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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 141.2
Character \(\chi\) \(=\) 184.141
Dual form 184.2.p.a.77.2

$q$-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+(-1.39285 + 0.244866i) q^{2} +(0.302712 + 0.138244i) q^{3} +(1.88008 - 0.682125i) q^{4} +(-2.42383 - 2.10026i) q^{5} +(-0.455485 - 0.118430i) q^{6} +(-3.31871 + 2.13281i) q^{7} +(-2.45165 + 1.41047i) q^{8} +(-1.89206 - 2.18355i) q^{9} +(3.89033 + 2.33185i) q^{10} +(1.71129 + 0.246047i) q^{11} +(0.663424 + 0.0534226i) q^{12} +(-2.84392 + 4.42522i) q^{13} +(4.10023 - 3.78333i) q^{14} +(-0.443376 - 0.970857i) q^{15} +(3.06941 - 2.56490i) q^{16} +(-4.25725 - 1.25004i) q^{17} +(3.17004 + 2.57807i) q^{18} +(-0.717267 - 2.44279i) q^{19} +(-5.98965 - 2.29531i) q^{20} +(-1.29946 + 0.186835i) q^{21} +(-2.44383 + 0.0763303i) q^{22} +(-1.06992 - 4.67496i) q^{23} +(-0.937133 + 0.0880401i) q^{24} +(0.752288 + 5.23228i) q^{25} +(2.87757 - 6.86006i) q^{26} +(-0.552156 - 1.88047i) q^{27} +(-4.78461 + 6.27363i) q^{28} +(0.815234 - 2.77643i) q^{29} +(0.855287 + 1.24369i) q^{30} +(0.976875 + 2.13906i) q^{31} +(-3.64718 + 4.32412i) q^{32} +(0.484015 + 0.311058i) q^{33} +(6.23581 + 0.698668i) q^{34} +(12.5235 + 1.80060i) q^{35} +(-5.04668 - 2.81464i) q^{36} +(2.53613 - 2.19757i) q^{37} +(1.59720 + 3.22681i) q^{38} +(-1.47265 + 0.946414i) q^{39} +(8.90475 + 1.73037i) q^{40} +(-5.17915 + 5.97706i) q^{41} +(1.76421 - 0.578428i) q^{42} +(7.18551 + 3.28151i) q^{43} +(3.38520 - 0.704727i) q^{44} +9.26639i q^{45} +(2.63499 + 6.24955i) q^{46} +8.61069 q^{47} +(1.28373 - 0.352099i) q^{48} +(3.55708 - 7.78893i) q^{49} +(-2.32903 - 7.10359i) q^{50} +(-1.11591 - 0.966942i) q^{51} +(-2.32824 + 10.2597i) q^{52} +(-6.99377 - 10.8825i) q^{53} +(1.22954 + 2.48402i) q^{54} +(-3.63113 - 4.19054i) q^{55} +(5.12806 - 9.90983i) q^{56} +(0.120575 - 0.838620i) q^{57} +(-0.455648 + 4.06678i) q^{58} +(-0.808852 + 1.25860i) q^{59} +(-1.49583 - 1.52285i) q^{60} +(-6.38861 + 2.91758i) q^{61} +(-1.88443 - 2.74019i) q^{62} +(10.9363 + 3.21119i) q^{63} +(4.02116 - 6.91594i) q^{64} +(16.1873 - 4.75302i) q^{65} +(-0.750330 - 0.314739i) q^{66} +(-3.42719 + 0.492755i) q^{67} +(-8.85665 + 0.553795i) q^{68} +(0.322407 - 1.56308i) q^{69} +(-17.8843 + 0.558595i) q^{70} +(1.54568 + 10.7504i) q^{71} +(7.71849 + 2.68462i) q^{72} +(-3.07056 + 0.901599i) q^{73} +(-2.99435 + 3.68190i) q^{74} +(-0.495605 + 1.68788i) q^{75} +(-3.01480 - 4.10337i) q^{76} +(-6.20406 + 2.83330i) q^{77} +(1.81944 - 1.67882i) q^{78} +(-11.4777 - 7.37629i) q^{79} +(-12.8267 - 