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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 133.20
Character \(\chi\) \(=\) 184.133
Dual form 184.2.p.a.101.20

$q$-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+(1.30998 + 0.532871i) q^{2} +(-1.38623 - 1.20117i) q^{3} +(1.43210 + 1.39610i) q^{4} +(2.42814 + 0.349114i) q^{5} +(-1.17586 - 2.31219i) q^{6} +(-0.504023 - 1.10366i) q^{7} +(1.13208 + 2.59199i) q^{8} +(0.0518652 + 0.360730i) q^{9} +(2.99478 + 1.75122i) q^{10} +(-0.0146685 - 0.0499564i) q^{11} +(-0.308254 - 3.65551i) q^{12} +(1.97157 + 0.900388i) q^{13} +(-0.0721540 - 1.71435i) q^{14} +(-2.94661 - 3.40056i) q^{15} +(0.101806 + 3.99870i) q^{16} +(-1.80878 + 1.16243i) q^{17} +(-0.124280 + 0.500187i) q^{18} +(-0.779449 + 1.21285i) q^{19} +(2.98993 + 3.88989i) q^{20} +(-0.626992 + 2.13534i) q^{21} +(0.00740483 - 0.0732584i) q^{22} +(-3.84939 - 2.86045i) q^{23} +(1.54411 - 4.95290i) q^{24} +(0.976513 + 0.286730i) q^{25} +(2.10293 + 2.23008i) q^{26} +(-2.61359 + 4.06683i) q^{27} +(0.819006 - 2.28421i) q^{28} +(-3.80111 - 5.91464i) q^{29} +(-2.04793 - 6.02483i) q^{30} +(-1.23571 - 1.42608i) q^{31} +(-1.99743 + 5.29247i) q^{32} +(-0.0396724 + 0.0868704i) q^{33} +(-2.98889 + 0.558917i) q^{34} +(-0.838537 - 2.85579i) q^{35} +(-0.429340 + 0.589010i) q^{36} +(-9.88119 + 1.42070i) q^{37} +(-1.66735 + 1.17346i) q^{38} +(-1.65153 - 3.61634i) q^{39} +(1.84395 + 6.68893i) q^{40} +(-1.47207 + 10.2385i) q^{41} +(-1.95921 + 2.46314i) q^{42} +(2.92273 + 2.53256i) q^{43} +(0.0487374 - 0.0920212i) q^{44} +0.894010i q^{45} +(-3.51838 - 5.79836i) q^{46} -2.07797 q^{47} +(4.66201 - 5.66540i) q^{48} +(3.62001 - 4.17771i) q^{49} +(1.12642 + 0.895966i) q^{50} +(3.90365 + 0.561261i) q^{51} +(1.56645 + 4.04196i) q^{52} +(5.87150 - 2.68142i) q^{53} +(-5.59085 + 3.93476i) q^{54} +(-0.0181768 - 0.126422i) q^{55} +(2.29007 - 2.55585i) q^{56} +(2.53733 - 0.745028i) q^{57} +(-1.82764 - 9.77357i) q^{58} +(10.5443 + 4.81543i) q^{59} +(0.527703 - 8.98370i) q^{60} +(6.59460 - 5.71425i) q^{61} +(-0.858835 - 2.52661i) q^{62} +(0.371981 - 0.239058i) q^{63} +(-5.43680 + 5.86867i) q^{64} +(4.47292 + 2.87457i) q^{65} +(-0.0982607 + 0.0926583i) q^{66} +(3.35638 - 11.4308i) q^{67} +(-4.21321 - 0.860521i) q^{68} +(1.90024 + 8.58902i) q^{69} +(0.423302 - 4.18787i) q^{70} +(11.0242 + 3.23699i) q^{71} +(-0.876293 + 0.542809i) q^{72} +(3.55961 + 2.28762i) q^{73} +(-13.7012 - 3.40431i) q^{74} +(-1.00926 - 1.57043i) q^{75} +(-2.80950 + 0.648725i) q^{76} +(-0.0477415 + 0.0413682i) q^{77} +(-0.236426 - 5.61739i) q^{78} +(1.56885 - 3.43530i) q^{79} +(-1.14880 + 9.74495i) q^{80} +(9.55703 - 2.80620i) q^{81} +(-7.38417 + 12.6278i) q^{82} +(-13.0840 + 1.88119i) q^{83} +(-3.87906 + 2.18267i) q^{84} +(-4.79778 + 2.19107i) q^{85} +(2.47919 + 4.87504i) q^{86} +(-1.83531 + 12.7648i) q^{87} +(0.112881 - 0.0945752i) q^{88} +(-9.62643 + 11.1095i) q^{89} +(-0.476392 + 1.17114i) q^{90} -2.62976i q^{91} +(-1.51924 - 9.47058i) q^{92} +3.46117i q^{93} +(-2.72209 - 1.10729i) q^{94} +(-2.31603 + 2.67284i) q^{95} +(9.12606 - 4.93731i) q^{96} +(1.06269 - 7.39115i) q^{97} +(6.96832 - 3.54372i) q^{98} +(0.0172600 - 0.00788239i) 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 \(\left(\mathbb{Z}/184\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(93\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{6}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30998 + 0.532871i 0.926296 + 0.376797i
\(3\) −1.38623 − 1.20117i −0.800338 − 0.693497i 0.155356 − 0.987859i \(-0.450348\pi\)
−0.955694 + 0.294361i \(0.904893\pi\)
\(4\) 1.43210 + 1.39610i 0.716049 + 0.698050i
\(5\) 2.42814 + 0.349114i 1.08590 + 0.156128i 0.661942 − 0.749555i \(-0.269732\pi\)
0.423955 + 0.905683i \(0.360642\pi\)
\(6\) −1.17586 − 2.31219i −0.480043 − 0.943948i
\(7\) −0.504023 − 1.10366i −0.190503 − 0.417143i 0.790146 − 0.612919i \(-0.210005\pi\)
−0.980649 + 0.195776i \(0.937278\pi\)
\(8\) 1.13208 + 2.59199i 0.400250 + 0.916406i
\(9\) 0.0518652 + 0.360730i 0.0172884 + 0.120243i
\(10\) 2.99478 + 1.75122i 0.947033 + 0.553783i
\(11\) −0.0146685 − 0.0499564i −0.00442273 − 0.0150624i 0.957252 − 0.289255i \(-0.0934076\pi\)
−0.961675 + 0.274193i \(0.911589\pi\)
\(12\) −0.308254 − 3.65551i −0.0889854 − 1.05525i
\(13\) 1.97157 + 0.900388i 0.546816 + 0.249723i 0.669614 − 0.742709i \(-0.266459\pi\)
−0.122798 + 0.992432i \(0.539187\pi\)
\(14\) −0.0721540 − 1.71435i −0.0192840 − 0.458179i
\(15\) −2.94661 − 3.40056i −0.760810 − 0.878022i
\(16\) 0.101806 + 3.99870i 0.0254515 + 0.999676i
\(17\) −1.80878 + 1.16243i −0.438693 + 0.281931i −0.741286 − 0.671189i \(-0.765784\pi\)
0.302593 + 0.953120i \(0.402148\pi\)
\(18\) −0.124280 + 0.500187i −0.0292931 + 0.117895i
\(19\) −0.779449 + 1.21285i −0.178818 + 0.278246i −0.919079 − 0.394073i \(-0.871066\pi\)
0.740261 + 0.672320i \(0.234702\pi\)
\(20\) 2.98993 + 3.88989i 0.668570 + 0.869806i
\(21\) −0.626992 + 2.13534i −0.136821 + 0.465969i
\(22\) 0.00740483 − 0.0732584i 0.00157872 − 0.0156187i
\(23\) −3.84939 − 2.86045i −0.802654 − 0.596445i
\(24\) 1.54411 − 4.95290i 0.315189 − 1.01101i
\(25\) 0.976513 + 0.286730i 0.195303 + 0.0573460i
\(26\) 2.10293 + 2.23008i 0.412419 + 0.437355i
\(27\) −2.61359 + 4.06683i −0.502986 + 0.782662i
\(28\) 0.819006 − 2.28421i 0.154778 − 0.431675i
