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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4800,2,Mod(1249,4800)] 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("4800.1249"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-4,0,0, 0,0,0,0,0,0,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.3281929702\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 192)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.3
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 4800.1249
Dual form 4800.2.d.j.1249.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.46410i q^{7} +1.00000 q^{9} -6.00000i q^{17} -4.00000i q^{19} -3.46410i q^{21} +6.92820i q^{23} -1.00000 q^{27} -3.46410i q^{29} +3.46410 q^{31} -6.92820 q^{37} -6.00000 q^{41} +4.00000 q^{43} -6.92820i q^{47} -5.00000 q^{49} +6.00000i q^{51} +3.46410 q^{53} +4.00000i q^{57} -12.0000i q^{59} -6.92820i q^{61} +3.46410i q^{63} -4.00000 q^{67} -6.92820i q^{69} -6.92820 q^{71} +2.00000i q^{73} +10.3923 q^{79} +1.00000 q^{81} +3.46410i q^{87} -6.00000 q^{89} -3.46410 q^{93} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{9} - 4 q^{27} - 24 q^{41} + 16 q^{43} - 20 q^{49} - 16 q^{67} + 4 q^{81} - 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(4351\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.46410i 1.30931i 0.755929 + 0.654654i \(0.227186\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.00000i − 1.45521i −0.685994 − 0.727607i \(-0.740633\pi\)
0.685994 − 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 − 0.458831i \(-0.848268\pi\)
0.888523 − 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) − 3.46410i − 0.755929i
\(22\) 0 0
\(23\) 6.92820i 1.44463i 0.691564 + 0.722315i \(0.256922\pi\)
−0.691564 + 0.722315i \(0.743078\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) − 3.46410i − 0.643268i −0.946864 − 0.321634i \(-0.895768\pi\)
0.946864 − 0.321634i \(-0.104232\pi\)
\(30\) 0 0
\(31\) 3.46410 0.622171 0.311086 − 0.950382i \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.92820 −1.13899 −0.569495 − 0.821995i \(-0.692861\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 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
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.92820i − 1.01058i −0.862949 − 0.505291i \(-0.831385\pi\)
0.862949 − 0.505291i \(-0.168615\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 6.00000i 0.840168i
\(52\) 0 0
\(53\) 3.46410 0.475831 0.237915 − 0.971286i \(-0.423536\pi\)
0.237915 + 0.971286i \(0.423536\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) − 12.0000i − 1.56227i −0.624364 − 0.781133i \(-0.714642\pi\)
0.624364 − 0.781133i \(-0.285358\pi\)
\(60\) 0 0
\(61\) − 6.92820i − 0.887066i −0.896258 − 0.443533i \(-0.853725\pi\)
0.896258 − 0.443533i \(-0.146275\pi\)
\(62\) 0 0
\(63\) 3.46410i 0.436436i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) − 6.92820i − 0.834058i
\(70\) 0 0
\(71\) −6.92820 −0.822226 −0.411113 − 0.911584i \(-0.634860\pi\)
−0.411113 + 0.911584i \(0.634860\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.3923 1.16923 0.584613 − 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.46410i 0.371391i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.46410 −0.359211
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 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 4800.2.d.j.1249.3 4
4.3 odd 2 4800.2.d.o.1249.1 4
5.2 odd 4 192.2.d.a.97.4 yes 4
5.3 odd 4 4800.2.k.j.2401.2 4
5.4 even 2 4800.2.d.o.1249.2 4
8.3 odd 2 inner 4800.2.d.j.1249.1 4
8.5 even 2 4800.2.d.o.1249.3 4
15.2 even 4 576.2.d.b.289.1 4
20.3 even 4 4800.2.k.j.2401.3 4
20.7 even 4 192.2.d.a.97.2 yes 4
20.19 odd 2 inner 4800.2.d.j.1249.4 4
40.3 even 4 4800.2.k.j.2401.1 4
40.13 odd 4 4800.2.k.j.2401.4 4
40.19 odd 2 4800.2.d.o.1249.4 4
40.27 even 4 192.2.d.a.97.3 yes 4
40.29 even 2 inner 4800.2.d.j.1249.2 4
40.37 odd 4 192.2.d.a.97.1 ✓ 4
60.47 odd 4 576.2.d.b.289.2 4
80.27 even 4 768.2.a.j.1.1 2
80.37 odd 4 768.2.a.k.1.1 2
80.67 even 4 768.2.a.k.1.2 2
80.77 odd 4 768.2.a.j.1.2 2
120.77 even 4 576.2.d.b.289.3 4
120.107 odd 4 576.2.d.b.289.4 4
240.77 even 4 2304.2.a.s.1.1 2
240.107 odd 4 2304.2.a.s.1.2 2
240.197 even 4 2304.2.a.u.1.2 2
240.227 odd 4 2304.2.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.2.d.a.97.1 ✓ 4 40.37 odd 4
192.2.d.a.97.2 yes 4 20.7 even 4
192.2.d.a.97.3 yes 4 40.27 even 4
192.2.d.a.97.4 yes 4 5.2 odd 4
576.2.d.b.289.1 4 15.2 even 4
576.2.d.b.289.2 4 60.47 odd 4
576.2.d.b.289.3 4 120.77 even 4
576.2.d.b.289.4 4 120.107 odd 4
768.2.a.j.1.1 2 80.27 even 4
768.2.a.j.1.2 2 80.77 odd 4
768.2.a.k.1.1 2 80.37 odd 4
768.2.a.k.1.2 2 80.67 even 4
2304.2.a.s.1.1 2 240.77 even 4
2304.2.a.s.1.2 2 240.107 odd 4
2304.2.a.u.1.1 2 240.227 odd 4
2304.2.a.u.1.2 2 240.197 even 4
4800.2.d.j.1249.1 4 8.3 odd 2 inner
4800.2.d.j.1249.2 4 40.29 even 2 inner
4800.2.d.j.1249.3 4 1.1 even 1 trivial
4800.2.d.j.1249.4 4 20.19 odd 2 inner
4800.2.d.o.1249.1 4 4.3 odd 2
4800.2.d.o.1249.2 4 5.4 even 2
4800.2.d.o.1249.3 4 8.5 even 2
4800.2.d.o.1249.4 4 40.19 odd 2
4800.2.k.j.2401.1 4 40.3 even 4
4800.2.k.j.2401.2 4 5.3 odd 4
4800.2.k.j.2401.3 4 20.3 even 4
4800.2.k.j.2401.4 4 40.13 odd 4