Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [384,2,Mod(1,384)] 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("384.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-1,0,4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 384.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -2.00000 q^{13} -4.00000 q^{15} -2.00000 q^{17} +8.00000 q^{19} +2.00000 q^{21} -4.00000 q^{23} +11.0000 q^{25} -1.00000 q^{27} +6.00000 q^{31} -4.00000 q^{33} -8.00000 q^{35} +2.00000 q^{37} +2.00000 q^{39} +6.00000 q^{41} +4.00000 q^{45} -4.00000 q^{47} -3.00000 q^{49} +2.00000 q^{51} +16.0000 q^{55} -8.00000 q^{57} -4.00000 q^{59} -14.0000 q^{61} -2.00000 q^{63} -8.00000 q^{65} +4.00000 q^{67} +4.00000 q^{69} -12.0000 q^{71} -10.0000 q^{73} -11.0000 q^{75} -8.00000 q^{77} -10.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -8.00000 q^{85} -14.0000 q^{89} +4.00000 q^{91} -6.00000 q^{93} +32.0000 q^{95} +10.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 4.00000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 4.00000 0.596285
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 16.0000 2.15744
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) −11.0000 −1.27017
\(76\) 0 0
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 32.0000 3.28313
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
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 384.2.a.d.1.1 yes 1
3.2 odd 2 1152.2.a.a.1.1 1
4.3 odd 2 384.2.a.h.1.1 yes 1
5.4 even 2 9600.2.a.bz.1.1 1
8.3 odd 2 384.2.a.a.1.1 ✓ 1
8.5 even 2 384.2.a.e.1.1 yes 1
12.11 even 2 1152.2.a.b.1.1 1
16.3 odd 4 768.2.d.c.385.2 2
16.5 even 4 768.2.d.f.385.2 2
16.11 odd 4 768.2.d.c.385.1 2
16.13 even 4 768.2.d.f.385.1 2
20.19 odd 2 9600.2.a.e.1.1 1
24.5 odd 2 1152.2.a.s.1.1 1
24.11 even 2 1152.2.a.t.1.1 1
40.19 odd 2 9600.2.a.bk.1.1 1
40.29 even 2 9600.2.a.t.1.1 1
48.5 odd 4 2304.2.d.o.1153.1 2
48.11 even 4 2304.2.d.f.1153.1 2
48.29 odd 4 2304.2.d.o.1153.2 2
48.35 even 4 2304.2.d.f.1153.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.a.a.1.1 ✓ 1 8.3 odd 2
384.2.a.d.1.1 yes 1 1.1 even 1 trivial
384.2.a.e.1.1 yes 1 8.5 even 2
384.2.a.h.1.1 yes 1 4.3 odd 2
768.2.d.c.385.1 2 16.11 odd 4
768.2.d.c.385.2 2 16.3 odd 4
768.2.d.f.385.1 2 16.13 even 4
768.2.d.f.385.2 2 16.5 even 4
1152.2.a.a.1.1 1 3.2 odd 2
1152.2.a.b.1.1 1 12.11 even 2
1152.2.a.s.1.1 1 24.5 odd 2
1152.2.a.t.1.1 1 24.11 even 2
2304.2.d.f.1153.1 2 48.11 even 4
2304.2.d.f.1153.2 2 48.35 even 4
2304.2.d.o.1153.1 2 48.5 odd 4
2304.2.d.o.1153.2 2 48.29 odd 4
9600.2.a.e.1.1 1 20.19 odd 2
9600.2.a.t.1.1 1 40.29 even 2
9600.2.a.bk.1.1 1 40.19 odd 2
9600.2.a.bz.1.1 1 5.4 even 2