LMFDB - Newform orbit 324.6.e.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,6,Mod(109,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("324.109");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	324.e (of order $3$, degree $2$, 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:	$51.9643576194$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 - \beta_{2} q^{5} + (29 \beta_1 - 29) q^{7} + ( - \beta_{3} + \beta_{2}) q^{11} - 329 \beta_1 q^{13} - 25 \beta_{3} q^{17} + 1799 q^{19} - 41 \beta_{2} q^{23} + (4651 \beta_1 - 4651) q^{25} + ( - 16 \beta_{3} + 16 \beta_{2}) q^{29}+ \cdots + (12917 \beta_1 - 12917) q^{97}+O(q^{100}) $ q - b2 * q^5 + (29b1 - 29) q^7 + (-b3 + b2) * q^11 - 329b1 q^13 - 25b3 q^17 + 1799 * q^19 - 41b2 q^23 + (4651b1 - 4651) q^25 + (-16b3 + 16b2) * q^29 - 5228b1 q^31 + 29b3 q^35 + 8783 * q^37 + 176b2 q^41 + (19976b1 - 19976) q^43 + (123b3 - 123b2) * q^47 + 15966b1 q^49 + 334b3 q^53 + 7776 * q^55 - 65b2 q^59 + (-1069b1 + 1069) q^61 + (-329b3 + 329b2) * q^65 + 62077b1 q^67 - 526b3 q^71 - 48079 * q^73 - 29b2 q^77 + (49979b1 - 49979) q^79 + (654b3 - 654b2) * q^83 + 194400b1 q^85 - 997b3 q^89 + 9541 * q^91 - 1799b2 q^95 + (12917b1 - 12917) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 58 q^{7} - 658 q^{13} + 7196 q^{19} - 9302 q^{25} - 10456 q^{31} + 35132 q^{37} - 39952 q^{43} + 31932 q^{49} + 31104 q^{55} + 2138 q^{61} + 124154 q^{67} - 192316 q^{73} - 99958 q^{79} + 388800 q^{85}+ \cdots - 25834 q^{97}+O(q^{100}) $ 4 * q - 58 * q^7 - 658 * q^13 + 7196 * q^19 - 9302 * q^25 - 10456 * q^31 + 35132 * q^37 - 39952 * q^43 + 31932 * q^49 + 31104 * q^55 + 2138 * q^61 + 124154 * q^67 - 192316 * q^73 - 99958 * q^79 + 388800 * q^85 + 38164 * q^91 - 25834 * q^97

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

$\beta_{1}$	$=$	$ ( \nu^{2} ) / 2 $ (v^2) / 2
$\beta_{2}$	$=$	$ 18\nu^{3} + 36\nu $ 18v^3 + 36v
$\beta_{3}$	$=$	$ -18\nu^{3} + 72\nu $ -18v^3 + 72v

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

$\nu$	$=$	$ ( \beta_{3} + \beta_{2} ) / 108 $ (b3 + b2) / 108
$\nu^{2}$	$=$	$ 2\beta_1 $ 2*b1
$\nu^{3}$	$=$	$ ( -\beta_{3} + 2\beta_{2} ) / 54 $ (-b3 + 2*b2) / 54

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

Character values

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

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

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} $

109.1

1.22474	−	0.707107i
−1.22474	+	0.707107i
1.22474	+	0.707107i
−1.22474	−	0.707107i

−44.0908

76.3675i

−14.5000

−

25.1147i

109.2

44.0908

−

76.3675i

−14.5000

−

25.1147i

217.1

−44.0908

−

76.3675i

−14.5000

25.1147i

217.2

44.0908

76.3675i

−14.5000

25.1147i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	324.6.e.e		4
3.b	odd	2	1	inner	324.6.e.e		4
9.c	even	3	1		108.6.a.d	✓	2
9.c	even	3	1	inner	324.6.e.e		4
9.d	odd	6	1		108.6.a.d	✓	2
9.d	odd	6	1	inner	324.6.e.e		4
36.f	odd	6	1		432.6.a.p		2
36.h	even	6	1		432.6.a.p		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
108.6.a.d	✓	2	9.c	even	3	1
108.6.a.d	✓	2	9.d	odd	6	1
324.6.e.e		4	1.a	even	1	1	trivial
324.6.e.e		4	3.b	odd	2	1	inner
324.6.e.e		4	9.c	even	3	1	inner
324.6.e.e		4	9.d	odd	6	1	inner
432.6.a.p		2	36.f	odd	6	1
432.6.a.p		2	36.h	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(324, [\chi])$:

