Show commands: Magma / Pari/GP / SageMath

Newspace parameters

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

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,2032] 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: \(79.9705665239\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 256.129
Dual form 256.8.b.f.129.1

$q$-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)
 
\(f(q)\) \(=\) \(q+12.0000i q^{3} -210.000i q^{5} +1016.00 q^{7} +2043.00 q^{9} -1092.00i q^{11} -1382.00i q^{13} +2520.00 q^{15} +14706.0 q^{17} -39940.0i q^{19} +12192.0i q^{21} +68712.0 q^{23} +34025.0 q^{25} +50760.0i q^{27} +102570. i q^{29} -227552. q^{31} +13104.0 q^{33} -213360. i q^{35} +160526. i q^{37} +16584.0 q^{39} -10842.0 q^{41} +630748. i q^{43} -429030. i q^{45} -472656. q^{47} +208713. q^{49} +176472. i q^{51} -1.49402e6i q^{53} -229320. q^{55} +479280. q^{57} -2.64066e6i q^{59} -827702. i q^{61} +2.07569e6 q^{63} -290220. q^{65} -126004. i q^{67} +824544. i q^{69} -1.41473e6 q^{71} -980282. q^{73} +408300. i q^{75} -1.10947e6i q^{77} +3.56680e6 q^{79} +3.85892e6 q^{81} +5.67289e6i q^{83} -3.08826e6i q^{85} -1.23084e6 q^{87} +1.19512e7 q^{89} -1.40411e6i q^{91} -2.73062e6i q^{93} -8.38740e6 q^{95} +8.68215e6 q^{97} -2.23096e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2032 q^{7} + 4086 q^{9} + 5040 q^{15} + 29412 q^{17} + 137424 q^{23} + 68050 q^{25} - 455104 q^{31} + 26208 q^{33} + 33168 q^{39} - 21684 q^{41} - 945312 q^{47} + 417426 q^{49} - 458640 q^{55} + 958560 q^{57}+ \cdots + 17364292 q^{97}+O(q^{100}) \) 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
\(3\) 12.0000i 0.256600i 0.991735 + 0.128300i \(0.0409521\pi\)
−0.991735 + 0.128300i \(0.959048\pi\)
\(4\) 0 0
\(5\) − 210.000i − 0.751319i −0.926758 − 0.375659i \(-0.877416\pi\)
0.926758 − 0.375659i \(-0.122584\pi\)
\(6\) 0 0
\(7\) 1016.00 1.11957 0.559784 − 0.828638i \(-0.310884\pi\)
0.559784 + 0.828638i \(0.310884\pi\)
\(8\) 0 0
\(9\) 2043.00 0.934156
\(10\) 0 0
\(11\) − 1092.00i − 0.247371i −0.992321 − 0.123685i \(-0.960529\pi\)
0.992321 − 0.123685i \(-0.0394713\pi\)
\(12\) 0 0
\(13\) − 1382.00i − 0.174464i −0.996188 − 0.0872321i \(-0.972198\pi\)
0.996188 − 0.0872321i \(-0.0278022\pi\)
\(14\) 0 0
\(15\) 2520.00 0.192789
\(16\) 0 0
\(17\) 14706.0 0.725978 0.362989 − 0.931793i \(-0.381756\pi\)
0.362989 + 0.931793i \(0.381756\pi\)
\(18\) 0 0
\(19\) − 39940.0i − 1.33589i −0.744211 − 0.667945i \(-0.767174\pi\)
0.744211 − 0.667945i \(-0.232826\pi\)
\(20\) 0 0
