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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 49.5
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.78987 - 0.817407i) q^{3} +(1.45070 + 1.25704i) q^{5} +(2.66148 - 1.71043i) q^{7} +(0.570905 + 0.658860i) q^{9} +(-2.15286 - 0.309535i) q^{11} +(0.495678 - 0.771289i) q^{13} +(-1.56906 - 3.43576i) q^{15} +(2.61973 + 0.769221i) q^{17} +(0.657885 + 2.24055i) q^{19} +(-6.16183 + 0.885937i) q^{21} +(4.60907 + 1.32532i) q^{23} +(-0.187187 - 1.30191i) q^{25} +(1.17979 + 4.01801i) q^{27} +(0.912416 - 3.10740i) q^{29} +(-2.51126 - 5.49888i) q^{31} +(3.60033 + 2.31379i) q^{33} +(6.01111 + 0.864267i) q^{35} +(3.15655 - 2.73516i) q^{37} +(-1.51766 + 0.975339i) q^{39} +(5.56883 - 6.42678i) q^{41} +(-10.2176 - 4.66623i) q^{43} +1.67346i q^{45} +8.82860 q^{47} +(1.25001 - 2.73713i) q^{49} +(-4.06021 - 3.51819i) q^{51} +(-2.20843 - 3.43639i) q^{53} +(-2.73407 - 3.15528i) q^{55} +(0.653913 - 4.54806i) q^{57} +(4.95086 - 7.70369i) q^{59} +(12.5779 - 5.74414i) q^{61} +(2.64639 + 0.777049i) q^{63} +(1.68863 - 0.495825i) q^{65} +(11.0584 - 1.58996i) q^{67} +(-7.16632 - 6.13965i) q^{69} +(1.57907 + 10.9827i) q^{71} +(-12.2637 + 3.60095i) q^{73} +(-0.729152 + 2.48327i) q^{75} +(-6.25924 + 2.85850i) q^{77} +(0.617855 + 0.397071i) q^{79} +(1.54488 - 10.7449i) q^{81} +(6.88908 - 5.96942i) q^{83} +(2.83351 + 4.40902i) q^{85} +(-4.17312 + 4.81604i) q^{87} +(-0.538653 + 1.17948i) q^{89} -2.90059i q^{91} +11.8950i q^{93} +(-1.86207 + 4.07737i) q^{95} +(-8.37752 + 9.66818i) q^{97} +(-1.02514 - 1.59515i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) 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
\(3\) −1.78987 − 0.817407i −1.03338 − 0.471930i −0.174800 − 0.984604i \(-0.555928\pi\)
−0.858583 + 0.512674i \(0.828655\pi\)
\(4\) 0 0
\(5\) 1.45070 + 1.25704i 0.648775 + 0.562167i 0.915856 − 0.401508i \(-0.131514\pi\)
−0.267081 + 0.963674i \(0.586059\pi\)
\(6\) 0 0
\(7\) 2.66148 − 1.71043i 1.00595 − 0.646482i 0.0696047 − 0.997575i \(-0.477826\pi\)
0.936341 + 0.351093i \(0.114190\pi\)
\(8\) 0 0
\(9\) 0.570905 + 0.658860i 0.190302 + 0.219620i
\(10\) 0 0
\(11\) −2.15286 − 0.309535i −0.649112 − 0.0933282i −0.190109 − 0.981763i \(-0.560884\pi\)
−0.459003 + 0.888435i \(0.651793\pi\)
\(12\) 0 0
\(13\) 0.495678 − 0.771289i 0.137476 − 0.213917i −0.765688 − 0.643212i \(-0.777601\pi\)
0.903164 + 0.429295i \(0.141238\pi\)
\(14\) 0 0
\(15\) −1.56906 − 3.43576i −0.405129 − 0.887110i
\(16\) 0 0
\(17\) 2.61973 + 0.769221i 0.635377 + 0.186564i 0.583530 − 0.812092i \(-0.301671\pi\)
0.0518470 + 0.998655i \(0.483489\pi\)
\(18\) 0 0
\(19\) 0.657885 + 2.24055i 0.150929 + 0.514018i 0.999896 − 0.0144286i \(-0.00459292\pi\)
−0.848967 + 0.528446i \(0.822775\pi\)
\(20\) 0 0
\(21\) −6.16183 + 0.885937i −1.34462 + 0.193327i
\(22\) 0 0
\(23\) 4.60907 + 1.32532i 0.961057 + 0.276349i
\(24\) 0 0
\(25\) −0.187187 − 1.30191i −0.0374374 − 0.260383i
\(26\) 0 0
\(27\) 1.17979 + 4.01801i 0.227052 + 0.773267i
\(28\) 0 0
\(29\) 0.912416 − 3.10740i 0.169431 − 0.577030i −0.830372 − 0.557209i \(-0.811872\pi\)
0.999804 − 0.0198210i \(-0.00630964\pi\)
\(30\) 0 0
\(31\) −2.51126 − 5.49888i −0.451035 − 0.987629i −0.989440 − 0.144943i \(-0.953700\pi\)
0.538405 − 0.842686i \(-0.319027\pi\)
\(32\) 0 0
\(33\) 3.60033 + 2.31379i 0.626737 + 0.402779i
\(34\) 0 0
\(35\) 6.01111 + 0.864267i 1.01606 + 0.146088i
\(36\) 0 0
\(37\) 3.15655 − 2.73516i 0.518933 − 0.449658i −0.355590 − 0.934642i \(-0.615720\pi\)
