Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2200,2,Mod(1,2200)] 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("2200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,2,0,0,0,-2,0,1,0,-1] 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: \(17.5670884447\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2200.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{3} -2.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +4.00000 q^{17} +4.00000 q^{19} -4.00000 q^{21} +6.00000 q^{23} -4.00000 q^{27} +2.00000 q^{29} +8.00000 q^{31} -2.00000 q^{33} +4.00000 q^{37} -6.00000 q^{41} +6.00000 q^{43} +2.00000 q^{47} -3.00000 q^{49} +8.00000 q^{51} +12.0000 q^{53} +8.00000 q^{57} +4.00000 q^{59} +14.0000 q^{61} -2.00000 q^{63} -10.0000 q^{67} +12.0000 q^{69} +8.00000 q^{71} -4.00000 q^{73} +2.00000 q^{77} -8.00000 q^{79} -11.0000 q^{81} -2.00000 q^{83} +4.00000 q^{87} -10.0000 q^{89} +16.0000 q^{93} +8.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 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
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 0 0
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −2.00000 −0.219529 −0.109764 − 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 16.0000 1.65912
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
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 2200.2.a.i.1.1 1
4.3 odd 2 4400.2.a.h.1.1 1
5.2 odd 4 440.2.b.a.89.1 ✓ 2
5.3 odd 4 440.2.b.a.89.2 yes 2
5.4 even 2 2200.2.a.c.1.1 1
15.2 even 4 3960.2.d.c.3169.2 2
15.8 even 4 3960.2.d.c.3169.1 2
20.3 even 4 880.2.b.b.529.1 2
20.7 even 4 880.2.b.b.529.2 2
20.19 odd 2 4400.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.b.a.89.1 ✓ 2 5.2 odd 4
440.2.b.a.89.2 yes 2 5.3 odd 4
880.2.b.b.529.1 2 20.3 even 4
880.2.b.b.529.2 2 20.7 even 4
2200.2.a.c.1.1 1 5.4 even 2
2200.2.a.i.1.1 1 1.1 even 1 trivial
3960.2.d.c.3169.1 2 15.8 even 4
3960.2.d.c.3169.2 2 15.2 even 4
4400.2.a.h.1.1 1 4.3 odd 2
4400.2.a.x.1.1 1 20.19 odd 2