$ T_{5}^{4} + 7776T_{5}^{2} + 60466176 $

T5^4 + 7776*T5^2 + 60466176

$ T_{7}^{2} + 29T_{7} + 841 $

T7^2 + 29*T7 + 841

$ T_{11}^{4} + 7776T_{11}^{2} + 60466176 $

T11^4 + 7776*T11^2 + 60466176

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} + 7776 T^{2} + 60466176 $ T^4 + 7776*T^2 + 60466176
$7$	$ (T^{2} + 29 T + 841)^{2} $ (T^2 + 29*T + 841)^2
$11$	$ T^{4} + 7776 T^{2} + 60466176 $ T^4 + 7776*T^2 + 60466176
$13$	$ (T^{2} + 329 T + 108241)^{2} $ (T^2 + 329*T + 108241)^2
$17$	$ (T^{2} - 4860000)^{2} $ (T^2 - 4860000)^2
$19$	$ (T - 1799)^{4} $ (T - 1799)^4
$23$	$ T^{4} + \cdots + 170862961959936 $ T^4 + 13071456*T^2 + 170862961959936
$29$	$ T^{4} + \cdots + 3962711310336 $ T^4 + 1990656*T^2 + 3962711310336
$31$	$ (T^{2} + 5228 T + 27331984)^{2} $ (T^2 + 5228*T + 27331984)^2
$37$	$ (T - 8783)^{4} $ (T - 8783)^4
$41$	$ T^{4} + \cdots + 58\!\cdots\!76 $ T^4 + 240869376*T^2 + 58018056294629376
$43$	$ (T^{2} + 19976 T + 399040576)^{2} $ (T^2 + 19976*T + 399040576)^2
$47$	$ T^{4} + \cdots + 13\!\cdots\!16 $ T^4 + 117643104*T^2 + 13839899918754816
$53$	$ (T^{2} - 867459456)^{2} $ (T^2 - 867459456)^2
$59$	$ T^{4} + \cdots + 10\!\cdots\!00 $ T^4 + 32853600*T^2 + 1079359032960000
$61$	$ (T^{2} - 1069 T + 1142761)^{2} $ (T^2 - 1069*T + 1142761)^2
$67$	$ (T^{2} - 62077 T + 3853553929)^{2} $ (T^2 - 62077*T + 3853553929)^2
$71$	$ (T^{2} - 2151432576)^{2} $ (T^2 - 2151432576)^2
$73$	$ (T + 48079)^{4} $ (T + 48079)^4
$79$	$ (T^{2} + 49979 T + 2497900441)^{2} $ (T^2 + 49979*T + 2497900441)^2
$83$	$ T^{4} + \cdots + 11\!\cdots\!56 $ T^4 + 3325919616*T^2 + 11061741292093587456
$89$	$ (T^{2} - 7729413984)^{2} $ (T^2 - 7729413984)^2
$97$	$ (T^{2} + 12917 T + 166848889)^{2} $ (T^2 + 12917*T + 166848889)^2
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Overview	Random
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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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{5} + (29 \beta_1 - 29) q^{7} + ( - \beta_{3} + \beta_{2}) q^{11} - 329 \beta_1 q^{13} - 25 \beta_{3} q^{17} + 1799 q^{19} - 41 \beta_{2} q^{23} + (4651 \beta_1 - 4651) q^{25} + ( - 16 \beta_{3} + 16 \beta_{2}) q^{29}+ \cdots + (12917 \beta_1 - 12917) q^{97}+O(q^{100}) \) q - b2 * q^5 + (29b1 - 29) q^7 + (-b3 + b2) * q^11 - 329b1 q^13 - 25b3 q^17 + 1799 * q^19 - 41b2 q^23 + (4651b1 - 4651) q^25 + (-16b3 + 16b2) * q^29 - 5228b1 q^31 + 29b3 q^35 + 8783 * q^37 + 176b2 q^41 + (19976b1 - 19976) q^43 + (123b3 - 123b2) * q^47 + 15966b1 q^49 + 334b3 q^53 + 7776 * q^55 - 65b2 q^59 + (-1069b1 + 1069) q^61 + (-329b3 + 329b2) * q^65 + 62077b1 q^67 - 526b3 q^71 - 48079 * q^73 - 29b2 q^77 + (49979b1 - 49979) q^79 + (654b3 - 654b2) * q^83 + 194400b1 q^85 - 997b3 q^89 + 9541 * q^91 - 1799b2 q^95 + (12917b1 - 12917) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 58 q^{7} - 658 q^{13} + 7196 q^{19} - 9302 q^{25} - 10456 q^{31} + 35132 q^{37} - 39952 q^{43} + 31932 q^{49} + 31104 q^{55} + 2138 q^{61} + 124154 q^{67} - 192316 q^{73} - 99958 q^{79} + 388800 q^{85}+ \cdots - 25834 q^{97}+O(q^{100}) \) 4 * q - 58 * q^7 - 658 * q^13 + 7196 * q^19 - 9302 * q^25 - 10456 * q^31 + 35132 * q^37 - 39952 * q^43 + 31932 * q^49 + 31104 * q^55 + 2138 * q^61 + 124154 * q^67 - 192316 * q^73 - 99958 * q^79 + 388800 * q^85 + 38164 * q^91 - 25834 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	324.6.e.e

