Properties

Label 24.11.h.a
Level 24
Weight 11
Character orbit 24.h
Self dual yes
Analytic conductor 15.249
Analytic rank 0
Dimension 1
CM discriminant -24
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 24.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(15.2485740642\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 32q^{2} + 243q^{3} + 1024q^{4} + 5282q^{5} - 7776q^{6} + 24950q^{7} - 32768q^{8} + 59049q^{9} + O(q^{10}) \) \( q - 32q^{2} + 243q^{3} + 1024q^{4} + 5282q^{5} - 7776q^{6} + 24950q^{7} - 32768q^{8} + 59049q^{9} - 169024q^{10} - 227050q^{11} + 248832q^{12} - 798400q^{14} + 1283526q^{15} + 1048576q^{16} - 1889568q^{18} + 5408768q^{20} + 6062850q^{21} + 7265600q^{22} - 7962624q^{24} + 18133899q^{25} + 14348907q^{27} + 25548800q^{28} - 36304750q^{29} - 41072832q^{30} - 8955802q^{31} - 33554432q^{32} - 55173150q^{33} + 131785900q^{35} + 60466176q^{36} - 173080576q^{40} - 194011200q^{42} - 232499200q^{44} + 311896818q^{45} + 254803968q^{48} + 340027251q^{49} - 580284768q^{50} + 617985986q^{53} - 459165024q^{54} - 1199278100q^{55} - 817561600q^{56} + 1161752000q^{58} - 588563050q^{59} + 1314330624q^{60} + 286585664q^{62} + 1473272550q^{63} + 1073741824q^{64} + 1765540800q^{66} - 4217148800q^{70} - 1934917632q^{72} + 4081435250q^{73} + 4406537457q^{75} - 5664897500q^{77} - 5863410298q^{79} + 5538578432q^{80} + 3486784401q^{81} - 7877173786q^{83} + 6208358400q^{84} - 8822054250q^{87} + 7439974400q^{88} - 9980698176q^{90} - 2176259886q^{93} - 8153726976q^{96} - 9031061950q^{97} - 10880872032q^{98} - 13407075450q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0
−32.0000 243.000 1024.00 5282.00 −7776.00 24950.0 −32768.0 59049.0 −169024.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.11.h.a 1
3.b odd 2 1 24.11.h.b yes 1
4.b odd 2 1 96.11.h.a 1
8.b even 2 1 24.11.h.b yes 1
8.d odd 2 1 96.11.h.b 1
12.b even 2 1 96.11.h.b 1
24.f even 2 1 96.11.h.a 1
24.h odd 2 1 CM 24.11.h.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.11.h.a 1 1.a even 1 1 trivial
24.11.h.a 1 24.h odd 2 1 CM
24.11.h.b yes 1 3.b odd 2 1
24.11.h.b yes 1 8.b even 2 1
96.11.h.a 1 4.b odd 2 1
96.11.h.a 1 24.f even 2 1
96.11.h.b 1 8.d odd 2 1
96.11.h.b 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 5282 \) acting on \(S_{11}^{\mathrm{new}}(24, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 32 T \)
$3$ \( 1 - 243 T \)
$5$ \( 1 - 5282 T + 9765625 T^{2} \)
$7$ \( 1 - 24950 T + 282475249 T^{2} \)
$11$ \( 1 + 227050 T + 25937424601 T^{2} \)
$13$ \( ( 1 - 371293 T )( 1 + 371293 T ) \)
$17$ \( ( 1 - 1419857 T )( 1 + 1419857 T ) \)
$19$ \( ( 1 - 2476099 T )( 1 + 2476099 T ) \)
$23$ \( ( 1 - 6436343 T )( 1 + 6436343 T ) \)
$29$ \( 1 + 36304750 T + 420707233300201 T^{2} \)
$31$ \( 1 + 8955802 T + 819628286980801 T^{2} \)
$37$ \( ( 1 - 69343957 T )( 1 + 69343957 T ) \)
$41$ \( ( 1 - 115856201 T )( 1 + 115856201 T ) \)
$43$ \( ( 1 - 147008443 T )( 1 + 147008443 T ) \)
$47$ \( ( 1 - 229345007 T )( 1 + 229345007 T ) \)
$53$ \( 1 - 617985986 T + 174887470365513049 T^{2} \)
$59$ \( 1 + 588563050 T + 511116753300641401 T^{2} \)
$61$ \( ( 1 - 844596301 T )( 1 + 844596301 T ) \)
$67$ \( ( 1 - 1350125107 T )( 1 + 1350125107 T ) \)
$71$ \( ( 1 - 1804229351 T )( 1 + 1804229351 T ) \)
$73$ \( 1 - 4081435250 T + 4297625829703557649 T^{2} \)
$79$ \( 1 + 5863410298 T + 9468276082626847201 T^{2} \)
$83$ \( 1 + 7877173786 T + 15516041187205853449 T^{2} \)
$89$ \( ( 1 - 5584059449 T )( 1 + 5584059449 T ) \)
$97$ \( 1 + 9031061950 T + 73742412689492826049 T^{2} \)
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