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Properties

Label 184.2.p.a.85.3
Level $184$
Weight $2$
Character 184.85
Analytic conductor $1.469$
Analytic rank $0$
Dimension $220$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

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 85.3
Character \(\chi\) \(=\) 184.85
Dual form 184.2.p.a.13.3

$q$-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+(-1.38192 - 0.300513i) q^{2} +(-1.16603 - 1.81437i) q^{3} +(1.81938 + 0.830568i) q^{4} +(-1.93519 + 0.883774i) q^{5} +(1.06611 + 2.85772i) q^{6} +(-1.41177 + 0.414532i) q^{7} +(-2.26464 - 1.69452i) q^{8} +(-0.686086 + 1.50232i) q^{9} +(2.93986 - 0.639750i) q^{10} +(-0.283071 + 0.245282i) q^{11} +(-0.614490 - 4.26950i) q^{12} +(-1.76591 + 6.01413i) q^{13} +(2.07551 - 0.148594i) q^{14} +(3.85998 + 2.48066i) q^{15} +(2.62031 + 3.02224i) q^{16} +(-0.532657 + 3.70471i) q^{17} +(1.39958 - 1.86990i) q^{18} +(3.16421 - 0.454945i) q^{19} +(-4.25490 + 0.000613575i) q^{20} +(2.39827 + 2.07812i) q^{21} +(0.464891 - 0.253893i) q^{22} +(-4.65550 - 1.15165i) q^{23} +(-0.433868 + 6.08476i) q^{24} +(-0.310380 + 0.358198i) q^{25} +(4.24766 - 7.78034i) q^{26} +(-2.87863 + 0.413885i) q^{27} +(-2.91284 - 0.418375i) q^{28} +(-5.13484 - 0.738279i) q^{29} +(-4.58870 - 4.58804i) q^{30} +(2.19128 + 1.40825i) q^{31} +(-2.71283 - 4.96392i) q^{32} +(0.775102 + 0.227590i) q^{33} +(1.84940 - 4.95953i) q^{34} +(2.36569 - 2.04988i) q^{35} +(-2.49603 + 2.16345i) q^{36} +(1.51885 + 0.693637i) q^{37} +(-4.50939 - 0.322191i) q^{38} +(12.9710 - 3.80862i) q^{39} +(5.88009 + 1.27780i) q^{40} +(-3.71473 - 8.13412i) q^{41} +(-2.68971 - 3.59249i) q^{42} +(-6.08220 - 9.46409i) q^{43} +(-0.718738 + 0.211153i) q^{44} -3.51362i q^{45} +(6.08743 + 2.99052i) q^{46} +2.05122 q^{47} +(2.42812 - 8.27824i) q^{48} +(-4.06753 + 2.61404i) q^{49} +(0.536563 - 0.401726i) q^{50} +(7.34282 - 3.35335i) q^{51} +(-8.20800 + 9.47530i) q^{52} +(1.55979 + 5.31217i) q^{53} +(4.10240 + 0.293112i) q^{54} +(0.331023 - 0.724840i) q^{55} +(3.89958 + 1.45351i) q^{56} +(-4.51499 - 5.21058i) q^{57} +(6.87406 + 2.56333i) q^{58} +(-2.85121 + 9.71032i) q^{59} +(4.96244 + 7.71925i) q^{60} +(4.28250 - 6.66369i) q^{61} +(-2.60496 - 2.60459i) q^{62} +(0.345834 - 2.40533i) q^{63} +(2.25718 + 7.67497i) q^{64} +(-1.89775 - 13.1992i) q^{65} +(-1.00273 - 0.547439i) q^{66} +(-2.23503 - 1.93667i) q^{67} +(-4.04612 + 6.29788i) q^{68} +(3.33893 + 9.78967i) q^{69} +(-3.88520 + 2.12184i) q^{70} +(-4.88134 + 5.63337i) q^{71} +(4.09945 - 2.23962i) q^{72} +(-0.758975 - 5.27878i) q^{73} +(-1.89048 - 1.41498i) q^{74} +(1.01182 + 0.145477i) q^{75} +(6.13477 + 1.80037i) q^{76} +(0.297953 - 