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Properties

Label 960.4.f.b.769.2
Level $960$
Weight $4$
Character 960.769
Analytic conductor $56.642$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [960,4,Mod(769,960)] 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("960.769"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-10,0,0,0,-18,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(56.6418336055\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 960.769
Dual form 960.4.f.b.769.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +(-5.00000 + 10.0000i) q^{5} -4.00000i q^{7} -9.00000 q^{9} +28.0000 q^{11} +16.0000i q^{13} +(-30.0000 - 15.0000i) q^{15} -108.000i q^{17} +32.0000 q^{19} +12.0000 q^{21} -28.0000i q^{23} +(-75.0000 - 100.000i) q^{25} -27.0000i q^{27} -238.000 q^{29} -180.000 q^{31} +84.0000i q^{33} +(40.0000 + 20.0000i) q^{35} -40.0000i q^{37} -48.0000 q^{39} +422.000 q^{41} -276.000i q^{43} +(45.0000 - 90.0000i) q^{45} -60.0000i q^{47} +327.000 q^{49} +324.000 q^{51} -220.000i q^{53} +(-140.000 + 280.000i) q^{55} +96.0000i q^{57} -804.000 q^{59} +358.000 q^{61} +36.0000i q^{63} +(-160.000 - 80.0000i) q^{65} -884.000i q^{67} +84.0000 q^{69} -64.0000 q^{71} -152.000i q^{73} +(300.000 - 225.000i) q^{75} -112.000i q^{77} +932.000 q^{79} +81.0000 q^{81} +1292.00i q^{83} +(1080.00 + 540.000i) q^{85} -714.000i q^{87} +1146.00 q^{89} +64.0000 q^{91} -540.000i q^{93} +(-160.000 + 320.000i) q^{95} -824.000i q^{97} -252.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{5} - 18 q^{9} + 56 q^{11} - 60 q^{15} + 64 q^{19} + 24 q^{21} - 150 q^{25} - 476 q^{29} - 360 q^{31} + 80 q^{35} - 96 q^{39} + 844 q^{41} + 90 q^{45} + 654 q^{49} + 648 q^{51} - 280 q^{55} - 1608 q^{59}+ \cdots - 504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) −5.00000 + 10.0000i −0.447214 + 0.894427i
\(6\) 0 0
\(7\) − 4.00000i − 0.215980i −0.994152 − 0.107990i \(-0.965559\pi\)
0.994152 − 0.107990i \(-0.0344414\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 28.0000 0.767483 0.383742 − 0.923440i \(-0.374635\pi\)
0.383742 + 0.923440i \(0.374635\pi\)
\(12\) 0 0
\(13\) 16.0000i 0.341354i 0.985327 + 0.170677i \(0.0545955\pi\)
−0.985327 + 0.170677i \(0.945405\pi\)
\(14\) 0 0
\(15\) −30.0000 − 15.0000i −0.516398 − 0.258199i
\(16\) 0 0
\(17\) − 108.000i − 1.54081i −0.637552 − 0.770407i \(-0.720053\pi\)
0.637552 − 0.770407i \(-0.279947\pi\)
\(18\) 0 0
\(19\) 32.0000 0.386384 0.193192 − 0.981161i \(-0.438116\pi\)
0.193192 + 0.981161i \(0.438116\pi\)
\(20\) 0 0
\(21\) 12.0000 0.124696
\(22\) 0 0
\(23\) − 28.0000i − 0.253844i −0.991913 − 0.126922i \(-0.959490\pi\)
0.991913 − 0.126922i \(-0.0405097\pi\)
\(24\) 0 0
\(25\) −75.0000 − 100.000i −0.600000 − 0.800000i
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) −238.000 −1.52398 −0.761991 − 0.647587i \(-0.775778\pi\)
−0.761991 + 0.647587i \(0.775778\pi\)
\(30\) 0 0
\(31\) −180.000 −1.04287 −0.521435 − 0.853291i \(-0.674603\pi\)
−0.521435 + 0.853291i \(0.674603\pi\)
\(32\) 0 0
\(33\) 84.0000i 0.443107i
\(34\) 0 0
\(35\) 40.0000 + 20.0000i 0.193178 + 0.0965891i
\(36\) 0 0
\(37\) − 40.0000i − 0.177729i −0.996044 − 0.0888643i \(-0.971676\pi\)
0.996044 − 0.0888643i \(-0.0283238\pi\)
\(38\) 0 0
\(39\) −48.0000 −0.197081
\(40\) 0 0
\(41\) 422.000 1.60745 0.803724 − 0.595003i \(-0.202849\pi\)
0.803724 + 0.595003i \(0.202849\pi\)
