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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 127.4
Root \(3.12250 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 256.127
Dual form 256.9.d.b.127.3

$q$-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)
 
\(f(q)\) \(=\) \(q+63.9200 q^{3} +217.680i q^{5} -1726.24i q^{7} -2475.24 q^{9} -5955.28 q^{11} +49121.2i q^{13} +13914.1i q^{15} -55631.9 q^{17} +202659. q^{19} -110341. i q^{21} -209565. i q^{23} +343240. q^{25} -577596. q^{27} -3082.26i q^{29} -1.36035e6i q^{31} -380661. q^{33} +375768. q^{35} +1.31420e6i q^{37} +3.13982e6i q^{39} -4.54489e6 q^{41} -3.48760e6 q^{43} -538809. i q^{45} -7.23361e6i q^{47} +2.78490e6 q^{49} -3.55599e6 q^{51} -2.48557e6i q^{53} -1.29634e6i q^{55} +1.29539e7 q^{57} -1.48790e7 q^{59} -4.13648e6i q^{61} +4.27285e6i q^{63} -1.06927e7 q^{65} +6.04481e6 q^{67} -1.33954e7i q^{69} -4.20242e7i q^{71} -9.00500e6 q^{73} +2.19399e7 q^{75} +1.02802e7i q^{77} +6.81741e6i q^{79} -2.06799e7 q^{81} -5.81780e7 q^{83} -1.21100e7i q^{85} -197018. i q^{87} -7.82499e7 q^{89} +8.47949e7 q^{91} -8.69535e7i q^{93} +4.41147e7i q^{95} +6.01423e7 q^{97} +1.47407e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 144 q^{3} + 18876 q^{9} - 40208 q^{11} - 391992 q^{17} + 664752 q^{19} + 791028 q^{25} - 2610144 q^{27} + 3084864 q^{33} + 3857984 q^{35} - 2953352 q^{41} + 3046768 q^{43} + 2839044 q^{49} + 31044576 q^{51}+ \cdots - 307632624 q^{99}+O(q^{100}) \) 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 \(-0.370894\pi\)
0.394568 + 0.918867i \(0.370894\pi\)
\(4\) 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 \(-0.882953\pi\)
0.933151 − 0.359483i \(-0.117047\pi\)
\(8\) 0 0
\(9\) −2475.24 −0.377265
\(10\) 0 0
\(11\) −5955.28 −0.406754 −0.203377 − 0.979101i \(-0.565192\pi\)
−0.203377 + 0.979101i \(0.565192\pi\)
\(12\) 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 \(-0.216469\pi\)
0.777537 + 0.628837i \(0.216469\pi\)
\(20\) 0 0
\(21\) − 110341.i − 0.567362i
\(22\) 0 0
\(23\) − 209565.i − 0.748872i −0.927253 − 0.374436i \(-0.877836\pi\)
0.927253 − 0.374436i \(-0.122164\pi\)
\(24\) 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 \(-0.736476\pi\)
0.676436 − 0.736502i \(-0.263524\pi\)
\(32\) 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 \(-0.670378\pi\)
−0.510063 + 0.860137i \(0.670378\pi\)
\(44\) 0 0
\(45\) − 538809.i − 0.131397i
\(46\) 0 0
\(47\) − 7.23361e6i − 1.48239i −0.671288 − 0.741197i \(-0.734259\pi\)
0.671288 − 0.741197i \(-0.265741\pi\)
\(48\) 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 \(-0.710423\pi\)
−0.613956 + 0.789340i \(0.710423\pi\)
\(60\) 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 \(-0.452077\pi\)
0.149987 + 0.988688i \(0.452077\pi\)
\(68\) 0 0
\(69\) − 1.33954e7i − 0.590962i
\(70\) 0 0
\(71\) − 4.20242e7i − 1.65374i −0.562397 − 0.826868i \(-0.690121\pi\)
0.562397 − 0.826868i \(-0.309879\pi\)
\(72\) 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 \(0.0278925\pi\)
−0.996163 + 0.0875147i \(0.972108\pi\)
\(80\) 0 0
\(81\) −2.06799e7 −0.480406
\(82\) 0 0
\(83\) −5.81780e7 −1.22587 −0.612937 − 0.790132i \(-0.710012\pi\)
−0.612937 + 0.790132i \(0.710012\pi\)
\(84\) 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.b.127.4 4
4.3 odd 2 256.9.d.h.127.2 4
8.3 odd 2 inner 256.9.d.b.127.3 4
8.5 even 2 256.9.d.h.127.1 4
16.3 odd 4 32.9.c.a.31.2 ✓ 4
16.5 even 4 64.9.c.f.63.2 4
16.11 odd 4 64.9.c.f.63.3 4
16.13 even 4 32.9.c.a.31.3 yes 4
48.29 odd 4 288.9.g.b.127.2 4
48.35 even 4 288.9.g.b.127.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.9.c.a.31.2 ✓ 4 16.3 odd 4
32.9.c.a.31.3 yes 4 16.13 even 4
64.9.c.f.63.2 4 16.5 even 4
64.9.c.f.63.3 4 16.11 odd 4
256.9.d.b.127.3 4 8.3 odd 2 inner
256.9.d.b.127.4 4 1.1 even 1 trivial
256.9.d.h.127.1 4 8.5 even 2
256.9.d.h.127.2 4 4.3 odd 2
288.9.g.b.127.1 4 48.35 even 4
288.9.g.b.127.2 4 48.29 odd 4