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Properties

Label 960.2.o.a.959.4
Level $960$
Weight $2$
Character 960.959
Analytic conductor $7.666$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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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(959,960)] 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("960.959"); 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([1, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 959.4
Root \(0.809017 + 1.40126i\) of defining polynomial
Character \(\chi\) \(=\) 960.959
Dual form 960.2.o.a.959.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +2.23607 q^{5} -3.00000 q^{9} +3.87298i q^{15} +4.47214 q^{17} +7.74597i q^{19} +3.46410i q^{23} +5.00000 q^{25} -5.19615i q^{27} +7.74597i q^{31} -6.70820 q^{45} -10.3923i q^{47} -7.00000 q^{49} +7.74597i q^{51} +4.47214 q^{53} -13.4164 q^{57} +2.00000 q^{61} -6.00000 q^{69} +8.66025i q^{75} +7.74597i q^{79} +9.00000 q^{81} +3.46410i q^{83} +10.0000 q^{85} -13.4164 q^{93} +17.3205i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 20 q^{25} - 28 q^{49} + 8 q^{61} - 24 q^{69} + 36 q^{81} + 40 q^{85}+O(q^{100}) \) 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
\(3\) 1.73205i 1.00000i
\(4\) 0 0
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 3.87298i 1.00000i
\(16\) 0 0
\(17\) 4.47214 1.08465 0.542326 − 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 0 0
\(19\) 7.74597i 1.77705i 0.458831 + 0.888523i \(0.348268\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410i 0.722315i 0.932505 + 0.361158i \(0.117618\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) − 5.19615i − 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 7.74597i 1.39122i 0.718421 + 0.695608i \(0.244865\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −6.70820 −1.00000
\(46\) 0 0
\(47\) − 10.3923i − 1.51587i −0.652328 − 0.757937i \(-0.726208\pi\)
0.652328 − 0.757937i \(-0.273792\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 7.74597i 1.08465i
\(52\) 0 0
\(53\) 4.47214 0.614295 0.307148 − 0.951662i \(-0.400625\pi\)
0.307148 + 0.951662i \(0.400625\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −13.4164 −1.77705
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 8.66025i 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7.74597i 0.871489i 0.900070 + 0.435745i \(0.143515\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 3.46410i 0.380235i 0.981761 + 0.190117i \(0.0608868\pi\)
−0.981761 + 0.190117i \(0.939113\pi\)
\(84\) 0 0
\(85\) 10.0000 1.08465
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −13.4164 −1.39122
\(94\) 0 0
\(95\) 17.3205i 1.77705i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 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.o.a.959.4 4
3.2 odd 2 inner 960.2.o.a.959.1 4
4.3 odd 2 inner 960.2.o.a.959.2 4
5.4 even 2 inner 960.2.o.a.959.1 4
8.3 odd 2 60.2.h.b.59.3 yes 4
8.5 even 2 60.2.h.b.59.4 yes 4
12.11 even 2 inner 960.2.o.a.959.3 4
15.14 odd 2 CM 960.2.o.a.959.4 4
20.19 odd 2 inner 960.2.o.a.959.3 4
24.5 odd 2 60.2.h.b.59.1 ✓ 4
24.11 even 2 60.2.h.b.59.2 yes 4
40.3 even 4 300.2.e.a.251.1 4
40.13 odd 4 300.2.e.a.251.3 4
40.19 odd 2 60.2.h.b.59.2 yes 4
40.27 even 4 300.2.e.a.251.4 4
40.29 even 2 60.2.h.b.59.1 ✓ 4
40.37 odd 4 300.2.e.a.251.2 4
60.59 even 2 inner 960.2.o.a.959.2 4
120.29 odd 2 60.2.h.b.59.4 yes 4
120.53 even 4 300.2.e.a.251.2 4
120.59 even 2 60.2.h.b.59.3 yes 4
120.77 even 4 300.2.e.a.251.3 4
120.83 odd 4 300.2.e.a.251.4 4
120.107 odd 4 300.2.e.a.251.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.2.h.b.59.1 ✓ 4 24.5 odd 2
60.2.h.b.59.1 ✓ 4 40.29 even 2
60.2.h.b.59.2 yes 4 24.11 even 2
60.2.h.b.59.2 yes 4 40.19 odd 2
60.2.h.b.59.3 yes 4 8.3 odd 2
60.2.h.b.59.3 yes 4 120.59 even 2
60.2.h.b.59.4 yes 4 8.5 even 2
60.2.h.b.59.4 yes 4 120.29 odd 2
300.2.e.a.251.1 4 40.3 even 4
300.2.e.a.251.1 4 120.107 odd 4
300.2.e.a.251.2 4 40.37 odd 4
300.2.e.a.251.2 4 120.53 even 4
300.2.e.a.251.3 4 40.13 odd 4
300.2.e.a.251.3 4 120.77 even 4
300.2.e.a.251.4 4 40.27 even 4
300.2.e.a.251.4 4 120.83 odd 4
960.2.o.a.959.1 4 3.2 odd 2 inner
960.2.o.a.959.1 4 5.4 even 2 inner
960.2.o.a.959.2 4 4.3 odd 2 inner
960.2.o.a.959.2 4 60.59 even 2 inner
960.2.o.a.959.3 4 12.11 even 2 inner
960.2.o.a.959.3 4 20.19 odd 2 inner
960.2.o.a.959.4 4 1.1 even 1 trivial
960.2.o.a.959.4 4 15.14 odd 2 CM