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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 141.22
Character \(\chi\) \(=\) 184.141
Dual form 184.2.p.a.77.22

$q$-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+(1.40542 - 0.157465i) q^{2} +(-0.302712 - 0.138244i) q^{3} +(1.95041 - 0.442609i) q^{4} +(2.42383 + 2.10026i) q^{5} +(-0.447207 - 0.146624i) q^{6} +(-3.31871 + 2.13281i) q^{7} +(2.67145 - 0.929173i) q^{8} +(-1.89206 - 2.18355i) q^{9} +(3.73722 + 2.57008i) q^{10} +(-1.71129 - 0.246047i) q^{11} +(-0.651601 - 0.135649i) q^{12} +(2.84392 - 4.42522i) q^{13} +(-4.32834 + 3.52007i) q^{14} +(-0.443376 - 0.970857i) q^{15} +(3.60819 - 1.72654i) q^{16} +(-4.25725 - 1.25004i) q^{17} +(-3.00297 - 2.77087i) q^{18} +(0.717267 + 2.44279i) q^{19} +(5.65707 + 3.02357i) 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} +(3.30008 - 6.66711i) q^{26} +(0.552156 + 1.88047i) q^{27} +(-5.52885 + 5.62874i) q^{28} +(-0.815234 + 2.77643i) q^{29} +(-0.776005 - 1.29465i) q^{30} +(0.976875 + 2.13906i) q^{31} +(4.79916 - 2.99467i) q^{32} +(0.484015 + 0.311058i) q^{33} +(-6.18005 - 1.08646i) q^{34} +(-12.5235 - 1.80060i) q^{35} +(-4.65675 - 3.42138i) q^{36} +(-2.53613 + 2.19757i) q^{37} +(1.39271 + 3.32020i) q^{38} +(-1.47265 + 0.946414i) q^{39} +(8.42666 + 3.35859i) q^{40} +(-5.17915 + 5.97706i) q^{41} +(1.79687 - 0.467201i) q^{42} +(-7.18551 - 3.28151i) q^{43} +(-3.44662 + 0.277541i) q^{44} -9.26639i q^{45} +(-2.23984 - 6.40181i) q^{46} +8.61069 q^{47} +(-1.33093 + 0.0238325i) q^{48} +(3.55708 - 7.78893i) q^{49} +(1.88118 + 7.23509i) q^{50} +(1.11591 + 0.966942i) q^{51} +(3.58816 - 9.88973i) q^{52} +(6.99377 + 10.8825i) q^{53} +(1.07212 + 2.55591i) q^{54} +(-3.63113 - 4.19054i) q^{55} +(-6.88403 + 8.78134i) q^{56} +(0.120575 - 0.838620i) q^{57} +(-0.708555 + 4.03042i) q^{58} +(0.808852 - 1.25860i) q^{59} +(-1.29447 - 1.69733i) q^{60} +(6.38861 - 2.91758i) q^{61} +(1.70975 + 2.85245i) q^{62} +(10.9363 + 3.21119i) q^{63} +(6.27328 - 4.96447i) q^{64} +(16.1873 - 4.75302i) q^{65} +(0.729225 + 0.360951i) 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} +(-7.08343 - 4.07520i) q^{72} +(-3.07056 + 0.901599i) q^{73} +(-3.21829 + 3.48786i) q^{74} +(0.495605 - 1.68788i) q^{75} +(2.48016 + 4.44696i) q^{76} +(6.20406 - 2.83330i) q^{77} +(-1.92066 + 1.56200i) q^{78} +(-11.4777 - 7.37629i) q^{79} +(12.3719 + 3.39332i) q^{80} +(-1.14073 + 7.93396i) q^{81} +(-6.33770 + 9.21581i) q^{82} +(7.72136 - 6.69059i) q^{83} +(2.45179 - 0.939558i) q^{84} +(-7.69344 - 11.9712i) q^{85} +(-10.6154 - 3.48043i) q^{86} +(0.630607 - 0.727759i) q^{87} +(-4.80025 + 0.932785i) q^{88} +(0.911256 - 1.99537i) q^{89} +(-1.45913 - 13.0232i) q^{90} +20.7516i q^{91} +(-4.15597 - 8.64453i) q^{92} -0.782567i q^{93} +(12.1016 - 1.35588i) q^{94} +(-3.39196 + 7.42736i) q^{95} +(-1.86676 + 0.243069i) q^{96} +(7.40796 - 8.54924i) q^{97} +(3.77271 - 11.5068i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.40542 − 0.157465i 0.993782 − 0.111345i
\(3\) −0.302712 − 0.138244i −0.174771 − 0.0798153i 0.326108 − 0.945332i \(-0.394263\pi\)
−0.500879 + 0.865517i \(0.666990\pi\)
\(4\) 1.95041 − 0.442609i 0.975205 − 0.221304i
\(5\) 2.42383 + 2.10026i 1.08397 + 0.939267i 0.998371 − 0.0570626i \(-0.0181735\pi\)
