Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1280,2,Mod(769,1280)] 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("1280.769"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-12,0,8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.2
Root \(-0.517638i\) of defining polynomial
Character \(\chi\) \(=\) 1280.769
Dual form 1280.2.c.h.769.4

$q$-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)
 
\(f(q)\) \(=\) \(q-2.44949i q^{3} +(1.73205 - 1.41421i) q^{5} +1.41421i q^{7} -3.00000 q^{9} +2.00000 q^{11} -5.65685i q^{13} +(-3.46410 - 4.24264i) q^{15} +4.89898i q^{17} -6.00000 q^{19} +3.46410 q^{21} -7.07107i q^{23} +(1.00000 - 4.89898i) q^{25} +6.92820 q^{29} -6.92820 q^{31} -4.89898i q^{33} +(2.00000 + 2.44949i) q^{35} -2.82843i q^{37} -13.8564 q^{39} +4.00000 q^{41} +2.44949i q^{43} +(-5.19615 + 4.24264i) q^{45} -4.24264i q^{47} +5.00000 q^{49} +12.0000 q^{51} +(3.46410 - 2.82843i) q^{55} +14.6969i q^{57} -2.00000 q^{59} -3.46410 q^{61} -4.24264i q^{63} +(-8.00000 - 9.79796i) q^{65} +2.44949i q^{67} -17.3205 q^{69} -6.92820 q^{71} -4.89898i q^{73} +(-12.0000 - 2.44949i) q^{75} +2.82843i q^{77} +6.92820 q^{79} -9.00000 q^{81} +12.2474i q^{83} +(6.92820 + 8.48528i) q^{85} -16.9706i q^{87} +2.00000 q^{89} +8.00000 q^{91} +16.9706i q^{93} +(-10.3923 + 8.48528i) q^{95} +14.6969i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 8 q^{11} - 24 q^{19} + 4 q^{25} + 8 q^{35} + 16 q^{41} + 20 q^{49} + 48 q^{51} - 8 q^{59} - 32 q^{65} - 48 q^{75} - 36 q^{81} + 8 q^{89} + 32 q^{91} - 24 q^{99}+O(q^{100}) \) 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
\(3\) − 2.44949i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(4\) 0 0
\(5\) 1.73205 − 1.41421i 0.774597 − 0.632456i
\(6\) 0 0
\(7\) 1.41421i 0.534522i 0.963624 + 0.267261i \(0.0861187\pi\)
−0.963624 + 0.267261i \(0.913881\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) − 5.65685i − 1.56893i −0.620174 − 0.784465i \(-0.712938\pi\)
0.620174 − 0.784465i \(-0.287062\pi\)
\(14\) 0 0
\(15\) −3.46410 − 4.24264i −0.894427 − 1.09545i
\(16\) 0 0
\(17\) 4.89898i 1.18818i 0.804400 + 0.594089i \(0.202487\pi\)
−0.804400 + 0.594089i \(0.797513\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 0 0
\(23\) − 7.07107i − 1.47442i −0.675664 − 0.737210i \(-0.736143\pi\)
0.675664 − 0.737210i \(-0.263857\pi\)
\(24\) 0 0
\(25\) 1.00000 − 4.89898i 0.200000 − 0.979796i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.92820 1.28654 0.643268 − 0.765641i \(-0.277578\pi\)
