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 \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.653952634465\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.4752.1
Defining polynomial: \( x^{4} + 3x^{2} - 6x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + \beta_{2} q^{3} + ( - \beta_{3} - \beta_{2} - 2) q^{4} + 2 \beta_{3} q^{5} + (\beta_{3} + \beta_1 - 2) q^{6} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 2) q^{7} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + \beta_{2} q^{3} + ( - \beta_{3} - \beta_{2} - 2) q^{4} + 2 \beta_{3} q^{5} + (\beta_{3} + \beta_1 - 2) q^{6} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 2) q^{7} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{8} + 3 q^{9} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{10} - 8 q^{11} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{12} + ( - 4 \beta_{2} + 8 \beta_1 - 4) q^{13} + (6 \beta_{2} + 2 \beta_1 + 8) q^{14} + ( - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 2) q^{15} + (2 \beta_{3} + 6 \beta_{2} + 4 \beta_1 - 4) q^{16} + ( - 8 \beta_{2} - 2) q^{17} + ( - 3 \beta_{2} + 3 \beta_1) q^{18} + ( - 4 \beta_{2} + 8) q^{19} + ( - 2 \beta_{3} - 6 \beta_{2} - 4 \beta_1 + 20) q^{20} + ( - 2 \beta_{3} + 4 \beta_{2} - 8 \beta_1 + 4) q^{21} + (8 \beta_{2} - 8 \beta_1) q^{22} + (4 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 4) q^{23} + (2 \beta_{2} - 6 \beta_1 + 12) q^{24} + (16 \beta_{2} - 11) q^{25} + ( - 4 \beta_{3} - 4 \beta_{2} - 24) q^{26} + 3 \beta_{2} q^{27} + (6 \beta_{3} - 10 \beta_{2} + 16 \beta_1 - 20) q^{28} + ( - 6 \beta_{3} + 8 \beta_{2} - 16 \beta_1 + 8) q^{29} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 16) q^{30} + (10 \beta_{3} + 6 \beta_{2} - 12 \beta_1 + 6) q^{31} + (8 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 24) q^{32} - 8 \beta_{2} q^{33} + ( - 8 \beta_{3} + 2 \beta_{2} - 10 \beta_1 + 16) q^{34} + (4 \beta_{2} + 24) q^{35} + ( - 3 \beta_{3} - 3 \beta_{2} - 6) q^{36} + ( - 4 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 4) q^{37} + ( - 4 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 8) q^{38} + (8 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 4) q^{39} + ( - 8 \beta_{3} - 12 \beta_{2} + 12 \beta_1 + 24) q^{40} + (24 \beta_{2} + 10) q^{41} + (2 \beta_{3} + 8 \beta_{2} + 2 \beta_1 + 20) q^{42} + ( - 12 \beta_{2} + 8) q^{43} + (8 \beta_{3} + 8 \beta_{2} + 16) q^{44} + 6 \beta_{3} q^{45} + ( - 12 \beta_{2} - 4 \beta_1 - 16) q^{46} + ( - 12 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 4) q^{47} + (2 \beta_{3} - 6 \beta_{2} + 8 \beta_1 + 20) q^{48} + ( - 32 \beta_{2} - 11) q^{49} + (16 \beta_{3} + 11 \beta_{2} + 5 \beta_1 - 32) q^{50} + ( - 2 \beta_{2} - 24) q^{51} + ( - 8 \beta_{3} + 32 \beta_{2} - 24 \beta_1) q^{52} + ( - 2 \beta_{3} - 8 \beta_{2} + 16 \beta_1 - 8) q^{53} + (3 \beta_{3} + 3 \beta_1 - 6) q^{54} - 16 \beta_{3} q^{55} + ( - 4 \beta_{3} - 8 \beta_{2} - 20 \beta_1 - 32) q^{56} + (8 \beta_{2} - 12) q^{57} + (2 \beta_{3} + 20 \beta_{2} + 6 \beta_1 + 36) q^{58} + (12 \beta_{2} - 32) q^{59} + ( - 2 \beta_{3} + 22 \beta_{2} - 8 \beta_1 - 20) q^{60} + (12 \beta_{3} + 4 \beta_{2} - 8 \beta_1 + 4) q^{61} + (16 \beta_{3} - 14 \beta_{2} - 10 \beta_1 + 56) q^{62} + ( - 6 \beta_{3} + 6 \beta_{2} - 12 \beta_1 + 6) q^{63} + (4 \beta_{3} + 4 \beta_{2} - 32 \beta_1 + 8) q^{64} + ( - 40 \beta_{2} + 24) q^{65} + ( - 8 \beta_{3} - 8 \beta_1 + 16) q^{66} + (12 \beta_{2} - 64) q^{67} + ( - 6 \beta_{3} + 10 \beta_{2} + 16 \beta_1 + 20) q^{68} + (4 \beta_{3} - 8 \beta_{2} + 16 \beta_1 - 8) q^{69} + (4 \beta_{3} - 24 \beta_{2} + 28 \beta_1 - 8) q^{70} + ( - 4 \beta_{3} + 20 \beta_{2} - 40 \beta_1 + 20) q^{71} + ( - 6 \beta_{3} + 12 \beta_{2} - 6 \beta_1) q^{72} + (32 \beta_{2} + 50) q^{73} + ( - 8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 32) q^{74} + ( - 11 \beta_{2} + 48) q^{75} + ( - 12 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 8) q^{76} + (16 \beta_{3} - 16 \beta_{2} + 32 \beta_1 - 16) q^{77} + (4 \beta_{3} - 20 \beta_{2} - 8 \beta_1 - 8) q^{78} + (2 \beta_{3} - 18 \beta_{2} + 36 \beta_1 - 18) q^{79} + ( - 20 \beta_{3} - 20 \beta_{2} + 32 \beta_1 - 40) q^{80} + 9 q^{81} + (24 \beta_{3} - 10 \beta_{2} + 34 \beta_1 - 48) q^{82} + ( - 16 \beta_{2} + 40) q^{83} + (10 \beta_{3} - 26 \beta_{2} + 28 \beta_1 - 20) q^{84} + (12 \beta_{3} + 16 \beta_{2} - 32 \beta_1 + 16) q^{85} + ( - 12 \beta_{3} - 8 \beta_{2} - 4 \beta_1 + 24) q^{86} + ( - 10 \beta_{3} + 14 \beta_{2} - 28 \beta_1 + 14) q^{87} + (16 \beta_{3} - 32 \beta_{2} + 16 \beta_1) q^{88} + (48 \beta_{2} - 50) q^{89} + (6 \beta_{3} - 12 \beta_{2} - 6 \beta_1 + 12) q^{90} + (56 \beta_{2} + 72) q^{91} + ( - 12 \beta_{3} + 20 \beta_{2} - 32 \beta_1 + 40) q^{92} + ( - 22 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 4) q^{93} + ( - 16 \beta_{3} + 20 \beta_{2} + 12 \beta_1 - 48) q^{94} + (24 \beta_{3} + 8 \beta_{2} - 16 \beta_1 + 8) q^{95} + ( - 4 \beta_{3} - 32 \beta_{2} + 20 \beta_1 - 16) q^{96} + ( - 48 \beta_{2} + 14) q^{97} + ( - 32 \beta_{3} + 11 \beta_{2} - 43 \beta_1 + 64) q^{98} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 8 q^{4} - 6 q^{6} - 4 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 8 q^{4} - 6 q^{6} - 4 q^{8} + 12 q^{9} + 12 q^{10} - 32 q^{11} - 12 q^{12} + 36 q^{14} - 8 q^{16} - 8 q^{17} + 6 q^{18} + 32 q^{19} + 72 q^{20} - 16 q^{22} + 36 q^{24} - 44 q^{25} - 96 q^{26} - 48 q^{28} - 60 q^{30} - 88 q^{32} + 44 q^{34} + 96 q^{35} - 24 q^{36} + 40 q^{38} + 120 q^{40} + 40 q^{41} + 84 q^{42} + 32 q^{43} + 64 q^{44} - 72 q^{46} + 96 q^{48} - 44 q^{49} - 118 q^{50} - 96 q^{51} - 48 q^{52} - 18 q^{54} - 168 q^{56} - 48 q^{57} + 156 q^{58} - 128 q^{59} - 96 q^{60} + 204 q^{62} - 32 q^{64} + 96 q^{65} + 48 q^{66} - 256 q^{67} + 112 q^{68} + 24 q^{70} - 12 q^{72} + 200 q^{73} - 120 q^{74} + 192 q^{75} - 16 q^{76} - 48 q^{78} - 96 q^{80} + 36 q^{81} - 124 q^{82} + 160 q^{83} - 24 q^{84} + 88 q^{86} + 32 q^{88} - 200 q^{89} + 36 q^{90} + 288 q^{91} + 96 q^{92} - 168 q^{94} - 24 q^{96} + 56 q^{97} + 170 q^{98} - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} - 6x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 3\nu^{2} + 4\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - \nu^{2} + 8\nu - 6 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + 2\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} - 7\beta_{2} + 2\beta _1 + 8 ) / 2 \) Copy content Toggle raw display

