Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1040,2,Mod(961,1040)] 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("1040.961"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9144576.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 961.5
Root \(2.26180i\) of defining polynomial
Character \(\chi\) \(=\) 1040.961
Dual form 1040.2.k.c.961.6

$q$-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)
 
\(f(q)\) \(=\) \(q+2.26180 q^{3} -1.00000i q^{5} -1.11575i q^{7} +2.11575 q^{9} -5.37755i q^{11} +(-3.37755 + 1.26180i) q^{13} -2.26180i q^{15} -7.90116i q^{19} -2.52360i q^{21} +6.49330 q^{23} -1.00000 q^{25} -2.00000 q^{27} +3.63935 q^{29} +3.14605i q^{31} -12.1630i q^{33} -1.11575 q^{35} +7.40786i q^{37} +(-7.63935 + 2.85395i) q^{39} -4.75510i q^{41} +4.78541 q^{43} -2.11575i q^{45} +6.16296i q^{47} +5.75510 q^{49} +0.292106 q^{53} -5.37755 q^{55} -17.8709i q^{57} +11.3776i q^{59} -5.34725 q^{61} -2.36065i q^{63} +(1.26180 + 3.37755i) q^{65} -11.8709i q^{67} +14.6866 q^{69} -3.43816i q^{71} +4.59214i q^{73} -2.26180 q^{75} -6.00000 q^{77} +10.8157 q^{79} -10.8709 q^{81} +7.40786i q^{83} +8.23150 q^{87} +14.8157i q^{89} +(1.40786 + 3.76850i) q^{91} +7.11575i q^{93} -7.90116 q^{95} -3.76850i q^{97} -11.3776i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{9} + 12 q^{23} - 6 q^{25} - 12 q^{27} - 12 q^{29} - 12 q^{39} - 12 q^{43} - 6 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{61} - 6 q^{65} - 36 q^{77} + 24 q^{79} - 18 q^{81} + 36 q^{87} - 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.26180 1.30585 0.652926 − 0.757422i \(-0.273541\pi\)
0.652926 + 0.757422i \(0.273541\pi\)
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) − 1.11575i − 0.421714i −0.977517 − 0.210857i \(-0.932375\pi\)
0.977517 − 0.210857i \(-0.0676254\pi\)
\(8\) 0 0
\(9\) 2.11575 0.705250
\(10\) 0 0
\(11\) − 5.37755i − 1.62139i −0.585467 − 0.810696i \(-0.699089\pi\)
0.585467 − 0.810696i \(-0.300911\pi\)
\(12\) 0 0
\(13\) −3.37755 + 1.26180i −0.936764 + 0.349961i
\(14\) 0 0
\(15\) − 2.26180i − 0.583995i
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) − 7.90116i − 1.81265i −0.422582 − 0.906325i \(-0.638876\pi\)
0.422582 − 0.906325i \(-0.361124\pi\)
\(20\) 0 0
\(21\) − 2.52360i − 0.550696i
\(22\) 0 0
\(23\) 6.49330 1.35395 0.676973 − 0.736007i \(-0.263291\pi\)
0.676973 + 0.736007i \(0.263291\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −2.00000 −0.384900
\(28\) 0 0
\(29\) 3.63935 0.675811 0.337906 − 0.941180i \(-0.390282\pi\)
0.337906 + 0.941180i \(0.390282\pi\)
\(30\) 0 0
\(31\) 3.14605i 0.565048i 0.959260 + 0.282524i \(0.0911716\pi\)
−0.959260 + 0.282524i \(0.908828\pi\)
\(32\) 0 0
\(33\) − 12.1630i − 2.11730i
\(34\) 0 0
\(35\) −1.11575 −0.188596
\(36\) 0 0
\(37\) 7.40786i 1.21784i 0.793230 + 0.608922i \(0.208398\pi\)
−0.793230 + 0.608922i \(0.791602\pi\)
\(38\) 0 0
\(39\) −7.63935 + 2.85395i −1.22328 + 0.456997i
\(40\) 0 0
\(41\) − 4.75510i − 0.742622i −0.928508 − 0.371311i \(-0.878908\pi\)
0.928508 − 0.371311i \(-0.121092\pi\)
\(42\) 0 0
