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.b.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)
 
\(f(q)\) \(=\) \(q-135.920 q^{3} +581.680i q^{5} -2670.24i q^{7} +11913.2 q^{9} -14148.7 q^{11} +42805.2i q^{13} -79061.9i q^{15} -140364. q^{17} +129717. q^{19} +362939. i q^{21} -196909. i q^{23} +52273.5 q^{25} -727476. q^{27} -355830. i q^{29} -3357.86i q^{31} +1.92309e6 q^{33} +1.55322e6 q^{35} -907325. i q^{37} -5.81808e6i q^{39} +3.06821e6 q^{41} +5.01099e6 q^{43} +6.92969e6i q^{45} +3.25359e6i q^{47} -1.36538e6 q^{49} +1.90783e7 q^{51} -234649. i q^{53} -8.23002e6i q^{55} -1.76312e7 q^{57} +7.78979e6 q^{59} -2.42160e7i q^{61} -3.18112e7i q^{63} -2.48989e7 q^{65} -6.87944e6 q^{67} +2.67639e7i q^{69} -5.76593e6i q^{71} +1.19654e7 q^{73} -7.10502e6 q^{75} +3.77805e7i q^{77} -3.55920e7i q^{79} +2.07158e7 q^{81} -2.03804e7 q^{83} -8.16470e7i q^{85} +4.83644e7i q^{87} +1.19259e7 q^{89} +1.14300e8 q^{91} +456400. i q^{93} +7.54539e7i q^{95} -3.19199e7 q^{97} -1.68557e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 144 q^{3} + 18876 q^{9} - 40208 q^{11} - 391992 q^{17} + 664752 q^{19} + 791028 q^{25} - 2610144 q^{27} + 3084864 q^{33} + 3857984 q^{35} - 2953352 q^{41} + 3046768 q^{43} + 2839044 q^{49} + 31044576 q^{51}+ \cdots - 307632624 q^{99}+O(q^{100}) \) 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
\(3\) −135.920 −1.67802 −0.839012 − 0.544113i \(-0.816866\pi\)
−0.839012 + 0.544113i \(0.816866\pi\)
\(4\) 0 0
\(5\) 581.680i 0.930688i 0.885130 + 0.465344i \(0.154069\pi\)
−0.885130 + 0.465344i \(0.845931\pi\)
\(6\) 0 0
\(7\) − 2670.24i − 1.11214i −0.831137 − 0.556068i \(-0.812309\pi\)
0.831137 − 0.556068i \(-0.187691\pi\)
\(8\) 0 0
\(9\) 11913.2 1.81577
\(10\) 0 0
\(11\) −14148.7 −0.966377 −0.483188 − 0.875516i \(-0.660521\pi\)
−0.483188 + 0.875516i \(0.660521\pi\)
\(12\) 0 0
\(13\) 42805.2i 1.49873i 0.662158 + 0.749364i \(0.269641\pi\)
−0.662158 + 0.749364i \(0.730359\pi\)
\(14\) 0 0
\(15\) − 79061.9i − 1.56172i
\(16\) 0 0
\(17\) −140364. −1.68058 −0.840292 − 0.542134i \(-0.817617\pi\)
−0.840292 + 0.542134i \(0.817617\pi\)
\(18\) 0 0
\(19\) 129717. 0.995367 0.497683 − 0.867359i \(-0.334184\pi\)
0.497683 + 0.867359i \(0.334184\pi\)
\(20\) 0 0
\(21\) 362939.i 1.86619i
\(22\) 0 0
\(23\) − 196909.i − 0.703647i −0.936066 − 0.351823i \(-0.885562\pi\)
0.936066 − 0.351823i \(-0.114438\pi\)
\(24\) 0 0
\(25\) 52273.5 0.133820
\(26\) 0 0
\(27\) −727476. −1.36887
\(28\) 0 0
\(29\) − 355830.i − 0.503096i −0.967845 − 0.251548i \(-0.919060\pi\)
0.967845 − 0.251548i \(-0.0809396\pi\)
\(30\) 0 0
\(31\) − 3357.86i − 0.00363593i −0.999998 − 0.00181797i \(-0.999421\pi\)
0.999998 − 0.00181797i \(-0.000578677\pi\)
\(32\) 0 0
\(33\) 1.92309e6 1.62160
\(34\) 0 0
\(35\) 1.55322e6 1.03505
\(36\) 0 0
\(37\) − 907325.i − 0.484123i −0.970261 − 0.242062i \(-0.922176\pi\)
0.970261 − 0.242062i \(-0.0778236\pi\)
\(38\) 0 0
