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Properties

Label 81.8.c.k
Level $81$
Weight $8$
Character orbit 81.c
Analytic conductor $25.303$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,8,Mod(28,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.28");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 81.c (of order \(3\), degree \(2\), not 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: \(25.3031870642\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 316x^{10} + 37872x^{8} + 2079550x^{6} + 47948824x^{4} + 251235828x^{2} + 43520409 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{21} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + (\beta_{7} + 73 \beta_{5} - \beta_{4} - 73) q^{4} + (\beta_{10} - 2 \beta_{9} + \cdots - 12 \beta_1) q^{5} + (\beta_{11} + 4 \beta_{7} + \cdots - \beta_{3}) q^{7} + ( - 7 \beta_{6} + 15 \beta_{2} - 50 \beta_1) q^{8}+ \cdots + ( - 19336 \beta_{6} + \cdots + 58635 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 438 q^{4} + 1932 q^{7} - 29844 q^{10} + 11886 q^{13} - 7314 q^{16} - 134328 q^{19} + 23364 q^{22} - 42324 q^{25} - 1386696 q^{28} + 470832 q^{31} + 1834866 q^{34} - 2052516 q^{37} + 3091374 q^{40}+ \cdots - 26307948 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 316x^{10} + 37872x^{8} + 2079550x^{6} + 47948824x^{4} + 251235828x^{2} + 43520409 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 833621 \nu^{10} + 198785657 \nu^{8} + 16148408049 \nu^{6} + 480550895879 \nu^{4} + \cdots + 1689641527131 ) / 61269381180 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1435883 \nu^{10} + 327336896 \nu^{8} + 25255750617 \nu^{6} + 699497730527 \nu^{4} + \cdots - 1127452322727 ) / 61269381180 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5065 \nu^{10} + 1194922 \nu^{8} + 96310629 \nu^{6} + 2874355369 \nu^{4} + 17863658362 \nu^{2} + 5239584909 ) / 38534202 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7766 \nu^{10} - 1843547 \nu^{8} - 149062212 \nu^{6} - 4403469689 \nu^{4} - 24037816568 \nu^{2} - 508469256 ) / 19267101 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14836 \nu^{11} - 3454537 \nu^{9} - 269430579 \nu^{7} - 7235286844 \nu^{5} + \cdots + 36943068060 ) / 73886136120 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1524618 \nu^{10} + 360683141 \nu^{8} + 29120962352 \nu^{6} + 863541161032 \nu^{4} + \cdots + 1271317687938 ) / 2269236340 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 67932346 \nu^{11} - 6034026680 \nu^{10} - 12856236727 \nu^{9} - 1432399148060 \nu^{8} + \cdots - 395070442526880 ) / 29940304269960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 281961325 \nu^{11} - 611044193 \nu^{10} - 68161527415 \nu^{9} - 145709886581 \nu^{8} + \cdots - 12\!\cdots\!23 ) / 89820912809880 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 841495310 \nu^{11} + 1052502239 \nu^{10} - 200330355590 \nu^{9} + 239937944768 \nu^{8} + \cdots - 826422552558891 ) / 89820912809880 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 619343425 \nu^{11} + 10057904946 \nu^{10} - 140680777105 \nu^{9} + 2379426681177 \nu^{8} + \cdots + 83\!\cdots\!86 ) / 29940304269960 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 431116992 \nu^{11} + 983850925 \nu^{10} + 102646131759 \nu^{9} + 232107623890 \nu^{8} + \cdots + 10\!\cdots\!05 ) / 14970152134980 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{11} - 4 \beta_{9} + 32 \beta_{8} + 4 \beta_{7} - 72 \beta_{5} - 2 \beta_{4} - \beta_{3} + \cdots + 36 ) / 108 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{6} + 10\beta_{4} - 8\beta_{3} + 13\beta_{2} + 203\beta _1 - 5688 ) / 108 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 90 \beta_{11} + 6 \beta_{10} + 50 \beta_{9} - 1102 \beta_{8} - 156 \beta_{7} - 3 \beta_{6} + \cdots - 1566 ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -309\beta_{6} - 802\beta_{4} + 872\beta_{3} - 1003\beta_{2} - 15197\beta _1 + 434016 ) / 108 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13858 \beta_{11} - 780 \beta_{10} - 944 \beta_{9} + 164332 \beta_{8} + 25676 \beta_{7} + 390 \beta_{6} + \cdots + 373644 ) / 108 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7905\beta_{6} + 