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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 49.17
Character \(\chi\) \(=\) 736.49
Dual form 736.2.x.a.721.17

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.57450 + 0.719052i) q^{3} +(-2.68037 - 2.32255i) q^{5} +(-0.314784 + 0.202299i) q^{7} +(-0.00255606 - 0.00294985i) q^{9} +(-3.35271 - 0.482046i) q^{11} +(-0.0151019 + 0.0234990i) q^{13} +(-2.55021 - 5.58419i) q^{15} +(-0.162865 - 0.0478215i) q^{17} +(-1.16810 - 3.97819i) q^{19} +(-0.641091 + 0.0921750i) q^{21} +(-4.32547 - 2.07130i) q^{23} +(1.07855 + 7.50147i) q^{25} +(-1.46488 - 4.98891i) q^{27} +(2.21620 - 7.54770i) q^{29} +(1.41823 + 3.10550i) q^{31} +(-4.93223 - 3.16975i) q^{33} +(1.31359 + 0.188865i) q^{35} +(-8.60491 + 7.45620i) q^{37} +(-0.0406750 + 0.0261402i) q^{39} +(4.85532 - 5.60334i) q^{41} +(-9.01825 - 4.11850i) q^{43} +0.0138433i q^{45} +5.96779 q^{47} +(-2.84974 + 6.24006i) q^{49} +(-0.222046 - 0.192404i) q^{51} +(-0.748843 - 1.16522i) q^{53} +(7.86691 + 9.07889i) q^{55} +(1.02134 - 7.10359i) q^{57} +(-1.06893 + 1.66328i) q^{59} +(7.17009 - 3.27447i) q^{61} +(0.00140136 + 0.000411476i) q^{63} +(0.0950563 - 0.0279111i) q^{65} +(7.10526 - 1.02158i) q^{67} +(-5.32110 - 6.37151i) q^{69} +(0.00700092 + 0.0486925i) q^{71} +(11.6526 - 3.42151i) q^{73} +(-3.69577 + 12.5866i) q^{75} +(1.15289 - 0.526509i) q^{77} +(5.97521 + 3.84003i) q^{79} +(1.27916 - 8.89679i) q^{81} +(4.52067 - 3.91718i) q^{83} +(0.325471 + 0.506442i) q^{85} +(8.91660 - 10.2903i) q^{87} +(-6.81956 + 14.9327i) q^{89} -0.0104522i q^{91} +5.90940i q^{93} +(-6.10860 + 13.3760i) q^{95} +(-3.32829 + 3.84105i) q^{97} +(0.00714776 + 0.0111221i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q + 22 q^{7} + 22 q^{15} - 18 q^{17} + 16 q^{23} - 4 q^{25} + 34 q^{31} - 30 q^{33} + 18 q^{39} - 18 q^{41} + 40 q^{47} - 28 q^{49} + 38 q^{55} - 30 q^{57} - 18 q^{63} - 38 q^{65} + 26 q^{71} - 18 q^{73}+ \cdots - 18 q^{97}+O(q^{100}) \) 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
\(3\) 1.57450 + 0.719052i 0.909040 + 0.415145i 0.814358 − 0.580363i \(-0.197089\pi\)
0.0946821 + 0.995508i \(0.469817\pi\)
\(4\) 0 0
\(5\) −2.68037 − 2.32255i −1.19870 − 1.03868i −0.998260 − 0.0589698i \(-0.981218\pi\)
−0.200437 − 0.979707i \(-0.564236\pi\)
\(6\) 0 0
\(7\) −0.314784 + 0.202299i −0.118977 + 0.0764619i −0.598776 − 0.800917i \(-0.704346\pi\)
0.479799 + 0.877378i \(0.340710\pi\)
\(8\) 0 0
