Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3200,2,Mod(1601,3200)] 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("3200.1601"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(33)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1601.2
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 3200.1601
Dual form 3200.2.d.p.1601.3

$q$-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)
 
\(f(q)\) \(=\) \(q-2.23607i q^{3} -2.00000 q^{9} -2.23607i q^{11} +4.00000i q^{13} +3.00000 q^{17} +2.23607i q^{19} -8.94427 q^{23} -2.23607i q^{27} +4.00000i q^{29} -8.94427 q^{31} -5.00000 q^{33} -8.00000i q^{37} +8.94427 q^{39} -5.00000 q^{41} -8.94427i q^{43} -8.94427 q^{47} -7.00000 q^{49} -6.70820i q^{51} -4.00000i q^{53} +5.00000 q^{57} +8.94427i q^{59} +8.00000i q^{61} -6.70820i q^{67} +20.0000i q^{69} +8.94427 q^{71} -9.00000 q^{73} -11.0000 q^{81} -6.70820i q^{83} +8.94427 q^{87} -15.0000 q^{89} +20.0000i q^{93} +2.00000 q^{97} +4.47214i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9} + 12 q^{17} - 20 q^{33} - 20 q^{41} - 28 q^{49} + 20 q^{57} - 36 q^{73} - 44 q^{81} - 60 q^{89} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 2.23607i − 1.29099i −0.763763 − 0.645497i \(-0.776650\pi\)
0.763763 − 0.645497i \(-0.223350\pi\)
\(4\) 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
\(11\) − 2.23607i − 0.674200i −0.941469 − 0.337100i \(-0.890554\pi\)
0.941469 − 0.337100i \(-0.109446\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 2.23607i 0.512989i 0.966546 + 0.256495i \(0.0825676\pi\)
−0.966546 + 0.256495i \(0.917432\pi\)
\(20\) 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
\(27\) − 2.23607i − 0.430331i
\(28\) 0 0
\(29\) 4.00000i 0.742781i 0.928477 + 0.371391i \(0.121119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 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
\(37\) − 8.00000i − 1.31519i −0.753371 − 0.657596i \(-0.771573\pi\)
0.753371 − 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 8.94427 1.43223
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 − 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) − 8.94427i − 1.36399i −0.731357 − 0.681994i \(-0.761113\pi\)
0.731357 − 0.681994i \(-0.238887\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.94427 −1.30466 −0.652328 − 0.757937i \(-0.726208\pi\)
−0.652328 + 0.757937i \(0.726208\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) − 6.70820i − 0.939336i
\(52\) 0 0
\(53\) − 4.00000i − 0.549442i −0.961524 − 0.274721i \(-0.911414\pi\)
0.961524 − 0.274721i \(-0.0885855\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.00000 0.662266
\(58\) 0 0
\(59\) 8.94427i 1.16445i 0.813029 + 0.582223i \(0.197817\pi\)
−0.813029 + 0.582223i \(0.802183\pi\)
\(60\) 0 0
\(61\) 8.00000i 1.02430i 0.858898 + 0.512148i \(0.171150\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 6.70820i − 0.819538i −0.912189 − 0.409769i \(-0.865609\pi\)
0.912189 − 0.409769i \(-0.134391\pi\)
\(68\) 0 0
\(69\) 20.0000i 2.40772i
\(70\) 0 0
\(71\) 8.94427 1.06149 0.530745 − 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) 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
\(83\) − 6.70820i − 0.736321i −0.929762 − 0.368161i \(-0.879988\pi\)
0.929762 − 0.368161i \(-0.120012\pi\)
\(84\) 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 \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 20.0000i 2.07390i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 4.47214i 0.449467i
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 3200.2.d.p.1601.2 yes 4
4.3 odd 2 inner 3200.2.d.p.1601.4 yes 4
5.2 odd 4 3200.2.f.n.449.1 4
5.3 odd 4 3200.2.f.m.449.3 4
5.4 even 2 3200.2.d.o.1601.3 yes 4
8.3 odd 2 inner 3200.2.d.p.1601.1 yes 4
8.5 even 2 inner 3200.2.d.p.1601.3 yes 4
16.3 odd 4 6400.2.a.br.1.1 2
16.5 even 4 6400.2.a.bt.1.1 2
16.11 odd 4 6400.2.a.bt.1.2 2
16.13 even 4 6400.2.a.br.1.2 2
20.3 even 4 3200.2.f.m.449.2 4
20.7 even 4 3200.2.f.n.449.4 4
20.19 odd 2 3200.2.d.o.1601.1 ✓ 4
40.3 even 4 3200.2.f.n.449.3 4
40.13 odd 4 3200.2.f.n.449.2 4
40.19 odd 2 3200.2.d.o.1601.4 yes 4
40.27 even 4 3200.2.f.m.449.1 4
40.29 even 2 3200.2.d.o.1601.2 yes 4
40.37 odd 4 3200.2.f.m.449.4 4
80.19 odd 4 6400.2.a.bs.1.2 2
80.29 even 4 6400.2.a.bs.1.1 2
80.59 odd 4 6400.2.a.bq.1.1 2
80.69 even 4 6400.2.a.bq.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3200.2.d.o.1601.1 ✓ 4 20.19 odd 2
3200.2.d.o.1601.2 yes 4 40.29 even 2
3200.2.d.o.1601.3 yes 4 5.4 even 2
3200.2.d.o.1601.4 yes 4 40.19 odd 2
3200.2.d.p.1601.1 yes 4 8.3 odd 2 inner
3200.2.d.p.1601.2 yes 4 1.1 even 1 trivial
3200.2.d.p.1601.3 yes 4 8.5 even 2 inner
3200.2.d.p.1601.4 yes 4 4.3 odd 2 inner
3200.2.f.m.449.1 4 40.27 even 4
3200.2.f.m.449.2 4 20.3 even 4
3200.2.f.m.449.3 4 5.3 odd 4
3200.2.f.m.449.4 4 40.37 odd 4
3200.2.f.n.449.1 4 5.2 odd 4
3200.2.f.n.449.2 4 40.13 odd 4
3200.2.f.n.449.3 4 40.3 even 4
3200.2.f.n.449.4 4 20.7 even 4
6400.2.a.bq.1.1 2 80.59 odd 4
6400.2.a.bq.1.2 2 80.69 even 4
6400.2.a.br.1.1 2 16.3 odd 4
6400.2.a.br.1.2 2 16.13 even 4
6400.2.a.bs.1.1 2 80.29 even 4
6400.2.a.bs.1.2 2 80.19 odd 4
6400.2.a.bt.1.1 2 16.5 even 4
6400.2.a.bt.1.2 2 16.11 odd 4