Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [256,7,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, 7); 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, 7, names="a")
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 256.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,48,0,0,0,0,0,3804] 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: \(58.8938454067\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.3
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 256.127
Dual form 256.7.d.g.127.4

$q$-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)
 
\(f(q)\) \(=\) \(q+51.1918 q^{3} -142.767i q^{5} +217.616i q^{7} +1891.60 q^{9} -25.7388 q^{11} -1390.16i q^{13} -7308.52i q^{15} +4004.09 q^{17} -129.021 q^{19} +11140.2i q^{21} -3697.19i q^{23} -4757.51 q^{25} +59515.8 q^{27} +40871.1i q^{29} -46174.6i q^{31} -1317.62 q^{33} +31068.5 q^{35} -65959.6i q^{37} -71165.0i q^{39} +41801.8 q^{41} +30093.2 q^{43} -270059. i q^{45} -79785.8i q^{47} +70292.1 q^{49} +204977. q^{51} -13717.2i q^{53} +3674.66i q^{55} -6604.80 q^{57} -300272. q^{59} +243442. i q^{61} +411644. i q^{63} -198470. q^{65} -406008. q^{67} -189266. i q^{69} -249585. i q^{71} +366195. q^{73} -243546. q^{75} -5601.18i q^{77} +452154. i q^{79} +1.66775e6 q^{81} -512869. q^{83} -571653. i q^{85} +2.09227e6i q^{87} +797498. q^{89} +302522. q^{91} -2.36376e6i q^{93} +18419.9i q^{95} -8788.19 q^{97} -48687.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 48 q^{3} + 3804 q^{9} - 2768 q^{11} - 5304 q^{17} - 20112 q^{19} - 36588 q^{25} + 158112 q^{27} + 71232 q^{33} - 65728 q^{35} + 209848 q^{41} - 116816 q^{43} + 95556 q^{49} + 771936 q^{51} + 526656 q^{57}+ \cdots - 125616 q^{99}+O(q^{100}) \) 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
\(3\) 51.1918 1.89599 0.947997 − 0.318279i \(-0.103105\pi\)
0.947997 + 0.318279i \(0.103105\pi\)
\(4\) 0 0
\(5\) − 142.767i − 1.14214i −0.820902 − 0.571069i \(-0.806529\pi\)
0.820902 − 0.571069i \(-0.193471\pi\)
\(6\) 0 0
\(7\) 217.616i 0.634450i 0.948350 + 0.317225i \(0.102751\pi\)
−0.948350 + 0.317225i \(0.897249\pi\)
\(8\) 0 0
\(9\) 1891.60 2.59479
\(10\) 0 0
\(11\) −25.7388 −0.0193379 −0.00966897 − 0.999953i \(-0.503078\pi\)
−0.00966897 + 0.999953i \(0.503078\pi\)
\(12\) 0 0
\(13\) − 1390.16i − 0.632755i −0.948633 − 0.316378i \(-0.897533\pi\)
0.948633 − 0.316378i \(-0.102467\pi\)
\(14\) 0 0
\(15\) − 7308.52i − 2.16549i
\(16\) 0 0
\(17\) 4004.09 0.814999 0.407499 − 0.913205i \(-0.366401\pi\)
0.407499 + 0.913205i \(0.366401\pi\)
\(18\) 0 0
\(19\) −129.021 −0.0188104 −0.00940520 − 0.999956i \(-0.502994\pi\)
−0.00940520 + 0.999956i \(0.502994\pi\)
\(20\) 0 0
\(21\) 11140.2i 1.20291i
\(22\) 0 0
\(23\) − 3697.19i − 0.303870i −0.988390 − 0.151935i \(-0.951449\pi\)
0.988390 − 0.151935i \(-0.0485505\pi\)
\(24\) 0 0
\(25\) −4757.51 −0.304481
\(26\) 0 0
\(27\) 59515.8 3.02372
\(28\) 0 0
\(29\) 40871.1i 1.67580i 0.545823 + 0.837901i \(0.316217\pi\)
−0.545823 + 0.837901i \(0.683783\pi\)
\(30\) 0 0
\(31\) − 46174.6i − 1.54995i −0.631991 − 0.774976i \(-0.717762\pi\)
0.631991 − 0.774976i \(-0.282238\pi\)
\(32\) 0 0
\(33\) −1317.62 −0.0366646
