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Properties

Label 960.4.a.t.1.1
Level $960$
Weight $4$
Character 960.1
Self dual yes
Analytic conductor $56.642$
Analytic rank $1$
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,-32,0,9,0,64] 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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -5.00000 q^{5} -32.0000 q^{7} +9.00000 q^{9} +64.0000 q^{11} +6.00000 q^{13} -15.0000 q^{15} +38.0000 q^{17} -116.000 q^{19} -96.0000 q^{21} +120.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +122.000 q^{29} -164.000 q^{31} +192.000 q^{33} +160.000 q^{35} -146.000 q^{37} +18.0000 q^{39} -238.000 q^{41} -148.000 q^{43} -45.0000 q^{45} +184.000 q^{47} +681.000 q^{49} +114.000 q^{51} -470.000 q^{53} -320.000 q^{55} -348.000 q^{57} -216.000 q^{59} -806.000 q^{61} -288.000 q^{63} -30.0000 q^{65} -732.000 q^{67} +360.000 q^{69} -264.000 q^{71} -638.000 q^{73} +75.0000 q^{75} -2048.00 q^{77} -596.000 q^{79} +81.0000 q^{81} -884.000 q^{83} -190.000 q^{85} +366.000 q^{87} +930.000 q^{89} -192.000 q^{91} -492.000 q^{93} +580.000 q^{95} +322.000 q^{97} +576.000 q^{99} +O(q^{100})\)

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 \(-0.831997\pi\)
−0.863919 + 0.503631i \(0.831997\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 64.0000 1.75425 0.877124 − 0.480264i \(-0.159459\pi\)
0.877124 + 0.480264i \(0.159459\pi\)
\(12\) 0 0
\(13\) 6.00000 0.128008 0.0640039 − 0.997950i \(-0.479613\pi\)
0.0640039 + 0.997950i \(0.479613\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) 38.0000 0.542138 0.271069 − 0.962560i \(-0.412623\pi\)
0.271069 + 0.962560i \(0.412623\pi\)
\(18\) 0 0
\(19\) −116.000 −1.40064 −0.700322 − 0.713827i \(-0.746960\pi\)
−0.700322 + 0.713827i \(0.746960\pi\)
\(20\) 0 0
\(21\) −96.0000 −0.997567
\(22\) 0 0
\(23\) 120.000 1.08790 0.543951 − 0.839117i \(-0.316928\pi\)
0.543951 + 0.839117i \(0.316928\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 122.000 0.781201 0.390601 − 0.920560i \(-0.372267\pi\)
0.390601 + 0.920560i \(0.372267\pi\)
\(30\) 0 0
\(31\) −164.000 −0.950170 −0.475085 − 0.879940i \(-0.657583\pi\)
−0.475085 + 0.879940i \(0.657583\pi\)
\(32\) 0 0
\(33\) 192.000 1.01282
\(34\) 0 0
\(35\) 160.000 0.772712
\(36\) 0 0
\(37\) −146.000 −0.648710 −0.324355 − 0.945936i \(-0.605147\pi\)
−0.324355 + 0.945936i \(0.605147\pi\)
\(38\) 0 0
\(39\) 18.0000 0.0739053
\(40\) 0 0
\(41\) −238.000 −0.906570 −0.453285 − 0.891366i \(-0.649748\pi\)
−0.453285 + 0.891366i \(0.649748\pi\)
\(42\) 0 0
\(43\) −148.000 −0.524879 −0.262439 − 0.964948i \(-0.584527\pi\)
−0.262439 + 0.964948i \(0.584527\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) 184.000 0.571046 0.285523 − 0.958372i \(-0.407833\pi\)
0.285523 + 0.958372i \(0.407833\pi\)
\(48\) 0 0
\(49\) 681.000 1.98542
\(50\) 0 0
\(51\) 114.000 0.313004
\(52\) 0 0
\(53\) −470.000 −1.21810 −0.609052 − 0.793131i \(-0.708450\pi\)
−0.609052 + 0.793131i \(0.708450\pi\)
\(54\) 0 0
\(55\) −320.000 −0.784523
\(56\) 0 0
\(57\) −348.000 −0.808662
\(58\) 0 0
\(59\) −216.000 −0.476624 −0.238312 − 0.971189i \(-0.576594\pi\)
−0.238312 + 0.971189i \(0.576594\pi\)
\(60\) 0 0
\(61\) −806.000 −1.69177 −0.845883 − 0.533369i \(-0.820926\pi\)
−0.845883 + 0.533369i \(0.820926\pi\)
\(62\) 0 0
\(63\) −288.000 −0.575946
\(64\) 0 0
\(65\) −30.0000 −0.0572468
\(66\) 0 0
\(67\) −732.000 −1.33475 −0.667373 − 0.744723i \(-0.732581\pi\)
−0.667373 + 0.744723i \(0.732581\pi\)
\(68\) 0 0
\(69\) 360.000 0.628100
\(70\) 0 0
\(71\) −264.000 −0.441282 −0.220641 − 0.975355i \(-0.570815\pi\)
−0.220641 + 0.975355i \(0.570815\pi\)
\(72\) 0 0
\(73\) −638.000 −1.02291 −0.511454 − 0.859311i \(-0.670893\pi\)
−0.511454 + 0.859311i \(0.670893\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) −2048.00 −3.03106
\(78\) 0 0
\(79\) −596.000 −0.848800 −0.424400 − 0.905475i \(-0.639515\pi\)
−0.424400 + 0.905475i \(0.639515\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −884.000 −1.16906 −0.584528 − 0.811374i \(-0.698720\pi\)
−0.584528 + 0.811374i \(0.698720\pi\)
\(84\) 0 0
\(85\) −190.000 −0.242452
\(86\) 0 0
\(87\) 366.000 0.451027
\(88\) 0 0
\(89\) 930.000 1.10764 0.553819 − 0.832637i \(-0.313170\pi\)
0.553819 + 0.832637i \(0.313170\pi\)
\(90\) 0 0
\(91\) −192.000 −0.221177
\(92\) 0 0
\(93\) −492.000 −0.548581
\(94\) 0 0
\(95\) 580.000 0.626387
\(96\) 0 0
\(97\) 322.000 0.337053 0.168527 − 0.985697i \(-0.446099\pi\)
0.168527 + 0.985697i \(0.446099\pi\)
\(98\) 0 0
\(99\) 576.000 0.584749
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.t.1.1 1
4.3 odd 2 960.4.a.i.1.1 1
8.3 odd 2 480.4.a.l.1.1 yes 1
8.5 even 2 480.4.a.c.1.1 ✓ 1
24.5 odd 2 1440.4.a.a.1.1 1
24.11 even 2 1440.4.a.j.1.1 1
40.19 odd 2 2400.4.a.a.1.1 1
40.29 even 2 2400.4.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.c.1.1 ✓ 1 8.5 even 2
480.4.a.l.1.1 yes 1 8.3 odd 2
960.4.a.i.1.1 1 4.3 odd 2
960.4.a.t.1.1 1 1.1 even 1 trivial
1440.4.a.a.1.1 1 24.5 odd 2
1440.4.a.j.1.1 1 24.11 even 2
2400.4.a.a.1.1 1 40.19 odd 2
2400.4.a.v.1.1 1 40.29 even 2