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Properties

Label 184.2.p.a.85.15
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 85.15
Character \(\chi\) \(=\) 184.85
Dual form 184.2.p.a.13.15

$q$-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+(0.494121 + 1.32508i) q^{2} +(1.16603 + 1.81437i) q^{3} +(-1.51169 + 1.30950i) q^{4} +(1.93519 - 0.883774i) q^{5} +(-1.82804 + 2.44160i) q^{6} +(-1.41177 + 0.414532i) q^{7} +(-2.48216 - 1.35606i) q^{8} +(-0.686086 + 1.50232i) q^{9} +(2.12729 + 2.12760i) q^{10} +(0.283071 - 0.245282i) q^{11} +(-4.13860 - 1.21585i) q^{12} +(1.76591 - 6.01413i) q^{13} +(-1.24687 - 1.66588i) q^{14} +(3.85998 + 2.48066i) q^{15} +(0.570401 - 3.95912i) q^{16} +(-0.532657 + 3.70471i) q^{17} +(-2.32971 - 0.166793i) q^{18} +(-3.16421 + 0.454945i) q^{19} +(-1.76811 + 3.87013i) 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} +(8.84179 - 0.631737i) q^{26} +(2.87863 - 0.413885i) q^{27} +(1.59132 - 2.47536i) q^{28} +(5.13484 + 0.738279i) q^{29} +(-1.37978 + 6.34055i) q^{30} +(2.19128 + 1.40825i) q^{31} +(5.52801 - 1.20046i) q^{32} +(0.775102 + 0.227590i) q^{33} +(-5.17224 + 1.12476i) q^{34} +(-2.36569 + 2.04988i) q^{35} +(-0.930143 - 3.16947i) q^{36} +(-1.51885 - 0.693637i) q^{37} +(-2.16634 - 3.96804i) q^{38} +(12.9710 - 3.80862i) q^{39} +(-6.00191 - 0.430570i) q^{40} +(-3.71473 - 8.13412i) q^{41} +(1.56864 - 4.20475i) q^{42} +(6.08220 + 9.46409i) q^{43} +(-0.106717 + 0.741473i) q^{44} +3.51362i q^{45} +(-0.774359 - 6.73798i) q^{46} +2.05122 q^{47} +(7.84843 - 3.58152i) q^{48} +(-4.06753 + 2.61404i) q^{49} +(-0.628008 - 0.234286i) q^{50} +(-7.34282 + 3.35335i) q^{51} +(5.20602 + 11.4039i) q^{52} +(-1.55979 - 5.31217i) q^{53} +(1.97082 + 3.60991i) q^{54} +(0.331023 - 0.724840i) q^{55} +(4.06636 + 0.885503i) q^{56} +(-4.51499 - 5.21058i) q^{57} +(1.55895 + 7.16889i) q^{58} +(2.85121 - 9.71032i) q^{59} +(-9.08353 + 1.30468i) q^{60} +(-4.28250 + 6.66369i) q^{61} +(-0.783288 + 3.59947i) q^{62} +(0.345834 - 2.40533i) q^{63} +(4.32221 + 6.73190i) q^{64} +(-1.89775 - 13.1992i) q^{65} +(0.0814183 + 1.13953i) 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} +(3.74020 - 2.79862i) q^{72} +(-0.758975 - 5.27878i) q^{73} +(0.168628 - 2.35535i) q^{74} +(-1.01182 - 0.145477i) q^{75} +(4.18754 - 4.83128i) q^{76} +(-0.297953 + 0.463623i) q^{77} +(11.4560 + 15.3057i) q^{78} +(-12.4558 - 3.65736i) q^{79} +(-2.39513 - 8.16578i) q^{80} +(7.35214 + 8.48482i) q^{81} +(8.94286 - 8.94157i) q^{82} +(-8.47618 - 3.87094i) q^{83} +(6.34674 + 0.000915229i) q^{84} +(2.24333 + 7.64008i) q^{85} +(-9.53536 + 12.7358i) q^{86} +(4.64785 + 10.1774i) q^{87} +(-1.03524 + 0.224969i) q^{88} +(-3.11818 + 2.00394i) q^{89} +(-4.65584 + 1.73616i) q^{90} +9.22257i q^{91} +(8.54575 - 4.35547i) q^{92} +5.61785i q^{93} +(1.01355 + 2.71804i) q^{94} +(-5.72129 + 3.67685i) q^{95} +(8.62389 + 8.63011i) q^{96} +(7.61767 + 16.6804i) q^{97} +(-5.47367 - 4.09816i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.494121 + 1.32508i 0.349397 + 0.936975i
\(3\) 1.16603 + 1.81437i 0.673206 + 1.04753i 0.994918 + 0.100691i \(0.0321054\pi\)
−0.321712 + 0.946838i \(0.604258\pi\)
\(4\) −1.51169 + 1.30950i −0.755844 + 0.654752i
