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Properties

Label 790.6.a.e
Level $790$
Weight $6$
Character orbit 790.a
Self dual yes
Analytic conductor $126.703$
Analytic rank $0$
Dimension $17$
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 = [17,-68,-22] 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: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 5 x^{16} - 2808 x^{15} + 13814 x^{14} + 3138420 x^{13} - 13599532 x^{12} + \cdots + 28\!\cdots\!40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 q^{2} + ( - \beta_1 - 1) q^{3} + 16 q^{4} + 25 q^{5} + (4 \beta_1 + 4) q^{6} + (\beta_{3} - \beta_1) q^{7} - 64 q^{8} + (\beta_{2} + 2 \beta_1 + 90) q^{9} - 100 q^{10} + ( - \beta_{10} + 25) q^{11}+ \cdots + (30 \beta_{16} + 75 \beta_{15} + \cdots + 7551) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 68 q^{2} - 22 q^{3} + 272 q^{4} + 425 q^{5} + 88 q^{6} - 4 q^{7} - 1088 q^{8} + 1537 q^{9} - 1700 q^{10} + 426 q^{11} - 352 q^{12} + 750 q^{13} + 16 q^{14} - 550 q^{15} + 4352 q^{16} + 2074 q^{17}+ \cdots + 126104 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 5 x^{16} - 2808 x^{15} + 13814 x^{14} + 3138420 x^{13} - 13599532 x^{12} + \cdots + 28\!\cdots\!40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 332 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!54 \nu^{16} + \cdots + 44\!\cdots\!40 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 82\!\cdots\!69 \nu^{16} + \cdots - 16\!\cdots\!40 ) / 76\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25\!\cdots\!09 \nu^{16} + \cdots + 81\!\cdots\!00 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 79\!\cdots\!43 \nu^{16} + \cdots - 11\!\cdots\!40 ) / 34\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33\!\cdots\!59 \nu^{16} + \cdots - 11\!\cdots\!20 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 71\!\cdots\!93 \nu^{16} + \cdots - 13\!\cdots\!52 ) / 22\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 73\!\cdots\!07 \nu^{16} + \cdots - 16\!\cdots\!20 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 52\!\cdots\!31 \nu^{16} + \cdots + 12\!\cdots\!00 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!87 \nu^{16} + \cdots - 44\!\cdots\!00 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 31\!\cdots\!65 \nu^{16} + \cdots + 38\!\cdots\!60 ) / 57\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11\!\cdots\!91 \nu^{16} + \cdots + 30\!\cdots\!00 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 96\!\cdots\!36 \nu^{16} + \cdots + 42\!\cdots\!80 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 23\!\cdots\!75 \nu^{16} + \cdots + 62\!\cdots\!80 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 24\!\cdots\!99 \nu^{16} + \cdots + 57\!\cdots\!80 ) / 22\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 332 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3 \beta_{15} - 3 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} - 6 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + \cdots - 126 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 36 \beta_{16} - 25 \beta_{15} + 68 \beta_{14} + 44 \beta_{13} + 47 \beta_{12} + 69 \beta_{11} + \cdots + 190028 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 530 \beta_{16} + 3166 \beta_{15} - 3453 \beta_{14} - 4387 \beta_{13} - 3171 \beta_{12} - 6090 \beta_{11} + \cdots - 543552 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 44473 \beta_{16} - 29259 \beta_{15} + 80306 \beta_{14} + 61311 \beta_{13} + 59874 \beta_{12} + \cdots + 125897372 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 671192 \beta_{16} + 2753317 \beta_{15} - 3078879 \beta_{14} - 3836027 \beta_{13} - 2753498 \beta_{12} + \cdots - 788258565 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 40890867 \beta_{16} - 29688684 \beta_{15} + 75411909 \beta_{14} + 62718371 \beta_{13} + \cdots + 89932511141 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 664417333 \beta_{16} + 2246195724 \beta_{15} - 2580966110 \beta_{14} - 3156376151 \beta_{13} + \cdots - 876671489478 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 34336140427 \beta_{16} - 28565506904 \beta_{15} + 66017374515 \beta_{14} + 57916800092 \beta_{13} + \cdots + 67058698530521 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 609408444136 \beta_{16} + 1791386180141 \beta_{15} - 2127978245523 \beta_{14} - 2553445390069 \beta_{13} + \cdots - 869061346367631 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 27955294582948 \beta_{16} - 26501697406735 \beta_{15} + 56231691694624 \beta_{14} + \cdots + 51\!