Properties

Label 5.24.a.a
Level $5$
Weight $24$
Character orbit 5.a
Self dual yes
Analytic conductor $16.760$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.7602018673\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - x^{2} - 215756x + 18660756 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 3^{3}\cdot 5 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 222) q^{2} + ( - \beta_{2} + 64 \beta_1 - 46476) q^{3} + (12 \beta_{2} - 334 \beta_1 - 3161172) q^{4} + 48828125 q^{5} + ( - 232 \beta_{2} - 247376 \beta_1 + 319966272) q^{6} + (24969 \beta_{2} + 134848 \beta_1 - 810894448) q^{7} + (7992 \beta_{2} - 9380284 \beta_1 - 4280140520) q^{8} + ( - 122796 \beta_{2} - 19484928 \beta_1 + 564361317) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 222) q^{2} + ( - \beta_{2} + 64 \beta_1 - 46476) q^{3} + (12 \beta_{2} - 334 \beta_1 - 3161172) q^{4} + 48828125 q^{5} + ( - 232 \beta_{2} - 247376 \beta_1 + 319966272) q^{6} + (24969 \beta_{2} + 134848 \beta_1 - 810894448) q^{7} + (7992 \beta_{2} - 9380284 \beta_1 - 4280140520) q^{8} + ( - 122796 \beta_{2} - 19484928 \beta_1 + 564361317) q^{9} + (48828125 \beta_1 + 10839843750) q^{10} + ( - 3176250 \beta_{2} - 369424000 \beta_1 - 55095343468) q^{11} + (5188096 \beta_{2} - 117716896 \beta_1 - 820308397248) q^{12} + (7666020 \beta_{2} - 1088653568 \beta_1 - 1184990244666) q^{13} + (26587176 \beta_{2} + 3241905268 \beta_1 + 546154822136) q^{14} + ( - 48828125 \beta_{2} + 3125000000 \beta_1 - 2269335937500) q^{15} + ( - 205234704 \beta_{2} + 5058297928 \beta_1 - 22995961492304) q^{16} + (392947092 \beta_{2} + 52011875072 \beta_1 - 138701923089838) q^{17} + ( - 356615136 \beta_{2} + \cdots - 100907890084746) q^{18}+ \cdots + (11\!\cdots\!78 \beta_{2} + \cdots + 65\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 666 q^{2} - 139428 q^{3} - 9483516 q^{4} + 146484375 q^{5} + 959898816 q^{6} - 2432683344 q^{7} - 12840421560 q^{8} + 1693083951 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 666 q^{2} - 139428 q^{3} - 9483516 q^{4} + 146484375 q^{5} + 959898816 q^{6} - 2432683344 q^{7} - 12840421560 q^{8} + 1693083951 q^{9} + 32519531250 q^{10} - 165286030404 q^{11} - 2460925191744 q^{12} - 3554970733998 q^{13} + 1638464466408 q^{14} - 6808007812500 q^{15} - 68987884476912 q^{16} - 416105769269514 q^{17} - 302723670254238 q^{18} - 975704043068460 q^{19} - 463062304687500 q^{20} - 51\!\cdots\!84 q^{21}+ \cdots + 19\!\cdots\!32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 215756x + 18660756 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{2} + 387\nu - 431642 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} - 129\beta _1 + 863026 ) / 6 \) 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
−502.392
89.8102
413.582
−2794.35 −370647. −580214. 4.88281e7 1.03572e9 2.05641e9 2.50620e10 4.32361e10 −1.36443e11
1.2 758.861 360571. −7.81274e6 4.88281e7 2.73623e8 −1.00441e10 −1.22946e10 3.58682e10 3.70538e10
1.3 2701.49 −129352. −1.09056e6 4.88281e7 −3.49442e8 5.55505e9 −2.56079e10 −7.74113e10 1.31909e11
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)

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 5.24.a.a 3
3.b odd 2 1 45.24.a.a 3
5.b even 2 1 25.24.a.b 3
5.c odd 4 2 25.24.b.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.24.a.a 3 1.a even 1 1 trivial
25.24.a.b 3 5.b even 2 1
25.24.b.b 6 5.c odd 4 2
45.24.a.a 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 666T_{2}^{2} - 7619376T_{2} + 5728572416 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 666 T^{2} + \cdots + 5728572416 \) Copy content Toggle raw display
$3$ \( T^{3} + 139428 T^{2} + \cdots - 17\!\cdots\!72 \) Copy content Toggle raw display
$5$ \( (T - 48828125)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2432683344 T^{2} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{3} + 165286030404 T^{2} + \cdots + 10\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{3} + 3554970733998 T^{2} + \cdots - 29\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{3} + 416105769269514 T^{2} + \cdots - 53\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{3} + 975704043068460 T^{2} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 80\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 60\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 54\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 12\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 53\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 40\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 55\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 15\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 19\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 27\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 11\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
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