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Properties

Label 185.2.bc.a.172.1
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 172.1
Character \(\chi\) \(=\) 185.172
Dual form 185.2.bc.a.128.1

$q$-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+(-2.66388 - 0.469713i) q^{2} +(-1.37304 + 1.96091i) q^{3} +(4.99622 + 1.81848i) q^{4} +(-0.214240 + 2.22578i) q^{5} +(4.57868 - 4.57868i) q^{6} +(3.60837 + 0.315692i) q^{7} +(-7.77001 - 4.48602i) q^{8} +(-0.933852 - 2.56574i) q^{9} +(1.61619 - 5.82857i) q^{10} +(1.47450 + 0.851304i) q^{11} +(-10.4259 + 7.30028i) q^{12} +(-1.15824 + 3.18223i) q^{13} +(-9.46397 - 2.53586i) q^{14} +(-4.07039 - 3.47620i) q^{15} +(10.4453 + 8.76463i) q^{16} +(3.99576 - 1.45434i) q^{17} +(1.28251 + 7.27345i) q^{18} +(-1.97777 + 2.82454i) q^{19} +(-5.11792 + 10.7309i) q^{20} +(-5.57349 + 6.64222i) q^{21} +(-3.52802 - 2.96036i) q^{22} +(-3.02928 + 1.74896i) q^{23} +(19.4652 - 9.07677i) q^{24} +(-4.90820 - 0.953705i) q^{25} +(4.58014 - 7.93304i) q^{26} +(-0.623382 - 0.167035i) q^{27} +(17.4541 + 8.13900i) q^{28} +(0.442747 + 1.65236i) q^{29} +(9.21020 + 11.1721i) q^{30} +(-5.32606 - 5.32606i) q^{31} +(-12.1739 - 14.5082i) q^{32} +(-3.69388 + 1.72248i) q^{33} +(-11.3273 + 1.99732i) q^{34} +(-1.47572 + 7.96381i) q^{35} -14.5172i q^{36} +(4.53250 - 4.05665i) q^{37} +(6.59525 - 6.59525i) q^{38} +(-4.64976 - 6.64054i) q^{39} +(11.6495 - 16.3332i) q^{40} +(-1.37504 + 3.77790i) q^{41} +(17.9670 - 15.0761i) q^{42} -4.73916i q^{43} +(5.81886 + 6.93465i) q^{44} +(5.91084 - 1.52887i) q^{45} +(8.89114 - 3.23611i) q^{46} +(0.620597 - 2.31610i) q^{47} +(-31.5284 + 8.44802i) q^{48} +(6.02703 + 1.06273i) q^{49} +(12.6269 + 4.84600i) q^{50} +(-2.63453 + 9.83218i) q^{51} +(-11.5736 + 13.7929i) q^{52} +(5.61442 - 0.491198i) q^{53} +(1.58215 + 0.737770i) q^{54} +(-2.21071 + 3.09953i) q^{55} +(-26.6209 - 18.6401i) q^{56} +(-2.82311 - 7.75643i) q^{57} +(-0.403291 - 4.60964i) q^{58} +(0.132493 + 1.51440i) q^{59} +(-14.0152 - 24.7697i) q^{60} +(2.41726 + 5.18382i) q^{61} +(11.6862 + 16.6897i) q^{62} +(-2.55970 - 9.55295i) q^{63} +(11.9796 + 20.7492i) q^{64} +(-6.83482 - 3.25975i) q^{65} +(10.6491 - 2.85342i) q^{66} +(-0.461270 + 5.27234i) q^{67} +22.6084 q^{68} +(0.729789 - 8.34153i) q^{69} +(7.67184 - 20.5214i) q^{70} +(-0.493958 - 2.80137i) q^{71} +(-4.25390 + 24.1251i) q^{72} +(5.50833 + 5.50833i) q^{73} +(-13.9795 + 8.67744i) q^{74} +(8.60929 - 8.31505i) q^{75} +(-15.0177 + 10.5155i) q^{76} +(5.05180 + 3.53731i) q^{77} +(9.26722 + 19.8736i) q^{78} +(-7.89820 - 0.691003i) q^{79} +(-21.7460 + 21.3712i) q^{80} +(7.45830 - 6.25825i) q^{81} +(5.43747 - 9.41797i) q^{82} +(-1.98344 + 4.25349i) q^{83} +(-39.9251 + 23.0508i) q^{84} +(2.38098 + 9.20527i) q^{85} +(-2.22605 + 12.6245i) q^{86} +(-3.84803 - 1.40057i) q^{87} +(-7.63792 - 13.2293i) q^{88} +(13.9431 - 1.21986i) q^{89} +(-16.4639 + 1.29631i) q^{90} +(-5.18396 + 11.1170i) q^{91} +(-18.3154 + 3.22950i) q^{92} +(17.7568 - 3.13100i) q^{93} +(-2.74110 + 5.87830i) q^{94} +(-5.86310 - 5.00721i) q^{95} +(45.1645 - 3.95138i) q^{96} +(3.92963 + 6.80632i) q^{97} +(-15.5561 - 5.66196i) q^{98} +(0.807256 - 4.57818i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.66388 − 0.469713i −1.88365 − 0.332137i −0.891079 − 0.453848i \(-0.850051\pi\)
−0.992566 + 0.121710i \(0.961162\pi\)
\(3\) −1.37304 + 1.96091i −0.792726 + 1.13213i 0.195735 + 0.980657i \(0.437291\pi\)
−0.988461 + 0.151473i \(0.951598\pi\)
\(4\) 4.99622 + 1.81848i 2.49811 + 0.909238i
\(5\) −0.214240 + 2.22578i −0.0958112 + 0.995400i
\(6\) 4.57868 − 4.57868i 1.86924 − 1.86924i
\(7\) 3.60837 + 0.315692i 1.36384 + 0.119320i 0.745454 − 0.666557i \(-0.232233\pi\)
0.618382 + 0.785877i \(0.287788\pi\)
\(8\) −7.77001 − 4.48602i −2.74711 − 1.58605i
\(9\) −0.933852 − 2.56574i −0.311284 − 0.855246i
\(10\) 1.61619 − 5.82857i 0.511084 − 1.84316i
\(11\) 1.47450 + 0.851304i 0.444579 + 0.256678i 0.705538 − 0.708672i \(-0.250705\pi\)
−0.260959 + 0.965350i \(0.584039\pi\)
\(12\) −10.4259 + 7.30028i −3.00969 + 2.10741i
\(13\) −1.15824 + 3.18223i −0.321238 + 0.882593i 0.669007 + 0.743256i \(0.266719\pi\)
−0.990245 + 0.139337i \(0.955503\pi\)
\(14\) −9.46397 − 2.53586i −2.52935 − 0.677738i
\(15\) −4.07039 − 3.47620i −1.05097 − 0.897550i
\(16\) 10.4453 + 8.76463i 2.61132 + 2.19116i
\(17\) 3.99576 − 1.45434i 0.969115 − 0.352729i 0.191516 − 0.981490i \(-0.438660\pi\)
0.777599 + 0.628761i \(0.216437\pi\)
\(18\) 1.28251 + 7.27345i 0.302289 + 1.71437i
\(19\) −1.97777 + 2.82454i −0.453731 + 0.647995i −0.979219 − 0.202806i \(-0.934994\pi\)
0.525488 + 0.850801i \(0.323883\pi\)
\(20\) −5.11792 + 10.7309i −1.14440 + 2.39950i
\(21\) −5.57349 + 6.64222i −1.21623 + 1.44945i
\(22\) −3.52802 − 2.96036i −0.752176 − 0.631151i
\(23\) −3.02928 + 1.74896i −0.631649 + 0.364682i −0.781390 − 0.624043i \(-0.785489\pi\)
0.149742 + 0.988725i \(0.452156\pi\)
\(24\) 19.4652 − 9.07677i 3.97332 − 1.85279i
\(25\) −4.90820 − 0.953705i −0.981640 − 0.190741i
\(26\) 4.58014 − 7.93304i 0.898240 − 1.55580i
\(27\) −0.623382 − 0.167035i −0.119970 − 0.0321458i
\(28\) 17.4541 + 8.13900i 3.29852 + 1.53813i