\(34\) 0 0
\(35\) 31068.5 0.724630
\(36\) 0 0
\(37\) − 65959.6i − 1.30219i −0.758998 − 0.651093i \(-0.774311\pi\)
0.758998 − 0.651093i \(-0.225689\pi\)
\(38\) 0 0
\(39\) − 71165.0i − 1.19970i
\(40\) 0 0
\(41\) 41801.8 0.606518 0.303259 − 0.952908i \(-0.401925\pi\)
0.303259 + 0.952908i \(0.401925\pi\)
\(42\) 0 0
\(43\) 30093.2 0.378498 0.189249 − 0.981929i \(-0.439395\pi\)
0.189249 + 0.981929i \(0.439395\pi\)
\(44\) 0 0
\(45\) − 270059.i − 2.96361i
\(46\) 0 0
\(47\) − 79785.8i − 0.768479i −0.923234 − 0.384239i \(-0.874464\pi\)
0.923234 − 0.384239i \(-0.125536\pi\)
\(48\) 0 0
\(49\) 70292.1 0.597473
\(50\) 0 0
\(51\) 204977. 1.54523
\(52\) 0 0
\(53\) − 13717.2i − 0.0921376i −0.998938 − 0.0460688i \(-0.985331\pi\)
0.998938 − 0.0460688i \(-0.0146693\pi\)
\(54\) 0 0
\(55\) 3674.66i 0.0220866i
\(56\) 0 0
\(57\) −6604.80 −0.0356644
\(58\) 0 0
\(59\) −300272. −1.46204 −0.731019 − 0.682357i \(-0.760955\pi\)
−0.731019 + 0.682357i \(0.760955\pi\)
\(60\) 0 0
\(61\) 243442.i 1.07252i 0.844052 + 0.536261i \(0.180164\pi\)
−0.844052 + 0.536261i \(0.819836\pi\)
\(62\) 0 0
\(63\) 411644.i 1.64627i
\(64\) 0 0
\(65\) −198470. −0.722694
\(66\) 0 0
\(67\) −406008. −1.34993 −0.674964 − 0.737851i \(-0.735841\pi\)
−0.674964 + 0.737851i \(0.735841\pi\)
\(68\) 0 0
\(69\) − 189266.i − 0.576137i
\(70\) 0 0
\(71\) − 249585.i − 0.697337i −0.937246 − 0.348669i \(-0.886634\pi\)
0.937246 − 0.348669i \(-0.113366\pi\)
\(72\) 0 0
\(73\) 366195. 0.941334 0.470667 − 0.882311i \(-0.344013\pi\)
0.470667 + 0.882311i \(0.344013\pi\)
\(74\) 0 0
\(75\) −243546. −0.577294
\(76\) 0 0
\(77\) − 5601.18i − 0.0122690i
\(78\) 0 0
\(79\) 452154.i 0.917076i 0.888675 + 0.458538i \(0.151627\pi\)
−0.888675 + 0.458538i \(0.848373\pi\)
\(80\) 0 0
\(81\) 1.66775e6 3.13816
\(82\) 0 0
\(83\) −512869. −0.896958 −0.448479 − 0.893793i \(-0.648034\pi\)
−0.448479 + 0.893793i \(0.648034\pi\)
\(84\) 0 0
\(85\) − 571653.i − 0.930842i
\(86\) 0 0
\(87\) 2.09227e6i 3.17731i
\(88\) 0 0
\(89\) 797498. 1.13125 0.565626 − 0.824662i \(-0.308635\pi\)
0.565626 + 0.824662i \(0.308635\pi\)
\(90\) 0 0
\(91\) 302522. 0.401452
\(92\) 0 0
\(93\) − 2.36376e6i − 2.93870i
\(94\) 0 0
\(95\) 18419.9i 0.0214841i
\(96\) 0 0
\(97\) −8788.19 −0.00962907 −0.00481453 − 0.999988i \(-0.501533\pi\)
−0.00481453 + 0.999988i \(0.501533\pi\)
\(98\) 0 0
\(99\) −48687.6 −0.0501779
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 256.7.d.g.127.3 4
4.3 odd 2 256.7.d.d.127.1 4
8.3 odd 2 inner 256.7.d.g.127.4 4
8.5 even 2 256.7.d.d.127.2 4
16.3 odd 4 64.7.c.e.63.1 4
16.5 even 4 32.7.c.b.31.1 ✓ 4
16.11 odd 4 32.7.c.b.31.4 yes 4
16.13 even 4 64.7.c.e.63.4 4
48.5 odd 4 288.7.g.b.127.1 4
48.11 even 4 288.7.g.b.127.2 4
48.29 odd 4 576.7.g.l.127.3 4
48.35 even 4 576.7.g.l.127.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.7.c.b.31.1 ✓ 4 16.5 even 4
32.7.c.b.31.4 yes 4 16.11 odd 4
64.7.c.e.63.1 4 16.3 odd 4
64.7.c.e.63.4 4 16.13 even 4
256.7.d.d.127.1 4 4.3 odd 2
256.7.d.d.127.2 4 8.5 even 2
256.7.d.g.127.3 4 1.1 even 1 trivial
256.7.d.g.127.4 4 8.3 odd 2 inner
288.7.g.b.127.1 4 48.5 odd 4
288.7.g.b.127.2 4 48.11 even 4
576.7.g.l.127.3 4 48.29 odd 4
576.7.g.l.127.4 4 48.35 even 4