0.643268 + 0.765641i \(0.277578\pi\)
\(30\) 0 0
\(31\) −6.92820 −1.24434 −0.622171 − 0.782881i \(-0.713749\pi\)
−0.622171 + 0.782881i \(0.713749\pi\)
\(32\) 0 0
\(33\) − 4.89898i − 0.852803i
\(34\) 0 0
\(35\) 2.00000 + 2.44949i 0.338062 + 0.414039i
\(36\) 0 0
\(37\) − 2.82843i − 0.464991i −0.972598 − 0.232495i \(-0.925311\pi\)
0.972598 − 0.232495i \(-0.0746890\pi\)
\(38\) 0 0
\(39\) −13.8564 −2.21880
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 2.44949i 0.373544i 0.982403 + 0.186772i \(0.0598025\pi\)
−0.982403 + 0.186772i \(0.940197\pi\)
\(44\) 0 0
\(45\) −5.19615 + 4.24264i −0.774597 + 0.632456i
\(46\) 0 0
\(47\) − 4.24264i − 0.618853i −0.950923 − 0.309426i \(-0.899863\pi\)
0.950923 − 0.309426i \(-0.100137\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 3.46410 − 2.82843i 0.467099 − 0.381385i
\(56\) 0 0
\(57\) 14.6969i 1.94666i
\(58\) 0 0
\(59\) −2.00000 −0.260378 −0.130189 − 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) −3.46410 −0.443533 −0.221766 − 0.975100i \(-0.571182\pi\)
−0.221766 + 0.975100i \(0.571182\pi\)
\(62\) 0 0
\(63\) − 4.24264i − 0.534522i
\(64\) 0 0
\(65\) −8.00000 − 9.79796i −0.992278 − 1.21529i
\(66\) 0 0
\(67\) 2.44949i 0.299253i 0.988743 + 0.149626i \(0.0478071\pi\)
−0.988743 + 0.149626i \(0.952193\pi\)
\(68\) 0 0
\(69\) −17.3205 −2.08514
\(70\) 0 0
\(71\) −6.92820 −0.822226 −0.411113 − 0.911584i \(-0.634860\pi\)
−0.411113 + 0.911584i \(0.634860\pi\)
\(72\) 0 0
\(73\) − 4.89898i − 0.573382i −0.958023 − 0.286691i \(-0.907445\pi\)
0.958023 − 0.286691i \(-0.0925553\pi\)
\(74\) 0 0
\(75\) −12.0000 − 2.44949i −1.38564 − 0.282843i
\(76\) 0 0
\(77\) 2.82843i 0.322329i
\(78\) 0 0
\(79\) 6.92820 0.779484 0.389742 − 0.920924i \(-0.372564\pi\)
0.389742 + 0.920924i \(0.372564\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 12.2474i 1.34433i 0.740400 + 0.672166i \(0.234636\pi\)
−0.740400 + 0.672166i \(0.765364\pi\)
\(84\) 0 0
\(85\) 6.92820 + 8.48528i 0.751469 + 0.920358i
\(86\) 0 0
\(87\) − 16.9706i − 1.81944i
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 16.9706i 1.75977i
\(94\) 0 0
\(95\) −10.3923 + 8.48528i −1.06623 + 0.870572i
\(96\) 0 0
\(97\) 14.6969i 1.49225i 0.665807 + 0.746124i \(0.268087\pi\)
−0.665807 + 0.746124i \(0.731913\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
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 1280.2.c.h.769.2 4
4.3 odd 2 1280.2.c.g.769.4 4
5.2 odd 4 6400.2.a.cu.1.1 4
5.3 odd 4 6400.2.a.cu.1.4 4
5.4 even 2 inner 1280.2.c.h.769.4 4
