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Properties

Label 790.6.a.d
Level $790$
Weight $6$
Character orbit 790.a
Self dual yes
Analytic conductor $126.703$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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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] = [790,6,Mod(1,790)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("790.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 790.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [15,60,-26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(126.703217652\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 4 x^{14} - 2364 x^{13} + 6706 x^{12} + 2211698 x^{11} - 3792234 x^{10} - 1046418217 x^{9} + \cdots - 64\!\cdots\!80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + (\beta_1 - 2) q^{3} + 16 q^{4} - 25 q^{5} + (4 \beta_1 - 8) q^{6} + (\beta_{4} + \beta_1 - 11) q^{7} + 64 q^{8} + (\beta_{5} - \beta_{4} - 3 \beta_1 + 77) q^{9} - 100 q^{10} + (\beta_{10} - \beta_{6} - 80) q^{11}+ \cdots + ( - 6 \beta_{14} + 25 \beta_{13} + \cdots - 35673) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 60 q^{2} - 26 q^{3} + 240 q^{4} - 375 q^{5} - 104 q^{6} - 156 q^{7} + 960 q^{8} + 1143 q^{9} - 1500 q^{10} - 1198 q^{11} - 416 q^{12} - 266 q^{13} - 624 q^{14} + 650 q^{15} + 3840 q^{16} - 3256 q^{17}+ \cdots - 534352 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 4 x^{14} - 2364 x^{13} + 6706 x^{12} + 2211698 x^{11} - 3792234 x^{10} - 1046418217 x^{9} + \cdots - 64\!\cdots\!80 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 57\!\cdots\!37 \nu^{14} + \cdots - 74\!\cdots\!96 ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39\!\cdots\!32 \nu^{14} + \cdots + 11\!\cdots\!88 ) / 58\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\!\cdots\!43 \nu^{14} + \cdots - 95\!\cdots\!40 ) / 58\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!43 \nu^{14} + \cdots - 11\!\cdots\!00 ) / 58\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21\!\cdots\!03 \nu^{14} + \cdots - 13\!\cdots\!60 ) / 58\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 39\!\cdots\!41 \nu^{14} + \cdots - 20\!\cdots\!00 ) / 97\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 87\!\cdots\!57 \nu^{14} + \cdots - 32\!\cdots\!80 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!43 \nu^{14} + \cdots - 71\!\cdots\!60 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 39\!\cdots\!29 \nu^{14} + \cdots - 19\!\cdots\!40 ) / 58\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 67\!\cdots\!39 \nu^{14} + \cdots + 33\!\cdots\!52 ) / 64\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13\!\cdots\!87 \nu^{14} + \cdots + 75\!\cdots\!00 ) / 97\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 22\!\cdots\!04 \nu^{14} + \cdots - 12\!\cdots\!80 ) / 14\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!01 \nu^{14} + \cdots + 55\!\cdots\!20 ) / 58\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta _1 + 316 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 4 \beta_{14} - \beta_{13} + 3 \beta_{12} + 2 \beta_{11} + \beta_{10} - 5 \beta_{9} - 2 \beta_{8} + \cdots + 418 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 86 \beta_{14} - 65 \beta_{13} - 48 \beta_{12} + 151 \beta_{11} + 38 \beta_{10} - 64 \beta_{9} + \cdots + 158016 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4583 \beta_{14} - 2129 \beta_{13} + 1815 \beta_{12} + 3490 \beta_{11} + 2708 \beta_{10} + \cdots + 415133 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 96031 \beta_{14} - 40825 \beta_{13} - 41514 \beta_{12} + 190943 \beta_{11} + 23230 \beta_{10} + \cdots + 92282282 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4156377 \beta_{14} - 2108412 \beta_{13} + 988677 \beta_{12} + 3951120 \beta_{11} + 3160023 \beta_{10} + \cdots + 389995539 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 86071251 \beta_{14} - 19182522 \beta_{13} - 28237635 \beta_{12} + 182791293 \beta_{11} + \cdots + 58559359117 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3471103423 \beta_{14} - 1698089596 \beta_{13} + 535684863 \beta_{12} + 3839372726 \beta_{11} + \cdots + 363474665092 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 72710849771 \beta_{14} - 7757564765 \beta_{13} - 17573567907 \beta_{12} + 158886746779 \beta_{11} + \cdots + 39253070572143 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2797627670522 \beta_{14} - 1264177826012 \beta_{13} + 291417517437 \beta_{12} + \cdots + 330987882267356 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 60102236441479 \beta_{14} - 2803343739880 \beta_{13} - 10433802782877 \beta_{12} + \cdots + 27\!\cdots\!60 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 22\!\cdots\!20 \beta_{14} - 911515102992906 \beta_{13} + 158895109555479 \beta_{12} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 49\!\cdots\!30 \beta_{14} + \cdots + 19\!\cdots\!33 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−25.8712
−23.5931
−21.1512
−15.9387
−11.7158
−11.6119
−6.74656
−1.51126
6.67741
8.17289
16.7596
17.9113
21.2182
23.2066
28.1938
4.00000 −27.8712 16.0000 −25.0000 −111.485 −174.710 64.0000 533.806 −100.000
1.2 4.00000 −25.5931 16.0000 −25.0000 −102.372 −185.720 64.0000 412.008 −100.000
1.3 4.00000 −23.1512 16.0000 −25.0000 −92.6050 18.7326 64.0000 292.980 −100.000
1.4 4.00000 −17.9387 16.0000 −25.0000 −71.7549 215.642 64.0000 78.7978 −100.000
1.5 4.00000 −13.7158 16.0000 −25.0000 −54.8631 60.9924 64.0000 −54.8776 −100.000
1.6 4.00000 −13.6119 16.0000 −25.0000 −54.4475 −81.4674 64.0000 −57.7169 −100.000
1.7 4.00000 −8.74656 16.0000 −25.0000 −34.9863 −48.9370 64.0000 −166.498 −100.000
1.8 4.00000 −3.51126 16.0000 −25.0000 −14.0450 −56.9903 64.0000 −230.671 −100.000
1.9 4.00000 4.67741 16.0000 −25.0000 18.7096 116.385 64.0000 −221.122 −100.000
1.10 4.00000 6.17289 16.0000 −25.0000 24.6916 210.502 64.0000 −204.895 −100.000
1.11 4.00000 14.7596 16.0000 −25.0000 59.0382 −145.811 64.0000 −25.1557 −100.000
1.12 4.00000 15.9113 16.0000 −25.0000 63.6453 −50.2931 64.0000 10.1702 −100.000
1.13 4.00000 19.2182 16.0000 −25.0000 76.8727 98.1823 64.0000 126.338 −100.000
1.14 4.00000 21.2066 16.0000 −25.0000 84.8263 −99.7550 64.0000 206.719 −100.000
1.15 4.00000 26.1938 16.0000 −25.0000 104.775 −32.7522 64.0000 443.117 −100.000
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(79\) \( -1 \)

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 790.6.a.d ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
790.6.a.d ✓ 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} + 26 T_{3}^{14} - 2056 T_{3}^{13} - 52574 T_{3}^{12} + 1645266 T_{3}^{11} + \cdots - 12\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(790))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T + 25)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 11\!\cdots\!20 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 10\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 55\!\cdots\!60 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 38\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 34\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 15\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 20\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 36\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 52\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 35\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 59\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T - 6241)^{15} \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 79\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 20\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 58\!\cdots\!84 \) Copy content Toggle raw display
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