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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [256,4,Mod(129,256)] 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("256.129"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.1044889615\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 256.129
Dual form 256.4.b.h.129.4

$q$-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)
 
\(f(q)\) \(=\) \(q-8.92820i q^{3} +11.8564i q^{5} +9.85641 q^{7} -52.7128 q^{9} +39.0718i q^{11} +91.5692i q^{13} +105.856 q^{15} -37.1384 q^{17} +46.4974i q^{19} -88.0000i q^{21} -120.708 q^{23} -15.5744 q^{25} +229.569i q^{27} -27.2820i q^{29} +81.1487 q^{31} +348.841 q^{33} +116.862i q^{35} -10.9948i q^{37} +817.549 q^{39} +205.426 q^{41} +115.359i q^{43} -624.985i q^{45} +312.841 q^{47} -245.851 q^{49} +331.580i q^{51} -90.9948i q^{53} -463.251 q^{55} +415.138 q^{57} -550.631i q^{59} +630.974i q^{61} -519.559 q^{63} -1085.68 q^{65} +661.041i q^{67} +1077.70i q^{69} +494.985 q^{71} -566.267 q^{73} +139.051i q^{75} +385.108i q^{77} -49.4153 q^{79} +626.395 q^{81} -564.067i q^{83} -440.328i q^{85} -243.580 q^{87} -1089.67 q^{89} +902.543i q^{91} -724.513i q^{93} -551.292 q^{95} +464.605 q^{97} -2059.58i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{7} - 100 q^{9} + 368 q^{15} + 184 q^{17} + 16 q^{23} - 284 q^{25} + 768 q^{31} + 176 q^{33} + 1552 q^{39} + 600 q^{41} + 32 q^{47} - 540 q^{49} + 752 q^{55} + 1328 q^{57} - 1136 q^{63} - 1904 q^{65}+ \cdots + 4408 q^{97}+O(q^{100}) \) 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
\(3\) − 8.92820i − 1.71823i −0.511780 − 0.859117i \(-0.671014\pi\)
0.511780 − 0.859117i \(-0.328986\pi\)
\(4\) 0 0
\(5\) 11.8564i 1.06047i 0.847851 + 0.530235i \(0.177896\pi\)
−0.847851 + 0.530235i \(0.822104\pi\)
\(6\) 0 0
\(7\) 9.85641 0.532196 0.266098 − 0.963946i \(-0.414266\pi\)
0.266098 + 0.963946i \(0.414266\pi\)
\(8\) 0 0
\(9\) −52.7128 −1.95233
\(10\) 0 0
\(11\) 39.0718i 1.07096i 0.844547 + 0.535481i \(0.179870\pi\)
−0.844547 + 0.535481i \(0.820130\pi\)
\(12\) 0 0
\(13\) 91.5692i 1.95359i 0.214166 + 0.976797i \(0.431297\pi\)
−0.214166 + 0.976797i \(0.568703\pi\)
\(14\) 0 0
\(15\) 105.856 1.82213
\(16\) 0 0
\(17\) −37.1384 −0.529847 −0.264923 − 0.964269i \(-0.585347\pi\)
−0.264923 + 0.964269i \(0.585347\pi\)
\(18\) 0 0
\(19\) 46.4974i 0.561434i 0.959791 + 0.280717i \(0.0905722\pi\)
−0.959791 + 0.280717i \(0.909428\pi\)
\(20\) 0 0
\(21\) − 88.0000i − 0.914437i
\(22\) 0 0
\(23\) −120.708 −1.09432 −0.547158 − 0.837029i \(-0.684290\pi\)
−0.547158 + 0.837029i \(0.684290\pi\)
\(24\) 0 0
\(25\) −15.5744 −0.124595
\(26\) 0 0
\(27\) 229.569i 1.63632i
\(28\) 0 0
\(29\) − 27.2820i − 0.174695i −0.996178 − 0.0873473i \(-0.972161\pi\)
0.996178 − 0.0873473i \(-0.0278390\pi\)
\(30\) 0 0
\(31\) 81.1487 0.470153 0.235077 − 0.971977i \(-0.424466\pi\)
0.235077 + 0.971977i \(0.424466\pi\)
\(32\) 0 0
\(33\) 348.841 1.84016
\(34\) 0 0
\(35\) 116.862i 0.564377i
\(36\) 0 0
\(37\) − 10.9948i − 0.0488525i −0.999702 − 0.0244262i \(-0.992224\pi\)
