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

$q$-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 \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) 1.73205 − 1.41421i 0.774597 − 0.632456i
\(6\) 0 0
\(7\) − 1.41421i − 0.534522i −0.963624 − 0.267261i \(-0.913881\pi\)
0.963624 − 0.267261i \(-0.0861187\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 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 \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 0 0
\(23\) 7.07107i 1.47442i 0.675664 + 0.737210i \(0.263857\pi\)
−0.675664 + 0.737210i \(0.736143\pi\)
\(24\) 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 \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) 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 \(-0.940197\pi\)
0.982403 − 0.186772i \(-0.0598025\pi\)
\(44\) 0 0
\(45\) −5.19615 + 4.24264i −0.774597 + 0.632456i
\(46\) 0 0
\(47\) 4.24264i 0.618853i 0.950923 + 0.309426i \(0.100137\pi\)
−0.950923 + 0.309426i \(0.899863\pi\)
\(48\) 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 \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 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 \(-0.952193\pi\)
0.988743 − 0.149626i \(-0.0478071\pi\)
\(68\) 0 0
\(69\) −17.3205 −2.08514
\(70\) 0 0
\(71\) 6.92820 0.822226 0.411113 − 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 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 \(-0.627436\pi\)
−0.389742 + 0.920924i \(0.627436\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) − 12.2474i − 1.34433i −0.740400 − 0.672166i \(-0.765364\pi\)
0.740400 − 0.672166i \(-0.234636\pi\)
\(84\) 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.g.769.4 4
4.3 odd 2 1280.2.c.h.769.2 4
5.2 odd 4 6400.2.a.ct.1.4 4
5.3 odd 4 6400.2.a.ct.1.1 4
5.4 even 2 inner 1280.2.c.g.769.2 4
8.3 odd 2 inner 1280.2.c.g.769.3 4
8.5 even 2 1280.2.c.h.769.1 4
16.3 odd 4 320.2.f.b.289.5 yes 8
16.5 even 4 320.2.f.b.289.8 yes 8
16.11 odd 4 320.2.f.b.289.4 yes 8
16.13 even 4 320.2.f.b.289.1 ✓ 8
20.3 even 4 6400.2.a.cu.1.4 4
20.7 even 4 6400.2.a.cu.1.1 4
20.19 odd 2 1280.2.c.h.769.4 4
40.3 even 4 6400.2.a.ct.1.2 4
40.13 odd 4 6400.2.a.cu.1.3 4
40.19 odd 2 inner 1280.2.c.g.769.1 4
40.27 even 4 6400.2.a.ct.1.3 4
40.29 even 2 1280.2.c.h.769.3 4
40.37 odd 4 6400.2.a.cu.1.2 4
48.5 odd 4 2880.2.d.g.289.2 8
48.11 even 4 2880.2.d.g.289.1 8
48.29 odd 4 2880.2.d.g.289.8 8
48.35 even 4 2880.2.d.g.289.7 8
80.3 even 4 1600.2.d.i.801.5 8
80.13 odd 4 1600.2.d.i.801.4 8
80.19 odd 4 320.2.f.b.289.3 yes 8
80.27 even 4 1600.2.d.i.801.8 8
80.29 even 4 320.2.f.b.289.7 yes 8
80.37 odd 4 1600.2.d.i.801.1 8
80.43 even 4 1600.2.d.i.801.2 8
80.53 odd 4 1600.2.d.i.801.7 8
80.59 odd 4 320.2.f.b.289.6 yes 8
80.67 even 4 1600.2.d.i.801.3 8
80.69 even 4 320.2.f.b.289.2 yes 8
80.77 odd 4 1600.2.d.i.801.6 8
240.29 odd 4 2880.2.d.g.289.3 8
240.59 even 4 2880.2.d.g.289.6 8
240.149 odd 4 2880.2.d.g.289.5 8
240.179 even 4 2880.2.d.g.289.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.f.b.289.1 ✓ 8 16.13 even 4
320.2.f.b.289.2 yes 8 80.69 even 4
320.2.f.b.289.3 yes 8 80.19 odd 4
320.2.f.b.289.4 yes 8 16.11 odd 4
320.2.f.b.289.5 yes 8 16.3 odd 4
320.2.f.b.289.6 yes 8 80.59 odd 4
320.2.f.b.289.7 yes 8 80.29 even 4
320.2.f.b.289.8 yes 8 16.5 even 4
1280.2.c.g.769.1 4 40.19 odd 2 inner
1280.2.c.g.769.2 4 5.4 even 2 inner
1280.2.c.g.769.3 4 8.3 odd 2 inner
1280.2.c.g.769.4 4 1.1 even 1 trivial
1280.2.c.h.769.1 4 8.5 even 2
1280.2.c.h.769.2 4 4.3 odd 2
1280.2.c.h.769.3 4 40.29 even 2
1280.2.c.h.769.4 4 20.19 odd 2
1600.2.d.i.801.1 8 80.37 odd 4
1600.2.d.i.801.2 8 80.43 even 4
1600.2.d.i.801.3 8 80.67 even 4
1600.2.d.i.801.4 8 80.13 odd 4
1600.2.d.i.801.5 8 80.3 even 4
1600.2.d.i.801.6 8 80.77 odd 4
1600.2.d.i.801.7 8 80.53 odd 4
1600.2.d.i.801.8 8 80.27 even 4
2880.2.d.g.289.1 8 48.11 even 4
2880.2.d.g.289.2 8 48.5 odd 4
2880.2.d.g.289.3 8 240.29 odd 4
2880.2.d.g.289.4 8 240.179 even 4
2880.2.d.g.289.5 8 240.149 odd 4
2880.2.d.g.289.6 8 240.59 even 4
2880.2.d.g.289.7 8 48.35 even 4
2880.2.d.g.289.8 8 48.29 odd 4
6400.2.a.ct.1.1 4 5.3 odd 4
6400.2.a.ct.1.2 4 40.3 even 4
6400.2.a.ct.1.3 4 40.27 even 4
6400.2.a.ct.1.4 4 5.2 odd 4
6400.2.a.cu.1.1 4 20.7 even 4
6400.2.a.cu.1.2 4 40.37 odd 4
6400.2.a.cu.1.3 4 40.13 odd 4
6400.2.a.cu.1.4 4 20.3 even 4