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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [184,2,Mod(13,184)] 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("184.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(184, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 11, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 184 = 2^{3} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 184.p (of order \(22\), degree \(10\), 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: \(1.46924739719\)
Analytic rank: \(0\)
Dimension: \(220\)
Relative dimension: \(22\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 141.1
Character \(\chi\) \(=\) 184.141
Dual form 184.2.p.a.77.1

$q$-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+(-1.41326 + 0.0518036i) q^{2} +(1.57450 + 0.719052i) q^{3} +(1.99463 - 0.146424i) q^{4} +(2.68037 + 2.32255i) q^{5} +(-2.26244 - 0.934645i) q^{6} +(0.314784 - 0.202299i) q^{7} +(-2.81136 + 0.310265i) q^{8} +(-0.00255606 - 0.00294985i) q^{9} +(-3.90838 - 3.14353i) q^{10} +(-3.35271 - 0.482046i) q^{11} +(3.24584 + 1.20370i) q^{12} +(0.0151019 - 0.0234990i) q^{13} +(-0.434393 + 0.302209i) q^{14} +(2.55021 + 5.58419i) q^{15} +(3.95712 - 0.584125i) q^{16} +(-0.162865 - 0.0478215i) q^{17} +(0.00376521 + 0.00403651i) q^{18} +(-1.16810 - 3.97819i) q^{19} +(5.68643 + 4.24017i) q^{20} +(0.641091 - 0.0921750i) q^{21} +(4.76323 + 0.507577i) q^{22} +(4.32547 + 2.07130i) q^{23} +(-4.64959 - 1.53300i) q^{24} +(1.07855 + 7.50147i) q^{25} +(-0.0201257 + 0.0339927i) q^{26} +(-1.46488 - 4.98891i) q^{27} +(0.598256 - 0.449604i) q^{28} +(-2.21620 + 7.54770i) q^{29} +(-3.89341 - 7.75982i) q^{30} +(-1.41823 - 3.10550i) q^{31} +(-5.56220 + 1.03052i) q^{32} +(-4.93223 - 3.16975i) q^{33} +(0.232649 + 0.0591475i) q^{34} +(1.31359 + 0.188865i) q^{35} +(-0.00553034 - 0.00550961i) q^{36} +(8.60491 - 7.45620i) q^{37} +(1.85692 + 5.56172i) q^{38} +(0.0406750 - 0.0261402i) q^{39} +(-8.25608 - 5.69790i) q^{40} +(4.85532 - 5.60334i) q^{41} +(-0.901257 + 0.163479i) q^{42} +(-9.01825 - 4.11850i) q^{43} +(-6.75800 - 0.470588i) q^{44} -0.0138433i q^{45} +(-6.22034 - 2.70322i) q^{46} -5.96779 q^{47} +(6.65052 + 1.92567i) q^{48} +(-2.84974 + 6.24006i) q^{49} +(-1.91288 - 10.5457i) q^{50} +(-0.222046 - 0.192404i) q^{51} +(0.0266819 - 0.0490832i) q^{52} +(0.748843 + 1.16522i) q^{53} +(2.32870 + 6.97476i) q^{54} +(-7.86691 - 9.07889i) q^{55} +(-0.822203 + 0.666402i) q^{56} +(1.02134 - 7.10359i) q^{57} +(2.74108 - 10.7817i) q^{58} +(-1.06893 + 1.66328i) q^{59} +(5.90440 + 10.7650i) q^{60} +(-7.17009 + 3.27447i) q^{61} +(2.16521 + 4.31542i) q^{62} +(-0.00140136 - 0.000411476i) q^{63} +(7.80747 - 1.74453i) q^{64} +(0.0950563 - 0.0279111i) q^{65} +(7.13475 + 4.22419i) q^{66} +(7.10526 - 1.02158i) q^{67} +(-0.331859 - 0.0715390i) q^{68} +(5.32110 + 6.37151i) q^{69} +(-1.86623 - 