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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 141.21
Character \(\chi\) \(=\) 184.141
Dual form 184.2.p.a.77.21

$q$-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+(1.37061 - 0.348457i) q^{2} +(-1.57450 - 0.719052i) q^{3} +(1.75715 - 0.955200i) q^{4} +(-2.68037 - 2.32255i) q^{5} +(-2.40859 - 0.436893i) q^{6} +(0.314784 - 0.202299i) q^{7} +(2.07553 - 1.92150i) q^{8} +(-0.00255606 - 0.00294985i) q^{9} +(-4.48305 - 2.24932i) q^{10} +(3.35271 + 0.482046i) q^{11} +(-3.45348 + 0.240481i) q^{12} +(-0.0151019 + 0.0234990i) q^{13} +(0.360954 - 0.386962i) q^{14} +(2.55021 + 5.58419i) q^{15} +(2.17519 - 3.35687i) q^{16} +(-0.162865 - 0.0478215i) q^{17} +(-0.00453127 - 0.00315243i) q^{18} +(1.16810 + 3.97819i) q^{19} +(-6.92832 - 1.52080i) 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} +(-0.0125104 + 0.0374704i) q^{26} +(1.46488 + 4.98891i) q^{27} +(0.359887 - 0.656152i) q^{28} +(2.21620 - 7.54770i) q^{29} +(5.44120 + 6.76511i) q^{30} +(-1.41823 - 3.10550i) q^{31} +(1.81161 - 5.35892i) q^{32} +(-4.93223 - 3.16975i) q^{33} +(-0.239889 - 0.00879319i) q^{34} +(-1.31359 - 0.188865i) q^{35} +(-0.00730910 - 0.00274180i) q^{36} +(-8.60491 + 7.45620i) q^{37} +(2.98724 + 5.04552i) q^{38} +(0.0406750 - 0.0261402i) q^{39} +(-10.0260 + 0.329804i) q^{40} +(4.85532 - 5.60334i) q^{41} +(-0.846569 + 0.349729i) q^{42} +(9.01825 + 4.11850i) q^{43} +(6.35168 - 2.35547i) q^{44} +0.0138433i q^{45} +(6.65030 + 1.33171i) q^{46} -5.96779 q^{47} +(-5.83860 + 3.72133i) q^{48} +(-2.84974 + 6.24006i) q^{49} +(4.09222 + 9.90578i) q^{50} +(0.222046 + 0.192404i) q^{51} +(-0.00409012 + 0.0557168i) q^{52} +(-0.748843 - 1.16522i) q^{53} +(3.74620 + 6.32741i) q^{54} +(-7.86691 - 9.07889i) q^{55} +(0.264625 - 1.02474i) q^{56} +(1.02134 - 7.10359i) q^{57} +(0.407504 - 11.1172i) q^{58} +(1.06893 - 1.66328i) q^{59} +(9.81513 + 7.37632i) q^{60} +(7.17009 - 3.27447i) q^{61} +(-3.02598 - 3.76224i) q^{62} +(-0.00140136 - 0.000411476i) q^{63} +(0.615658 - 7.97628i) q^{64} +(0.0950563 - 0.0279111i) q^{65} +(-7.86470 - 2.62583i) 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} +(-0.0109733 - 0.00121103i) q^{72} +(11.6526 - 3.42151i) q^{73} +(-9.19583 + 13.2180i) q^{74} +(3.69577 - 12.5866i) q^{75} +(5.85250 + 5.87452i) q^{76} +(1.15289 - 0.526509i) q^{77} +(0.0466409 - 0.0500016i) q^{78} +(-5.97521 - 3.84003i) q^{79} +(-13.6268 + 3.94566i) q^{80} +(1.27916 - 