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Properties

Label 185.2.bc.a.172.3
Level $185$
Weight $2$
Character 185.172
Analytic conductor $1.477$
Analytic rank $0$
Dimension $204$
Inner twists $2$

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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] = [185,2,Mod(2,185)] 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("185.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([9, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.bc (of order \(36\), degree \(12\), 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: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(17\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 172.3
Character \(\chi\) \(=\) 185.172
Dual form 185.2.bc.a.128.3

$q$-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+(-1.82148 - 0.321176i) q^{2} +(-1.59505 + 2.27797i) q^{3} +(1.33525 + 0.485992i) q^{4} +(-1.23548 - 1.86376i) q^{5} +(3.63698 - 3.63698i) q^{6} +(-3.54263 - 0.309940i) q^{7} +(0.927519 + 0.535504i) q^{8} +(-1.61890 - 4.44788i) q^{9} +(1.65180 + 3.79161i) q^{10} +(4.83320 + 2.79045i) q^{11} +(-3.23687 + 2.26648i) q^{12} +(1.08466 - 2.98008i) q^{13} +(6.35329 + 1.70236i) q^{14} +(6.21623 + 0.158418i) q^{15} +(-3.69448 - 3.10004i) q^{16} +(2.83670 - 1.03247i) q^{17} +(1.52023 + 8.62168i) q^{18} +(-0.0922365 + 0.131727i) q^{19} +(-0.743898 - 3.08902i) q^{20} +(6.35671 - 7.57564i) q^{21} +(-7.90736 - 6.63506i) q^{22} +(5.58214 - 3.22285i) q^{23} +(-2.69930 + 1.25871i) q^{24} +(-1.94720 + 4.60526i) q^{25} +(-2.93282 + 5.07980i) q^{26} +(4.65596 + 1.24756i) q^{27} +(-4.57968 - 2.13554i) q^{28} +(-2.02173 - 7.54520i) q^{29} +(-11.2719 - 2.28506i) q^{30} +(-3.60337 - 3.60337i) q^{31} +(4.35691 + 5.19236i) q^{32} +(-14.0658 + 6.55898i) q^{33} +(-5.49859 + 0.969550i) q^{34} +(3.79918 + 6.98554i) q^{35} -6.72581i q^{36} +(5.82563 - 1.74988i) q^{37} +(0.210315 - 0.210315i) q^{38} +(5.05845 + 7.22421i) q^{39} +(-0.147877 - 2.39028i) q^{40} +(0.214432 - 0.589148i) q^{41} +(-14.0117 + 11.7572i) q^{42} +0.198014i q^{43} +(5.09741 + 6.07486i) q^{44} +(-6.28967 + 8.51248i) q^{45} +(-11.2029 + 4.07751i) q^{46} +(0.138518 - 0.516958i) q^{47} +(12.9547 - 3.47120i) q^{48} +(5.56052 + 0.980470i) q^{49} +(5.02589 - 7.76299i) q^{50} +(-2.17273 + 8.10875i) q^{51} +(2.89659 - 3.45203i) q^{52} +(-2.42823 + 0.212443i) q^{53} +(-8.08006 - 3.76779i) q^{54} +(-0.770571 - 12.4555i) q^{55} +(-3.11989 - 2.18457i) q^{56} +(-0.152949 - 0.420224i) q^{57} +(1.25920 + 14.3928i) q^{58} +(0.179432 + 2.05092i) q^{59} +(8.22325 + 3.23257i) q^{60} +(-0.187756 - 0.402643i) q^{61} +(5.40616 + 7.72080i) q^{62} +(4.35657 + 16.2590i) q^{63} +(-1.44556 - 2.50378i) q^{64} +(-6.89423 + 1.66027i) q^{65} +(27.7271 - 7.42946i) q^{66} +(1.22935 - 14.0515i) q^{67} +4.28948 q^{68} +(-1.56225 + 17.8566i) q^{69} +(-4.67654 - 13.9442i) q^{70} +(0.219398 + 1.24427i) q^{71} +(0.880298 - 4.99242i) q^{72} +(-6.87081 - 6.87081i) q^{73} +(-11.1733 + 1.31631i) q^{74} +(-7.38475 - 11.7813i) q^{75} +(-0.187177 + 0.131063i) q^{76} +(-16.2574 - 11.3835i) q^{77} +(-6.89362 - 14.7834i) q^{78} +(6.64942 + 0.581749i) q^{79} +(-1.21329 + 10.7157i) q^{80} +(0.609426 - 0.511369i) q^{81} +(-0.579805 + 1.00425i) q^{82} +(4.10949 - 8.81283i) q^{83} +(12.1695 - 7.02607i) q^{84} +(-5.42895 - 4.01133i) q^{85} +(0.0635975 - 0.360680i) q^{86} +(20.4125 + 7.42954i) q^{87} +(2.98859 + 5.17640i) q^{88} +(5.12449 - 0.448334i) q^{89} +(14.1905 - 13.4852i) q^{90} +(-4.76620 + 10.2212i) q^{91} +(9.01985 - 1.59044i) q^{92} +(13.9559 - 2.46081i) q^{93} +(-0.418343 + 0.897140i) q^{94} +(0.359464 + 0.00916080i) q^{95} +(-18.7775 + 1.64282i) q^{96} +(3.68600 + 6.38433i) q^{97} +(-9.81348 - 3.57181i) q^{98} +(4.58714 - 26.0150i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 204 q - 12 q^{2} - 18 q^{3} - 12 q^{5} - 24 q^{6} - 12 q^{7} - 18 q^{8} - 6 q^{10} - 36 q^{11} - 12 q^{12} - 12 q^{13} + 24 q^{14} - 12 q^{15} - 24 q^{16} + 12 q^{17} + 66 q^{18} - 108 q^{20} - 24 q^{21}+ \cdots - 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\)
\(\chi(n)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.82148 − 0.321176i −1.28798 − 0.227106i −0.512616 − 0.858618i \(-0.671323\pi\)
−0.775366 + 0.631512i \(0.782434\pi\)
\(3\) −1.59505 + 2.27797i −0.920903 + 1.31519i 0.0279962 + 0.999608i \(0.491087\pi\)
−0.948900 + 0.315578i \(0.897802\pi\)
\(4\) 1.33525 + 0.485992i 0.667626 + 0.242996i
\(5\) −1.23548 − 1.86376i −0.552521 − 0.833499i
\(6\) 3.63698 − 3.63698i 1.48479 − 1.48479i
\(7\) −3.54263 − 0.309940i −1.33899 − 0.117146i −0.604955 − 0.796259i \(-0.706809\pi\)
−0.734034 + 0.679113i \(0.762365\pi\)
\(8\) 0.927519 + 0.535504i 0.327928 + 0.189329i
\(9\) −1.61890 − 4.44788i −0.539632 − 1.48263i
\(10\) 1.65180 + 3.79161i 0.522345 + 1.19901i
\(11\) 4.83320 + 2.79045i 1.45727 + 0.841353i 0.998876 − 0.0473983i \(-0.0150930\pi\)
0.458390 + 0.888751i \(0.348426\pi\)
\(12\) −3.23687 + 2.26648i −0.934404 + 0.654277i
\(13\) 1.08466 − 2.98008i 0.300831 − 0.826526i −0.693525 − 0.720432i \(-0.743943\pi\)
0.994356 − 0.106094i \(-0.0338344\pi\)
\(14\) 6.35329 + 1.70236i 1.69799 + 0.454974i
\(15\) 6.21623 + 0.158418i 1.60502 + 0.0409034i
\(16\) −3.69448 − 3.10004i −0.923621 − 0.775010i
\(17\) 2.83670 − 1.03247i 0.688000 − 0.250411i 0.0257213 − 0.999669i \(-0.491812\pi\)
0.662279 + 0.749258i \(0.269590\pi\)
\(18\) 1.52023 + 8.62168i 0.358323 + 2.03215i
\(19\) −0.0922365 + 0.131727i −0.0211605 + 0.0302203i −0.829593 − 0.558369i \(-0.811427\pi\)
0.808432 + 0.588589i \(0.200316\pi\)
\(20\) −0.743898 − 3.08902i −0.166341 − 0.690726i
\(21\) 6.35671 − 7.57564i 1.38715 − 1.65314i
