Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6400,2,Mod(1,6400)] 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("6400.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,12,0,-8,0,0,0,0,0,0,0,-24,0,0,0,0,0,0,0,0,0, 0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(51.1042572936\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 320)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.93185\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.44949 q^{3} +1.41421 q^{7} +3.00000 q^{9} -2.00000 q^{11} -5.65685 q^{13} -4.89898 q^{17} -6.00000 q^{19} +3.46410 q^{21} +7.07107 q^{23} -6.92820 q^{29} +6.92820 q^{31} -4.89898 q^{33} +2.82843 q^{37} -13.8564 q^{39} +4.00000 q^{41} -2.44949 q^{43} -4.24264 q^{47} -5.00000 q^{49} -12.0000 q^{51} -14.6969 q^{57} -2.00000 q^{59} -3.46410 q^{61} +4.24264 q^{63} +2.44949 q^{67} +17.3205 q^{69} +6.92820 q^{71} -4.89898 q^{73} -2.82843 q^{77} +6.92820 q^{79} -9.00000 q^{81} -12.2474 q^{83} -16.9706 q^{87} -2.00000 q^{89} -8.00000 q^{91} +16.9706 q^{93} -14.6969 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9} - 8 q^{11} - 24 q^{19} + 16 q^{41} - 20 q^{49} - 48 q^{51} - 8 q^{59} - 36 q^{81} - 8 q^{89} - 32 q^{91} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2.44949 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.41421 0.534522 0.267261 − 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(8\) 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
\(13\) −5.65685 −1.56893 −0.784465 − 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.89898 −1.18818 −0.594089 − 0.804400i \(-0.702487\pi\)
−0.594089 + 0.804400i \(0.702487\pi\)
\(18\) 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
\(23\) 7.07107 1.47442 0.737210 − 0.675664i \(-0.236143\pi\)
0.737210 + 0.675664i \(0.236143\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.92820 −1.28654 −0.643268 − 0.765641i \(-0.722422\pi\)
−0.643268 + 0.765641i \(0.722422\pi\)
\(30\) 0 0
\(31\) 6.92820 1.24434 0.622171 − 0.782881i \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) 0 0
\(33\) −4.89898 −0.852803
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.82843 0.464991 0.232495 − 0.972598i \(-0.425311\pi\)
0.232495 + 0.972598i \(0.425311\pi\)
\(38\) 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
\(43\) −2.44949 −0.373544 −0.186772 − 0.982403i \(-0.559803\pi\)
−0.186772 + 0.982403i \(0.559803\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.24264 −0.618853 −0.309426 − 0.950923i \(-0.600137\pi\)
−0.309426 + 0.950923i \(0.600137\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −14.6969 −1.94666
\(58\) 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
\(63\) 4.24264 0.534522
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.44949 0.299253 0.149626 − 0.988743i \(-0.452193\pi\)
0.149626 + 0.988743i \(0.452193\pi\)
\(68\) 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
\(73\) −4.89898 −0.573382 −0.286691 − 0.958023i \(-0.592555\pi\)
−0.286691 + 0.958023i \(0.592555\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.82843 −0.322329
