Properties

Label 117.2.q.d
Level $117$
Weight $2$
Character orbit 117.q
Analytic conductor $0.934$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(10,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.q (of order \(6\), degree \(2\), 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: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 \beta_{2} q^{4} - \beta_{3} q^{5} + (2 \beta_{2} - 4) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 \beta_{2} q^{4} - \beta_{3} q^{5} + (2 \beta_{2} - 4) q^{7} + \beta_{3} q^{8} + ( - 5 \beta_{2} + 5) q^{10} - 2 \beta_1 q^{11} + (3 \beta_{2} + 1) q^{13} + (2 \beta_{3} - 4 \beta_1) q^{14} + ( - \beta_{2} + 1) q^{16} + (\beta_{3} + \beta_1) q^{17} + (2 \beta_{2} - 4) q^{19} + ( - 3 \beta_{3} + 3 \beta_1) q^{20} - 10 \beta_{2} q^{22} + ( - 4 \beta_{3} + 2 \beta_1) q^{23} + (3 \beta_{3} + \beta_1) q^{26} + ( - 6 \beta_{2} - 6) q^{28} + (2 \beta_{3} - \beta_1) q^{29} + ( - 3 \beta_{3} + 3 \beta_1) q^{32} + (10 \beta_{2} - 5) q^{34} + (2 \beta_{3} + 2 \beta_1) q^{35} + (\beta_{2} + 1) q^{37} + (2 \beta_{3} - 4 \beta_1) q^{38} + 5 q^{40} + \beta_1 q^{41} - 2 \beta_{2} q^{43} - 6 \beta_{3} q^{44} + ( - 10 \beta_{2} + 20) q^{46} + 2 \beta_{3} q^{47} + ( - 5 \beta_{2} + 5) q^{49} + (12 \beta_{2} - 9) q^{52} + (3 \beta_{3} - 6 \beta_1) q^{53} + (10 \beta_{2} - 10) q^{55} + ( - 2 \beta_{3} - 2 \beta_1) q^{56} + (5 \beta_{2} - 10) q^{58} + (4 \beta_{3} - 4 \beta_1) q^{59} + 7 \beta_{2} q^{61} + 13 q^{64} + ( - 4 \beta_{3} + 3 \beta_1) q^{65} + ( - 2 \beta_{2} - 2) q^{67} + (6 \beta_{3} - 3 \beta_1) q^{68} + (20 \beta_{2} - 10) q^{70} + ( - 2 \beta_{3} + 2 \beta_1) q^{71} + ( - 18 \beta_{2} + 9) q^{73} + (\beta_{3} + \beta_1) q^{74} + ( - 6 \beta_{2} - 6) q^{76} + ( - 4 \beta_{3} + 8 \beta_1) q^{77} + 8 q^{79} - \beta_1 q^{80} + 5 \beta_{2} q^{82} + 2 \beta_{3} q^{83} + ( - 5 \beta_{2} + 10) q^{85} - 2 \beta_{3} q^{86} + ( - 10 \beta_{2} + 10) q^{88} - 2 \beta_1 q^{89} + ( - 4 \beta_{2} - 10) q^{91} + ( - 6 \beta_{3} + 12 \beta_1) q^{92} + (10 \beta_{2} - 10) q^{94} + (2 \beta_{3} + 2 \beta_1) q^{95} + ( - 4 \beta_{2} + 8) q^{97} + ( - 5 \beta_{3} + 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{4} - 12 q^{7} + 10 q^{10} + 10 q^{13} + 2 q^{16} - 12 q^{19} - 20 q^{22} - 36 q^{28} + 6 q^{37} + 20 q^{40} - 4 q^{43} + 60 q^{46} + 10 q^{49} - 12 q^{52} - 20 q^{55} - 30 q^{58} + 14 q^{61} + 52 q^{64} - 12 q^{67} - 36 q^{76} + 32 q^{79} + 10 q^{82} + 30 q^{85} + 20 q^{88} - 48 q^{91} - 20 q^{94} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) : Copy content Toggle raw display

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

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−1.93649 + 1.11803i
1.93649 1.11803i
−1.93649 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i 0 1.50000 2.59808i 2.23607i 0 −3.00000 1.73205i 2.23607i 0 2.50000 + 4.33013i
10.2 1.93649 1.11803i 0 1.50000 2.59808i 2.23607i 0 −3.00000 1.73205i 2.23607i 0 2.50000 + 4.33013i
82.1 −1.93649 1.11803i 0 1.50000 + 2.59808i 2.23607i 0 −3.00000 + 1.73205i 2.23607i 0 2.50000 4.33013i
82.2 1.93649 + 1.11803i 0 1.50000 + 2.59808i 2.23607i 0 −3.00000 + 1.73205i 2.23607i 0 2.50000 4.33013i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.q.d 4
3.b odd 2 1 inner 117.2.q.d 4
4.b odd 2 1 1872.2.by.l 4
12.b even 2 1 1872.2.by.l 4
13.c even 3 1 1521.2.b.g 4
13.e even 6 1 inner 117.2.q.d 4
13.e even 6 1 1521.2.b.g 4
13.f odd 12 2 1521.2.a.u 4
39.h odd 6 1 inner 117.2.q.d 4
39.h odd 6 1 1521.2.b.g 4
39.i odd 6 1 1521.2.b.g 4
39.k even 12 2 1521.2.a.u 4
52.i odd 6 1 1872.2.by.l 4
156.r even 6 1 1872.2.by.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.q.d 4 1.a even 1 1 trivial
117.2.q.d 4 3.b odd 2 1 inner
117.2.q.d 4 13.e even 6 1 inner
117.2.q.d 4 39.h odd 6 1 inner
1521.2.a.u 4 13.f odd 12 2
1521.2.a.u 4 39.k even 12 2
1521.2.b.g 4 13.c even 3 1
1521.2.b.g 4 13.e even 6 1
1521.2.b.g 4 39.h odd 6 1
1521.2.b.g 4 39.i odd 6 1
1872.2.by.l 4 4.b odd 2 1
1872.2.by.l 4 12.b even 2 1
1872.2.by.l 4 52.i odd 6 1
1872.2.by.l 4 156.r even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\):

\( T_{2}^{4} - 5T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{5}^{2} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5T^{2} + 25 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 20T^{2} + 400 \) Copy content Toggle raw display
$13$ \( (T^{2} - 5 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 15T^{2} + 225 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 60T^{2} + 3600 \) Copy content Toggle raw display
$29$ \( T^{4} + 15T^{2} + 225 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 5T^{2} + 25 \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 135)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 80T^{2} + 6400 \) Copy content Toggle raw display
$61$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 20T^{2} + 400 \) Copy content Toggle raw display
$73$ \( (T^{2} + 243)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 20T^{2} + 400 \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T + 48)^{2} \) Copy content Toggle raw display
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