Properties

Label 81.10.c.j
Level $81$
Weight $10$
Character orbit 81.c
Analytic conductor $41.718$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 83x^{6} + 5449x^{4} + 119520x^{2} + 2073600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{16} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{6} + \beta_{5} + 559 \beta_1 - 559) q^{4} + ( - \beta_{7} + 10 \beta_{3}) q^{5} + ( - 4 \beta_{6} - 2963 \beta_1) q^{7} + ( - 10 \beta_{7} + \cdots - 818 \beta_{2}) q^{8}+ \cdots + ( - 237040 \beta_{7} + \cdots + 129798 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2236 q^{4} - 11852 q^{7} - 83520 q^{10} - 179684 q^{13} - 2348680 q^{16} + 2022856 q^{19} - 3473568 q^{22} - 2554060 q^{25} + 39587848 q^{28} - 889136 q^{31} + 43111008 q^{34} - 7610312 q^{37} + 133649280 q^{40}+ \cdots - 1935734516 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 83x^{6} + 5449x^{4} + 119520x^{2} + 2073600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 83\nu^{6} + 5449\nu^{4} + 452267\nu^{2} + 9920160 ) / 7846560 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 71195\nu ) / 21796 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 293\nu^{7} + 38143\nu^{5} + 1596557\nu^{3} + 35019360\nu ) / 6277248 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -21\nu^{7} + 5026047\nu ) / 10898 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -54\nu^{6} + 5757129 ) / 5449 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -12027\nu^{6} - 1356801\nu^{4} - 65535123\nu^{2} - 1437467040 ) / 871840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28183\nu^{7} + 2511989\nu^{5} + 153569167\nu^{3} + 3368432160\nu ) / 5231040 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - 42\beta_{2} ) / 324 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} + 2241\beta _1 - 2241 ) / 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 35\beta_{7} - 35\beta_{4} - 2766\beta_{3} + 2766\beta_{2} ) / 324 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -83\beta_{6} - 108243\beta_1 ) / 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1465\beta_{7} + 169098\beta_{3} ) / 324 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5449\beta_{5} + 5757129 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 71195\beta_{4} - 10052094\beta_{2} ) / 324 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
3.81773 + 6.61251i
−2.48494 4.30404i
2.48494 + 4.30404i
−3.81773 6.61251i
3.81773 6.61251i
−2.48494 + 4.30404i
2.48494 4.30404i
−3.81773 + 6.61251i
−22.2386 38.5183i 0 −733.108 + 1269.78i 525.311 909.865i 0 −3295.93 5708.72i 42440.8 0 −46728.6
28.2 −6.39890 11.0832i 0 174.108 301.564i −1009.89 + 1749.17i 0 332.932 + 576.655i −11008.9 0 25848.6
28.3 6.39890 + 11.0832i 0 174.108 301.564i 1009.89 1749.17i 0 332.932 + 576.655i 11008.9 0 25848.6
28.4 22.2386 + 38.5183i 0 −733.108 + 1269.78i −525.311 + 909.865i 0 −3295.93 5708.72i −42440.8 0 −46728.6
55.1 −22.2386 + 38.5183i 0 −733.108 1269.78i 525.311 + 909.865i 0 −3295.93 + 5708.72i 42440.8 0 −46728.6
55.2 −6.39890 + 11.0832i 0 174.108 + 301.564i −1009.89 1749.17i 0 332.932 576.655i −11008.9 0 25848.6
55.3 6.39890 11.0832i 0 174.108 + 301.564i 1009.89 + 1749.17i 0 332.932 576.655i 11008.9 0 25848.6
55.4 22.2386 38.5183i 0 −733.108 1269.78i −525.311 909.865i 0 −3295.93 + 5708.72i −42440.8 0 −46728.6
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.4
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 81.10.c.j 8
3.b odd 2 1 inner 81.10.c.j 8
9.c even 3 1 27.10.a.d 4
9.c even 3 1 inner 81.10.c.j 8
9.d odd 6 1 27.10.a.d 4
9.d odd 6 1 inner 81.10.c.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.10.a.d 4 9.c even 3 1
27.10.a.d 4 9.d odd 6 1
81.10.c.j 8 1.a even 1 1 trivial
81.10.c.j 8 3.b odd 2 1 inner
81.10.c.j 8 9.c even 3 1 inner
81.10.c.j 8 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 2142T_{2}^{6} + 4264164T_{2}^{4} + 694008000T_{2}^{2} + 104976000000 \) acting on \(S_{10}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 104976000000 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 19265840368369)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 18\!\cdots\!25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 505714 T - 484090125647)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 8057965728479)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 80\!\cdots\!44)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 62\!\cdots\!69)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 17\!\cdots\!25)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots + 69\!\cdots\!53)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 15\!\cdots\!25)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 62\!\cdots\!09)^{2} \) Copy content Toggle raw display
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