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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 49.18
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.18

$q$-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 0.858583 − 0.512674i \(-0.171345\pi\)
0.174800 + 0.984604i \(0.444072\pi\)
\(4\) 0 0
\(5\) −1.45070 − 1.25704i −0.648775 − 0.562167i 0.267081 − 0.963674i \(-0.413941\pi\)
−0.915856 + 0.401508i \(0.868486\pi\)
\(6\) 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 0.459003 − 0.888435i \(-0.348207\pi\)
0.190109 + 0.981763i \(0.439116\pi\)
\(12\) 0 0
\(13\) −0.495678 + 0.771289i −0.137476 + 0.213917i −0.903164 − 0.429295i \(-0.858762\pi\)
0.765688 + 0.643212i \(0.222399\pi\)
\(14\) 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 0.848967 − 0.528446i \(-0.177225\pi\)
−0.999896 + 0.0144286i \(0.995407\pi\)
\(20\) 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 \(0.188128\pi\)
−0.999804 + 0.0198210i \(0.993690\pi\)
\(30\) 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 −0.874523 − 0.484984i \(-0.838825\pi\)
0.355590 + 0.934642i \(0.384280\pi\)
\(38\) 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 0.993512 − 0.113730i \(-0.0362798\pi\)
0.564661 + 0.825323i \(0.309007\pi\)
\(44\) 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 0.959145 − 0.282917i \(-0.0913020\pi\)
−0.655793 + 0.754941i \(0.727666\pi\)
\(54\) 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 \(0.385100\pi\)
−0.997727 + 0.0673795i \(0.978536\pi\)
\(60\) 0 0
\(61\) −12.5779 + 5.74414i −1.61044 + 0.735462i −0.998463 − 0.0554171i \(-0.982351\pi\)
−0.611973 + 0.790879i \(0.709624\pi\)
\(62\) 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 −0.779524 − 0.626372i \(-0.784539\pi\)
−0.571478 + 0.820617i \(0.693630\pi\)
\(68\) 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 −0.945107 − 0.326761i \(-0.894043\pi\)
0.188932 + 0.981990i \(0.439497\pi\)
\(84\) 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.49.18 220
4.3 odd 2 184.2.p.a.141.14 yes 220
8.3 odd 2 184.2.p.a.141.11 yes 220
8.5 even 2 inner 736.2.x.a.49.5 220
23.8 even 11 inner 736.2.x.a.721.5 220
92.31 odd 22 184.2.p.a.77.11 ✓ 220
184.77 even 22 inner 736.2.x.a.721.18 220
184.123 odd 22 184.2.p.a.77.14 yes 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.11 ✓ 220 92.31 odd 22
184.2.p.a.77.14 yes 220 184.123 odd 22
184.2.p.a.141.11 yes 220 8.3 odd 2
184.2.p.a.141.14 yes 220 4.3 odd 2
736.2.x.a.49.5 220 8.5 even 2 inner
736.2.x.a.49.18 220 1.1 even 1 trivial
736.2.x.a.721.5 220 23.8 even 11 inner
736.2.x.a.721.18 220 184.77 even 22 inner