\(29\) 0.442747 + 1.65236i 0.0822161 + 0.306835i 0.994772 − 0.102116i \(-0.0325614\pi\)
−0.912556 + 0.408951i \(0.865895\pi\)
\(30\) 9.21020 + 11.1721i 1.68154 + 2.03973i
\(31\) −5.32606 − 5.32606i −0.956588 − 0.956588i 0.0425081 − 0.999096i \(-0.486465\pi\)
−0.999096 + 0.0425081i \(0.986465\pi\)
\(32\) −12.1739 − 14.5082i −2.15205 − 2.56472i
\(33\) −3.69388 + 1.72248i −0.643022 + 0.299846i
\(34\) −11.3273 + 1.99732i −1.94262 + 0.342537i
\(35\) −1.47572 + 7.96381i −0.249442 + 1.34613i
\(36\) − 14.5172i − 2.41953i
\(37\) 4.53250 − 4.05665i 0.745139 − 0.666909i
\(38\) 6.59525 − 6.59525i 1.06989 − 1.06989i
\(39\) −4.64976 − 6.64054i −0.744557 − 1.06334i
\(40\) 11.6495 − 16.3332i 1.84195 − 2.58251i
\(41\) −1.37504 + 3.77790i −0.214745 + 0.590008i −0.999558 − 0.0297249i \(-0.990537\pi\)
0.784813 + 0.619733i \(0.212759\pi\)
\(42\) 17.9670 − 15.0761i 2.77237 − 2.32630i
\(43\) − 4.73916i − 0.722716i −0.932427 − 0.361358i \(-0.882313\pi\)
0.932427 − 0.361358i \(-0.117687\pi\)
\(44\) 5.81886 + 6.93465i 0.877226 + 1.04544i
\(45\) 5.91084 − 1.52887i 0.881136 − 0.227910i
\(46\) 8.89114 − 3.23611i 1.31093 − 0.477138i
\(47\) 0.620597 − 2.31610i 0.0905234 − 0.337838i −0.905779 − 0.423750i \(-0.860714\pi\)
0.996303 + 0.0859118i \(0.0273803\pi\)
\(48\) −31.5284 + 8.44802i −4.55074 + 1.21937i
\(49\) 6.02703 + 1.06273i 0.861005 + 0.151818i
\(50\) 12.6269 + 4.84600i 1.78571 + 0.685328i
\(51\) −2.63453 + 9.83218i −0.368907 + 1.37678i
\(52\) −11.5736 + 13.7929i −1.60497 + 1.91273i
\(53\) 5.61442 − 0.491198i 0.771200 − 0.0674712i 0.305238 − 0.952276i \(-0.401264\pi\)
0.465962 + 0.884805i \(0.345708\pi\)
\(54\) 1.58215 + 0.737770i 0.215304 + 0.100398i
\(55\) −2.21071 + 3.09953i −0.298093 + 0.417941i
\(56\) −26.6209 − 18.6401i −3.55736 − 2.49089i
\(57\) −2.82311 − 7.75643i −0.373930 − 1.02736i
\(58\) −0.403291 − 4.60964i −0.0529547 − 0.605275i
\(59\) 0.132493 + 1.51440i 0.0172491 + 0.197158i 0.999930 + 0.0118511i \(0.00377241\pi\)
−0.982681 + 0.185307i \(0.940672\pi\)
\(60\) −14.0152 − 24.7697i −1.80935 − 3.19776i
\(61\) 2.41726 + 5.18382i 0.309498 + 0.663721i 0.998113 − 0.0614066i \(-0.0195586\pi\)
−0.688615 + 0.725127i \(0.741781\pi\)
\(62\) 11.6862 + 16.6897i 1.48415 + 2.11959i
\(63\) −2.55970 − 9.55295i −0.322492 − 1.20356i
\(64\) 11.9796 + 20.7492i 1.49745 + 2.59366i
\(65\) −6.83482 − 3.25975i −0.847754 − 0.404322i
\(66\) 10.6491 − 2.85342i 1.31082 − 0.351232i
\(67\) −0.461270 + 5.27234i −0.0563531 + 0.644118i 0.914523 + 0.404535i \(0.132567\pi\)
−0.970876 + 0.239584i \(0.922989\pi\)
\(68\) 22.6084 2.74167
\(69\) 0.729789 − 8.34153i 0.0878563 − 1.00420i
