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Properties

Label 116.6.c.a
Level $116$
Weight $6$
Character orbit 116.c
Analytic conductor $18.605$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [116,6,Mod(57,116)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(116, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("116.57");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 116 = 2^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 116.c (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: \(18.6045230983\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 1783 x^{10} + 1098222 x^{8} + 308936830 x^{6} + 41608294477 x^{4} + 2492454459339 x^{2} + 53357557623300 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{4} - 1) q^{5} + (\beta_{3} + 6) q^{7} + (\beta_{2} - 54) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{4} - 1) q^{5} + (\beta_{3} + 6) q^{7} + (\beta_{2} - 54) q^{9} + (\beta_{6} - 2 \beta_1) q^{11} + (\beta_{5} + \beta_{4} - 16) q^{13} + ( - \beta_{7} + 3 \beta_1) q^{15} + (\beta_{10} + 3 \beta_1) q^{17} + ( - \beta_{11} + \beta_{7} - 3 \beta_1) q^{19} + (\beta_{11} + \beta_{10} + \cdots - 11 \beta_1) q^{21}+ \cdots + ( - 10 \beta_{11} - 34 \beta_{10} + \cdots - 215 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{5} + 76 q^{7} - 650 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{5} + 76 q^{7} - 650 q^{9} - 194 q^{13} + 3996 q^{23} + 1782 q^{25} - 1900 q^{29} + 7266 q^{33} + 12348 q^{35} - 13988 q^{45} - 8396 q^{49} - 10788 q^{51} - 28626 q^{53} + 11140 q^{57} + 27356 q^{59} + 55392 q^{63} - 55126 q^{65} - 13184 q^{67} + 44352 q^{71} + 74264 q^{81} - 192628 q^{83} + 98908 q^{87} + 13580 q^{91} + 59710 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 1783 x^{10} + 1098222 x^{8} + 308936830 x^{6} + 41608294477 x^{4} + 2492454459339 x^{2} + 53357557623300 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 297 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 48209459 \nu^{10} + 121039928174 \nu^{8} + 100997914312200 \nu^{6} + \cdots + 13\!\cdots\!40 ) / 73\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5775043 \nu^{10} + 9955225678 \nu^{8} + 5820934127226 \nu^{6} + \cdots + 61\!\cdots\!60 ) / 59\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1482631577 \nu^{10} - 2498834338946 \nu^{8} + \cdots - 11\!\cdots\!08 ) / 65\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1381697279 \nu^{11} - 2191158680282 \nu^{9} + \cdots - 32\!\cdots\!06 \nu ) / 19\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5775043 \nu^{11} + 9955225678 \nu^{9} + 5820934127226 \nu^{7} + \cdots + 61\!\cdots\!92 \nu ) / 59\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 68464062773 \nu^{10} + 108750178041758 \nu^{8} + \cdots + 28\!\cdots\!60 ) / 65\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41713713533 \nu^{11} + 68614689449702 \nu^{9} + \cdots + 26\!\cdots\!28 \nu ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 263078221489 \nu^{11} - 428062061710294 \nu^{9} + \cdots - 16\!\cdots\!60 \nu ) / 85\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49207072040 \nu^{11} + 85427602121411 \nu^{9} + \cdots + 46\!\cdots\!02 \nu ) / 70\!\cdots\!10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 297 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{10} - 3\beta_{9} + 2\beta_{7} - \beta_{6} - 552\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{8} - 79\beta_{5} - 418\beta_{4} - 194\beta_{3} - 919\beta_{2} + 163616 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{11} + 1738\beta_{10} + 3219\beta_{9} - 2632\beta_{7} + 2300\beta_{6} + 405113\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14184\beta_{8} + 96338\beta_{5} + 566534\beta_{4} + 224368\beta_{3} + 796555\beta_{2} - 119970391 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -25044\beta_{11} - 1473930\beta_{10} - 2942649\beta_{9} + 2626776\beta_{7} - 2438757\beta_{6} - 331852072\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 13400010 \beta_{8} - 92305857 \beta_{5} - 568184160 \beta_{4} - 207877560 \beta_{3} + \cdots + 98244915612 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 30334194 \beta_{11} + 1269799730 \beta_{10} + 2594834139 \beta_{9} - 2402230616 \beta_{7} + \cdots + 282690490725 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 11969933340 \beta_{8} + 82867518712 \beta_{5} + 519783474580 \beta_{4} + 183399383900 \beta_{3} + \cdots - 83682565291973 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 30215386884 \beta_{11} - 1101413393026 \beta_{10} - 2265758546703 \beta_{9} + \cdots - 244089596964296 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\) \(59\) \(89\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
− 29.4895i
− 17.8714i
− 16.9196i
− 13.6880i
− 8.15074i
− 7.34253i
7.34253i
8.15074i
13.6880i
16.9196i
17.8714i
29.4895i
0 − 29.4895i 0 28.0043 0 −60.4647 0 −626.633 0
57.2 0 − 17.8714i 0 −82.5785 0 −124.027 0 −76.3873 0
57.3 0 − 16.9196i 0 −26.9669 0 194.819 0 −43.2724 0
57.4 0 − 13.6880i 0 81.0215 0 136.665 0 55.6395 0
57.5 0 − 8.15074i 0 46.4210 0 −141.197 0 176.565 0
57.6 0 − 7.34253i 0 −50.9013 0 32.2054 0 189.087 0
57.7 0 7.34253i 0 −50.9013 0 32.2054 0 189.087 0
57.8 0 8.15074i 0 46.4210 0 −141.197 0 176.565 0
57.9 0 13.6880i 0 81.0215 0 136.665 0 55.6395 0
57.10 0 16.9196i 0 −26.9669 0 194.819 0 −43.2724 0
57.11 0 17.8714i 0 −82.5785 0 −124.027 0 −76.3873 0
57.12 0 29.4895i 0 28.0043 0 −60.4647 0 −626.633 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 116.6.c.a ✓ 12
3.b odd 2 1 1044.6.h.a 12
4.b odd 2 1 464.6.e.a 12
29.b even 2 1 inner 116.6.c.a ✓ 12
87.d odd 2 1 1044.6.h.a 12
116.d odd 2 1 464.6.e.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.6.c.a ✓ 12 1.a even 1 1 trivial
116.6.c.a ✓ 12 29.b even 2 1 inner
464.6.e.a 12 4.b odd 2 1
464.6.e.a 12 116.d odd 2 1
1044.6.h.a 12 3.b odd 2 1
1044.6.h.a 12 87.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(116, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 53357557623300 \) Copy content Toggle raw display
$5$ \( (T^{6} + 5 T^{5} + \cdots - 11938955386)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 38 T^{5} + \cdots - 907942798080)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 18\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 91\!\cdots\!50)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 28\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 52\!\cdots\!08)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 74\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 28\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 10\!\cdots\!90)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 13\!\cdots\!60)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 34\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 53\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 83\!\cdots\!52 \) Copy content Toggle raw display
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