\(43\) 4.78541 0.729768 0.364884 − 0.931053i \(-0.381109\pi\)
0.364884 + 0.931053i \(0.381109\pi\)
\(44\) 0 0
\(45\) − 2.11575i − 0.315397i
\(46\) 0 0
\(47\) 6.16296i 0.898960i 0.893290 + 0.449480i \(0.148391\pi\)
−0.893290 + 0.449480i \(0.851609\pi\)
\(48\) 0 0
\(49\) 5.75510 0.822158
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.292106 0.0401238 0.0200619 − 0.999799i \(-0.493614\pi\)
0.0200619 + 0.999799i \(0.493614\pi\)
\(54\) 0 0
\(55\) −5.37755 −0.725109
\(56\) 0 0
\(57\) − 17.8709i − 2.36705i
\(58\) 0 0
\(59\) 11.3776i 1.48123i 0.671929 + 0.740616i \(0.265466\pi\)
−0.671929 + 0.740616i \(0.734534\pi\)
\(60\) 0 0
\(61\) −5.34725 −0.684645 −0.342322 − 0.939583i \(-0.611214\pi\)
−0.342322 + 0.939583i \(0.611214\pi\)
\(62\) 0 0
\(63\) − 2.36065i − 0.297413i
\(64\) 0 0
\(65\) 1.26180 + 3.37755i 0.156507 + 0.418934i
\(66\) 0 0
\(67\) − 11.8709i − 1.45026i −0.688614 − 0.725128i \(-0.741781\pi\)
0.688614 − 0.725128i \(-0.258219\pi\)
\(68\) 0 0
\(69\) 14.6866 1.76805
\(70\) 0 0
\(71\) − 3.43816i − 0.408034i −0.978967 − 0.204017i \(-0.934600\pi\)
0.978967 − 0.204017i \(-0.0653998\pi\)
\(72\) 0 0
\(73\) 4.59214i 0.537470i 0.963214 + 0.268735i \(0.0866056\pi\)
−0.963214 + 0.268735i \(0.913394\pi\)
\(74\) 0 0
\(75\) −2.26180 −0.261170
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 10.8157 1.21686 0.608431 − 0.793607i \(-0.291799\pi\)
0.608431 + 0.793607i \(0.291799\pi\)
\(80\) 0 0
\(81\) −10.8709 −1.20787
\(82\) 0 0
\(83\) 7.40786i 0.813118i 0.913625 + 0.406559i \(0.133271\pi\)
−0.913625 + 0.406559i \(0.866729\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.23150 0.882509
\(88\) 0 0
\(89\) 14.8157i 1.57046i 0.619203 + 0.785231i \(0.287456\pi\)
−0.619203 + 0.785231i \(0.712544\pi\)
\(90\) 0 0
\(91\) 1.40786 + 3.76850i 0.147583 + 0.395046i
\(92\) 0 0
\(93\) 7.11575i 0.737869i
\(94\) 0 0
\(95\) −7.90116 −0.810642
\(96\) 0 0
\(97\) − 3.76850i − 0.382633i −0.981528 − 0.191317i \(-0.938724\pi\)
0.981528 − 0.191317i \(-0.0612757\pi\)
\(98\) 0 0
\(99\) − 11.3776i − 1.14349i
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 1040.2.k.c.961.5 6
4.3 odd 2 260.2.f.a.181.1 ✓ 6
12.11 even 2 2340.2.c.d.181.5 6
13.12 even 2 inner 1040.2.k.c.961.6 6
20.3 even 4 1300.2.d.d.649.2 6
20.7 even 4 1300.2.d.c.649.5 6
20.19 odd 2 1300.2.f.e.701.5 6
52.31 even 4 3380.2.a.m.1.1 3
52.47 even 4 3380.2.a.n.1.1 3
52.51 odd 2 260.2.f.a.181.2 yes 6
156.155 even 2 2340.2.c.d.181.2 6
260.103 even 4 1300.2.d.c.649.2 6
260.207 even 4 1300.2.d.d.649.5 6
260.259 odd 2 1300.2.f.e.701.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.f.a.181.1 ✓ 6 4.3 odd 2
260.2.f.a.181.2 yes 6 52.51 odd 2
1040.2.k.c.961.5 6 1.1 even 1 trivial
1040.2.k.c.961.6 6 13.12 even 2 inner
1300.2.d.c.649.2 6 260.103 even 4
1300.2.d.c.649.5 6 20.7 even 4
1300.2.d.d.649.2 6 20.3 even 4
1300.2.d.d.649.5 6 260.207 even 4
1300.2.f.e.701.5 6 20.19 odd 2
1300.2.f.e.701.6 6 260.259 odd 2
2340.2.c.d.181.2 6 156.155 even 2
2340.2.c.d.181.5 6 12.11 even 2
3380.2.a.m.1.1 3 52.31 even 4
3380.2.a.n.1.1 3 52.47 even 4