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Properties

Label 117.3.d.a
Level $117$
Weight $3$
Character orbit 117.d
Analytic conductor $3.188$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,3,Mod(116,117)]
 
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([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.116");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 117.d (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: \(3.18801909302\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.138169810944.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2x^{6} - 3x^{4} + 18x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{6} q^{5} - \beta_{4} q^{7} + (\beta_{6} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{6} q^{5} - \beta_{4} q^{7} + (\beta_{6} - \beta_1) q^{8} + (\beta_{2} + 2) q^{10} + (3 \beta_{6} + 2 \beta_1) q^{11} + ( - \beta_{5} - \beta_{4} - \beta_{2} - 3) q^{13} - \beta_{3} q^{14} + ( - 2 \beta_{2} + 3) q^{16} + ( - \beta_{7} - \beta_{3}) q^{17} + ( - 2 \beta_{5} + 3 \beta_{4}) q^{19} + ( - 3 \beta_{6} - 6 \beta_1) q^{20} + (\beta_{2} - 4) q^{22} + ( - \beta_{7} - 2 \beta_{3}) q^{23} + ( - 2 \beta_{2} - 11) q^{25} + (\beta_{7} - \beta_{6} + \cdots + 7 \beta_1) q^{26}+ \cdots + (6 \beta_{6} - 43 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} + 16 q^{10} - 24 q^{13} + 24 q^{16} - 32 q^{22} - 88 q^{25} + 128 q^{40} + 16 q^{43} + 152 q^{49} - 200 q^{52} + 304 q^{55} - 240 q^{61} - 552 q^{64} + 576 q^{79} - 176 q^{82} + 272 q^{88} - 336 q^{91} + 1024 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 2x^{6} - 3x^{4} + 18x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 2\nu^{5} + 3\nu^{3} + 9\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 2\nu^{4} + 12\nu^{2} - 9 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} + 6\nu^{2} - 27 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} + \nu^{5} - 21\nu^{3} + 81\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} + 10\nu^{5} + 24\nu^{3} + 108\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{7} + 5\nu^{5} - 3\nu^{3} - 63\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{6} + 4\nu^{4} + 24\nu^{2} + 162 ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{4} + 6\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{3} + 6\beta_{2} - 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{6} + 2\beta_{5} - 5\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + 20\beta_{3} - 12\beta_{2} + 30 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 30\beta_{6} + 13\beta_{5} + 8\beta_{4} - 18\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{7} - 8\beta_{3} - 6\beta_{2} - 120 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -78\beta_{6} - 5\beta_{5} - 28\beta_{4} - 234\beta_1 ) / 12 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
1.55647 − 0.759866i
1.55647 + 0.759866i
0.278201 + 1.70956i
0.278201 − 1.70956i
−0.278201 + 1.70956i
−0.278201 − 1.70956i
−1.55647 − 0.759866i
−1.55647 + 0.759866i
−3.11294 0 5.69042 −2.14923 0 − 1.36290i −5.26217 0 6.69042
116.2 −3.11294 0 5.69042 −2.14923 0 1.36290i −5.26217 0 6.69042
116.3 −0.556403 0 −3.69042 4.83537 0 − 7.62512i 4.27897 0 −2.69042
116.4 −0.556403 0 −3.69042 4.83537 0 7.62512i 4.27897 0 −2.69042
116.5 0.556403 0 −3.69042 −4.83537 0 − 7.62512i −4.27897 0 −2.69042
116.6 0.556403 0 −3.69042 −4.83537 0 7.62512i −4.27897 0 −2.69042
116.7 3.11294 0 5.69042 2.14923 0 − 1.36290i 5.26217 0 6.69042
116.8 3.11294 0 5.69042 2.14923 0 1.36290i 5.26217 0 6.69042
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.3.d.a ✓ 8
3.b odd 2 1 inner 117.3.d.a ✓ 8
4.b odd 2 1 1872.3.l.d 8
12.b even 2 1 1872.3.l.d 8
13.b even 2 1 inner 117.3.d.a ✓ 8
39.d odd 2 1 inner 117.3.d.a ✓ 8
52.b odd 2 1 1872.3.l.d 8
156.h even 2 1 1872.3.l.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.3.d.a ✓ 8 1.a even 1 1 trivial
117.3.d.a ✓ 8 3.b odd 2 1 inner
117.3.d.a ✓ 8 13.b even 2 1 inner
117.3.d.a ✓ 8 39.d odd 2 1 inner
1872.3.l.d 8 4.b odd 2 1
1872.3.l.d 8 12.b even 2 1
1872.3.l.d 8 52.b odd 2 1
1872.3.l.d 8 156.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(117, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 10 T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 28 T^{2} + 108)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 60 T^{2} + 108)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 244 T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 12 T^{3} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 828 T^{2} + 142884)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 924 T^{2} + 117612)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 936 T^{2} + 104976)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 3492 T^{2} + 2022084)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1428 T^{2} + 375948)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2040 T^{2} + 657072)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 220 T^{2} + 1452)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 788)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 6580 T^{2} + 2312652)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 5148 T^{2} + 3175524)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 3484 T^{2} + 2027052)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 60 T + 548)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 11412 T^{2} + 32551308)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 23452 T^{2} + 100433388)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 17064 T^{2} + 29428272)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 144 T + 5096)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 17596 T^{2} + 77358252)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 11068 T^{2} + 4869228)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 28968 T^{2} + 197998128)^{2} \) Copy content Toggle raw display
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