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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 2737.75
Character \(\chi\) \(=\) 3552.2737
Dual form 3552.2.o.a.2737.76

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -2.44038 q^{5} -4.61327 q^{7} -1.00000 q^{9} -5.06341i q^{11} -0.712581 q^{13} +2.44038i q^{15} -7.27568i q^{17} +0.0331828 q^{19} +4.61327i q^{21} -4.39595i q^{23} +0.955437 q^{25} +1.00000i q^{27} -0.282192 q^{29} -7.20229i q^{31} -5.06341 q^{33} +11.2581 q^{35} +(6.04784 - 0.650853i) q^{37} +0.712581i q^{39} -10.2375 q^{41} -5.99710 q^{43} +2.44038 q^{45} -8.51773 q^{47} +14.2822 q^{49} -7.27568 q^{51} -3.34248i q^{53} +12.3566i q^{55} -0.0331828i q^{57} +2.99492 q^{59} -10.1290 q^{61} +4.61327 q^{63} +1.73897 q^{65} +12.4307i q^{67} -4.39595 q^{69} +12.9471 q^{71} +5.32731 q^{73} -0.955437i q^{75} +23.3588i q^{77} +1.15117i q^{79} +1.00000 q^{81} +3.97545i q^{83} +17.7554i q^{85} +0.282192i q^{87} -4.16072i q^{89} +3.28733 q^{91} -7.20229 q^{93} -0.0809786 q^{95} -13.9820i q^{97} +5.06341i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q - 8 q^{7} - 76 q^{9} + 84 q^{25} - 8 q^{41} + 60 q^{49} + 8 q^{63} + 48 q^{65} + 16 q^{71} + 24 q^{73} + 76 q^{81}+O(q^{100}) \) 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
\(5\) −2.44038 −1.09137 −0.545685 − 0.837991i \(-0.683730\pi\)
−0.545685 + 0.837991i \(0.683730\pi\)
\(6\) 0 0
\(7\) −4.61327 −1.74365 −0.871826 − 0.489817i \(-0.837064\pi\)
−0.871826 + 0.489817i \(0.837064\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 5.06341i − 1.52667i −0.646000 − 0.763337i \(-0.723560\pi\)
0.646000 − 0.763337i \(-0.276440\pi\)
\(12\) 0 0
\(13\) −0.712581 −0.197634 −0.0988172 − 0.995106i \(-0.531506\pi\)
−0.0988172 + 0.995106i \(0.531506\pi\)
\(14\) 0 0
\(15\) 2.44038i 0.630102i
\(16\) 0 0
\(17\) − 7.27568i − 1.76461i −0.470678 − 0.882305i \(-0.655991\pi\)
0.470678 − 0.882305i \(-0.344009\pi\)
\(18\) 0 0
\(19\) 0.0331828 0.00761267 0.00380633 − 0.999993i \(-0.498788\pi\)
0.00380633 + 0.999993i \(0.498788\pi\)
\(20\) 0 0
\(21\) 4.61327i 1.00670i
\(22\) 0 0
\(23\) − 4.39595i − 0.916620i −0.888792 − 0.458310i \(-0.848455\pi\)
0.888792 − 0.458310i \(-0.151545\pi\)
\(24\) 0 0
\(25\) 0.955437 0.191087
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −0.282192 −0.0524018 −0.0262009 − 0.999657i \(-0.508341\pi\)
−0.0262009 + 0.999657i \(0.508341\pi\)
\(30\) 0 0
\(31\) − 7.20229i − 1.29357i −0.762673 − 0.646784i \(-0.776113\pi\)
0.762673 − 0.646784i \(-0.223887\pi\)
\(32\) 0 0
\(33\) −5.06341 −0.881426
\(34\) 0 0
\(35\) 11.2581 1.90297
\(36\) 0 0
\(37\) 6.04784 − 0.650853i 0.994259 − 0.107000i
\(38\) 0 0
\(39\) 0.712581i 0.114104i
\(40\) 0 0
\(41\) −10.2375 −1.59883 −0.799416 − 0.600778i \(-0.794858\pi\)
