Properties

Label 24.2.d.a
Level $24$
Weight $2$
Character orbit 24.d
Analytic conductor $0.192$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + i ) q^{2} + i q^{3} -2 i q^{4} -2 i q^{5} + ( -1 - i ) q^{6} -2 q^{7} + ( 2 + 2 i ) q^{8} - q^{9} +O(q^{10})\) \( q + ( -1 + i ) q^{2} + i q^{3} -2 i q^{4} -2 i q^{5} + ( -1 - i ) q^{6} -2 q^{7} + ( 2 + 2 i ) q^{8} - q^{9} + ( 2 + 2 i ) q^{10} + 2 q^{12} + 4 i q^{13} + ( 2 - 2 i ) q^{14} + 2 q^{15} -4 q^{16} -2 q^{17} + ( 1 - i ) q^{18} -4 i q^{19} -4 q^{20} -2 i q^{21} + 4 q^{23} + ( -2 + 2 i ) q^{24} + q^{25} + ( -4 - 4 i ) q^{26} -i q^{27} + 4 i q^{28} + 6 i q^{29} + ( -2 + 2 i ) q^{30} + 2 q^{31} + ( 4 - 4 i ) q^{32} + ( 2 - 2 i ) q^{34} + 4 i q^{35} + 2 i q^{36} -8 i q^{37} + ( 4 + 4 i ) q^{38} -4 q^{39} + ( 4 - 4 i ) q^{40} + 2 q^{41} + ( 2 + 2 i ) q^{42} + 4 i q^{43} + 2 i q^{45} + ( -4 + 4 i ) q^{46} -12 q^{47} -4 i q^{48} -3 q^{49} + ( -1 + i ) q^{50} -2 i q^{51} + 8 q^{52} -6 i q^{53} + ( 1 + i ) q^{54} + ( -4 - 4 i ) q^{56} + 4 q^{57} + ( -6 - 6 i ) q^{58} -4 i q^{59} -4 i q^{60} + ( -2 + 2 i ) q^{62} + 2 q^{63} + 8 i q^{64} + 8 q^{65} + 12 i q^{67} + 4 i q^{68} + 4 i q^{69} + ( -4 - 4 i ) q^{70} + 12 q^{71} + ( -2 - 2 i ) q^{72} -6 q^{73} + ( 8 + 8 i ) q^{74} + i q^{75} -8 q^{76} + ( 4 - 4 i ) q^{78} + 10 q^{79} + 8 i q^{80} + q^{81} + ( -2 + 2 i ) q^{82} -16 i q^{83} -4 q^{84} + 4 i q^{85} + ( -4 - 4 i ) q^{86} -6 q^{87} -10 q^{89} + ( -2 - 2 i ) q^{90} -8 i q^{91} -8 i q^{92} + 2 i q^{93} + ( 12 - 12 i ) q^{94} -8 q^{95} + ( 4 + 4 i ) q^{96} -2 q^{97} + ( 3 - 3 i ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{6} - 4q^{7} + 4q^{8} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{6} - 4q^{7} + 4q^{8} - 2q^{9} + 4q^{10} + 4q^{12} + 4q^{14} + 4q^{15} - 8q^{16} - 4q^{17} + 2q^{18} - 8q^{20} + 8q^{23} - 4q^{24} + 2q^{25} - 8q^{26} - 4q^{30} + 4q^{31} + 8q^{32} + 4q^{34} + 8q^{38} - 8q^{39} + 8q^{40} + 4q^{41} + 4q^{42} - 8q^{46} - 24q^{47} - 6q^{49} - 2q^{50} + 16q^{52} + 2q^{54} - 8q^{56} + 8q^{57} - 12q^{58} - 4q^{62} + 4q^{63} + 16q^{65} - 8q^{70} + 24q^{71} - 4q^{72} - 12q^{73} + 16q^{74} - 16q^{76} + 8q^{78} + 20q^{79} + 2q^{81} - 4q^{82} - 8q^{84} - 8q^{86} - 12q^{87} - 20q^{89} - 4q^{90} + 24q^{94} - 16q^{95} + 8q^{96} - 4q^{97} + 6q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.00000i
1.00000i
−1.00000 1.00000i 1.00000i 2.00000i 2.00000i −1.00000 + 1.00000i −2.00000 2.00000 2.00000i −1.00000 2.00000 2.00000i
13.2 −1.00000 + 1.00000i 1.00000i 2.00000i 2.00000i −1.00000 1.00000i −2.00000 2.00000 + 2.00000i −1.00000 2.00000 + 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b 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.d.a 2
3.b odd 2 1 72.2.d.b 2
4.b odd 2 1 96.2.d.a 2
5.b even 2 1 600.2.k.b 2
