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Properties

Label 184.2.p.a.85.10
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 85.10
Character \(\chi\) \(=\) 184.85
Dual form 184.2.p.a.13.10

$q$-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+(-0.329997 - 1.37517i) q^{2} +(0.583943 + 0.908634i) q^{3} +(-1.78220 + 0.907607i) q^{4} +(0.293263 - 0.133929i) q^{5} +(1.05683 - 1.10287i) q^{6} +(1.64760 - 0.483779i) q^{7} +(1.83624 + 2.15133i) q^{8} +(0.761620 - 1.66772i) q^{9} +(-0.280951 - 0.359091i) q^{10} +(4.72619 - 4.09527i) q^{11} +(-1.86539 - 1.08938i) q^{12} +(-0.960051 + 3.26963i) q^{13} +(-1.20898 - 2.10609i) q^{14} +(0.292941 + 0.188262i) q^{15} +(2.35250 - 3.23508i) q^{16} +(-0.348661 + 2.42499i) q^{17} +(-2.54473 - 0.497018i) q^{18} +(0.170399 - 0.0244997i) q^{19} +(-0.401100 + 0.504856i) q^{20} +(1.40168 + 1.21457i) q^{21} +(-7.19133 - 5.14790i) q^{22} +(-4.35783 + 2.00234i) q^{23} +(-0.882511 + 2.92472i) q^{24} +(-3.20624 + 3.70020i) q^{25} +(4.81313 + 0.241266i) q^{26} +(5.16739 - 0.742958i) q^{27} +(-2.49728 + 2.35757i) q^{28} +(-1.83955 - 0.264487i) q^{29} +(0.162223 - 0.464971i) q^{30} +(-7.35378 - 4.72599i) q^{31} +(-5.22512 - 2.16752i) q^{32} +(6.48093 + 1.90297i) q^{33} +(3.44984 - 0.320771i) q^{34} +(0.418389 - 0.362536i) q^{35} +(0.156269 + 3.66346i) q^{36} +(-2.57970 - 1.17811i) q^{37} +(-0.0899227 - 0.226244i) q^{38} +(-3.53151 + 1.03695i) q^{39} +(0.826627 + 0.384981i) q^{40} +(1.22872 + 2.69052i) q^{41} +(1.20769 - 2.32836i) q^{42} +(0.434032 + 0.675367i) q^{43} +(-4.70614 + 11.5881i) q^{44} -0.591082i q^{45} +(4.19163 + 5.33200i) q^{46} +9.31419 q^{47} +(4.31323 + 0.248454i) q^{48} +(-3.40823 + 2.19034i) q^{49} +(6.14646 + 3.18808i) q^{50} +(-2.40703 + 1.09925i) q^{51} +(-1.25654 - 6.69850i) q^{52} +(1.61528 + 5.50113i) q^{53} +(-2.72692 - 6.86088i) q^{54} +(0.837543 - 1.83396i) q^{55} +(4.06616 + 2.65620i) q^{56} +(0.121765 + 0.140524i) q^{57} +(0.243331 + 2.61698i) q^{58} +(-0.382589 + 1.30298i) q^{59} +(-0.692949 - 0.0696454i) q^{60} +(-6.21682 + 9.67356i) q^{61} +(-4.07232 + 11.6723i) q^{62} +(0.448040 - 3.11618i) q^{63} +(-1.25645 + 7.90072i) q^{64} +(0.156350 + 1.08744i) q^{65} +(0.478227 - 9.54037i) q^{66} +(-9.79045 - 8.48347i) q^{67} +(-1.57955 - 4.63827i) q^{68} +(-4.36411 - 2.79042i) q^{69} +(-0.636617 - 0.455721i) q^{70} +(3.39541 - 3.91851i) q^{71} +(4.98632 - 1.42383i) q^{72} +(-1.89100 - 13.1522i) q^{73} +(-0.768810 + 3.93630i) q^{74} +(-5.23438 - 0.752591i) q^{75} +(-0.281450 + 0.198319i) q^{76} +(5.80567 - 9.03380i) q^{77} +(2.59137 + 4.51425i) q^{78} +(-6.04520 - 1.77503i) q^{79} +(0.256630 - 1.26380i) q^{80} +(0.0906820 + 0.104653i) q^{81} +(3.29446 - 2.57757i) q^{82} +(4.83071 + 2.20611i) q^{83} +(-3.60043 - 0.892425i) q^{84} +(0.222527 + 0.757856i) q^{85} +(0.785517 - 0.819739i) q^{86} +(-0.833871 - 1.82592i) q^{87} +(17.4887 + 2.64771i) q^{88} +(7.15465 - 4.59801i) q^{89} +(-0.812840 + 0.195056i) q^{90} +5.85150i q^{91} +(5.94920 - 7.52377i) q^{92} -9.44160i q^{93} +(-3.07366 - 12.8086i) q^{94} +(0.0466906 - 0.0300062i) q^{95} +(-1.08169 - 6.01343i) q^{96} +(3.69026 + 8.08055i) q^{97} +(4.13680 + 3.96410i) q^{98} +(-3.23018 - 11.0010i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.329997 − 1.37517i −0.233343 − 0.972394i
\(3\) 0.583943 + 0.908634i 0.337140 + 0.524600i 0.967886 − 0.251389i \(-0.0808875\pi\)
−0.630746 + 0.775989i \(0.717251\pi\)
\(4\) −1.78220 + 0.907607i −0.891102 + 0.453804i
\(5\) 0.293263 − 0.133929i 0.131151 − 0.0598948i −0.348759 − 0.937213i \(-0.613397\pi\)
0.479910 + 0.877318i \(0.340669\pi\)
\(6\) 1.05683 − 1.10287i 0.431449 − 0.450245i
\(7\) 1.64760 − 0.483779i 0.622734 − 0.182851i 0.0448816 − 0.998992i \(-0.485709\pi\)
0.577853 + 0.816141i \(0.303891\pi\)
\(8\) 1.83624 + 2.15133i 0.649209 + 0.760610i
\(9\) 0.761620 − 1.66772i 0.253873 − 0.555905i
\(10\) −0.280951 − 0.359091i −0.0888446 − 0.113555i
\(11\) 4.72619 − 4.09527i 1.42500 − 1.23477i 0.494146 − 0.869379i \(-0.335481\pi\)
0.930854 − 0.365391i \(-0.119065\pi\)
\(12\) −1.86539 − 1.08938i −0.538491 − 0.314477i
\(13\) −0.960051 + 3.26963i −0.266270 + 0.906833i 0.712465 + 0.701707i \(0.247579\pi\)
−0.978736 + 0.205126i \(0.934240\pi\)
\(14\) −1.20898 − 2.10609i −0.323115 − 0.562876i
\(15\) 0.292941 + 0.188262i 0.0756371 + 0.0486090i
\(16\) 2.35250 − 3.23508i 0.588125 − 0.808770i
\(17\) −0.348661 + 2.42499i −0.0845627 + 0.588146i 0.902847 + 0.429962i \(0.141473\pi\)
−0.987410 + 0.158184i \(0.949436\pi\)
\(18\) −2.54473 − 0.497018i −0.599799 − 0.117148i
\(19\) 0.170399 − 0.0244997i 0.0390923 − 0.00562062i −0.122741 − 0.992439i \(-0.539168\pi\)
0.161833 + 0.986818i \(0.448259\pi\)
\(20\) −0.401100 + 0.504856i −0.0896886 + 0.112889i
\(21\) 1.40168 + 1.21457i 0.305872 + 0.265040i
\(22\) −7.19133 − 5.14790i −1.53320 − 1.09754i
\(23\) −4.35783 + 2.00234i −0.908669 + 0.417516i
\(24\) −0.882511 + 2.92472i −0.180142 + 0.597007i
\(25\) −3.20624 + 3.70020i −0.641247 + 0.740039i
\(26\) 4.81313 + 0.241266i 0.943932 + 0.0473161i
\(27\) 5.16739 − 0.742958i 0.994464 − 0.142982i
\(28\) −2.49728 + 2.35757i −0.471941 + 0.445538i
\(29\) −1.83955 − 0.264487i −0.341596 − 0.0491141i −0.0306178 − 0.999531i \(-0.509747\pi\)
−0.310978 + 0.950417i \(0.600657\pi\)