\(5\) 1.93519 − 0.883774i 0.865445 − 0.395236i 0.0673232 − 0.997731i \(-0.478554\pi\)
0.798122 + 0.602496i \(0.205827\pi\)
\(6\) −1.82804 + 2.44160i −0.746292 + 0.996780i
\(7\) −1.41177 + 0.414532i −0.533598 + 0.156678i −0.537421 − 0.843314i \(-0.680601\pi\)
0.00382311 + 0.999993i \(0.498783\pi\)
\(8\) −2.48216 − 1.35606i −0.877575 − 0.479439i
\(9\) −0.686086 + 1.50232i −0.228695 + 0.500773i
\(10\) 2.12729 + 2.12760i 0.672710 + 0.672807i
\(11\) 0.283071 − 0.245282i 0.0853491 − 0.0739554i −0.611141 − 0.791521i \(-0.709289\pi\)
0.696490 + 0.717566i \(0.254744\pi\)
\(12\) −4.13860 − 1.21585i −1.19471 − 0.350986i
\(13\) 1.76591 − 6.01413i 0.489775 − 1.66802i −0.229512 − 0.973306i \(-0.573713\pi\)
0.719287 − 0.694713i \(-0.244469\pi\)
\(14\) −1.24687 − 1.66588i −0.333241 − 0.445225i
\(15\) 3.85998 + 2.48066i 0.996644 + 0.640504i
\(16\) 0.570401 − 3.95912i 0.142600 − 0.989780i
\(17\) −0.532657 + 3.70471i −0.129188 + 0.898524i 0.817398 + 0.576073i \(0.195416\pi\)
−0.946586 + 0.322451i \(0.895493\pi\)
\(18\) −2.32971 − 0.166793i −0.549117 − 0.0393134i
\(19\) −3.16421 + 0.454945i −0.725919 + 0.104371i −0.495361 − 0.868687i \(-0.664964\pi\)
−0.230558 + 0.973059i \(0.574055\pi\)
\(20\) −1.76811 + 3.87013i −0.395361 + 0.865388i
\(21\) −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
\(26\) 8.84179 − 0.631737i 1.73402 − 0.123894i
\(27\) 2.87863 − 0.413885i 0.553993 − 0.0796521i
\(28\) 1.59132 − 2.47536i 0.300731 − 0.467798i
\(29\) 5.13484 + 0.738279i 0.953516 + 0.137095i 0.601477 − 0.798890i \(-0.294579\pi\)
0.352039 + 0.935985i \(0.385488\pi\)
\(30\) −1.37978 + 6.34055i −0.251912 + 1.15762i
\(31\) 2.19128 + 1.40825i 0.393565 + 0.252929i 0.722424 − 0.691450i \(-0.243028\pi\)
−0.328859 + 0.944379i \(0.606664\pi\)
\(32\) 5.52801 − 1.20046i 0.977223 − 0.212213i
\(33\) 0.775102 + 0.227590i 0.134928 + 0.0396184i
\(34\) −5.17224 + 1.12476i −0.887032 + 0.192895i
\(35\) −2.36569 + 2.04988i −0.399875 + 0.346493i
\(36\) −0.930143 − 3.16947i −0.155024 − 0.528245i
\(37\) −1.51885 − 0.693637i −0.249698 − 0.114033i 0.286635 − 0.958040i \(-0.407463\pi\)
−0.536333 + 0.844007i \(0.680191\pi\)
\(38\) −2.16634 − 3.96804i −0.351427 − 0.643701i
\(39\) 12.9710 − 3.80862i 2.07702 − 0.609867i
\(40\) −6.00191 − 0.430570i −0.948985 − 0.0680791i
\(41\) −3.71473 − 8.13412i −0.580143 − 1.27034i −0.941218 − 0.337799i \(-0.890317\pi\)
0.361075 − 0.932537i \(-0.382410\pi\)
\(42\) 1.56864 − 4.20475i 0.242046 − 0.648807i
\(43\) 6.08220 + 9.46409i 0.927527 + 1.44326i 0.896149 + 0.443753i \(0.146353\pi\)
0.0313780 + 0.999508i \(0.490010\pi\)
\(44\) −0.106717 + 0.741473i −0.0160882 + 0.111781i
\(45\) 3.51362i 0.523780i
\(46\) −0.774359 − 6.73798i −0.114173 − 0.993461i
\(47\) 2.05122 0.299202 0.149601 − 0.988746i \(-0.452201\pi\)
0.149601 + 0.988746i \(0.452201\pi\)
\(48\) 7.84843 − 3.58152i 1.13282 − 0.516948i
\(49\) −4.06753 + 2.61404i −0.581075 + 0.373434i
\(50\) −0.628008 − 0.234286i −0.0888137 − 0.0331331i
\(51\) −7.34282 + 3.35335i −1.02820 + 0.469563i
\(52\) 5.20602 + 11.4039i 0.721945 + 1.58144i
\(53\) −1.55979 − 5.31217i −0.214254 − 0.729682i −0.994549 − 0.104270i \(-0.966749\pi\)
0.780295 − 0.625412i \(-0.215069\pi\)
\(54\) 1.97082 + 3.60991i 0.268195 + 0.491247i
\(55\) 0.331023 − 0.724840i 0.0446352 − 0.0977374i
