↑

Properties

Label 117.12.b.e
Level $117$
Weight $12$
Character orbit 117.b
Analytic conductor $89.896$
Analytic rank $0$
Dimension $20$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,12,Mod(64,117)]
 
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, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.64");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 117.b (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: \(89.8961521255\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 6409296870 x^{18} + \cdots + 25\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{56}\cdot 3^{31}\cdot 5^{4}\cdot 11^{4}\cdot 13^{8} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + ( - \beta_1 - 614) q^{4} + ( - \beta_{8} - 42 \beta_{7}) q^{5} - \beta_{9} q^{7} + (\beta_{14} - 3 \beta_{8} + 1082 \beta_{7}) q^{8} + (5 \beta_{2} - 25 \beta_1 - 110600) q^{10} + (\beta_{15} + \beta_{14} + \cdots - 1670 \beta_{7}) q^{11}+ \cdots + ( - 18990 \beta_{16} + \cdots + 920036601 \beta_{7}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 12288 q^{4} - 2212240 q^{10} - 1708564 q^{13} + 32533888 q^{16} - 89284144 q^{22} + 209701380 q^{25} + 309455680 q^{40} + 2936996032 q^{43} - 10965350212 q^{49} + 4282707936 q^{52} + 6622126880 q^{55}+ \cdots + 330546354704 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 6409296870 x^{18} + \cdots + 25\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 70\!\cdots\!65 \nu^{18} + \cdots - 27\!\cdots\!21 ) / 11\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 48\!\cdots\!95 \nu^{18} + \cdots - 84\!\cdots\!39 ) / 90\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 31\!\cdots\!91 \nu^{18} + \cdots - 62\!\cdots\!45 ) / 90\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31\!\cdots\!25 \nu^{18} + \cdots + 16\!\cdots\!49 ) / 17\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 48\!\cdots\!76 \nu^{18} + \cdots - 16\!\cdots\!97 ) / 17\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 28\!\cdots\!00 \nu^{18} + \cdots + 52\!\cdots\!79 ) / 27\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17\!\cdots\!67 \nu^{19} + \cdots + 11\!\cdots\!47 \nu ) / 98\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 84\!\cdots\!53 \nu^{19} + \cdots + 84\!\cdots\!67 \nu ) / 68\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35\!\cdots\!66 \nu^{19} + \cdots - 65\!\cdots\!81 \nu ) / 60\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 47\!\cdots\!43 \nu^{19} + \cdots + 67\!\cdots\!90 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 47\!\cdots\!43 \nu^{19} + \cdots + 67\!\cdots\!90 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30\!\cdots\!63 \nu^{19} + \cdots + 77\!\cdots\!25 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 30\!\cdots\!63 \nu^{19} + \cdots + 32\!\cdots\!35 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 60\!\cdots\!39 \nu^{19} + \cdots - 35\!\cdots\!33 \nu ) / 13\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 68\!\cdots\!21 \nu^{19} + \cdots + 93\!\cdots\!21 \nu ) / 13\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 44\!\cdots\!61 \nu^{19} + \cdots - 23\!\cdots\!96 \nu ) / 68\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 40\!\cdots\!53 \nu^{19} + \cdots + 10\!\cdots\!12 \nu ) / 52\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 20\!\cdots\!91 \nu^{19} + \cdots + 12\!\cdots\!84 \nu ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 58\!\cdots\!92 \nu^{19} + \cdots - 67\!\cdots\!15 \nu ) / 40\!\cdots\!74 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + 41\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 269 \beta_{11} + \cdots - 2563668808 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 454386 \beta_{19} + 202489 \beta_{18} + 2308253 \beta_{17} - 6332590 \beta_{16} + \cdots - 70207130152 \beta_{7} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1043403323 \beta_{16} + 1043403323 \beta_{15} - 1043403323 \beta_{14} - 1981056679 \beta_{13} + \cdots + 24\!\cdots\!96 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 762313357853717 \beta_{19} - 391166528334173 \beta_{18} + \cdots - 62\!\cdots\!70 \beta_{7} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 26\!\cdots\!37 \beta_{16} + \cdots - 26\!