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Properties

Label 117.6.g.e
Level $117$
Weight $6$
Character orbit 117.g
Analytic conductor $18.765$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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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: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 256 x^{18} + 42243 x^{16} + 4153796 x^{14} + 297063177 x^{12} + 14013245062 x^{10} + \cdots + 49\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{5} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{7} + \beta_{4} - 19 \beta_{2}) q^{4} + \beta_{13} q^{5} + ( - \beta_{7} - \beta_{6} + \cdots - 22 \beta_{2}) q^{7} + (\beta_{14} - \beta_{13} + \cdots + 10 \beta_{3}) q^{8}+ \cdots + (20 \beta_{17} - 77 \beta_{14} + \cdots + 1484 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 192 q^{4} - 222 q^{7} - 204 q^{10} + 1042 q^{13} + 1012 q^{16} + 448 q^{19} + 3752 q^{22} + 9100 q^{25} + 1804 q^{28} - 6204 q^{31} + 23480 q^{34} + 20668 q^{37} - 36536 q^{40} - 16230 q^{43} + 272 q^{46}+ \cdots + 531382 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 256 x^{18} + 42243 x^{16} + 4153796 x^{14} + 297063177 x^{12} + 14013245062 x^{10} + \cdots + 49\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!04 \nu^{18} + \cdots - 33\!\cdots\!64 ) / 49\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!04 \nu^{19} + \cdots + 33\!\cdots\!64 \nu ) / 49\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!03 \nu^{18} + \cdots + 27\!\cdots\!60 ) / 68\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!29 \nu^{18} + \cdots - 95\!\cdots\!12 ) / 51\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27\!\cdots\!84 \nu^{18} + \cdots + 37\!\cdots\!44 ) / 68\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!27 \nu^{18} + \cdots - 12\!\cdots\!08 ) / 16\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 46\!\cdots\!69 \nu^{19} + \cdots + 77\!\cdots\!76 \nu ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 45\!\cdots\!11 \nu^{18} + \cdots - 46\!\cdots\!68 ) / 25\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13\!\cdots\!19 \nu^{19} + \cdots - 43\!\cdots\!28 \nu ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 63\!\cdots\!33 \nu^{19} + \cdots - 35\!\cdots\!28 \nu ) / 42\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12\!\cdots\!07 \nu^{18} + \cdots - 93\!\cdots\!48 ) / 33\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 44\!\cdots\!12 \nu^{19} + \cdots - 14\!\cdots\!92 \nu ) / 25\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19\!\cdots\!41 \nu^{19} + \cdots - 19\!\cdots\!88 \nu ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21\!\cdots\!21 \nu^{18} + \cdots - 13\!\cdots\!84 ) / 51\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 32\!\cdots\!77 \nu^{18} + \cdots + 18\!\cdots\!72 ) / 51\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 30\!\cdots\!83 \nu^{19} + \cdots - 90\!\cdots\!12 \nu ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 14\!\cdots\!44 \nu^{19} + \cdots + 51\!\cdots\!04 \nu ) / 25\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 35\!\cdots\!67 \nu^{19} + \cdots + 39\!\cdots\!20 \nu ) / 42\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{4} - 51\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} - \beta_{11} + 74\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{15} - 2\beta_{12} + 6\beta_{9} - 107\beta_{7} + 8\beta_{5} + 3767\beta_{2} - 3761 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{17} - 131\beta_{14} + 4\beta_{10} - 175\beta_{8} - 6276\beta_{3} - 6276\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 374 \beta_{16} + 962 \beta_{15} + 374 \beta_{12} - 1550 \beta_{9} + 1464 \beta_{6} - 1464 \beta_{5} + \cdots + 317999 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 716 \beta_{19} + 1464 \beta_{18} + 1464 \beta_{17} + 22083 \beta_{13} + 14815 \beta_{11} + \cdots + 568080 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 48846 \beta_{16} - 185862 \beta_{15} + 68508 \beta_{9} + 1022367 \beta_{7} - 193720 \beta_{6} + \cdots - 117354 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 96028 \beta_{19} - 193720 \beta_{18} + 1603527 \beta_{14} - 2503035 \beta_{13} + \cdots + 53660120 \beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 7460188 \beta_{15} - 5602206 \beta_{12} + 13062394 \beta_{9} - 101814319 \beta_{7} + 22728216 \beta_{5} + \cdots - 2687636965 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 22728216 \beta_{17} - 169998967 \beta_{14} + 11523804 \beta_{10} - 270782155 \beta_{8} + \cdots - 5221900808 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 606874430 \beta_{16} + 1397147898 \beta_{15} + 606874430 \beta_{12} - 2187421366 \beta_{9} + \cdots + 262310367455 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1305325372 \beta_{19} + 2519074232 \beta_{18} + 2519074232 \beta_{17} + 28627266859 \beta_{13} + \cdots + 518604659496 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 63987567806 \beta_{16} - 229201706102 \beta_{15} + 82607069148 \beta_{9} + 1042145919439 \beta_{7} + \cdots - 146594636954 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 142919349596 \beta_{19} - 270894485208 \beta_{18} + 1854829375063 \beta_{14} + \cdots + 52201624434408 \beta_{3} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 8575573403772 \beta_{15} - 6658880465790 \beta_{12} + 15234453869562 \beta_{9} - 106410526730799 \beta_{7} + \cdots - 26\!