8.3 odd 2 inner 1280.2.c.h.769.1 4
8.5 even 2 1280.2.c.g.769.3 4
16.3 odd 4 320.2.f.b.289.1 ✓ 8
16.5 even 4 320.2.f.b.289.4 yes 8
16.11 odd 4 320.2.f.b.289.8 yes 8
16.13 even 4 320.2.f.b.289.5 yes 8
20.3 even 4 6400.2.a.ct.1.1 4
20.7 even 4 6400.2.a.ct.1.4 4
20.19 odd 2 1280.2.c.g.769.2 4
40.3 even 4 6400.2.a.cu.1.3 4
40.13 odd 4 6400.2.a.ct.1.2 4
40.19 odd 2 inner 1280.2.c.h.769.3 4
40.27 even 4 6400.2.a.cu.1.2 4
40.29 even 2 1280.2.c.g.769.1 4
40.37 odd 4 6400.2.a.ct.1.3 4
48.5 odd 4 2880.2.d.g.289.1 8
48.11 even 4 2880.2.d.g.289.2 8
48.29 odd 4 2880.2.d.g.289.7 8
48.35 even 4 2880.2.d.g.289.8 8
80.3 even 4 1600.2.d.i.801.4 8
80.13 odd 4 1600.2.d.i.801.5 8
80.19 odd 4 320.2.f.b.289.7 yes 8
80.27 even 4 1600.2.d.i.801.1 8
80.29 even 4 320.2.f.b.289.3 yes 8
80.37 odd 4 1600.2.d.i.801.8 8
80.43 even 4 1600.2.d.i.801.7 8
80.53 odd 4 1600.2.d.i.801.2 8
80.59 odd 4 320.2.f.b.289.2 yes 8
80.67 even 4 1600.2.d.i.801.6 8
80.69 even 4 320.2.f.b.289.6 yes 8
80.77 odd 4 1600.2.d.i.801.3 8
240.29 odd 4 2880.2.d.g.289.4 8
240.59 even 4 2880.2.d.g.289.5 8
240.149 odd 4 2880.2.d.g.289.6 8
240.179 even 4 2880.2.d.g.289.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.f.b.289.1 ✓ 8 16.3 odd 4
320.2.f.b.289.2 yes 8 80.59 odd 4
320.2.f.b.289.3 yes 8 80.29 even 4
320.2.f.b.289.4 yes 8 16.5 even 4
320.2.f.b.289.5 yes 8 16.13 even 4
320.2.f.b.289.6 yes 8 80.69 even 4
320.2.f.b.289.7 yes 8 80.19 odd 4
320.2.f.b.289.8 yes 8 16.11 odd 4
1280.2.c.g.769.1 4 40.29 even 2
1280.2.c.g.769.2 4 20.19 odd 2
1280.2.c.g.769.3 4 8.5 even 2
1280.2.c.g.769.4 4 4.3 odd 2
1280.2.c.h.769.1 4 8.3 odd 2 inner
1280.2.c.h.769.2 4 1.1 even 1 trivial
1280.2.c.h.769.3 4 40.19 odd 2 inner
1280.2.c.h.769.4 4 5.4 even 2 inner
1600.2.d.i.801.1 8 80.27 even 4
1600.2.d.i.801.2 8 80.53 odd 4
1600.2.d.i.801.3 8 80.77 odd 4
1600.2.d.i.801.4 8 80.3 even 4
1600.2.d.i.801.5 8 80.13 odd 4
1600.2.d.i.801.6 8 80.67 even 4
1600.2.d.i.801.7 8 80.43 even 4
1600.2.d.i.801.8 8 80.37 odd 4
2880.2.d.g.289.1 8 48.5 odd 4
2880.2.d.g.289.2 8 48.11 even 4
2880.2.d.g.289.3 8 240.179 even 4
2880.2.d.g.289.4 8 240.29 odd 4
2880.2.d.g.289.5 8 240.59 even 4
2880.2.d.g.289.6 8 240.149 odd 4
2880.2.d.g.289.7 8 48.29 odd 4
2880.2.d.g.289.8 8 48.35 even 4
6400.2.a.ct.1.1 4 20.3 even 4
6400.2.a.ct.1.2 4 40.13 odd 4
6400.2.a.ct.1.3 4 40.37 odd 4
6400.2.a.ct.1.4 4 20.7 even 4
6400.2.a.cu.1.1 4 5.2 odd 4
6400.2.a.cu.1.2 4 40.27 even 4
6400.2.a.cu.1.3 4 40.3 even 4
6400.2.a.cu.1.4 4 5.3 odd 4