Properties

Label 117.4.b.c
Level $117$
Weight $4$
Character orbit 117.b
Analytic conductor $6.903$
Analytic rank $0$
Dimension $4$
CM discriminant -39
Inner twists $4$

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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: \( x^{4} + 5x^{2} + 3 \) Copy content Toggle raw display
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 5x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -6\nu^{3} - 18\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 6\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} - 3\beta_1 ) / 3 \) Copy content Toggle raw display

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 0 −11.6056 20.3925i 0 0 15.9647i 0 90.2944
64.2 3.52058i 0 −4.39445 9.17304i 0 0 12.6936i 0 −32.2944
64.3 3.52058i 0 −4.39445 9.17304i 0 0 12.6936i 0 −32.2944
64.4 4.42782i 0 −11.6056 20.3925i 0 0 15.9647i 0 90.2944
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 \( T_{2}^{4} + 32T_{2}^{2} + 243 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 32T^{2} + 243 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 500 T^{2} + 34992 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 5324 T^{2} + 2056752 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2197)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 9184890672 \) Copy content Toggle raw display
$43$ \( (T - 452)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 2077595568 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 163107544752 \) Copy content Toggle raw display
$61$ \( (T^{2} - 892372)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 21709693872 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 174928)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 20406871728 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 67381252272 \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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