0.198868i) q^{70} +(-0.00700092 - 0.0486925i) q^{71} +(0.00810125 + 0.00750004i) q^{72} +(11.6526 - 3.42151i) q^{73} +(-11.7748 + 10.9833i) q^{74} +(-3.69577 + 12.5866i) q^{75} +(-2.91244 - 7.76398i) q^{76} +(-1.15289 + 0.526509i) q^{77} +(-0.0561304 + 0.0390502i) q^{78} +(-5.97521 - 3.84003i) q^{79} +(11.9632 + 7.62494i) q^{80} +(1.27916 - 8.89679i) q^{81} +(-6.57158 + 8.17052i) q^{82} +(4.52067 - 3.91718i) q^{83} +(1.26525 - 0.277727i) q^{84} +(-0.325471 - 0.506442i) q^{85} +(12.9585 + 5.35335i) q^{86} +(-8.91660 + 10.2903i) q^{87} +(9.57522 + 0.314977i) q^{88} +(-6.81956 + 14.9327i) q^{89} +(0.000717131 + 0.0195642i) q^{90} -0.0104522i q^{91} +(8.93102 + 3.49814i) q^{92} -5.90940i q^{93} +(8.43407 - 0.309153i) q^{94} +(6.10860 - 13.3760i) q^{95} +(-9.49869 - 2.37696i) q^{96} +(-3.32829 + 3.84105i) q^{97} +(3.70418 - 8.96649i) q^{98} +(0.00714776 + 0.0111221i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 220 q - 11 q^{2} - 11 q^{4} - 14 q^{6} - 22 q^{7} - 14 q^{8} - 9 q^{10} - 12 q^{12} - 3 q^{14} - 22 q^{15} + 5 q^{16} - 18 q^{17} - 4 q^{18} - 27 q^{20} - 42 q^{22} - 16 q^{23} - 22 q^{24} - 4 q^{25} - 16 q^{26}+ \cdots + 109 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/184\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(93\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{8}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41326 + 0.0518036i −0.999329 + 0.0366307i
\(3\) 1.57450 + 0.719052i 0.909040 + 0.415145i 0.814358 − 0.580363i \(-0.197089\pi\)
0.0946821 + 0.995508i \(0.469817\pi\)
\(4\) 1.99463 − 0.146424i 0.997316 − 0.0732121i
\(5\) 2.68037 + 2.32255i 1.19870 + 1.03868i 0.998260 + 0.0589698i \(0.0187816\pi\)
0.200437 + 0.979707i \(0.435764\pi\)
\(6\) −2.26244 − 0.934645i −0.923637 − 0.381567i
\(7\) 0.314784 − 0.202299i 0.118977 − 0.0764619i −0.479799 − 0.877378i \(-0.659290\pi\)
0.598776 + 0.800917i \(0.295654\pi\)
\(8\) −2.81136 + 0.310265i −0.993965 + 0.109695i
\(9\) −0.00255606 − 0.00294985i −0.000852021 − 0.000983285i
\(10\) −3.90838 − 3.14353i −1.23594 − 0.994070i
\(11\) −3.35271 − 0.482046i −1.01088 − 0.145342i −0.383078 − 0.923716i \(-0.625136\pi\)
−0.627801 + 0.778374i \(0.716045\pi\)
\(12\) 3.24584 + 1.20370i 0.936994 + 0.347478i
\(13\) 0.0151019 − 0.0234990i 0.00418852 − 0.00651746i −0.839153 − 0.543895i \(-0.816949\pi\)
0.843341 + 0.537378i \(0.180585\pi\)
\(14\) −0.434393 + 0.302209i −0.116096 + 0.0807688i
\(15\) 2.55021 + 5.58419i 0.658462 + 1.44183i
\(16\) 3.95712 − 0.584125i 0.989280 − 0.146031i
\(17\) −0.162865 − 0.0478215i −0.0395006 − 0.0115984i 0.261923 − 0.965089i \(-0.415643\pi\)
−0.301423 + 0.953490i \(0.597462\pi\)
\(18\) 0.00376521 + 0.00403651i 0.000887468 + 0.000951415i
\(19\) −1.16810 − 3.97819i −0.267981 − 0.912658i −0.978023 − 0.208499i \(-0.933142\pi\)
0.710042 − 0.704159i \(-0.248676\pi\)
\(20\) 5.68643 + 4.24017i 1.27152 + 0.948130i