\cdots\!37 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 28637995618552 \beta_{17} - 192324513586455 \beta_{14} + 15320234686972 \beta_{10} + \cdots - 53\!\cdots\!28 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 688625240391102 \beta_{16} + \cdots + 26\!\cdots\!19 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 16\!\cdots\!00 \beta_{19} + \cdots + 54\!\cdots\!32 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
−5.07546 − 8.79096i
−4.04655 − 7.00882i
−3.88850 − 6.73507i
−2.04296 − 3.53851i
−1.60346 − 2.77727i
1.60346 + 2.77727i
2.04296 + 3.53851i
3.88850 + 6.73507i
4.04655 + 7.00882i
5.07546 + 8.79096i
−5.07546 + 8.79096i
−4.04655 + 7.00882i
−3.88850 + 6.73507i
−2.04296 + 3.53851i
−1.60346 + 2.77727i
1.60346 − 2.77727i
2.04296 − 3.53851i
3.88850 − 6.73507i
4.04655 − 7.00882i
5.07546 − 8.79096i
−5.07546 − 8.79096i 0 −35.5206 + 61.5235i −29.9388 0 −64.0938 + 111.014i 396.304 0 151.953 + 263.190i
55.2 −4.04655 − 7.00882i 0 −16.7491 + 29.0102i −27.3241 0 84.4772 − 146.319i 12.1246 0 110.568 + 191.510i
55.3 −3.88850 − 6.73507i 0 −14.2408 + 24.6658i 91.3456 0 16.8261 − 29.1437i −27.3625 0 −355.197 − 615.219i
55.4 −2.04296 − 3.53851i 0 7.65263 − 13.2547i 41.3975 0 −49.6097 + 85.9266i −193.285 0 −84.5735 − 146.486i
55.5 −1.60346 − 2.77727i 0 10.8578 − 18.8063i −78.7356 0 −43.0999 + 74.6511i −172.262 0 126.249 + 218.670i
55.6 1.60346 + 2.77727i 0 10.8578 − 18.8063i 78.7356 0 −43.0999 + 74.6511i 172.262 0 126.249 + 218.670i
55.7 2.04296 + 3.53851i 0 7.65263 − 13.2547i −41.3975 0 −49.6097 + 85.9266i 193.285 0 −84.5735 − 146.486i
55.8 3.88850 + 6.73507i 0 −14.2408 + 24.6658i −91.3456 0 16.8261 − 29.1437i 27.3625 0 −355.197 − 615.219i
55.9 4.04655 + 7.00882i 0 −16.7491 + 29.0102i 27.3241 0 84.4772 − 146.319i −12.1246 0 110.568 + 191.510i
55.10 5.07546 + 8.79096i 0 −35.5206 + 61.5235i 29.9388 0 −64.0938 + 111.014i −396.304 0 151.953 + 263.190i
100.1 −5.07546 + 8.79096i 0 −35.5206 − 61.5235i −29.9388 0 −64.0938 − 111.014i 396.304 0 151.953 − 263.190i
100.2 −4.04655 + 7.00882i 0 −16.7491 − 29.0102i −27.3241 0 84.4772 + 146.319i 12.1246 0 110.568 − 191.510i
100.3 −3.88850 + 6.73507i 0 −14.2408 − 24.6658i 91.3456 0 16.8261 + 29.1437i −27.3625 0 −355.197 + 615.219i
100.4 −2.04296 + 3.53851i 0 7.65263 + 13.2547i 41.3975 0 −49.6097 − 85.9266i −193.285 0 −84.5735 + 146.486i
100.5 −1.60346 + 2.77727i 0 10.8578 + 18.8063i −78.7356 0 −43.0999 − 74.6511i −172.262 0 126.249 − 218.670i
100.6 1.60346 − 2.77727i 0 10.8578 + 18.8063i 78.7356 0 −43.0999 − 74.6511i 172.262 0 126.249 − 218.670i
100.7 2.04296 − 3.53851i 0 7.65263 + 13.2547i −41.3975 0 −49.6097 − 85.9266i 193.285 0 −84.5735 + 146.486i
100.8 3.88850 − 6.73507i 0 −14.2408 − 24.6658i −91.3456 0 16.8261 + 29.1437i 27.3625 0 −355.197 + 615.219i
100.9 4.04655 − 7.00882i 0 −16.7491 − 29.0102i 27.3241 0 84.4772 + 146.319i −12.1246 0 110.568 − 191.510i
100.10 5.07546 − 8.79096i 0 −35.5206 − 61.5235i 29.9388 0 −64.0938 − 111.014i −396.304 0 151.953 − 263.190i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.c even 3 1 inner
39.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.6.g.e ✓ 20
3.b odd 2 1 inner 117.6.g.e ✓ 20
13.c even 3 1 inner 117.6.g.e ✓ 20
39.i odd 6 1 inner 117.6.g.e ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.6.g.e ✓ 20 1.a even 1 1 trivial
117.6.g.e ✓ 20 3.b odd 2 1 inner
117.6.g.e ✓ 20 13.c even 3 1 inner
117.6.g.e ✓ 20 39.i odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 256 T_{2}^{18} + 42243 T_{2}^{16} + 4153796 T_{2}^{14} + 297063177 T_{2}^{12} + \cdots + 49\!\cdots\!96 \) acting on \(S_{6}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 49\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots - 59\!\cdots\!96)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 38\!\cdots\!56)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 70\!\cdots\!93)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 66\!\cdots\!04)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 60\!\cdots\!52)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 35\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 29\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 21\!\cdots\!16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 15\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 48\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 25\!\cdots\!69)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 24\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 31\!\cdots\!23)^{4} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 59\!\cdots\!36)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 23\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 17\!\cdots\!36)^{2} \) Copy content Toggle raw display
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