\(21\) 12192.0i 0.287281i
\(22\) 0 0
\(23\) 68712.0 1.17757 0.588783 − 0.808291i \(-0.299607\pi\)
0.588783 + 0.808291i \(0.299607\pi\)
\(24\) 0 0
\(25\) 34025.0 0.435520
\(26\) 0 0
\(27\) 50760.0i 0.496305i
\(28\) 0 0
\(29\) 102570.i 0.780957i 0.920612 + 0.390479i \(0.127690\pi\)
−0.920612 + 0.390479i \(0.872310\pi\)
\(30\) 0 0
\(31\) −227552. −1.37188 −0.685938 − 0.727660i \(-0.740608\pi\)
−0.685938 + 0.727660i \(0.740608\pi\)
\(32\) 0 0
\(33\) 13104.0 0.0634753
\(34\) 0 0
\(35\) − 213360.i − 0.841153i
\(36\) 0 0
\(37\) 160526.i 0.521002i 0.965474 + 0.260501i \(0.0838877\pi\)
−0.965474 + 0.260501i \(0.916112\pi\)
\(38\) 0 0
\(39\) 16584.0 0.0447675
\(40\) 0 0
\(41\) −10842.0 −0.0245678 −0.0122839 − 0.999925i \(-0.503910\pi\)
−0.0122839 + 0.999925i \(0.503910\pi\)
\(42\) 0 0
\(43\) 630748.i 1.20981i 0.796299 + 0.604904i \(0.206788\pi\)
−0.796299 + 0.604904i \(0.793212\pi\)
\(44\) 0 0
\(45\) − 429030.i − 0.701849i
\(46\) 0 0
\(47\) −472656. −0.664053 −0.332026 − 0.943270i \(-0.607732\pi\)
−0.332026 + 0.943270i \(0.607732\pi\)
\(48\) 0 0
\(49\) 208713. 0.253433
\(50\) 0 0
\(51\) 176472.i 0.186286i
\(52\) 0 0
\(53\) − 1.49402e6i − 1.37845i −0.724548 − 0.689224i \(-0.757952\pi\)
0.724548 − 0.689224i \(-0.242048\pi\)
\(54\) 0 0
\(55\) −229320. −0.185854
\(56\) 0 0
\(57\) 479280. 0.342789
\(58\) 0 0
\(59\) − 2.64066e6i − 1.67390i −0.547277 − 0.836952i \(-0.684335\pi\)
0.547277 − 0.836952i \(-0.315665\pi\)
\(60\) 0 0
\(61\) − 827702.i − 0.466895i −0.972369 − 0.233448i \(-0.924999\pi\)
0.972369 − 0.233448i \(-0.0750008\pi\)
\(62\) 0 0
\(63\) 2.07569e6 1.04585
\(64\) 0 0
\(65\) −290220. −0.131078
\(66\) 0 0
\(67\) − 126004.i − 0.0511826i −0.999672 − 0.0255913i \(-0.991853\pi\)
0.999672 − 0.0255913i \(-0.00814686\pi\)
\(68\) 0 0
\(69\) 824544.i 0.302164i
\(70\) 0 0
\(71\) −1.41473e6 −0.469104 −0.234552 − 0.972104i \(-0.575362\pi\)
−0.234552 + 0.972104i \(0.575362\pi\)
\(72\) 0 0
\(73\) −980282. −0.294931 −0.147466 − 0.989067i \(-0.547112\pi\)
−0.147466 + 0.989067i \(0.547112\pi\)
\(74\) 0 0
\(75\) 408300.i 0.111754i
\(76\) 0 0
\(77\) − 1.10947e6i − 0.276948i
\(78\) 0 0
\(79\) 3.56680e6 0.813924 0.406962 − 0.913445i \(-0.366588\pi\)
0.406962 + 0.913445i \(0.366588\pi\)
\(80\) 0 0
\(81\) 3.85892e6 0.806805
\(82\) 0 0
\(83\) 5.67289e6i 1.08901i 0.838758 + 0.544504i \(0.183282\pi\)
−0.838758 + 0.544504i \(0.816718\pi\)
\(84\) 0 0
\(85\) − 3.08826e6i − 0.545441i
\(86\) 0 0
\(87\) −1.23084e6 −0.200394
\(88\) 0 0
\(89\) 1.19512e7 1.79699 0.898496 − 0.438982i \(-0.144661\pi\)