\(21\) 0.641091 − 0.0921750i 0.139898 − 0.0201142i
\(22\) 4.76323 + 0.507577i 1.01552 + 0.108216i
\(23\) 4.32547 + 2.07130i 0.901923 + 0.431897i
\(24\) −4.64959 − 1.53300i −0.949094 − 0.312922i
\(25\) 1.07855 + 7.50147i 0.215710 + 1.50029i
\(26\) −0.0201257 + 0.0339927i −0.00394697 + 0.00666651i
\(27\) −1.46488 − 4.98891i −0.281916 − 0.960116i
\(28\) 0.598256 − 0.449604i 0.113060 − 0.0849673i
\(29\) −2.21620 + 7.54770i −0.411539 + 1.40157i 0.449613 + 0.893224i \(0.351562\pi\)
−0.861151 + 0.508348i \(0.830256\pi\)
\(30\) −3.89341 − 7.75982i −0.710835 − 1.41674i
\(31\) −1.41823 − 3.10550i −0.254722 − 0.557764i 0.738465 − 0.674291i \(-0.235551\pi\)
−0.993187 + 0.116528i \(0.962824\pi\)
\(32\) −5.56220 + 1.03052i −0.983267 + 0.182171i
\(33\) −4.93223 − 3.16975i −0.858591 − 0.551783i
\(34\) 0.232649 + 0.0591475i 0.0398990 + 0.0101437i
\(35\) 1.31359 + 0.188865i 0.222036 + 0.0319240i
\(36\) −0.00553034 − 0.00550961i −0.000921723 − 0.000918268i
\(37\) 8.60491 − 7.45620i 1.41464 − 1.22579i 0.476664 − 0.879085i \(-0.341846\pi\)
0.937974 − 0.346706i \(-0.112700\pi\)
\(38\) 1.85692 + 5.56172i 0.301232 + 0.902230i
\(39\) 0.0406750 − 0.0261402i 0.00651322 − 0.00418579i
\(40\) −8.25608 − 5.69790i −1.30540 − 0.900917i
\(41\) 4.85532 − 5.60334i 0.758274 − 0.875094i −0.237069 − 0.971493i \(-0.576187\pi\)
0.995343 + 0.0963985i \(0.0307323\pi\)
\(42\) −0.901257 + 0.163479i −0.139067 + 0.0252253i
\(43\) −9.01825 − 4.11850i −1.37527 − 0.628065i −0.415692 − 0.909506i \(-0.636460\pi\)
−0.959579 + 0.281441i \(0.909188\pi\)
\(44\) −6.75800 − 0.470588i −1.01881 − 0.0709438i
\(45\) − 0.0138433i − 0.00206363i
\(46\) −6.22034 − 2.70322i −0.917138 − 0.398569i
\(47\) −5.96779 −0.870492 −0.435246 − 0.900312i \(-0.643339\pi\)
−0.435246 + 0.900312i \(0.643339\pi\)
\(48\) 6.65052 + 1.92567i 0.959919 + 0.277946i
\(49\) −2.84974 + 6.24006i −0.407106 + 0.891438i
\(50\) −1.91288 − 10.5457i −0.270522 − 1.49139i
\(51\) −0.222046 − 0.192404i −0.0310926 − 0.0269419i
\(52\) 0.0266819 − 0.0490832i 0.00370012 − 0.00680662i
\(53\) 0.748843 + 1.16522i 0.102861 + 0.160056i 0.888869 − 0.458161i \(-0.151492\pi\)
−0.786008 + 0.618217i \(0.787855\pi\)
\(54\) 2.32870 + 6.97476i 0.316896 + 0.949145i
\(55\) −7.86691 − 9.07889i −1.06077 − 1.22420i
\(56\) −0.822203 + 0.666402i −0.109872 + 0.0890517i
\(57\) 1.02134 − 7.10359i 0.135280 − 0.940894i
\(58\) 2.74108 − 10.7817i 0.359922 − 1.41571i
\(59\) −1.06893 + 1.66328i −0.139162 + 0.216541i −0.903838 − 0.427874i \(-0.859263\pi\)
0.764676 + 0.644415i \(0.222899\pi\)
\(60\) 5.90440 + 10.7650i 0.762255 + 1.38975i
\(61\) −7.17009 + 3.27447i −0.918036 + 0.419253i −0.817663 − 0.575697i \(-0.804731\pi\)
−0.100373 + 0.994950i \(0.532003\pi\)
\(62\) 2.16521 + 4.31542i 0.274982 + 0.548059i
\(63\) −0.00140136 0.000411476i −0.000176555 5.18412e-5i
\(64\) 7.80747 − 1.74453i 0.975934 − 0.218067i
