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Properties

Label 24.3.h.b.5.1
Level $24$
Weight $3$
Character 24.5
Self dual yes
Analytic conductor $0.654$
Analytic rank $0$
Dimension $1$
CM discriminant -24
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [24,3,Mod(5,24)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("24.5"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(24, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 24.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.653952634465\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 5.1
Character \(\chi\) \(=\) 24.5

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -2.00000 q^{5} -6.00000 q^{6} -10.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -4.00000 q^{10} +10.0000 q^{11} -12.0000 q^{12} -20.0000 q^{14} +6.00000 q^{15} +16.0000 q^{16} +18.0000 q^{18} -8.00000 q^{20} +30.0000 q^{21} +20.0000 q^{22} -24.0000 q^{24} -21.0000 q^{25} -27.0000 q^{27} -40.0000 q^{28} -50.0000 q^{29} +12.0000 q^{30} +38.0000 q^{31} +32.0000 q^{32} -30.0000 q^{33} +20.0000 q^{35} +36.0000 q^{36} -16.0000 q^{40} +60.0000 q^{42} +40.0000 q^{44} -18.0000 q^{45} -48.0000 q^{48} +51.0000 q^{49} -42.0000 q^{50} +94.0000 q^{53} -54.0000 q^{54} -20.0000 q^{55} -80.0000 q^{56} -100.000 q^{58} +10.0000 q^{59} +24.0000 q^{60} +76.0000 q^{62} -90.0000 q^{63} +64.0000 q^{64} -60.0000 q^{66} +40.0000 q^{70} +72.0000 q^{72} +50.0000 q^{73} +63.0000 q^{75} -100.000 q^{77} -58.0000 q^{79} -32.0000 q^{80} +81.0000 q^{81} -134.000 q^{83} +120.000 q^{84} +150.000 q^{87} +80.0000 q^{88} -36.0000 q^{90} -114.000 q^{93} -96.0000 q^{96} -190.000 q^{97} +102.000 q^{98} +90.0000 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.00000
\(3\) −3.00000 −1.00000
\(4\) 4.00000 1.00000
\(5\) −2.00000 −0.400000 −0.200000 − 0.979796i \(-0.564094\pi\)
−0.200000 + 0.979796i \(0.564094\pi\)
\(6\) −6.00000 −1.00000
\(7\) −10.0000 −1.42857 −0.714286 − 0.699854i \(-0.753248\pi\)
−0.714286 + 0.699854i \(0.753248\pi\)
\(8\) 8.00000 1.00000
\(9\) 9.00000 1.00000
\(10\) −4.00000 −0.400000
\(11\) 10.0000 0.909091 0.454545 − 0.890724i \(-0.349802\pi\)
0.454545 + 0.890724i \(0.349802\pi\)
\(12\) −12.0000 −1.00000
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −20.0000 −1.42857
\(15\) 6.00000 0.400000
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 18.0000 1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −8.00000 −0.400000
\(21\) 30.0000 1.42857
\(22\) 20.0000 0.909091
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −24.0000 −1.00000
\(25\) −21.0000 −0.840000
\(26\) 0 0
\(27\) −27.0000 −1.00000
\(28\) −40.0000 −1.42857
\(29\) −50.0000 −1.72414 −0.862069 − 0.506791i \(-0.830832\pi\)
−0.862069 + 0.506791i \(0.830832\pi\)
\(30\) 12.0000 0.400000
\(31\) 38.0000 1.22581 0.612903 − 0.790158i \(-0.290002\pi\)
0.612903 + 0.790158i \(0.290002\pi\)
\(32\) 32.0000 1.00000
\(33\) −30.0000 −0.909091
\(34\) 0 0
\(35\) 20.0000 0.571429
