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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3552,2,Mod(2737,3552)] 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("3552.2737"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3552 = 2^{5} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3552.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.3628627980\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: no (minimal twist has level 888)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2737.16
Character \(\chi\) \(=\) 3552.2737
Dual form 3552.2.o.a.2737.15

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -0.598100 q^{5} +4.50845 q^{7} -1.00000 q^{9} +1.30412i q^{11} -3.44826 q^{13} -0.598100i q^{15} -4.39922i q^{17} -3.50187 q^{19} +4.50845i q^{21} +3.10879i q^{23} -4.64228 q^{25} -1.00000i q^{27} -5.02900 q^{29} +5.05333i q^{31} -1.30412 q^{33} -2.69651 q^{35} +(-3.40580 + 5.03990i) q^{37} -3.44826i q^{39} +1.71921 q^{41} -9.32710 q^{43} +0.598100 q^{45} -10.9753 q^{47} +13.3262 q^{49} +4.39922 q^{51} +9.44849i q^{53} -0.779996i q^{55} -3.50187i q^{57} +3.55467 q^{59} +7.87202 q^{61} -4.50845 q^{63} +2.06240 q^{65} +7.87891i q^{67} -3.10879 q^{69} -1.68222 q^{71} -3.11875 q^{73} -4.64228i q^{75} +5.87958i q^{77} +16.6650i q^{79} +1.00000 q^{81} -16.7465i q^{83} +2.63117i q^{85} -5.02900i q^{87} -5.64431i q^{89} -15.5463 q^{91} -5.05333 q^{93} +2.09447 q^{95} +7.07085i q^{97} -1.30412i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q - 8 q^{7} - 76 q^{9} + 84 q^{25} - 8 q^{41} + 60 q^{49} + 8 q^{63} + 48 q^{65} + 16 q^{71} + 24 q^{73} + 76 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3552\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(2369\) \(3073\) \(3109\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −0.598100 −0.267478 −0.133739 − 0.991017i \(-0.542698\pi\)
−0.133739 + 0.991017i \(0.542698\pi\)
\(6\) 0 0
\(7\) 4.50845 1.70404 0.852018 − 0.523513i \(-0.175379\pi\)
0.852018 + 0.523513i \(0.175379\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.30412i 0.393208i 0.980483 + 0.196604i \(0.0629914\pi\)
−0.980483 + 0.196604i \(0.937009\pi\)
\(12\) 0 0
\(13\) −3.44826 −0.956375 −0.478188 − 0.878258i \(-0.658706\pi\)
−0.478188 + 0.878258i \(0.658706\pi\)
\(14\) 0 0
\(15\) − 0.598100i − 0.154429i
\(16\) 0 0
\(17\) − 4.39922i − 1.06697i −0.845811 − 0.533483i \(-0.820883\pi\)
0.845811 − 0.533483i \(-0.179117\pi\)
\(18\) 0 0
\(19\) −3.50187 −0.803384 −0.401692 − 0.915775i \(-0.631578\pi\)
−0.401692 + 0.915775i \(0.631578\pi\)
\(20\) 0 0
\(21\) 4.50845i 0.983825i
\(22\) 0 0
\(23\) 3.10879i 0.648228i 0.946018 + 0.324114i \(0.105066\pi\)
−0.946018 + 0.324114i \(0.894934\pi\)
\(24\) 0 0
\(25\) −4.64228 −0.928455
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) −5.02900 −0.933862 −0.466931 − 0.884294i \(-0.654640\pi\)
−0.466931 + 0.884294i \(0.654640\pi\)
\(30\) 0 0
\(31\) 5.05333i 0.907606i 0.891102 + 0.453803i \(0.149933\pi\)
−0.891102 + 0.453803i \(0.850067\pi\)
