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Properties

Label 117.3.o.a
Level $117$
Weight $3$
Character orbit 117.o
Analytic conductor $3.188$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,3,Mod(17,117)]
 
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([3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 117.o (of order \(6\), 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: \(3.18801909302\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 32 x^{18} + 690 x^{16} + 8192 x^{14} + 70099 x^{12} + 319704 x^{10} + 1042578 x^{8} + \cdots + 46656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{10} - 2 \beta_{5}) q^{4} + ( - \beta_{16} + \beta_{14} + \beta_{12}) q^{5} + (\beta_{8} - \beta_{5} + \beta_{3} + 2) q^{7} + ( - \beta_{17} - 3 \beta_{7} - 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{10} - 2 \beta_{5}) q^{4} + ( - \beta_{16} + \beta_{14} + \beta_{12}) q^{5} + (\beta_{8} - \beta_{5} + \beta_{3} + 2) q^{7} + ( - \beta_{17} - 3 \beta_{7} - 3 \beta_1) q^{8} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_{2}) q^{10}+ \cdots + (\beta_{19} + 2 \beta_{18} + \cdots - 5 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 24 q^{4} + 42 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 24 q^{4} + 42 q^{7} - 4 q^{10} + 2 q^{13} - 104 q^{16} + 68 q^{22} + 228 q^{25} - 420 q^{28} - 72 q^{37} + 232 q^{40} - 62 q^{43} + 588 q^{46} + 188 q^{49} - 72 q^{52} - 256 q^{55} - 348 q^{58} - 218 q^{61} + 320 q^{64} - 762 q^{67} - 588 q^{76} + 100 q^{79} + 848 q^{82} + 204 q^{85} + 1084 q^{88} + 18 q^{91} + 356 q^{94} - 894 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 32 x^{18} + 690 x^{16} + 8192 x^{14} + 70099 x^{12} + 319704 x^{10} + 1042578 x^{8} + \cdots + 46656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3835 \nu^{18} + 106568 \nu^{16} + 2278851 \nu^{14} + 24427592 \nu^{12} + 225387670 \nu^{10} + \cdots - 5176219464 ) / 5628504078 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1844098703095 \nu^{18} + 56324410539800 \nu^{16} + \cdots - 12\!\cdots\!98 ) / 21\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1844098703095 \nu^{19} + 56324410539800 \nu^{17} + \cdots - 23\!\cdots\!33 \nu ) / 21\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 64802610521457 \nu^{18} + \cdots - 24\!\cdots\!36 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10640978280640 \nu^{18} - 328493352429008 \nu^{16} + \cdots - 31\!\cdots\!68 ) / 14\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 64802610521457 \nu^{19} + \cdots - 20\!\cdots\!32 \nu ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 402523923538267 \nu^{18} + \cdots - 24\!\cdots\!00 ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 258049398711479 \nu^{19} + \cdots - 39\!\cdots\!34 \nu ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 64802610521457 \nu^{18} + \cdots - 24\!\cdots\!36 ) / 29\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 680091530147347 \nu^{18} + \cdots + 16\!\cdots\!00 ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 547748998456227 \nu^{19} + \cdots + 19\!\cdots\!48 \nu ) / 35\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 971500763056744 \nu^{18} + \cdots + 15\!\cdots\!88 ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 7987993 \nu^{19} - 255891896 \nu^{17} - 5519388066 \nu^{15} - 65601715928 \nu^{13} + \cdots - 5267949492528 \nu ) / 405252293616 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14\!\cdots\!05 \nu^{18} + \cdots + 20\!\cdots\!32 ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 27\!\cdots\!43 \nu^{19} + \cdots - 54\!