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 = [2,0,0,0,0,0,0,0,-2,0,0,0,-12,0,0,0,12,0,0,0,12,0,0,0,0,0,0,0, 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 640)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) 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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{3} +4.24264 q^{7} -1.00000 q^{9} -2.82843 q^{11} -6.00000 q^{13} +6.00000 q^{17} +6.00000 q^{21} +4.24264 q^{23} -5.65685 q^{27} +8.48528 q^{31} -4.00000 q^{33} -6.00000 q^{37} -8.48528 q^{39} +4.24264 q^{43} +4.24264 q^{47} +11.0000 q^{49} +8.48528 q^{51} -6.00000 q^{53} +11.3137 q^{59} +6.00000 q^{61} -4.24264 q^{63} +12.7279 q^{67} +6.00000 q^{69} +8.48528 q^{71} -2.00000 q^{73} -12.0000 q^{77} -5.00000 q^{81} +9.89949 q^{83} +6.00000 q^{89} -25.4558 q^{91} +12.0000 q^{93} +10.0000 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} - 12 q^{13} + 12 q^{17} + 12 q^{21} - 8 q^{33} - 12 q^{37} + 22 q^{49} - 12 q^{53} + 12 q^{61} + 12 q^{69} - 4 q^{73} - 24 q^{77} - 10 q^{81} + 12 q^{89} + 24 q^{93} + 20 q^{97}+O(q^{100}) \) 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
\(3\) 1.41421 0.816497 0.408248 − 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.24264 1.60357 0.801784 − 0.597614i \(-0.203885\pi\)
0.801784 + 0.597614i \(0.203885\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.82843 −0.852803 −0.426401 − 0.904534i \(-0.640219\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) 4.24264 0.884652 0.442326 − 0.896854i \(-0.354153\pi\)
0.442326 + 0.896854i \(0.354153\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 8.48528 1.52400 0.762001 − 0.647576i \(-0.224217\pi\)
0.762001 + 0.647576i \(0.224217\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −8.48528 −1.35873
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 4.24264 0.646997 0.323498 − 0.946229i \(-0.395141\pi\)
0.323498 + 0.946229i \(0.395141\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.24264 0.618853 0.309426 − 0.950923i \(-0.399863\pi\)
0.309426 + 0.950923i \(0.399863\pi\)
\(48\) 0 0
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) 8.48528 1.18818
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.3137 1.47292 0.736460 − 0.676481i \(-0.236496\pi\)
0.736460 + 0.676481i \(0.236496\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) −4.24264 −0.534522
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.7279 1.55496 0.777482 − 0.628906i \(-0.216497\pi\)
0.777482 + 0.628906i \(0.216497\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 8.48528 1.00702 0.503509 − 0.863990i \(-0.332042\pi\)
0.503509 + 0.863990i \(0.332042\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 9.89949 1.08661 0.543305 − 0.839535i \(-0.317173\pi\)
0.543305 + 0.839535i \(0.317173\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −25.4558 −2.66850
\(92\) 0 0
\(93\) 12.0000 1.24434
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
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.bk.1.2 2
4.3 odd 2 inner 6400.2.a.bk.1.1 2
5.4 even 2 1280.2.a.g.1.1 2
8.3 odd 2 6400.2.a.bp.1.2 2
8.5 even 2 6400.2.a.bp.1.1 2
16.3 odd 4 3200.2.d.v.1601.2 4
16.5 even 4 3200.2.d.v.1601.1 4
16.11 odd 4 3200.2.d.v.1601.4 4
16.13 even 4 3200.2.d.v.1601.3 4
20.19 odd 2 1280.2.a.g.1.2 2
40.19 odd 2 1280.2.a.i.1.1 2
40.29 even 2 1280.2.a.i.1.2 2
80.3 even 4 3200.2.f.l.449.3 4
80.13 odd 4 3200.2.f.l.449.2 4
80.19 odd 4 640.2.d.b.321.4 yes 4
80.27 even 4 3200.2.f.l.449.4 4
80.29 even 4 640.2.d.b.321.2 yes 4
80.37 odd 4 3200.2.f.l.449.1 4
80.43 even 4 3200.2.f.g.449.1 4
80.53 odd 4 3200.2.f.g.449.4 4
80.59 odd 4 640.2.d.b.321.1 ✓ 4
80.67 even 4 3200.2.f.g.449.2 4
80.69 even 4 640.2.d.b.321.3 yes 4
80.77 odd 4 3200.2.f.g.449.3 4
240.29 odd 4 5760.2.k.v.2881.2 4
240.59 even 4 5760.2.k.v.2881.3 4
240.149 odd 4 5760.2.k.v.2881.4 4
240.179 even 4 5760.2.k.v.2881.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.d.b.321.1 ✓ 4 80.59 odd 4
640.2.d.b.321.2 yes 4 80.29 even 4
640.2.d.b.321.3 yes 4 80.69 even 4
640.2.d.b.321.4 yes 4 80.19 odd 4
1280.2.a.g.1.1 2 5.4 even 2
1280.2.a.g.1.2 2 20.19 odd 2
1280.2.a.i.1.1 2 40.19 odd 2
1280.2.a.i.1.2 2 40.29 even 2
3200.2.d.v.1601.1 4 16.5 even 4
3200.2.d.v.1601.2 4 16.3 odd 4
3200.2.d.v.1601.3 4 16.13 even 4
3200.2.d.v.1601.4 4 16.11 odd 4
3200.2.f.g.449.1 4 80.43 even 4
3200.2.f.g.449.2 4 80.67 even 4
3200.2.f.g.449.3 4 80.77 odd 4
3200.2.f.g.449.4 4 80.53 odd 4
3200.2.f.l.449.1 4 80.37 odd 4
3200.2.f.l.449.2 4 80.13 odd 4
3200.2.f.l.449.3 4 80.3 even 4
3200.2.f.l.449.4 4 80.27 even 4
5760.2.k.v.2881.1 4 240.179 even 4
5760.2.k.v.2881.2 4 240.29 odd 4
5760.2.k.v.2881.3 4 240.59 even 4
5760.2.k.v.2881.4 4 240.149 odd 4
6400.2.a.bk.1.1 2 4.3 odd 2 inner
6400.2.a.bk.1.2 2 1.1 even 1 trivial
6400.2.a.bp.1.1 2 8.5 even 2
6400.2.a.bp.1.2 2 8.3 odd 2