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Properties

Label 2368.2.a.q.1.1
Level $2368$
Weight $2$
Character 2368.1
Self dual yes
Analytic conductor $18.909$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [2368,2,Mod(1,2368)] 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("2368.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,3,0,2,0,-1,0,6,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 37)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2368.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +5.00000 q^{11} +2.00000 q^{13} +6.00000 q^{15} -3.00000 q^{21} +2.00000 q^{23} -1.00000 q^{25} +9.00000 q^{27} -6.00000 q^{29} -4.00000 q^{31} +15.0000 q^{33} -2.00000 q^{35} +1.00000 q^{37} +6.00000 q^{39} -9.00000 q^{41} -2.00000 q^{43} +12.0000 q^{45} -9.00000 q^{47} -6.00000 q^{49} -1.00000 q^{53} +10.0000 q^{55} -8.00000 q^{59} +8.00000 q^{61} -6.00000 q^{63} +4.00000 q^{65} -8.00000 q^{67} +6.00000 q^{69} +9.00000 q^{71} -1.00000 q^{73} -3.00000 q^{75} -5.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} +15.0000 q^{83} -18.0000 q^{87} +4.00000 q^{89} -2.00000 q^{91} -12.0000 q^{93} +4.00000 q^{97} +30.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 6.00000 1.54919
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 15.0000 2.61116
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 12.0000 1.78885
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.00000 −0.137361 −0.0686803 − 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) 0 0
\(55\) 10.0000 1.34840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) −6.00000 −0.755929
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 − 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 0 0
\(75\) −3.00000 −0.346410
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 15.0000 1.64646 0.823232 − 0.567705i \(-0.192169\pi\)
0.823232 + 0.567705i \(0.192169\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −18.0000 −1.92980
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) −12.0000 −1.24434
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.00000 0.406138 0.203069 − 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) 30.0000 3.01511
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 2368.2.a.q.1.1 1
4.3 odd 2 2368.2.a.b.1.1 1
8.3 odd 2 592.2.a.e.1.1 1
8.5 even 2 37.2.a.a.1.1 ✓ 1
24.5 odd 2 333.2.a.d.1.1 1
24.11 even 2 5328.2.a.r.1.1 1
40.13 odd 4 925.2.b.b.149.2 2
40.29 even 2 925.2.a.e.1.1 1
40.37 odd 4 925.2.b.b.149.1 2
56.13 odd 2 1813.2.a.a.1.1 1
88.21 odd 2 4477.2.a.b.1.1 1
104.77 even 2 6253.2.a.c.1.1 1
120.29 odd 2 8325.2.a.e.1.1 1
296.117 odd 4 1369.2.b.c.1368.2 2
296.221 even 2 1369.2.a.e.1.1 1
296.253 odd 4 1369.2.b.c.1368.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
37.2.a.a.1.1 ✓ 1 8.5 even 2
333.2.a.d.1.1 1 24.5 odd 2
592.2.a.e.1.1 1 8.3 odd 2
925.2.a.e.1.1 1 40.29 even 2
925.2.b.b.149.1 2 40.37 odd 4
925.2.b.b.149.2 2 40.13 odd 4
1369.2.a.e.1.1 1 296.221 even 2
1369.2.b.c.1368.1 2 296.253 odd 4
1369.2.b.c.1368.2 2 296.117 odd 4
1813.2.a.a.1.1 1 56.13 odd 2
2368.2.a.b.1.1 1 4.3 odd 2
2368.2.a.q.1.1 1 1.1 even 1 trivial
4477.2.a.b.1.1 1 88.21 odd 2
5328.2.a.r.1.1 1 24.11 even 2
6253.2.a.c.1.1 1 104.77 even 2
8325.2.a.e.1.1 1 120.29 odd 2