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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 141.14
Character \(\chi\) \(=\) 184.141
Dual form 184.2.p.a.77.14

$q$-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+(0.603888 + 1.27880i) q^{2} +(-1.78987 - 0.817407i) q^{3} +(-1.27064 + 1.54450i) q^{4} +(-1.45070 - 1.25704i) q^{5} +(-0.0355856 - 2.78250i) q^{6} +(-2.66148 + 1.71043i) q^{7} +(-2.74242 - 0.692182i) q^{8} +(0.570905 + 0.658860i) q^{9} +(0.731438 - 2.61427i) q^{10} +(-2.15286 - 0.309535i) q^{11} +(3.53676 - 1.72583i) q^{12} +(-0.495678 + 0.771289i) q^{13} +(-3.79453 - 2.37058i) q^{14} +(1.56906 + 3.43576i) q^{15} +(-0.770957 - 3.92500i) q^{16} +(2.61973 + 0.769221i) q^{17} +(-0.497784 + 1.12795i) q^{18} +(0.657885 + 2.24055i) q^{19} +(3.78482 - 0.643366i) q^{20} +(6.16183 - 0.885937i) q^{21} +(-0.904256 - 2.93999i) q^{22} +(-4.60907 - 1.32532i) q^{23} +(4.34279 + 3.48059i) q^{24} +(-0.187187 - 1.30191i) q^{25} +(-1.28566 - 0.168098i) q^{26} +(1.17979 + 4.01801i) q^{27} +(0.740021 - 6.28399i) q^{28} +(-0.912416 + 3.10740i) q^{29} +(-3.44610 + 4.08132i) q^{30} +(2.51126 + 5.49888i) q^{31} +(4.55370 - 3.35616i) q^{32} +(3.60033 + 2.31379i) q^{33} +(0.598345 + 3.81462i) q^{34} +(6.01111 + 0.864267i) q^{35} +(-1.74302 + 0.0445905i) q^{36} +(-3.15655 + 2.73516i) q^{37} +(-2.46792 + 2.19434i) q^{38} +(1.51766 - 0.975339i) q^{39} +(3.10834 + 4.45149i) q^{40} +(5.56883 - 6.42678i) q^{41} +(4.85399 + 7.34471i) q^{42} +(-10.2176 - 4.66623i) q^{43} +(3.21358 - 2.93179i) q^{44} -1.67346i q^{45} +(-1.08854 - 6.69441i) q^{46} -8.82860 q^{47} +(-1.82841 + 7.65543i) q^{48} +(1.25001 - 2.73713i) q^{49} +(1.55184 - 1.02558i) q^{50} +(-4.06021 - 3.51819i) q^{51} +(-0.561429 - 1.74560i) q^{52} +(2.20843 + 3.43639i) q^{53} +(-4.42575 + 3.93515i) q^{54} +(2.73407 + 3.15528i) q^{55} +(8.48284 - 2.84849i) q^{56} +(0.653913 - 4.54806i) q^{57} +(-4.52473 + 0.709730i) q^{58} +(4.95086 - 7.70369i) q^{59} +(-7.30024 - 1.94220i) q^{60} +(-12.5779 + 5.74414i) q^{61} +(-5.51543 + 6.53210i) q^{62} +(-2.64639 - 0.777049i) q^{63} +(7.04177 + 3.79651i) q^{64} +(1.68863 - 0.495825i) q^{65} +(-0.784671 + 6.00136i) q^{66} +(11.0584 - 1.58996i) q^{67} +(-4.51679 + 3.06876i) q^{68} +(7.16632 + 6.13965i) q^{69} +(2.52482 + 8.20890i) q^{70} +(-1.57907 - 10.9827i) q^{71} +(-1.10961 - 2.20204i) q^{72} +(-12.2637 + 3.60095i) q^{73} +(-5.40392 - 2.38485i) q^{74} +(-0.729152 + 2.48327i) q^{75} +(-4.29647 - 1.83083i) q^{76} +(6.25924 - 2.85850i) q^{77} +(2.16375 + 1.35178i) q^{78} +(-0.617855 - 0.397071i) q^{79} +(-3.81546 + 