LMFDB - Newform orbit 324.2.h.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [324,2,Mod(107,324)] 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("324.107"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	324.h (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$2.58715302549$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{3} q^{2} + ( - 2 \beta_1 + 2) q^{4} + ( - \beta_{6} + \beta_{5}) q^{5} + (2 \beta_{5} - 2 \beta_{3}) q^{8} + (\beta_{4} - 1) q^{10} + ( - \beta_{4} + \beta_{2} + 2 \beta_1 - 2) q^{13} - 4 \beta_1 q^{16}+ \cdots + ( - 7 \beta_{5} + 7 \beta_{3}) q^{98}+O(q^{100}) $ q - b3 * q^2 + (-2b1 + 2) q^4 + (-b6 + b5) * q^5 + (2b5 - 2b3) * q^8 + (b4 - 1) * q^10 + (-b4 + b2 + 2b1 - 2) q^13 - 4b1 q^16 + (b7 + 2b5 - 2b3) * q^17 + (2b7 - 2b6 + 2b3) q^20 + (-b2 + 9b1) q^25 + (-2b7 - b5 + b3) q^26 + (-b7 + b6 - 4b3) q^29 + 4b5 q^32 + (-b2 - 5b1) q^34 + (-2b4 - 1) q^37 + (2b4 - 2b2 + 2b1 - 2) q^40 - b5 * q^41 + (7b1 - 7) q^49 + (2b6 - 10b5) * q^50 + (2b2 + 4b1) * q^52 + (-5b5 + 5b3) * q^53 + (-b4 + b2 - 7b1 + 7) q^58 + (2b2 - 5b1) * q^61 - 8 * q^64 + (-b7 + b6 + 12b3) q^65 + (2b6 + 4b5) * q^68 + (b4 + 8) * q^73 + (-4b7 + 4b6 - b3) * q^74 + (4b7 - 4b5 + 4b3) q^80 + 2 * q^82 + (2b4 - 2b2 - 11b1 + 11) q^85 + (b7 - 7b5 + 7b3) * q^89 - 8b1 q^97 + (-7b5 + 7b3) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4} - 8 q^{10} - 8 q^{13} - 16 q^{16} + 36 q^{25} - 20 q^{34} - 8 q^{37} - 8 q^{40} - 28 q^{49} + 16 q^{52} + 28 q^{58} - 20 q^{61} - 64 q^{64} + 64 q^{73} + 16 q^{82} + 44 q^{85} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^4 - 8 * q^10 - 8 * q^13 - 16 * q^16 + 36 * q^25 - 20 * q^34 - 8 * q^37 - 8 * q^40 - 28 * q^49 + 16 * q^52 + 28 * q^58 - 20 * q^61 - 64 * q^64 + 64 * q^73 + 16 * q^82 + 44 * q^85 - 32 * q^97

Basis of coefficient ring

$\beta_{1}$	$=$	$ \zeta_{24}^{4} $ v^4
$\beta_{2}$	$=$	$ 3\zeta_{24}^{6} + 3\zeta_{24}^{2} $ 3v^6 + 3v^2
$\beta_{3}$	$=$	$ -\zeta_{24}^{7} + \zeta_{24} $ -v^7 + v
$\beta_{4}$	$=$	$ -3\zeta_{24}^{6} + 6\zeta_{24}^{2} $ -3v^6 + 6v^2
$\beta_{5}$	$=$	$ -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} $ -v^7 + v^5 + v^3
$\beta_{6}$	$=$	$ \zeta_{24}^{7} - \zeta_{24}^{5} + 2\zeta_{24}^{3} + 3\zeta_{24} $ v^7 - v^5 + 2v^3 + 3v
$\beta_{7}$	$=$	$ 3\zeta_{24}^{7} + 2\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} $ 3v^7 + 2v^5 - v^3 + v

Display $\zeta_{24}^j$ in terms of $\beta_i$

$\zeta_{24}$	$=$	$ ( \beta_{7} + \beta_{6} - \beta_{5} + 5\beta_{3} ) / 9 $ (b7 + b6 - b5 + 5*b3) / 9
$\zeta_{24}^{2}$	$=$	$ ( \beta_{4} + \beta_{2} ) / 9 $ (b4 + b2) / 9
$\zeta_{24}^{3}$	$=$	$ ( -\beta_{7} + 2\beta_{6} + 4\beta_{5} - 5\beta_{3} ) / 9 $ (-b7 + 2b6 + 4b5 - 5*b3) / 9
$\zeta_{24}^{4}$	$=$	$ \beta_1 $ b1
$\zeta_{24}^{5}$	$=$	$ ( 2\beta_{7} - \beta_{6} + 4\beta_{5} + \beta_{3} ) / 9 $ (2b7 - b6 + 4b5 + b3) / 9
$\zeta_{24}^{6}$	$=$	$ ( -\beta_{4} + 2\beta_{2} ) / 9 $ (-b4 + 2*b2) / 9
$\zeta_{24}^{7}$	$=$	$ ( \beta_{7} + \beta_{6} - \beta_{5} - 4\beta_{3} ) / 9 $ (b7 + b6 - b5 - 4*b3) / 9

