Properties

Label 24.3.b.a
Level 24
Weight 3
Character orbit 24.b
Analytic conductor 0.654
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.653952634465\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.4752.1
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.

\(f(q)\) \(=\) \( q\) \( + ( \beta_{1} - \beta_{2} ) q^{2} \) \( + \beta_{2} q^{3} \) \( + ( -2 - \beta_{2} - \beta_{3} ) q^{4} \) \( + 2 \beta_{3} q^{5} \) \( + ( -2 + \beta_{1} + \beta_{3} ) q^{6} \) \( + ( 2 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} \) \( + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{8} \) \( + 3 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( \beta_{1} - \beta_{2} ) q^{2} \) \( + \beta_{2} q^{3} \) \( + ( -2 - \beta_{2} - \beta_{3} ) q^{4} \) \( + 2 \beta_{3} q^{5} \) \( + ( -2 + \beta_{1} + \beta_{3} ) q^{6} \) \( + ( 2 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} \) \( + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{8} \) \( + 3 q^{9} \) \( + ( 4 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{10} \) \( -8 q^{11} \) \( + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{12} \) \( + ( -4 + 8 \beta_{1} - 4 \beta_{2} ) q^{13} \) \( + ( 8 + 2 \beta_{1} + 6 \beta_{2} ) q^{14} \) \( + ( -2 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{15} \) \( + ( -4 + 4 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{16} \) \( + ( -2 - 8 \beta_{2} ) q^{17} \) \( + ( 3 \beta_{1} - 3 \beta_{2} ) q^{18} \) \( + ( 8 - 4 \beta_{2} ) q^{19} \) \( + ( 20 - 4 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{20} \) \( + ( 4 - 8 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{21} \) \( + ( -8 \beta_{1} + 8 \beta_{2} ) q^{22} \) \( + ( -4 + 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{23} \) \( + ( 12 - 6 \beta_{1} + 2 \beta_{2} ) q^{24} \) \( + ( -11 + 16 \beta_{2} ) q^{25} \) \( + ( -24 - 4 \beta_{2} - 4 \beta_{3} ) q^{26} \) \( + 3 \beta_{2} q^{27} \) \( + ( -20 + 16 \beta_{1} - 10 \beta_{2} + 6 \beta_{3} ) q^{28} \) \( + ( 8 - 16 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} ) q^{29} \) \( + ( -16 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{30} \) \( + ( 6 - 12 \beta_{1} + 6 \beta_{2} + 10 \beta_{3} ) q^{31} \) \( + ( -24 + 4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{32} \) \( -8 \beta_{2} q^{33} \) \( + ( 16 - 10 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{34} \) \( + ( 24 + 4 \beta_{2} ) q^{35} \) \( + ( -6 - 3 \beta_{2} - 3 \beta_{3} ) q^{36} \) \( + ( -4 + 8 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{37} \) \( + ( 8 + 4 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{38} \) \( + ( -4 + 8 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{39} \) \( + ( 24 + 12 \beta_{1} - 12 \beta_{2} - 8 \beta_{3} ) q^{40} \) \( + ( 10 + 24 \beta_{2} ) q^{41} \) \( + ( 20 + 2 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{42} \) \( + ( 8 - 12 \beta_{2} ) q^{43} \) \( + ( 16 + 8 \beta_{2} + 8 \beta_{3} ) q^{44} \) \( + 6 \beta_{3} q^{45} \) \( + ( -16 - 4 \beta_{1} - 12 \beta_{2} ) q^{46} \) \( + ( -4 + 8 \beta_{1} - 4 \beta_{2} - 12 \beta_{3} ) q^{47} \) \( + ( 20 + 8 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{48} \) \( + ( -11 - 32 \beta_{2} ) q^{49} \) \( + ( -32 + 5 \beta_{1} + 11 \beta_{2} + 16 \beta_{3} ) q^{50} \) \( + ( -24 - 2 \beta_{2} ) q^{51} \) \( + ( -24 \beta_{1} + 32 \beta_{2} - 8 \beta_{3} ) q^{52} \) \( + ( -8 + 16 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{53} \) \( + ( -6 + 3 \beta_{1} + 3 \beta_{3} ) q^{54} \) \( -16 \beta_{3} q^{55} \) \( + ( -32 - 20 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{56} \) \( + ( -12 + 8 \beta_{2} ) q^{57} \) \( + ( 36 + 6 \beta_{1} + 20 \beta_{2} + 2 \beta_{3} ) q^{58} \) \( + ( -32 + 12 \beta_{2} ) q^{59} \) \( + ( -20 - 8 \beta_{1} + 22 \beta_{2} - 2 \beta_{3} ) q^{60} \) \( + ( 4 - 8 \beta_{1} + 4 \beta_{2} + 12 \beta_{3} ) q^{61} \) \( + ( 56 - 10 \beta_{1} - 14 \beta_{2} + 16 \beta_{3} ) q^{62} \) \( + ( 6 - 12 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{63} \) \( + ( 8 - 32 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{64} \) \( + ( 24 - 40 \beta_{2} ) q^{65} \) \( + ( 16 - 8 \beta_{1} - 8 \beta_{3} ) q^{66} \) \( + ( -64 + 12 \beta_{2} ) q^{67} \) \( + ( 20 + 16 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} ) q^{68} \) \( + ( -8 + 16 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{69} \) \( + ( -8 + 28 \beta_{1} - 24 \beta_{2} + 4 \beta_{3} ) q^{70} \) \( + ( 20 - 40 \beta_{1} + 20 \beta_{2} - 4 \beta_{3} ) q^{71} \) \( + ( -6 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} ) q^{72} \) \( + ( 50 + 32 \beta_{2} ) q^{73} \) \( + ( -32 + 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{74} \) \( + ( 48 - 11 \beta_{2} ) q^{75} \) \( + ( -8 + 8 \beta_{1} - 4 \beta_{2} - 12 \beta_{3} ) q^{76} \) \( + ( -16 + 32 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} ) q^{77} \) \( + ( -8 - 8 \beta_{1} - 20 \beta_{2} + 4 \beta_{3} ) q^{78} \) \( + ( -18 + 36 \beta_{1} - 18 \beta_{2} + 2 \beta_{3} ) q^{79} \) \( + ( -40 + 32 \beta_{1} - 20 \beta_{2} - 20 \beta_{3} ) q^{80} \) \( + 9 q^{81} \) \( + ( -48 + 34 \beta_{1} - 10 \beta_{2} + 24 \beta_{3} ) q^{82} \) \( + ( 40 - 16 \beta_{2} ) q^{83} \) \( + ( -20 + 28 \beta_{1} - 26 \beta_{2} + 10 \beta_{3} ) q^{84} \) \( + ( 16 - 32 \beta_{1} + 16 \beta_{2} + 12 \beta_{3} ) q^{85} \) \( + ( 24 - 4 \beta_{1} - 8 \beta_{2} - 12 \beta_{3} ) q^{86} \) \( + ( 14 - 28 \beta_{1} + 14 \beta_{2} - 10 \beta_{3} ) q^{87} \) \( + ( 16 \beta_{1} - 32 \beta_{2} + 16 \beta_{3} ) q^{88} \) \( + ( -50 + 48 \beta_{2} ) q^{89} \) \( + ( 12 - 6 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} ) q^{90} \) \( + ( 72 + 56 \beta_{2} ) q^{91} \) \( + ( 40 - 32 \beta_{1} + 20 \beta_{2} - 12 \beta_{3} ) q^{92} \) \( + ( -4 + 8 \beta_{1} - 4 \beta_{2} - 22 \beta_{3} ) q^{93} \) \( + ( -48 + 12 \beta_{1} + 20 \beta_{2} - 16 \beta_{3} ) q^{94} \) \( + ( 8 - 16 \beta_{1} + 8 \beta_{2} + 24 \beta_{3} ) q^{95} \) \( + ( -16 + 20 \beta_{1} - 32 \beta_{2} - 4 \beta_{3} ) q^{96} \) \( + ( 14 - 48 \beta_{2} ) q^{97} \) \( + ( 64 - 43 \beta_{1} + 11 \beta_{2} - 32 \beta_{3} ) q^{98} \) \( -24 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 4q^{8} \) \(\mathstrut +\mathstrut 12q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 4q^{8} \) \(\mathstrut +\mathstrut 12q^{9} \) \(\mathstrut +\mathstrut 12q^{10} \) \(\mathstrut -\mathstrut 32q^{11} \) \(\mathstrut -\mathstrut 12q^{12} \) \(\mathstrut +\mathstrut 36q^{14} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 8q^{17} \) \(\mathstrut +\mathstrut 6q^{18} \) \(\mathstrut +\mathstrut 32q^{19} \) \(\mathstrut +\mathstrut 