0.898496 + 0.438982i \(0.144661\pi\)
\(90\) 0 0
\(91\) − 1.40411e6i − 0.195325i
\(92\) 0 0
\(93\) − 2.73062e6i − 0.352023i
\(94\) 0 0
\(95\) −8.38740e6 −1.00368
\(96\) 0 0
\(97\) 8.68215e6 0.965886 0.482943 − 0.875652i \(-0.339568\pi\)
0.482943 + 0.875652i \(0.339568\pi\)
\(98\) 0 0
\(99\) − 2.23096e6i − 0.231083i
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 256.8.b.f.129.2 2
4.3 odd 2 256.8.b.b.129.1 2
8.3 odd 2 256.8.b.b.129.2 2
8.5 even 2 inner 256.8.b.f.129.1 2
16.3 odd 4 2.8.a.a.1.1 ✓ 1
16.5 even 4 64.8.a.e.1.1 1
16.11 odd 4 64.8.a.c.1.1 1
16.13 even 4 16.8.a.b.1.1 1
48.5 odd 4 576.8.a.f.1.1 1
48.11 even 4 576.8.a.g.1.1 1
48.29 odd 4 144.8.a.i.1.1 1
48.35 even 4 18.8.a.b.1.1 1
80.3 even 4 50.8.b.c.49.2 2
80.13 odd 4 400.8.c.j.49.1 2
80.19 odd 4 50.8.a.g.1.1 1
80.29 even 4 400.8.a.l.1.1 1
80.67 even 4 50.8.b.c.49.1 2
80.77 odd 4 400.8.c.j.49.2 2
112.3 even 12 98.8.c.e.79.1 2
112.19 even 12 98.8.c.e.67.1 2
112.51 odd 12 98.8.c.d.67.1 2
112.67 odd 12 98.8.c.d.79.1 2
112.83 even 4 98.8.a.a.1.1 1
144.67 odd 12 162.8.c.l.55.1 2
144.83 even 12 162.8.c.a.109.1 2
144.115 odd 12 162.8.c.l.109.1 2
144.131 even 12 162.8.c.a.55.1 2
176.131 even 4 242.8.a.e.1.1 1
208.51 odd 4 338.8.a.d.1.1 1
208.83 even 4 338.8.b.d.337.2 2
208.99 even 4 338.8.b.d.337.1 2
240.83 odd 4 450.8.c.g.199.1 2
240.179 even 4 450.8.a.c.1.1 1
240.227 odd 4 450.8.c.g.199.2 2
272.67 odd 4 578.8.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 ✓ 1 16.3 odd 4
16.8.a.b.1.1 1 16.13 even 4
18.8.a.b.1.1 1 48.35 even 4
50.8.a.g.1.1 1 80.19 odd 4
50.8.b.c.49.1 2 80.67 even 4
50.8.b.c.49.2 2 80.3 even 4
64.8.a.c.1.1 1 16.11 odd 4
64.8.a.e.1.1 1 16.5 even 4
98.8.a.a.1.1 1 112.83 even 4
98.8.c.d.67.1 2 112.51 odd 12
98.8.c.d.79.1 2 112.67 odd 12
98.8.c.e.67.1 2 112.19 even 12
98.8.c.e.79.1 2 112.3 even 12
144.8.a.i.1.1 1 48.29 odd 4
162.8.c.a.55.1 2 144.131 even 12
162.8.c.a.109.1 2 144.83 even 12
162.8.c.l.55.1 2 144.67 odd 12
162.8.c.l.109.1 2 144.115 odd 12
242.8.a.e.1.1 1 176.131 even 4
256.8.b.b.129.1 2 4.3 odd 2
256.8.b.b.129.2 2 8.3 odd 2
256.8.b.f.129.1 2 8.5 even 2 inner
256.8.b.f.129.2 2 1.1 even 1 trivial
338.8.a.d.1.1 1 208.51 odd 4
338.8.b.d.337.1 2 208.99 even 4
338.8.b.d.337.2 2 208.83 even 4
400.8.a.l.1.1 1 80.29 even 4
400.8.c.j.49.1 2 80.13 odd 4
400.8.c.j.49.2 2 80.77 odd 4
450.8.a.c.1.1 1 240.179 even 4
450.8.c.g.199.1 2 240.83 odd 4
450.8.c.g.199.2 2 240.227 odd 4
576.8.a.f.1.1 1 48.5 odd 4
576.8.a.g.1.1 1 48.11 even 4
578.8.a.b.1.1 1 272.67 odd 4