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 Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.3.b.a 4
3.b odd 2 1 72.3.b.b 4
4.b odd 2 1 96.3.b.a 4
5.b even 2 1 600.3.g.a 4
5.c odd 4 2 600.3.p.a 8
8.b even 2 1 96.3.b.a 4
8.d odd 2 1 inner 24.3.b.a 4
12.b even 2 1 288.3.b.b 4
16.e even 4 2 768.3.g.h 8
16.f odd 4 2 768.3.g.h 8
20.d odd 2 1 2400.3.g.a 4
20.e even 4 2 2400.3.p.a 8
24.f even 2 1 72.3.b.b 4
24.h odd 2 1 288.3.b.b 4
40.e odd 2 1 600.3.g.a 4
40.f even 2 1 2400.3.g.a 4
40.i odd 4 2 2400.3.p.a 8
40.k even 4 2 600.3.p.a 8
48.i odd 4 2 2304.3.g.z 8
48.k even 4 2 2304.3.g.z 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.b.a 4 1.a even 1 1 trivial
24.3.b.a 4 8.d odd 2 1 inner
72.3.b.b 4 3.b odd 2 1
72.3.b.b 4 24.f even 2 1
96.3.b.a 4 4.b odd 2 1
96.3.b.a 4 8.b even 2 1
288.3.b.b 4 12.b even 2 1
288.3.b.b 4 24.h odd 2 1
600.3.g.a 4 5.b even 2 1
600.3.g.a 4 40.e odd 2 1
600.3.p.a 8 5.c odd 4 2
600.3.p.a 8 40.k even 4 2
768.3.g.h 8 16.e even 4 2
768.3.g.h 8 16.f odd 4 2
2304.3.g.z 8 48.i odd 4 2
2304.3.g.z 8 48.k even 4 2
2400.3.g.a 4 20.d odd 2 1
2400.3.g.a 4 40.f even 2 1
2400.3.p.a 8 20.e even 4 2
2400.3.p.a 8 40.i odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(24, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 6 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 72T^{2} + 528 \) Copy content Toggle raw display
$7$ \( T^{4} + 120T^{2} + 528 \) Copy content Toggle raw display
$11$ \( (T + 8)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 384 T^{2} + 33792 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 188)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 480T^{2} + 8448 \) Copy content Toggle raw display
$29$ \( T^{4} + 1608T^{2} + 528 \) Copy content Toggle raw display
$31$ \( T^{4} + 3384 T^{2} + 279312 \) Copy content Toggle raw display
$37$ \( T^{4} + 864 T^{2} + 76032 \) Copy content Toggle raw display
$41$ \( (T^{2} - 20 T - 1628)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 16 T - 368)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 3552 T^{2} + 8448 \) Copy content Toggle raw display
$53$ \( T^{4} + 1800 T^{2} + 803088 \) Copy content Toggle raw display
$59$ \( (T^{2} + 64 T + 592)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 3552 T^{2} + 8448 \) Copy content Toggle raw display
$67$ \( (T^{2} + 128 T + 3664)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 8928 T^{2} + \cdots + 12849408 \) Copy content Toggle raw display
$73$ \( (T^{2} - 100 T - 572)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 7416 T^{2} + \cdots + 10797072 \) Copy content Toggle raw display
$83$ \( (T^{2} - 80 T + 832)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 100 T - 4412)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 28 T - 6716)^{2} \) Copy content Toggle raw display
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