\(32\) 0 0
\(33\) −1.30412 −0.227019
\(34\) 0 0
\(35\) −2.69651 −0.455793
\(36\) 0 0
\(37\) −3.40580 + 5.03990i −0.559910 + 0.828554i
\(38\) 0 0
\(39\) − 3.44826i − 0.552164i
\(40\) 0 0
\(41\) 1.71921 0.268496 0.134248 − 0.990948i \(-0.457138\pi\)
0.134248 + 0.990948i \(0.457138\pi\)
\(42\) 0 0
\(43\) −9.32710 −1.42237 −0.711185 − 0.703005i \(-0.751841\pi\)
−0.711185 + 0.703005i \(0.751841\pi\)
\(44\) 0 0
\(45\) 0.598100 0.0891595
\(46\) 0 0
\(47\) −10.9753 −1.60092 −0.800458 − 0.599388i \(-0.795411\pi\)
−0.800458 + 0.599388i \(0.795411\pi\)
\(48\) 0 0
\(49\) 13.3262 1.90374
\(50\) 0 0
\(51\) 4.39922 0.616014
\(52\) 0 0
\(53\) 9.44849i 1.29785i 0.760853 + 0.648925i \(0.224781\pi\)
−0.760853 + 0.648925i \(0.775219\pi\)
\(54\) 0 0
\(55\) − 0.779996i − 0.105175i
\(56\) 0 0
\(57\) − 3.50187i − 0.463834i
\(58\) 0 0
\(59\) 3.55467 0.462779 0.231389 − 0.972861i \(-0.425673\pi\)
0.231389 + 0.972861i \(0.425673\pi\)
\(60\) 0 0
\(61\) 7.87202 1.00791 0.503954 − 0.863730i \(-0.331878\pi\)
0.503954 + 0.863730i \(0.331878\pi\)
\(62\) 0 0
\(63\) −4.50845 −0.568012
\(64\) 0 0
\(65\) 2.06240 0.255810
\(66\) 0 0
\(67\) 7.87891i 0.962562i 0.876566 + 0.481281i \(0.159828\pi\)
−0.876566 + 0.481281i \(0.840172\pi\)
\(68\) 0 0
\(69\) −3.10879 −0.374254
\(70\) 0 0
\(71\) −1.68222 −0.199643 −0.0998213 − 0.995005i \(-0.531827\pi\)
−0.0998213 + 0.995005i \(0.531827\pi\)
\(72\) 0 0
\(73\) −3.11875 −0.365023 −0.182511 − 0.983204i \(-0.558423\pi\)
−0.182511 + 0.983204i \(0.558423\pi\)
\(74\) 0 0
\(75\) − 4.64228i − 0.536044i
\(76\) 0 0
\(77\) 5.87958i 0.670041i
\(78\) 0 0
\(79\) 16.6650i 1.87496i 0.348038 + 0.937480i \(0.386848\pi\)
−0.348038 + 0.937480i \(0.613152\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 16.7465i − 1.83816i −0.394067 − 0.919082i \(-0.628932\pi\)
0.394067 − 0.919082i \(-0.371068\pi\)
\(84\) 0 0
\(85\) 2.63117i 0.285390i
\(86\) 0 0
\(87\) − 5.02900i − 0.539165i
\(88\) 0 0
\(89\) − 5.64431i − 0.598296i −0.954207 − 0.299148i \(-0.903298\pi\)
0.954207 − 0.299148i \(-0.0967024\pi\)
\(90\) 0 0
\(91\) −15.5463 −1.62970
\(92\) 0 0
\(93\) −5.05333 −0.524006
\(94\) 0 0
\(95\) 2.09447 0.214888
\(96\) 0 0
\(97\) 7.07085i 0.717936i 0.933350 + 0.358968i \(0.116871\pi\)
−0.933350 + 0.358968i \(0.883129\pi\)
\(98\) 0 0
\(99\) − 1.30412i − 0.131069i
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 3552.2.o.a.2737.16 76
4.3 odd 2 888.2.o.a.517.67 yes 76
8.3 odd 2 888.2.o.a.517.9 ✓ 76
8.5 even 2 inner 3552.2.o.a.2737.57 76
37.36 even 2 inner 3552.2.o.a.2737.58 76
148.147 odd 2 888.2.o.a.517.10 yes 76
296.147 odd 2 888.2.o.a.517.68 yes 76
296.221 even 2 inner 3552.2.o.a.2737.15 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.9 ✓ 76 8.3 odd 2
888.2.o.a.517.10 yes 76 148.147 odd 2
888.2.o.a.517.67 yes 76 4.3 odd 2
888.2.o.a.517.68 yes 76 296.147 odd 2
3552.2.o.a.2737.15 76 296.221 even 2 inner
3552.2.o.a.2737.16 76 1.1 even 1 trivial
3552.2.o.a.2737.57 76 8.5 even 2 inner
3552.2.o.a.2737.58 76 37.36 even 2 inner