Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [256,9,Mod(127,256)] 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("256.127"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 256.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,144,0,0,0,0,0,18876] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(104.288924176\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{39})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 19x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.2
Root \(-3.12250 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 256.127
Dual form 256.9.d.h.127.1

$q$-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)
 
\(f(q)\) \(=\) \(q-63.9200 q^{3} +217.680i q^{5} +1726.24i q^{7} -2475.24 q^{9} +5955.28 q^{11} +49121.2i q^{13} -13914.1i q^{15} -55631.9 q^{17} -202659. q^{19} -110341. i q^{21} +209565. i q^{23} +343240. q^{25} +577596. q^{27} -3082.26i q^{29} +1.36035e6i q^{31} -380661. q^{33} -375768. q^{35} +1.31420e6i q^{37} -3.13982e6i q^{39} -4.54489e6 q^{41} +3.48760e6 q^{43} -538809. i q^{45} +7.23361e6i q^{47} +2.78490e6 q^{49} +3.55599e6 q^{51} -2.48557e6i q^{53} +1.29634e6i q^{55} +1.29539e7 q^{57} +1.48790e7 q^{59} -4.13648e6i q^{61} -4.27285e6i q^{63} -1.06927e7 q^{65} -6.04481e6 q^{67} -1.33954e7i q^{69} +4.20242e7i q^{71} -9.00500e6 q^{73} -2.19399e7 q^{75} +1.02802e7i q^{77} -6.81741e6i q^{79} -2.06799e7 q^{81} +5.81780e7 q^{83} -1.21100e7i q^{85} +197018. i q^{87} -7.82499e7 q^{89} -8.47949e7 q^{91} -8.69535e7i q^{93} -4.41147e7i q^{95} +6.01423e7 q^{97} -1.47407e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 144 q^{3} + 18876 q^{9} + 40208 q^{11} - 391992 q^{17} - 664752 q^{19} + 791028 q^{25} + 2610144 q^{27} + 3084864 q^{33} - 3857984 q^{35} - 2953352 q^{41} - 3046768 q^{43} + 2839044 q^{49} - 31044576 q^{51}+ \cdots + 307632624 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(-1\)