\(22\) −7.90736 − 6.63506i −1.68586 − 1.41460i
\(23\) 5.58214 − 3.22285i 1.16396 − 0.672011i 0.211708 − 0.977333i \(-0.432097\pi\)
0.952249 + 0.305322i \(0.0987641\pi\)
\(24\) −2.69930 + 1.25871i −0.550993 + 0.256932i
\(25\) −1.94720 + 4.60526i −0.389440 + 0.921052i
\(26\) −2.93282 + 5.07980i −0.575174 + 0.996230i
\(27\) 4.65596 + 1.24756i 0.896040 + 0.240093i
\(28\) −4.57968 − 2.13554i −0.865478 − 0.403579i
\(29\) −2.02173 − 7.54520i −0.375426 − 1.40111i −0.852722 − 0.522365i \(-0.825050\pi\)
0.477296 − 0.878743i \(-0.341617\pi\)
\(30\) −11.2719 − 2.28506i −2.05795 − 0.417193i
\(31\) −3.60337 − 3.60337i −0.647185 − 0.647185i 0.305127 − 0.952312i \(-0.401301\pi\)
−0.952312 + 0.305127i \(0.901301\pi\)
\(32\) 4.35691 + 5.19236i 0.770200 + 0.917889i
\(33\) −14.0658 + 6.55898i −2.44854 + 1.14177i
\(34\) −5.49859 + 0.969550i −0.943001 + 0.166276i
\(35\) 3.79918 + 6.98554i 0.642178 + 1.18077i
\(36\) − 6.72581i − 1.12097i
\(37\) 5.82563 − 1.74988i 0.957727 − 0.287678i
\(38\) 0.210315 − 0.210315i 0.0341175 − 0.0341175i
\(39\) 5.05845 + 7.22421i 0.810000 + 1.15680i
\(40\) −0.147877 − 2.39028i −0.0233814 − 0.377936i
\(41\) 0.214432 − 0.589148i 0.0334887 − 0.0920094i −0.921821 − 0.387615i \(-0.873299\pi\)
0.955310 + 0.295605i \(0.0955213\pi\)
\(42\) −14.0117 + 11.7572i −2.16206 + 1.81418i
\(43\) 0.198014i 0.0301969i 0.999886 + 0.0150985i \(0.00480617\pi\)
−0.999886 + 0.0150985i \(0.995194\pi\)
\(44\) 5.09741 + 6.07486i 0.768463 + 0.915819i
\(45\) −6.28967 + 8.51248i −0.937609 + 1.26897i
\(46\) −11.2029 + 4.07751i −1.65177 + 0.601196i
\(47\) 0.138518 − 0.516958i 0.0202050 − 0.0754061i −0.955087 − 0.296325i \(-0.904239\pi\)
0.975292 + 0.220919i \(0.0709056\pi\)
\(48\) 12.9547 − 3.47120i 1.86985 − 0.501024i
\(49\) 5.56052 + 0.980470i 0.794360 + 0.140067i
\(50\) 5.02589 − 7.76299i 0.710768 − 1.09785i
\(51\) −2.17273 + 8.10875i −0.304244 + 1.13545i
\(52\) 2.89659 − 3.45203i 0.401685 − 0.478710i
\(53\) −2.42823 + 0.212443i −0.333544 + 0.0291813i −0.252698 − 0.967545i \(-0.581318\pi\)
−0.0808460 + 0.996727i \(0.525762\pi\)
\(54\) −8.08006 − 3.76779i −1.09956 − 0.512732i
\(55\) −0.770571 − 12.4555i −0.103904 − 1.67949i
\(56\) −3.11989 − 2.18457i −0.416912 − 0.291925i
\(57\) −0.152949 − 0.420224i −0.0202586 − 0.0556600i
\(58\) 1.25920 + 14.3928i 0.165341 + 1.88986i
\(59\) 0.179432 + 2.05092i 0.0233601 + 0.267007i 0.998889 + 0.0471341i \(0.0150088\pi\)
−0.975528 + 0.219873i \(0.929436\pi\)
\(60\) 8.22325 + 3.23257i 1.06162 + 0.417323i
\(61\) −0.187756 − 0.402643i −0.0240396 − 0.0515532i 0.893921 − 0.448224i \(-0.147943\pi\)
−0.917961 + 0.396671i \(0.870165\pi\)
\(62\) 5.40616 + 7.72080i 0.686583 + 0.980542i
\(63\) 4.35657 + 16.2590i 0.548877 + 2.04844i
\(64\) −1.44556 − 2.50378i −0.180695 − 0.312973i
\(65\) −6.89423 + 1.66027i −0.855124 + 0.205931i
\(66\) 27.7271 − 7.42946i 3.41297 − 0.914503i