\cdots\!93 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 540584521017593 \beta_{16} + \cdots - 81\!\cdots\!17 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 22\!\cdots\!55 \beta_{16} + \cdots + 40\!\cdots\!92 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 47\!\cdots\!33 \beta_{16} + \cdots - 72\!\cdots\!14 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 18\!\cdots\!38 \beta_{16} + \cdots + 31\!\cdots\!93 \) 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
25.5368
25.4047
24.9957
19.4804
19.4007
13.7621
4.92702
1.87160
−0.776856
−1.27638
−7.72047
−10.9435
−16.1598
−17.2980
−18.6685
−28.7650
−28.7705
−4.00000 −26.5368 16.0000 25.0000 106.147 −204.190 −64.0000 461.201 −100.000
1.2 −4.00000 −26.4047 16.0000 25.0000 105.619 78.5735 −64.0000 454.210 −100.000
1.3 −4.00000 −25.9957 16.0000 25.0000 103.983 −179.953 −64.0000 432.778 −100.000
1.4 −4.00000 −20.4804 16.0000 25.0000 81.9215 210.744 −64.0000 176.445 −100.000
1.5 −4.00000 −20.4007 16.0000 25.0000 81.6030 14.1227 −64.0000 173.190 −100.000
1.6 −4.00000 −14.7621 16.0000 25.0000 59.0483 182.739 −64.0000 −25.0812 −100.000
1.7 −4.00000 −5.92702 16.0000 25.0000 23.7081 12.1582 −64.0000 −207.870 −100.000
1.8 −4.00000 −2.87160 16.0000 25.0000 11.4864 −101.453 −64.0000 −234.754 −100.000
1.9 −4.00000 −0.223144 16.0000 25.0000 0.892576 148.132 −64.0000 −242.950 −100.000
1.10 −4.00000 0.276377 16.0000 25.0000 −1.10551 −19.4017 −64.0000 −242.924 −100.000
1.11 −4.00000 6.72047 16.0000 25.0000 −26.8819 −229.263 −64.0000 −197.835 −100.000
1.12 −4.00000 9.94353 16.0000 25.0000 −39.7741 −212.527 −64.0000 −144.126 −100.000
1.13 −4.00000 15.1598 16.0000 25.0000 −60.6393 182.890 −64.0000 −13.1799 −100.000
1.14 −4.00000 16.2980 16.0000 25.0000 −65.1921 −85.5323 −64.0000 22.6257 −100.000
1.15 −4.00000 17.6685 16.0000 25.0000 −70.6740 −32.5000 −64.0000 69.1759 −100.000
1.16 −4.00000 27.7650 16.0000 25.0000 −111.060 163.103 −64.0000 527.893 −100.000
1.17 −4.00000 27.7705 16.0000 25.0000 −111.082 68.3564 −64.0000 528.201 −100.000
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.17
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.e ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
790.6.a.e ✓ 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{17} + 22 T_{3}^{16} - 2592 T_{3}^{15} - 54654 T_{3}^{14} + 2655364 T_{3}^{13} + \cdots + 26\!\cdots\!48 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(790))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{17} \) Copy content Toggle raw display
$3$ \( T^{17} + \cdots + 26\!\cdots\!48 \) Copy content Toggle raw display
$5$ \( (T - 25)^{17} \) Copy content Toggle raw display
$7$ \( T^{17} + \cdots - 15\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots - 28\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{17} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 30\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots + 87\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 25\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 34\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 24\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 43\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 43\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 14\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T + 6241)^{17} \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 55\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
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