.1.ce.a		4
680.cr	odd	16	2		3400.1.br.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
136.1.p.a	✓	4	17.e	odd	16	2
136.1.p.a	✓	4	136.s	even	16	2
544.1.bl.a		4	68.i	even	16	2
544.1.bl.a		4	136.q	odd	16	2
1224.1.bv.a		4	51.i	even	16	2
1224.1.bv.a		4	408.bg	odd	16	2
2312.1.e.b		4	1.a	even	1	1	trivial
2312.1.e.b		4	8.d	odd	2	1	CM
2312.1.e.b		4	17.b	even	2	1	inner
2312.1.e.b		4	136.e	odd	2	1	inner
2312.1.f.c		4	17.c	even	4	2
2312.1.f.c		4	136.j	odd	4	2
2312.1.j.c		8	17.d	even	8	4
2312.1.j.c		8	136.p	odd	8	4
2312.1.p.a		4	17.e	odd	16	2
2312.1.p.a		4	136.s	even	16	2
2312.1.p.b		4	17.e	odd	16	2
2312.1.p.b		4	136.s	even	16	2
2312.1.p.d		4	17.e	odd	16	2
2312.1.p.d		4	136.s	even	16	2
3400.1.br.a		4	85.o	even	16	2
3400.1.br.a		4	680.cr	odd	16	2
3400.1.br.b		4	85.r	even	16	2
3400.1.br.b		4	680.ch	odd	16	2
3400.1.ce.a		4	85.p	odd	16	2
3400.1.ce.a		4	680.co	even	16	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{4} + 4T_{3}^{2} + 2 $ T3^4 + 4*T3^2 + 2 acting on $S_{1}^{\mathrm{new}}(2312, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T - 1)^{4} $ (T - 1)^4
$3$	$ T^{4} + 4T^{2} + 2 $ T^4 + 4*T^2 + 2
$5$	$ T^{4} $ T^4
$7$	$ T^{4} $ T^4
$11$	$ T^{4} + 4T^{2} + 2 $ T^4 + 4*T^2 + 2
$13$	$ T^{4} $ T^4
$17$	$ T^{4} $ T^4
$19$	$ T^{4} $ T^4
$23$	$ T^{4} $ T^4
$29$	$ T^{4} $ T^4
$31$	$ T^{4} $ T^4
$37$	$ T^{4} $ T^4
$41$	$ T^{4} + 4T^{2} + 2 $ T^4 + 4*T^2 + 2
$43$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$47$	$ T^{4} $ T^4
$53$	$ T^{4} $ T^4
$59$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$61$	$ T^{4} $ T^4
$67$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$71$	$ T^{4} $ T^4
$73$	$ T^{4} + 4T^{2} + 2 $ T^4 + 4*T^2 + 2
$79$	$ T^{4} $ T^4
$83$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$89$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$97$	$ T^{4} + 4T^{2} + 2 $ T^4 + 4*T^2 + 2
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Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2312.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + q^{8} + (\beta_{2} - 1) q^{9} + \beta_{3} q^{11} - \beta_1 q^{12} + q^{16} + (\beta_{2} - 1) q^{18} + \beta_{3} q^{22} - \beta_1 q^{24} - q^{25}+ \cdots + \beta_1 q^{99}+O(q^{100}) \) q + q^2 - b1 * q^3 + q^4 - b1 * q^6 + q^8 + (b2 - 1) * q^9 + b3 * q^11 - b1 * q^12 + q^16 + (b2 - 1) * q^18 + b3 * q^22 - b1 * q^24 - q^25 + (-b3 + b1) * q^27 + q^32 + b2 * q^33 + (b2 - 1) * q^36 - b3 * q^41 - b2 * q^43 + b3 * q^44 - b1 * q^48 - q^49 - q^50 + (-b3 + b1) * q^54 - b2 * q^59 + q^64 + b2 * q^66 - b2 * q^67 + (b2 - 1) * q^72 - b3 * q^73 + b1 * q^75 + (-b2 + 1) * q^81 - b3 * q^82 + b2 * q^83 - b2 * q^86 + b3 * q^88 + b2 * q^89 - b1 * q^96 + b1 * q^97 - q^98 + b1 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 4 q^{9} + 4 q^{16} - 4 q^{18} - 4 q^{25} + 4 q^{32} - 4 q^{36} - 4 q^{49} - 4 q^{50} + 4 q^{64} - 4 q^{72} + 4 q^{81} - 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 4 * q^9 + 4 * q^16 - 4 * q^18 - 4 * q^25 + 4 * q^32 - 4 * q^36 - 4 * q^49 - 4 * q^50 + 4 * q^64 - 4 * q^72 + 4 * q^81 - 4 * q^98

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	2312.1.e.b

Level	$2312$
Weight	$1$
Character orbit	2312.e
Analytic conductor	$1.154$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{8}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.15383830921\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{-2 + \sqrt{2}})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} + 4x^{2} + 2 \) x^4 + 4*x^2 + 2
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 136)
Projective image:	\(D_{8}\)
Projective field:	Galois closure of 8.0.1680747204608.3

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} + 2 \) v^2 + 2
\(\beta_{3}\)	\(=\)	\( \nu^{3} + 3\nu \) v^3 + 3*v

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} - 2 \) b2 - 2
\(\nu^{3}\)	\(=\)	\( \beta_{3} - 3\beta_1 \) b3 - 3*b1

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
17.b	even	2	1	inner
136.e	odd	2	1	inner

$p$	$F_p(T)$
$2$	\( (T - 1)^{4} \) (T - 1)^4
$3$	\( T^{4} + 4T^{2} + 2 \) T^4 + 4*T^2 + 2
$5$	\( T^{4} \) T^4
$7$	\( T^{4} \) T^4
$11$	\( T^{4} + 4T^{2} + 2 \) T^4 + 4*T^2 + 2
$13$	\( T^{4} \) T^4
$17$	\( T^{4} \) T^4
$19$	\( T^{4} \) T^4
$23$	\( T^{4} \) T^4
$29$	\( T^{4} \) T^4
$31$	\( T^{4} \) T^4
$37$	\( T^{4} \) T^4
$41$	\( T^{4} + 4T^{2} + 2 \) T^4 + 4*T^2 + 2
$43$	\( (T^{2} - 2)^{2} \) (T^2 - 2)^2
$47$	\( T^{4} \) T^4
$53$	\( T^{4} \) T^4
$59$	\( (T^{2} - 2)^{2} \) (T^2 - 2)^2
$61$	\( T^{4} \) T^4
$67$	\( (T^{2} - 2)^{2} \) (T^2 - 2)^2
$71$	\( T^{4} \) T^4
$73$	\( T^{4} + 4T^{2} + 2 \) T^4 + 4*T^2 + 2
$79$	\( T^{4} \) T^4
$83$	\( (T^{2} - 2)^{2} \) (T^2 - 2)^2
$89$	\( (T^{2} - 2)^{2} \) (T^2 - 2)^2
$97$	\( T^{4} + 4T^{2} + 2 \) T^4 + 4*T^2 + 2
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\(n\)	\(1157\)	\(1735\)	\(1737\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 80-90
Significant digits:
Format: