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Properties

Label 256.8.b.i.129.1
Level $256$
Weight $8$
Character 256.129
Analytic conductor $79.971$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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 = [4,0,0,0,0,0,0] 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: \(4\)
Coefficient field: \(\Q(i, \sqrt{15})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 129.1
Root \(1.93649 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 256.129
Dual form 256.8.b.i.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)
 
\(f(q)\) \(=\) \(q-61.9677i q^{3} -70.0000i q^{5} +1115.42 q^{7} -1653.00 q^{9} -7250.22i q^{11} +13758.0i q^{13} -4337.74 q^{15} +16994.0 q^{17} -34020.3i q^{19} -69120.0i q^{21} -32347.2 q^{23} +73225.0 q^{25} -33090.8i q^{27} +34190.0i q^{29} +120465. q^{31} -449280. q^{33} -78079.3i q^{35} -35206.0i q^{37} +852552. q^{39} +484550. q^{41} -672040. i q^{43} +115710. i q^{45} +1.20688e6 q^{47} +420617. q^{49} -1.05308e6i q^{51} -851702. i q^{53} -507516. q^{55} -2.10816e6 q^{57} +695464. i q^{59} +71630.0i q^{61} -1.84379e6 q^{63} +963060. q^{65} +307298. i q^{67} +2.00448e6i q^{69} -757370. q^{71} -3.91204e6 q^{73} -4.53759e6i q^{75} -8.08704e6i q^{77} -314548. q^{79} -5.66567e6 q^{81} +1.53649e6i q^{83} -1.18958e6i q^{85} +2.11868e6 q^{87} +2.51063e6 q^{89} +1.53459e7i q^{91} -7.46496e6i q^{93} -2.38142e6 q^{95} -50094.0 q^{97} +1.19846e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6612 q^{9} + 67976 q^{17} + 292900 q^{25} - 1797120 q^{33} + 1938200 q^{41} + 1682468 q^{49} - 8432640 q^{57} + 3852240 q^{65} - 15648168 q^{73} - 22662684 q^{81} + 10042520 q^{89} - 200376 q^{97}+O(q^{100}) \) 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
\(3\) − 61.9677i − 1.32508i −0.749028 − 0.662539i \(-0.769479\pi\)
0.749028 − 0.662539i \(-0.230521\pi\)
\(4\) 0 0
\(5\) − 70.0000i − 0.250440i −0.992129 − 0.125220i \(-0.960036\pi\)
0.992129 − 0.125220i \(-0.0399636\pi\)
\(6\) 0 0
\(7\) 1115.42 1.22912 0.614561 − 0.788869i \(-0.289333\pi\)
0.614561 + 0.788869i \(0.289333\pi\)
\(8\) 0 0
\(9\) −1653.00 −0.755830
\(10\) 0 0
\(11\) − 7250.22i − 1.64239i −0.570646 − 0.821196i \(-0.693307\pi\)
0.570646 − 0.821196i \(-0.306693\pi\)
\(12\) 0 0
\(13\) 13758.0i 1.73682i 0.495851 + 0.868408i \(0.334856\pi\)
−0.495851 + 0.868408i \(0.665144\pi\)
\(14\) 0 0
\(15\) −4337.74 −0.331852
\(16\) 0 0
\(17\) 16994.0 0.838927 0.419464 − 0.907772i \(-0.362218\pi\)
0.419464 + 0.907772i \(0.362218\pi\)
\(18\) 0 0
\(19\) − 34020.3i − 1.13789i −0.822376 − 0.568945i \(-0.807352\pi\)
0.822376 − 0.568945i \(-0.192648\pi\)
\(20\) 0 0
\(21\) − 69120.0i − 1.62868i
\(22\) 0 0
\(23\) −32347.2 −0.554356 −0.277178 − 0.960819i \(-0.589399\pi\)
−0.277178 + 0.960819i \(0.589399\pi\)
\(24\) 0 0
\(25\) 73225.0 0.937280
\(26\) 0 0
\(27\) − 33090.8i − 0.323544i
\(28\) 0 0
\(29\) 34190.0i 0.260319i 0.991493 + 0.130160i \(0.0415490\pi\)
−0.991493 + 0.130160i \(0.958451\pi\)
\(30\) 0 0
\(31\) 120465. 0.726266 0.363133 − 0.931737i \(-0.381707\pi\)
0.363133 + 0.931737i \(0.381707\pi\)
\(32\) 0 0
\(33\) −449280. −2.17630
\(34\) 0 0
\(35\) − 78079.3i − 0.307821i
\(36\) 0 0
\(37\) − 35206.0i − 0.114264i −0.998367 − 0.0571322i \(-0.981804\pi\)
0.998367 − 0.0571322i \(-0.0181956\pi\)
\(38\) 0 0
\(39\) 852552. 2.30141
\(40\) 0 0
\(41\) 484550. 1.09798 0.548991 − 0.835828i \(-0.315012\pi\)
0.548991 + 0.835828i \(0.315012\pi\)
\(42\) 0 0
\(43\) − 672040.i − 1.28901i −0.764601 − 0.644504i \(-0.777064\pi\)
0.764601 − 0.644504i \(-0.222936\pi\)
\(44\) 0 0
\(45\) 115710.i 0.189290i
\(46\) 0 0
\(47\) 1.20688e6 1.69560 0.847799 − 0.530318i \(-0.177927\pi\)
0.847799 + 0.530318i \(0.177927\pi\)
\(48\) 0 0
\(49\) 420617. 0.510741
\(50\) 0 0
\(51\) − 1.05308e6i − 1.11164i
\(52\) 0 0
\(53\) − 851702.i − 0.785818i −0.919577 − 0.392909i \(-0.871469\pi\)
0.919577 − 0.392909i \(-0.128531\pi\)
\(54\) 0 0
\(55\) −507516. −0.411320
\(56\) 0 0
\(57\) −2.10816e6 −1.50779
\(58\) 0 0
\(59\) 695464.i 0.440852i 0.975404 + 0.220426i \(0.0707447\pi\)
−0.975404 + 0.220426i \(0.929255\pi\)
\(60\) 0 0
\(61\) 71630.0i 0.0404055i 0.999796 + 0.0202028i \(0.00643117\pi\)
−0.999796 + 0.0202028i \(0.993569\pi\)
\(62\) 0 0
\(63\) −1.84379e6 −0.929007
\(64\) 0 0
\(65\) 963060. 0.434967
\(66\) 0 0
\(67\) 307298.i 0.124824i 0.998050 + 0.0624120i \(0.0198793\pi\)
−0.998050 + 0.0624120i \(0.980121\pi\)
\(68\) 0 0
\(69\) 2.00448e6i 0.734564i
\(70\) 0 0
\(71\) −757370. −0.251133 −0.125566 − 0.992085i \(-0.540075\pi\)
−0.125566 + 0.992085i \(0.540075\pi\)
\(72\) 0 0
\(73\) −3.91204e6 −1.17699 −0.588496 − 0.808500i \(-0.700280\pi\)
−0.588496 + 0.808500i \(0.700280\pi\)
\(74\) 0 0
\(75\) − 4.53759e6i − 1.24197i
\(76\) 0 0
\(77\) − 8.08704e6i − 2.01870i
\(78\) 0 0
\(79\) −314548. −0.0717782 −0.0358891 − 0.999356i \(-0.511426\pi\)
−0.0358891 + 0.999356i \(0.511426\pi\)
\(80\) 0 0
\(81\) −5.66567e6 −1.18455
\(82\) 0 0
\(83\) 1.53649e6i 0.294955i 0.989065 + 0.147478i \(0.0471155\pi\)
−0.989065 + 0.147478i \(0.952885\pi\)
\(84\) 0 0
\(85\) − 1.18958e6i − 0.210101i
\(86\) 0 0
\(87\) 2.11868e6 0.344943
\(88\) 0 0
\(89\) 2.51063e6 0.377501 0.188750 − 0.982025i \(-0.439556\pi\)
0.188750 + 0.982025i \(0.439556\pi\)
\(90\) 0 0
\(91\) 1.53459e7i 2.13476i
\(92\) 0 0
\(93\) − 7.46496e6i − 0.962359i
\(94\) 0 0
\(95\) −2.38142e6 −0.284973
\(96\) 0 0
\(97\) −50094.0 −0.00557294 −0.00278647 − 0.999996i \(-0.500887\pi\)
−0.00278647 + 0.999996i \(0.500887\pi\)
\(98\) 0 0
\(99\) 1.19846e7i 1.24137i
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.i.129.1 4
4.3 odd 2 inner 256.8.b.i.129.3 4
8.3 odd 2 inner 256.8.b.i.129.2 4
8.5 even 2 inner 256.8.b.i.129.4 4
16.3 odd 4 64.8.a.i.1.1 2
16.5 even 4 32.8.a.c.1.1 ✓ 2
16.11 odd 4 32.8.a.c.1.2 yes 2
16.13 even 4 64.8.a.i.1.2 2
48.5 odd 4 288.8.a.k.1.1 2
48.11 even 4 288.8.a.k.1.2 2
48.29 odd 4 576.8.a.bk.1.1 2
48.35 even 4 576.8.a.bk.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.8.a.c.1.1 ✓ 2 16.5 even 4
32.8.a.c.1.2 yes 2 16.11 odd 4
64.8.a.i.1.1 2 16.3 odd 4
64.8.a.i.1.2 2 16.13 even 4
256.8.b.i.129.1 4 1.1 even 1 trivial
256.8.b.i.129.2 4 8.3 odd 2 inner
256.8.b.i.129.3 4 4.3 odd 2 inner
256.8.b.i.129.4 4 8.5 even 2 inner
288.8.a.k.1.1 2 48.5 odd 4
288.8.a.k.1.2 2 48.11 even 4
576.8.a.bk.1.1 2 48.29 odd 4
576.8.a.bk.1.2 2 48.35 even 4