72q^{20} \) \(\mathstrut -\mathstrut 16q^{22} \) \(\mathstrut +\mathstrut 36q^{24} \) \(\mathstrut -\mathstrut 44q^{25} \) \(\mathstrut -\mathstrut 96q^{26} \) \(\mathstrut -\mathstrut 48q^{28} \) \(\mathstrut -\mathstrut 60q^{30} \) \(\mathstrut -\mathstrut 88q^{32} \) \(\mathstrut +\mathstrut 44q^{34} \) \(\mathstrut +\mathstrut 96q^{35} \) \(\mathstrut -\mathstrut 24q^{36} \) \(\mathstrut +\mathstrut 40q^{38} \) \(\mathstrut +\mathstrut 120q^{40} \) \(\mathstrut +\mathstrut 40q^{41} \) \(\mathstrut +\mathstrut 84q^{42} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut +\mathstrut 64q^{44} \) \(\mathstrut -\mathstrut 72q^{46} \) \(\mathstrut +\mathstrut 96q^{48} \) \(\mathstrut -\mathstrut 44q^{49} \) \(\mathstrut -\mathstrut 118q^{50} \) \(\mathstrut -\mathstrut 96q^{51} \) \(\mathstrut -\mathstrut 48q^{52} \) \(\mathstrut -\mathstrut 18q^{54} \) \(\mathstrut -\mathstrut 168q^{56} \) \(\mathstrut -\mathstrut 48q^{57} \) \(\mathstrut +\mathstrut 156q^{58} \) \(\mathstrut -\mathstrut 128q^{59} \) \(\mathstrut -\mathstrut 96q^{60} \) \(\mathstrut +\mathstrut 204q^{62} \) \(\mathstrut -\mathstrut 32q^{64} \) \(\mathstrut +\mathstrut 96q^{65} \) \(\mathstrut +\mathstrut 48q^{66} \) \(\mathstrut -\mathstrut 256q^{67} \) \(\mathstrut +\mathstrut 112q^{68} \) \(\mathstrut +\mathstrut 24q^{70} \) \(\mathstrut -\mathstrut 12q^{72} \) \(\mathstrut +\mathstrut 200q^{73} \) \(\mathstrut -\mathstrut 120q^{74} \) \(\mathstrut +\mathstrut 192q^{75} \) \(\mathstrut -\mathstrut 16q^{76} \) \(\mathstrut -\mathstrut 48q^{78} \) \(\mathstrut -\mathstrut 96q^{80} \) \(\mathstrut +\mathstrut 36q^{81} \) \(\mathstrut -\mathstrut 124q^{82} \) \(\mathstrut +\mathstrut 160q^{83} \) \(\mathstrut -\mathstrut 24q^{84} \) \(\mathstrut +\mathstrut 88q^{86} \) \(\mathstrut +\mathstrut 32q^{88} \) \(\mathstrut -\mathstrut 200q^{89} \) \(\mathstrut +\mathstrut 36q^{90} \) \(\mathstrut +\mathstrut 288q^{91} \) \(\mathstrut +\mathstrut 96q^{92} \) \(\mathstrut -\mathstrut 168q^{94} \) \(\mathstrut -\mathstrut 24q^{96} \) \(\mathstrut +\mathstrut 56q^{97} \) \(\mathstrut +\mathstrut 170q^{98} \) \(\mathstrut -\mathstrut 96q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(3\) \(x^{2}\mathstrut -\mathstrut \) \(6\) \(x\mathstrut +\mathstrut \) \(6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 3 \nu^{2} + 4 \nu + 2 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 6 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - \nu^{2} + 8 \nu - 6 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut -\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.866025 0.719687i
0.866025 + 0.719687i
−0.866025 + 1.99551i
−0.866025 1.99551i
−0.366025 1.96622i 1.73205 −3.73205 + 1.43937i 2.87875i −0.633975 3.40559i 10.7436i 4.19615 + 6.81119i 3.00000 −5.66025 + 1.05369i
19.2 −0.366025 + 1.96622i 1.73205 −3.73205 1.43937i 2.87875i −0.633975 + 3.40559i 10.7436i 4.19615 6.81119i 3.00000 −5.66025 1.05369i
19.3 1.36603 1.46081i −1.73205 −0.267949 3.99102i 7.98203i −2.36603 + 2.53020i 2.13878i −6.19615 5.06040i 3.00000 11.6603 + 10.9037i
19.4 1.36603 + 1.46081i −1.73205 −0.267949 + 3.99102i 7.98203i −2.36603 2.53020i 2.13878i −6.19615 + 5.06040i 3.00000 11.6603 10.9037i
\(n\): 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
8.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(24, [\chi])\).