0.229684i) q^{80} +(-1.14073 + 7.93396i) q^{81} +(5.75022 - 9.59336i) q^{82} +(-7.72136 + 6.69059i) q^{83} +(-2.31565 + 1.23766i) q^{84} +(7.69344 + 11.9712i) q^{85} +(-10.8119 - 2.81118i) q^{86} +(0.630607 - 0.727759i) q^{87} +(-4.54253 + 1.81050i) q^{88} +(0.911256 - 1.99537i) q^{89} +(-2.26902 - 12.9067i) q^{90} -20.7516i q^{91} +(-5.20045 - 8.05949i) q^{92} +0.782567i q^{93} +(-11.9934 + 2.10846i) q^{94} +(-3.39196 + 7.42736i) q^{95} +(-1.70183 + 0.804764i) q^{96} +(7.40796 - 8.54924i) q^{97} +(-3.04725 + 11.7198i) q^{98} +(-2.70061 - 4.20223i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39285 + 0.244866i −0.984896 + 0.173146i
\(3\) 0.302712 + 0.138244i 0.174771 + 0.0798153i 0.500879 − 0.865517i \(-0.333010\pi\)
−0.326108 + 0.945332i \(0.605737\pi\)
\(4\) 1.88008 − 0.682125i 0.940041 − 0.341062i
\(5\) −2.42383 − 2.10026i −1.08397 − 0.939267i −0.0856012 − 0.996329i \(-0.527281\pi\)
−0.998371 + 0.0570626i \(0.981827\pi\)
\(6\) −0.455485 − 0.118430i −0.185951 − 0.0483488i
\(7\) −3.31871 + 2.13281i −1.25436 + 0.806125i −0.987501 − 0.157611i \(-0.949621\pi\)
−0.266854 + 0.963737i \(0.585984\pi\)
\(8\) −2.45165 + 1.41047i −0.866789 + 0.498676i
\(9\) −1.89206 − 2.18355i −0.630686 − 0.727851i
\(10\) 3.89033 + 2.33185i 1.23023 + 0.737395i
\(11\) 1.71129 + 0.246047i 0.515974 + 0.0741859i 0.395383 − 0.918516i \(-0.370612\pi\)
0.120591 + 0.992702i \(0.461521\pi\)
\(12\) 0.663424 + 0.0534226i 0.191514 + 0.0154218i
\(13\) −2.84392 + 4.42522i −0.788760 + 1.22734i 0.181050 + 0.983474i \(0.442050\pi\)
−0.969810 + 0.243861i \(0.921586\pi\)
\(14\) 4.10023 − 3.78333i 1.09583 − 1.01114i
\(15\) −0.443376 − 0.970857i −0.114479 − 0.250674i
\(16\) 3.06941 − 2.56490i 0.767353 − 0.641225i
\(17\) −4.25725 − 1.25004i −1.03253 − 0.303179i −0.278793 − 0.960351i \(-0.589934\pi\)
−0.753741 + 0.657172i \(0.771753\pi\)
\(18\) 3.17004 + 2.57807i 0.747185 + 0.607656i
\(19\) −0.717267 − 2.44279i −0.164552 − 0.560414i −0.999942 − 0.0107557i \(-0.996576\pi\)
0.835390 − 0.549658i \(-0.185242\pi\)
\(20\) −5.98965 − 2.29531i −1.33933 − 0.513247i
\(21\) −1.29946 + 0.186835i −0.283566 + 0.0407707i
\(22\) −2.44383 + 0.0763303i −0.521026 + 0.0162737i
\(23\) −1.06992 − 4.67496i −0.223095 − 0.974797i
\(24\) −0.937133 + 0.0880401i −0.191292 + 0.0179711i
\(25\) 0.752288 + 5.23228i 0.150458 + 1.04646i
\(26\) 2.87757 − 6.86006i 0.564338 − 1.34537i
\(27\) −0.552156 − 1.88047i −0.106262 − 0.361897i
\(28\) −4.78461 + 6.27363i −0.904206 + 1.18560i
\(29\) 0.815234 − 2.77643i 0.151385 − 0.515570i −0.848523 − 0.529159i \(-0.822507\pi\)
0.999908 + 0.0135889i \(0.00432562\pi\)
