LMFDB - Newform orbit 2368.2.g.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(961,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("2368.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.g (of order $2$, degree $1$, 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:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3356224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 8x^{4} + 16x^{2} + 1 $ x^6 + 8x^4 + 16x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{3} - 1) q^{3} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{4} + \beta_{3} + 2) q^{7} + (\beta_{4} + 1) q^{9} + ( - \beta_{4} - 4) q^{11} + (2 \beta_{2} + \beta_1) q^{13} + ( - \beta_{5} - 2 \beta_{2} + \beta_1) q^{15}+ \cdots + ( - 5 \beta_{4} + \beta_{3} - 7) q^{99}+O(q^{100}) $ q + (b3 - 1) * q^3 + (-b5 + b2 - b1) * q^5 + (-b4 + b3 + 2) * q^7 + (b4 + 1) * q^9 + (-b4 - 4) * q^11 + (2b2 + b1) q^13 + (-b5 - 2b2 + b1) q^15 + (-2b5 + b2 + b1) q^17 - 2b5 q^19 + (3b4 + 3b3 + 2) * q^21 + (-2b5 - b2 - 2b1) * q^23 + (3b4 + 2b3 - 6) * q^25 + (-2b4 - 2b3 + 1) * q^27 + (-2b2 - b1) q^29 + (-b5 + 3b2 + 3b1) * q^31 + (2b4 - 4b3 + 5) * q^33 + (-6b5 + 4b2) * q^35 + (b5 + 3b4 + b3 + b2 + 2b1) * q^37 + (3b5 - 2b2) * q^39 + (-2b4 - 3b3 + 1) * q^41 + (3b2 - b1) q^43 + (b5 - 2b2 - 3b1) * q^45 + (3b4 + 3b3 + 2) * q^47 + (-b4 + 5b3 + 5) q^49 + (-3b2 - 5b1) * q^51 + (-3b4 + b3) q^53 + (2b5 - b2 + 6b1) * q^55 + (-2b5 - 2b2 - 4b1) q^57 + (4b5 - 3b2 - b1) * q^59 + (-3b5 - 2b2 - 2b1) q^61 + (2b3 - 2) q^63 + (7b4 + 5b3 - 5) * q^65 + (-6b4 - 3b3) * q^67 + (-5b5 - b2 - b1) q^69 + (-3b4 - b3 + 10) q^71 + (-b4 + 4b3 - 2) q^73 + (-4b4 - 4b3 + 9) * q^75 + (3b4 - 5b3 - 4) * q^77 + (4b5 - 4b2 - 3b1) q^79 + (-b4 - b3 - 8) * q^81 + (5b4 + 3b3 - 4) * q^83 + (6b4 - 2b3 - 2) * q^85 + (-3b5 + 2b2) * q^87 + (2b5 - 4b2 - 6b1) q^89 + (2b5 + 7b2 + 3b1) q^91 + (5b5 - 4b2 - 5b1) q^93 + (2b4 - 4b3 - 2) * q^95 + (4b5 + 3b2 - 3b1) q^97 + (-5b4 + b3 - 7) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 14 q^{7} + 6 q^{9} - 24 q^{11} + 18 q^{21} - 32 q^{25} + 2 q^{27} + 22 q^{33} + 2 q^{37} + 18 q^{47} + 40 q^{49} + 2 q^{53} - 8 q^{63} - 20 q^{65} - 6 q^{67} + 58 q^{71} - 4 q^{73} + 46 q^{75}+ \cdots - 40 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + 14 * q^7 + 6 * q^9 - 24 * q^11 + 18 * q^21 - 32 * q^25 + 2 * q^27 + 22 * q^33 + 2 * q^37 + 18 * q^47 + 40 * q^49 + 2 * q^53 - 8 * q^63 - 20 * q^65 - 6 * q^67 + 58 * q^71 - 4 * q^73 + 46 * q^75 - 34 * q^77 - 50 * q^81 - 18 * q^83 - 16 * q^85 - 20 * q^95 - 40 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 8x^{4} + 16x^{2} + 1 $ x^6 + 8*x^4 + 16*x^2 + 1 :

$\beta_{1}$	$=$	$ \nu^{3} + 3\nu $ v^3 + 3*v
$\beta_{2}$	$=$	$ \nu^{3} + 5\nu $ v^3 + 5*v
$\beta_{3}$	$=$	$ \nu^{2} + 3 $ v^2 + 3
$\beta_{4}$	$=$	$ \nu^{4} + 4\nu^{2} $ v^4 + 4*v^2
$\beta_{5}$	$=$	$ \nu^{5} + 7\nu^{3} + 12\nu $ v^5 + 7v^3 + 12v

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

$\nu$	$=$	$ ( \beta_{2} - \beta_1 ) / 2 $ (b2 - b1) / 2
$\nu^{2}$	$=$	$ \beta_{3} - 3 $ b3 - 3
$\nu^{3}$	$=$	$ ( -3\beta_{2} + 5\beta_1 ) / 2 $ (-3b2 + 5b1) / 2
$\nu^{4}$	$=$	$ \beta_{4} - 4\beta_{3} + 12 $ b4 - 4*b3 + 12
$\nu^{5}$	$=$	$ ( 2\beta_{5} + 9\beta_{2} - 23\beta_1 ) / 2 $ (2b5 + 9b2 - 23*b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times$.

