Properties

Label 24.3.h.c
Level 24
Weight 3
Character orbit 24.h
Analytic conductor 0.654
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.653952634465\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-7})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{2} - \beta_{3} ) q^{3} \) \( + ( -3 + \beta_{3} ) q^{4} \) \( + 4 \beta_{2} q^{5} \) \( + ( 1 - \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{6} \) \( + 4 q^{7} \) \( + ( -2 \beta_{1} + 4 \beta_{2} ) q^{8} \) \( + ( -5 - 4 \beta_{1} - 2 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{2} - \beta_{3} ) q^{3} \) \( + ( -3 + \beta_{3} ) q^{4} \) \( + 4 \beta_{2} q^{5} \) \( + ( 1 - \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{6} \) \( + 4 q^{7} \) \( + ( -2 \beta_{1} + 4 \beta_{2} ) q^{8} \) \( + ( -5 - 4 \beta_{1} - 2 \beta_{2} ) q^{9} \) \( + ( -4 - 4 \beta_{3} ) q^{10} \) \( -6 \beta_{2} q^{11} \) \( + ( 7 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{12} \) \( + 4 \beta_{3} q^{13} \) \( + 4 \beta_{1} q^{14} \) \( + ( -8 + 8 \beta_{1} + 4 \beta_{2} ) q^{15} \) \( + ( 2 - 6 \beta_{3} ) q^{16} \) \( + ( 8 \beta_{1} + 4 \beta_{2} ) q^{17} \) \( + ( 14 - 5 \beta_{1} - 2 \beta_{3} ) q^{18} \) \( + 2 \beta_{3} q^{19} \) \( + ( -8 \beta_{1} - 16 \beta_{2} ) q^{20} \) \( + ( -4 \beta_{2} - 4 \beta_{3} ) q^{21} \) \( + ( 6 + 6 \beta_{3} ) q^{22} \) \( + ( -16 \beta_{1} - 8 \beta_{2} ) q^{23} \) \( + ( -10 + 10 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} ) q^{24} \) \( + 7 q^{25} \) \( + ( 4 \beta_{1} + 16 \beta_{2} ) q^{26} \) \( + ( 19 \beta_{2} + \beta_{3} ) q^{27} \) \( + ( -12 + 4 \beta_{3} ) q^{28} \) \( -12 \beta_{2} q^{29} \) \( + ( -28 - 8 \beta_{1} + 4 \beta_{3} ) q^{30} \) \( -4 q^{31} \) \( + ( -4 \beta_{1} - 24 \beta_{2} ) q^{32} \) \( + ( 12 - 12 \beta_{1} - 6 \beta_{2} ) q^{33} \) \( + ( -28 + 4 \beta_{3} ) q^{34} \) \( + 16 \beta_{2} q^{35} \) \( + ( 15 + 12 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} ) q^{36} \) \( -20 \beta_{3} q^{37} \) \( + ( 2 \beta_{1} + 8 \beta_{2} ) q^{38} \) \( + ( 28 + 8 \beta_{1} + 4 \beta_{2} ) q^{39} \) \( + ( 40 + 8 \beta_{3} ) q^{40} \) \( + ( 16 \beta_{1} + 8 \beta_{2} ) q^{41} \) \( + ( 4 - 4 \beta_{1} - 16 \beta_{2} + 4 \beta_{3} ) q^{42} \) \( -2 \beta_{3} q^{43} \) \( + ( 12 \beta_{1} + 24 \beta_{2} ) q^{44} \) \( + ( -20 \beta_{2} + 16 \beta_{3} ) q^{45} \) \( + ( 56 - 8 \beta_{3} ) q^{46} \) \( + ( -42 - 12 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{48} \) \( -33 q^{49} \) \( + 7 \beta_{1} q^{50} \) \( + ( -28 \beta_{2} + 8 \beta_{3} ) q^{51} \) \( + ( -28 - 12 \beta_{3} ) q^{52} \) \( + 36 \beta_{2} q^{53} \) \( + ( -19 + \beta_{1} + 4 \beta_{2} - 19 \beta_{3} ) q^{54} \) \( -48 q^{55} \) \( + ( -8 \beta_{1} + 16 \beta_{2} ) q^{56} \) \( + ( 14 + 4 \beta_{1} + 2 \beta_{2} ) q^{57} \) \( + ( 12 + 12 \beta_{3} ) q^{58} \) \( -34 \beta_{2} q^{59} \) \( + ( 24 - 24 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{60} \) \( + 36 \beta_{3} q^{61} \) \( -4 \beta_{1} q^{62} \) \( + ( -20 - 16 \beta_{1} - 8 \beta_{2} ) q^{63} \) \( + ( 36 + 20 \beta_{3} ) q^{64} \) \( + ( -32 \beta_{1} - 16 \beta_{2} ) q^{65} \) \( + ( 42 + 12 \beta_{1} - 6 \beta_{3} ) q^{66} \) \( -18 \beta_{3} q^{67} \) \( + ( -24 \beta_{1} + 16 \beta_{2} ) q^{68} \) \( + ( 56 \beta_{2} - 16 \beta_{3} ) q^{69} \) \( + ( -16 - 16 \beta_{3} ) q^{70} \) \( + ( 48 \beta_{1} + 24 \beta_{2} ) q^{71} \) \( + ( -28 + 10 \beta_{1} - 20 \beta_{2} + 20 \beta_{3} ) q^{72} \) \( -6 q^{73} \) \( + ( -20 \beta_{1} - 80 \beta_{2} ) q^{74} \) \( + ( -7 \beta_{2} - 7 \beta_{3} ) q^{75} \) \( + ( -14 - 6 \beta_{3} ) q^{76} \) \( -24 \beta_{2} q^{77} \) \( + ( -28 + 28 \beta_{1} + 4 \beta_{3} ) q^{78} \) \( + 124 q^{79} \) \( + ( 48 \beta_{1} + 32 \beta_{2} ) q^{80} \) \( + ( -31 + 40 \beta_{1} + 20 \beta_{2} ) q^{81} \) \( + ( -56 + 8 \beta_{3} ) q^{82} \) \( + 2 \beta_{2} q^{83} \) \( + ( 28 + 8 \beta_{1} + 16 \beta_{2} + 12 \beta_{3} ) q^{84} \) \( -32 \beta_{3} q^{85} \) \( + ( -2 \beta_{1} - 8 \beta_{2} ) q^{86} \) \( + ( 24 - 24 \beta_{1} - 12 \beta_{2} ) q^{87} \) \( + ( -60 - 12 \beta_{3} ) q^{88} \) \( + ( -56 \beta_{1} - 28 \beta_{2} ) q^{89} \) \( + ( 20 + 16 \beta_{1} + 64 \beta_{2} + 20 \beta_{3} ) q^{90} \) \( + 16 \beta_{3} q^{91} \) \( + ( 48 \beta_{1} - 32 \beta_{2} ) q^{92} \) \( + ( 4 \beta_{2} + 4 \beta_{3} ) q^{93} \) \( + ( -16 \beta_{1} - 8 \beta_{2} ) q^{95} \) \( + ( 44 - 44 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{96} \) \( + 118 q^{97} \) \( -33 \beta_{1} q^{98} \) \( + ( 30 \beta_{2} - 24 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 16q^{7} \) \(\mathstrut -\mathstrut 20q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 16q^{7} \) \(\mathstrut -\mathstrut 20q^{9} \) \(\mathstrut -\mathstrut 16q^{10} \) \(\mathstrut +\mathstrut 28q^{12} \) \(\mathstrut -\mathstrut 32q^{15} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut +\mathstrut 56q^{18} \) \(\mathstrut +\mathstrut 24q^{22} \) \(\mathstrut -\mathstrut 40q^{24} \) \(\mathstrut +\mathstrut 28q^{25} \) \(\mathstrut -\mathstrut 48q^{28} \) \(\mathstrut -\mathstrut 112q^{30} \) \(\mathstrut -\mathstrut 16q^{31} \) \(\mathstrut +\mathstrut 48q^{33} \) \(\mathstrut -\mathstrut 112q^{34} \) \(\mathstrut +\mathstrut 60q^{36} \) \(\mathstrut +\mathstrut 112q^{39} \) \(\mathstrut +\mathstrut 160q^{40} \) \(\mathstrut +\mathstrut 16q^{42} \) \(\mathstrut +\mathstrut 224q^{46} \) \(\mathstrut -\mathstrut 168q^{48} \) \(\mathstrut -\mathstrut 132q^{49} \) \(\mathstrut -\mathstrut 112q^{52} \) \(\mathstrut -\mathstrut 76q^{54} \) \(\mathstrut -\mathstrut 192q^{55} \) \(\mathstrut +\mathstrut 56q^{57} \) \(\mathstrut +\mathstrut 48q^{58} \) \(\mathstrut +\mathstrut 96q^{60} \) \(\mathstrut -\mathstrut 80q^{63} \) \(\mathstrut +\mathstrut 144q^{64} \) \(\mathstrut +\mathstrut 168q^{66} \) \(\mathstrut -\mathstrut 64q^{70} \) \(\mathstrut -\mathstrut 112q^{72} \) \(\mathstrut -\mathstrut 24q^{73} \) \(\mathstrut -\mathstrut 56q^{76} \) \(\mathstrut -\mathstrut 112q^{78} \) \(\mathstrut +\mathstrut 496q^{79} \) \(\mathstrut -\mathstrut 124q^{81} \) \(\mathstrut -\mathstrut 224q^{82} \) \(\mathstrut +\mathstrut 112q^{84} \) \(\mathstrut +\mathstrut 96q^{87} \) \(\mathstrut -\mathstrut 240q^{88} \) \(\mathstrut +\mathstrut 80q^{90} \) \(\mathstrut +\mathstrut 176q^{96} \) \(\mathstrut +\mathstrut 472q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(6\) \(x^{2}\mathstrut +\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut -\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\)

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
−0.707107 1.87083i
−0.707107 + 1.87083i
0.707107 1.87083i
0.707107 + 1.87083i
−0.707107 1.87083i −1.41421 2.64575i −3.00000 + 2.64575i 5.65685 −3.94975 + 4.51658i 4.00000 7.07107 + 3.74166i −5.00000 + 7.48331i −4.00000 10.5830i
5.2 −0.707107 + 1.87083i −1.41421 + 2.64575i −3.00000 2.64575i 5.65685 −3.94975 4.51658i 4.00000 7.07107 3.74166i −5.00000 7.48331i −4.00000 + 10.5830i
5.3 0.707107 1.87083i 1.41421 + 2.64575i −3.00000 2.64575i −5.65685 5.94975 0.774923i 4.00000 −7.07107 + 3.74166i −5.00000 + 7.48331i −4.00000 + 10.5830i
5.4 0.707107 + 1.87083i 1.41421 2.64575i −3.00000 + 2.64575i −5.65685 5.94975 + 0.774923i 4.00000 −7.07107 3.74166i −5.00000 7.48331i −4.00000 10.5830i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
8.b Even 1 yes
24.h Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5}^{2} \) \(\mathstrut -\mathstrut 32 \) acting on \(S_{3}^{\mathrm{new}}(24, [\chi])\).