\(42\) 0 0
\(43\) − 276.000i − 0.978828i −0.872052 − 0.489414i \(-0.837211\pi\)
0.872052 − 0.489414i \(-0.162789\pi\)
\(44\) 0 0
\(45\) 45.0000 − 90.0000i 0.149071 − 0.298142i
\(46\) 0 0
\(47\) − 60.0000i − 0.186211i −0.995656 − 0.0931053i \(-0.970321\pi\)
0.995656 − 0.0931053i \(-0.0296793\pi\)
\(48\) 0 0
\(49\) 327.000 0.953353
\(50\) 0 0
\(51\) 324.000 0.889590
\(52\) 0 0
\(53\) − 220.000i − 0.570176i −0.958501 − 0.285088i \(-0.907977\pi\)
0.958501 − 0.285088i \(-0.0920228\pi\)
\(54\) 0 0
\(55\) −140.000 + 280.000i −0.343229 + 0.686458i
\(56\) 0 0
\(57\) 96.0000i 0.223079i
\(58\) 0 0
\(59\) −804.000 −1.77410 −0.887050 − 0.461674i \(-0.847249\pi\)
−0.887050 + 0.461674i \(0.847249\pi\)
\(60\) 0 0
\(61\) 358.000 0.751430 0.375715 − 0.926735i \(-0.377397\pi\)
0.375715 + 0.926735i \(0.377397\pi\)
\(62\) 0 0
\(63\) 36.0000i 0.0719932i
\(64\) 0 0
\(65\) −160.000 − 80.0000i −0.305316 − 0.152658i
\(66\) 0 0
\(67\) − 884.000i − 1.61191i −0.591979 − 0.805954i \(-0.701653\pi\)
0.591979 − 0.805954i \(-0.298347\pi\)
\(68\) 0 0
\(69\) 84.0000 0.146557
\(70\) 0 0
\(71\) −64.0000 −0.106978 −0.0534888 − 0.998568i \(-0.517034\pi\)
−0.0534888 + 0.998568i \(0.517034\pi\)
\(72\) 0 0
\(73\) − 152.000i − 0.243702i −0.992548 − 0.121851i \(-0.961117\pi\)
0.992548 − 0.121851i \(-0.0388830\pi\)
\(74\) 0 0
\(75\) 300.000 − 225.000i 0.461880 − 0.346410i
\(76\) 0 0
\(77\) − 112.000i − 0.165761i
\(78\) 0 0
\(79\) 932.000 1.32732 0.663659 − 0.748035i \(-0.269002\pi\)
0.663659 + 0.748035i \(0.269002\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1292.00i 1.70862i 0.519764 + 0.854310i \(0.326020\pi\)
−0.519764 + 0.854310i \(0.673980\pi\)
\(84\) 0 0
\(85\) 1080.00 + 540.000i 1.37815 + 0.689073i
\(86\) 0 0
\(87\) − 714.000i − 0.879872i
\(88\) 0 0
\(89\) 1146.00 1.36490 0.682448 − 0.730934i \(-0.260915\pi\)
0.682448 + 0.730934i \(0.260915\pi\)
\(90\) 0 0
\(91\) 64.0000 0.0737255
\(92\) 0 0
\(93\) − 540.000i − 0.602101i
\(94\) 0 0
\(95\) −160.000 + 320.000i −0.172796 + 0.345593i
\(96\) 0 0
\(97\) − 824.000i − 0.862521i −0.902227 − 0.431260i \(-0.858069\pi\)
0.902227 − 0.431260i \(-0.141931\pi\)
\(98\) 0 0
\(99\) −252.000 −0.255828
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 960.4.f.b.769.2 2
4.3 odd 2 960.4.f.a.769.1 2
5.4 even 2 inner 960.4.f.b.769.1 2
8.3 odd 2 240.4.f.e.49.2 2
8.5 even 2 120.4.f.c.49.1 ✓ 2
20.19 odd 2 960.4.f.a.769.2 2
24.5 odd 2 360.4.f.a.289.2 2
24.11 even 2 720.4.f.b.289.2 2
40.3 even 4 1200.4.a.l.1.1 1
40.13 odd 4 600.4.a.k.1.1 1
40.19 odd 2 240.4.f.e.49.1 2
40.27 even 4 1200.4.a.z.1.1 1
40.29 even 2 120.4.f.c.49.2 yes 2
40.37 odd 4 600.4.a.f.1.1 1
120.29 odd 2 360.4.f.a.289.1 2
120.53 even 4 1800.4.a.o.1.1 1
120.59 even 2 720.4.f.b.289.1 2
120.77 even 4 1800.4.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.f.c.49.1 ✓ 2 8.5 even 2
120.4.f.c.49.2 yes 2 40.29 even 2
240.4.f.e.49.1 2 40.19 odd 2
240.4.f.e.49.2 2 8.3 odd 2
360.4.f.a.289.1 2 120.29 odd 2
360.4.f.a.289.2 2 24.5 odd 2
600.4.a.f.1.1 1 40.37 odd 4
600.4.a.k.1.1 1 40.13 odd 4
720.4.f.b.289.1 2 120.59 even 2
720.4.f.b.289.2 2 24.11 even 2
960.4.f.a.769.1 2 4.3 odd 2
960.4.f.a.769.2 2 20.19 odd 2
960.4.f.b.769.1 2 5.4 even 2 inner
960.4.f.b.769.2 2 1.1 even 1 trivial
1200.4.a.l.1.1 1 40.3 even 4
1200.4.a.z.1.1 1 40.27 even 4
1800.4.a.o.1.1 1 120.53 even 4
1800.4.a.u.1.1 1 120.77 even 4