0.874523 + 0.484984i \(0.161175\pi\)
\(38\) 0 0
\(39\) −1.51766 + 0.975339i −0.243020 + 0.156179i
\(40\) 0 0
\(41\) 5.56883 − 6.42678i 0.869706 − 1.00369i −0.130220 − 0.991485i \(-0.541568\pi\)
0.999925 − 0.0122088i \(-0.00388628\pi\)
\(42\) 0 0
\(43\) −10.2176 − 4.66623i −1.55817 − 0.711593i −0.564661 − 0.825323i \(-0.690993\pi\)
−0.993512 + 0.113730i \(0.963720\pi\)
\(44\) 0 0
\(45\) 1.67346i 0.249465i
\(46\) 0 0
\(47\) 8.82860 1.28778 0.643892 − 0.765116i \(-0.277319\pi\)
0.643892 + 0.765116i \(0.277319\pi\)
\(48\) 0 0
\(49\) 1.25001 − 2.73713i 0.178572 − 0.391019i
\(50\) 0 0
\(51\) −4.06021 − 3.51819i −0.568543 − 0.492645i
\(52\) 0 0
\(53\) −2.20843 − 3.43639i −0.303351 − 0.472024i 0.655793 − 0.754941i \(-0.272334\pi\)
−0.959145 + 0.282917i \(0.908698\pi\)
\(54\) 0 0
\(55\) −2.73407 − 3.15528i −0.368662 − 0.425458i
\(56\) 0 0
\(57\) 0.653913 − 4.54806i 0.0866128 − 0.602405i
\(58\) 0 0
\(59\) 4.95086 − 7.70369i 0.644547 − 1.00293i −0.353180 − 0.935555i \(-0.614900\pi\)
0.997727 − 0.0673795i \(-0.0214638\pi\)
\(60\) 0 0
\(61\) 12.5779 − 5.74414i 1.61044 − 0.735462i 0.611973 − 0.790879i \(-0.290376\pi\)
0.998463 + 0.0554171i \(0.0176488\pi\)
\(62\) 0 0
\(63\) 2.64639 + 0.777049i 0.333413 + 0.0978990i
\(64\) 0 0
\(65\) 1.68863 − 0.495825i 0.209448 − 0.0614995i
\(66\) 0 0
\(67\) 11.0584 − 1.58996i 1.35100 − 0.194245i 0.571478 − 0.820617i \(-0.306370\pi\)
0.779524 + 0.626372i \(0.215461\pi\)
\(68\) 0 0
\(69\) −7.16632 − 6.13965i −0.862723 − 0.739126i
\(70\) 0 0
\(71\) 1.57907 + 10.9827i 0.187401 + 1.30341i 0.838704 + 0.544588i \(0.183314\pi\)
−0.651303 + 0.758818i \(0.725777\pi\)
\(72\) 0 0
\(73\) −12.2637 + 3.60095i −1.43536 + 0.421459i −0.904672 − 0.426109i \(-0.859884\pi\)
−0.530687 + 0.847568i \(0.678066\pi\)
\(74\) 0 0
\(75\) −0.729152 + 2.48327i −0.0841953 + 0.286743i
\(76\) 0 0
\(77\) −6.25924 + 2.85850i −0.713306 + 0.325756i
\(78\) 0 0
\(79\) 0.617855 + 0.397071i 0.0695141 + 0.0446740i 0.574936 − 0.818198i \(-0.305027\pi\)
−0.505422 + 0.862872i \(0.668663\pi\)
\(80\) 0 0
\(81\) 1.54488 − 10.7449i 0.171653 − 1.19387i
\(82\) 0 0
\(83\) 6.88908 − 5.96942i 0.756175 − 0.655229i −0.188932 − 0.981990i \(-0.560503\pi\)
0.945107 + 0.326761i \(0.105957\pi\)
\(84\) 0 0
\(85\) 2.83351 + 4.40902i 0.307337 + 0.478225i
\(86\) 0 0
\(87\) −4.17312 + 4.81604i −0.447405 + 0.516333i
\(88\) 0 0
\(89\) −0.538653 + 1.17948i −0.0570971 + 0.125025i −0.936030 − 0.351920i \(-0.885529\pi\)
0.878933 + 0.476945i \(0.158256\pi\)
\(90\) 0 0
\(91\) − 2.90059i − 0.304065i
\(92\) 0 0
\(93\) 11.8950i 1.23346i
\(94\) 0 0
\(95\) −1.86207 + 4.07737i −0.191045 + 0.418329i
\(96\) 0 0
\(97\) −8.37752 + 9.66818i −0.850609 + 0.981655i −0.999975 − 0.00712715i \(-0.997731\pi\)
0.149366 + 0.988782i \(0.452277\pi\)
\(98\) 0 0
\(99\) −1.02514 − 1.59515i −0.103030 − 0.160318i
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.49.5 220
4.3 odd 2 184.2.p.a.141.11 yes 220
8.3 odd 2 184.2.p.a.141.14 yes 220
8.5 even 2 inner 736.2.x.a.49.18 220
23.8 even 11 inner 736.2.x.a.721.18 220
92.31 odd 22 184.2.p.a.77.14 yes 220
184.77 even 22 inner 736.2.x.a.721.5 220
184.123 odd 22 184.2.p.a.77.11 ✓ 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.11 ✓ 220 184.123 odd 22
184.2.p.a.77.14 yes 220 92.31 odd 22
184.2.p.a.141.11 yes 220 4.3 odd 2
184.2.p.a.141.14 yes 220 8.3 odd 2
736.2.x.a.49.5 220 1.1 even 1 trivial
736.2.x.a.49.18 220 8.5 even 2 inner
736.2.x.a.721.5 220 184.77 even 22 inner
736.2.x.a.721.18 220 23.8 even 11 inner