Display $\beta_i$ in terms of $\zeta_{24}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$-1$	$1 - \beta_{1}$

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} $

107.1

0.965926	−	0.258819i
0.258819	+	0.965926i
−0.258819	−	0.965926i
−0.965926	+	0.258819i
0.965926	+	0.258819i
0.258819	−	0.965926i
−0.258819	+	0.965926i
−0.965926	−	0.258819i

−1.22474

−

0.707107i

1.00000

1.73205i

−2.56961

1.48356i

−

2.82843i

4.19615

107.2

−1.22474

−

0.707107i

1.00000

1.73205i

3.79435

−

2.19067i

−

2.82843i

−6.19615

107.3

1.22474

0.707107i

1.00000

1.73205i

−3.79435

2.19067i

2.82843i

−6.19615

107.4

1.22474

0.707107i

1.00000

1.73205i

2.56961

−

1.48356i

2.82843i

4.19615

215.1

−1.22474

0.707107i

1.00000

−

1.73205i

−2.56961

−

1.48356i

2.82843i

4.19615

215.2

−1.22474

0.707107i

1.00000

−

1.73205i

3.79435

2.19067i

2.82843i

−6.19615

215.3

1.22474

−

0.707107i

1.00000

−

1.73205i

−3.79435

−

2.19067i

−

2.82843i

−6.19615

215.4

1.22474

−

0.707107i

1.00000

−

1.73205i

2.56961

1.48356i

−

2.82843i

4.19615

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
3.b	odd	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner
12.b	even	2	1	inner
36.f	odd	6	1	inner
36.h	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	324.2.h.e		8
3.b	odd	2	1	inner	324.2.h.e		8
4.b	odd	2	1	CM	324.2.h.e		8
9.c	even	3	1		324.2.b.a	✓	4
9.c	even	3	1	inner	324.2.h.e		8
9.d	odd	6	1		324.2.b.a	✓	4
9.d	odd	6	1	inner	324.2.h.e		8
12.b	even	2	1	inner	324.2.h.e		8
36.f	odd	6	1		324.2.b.a	✓	4
36.f	odd	6	1	inner	324.2.h.e		8
36.h	even	6	1		324.2.b.a	✓	4
36.h	even	6	1	inner	324.2.h.e		8
72.j	odd	6	1		5184.2.c.e		4
72.l	even	6	1		5184.2.c.e		4
72.n	even	6	1		5184.2.c.e		4
72.p	odd	6	1		5184.2.c.e		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
324.2.b.a	✓	4	9.c	even	3	1
324.2.b.a	✓	4	9.d	odd	6	1
324.2.b.a	✓	4	36.f	odd	6	1
324.2.b.a	✓	4	36.h	even	6	1
324.2.h.e		8	1.a	even	1	1	trivial
324.2.h.e		8	3.b	odd	2	1	inner
324.2.h.e		8	4.b	odd	2	1	CM
324.2.h.e		8	9.c	even	3	1	inner
324.2.h.e		8	9.d	odd	6	1	inner
324.2.h.e		8	12.b	even	2	1	inner
324.2.h.e		8	36.f	odd	6	1	inner
324.2.h.e		8	36.h	even	6	1	inner
5184.2.c.e		4	72.j	odd	6	1
5184.2.c.e		4	72.l	even	6	1
5184.2.c.e		4	72.n	even	6	1
5184.2.c.e		4	72.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(324, [\chi])$:

