LMFDB - Newform orbit 117.4.b.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(64,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("117.64");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.b (of order $2$, degree $1$, 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:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.8112.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 5x^{2} + 3 $ x^4 + 5*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

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

$f(q)$	$=$	$ q + \beta_1 q^{2} + ( - \beta_{2} - 8) q^{4} + (\beta_{3} - 2 \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{8}+O(q^{10}) $ q + b1 * q^2 + (-b2 - 8) * q^4 + (b3 - 2b1) q^5 + (b3 - b1) * q^8	\( q + \beta_1 q^{2} + ( - \beta_{2} - 8) q^{4} + (\beta_{3} - 2 \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{8} + (17 \beta_{2} + 29) q^{10} + (3 \beta_{3} - 8 \beta_1) q^{11} + 13 \beta_{2} q^{13} + (8 \beta_{2} - 51) q^{16} + ( - 9 \beta_{3} + 30 \beta_1) q^{20} + (53 \beta_{2} + 119) q^{22} + ( - 46 \beta_{2} - 125) q^{25} + ( - 13 \beta_{3} + 13 \beta_1) q^{26} - 51 \beta_1 q^{32} + ( - 29 \beta_{2} - 221) q^{40} + (23 \beta_{3} - 50 \beta_1) q^{41} + 452 q^{43} + ( - 29 \beta_{3} + 108 \beta_1) q^{44} + (23 \beta_{3} + 76 \beta_1) q^{47} + 343 q^{49} + (46 \beta_{3} - 171 \beta_1) q^{50} + ( - 104 \beta_{2} - 169) q^{52} + ( - 172 \beta_{2} - 808) q^{55} + (\beta_{3} + 160 \beta_1) q^{59} - 262 \beta_{2} q^{61} + (115 \beta_{2} + 408) q^{64} + (13 \beta_{3} - 182 \beta_1) q^{65} + ( - 43 \beta_{3} - 140 \beta_1) q^{71} + 116 \beta_{2} q^{79} + ( - 43 \beta_{3} - 10 \beta_1) q^{80} + (395 \beta_{2} + 731) q^{82} + ( - 51 \beta_{3} + 208 \beta_1) q^{83} + 452 \beta_1 q^{86} + ( - 119 \beta_{2} - 689) q^{88} + ( - 61 \beta_{3} - 194 \beta_1) q^{89} + (269 \beta_{2} - 1285) q^{94} + 343 \beta_1 q^{98}+O(q^{100}) \) q + b1 * q^2 + (-b2 - 8) * q^4 + (b3 - 2b1) q^5 + (b3 - b1) * q^8 + (17b2 + 29) q^10 + (3b3 - 8b1) * q^11 + 13b2 q^13 + (8b2 - 51) q^16 + (-9b3 + 30b1) * q^20 + (53b2 + 119) q^22 + (-46b2 - 125) q^25 + (-13b3 + 13b1) * q^26 - 51b1 q^32 + (-29b2 - 221) q^40 + (23b3 - 50b1) * q^41 + 452 * q^43 + (-29b3 + 108b1) * q^44 + (23b3 + 76b1) * q^47 + 343 * q^49 + (46b3 - 171b1) * q^50 + (-104b2 - 169) q^52 + (-172b2 - 808) q^55 + (b3 + 160b1) q^59 - 262b2 q^61 + (115b2 + 408) q^64 + (13b3 - 182b1) * q^65 + (-43b3 - 140b1) * q^71 + 116b2 q^79 + (-43b3 - 10b1) * q^80 + (395b2 + 731) q^82 + (-51b3 + 208b1) * q^83 + 452b1 q^86 + (-119b2 - 689) q^88 + (-61b3 - 194b1) * q^89 + (269b2 - 1285) q^94 + 343b1 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4}+O(q^{10}) $ 4 * q - 32 * q^4	$ 4 q - 32 q^{4} + 116 q^{10} - 204 q^{16} + 476 q^{22} - 500 q^{25} - 884 q^{40} + 1808 q^{43} + 1372 q^{49} - 676 q^{52} - 3232 q^{55} + 1632 q^{64} + 2924 q^{82} - 2756 q^{88} - 5140 q^{94}+O(q^{100}) $ 4 * q - 32 * q^4 + 116 * q^10 - 204 * q^16 + 476 * q^22 - 500 * q^25 - 884 * q^40 + 1808 * q^43 + 1372 * q^49 - 676 * q^52 - 3232 * q^55 + 1632 * q^64 + 2924 * q^82 - 2756 * q^88 - 5140 * q^94

