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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [736,2,Mod(49,736)] 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("736.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 11, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 736.x (of order \(22\), degree \(10\), 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: \(5.87698958877\)
Analytic rank: \(0\)
Dimension: \(220\)
Relative dimension: \(22\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 49.20
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.20

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.23415 + 1.02030i) q^{3} +(-0.183115 - 0.158670i) q^{5} +(0.661458 - 0.425093i) q^{7} +(1.98583 + 2.29177i) q^{9} +(3.33335 + 0.479263i) q^{11} +(0.940115 - 1.46285i) q^{13} +(-0.247215 - 0.541325i) q^{15} +(-0.130801 - 0.0384068i) q^{17} +(0.714884 + 2.43467i) q^{19} +(1.91152 - 0.274835i) q^{21} +(-0.802589 + 4.72820i) q^{23} +(-0.703219 - 4.89100i) q^{25} +(0.0224478 + 0.0764500i) q^{27} +(0.320194 - 1.09048i) q^{29} +(3.94341 + 8.63486i) q^{31} +(6.95821 + 4.47177i) q^{33} +(-0.188572 - 0.0271126i) q^{35} +(5.85290 - 5.07156i) q^{37} +(3.59290 - 2.30902i) q^{39} +(-3.39667 + 3.91997i) q^{41} +(-7.77106 - 3.54892i) q^{43} -0.734747i q^{45} -3.04973 q^{47} +(-2.65108 + 5.80506i) q^{49} +(-0.253044 - 0.219263i) q^{51} +(-2.38915 - 3.71759i) q^{53} +(-0.534341 - 0.616662i) q^{55} +(-0.886942 + 6.16882i) q^{57} +(6.72140 - 10.4587i) q^{59} +(-1.43210 + 0.654019i) q^{61} +(2.28776 + 0.671746i) q^{63} +(-0.404259 + 0.118701i) q^{65} +(-7.04917 + 1.01352i) q^{67} +(-6.61729 + 9.74462i) q^{69} +(1.23625 + 8.59834i) q^{71} +(-2.95353 + 0.867234i) q^{73} +(3.41920 - 11.6447i) q^{75} +(2.40860 - 1.09997i) q^{77} +(-13.5072 - 8.68054i) q^{79} +(1.26683 - 8.81102i) q^{81} +(-6.88920 + 5.96952i) q^{83} +(0.0178577 + 0.0277871i) q^{85} +(1.82798 - 2.10960i) q^{87} +(-5.14366 + 11.2630i) q^{89} -1.36725i q^{91} +23.3150i q^{93} +(0.255403 - 0.559255i) q^{95} +(11.1898 - 12.9137i) q^{97} +(5.52109 + 8.59099i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/736\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(415\) \(645\)
\(\chi(n)\) \(e\left(\frac{8}{11}\right)\) \(1\) \(-1\)

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
\(3\) 2.23415 + 1.02030i 1.28989 + 0.589072i 0.937891 − 0.346930i \(-0.112776\pi\)
0.351996 + 0.936002i \(0.385503\pi\)
\(4\) 0 0
\(5\) −0.183115 − 0.158670i −0.0818914 − 0.0709593i 0.612953 − 0.790119i \(-0.289981\pi\)