\(9\) −0.00255606 − 0.00294985i −0.000852021 − 0.000983285i
\(10\) 0 0
\(11\) −3.35271 − 0.482046i −1.01088 − 0.145342i −0.383078 − 0.923716i \(-0.625136\pi\)
−0.627801 + 0.778374i \(0.716045\pi\)
\(12\) 0 0
\(13\) −0.0151019 + 0.0234990i −0.00418852 + 0.00651746i −0.843341 − 0.537378i \(-0.819415\pi\)
0.839153 + 0.543895i \(0.183051\pi\)
\(14\) 0 0
\(15\) −2.55021 − 5.58419i −0.658462 − 1.44183i
\(16\) 0 0
\(17\) −0.162865 − 0.0478215i −0.0395006 − 0.0115984i 0.261923 − 0.965089i \(-0.415643\pi\)
−0.301423 + 0.953490i \(0.597462\pi\)
\(18\) 0 0
\(19\) −1.16810 − 3.97819i −0.267981 − 0.912658i −0.978023 − 0.208499i \(-0.933142\pi\)
0.710042 − 0.704159i \(-0.248676\pi\)
\(20\) 0 0
\(21\) −0.641091 + 0.0921750i −0.139898 + 0.0201142i
\(22\) 0 0
\(23\) −4.32547 − 2.07130i −0.901923 − 0.431897i
\(24\) 0 0
\(25\) 1.07855 + 7.50147i 0.215710 + 1.50029i
\(26\) 0 0
\(27\) −1.46488 − 4.98891i −0.281916 − 0.960116i
\(28\) 0 0
\(29\) 2.21620 − 7.54770i 0.411539 − 1.40157i −0.449613 − 0.893224i \(-0.648438\pi\)
0.861151 − 0.508348i \(-0.169744\pi\)
\(30\) 0 0
\(31\) 1.41823 + 3.10550i 0.254722 + 0.557764i 0.993187 − 0.116528i \(-0.0371764\pi\)
−0.738465 + 0.674291i \(0.764449\pi\)
\(32\) 0 0
\(33\) −4.93223 − 3.16975i −0.858591 − 0.551783i
\(34\) 0 0
\(35\) 1.31359 + 0.188865i 0.222036 + 0.0319240i
\(36\) 0 0
\(37\) −8.60491 + 7.45620i −1.41464 + 1.22579i −0.476664 + 0.879085i \(0.658154\pi\)
−0.937974 + 0.346706i \(0.887300\pi\)
\(38\) 0 0
\(39\) −0.0406750 + 0.0261402i −0.00651322 + 0.00418579i
\(40\) 0 0
\(41\) 4.85532 − 5.60334i 0.758274 − 0.875094i −0.237069 − 0.971493i \(-0.576187\pi\)
0.995343 + 0.0963985i \(0.0307323\pi\)
\(42\) 0 0
\(43\) −9.01825 − 4.11850i −1.37527 − 0.628065i −0.415692 − 0.909506i \(-0.636460\pi\)
−0.959579 + 0.281441i \(0.909188\pi\)
\(44\) 0 0
\(45\) 0.0138433i 0.00206363i
\(46\) 0 0
\(47\) 5.96779 0.870492 0.435246 − 0.900312i \(-0.356661\pi\)
0.435246 + 0.900312i \(0.356661\pi\)
\(48\) 0 0
\(49\) −2.84974 + 6.24006i −0.407106 + 0.891438i
\(50\) 0 0
\(51\) −0.222046 − 0.192404i −0.0310926 − 0.0269419i
\(52\) 0 0
\(53\) −0.748843 − 1.16522i −0.102861 − 0.160056i 0.786008 − 0.618217i \(-0.212145\pi\)
−0.888869 + 0.458161i \(0.848508\pi\)
\(54\) 0 0
\(55\) 7.86691 + 9.07889i 1.06077 + 1.22420i
\(56\) 0 0
\(57\) 1.02134 − 7.10359i 0.135280 − 0.940894i
\(58\) 0 0
\(59\) −1.06893 + 1.66328i −0.139162 + 0.216541i −0.903838 − 0.427874i \(-0.859263\pi\)
0.764676 + 0.644415i \(0.222899\pi\)