\cdots\!88 \nu ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 712828715736027 \nu^{19} + \cdots + 27\!\cdots\!96 \nu ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 16\!\cdots\!31 \nu^{19} + \cdots + 11\!\cdots\!16 \nu ) / 11\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 23\!\cdots\!43 \nu^{19} + \cdots - 38\!\cdots\!12 \nu ) / 10\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 6\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} - 11\beta_{7} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{15} - 15\beta_{10} + \beta_{8} + \beta_{6} + 66\beta_{5} - 14\beta_{3} - 66 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16\beta_{17} - 2\beta_{16} + 4\beta_{14} + 137\beta_{7} - 16\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16\beta_{11} - 16\beta_{8} - 2\beta_{6} + 185\beta_{3} - 2\beta_{2} + 822 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{19} - 2\beta_{18} - 42\beta_{16} - 42\beta_{14} - 8\beta_{9} + 219\beta_{4} + 1763\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 205 \beta_{15} + 24 \beta_{13} - 205 \beta_{11} + 2643 \beta_{10} - 12 \beta_{8} - 263 \beta_{6} + \cdots + 116 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 116 \beta_{19} - 58 \beta_{18} - 3022 \beta_{17} + 1760 \beta_{16} - 1072 \beta_{14} + \cdots - 22901 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2464 \beta_{15} - 384 \beta_{13} - 34641 \beta_{10} + 3232 \beta_{8} + 4840 \beta_{6} + 137406 \beta_{5} + \cdots - 137406 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1188 \beta_{19} + 2376 \beta_{18} + 40669 \beta_{17} - 18764 \beta_{16} + 29248 \beta_{14} + \cdots - 39481 \beta_{4} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -8280\beta_{13} + 28825\beta_{11} - 37105\beta_{8} - 21140\beta_{6} + 408614\beta_{3} - 21140\beta_{2} + 1790706 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 21140 \beta_{19} - 21140 \beta_{18} - 155070 \beta_{16} - 155070 \beta_{14} - 109400 \beta_{9} + \cdots + 3895721 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 331144 \beta_{15} + 303360 \beta_{13} - 331144 \beta_{11} + 5953857 \beta_{10} - 151680 \beta_{8} + \cdots + 698060 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 698060 \beta_{19} - 349030 \beta_{18} - 7332091 \beta_{17} + 7060736 \beta_{16} + \cdots - 50908955 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3735781 \beta_{15} - 2549220 \beta_{13} - 78143139 \beta_{10} + 8834221 \beta_{8} + 14741857 \beta_{6} + \cdots - 305453730 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 5503038 \beta_{19} + 11006076 \beta_{18} + 98388034 \beta_{17} - 73103420 \beta_{16} + \cdots - 92884996 \beta_{4} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 40665888 \beta_{13} + 41213032 \beta_{11} - 81878920 \beta_{8} - 84109496 \beta_{6} + \cdots + 3995585886 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 84109496 \beta_{19} - 84109496 \beta_{18} - 462307656 \beta_{16} - 462307656 \beta_{14} + \cdots + 8719062875 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(\beta_{5}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−1.82939 + 3.16860i
−1.77247 + 3.07001i
−0.977132 + 1.69244i
−0.718446 + 1.24438i
−0.201764 + 0.349465i
0.201764 − 0.349465i
0.718446 − 1.24438i
0.977132 − 1.69244i
1.77247 − 3.07001i
1.82939 − 3.16860i
−1.82939 − 3.16860i
−1.77247 − 3.07001i
−0.977132 − 1.69244i
−0.718446 − 1.24438i
−0.201764 − 0.349465i
0.201764 + 0.349465i
0.718446 + 1.24438i
0.977132 + 1.69244i
1.77247 + 3.07001i
1.82939 + 3.16860i
−1.82939 + 3.16860i 0 −4.69337 − 8.12915i −5.80322 0 3.12507 − 1.80426i 19.7089 0 10.6164 − 18.3881i
17.2 −1.77247 + 3.07001i 0 −4.28332 − 7.41892i 8.67010 0 7.13217 − 4.11776i 16.1885 0 −15.3675 + 26.6173i
