Properties

Label 24.5.h.a
Level 24
Weight 5
Character orbit 24.h
Self dual yes
Analytic conductor 2.481
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: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 24.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(2.48087911401\)
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

\(f(q)\) \(=\) \( q - 4q^{2} - 9q^{3} + 16q^{4} + 46q^{5} + 36q^{6} + 2q^{7} - 64q^{8} + 81q^{9} + O(q^{10}) \) \( q - 4q^{2} - 9q^{3} + 16q^{4} + 46q^{5} + 36q^{6} + 2q^{7} - 64q^{8} + 81q^{9} - 184q^{10} + 142q^{11} - 144q^{12} - 8q^{14} - 414q^{15} + 256q^{16} - 324q^{18} + 736q^{20} - 18q^{21} - 568q^{22} + 576q^{24} + 1491q^{25} - 729q^{27} + 32q^{28} - 818q^{29} + 1656q^{30} - 478q^{31} - 1024q^{32} - 1278q^{33} + 92q^{35} + 1296q^{36} - 2944q^{40} + 72q^{42} + 2272q^{44} + 3726q^{45} - 2304q^{48} - 2397q^{49} - 5964q^{50} - 3218q^{53} + 2916q^{54} + 6532q^{55} - 128q^{56} + 3272q^{58} + 6862q^{59} - 6624q^{60} + 1912q^{62} + 162q^{63} + 4096q^{64} + 5112q^{66} - 368q^{70} - 5184q^{72} - 8158q^{73} - 13419q^{75} + 284q^{77} - 9118q^{79} + 11776q^{80} + 6561q^{81} - 4178q^{83} - 288q^{84} + 7362q^{87} - 9088q^{88} - 14904q^{90} + 4302q^{93} + 9216q^{96} + 17282q^{97} + 9588q^{98} + 11502q^{99} + O(q^{100}) \)

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
−4.00000 −9.00000 16.0000 46.0000 36.0000 2.00000 −64.0000 81.0000 −184.000
\(n\): 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 24.5.h.a 1
3.b odd 2 1 24.5.h.b yes 1
4.b odd 2 1 96.5.h.b 1
8.b even 2 1 24.5.h.b yes 1
8.d odd 2 1 96.5.h.a 1
12.b even 2 1 96.5.h.a 1
24.f even 2 1 96.5.h.b 1
24.h odd 2 1 CM 24.5.h.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.5.h.a 1 1.a even 1 1 trivial
24.5.h.a 1 24.h odd 2 1 CM
24.5.h.b yes 1 3.b odd 2 1
24.5.h.b yes 1 8.b even 2 1
96.5.h.a 1 8.d odd 2 1
96.5.h.a 1 12.b even 2 1
96.5.h.b 1 4.b odd 2 1
96.5.h.b 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 46 \) acting on \(S_{5}^{\mathrm{new}}(24, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T \)
$3$ \( 1 + 9 T \)
$5$ \( 1 - 46 T + 625 T^{2} \)
$7$ \( 1 - 2 T + 2401 T^{2} \)
$11$ \( 1 - 142 T + 14641 T^{2} \)
$13$ \( ( 1 - 169 T )( 1 + 169 T ) \)
$17$ \( ( 1 - 289 T )( 1 + 289 T ) \)
$19$ \( ( 1 - 361 T )( 1 + 361 T ) \)
$23$ \( ( 1 - 529 T )( 1 + 529 T ) \)
$29$ \( 1 + 818 T + 707281 T^{2} \)
$31$ \( 1 + 478 T + 923521 T^{2} \)
$37$ \( ( 1 - 1369 T )( 1 + 1369 T ) \)
$41$ \( ( 1 - 1681 T )( 1 + 1681 T ) \)
$43$ \( ( 1 - 1849 T )( 1 + 1849 T ) \)
$47$ \( ( 1 - 2209 T )( 1 + 2209 T ) \)
$53$ \( 1 + 3218 T + 7890481 T^{2} \)
$59$ \( 1 - 6862 T + 12117361 T^{2} \)
$61$ \( ( 1 - 3721 T )( 1 + 3721 T ) \)
$67$ \( ( 1 - 4489 T )( 1 + 4489 T ) \)
$71$ \( ( 1 - 5041 T )( 1 + 5041 T ) \)
$73$ \( 1 + 8158 T + 28398241 T^{2} \)
$79$ \( 1 + 9118 T + 38950081 T^{2} \)
$83$ \( 1 + 4178 T + 47458321 T^{2} \)
$89$ \( ( 1 - 7921 T )( 1 + 7921 T ) \)
$97$ \( 1 - 17282 T + 88529281 T^{2} \)
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