Properties

Label 24.2.f.a
Level 24
Weight 2
Character orbit 24.f
Analytic conductor 0.192
Analytic rank 0
Dimension 2
CM discriminant -8
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.191640964851\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -1 - \beta ) q^{3} -2 q^{4} + ( 2 - \beta ) q^{6} -2 \beta q^{8} + ( -1 + 2 \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{2} + ( -1 - \beta ) q^{3} -2 q^{4} + ( 2 - \beta ) q^{6} -2 \beta q^{8} + ( -1 + 2 \beta ) q^{9} + 2 \beta q^{11} + ( 2 + 2 \beta ) q^{12} + 4 q^{16} -4 \beta q^{17} + ( -4 - \beta ) q^{18} + 2 q^{19} -4 q^{22} + ( -4 + 2 \beta ) q^{24} -5 q^{25} + ( 5 - \beta ) q^{27} + 4 \beta q^{32} + ( 4 - 2 \beta ) q^{33} + 8 q^{34} + ( 2 - 4 \beta ) q^{36} + 2 \beta q^{38} + 8 \beta q^{41} -10 q^{43} -4 \beta q^{44} + ( -4 - 4 \beta ) q^{48} + 7 q^{49} -5 \beta q^{50} + ( -8 + 4 \beta ) q^{51} + ( 2 + 5 \beta ) q^{54} + ( -2 - 2 \beta ) q^{57} -10 \beta q^{59} -8 q^{64} + ( 4 + 4 \beta ) q^{66} + 14 q^{67} + 8 \beta q^{68} + ( 8 + 2 \beta ) q^{72} + 2 q^{73} + ( 5 + 5 \beta ) q^{75} -4 q^{76} + ( -7 - 4 \beta ) q^{81} -16 q^{82} + 2 \beta q^{83} -10 \beta q^{86} + 8 q^{88} -4 \beta q^{89} + ( 8 - 4 \beta ) q^{96} -10 q^{97} + 7 \beta q^{98} + ( -8 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 4q^{4} + 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 4q^{4} + 4q^{6} - 2q^{9} + 4q^{12} + 8q^{16} - 8q^{18} + 4q^{19} - 8q^{22} - 8q^{24} - 10q^{25} + 10q^{27} + 8q^{33} + 16q^{34} + 4q^{36} - 20q^{43} - 8q^{48} + 14q^{49} - 16q^{51} + 4q^{54} - 4q^{57} - 16q^{64} + 8q^{66} + 28q^{67} + 16q^{72} + 4q^{73} + 10q^{75} - 8q^{76} - 14q^{81} - 32q^{82} + 16q^{88} + 16q^{96} - 20q^{97} - 16q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
1.41421i
1.41421i
1.41421i −1.00000 + 1.41421i −2.00000 0 2.00000 + 1.41421i 0 2.82843i −1.00000 2.82843i 0
11.2 1.41421i −1.00000 1.41421i −2.00000 0 2.00000 1.41421i 0 2.82843i −1.00000 + 2.82843i 0
\(n\): 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 CM by \(\Q(\sqrt{-2}) \)
3.b odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.2.f.a 2
3.b odd 2 1 inner 24.2.f.a 2
4.b odd 2 1 96.2.f.a 2
5.b even 2 1 600.2.b.a 2
5.c odd 4 2 600.2.m.a 4
8.b even 2 1 96.2.f.a 2
8.d odd 2 1 CM 24.2.f.a 2
9.c even 3 2 648.2.l.b 4
9.d odd 6 2 648.2.l.b 4
12.b even 2 1 96.2.f.a 2
15.d odd 2 1 600.2.b.a 2
15.e even 4 2 600.2.m.a 4
16.e even 4 2 768.2.c.h 4
16.f odd 4 2 768.2.c.h 4
20.d odd 2 1 2400.2.b.a 2
20.e even 4 2 2400.2.m.a 4
24.f even 2 1 inner 24.2.f.a 2
24.h odd 2 1 96.2.f.a 2
36.f odd 6 2 2592.2.p.b 4
36.h even 6 2 2592.2.p.b 4
40.e odd 2 1 600.2.b.a 2
40.f even 2 1 2400.2.b.a 2
40.i odd 4 2 2400.2.m.a 4
40.k even 4 2 600.2.m.a 4
48.i odd 4 2 768.2.c.h 4
48.k even 4 2 768.2.c.h 4
60.h even 2 1 2400.2.b.a 2
60.l odd 4 2 2400.2.m.a 4
72.j odd 6 2 2592.2.p.b 4
72.l even 6 2 648.2.l.b 4
72.n even 6 2 2592.2.p.b 4
72.p odd 6 2 648.2.l.b 4
120.i odd 2 1 2400.2.b.a 2
120.m even 2 1 600.2.b.a 2
120.q odd 4 2 600.2.m.a 4
120.w even 4 2 2400.2.m.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.f.a 2 1.a even 1 1 trivial
24.2.f.a 2 3.b odd 2 1 inner
24.2.f.a 2 8.d odd 2 1 CM
24.2.f.a 2 24.f even 2 1 inner
96.2.f.a 2 4.b odd 2 1
96.2.f.a 2 8.b even 2 1
96.2.f.a 2 12.b even 2 1
96.2.f.a 2 24.h odd 2 1
600.2.b.a 2 5.b even 2 1
600.2.b.a 2 15.d odd 2 1
600.2.b.a 2 40.e odd 2 1
600.2.b.a 2 120.m even 2 1
600.2.m.a 4 5.c odd 4 2
600.2.m.a 4 15.e even 4 2
600.2.m.a 4 40.k even 4 2
600.2.m.a 4 120.q odd 4 2
648.2.l.b 4 9.c even 3 2
648.2.l.b 4 9.d odd 6 2
648.2.l.b 4 72.l even 6 2
648.2.l.b 4 72.p odd 6 2
768.2.c.h 4 16.e even 4 2
768.2.c.h 4 16.f odd 4 2
768.2.c.h 4 48.i odd 4 2
768.2.c.h 4 48.k even 4 2
2400.2.b.a 2 20.d odd 2 1
2400.2.b.a 2 40.f even 2 1
2400.2.b.a 2 60.h even 2 1
2400.2.b.a 2 120.i odd 2 1
2400.2.m.a 4 20.e even 4 2
2400.2.m.a 4 40.i odd 4 2
2400.2.m.a 4 60.l odd 4 2
2400.2.m.a 4 120.w even 4 2
2592.2.p.b 4 36.f odd 6 2
2592.2.p.b 4 36.h even 6 2
2592.2.p.b 4 72.j odd 6 2
2592.2.p.b 4 72.n even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(24, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} \)
$3$ \( 1 + 2 T + 3 T^{2} \)
$5$ \( ( 1 + 5 T^{2} )^{2} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} ) \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} ) \)
$19$ \( ( 1 - 2 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{2} \)
$37$ \( ( 1 - 37 T^{2} )^{2} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )( 1 + 6 T + 41 T^{2} ) \)
$43$ \( ( 1 + 10 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 6 T + 59 T^{2} )( 1 + 6 T + 59 T^{2} ) \)
$61$ \( ( 1 - 61 T^{2} )^{2} \)
$67$ \( ( 1 - 14 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 2 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 79 T^{2} )^{2} \)
$83$ \( ( 1 - 18 T + 83 T^{2} )( 1 + 18 T + 83 T^{2} ) \)
$89$ \( ( 1 - 18 T + 89 T^{2} )( 1 + 18 T + 89 T^{2} ) \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{2} \)
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