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Properties

Label 960.2.a.e.1.1
Level $960$
Weight $2$
Character 960.1
Self dual yes
Analytic conductor $7.666$
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] = [960,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-1,0,1,0,-4,0,1,0,0] 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: \(7.66563859404\)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} -2.00000 q^{13} -1.00000 q^{15} +6.00000 q^{17} +4.00000 q^{19} +4.00000 q^{21} +1.00000 q^{25} -1.00000 q^{27} +6.00000 q^{29} +8.00000 q^{31} -4.00000 q^{35} -2.00000 q^{37} +2.00000 q^{39} -6.00000 q^{41} +4.00000 q^{43} +1.00000 q^{45} +9.00000 q^{49} -6.00000 q^{51} +6.00000 q^{53} -4.00000 q^{57} +10.0000 q^{61} -4.00000 q^{63} -2.00000 q^{65} +4.00000 q^{67} +2.00000 q^{73} -1.00000 q^{75} +8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +6.00000 q^{85} -6.00000 q^{87} +18.0000 q^{89} +8.00000 q^{91} -8.00000 q^{93} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
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 960.2.a.e.1.1 1
3.2 odd 2 2880.2.a.a.1.1 1
4.3 odd 2 960.2.a.p.1.1 1
5.2 odd 4 4800.2.f.p.3649.2 2
5.3 odd 4 4800.2.f.p.3649.1 2
5.4 even 2 4800.2.a.cq.1.1 1
8.3 odd 2 240.2.a.b.1.1 1
8.5 even 2 30.2.a.a.1.1 ✓ 1
12.11 even 2 2880.2.a.q.1.1 1
16.3 odd 4 3840.2.k.f.1921.2 2
16.5 even 4 3840.2.k.y.1921.2 2
16.11 odd 4 3840.2.k.f.1921.1 2
16.13 even 4 3840.2.k.y.1921.1 2
20.3 even 4 4800.2.f.w.3649.2 2
20.7 even 4 4800.2.f.w.3649.1 2
20.19 odd 2 4800.2.a.d.1.1 1
24.5 odd 2 90.2.a.c.1.1 1
24.11 even 2 720.2.a.j.1.1 1
40.3 even 4 1200.2.f.e.49.1 2
40.13 odd 4 150.2.c.a.49.2 2
40.19 odd 2 1200.2.a.k.1.1 1
40.27 even 4 1200.2.f.e.49.2 2
40.29 even 2 150.2.a.b.1.1 1
40.37 odd 4 150.2.c.a.49.1 2
56.5 odd 6 1470.2.i.q.361.1 2
56.13 odd 2 1470.2.a.d.1.1 1
56.37 even 6 1470.2.i.o.361.1 2
56.45 odd 6 1470.2.i.q.961.1 2
56.53 even 6 1470.2.i.o.961.1 2
72.5 odd 6 810.2.e.b.541.1 2
72.13 even 6 810.2.e.l.541.1 2
72.29 odd 6 810.2.e.b.271.1 2
72.61 even 6 810.2.e.l.271.1 2
88.21 odd 2 3630.2.a.w.1.1 1
104.5 odd 4 5070.2.b.k.1351.2 2
104.21 odd 4 5070.2.b.k.1351.1 2
104.77 even 2 5070.2.a.w.1.1 1
120.29 odd 2 450.2.a.d.1.1 1
120.53 even 4 450.2.c.b.199.1 2
120.59 even 2 3600.2.a.f.1.1 1
120.77 even 4 450.2.c.b.199.2 2
120.83 odd 4 3600.2.f.i.2449.1 2
120.107 odd 4 3600.2.f.i.2449.2 2
136.101 even 2 8670.2.a.g.1.1 1
168.125 even 2 4410.2.a.z.1.1 1
280.69 odd 2 7350.2.a.ct.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 ✓ 1 8.5 even 2
90.2.a.c.1.1 1 24.5 odd 2
150.2.a.b.1.1 1 40.29 even 2
150.2.c.a.49.1 2 40.37 odd 4
150.2.c.a.49.2 2 40.13 odd 4
240.2.a.b.1.1 1 8.3 odd 2
450.2.a.d.1.1 1 120.29 odd 2
450.2.c.b.199.1 2 120.53 even 4
450.2.c.b.199.2 2 120.77 even 4
720.2.a.j.1.1 1 24.11 even 2
810.2.e.b.271.1 2 72.29 odd 6
810.2.e.b.541.1 2 72.5 odd 6
810.2.e.l.271.1 2 72.61 even 6
810.2.e.l.541.1 2 72.13 even 6
960.2.a.e.1.1 1 1.1 even 1 trivial
960.2.a.p.1.1 1 4.3 odd 2
1200.2.a.k.1.1 1 40.19 odd 2
1200.2.f.e.49.1 2 40.3 even 4
1200.2.f.e.49.2 2 40.27 even 4
1470.2.a.d.1.1 1 56.13 odd 2
1470.2.i.o.361.1 2 56.37 even 6
1470.2.i.o.961.1 2 56.53 even 6
1470.2.i.q.361.1 2 56.5 odd 6
1470.2.i.q.961.1 2 56.45 odd 6
2880.2.a.a.1.1 1 3.2 odd 2
2880.2.a.q.1.1 1 12.11 even 2
3600.2.a.f.1.1 1 120.59 even 2
3600.2.f.i.2449.1 2 120.83 odd 4
3600.2.f.i.2449.2 2 120.107 odd 4
3630.2.a.w.1.1 1 88.21 odd 2
3840.2.k.f.1921.1 2 16.11 odd 4
3840.2.k.f.1921.2 2 16.3 odd 4
3840.2.k.y.1921.1 2 16.13 even 4
3840.2.k.y.1921.2 2 16.5 even 4
4410.2.a.z.1.1 1 168.125 even 2
4800.2.a.d.1.1 1 20.19 odd 2
4800.2.a.cq.1.1 1 5.4 even 2
4800.2.f.p.3649.1 2 5.3 odd 4
4800.2.f.p.3649.2 2 5.2 odd 4
4800.2.f.w.3649.1 2 20.7 even 4
4800.2.f.w.3649.2 2 20.3 even 4
5070.2.a.w.1.1 1 104.77 even 2
5070.2.b.k.1351.1 2 104.21 odd 4
5070.2.b.k.1351.2 2 104.5 odd 4
7350.2.a.ct.1.1 1 280.69 odd 2
8670.2.a.g.1.1 1 136.101 even 2