Show commands: Magma / Pari/GP / SageMath

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 49.19
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.19

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.00125 + 0.913942i) q^{3} +(2.16668 + 1.87744i) q^{5} +(1.81284 - 1.16504i) q^{7} +(1.20514 + 1.39081i) q^{9} +(3.06381 + 0.440509i) q^{11} +(-3.74765 + 5.83146i) q^{13} +(2.62021 + 5.73746i) q^{15} +(-3.59290 - 1.05497i) q^{17} +(-1.11227 - 3.78806i) q^{19} +(4.69272 - 0.674711i) q^{21} +(-1.82766 - 4.43392i) q^{23} +(0.458156 + 3.18654i) q^{25} +(-0.718816 - 2.44806i) q^{27} +(1.73029 - 5.89282i) q^{29} +(0.388858 + 0.851480i) q^{31} +(5.72885 + 3.68171i) q^{33} +(6.11513 + 0.879224i) q^{35} +(3.19871 - 2.77170i) q^{37} +(-12.8296 + 8.24509i) q^{39} +(-5.09055 + 5.87481i) q^{41} +(-1.23784 - 0.565304i) q^{43} +5.27603i q^{45} -1.65604 q^{47} +(-0.978848 + 2.14338i) q^{49} +(-6.22611 - 5.39496i) q^{51} +(-0.0201065 - 0.0312863i) q^{53} +(5.81127 + 6.70656i) q^{55} +(1.23612 - 8.59741i) q^{57} +(-4.60522 + 7.16586i) q^{59} +(1.04835 - 0.478765i) q^{61} +(3.80507 + 1.11727i) q^{63} +(-19.0682 + 5.59893i) q^{65} +(13.3803 - 1.92379i) q^{67} +(0.394740 - 10.5438i) q^{69} +(1.20781 + 8.40048i) q^{71} +(13.6976 - 4.02198i) q^{73} +(-1.99543 + 6.79581i) q^{75} +(6.06739 - 2.77088i) q^{77} +(-9.09086 - 5.84234i) q^{79} +(1.58456 - 11.0209i) q^{81} +(0.881386 - 0.763725i) q^{83} +(-5.80403 - 9.03124i) q^{85} +(8.84844 - 10.2116i) q^{87} +(3.82553 - 8.37675i) q^{89} +14.9376i q^{91} +2.05942i q^{93} +(4.70191 - 10.2958i) q^{95} +(-7.61497 + 8.78814i) q^{97} +(3.07966 + 4.79204i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) 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
\(3\) 2.00125 + 0.913942i 1.15542 + 0.527664i 0.898589 − 0.438791i \(-0.144593\pi\)
0.256835 + 0.966455i \(0.417320\pi\)
\(4\) 0 0
\(5\) 2.16668 + 1.87744i 0.968971 + 0.839618i 0.987082 − 0.160218i \(-0.0512198\pi\)
−0.0181110 + 0.999836i \(0.505765\pi\)
\(6\) 0 0
\(7\) 1.81284 − 1.16504i 0.685187 − 0.440343i −0.151185 − 0.988506i \(-0.548309\pi\)
0.836372 + 0.548162i \(0.184672\pi\)
\(8\) 0 0
\(9\) 1.20514 + 1.39081i 0.401714 + 0.463603i
\(10\) 0 0
\(11\) 3.06381 + 0.440509i 0.923772 + 0.132818i 0.587756 − 0.809038i \(-0.300011\pi\)
0.336016 + 0.941856i \(0.390920\pi\)
\(12\) 0 0
\(13\) −3.74765 + 5.83146i −1.03941 + 1.61736i −0.287871 + 0.957669i \(0.592947\pi\)
−0.751541 + 0.659687i \(0.770689\pi\)
\(14\) 0 0
\(15\) 2.62021 + 5.73746i 0.676535 + 1.48141i
\(16\) 0 0
\(17\) −3.59290 − 1.05497i −0.871406 − 0.255868i −0.184692 − 0.982796i \(-0.559129\pi\)
−0.686713 + 0.726929i \(0.740947\pi\)
\(18\) 0 0
\(19\) −1.11227 − 3.78806i −0.255173 − 0.869040i −0.983051 − 0.183333i \(-0.941311\pi\)
0.727878 − 0.685707i \(-0.240507\pi\)
\(20\) 0 0
\(21\) 4.69272 − 0.674711i 1.02404 − 0.147234i
\(22\) 0 0
\(23\) −1.82766 − 4.43392i −0.381093 − 0.924537i
\(24\) 0 0
\(25\) 0.458156 + 3.18654i 0.0916311 + 0.637309i
\(26\) 0 0
\(27\) −0.718816 − 2.44806i −0.138336 − 0.471130i
\(28\) 0 0
\(29\) 1.73029 − 5.89282i 0.321306 − 1.09427i −0.627554 − 0.778573i \(-0.715944\pi\)
0.948860 − 0.315696i \(-0.102238\pi\)
\(30\) 0 0
\(31\) 0.388858 + 0.851480i 0.0698409 + 0.152930i 0.941333 − 0.337480i \(-0.109574\pi\)
−0.871492 + 0.490410i \(0.836847\pi\)
\(32\) 0 0
\(33\) 5.72885 + 3.68171i 0.997265 + 0.640903i
\(34\) 0 0
\(35\) 6.11513 + 0.879224i 1.03365 + 0.148616i
\(36\) 0 0
\(37\) 3.19871 − 2.77170i 0.525865 − 0.455664i −0.351018 − 0.936369i \(-0.614164\pi\)