6.66314i) q^{80} +(1.54488 - 10.7449i) q^{81} +(11.5815 + 3.24035i) q^{82} +(6.88908 - 5.96942i) q^{83} +(-6.46113 + 10.6426i) q^{84} +(-2.83351 - 4.40902i) q^{85} +(-0.203143 - 15.8841i) q^{86} +(4.17312 - 4.81604i) q^{87} +(5.68980 + 2.33905i) q^{88} +(-0.538653 + 1.17948i) q^{89} +(2.14002 - 1.01058i) q^{90} -2.90059i q^{91} +(7.90342 - 5.43470i) q^{92} -11.8950i q^{93} +(-5.33149 - 11.2900i) q^{94} +(1.86207 - 4.07737i) q^{95} +(-10.8939 + 2.28486i) q^{96} +(-8.37752 + 9.66818i) q^{97} +(4.25510 - 0.0544186i) q^{98} +(-1.02514 - 1.59515i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.603888 + 1.27880i 0.427013 + 0.904245i
\(3\) −1.78987 − 0.817407i −1.03338 − 0.471930i −0.174800 − 0.984604i \(-0.555928\pi\)
−0.858583 + 0.512674i \(0.828655\pi\)
\(4\) −1.27064 + 1.54450i −0.635319 + 0.772250i
\(5\) −1.45070 − 1.25704i −0.648775 − 0.562167i 0.267081 − 0.963674i \(-0.413941\pi\)
−0.915856 + 0.401508i \(0.868486\pi\)
\(6\) −0.0355856 − 2.78250i −0.0145277 − 1.13595i
\(7\) −2.66148 + 1.71043i −1.00595 + 0.646482i −0.936341 − 0.351093i \(-0.885810\pi\)
−0.0696047 + 0.997575i \(0.522174\pi\)
\(8\) −2.74242 − 0.692182i −0.969593 − 0.244723i
\(9\) 0.570905 + 0.658860i 0.190302 + 0.219620i
\(10\) 0.731438 − 2.61427i 0.231301 − 0.826704i
\(11\) −2.15286 − 0.309535i −0.649112 − 0.0933282i −0.190109 − 0.981763i \(-0.560884\pi\)
−0.459003 + 0.888435i \(0.651793\pi\)
\(12\) 3.53676 − 1.72583i 1.02098 − 0.498204i
\(13\) −0.495678 + 0.771289i −0.137476 + 0.213917i −0.903164 − 0.429295i \(-0.858762\pi\)
0.765688 + 0.643212i \(0.222399\pi\)
\(14\) −3.79453 − 2.37058i −1.01413 − 0.633565i
\(15\) 1.56906 + 3.43576i 0.405129 + 0.887110i
\(16\) −0.770957 − 3.92500i −0.192739 − 0.981250i
\(17\) 2.61973 + 0.769221i 0.635377 + 0.186564i 0.583530 − 0.812092i \(-0.301671\pi\)
0.0518470 + 0.998655i \(0.483489\pi\)
\(18\) −0.497784 + 1.12795i −0.117329 + 0.265860i
\(19\) 0.657885 + 2.24055i 0.150929 + 0.514018i 0.999896 − 0.0144286i \(-0.00459292\pi\)
−0.848967 + 0.528446i \(0.822775\pi\)
\(20\) 3.78482 − 0.643366i 0.846312 − 0.143861i
\(21\) 6.16183 − 0.885937i 1.34462 − 0.193327i
\(22\) −0.904256 − 2.93999i −0.192788 − 0.626809i
\(23\) −4.60907 − 1.32532i −0.961057 − 0.276349i
\(24\) 4.34279 + 3.48059i 0.886469 + 0.710473i
\(25\) −0.187187 − 1.30191i −0.0374374 − 0.260383i
\(26\) −1.28566 − 0.168098i −0.252138 − 0.0329667i
\(27\) 1.17979 + 4.01801i 0.227052 + 0.773267i
\(28\) 0.740021 − 6.28399i 0.139851 − 1.18756i
\(29\) −0.912416 + 3.10740i −0.169431 + 0.577030i 0.830372 + 0.557209i \(0.188128\pi\)
−0.999804 + 0.0198210i \(0.993690\pi\)