\(30\) 0.855287 + 1.24369i 0.156153 + 0.227066i
\(31\) 0.976875 + 2.13906i 0.175452 + 0.384186i 0.976844 − 0.213954i \(-0.0686341\pi\)
−0.801392 + 0.598140i \(0.795907\pi\)
\(32\) −3.64718 + 4.32412i −0.644737 + 0.764404i
\(33\) 0.484015 + 0.311058i 0.0842562 + 0.0541482i
\(34\) 6.23581 + 0.698668i 1.06943 + 0.119821i
\(35\) 12.5235 + 1.80060i 2.11685 + 0.304358i
\(36\) −5.04668 − 2.81464i −0.841113 − 0.469106i
\(37\) 2.53613 − 2.19757i 0.416937 − 0.361278i −0.420985 − 0.907068i \(-0.638316\pi\)
0.837922 + 0.545789i \(0.183770\pi\)
\(38\) 1.59720 + 3.22681i 0.259100 + 0.523457i
\(39\) −1.47265 + 0.946414i −0.235813 + 0.151548i
\(40\) 8.90475 + 1.73037i 1.40796 + 0.273596i
\(41\) −5.17915 + 5.97706i −0.808847 + 0.933460i −0.998832 − 0.0483275i \(-0.984611\pi\)
0.189984 + 0.981787i \(0.439156\pi\)
\(42\) 1.76421 − 0.578428i 0.272224 − 0.0892533i
\(43\) 7.18551 + 3.28151i 1.09578 + 0.500426i 0.879497 − 0.475905i \(-0.157879\pi\)
0.216283 + 0.976331i \(0.430607\pi\)
\(44\) 3.38520 − 0.704727i 0.510339 − 0.106242i
\(45\) 9.26639i 1.38135i
\(46\) 2.63499 + 6.24955i 0.388508 + 0.921446i
\(47\) 8.61069 1.25600 0.627999 − 0.778214i \(-0.283874\pi\)
0.627999 + 0.778214i \(0.283874\pi\)
\(48\) 1.28373 − 0.352099i 0.185291 − 0.0508211i
\(49\) 3.55708 − 7.78893i 0.508155 − 1.11270i
\(50\) −2.32903 − 7.10359i −0.329375 − 1.00460i
\(51\) −1.11591 − 0.966942i −0.156259 − 0.135399i
\(52\) −2.32824 + 10.2597i −0.322869 + 1.42276i
\(53\) −6.99377 − 10.8825i −0.960668 − 1.49483i −0.866442 − 0.499278i \(-0.833599\pi\)
−0.0942263 − 0.995551i \(-0.530038\pi\)
\(54\) 1.22954 + 2.48402i 0.167319 + 0.338032i
\(55\) −3.63113 − 4.19054i −0.489621 − 0.565053i
\(56\) 5.12806 − 9.90983i 0.685266 − 1.32426i
\(57\) 0.120575 − 0.838620i 0.0159706 − 0.111078i
\(58\) −0.455648 + 4.06678i −0.0598295 + 0.533995i
\(59\) −0.808852 + 1.25860i −0.105304 + 0.163855i −0.889906 − 0.456143i \(-0.849230\pi\)
0.784603 + 0.619999i \(0.212867\pi\)
\(60\) −1.49583 − 1.52285i −0.193111 − 0.196599i
\(61\) −6.38861 + 2.91758i −0.817978 + 0.373558i −0.780046 − 0.625722i \(-0.784804\pi\)
−0.0379320 + 0.999280i \(0.512077\pi\)
\(62\) −1.88443 − 2.74019i −0.239322 − 0.348004i
\(63\) 10.9363 + 3.21119i 1.37784 + 0.404571i
\(64\) 4.02116 − 6.91594i 0.502645 − 0.864493i
\(65\) 16.1873 − 4.75302i 2.00779 − 0.589540i
\(66\) −0.750330 − 0.314739i −0.0923592 − 0.0387417i
\(67\) −3.42719 + 0.492755i −0.418698 + 0.0601996i −0.348444 − 0.937330i \(-0.613290\pi\)
−0.0702535 + 0.997529i \(0.522381\pi\)
\(68\) −8.85665 + 0.553795i −1.07403 + 0.0671576i
\(69\) 0.322407 − 1.56308i 0.0388132 − 0.188173i
