Properties

Label 117.4.a.c
Level $117$
Weight $4$
Character orbit 117.a
Self dual yes
Analytic conductor $6.903$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,4,Mod(1,117)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)

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: yes
Analytic conductor: \(6.90322347067\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + ( - 2 \beta + 7) q^{4} + ( - 2 \beta - 12) q^{5} + 2 \beta q^{7} + (\beta - 27) q^{8} + ( - 10 \beta - 16) q^{10} + ( - 12 \beta + 22) q^{11} - 13 q^{13} + ( - 2 \beta + 28) q^{14}+ \cdots + ( - 287 \beta + 287) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8} - 32 q^{10} + 44 q^{11} - 26 q^{13} + 56 q^{14} - 30 q^{16} - 164 q^{17} + 48 q^{19} - 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 26 q^{26} - 112 q^{28}+ \cdots + 574 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.74166
3.74166
−4.74166 0 14.4833 −4.51669 0 −7.48331 −30.7417 0 21.4166
1.2 2.74166 0 −0.483315 −19.4833 0 7.48331 −23.2583 0 −53.4166
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.a.c 2
3.b odd 2 1 39.4.a.b 2
4.b odd 2 1 1872.4.a.t 2
12.b even 2 1 624.4.a.r 2
13.b even 2 1 1521.4.a.s 2
15.d odd 2 1 975.4.a.j 2
21.c even 2 1 1911.4.a.h 2
24.f even 2 1 2496.4.a.s 2
24.h odd 2 1 2496.4.a.bc 2
39.d odd 2 1 507.4.a.f 2
39.f even 4 2 507.4.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.b 2 3.b odd 2 1
117.4.a.c 2 1.a even 1 1 trivial
507.4.a.f 2 39.d odd 2 1
507.4.b.f 4 39.f even 4 2
624.4.a.r 2 12.b even 2 1
975.4.a.j 2 15.d odd 2 1
1521.4.a.s 2 13.b even 2 1
1872.4.a.t 2 4.b odd 2 1
1911.4.a.h 2 21.c even 2 1
2496.4.a.s 2 24.f even 2 1
2496.4.a.bc 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 13 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(117))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 13 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 24T + 88 \) Copy content Toggle raw display
$7$ \( T^{2} - 56 \) Copy content Toggle raw display
$11$ \( T^{2} - 44T - 1532 \) Copy content Toggle raw display
$13$ \( (T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 164T + 6500 \) Copy content Toggle raw display
$19$ \( T^{2} - 48T + 520 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T - 32240 \) Copy content Toggle raw display
$29$ \( T^{2} + 404T + 32740 \) Copy content Toggle raw display
$31$ \( T^{2} - 40T - 9064 \) Copy content Toggle raw display
$37$ \( T^{2} + 100T - 8476 \) Copy content Toggle raw display
$41$ \( T^{2} + 200T - 113704 \) Copy content Toggle raw display
$43$ \( T^{2} + 616T + 57008 \) Copy content Toggle raw display
$47$ \( T^{2} - 324T + 11908 \) Copy content Toggle raw display
$53$ \( T^{2} - 164T - 194876 \) Copy content Toggle raw display
$59$ \( T^{2} + 140T - 17500 \) Copy content Toggle raw display
$61$ \( T^{2} - 628T - 160348 \) Copy content Toggle raw display
$67$ \( T^{2} + 472T - 348904 \) Copy content Toggle raw display
$71$ \( T^{2} + 428T - 52988 \) Copy content Toggle raw display
$73$ \( T^{2} + 900T + 121636 \) Copy content Toggle raw display
$79$ \( T^{2} + 432T - 61760 \) Copy content Toggle raw display
$83$ \( T^{2} - 1388 T + 424292 \) Copy content Toggle raw display
$89$ \( T^{2} + 960T - 275000 \) Copy content Toggle raw display
$97$ \( T^{2} + 532T - 606844 \) Copy content Toggle raw display
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