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Properties

Label 117.5.j.b
Level $117$
Weight $5$
Character orbit 117.j
Analytic conductor $12.094$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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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: \( x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{13} + 10 \beta_{4}) q^{4} + ( - \beta_{12} - \beta_{4} - 1) q^{5} + ( - \beta_{14} + \beta_{4} - \beta_{3} - 1) q^{7} + (\beta_{17} + \beta_{15} + \cdots - 10 \beta_{3}) q^{8}+ \cdots + (12 \beta_{19} + 18 \beta_{17} + \cdots - 638) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!03 \nu^{19} + \cdots - 34\!\cdots\!88 ) / 13\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 82\!\cdots\!67 \nu^{19} + \cdots - 11\!\cdots\!60 ) / 56\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!35 \nu^{19} + \cdots - 14\!\cdots\!52 ) / 11\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26\!\cdots\!69 \nu^{19} + \cdots - 77\!\cdots\!96 ) / 67\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 43\!\cdots\!75 \nu^{19} + \cdots + 20\!\cdots\!84 ) / 67\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!77 \nu^{19} + \cdots + 44\!\cdots\!76 ) / 12\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 68\!\cdots\!08 \nu^{19} + \cdots - 23\!\cdots\!88 ) / 64\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31\!\cdots\!73 \nu^{19} + \cdots + 87\!\cdots\!92 ) / 19\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 87\!\cdots\!77 \nu^{19} + \cdots + 24\!\cdots\!64 ) / 45\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!20 \nu^{19} + \cdots - 45\!\cdots\!32 ) / 64\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 33\!\cdots\!81 \nu^{19} + \cdots + 92\!\cdots\!00 ) / 16\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11\!\cdots\!35 \nu^{19} + \cdots + 14\!\cdots\!52 ) / 45\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16\!\cdots\!36 \nu^{19} + \cdots - 32\!\cdots\!92 ) / 64\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 19\!\cdots\!31 \nu^{19} + \cdots + 26\!\cdots\!68 ) / 67\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 52\!\cdots\!99 \nu^{19} + \cdots - 14\!\cdots\!72 ) / 13\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 17\!\cdots\!28 \nu^{19} + \cdots - 55\!\cdots\!16 ) / 37\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 58\!\cdots\!41 \nu^{19} + \cdots - 16\!\cdots\!84 ) / 79\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 24\!\cdots\!87 \nu^{19} + \cdots - 32\!\cdots\!60 ) / 22\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + 26\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{15} - \beta_{11} + \beta_{6} - 42\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{16} + 3 \beta_{14} + 3 \beta_{12} + \beta_{11} + 3 \beta_{9} - 3 \beta_{7} + 3 \beta_{6} + \cdots - 1113 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 6 \beta_{19} - 71 \beta_{18} + 64 \beta_{16} - 71 \beta_{15} + 9 \beta_{13} - 22 \beta_{12} + \cdots + 377 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 48 \beta_{19} + 18 \beta_{18} + 18 \beta_{17} - 84 \beta_{16} - 280 \beta_{15} - 244 \beta_{14} + \cdots + 1456 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 612 \beta_{19} - 4327 \beta_{17} - 4267 \beta_{15} + 2120 \beta_{14} + 978 \beta_{13} + 3551 \beta_{11} + \cdots - 38660 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2178 \beta_{18} + 2178 \beta_{17} + 5693 \beta_{16} - 16405 \beta_{14} - 18093 \beta_{12} + \cdots + 2651299 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 45468 \beta_{19} + 253459 \beta_{18} - 192438 \beta_{16} + 242425 \beta_{15} - 77889 \beta_{13} + \cdots - 2930125 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4210 \beta_{19} - 192120 \beta_{18} - 192120 \beta_{17} + 363150 \beta_{16} + 1129474 \beta_{15} + \cdots - 7528124 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3000672 \beta_{19} + 14603735 \beta_{17} + 13409411 \beta_{15} - 10308052 \beta_{14} - 5580402 \beta_{13} + \cdots + 198445014 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 14864580 \beta_{18} - 14864580 \beta_{17} - 22611365 \beta_{16} + 64524627 \beta_{14} + \cdots - 7238942981 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 186968658 \beta_{19} - 834317419 \beta_{18} + 571043708 \beta_{16} - 731327311 \beta_{15} + \cdots + 12748032461 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 462121580 \beta_{19} + 1067886438 \beta_{18} + 1067886438 \beta_{17} - 1389237200 \beta_{16} + \cdots + 30090976712 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 11286969396 \beta_{19} - 47446397607 \beta_{17} - 39589952451 \beta_{15} + 40253901712 \beta_{14} + \cdots - 796558371952 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 73104401742 \beta_{18} + 73104401742 \beta_{17} + 84582095837 \beta_{16} - 236049612449 \beta_{14} + \cdots + 21249258265383 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 668742068952 \beta_{19} + 2691935807403 \beta_{18} - 1750626100362 \beta_{16} + 2135598631389 \beta_{15} + \cdots - 48995546053117 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 2746950575330 \beta_{19} - 4838343346236 \beta_{18} - 4838343346236 \beta_{17} + \cdots - 112009570267140 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 39167603966496 \beta_{19} + 152593590865935 \beta_{17} + 115067838171795 \beta_{15} + \cdots + 29\!