\(29\) −3.80111 − 5.91464i −0.705849 − 1.09832i −0.990207 − 0.139604i \(-0.955417\pi\)
0.284359 − 0.958718i \(-0.408219\pi\)
\(30\) −2.04793 − 6.02483i −0.373900 − 1.09998i
\(31\) −1.23571 − 1.42608i −0.221940 − 0.256132i 0.633850 − 0.773456i \(-0.281474\pi\)
−0.855790 + 0.517324i \(0.826928\pi\)
\(32\) −1.99743 + 5.29247i −0.353099 + 0.935586i
\(33\) −0.0396724 + 0.0868704i −0.00690607 + 0.0151222i
\(34\) −2.98889 + 0.558917i −0.512590 + 0.0958534i
\(35\) −0.838537 − 2.85579i −0.141739 − 0.482717i
\(36\) −0.429340 + 0.589010i −0.0715567 + 0.0981684i
\(37\) −9.88119 + 1.42070i −1.62446 + 0.233562i −0.893566 − 0.448931i \(-0.851805\pi\)
−0.730892 + 0.682493i \(0.760896\pi\)
\(38\) −1.66735 + 1.17346i −0.270481 + 0.190360i
\(39\) −1.65153 − 3.61634i −0.264456 − 0.579078i
\(40\) 1.84395 + 6.68893i 0.291553 + 1.05761i
\(41\) −1.47207 + 10.2385i −0.229899 + 1.59898i 0.468627 + 0.883396i \(0.344749\pi\)
−0.698526 + 0.715585i \(0.746160\pi\)
\(42\) −1.95921 + 2.46314i −0.302312 + 0.380072i
\(43\) 2.92273 + 2.53256i 0.445712 + 0.386211i 0.848598 − 0.529039i \(-0.177447\pi\)
−0.402886 + 0.915250i \(0.631993\pi\)
\(44\) 0.0487374 − 0.0920212i 0.00734745 − 0.0138727i
\(45\) 0.894010i 0.133271i
\(46\) −3.51838 − 5.79836i −0.518757 − 0.854922i
\(47\) −2.07797 −0.303102 −0.151551 − 0.988449i \(-0.548427\pi\)
−0.151551 + 0.988449i \(0.548427\pi\)
\(48\) 4.66201 − 5.66540i 0.672903 − 0.817730i
\(49\) 3.62001 − 4.17771i 0.517144 − 0.596816i
\(50\) 1.12642 + 0.895966i 0.159300 + 0.126709i
\(51\) 3.90365 + 0.561261i 0.546621 + 0.0785922i
\(52\) 1.56645 + 4.04196i 0.217228 + 0.560519i
\(53\) 5.87150 − 2.68142i 0.806512 − 0.368322i 0.0308930 − 0.999523i \(-0.490165\pi\)
0.775619 + 0.631201i \(0.217438\pi\)
\(54\) −5.59085 + 3.93476i −0.760819 + 0.535453i
\(55\) −0.0181768 − 0.126422i −0.00245095 − 0.0170468i
\(56\) 2.29007 − 2.55585i 0.306024 − 0.341540i
\(57\) 2.53733 − 0.745028i 0.336078 − 0.0986813i
\(58\) −1.82764 − 9.77357i −0.239981 − 1.28333i
\(59\) 10.5443 + 4.81543i 1.37275 + 0.626916i 0.958983 − 0.283464i \(-0.0914838\pi\)
0.413772 + 0.910381i \(0.364211\pi\)
\(60\) 0.527703 − 8.98370i 0.0681262 − 1.15979i
\(61\) 6.59460 − 5.71425i 0.844352 − 0.731635i −0.120982 − 0.992655i \(-0.538604\pi\)
0.965333 + 0.261020i \(0.0840588\pi\)
\(62\) −0.858835 − 2.52661i −0.109072 − 0.320880i
\(63\) 0.371981 − 0.239058i 0.0468652 − 0.0301185i
\(64\) −5.43680 + 5.86867i −0.679600 + 0.733583i
\(65\) 4.47292 + 2.87457i 0.554797 + 0.356546i
\(66\) −0.0982607 + 0.0926583i −0.0120951 + 0.0114054i
\(67\) 3.35638 − 11.4308i 0.410047 − 1.39649i −0.453064 − 0.891478i \(-0.649669\pi\)
0.863111 − 0.505014i \(-0.168513\pi\)
\(68\) −4.21321 − 0.860521i −0.510927 − 0.104354i
