Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [720,2,Mod(1,720)] 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("720.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,-1,0,-2,0,0,0,6,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 720.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} -2.00000 q^{7} +6.00000 q^{11} -4.00000 q^{13} +6.00000 q^{17} +4.00000 q^{19} +1.00000 q^{25} +6.00000 q^{29} +4.00000 q^{31} +2.00000 q^{35} +8.00000 q^{37} -8.00000 q^{43} -3.00000 q^{49} +6.00000 q^{53} -6.00000 q^{55} +6.00000 q^{59} +2.00000 q^{61} +4.00000 q^{65} +4.00000 q^{67} -12.0000 q^{71} -10.0000 q^{73} -12.0000 q^{77} +4.00000 q^{79} +12.0000 q^{83} -6.00000 q^{85} -12.0000 q^{89} +8.00000 q^{91} -4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 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
\(75\) 0 0
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
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 720.2.a.b.1.1 1
3.2 odd 2 720.2.a.g.1.1 1
4.3 odd 2 90.2.a.b.1.1 yes 1
5.2 odd 4 3600.2.f.u.2449.1 2
5.3 odd 4 3600.2.f.u.2449.2 2
5.4 even 2 3600.2.a.bj.1.1 1
8.3 odd 2 2880.2.a.bf.1.1 1
8.5 even 2 2880.2.a.u.1.1 1
12.11 even 2 90.2.a.a.1.1 ✓ 1
15.2 even 4 3600.2.f.a.2449.1 2
15.8 even 4 3600.2.f.a.2449.2 2
15.14 odd 2 3600.2.a.ba.1.1 1
20.3 even 4 450.2.c.a.199.1 2
20.7 even 4 450.2.c.a.199.2 2
20.19 odd 2 450.2.a.a.1.1 1
24.5 odd 2 2880.2.a.h.1.1 1
24.11 even 2 2880.2.a.k.1.1 1
28.27 even 2 4410.2.a.bf.1.1 1
36.7 odd 6 810.2.e.e.271.1 2
36.11 even 6 810.2.e.h.271.1 2
36.23 even 6 810.2.e.h.541.1 2
36.31 odd 6 810.2.e.e.541.1 2
60.23 odd 4 450.2.c.d.199.2 2
60.47 odd 4 450.2.c.d.199.1 2
60.59 even 2 450.2.a.e.1.1 1
84.83 odd 2 4410.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.a.a.1.1 ✓ 1 12.11 even 2
90.2.a.b.1.1 yes 1 4.3 odd 2
450.2.a.a.1.1 1 20.19 odd 2
450.2.a.e.1.1 1 60.59 even 2
450.2.c.a.199.1 2 20.3 even 4
450.2.c.a.199.2 2 20.7 even 4
450.2.c.d.199.1 2 60.47 odd 4
450.2.c.d.199.2 2 60.23 odd 4
720.2.a.b.1.1 1 1.1 even 1 trivial
720.2.a.g.1.1 1 3.2 odd 2
810.2.e.e.271.1 2 36.7 odd 6
810.2.e.e.541.1 2 36.31 odd 6
810.2.e.h.271.1 2 36.11 even 6
810.2.e.h.541.1 2 36.23 even 6
2880.2.a.h.1.1 1 24.5 odd 2
2880.2.a.k.1.1 1 24.11 even 2
2880.2.a.u.1.1 1 8.5 even 2
2880.2.a.bf.1.1 1 8.3 odd 2
3600.2.a.ba.1.1 1 15.14 odd 2
3600.2.a.bj.1.1 1 5.4 even 2
3600.2.f.a.2449.1 2 15.2 even 4
3600.2.f.a.2449.2 2 15.8 even 4
3600.2.f.u.2449.1 2 5.2 odd 4
3600.2.f.u.2449.2 2 5.3 odd 4
4410.2.a.k.1.1 1 84.83 odd 2
4410.2.a.bf.1.1 1 28.27 even 2