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Properties

Label 184.2.p.a.85.21
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 85.21
Character \(\chi\) \(=\) 184.85
Dual form 184.2.p.a.13.21

$q$-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+(1.32617 - 0.491199i) q^{2} +(1.35779 + 2.11277i) q^{3} +(1.51745 - 1.30283i) q^{4} +(-3.81860 + 1.74390i) q^{5} +(2.83845 + 2.13494i) q^{6} +(1.23190 - 0.361720i) q^{7} +(1.37244 - 2.47314i) q^{8} +(-1.37394 + 3.00851i) q^{9} +(-4.20751 + 4.18839i) q^{10} +(2.10280 - 1.82209i) q^{11} +(4.81295 + 1.43705i) q^{12} +(0.900873 - 3.06809i) q^{13} +(1.45604 - 1.08481i) q^{14} +(-8.86932 - 5.69997i) q^{15} +(0.605290 - 3.95394i) q^{16} +(-0.200666 + 1.39566i) q^{17} +(-0.344300 + 4.66467i) q^{18} +(-5.93769 + 0.853711i) q^{19} +(-3.52253 + 7.62124i) q^{20} +(2.43690 + 2.11159i) q^{21} +(1.89366 - 3.44929i) q^{22} +(-4.77284 - 0.469068i) q^{23} +(7.08866 - 0.458352i) q^{24} +(8.26623 - 9.53973i) q^{25} +(-0.312334 - 4.51131i) q^{26} +(-0.764149 + 0.109868i) q^{27} +(1.39809 - 2.15385i) q^{28} +(-5.36311 - 0.771098i) q^{29} +(-14.5620 - 3.20251i) q^{30} +(2.59408 + 1.66712i) q^{31} +(-1.13945 - 5.54091i) q^{32} +(6.70482 + 1.96871i) q^{33} +(0.419430 + 1.94945i) q^{34} +(-4.07335 + 3.52958i) q^{35} +(1.83468 + 6.35526i) q^{36} +(6.97049 + 3.18332i) q^{37} +(-7.45504 + 4.04876i) q^{38} +(7.70537 - 2.26250i) q^{39} +(-0.927922 + 11.8373i) q^{40} +(0.478975 + 1.04881i) q^{41} +(4.26895 + 1.60332i) q^{42} +(-0.886529 - 1.37947i) q^{43} +(0.817026 - 5.50450i) q^{44} -13.8843i q^{45} +(-6.55999 + 1.72235i) q^{46} -2.61387 q^{47} +(9.17561 - 4.08979i) q^{48} +(-4.50203 + 2.89328i) q^{49} +(6.27650 - 16.7117i) q^{50} +(-3.22117 + 1.47106i) q^{51} +(-2.63016 - 5.82935i) q^{52} +(3.24616 + 11.0554i) q^{53} +(-0.959424 + 0.521053i) q^{54} +(-4.85222 + 10.6249i) q^{55} +(0.796138 - 3.54311i) q^{56} +(-9.86586 - 11.3858i) q^{57} +(-7.49115 + 1.61175i) q^{58} +(-2.45402 + 8.35762i) q^{59} +(-20.8848 + 2.90578i) q^{60} +(4.47023 - 6.95581i) q^{61} +(4.25908 + 0.936666i) q^{62} +(-0.604326 + 4.20318i) q^{63} +(-4.23280 - 6.78848i) q^{64} +(1.91036 + 13.2868i) q^{65} +(9.85875 - 0.682556i) q^{66} +(-6.96808 - 6.03788i) q^{67} +(1.51380 + 2.37927i) q^{68} +(-5.48950 - 10.7208i) q^{69} +(-3.66822 + 6.68164i) q^{70} +(-3.19046 + 3.68199i) q^{71} +(5.55480 + 7.52696i) q^{72} +(1.59620 + 11.1018i) q^{73} +(10.8077 + 0.797716i) q^{74} +(31.3791 + 4.51163i) q^{75} +(-7.89790 + 9.03124i) q^{76} +(1.93136 - 3.00526i) q^{77} +(9.10728 - 6.78532i) q^{78} +(3.30955 + 