16786\beta_{4} - 20375\beta_{3} + 16465\beta_{2} + 275405\beta _1 - 8507205 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1069658 \beta_{11} + 33048 \beta_{10} - 368004 \beta_{9} - 12496440 \beta_{8} - 2114260 \beta_{7} + \cdots - 39803868 ) / 108 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -3076275\beta_{6} - 5695714\beta_{4} + 7315712\beta_{3} - 4540477\beta_{2} - 79680299\beta _1 + 2692218240 ) / 108 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 41853630 \beta_{11} + 207966 \beta_{10} + 30393730 \beta_{9} + 480247498 \beta_{8} + 86925492 \beta_{7} + \cdots + 1963504962 ) / 54 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 289269777 \beta_{6} + 486840970 \beta_{4} - 642005108 \beta_{3} + 342201127 \beta_{2} + \cdots - 214437779208 ) / 108 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6639449602 \beta_{11} - 327578052 \beta_{10} - 7142121184 \beta_{9} - 74516362780 \beta_{8} + \cdots - 370267332372 ) / 108 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
8.84010i
8.08899i
− 8.93436i
− 9.24450i
− 2.63779i
0.423455i
− 8.84010i
− 8.08899i
8.93436i
9.24450i
2.63779i
− 0.423455i
−9.88711 − 17.1250i 0 −131.510 + 227.782i 236.803 − 410.154i 0 513.424 + 889.277i 2669.91 0 −9365.17
28.2 −6.69159 − 11.5902i 0 −25.5547 + 44.2621i −36.7045 + 63.5741i 0 242.085 + 419.304i −1029.04 0 982.446
28.3 −2.86666 − 4.96520i 0 47.5645 − 82.3842i −80.3835 + 139.228i 0 −272.509 − 472.000i −1279.27 0 921.729
28.4 2.86666 + 4.96520i 0 47.5645 − 82.3842i 80.3835 − 139.228i 0 −272.509 − 472.000i 1279.27 0 921.729
28.5 6.69159 + 11.5902i 0 −25.5547 + 44.2621i 36.7045 − 63.5741i 0 242.085 + 419.304i 1029.04 0 982.446
28.6 9.88711 + 17.1250i 0 −131.510 + 227.782i −236.803 + 410.154i 0 513.424 + 889.277i −2669.91 0 −9365.17
55.1 −9.88711 + 17.1250i 0 −131.510 − 227.782i 236.803 + 410.154i 0 513.424 − 889.277i 2669.91 0 −9365.17
55.2 −6.69159 + 11.5902i 0 −25.5547 − 44.2621i −36.7045 − 63.5741i 0 242.085 − 419.304i −1029.04 0 982.446
55.3 −2.86666 + 4.96520i 0 47.5645 + 82.3842i −80.3835 − 139.228i 0 −272.509 + 472.000i −1279.27 0 921.729
55.4 2.86666 − 4.96520i 0 47.5645 + 82.3842i 80.3835 + 139.228i 0 −272.509 + 472.000i 1279.27 0 921.729
55.5 6.69159 − 11.5902i 0 −25.5547 − 44.2621i 36.7045 + 63.5741i 0 242.085 − 419.304i 1029.04 0 982.446
55.6 9.88711 − 17.1250i 0 −131.510 − 227.782i −236.803 − 410.154i 0 513.424 − 889.277i −2669.91 0 −9365.17
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.8.c.k 12
3.b odd 2 1 inner 81.8.c.k 12
9.c even 3 1 81.8.a.d ✓ 6
9.c even 3 1 inner 81.8.c.k 12
9.d odd 6 1 81.8.a.d ✓ 6
9.d odd 6 1 inner 81.8.c.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.8.a.d ✓ 6 9.c even 3 1
81.8.a.d ✓ 6 9.d odd 6 1
81.8.c.k 12 1.a even 1 1 trivial
81.8.c.k 12 3.b odd 2 1 inner
81.8.c.k 12 9.c even 3 1 inner
81.8.c.k 12 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 603 T_{2}^{10} + 274833 T_{2}^{8} + 48927672 T_{2}^{6} + 6492994992 T_{2}^{4} + \cdots + 5299793328384 \) acting on \(S_{8}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 5299793328384 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 97\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 73\!\cdots\!76)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 43\!\cdots\!49)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 10\!\cdots\!87)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots - 91889736662600)^{4} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 35\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 12\!\cdots\!69 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 59\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots - 22\!\cdots\!69)^{4} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 38\!\cdots\!69)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 88\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots - 63\!\cdots\!81)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 51\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 12\!\cdots\!03)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 43\!\cdots\!96)^{2} \) Copy content Toggle raw display
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