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Properties

Label 117.5.d.a
Level $117$
Weight $5$
Character orbit 117.d
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(116,117)]
 
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([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.116");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 117.d (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: \(12.0942856808\)
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: \( x^{20} - 236 x^{18} + 24058 x^{16} - 1366148 x^{14} + 47062921 x^{12} - 994844864 x^{10} + \cdots + 2213572644864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{23} \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_1 + 10) q^{4} + (\beta_{15} - \beta_{4}) q^{5} + \beta_{3} q^{7} + (\beta_{12} + 9 \beta_{4}) q^{8} + (\beta_{2} - 14) q^{10} + (\beta_{19} + \beta_{15} + \cdots - 3 \beta_{4}) q^{11}+ \cdots + (2 \beta_{17} + 2 \beta_{15} + \cdots - 1863 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 192 q^{4} - 280 q^{10} + 584 q^{13} + 1768 q^{16} - 1312 q^{22} + 3108 q^{25} + 1264 q^{40} - 10160 q^{43} - 14764 q^{49} + 10200 q^{52} + 7280 q^{55} - 1544 q^{61} + 32024 q^{64} - 20048 q^{79}+ \cdots + 27008 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 236 x^{18} + 24058 x^{16} - 1366148 x^{14} + 47062921 x^{12} - 994844864 x^{10} + \cdots + 2213572644864 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 63\!\cdots\!77 \nu^{18} + \cdots - 19\!\cdots\!96 ) / 78\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12\!\cdots\!61 \nu^{18} + \cdots + 67\!\cdots\!04 ) / 31\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31\!\cdots\!63 \nu^{18} + \cdots + 25\!\cdots\!36 ) / 37\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!35 \nu^{19} + \cdots + 28\!\cdots\!08 \nu ) / 16\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 63\!\cdots\!77 \nu^{18} + \cdots + 42\!\cdots\!68 ) / 58\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 90\!\cdots\!45 \nu^{18} + \cdots + 11\!\cdots\!28 ) / 31\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!55 \nu^{18} + \cdots - 48\!\cdots\!76 ) / 43\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 35\!\cdots\!99 \nu^{18} + \cdots - 36\!\cdots\!44 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 40\!\cdots\!21 \nu^{18} + \cdots + 31\!\cdots\!88 ) / 29\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\!\cdots\!10 \nu^{19} + \cdots + 92\!\cdots\!48 \nu ) / 90\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!35 \nu^{19} + \cdots + 19\!\cdots\!16 \nu ) / 60\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 34\!\cdots\!13 \nu^{19} + \cdots - 13\!\cdots\!32 \nu ) / 16\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 32\!\cdots\!55 \nu^{18} + \cdots + 45\!\cdots\!20 ) / 94\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 88\!\cdots\!27 \nu^{19} + \cdots - 65\!\cdots\!76 \nu ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 84\!\cdots\!91 \nu^{19} + \cdots - 94\!\cdots\!40 \nu ) / 16\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 19\!\cdots\!84 \nu^{19} + \cdots + 19\!\cdots\!60 \nu ) / 90\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 86\!\cdots\!87 \nu^{19} + \cdots + 87\!\cdots\!32 \nu ) / 32\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 50\!\cdots\!85 \nu^{19} + \cdots + 88\!\cdots\!44 \nu ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 12\!\cdots\!65 \nu^{19} + \cdots + 12\!\cdots\!64 \nu ) / 32\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - 27\beta_{4} ) / 27 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 27\beta _1 + 648 ) / 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{16} + 3\beta_{14} - 27\beta_{12} + 73\beta_{11} + 3\beta_{10} - 945\beta_{4} ) / 27 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{13} - 36 \beta_{9} - 24 \beta_{8} - 24 \beta_{7} + 27 \beta_{6} + 148 \beta_{5} + 100 \beta_{3} + \cdots + 20979 ) / 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 27 \beta_{19} + 45 \beta_{18} - 135 \beta_{17} - 510 \beta_{16} - 297 \beta_{15} + \cdots - 31428 \beta_{4} ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 344 \beta_{13} - 3582 \beta_{9} - 1956 \beta_{8} - 2820 \beta_{7} + 1566 \beta_{6} + 9372 \beta_{5} + \cdots + 584118 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1350 \beta_{19} + 4158 \beta_{18} - 7614 \beta_{17} - 34146 \beta_{16} - 20250 \beta_{15} + \cdots - 708993 \beta_{4} ) / 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 21480 \beta_{13} - 255960 \beta_{9} - 156528 \beta_{8} - 192816 \beta_{7} + 37179 \beta_{6} + \cdots + 3675699 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3645 \beta_{19} + 294057 \beta_{18} - 66447 \beta_{17} - 2046642 \beta_{16} + \cdots + 20452662 \beta_{4} ) / 27 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1177732 \beta_{13} - 15403806 \beta_{9} - 11244804 \beta_{8} - 10402404 \beta_{7} - 2452896 \beta_{6} + \cdots - 1441765656 ) / 27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5095440 \beta_{19} + 17450928 \beta_{18} + 26478576 \beta_{17} - 110209110 \beta_{16} + \cdots + 4684735575 \beta_{4} ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 57232412 \beta_{13} - 793262988 \beta_{9} - 695719848 \beta_{8} - 446196648 \beta_{7} + \cdots - 174473570673 ) / 27 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 509689449 \beta_{19} + 854544249 \beta_{18} + 3263797773 \beta_{17} - 5094163854 \beta_{16} + \cdots + 457259923656 \beta_{4} ) / 27 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2287986672 \beta_{13} - 32934627702 \beta_{9} - 36455638644 \beta_{8} - 11838167892 \beta_{7} + \cdots - 14631583645206 ) / 27 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 37450013526 \beta_{19} + 30647453922 \beta_{18} + 267737897934 \beta_{17} + \cdots + 34936143233787 \beta_{4} ) / 27 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 53006294032 \beta_{13} - 804499751088 \beta_{9} - 1517927747712 \beta_{8} + 307225521984 \beta_{7} + \cdots - 10\!\cdots\!37 ) / 27 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 2370841991145 \beta_{19} + 259647926157 \beta_{18} + 18057167500365 \beta_{17} + \cdots + 22\!\cdots\!78 \beta_{4} ) / 27 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 2334963874900 \beta_{13} + 30438203738490 \beta_{9} - 35916975682932 \beta_{8} + \cdots - 63\!\cdots\!96 ) / 27 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 133252263931716 \beta_{19} - 85111567723620 \beta_{18} + \cdots + 13\!\cdots\!23 \beta_{4} ) / 27 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
7.67147 − 1.41421i
7.67147 + 1.41421i
5.58288 + 1.41421i
5.58288 − 1.41421i
5.25893 + 1.41421i
5.25893 − 1.41421i
3.20990 − 1.41421i
3.20990 + 1.41421i
0.142298 − 1.41421i
0.142298 + 1.41421i
−0.142298 + 1.41421i
−0.142298 − 1.41421i
−3.20990 + 1.41421i
−3.20990 − 1.41421i
−5.25893 − 1.41421i
−5.25893 + 1.41421i
−5.58288 − 1.41421i
−5.58288 + 1.41421i
−7.67147 + 1.41421i
−7.67147 − 1.41421i
−7.67147 0 42.8514 −4.25231 0 − 71.7676i −205.990 0 32.6214
116.2 −7.67147 0 42.8514 −4.25231 0 71.7676i −205.990 0 32.6214
116.3 −5.58288 0 15.1685 46.7691 0 − 61.8801i 4.64207 0 −261.106
116.4 −5.58288 0 15.1685 46.7691 0 61.8801i 4.64207 0 −261.106
116.5 −5.25893 0 11.6564 −28.6948 0 − 60.3558i 22.8428 0 150.904
116.6 −5.25893 0 11.6564 −28.6948 0 60.3558i 22.8428 0 150.904
116.7 −3.20990 0 −5.69654 −3.66150 0 − 49.7677i 69.6437 0 11.7531
116.8 −3.20990 0 −5.69654 −3.66150 0 49.7677i 69.6437 0 11.7531
116.9 −0.142298 0 −15.9798 29.3219 0 − 24.4254i 4.55066 0 −4.17246
116.10 −0.142298 0 −15.9798 29.3219 0 24.4254i 4.55066 0 −4.17246
116.11 0.142298 0 −15.9798 −29.3219 0 − 24.4254i −4.55066 0 −4.17246
116.12 0.142298 0 −15.9798 −29.3219 0 24.4254i −4.55066 0 −4.17246
116.13 3.20990 0 −5.69654 3.66150 0 − 49.7677i −69.6437 0 11.7531
116.14 3.20990 0 −5.69654 3.66150 0 49.7677i −69.6437 0 11.7531
116.15 5.25893 0 11.6564 28.6948 0 − 60.3558i −22.8428 0 150.904
116.16 5.25893 0 11.6564 28.6948 0 60.3558i −22.8428 0 150.904
116.17 5.58288 0 15.1685 −46.7691 0 − 61.8801i −4.64207 0 −261.106
116.18 5.58288 0 15.1685 −46.7691 0 61.8801i −4.64207 0 −261.106
116.19 7.67147 0 42.8514 4.25231 0 − 71.7676i 205.990 0 32.6214
116.20 7.67147 0 42.8514 4.25231 0 71.7676i 205.990 0 32.6214
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.20
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 117.5.d.a ✓ 20
3.b odd 2 1 inner 117.5.d.a ✓ 20
13.b even 2 1 inner 117.5.d.a ✓ 20
39.d odd 2 1 inner 117.5.d.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.5.d.a ✓ 20 1.a even 1 1 trivial
117.5.d.a ✓ 20 3.b odd 2 1 inner
117.5.d.a ✓ 20 13.b even 2 1 inner
117.5.d.a ✓ 20 39.d odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(117, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - 128 T^{8} + \cdots - 10584)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots - 375384098304)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 10\!\cdots\!92)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 72\!\cdots\!44)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 19\!\cdots\!01)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 19\!\cdots\!68)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 10\!\cdots\!72)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 34\!\cdots\!68)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 43\!\cdots\!68)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 46\!\cdots\!48)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 38\!\cdots\!68)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 13\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 23\!\cdots\!28)^{4} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 59\!\cdots\!04)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 70\!\cdots\!88)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 45\!\cdots\!64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 31\!\cdots\!12)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 77\!\cdots\!08)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 29\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 29\!\cdots\!92)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 10\!\cdots\!12)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 13\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 65\!\cdots\!28)^{2} \) Copy content Toggle raw display
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