\(30\) −3.44610 + 4.08132i −0.629169 + 0.745144i
\(31\) 2.51126 + 5.49888i 0.451035 + 0.987629i 0.989440 + 0.144943i \(0.0462997\pi\)
−0.538405 + 0.842686i \(0.680973\pi\)
\(32\) 4.55370 − 3.35616i 0.804988 − 0.593291i
\(33\) 3.60033 + 2.31379i 0.626737 + 0.402779i
\(34\) 0.598345 + 3.81462i 0.102615 + 0.654202i
\(35\) 6.01111 + 0.864267i 1.01606 + 0.146088i
\(36\) −1.74302 + 0.0445905i −0.290504 + 0.00743175i
\(37\) −3.15655 + 2.73516i −0.518933 + 0.449658i −0.874523 − 0.484984i \(-0.838825\pi\)
0.355590 + 0.934642i \(0.384280\pi\)
\(38\) −2.46792 + 2.19434i −0.400349 + 0.355970i
\(39\) 1.51766 − 0.975339i 0.243020 − 0.156179i
\(40\) 3.10834 + 4.45149i 0.491472 + 0.703843i
\(41\) 5.56883 − 6.42678i 0.869706 − 1.00369i −0.130220 − 0.991485i \(-0.541568\pi\)
0.999925 − 0.0122088i \(-0.00388628\pi\)
\(42\) 4.85399 + 7.34471i 0.748987 + 1.13331i
\(43\) −10.2176 − 4.66623i −1.55817 − 0.711593i −0.564661 − 0.825323i \(-0.690993\pi\)
−0.993512 + 0.113730i \(0.963720\pi\)
\(44\) 3.21358 − 2.93179i 0.484466 − 0.441983i
\(45\) − 1.67346i − 0.249465i
\(46\) −1.08854 − 6.69441i −0.160497 − 0.987036i
\(47\) −8.82860 −1.28778 −0.643892 − 0.765116i \(-0.722681\pi\)
−0.643892 + 0.765116i \(0.722681\pi\)
\(48\) −1.82841 + 7.65543i −0.263908 + 1.10497i
\(49\) 1.25001 − 2.73713i 0.178572 − 0.391019i
\(50\) 1.55184 − 1.02558i 0.219463 − 0.145039i
\(51\) −4.06021 − 3.51819i −0.568543 − 0.492645i
\(52\) −0.561429 − 1.74560i −0.0778562 − 0.242072i
\(53\) 2.20843 + 3.43639i 0.303351 + 0.472024i 0.959145 − 0.282917i \(-0.0913020\pi\)
−0.655793 + 0.754941i \(0.727666\pi\)
\(54\) −4.42575 + 3.93515i −0.602269 + 0.535506i
\(55\) 2.73407 + 3.15528i 0.368662 + 0.425458i
\(56\) 8.48284 − 2.84849i 1.13357 − 0.380646i
\(57\) 0.653913 − 4.54806i 0.0866128 − 0.602405i
\(58\) −4.52473 + 0.709730i −0.594126 + 0.0931921i
\(59\) 4.95086 − 7.70369i 0.644547 − 1.00293i −0.353180 − 0.935555i \(-0.614900\pi\)
0.997727 − 0.0673795i \(-0.0214638\pi\)
\(60\) −7.30024 − 1.94220i −0.942457 − 0.250737i
\(61\) −12.5779 + 5.74414i −1.61044 + 0.735462i −0.998463 − 0.0554171i \(-0.982351\pi\)
−0.611973 + 0.790879i \(0.709624\pi\)
\(62\) −5.51543 + 6.53210i −0.700461 + 0.829577i
\(63\) −2.64639 − 0.777049i −0.333413 − 0.0978990i
\(64\) 7.04177 + 3.79651i 0.880221 + 0.474564i
\(65\) 1.68863 − 0.495825i 0.209448 − 0.0614995i
\(66\) −0.784671 + 6.00136i −0.0965863 + 0.738716i
\(67\) 11.0584 − 1.58996i 1.35100 − 0.194245i 0.571478 − 0.820617i \(-0.306370\pi\)
0.779524 + 0.626372i \(0.215461\pi\)
\(68\) −4.51679 + 3.06876i −0.547741 + 0.372142i