0.463623i) q^{77} +(-19.0693 + 1.36525i) q^{78} +(-12.4558 - 3.65736i) q^{79} +(-7.74180 - 3.53286i) q^{80} +(7.35214 + 8.48482i) q^{81} +(2.68903 + 12.3570i) q^{82} +(8.47618 + 3.87094i) q^{83} +(2.63736 + 5.77282i) q^{84} +(-2.24333 - 7.64008i) q^{85} +(5.56101 + 14.9064i) q^{86} +(4.64785 + 10.1774i) q^{87} +(1.05669 - 0.0758057i) q^{88} +(-3.11818 + 2.00394i) q^{89} +(-1.05589 + 4.85553i) q^{90} -9.22257i q^{91} +(-7.51363 - 5.96200i) q^{92} -5.61785i q^{93} +(-2.83462 - 0.616419i) q^{94} +(-5.72129 + 3.67685i) q^{95} +(-5.84318 + 10.7102i) q^{96} +(7.61767 + 16.6804i) q^{97} +(6.40653 - 2.39004i) q^{98} +(-0.174281 - 0.593547i) 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 \(\left(\mathbb{Z}/184\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(93\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{4}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38192 − 0.300513i −0.977162 − 0.212495i
\(3\) −1.16603 − 1.81437i −0.673206 − 1.04753i −0.994918 − 0.100691i \(-0.967895\pi\)
0.321712 − 0.946838i \(-0.395742\pi\)
\(4\) 1.81938 + 0.830568i 0.909692 + 0.415284i
\(5\) −1.93519 + 0.883774i −0.865445 + 0.395236i −0.798122 − 0.602496i \(-0.794173\pi\)
−0.0673232 + 0.997731i \(0.521446\pi\)
\(6\) 1.06611 + 2.85772i 0.435237 + 1.16666i
\(7\) −1.41177 + 0.414532i −0.533598 + 0.156678i −0.537421 − 0.843314i \(-0.680601\pi\)
0.00382311 + 0.999993i \(0.498783\pi\)
\(8\) −2.26464 − 1.69452i −0.800671 − 0.599104i
\(9\) −0.686086 + 1.50232i −0.228695 + 0.500773i
\(10\) 2.93986 − 0.639750i 0.929666 − 0.202307i
\(11\) −0.283071 + 0.245282i −0.0853491 + 0.0739554i −0.696490 − 0.717566i \(-0.745256\pi\)
0.611141 + 0.791521i \(0.290711\pi\)
\(12\) −0.614490 − 4.26950i −0.177388 − 1.23250i
\(13\) −1.76591 + 6.01413i −0.489775 + 1.66802i 0.229512 + 0.973306i \(0.426287\pi\)
−0.719287 + 0.694713i \(0.755531\pi\)
\(14\) 2.07551 − 0.148594i 0.554705 − 0.0397135i
\(15\) 3.85998 + 2.48066i 0.996644 + 0.640504i
\(16\) 2.62031 + 3.02224i 0.655079 + 0.755561i
\(17\) −0.532657 + 3.70471i −0.129188 + 0.898524i 0.817398 + 0.576073i \(0.195416\pi\)
−0.946586 + 0.322451i \(0.895493\pi\)
\(18\) 1.39958 − 1.86990i 0.329884 − 0.440740i
\(19\) 3.16421 − 0.454945i 0.725919 − 0.104371i 0.230558 − 0.973059i \(-0.425945\pi\)
0.495361 + 0.868687i \(0.335036\pi\)
\(20\) −4.25490 0.000613575i −0.951424 0.000137200i
\(21\) 2.39827 + 2.07812i 0.523346 + 0.453482i
\(22\) 0.464891 − 0.253893i 0.0991150 − 0.0541302i
\(23\) −4.65550 − 1.15165i −0.970740 − 0.240135i
\(24\) −0.433868 + 6.08476i −0.0885629 + 1.24205i
\(25\) −0.310380 + 0.358198i −0.0620761 + 0.0716396i
\(26\) 4.24766 − 7.78034i 0.833035 − 1.52585i
\(27\) −2.87863 + 0.413885i −0.553993 + 0.0796521i
