↑

Properties

Label 960.4.a.ba.1.1
Level $960$
Weight $4$
Character 960.1
Self dual yes
Analytic conductor $56.642$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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(1,960)] 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("960.1"); 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, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,3,0,-5,0,24,0,9,0,52] 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: yes
Analytic conductor: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 960.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -5.00000 q^{5} +24.0000 q^{7} +9.00000 q^{9} +52.0000 q^{11} -22.0000 q^{13} -15.0000 q^{15} -14.0000 q^{17} -20.0000 q^{19} +72.0000 q^{21} +168.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -230.000 q^{29} +288.000 q^{31} +156.000 q^{33} -120.000 q^{35} +34.0000 q^{37} -66.0000 q^{39} +122.000 q^{41} -188.000 q^{43} -45.0000 q^{45} -256.000 q^{47} +233.000 q^{49} -42.0000 q^{51} +338.000 q^{53} -260.000 q^{55} -60.0000 q^{57} +100.000 q^{59} -742.000 q^{61} +216.000 q^{63} +110.000 q^{65} -84.0000 q^{67} +504.000 q^{69} +328.000 q^{71} -38.0000 q^{73} +75.0000 q^{75} +1248.00 q^{77} +240.000 q^{79} +81.0000 q^{81} +1212.00 q^{83} +70.0000 q^{85} -690.000 q^{87} +330.000 q^{89} -528.000 q^{91} +864.000 q^{93} +100.000 q^{95} +866.000 q^{97} +468.000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 24.0000 1.29588 0.647939 − 0.761692i \(-0.275631\pi\)
0.647939 + 0.761692i \(0.275631\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 52.0000 1.42533 0.712663 − 0.701506i \(-0.247489\pi\)
0.712663 + 0.701506i \(0.247489\pi\)
\(12\) 0 0
\(13\) −22.0000 −0.469362 −0.234681 − 0.972072i \(-0.575405\pi\)
−0.234681 + 0.972072i \(0.575405\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) −14.0000 −0.199735 −0.0998676 − 0.995001i \(-0.531842\pi\)
−0.0998676 + 0.995001i \(0.531842\pi\)
\(18\) 0 0
\(19\) −20.0000 −0.241490 −0.120745 − 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) 0 0
\(21\) 72.0000 0.748176
\(22\) 0 0
\(23\) 168.000 1.52306 0.761531 − 0.648129i \(-0.224448\pi\)
0.761531 + 0.648129i \(0.224448\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −230.000 −1.47276 −0.736378 − 0.676570i \(-0.763465\pi\)
−0.736378 + 0.676570i \(0.763465\pi\)
\(30\) 0 0
\(31\) 288.000 1.66859 0.834296 − 0.551317i \(-0.185875\pi\)
0.834296 + 0.551317i \(0.185875\pi\)
\(32\) 0 0
\(33\) 156.000 0.822913
\(34\) 0 0
\(35\) −120.000 −0.579534
\(36\) 0 0
\(37\) 34.0000 0.151069 0.0755347 − 0.997143i \(-0.475934\pi\)
0.0755347 + 0.997143i \(0.475934\pi\)
\(38\) 0 0
\(39\) −66.0000 −0.270986
\(40\) 0 0
\(41\) 122.000 0.464712 0.232356 − 0.972631i \(-0.425357\pi\)
0.232356 + 0.972631i \(0.425357\pi\)
\(42\) 0 0
\(43\) −188.000 −0.666738 −0.333369 − 0.942796i \(-0.608185\pi\)
−0.333369 + 0.942796i \(0.608185\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) −256.000 −0.794499 −0.397249 − 0.917711i \(-0.630035\pi\)
−0.397249 + 0.917711i \(0.630035\pi\)
\(48\) 0 0