\(39\) − 5.81808e6i − 2.51490i
\(40\) 0 0
\(41\) 3.06821e6 1.08580 0.542900 − 0.839797i \(-0.317326\pi\)
0.542900 + 0.839797i \(0.317326\pi\)
\(42\) 0 0
\(43\) 5.01099e6 1.46572 0.732858 − 0.680382i \(-0.238186\pi\)
0.732858 + 0.680382i \(0.238186\pi\)
\(44\) 0 0
\(45\) 6.92969e6i 1.68991i
\(46\) 0 0
\(47\) 3.25359e6i 0.666764i 0.942792 + 0.333382i \(0.108190\pi\)
−0.942792 + 0.333382i \(0.891810\pi\)
\(48\) 0 0
\(49\) −1.36538e6 −0.236847
\(50\) 0 0
\(51\) 1.90783e7 2.82006
\(52\) 0 0
\(53\) − 234649.i − 0.0297382i −0.999889 − 0.0148691i \(-0.995267\pi\)
0.999889 − 0.0148691i \(-0.00473315\pi\)
\(54\) 0 0
\(55\) − 8.23002e6i − 0.899395i
\(56\) 0 0
\(57\) −1.76312e7 −1.67025
\(58\) 0 0
\(59\) 7.78979e6 0.642862 0.321431 − 0.946933i \(-0.395836\pi\)
0.321431 + 0.946933i \(0.395836\pi\)
\(60\) 0 0
\(61\) − 2.42160e7i − 1.74897i −0.485049 − 0.874487i \(-0.661198\pi\)
0.485049 − 0.874487i \(-0.338802\pi\)
\(62\) 0 0
\(63\) − 3.18112e7i − 2.01938i
\(64\) 0 0
\(65\) −2.48989e7 −1.39485
\(66\) 0 0
\(67\) −6.87944e6 −0.341392 −0.170696 − 0.985324i \(-0.554602\pi\)
−0.170696 + 0.985324i \(0.554602\pi\)
\(68\) 0 0
\(69\) 2.67639e7i 1.18074i
\(70\) 0 0
\(71\) − 5.76593e6i − 0.226901i −0.993544 − 0.113450i \(-0.963810\pi\)
0.993544 − 0.113450i \(-0.0361903\pi\)
\(72\) 0 0
\(73\) 1.19654e7 0.421343 0.210672 − 0.977557i \(-0.432435\pi\)
0.210672 + 0.977557i \(0.432435\pi\)
\(74\) 0 0
\(75\) −7.10502e6 −0.224554
\(76\) 0 0
\(77\) 3.77805e7i 1.07474i
\(78\) 0 0
\(79\) − 3.55920e7i − 0.913784i −0.889522 − 0.456892i \(-0.848963\pi\)
0.889522 − 0.456892i \(-0.151037\pi\)
\(80\) 0 0
\(81\) 2.07158e7 0.481239
\(82\) 0 0
\(83\) −2.03804e7 −0.429438 −0.214719 − 0.976676i \(-0.568884\pi\)
−0.214719 + 0.976676i \(0.568884\pi\)
\(84\) 0 0
\(85\) − 8.16470e7i − 1.56410i
\(86\) 0 0
\(87\) 4.83644e7i 0.844207i
\(88\) 0 0
\(89\) 1.19259e7 0.190078 0.0950390 − 0.995474i \(-0.469702\pi\)
0.0950390 + 0.995474i \(0.469702\pi\)
\(90\) 0 0
\(91\) 1.14300e8 1.66679
\(92\) 0 0
\(93\) 456400.i 0.00610119i
\(94\) 0 0
\(95\) 7.54539e7i 0.926376i
\(96\) 0 0
\(97\) −3.19199e7 −0.360558 −0.180279 − 0.983616i \(-0.557700\pi\)
−0.180279 + 0.983616i \(0.557700\pi\)
\(98\) 0 0
\(99\) −1.68557e8 −1.75471
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.127.2 4
4.3 odd 2 256.9.d.h.127.4 4
8.3 odd 2 inner 256.9.d.b.127.1 4
8.5 even 2 256.9.d.h.127.3 4
16.3 odd 4 64.9.c.f.63.4 4
16.5 even 4 32.9.c.a.31.4 yes 4
16.11 odd 4 32.9.c.a.31.1 ✓ 4
16.13 even 4 64.9.c.f.63.1 4
48.5 odd 4 288.9.g.b.127.4 4
48.11 even 4 288.9.g.b.127.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.9.c.a.31.1 ✓ 4 16.11 odd 4
32.9.c.a.31.4 yes 4 16.5 even 4
64.9.c.f.63.1 4 16.13 even 4
64.9.c.f.63.4 4 16.3 odd 4
256.9.d.b.127.1 4 8.3 odd 2 inner
256.9.d.b.127.2 4 1.1 even 1 trivial
256.9.d.h.127.3 4 8.5 even 2
256.9.d.h.127.4 4 4.3 odd 2
288.9.g.b.127.3 4 48.11 even 4
288.9.g.b.127.4 4 48.5 odd 4