8.89679i) q^{81} +(4.70223 - 9.37187i) q^{82} +(-4.52067 + 3.91718i) q^{83} +(-1.03845 + 0.774336i) q^{84} +(0.325471 + 0.506442i) q^{85} +(13.7956 + 2.50239i) q^{86} +(-8.91660 + 10.2903i) q^{87} +(7.88490 - 5.44173i) q^{88} +(-6.81956 + 14.9327i) q^{89} +(0.00482379 + 0.0189738i) q^{90} +0.0104522i q^{91} +(9.57903 - 0.492090i) q^{92} +5.90940i q^{93} +(-8.17953 + 2.07952i) q^{94} +(6.10860 - 13.3760i) q^{95} +(-6.70573 + 7.13500i) q^{96} +(-3.32829 + 3.84105i) q^{97} +(-1.73149 + 9.54572i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.37061 − 0.348457i 0.969169 − 0.246397i
\(3\) −1.57450 − 0.719052i −0.909040 − 0.415145i −0.0946821 − 0.995508i \(-0.530183\pi\)
−0.814358 + 0.580363i \(0.802911\pi\)
\(4\) 1.75715 − 0.955200i 0.878577 − 0.477600i
\(5\) −2.68037 − 2.32255i −1.19870 − 1.03868i −0.998260 − 0.0589698i \(-0.981218\pi\)
−0.200437 − 0.979707i \(-0.564236\pi\)
\(6\) −2.40859 − 0.436893i −0.983304 − 0.178361i
\(7\) 0.314784 − 0.202299i 0.118977 − 0.0764619i −0.479799 − 0.877378i \(-0.659290\pi\)
0.598776 + 0.800917i \(0.295654\pi\)
\(8\) 2.07553 − 1.92150i 0.733811 − 0.679354i
\(9\) −0.00255606 − 0.00294985i −0.000852021 − 0.000983285i
\(10\) −4.48305 − 2.24932i −1.41767 − 0.711298i
\(11\) 3.35271 + 0.482046i 1.01088 + 0.145342i 0.627801 − 0.778374i \(-0.283955\pi\)
0.383078 + 0.923716i \(0.374864\pi\)
\(12\) −3.45348 + 0.240481i −0.996935 + 0.0694208i
\(13\) −0.0151019 + 0.0234990i −0.00418852 + 0.00651746i −0.843341 − 0.537378i \(-0.819415\pi\)
0.839153 + 0.543895i \(0.183051\pi\)
\(14\) 0.360954 − 0.386962i 0.0964689 − 0.103420i
\(15\) 2.55021 + 5.58419i 0.658462 + 1.44183i
\(16\) 2.17519 − 3.35687i 0.543797 − 0.839217i
\(17\) −0.162865 − 0.0478215i −0.0395006 − 0.0115984i 0.261923 − 0.965089i \(-0.415643\pi\)
−0.301423 + 0.953490i \(0.597462\pi\)
\(18\) −0.00453127 − 0.00315243i −0.00106803 − 0.000743034i
\(19\) 1.16810 + 3.97819i 0.267981 + 0.912658i 0.978023 + 0.208499i \(0.0668578\pi\)
−0.710042 + 0.704159i \(0.751324\pi\)
\(20\) −6.92832 − 1.52080i −1.54922 − 0.340060i
\(21\) −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
\(26\) −0.0125104 + 0.0374704i −0.00245350 + 0.00734855i
\(27\) 1.46488 + 4.98891i 0.281916 + 0.960116i
\(28\) 0.359887 − 0.656152i 0.0680123 − 0.124001i
\(29\) 2.21620 − 7.54770i 0.411539 − 1.40157i −0.449613 − 0.893224i \(-0.648438\pi\)
0.861151 − 0.508348i \(-0.169744\pi\)
\(30\) 5.44120 + 6.76511i 0.993423 + 1.23514i
