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Properties

Label 117.5.j.c
Level $117$
Weight $5$
Character orbit 117.j
Analytic conductor $12.094$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,5,Mod(73,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.73");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 117.j (of order \(4\), 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: \(12.0942856808\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 5446x^{16} + 8151009x^{12} + 2985553228x^{8} + 106128756016x^{4} + 342225000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + ( - \beta_{9} + 10 \beta_{7}) q^{4} + (\beta_{15} - \beta_{8}) q^{5} + (\beta_{10} + \beta_{7} - 1) q^{7} + ( - \beta_{3} - 11 \beta_1) q^{8} + (\beta_{12} - \beta_{11} + \cdots + 28 \beta_{7}) q^{10}+ \cdots + (20 \beta_{18} - 20 \beta_{17} + \cdots + 3667 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 24 q^{7} + 184 q^{13} - 2328 q^{16} - 1032 q^{19} - 4096 q^{22} + 3656 q^{28} + 2840 q^{31} - 6624 q^{34} + 452 q^{37} - 12096 q^{40} + 4032 q^{46} + 29824 q^{52} - 15808 q^{55} - 10648 q^{58} + 8408 q^{61}+ \cdots - 89812 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 5446x^{16} + 8151009x^{12} + 2985553228x^{8} + 106128756016x^{4} + 342225000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 305729 \nu^{16} + 1694019378 \nu^{12} + 2668722219737 \nu^{8} + \cdots + 23\!\cdots\!44 ) / 97\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 305729 \nu^{17} - 1694019378 \nu^{13} - 2668722219737 \nu^{9} + \cdots - 40\!\cdots\!92 \nu ) / 97\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4013082751 \nu^{16} - 21660542309502 \nu^{12} + \cdots - 10\!\cdots\!32 ) / 13\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 112300148373 \nu^{17} - 608371660992058 \nu^{13} + \cdots - 37\!\cdots\!68 \nu ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 716806896243 \nu^{17} + \cdots + 63\!\cdots\!88 \nu ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11682588607 \nu^{18} - 63712803286222 \nu^{14} + \cdots - 16\!\cdots\!12 \nu^{2} ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11682588607 \nu^{19} + 63712803286222 \nu^{15} + \cdots + 16\!\cdots\!12 \nu^{3} ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11682588607 \nu^{18} - 63712803286222 \nu^{14} + \cdots - 15\!\cdots\!12 \nu^{2} ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 52739561610353 \nu^{18} + 210694727714250 \nu^{16} + \cdots + 65\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 52739561610353 \nu^{18} - 210694727714250 \nu^{16} + \cdots - 65\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!53 \nu^{18} + \cdots + 68\!\cdots\!48 \nu^{2} ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!17 \nu^{18} + \cdots - 32\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!17 \nu^{18} + \cdots + 32\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 43\!\cdots\!27 \nu^{19} + \cdots + 24\!\cdots\!32 \nu^{3} ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 502351310101 \nu^{19} + \cdots + 67\!\cdots\!16 \nu^{3} ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 52\!\cdots\!43 \nu^{19} + \cdots + 77\!\cdots\!00 \nu ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 52\!\cdots\!43 \nu^{19} + \cdots - 77\!\cdots\!00 \nu ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 52\!\cdots\!07 \nu^{19} + \cdots - 48\!\cdots\!12 \nu^{3} ) / 54\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 26\beta_{7} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{16} + 43\beta_{8} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} - \beta_{4} + 56\beta_{2} - 1112 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{18} - \beta_{17} - 5\beta_{6} - 40\beta_{5} - 66\beta_{3} - 2082\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 72\beta_{14} + 72\beta_{13} + 108\beta_{12} - 124\beta_{11} - 124\beta_{10} - 2951\beta_{9} + 53800\beta_{7} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 396\beta_{19} - 124\beta_{18} - 124\beta_{17} + 3847\beta_{16} + 3976\beta_{15} - 106023\beta_{8} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4143\beta_{14} - 4143\beta_{13} - 10503\beta_{11} + 10503\beta_{10} + 8071\beta_{4} - 155830\beta_{2} + 2739484 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -10503\beta_{18} + 10503\beta_{17} + 24643\beta_{6} + 292168\beta_{5} + 217836\beta_{3} + 5557072\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 225110 \beta_{14} - 225110 \beta_{13} - 531010 \beta_{12} + 754802 \beta_{11} + \cdots - 143593144 \beta_{7} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1431450 \beta_{19} + 754802 \beta_{18} + 754802 \beta_{17} - 12231653 \beta_{16} + \cdots + 296767669 \beta_{8} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 12075345 \beta_{14} + 12075345 \beta_{13} + 49698929 \beta_{11} - 49698929 \beta_{10} + \cdots - 7668884084 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 49698929 \beta_{18} - 49698929 \beta_{17} - 81238357 \beta_{6} - 1193035752 \beta_{5} + \cdots - 16057123394 