\(60\) 0 0
\(61\) 7.17009 − 3.27447i 0.918036 − 0.419253i 0.100373 − 0.994950i \(-0.467997\pi\)
0.817663 + 0.575697i \(0.195269\pi\)
\(62\) 0 0
\(63\) 0.00140136 0.000411476i 0.000176555 5.18412e-5i
\(64\) 0 0
\(65\) 0.0950563 − 0.0279111i 0.0117903 − 0.00346194i
\(66\) 0 0
\(67\) 7.10526 − 1.02158i 0.868045 − 0.124806i 0.306119 − 0.951993i \(-0.400970\pi\)
0.561926 + 0.827187i \(0.310060\pi\)
\(68\) 0 0
\(69\) −5.32110 − 6.37151i −0.640585 − 0.767040i
\(70\) 0 0
\(71\) 0.00700092 + 0.0486925i 0.000830856 + 0.00577873i 0.990233 − 0.139425i \(-0.0445254\pi\)
−0.989402 + 0.145204i \(0.953616\pi\)
\(72\) 0 0
\(73\) 11.6526 − 3.42151i 1.36383 − 0.400457i 0.483719 − 0.875223i \(-0.339286\pi\)
0.880112 + 0.474766i \(0.157467\pi\)
\(74\) 0 0
\(75\) −3.69577 + 12.5866i −0.426750 + 1.45338i
\(76\) 0 0
\(77\) 1.15289 − 0.526509i 0.131385 − 0.0600013i
\(78\) 0 0
\(79\) 5.97521 + 3.84003i 0.672263 + 0.432037i 0.831741 − 0.555164i \(-0.187345\pi\)
−0.159477 + 0.987202i \(0.550981\pi\)
\(80\) 0 0
\(81\) 1.27916 − 8.89679i 0.142129 − 0.988532i
\(82\) 0 0
\(83\) 4.52067 − 3.91718i 0.496208 − 0.429967i −0.370463 − 0.928847i \(-0.620801\pi\)
0.866671 + 0.498881i \(0.166255\pi\)
\(84\) 0 0
\(85\) 0.325471 + 0.506442i 0.0353022 + 0.0549314i
\(86\) 0 0
\(87\) 8.91660 − 10.2903i 0.955960 − 1.10324i
\(88\) 0 0
\(89\) −6.81956 + 14.9327i −0.722871 + 1.58287i 0.0869645 + 0.996211i \(0.472283\pi\)
−0.809836 + 0.586656i \(0.800444\pi\)
\(90\) 0 0
\(91\) − 0.0104522i − 0.00109569i
\(92\) 0 0
\(93\) 5.90940i 0.612776i
\(94\) 0 0
\(95\) −6.10860 + 13.3760i −0.626729 + 1.37235i
\(96\) 0 0
\(97\) −3.32829 + 3.84105i −0.337936 + 0.389999i −0.899127 − 0.437687i \(-0.855798\pi\)
0.561191 + 0.827686i \(0.310343\pi\)
\(98\) 0 0
\(99\) 0.00714776 + 0.0111221i 0.000718377 + 0.00111782i
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.49.17 220
4.3 odd 2 184.2.p.a.141.21 yes 220
8.3 odd 2 184.2.p.a.141.1 yes 220
8.5 even 2 inner 736.2.x.a.49.6 220
23.8 even 11 inner 736.2.x.a.721.6 220
92.31 odd 22 184.2.p.a.77.1 ✓ 220
184.77 even 22 inner 736.2.x.a.721.17 220
184.123 odd 22 184.2.p.a.77.21 yes 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.1 ✓ 220 92.31 odd 22
184.2.p.a.77.21 yes 220 184.123 odd 22
184.2.p.a.141.1 yes 220 8.3 odd 2
184.2.p.a.141.21 yes 220 4.3 odd 2
736.2.x.a.49.6 220 8.5 even 2 inner
736.2.x.a.49.17 220 1.1 even 1 trivial
736.2.x.a.721.6 220 23.8 even 11 inner
736.2.x.a.721.17 220 184.77 even 22 inner