17.3 −0.977132 + 1.69244i 0 0.0904276 + 0.156625i −5.01168 0 −1.22684 + 0.708319i −8.17049 0 4.89707 − 8.48198i
17.4 −0.718446 + 1.24438i 0 0.967671 + 1.67606i 3.30575 0 10.5080 − 6.06680i −8.52844 0 −2.37500 + 4.11362i
17.5 −0.201764 + 0.349465i 0 1.91858 + 3.32308i −6.09157 0 −9.03840 + 5.21832i −3.16251 0 1.22906 − 2.12879i
17.6 0.201764 − 0.349465i 0 1.91858 + 3.32308i 6.09157 0 −9.03840 + 5.21832i 3.16251 0 1.22906 − 2.12879i
17.7 0.718446 − 1.24438i 0 0.967671 + 1.67606i −3.30575 0 10.5080 − 6.06680i 8.52844 0 −2.37500 + 4.11362i
17.8 0.977132 − 1.69244i 0 0.0904276 + 0.156625i 5.01168 0 −1.22684 + 0.708319i 8.17049 0 4.89707 − 8.48198i
17.9 1.77247 − 3.07001i 0 −4.28332 − 7.41892i −8.67010 0 7.13217 − 4.11776i −16.1885 0 −15.3675 + 26.6173i
17.10 1.82939 − 3.16860i 0 −4.69337 − 8.12915i 5.80322 0 3.12507 − 1.80426i −19.7089 0 10.6164 − 18.3881i
62.1 −1.82939 − 3.16860i 0 −4.69337 + 8.12915i −5.80322 0 3.12507 + 1.80426i 19.7089 0 10.6164 + 18.3881i
62.2 −1.77247 − 3.07001i 0 −4.28332 + 7.41892i 8.67010 0 7.13217 + 4.11776i 16.1885 0 −15.3675 − 26.6173i
62.3 −0.977132 − 1.69244i 0 0.0904276 − 0.156625i −5.01168 0 −1.22684 − 0.708319i −8.17049 0 4.89707 + 8.48198i
62.4 −0.718446 − 1.24438i 0 0.967671 − 1.67606i 3.30575 0 10.5080 + 6.06680i −8.52844 0 −2.37500 − 4.11362i
62.5 −0.201764 − 0.349465i 0 1.91858 − 3.32308i −6.09157 0 −9.03840 − 5.21832i −3.16251 0 1.22906 + 2.12879i
62.6 0.201764 + 0.349465i 0 1.91858 − 3.32308i 6.09157 0 −9.03840 − 5.21832i 3.16251 0 1.22906 + 2.12879i
62.7 0.718446 + 1.24438i 0 0.967671 − 1.67606i −3.30575 0 10.5080 + 6.06680i 8.52844 0 −2.37500 − 4.11362i
62.8 0.977132 + 1.69244i 0 0.0904276 − 0.156625i 5.01168 0 −1.22684 − 0.708319i 8.17049 0 4.89707 + 8.48198i
62.9 1.77247 + 3.07001i 0 −4.28332 + 7.41892i −8.67010 0 7.13217 + 4.11776i −16.1885 0 −15.3675 − 26.6173i
62.10 1.82939 + 3.16860i 0 −4.69337 + 8.12915i 5.80322 0 3.12507 + 1.80426i −19.7089 0 10.6164 + 18.3881i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.3.o.a ✓ 20
3.b odd 2 1 inner 117.3.o.a ✓ 20
13.e even 6 1 inner 117.3.o.a ✓ 20
39.h odd 6 1 inner 117.3.o.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.3.o.a ✓ 20 1.a even 1 1 trivial
117.3.o.a ✓ 20 3.b odd 2 1 inner
117.3.o.a ✓ 20 13.e even 6 1 inner
117.3.o.a ✓ 20 39.h odd 6 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 32 T^{18} + \cdots + 46656 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} - 182 T^{8} + \cdots - 25783974)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} - 21 T^{9} + \cdots + 28422252)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 8742808166976 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 137858491849)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{10} - 510 T^{8} + \cdots + 10115677872)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 11\!\cdots\!72)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 2683261093932)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{10} + 31 T^{9} + \cdots + 36003544516)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 56\!\cdots\!76)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 503364593584008)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 10\!\cdots\!61)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 80\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 50\!\cdots\!43)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 25 T^{4} + \cdots + 4595264)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 43\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
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