Properties

Label 81.10.a.d
Level $81$
Weight $10$
Character orbit 81.a
Self dual yes
Analytic conductor $41.718$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,10,Mod(1,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 81.a (trivial)

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: yes
Analytic conductor: \(41.7179027293\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2930 x^{6} - 1276 x^{5} + 2487472 x^{4} + 3423248 x^{3} - 586568096 x^{2} + \cdots + 965565184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{18}\cdot 17 \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + (\beta_{2} - 2 \beta_1 + 224) q^{4} + (\beta_{5} + \beta_{2} + 56) q^{5} + (\beta_{5} + \beta_{3} - 67 \beta_1 + 51) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots + 930) q^{8}+ \cdots + (18719 \beta_{7} + 35179 \beta_{6} + \cdots - 302145943) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 15 q^{2} + 1793 q^{4} + 453 q^{5} + 343 q^{7} + 7239 q^{8} + 510 q^{10} + 99150 q^{11} - 32435 q^{13} + 394824 q^{14} + 328193 q^{16} + 415539 q^{17} - 85277 q^{19} + 1855164 q^{20} - 529359 q^{22}+ \cdots - 2413650159 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2930 x^{6} - 1276 x^{5} + 2487472 x^{4} + 3423248 x^{3} - 586568096 x^{2} + \cdots + 965565184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 732 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 503 \nu^{7} + 61083 \nu^{6} + 2528604 \nu^{5} - 160366700 \nu^{4} - 8380255800 \nu^{3} + \cdots + 7794440754048 ) / 2707592448 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15021 \nu^{7} + 418759 \nu^{6} + 58051644 \nu^{5} - 982462380 \nu^{4} + \cdots + 60735117255808 ) / 10830369792 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17033 \nu^{7} + 174427 \nu^{6} + 47937228 \nu^{5} - 340995580 \nu^{4} + \cdots + 21174648020608 ) / 10830369792 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 26447 \nu^{7} - 408067 \nu^{6} + 76309476 \nu^{5} + 1090595804 \nu^{4} + \cdots + 82897833766784 ) / 5415184896 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18075 \nu^{7} + 103961 \nu^{6} + 50268732 \nu^{5} - 301142724 \nu^{4} + \cdots - 17752715711872 ) / 2707592448 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 732 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} - \beta_{3} + 3\beta_{2} + 1234\beta _1 + 1422 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -20\beta_{7} + 4\beta_{6} + 61\beta_{5} + 11\beta_{4} - 15\beta_{3} + 1685\beta_{2} + 5262\beta _1 + 903286 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 60 \beta_{7} + 140 \beta_{6} - 2441 \beta_{5} + 2337 \beta_{4} - 1693 \beta_{3} + 13367 \beta_{2} + \cdots + 3788658 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 47892 \beta_{7} + 1956 \beta_{6} + 163181 \beta_{5} + 33115 \beta_{4} - 40879 \beta_{3} + \cdots + 1270802438 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 258908 \beta_{7} + 333964 \beta_{6} - 4757753 \beta_{5} + 4397969 \beta_{4} - 2584941 \beta_{3} + \cdots + 9953357362 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
41.7589
27.6540
23.4832
0.320476
−5.66432
−15.7185
−32.8660
−37.9678
−39.7589 0 1068.77 846.447 0 −7043.99 −22136.7 0 −33653.8
1.2 −25.6540 0 146.127 −706.451 0 −4568.42 9386.09 0 18123.3
1.3 −21.4832 0 −50.4713 44.8401 0 9345.55 12083.7 0 −963.310
1.4 1.67952 0 −509.179 1538.61 0 5656.94 −1715.10 0 2584.13
1.5 7.66432 0 −453.258 −2210.95 0 −6233.15 −7398.05 0 −16945.4
1.6 17.7185 0 −198.054 739.877 0 −6027.34 −12581.1 0 13109.5
1.7 34.8660 0 703.639 −2006.52 0 4867.88 6681.68 0 −69959.4
1.8 39.9678 0 1085.42 2207.15 0 4345.52 22918.5 0 88215.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.10.a.d 8
3.b odd 2 1 81.10.a.c 8
9.c even 3 2 27.10.c.a 16
9.d odd 6 2 9.10.c.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.10.c.a 16 9.d odd 6 2
27.10.c.a 16 9.c even 3 2
81.10.a.c 8 3.b odd 2 1
81.10.a.d 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 15 T_{2}^{7} - 2832 T_{2}^{6} + 36072 T_{2}^{5} + 2299752 T_{2}^{4} - 22804416 T_{2}^{3} + \cdots - 6964475904 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(81))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots - 6964475904 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 14\!\cdots\!71 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 52\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots - 82\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 88\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 32\!\cdots\!68 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 26\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 99\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 81\!\cdots\!05 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 12\!\cdots\!11 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 47\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots - 12\!\cdots\!97 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 87\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 93\!\cdots\!13 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 16\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 62\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 89\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 44\!\cdots\!49 \) Copy content Toggle raw display
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