Properties

Label 24.3.h.b
Level 24
Weight 3
Character orbit 24.h
Self dual yes
Analytic conductor 0.654
Analytic rank 0
Dimension 1
CM discriminant -24
Inner twists 2

Related objects

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.653952634465\)
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

\(f(q)\) \(=\) \( q + 2q^{2} - 3q^{3} + 4q^{4} - 2q^{5} - 6q^{6} - 10q^{7} + 8q^{8} + 9q^{9} + O(q^{10}) \) \( q + 2q^{2} - 3q^{3} + 4q^{4} - 2q^{5} - 6q^{6} - 10q^{7} + 8q^{8} + 9q^{9} - 4q^{10} + 10q^{11} - 12q^{12} - 20q^{14} + 6q^{15} + 16q^{16} + 18q^{18} - 8q^{20} + 30q^{21} + 20q^{22} - 24q^{24} - 21q^{25} - 27q^{27} - 40q^{28} - 50q^{29} + 12q^{30} + 38q^{31} + 32q^{32} - 30q^{33} + 20q^{35} + 36q^{36} - 16q^{40} + 60q^{42} + 40q^{44} - 18q^{45} - 48q^{48} + 51q^{49} - 42q^{50} + 94q^{53} - 54q^{54} - 20q^{55} - 80q^{56} - 100q^{58} + 10q^{59} + 24q^{60} + 76q^{62} - 90q^{63} + 64q^{64} - 60q^{66} + 40q^{70} + 72q^{72} + 50q^{73} + 63q^{75} - 100q^{77} - 58q^{79} - 32q^{80} + 81q^{81} - 134q^{83} + 120q^{84} + 150q^{87} + 80q^{88} - 36q^{90} - 114q^{93} - 96q^{96} - 190q^{97} + 102q^{98} + 90q^{99} + O(q^{100}) \)

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
2.00000 −3.00000 4.00000 −2.00000 −6.00000 −10.0000 8.00000 9.00000 −4.00000
\(n\): 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 24.3.h.b yes 1
3.b odd 2 1 24.3.h.a 1
4.b odd 2 1 96.3.h.b 1
8.b even 2 1 24.3.h.a 1
8.d odd 2 1 96.3.h.a 1
12.b even 2 1 96.3.h.a 1
16.e even 4 2 768.3.e.d 2
16.f odd 4 2 768.3.e.c 2
24.f even 2 1 96.3.h.b 1
24.h odd 2 1 CM 24.3.h.b yes 1
48.i odd 4 2 768.3.e.d 2
48.k even 4 2 768.3.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.h.a 1 3.b odd 2 1
24.3.h.a 1 8.b even 2 1
24.3.h.b yes 1 1.a even 1 1 trivial
24.3.h.b yes 1 24.h odd 2 1 CM
96.3.h.a 1 8.d odd 2 1
96.3.h.a 1 12.b even 2 1
96.3.h.b 1 4.b odd 2 1
96.3.h.b 1 24.f even 2 1
768.3.e.c 2 16.f odd 4 2
768.3.e.c 2 48.k even 4 2
768.3.e.d 2 16.e even 4 2
768.3.e.d 2 48.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 2 \) acting on \(S_{3}^{\mathrm{new}}(24, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T \)
$3$ \( 1 + 3 T \)
$5$ \( 1 + 2 T + 25 T^{2} \)
$7$ \( 1 + 10 T + 49 T^{2} \)
$11$ \( 1 - 10 T + 121 T^{2} \)
$13$ \( ( 1 - 13 T )( 1 + 13 T ) \)
$17$ \( ( 1 - 17 T )( 1 + 17 T ) \)
$19$ \( ( 1 - 19 T )( 1 + 19 T ) \)
$23$ \( ( 1 - 23 T )( 1 + 23 T ) \)
$29$ \( 1 + 50 T + 841 T^{2} \)
$31$ \( 1 - 38 T + 961 T^{2} \)
$37$ \( ( 1 - 37 T )( 1 + 37 T ) \)
$41$ \( ( 1 - 41 T )( 1 + 41 T ) \)
$43$ \( ( 1 - 43 T )( 1 + 43 T ) \)
$47$ \( ( 1 - 47 T )( 1 + 47 T ) \)
$53$ \( 1 - 94 T + 2809 T^{2} \)
$59$ \( 1 - 10 T + 3481 T^{2} \)
$61$ \( ( 1 - 61 T )( 1 + 61 T ) \)
$67$ \( ( 1 - 67 T )( 1 + 67 T ) \)
$71$ \( ( 1 - 71 T )( 1 + 71 T ) \)
$73$ \( 1 - 50 T + 5329 T^{2} \)
$79$ \( 1 + 58 T + 6241 T^{2} \)
$83$ \( 1 + 134 T + 6889 T^{2} \)
$89$ \( ( 1 - 89 T )( 1 + 89 T ) \)
$97$ \( 1 + 190 T + 9409 T^{2} \)
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