5.c odd 4 1 600.2.d.b 2
5.c odd 4 1 600.2.d.c 2
7.b odd 2 1 1176.2.c.a 2
8.b even 2 1 inner 24.2.d.a 2
8.d odd 2 1 96.2.d.a 2
9.c even 3 2 648.2.n.k 4
9.d odd 6 2 648.2.n.c 4
12.b even 2 1 288.2.d.b 2
15.d odd 2 1 1800.2.k.a 2
15.e even 4 1 1800.2.d.b 2
15.e even 4 1 1800.2.d.i 2
16.e even 4 1 768.2.a.a 1
16.e even 4 1 768.2.a.h 1
16.f odd 4 1 768.2.a.d 1
16.f odd 4 1 768.2.a.e 1
20.d odd 2 1 2400.2.k.a 2
20.e even 4 1 2400.2.d.b 2
20.e even 4 1 2400.2.d.c 2
24.f even 2 1 288.2.d.b 2
24.h odd 2 1 72.2.d.b 2
28.d even 2 1 4704.2.c.a 2
36.f odd 6 2 2592.2.r.f 4
36.h even 6 2 2592.2.r.g 4
40.e odd 2 1 2400.2.k.a 2
40.f even 2 1 600.2.k.b 2
40.i odd 4 1 600.2.d.b 2
40.i odd 4 1 600.2.d.c 2
40.k even 4 1 2400.2.d.b 2
40.k even 4 1 2400.2.d.c 2
48.i odd 4 1 2304.2.a.e 1
48.i odd 4 1 2304.2.a.o 1
48.k even 4 1 2304.2.a.b 1
48.k even 4 1 2304.2.a.l 1
56.e even 2 1 4704.2.c.a 2
56.h odd 2 1 1176.2.c.a 2
60.h even 2 1 7200.2.k.d 2
60.l odd 4 1 7200.2.d.d 2
60.l odd 4 1 7200.2.d.g 2
72.j odd 6 2 648.2.n.c 4
72.l even 6 2 2592.2.r.g 4
72.n even 6 2 648.2.n.k 4
72.p odd 6 2 2592.2.r.f 4
120.i odd 2 1 1800.2.k.a 2
120.m even 2 1 7200.2.k.d 2
120.q odd 4 1 7200.2.d.d 2
120.q odd 4 1 7200.2.d.g 2
120.w even 4 1 1800.2.d.b 2
120.w even 4 1 1800.2.d.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.d.a 2 1.a even 1 1 trivial
24.2.d.a 2 8.b even 2 1 inner
72.2.d.b 2 3.b odd 2 1
72.2.d.b 2 24.h odd 2 1
96.2.d.a 2 4.b odd 2 1
96.2.d.a 2 8.d odd 2 1
288.2.d.b 2 12.b even 2 1
288.2.d.b 2 24.f even 2 1
600.2.d.b 2 5.c odd 4 1
600.2.d.b 2 40.i odd 4 1
600.2.d.c 2 5.c odd 4 1
600.2.d.c 2 40.i odd 4 1
600.2.k.b 2 5.b even 2 1
600.2.k.b 2 40.f even 2 1
648.2.n.c 4 9.d odd 6 2
648.2.n.c 4 72.j odd 6 2
648.2.n.k 4 9.c even 3 2
648.2.n.k 4 72.n even 6 2
768.2.a.a 1 16.e even 4 1
768.2.a.d 1 16.f odd 4 1
768.2.a.e 1 16.f odd 4 1
768.2.a.h 1 16.e even 4 1
1176.2.c.a 2 7.b odd 2 1
1176.2.c.a 2 56.h odd 2 1
1800.2.d.b 2 15.e even 4 1
1800.2.d.b 2 120.w even 4 1
1800.2.d.i 2 15.e even 4 1
1800.2.d.i 2 120.w even 4 1
1800.2.k.a 2 15.d odd 2 1
1800.2.k.a 2 120.i odd 2 1
2304.2.a.b 1 48.k even 4 1
2304.2.a.e 1 48.i odd 4 1
2304.2.a.l 1 48.k even 4 1
2304.2.a.o 1 48.i odd 4 1
2400.2.d.b 2 20.e even 4 1
2400.2.d.b 2 40.k even 4 1
2400.2.d.c 2 20.e even 4 1
2400.2.d.c 2 40.k even 4 1
2400.2.k.a 2 20.d odd 2 1
2400.2.k.a 2 40.e odd 2 1
2592.2.r.f 4 36.f odd 6 2
2592.2.r.f 4 72.p odd 6 2
2592.2.r.g 4 36.h even 6 2
2592.2.r.g 4 72.l even 6 2
4704.2.c.a 2 28.d even 2 1
4704.2.c.a 2 56.e even 2 1
7200.2.d.d 2 60.l odd 4 1
7200.2.d.d 2 120.q odd 4 1
7200.2.d.g 2 60.l odd 4 1
7200.2.d.g 2 120.q odd 4 1
7200.2.k.d 2 60.h even 2 1
7200.2.k.d 2 120.m even 2 1

Hecke kernels

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