Level	$324$
Weight	$6$
Character orbit	324.e
Analytic conductor	$51.964$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(51.9643576194\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} - 2x^{2} + 4 \) x^4 - 2*x^2 + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4}\cdot 3^{5} \)
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{2} ) / 2 \) (v^2) / 2
\(\beta_{2}\)	\(=\)	\( 18\nu^{3} + 36\nu \) 18v^3 + 36v
\(\beta_{3}\)	\(=\)	\( -18\nu^{3} + 72\nu \) -18v^3 + 72v

\(\nu\)	\(=\)	\( ( \beta_{3} + \beta_{2} ) / 108 \) (b3 + b2) / 108
\(\nu^{2}\)	\(=\)	\( 2\beta_1 \) 2*b1
\(\nu^{3}\)	\(=\)	\( ( -\beta_{3} + 2\beta_{2} ) / 54 \) (-b3 + 2*b2) / 54

$p$	$F_p(T)$
$2$	\( T^{4} \) T^4
$3$	\( T^{4} \) T^4
$5$	\( T^{4} + 7776 T^{2} + 60466176 \) T^4 + 7776*T^2 + 60466176
$7$	\( (T^{2} + 29 T + 841)^{2} \) (T^2 + 29*T + 841)^2
$11$	\( T^{4} + 7776 T^{2} + 60466176 \) T^4 + 7776*T^2 + 60466176
$13$	\( (T^{2} + 329 T + 108241)^{2} \) (T^2 + 329*T + 108241)^2
$17$	\( (T^{2} - 4860000)^{2} \) (T^2 - 4860000)^2
$19$	\( (T - 1799)^{4} \) (T - 1799)^4
$23$	\( T^{4} + \cdots + 170862961959936 \) T^4 + 13071456*T^2 + 170862961959936
$29$	\( T^{4} + \cdots + 3962711310336 \) T^4 + 1990656*T^2 + 3962711310336
$31$	\( (T^{2} + 5228 T + 27331984)^{2} \) (T^2 + 5228*T + 27331984)^2
$37$	\( (T - 8783)^{4} \) (T - 8783)^4
$41$	\( T^{4} + \cdots + 58\!\cdots\!76 \) T^4 + 240869376*T^2 + 58018056294629376
$43$	\( (T^{2} + 19976 T + 399040576)^{2} \) (T^2 + 19976*T + 399040576)^2
$47$	\( T^{4} + \cdots + 13\!\cdots\!16 \) T^4 + 117643104*T^2 + 13839899918754816
$53$	\( (T^{2} - 867459456)^{2} \) (T^2 - 867459456)^2
$59$	\( T^{4} + \cdots + 10\!\cdots\!00 \) T^4 + 32853600*T^2 + 1079359032960000
$61$	\( (T^{2} - 1069 T + 1142761)^{2} \) (T^2 - 1069*T + 1142761)^2
$67$	\( (T^{2} - 62077 T + 3853553929)^{2} \) (T^2 - 62077*T + 3853553929)^2
$71$	\( (T^{2} - 2151432576)^{2} \) (T^2 - 2151432576)^2
$73$	\( (T + 48079)^{4} \) (T + 48079)^4
$79$	\( (T^{2} + 49979 T + 2497900441)^{2} \) (T^2 + 49979*T + 2497900441)^2
$83$	\( T^{4} + \cdots + 11\!\cdots\!56 \) T^4 + 3325919616*T^2 + 11061741292093587456
$89$	\( (T^{2} - 7729413984)^{2} \) (T^2 - 7729413984)^2
$97$	\( (T^{2} + 12917 T + 166848889)^{2} \) (T^2 + 12917*T + 166848889)^2
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\(n\)	\(163\)	\(245\)
\(\chi(n)\)	\(1\)	\(-\beta_{1}\)

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