\(31\) −1.41823 − 3.10550i −0.254722 − 0.557764i 0.738465 − 0.674291i \(-0.235551\pi\)
−0.993187 + 0.116528i \(0.962824\pi\)
\(32\) 1.81161 − 5.35892i 0.320251 − 0.947333i
\(33\) −4.93223 − 3.16975i −0.858591 − 0.551783i
\(34\) −0.239889 − 0.00879319i −0.0411406 − 0.00150802i
\(35\) −1.31359 − 0.188865i −0.222036 − 0.0319240i
\(36\) −0.00730910 − 0.00274180i −0.00121818 − 0.000456967i
\(37\) −8.60491 + 7.45620i −1.41464 + 1.22579i −0.476664 + 0.879085i \(0.658154\pi\)
−0.937974 + 0.346706i \(0.887300\pi\)
\(38\) 2.98724 + 5.04552i 0.484595 + 0.818491i
\(39\) 0.0406750 − 0.0261402i 0.00651322 − 0.00418579i
\(40\) −10.0260 + 0.329804i −1.58525 + 0.0521466i
\(41\) 4.85532 − 5.60334i 0.758274 − 0.875094i −0.237069 − 0.971493i \(-0.576187\pi\)
0.995343 + 0.0963985i \(0.0307323\pi\)
\(42\) −0.846569 + 0.349729i −0.130628 + 0.0539644i
\(43\) 9.01825 + 4.11850i 1.37527 + 0.628065i 0.959579 − 0.281441i \(-0.0908122\pi\)
0.415692 + 0.909506i \(0.363540\pi\)
\(44\) 6.35168 − 2.35547i 0.957551 − 0.355101i
\(45\) 0.0138433i 0.00206363i
\(46\) 6.65030 + 1.33171i 0.980534 + 0.196350i
\(47\) −5.96779 −0.870492 −0.435246 − 0.900312i \(-0.643339\pi\)
−0.435246 + 0.900312i \(0.643339\pi\)
\(48\) −5.83860 + 3.72133i −0.842729 + 0.537128i
\(49\) −2.84974 + 6.24006i −0.407106 + 0.891438i
\(50\) 4.09222 + 9.90578i 0.578727 + 1.40089i
\(51\) 0.222046 + 0.192404i 0.0310926 + 0.0269419i
\(52\) −0.00409012 + 0.0557168i −0.000567198 + 0.00772652i
\(53\) −0.748843 − 1.16522i −0.102861 − 0.160056i 0.786008 − 0.618217i \(-0.212145\pi\)
−0.888869 + 0.458161i \(0.848508\pi\)
\(54\) 3.74620 + 6.32741i 0.509793 + 0.861052i
\(55\) −7.86691 − 9.07889i −1.06077 − 1.22420i
\(56\) 0.264625 − 1.02474i 0.0353620 − 0.136936i
\(57\) 1.02134 − 7.10359i 0.135280 − 0.940894i
\(58\) 0.407504 − 11.1172i 0.0535079 − 1.45976i
\(59\) 1.06893 − 1.66328i 0.139162 − 0.216541i −0.764676 − 0.644415i \(-0.777101\pi\)
0.903838 + 0.427874i \(0.140737\pi\)
\(60\) 9.81513 + 7.37632i 1.26713 + 0.952279i
\(61\) 7.17009 − 3.27447i 0.918036 − 0.419253i 0.100373 − 0.994950i \(-0.467997\pi\)
0.817663 + 0.575697i \(0.195269\pi\)
\(62\) −3.02598 − 3.76224i −0.384300 − 0.477805i
\(63\) −0.00140136 0.000411476i −0.000176555 5.18412e-5i
\(64\) 0.615658 − 7.97628i 0.0769573 − 0.997034i
\(65\) 0.0950563 − 0.0279111i 0.0117903 − 0.00346194i
\(66\) −7.86470 − 2.62583i −0.968078 − 0.323217i
\(67\) −7.10526 + 1.02158i −0.868045 + 0.124806i −0.561926 − 0.827187i \(-0.689940\pi\)