\(36\) 36.0000 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −16.0000 −0.400000
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 60.0000 1.42857
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 40.0000 0.909091
\(45\) −18.0000 −0.400000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −48.0000 −1.00000
\(49\) 51.0000 1.04082
\(50\) −42.0000 −0.840000
\(51\) 0 0
\(52\) 0 0
\(53\) 94.0000 1.77358 0.886792 − 0.462168i \(-0.152928\pi\)
0.886792 + 0.462168i \(0.152928\pi\)
\(54\) −54.0000 −1.00000
\(55\) −20.0000 −0.363636
\(56\) −80.0000 −1.42857
\(57\) 0 0
\(58\) −100.000 −1.72414
\(59\) 10.0000 0.169492 0.0847458 − 0.996403i \(-0.472992\pi\)
0.0847458 + 0.996403i \(0.472992\pi\)
\(60\) 24.0000 0.400000
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 76.0000 1.22581
\(63\) −90.0000 −1.42857
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) −60.0000 −0.909091
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 40.0000 0.571429
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 72.0000 1.00000
\(73\) 50.0000 0.684932 0.342466 − 0.939530i \(-0.388738\pi\)
0.342466 + 0.939530i \(0.388738\pi\)
\(74\) 0 0
\(75\) 63.0000 0.840000
\(76\) 0 0
\(77\) −100.000 −1.29870
\(78\) 0 0
\(79\) −58.0000 −0.734177 −0.367089 − 0.930186i \(-0.619645\pi\)
−0.367089 + 0.930186i \(0.619645\pi\)
\(80\) −32.0000 −0.400000
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) −134.000 −1.61446 −0.807229 − 0.590238i \(-0.799034\pi\)
−0.807229 + 0.590238i \(0.799034\pi\)
\(84\) 120.000 1.42857
\(85\) 0 0
\(86\) 0 0
\(87\) 150.000 1.72414
\(88\) 80.0000 0.909091
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) −36.0000 −0.400000
\(91\) 0 0
\(92\) 0 0
\(93\) −114.000 −1.22581
\(94\) 0 0
\(95\) 0 0
\(96\) −96.0000 −1.00000
\(97\) −190.000 −1.95876 −0.979381 − 0.202020i \(-0.935249\pi\)
−0.979381 + 0.202020i \(0.935249\pi\)
\(98\) 102.000 1.04082
\(99\) 90.0000 0.909091
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 24.3.h.b.5.1 yes 1
3.2 odd 2 24.3.h.a.5.1 ✓ 1
4.3 odd 2 96.3.h.b.17.1 1
8.3 odd 2 96.3.h.a.17.1 1
8.5 even 2 24.3.h.a.5.1 ✓ 1
12.11 even 2 96.3.h.a.17.1 1
16.3 odd 4 768.3.e.c.257.2 2
16.5 even 4 768.3.e.d.257.2 2
16.11 odd 4 768.3.e.c.257.1 2
16.13 even 4 768.3.e.d.257.1 2
24.5 odd 2 CM 24.3.h.b.5.1 yes 1
24.11 even 2 96.3.h.b.17.1 1
48.5 odd 4 768.3.e.d.257.1 2
48.11 even 4 768.3.e.c.257.2 2
48.29 odd 4 768.3.e.d.257.2 2
48.35 even 4 768.3.e.c.257.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.3.h.a.5.1 ✓ 1 3.2 odd 2
24.3.h.a.5.1 ✓ 1 8.5 even 2
24.3.h.b.5.1 yes 1 1.1 even 1 trivial
24.3.h.b.5.1 yes 1 24.5 odd 2 CM
96.3.h.a.17.1 1 8.3 odd 2
96.3.h.a.17.1 1 12.11 even 2
96.3.h.b.17.1 1 4.3 odd 2
96.3.h.b.17.1 1 24.11 even 2
768.3.e.c.257.1 2 16.11 odd 4
768.3.e.c.257.1 2 48.35 even 4
768.3.e.c.257.2 2 16.3 odd 4
768.3.e.c.257.2 2 48.11 even 4
768.3.e.d.257.1 2 16.13 even 4
768.3.e.d.257.1 2 48.5 odd 4
768.3.e.d.257.2 2 16.5 even 4
768.3.e.d.257.2 2 48.29 odd 4