\(67\) 1.22935 − 14.0515i 0.150189 − 1.71667i −0.430542 − 0.902570i \(-0.641678\pi\)
0.580731 − 0.814095i \(-0.302767\pi\)
\(68\) 4.28948 0.520176
\(69\) −1.56225 + 17.8566i −0.188072 + 2.14968i
\(70\) −4.67654 − 13.9442i −0.558954 − 1.66665i
\(71\) 0.219398 + 1.24427i 0.0260378 + 0.147668i 0.995055 − 0.0993256i \(-0.0316685\pi\)
−0.969017 + 0.246993i \(0.920557\pi\)
\(72\) 0.880298 − 4.99242i 0.103744 − 0.588362i
\(73\) −6.87081 − 6.87081i −0.804168 − 0.804168i 0.179576 − 0.983744i \(-0.442527\pi\)
−0.983744 + 0.179576i \(0.942527\pi\)
\(74\) −11.1733 + 1.31631i −1.29887 + 0.153018i
\(75\) −7.38475 − 11.7813i −0.852717 − 1.36039i
\(76\) −0.187177 + 0.131063i −0.0214707 + 0.0150340i
\(77\) −16.2574 − 11.3835i −1.85270 − 1.29728i
\(78\) −6.89362 − 14.7834i −0.780549 − 1.67389i
\(79\) 6.64942 + 0.581749i 0.748118 + 0.0654519i 0.454837 − 0.890574i \(-0.349697\pi\)
0.293281 + 0.956026i \(0.405253\pi\)
\(80\) −1.21329 + 10.7157i −0.135650 + 1.19805i
\(81\) 0.609426 − 0.511369i 0.0677140 − 0.0568188i
\(82\) −0.579805 + 1.00425i −0.0640287 + 0.110901i
\(83\) 4.10949 − 8.81283i 0.451075 − 0.967333i −0.540922 − 0.841073i \(-0.681925\pi\)
0.991997 − 0.126261i \(-0.0402976\pi\)
\(84\) 12.1695 − 7.02607i 1.32780 − 0.766607i
\(85\) −5.42895 − 4.01133i −0.588852 − 0.435089i
\(86\) 0.0635975 − 0.360680i 0.00685790 − 0.0388931i
\(87\) 20.4125 + 7.42954i 2.18845 + 0.796530i
\(88\) 2.98859 + 5.17640i 0.318585 + 0.551806i
\(89\) 5.12449 − 0.448334i 0.543194 − 0.0475234i 0.187743 − 0.982218i \(-0.439883\pi\)
0.355451 + 0.934695i \(0.384327\pi\)
\(90\) 14.1905 − 13.4852i 1.49581 − 1.42147i
\(91\) −4.76620 + 10.2212i −0.499634 + 1.07147i
\(92\) 9.01985 − 1.59044i 0.940384 − 0.165815i
\(93\) 13.9559 − 2.46081i 1.44716 − 0.255174i
\(94\) −0.418343 + 0.897140i −0.0431488 + 0.0925329i
\(95\) 0.359464 + 0.00916080i 0.0368802 + 0.000939878i
\(96\) −18.7775 + 1.64282i −1.91647 + 0.167670i
\(97\) 3.68600 + 6.38433i 0.374256 + 0.648231i 0.990215 − 0.139547i \(-0.0445648\pi\)
−0.615959 + 0.787778i \(0.711231\pi\)
\(98\) −9.81348 − 3.57181i −0.991311 − 0.360808i
\(99\) 4.58714 − 26.0150i 0.461025 − 2.61460i
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 185.2.bc.a.172.3 yes 204
5.2 odd 4 925.2.bn.b.468.15 204
5.3 odd 4 185.2.z.a.98.3 yes 204
5.4 even 2 925.2.bq.b.357.15 204
37.17 odd 36 185.2.z.a.17.3 ✓ 204
185.17 even 36 925.2.bq.b.868.15 204
185.54 odd 36 925.2.bn.b.757.15 204
185.128 even 36 inner 185.2.bc.a.128.3 yes 204
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.z.a.17.3 ✓ 204 37.17 odd 36
185.2.z.a.98.3 yes 204 5.3 odd 4
185.2.bc.a.128.3 yes 204 185.128 even 36 inner
185.2.bc.a.172.3 yes 204 1.1 even 1 trivial
925.2.bn.b.468.15 204 5.2 odd 4
925.2.bn.b.757.15 204 185.54 odd 36
925.2.bq.b.357.15 204 5.4 even 2
925.2.bq.b.868.15 204 185.17 even 36