\(30\) 0.162223 − 0.464971i 0.0296177 − 0.0848917i
\(31\) −7.35378 − 4.72599i −1.32078 − 0.848813i −0.325468 − 0.945553i \(-0.605522\pi\)
−0.995310 + 0.0967403i \(0.969158\pi\)
\(32\) −5.22512 − 2.16752i −0.923679 − 0.383168i
\(33\) 6.48093 + 1.90297i 1.12818 + 0.331265i
\(34\) 3.44984 − 0.320771i 0.591642 − 0.0550118i
\(35\) 0.418389 − 0.362536i 0.0707206 − 0.0612797i
\(36\) 0.156269 + 3.66346i 0.0260448 + 0.610577i
\(37\) −2.57970 − 1.17811i −0.424100 − 0.193680i 0.191920 − 0.981411i \(-0.438529\pi\)
−0.616020 + 0.787731i \(0.711256\pi\)
\(38\) −0.0899227 − 0.226244i −0.0145874 − 0.0367016i
\(39\) −3.53151 + 1.03695i −0.565495 + 0.166044i
\(40\) 0.826627 + 0.384981i 0.130701 + 0.0608708i
\(41\) 1.22872 + 2.69052i 0.191894 + 0.420188i 0.980984 − 0.194089i \(-0.0621750\pi\)
−0.789090 + 0.614277i \(0.789448\pi\)
\(42\) 1.20769 − 2.32836i 0.186350 − 0.359274i
\(43\) 0.434032 + 0.675367i 0.0661893 + 0.102993i 0.872771 − 0.488131i \(-0.162321\pi\)
−0.806581 + 0.591123i \(0.798685\pi\)
\(44\) −4.70614 + 11.5881i −0.709477 + 1.74698i
\(45\) − 0.591082i − 0.0881133i
\(46\) 4.19163 + 5.33200i 0.618022 + 0.786160i
\(47\) 9.31419 1.35861 0.679307 − 0.733854i \(-0.262281\pi\)
0.679307 + 0.733854i \(0.262281\pi\)
\(48\) 4.31323 + 0.248454i 0.622561 + 0.0358613i
\(49\) −3.40823 + 2.19034i −0.486890 + 0.312905i
\(50\) 6.14646 + 3.18808i 0.869241 + 0.450862i
\(51\) −2.40703 + 1.09925i −0.337051 + 0.153926i
\(52\) −1.25654 − 6.69850i −0.174250 − 0.928915i
\(53\) 1.61528 + 5.50113i 0.221875 + 0.755638i 0.992912 + 0.118848i \(0.0379201\pi\)
−0.771037 + 0.636790i \(0.780262\pi\)
\(54\) −2.72692 − 6.86088i −0.371087 − 0.933647i
\(55\) 0.837543 − 1.83396i 0.112934 − 0.247292i
\(56\) 4.06616 + 2.65620i 0.543363 + 0.354950i
\(57\) 0.121765 + 0.140524i 0.0161281 + 0.0186129i
\(58\) 0.243331 + 2.61698i 0.0319509 + 0.343626i
\(59\) −0.382589 + 1.30298i −0.0498088 + 0.169633i −0.980641 − 0.195813i \(-0.937265\pi\)
0.930832 + 0.365446i \(0.119084\pi\)
\(60\) −0.692949 − 0.0696454i −0.0894593 − 0.00899118i
\(61\) −6.21682 + 9.67356i −0.795982 + 1.23857i 0.171388 + 0.985204i \(0.445175\pi\)
−0.967370 + 0.253369i \(0.918461\pi\)
\(62\) −4.07232 + 11.6723i −0.517186 + 1.48238i
\(63\) 0.448040 − 3.11618i 0.0564477 − 0.392602i
\(64\) −1.25645 + 7.90072i −0.157056 + 0.987590i
\(65\) 0.156350 + 1.08744i 0.0193929 + 0.134880i
\(66\) 0.478227 − 9.54037i 0.0588657 − 1.17434i
\(67\) −9.79045 − 8.48347i −1.19609 − 1.03642i −0.998422 − 0.0561529i \(-0.982117\pi\)
−0.197672 − 0.980268i \(-0.563338\pi\)
\(68\) −1.57955 − 4.63827i −0.191549 − 0.562473i
\(69\) −4.36411 − 2.79042i −0.525378 − 0.335927i
