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Properties

Label 117.6.g.b
Level $117$
Weight $6$
Character orbit 117.g
Analytic conductor $18.765$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.55");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(18.7649069181\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 70x^{6} - 133x^{5} + 4766x^{4} - 6777x^{3} + 16825x^{2} + 9696x + 9216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 13)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1 + 1) q^{2} + ( - \beta_{7} + \beta_{4} + \cdots + \beta_1) q^{4} + (\beta_{6} - \beta_{4} - 2 \beta_{2} + 2) q^{5} + (2 \beta_{7} + \beta_{6} + \cdots - 4 \beta_1) q^{7} + ( - 4 \beta_{6} + 3 \beta_{4} + \cdots - 2) q^{8}+ \cdots + (1049 \beta_{7} + 502 \beta_{6} + \cdots + 8633 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5 q^{2} - 17 q^{4} + 20 q^{5} + 68 q^{7} - 18 q^{8} + 303 q^{10} + 480 q^{11} - 234 q^{13} + 1324 q^{14} + 351 q^{16} - 60 q^{17} + 520 q^{19} - 4429 q^{20} - 3926 q^{22} - 6644 q^{23} - 13572 q^{25}+ \cdots + 53863 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 70x^{6} - 133x^{5} + 4766x^{4} - 6777x^{3} + 16825x^{2} + 9696x + 9216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 319677 \nu^{7} + 174777 \nu^{6} + 21464689 \nu^{5} - 9316963 \nu^{4} + 1509169756 \nu^{3} + \cdots + 3235111584 ) / 5480995715 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 33699079 \nu^{7} - 64388071 \nu^{6} + 2342156938 \nu^{5} - 6542587651 \nu^{4} + \cdots + 318334817568 ) / 526175588640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 814131 \nu^{7} - 737924 \nu^{6} + 54664767 \nu^{5} - 23727789 \nu^{4} + 3798492578 \nu^{3} + \cdots + 192123818377 ) / 5480995715 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 163476431 \nu^{7} + 338880779 \nu^{6} - 12327003362 \nu^{5} + 33519877559 \nu^{4} + \cdots + 1093886512128 ) / 263087794320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 319677 \nu^{7} + 174777 \nu^{6} + 21464689 \nu^{5} - 9316963 \nu^{4} + 1424846745 \nu^{3} + \cdots + 6608032024 ) / 337292044 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 87076937 \nu^{7} + 166612553 \nu^{6} - 6060637334 \nu^{5} + 17508240653 \nu^{4} + \cdots + 537815172096 ) / 40475045280 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{4} - 35\beta_{3} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{6} + 65\beta_{2} + 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 69\beta_{7} - 4\beta_{5} + 2279\beta_{3} + 105\beta _1 - 2279 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -32\beta_{7} + 280\beta_{6} - 280\beta_{5} + 32\beta_{4} - 4016\beta_{3} - 4385\beta_{2} - 4385\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 152\beta_{6} - 4633\beta_{4} + 9329\beta_{2} + 153915 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4544\beta_{7} + 18684\beta_{5} + 349528\beta_{3} + 297601\beta _1 - 349528 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-\beta_{3}\) \(1\)

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
−4.21857 − 7.30677i
−0.329369 − 0.570484i
1.09142 + 1.89039i
3.95652 + 6.85289i
−4.21857 + 7.30677i
−0.329369 + 0.570484i
1.09142 − 1.89039i
3.95652 − 6.85289i
−3.71857 − 6.44074i 0 −11.6555 + 20.1878i 9.82751 0 17.3506 − 30.0522i −64.6219 0 −36.5442 − 63.2965i
55.2 0.170631 + 0.295541i 0 15.9418 − 27.6120i −13.9092 0 −18.2258 + 31.5679i 21.8010 0 −2.37335 − 4.11076i
55.3 1.59142 + 2.75641i 0 10.9348 − 18.9396i −44.5576 0 −27.1222 + 46.9770i 171.458 0 −70.9097 − 122.819i
55.4 4.45652 + 7.71892i 0 −23.7211 + 41.0862i 58.6393 0 61.9973 − 107.382i −137.637 0 261.327 + 452.632i
100.1 −3.71857 + 6.44074i 0 −11.6555 − 20.1878i 9.82751 0 17.3506 + 30.0522i −64.6219 0 −36.5442 + 63.2965i
100.2 0.170631 − 0.295541i 0 15.9418 + 27.6120i −13.9092 0 −18.2258 − 31.5679i 21.8010 0 −2.37335 + 4.11076i
100.3 1.59142 − 2.75641i 0 10.9348 + 18.9396i −44.5576 0 −27.1222 − 46.9770i 171.458 0 −70.9097 + 122.819i
100.4 4.45652 − 7.71892i 0 −23.7211 − 41.0862i 58.6393 0 61.9973 + 107.382i −137.637 0 261.327 − 452.632i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.4
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 117.6.g.b 8
3.b odd 2 1 13.6.c.a ✓ 8
12.b even 2 1 208.6.i.b 8
13.c even 3 1 inner 117.6.g.b 8
39.h odd 6 1 169.6.a.c 4
39.i odd 6 1 13.6.c.a ✓ 8
39.i odd 6 1 169.6.a.d 4
39.k even 12 2 169.6.b.c 8
156.p even 6 1 208.6.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.6.c.a ✓ 8 3.b odd 2 1
13.6.c.a ✓ 8 39.i odd 6 1
117.6.g.b 8 1.a even 1 1 trivial
117.6.g.b 8 13.c even 3 1 inner
169.6.a.c 4 39.h odd 6 1
169.6.a.d 4 39.i odd 6 1
169.6.b.c 8 39.k even 12 2
208.6.i.b 8 12.b even 2 1
208.6.i.b 8 156.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 5T_{2}^{7} + 85T_{2}^{6} - 164T_{2}^{5} + 4832T_{2}^{4} - 14640T_{2}^{3} + 49504T_{2}^{2} - 16704T_{2} + 5184 \) acting on \(S_{6}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 5 T^{7} + \cdots + 5184 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 10 T^{3} + \cdots + 357156)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 72383069214976 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 40\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 14\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 18\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 4693845913600)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 28\!\cdots\!09 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 96\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 63\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 24\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 38\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 13\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 69\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 34\!\cdots\!48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 30\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 91\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
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