\(69\) 7.16632 + 6.13965i 0.862723 + 0.739126i
\(70\) 2.52482 + 8.20890i 0.301773 + 0.981151i
\(71\) −1.57907 − 10.9827i −0.187401 − 1.30341i −0.838704 − 0.544588i \(-0.816686\pi\)
0.651303 − 0.758818i \(-0.274223\pi\)
\(72\) −1.10961 − 2.20204i −0.130769 − 0.259513i
\(73\) −12.2637 + 3.60095i −1.43536 + 0.421459i −0.904672 − 0.426109i \(-0.859884\pi\)
−0.530687 + 0.847568i \(0.678066\pi\)
\(74\) −5.40392 − 2.38485i −0.628193 − 0.277233i
\(75\) −0.729152 + 2.48327i −0.0841953 + 0.286743i
\(76\) −4.29647 − 1.83083i −0.492838 − 0.210010i
\(77\) 6.25924 − 2.85850i 0.713306 − 0.325756i
\(78\) 2.16375 + 1.35178i 0.244997 + 0.153059i
\(79\) −0.617855 − 0.397071i −0.0695141 − 0.0446740i 0.505422 − 0.862872i \(-0.331337\pi\)
−0.574936 + 0.818198i \(0.694973\pi\)
\(80\) −3.81546 + 6.66314i −0.426581 + 0.744962i
\(81\) 1.54488 − 10.7449i 0.171653 − 1.19387i
\(82\) 11.5815 + 3.24035i 1.27896 + 0.357837i
\(83\) 6.88908 − 5.96942i 0.756175 − 0.655229i −0.188932 − 0.981990i \(-0.560503\pi\)
0.945107 + 0.326761i \(0.105957\pi\)
\(84\) −6.46113 + 10.6426i −0.704967 + 1.16121i
\(85\) −2.83351 − 4.40902i −0.307337 − 0.478225i
\(86\) −0.203143 − 15.8841i −0.0219055 − 1.71283i
\(87\) 4.17312 − 4.81604i 0.447405 − 0.516333i
\(88\) 5.68980 + 2.33905i 0.606535 + 0.249343i
\(89\) −0.538653 + 1.17948i −0.0570971 + 0.125025i −0.936030 − 0.351920i \(-0.885529\pi\)
0.878933 + 0.476945i \(0.158256\pi\)
\(90\) 2.14002 − 1.01058i 0.225578 − 0.106525i
\(91\) − 2.90059i − 0.304065i
\(92\) 7.90342 − 5.43470i 0.823989 − 0.566607i
\(93\) − 11.8950i − 1.23346i
\(94\) −5.33149 − 11.2900i −0.549901 − 1.16447i
\(95\) 1.86207 − 4.07737i 0.191045 − 0.418329i
\(96\) −10.8939 + 2.28486i −1.11185 + 0.233198i
\(97\) −8.37752 + 9.66818i −0.850609 + 0.981655i −0.999975 − 0.00712715i \(-0.997731\pi\)
0.149366 + 0.988782i \(0.452277\pi\)
\(98\) 4.25510 − 0.0544186i 0.429830 − 0.00549711i
\(99\) −1.02514 − 1.59515i −0.103030 − 0.160318i
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.141.14 yes 220
4.3 odd 2 736.2.x.a.49.18 220
8.3 odd 2 736.2.x.a.49.5 220
8.5 even 2 inner 184.2.p.a.141.11 yes 220
23.8 even 11 inner 184.2.p.a.77.11 ✓ 220
92.31 odd 22 736.2.x.a.721.5 220
184.77 even 22 inner 184.2.p.a.77.14 yes 220
184.123 odd 22 736.2.x.a.721.18 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.11 ✓ 220 23.8 even 11 inner
184.2.p.a.77.14 yes 220 184.77 even 22 inner
184.2.p.a.141.11 yes 220 8.5 even 2 inner
184.2.p.a.141.14 yes 220 1.1 even 1 trivial
736.2.x.a.49.5 220 8.3 odd 2
736.2.x.a.49.18 220 4.3 odd 2
736.2.x.a.721.5 220 92.31 odd 22
736.2.x.a.721.18 220 184.123 odd 22