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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 2737.30
Character \(\chi\) \(=\) 3552.2737
Dual form 3552.2.o.a.2737.29

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -1.62102 q^{5} +0.682716 q^{7} -1.00000 q^{9} +1.65068i q^{11} +6.88190 q^{13} -1.62102i q^{15} -5.88377i q^{17} -5.78573 q^{19} +0.682716i q^{21} +4.15036i q^{23} -2.37231 q^{25} -1.00000i q^{27} +5.64652 q^{29} -9.48802i q^{31} -1.65068 q^{33} -1.10669 q^{35} +(-0.243550 - 6.07788i) q^{37} +6.88190i q^{39} +2.20152 q^{41} +5.62302 q^{43} +1.62102 q^{45} +2.88131 q^{47} -6.53390 q^{49} +5.88377 q^{51} +4.56700i q^{53} -2.67578i q^{55} -5.78573i q^{57} +6.60560 q^{59} +0.499819 q^{61} -0.682716 q^{63} -11.1557 q^{65} -10.9456i q^{67} -4.15036 q^{69} +1.34402 q^{71} -9.65894 q^{73} -2.37231i q^{75} +1.12695i q^{77} -3.25589i q^{79} +1.00000 q^{81} +12.4949i q^{83} +9.53768i q^{85} +5.64652i q^{87} +10.7654i q^{89} +4.69838 q^{91} +9.48802 q^{93} +9.37875 q^{95} -11.0332i q^{97} -1.65068i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q - 8 q^{7} - 76 q^{9} + 84 q^{25} - 8 q^{41} + 60 q^{49} + 8 q^{63} + 48 q^{65} + 16 q^{71} + 24 q^{73} + 76 q^{81}+O(q^{100}) \) 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
\(5\) −1.62102 −0.724940 −0.362470 − 0.931995i \(-0.618067\pi\)
−0.362470 + 0.931995i \(0.618067\pi\)
\(6\) 0 0
\(7\) 0.682716 0.258042 0.129021 − 0.991642i \(-0.458816\pi\)
0.129021 + 0.991642i \(0.458816\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.65068i 0.497699i 0.968542 + 0.248850i \(0.0800525\pi\)
−0.968542 + 0.248850i \(0.919948\pi\)
\(12\) 0 0
\(13\) 6.88190 1.90870 0.954348 − 0.298698i \(-0.0965522\pi\)
0.954348 + 0.298698i \(0.0965522\pi\)
\(14\) 0 0
\(15\) − 1.62102i − 0.418545i
\(16\) 0 0
\(17\) − 5.88377i − 1.42702i −0.700644 − 0.713511i \(-0.747104\pi\)
0.700644 − 0.713511i \(-0.252896\pi\)
\(18\) 0 0
\(19\) −5.78573 −1.32734 −0.663668 − 0.748027i \(-0.731001\pi\)
−0.663668 + 0.748027i \(0.731001\pi\)
\(20\) 0 0
\(21\) 0.682716i 0.148981i
\(22\) 0 0
\(23\) 4.15036i 0.865411i 0.901535 + 0.432705i \(0.142441\pi\)
−0.901535 + 0.432705i \(0.857559\pi\)
\(24\) 0 0
\(25\) −2.37231 −0.474461
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 5.64652 1.04853 0.524266 − 0.851554i \(-0.324340\pi\)
0.524266 + 0.851554i \(0.324340\pi\)
\(30\) 0 0
\(31\) − 9.48802i − 1.70410i −0.523461 − 0.852050i \(-0.675359\pi\)
0.523461 − 0.852050i \(-0.324641\pi\)
\(32\) 0 0
\(33\) −1.65068 −0.287347
\(34\) 0 0
\(35\) −1.10669 −0.187065
\(36\) 0 0
\(37\) −0.243550 − 6.07788i −0.0400393 − 0.999198i
\(38\) 0 0
\(39\) 6.88190i 1.10199i
\(40\) 0 0
\(41\) 2.20152 0.343820 0.171910 − 0.985113i \(-0.445006\pi\)