\(49\) 233.000 0.679300
\(50\) 0 0
\(51\) −42.0000 −0.115317
\(52\) 0 0
\(53\) 338.000 0.875998 0.437999 − 0.898976i \(-0.355687\pi\)
0.437999 + 0.898976i \(0.355687\pi\)
\(54\) 0 0
\(55\) −260.000 −0.637425
\(56\) 0 0
\(57\) −60.0000 −0.139424
\(58\) 0 0
\(59\) 100.000 0.220659 0.110330 − 0.993895i \(-0.464809\pi\)
0.110330 + 0.993895i \(0.464809\pi\)
\(60\) 0 0
\(61\) −742.000 −1.55743 −0.778716 − 0.627376i \(-0.784129\pi\)
−0.778716 + 0.627376i \(0.784129\pi\)
\(62\) 0 0
\(63\) 216.000 0.431959
\(64\) 0 0
\(65\) 110.000 0.209905
\(66\) 0 0
\(67\) −84.0000 −0.153168 −0.0765838 − 0.997063i \(-0.524401\pi\)
−0.0765838 + 0.997063i \(0.524401\pi\)
\(68\) 0 0
\(69\) 504.000 0.879340
\(70\) 0 0
\(71\) 328.000 0.548260 0.274130 − 0.961693i \(-0.411610\pi\)
0.274130 + 0.961693i \(0.411610\pi\)
\(72\) 0 0
\(73\) −38.0000 −0.0609255 −0.0304628 − 0.999536i \(-0.509698\pi\)
−0.0304628 + 0.999536i \(0.509698\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) 1248.00 1.84705
\(78\) 0 0
\(79\) 240.000 0.341799 0.170899 − 0.985288i \(-0.445333\pi\)
0.170899 + 0.985288i \(0.445333\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1212.00 1.60282 0.801411 − 0.598114i \(-0.204083\pi\)
0.801411 + 0.598114i \(0.204083\pi\)
\(84\) 0 0
\(85\) 70.0000 0.0893243
\(86\) 0 0
\(87\) −690.000 −0.850296
\(88\) 0 0
\(89\) 330.000 0.393033 0.196516 − 0.980501i \(-0.437037\pi\)
0.196516 + 0.980501i \(0.437037\pi\)
\(90\) 0 0
\(91\) −528.000 −0.608236
\(92\) 0 0
\(93\) 864.000 0.963362
\(94\) 0 0
\(95\) 100.000 0.107998
\(96\) 0 0
\(97\) 866.000 0.906484 0.453242 − 0.891387i \(-0.350267\pi\)
0.453242 + 0.891387i \(0.350267\pi\)
\(98\) 0 0
\(99\) 468.000 0.475109
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.a.ba.1.1 1
4.3 odd 2 960.4.a.b.1.1 1
8.3 odd 2 15.4.a.a.1.1 ✓ 1
8.5 even 2 240.4.a.e.1.1 1
24.5 odd 2 720.4.a.n.1.1 1
24.11 even 2 45.4.a.c.1.1 1
40.3 even 4 75.4.b.b.49.1 2
40.13 odd 4 1200.4.f.b.49.1 2
40.19 odd 2 75.4.a.b.1.1 1
40.27 even 4 75.4.b.b.49.2 2
40.29 even 2 1200.4.a.t.1.1 1
40.37 odd 4 1200.4.f.b.49.2 2
56.27 even 2 735.4.a.e.1.1 1
72.11 even 6 405.4.e.i.271.1 2
72.43 odd 6 405.4.e.g.271.1 2
72.59 even 6 405.4.e.i.136.1 2
72.67 odd 6 405.4.e.g.136.1 2
88.43 even 2 1815.4.a.e.1.1 1
120.59 even 2 225.4.a.f.1.1 1
120.83 odd 4 225.4.b.e.199.2 2
120.107 odd 4 225.4.b.e.199.1 2
168.83 odd 2 2205.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 ✓ 1 8.3 odd 2
45.4.a.c.1.1 1 24.11 even 2
75.4.a.b.1.1 1 40.19 odd 2
75.4.b.b.49.1 2 40.3 even 4
75.4.b.b.49.2 2 40.27 even 4
225.4.a.f.1.1 1 120.59 even 2
225.4.b.e.199.1 2 120.107 odd 4
225.4.b.e.199.2 2 120.83 odd 4
240.4.a.e.1.1 1 8.5 even 2
405.4.e.g.136.1 2 72.67 odd 6
405.4.e.g.271.1 2 72.43 odd 6
405.4.e.i.136.1 2 72.59 even 6
405.4.e.i.271.1 2 72.11 even 6
720.4.a.n.1.1 1 24.5 odd 2
735.4.a.e.1.1 1 56.27 even 2
960.4.a.b.1.1 1 4.3 odd 2
960.4.a.ba.1.1 1 1.1 even 1 trivial
1200.4.a.t.1.1 1 40.29 even 2
1200.4.f.b.49.1 2 40.13 odd 4
1200.4.f.b.49.2 2 40.37 odd 4
1815.4.a.e.1.1 1 88.43 even 2
2205.4.a.l.1.1 1 168.83 odd 2