$n$	$705$	$1407$	$1925$
$\chi(n)$	$-1$	$1$	$1$

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} $

961.1

	−	2.11491i
		2.11491i
	−	1.86081i
		1.86081i
		0.254102i
	−	0.254102i

−2.47283

−

2.75698i

−1.58774

3.11491

961.2

−2.47283

2.75698i

−1.58774

3.11491

961.3

−1.46260

−

4.18421i

3.39821

−0.860806

961.4

−1.46260

4.18421i

3.39821

−0.860806

961.5

1.93543

−

2.42723i

5.18953

0.745898

961.6

1.93543

2.42723i

5.18953

0.745898

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2368.2.g.k		6
4.b	odd	2	1		2368.2.g.l		6
8.b	even	2	1		1184.2.g.f	yes	6
8.d	odd	2	1		1184.2.g.e	✓	6
37.b	even	2	1	inner	2368.2.g.k		6
148.b	odd	2	1		2368.2.g.l		6
296.e	even	2	1		1184.2.g.f	yes	6
296.h	odd	2	1		1184.2.g.e	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1184.2.g.e	✓	6	8.d	odd	2	1
1184.2.g.e	✓	6	296.h	odd	2	1
1184.2.g.f	yes	6	8.b	even	2	1
1184.2.g.f	yes	6	296.e	even	2	1
2368.2.g.k		6	1.a	even	1	1	trivial
2368.2.g.k		6	37.b	even	2	1	inner
2368.2.g.l		6	4.b	odd	2	1
2368.2.g.l		6	148.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2368, [\chi])$:

