Properties

Label 81.10.c.h
Level $81$
Weight $10$
Character orbit 81.c
Analytic conductor $41.718$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.28");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 10 \)
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: \(41.7179027293\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 119x^{4} - 154x^{3} + 14060x^{2} - 16048x + 18496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 27)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} + 1) q^{2} + (\beta_{5} - 2 \beta_{3} + \cdots + 2 \beta_1) q^{4} + (3 \beta_{5} + 7 \beta_{3} + \cdots - 7 \beta_1) q^{5} + (5 \beta_{5} - 5 \beta_{4} + \cdots + 1231) q^{7}+ \cdots + (1523280 \beta_{4} - 25616490 \beta_1 - 864060762) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 597 q^{4} - 1983 q^{5} + 3693 q^{7} + 9006 q^{8} - 37962 q^{10} - 16863 q^{11} - 116916 q^{13} - 503463 q^{14} + 239919 q^{16} + 2028096 q^{17} - 30444 q^{19} - 2548407 q^{20} - 305721 q^{22}+ \cdots - 5184364572 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 119x^{4} - 154x^{3} + 14060x^{2} - 16048x + 18496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 119\nu^{4} - 14161\nu^{3} + 14060\nu^{2} - 16048\nu + 1357348 ) / 552364 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 413\nu^{5} - 409\nu^{4} + 48671\nu^{3} - 6958\nu^{2} + 5750540\nu - 6563632 ) / 6628368 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -413\nu^{5} + 409\nu^{4} - 48671\nu^{3} + 6958\nu^{2} + 14134564\nu - 64736 ) / 6628368 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -88\nu^{5} + 10472\nu^{4} - 3349\nu^{3} + 1237280\nu^{2} - 1412224\nu + 97490155 ) / 138091 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -294445\nu^{5} + 288761\nu^{4} - 34362559\nu^{3} + 64278926\nu^{2} - 4059979660\nu + 4634036528 ) / 6628368 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{3} + 711\beta_{2} + 2\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{4} - 352\beta _1 + 159 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -119\beta_{5} + 119\beta_{4} + 292\beta_{3} - 83331\beta_{2} - 83331 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -101\beta_{5} - 41516\beta_{3} + 32127\beta_{2} + 41516\beta_1 ) / 9 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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
−5.46600 9.46739i
0.577142 + 0.999640i
5.38886 + 9.33377i
−5.46600 + 9.46739i
0.577142 0.999640i
5.38886 9.33377i
−16.3980 28.4022i 0 −281.788 + 488.072i −657.691 + 1139.16i 0 −2579.27 4467.43i 1691.52 0 43139.3
28.2 1.73143 + 2.99892i 0 250.004 433.020i 702.004 1215.91i 0 −832.785 1442.43i 3504.44 0 4861.88
28.3 16.1666 + 28.0013i 0 −266.716 + 461.965i −1035.81 + 1794.08i 0 5258.56 + 9108.09i −692.954 0 −66982.2
55.1 −16.3980 + 28.4022i 0 −281.788 488.072i −657.691 1139.16i 0 −2579.27 + 4467.43i 1691.52 0 43139.3
55.2 1.73143 2.99892i 0 250.004 + 433.020i 702.004 + 1215.91i 0 −832.785 + 1442.43i 3504.44 0 4861.88
55.3 16.1666 28.0013i 0 −266.716 461.965i −1035.81 1794.08i 0 5258.56 9108.09i −692.954 0 −66982.2
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 3T_{2}^{5} + 1071T_{2}^{4} - 4158T_{2}^{3} + 1138860T_{2}^{2} - 3899664T_{2} + 13483584 \) acting on \(S_{10}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3 T^{5} + \cdots + 13483584 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 81\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 92\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{3} + \cdots - 78\!\cdots\!84)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots + 44\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 39\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots - 10\!\cdots\!80)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{3} + \cdots + 13\!\cdots\!97)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots + 98\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots - 64\!\cdots\!45)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 69\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 45\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 67\!\cdots\!25 \) Copy content Toggle raw display
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