Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1776,2,Mod(1,1776)] 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("1776.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1776, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1776.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,-4,0,4,0,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.1814313990\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 888)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1776.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.585786 q^{5} +4.82843 q^{7} +1.00000 q^{9} -2.82843 q^{11} -4.82843 q^{13} +0.585786 q^{15} -3.41421 q^{17} -5.65685 q^{19} -4.82843 q^{21} +5.41421 q^{23} -4.65685 q^{25} -1.00000 q^{27} -1.75736 q^{29} +6.82843 q^{31} +2.82843 q^{33} -2.82843 q^{35} -1.00000 q^{37} +4.82843 q^{39} -11.6569 q^{41} +2.82843 q^{43} -0.585786 q^{45} -5.65685 q^{47} +16.3137 q^{49} +3.41421 q^{51} -12.8284 q^{53} +1.65685 q^{55} +5.65685 q^{57} +4.24264 q^{59} +13.3137 q^{61} +4.82843 q^{63} +2.82843 q^{65} -4.82843 q^{67} -5.41421 q^{69} -8.00000 q^{71} +8.00000 q^{73} +4.65685 q^{75} -13.6569 q^{77} -13.6569 q^{79} +1.00000 q^{81} +0.485281 q^{83} +2.00000 q^{85} +1.75736 q^{87} -13.0711 q^{89} -23.3137 q^{91} -6.82843 q^{93} +3.31371 q^{95} -12.1421 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{13} + 4 q^{15} - 4 q^{17} - 4 q^{21} + 8 q^{23} + 2 q^{25} - 2 q^{27} - 12 q^{29} + 8 q^{31} - 2 q^{37} + 4 q^{39} - 12 q^{41} - 4 q^{45} + 10 q^{49} + 4 q^{51}+ \cdots + 4 q^{97}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −0.585786 −0.261972 −0.130986 − 0.991384i \(-0.541814\pi\)
−0.130986 + 0.991384i \(0.541814\pi\)
\(6\) 0 0
\(7\) 4.82843 1.82497 0.912487 − 0.409106i \(-0.134159\pi\)
0.912487 + 0.409106i \(0.134159\pi\)
\(8\) 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
\(13\) −4.82843 −1.33916 −0.669582 − 0.742738i \(-0.733527\pi\)
−0.669582 + 0.742738i \(0.733527\pi\)
\(14\) 0 0
\(15\) 0.585786 0.151249
\(16\) 0 0
\(17\) −3.41421 −0.828068 −0.414034 − 0.910261i \(-0.635881\pi\)
−0.414034 + 0.910261i \(0.635881\pi\)
\(18\) 0 0
\(19\) −5.65685 −1.29777 −0.648886 − 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 0 0
\(21\) −4.82843 −1.05365
\(22\) 0 0
\(23\) 5.41421 1.12894 0.564471 − 0.825453i \(-0.309080\pi\)
0.564471 + 0.825453i \(0.309080\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.75736 −0.326333 −0.163167 − 0.986599i \(-0.552171\pi\)
−0.163167 + 0.986599i \(0.552171\pi\)
\(30\) 0 0
\(31\) 6.82843 1.22642 0.613211 − 0.789919i \(-0.289878\pi\)
0.613211 + 0.789919i \(0.289878\pi\)
\(32\) 0 0
\(33\) 2.82843 0.492366
\(34\) 0 0
\(35\) −2.82843 −0.478091
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 4.82843 0.773167
\(40\) 0 0
\(41\) −11.6569 −1.82049 −0.910247 − 0.414065i \(-0.864109\pi\)
−0.910247 + 0.414065i \(0.864109\pi\)
\(42\) 0 0
\(43\) 2.82843 0.431331 0.215666 − 0.976467i \(-0.430808\pi\)
0.215666 + 0.976467i \(0.430808\pi\)
\(44\) 0 0
\(45\) −0.585786 −0.0873239
\(46\) 0 0
\(47\) −5.65685 −0.825137 −0.412568 − 0.910927i \(-0.635368\pi\)
−0.412568 + 0.910927i \(0.635368\pi\)
\(48\) 0 0
\(49\) 16.3137 2.33053
\(50\) 0 0
\(51\) 3.41421 0.478086
\(52\) 0 0
\(53\) −12.8284 −1.76212 −0.881060 − 0.473005i \(-0.843169\pi\)
−0.881060 + 0.473005i \(0.843169\pi\)
\(54\) 0 0
\(55\) 1.65685 0.223410
\(56\) 0 0
\(57\) 5.65685 0.749269
\(58\) 0 0
\(59\) 4.24264 0.552345 0.276172 − 0.961108i \(-0.410934\pi\)
0.276172 + 0.961108i \(0.410934\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 − 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0 0
\(63\) 4.82843 0.608325
\(64\) 0 0
\(65\) 2.82843 0.350823
\(66\) 0 0
\(67\) −4.82843 −0.589886 −0.294943 − 0.955515i \(-0.595301\pi\)
−0.294943 + 0.955515i \(0.595301\pi\)
\(68\) 0 0
\(69\) −5.41421 −0.651795
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 4.65685 0.537727
\(76\) 0 0
\(77\) −13.6569 −1.55634
\(78\) 0 0
\(79\) −13.6569 −1.53652 −0.768258 − 0.640140i \(-0.778876\pi\)
−0.768258 + 0.640140i \(0.778876\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.485281 0.0532666 0.0266333 − 0.999645i \(-0.491521\pi\)
0.0266333 + 0.999645i \(0.491521\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 1.75736 0.188409
\(88\) 0 0
\(89\) −13.0711 −1.38553 −0.692765 − 0.721163i \(-0.743608\pi\)
−0.692765 + 0.721163i \(0.743608\pi\)
\(90\) 0 0
\(91\) −23.3137 −2.44394
\(92\) 0 0
\(93\) −6.82843 −0.708075
\(94\) 0 0
\(95\) 3.31371 0.339979
\(96\) 0 0
\(97\) −12.1421 −1.23285 −0.616424 − 0.787415i \(-0.711419\pi\)
−0.616424 + 0.787415i \(0.711419\pi\)
\(98\) 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 1776.2.a.l.1.2 2
3.2 odd 2 5328.2.a.bi.1.1 2
4.3 odd 2 888.2.a.g.1.2 ✓ 2
8.3 odd 2 7104.2.a.bh.1.1 2
8.5 even 2 7104.2.a.bn.1.1 2
12.11 even 2 2664.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.a.g.1.2 ✓ 2 4.3 odd 2
1776.2.a.l.1.2 2 1.1 even 1 trivial
2664.2.a.l.1.1 2 12.11 even 2
5328.2.a.bi.1.1 2 3.2 odd 2
7104.2.a.bh.1.1 2 8.3 odd 2
7104.2.a.bn.1.1 2 8.5 even 2