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 = [2,0,0,0,0,0,0,0,4,0,0,0,8,0,0,0,-6,0,0,0,0,0,0,0,0,0,0,0,-8, 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: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3200)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) 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)
 
\(f(q)\) \(=\) \(q-2.23607 q^{3} +2.00000 q^{9} -2.23607 q^{11} +4.00000 q^{13} -3.00000 q^{17} -2.23607 q^{19} -8.94427 q^{23} +2.23607 q^{27} -4.00000 q^{29} -8.94427 q^{31} +5.00000 q^{33} +8.00000 q^{37} -8.94427 q^{39} +5.00000 q^{41} +8.94427 q^{43} +8.94427 q^{47} -7.00000 q^{49} +6.70820 q^{51} +4.00000 q^{53} +5.00000 q^{57} +8.94427 q^{59} -8.00000 q^{61} -6.70820 q^{67} +20.0000 q^{69} -8.94427 q^{71} -9.00000 q^{73} -11.0000 q^{81} -6.70820 q^{83} +8.94427 q^{87} +15.0000 q^{89} +20.0000 q^{93} -2.00000 q^{97} -4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9} + 8 q^{13} - 6 q^{17} - 8 q^{29} + 10 q^{33} + 16 q^{37} + 10 q^{41} - 14 q^{49} + 8 q^{53} + 10 q^{57} - 16 q^{61} + 40 q^{69} - 18 q^{73} - 22 q^{81} + 30 q^{89} + 40 q^{93} - 4 q^{97}+O(q^{100}) \) 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
\(3\) −2.23607 −1.29099 −0.645497 − 0.763763i \(-0.723350\pi\)
−0.645497 + 0.763763i \(0.723350\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −2.23607 −0.674200 −0.337100 − 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −2.23607 −0.512989 −0.256495 − 0.966546i \(-0.582568\pi\)
−0.256495 + 0.966546i \(0.582568\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.94427 −1.86501 −0.932505 − 0.361158i \(-0.882382\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.23607 0.430331
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −8.94427 −1.60644 −0.803219 − 0.595683i \(-0.796881\pi\)
−0.803219 + 0.595683i \(0.796881\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) −8.94427 −1.43223
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 − 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 8.94427 1.36399 0.681994 − 0.731357i \(-0.261113\pi\)
0.681994 + 0.731357i \(0.261113\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.94427 1.30466 0.652328 − 0.757937i \(-0.273792\pi\)
0.652328 + 0.757937i \(0.273792\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 6.70820 0.939336
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.00000 0.662266
\(58\) 0 0
\(59\) 8.94427 1.16445 0.582223 − 0.813029i \(-0.302183\pi\)
0.582223 + 0.813029i \(0.302183\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.70820 −0.819538 −0.409769 − 0.912189i \(-0.634391\pi\)
−0.409769 + 0.912189i \(0.634391\pi\)
\(68\) 0 0
\(69\) 20.0000 2.40772
\(70\) 0 0
\(71\) −8.94427 −1.06149 −0.530745 − 0.847532i \(-0.678088\pi\)
−0.530745 + 0.847532i \(0.678088\pi\)
\(72\) 0 0
\(73\) −9.00000 −1.05337 −0.526685 − 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −6.70820 −0.736321 −0.368161 − 0.929762i \(-0.620012\pi\)
−0.368161 + 0.929762i \(0.620012\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.94427 0.958927
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 − 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 20.0000 2.07390
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −4.47214 −0.449467
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.bs.1.1 2
4.3 odd 2 inner 6400.2.a.bs.1.2 2
5.4 even 2 6400.2.a.br.1.2 2
8.3 odd 2 6400.2.a.bq.1.1 2
8.5 even 2 6400.2.a.bq.1.2 2
16.3 odd 4 3200.2.d.o.1601.4 yes 4
16.5 even 4 3200.2.d.o.1601.3 yes 4
16.11 odd 4 3200.2.d.o.1601.1 ✓ 4
16.13 even 4 3200.2.d.o.1601.2 yes 4
20.19 odd 2 6400.2.a.br.1.1 2
40.19 odd 2 6400.2.a.bt.1.2 2
40.29 even 2 6400.2.a.bt.1.1 2
80.3 even 4 3200.2.f.m.449.1 4
80.13 odd 4 3200.2.f.m.449.4 4
80.19 odd 4 3200.2.d.p.1601.1 yes 4
80.27 even 4 3200.2.f.m.449.2 4
80.29 even 4 3200.2.d.p.1601.3 yes 4
80.37 odd 4 3200.2.f.m.449.3 4
80.43 even 4 3200.2.f.n.449.4 4
80.53 odd 4 3200.2.f.n.449.1 4
80.59 odd 4 3200.2.d.p.1601.4 yes 4
80.67 even 4 3200.2.f.n.449.3 4
80.69 even 4 3200.2.d.p.1601.2 yes 4
80.77 odd 4 3200.2.f.n.449.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3200.2.d.o.1601.1 ✓ 4 16.11 odd 4
3200.2.d.o.1601.2 yes 4 16.13 even 4
3200.2.d.o.1601.3 yes 4 16.5 even 4
3200.2.d.o.1601.4 yes 4 16.3 odd 4
3200.2.d.p.1601.1 yes 4 80.19 odd 4
3200.2.d.p.1601.2 yes 4 80.69 even 4
3200.2.d.p.1601.3 yes 4 80.29 even 4
3200.2.d.p.1601.4 yes 4 80.59 odd 4
3200.2.f.m.449.1 4 80.3 even 4
3200.2.f.m.449.2 4 80.27 even 4
3200.2.f.m.449.3 4 80.37 odd 4
3200.2.f.m.449.4 4 80.13 odd 4
3200.2.f.n.449.1 4 80.53 odd 4
3200.2.f.n.449.2 4 80.77 odd 4
3200.2.f.n.449.3 4 80.67 even 4
3200.2.f.n.449.4 4 80.43 even 4
6400.2.a.bq.1.1 2 8.3 odd 2
6400.2.a.bq.1.2 2 8.5 even 2
6400.2.a.br.1.1 2 20.19 odd 2
6400.2.a.br.1.2 2 5.4 even 2
6400.2.a.bs.1.1 2 1.1 even 1 trivial
6400.2.a.bs.1.2 2 4.3 odd 2 inner
6400.2.a.bt.1.1 2 40.29 even 2
6400.2.a.bt.1.2 2 40.19 odd 2