\(28\) −2.91284 − 0.418375i −0.550475 − 0.0790654i
\(29\) −5.13484 − 0.738279i −0.953516 − 0.137095i −0.352039 − 0.935985i \(-0.614512\pi\)
−0.601477 + 0.798890i \(0.705421\pi\)
\(30\) −4.58870 − 4.58804i −0.837779 − 0.837658i
\(31\) 2.19128 + 1.40825i 0.393565 + 0.252929i 0.722424 − 0.691450i \(-0.243028\pi\)
−0.328859 + 0.944379i \(0.606664\pi\)
\(32\) −2.71283 − 4.96392i −0.479565 − 0.877506i
\(33\) 0.775102 + 0.227590i 0.134928 + 0.0396184i
\(34\) 1.84940 − 4.95953i 0.317170 − 0.850552i
\(35\) 2.36569 − 2.04988i 0.399875 − 0.346493i
\(36\) −2.49603 + 2.16345i −0.416005 + 0.360575i
\(37\) 1.51885 + 0.693637i 0.249698 + 0.114033i 0.536333 − 0.844007i \(-0.319809\pi\)
−0.286635 + 0.958040i \(0.592537\pi\)
\(38\) −4.50939 − 0.322191i −0.731519 − 0.0522663i
\(39\) 12.9710 − 3.80862i 2.07702 − 0.609867i
\(40\) 5.88009 + 1.27780i 0.929724 + 0.202039i
\(41\) −3.71473 − 8.13412i −0.580143 − 1.27034i −0.941218 − 0.337799i \(-0.890317\pi\)
0.361075 − 0.932537i \(-0.382410\pi\)
\(42\) −2.68971 − 3.59249i −0.415031 − 0.554334i
\(43\) −6.08220 − 9.46409i −0.927527 − 1.44326i −0.896149 − 0.443753i \(-0.853647\pi\)
−0.0313780 − 0.999508i \(-0.509990\pi\)
\(44\) −0.718738 + 0.211153i −0.108354 + 0.0318325i
\(45\) − 3.51362i − 0.523780i
\(46\) 6.08743 + 2.99052i 0.897543 + 0.440928i
\(47\) 2.05122 0.299202 0.149601 − 0.988746i \(-0.452201\pi\)
0.149601 + 0.988746i \(0.452201\pi\)
\(48\) 2.42812 − 8.27824i 0.350469 − 1.19486i
\(49\) −4.06753 + 2.61404i −0.581075 + 0.373434i
\(50\) 0.536563 − 0.401726i 0.0758815 − 0.0568127i
\(51\) 7.34282 − 3.35335i 1.02820 − 0.469563i
\(52\) −8.20800 + 9.47530i −1.13825 + 1.31399i
\(53\) 1.55979 + 5.31217i 0.214254 + 0.729682i 0.994549 + 0.104270i \(0.0332508\pi\)
−0.780295 + 0.625412i \(0.784931\pi\)
\(54\) 4.10240 + 0.293112i 0.558266 + 0.0398875i
\(55\) 0.331023 − 0.724840i 0.0446352 − 0.0977374i
\(56\) 3.89958 + 1.45351i 0.521103 + 0.194233i
\(57\) −4.51499 − 5.21058i −0.598025 − 0.690158i
\(58\) 6.87406 + 2.56333i 0.902608 + 0.336581i
\(59\) −2.85121 + 9.71032i −0.371196 + 1.26418i 0.536269 + 0.844047i \(0.319833\pi\)
−0.907464 + 0.420129i \(0.861985\pi\)
\(60\) 4.96244 + 7.71925i 0.640648 + 0.996551i
\(61\) 4.28250 − 6.66369i 0.548317 − 0.853198i −0.450907 − 0.892571i \(-0.648899\pi\)
0.999225 + 0.0393724i \(0.0125359\pi\)
\(62\) −2.60496 − 2.60459i −0.330831 − 0.330783i
\(63\) 0.345834 − 2.40533i 0.0435710 − 0.303043i
\(64\) 2.25718 + 7.67497i 0.282148 + 0.959371i
\(65\) −1.89775 − 13.1992i −0.235388 − 1.63716i
\(66\) −1.00273 − 0.547439i −0.123428 − 0.0673851i
\(67\) −2.23503 − 1.93667i −0.273053 − 0.236602i 0.507560 − 0.861616i \(-0.330548\pi\)
