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Properties

Label 117.2.g.c
Level $117$
Weight $2$
Character orbit 117.g
Analytic conductor $0.934$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(55,117)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.g (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + (\beta_{3} + 2) q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{3} - 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + (\beta_{3} + 2) q^{5} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{3} - 4) q^{8} + ( - 4 \beta_{2} + \beta_1) q^{10} - 2 \beta_{2} q^{11} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{13} + ( - 2 \beta_{3} + 4) q^{14} - 3 \beta_1 q^{16} + ( - \beta_{3} - \beta_1) q^{17} + ( - 2 \beta_{3} + 4 \beta_{2} + \cdots - 4) q^{19}+ \cdots + ( - \beta_{3} - 12 \beta_{2} + \cdots + 12) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 5 q^{4} + 6 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 5 q^{4} + 6 q^{5} + 3 q^{7} - 18 q^{8} - 7 q^{10} - 4 q^{11} + 2 q^{13} + 20 q^{14} - 3 q^{16} + q^{17} - 6 q^{19} + q^{20} + 2 q^{22} + 4 q^{23} + 6 q^{25} - 25 q^{26} + 16 q^{28} + q^{29} + 2 q^{31} + 9 q^{32} + 18 q^{34} - 4 q^{35} - 11 q^{37} + 28 q^{38} - 10 q^{40} + q^{41} - 5 q^{43} + 20 q^{44} - 2 q^{46} + q^{49} - 24 q^{50} - 28 q^{52} - 22 q^{53} - 6 q^{55} - 22 q^{56} + 25 q^{58} - 14 q^{59} - 16 q^{61} - 8 q^{62} + 14 q^{64} + 3 q^{65} - 5 q^{67} + 11 q^{68} - 4 q^{70} + 28 q^{71} - 24 q^{73} - 3 q^{74} + 2 q^{76} - 12 q^{77} + 30 q^{79} + 21 q^{80} - 9 q^{82} + 20 q^{83} - 7 q^{85} + 12 q^{86} + 18 q^{88} + 18 q^{89} + 27 q^{91} - 20 q^{92} + 34 q^{94} - 26 q^{95} + 13 q^{97} + 25 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−0.780776 − 1.35234i
1.28078 + 2.21837i
−0.780776 + 1.35234i
1.28078 − 2.21837i
−0.780776 − 1.35234i 0 −0.219224 + 0.379706i 3.56155 0 −0.280776 + 0.486319i −2.43845 0 −2.78078 − 4.81645i
55.2 1.28078 + 2.21837i 0 −2.28078 + 3.95042i −0.561553 0 1.78078 − 3.08440i −6.56155 0 −0.719224 − 1.24573i
100.1 −0.780776 + 1.35234i 0 −0.219224 − 0.379706i 3.56155 0 −0.280776 − 0.486319i −2.43845 0 −2.78078 + 4.81645i
100.2 1.28078 − 2.21837i 0 −2.28078 − 3.95042i −0.561553 0 1.78078 + 3.08440i −6.56155 0 −0.719224 + 1.24573i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.g.c 4
3.b odd 2 1 39.2.e.b ✓ 4
4.b odd 2 1 1872.2.t.r 4
12.b even 2 1 624.2.q.h 4
13.c even 3 1 inner 117.2.g.c 4
13.c even 3 1 1521.2.a.g 2
13.e even 6 1 1521.2.a.m 2
13.f odd 12 2 1521.2.b.h 4
15.d odd 2 1 975.2.i.k 4
15.e even 4 2 975.2.bb.i 8
39.d odd 2 1 507.2.e.g 4
39.f even 4 2 507.2.j.g 8
39.h odd 6 1 507.2.a.d 2
39.h odd 6 1 507.2.e.g 4
39.i odd 6 1 39.2.e.b ✓ 4
39.i odd 6 1 507.2.a.g 2
39.k even 12 2 507.2.b.d 4
39.k even 12 2 507.2.j.g 8
52.j odd 6 1 1872.2.t.r 4
156.p even 6 1 624.2.q.h 4
156.p even 6 1 8112.2.a.bk 2
156.r even 6 1 8112.2.a.bo 2
195.x odd 6 1 975.2.i.k 4
195.bl even 12 2 975.2.bb.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b ✓ 4 3.b odd 2 1
39.2.e.b ✓ 4 39.i odd 6 1
117.2.g.c 4 1.a even 1 1 trivial
117.2.g.c 4 13.c even 3 1 inner
507.2.a.d 2 39.h odd 6 1
507.2.a.g 2 39.i odd 6 1
507.2.b.d 4 39.k even 12 2
507.2.e.g 4 39.d odd 2 1
507.2.e.g 4 39.h odd 6 1
507.2.j.g 8 39.f even 4 2
507.2.j.g 8 39.k even 12 2
624.2.q.h 4 12.b even 2 1
624.2.q.h 4 156.p even 6 1
975.2.i.k 4 15.d odd 2 1
975.2.i.k 4 195.x odd 6 1
975.2.bb.i 8 15.e even 4 2
975.2.bb.i 8 195.bl even 12 2
1521.2.a.g 2 13.c even 3 1
1521.2.a.m 2 13.e even 6 1
1521.2.b.h 4 13.f odd 12 2
1872.2.t.r 4 4.b odd 2 1
1872.2.t.r 4 52.j odd 6 1
8112.2.a.bk 2 156.p even 6 1
8112.2.a.bo 2 156.r even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} + 5T_{2}^{2} + 4T_{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - T^{3} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - T^{3} + \cdots + 1444 \) Copy content Toggle raw display
$31$ \( (T^{2} - T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 11 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$41$ \( T^{4} - T^{3} + \cdots + 16 \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$47$ \( (T^{2} - 68)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 11 T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 14 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$61$ \( T^{4} + 16 T^{3} + \cdots + 2209 \) Copy content Toggle raw display
$67$ \( T^{4} + 5 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$71$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 19)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 15 T + 52)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 10 T + 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 18 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$97$ \( T^{4} - 13 T^{3} + \cdots + 1444 \) Copy content Toggle raw display
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