$ T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 7 $

$ T_{5}^{6} + 31T_{5}^{4} + 281T_{5}^{2} + 784 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ (T^{3} + 2 T^{2} - 4 T - 7)^{2} $ (T^3 + 2T^2 - 4T - 7)^2
$5$	$ T^{6} + 31 T^{4} + \cdots + 784 $ T^6 + 31T^4 + 281T^2 + 784
$7$	$ (T^{3} - 7 T^{2} + 4 T + 28)^{2} $ (T^3 - 7T^2 + 4T + 28)^2
$11$	$ (T^{3} + 12 T^{2} + \cdots + 49)^{2} $ (T^3 + 12T^2 + 44T + 49)^2
$13$	$ T^{6} + 35 T^{4} + \cdots + 196 $ T^6 + 35T^4 + 277T^2 + 196
$17$	$ T^{6} + 48 T^{4} + \cdots + 3136 $ T^6 + 48T^4 + 704T^2 + 3136
$19$	$ T^{6} + 44 T^{4} + \cdots + 256 $ T^6 + 44T^4 + 336T^2 + 256
$23$	$ T^{6} + 79 T^{4} + \cdots + 16 $ T^6 + 79T^4 + 353T^2 + 16
$29$	$ T^{6} + 35 T^{4} + \cdots + 196 $ T^6 + 35T^4 + 277T^2 + 196
$31$	$ T^{6} + 107 T^{4} + \cdots + 21904 $ T^6 + 107T^4 + 3249T^2 + 21904
$37$	$ T^{6} - 2 T^{5} + \cdots + 50653 $ T^6 - 2T^5 - 17T^4 + 364T^3 - 629T^2 - 2738*T + 50653
$41$	$ (T^{3} - 46 T + 53)^{2} $ (T^3 - 46*T + 53)^2
$43$	$ T^{6} + 140 T^{4} + \cdots + 33856 $ T^6 + 140T^4 + 4912T^2 + 33856
$47$	$ (T^{3} - 9 T^{2} + \cdots + 196)^{2} $ (T^3 - 9T^2 - 30T + 196)^2
$53$	$ (T^{3} - T^{2} - 50 T + 148)^{2} $ (T^3 - T^2 - 50*T + 148)^2
$59$	$ T^{6} + 176 T^{4} + \cdots + 153664 $ T^6 + 176T^4 + 9408T^2 + 153664
$61$	$ T^{6} + 171 T^{4} + \cdots + 196 $ T^6 + 171T^4 + 1149T^2 + 196
$67$	$ (T^{3} + 3 T^{2} + \cdots - 756)^{2} $ (T^3 + 3T^2 - 135T - 756)^2
$71$	$ (T^{3} - 29 T^{2} + \cdots - 644)^{2} $ (T^3 - 29T^2 + 248T - 644)^2
$73$	$ (T^{3} + 2 T^{2} + \cdots - 199)^{2} $ (T^3 + 2T^2 - 100T - 199)^2
$79$	$ T^{6} + 251 T^{4} + \cdots + 268324 $ T^6 + 251T^4 + 17973T^2 + 268324
$83$	$ (T^{3} + 9 T^{2} + \cdots + 112)^{2} $ (T^3 + 9T^2 - 76T + 112)^2
$89$	$ T^{6} + 360 T^{4} + \cdots + 200704 $ T^6 + 360T^4 + 19856T^2 + 200704
$97$	$ T^{6} + 608 T^{4} + \cdots + 5271616 $ T^6 + 608T^4 + 105984T^2 + 5271616
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Level:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{3} - 1) q^{3} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{4} + \beta_{3} + 2) q^{7} + (\beta_{4} + 1) q^{9} + ( - \beta_{4} - 4) q^{11} + (2 \beta_{2} + \beta_1) q^{13} + ( - \beta_{5} - 2 \beta_{2} + \beta_1) q^{15}+ \cdots + ( - 5 \beta_{4} + \beta_{3} - 7) q^{99}+O(q^{100}) \) q + (b3 - 1) * q^3 + (-b5 + b2 - b1) * q^5 + (-b4 + b3 + 2) * q^7 + (b4 + 1) * q^9 + (-b4 - 4) * q^11 + (2b2 + b1) q^13 + (-b5 - 2b2 + b1) q^15 + (-2b5 + b2 + b1) q^17 - 2b5 q^19 + (3b4 + 3b3 + 2) * q^21 + (-2b5 - b2 - 2b1) * q^23 + (3b4 + 2b3 - 6) * q^25 + (-2b4 - 2b3 + 1) * q^27 + (-2b2 - b1) q^29 + (-b5 + 3b2 + 3b1) * q^31 + (2b4 - 4b3 + 5) * q^33 + (-6b5 + 4b2) * q^35 + (b5 + 3b4 + b3 + b2 + 2b1) * q^37 + (3b5 - 2b2) * q^39 + (-2b4 - 3b3 + 1) * q^41 + (3b2 - b1) q^43 + (b5 - 2b2 - 3b1) * q^45 + (3b4 + 3b3 + 2) * q^47 + (-b4 + 5b3 + 5) q^49 + (-3b2 - 5b1) * q^51 + (-3b4 + b3) q^53 + (2b5 - b2 + 6b1) * q^55 + (-2b5 - 2b2 - 4b1) q^57 + (4b5 - 3b2 - b1) * q^59 + (-3b5 - 2b2 - 2b1) q^61 + (2b3 - 2) q^63 + (7b4 + 5b3 - 5) * q^65 + (-6b4 - 3b3) * q^67 + (-5b5 - b2 - b1) q^69 + (-3b4 - b3 + 10) q^71 + (-b4 + 4b3 - 2) q^73 + (-4b4 - 4b3 + 9) * q^75 + (3b4 - 5b3 - 4) * q^77 + (4b5 - 4b2 - 3b1) q^79 + (-b4 - b3 - 8) * q^81 + (5b4 + 3b3 - 4) * q^83 + (6b4 - 2b3 - 2) * q^85 + (-3b5 + 2b2) * q^87 + (2b5 - 4b2 - 6b1) q^89 + (2b5 + 7b2 + 3b1) q^91 + (5b5 - 4b2 - 5b1) q^93 + (2b4 - 4b3 - 2) * q^95 + (4b5 + 3b2 - 3b1) q^97 + (-5b4 + b3 - 7) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 14 q^{7} + 6 q^{9} - 24 q^{11} + 18 q^{21} - 32 q^{25} + 2 q^{27} + 22 q^{33} + 2 q^{37} + 18 q^{47} + 40 q^{49} + 2 q^{53} - 8 q^{63} - 20 q^{65} - 6 q^{67} + 58 q^{71} - 4 q^{73} + 46 q^{75}+ \cdots - 40 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + 14 * q^7 + 6 * q^9 - 24 * q^11 + 18 * q^21 - 32 * q^25 + 2 * q^27 + 22 * q^33 + 2 * q^37 + 18 * q^47 + 40 * q^49 + 2 * q^53 - 8 * q^63 - 20 * q^65 - 6 * q^67 + 58 * q^71 - 4 * q^73 + 46 * q^75 - 34 * q^77 - 50 * q^81 - 18 * q^83 - 16 * q^85 - 20 * q^95 - 40 * q^99

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