−0.694845 + 0.719160i \(0.744527\pi\)
\(6\) 0 0
\(7\) 0.661458 − 0.425093i 0.250008 − 0.160670i −0.409635 − 0.912249i \(-0.634344\pi\)
0.659643 + 0.751579i \(0.270708\pi\)
\(8\) 0 0
\(9\) 1.98583 + 2.29177i 0.661942 + 0.763922i
\(10\) 0 0
\(11\) 3.33335 + 0.479263i 1.00504 + 0.144503i 0.625131 − 0.780520i \(-0.285045\pi\)
0.379911 + 0.925023i \(0.375955\pi\)
\(12\) 0 0
\(13\) 0.940115 − 1.46285i 0.260741 − 0.405721i −0.686058 − 0.727547i \(-0.740660\pi\)
0.946799 + 0.321826i \(0.104297\pi\)
\(14\) 0 0
\(15\) −0.247215 − 0.541325i −0.0638306 − 0.139769i
\(16\) 0 0
\(17\) −0.130801 − 0.0384068i −0.0317240 − 0.00931501i 0.265832 − 0.964019i \(-0.414353\pi\)
−0.297556 + 0.954704i \(0.596172\pi\)
\(18\) 0 0
\(19\) 0.714884 + 2.43467i 0.164006 + 0.558552i 0.999953 + 0.00974164i \(0.00310091\pi\)
−0.835947 + 0.548810i \(0.815081\pi\)
\(20\) 0 0
\(21\) 1.91152 − 0.274835i 0.417128 − 0.0599740i
\(22\) 0 0
\(23\) −0.802589 + 4.72820i −0.167351 + 0.985897i
\(24\) 0 0
\(25\) −0.703219 − 4.89100i −0.140644 − 0.978200i
\(26\) 0 0
\(27\) 0.0224478 + 0.0764500i 0.00432007 + 0.0147128i
\(28\) 0 0
\(29\) 0.320194 − 1.09048i 0.0594586 − 0.202497i −0.924407 − 0.381408i \(-0.875439\pi\)
0.983865 + 0.178911i \(0.0572574\pi\)
\(30\) 0 0
\(31\) 3.94341 + 8.63486i 0.708257 + 1.55087i 0.829663 + 0.558264i \(0.188532\pi\)
−0.121406 + 0.992603i \(0.538740\pi\)
\(32\) 0 0
\(33\) 6.95821 + 4.47177i 1.21127 + 0.778435i
\(34\) 0 0
\(35\) −0.188572 − 0.0271126i −0.0318745 − 0.00458287i
\(36\) 0 0
\(37\) 5.85290 − 5.07156i 0.962210 − 0.833760i −0.0239231 − 0.999714i \(-0.507616\pi\)
0.986134 + 0.165954i \(0.0530702\pi\)
\(38\) 0 0
\(39\) 3.59290 − 2.30902i 0.575325 − 0.369739i
\(40\) 0 0
\(41\) −3.39667 + 3.91997i −0.530471 + 0.612197i −0.956221 − 0.292644i \(-0.905465\pi\)
0.425750 + 0.904841i \(0.360010\pi\)
\(42\) 0 0
\(43\) −7.77106 − 3.54892i −1.18508 − 0.541206i −0.277349 − 0.960769i \(-0.589456\pi\)
−0.907726 + 0.419563i \(0.862183\pi\)
\(44\) 0 0
\(45\) − 0.734747i − 0.109530i
\(46\) 0 0
\(47\) −3.04973 −0.444849 −0.222425 − 0.974950i \(-0.571397\pi\)
−0.222425 + 0.974950i \(0.571397\pi\)
\(48\) 0 0
\(49\) −2.65108 + 5.80506i −0.378726 + 0.829294i
\(50\) 0 0
\(51\) −0.253044 − 0.219263i −0.0354332 − 0.0307030i
\(52\) 0 0
\(53\) −2.38915 − 3.71759i −0.328175 − 0.510650i 0.637484 − 0.770464i \(-0.279975\pi\)
−0.965659 + 0.259813i \(0.916339\pi\)
\(54\) 0 0
\(55\) −0.534341 − 0.616662i −0.0720505 − 0.0831507i
\(56\) 0 0
\(57\) −0.886942 + 6.16882i −0.117478 + 0.817080i
\(58\) 0 0