\(70\) −17.8843 + 0.558595i −2.13758 + 0.0667649i
\(71\) 1.54568 + 10.7504i 0.183438 + 1.27584i 0.848558 + 0.529103i \(0.177471\pi\)
−0.665120 + 0.746737i \(0.731619\pi\)
\(72\) 7.71849 + 2.68462i 0.909633 + 0.316385i
\(73\) −3.07056 + 0.901599i −0.359382 + 0.105524i −0.456439 − 0.889755i \(-0.650875\pi\)
0.0970567 + 0.995279i \(0.469057\pi\)
\(74\) −2.99435 + 3.68190i −0.348086 + 0.428013i
\(75\) −0.495605 + 1.68788i −0.0572276 + 0.194899i
\(76\) −3.01480 − 4.10337i −0.345822 − 0.470689i
\(77\) −6.20406 + 2.83330i −0.707018 + 0.322884i
\(78\) 1.81944 − 1.67882i 0.206011 − 0.190089i
\(79\) −11.4777 − 7.37629i −1.29135 − 0.829898i −0.299104 − 0.954220i \(-0.596688\pi\)
−0.992242 + 0.124323i \(0.960324\pi\)
\(80\) −12.8267 − 0.229684i −1.43407 − 0.0256794i
\(81\) −1.14073 + 7.93396i −0.126748 + 0.881551i
\(82\) 5.75022 − 9.59336i 0.635005 − 1.05941i
\(83\) −7.72136 + 6.69059i −0.847529 + 0.734388i −0.965992 − 0.258570i \(-0.916749\pi\)
0.118463 + 0.992958i \(0.462203\pi\)
\(84\) −2.31565 + 1.23766i −0.252658 + 0.135040i
\(85\) 7.69344 + 11.9712i 0.834471 + 1.29846i
\(86\) −10.8119 − 2.81118i −1.16588 − 0.303137i
\(87\) 0.630607 − 0.727759i 0.0676081 − 0.0780239i
\(88\) −4.54253 + 1.81050i −0.484235 + 0.193000i
\(89\) 0.911256 − 1.99537i 0.0965929 − 0.211509i −0.855167 − 0.518352i \(-0.826546\pi\)
0.951760 + 0.306844i \(0.0992728\pi\)
\(90\) −2.26902 − 12.9067i −0.239176 − 1.36049i
\(91\) − 20.7516i − 2.17535i
\(92\) −5.20045 − 8.05949i −0.542185 − 0.840259i
\(93\) 0.782567i 0.0811484i
\(94\) −11.9934 + 2.10846i −1.23703 + 0.217472i
\(95\) −3.39196 + 7.42736i −0.348008 + 0.762031i
\(96\) −1.70183 + 0.804764i −0.173693 + 0.0821359i
\(97\) 7.40796 − 8.54924i 0.752164 − 0.868044i −0.242612 − 0.970123i \(-0.578004\pi\)
0.994776 + 0.102080i \(0.0325496\pi\)
\(98\) −3.04725 + 11.7198i −0.307819 + 1.18388i
\(99\) −2.70061 − 4.20223i −0.271422 − 0.422340i
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.141.2 yes 220
4.3 odd 2 736.2.x.a.49.10 220
8.3 odd 2 736.2.x.a.49.13 220
8.5 even 2 inner 184.2.p.a.141.22 yes 220
23.8 even 11 inner 184.2.p.a.77.22 yes 220
92.31 odd 22 736.2.x.a.721.13 220
184.77 even 22 inner 184.2.p.a.77.2 ✓ 220
184.123 odd 22 736.2.x.a.721.10 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.2 ✓ 220 184.77 even 22 inner
184.2.p.a.77.22 yes 220 23.8 even 11 inner
184.2.p.a.141.2 yes 220 1.1 even 1 trivial
184.2.p.a.141.22 yes 220 8.5 even 2 inner
736.2.x.a.49.10 220 4.3 odd 2
736.2.x.a.49.13 220 8.3 odd 2
736.2.x.a.721.10 220 184.123 odd 22
736.2.x.a.721.13 220 92.31 odd 22