0.0856012 + 0.996329i \(0.472719\pi\)
\(6\) −0.447207 − 0.146624i −0.182571 − 0.0598592i
\(7\) −3.31871 + 2.13281i −1.25436 + 0.806125i −0.987501 − 0.157611i \(-0.949621\pi\)
−0.266854 + 0.963737i \(0.585984\pi\)
\(8\) 2.67145 − 0.929173i 0.944500 − 0.328512i
\(9\) −1.89206 − 2.18355i −0.630686 − 0.727851i
\(10\) 3.73722 + 2.57008i 1.18181 + 0.812732i
\(11\) −1.71129 − 0.246047i −0.515974 − 0.0741859i −0.120591 − 0.992702i \(-0.538479\pi\)
−0.395383 + 0.918516i \(0.629388\pi\)
\(12\) −0.651601 − 0.135649i −0.188101 − 0.0391586i
\(13\) 2.84392 − 4.42522i 0.788760 − 1.22734i −0.181050 − 0.983474i \(-0.557950\pi\)
0.969810 − 0.243861i \(-0.0784141\pi\)
\(14\) −4.32834 + 3.52007i −1.15680 + 0.940778i
\(15\) −0.443376 − 0.970857i −0.114479 − 0.250674i
\(16\) 3.60819 − 1.72654i 0.902049 − 0.431634i
\(17\) −4.25725 − 1.25004i −1.03253 − 0.303179i −0.278793 − 0.960351i \(-0.589934\pi\)
−0.753741 + 0.657172i \(0.771753\pi\)
\(18\) −3.00297 − 2.77087i −0.707807 − 0.653101i
\(19\) 0.717267 + 2.44279i 0.164552 + 0.560414i 0.999942 + 0.0107557i \(0.00342372\pi\)
−0.835390 + 0.549658i \(0.814758\pi\)
\(20\) 5.65707 + 3.02357i 1.26496 + 0.676090i
\(21\) 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
\(26\) 3.30008 − 6.66711i 0.647199 − 1.30753i
\(27\) 0.552156 + 1.88047i 0.106262 + 0.361897i
\(28\) −5.52885 + 5.62874i −1.04485 + 1.06373i
\(29\) −0.815234 + 2.77643i −0.151385 + 0.515570i −0.999908 − 0.0135889i \(-0.995674\pi\)
0.848523 + 0.529159i \(0.177493\pi\)
\(30\) −0.776005 − 1.29465i −0.141678 − 0.236369i
\(31\) 0.976875 + 2.13906i 0.175452 + 0.384186i 0.976844 − 0.213954i \(-0.0686341\pi\)
−0.801392 + 0.598140i \(0.795907\pi\)
\(32\) 4.79916 − 2.99467i 0.848379 − 0.529389i
\(33\) 0.484015 + 0.311058i 0.0842562 + 0.0541482i
\(34\) −6.18005 − 1.08646i −1.05987 − 0.186327i
\(35\) −12.5235 − 1.80060i −2.11685 − 0.304358i
\(36\) −4.65675 − 3.42138i −0.776125 − 0.570230i
\(37\) −2.53613 + 2.19757i −0.416937 + 0.361278i −0.837922 − 0.545789i \(-0.816230\pi\)
0.420985 + 0.907068i \(0.361684\pi\)
\(38\) 1.39271 + 3.32020i 0.225928 + 0.538607i
\(39\) −1.47265 + 0.946414i −0.235813 + 0.151548i
\(40\) 8.42666 + 3.35859i 1.33237 + 0.531040i
\(41\) −5.17915 + 5.97706i −0.808847 + 0.933460i −0.998832 − 0.0483275i \(-0.984611\pi\)
0.189984 + 0.981787i \(0.439156\pi\)
\(42\) 1.79687 − 0.467201i 0.277263 − 0.0720907i
\(43\) −7.18551 − 3.28151i −1.09578 − 0.500426i −0.216283 − 0.976331i \(-0.569393\pi\)
−0.879497 + 0.475905i \(0.842121\pi\)
\(44\) −3.44662 + 0.277541i −0.519598 + 0.0418409i
\(45\) − 9.26639i − 1.38135i
\(46\) −2.23984 − 6.40181i −0.330246 − 0.943895i
\(47\) 8.61069 1.25600 0.627999 − 0.778214i \(-0.283874\pi\)
0.627999 + 0.778214i \(0.283874\pi\)
\(48\) −1.33093 + 0.0238325i −0.192103 + 0.00343992i
\(49\) 3.55708 − 7.78893i 0.508155 − 1.11270i
\(50\) 1.88118 + 7.23509i 0.266039 + 1.02320i
\(51\) 1.11591 + 0.966942i 0.156259 + 0.135399i
\(52\) 3.58816 − 9.88973i 0.497588 − 1.37146i
\(53\) 6.99377 + 10.8825i 0.960668 + 1.49483i 0.866442 + 0.499278i \(0.166401\pi\)
0.0942263 + 0.995551i \(0.469962\pi\)
\(54\) 1.07212 + 2.55591i 0.145897 + 0.347815i
\(55\) −3.63113 − 4.19054i −0.489621 − 0.565053i