\(59\) 6.72140 − 10.4587i 0.875051 − 1.36161i −0.0566550 − 0.998394i \(-0.518044\pi\)
0.931706 − 0.363213i \(-0.118320\pi\)
\(60\) 0 0
\(61\) −1.43210 + 0.654019i −0.183362 + 0.0837385i −0.504981 − 0.863131i \(-0.668500\pi\)
0.321619 + 0.946869i \(0.395773\pi\)
\(62\) 0 0
\(63\) 2.28776 + 0.671746i 0.288230 + 0.0846320i
\(64\) 0 0
\(65\) −0.404259 + 0.118701i −0.0501422 + 0.0147231i
\(66\) 0 0
\(67\) −7.04917 + 1.01352i −0.861193 + 0.123821i −0.558740 − 0.829343i \(-0.688715\pi\)
−0.302454 + 0.953164i \(0.597806\pi\)
\(68\) 0 0
\(69\) −6.61729 + 9.74462i −0.796628 + 1.17311i
\(70\) 0 0
\(71\) 1.23625 + 8.59834i 0.146716 + 1.02043i 0.921547 + 0.388266i \(0.126926\pi\)
−0.774831 + 0.632169i \(0.782165\pi\)
\(72\) 0 0
\(73\) −2.95353 + 0.867234i −0.345684 + 0.101502i −0.449966 − 0.893046i \(-0.648564\pi\)
0.104282 + 0.994548i \(0.466746\pi\)
\(74\) 0 0
\(75\) 3.41920 − 11.6447i 0.394815 − 1.34462i
\(76\) 0 0
\(77\) 2.40860 − 1.09997i 0.274486 − 0.125353i
\(78\) 0 0
\(79\) −13.5072 − 8.68054i −1.51968 − 0.976637i −0.991868 − 0.127274i \(-0.959377\pi\)
−0.527809 − 0.849363i \(-0.676986\pi\)
\(80\) 0 0
\(81\) 1.26683 − 8.81102i 0.140759 − 0.979002i
\(82\) 0 0
\(83\) −6.88920 + 5.96952i −0.756188 + 0.655240i −0.945110 − 0.326752i \(-0.894046\pi\)
0.188923 + 0.981992i \(0.439500\pi\)
\(84\) 0 0
\(85\) 0.0178577 + 0.0277871i 0.00193694 + 0.00301393i
\(86\) 0 0
\(87\) 1.82798 − 2.10960i 0.195980 − 0.226173i
\(88\) 0 0
\(89\) −5.14366 + 11.2630i −0.545227 + 1.19388i 0.413749 + 0.910391i \(0.364219\pi\)
−0.958975 + 0.283489i \(0.908508\pi\)
\(90\) 0 0
\(91\) − 1.36725i − 0.143327i
\(92\) 0 0
\(93\) 23.3150i 2.41766i
\(94\) 0 0
\(95\) 0.255403 − 0.559255i 0.0262038 − 0.0573784i
\(96\) 0 0
\(97\) 11.1898 − 12.9137i 1.13615 − 1.31119i 0.192103 − 0.981375i \(-0.438469\pi\)
0.944047 − 0.329812i \(-0.106985\pi\)
\(98\) 0 0
\(99\) 5.52109 + 8.59099i 0.554891 + 0.863427i
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 736.2.x.a.49.20 220
4.3 odd 2 184.2.p.a.141.13 yes 220
8.3 odd 2 184.2.p.a.141.7 yes 220
8.5 even 2 inner 736.2.x.a.49.3 220
23.8 even 11 inner 736.2.x.a.721.3 220
92.31 odd 22 184.2.p.a.77.7 ✓ 220
184.77 even 22 inner 736.2.x.a.721.20 220
184.123 odd 22 184.2.p.a.77.13 yes 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.7 ✓ 220 92.31 odd 22
184.2.p.a.77.13 yes 220 184.123 odd 22
184.2.p.a.141.7 yes 220 8.3 odd 2
184.2.p.a.141.13 yes 220 4.3 odd 2
736.2.x.a.49.3 220 8.5 even 2 inner
736.2.x.a.49.20 220 1.1 even 1 trivial
736.2.x.a.721.3 220 23.8 even 11 inner
736.2.x.a.721.20 220 184.77 even 22 inner