0.971771i) q^{79} +(4.58390 + 16.1541i) q^{80} +(5.22797 + 6.03339i) q^{81} +(1.15038 + 1.15563i) q^{82} +(-6.86240 - 3.13395i) q^{83} +(6.44890 + 0.0293632i) q^{84} +(-1.66762 - 5.67941i) q^{85} +(-1.85328 - 1.39394i) q^{86} +(-5.65284 - 12.3780i) q^{87} +(-1.62029 - 7.70122i) q^{88} +(4.04182 - 2.59752i) q^{89} +(-6.81996 - 18.4129i) q^{90} -4.10546i q^{91} +(-7.85364 + 5.50639i) q^{92} +7.74430i q^{93} +(-3.46643 + 1.28393i) q^{94} +(21.1849 - 13.6147i) q^{95} +(10.1595 - 9.93081i) q^{96} +(-2.85178 - 6.24453i) q^{97} +(-4.54927 + 6.04836i) q^{98} +(2.59265 + 8.82974i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.32617 − 0.491199i 0.937743 − 0.347330i
\(3\) 1.35779 + 2.11277i 0.783923 + 1.21981i 0.971379 + 0.237535i \(0.0763393\pi\)
−0.187456 + 0.982273i \(0.560024\pi\)
\(4\) 1.51745 − 1.30283i 0.758723 − 0.651413i
\(5\) −3.81860 + 1.74390i −1.70773 + 0.779894i −0.710648 + 0.703547i \(0.751598\pi\)
−0.997081 + 0.0763466i \(0.975674\pi\)
\(6\) 2.83845 + 2.13494i 1.15879 + 0.871586i
\(7\) 1.23190 − 0.361720i 0.465616 − 0.136717i −0.0405055 − 0.999179i \(-0.512897\pi\)
0.506122 + 0.862462i \(0.331079\pi\)
\(8\) 1.37244 − 2.47314i 0.485232 − 0.874385i
\(9\) −1.37394 + 3.00851i −0.457980 + 1.00284i
\(10\) −4.20751 + 4.18839i −1.33053 + 1.32449i
\(11\) 2.10280 − 1.82209i 0.634018 − 0.549380i −0.277455 − 0.960738i \(-0.589491\pi\)
0.911474 + 0.411359i \(0.134946\pi\)
\(12\) 4.81295 + 1.43705i 1.38938 + 0.414839i
\(13\) 0.900873 − 3.06809i 0.249857 − 0.850935i −0.735074 − 0.677987i \(-0.762852\pi\)
0.984931 − 0.172948i \(-0.0553293\pi\)
\(14\) 1.45604 − 1.08481i 0.389142 − 0.289928i
\(15\) −8.86932 − 5.69997i −2.29005 − 1.47172i
\(16\) 0.605290 − 3.95394i 0.151322 − 0.988484i
\(17\) −0.200666 + 1.39566i −0.0486686 + 0.338497i 0.950910 + 0.309468i \(0.100151\pi\)
−0.999579 + 0.0290297i \(0.990758\pi\)
\(18\) −0.344300 + 4.66467i −0.0811522 + 1.09947i
\(19\) −5.93769 + 0.853711i −1.36220 + 0.195855i −0.784366 − 0.620299i \(-0.787011\pi\)
−0.577835 + 0.816154i \(0.696102\pi\)
\(20\) −3.52253 + 7.62124i −0.787662 + 1.70416i
\(21\) 2.43690 + 2.11159i 0.531776 + 0.460786i
\(22\) 1.89366 − 3.44929i 0.403730 − 0.735391i
\(23\) −4.77284 − 0.469068i −0.995205 − 0.0978075i
\(24\) 7.08866 − 0.458352i 1.44697 − 0.0935608i
\(25\) 8.26623 − 9.53973i 1.65325 − 1.90795i
\(26\) −0.312334 − 4.51131i −0.0612538 − 0.884742i
\(27\) −0.764149 + 0.109868i −0.147061 + 0.0211441i
\(28\) 1.39809 − 2.15385i 0.264215 − 0.407039i
\(29\) −5.36311 − 0.771098i −0.995904 − 0.143189i −0.374957 − 0.927042i \(-0.622343\pi\)
−0.620947 + 0.783853i \(0.713252\pi\)