$ T_{5}^{8} - 28T_{5}^{6} + 615T_{5}^{4} - 4732T_{5}^{2} + 28561 $

T5^8 - 28*T5^6 + 615*T5^4 - 4732*T5^2 + 28561

$ T_{7} $

T7

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} - 2 T^{2} + 4)^{2} $ (T^4 - 2*T^2 + 4)^2
$3$	$ T^{8} $ T^8
$5$	$ T^{8} - 28 T^{6} + \cdots + 28561 $ T^8 - 28T^6 + 615T^4 - 4732*T^2 + 28561
$7$	$ T^{8} $ T^8
$11$	$ T^{8} $ T^8
$13$	$ (T^{4} + 4 T^{3} + \cdots + 529)^{2} $ (T^4 + 4T^3 + 39T^2 - 92*T + 529)^2
$17$	$ (T^{4} + 52 T^{2} + 1)^{2} $ (T^4 + 52*T^2 + 1)^2
$19$	$ T^{8} $ T^8
$23$	$ T^{8} $ T^8
$29$	$ T^{8} - 76 T^{6} + \cdots + 14641 $ T^8 - 76T^6 + 5655T^4 - 9196*T^2 + 14641
$31$	$ T^{8} $ T^8
$37$	$ (T^{2} + 2 T - 107)^{4} $ (T^2 + 2*T - 107)^4
$41$	$ (T^{4} - 2 T^{2} + 4)^{2} $ (T^4 - 2*T^2 + 4)^2
$43$	$ T^{8} $ T^8
$47$	$ T^{8} $ T^8
$53$	$ (T^{2} + 50)^{4} $ (T^2 + 50)^4
$59$	$ T^{8} $ T^8
$61$	$ (T^{4} + 10 T^{3} + \cdots + 6889)^{2} $ (T^4 + 10T^3 + 183T^2 - 830*T + 6889)^2
$67$	$ T^{8} $ T^8
$71$	$ T^{8} $ T^8
$73$	$ (T^{2} - 16 T + 37)^{4} $ (T^2 - 16*T + 37)^4
$79$	$ T^{8} $ T^8
$83$	$ T^{8} $ T^8
$89$	$ (T^{4} + 196 T^{2} + 5041)^{2} $ (T^4 + 196*T^2 + 5041)^2
$97$	$ (T^{2} + 8 T + 64)^{4} $ (T^2 + 8*T + 64)^4
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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

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	324.h (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{2} + ( - 2 \beta_1 + 2) q^{4} + ( - \beta_{6} + \beta_{5}) q^{5} + (2 \beta_{5} - 2 \beta_{3}) q^{8} + (\beta_{4} - 1) q^{10} + ( - \beta_{4} + \beta_{2} + 2 \beta_1 - 2) q^{13} - 4 \beta_1 q^{16}+ \cdots + ( - 7 \beta_{5} + 7 \beta_{3}) q^{98}+O(q^{100}) \) q - b3 * q^2 + (-2b1 + 2) q^4 + (-b6 + b5) * q^5 + (2b5 - 2b3) * q^8 + (b4 - 1) * q^10 + (-b4 + b2 + 2b1 - 2) q^13 - 4b1 q^16 + (b7 + 2b5 - 2b3) * q^17 + (2b7 - 2b6 + 2b3) q^20 + (-b2 + 9b1) q^25 + (-2b7 - b5 + b3) q^26 + (-b7 + b6 - 4b3) q^29 + 4b5 q^32 + (-b2 - 5b1) q^34 + (-2b4 - 1) q^37 + (2b4 - 2b2 + 2b1 - 2) q^40 - b5 * q^41 + (7b1 - 7) q^49 + (2b6 - 10b5) * q^50 + (2b2 + 4b1) * q^52 + (-5b5 + 5b3) * q^53 + (-b4 + b2 - 7b1 + 7) q^58 + (2b2 - 5b1) * q^61 - 8 * q^64 + (-b7 + b6 + 12b3) q^65 + (2b6 + 4b5) * q^68 + (b4 + 8) * q^73 + (-4b7 + 4b6 - b3) * q^74 + (4b7 - 4b5 + 4b3) q^80 + 2 * q^82 + (2b4 - 2b2 - 11b1 + 11) q^85 + (b7 - 7b5 + 7b3) * q^89 - 8b1 q^97 + (-7b5 + 7b3) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} - 8 q^{10} - 8 q^{13} - 16 q^{16} + 36 q^{25} - 20 q^{34} - 8 q^{37} - 8 q^{40} - 28 q^{49} + 16 q^{52} + 28 q^{58} - 20 q^{61} - 64 q^{64} + 64 q^{73} + 16 q^{82} + 44 q^{85} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^4 - 8 * q^10 - 8 * q^13 - 16 * q^16 + 36 * q^25 - 20 * q^34 - 8 * q^37 - 8 * q^40 - 28 * q^49 + 16 * q^52 + 28 * q^58 - 20 * q^61 - 64 * q^64 + 64 * q^73 + 16 * q^82 + 44 * q^85 - 32 * q^97

Introduction

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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	324.2.h.e

Level	$324$
Weight	$2$
Character orbit	324.h
Analytic conductor	$2.587$
Analytic rank	$0$
Dimension	$8$
CM discriminant	-4
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(2.58715302549\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{24})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

