↑

Properties

Label 40.2.a.a.1.1
Level $40$
Weight $2$
Character 40.1
Self dual yes
Analytic conductor $0.319$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [40,2,Mod(1,40)] 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("40.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 40.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.319401608085\)
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 \(\chi\) \(=\) 40.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} -4.00000 q^{7} -3.00000 q^{9} +4.00000 q^{11} -2.00000 q^{13} +2.00000 q^{17} +4.00000 q^{19} +4.00000 q^{23} +1.00000 q^{25} -2.00000 q^{29} -8.00000 q^{31} -4.00000 q^{35} +6.00000 q^{37} -6.00000 q^{41} -8.00000 q^{43} -3.00000 q^{45} +4.00000 q^{47} +9.00000 q^{49} +6.00000 q^{53} +4.00000 q^{55} -4.00000 q^{59} -2.00000 q^{61} +12.0000 q^{63} -2.00000 q^{65} +8.00000 q^{67} -6.00000 q^{73} -16.0000 q^{77} +9.00000 q^{81} -16.0000 q^{83} +2.00000 q^{85} -6.00000 q^{89} +8.00000 q^{91} +4.00000 q^{95} -14.0000 q^{97} -12.0000 q^{99} +O(q^{100})\)

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
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 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
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 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
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 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
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 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
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 12.0000 1.51186
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −16.0000 −1.82337
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 − 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
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 40.2.a.a.1.1 ✓ 1
3.2 odd 2 360.2.a.a.1.1 1
4.3 odd 2 80.2.a.a.1.1 1
5.2 odd 4 200.2.c.b.49.1 2
5.3 odd 4 200.2.c.b.49.2 2
5.4 even 2 200.2.a.c.1.1 1
7.2 even 3 1960.2.q.h.361.1 2
7.3 odd 6 1960.2.q.i.961.1 2
7.4 even 3 1960.2.q.h.961.1 2
7.5 odd 6 1960.2.q.i.361.1 2
7.6 odd 2 1960.2.a.g.1.1 1
8.3 odd 2 320.2.a.d.1.1 1
8.5 even 2 320.2.a.c.1.1 1
9.2 odd 6 3240.2.q.x.1081.1 2
9.4 even 3 3240.2.q.k.2161.1 2
9.5 odd 6 3240.2.q.x.2161.1 2
9.7 even 3 3240.2.q.k.1081.1 2
11.10 odd 2 4840.2.a.f.1.1 1
12.11 even 2 720.2.a.e.1.1 1
13.12 even 2 6760.2.a.i.1.1 1
15.2 even 4 1800.2.f.a.649.1 2
15.8 even 4 1800.2.f.a.649.2 2
15.14 odd 2 1800.2.a.v.1.1 1
16.3 odd 4 1280.2.d.a.641.1 2
16.5 even 4 1280.2.d.j.641.2 2
16.11 odd 4 1280.2.d.a.641.2 2
16.13 even 4 1280.2.d.j.641.1 2
20.3 even 4 400.2.c.d.49.1 2
20.7 even 4 400.2.c.d.49.2 2
20.19 odd 2 400.2.a.e.1.1 1
24.5 odd 2 2880.2.a.t.1.1 1
24.11 even 2 2880.2.a.bg.1.1 1
28.27 even 2 3920.2.a.s.1.1 1
35.34 odd 2 9800.2.a.x.1.1 1
40.3 even 4 1600.2.c.m.449.1 2
40.13 odd 4 1600.2.c.k.449.2 2
40.19 odd 2 1600.2.a.k.1.1 1
40.27 even 4 1600.2.c.m.449.2 2
40.29 even 2 1600.2.a.o.1.1 1
40.37 odd 4 1600.2.c.k.449.1 2
44.43 even 2 9680.2.a.q.1.1 1
60.23 odd 4 3600.2.f.t.2449.1 2
60.47 odd 4 3600.2.f.t.2449.2 2
60.59 even 2 3600.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
80.2.a.a.1.1 1 4.3 odd 2
200.2.a.c.1.1 1 5.4 even 2
200.2.c.b.49.1 2 5.2 odd 4
200.2.c.b.49.2 2 5.3 odd 4
320.2.a.c.1.1 1 8.5 even 2
320.2.a.d.1.1 1 8.3 odd 2
360.2.a.a.1.1 1 3.2 odd 2
400.2.a.e.1.1 1 20.19 odd 2
400.2.c.d.49.1 2 20.3 even 4
400.2.c.d.49.2 2 20.7 even 4
720.2.a.e.1.1 1 12.11 even 2
1280.2.d.a.641.1 2 16.3 odd 4
1280.2.d.a.641.2 2 16.11 odd 4
1280.2.d.j.641.1 2 16.13 even 4
1280.2.d.j.641.2 2 16.5 even 4
1600.2.a.k.1.1 1 40.19 odd 2
1600.2.a.o.1.1 1 40.29 even 2
1600.2.c.k.449.1 2 40.37 odd 4
1600.2.c.k.449.2 2 40.13 odd 4
1600.2.c.m.449.1 2 40.3 even 4
1600.2.c.m.449.2 2 40.27 even 4
1800.2.a.v.1.1 1 15.14 odd 2
1800.2.f.a.649.1 2 15.2 even 4
1800.2.f.a.649.2 2 15.8 even 4
1960.2.a.g.1.1 1 7.6 odd 2
1960.2.q.h.361.1 2 7.2 even 3
1960.2.q.h.961.1 2 7.4 even 3
1960.2.q.i.361.1 2 7.5 odd 6
1960.2.q.i.961.1 2 7.3 odd 6
2880.2.a.t.1.1 1 24.5 odd 2
2880.2.a.bg.1.1 1 24.11 even 2
3240.2.q.k.1081.1 2 9.7 even 3
3240.2.q.k.2161.1 2 9.4 even 3
3240.2.q.x.1081.1 2 9.2 odd 6
3240.2.q.x.2161.1 2 9.5 odd 6
3600.2.a.h.1.1 1 60.59 even 2
3600.2.f.t.2449.1 2 60.23 odd 4
3600.2.f.t.2449.2 2 60.47 odd 4
3920.2.a.s.1.1 1 28.27 even 2
4840.2.a.f.1.1 1 11.10 odd 2
6760.2.a.i.1.1 1 13.12 even 2
9680.2.a.q.1.1 1 44.43 even 2
9800.2.a.x.1.1 1 35.34 odd 2