Properties

Label 7.20.a.a
Level $7$
Weight $20$
Character orbit 7.a
Self dual yes
Analytic conductor $16.017$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 7.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0171687589\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 330513x^{2} - 30288715x + 14876898628 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 86) q^{2} + ( - \beta_{2} - 8 \beta_1 - 7378) q^{3} + (2 \beta_{3} + 106 \beta_1 + 143996) q^{4} + ( - 19 \beta_{3} + 37 \beta_{2} - 596 \beta_1 - 621336) q^{5} + ( - 36 \beta_{3} + 816 \beta_{2} - 7026 \beta_1 - 4327140) q^{6} + 40353607 q^{7} + (948 \beta_{3} + 2688 \beta_{2} - 133804 \beta_1 + 102266936) q^{8} + (1989 \beta_{3} + 12390 \beta_{2} - 992940 \beta_1 - 83323443) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 86) q^{2} + ( - \beta_{2} - 8 \beta_1 - 7378) q^{3} + (2 \beta_{3} + 106 \beta_1 + 143996) q^{4} + ( - 19 \beta_{3} + 37 \beta_{2} - 596 \beta_1 - 621336) q^{5} + ( - 36 \beta_{3} + 816 \beta_{2} - 7026 \beta_1 - 4327140) q^{6} + 40353607 q^{7} + (948 \beta_{3} + 2688 \beta_{2} - 133804 \beta_1 + 102266936) q^{8} + (1989 \beta_{3} + 12390 \beta_{2} - 992940 \beta_1 - 83323443) q^{9} + ( - 7444 \beta_{3} - 55728 \beta_{2} - 2953916 \beta_1 - 347819336) q^{10} + ( - 14837 \beta_{3} - 106078 \beta_{2} + \cdots - 1304964084) q^{11}+ \cdots + (18714686571927 \beta_{3} + \cdots + 22\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 342 q^{2} - 29526 q^{3} + 576196 q^{4} - 2486610 q^{5} - 17324244 q^{6} + 161414428 q^{7} + 408794760 q^{8} - 335304432 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 342 q^{2} - 29526 q^{3} + 576196 q^{4} - 2486610 q^{5} - 17324244 q^{6} + 161414428 q^{7} + 408794760 q^{8} - 335304432 q^{9} - 1397073720 q^{10} - 5232894012 q^{11} - 2652921096 q^{12} - 24071694934 q^{13} - 13800933594 q^{14} - 150674677560 q^{15} - 695063798768 q^{16} - 1122693554556 q^{17} - 2614443305094 q^{18} - 1689034371682 q^{19} - 6308977253040 q^{20} - 1191480600282 q^{21} - 16919940544224 q^{22} - 20343270469752 q^{23} - 9188544894480 q^{24} - 12146712350000 q^{25} - 5032107987984 q^{26} + 11404928663100 q^{27} + 23251586938972 q^{28} + 17794845083772 q^{29} + 274558228864560 q^{30} + 438619343652812 q^{31} + 172526736764448 q^{32} + 560272505144688 q^{33} + 582762872846028 q^{34} - 100343682702270 q^{35} + 218182607781732 q^{36} + 371101054682492 q^{37} + 269634109145940 q^{38} - 34\!\cdots\!08 q^{39}+ \cdots + 89\!\cdots\!76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 330513x^{2} - 30288715x + 14876898628 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 366\nu^{2} - 190203\nu + 37566932 ) / 336 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 278\nu - 330444 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 139\beta _1 + 330444 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 366\beta_{3} + 672\beta_{2} + 241077\beta _1 + 45808640 ) / 2 \) Copy content Toggle raw display

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−408.172
−351.370
178.094
582.447
−902.343 40157.9 289935. −3.86047e6 −3.62362e7 4.03536e7 2.11467e8 4.50396e8 3.48347e9
1.2 −788.739 −48874.0 97821.5 1.27186e6 3.85488e7 4.03536e7 3.36371e8 1.22641e9 −1.00316e9
1.3 270.188 −3480.67 −451287. 4.93061e6 −940434. 4.03536e7 −2.63588e8 −1.15015e9 1.33219e9
1.4 1078.89 −17329.2 639726. −4.82861e6 −1.86964e7 4.03536e7 1.24545e8 −8.61959e8 −5.20956e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)

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 7.20.a.a 4
3.b odd 2 1 63.20.a.b 4
7.b odd 2 1 49.20.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.20.a.a 4 1.a even 1 1 trivial
49.20.a.c 4 7.b odd 2 1
63.20.a.b 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 342T_{2}^{3} - 1278192T_{2}^{2} - 467202816T_{2} + 207467274240 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(7))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 342 T^{3} + \cdots + 207467274240 \) Copy content Toggle raw display
$3$ \( T^{4} + 29526 T^{3} + \cdots - 11\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( T^{4} + 2486610 T^{3} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T - 40353607)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 5232894012 T^{3} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{4} + 24071694934 T^{3} + \cdots + 14\!\cdots\!80 \) Copy content Toggle raw display
$17$ \( T^{4} + 1122693554556 T^{3} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{4} + 1689034371682 T^{3} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + 20343270469752 T^{3} + \cdots - 14\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{4} - 17794845083772 T^{3} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} - 438619343652812 T^{3} + \cdots - 39\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{4} - 371101054682492 T^{3} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 16\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 72\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 28\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 87\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 26\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 55\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 98\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 44\!\cdots\!68 \) Copy content Toggle raw display
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