\(\beta_{1}\)	\(=\)	\( \zeta_{24}^{4} \) v^4
\(\beta_{2}\)	\(=\)	\( 3\zeta_{24}^{6} + 3\zeta_{24}^{2} \) 3v^6 + 3v^2
\(\beta_{3}\)	\(=\)	\( -\zeta_{24}^{7} + \zeta_{24} \) -v^7 + v
\(\beta_{4}\)	\(=\)	\( -3\zeta_{24}^{6} + 6\zeta_{24}^{2} \) -3v^6 + 6v^2
\(\beta_{5}\)	\(=\)	\( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) -v^7 + v^5 + v^3
\(\beta_{6}\)	\(=\)	\( \zeta_{24}^{7} - \zeta_{24}^{5} + 2\zeta_{24}^{3} + 3\zeta_{24} \) v^7 - v^5 + 2v^3 + 3v
\(\beta_{7}\)	\(=\)	\( 3\zeta_{24}^{7} + 2\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) 3v^7 + 2v^5 - v^3 + v

\(\zeta_{24}\)	\(=\)	\( ( \beta_{7} + \beta_{6} - \beta_{5} + 5\beta_{3} ) / 9 \) (b7 + b6 - b5 + 5*b3) / 9
\(\zeta_{24}^{2}\)	\(=\)	\( ( \beta_{4} + \beta_{2} ) / 9 \) (b4 + b2) / 9
\(\zeta_{24}^{3}\)	\(=\)	\( ( -\beta_{7} + 2\beta_{6} + 4\beta_{5} - 5\beta_{3} ) / 9 \) (-b7 + 2b6 + 4b5 - 5*b3) / 9
\(\zeta_{24}^{4}\)	\(=\)	\( \beta_1 \) b1
\(\zeta_{24}^{5}\)	\(=\)	\( ( 2\beta_{7} - \beta_{6} + 4\beta_{5} + \beta_{3} ) / 9 \) (2b7 - b6 + 4b5 + b3) / 9
\(\zeta_{24}^{6}\)	\(=\)	\( ( -\beta_{4} + 2\beta_{2} ) / 9 \) (-b4 + 2*b2) / 9
\(\zeta_{24}^{7}\)	\(=\)	\( ( \beta_{7} + \beta_{6} - \beta_{5} - 4\beta_{3} ) / 9 \) (b7 + b6 - b5 - 4*b3) / 9

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 107.4
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
3.b	odd	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner
12.b	even	2	1	inner
36.f	odd	6	1	inner
36.h	even	6	1	inner

$p$	$F_p(T)$
$2$	\( (T^{4} - 2 T^{2} + 4)^{2} \) (T^4 - 2*T^2 + 4)^2
$3$	\( T^{8} \) T^8
$5$	\( T^{8} - 28 T^{6} + \cdots + 28561 \) T^8 - 28T^6 + 615T^4 - 4732*T^2 + 28561
$7$	\( T^{8} \) T^8
$11$	\( T^{8} \) T^8
$13$	\( (T^{4} + 4 T^{3} + \cdots + 529)^{2} \) (T^4 + 4T^3 + 39T^2 - 92*T + 529)^2
$17$	\( (T^{4} + 52 T^{2} + 1)^{2} \) (T^4 + 52*T^2 + 1)^2
$19$	\( T^{8} \) T^8
$23$	\( T^{8} \) T^8
$29$	\( T^{8} - 76 T^{6} + \cdots + 14641 \) T^8 - 76T^6 + 5655T^4 - 9196*T^2 + 14641
$31$	\( T^{8} \) T^8
$37$	\( (T^{2} + 2 T - 107)^{4} \) (T^2 + 2*T - 107)^4
$41$	\( (T^{4} - 2 T^{2} + 4)^{2} \) (T^4 - 2*T^2 + 4)^2
$43$	\( T^{8} \) T^8
$47$	\( T^{8} \) T^8
$53$	\( (T^{2} + 50)^{4} \) (T^2 + 50)^4
$59$	\( T^{8} \) T^8
$61$	\( (T^{4} + 10 T^{3} + \cdots + 6889)^{2} \) (T^4 + 10T^3 + 183T^2 - 830*T + 6889)^2
$67$	\( T^{8} \) T^8
$71$	\( T^{8} \) T^8
$73$	\( (T^{2} - 16 T + 37)^{4} \) (T^2 - 16*T + 37)^4
$79$	\( T^{8} \) T^8
$83$	\( T^{8} \) T^8
$89$	\( (T^{4} + 196 T^{2} + 5041)^{2} \) (T^4 + 196*T^2 + 5041)^2
$97$	\( (T^{2} + 8 T + 64)^{4} \) (T^2 + 8*T + 64)^4
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\(n\)	\(163\)	\(245\)
\(\chi(n)\)	\(-1\)	\(1 - \beta_{1}\)