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 649088940 \beta_{14} + 649088940 \beta_{13} + 1977841552 \beta_{12} - 3107536992 \beta_{11} + \cdots + 414963649232 \beta_{7} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 4574197312 \beta_{19} - 3107536992 \beta_{18} - 3107536992 \beta_{17} + 38408611571 \beta_{16} + \cdots - 877161368091 \beta_{8} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 35126858227 \beta_{14} - 35126858227 \beta_{13} - 188079332635 \beta_{11} + 188079332635 \beta_{10} + \cdots + 22669639468892 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 188079332635 \beta_{18} + 188079332635 \beta_{17} + 256939198375 \beta_{6} + \cdots + 48263053495156 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1915256223666 \beta_{14} - 1915256223666 \beta_{13} - 6765331411726 \beta_{12} + \cdots - 12\!\cdots\!60 \beta_{7} \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 14426356306390 \beta_{19} + 11141515663486 \beta_{18} + 11141515663486 \beta_{17} + \cdots + 26\!\cdots\!21 \beta_{8} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
−5.30408 + 5.30408i
−4.58096 + 4.58096i
−3.30845 + 3.30845i
−1.72900 + 1.72900i
−0.972788 + 0.972788i
0.972788 − 0.972788i
1.72900 − 1.72900i
3.30845 − 3.30845i
4.58096 − 4.58096i
5.30408 − 5.30408i
−5.30408 − 5.30408i
−4.58096 − 4.58096i
−3.30845 − 3.30845i
−1.72900 − 1.72900i
−0.972788 − 0.972788i
0.972788 + 0.972788i
1.72900 + 1.72900i
3.30845 + 3.30845i
4.58096 + 4.58096i
5.30408 + 5.30408i
−5.30408 − 5.30408i 0 40.2665i −24.6751 − 24.6751i 0 56.6657 − 56.6657i 128.711 − 128.711i 0 261.758i
73.2 −4.58096 − 4.58096i 0 25.9704i 17.3470 + 17.3470i 0 −65.5353 + 65.5353i 45.6738 − 45.6738i 0 − 158.931i
73.3 −3.30845 − 3.30845i 0 5.89171i 9.29067 + 9.29067i 0 26.5597 − 26.5597i −33.4428 + 33.4428i 0 − 61.4755i
73.4 −1.72900 − 1.72900i 0 − 10.0211i −30.0548 − 30.0548i 0 −38.2846 + 38.2846i −44.9905 + 44.9905i 0 103.929i
73.5 −0.972788 − 0.972788i 0 − 14.1074i 5.79781 + 5.79781i 0 14.5945 − 14.5945i −29.2881 + 29.2881i 0 − 11.2801i
73.6 0.972788 + 0.972788i 0 − 14.1074i −5.79781 − 5.79781i 0 14.5945 − 14.5945i 29.2881 − 29.2881i 0 − 11.2801i
73.7 1.72900 + 1.72900i 0 − 10.0211i 30.0548 + 30.0548i 0 −38.2846 + 38.2846i 44.9905 − 44.9905i 0 103.929i
73.8 3.30845 + 3.30845i 0 5.89171i −9.29067 − 9.29067i 0 26.5597 − 26.5597i 33.4428 − 33.4428i 0 − 61.4755i
73.9 4.58096 + 4.58096i 0 25.9704i −17.3470 − 17.3470i 0 −65.5353 + 65.5353i −45.6738 + 45.6738i 0 − 158.931i
73.10 5.30408 + 5.30408i 0 40.2665i 24.6751 + 24.6751i 0 56.6657 − 56.6657i −128.711 + 128.711i 0 261.758i
109.1 −5.30408 + 5.30408i 0 − 40.2665i −24.6751 + 24.6751i 0 56.6657 + 56.6657i 128.711 + 128.711i 0 − 261.758i
109.2 −4.58096 + 4.58096i 0 − 25.9704i 17.3470 − 17.3470i 0 −65.5353 − 65.5353i 45.6738 + 45.6738i 0 158.931i
109.3 −3.30845 + 3.30845i 0 − 5.89171i 9.29067 − 9.29067i 0 26.5597 + 26.5597i −33.4428 − 33.4428i 0 61.4755i
109.4 −1.72900 + 1.72900i 0 10.0211i −30.0548 + 30.0548i 0 −38.2846 − 38.2846i −44.9905 − 44.9905i 0 − 103.929i
109.5 −0.972788 + 0.972788i 0 14.1074i 5.79781 − 5.79781i 0 14.5945 + 14.5945i −29.2881 − 29.2881i 0 11.2801i
109.6 0.972788 − 0.972788i 0 14.1074i −5.79781 + 5.79781i 0 14.5945 + 14.5945i 29.2881 + 29.2881i 0 11.2801i
109.7 1.72900 − 1.72900i 0 10.0211i 30.0548 − 30.0548i 0 −38.2846 − 38.2846i 44.9905 + 44.9905i 0 − 103.929i
109.8 3.30845 − 3.30845i 0 − 5.89171i −9.29067 + 9.29067i 0 26.5597 + 26.5597i 33.4428 + 33.4428i 0 61.4755i
109.9 4.58096 − 4.58096i 0 − 25.9704i −17.3470 + 17.3470i 0 −65.5353 − 65.5353i −45.6738 − 45.6738i 0 158.931i
109.10 5.30408 − 5.30408i 0 − 40.2665i 24.6751 − 24.6751i 0 56.6657 + 56.6657i −128.711 − 128.711i 0 − 261.758i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.5.j.c ✓ 20
3.b odd 2 1 inner 117.5.j.c ✓ 20
13.d odd 4 1 inner 117.5.j.c ✓ 20
39.f even 4 1 inner 117.5.j.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.5.j.c ✓ 20 1.a even 1 1 trivial
117.5.j.c ✓ 20 3.b odd 2 1 inner
117.5.j.c ✓ 20 13.d odd 4 1 inner
117.5.j.c ✓ 20 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 5446T_{2}^{16} + 8151009T_{2}^{12} + 2985553228T_{2}^{8} + 106128756016T_{2}^{4} + 342225000000 \) acting on \(S_{5}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 342225000000 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 97\!\cdots\!32)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 19\!\cdots\!01)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 28\!\cdots\!52)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 19\!\cdots\!48)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 17\!\cdots\!88)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 41\!\cdots\!40)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 32\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 40\!\cdots\!68)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 77\!\cdots\!60)^{4} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 72\!\cdots\!28)^{2} \) Copy content Toggle raw display
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