\(30\) −14.5620 − 3.20251i −2.65865 − 0.584696i
\(31\) 2.59408 + 1.66712i 0.465911 + 0.299423i 0.752454 − 0.658644i \(-0.228870\pi\)
−0.286543 + 0.958067i \(0.592506\pi\)
\(32\) −1.13945 − 5.54091i −0.201429 − 0.979503i
\(33\) 6.70482 + 1.96871i 1.16716 + 0.342709i
\(34\) 0.419430 + 1.94945i 0.0719317 + 0.334327i
\(35\) −4.07335 + 3.52958i −0.688522 + 0.596607i
\(36\) 1.83468 + 6.35526i 0.305781 + 1.05921i
\(37\) 6.97049 + 3.18332i 1.14594 + 0.523334i 0.895616 − 0.444827i \(-0.146735\pi\)
0.250326 + 0.968162i \(0.419462\pi\)
\(38\) −7.45504 + 4.04876i −1.20937 + 0.656795i
\(39\) 7.70537 − 2.26250i 1.23385 − 0.362290i
\(40\) −0.927922 + 11.8373i −0.146717 + 1.87164i
\(41\) 0.478975 + 1.04881i 0.0748033 + 0.163796i 0.943339 − 0.331830i \(-0.107666\pi\)
−0.868536 + 0.495626i \(0.834939\pi\)
\(42\) 4.26895 + 1.60332i 0.658714 + 0.247397i
\(43\) −0.886529 − 1.37947i −0.135194 − 0.210367i 0.767055 − 0.641582i \(-0.221721\pi\)
−0.902249 + 0.431215i \(0.858085\pi\)
\(44\) 0.817026 − 5.50450i 0.123171 − 0.829835i
\(45\) − 13.8843i − 2.06975i
\(46\) −6.55999 + 1.72235i −0.967218 + 0.253947i
\(47\) −2.61387 −0.381272 −0.190636 − 0.981661i \(-0.561055\pi\)
−0.190636 + 0.981661i \(0.561055\pi\)
\(48\) 9.17561 − 4.08979i 1.32439 − 0.590311i
\(49\) −4.50203 + 2.89328i −0.643147 + 0.413325i
\(50\) 6.27650 − 16.7117i 0.887631 − 2.36339i
\(51\) −3.22117 + 1.47106i −0.451054 + 0.205989i
\(52\) −2.63016 − 5.82935i −0.364738 − 0.808385i
\(53\) 3.24616 + 11.0554i 0.445894 + 1.51858i 0.809555 + 0.587045i \(0.199709\pi\)
−0.363660 + 0.931532i \(0.618473\pi\)
\(54\) −0.959424 + 0.521053i −0.130561 + 0.0709063i
\(55\) −4.85222 + 10.6249i −0.654273 + 1.43266i
\(56\) 0.796138 − 3.54311i 0.106388 − 0.473468i
\(57\) −9.86586 − 11.3858i −1.30676 − 1.50809i
\(58\) −7.49115 + 1.61175i −0.983636 + 0.211633i
\(59\) −2.45402 + 8.35762i −0.319486 + 1.08807i 0.630607 + 0.776102i \(0.282806\pi\)
−0.950093 + 0.311967i \(0.899012\pi\)
\(60\) −20.8848 + 2.90578i −2.69621 + 0.375135i
\(61\) 4.47023 − 6.95581i 0.572354 − 0.890600i −0.427556 − 0.903989i \(-0.640625\pi\)
0.999910 + 0.0133885i \(0.00426182\pi\)
\(62\) 4.25908 + 0.936666i 0.540903 + 0.118957i
\(63\) −0.604326 + 4.20318i −0.0761380 + 0.529551i
\(64\) −4.23280 − 6.78848i −0.529100 − 0.848560i
\(65\) 1.91036 + 13.2868i 0.236951 + 1.64803i
\(66\) 9.85875 − 0.682556i 1.21353 − 0.0840168i
\(67\) −6.96808 − 6.03788i −0.851287 − 0.737644i 0.115479 − 0.993310i \(-0.463160\pi\)
−0.966765 + 0.255666i \(0.917705\pi\)
\(68\) 1.51380 + 2.37927i 0.183575 + 0.288529i