−0.799416 + 0.600778i \(0.794858\pi\)
\(42\) 0 0
\(43\) −5.99710 −0.914549 −0.457274 − 0.889326i \(-0.651174\pi\)
−0.457274 + 0.889326i \(0.651174\pi\)
\(44\) 0 0
\(45\) 2.44038 0.363790
\(46\) 0 0
\(47\) −8.51773 −1.24244 −0.621219 − 0.783637i \(-0.713362\pi\)
−0.621219 + 0.783637i \(0.713362\pi\)
\(48\) 0 0
\(49\) 14.2822 2.04032
\(50\) 0 0
\(51\) −7.27568 −1.01880
\(52\) 0 0
\(53\) − 3.34248i − 0.459125i −0.973294 − 0.229562i \(-0.926271\pi\)
0.973294 − 0.229562i \(-0.0737295\pi\)
\(54\) 0 0
\(55\) 12.3566i 1.66617i
\(56\) 0 0
\(57\) − 0.0331828i − 0.00439518i
\(58\) 0 0
\(59\) 2.99492 0.389906 0.194953 − 0.980813i \(-0.437545\pi\)
0.194953 + 0.980813i \(0.437545\pi\)
\(60\) 0 0
\(61\) −10.1290 −1.29689 −0.648445 − 0.761262i \(-0.724580\pi\)
−0.648445 + 0.761262i \(0.724580\pi\)
\(62\) 0 0
\(63\) 4.61327 0.581217
\(64\) 0 0
\(65\) 1.73897 0.215692
\(66\) 0 0
\(67\) 12.4307i 1.51865i 0.650710 + 0.759326i \(0.274471\pi\)
−0.650710 + 0.759326i \(0.725529\pi\)
\(68\) 0 0
\(69\) −4.39595 −0.529211
\(70\) 0 0
\(71\) 12.9471 1.53654 0.768268 − 0.640128i \(-0.221119\pi\)
0.768268 + 0.640128i \(0.221119\pi\)
\(72\) 0 0
\(73\) 5.32731 0.623514 0.311757 − 0.950162i \(-0.399083\pi\)
0.311757 + 0.950162i \(0.399083\pi\)
\(74\) 0 0
\(75\) − 0.955437i − 0.110324i
\(76\) 0 0
\(77\) 23.3588i 2.66199i
\(78\) 0 0
\(79\) 1.15117i 0.129517i 0.997901 + 0.0647584i \(0.0206277\pi\)
−0.997901 + 0.0647584i \(0.979372\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.97545i 0.436363i 0.975908 + 0.218181i \(0.0700125\pi\)
−0.975908 + 0.218181i \(0.929988\pi\)
\(84\) 0 0
\(85\) 17.7554i 1.92584i
\(86\) 0 0
\(87\) 0.282192i 0.0302542i
\(88\) 0 0
\(89\) − 4.16072i − 0.441036i −0.975383 − 0.220518i \(-0.929225\pi\)
0.975383 − 0.220518i \(-0.0707748\pi\)
\(90\) 0 0
\(91\) 3.28733 0.344605
\(92\) 0 0
\(93\) −7.20229 −0.746842
\(94\) 0 0
\(95\) −0.0809786 −0.00830823
\(96\) 0 0
\(97\) − 13.9820i − 1.41966i −0.704373 − 0.709830i \(-0.748772\pi\)
0.704373 − 0.709830i \(-0.251228\pi\)
\(98\) 0 0
\(99\) 5.06341i 0.508891i
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.2737.75 76
4.3 odd 2 888.2.o.a.517.74 yes 76
8.3 odd 2 888.2.o.a.517.4 yes 76
8.5 even 2 inner 3552.2.o.a.2737.60 76
37.36 even 2 inner 3552.2.o.a.2737.59 76
148.147 odd 2 888.2.o.a.517.3 ✓ 76
296.147 odd 2 888.2.o.a.517.73 yes 76
296.221 even 2 inner 3552.2.o.a.2737.76 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.3 ✓ 76 148.147 odd 2
888.2.o.a.517.4 yes 76 8.3 odd 2
888.2.o.a.517.73 yes 76 296.147 odd 2
888.2.o.a.517.74 yes 76 4.3 odd 2
3552.2.o.a.2737.59 76 37.36 even 2 inner
3552.2.o.a.2737.60 76 8.5 even 2 inner
3552.2.o.a.2737.75 76 1.1 even 1 trivial
3552.2.o.a.2737.76 76 296.221 even 2 inner