0.999702 − 0.0244262i \(-0.00777589\pi\)
\(38\) 0 0
\(39\) 817.549 3.35673
\(40\) 0 0
\(41\) 205.426 0.782490 0.391245 − 0.920287i \(-0.372044\pi\)
0.391245 + 0.920287i \(0.372044\pi\)
\(42\) 0 0
\(43\) 115.359i 0.409118i 0.978854 + 0.204559i \(0.0655760\pi\)
−0.978854 + 0.204559i \(0.934424\pi\)
\(44\) 0 0
\(45\) − 624.985i − 2.07038i
\(46\) 0 0
\(47\) 312.841 0.970905 0.485453 − 0.874263i \(-0.338655\pi\)
0.485453 + 0.874263i \(0.338655\pi\)
\(48\) 0 0
\(49\) −245.851 −0.716767
\(50\) 0 0
\(51\) 331.580i 0.910400i
\(52\) 0 0
\(53\) − 90.9948i − 0.235832i −0.993024 − 0.117916i \(-0.962379\pi\)
0.993024 − 0.117916i \(-0.0376214\pi\)
\(54\) 0 0
\(55\) −463.251 −1.13572
\(56\) 0 0
\(57\) 415.138 0.964674
\(58\) 0 0
\(59\) − 550.631i − 1.21502i −0.794313 − 0.607509i \(-0.792169\pi\)
0.794313 − 0.607509i \(-0.207831\pi\)
\(60\) 0 0
\(61\) 630.974i 1.32439i 0.749330 + 0.662196i \(0.230376\pi\)
−0.749330 + 0.662196i \(0.769624\pi\)
\(62\) 0 0
\(63\) −519.559 −1.03902
\(64\) 0 0
\(65\) −1085.68 −2.07173
\(66\) 0 0
\(67\) 661.041i 1.20536i 0.797984 + 0.602679i \(0.205900\pi\)
−0.797984 + 0.602679i \(0.794100\pi\)
\(68\) 0 0
\(69\) 1077.70i 1.88029i
\(70\) 0 0
\(71\) 494.985 0.827378 0.413689 − 0.910418i \(-0.364240\pi\)
0.413689 + 0.910418i \(0.364240\pi\)
\(72\) 0 0
\(73\) −566.267 −0.907897 −0.453949 − 0.891028i \(-0.649985\pi\)
−0.453949 + 0.891028i \(0.649985\pi\)
\(74\) 0 0
\(75\) 139.051i 0.214083i
\(76\) 0 0
\(77\) 385.108i 0.569962i
\(78\) 0 0
\(79\) −49.4153 −0.0703754 −0.0351877 − 0.999381i \(-0.511203\pi\)
−0.0351877 + 0.999381i \(0.511203\pi\)
\(80\) 0 0
\(81\) 626.395 0.859252
\(82\) 0 0
\(83\) − 564.067i − 0.745956i −0.927840 − 0.372978i \(-0.878337\pi\)
0.927840 − 0.372978i \(-0.121663\pi\)
\(84\) 0 0
\(85\) − 440.328i − 0.561886i
\(86\) 0 0
\(87\) −243.580 −0.300166
\(88\) 0 0
\(89\) −1089.67 −1.29781 −0.648904 − 0.760870i \(-0.724772\pi\)
−0.648904 + 0.760870i \(0.724772\pi\)
\(90\) 0 0
\(91\) 902.543i 1.03970i
\(92\) 0 0
\(93\) − 724.513i − 0.807833i
\(94\) 0 0
\(95\) −551.292 −0.595383
\(96\) 0 0
\(97\) 464.605 0.486325 0.243162 − 0.969986i \(-0.421815\pi\)
0.243162 + 0.969986i \(0.421815\pi\)
\(98\) 0 0
\(99\) − 2059.58i − 2.09087i
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 256.4.b.h.129.1 4
4.3 odd 2 256.4.b.i.129.4 4
8.3 odd 2 256.4.b.i.129.1 4
8.5 even 2 inner 256.4.b.h.129.4 4
16.3 odd 4 128.4.a.e.1.1 ✓ 2
16.5 even 4 128.4.a.f.1.1 yes 2
16.11 odd 4 128.4.a.h.1.2 yes 2
16.13 even 4 128.4.a.g.1.2 yes 2
48.5 odd 4 1152.4.a.r.1.2 2
48.11 even 4 1152.4.a.q.1.2 2
48.29 odd 4 1152.4.a.t.1.1 2
48.35 even 4 1152.4.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.a.e.1.1 ✓ 2 16.3 odd 4
128.4.a.f.1.1 yes 2 16.5 even 4
128.4.a.g.1.2 yes 2 16.13 even 4
128.4.a.h.1.2 yes 2 16.11 odd 4
256.4.b.h.129.1 4 1.1 even 1 trivial
256.4.b.h.129.4 4 8.5 even 2 inner
256.4.b.i.129.1 4 8.3 odd 2
256.4.b.i.129.4 4 4.3 odd 2
1152.4.a.q.1.2 2 48.11 even 4
1152.4.a.r.1.2 2 48.5 odd 4
1152.4.a.s.1.1 2 48.35 even 4
1152.4.a.t.1.1 2 48.29 odd 4