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} + 5x^{2} + 3 $ x^4 + 5*x^2 + 3 :

$\beta_{1}$	$=$	$ \nu^{3} + 6\nu $ v^3 + 6*v
$\beta_{2}$	$=$	$ 2\nu^{2} + 5 $ 2*v^2 + 5
$\beta_{3}$	$=$	$ -6\nu^{3} - 18\nu $ -6v^3 - 18v

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

$\nu$	$=$	$ ( \beta_{3} + 6\beta_1 ) / 18 $ (b3 + 6*b1) / 18
$\nu^{2}$	$=$	$ ( \beta_{2} - 5 ) / 2 $ (b2 - 5) / 2
$\nu^{3}$	$=$	$ ( -\beta_{3} - 3\beta_1 ) / 3 $ (-b3 - 3*b1) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$-1$	$1$

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} $

64.1

	−	0.835000i
	−	2.07431i
		2.07431i
		0.835000i

−

4.42782i

−11.6056

20.3925i

15.9647i

90.2944

64.2

−

3.52058i

−4.39445

−

9.17304i

−

12.6936i

−32.2944

64.3

3.52058i

−4.39445

9.17304i

12.6936i

−32.2944

64.4

4.42782i

−11.6056

−

20.3925i

−

15.9647i

90.2944

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
39.d	odd	2	1	CM by $\Q(\sqrt{-39}) $
3.b	odd	2	1	inner
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	117.4.b.c	✓	4
3.b	odd	2	1	inner	117.4.b.c	✓	4
13.b	even	2	1	inner	117.4.b.c	✓	4
13.d	odd	4	2		1521.4.a.y		4
39.d	odd	2	1	CM	117.4.b.c	✓	4
39.f	even	4	2		1521.4.a.y		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.4.b.c	✓	4	1.a	even	1	1	trivial
117.4.b.c	✓	4	3.b	odd	2	1	inner
117.4.b.c	✓	4	13.b	even	2	1	inner
117.4.b.c	✓	4	39.d	odd	2	1	CM
1521.4.a.y		4	13.d	odd	4	2
1521.4.a.y		4	39.f	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{4} + 32T_{2}^{2} + 243 $ T2^4 + 32*T2^2 + 243 acting on $S_{4}^{\mathrm{new}}(117, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} + 32T^{2} + 243 $ T^4 + 32*T^2 + 243
$3$	$ T^{4} $ T^4
$5$	$ T^{4} + 500 T^{2} + 34992 $ T^4 + 500*T^2 + 34992
$7$	$ T^{4} $ T^4
$11$	$ T^{4} + 5324 T^{2} + \cdots + 2056752 $ T^4 + 5324*T^2 + 2056752
$13$	$ (T^{2} - 2197)^{2} $ (T^2 - 2197)^2
$17$	$ 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} + 275684 T^{2} + \cdots + 9184890672 $ T^4 + 275684*T^2 + 9184890672
$43$	$ (T - 452)^{4} $ (T - 452)^4
$47$	$ T^{4} + 415292 T^{2} + \cdots + 2077595568 $ T^4 + 415292*T^2 + 2077595568
$53$	$ T^{4} $ T^4
$59$	$ T^{4} + 821516 T^{2} + \cdots + 163107544752 $ T^4 + 821516*T^2 + 163107544752
$61$	$ (T^{2} - 892372)^{2} $ (T^2 - 892372)^2
$67$	$ T^{4} $ T^4
$71$	$ T^{4} + 1431644 T^{2} + \cdots + 21709693872 $ T^4 + 1431644*T^2 + 21709693872
$73$	$ T^{4} $ T^4
$79$	$ (T^{2} - 174928)^{2} $ (T^2 - 174928)^2
$83$	$ T^{4} + 2287148 T^{2} + \cdots + 20406871728 $ T^4 + 2287148*T^2 + 20406871728
$89$	$ T^{4} + 2819876 T^{2} + \cdots + 67381252272 $ T^4 + 2819876*T^2 + 67381252272
$97$	$ T^{4} $ T^4