−0.780613 + 0.625015i \(0.785093\pi\)
\(68\) −4.04612 + 6.29788i −0.490664 + 0.763730i
\(69\) 3.33893 + 9.78967i 0.401960 + 1.17854i
\(70\) −3.88520 + 2.12184i −0.464371 + 0.253609i
\(71\) −4.88134 + 5.63337i −0.579309 + 0.668558i −0.967456 − 0.253039i \(-0.918570\pi\)
0.388147 + 0.921597i \(0.373115\pi\)
\(72\) 4.09945 − 2.23962i 0.483125 − 0.263942i
\(73\) −0.758975 − 5.27878i −0.0888313 − 0.617835i −0.984797 − 0.173711i \(-0.944424\pi\)
0.895965 − 0.444124i \(-0.146485\pi\)
\(74\) −1.89048 − 1.41498i −0.219764 − 0.164488i
\(75\) 1.01182 + 0.145477i 0.116835 + 0.0167983i
\(76\) 6.13477 + 1.80037i 0.703707 + 0.206517i
\(77\) 0.297953 − 0.463623i 0.0339548 − 0.0528348i
\(78\) −19.0693 + 1.36525i −2.15918 + 0.154584i
\(79\) −12.4558 − 3.65736i −1.40139 − 0.411485i −0.508229 − 0.861222i \(-0.669700\pi\)
−0.893161 + 0.449736i \(0.851518\pi\)
\(80\) −7.74180 − 3.53286i −0.865559 − 0.394986i
\(81\) 7.35214 + 8.48482i 0.816904 + 0.942758i
\(82\) 2.68903 + 12.3570i 0.296954 + 1.36460i
\(83\) 8.47618 + 3.87094i 0.930381 + 0.424891i 0.822173 − 0.569237i \(-0.192761\pi\)
0.108208 + 0.994128i \(0.465489\pi\)
\(84\) 2.63736 + 5.77282i 0.287760 + 0.629866i
\(85\) −2.24333 − 7.64008i −0.243323 − 0.828683i
\(86\) 5.56101 + 14.9064i 0.599659 + 1.60739i
\(87\) 4.64785 + 10.1774i 0.498302 + 1.09113i
\(88\) 1.05669 − 0.0758057i 0.112644 − 0.00808091i
\(89\) −3.11818 + 2.00394i −0.330527 + 0.212417i −0.695369 − 0.718653i \(-0.744759\pi\)
0.364842 + 0.931069i \(0.381123\pi\)
\(90\) −1.05589 + 4.85553i −0.111301 + 0.511818i
\(91\) − 9.22257i − 0.966788i
\(92\) −7.51363 − 5.96200i −0.783350 − 0.621581i
\(93\) − 5.61785i − 0.582544i
\(94\) −2.83462 − 0.616419i −0.292369 − 0.0635788i
\(95\) −5.72129 + 3.67685i −0.586992 + 0.377237i
\(96\) −5.84318 + 10.7102i −0.596367 + 1.09310i
\(97\) 7.61767 + 16.6804i 0.773457 + 1.69363i 0.718890 + 0.695124i \(0.244651\pi\)
0.0545672 + 0.998510i \(0.482622\pi\)
\(98\) 6.40653 − 2.39004i 0.647158 − 0.241430i
\(99\) −0.174281 − 0.593547i −0.0175159 − 0.0596537i
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.85.3 yes 220
4.3 odd 2 736.2.x.a.177.18 220
8.3 odd 2 736.2.x.a.177.5 220
8.5 even 2 inner 184.2.p.a.85.15 yes 220
23.13 even 11 inner 184.2.p.a.13.15 yes 220
92.59 odd 22 736.2.x.a.657.5 220
184.13 even 22 inner 184.2.p.a.13.3 ✓ 220
184.59 odd 22 736.2.x.a.657.18 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.13.3 ✓ 220 184.13 even 22 inner
184.2.p.a.13.15 yes 220 23.13 even 11 inner
184.2.p.a.85.3 yes 220 1.1 even 1 trivial
184.2.p.a.85.15 yes 220 8.5 even 2 inner
736.2.x.a.177.5 220 8.3 odd 2
736.2.x.a.177.18 220 4.3 odd 2
736.2.x.a.657.5 220 92.59 odd 22
736.2.x.a.657.18 220 184.59 odd 22