\(56\) 4.06636 + 0.885503i 0.543390 + 0.118330i
\(57\) −4.51499 − 5.21058i −0.598025 − 0.690158i
\(58\) 1.55895 + 7.16889i 0.204701 + 0.941321i
\(59\) 2.85121 − 9.71032i 0.371196 − 1.26418i −0.536269 − 0.844047i \(-0.680167\pi\)
0.907464 − 0.420129i \(-0.138015\pi\)
\(60\) −9.08353 + 1.30468i −1.17268 + 0.168433i
\(61\) −4.28250 + 6.66369i −0.548317 + 0.853198i −0.999225 − 0.0393724i \(-0.987464\pi\)
0.450907 + 0.892571i \(0.351101\pi\)
\(62\) −0.783288 + 3.59947i −0.0994777 + 0.457133i
\(63\) 0.345834 − 2.40533i 0.0435710 − 0.303043i
\(64\) 4.32221 + 6.73190i 0.540277 + 0.841487i
\(65\) −1.89775 − 13.1992i −0.235388 − 1.63716i
\(66\) 0.0814183 + 1.13953i 0.0100219 + 0.140267i
\(67\) 2.23503 + 1.93667i 0.273053 + 0.236602i 0.780613 − 0.625015i \(-0.214907\pi\)
−0.507560 + 0.861616i \(0.669452\pi\)
\(68\) −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 \(0.373115\pi\)
\(72\) 3.74020 − 2.79862i 0.440787 − 0.329820i
\(73\) −0.758975 − 5.27878i −0.0888313 − 0.617835i −0.984797 − 0.173711i \(-0.944424\pi\)
0.895965 − 0.444124i \(-0.146485\pi\)
\(74\) 0.168628 − 2.35535i 0.0196027 − 0.273803i
\(75\) −1.01182 − 0.145477i −0.116835 − 0.0167983i
\(76\) 4.18754 − 4.83128i 0.480344 − 0.554185i
\(77\) −0.297953 + 0.463623i −0.0339548 + 0.0528348i
\(78\) 11.4560 + 15.3057i 1.29713 + 1.73303i
\(79\) −12.4558 − 3.65736i −1.40139 − 0.411485i −0.508229 − 0.861222i \(-0.669700\pi\)
−0.893161 + 0.449736i \(0.851518\pi\)
\(80\) −2.39513 − 8.16578i −0.267784 − 0.912962i
\(81\) 7.35214 + 8.48482i 0.816904 + 0.942758i
\(82\) 8.94286 − 8.94157i 0.987573 − 0.987431i
\(83\) −8.47618 − 3.87094i −0.930381 − 0.424891i −0.108208 − 0.994128i \(-0.534511\pi\)
−0.822173 + 0.569237i \(0.807239\pi\)
\(84\) 6.34674 0.000915229i 0.692486 9.98597e-5i
\(85\) 2.24333 + 7.64008i 0.243323 + 0.828683i
\(86\) −9.53536 + 12.7358i −1.02822 + 1.37334i
\(87\) 4.64785 + 10.1774i 0.498302 + 1.09113i
\(88\) −1.03524 + 0.224969i −0.110357 + 0.0239818i
\(89\) −3.11818 + 2.00394i −0.330527 + 0.212417i −0.695369 − 0.718653i \(-0.744759\pi\)
0.364842 + 0.931069i \(0.381123\pi\)
\(90\) −4.65584 + 1.73616i −0.490769 + 0.183007i
\(91\) 9.22257i 0.966788i
\(92\) 8.54575 − 4.35547i 0.890956 − 0.454089i
\(93\) 5.61785i 0.582544i
\(94\) 1.01355 + 2.71804i 0.104540 + 0.280344i
\(95\) −5.72129 + 3.67685i −0.586992 + 0.377237i
\(96\) 8.62389 + 8.63011i 0.880172 + 0.880807i
\(97\) 7.61767 + 16.6804i 0.773457 + 1.69363i 0.718890 + 0.695124i \(0.244651\pi\)
0.0545672 + 0.998510i \(0.482622\pi\)
\(98\) −5.47367 − 4.09816i −0.552924 − 0.413976i
\(99\) 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.85.15 yes 220
4.3 odd 2 736.2.x.a.177.5 220
8.3 odd 2 736.2.x.a.177.18 220
8.5 even 2 inner 184.2.p.a.85.3 yes 220
23.13 even 11 inner 184.2.p.a.13.3 ✓ 220
92.59 odd 22 736.2.x.a.657.18 220
184.13 even 22 inner 184.2.p.a.13.15 yes 220
184.59 odd 22 736.2.x.a.657.5 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.13.3 ✓ 220 23.13 even 11 inner
184.2.p.a.13.15 yes 220 184.13 even 22 inner
184.2.p.a.85.3 yes 220 8.5 even 2 inner
184.2.p.a.85.15 yes 220 1.1 even 1 trivial
736.2.x.a.177.5 220 4.3 odd 2
736.2.x.a.177.18 220 8.3 odd 2
736.2.x.a.657.5 220 184.59 odd 22
736.2.x.a.657.18 220 92.59 odd 22