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Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + ( - \beta_{2} - 8) q^{4} + (\beta_{3} - 2 \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{8}+O(q^{10}) \) q + b1 * q^2 + (-b2 - 8) * q^4 + (b3 - 2b1) q^5 + (b3 - b1) * q^8	\( q + \beta_1 q^{2} + ( - \beta_{2} - 8) q^{4} + (\beta_{3} - 2 \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{8} + (17 \beta_{2} + 29) q^{10} + (3 \beta_{3} - 8 \beta_1) q^{11} + 13 \beta_{2} q^{13} + (8 \beta_{2} - 51) q^{16} + ( - 9 \beta_{3} + 30 \beta_1) q^{20} + (53 \beta_{2} + 119) q^{22} + ( - 46 \beta_{2} - 125) q^{25} + ( - 13 \beta_{3} + 13 \beta_1) q^{26} - 51 \beta_1 q^{32} + ( - 29 \beta_{2} - 221) q^{40} + (23 \beta_{3} - 50 \beta_1) q^{41} + 452 q^{43} + ( - 29 \beta_{3} + 108 \beta_1) q^{44} + (23 \beta_{3} + 76 \beta_1) q^{47} + 343 q^{49} + (46 \beta_{3} - 171 \beta_1) q^{50} + ( - 104 \beta_{2} - 169) q^{52} + ( - 172 \beta_{2} - 808) q^{55} + (\beta_{3} + 160 \beta_1) q^{59} - 262 \beta_{2} q^{61} + (115 \beta_{2} + 408) q^{64} + (13 \beta_{3} - 182 \beta_1) q^{65} + ( - 43 \beta_{3} - 140 \beta_1) q^{71} + 116 \beta_{2} q^{79} + ( - 43 \beta_{3} - 10 \beta_1) q^{80} + (395 \beta_{2} + 731) q^{82} + ( - 51 \beta_{3} + 208 \beta_1) q^{83} + 452 \beta_1 q^{86} + ( - 119 \beta_{2} - 689) q^{88} + ( - 61 \beta_{3} - 194 \beta_1) q^{89} + (269 \beta_{2} - 1285) q^{94} + 343 \beta_1 q^{98}+O(q^{100}) \) q + b1 * q^2 + (-b2 - 8) * q^4 + (b3 - 2b1) q^5 + (b3 - b1) * q^8 + (17b2 + 29) q^10 + (3b3 - 8b1) * q^11 + 13b2 q^13 + (8b2 - 51) q^16 + (-9b3 + 30b1) * q^20 + (53b2 + 119) q^22 + (-46b2 - 125) q^25 + (-13b3 + 13b1) * q^26 - 51b1 q^32 + (-29b2 - 221) q^40 + (23b3 - 50b1) * q^41 + 452 * q^43 + (-29b3 + 108b1) * q^44 + (23b3 + 76b1) * q^47 + 343 * q^49 + (46b3 - 171b1) * q^50 + (-104b2 - 169) q^52 + (-172b2 - 808) q^55 + (b3 + 160b1) q^59 - 262b2 q^61 + (115b2 + 408) q^64 + (13b3 - 182b1) * q^65 + (-43b3 - 140b1) * q^71 + 116b2 q^79 + (-43b3 - 10b1) * q^80 + (395b2 + 731) q^82 + (-51b3 + 208b1) * q^83 + 452b1 q^86 + (-119b2 - 689) q^88 + (-61b3 - 194b1) * q^89 + (269b2 - 1285) q^94 + 343b1 q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4}+O(q^{10}) \) 4 * q - 32 * q^4	\( 4 q - 32 q^{4} + 116 q^{10} - 204 q^{16} + 476 q^{22} - 500 q^{25} - 884 q^{40} + 1808 q^{43} + 1372 q^{49} - 676 q^{52} - 3232 q^{55} + 1632 q^{64} + 2924 q^{82} - 2756 q^{88} - 5140 q^{94}+O(q^{100}) \) 4 * q - 32 * q^4 + 116 * q^10 - 204 * q^16 + 476 * q^22 - 500 * q^25 - 884 * q^40 + 1808 * q^43 + 1372 * q^49 - 676 * q^52 - 3232 * q^55 + 1632 * q^64 + 2924 * q^82 - 2756 * q^88 - 5140 * q^94

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