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
\(3\) −63.9200 −0.789135 −0.394568 − 0.918867i \(-0.629106\pi\)
−0.394568 + 0.918867i \(0.629106\pi\)
\(4\) 0 0
\(5\) 217.680i 0.348288i 0.984720 + 0.174144i \(0.0557158\pi\)
−0.984720 + 0.174144i \(0.944284\pi\)
\(6\) 0 0
\(7\) 1726.24i 0.718967i 0.933151 + 0.359483i \(0.117047\pi\)
−0.933151 + 0.359483i \(0.882953\pi\)
\(8\) 0 0
\(9\) −2475.24 −0.377265
\(10\) 0 0
\(11\) 5955.28 0.406754 0.203377 − 0.979101i \(-0.434808\pi\)
0.203377 + 0.979101i \(0.434808\pi\)
\(12\) 0 0
\(13\) 49121.2i 1.71987i 0.510404 + 0.859935i \(0.329496\pi\)
−0.510404 + 0.859935i \(0.670504\pi\)
\(14\) 0 0
\(15\) − 13914.1i − 0.274846i
\(16\) 0 0
\(17\) −55631.9 −0.666083 −0.333042 − 0.942912i \(-0.608075\pi\)
−0.333042 + 0.942912i \(0.608075\pi\)
\(18\) 0 0
\(19\) −202659. −1.55507 −0.777537 − 0.628837i \(-0.783531\pi\)
−0.777537 + 0.628837i \(0.783531\pi\)
\(20\) 0 0
\(21\) − 110341.i − 0.567362i
\(22\) 0 0
\(23\) 209565.i 0.748872i 0.927253 + 0.374436i \(0.122164\pi\)
−0.927253 + 0.374436i \(0.877836\pi\)
\(24\) 0 0
\(25\) 343240. 0.878696
\(26\) 0 0
\(27\) 577596. 1.08685
\(28\) 0 0
\(29\) − 3082.26i − 0.00435790i −0.999998 − 0.00217895i \(-0.999306\pi\)
0.999998 − 0.00217895i \(-0.000693582\pi\)
\(30\) 0 0
\(31\) 1.36035e6i 1.47300i 0.676436 + 0.736502i \(0.263524\pi\)
−0.676436 + 0.736502i \(0.736476\pi\)
\(32\) 0 0
\(33\) −380661. −0.320984
\(34\) 0 0
\(35\) −375768. −0.250407
\(36\) 0 0
\(37\) 1.31420e6i 0.701220i 0.936522 + 0.350610i \(0.114026\pi\)
−0.936522 + 0.350610i \(0.885974\pi\)
\(38\) 0 0
\(39\) − 3.13982e6i − 1.35721i
\(40\) 0 0
\(41\) −4.54489e6 −1.60838 −0.804189 − 0.594374i \(-0.797400\pi\)
−0.804189 + 0.594374i \(0.797400\pi\)
\(42\) 0 0
\(43\) 3.48760e6 1.02013 0.510063 − 0.860137i \(-0.329622\pi\)
0.510063 + 0.860137i \(0.329622\pi\)
\(44\) 0 0
\(45\) − 538809.i − 0.131397i
\(46\) 0 0
\(47\) 7.23361e6i 1.48239i 0.671288 + 0.741197i \(0.265741\pi\)
−0.671288 + 0.741197i \(0.734259\pi\)
\(48\) 0 0
\(49\) 2.78490e6 0.483087
\(50\) 0 0
\(51\) 3.55599e6 0.525630
\(52\) 0 0
\(53\) − 2.48557e6i − 0.315009i −0.987518 − 0.157505i \(-0.949655\pi\)
0.987518 − 0.157505i \(-0.0503449\pi\)
\(54\) 0 0
\(55\) 1.29634e6i 0.141667i
\(56\) 0 0
\(57\) 1.29539e7 1.22716
\(58\) 0 0
\(59\) 1.48790e7 1.22791 0.613956 − 0.789340i \(-0.289577\pi\)
0.613956 + 0.789340i \(0.289577\pi\)
\(60\) 0 0
\(61\) − 4.13648e6i − 0.298752i −0.988780 − 0.149376i \(-0.952273\pi\)
0.988780 − 0.149376i \(-0.0477266\pi\)
\(62\) 0 0
\(63\) − 4.27285e6i − 0.271241i
\(64\) 0 0
\(65\) −1.06927e7 −0.599009
\(66\) 0 0
\(67\) −6.04481e6 −0.299974 −0.149987 − 0.988688i \(-0.547923\pi\)
−0.149987 + 0.988688i \(0.547923\pi\)
\(68\) 0 0
\(69\) − 1.33954e7i − 0.590962i
\(70\) 0 0
\(71\) 4.20242e7i 1.65374i 0.562397 + 0.826868i \(0.309879\pi\)
−0.562397 + 0.826868i \(0.690121\pi\)
\(72\) 0 0
\(73\) −9.00500e6 −0.317097 −0.158548 − 0.987351i \(-0.550681\pi\)
−0.158548 + 0.987351i \(0.550681\pi\)
\(74\) 0 0
\(75\) −2.19399e7 −0.693410
\(76\) 0 0
\(77\) 1.02802e7i 0.292442i
\(78\) 0 0
\(79\) − 6.81741e6i − 0.175029i −0.996163 − 0.0875147i \(-0.972108\pi\)
0.996163 − 0.0875147i \(-0.0278925\pi\)
\(80\) 0 0
\(81\) −2.06799e7 −0.480406
\(82\) 0 0
\(83\) 5.81780e7 1.22587 0.612937 − 0.790132i \(-0.289988\pi\)
0.612937 + 0.790132i \(0.289988\pi\)
\(84\) 0 0
\(85\) − 1.21100e7i − 0.231989i
\(86\) 0 0
\(87\) 197018.i 0.00343898i
\(88\) 0 0
\(89\) −7.82499e7 −1.24716 −0.623582 − 0.781758i \(-0.714323\pi\)
−0.623582 + 0.781758i \(0.714323\pi\)
\(90\) 0 0
\(91\) −8.47949e7 −1.23653
\(92\) 0 0
\(93\) − 8.69535e7i − 1.16240i
\(94\) 0 0
\(95\) − 4.41147e7i − 0.541613i
\(96\) 0 0
\(97\) 6.01423e7 0.679350 0.339675 − 0.940543i \(-0.389683\pi\)
0.339675 + 0.940543i \(0.389683\pi\)
\(98\) 0 0
\(99\) −1.47407e7 −0.153454
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 256.9.d.h.127.2 4
4.3 odd 2 256.9.d.b.127.4 4
8.3 odd 2 inner 256.9.d.h.127.1 4
8.5 even 2 256.9.d.b.127.3 4
16.3 odd 4 32.9.c.a.31.3 yes 4
16.5 even 4 64.9.c.f.63.3 4
16.11 odd 4 64.9.c.f.63.2 4
16.13 even 4 32.9.c.a.31.2 ✓ 4
48.29 odd 4 288.9.g.b.127.1 4
48.35 even 4 288.9.g.b.127.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.9.c.a.31.2 ✓ 4 16.13 even 4
32.9.c.a.31.3 yes 4 16.3 odd 4
64.9.c.f.63.2 4 16.11 odd 4
64.9.c.f.63.3 4 16.5 even 4
256.9.d.b.127.3 4 8.5 even 2
256.9.d.b.127.4 4 4.3 odd 2
256.9.d.h.127.1 4 8.3 odd 2 inner
256.9.d.h.127.2 4 1.1 even 1 trivial
288.9.g.b.127.1 4 48.29 odd 4
288.9.g.b.127.2 4 48.35 even 4