\cdots\!84 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 54\!\cdots\!50 \beta_{19} + \cdots + 10\!\cdots\!84 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 17\!\cdots\!00 \beta_{16} + \cdots + 31\!\cdots\!12 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 14\!\cdots\!24 \beta_{19} + \cdots - 38\!\cdots\!28 \beta_{7} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 75\!\cdots\!63 \beta_{16} + \cdots - 79\!\cdots\!80 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 19\!\cdots\!11 \beta_{19} + \cdots + 64\!\cdots\!54 \beta_{7} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 16\!\cdots\!56 \beta_{16} + \cdots + 12\!\cdots\!05 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 67\!\cdots\!94 \beta_{19} + \cdots - 25\!\cdots\!93 \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 10\!\cdots\!44 \beta_{16} + \cdots - 66\!\cdots\!00 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 36\!\cdots\!14 \beta_{19} + \cdots + 16\!\cdots\!76 \beta_{7} ) / 8 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 16\!\cdots\!00 \beta_{16} + \cdots + 87\!\cdots\!48 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 48\!\cdots\!17 \beta_{19} + \cdots - 24\!\cdots\!90 \beta_{7} ) / 8 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 24\!\cdots\!89 \beta_{16} + \cdots - 11\!\cdots\!00 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 32\!\cdots\!58 \beta_{19} + \cdots + 18\!\cdots\!32 \beta_{7} ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
29430.1i
− 29001.2i
21451.0i
− 12976.2i
35954.6i
− 37188.3i
23849.3i
− 27176.5i
9478.47i
818.316i
− 818.316i
− 9478.47i
27176.5i
− 23849.3i
37188.3i
− 35954.6i
12976.2i
− 21451.0i
29001.2i
− 29430.1i
− 84.2684i 0 −5053.16 − 513.228i 0 − 58431.3i 253240.i 0 −43248.9
64.2 − 84.2684i 0 −5053.16 − 513.228i 0 58431.3i 253240.i 0 −43248.9
64.3 − 63.1311i 0 −1937.54 − 8537.94i 0 − 34427.2i − 6973.77i 0 −539010.
64.4 − 63.1311i 0 −1937.54 − 8537.94i 0 34427.2i − 6973.77i 0 −539010.
64.5 − 38.6066i 0 557.529 1195.10i 0 − 73143.0i − 100591.i 0 46138.6
64.6 − 38.6066i 0 557.529 1195.10i 0 73143.0i − 100591.i 0 46138.6
64.7 − 25.2948i 0 1408.18 3301.89i 0 − 51025.8i − 87423.1i 0 83520.4
64.8 − 25.2948i 0 1408.18 3301.89i 0 51025.8i − 87423.1i 0 83520.4
64.9 − 9.74725i 0 1952.99 − 10306.5i 0 − 8660.16i − 38998.7i 0 −100460.
64.10 − 9.74725i 0 1952.99 − 10306.5i 0 8660.16i − 38998.7i 0 −100460.
64.11 9.74725i 0 1952.99 10306.5i 0 − 8660.16i 38998.7i 0 −100460.
64.12 9.74725i 0 1952.99 10306.5i 0 8660.16i 38998.7i 0 −100460.
64.13 25.2948i 0 1408.18 − 3301.89i 0 − 51025.8i 87423.1i 0 83520.4
64.14 25.2948i 0 1408.18 − 3301.89i 0 51025.8i 87423.1i 0 83520.4
64.15 38.6066i 0 557.529 − 1195.10i 0 − 73143.0i 100591.i 0 46138.6
64.16 38.6066i 0 557.529 − 1195.10i 0 73143.0i 100591.i 0 46138.6
64.17 63.1311i 0 −1937.54 8537.94i 0 − 34427.2i 6973.77i 0 −539010.
64.18 63.1311i 0 −1937.54 8537.94i 0 34427.2i 6973.77i 0 −539010.
64.19 84.2684i 0 −5053.16 513.228i 0 − 58431.3i − 253240.i 0 −43248.9
64.20 84.2684i 0 −5053.16 513.228i 0 58431.3i − 253240.i 0 −43248.9
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.20
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 117.12.b.e ✓ 20
3.b odd 2 1 inner 117.12.b.e ✓ 20
13.b even 2 1 inner 117.12.b.e ✓ 20
39.d odd 2 1 inner 117.12.b.e ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.12.b.e ✓ 20 1.a even 1 1 trivial
117.12.b.e ✓ 20 3.b odd 2 1 inner
117.12.b.e ✓ 20 13.b even 2 1 inner
117.12.b.e ✓ 20 39.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 13312T_{2}^{8} + 54129232T_{2}^{6} + 75887622528T_{2}^{4} + 33722566705152T_{2}^{2} + 2564274585600000 \) acting on \(S_{12}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 18\!\cdots\!57)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 94\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 59\!\cdots\!00)^{4} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 90\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 93\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 26\!\cdots\!84)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 34\!\cdots\!68)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 15\!\cdots\!60)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less