\(78\) 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
\(83\) −12.2474 −1.34433 −0.672166 − 0.740400i \(-0.734636\pi\)
−0.672166 + 0.740400i \(0.734636\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −16.9706 −1.81944
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 16.9706 1.75977
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.6969 −1.49225 −0.746124 − 0.665807i \(-0.768087\pi\)
−0.746124 + 0.665807i \(0.768087\pi\)
\(98\) 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 6400.2.a.ct.1.4 4
4.3 odd 2 6400.2.a.cu.1.1 4
5.2 odd 4 1280.2.c.g.769.2 4
5.3 odd 4 1280.2.c.g.769.4 4
5.4 even 2 inner 6400.2.a.ct.1.1 4
8.3 odd 2 inner 6400.2.a.ct.1.3 4
8.5 even 2 6400.2.a.cu.1.2 4
16.3 odd 4 1600.2.d.i.801.3 8
16.5 even 4 1600.2.d.i.801.1 8
16.11 odd 4 1600.2.d.i.801.8 8
16.13 even 4 1600.2.d.i.801.6 8
20.3 even 4 1280.2.c.h.769.2 4
20.7 even 4 1280.2.c.h.769.4 4
20.19 odd 2 6400.2.a.cu.1.4 4
40.3 even 4 1280.2.c.g.769.3 4
40.13 odd 4 1280.2.c.h.769.1 4
40.19 odd 2 inner 6400.2.a.ct.1.2 4
40.27 even 4 1280.2.c.g.769.1 4
40.29 even 2 6400.2.a.cu.1.3 4
40.37 odd 4 1280.2.c.h.769.3 4
80.3 even 4 320.2.f.b.289.5 yes 8
80.13 odd 4 320.2.f.b.289.1 ✓ 8
80.19 odd 4 1600.2.d.i.801.5 8
80.27 even 4 320.2.f.b.289.6 yes 8
80.29 even 4 1600.2.d.i.801.4 8
80.37 odd 4 320.2.f.b.289.2 yes 8
80.43 even 4 320.2.f.b.289.4 yes 8
80.53 odd 4 320.2.f.b.289.8 yes 8
80.59 odd 4 1600.2.d.i.801.2 8
80.67 even 4 320.2.f.b.289.3 yes 8
80.69 even 4 1600.2.d.i.801.7 8
80.77 odd 4 320.2.f.b.289.7 yes 8
240.53 even 4 2880.2.d.g.289.2 8
240.77 even 4 2880.2.d.g.289.3 8
240.83 odd 4 2880.2.d.g.289.7 8
240.107 odd 4 2880.2.d.g.289.6 8
240.173 even 4 2880.2.d.g.289.8 8
240.197 even 4 2880.2.d.g.289.5 8
240.203 odd 4 2880.2.d.g.289.1 8
240.227 odd 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 80.13 odd 4
320.2.f.b.289.2 yes 8 80.37 odd 4
320.2.f.b.289.3 yes 8 80.67 even 4
320.2.f.b.289.4 yes 8 80.43 even 4
320.2.f.b.289.5 yes 8 80.3 even 4
320.2.f.b.289.6 yes 8 80.27 even 4
320.2.f.b.289.7 yes 8 80.77 odd 4
320.2.f.b.289.8 yes 8 80.53 odd 4
1280.2.c.g.769.1 4 40.27 even 4
1280.2.c.g.769.2 4 5.2 odd 4
1280.2.c.g.769.3 4 40.3 even 4
1280.2.c.g.769.4 4 5.3 odd 4
1280.2.c.h.769.1 4 40.13 odd 4
1280.2.c.h.769.2 4 20.3 even 4
1280.2.c.h.769.3 4 40.37 odd 4
1280.2.c.h.769.4 4 20.7 even 4
1600.2.d.i.801.1 8 16.5 even 4
1600.2.d.i.801.2 8 80.59 odd 4
1600.2.d.i.801.3 8 16.3 odd 4
1600.2.d.i.801.4 8 80.29 even 4
1600.2.d.i.801.5 8 80.19 odd 4
1600.2.d.i.801.6 8 16.13 even 4
1600.2.d.i.801.7 8 80.69 even 4
1600.2.d.i.801.8 8 16.11 odd 4
2880.2.d.g.289.1 8 240.203 odd 4
2880.2.d.g.289.2 8 240.53 even 4
2880.2.d.g.289.3 8 240.77 even 4
2880.2.d.g.289.4 8 240.227 odd 4
2880.2.d.g.289.5 8 240.197 even 4
2880.2.d.g.289.6 8 240.107 odd 4
2880.2.d.g.289.7 8 240.83 odd 4
2880.2.d.g.289.8 8 240.173 even 4
6400.2.a.ct.1.1 4 5.4 even 2 inner
6400.2.a.ct.1.2 4 40.19 odd 2 inner
6400.2.a.ct.1.3 4 8.3 odd 2 inner
6400.2.a.ct.1.4 4 1.1 even 1 trivial
6400.2.a.cu.1.1 4 4.3 odd 2
6400.2.a.cu.1.2 4 8.5 even 2
6400.2.a.cu.1.3 4 40.29 even 2
6400.2.a.cu.1.4 4 20.19 odd 2