0.171910 + 0.985113i \(0.445006\pi\)
\(42\) 0 0
\(43\) 5.62302 0.857502 0.428751 − 0.903423i \(-0.358954\pi\)
0.428751 + 0.903423i \(0.358954\pi\)
\(44\) 0 0
\(45\) 1.62102 0.241647
\(46\) 0 0
\(47\) 2.88131 0.420282 0.210141 − 0.977671i \(-0.432608\pi\)
0.210141 + 0.977671i \(0.432608\pi\)
\(48\) 0 0
\(49\) −6.53390 −0.933414
\(50\) 0 0
\(51\) 5.88377 0.823892
\(52\) 0 0
\(53\) 4.56700i 0.627326i 0.949534 + 0.313663i \(0.101556\pi\)
−0.949534 + 0.313663i \(0.898444\pi\)
\(54\) 0 0
\(55\) − 2.67578i − 0.360802i
\(56\) 0 0
\(57\) − 5.78573i − 0.766338i
\(58\) 0 0
\(59\) 6.60560 0.859975 0.429988 − 0.902835i \(-0.358518\pi\)
0.429988 + 0.902835i \(0.358518\pi\)
\(60\) 0 0
\(61\) 0.499819 0.0639953 0.0319976 − 0.999488i \(-0.489813\pi\)
0.0319976 + 0.999488i \(0.489813\pi\)
\(62\) 0 0
\(63\) −0.682716 −0.0860141
\(64\) 0 0
\(65\) −11.1557 −1.38369
\(66\) 0 0
\(67\) − 10.9456i − 1.33722i −0.743612 − 0.668611i \(-0.766889\pi\)
0.743612 − 0.668611i \(-0.233111\pi\)
\(68\) 0 0
\(69\) −4.15036 −0.499645
\(70\) 0 0
\(71\) 1.34402 0.159506 0.0797531 − 0.996815i \(-0.474587\pi\)
0.0797531 + 0.996815i \(0.474587\pi\)
\(72\) 0 0
\(73\) −9.65894 −1.13049 −0.565246 − 0.824922i \(-0.691219\pi\)
−0.565246 + 0.824922i \(0.691219\pi\)
\(74\) 0 0
\(75\) − 2.37231i − 0.273930i
\(76\) 0 0
\(77\) 1.12695i 0.128427i
\(78\) 0 0
\(79\) − 3.25589i − 0.366316i −0.983083 − 0.183158i \(-0.941368\pi\)
0.983083 − 0.183158i \(-0.0586321\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.4949i 1.37149i 0.727842 + 0.685745i \(0.240523\pi\)
−0.727842 + 0.685745i \(0.759477\pi\)
\(84\) 0 0
\(85\) 9.53768i 1.03451i
\(86\) 0 0
\(87\) 5.64652i 0.605371i
\(88\) 0 0
\(89\) 10.7654i 1.14113i 0.821254 + 0.570563i \(0.193275\pi\)
−0.821254 + 0.570563i \(0.806725\pi\)
\(90\) 0 0
\(91\) 4.69838 0.492524
\(92\) 0 0
\(93\) 9.48802 0.983862
\(94\) 0 0
\(95\) 9.37875 0.962240
\(96\) 0 0
\(97\) − 11.0332i − 1.12026i −0.828406 − 0.560128i \(-0.810752\pi\)
0.828406 − 0.560128i \(-0.189248\pi\)
\(98\) 0 0
\(99\) − 1.65068i − 0.165900i
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.2737.30 76
4.3 odd 2 888.2.o.a.517.72 yes 76
8.3 odd 2 888.2.o.a.517.6 yes 76
8.5 even 2 inner 3552.2.o.a.2737.27 76
37.36 even 2 inner 3552.2.o.a.2737.28 76
148.147 odd 2 888.2.o.a.517.5 ✓ 76
296.147 odd 2 888.2.o.a.517.71 yes 76
296.221 even 2 inner 3552.2.o.a.2737.29 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.5 ✓ 76 148.147 odd 2
888.2.o.a.517.6 yes 76 8.3 odd 2
888.2.o.a.517.71 yes 76 296.147 odd 2
888.2.o.a.517.72 yes 76 4.3 odd 2
3552.2.o.a.2737.27 76 8.5 even 2 inner
3552.2.o.a.2737.28 76 37.36 even 2 inner
3552.2.o.a.2737.29 76 296.221 even 2 inner
3552.2.o.a.2737.30 76 1.1 even 1 trivial