\(70\) −0.636617 − 0.455721i −0.0760902 − 0.0544691i
\(71\) 3.39541 − 3.91851i 0.402961 − 0.465042i −0.517610 − 0.855617i \(-0.673178\pi\)
0.920571 + 0.390575i \(0.127724\pi\)
\(72\) 4.98632 − 1.42383i 0.587644 − 0.167800i
\(73\) −1.89100 − 13.1522i −0.221324 − 1.53934i −0.733037 − 0.680188i \(-0.761898\pi\)
0.511713 − 0.859156i \(-0.329011\pi\)
\(74\) −0.768810 + 3.93630i −0.0893723 + 0.457586i
\(75\) −5.23438 − 0.752591i −0.604414 − 0.0869017i
\(76\) −0.281450 + 0.198319i −0.0322845 + 0.0227488i
\(77\) 5.80567 − 9.03380i 0.661618 − 1.02950i
\(78\) 2.59137 + 4.51425i 0.293415 + 0.511139i
\(79\) −6.04520 − 1.77503i −0.680138 − 0.199707i −0.0766255 − 0.997060i \(-0.524415\pi\)
−0.603513 + 0.797353i \(0.706233\pi\)
\(80\) 0.256630 − 1.26380i 0.0286922 − 0.141297i
\(81\) 0.0906820 + 0.104653i 0.0100758 + 0.0116281i
\(82\) 3.29446 − 2.57757i 0.363812 − 0.284645i
\(83\) 4.83071 + 2.20611i 0.530240 + 0.242152i 0.662499 − 0.749063i \(-0.269496\pi\)
−0.132259 + 0.991215i \(0.542223\pi\)
\(84\) −3.60043 − 0.892425i −0.392839 − 0.0973715i
\(85\) 0.222527 + 0.757856i 0.0241364 + 0.0822010i
\(86\) 0.785517 − 0.819739i 0.0847045 − 0.0883947i
\(87\) −0.833871 − 1.82592i −0.0894003 − 0.195759i
\(88\) 17.4887 + 2.64771i 1.86430 + 0.282246i
\(89\) 7.15465 − 4.59801i 0.758391 − 0.487388i −0.103407 − 0.994639i \(-0.532975\pi\)
0.861799 + 0.507251i \(0.169338\pi\)
\(90\) −0.812840 + 0.195056i −0.0856809 + 0.0205607i
\(91\) 5.85150i 0.613404i
\(92\) 5.94920 − 7.52377i 0.620247 − 0.784407i
\(93\) − 9.44160i − 0.979048i
\(94\) −3.07366 − 12.8086i −0.317024 − 1.32111i
\(95\) 0.0466906 − 0.0300062i 0.00479036 − 0.00307857i
\(96\) −1.08169 − 6.01343i −0.110399 − 0.613743i
\(97\) 3.69026 + 8.08055i 0.374690 + 0.820456i 0.999221 + 0.0394562i \(0.0125626\pi\)
−0.624532 + 0.780999i \(0.714710\pi\)
\(98\) 4.13680 + 3.96410i 0.417880 + 0.400435i
\(99\) −3.23018 − 11.0010i −0.324645 − 1.10564i
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.85.10 yes 220
4.3 odd 2 736.2.x.a.177.8 220
8.3 odd 2 736.2.x.a.177.15 220
8.5 even 2 inner 184.2.p.a.85.22 yes 220
23.13 even 11 inner 184.2.p.a.13.22 yes 220
92.59 odd 22 736.2.x.a.657.15 220
184.13 even 22 inner 184.2.p.a.13.10 ✓ 220
184.59 odd 22 736.2.x.a.657.8 220
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.p.a.13.10 ✓ 220 184.13 even 22 inner
184.2.p.a.13.22 yes 220 23.13 even 11 inner
184.2.p.a.85.10 yes 220 1.1 even 1 trivial
184.2.p.a.85.22 yes 220 8.5 even 2 inner
736.2.x.a.177.8 220 4.3 odd 2
736.2.x.a.177.15 220 8.3 odd 2
736.2.x.a.657.8 220 184.59 odd 22
736.2.x.a.657.15 220 92.59 odd 22