\cdots\!86 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(\beta_{4}\) \(1\)

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
−5.39509 − 5.39509i
−4.67933 − 4.67933i
−2.89776 − 2.89776i
−1.66937 − 1.66937i
−0.755803 − 0.755803i
−0.0868646 − 0.0868646i
1.99524 + 1.99524i
4.05053 + 4.05053i
4.32919 + 4.32919i
5.10926 + 5.10926i
−5.39509 + 5.39509i
−4.67933 + 4.67933i
−2.89776 + 2.89776i
−1.66937 + 1.66937i
−0.755803 + 0.755803i
−0.0868646 + 0.0868646i
1.99524 − 1.99524i
4.05053 − 4.05053i
4.32919 − 4.32919i
5.10926 − 5.10926i
−5.39509 − 5.39509i 0 42.2140i 17.6256 + 17.6256i 0 26.0542 − 26.0542i 141.427 − 141.427i 0 − 190.184i
73.2 −4.67933 − 4.67933i 0 27.7923i −13.1418 − 13.1418i 0 −30.2175 + 30.2175i 55.1800 − 55.1800i 0 122.989i
73.3 −2.89776 − 2.89776i 0 0.794047i −18.6373 − 18.6373i 0 −15.2549 + 15.2549i −44.0632 + 44.0632i 0 108.013i
73.4 −1.66937 − 1.66937i 0 − 10.4264i 28.4160 + 28.4160i 0 −17.1341 + 17.1341i −44.1155 + 44.1155i 0 − 94.8736i
73.5 −0.755803 − 0.755803i 0 − 14.8575i −25.9238 − 25.9238i 0 55.8718 − 55.8718i −23.3222 + 23.3222i 0 39.1866i
73.6 −0.0868646 − 0.0868646i 0 − 15.9849i 14.6130 + 14.6130i 0 21.4788 − 21.4788i −2.77836 + 2.77836i 0 − 2.53870i
73.7 1.99524 + 1.99524i 0 − 8.03802i −3.82538 − 3.82538i 0 −42.4180 + 42.4180i 47.9617 − 47.9617i 0 − 15.2651i
73.8 4.05053 + 4.05053i 0 16.8136i −33.5573 − 33.5573i 0 23.7132 − 23.7132i −3.29565 + 3.29565i 0 − 271.850i
73.9 4.32919 + 4.32919i 0 21.4837i 26.0804 + 26.0804i 0 −16.9579 + 16.9579i −23.7399 + 23.7399i 0 225.814i
73.10 5.10926 + 5.10926i 0 36.2092i −3.64937 − 3.64937i 0 −17.1355 + 17.1355i −103.254 + 103.254i 0 − 37.2912i
109.1 −5.39509 + 5.39509i 0 − 42.2140i 17.6256 − 17.6256i 0 26.0542 + 26.0542i 141.427 + 141.427i 0 190.184i
109.2 −4.67933 + 4.67933i 0 − 27.7923i −13.1418 + 13.1418i 0 −30.2175 − 30.2175i 55.1800 + 55.1800i 0 − 122.989i
109.3 −2.89776 + 2.89776i 0 − 0.794047i −18.6373 + 18.6373i 0 −15.2549 − 15.2549i −44.0632 − 44.0632i 0 − 108.013i
109.4 −1.66937 + 1.66937i 0 10.4264i 28.4160 − 28.4160i 0 −17.1341 − 17.1341i −44.1155 − 44.1155i 0 94.8736i
109.5 −0.755803 + 0.755803i 0 14.8575i −25.9238 + 25.9238i 0 55.8718 + 55.8718i −23.3222 − 23.3222i 0 − 39.1866i
109.6 −0.0868646 + 0.0868646i 0 15.9849i 14.6130 − 14.6130i 0 21.4788 + 21.4788i −2.77836 − 2.77836i 0 2.53870i
109.7 1.99524 − 1.99524i 0 8.03802i −3.82538 + 3.82538i 0 −42.4180 − 42.4180i 47.9617 + 47.9617i 0 15.2651i
109.8 4.05053 − 4.05053i 0 − 16.8136i −33.5573 + 33.5573i 0 23.7132 + 23.7132i −3.29565 − 3.29565i 0 271.850i
109.9 4.32919 − 4.32919i 0 − 21.4837i 26.0804 − 26.0804i 0 −16.9579 − 16.9579i −23.7399 − 23.7399i 0 − 225.814i
109.10 5.10926 − 5.10926i 0 − 36.2092i −3.64937 + 3.64937i 0 −17.1355 − 17.1355i −103.254 − 103.254i 0 37.2912i
\(n\): 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
13.d odd 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.b 20
3.b odd 2 1 39.5.g.a ✓ 20
13.d odd 4 1 inner 117.5.j.b 20
39.f even 4 1 39.5.g.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.5.g.a ✓ 20 3.b odd 2 1
39.5.g.a ✓ 20 39.f even 4 1
117.5.j.b 20 1.a even 1 1 trivial
117.5.j.b 20 13.d odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 5446 T_{2}^{16} - 1452 T_{2}^{15} + 106320 T_{2}^{13} + 8376897 T_{2}^{12} + \cdots + 2103506496 \) acting on \(S_{5}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 2103506496 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 53\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 84\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 42\!\cdots\!64)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 57\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 10\!\cdots\!52)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 46\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 14\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 71\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 88\!\cdots\!36 \) Copy content Toggle raw display
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