\(56\) −6.88403 + 8.78134i −0.919917 + 1.17346i
\(57\) 0.120575 − 0.838620i 0.0159706 − 0.111078i
\(58\) −0.708555 + 4.03042i −0.0930378 + 0.529220i
\(59\) 0.808852 − 1.25860i 0.105304 − 0.163855i −0.784603 − 0.619999i \(-0.787133\pi\)
0.889906 + 0.456143i \(0.150770\pi\)
\(60\) −1.29447 − 1.69733i −0.167116 − 0.219124i
\(61\) 6.38861 − 2.91758i 0.817978 − 0.373558i 0.0379320 − 0.999280i \(-0.487923\pi\)
0.780046 + 0.625722i \(0.215196\pi\)
\(62\) 1.70975 + 2.85245i 0.217138 + 0.362262i
\(63\) 10.9363 + 3.21119i 1.37784 + 0.404571i
\(64\) 6.27328 − 4.96447i 0.784160 − 0.620559i
\(65\) 16.1873 − 4.75302i 2.00779 − 0.589540i
\(66\) 0.729225 + 0.360951i 0.0897614 + 0.0444300i
\(67\) 3.42719 − 0.492755i 0.418698 − 0.0601996i 0.0702535 − 0.997529i \(-0.477619\pi\)
0.348444 + 0.937330i \(0.386710\pi\)
\(68\) −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 \(0.731619\pi\)
\(72\) −7.08343 − 4.07520i −0.834791 − 0.480267i
\(73\) −3.07056 + 0.901599i −0.359382 + 0.105524i −0.456439 − 0.889755i \(-0.650875\pi\)
0.0970567 + 0.995279i \(0.469057\pi\)
\(74\) −3.21829 + 3.48786i −0.374118 + 0.405455i
\(75\) 0.495605 − 1.68788i 0.0572276 − 0.194899i
\(76\) 2.48016 + 4.44696i 0.284494 + 0.510102i
\(77\) 6.20406 − 2.83330i 0.707018 − 0.322884i
\(78\) −1.92066 + 1.56200i −0.217472 + 0.176862i
\(79\) −11.4777 − 7.37629i −1.29135 − 0.829898i −0.299104 − 0.954220i \(-0.596688\pi\)
−0.992242 + 0.124323i \(0.960324\pi\)
\(80\) 12.3719 + 3.39332i 1.38322 + 0.379385i
\(81\) −1.14073 + 7.93396i −0.126748 + 0.881551i
\(82\) −6.33770 + 9.21581i −0.699882 + 1.01772i
\(83\) 7.72136 − 6.69059i 0.847529 − 0.734388i −0.118463 − 0.992958i \(-0.537797\pi\)
0.965992 + 0.258570i \(0.0832514\pi\)
\(84\) 2.45179 − 0.939558i 0.267512 − 0.102514i
\(85\) −7.69344 − 11.9712i −0.834471 − 1.29846i
\(86\) −10.6154 − 3.48043i −1.14469 − 0.375305i
\(87\) 0.630607 − 0.727759i 0.0676081 − 0.0780239i
\(88\) −4.80025 + 0.932785i −0.511708 + 0.0994352i
\(89\) 0.911256 − 1.99537i 0.0965929 − 0.211509i −0.855167 − 0.518352i \(-0.826546\pi\)
0.951760 + 0.306844i \(0.0992728\pi\)
\(90\) −1.45913 − 13.0232i −0.153806 − 1.37276i
\(91\) 20.7516i 2.17535i
\(92\) −4.15597 − 8.64453i −0.433290 − 0.901255i
\(93\) − 0.782567i − 0.0811484i
\(94\) 12.1016 − 1.35588i 1.24819 − 0.139849i
\(95\) −3.39196 + 7.42736i −0.348008 + 0.762031i
\(96\) −1.86676 + 0.243069i −0.190526 + 0.0248082i
\(97\) 7.40796 − 8.54924i 0.752164 − 0.868044i −0.242612 − 0.970123i \(-0.578004\pi\)
0.994776 + 0.102080i \(0.0325496\pi\)
\(98\) 3.77271 − 11.5068i 0.381101 − 1.16237i
\(99\) 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.141.22 yes 220
4.3 odd 2 736.2.x.a.49.13 220
8.3 odd 2 736.2.x.a.49.10 220
8.5 even 2 inner 184.2.p.a.141.2 yes 220
23.8 even 11 inner 184.2.p.a.77.2 ✓ 220
92.31 odd 22 736.2.x.a.721.10 220
184.77 even 22 inner 184.2.p.a.77.22 yes 220
184.123 odd 22 736.2.x.a.721.13 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.77.2 ✓ 220 23.8 even 11 inner
184.2.p.a.77.22 yes 220 184.77 even 22 inner
184.2.p.a.141.2 yes 220 8.5 even 2 inner
184.2.p.a.141.22 yes 220 1.1 even 1 trivial
736.2.x.a.49.10 220 8.3 odd 2
736.2.x.a.49.13 220 4.3 odd 2
736.2.x.a.721.10 220 92.31 odd 22
736.2.x.a.721.13 220 184.123 odd 22