\(70\) 7.67184 − 20.5214i 0.916961 − 2.45278i
\(71\) −0.493958 − 2.80137i −0.0586220 − 0.332462i 0.941366 − 0.337387i \(-0.109543\pi\)
−0.999988 + 0.00492555i \(0.998432\pi\)
\(72\) −4.25390 + 24.1251i −0.501327 + 2.84317i
\(73\) 5.50833 + 5.50833i 0.644701 + 0.644701i 0.951707 − 0.307006i \(-0.0993273\pi\)
−0.307006 + 0.951707i \(0.599327\pi\)
\(74\) −13.9795 + 8.67744i −1.62508 + 1.00873i
\(75\) 8.60929 − 8.31505i 0.994116 − 0.960140i
\(76\) −15.0177 + 10.5155i −1.72265 + 1.20621i
\(77\) 5.05180 + 3.53731i 0.575706 + 0.403114i
\(78\) 9.26722 + 19.8736i 1.04931 + 2.25025i
\(79\) −7.89820 − 0.691003i −0.888617 − 0.0777439i −0.366301 − 0.930496i \(-0.619376\pi\)
−0.522315 + 0.852752i \(0.674932\pi\)
\(80\) −21.7460 + 21.3712i −2.43127 + 2.38937i
\(81\) 7.45830 − 6.25825i 0.828699 − 0.695361i
\(82\) 5.43747 − 9.41797i 0.600468 − 1.04004i
\(83\) −1.98344 + 4.25349i −0.217710 + 0.466881i −0.984583 − 0.174918i \(-0.944034\pi\)
0.766873 + 0.641799i \(0.221812\pi\)
\(84\) −39.9251 + 23.0508i −4.35619 + 2.51505i
\(85\) 2.38098 + 9.20527i 0.258254 + 0.998452i
\(86\) −2.22605 + 12.6245i −0.240041 + 1.36134i
\(87\) −3.84803 − 1.40057i −0.412552 − 0.150157i
\(88\) −7.63792 − 13.2293i −0.814205 − 1.41024i
\(89\) 13.9431 − 1.21986i 1.47797 − 0.129305i 0.680535 − 0.732716i \(-0.261747\pi\)
0.797431 + 0.603410i \(0.206192\pi\)
\(90\) −16.4639 + 1.29631i −1.73544 + 0.136643i
\(91\) −5.18396 + 11.1170i −0.543427 + 1.16538i
\(92\) −18.3154 + 3.22950i −1.90951 + 0.336698i
\(93\) 17.7568 − 3.13100i 1.84129 − 0.324670i
\(94\) −2.74110 + 5.87830i −0.282722 + 0.606300i
\(95\) −5.86310 − 5.00721i −0.601541 − 0.513729i
\(96\) 45.1645 − 3.95138i 4.60958 − 0.403286i
\(97\) 3.92963 + 6.80632i 0.398993 + 0.691077i 0.993602 − 0.112939i \(-0.0360263\pi\)
−0.594609 + 0.804015i \(0.702693\pi\)
\(98\) −15.5561 − 5.66196i −1.57140 − 0.571944i
\(99\) 0.807256 − 4.57818i 0.0811323 − 0.460124i
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.172.1 yes 204
5.2 odd 4 925.2.bn.b.468.17 204
5.3 odd 4 185.2.z.a.98.1 yes 204
5.4 even 2 925.2.bq.b.357.17 204
37.17 odd 36 185.2.z.a.17.1 ✓ 204
185.17 even 36 925.2.bq.b.868.17 204
185.54 odd 36 925.2.bn.b.757.17 204
185.128 even 36 inner 185.2.bc.a.128.1 yes 204
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.z.a.17.1 ✓ 204 37.17 odd 36
185.2.z.a.98.1 yes 204 5.3 odd 4
185.2.bc.a.128.1 yes 204 185.128 even 36 inner
185.2.bc.a.172.1 yes 204 1.1 even 1 trivial
925.2.bn.b.468.17 204 5.2 odd 4
925.2.bn.b.757.17 204 185.54 odd 36
925.2.bq.b.357.17 204 5.4 even 2
925.2.bq.b.868.17 204 185.17 even 36