\(69\) 1.90024 + 8.58902i 0.228762 + 1.03400i
\(70\) 0.423302 − 4.18787i 0.0505943 − 0.500546i
\(71\) 11.0242 + 3.23699i 1.30833 + 0.384160i 0.860268 − 0.509842i \(-0.170296\pi\)
0.448062 + 0.894002i \(0.352114\pi\)
\(72\) −0.876293 + 0.542809i −0.103272 + 0.0639707i
\(73\) 3.55961 + 2.28762i 0.416621 + 0.267746i 0.732114 − 0.681182i \(-0.238534\pi\)
−0.315494 + 0.948928i \(0.602170\pi\)
\(74\) −13.7012 − 3.40431i −1.59273 − 0.395743i
\(75\) −1.00926 − 1.57043i −0.116539 − 0.181338i
\(76\) −2.80950 + 0.648725i −0.322272 + 0.0744139i
\(77\) −0.0477415 + 0.0413682i −0.00544065 + 0.00471435i
\(78\) −0.236426 − 5.61739i −0.0267700 − 0.636044i
\(79\) 1.56885 − 3.43530i 0.176509 − 0.386501i −0.800612 − 0.599182i \(-0.795492\pi\)
0.977122 + 0.212681i \(0.0682197\pi\)
\(80\) −1.14880 + 9.74495i −0.128440 + 1.08952i
\(81\) 9.55703 − 2.80620i 1.06189 − 0.311800i
\(82\) −7.38417 + 12.6278i −0.815445 + 1.39451i
\(83\) −13.0840 + 1.88119i −1.43615 + 0.206488i −0.816027 − 0.578013i \(-0.803828\pi\)
−0.620127 + 0.784501i \(0.712919\pi\)
\(84\) −3.87906 + 2.18267i −0.423240 + 0.238149i
\(85\) −4.79778 + 2.19107i −0.520393 + 0.237655i
\(86\) 2.47919 + 4.87504i 0.267338 + 0.525689i
\(87\) −1.83531 + 12.7648i −0.196765 + 1.36853i
\(88\) 0.112881 − 0.0945752i 0.0120331 − 0.0100818i
\(89\) −9.62643 + 11.1095i −1.02040 + 1.17760i −0.0364178 + 0.999337i \(0.511595\pi\)
−0.983982 + 0.178268i \(0.942951\pi\)
\(90\) −0.476392 + 1.17114i −0.0502161 + 0.123449i
\(91\) − 2.62976i − 0.275673i
\(92\) −1.51924 − 9.47058i −0.158391 − 0.987376i
\(93\) 3.46117i 0.358907i
\(94\) −2.72209 − 1.10729i −0.280763 − 0.114208i
\(95\) −2.31603 + 2.67284i −0.237620 + 0.274228i
\(96\) 9.12606 − 4.93731i 0.931425 − 0.503912i
\(97\) 1.06269 − 7.39115i 0.107899 − 0.750457i −0.861992 − 0.506921i \(-0.830783\pi\)
0.969892 − 0.243536i \(-0.0783074\pi\)
\(98\) 6.96832 − 3.54372i 0.703906 − 0.357970i
\(99\) 0.0172600 − 0.00788239i 0.00173470 − 0.000792210i
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.133.20 yes 220
4.3 odd 2 736.2.x.a.593.16 220
8.3 odd 2 736.2.x.a.593.7 220
8.5 even 2 inner 184.2.p.a.133.22 yes 220
23.9 even 11 inner 184.2.p.a.101.22 yes 220
92.55 odd 22 736.2.x.a.561.7 220
184.101 even 22 inner 184.2.p.a.101.20 ✓ 220
184.147 odd 22 736.2.x.a.561.16 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.101.20 ✓ 220 184.101 even 22 inner
184.2.p.a.101.22 yes 220 23.9 even 11 inner
184.2.p.a.133.20 yes 220 1.1 even 1 trivial
184.2.p.a.133.22 yes 220 8.5 even 2 inner
736.2.x.a.561.7 220 92.55 odd 22
736.2.x.a.561.16 220 184.147 odd 22
736.2.x.a.593.7 220 8.3 odd 2
736.2.x.a.593.16 220 4.3 odd 2