\(65\) 0.0950563 − 0.0279111i 0.0117903 − 0.00346194i
\(66\) 7.13475 + 4.22419i 0.878227 + 0.519962i
\(67\) 7.10526 − 1.02158i 0.868045 − 0.124806i 0.306119 − 0.951993i \(-0.400970\pi\)
0.561926 + 0.827187i \(0.310060\pi\)
\(68\) −0.331859 − 0.0715390i −0.0402438 − 0.00867538i
\(69\) 5.32110 + 6.37151i 0.640585 + 0.767040i
\(70\) −1.86623 − 0.198868i −0.223057 − 0.0237693i
\(71\) −0.00700092 − 0.0486925i −0.000830856 − 0.00577873i 0.989402 − 0.145204i \(-0.0463837\pi\)
−0.990233 + 0.139425i \(0.955475\pi\)
\(72\) 0.00810125 + 0.00750004i 0.000954741 + 0.000883888i
\(73\) 11.6526 − 3.42151i 1.36383 − 0.400457i 0.483719 − 0.875223i \(-0.339286\pi\)
0.880112 + 0.474766i \(0.157467\pi\)
\(74\) −11.7748 + 10.9833i −1.36879 + 1.27679i
\(75\) −3.69577 + 12.5866i −0.426750 + 1.45338i
\(76\) −2.91244 − 7.76398i −0.334079 − 0.890590i
\(77\) −1.15289 + 0.526509i −0.131385 + 0.0600013i
\(78\) −0.0561304 + 0.0390502i −0.00635552 + 0.00442156i
\(79\) −5.97521 − 3.84003i −0.672263 − 0.432037i 0.159477 − 0.987202i \(-0.449019\pi\)
−0.831741 + 0.555164i \(0.812655\pi\)
\(80\) 11.9632 + 7.62494i 1.33753 + 0.852495i
\(81\) 1.27916 − 8.89679i 0.142129 − 0.988532i
\(82\) −6.57158 + 8.17052i −0.725709 + 0.902283i
\(83\) 4.52067 − 3.91718i 0.496208 − 0.429967i −0.370463 − 0.928847i \(-0.620801\pi\)
0.866671 + 0.498881i \(0.166255\pi\)
\(84\) 1.26525 − 0.277727i 0.138050 − 0.0303025i
\(85\) −0.325471 − 0.506442i −0.0353022 − 0.0549314i
\(86\) 12.9585 + 5.35335i 1.39735 + 0.577266i
\(87\) −8.91660 + 10.2903i −0.955960 + 1.10324i
\(88\) 9.57522 + 0.314977i 1.02072 + 0.0335766i
\(89\) −6.81956 + 14.9327i −0.722871 + 1.58287i 0.0869645 + 0.996211i \(0.472283\pi\)
−0.809836 + 0.586656i \(0.800444\pi\)
\(90\) 0.000717131 0.0195642i 7.55923e−5 0.00206225i
\(91\) − 0.0104522i − 0.00109569i
\(92\) 8.93102 + 3.49814i 0.931123 + 0.364706i
\(93\) − 5.90940i − 0.612776i
\(94\) 8.43407 − 0.309153i 0.869908 − 0.0318867i
\(95\) 6.10860 − 13.3760i 0.626729 − 1.37235i
\(96\) −9.49869 − 2.37696i −0.969456 − 0.242597i
\(97\) −3.32829 + 3.84105i −0.337936 + 0.389999i −0.899127 − 0.437687i \(-0.855798\pi\)
0.561191 + 0.827686i \(0.310343\pi\)
\(98\) 3.70418 − 8.96649i 0.374179 − 0.905752i
\(99\) 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 184.2.p.a.141.1 yes 220
4.3 odd 2 736.2.x.a.49.6 220
8.3 odd 2 736.2.x.a.49.17 220
8.5 even 2 inner 184.2.p.a.141.21 yes 220
23.8 even 11 inner 184.2.p.a.77.21 yes 220
92.31 odd 22 736.2.x.a.721.17 220
184.77 even 22 inner 184.2.p.a.77.1 ✓ 220
184.123 odd 22 736.2.x.a.721.6 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.1 ✓ 220 184.77 even 22 inner
184.2.p.a.77.21 yes 220 23.8 even 11 inner
184.2.p.a.141.1 yes 220 1.1 even 1 trivial
184.2.p.a.141.21 yes 220 8.5 even 2 inner
736.2.x.a.49.6 220 4.3 odd 2
736.2.x.a.49.17 220 8.3 odd 2
736.2.x.a.721.6 220 184.123 odd 22
736.2.x.a.721.17 220 92.31 odd 22