\(69\) −5.48950 − 10.7208i −0.660858 − 1.29063i
\(70\) −3.66822 + 6.68164i −0.438437 + 0.798609i
\(71\) −3.19046 + 3.68199i −0.378638 + 0.436972i −0.912798 − 0.408412i \(-0.866083\pi\)
0.534159 + 0.845384i \(0.320628\pi\)
\(72\) 5.55480 + 7.52696i 0.654639 + 0.887060i
\(73\) 1.59620 + 11.1018i 0.186821 + 1.29937i 0.840175 + 0.542315i \(0.182452\pi\)
−0.653354 + 0.757052i \(0.726639\pi\)
\(74\) 10.8077 + 0.797716i 1.25637 + 0.0927326i
\(75\) 31.3791 + 4.51163i 3.62334 + 0.520958i
\(76\) −7.89790 + 9.03124i −0.905951 + 1.03595i
\(77\) 1.93136 − 3.00526i 0.220099 − 0.342481i
\(78\) 9.10728 − 6.78532i 1.03120 − 0.768287i
\(79\) 3.30955 + 0.971771i 0.372353 + 0.109333i 0.462555 − 0.886591i \(-0.346933\pi\)
−0.0902014 + 0.995924i \(0.528751\pi\)
\(80\) 4.58390 + 16.1541i 0.512495 + 1.80608i
\(81\) 5.22797 + 6.03339i 0.580885 + 0.670377i
\(82\) 1.15038 + 1.15563i 0.127038 + 0.127617i
\(83\) −6.86240 − 3.13395i −0.753246 − 0.343996i 0.00151229 − 0.999999i \(-0.499519\pi\)
−0.754758 + 0.656003i \(0.772246\pi\)
\(84\) 6.44890 + 0.0293632i 0.703633 + 0.00320379i
\(85\) −1.66762 − 5.67941i −0.180879 − 0.616018i
\(86\) −1.85328 − 1.39394i −0.199844 − 0.150313i
\(87\) −5.65284 − 12.3780i −0.606048 − 1.32706i
\(88\) −1.62029 − 7.70122i −0.172724 − 0.820953i
\(89\) 4.04182 − 2.59752i 0.428432 − 0.275337i −0.308603 − 0.951191i \(-0.599861\pi\)
0.737035 + 0.675854i \(0.236225\pi\)
\(90\) −6.81996 − 18.4129i −0.718887 − 1.94089i
\(91\) − 4.10546i − 0.430369i
\(92\) −7.85364 + 5.50639i −0.818799 + 0.574081i
\(93\) 7.74430i 0.803046i
\(94\) −3.46643 + 1.28393i −0.357535 + 0.132427i
\(95\) 21.1849 − 13.6147i 2.17352 − 1.39684i
\(96\) 10.1595 − 9.93081i 1.03690 − 1.01356i
\(97\) −2.85178 − 6.24453i −0.289555 − 0.634036i 0.707824 − 0.706388i \(-0.249677\pi\)
−0.997379 + 0.0723520i \(0.976949\pi\)
\(98\) −4.54927 + 6.04836i −0.459546 + 0.610977i
\(99\) 2.59265 + 8.82974i 0.260571 + 0.887422i
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.85.21 yes 220
4.3 odd 2 736.2.x.a.177.3 220
8.3 odd 2 736.2.x.a.177.20 220
8.5 even 2 inner 184.2.p.a.85.13 yes 220
23.13 even 11 inner 184.2.p.a.13.13 ✓ 220
92.59 odd 22 736.2.x.a.657.20 220
184.13 even 22 inner 184.2.p.a.13.21 yes 220
184.59 odd 22 736.2.x.a.657.3 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.13.13 ✓ 220 23.13 even 11 inner
184.2.p.a.13.21 yes 220 184.13 even 22 inner
184.2.p.a.85.13 yes 220 8.5 even 2 inner
184.2.p.a.85.21 yes 220 1.1 even 1 trivial
736.2.x.a.177.3 220 4.3 odd 2
736.2.x.a.177.20 220 8.3 odd 2
736.2.x.a.657.3 220 184.59 odd 22
736.2.x.a.657.20 220 92.59 odd 22