−0.306119 + 0.951993i \(0.599030\pi\)
\(68\) −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 \(0.955475\pi\)
\(72\) −0.0109733 − 0.00121103i −0.00129322 − 0.000142722i
\(73\) 11.6526 − 3.42151i 1.36383 − 0.400457i 0.483719 − 0.875223i \(-0.339286\pi\)
0.880112 + 0.474766i \(0.157467\pi\)
\(74\) −9.19583 + 13.2180i −1.06899 + 1.53656i
\(75\) 3.69577 − 12.5866i 0.426750 − 1.45338i
\(76\) 5.85250 + 5.87452i 0.671327 + 0.673854i
\(77\) 1.15289 − 0.526509i 0.131385 − 0.0600013i
\(78\) 0.0466409 − 0.0500016i 0.00528104 − 0.00566157i
\(79\) −5.97521 − 3.84003i −0.672263 − 0.432037i 0.159477 − 0.987202i \(-0.449019\pi\)
−0.831741 + 0.555164i \(0.812655\pi\)
\(80\) −13.6268 + 3.94566i −1.52352 + 0.441138i
\(81\) 1.27916 − 8.89679i 0.142129 − 0.988532i
\(82\) 4.70223 − 9.37187i 0.519275 − 1.03495i
\(83\) −4.52067 + 3.91718i −0.496208 + 0.429967i −0.866671 − 0.498881i \(-0.833745\pi\)
0.370463 + 0.928847i \(0.379199\pi\)
\(84\) −1.03845 + 0.774336i −0.113304 + 0.0844870i
\(85\) 0.325471 + 0.506442i 0.0353022 + 0.0549314i
\(86\) 13.7956 + 2.50239i 1.48762 + 0.269839i
\(87\) −8.91660 + 10.2903i −0.955960 + 1.10324i
\(88\) 7.88490 − 5.44173i 0.840533 − 0.580090i
\(89\) −6.81956 + 14.9327i −0.722871 + 1.58287i 0.0869645 + 0.996211i \(0.472283\pi\)
−0.809836 + 0.586656i \(0.800444\pi\)
\(90\) 0.00482379 + 0.0189738i 0.000508473 + 0.00200001i
\(91\) 0.0104522i 0.00109569i
\(92\) 9.57903 − 0.492090i 0.998683 − 0.0513039i
\(93\) 5.90940i 0.612776i
\(94\) −8.17953 + 2.07952i −0.843654 + 0.214486i
\(95\) 6.10860 − 13.3760i 0.626729 − 1.37235i
\(96\) −6.70573 + 7.13500i −0.684401 + 0.728213i
\(97\) −3.32829 + 3.84105i −0.337936 + 0.389999i −0.899127 − 0.437687i \(-0.855798\pi\)
0.561191 + 0.827686i \(0.310343\pi\)
\(98\) −1.73149 + 9.54572i −0.174907 + 0.964263i
\(99\) −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.141.21 yes 220
4.3 odd 2 736.2.x.a.49.17 220
8.3 odd 2 736.2.x.a.49.6 220
8.5 even 2 inner 184.2.p.a.141.1 yes 220
23.8 even 11 inner 184.2.p.a.77.1 ✓ 220
92.31 odd 22 736.2.x.a.721.6 220
184.77 even 22 inner 184.2.p.a.77.21 yes 220
184.123 odd 22 736.2.x.a.721.17 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.1 ✓ 220 23.8 even 11 inner
184.2.p.a.77.21 yes 220 184.77 even 22 inner
184.2.p.a.141.1 yes 220 8.5 even 2 inner
184.2.p.a.141.21 yes 220 1.1 even 1 trivial
736.2.x.a.49.6 220 8.3 odd 2
736.2.x.a.49.17 220 4.3 odd 2
736.2.x.a.721.6 220 92.31 odd 22
736.2.x.a.721.17 220 184.123 odd 22