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Properties

Label 81.10.c.k
Level $81$
Weight $10$
Character orbit 81.c
Analytic conductor $41.718$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,10,Mod(28,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [8,33] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content 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: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 877x^{6} + 211575x^{4} + 13362487x^{2} + 164814244 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (8 \beta_{3} + \beta_{2}) q^{2} + ( - \beta_{7} + 6 \beta_{4} + \cdots - 212) q^{4} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots + 144) q^{5} + (\beta_{7} - 3 \beta_{6} + \cdots - 12 \beta_{2}) q^{7}+ \cdots + ( - 326672 \beta_{5} + \cdots - 189880696) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 33 q^{2} - 853 q^{4} + 570 q^{5} + 3238 q^{7} - 9582 q^{8} - 19446 q^{10} - 96690 q^{11} + 141118 q^{13} + 3036 q^{14} + 244463 q^{16} - 570312 q^{17} - 930332 q^{19} - 1041711 q^{20} + 2244480 q^{22}+ \cdots - 1471002642 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 877x^{6} + 211575x^{4} + 13362487x^{2} + 164814244 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 1722\nu^{4} + 844569\nu^{2} + 75740572 ) / 91344 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 641 \nu^{7} - 6419 \nu^{6} - 555738 \nu^{5} - 11053518 \nu^{4} - 124566057 \nu^{3} + \cdots - 98609884772 ) / 14072091264 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -641\nu^{7} - 555738\nu^{5} - 124566057\nu^{3} - 4903077164\nu + 7036045632 ) / 14072091264 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 641 \nu^{7} + 6419 \nu^{6} - 555738 \nu^{5} + 11053518 \nu^{4} - 124566057 \nu^{3} + \cdots + 98609884772 ) / 14072091264 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -221\nu^{6} - 175038\nu^{4} - 33397353\nu^{2} - 909847340 ) / 274032 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 140605 \nu^{7} + 5674396 \nu^{6} + 64275042 \nu^{5} + 4494275688 \nu^{4} + \cdots + 23361240301840 ) / 14072091264 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 99188 \nu^{7} + 19257 \nu^{6} - 81878352 \nu^{5} + 33160554 \nu^{4} - 17464134996 \nu^{3} + \cdots + 1458536195004 ) / 3518022816 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{3} + \beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -12\beta_{4} + 12\beta_{2} + \beta _1 - 661 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -28\beta_{7} - 8\beta_{6} - 4\beta_{5} - 1169\beta_{4} + 17914\beta_{3} - 1169\beta_{2} + 14\beta _1 - 8957 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{5} + 8948\beta_{4} - 8948\beta_{2} - 451\beta _1 + 261835 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 20012 \beta_{7} + 3520 \beta_{6} + 1760 \beta_{5} + 529603 \beta_{4} - 12673670 \beta_{3} + \cdots + 6336835 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2296\beta_{5} - 1757876\beta_{4} + 1757876\beta_{2} + 68695\beta _1 - 39947159 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3969620 \beta_{7} - 499048 \beta_{6} - 249524 \beta_{5} - 84977651 \beta_{4} + 2451655390 \beta_{3} + \cdots - 1225827695 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{3}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
− 22.7133i
− 4.02852i
8.49648i
16.5133i
22.7133i
4.02852i
− 8.49648i
− 16.5133i
−15.1703 − 26.2757i 0 −204.274 + 353.814i 115.867 − 200.688i 0 849.673 + 1471.68i −3138.77 0 −7030.96
28.2 1.01120 + 1.75145i 0 253.955 − 439.863i −321.605 + 557.036i 0 −1476.29 − 2557.01i 2062.66 0 −1300.83
28.3 11.8582 + 20.5389i 0 −25.2322 + 43.7034i 1379.01 − 2388.52i 0 4118.85 + 7134.05i 10945.9 0 65410.1
28.4 18.8009 + 32.5641i 0 −450.948 + 781.066i −888.273 + 1538.53i 0 −1873.23 − 3244.52i −14660.8 0 −66801.3
55.1 −15.1703 + 26.2757i 0 −204.274 − 353.814i 115.867 + 200.688i 0 849.673 − 1471.68i −3138.77 0 −7030.96
55.2 1.01120 − 1.75145i 0 253.955 + 439.863i −321.605 − 557.036i 0 −1476.29 + 2557.01i 2062.66 0 −1300.83
55.3 11.8582 − 20.5389i 0 −25.2322 − 43.7034i 1379.01 + 2388.52i 0 4118.85 − 7134.05i 10945.9 0 65410.1
55.4 18.8009 − 32.5641i 0 −450.948 − 781.066i −888.273 − 1538.53i 0 −1873.23 + 3244.52i −14660.8 0 −66801.3
\(n\): 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
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.k 8
3.b odd 2 1 81.10.c.i 8
9.c even 3 1 81.10.a.a ✓ 4
9.c even 3 1 inner 81.10.c.k 8
9.d odd 6 1 81.10.a.b yes 4
9.d odd 6 1 81.10.c.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.10.a.a ✓ 4 9.c even 3 1
81.10.a.b yes 4 9.d odd 6 1
81.10.c.i 8 3.b odd 2 1
81.10.c.i 8 9.d odd 6 1
81.10.c.k 8 1.a even 1 1 trivial
81.10.c.k 8 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 33 T_{2}^{7} + 1995 T_{2}^{6} - 28134 T_{2}^{5} + 1833084 T_{2}^{4} - 29900016 T_{2}^{3} + \cdots + 2994278400 \) acting on \(S_{10}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 2994278400 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 53\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 75\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 16\!\cdots\!73)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 61\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 39\!\cdots\!35)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 26\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 11\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 57\!\cdots\!23)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 71\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 13\!\cdots\!45)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 78\!\cdots\!16 \) Copy content Toggle raw display
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