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Properties

Label 960.4.a.j.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 $1$

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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(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,-5,0,32,0,9,0,60] 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 30)
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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -5.00000 q^{5} +32.0000 q^{7} +9.00000 q^{9} +60.0000 q^{11} +34.0000 q^{13} +15.0000 q^{15} +42.0000 q^{17} +76.0000 q^{19} -96.0000 q^{21} +25.0000 q^{25} -27.0000 q^{27} -6.00000 q^{29} -232.000 q^{31} -180.000 q^{33} -160.000 q^{35} -134.000 q^{37} -102.000 q^{39} +234.000 q^{41} +412.000 q^{43} -45.0000 q^{45} -360.000 q^{47} +681.000 q^{49} -126.000 q^{51} -222.000 q^{53} -300.000 q^{55} -228.000 q^{57} -660.000 q^{59} +490.000 q^{61} +288.000 q^{63} -170.000 q^{65} -812.000 q^{67} +120.000 q^{71} +746.000 q^{73} -75.0000 q^{75} +1920.00 q^{77} +152.000 q^{79} +81.0000 q^{81} +804.000 q^{83} -210.000 q^{85} +18.0000 q^{87} -678.000 q^{89} +1088.00 q^{91} +696.000 q^{93} -380.000 q^{95} +194.000 q^{97} +540.000 q^{99} +O(q^{100})\)

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
\(7\) 32.0000 1.72784 0.863919 − 0.503631i \(-0.168003\pi\)
0.863919 + 0.503631i \(0.168003\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 60.0000 1.64461 0.822304 − 0.569049i \(-0.192689\pi\)
0.822304 + 0.569049i \(0.192689\pi\)
\(12\) 0 0
\(13\) 34.0000 0.725377 0.362689 − 0.931910i \(-0.381859\pi\)
0.362689 + 0.931910i \(0.381859\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) 42.0000 0.599206 0.299603 − 0.954064i \(-0.403146\pi\)
0.299603 + 0.954064i \(0.403146\pi\)
\(18\) 0 0
\(19\) 76.0000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −96.0000 −0.997567
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −0.0384197 −0.0192099 − 0.999815i \(-0.506115\pi\)
−0.0192099 + 0.999815i \(0.506115\pi\)
\(30\) 0 0
\(31\) −232.000 −1.34414 −0.672071 − 0.740486i \(-0.734595\pi\)
−0.672071 + 0.740486i \(0.734595\pi\)
\(32\) 0 0
\(33\) −180.000 −0.949514
\(34\) 0 0
\(35\) −160.000 −0.772712
\(36\) 0 0
\(37\) −134.000 −0.595391 −0.297695 − 0.954661i \(-0.596218\pi\)
−0.297695 + 0.954661i \(0.596218\pi\)
\(38\) 0 0
\(39\) −102.000 −0.418797
\(40\) 0 0
\(41\) 234.000 0.891333 0.445667 − 0.895199i \(-0.352967\pi\)
0.445667 + 0.895199i \(0.352967\pi\)
\(42\) 0 0
\(43\) 412.000 1.46115 0.730575 − 0.682833i \(-0.239252\pi\)
0.730575 + 0.682833i \(0.239252\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) −360.000 −1.11726 −0.558632 − 0.829416i \(-0.688674\pi\)
−0.558632 + 0.829416i \(0.688674\pi\)
\(48\) 0 0
\(49\) 681.000 1.98542
\(50\) 0 0
\(51\) −126.000 −0.345952
\(52\) 0 0
\(53\) −222.000 −0.575359 −0.287680 − 0.957727i \(-0.592884\pi\)
−0.287680 + 0.957727i \(0.592884\pi\)
\(54\) 0 0
\(55\) −300.000 −0.735491
\(56\) 0 0
\(57\) −228.000 −0.529813
\(58\) 0 0
\(59\) −660.000 −1.45635 −0.728175 − 0.685391i \(-0.759631\pi\)
−0.728175 + 0.685391i \(0.759631\pi\)
\(60\) 0 0
\(61\) 490.000 1.02849 0.514246 − 0.857642i \(-0.328072\pi\)
0.514246 + 0.857642i \(0.328072\pi\)
\(62\) 0 0
\(63\) 288.000 0.575946
\(64\) 0 0
\(65\) −170.000 −0.324399
\(66\) 0 0
\(67\) −812.000 −1.48062 −0.740310 − 0.672265i \(-0.765321\pi\)
−0.740310 + 0.672265i \(0.765321\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 120.000 0.200583 0.100291 − 0.994958i \(-0.468022\pi\)
0.100291 + 0.994958i \(0.468022\pi\)
\(72\) 0 0
\(73\) 746.000 1.19606 0.598032 − 0.801472i \(-0.295949\pi\)
0.598032 + 0.801472i \(0.295949\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) 0 0
\(77\) 1920.00 2.84161
\(78\) 0 0
\(79\) 152.000 0.216473 0.108236 − 0.994125i \(-0.465480\pi\)
0.108236 + 0.994125i \(0.465480\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 804.000 1.06326 0.531629 − 0.846977i \(-0.321580\pi\)
0.531629 + 0.846977i \(0.321580\pi\)
\(84\) 0 0
\(85\) −210.000 −0.267973
\(86\) 0 0
\(87\) 18.0000 0.0221816
\(88\) 0 0
\(89\) −678.000 −0.807504 −0.403752 − 0.914868i \(-0.632294\pi\)
−0.403752 + 0.914868i \(0.632294\pi\)
\(90\) 0 0
\(91\) 1088.00 1.25333
\(92\) 0 0
\(93\) 696.000 0.776041
\(94\) 0 0
\(95\) −380.000 −0.410391
\(96\) 0 0
\(97\) 194.000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 540.000 0.548202
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.j.1.1 1
4.3 odd 2 960.4.a.s.1.1 1
8.3 odd 2 240.4.a.c.1.1 1
8.5 even 2 30.4.a.a.1.1 ✓ 1
24.5 odd 2 90.4.a.d.1.1 1
24.11 even 2 720.4.a.b.1.1 1
40.3 even 4 1200.4.f.u.49.1 2
40.13 odd 4 150.4.c.a.49.2 2
40.19 odd 2 1200.4.a.bk.1.1 1
40.27 even 4 1200.4.f.u.49.2 2
40.29 even 2 150.4.a.e.1.1 1
40.37 odd 4 150.4.c.a.49.1 2
56.13 odd 2 1470.4.a.a.1.1 1
72.5 odd 6 810.4.e.e.541.1 2
72.13 even 6 810.4.e.m.541.1 2
72.29 odd 6 810.4.e.e.271.1 2
72.61 even 6 810.4.e.m.271.1 2
120.29 odd 2 450.4.a.b.1.1 1
120.53 even 4 450.4.c.k.199.1 2
120.77 even 4 450.4.c.k.199.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.4.a.a.1.1 ✓ 1 8.5 even 2
90.4.a.d.1.1 1 24.5 odd 2
150.4.a.e.1.1 1 40.29 even 2
150.4.c.a.49.1 2 40.37 odd 4
150.4.c.a.49.2 2 40.13 odd 4
240.4.a.c.1.1 1 8.3 odd 2
450.4.a.b.1.1 1 120.29 odd 2
450.4.c.k.199.1 2 120.53 even 4
450.4.c.k.199.2 2 120.77 even 4
720.4.a.b.1.1 1 24.11 even 2
810.4.e.e.271.1 2 72.29 odd 6
810.4.e.e.541.1 2 72.5 odd 6
810.4.e.m.271.1 2 72.61 even 6
810.4.e.m.541.1 2 72.13 even 6
960.4.a.j.1.1 1 1.1 even 1 trivial
960.4.a.s.1.1 1 4.3 odd 2
1200.4.a.bk.1.1 1 40.19 odd 2
1200.4.f.u.49.1 2 40.3 even 4
1200.4.f.u.49.2 2 40.27 even 4
1470.4.a.a.1.1 1 56.13 odd 2