0.876883 + 0.480704i \(0.159619\pi\)
\(38\) 0 0
\(39\) −12.8296 + 8.24509i −2.05438 + 1.32027i
\(40\) 0 0
\(41\) −5.09055 + 5.87481i −0.795011 + 0.917491i −0.998097 − 0.0616698i \(-0.980357\pi\)
0.203086 + 0.979161i \(0.434903\pi\)
\(42\) 0 0
\(43\) −1.23784 − 0.565304i −0.188769 − 0.0862080i 0.318788 − 0.947826i \(-0.396724\pi\)
−0.507557 + 0.861618i \(0.669451\pi\)
\(44\) 0 0
\(45\) 5.27603i 0.786504i
\(46\) 0 0
\(47\) −1.65604 −0.241559 −0.120779 − 0.992679i \(-0.538539\pi\)
−0.120779 + 0.992679i \(0.538539\pi\)
\(48\) 0 0
\(49\) −0.978848 + 2.14338i −0.139835 + 0.306197i
\(50\) 0 0
\(51\) −6.22611 − 5.39496i −0.871830 − 0.755445i
\(52\) 0 0
\(53\) −0.0201065 − 0.0312863i −0.00276184 − 0.00429751i 0.839870 − 0.542788i \(-0.182631\pi\)
−0.842632 + 0.538490i \(0.818995\pi\)
\(54\) 0 0
\(55\) 5.81127 + 6.70656i 0.783592 + 0.904313i
\(56\) 0 0
\(57\) 1.23612 − 8.59741i 0.163728 − 1.13875i
\(58\) 0 0
\(59\) −4.60522 + 7.16586i −0.599548 + 0.932915i 0.400315 + 0.916378i \(0.368901\pi\)
−0.999863 + 0.0165377i \(0.994736\pi\)
\(60\) 0 0
\(61\) 1.04835 − 0.478765i 0.134227 − 0.0612995i −0.347169 − 0.937803i \(-0.612857\pi\)
0.481396 + 0.876503i \(0.340130\pi\)
\(62\) 0 0
\(63\) 3.80507 + 1.11727i 0.479394 + 0.140763i
\(64\) 0 0
\(65\) −19.0682 + 5.59893i −2.36512 + 0.694462i
\(66\) 0 0
\(67\) 13.3803 − 1.92379i 1.63466 − 0.235029i 0.737057 − 0.675830i \(-0.236215\pi\)
0.897605 + 0.440801i \(0.145306\pi\)
\(68\) 0 0
\(69\) 0.394740 − 10.5438i 0.0475211 − 1.26932i
\(70\) 0 0
\(71\) 1.20781 + 8.40048i 0.143340 + 0.996954i 0.926812 + 0.375526i \(0.122538\pi\)
−0.783472 + 0.621428i \(0.786553\pi\)
\(72\) 0 0
\(73\) 13.6976 − 4.02198i 1.60318 − 0.470737i 0.646753 − 0.762699i \(-0.276126\pi\)
0.956430 + 0.291962i \(0.0943081\pi\)
\(74\) 0 0
\(75\) −1.99543 + 6.79581i −0.230412 + 0.784712i
\(76\) 0 0
\(77\) 6.06739 − 2.77088i 0.691443 − 0.315771i
\(78\) 0 0
\(79\) −9.09086 − 5.84234i −1.02280 − 0.657315i −0.0821264 − 0.996622i \(-0.526171\pi\)
−0.940676 + 0.339307i \(0.889807\pi\)
\(80\) 0 0
\(81\) 1.58456 − 11.0209i 0.176062 − 1.22454i
\(82\) 0 0
\(83\) 0.881386 − 0.763725i 0.0967447 − 0.0838297i −0.605146 − 0.796115i \(-0.706885\pi\)
0.701890 + 0.712285i \(0.252340\pi\)
\(84\) 0 0
\(85\) −5.80403 − 9.03124i −0.629535 − 0.979576i
\(86\) 0 0
\(87\) 8.84844 − 10.2116i 0.948652 − 1.09480i
\(88\) 0 0
\(89\) 3.82553 − 8.37675i 0.405506 − 0.887934i −0.591177 − 0.806542i \(-0.701336\pi\)
0.996682 − 0.0813914i \(-0.0259364\pi\)
\(90\) 0 0
\(91\) 14.9376i 1.56589i
\(92\) 0 0
\(93\) 2.05942i 0.213552i
\(94\) 0 0
\(95\) 4.70191 − 10.2958i 0.482406 − 1.05632i
\(96\) 0 0
\(97\) −7.61497 + 8.78814i −0.773183 + 0.892300i −0.996597 − 0.0824231i \(-0.973734\pi\)
0.223415 + 0.974723i \(0.428280\pi\)
\(98\) 0 0
\(99\) 3.07966 + 4.79204i 0.309517 + 0.481618i
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.49.19 220
4.3 odd 2 184.2.p.a.141.8 yes 220
8.3 odd 2 184.2.p.a.141.17 yes 220
8.5 even 2 inner 736.2.x.a.49.4 220
23.8 even 11 inner 736.2.x.a.721.4 220
92.31 odd 22 184.2.p.a.77.17 yes 220
184.77 even 22 inner 736.2.x.a.721.19 220
184.123 odd 22 184.2.p.a.77.8 ✓ 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.8 ✓ 220 184.123 odd 22
184.2.p.a.77.17 yes 220 92.31 odd 22
184.2.p.a.141.8 yes 220 4.3 odd 2
184.2.p.a.141.17 yes 220 8.3 odd 2
736.2.x.a.49.4 220 8.5 even 2 inner
736.2.x.a.49.19 220 1.1 even 1 trivial
736.2.x.a.721.4 220 23.8 even 11 inner
736.2.x.a.721.19 220 184.77 even 22 inner