Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1280,1,Mod(639,1280)] 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("1280.639"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1280.e (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(0.638803216170\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{5})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.419430400.3

Embedding invariants

Embedding label 639.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1280.639
Dual form 1280.1.e.a.639.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{5} +1.00000 q^{9} -1.00000 q^{25} +2.00000i q^{29} +2.00000 q^{41} +1.00000i q^{45} -1.00000 q^{49} -2.00000i q^{61} +1.00000 q^{81} -2.00000 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{9} - 2 q^{25} + 4 q^{41} - 2 q^{49} + 2 q^{81} - 4 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 1.00000i 1.00000i
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 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 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 1.00000i 1.00000i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 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 1280.1.e.a.639.2 2
4.3 odd 2 CM 1280.1.e.a.639.2 2
5.4 even 2 RM 1280.1.e.a.639.2 2
8.3 odd 2 inner 1280.1.e.a.639.1 2
8.5 even 2 inner 1280.1.e.a.639.1 2
16.3 odd 4 320.1.h.a.319.1 1
16.5 even 4 80.1.h.a.79.1 ✓ 1
16.11 odd 4 80.1.h.a.79.1 ✓ 1
16.13 even 4 320.1.h.a.319.1 1
20.19 odd 2 CM 1280.1.e.a.639.2 2
40.19 odd 2 inner 1280.1.e.a.639.1 2
40.29 even 2 inner 1280.1.e.a.639.1 2
48.5 odd 4 720.1.j.a.559.1 1
48.11 even 4 720.1.j.a.559.1 1
48.29 odd 4 2880.1.j.a.1279.1 1
48.35 even 4 2880.1.j.a.1279.1 1
80.3 even 4 1600.1.b.a.1151.1 1
80.13 odd 4 1600.1.b.a.1151.1 1
80.19 odd 4 320.1.h.a.319.1 1
80.27 even 4 400.1.b.a.351.1 1
80.29 even 4 320.1.h.a.319.1 1
80.37 odd 4 400.1.b.a.351.1 1
80.43 even 4 400.1.b.a.351.1 1
80.53 odd 4 400.1.b.a.351.1 1
80.59 odd 4 80.1.h.a.79.1 ✓ 1
80.67 even 4 1600.1.b.a.1151.1 1
80.69 even 4 80.1.h.a.79.1 ✓ 1
80.77 odd 4 1600.1.b.a.1151.1 1
112.5 odd 12 3920.1.bt.a.1439.1 2
112.11 odd 12 3920.1.bt.b.79.1 2
112.27 even 4 3920.1.j.a.3039.1 1
112.37 even 12 3920.1.bt.b.1439.1 2
112.53 even 12 3920.1.bt.b.79.1 2
112.59 even 12 3920.1.bt.a.79.1 2
112.69 odd 4 3920.1.j.a.3039.1 1
112.75 even 12 3920.1.bt.a.1439.1 2
112.101 odd 12 3920.1.bt.a.79.1 2
112.107 odd 12 3920.1.bt.b.1439.1 2
240.29 odd 4 2880.1.j.a.1279.1 1
240.53 even 4 3600.1.e.a.3151.1 1
240.59 even 4 720.1.j.a.559.1 1
240.107 odd 4 3600.1.e.a.3151.1 1
240.149 odd 4 720.1.j.a.559.1 1
240.179 even 4 2880.1.j.a.1279.1 1
240.197 even 4 3600.1.e.a.3151.1 1
240.203 odd 4 3600.1.e.a.3151.1 1
560.59 even 12 3920.1.bt.a.79.1 2
560.69 odd 4 3920.1.j.a.3039.1 1
560.139 even 4 3920.1.j.a.3039.1 1
560.149 even 12 3920.1.bt.b.1439.1 2
560.219 odd 12 3920.1.bt.b.1439.1 2
560.229 odd 12 3920.1.bt.a.1439.1 2
560.299 even 12 3920.1.bt.a.1439.1 2
560.389 even 12 3920.1.bt.b.79.1 2
560.459 odd 12 3920.1.bt.b.79.1 2
560.549 odd 12 3920.1.bt.a.79.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.1.h.a.79.1 ✓ 1 16.5 even 4
80.1.h.a.79.1 ✓ 1 16.11 odd 4
80.1.h.a.79.1 ✓ 1 80.59 odd 4
80.1.h.a.79.1 ✓ 1 80.69 even 4
320.1.h.a.319.1 1 16.3 odd 4
320.1.h.a.319.1 1 16.13 even 4
320.1.h.a.319.1 1 80.19 odd 4
320.1.h.a.319.1 1 80.29 even 4
400.1.b.a.351.1 1 80.27 even 4
400.1.b.a.351.1 1 80.37 odd 4
400.1.b.a.351.1 1 80.43 even 4
400.1.b.a.351.1 1 80.53 odd 4
720.1.j.a.559.1 1 48.5 odd 4
720.1.j.a.559.1 1 48.11 even 4
720.1.j.a.559.1 1 240.59 even 4
720.1.j.a.559.1 1 240.149 odd 4
1280.1.e.a.639.1 2 8.3 odd 2 inner
1280.1.e.a.639.1 2 8.5 even 2 inner
1280.1.e.a.639.1 2 40.19 odd 2 inner
1280.1.e.a.639.1 2 40.29 even 2 inner
1280.1.e.a.639.2 2 1.1 even 1 trivial
1280.1.e.a.639.2 2 4.3 odd 2 CM
1280.1.e.a.639.2 2 5.4 even 2 RM
1280.1.e.a.639.2 2 20.19 odd 2 CM
1600.1.b.a.1151.1 1 80.3 even 4
1600.1.b.a.1151.1 1 80.13 odd 4
1600.1.b.a.1151.1 1 80.67 even 4
1600.1.b.a.1151.1 1 80.77 odd 4
2880.1.j.a.1279.1 1 48.29 odd 4
2880.1.j.a.1279.1 1 48.35 even 4
2880.1.j.a.1279.1 1 240.29 odd 4
2880.1.j.a.1279.1 1 240.179 even 4
3600.1.e.a.3151.1 1 240.53 even 4
3600.1.e.a.3151.1 1 240.107 odd 4
3600.1.e.a.3151.1 1 240.197 even 4
3600.1.e.a.3151.1 1 240.203 odd 4
3920.1.j.a.3039.1 1 112.27 even 4
3920.1.j.a.3039.1 1 112.69 odd 4
3920.1.j.a.3039.1 1 560.69 odd 4
3920.1.j.a.3039.1 1 560.139 even 4
3920.1.bt.a.79.1 2 112.59 even 12
3920.1.bt.a.79.1 2 112.101 odd 12
3920.1.bt.a.79.1 2 560.59 even 12
3920.1.bt.a.79.1 2 560.549 odd 12
3920.1.bt.a.1439.1 2 112.5 odd 12
3920.1.bt.a.1439.1 2 112.75 even 12
3920.1.bt.a.1439.1 2 560.229 odd 12
3920.1.bt.a.1439.1 2 560.299 even 12
3920.1.bt.b.79.1 2 112.11 odd 12
3920.1.bt.b.79.1 2 112.53 even 12
3920.1.bt.b.79.1 2 560.389 even 12
3920.1.bt.b.79.1 2 560.459 odd 12
3920.1.bt.b.1439.1 2 112.37 even 12
3920.1.bt.b.1439.1 2 112.107 odd 12
3920.1.bt.b.1439.1 2 560.149 even 12
3920.1.bt.b.1439.1 2 560.219 odd 12