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Properties

Label 185.2.v.a.139.9
Level $185$
Weight $2$
Character 185.139
Analytic conductor $1.477$
Analytic rank $0$
Dimension $96$
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] = [185,2,Mod(4,185)] 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("185.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(18)) 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.v (of order \(18\), degree \(6\), 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: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 139.9
Character \(\chi\) \(=\) 185.139
Dual form 185.2.v.a.4.9

$q$-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+(0.0322691 + 0.183007i) q^{2} +(-0.566572 - 0.0999019i) q^{3} +(1.84693 - 0.672229i) q^{4} +(-1.07497 - 1.96072i) q^{5} -0.106911i q^{6} +(1.33240 - 1.58789i) q^{7} +(0.368452 + 0.638178i) q^{8} +(-2.50805 - 0.912857i) q^{9} +(0.324138 - 0.259999i) q^{10} +(0.238483 + 0.413065i) q^{11} +(-1.11358 + 0.196354i) q^{12} +(1.29130 - 0.469995i) q^{13} +(0.333590 + 0.192598i) q^{14} +(0.413170 + 1.21828i) q^{15} +(2.90637 - 2.43873i) q^{16} +(3.67129 + 1.33624i) q^{17} +(0.0861268 - 0.488450i) q^{18} +(2.47154 + 0.435800i) q^{19} +(-3.30346 - 2.89870i) q^{20} +(-0.913531 + 0.766543i) q^{21} +(-0.0678984 + 0.0569735i) q^{22} +(-1.93058 + 3.34387i) q^{23} +(-0.145000 - 0.398383i) q^{24} +(-2.68886 + 4.21545i) q^{25} +(0.127682 + 0.221151i) q^{26} +(2.82450 + 1.63073i) q^{27} +(1.39342 - 3.82840i) q^{28} +(-6.06811 + 3.50343i) q^{29} +(-0.209622 + 0.114926i) q^{30} -1.94048i q^{31} +(1.66910 + 1.40054i) q^{32} +(-0.0938520 - 0.257856i) q^{33} +(-0.126072 + 0.714992i) q^{34} +(-4.54570 - 0.905520i) q^{35} -5.24586 q^{36} +(5.95714 + 1.22985i) q^{37} +0.466374i q^{38} +(-0.778569 + 0.137283i) q^{39} +(0.855214 - 1.40846i) q^{40} +(-5.79284 + 2.10842i) q^{41} +(-0.169762 - 0.142447i) q^{42} -10.6440 q^{43} +(0.718138 + 0.602589i) q^{44} +(0.906234 + 5.89890i) q^{45} +(-0.674251 - 0.245407i) q^{46} +(8.85447 + 5.11213i) q^{47} +(-1.89030 + 1.09137i) q^{48} +(0.469429 + 2.66227i) q^{49} +(-0.858226 - 0.356053i) q^{50} +(-1.94656 - 1.12385i) q^{51} +(2.06901 - 1.73610i) q^{52} +(-3.57349 - 4.25872i) q^{53} +(-0.207291 + 0.569527i) q^{54} +(0.553543 - 0.911634i) q^{55} +(1.50428 + 0.265245i) q^{56} +(-1.35677 - 0.493824i) q^{57} +(-0.836966 - 0.997457i) q^{58} +(-1.93057 - 2.30076i) q^{59} +(1.58206 + 1.97234i) q^{60} +(4.47819 + 12.3037i) q^{61} +(0.355122 - 0.0626176i) q^{62} +(-4.79123 + 2.76622i) q^{63} +(3.59155 - 6.22074i) q^{64} +(-2.30965 - 2.02665i) q^{65} +(0.0441611 - 0.0254964i) q^{66} +(1.43708 - 1.71265i) q^{67} +7.67889 q^{68} +(1.42787 - 1.70167i) q^{69} +(0.0190312 - 0.861116i) q^{70} +(1.33200 - 7.55415i) q^{71} +(-0.341533 - 1.93693i) q^{72} -0.472707i q^{73} +(-0.0328408 + 1.12989i) q^{74} +(1.94457 - 2.11973i) q^{75} +(4.85774 - 0.856550i) q^{76} +(0.973655 + 0.171682i) q^{77} +(-0.0502475 - 0.138054i) q^{78} +(5.91818 - 7.05302i) q^{79} +(-7.90595 - 3.07701i) q^{80} +(4.69638 + 3.94073i) q^{81} +(-0.572787 - 0.992096i) q^{82} +(2.68018 - 7.36374i) q^{83} +(-1.17194 + 2.02986i) q^{84} +(-1.32654 - 8.63480i) q^{85} +(-0.343474 - 1.94794i) q^{86} +(3.78802 - 1.37873i) q^{87} +(-0.175740 + 0.304390i) q^{88} +(5.21281 + 6.21239i) q^{89} +(-1.05030 + 0.356200i) q^{90} +(0.974225 - 2.67666i) q^{91} +(-1.31782 + 7.47370i) q^{92} +(-0.193858 + 1.09942i) q^{93} +(-0.649831 + 1.78540i) q^{94} +(-1.80236 - 5.31448i) q^{95} +(-0.805746 - 0.960251i) q^{96} +(-6.94919 + 12.0364i) q^{97} +(-0.472066 + 0.171818i) q^{98} +(-0.221060 - 1.25369i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 12 q^{4} - 12 q^{5} - 6 q^{9} - 3 q^{10} - 30 q^{11} - 36 q^{14} - 21 q^{15} - 18 q^{19} - 39 q^{20} - 24 q^{21} + 96 q^{24} + 36 q^{25} + 48 q^{26} - 18 q^{29} - 30 q^{30} - 54 q^{34} + 6 q^{35}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(76\) \(112\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0322691 + 0.183007i 0.0228177 + 0.129406i 0.994089 − 0.108570i \(-0.0346272\pi\)
−0.971271 + 0.237976i \(0.923516\pi\)
\(3\) −0.566572 − 0.0999019i −0.327110 − 0.0576784i 0.00768141 − 0.999970i \(-0.497555\pi\)
−0.334792 + 0.942292i \(0.608666\pi\)
\(4\) 1.84693 − 0.672229i 0.923467 − 0.336115i
\(5\) −1.07497 − 1.96072i −0.480743 − 0.876862i
\(6\) − 0.106911i − 0.0436461i
\(7\) 1.33240 − 1.58789i 0.503598 − 0.600165i −0.453023 − 0.891499i \(-0.649655\pi\)
0.956621 + 0.291334i \(0.0940990\pi\)
\(8\) 0.368452 + 0.638178i 0.130268 + 0.225630i
\(9\) −2.50805 − 0.912857i −0.836018 − 0.304286i
\(10\) 0.324138 − 0.259999i 0.102501 − 0.0822189i
\(11\) 0.238483 + 0.413065i 0.0719055 + 0.124544i 0.899736 − 0.436434i \(-0.143759\pi\)
−0.827831 + 0.560978i \(0.810425\pi\)
\(12\) −1.11358 + 0.196354i −0.321462 + 0.0566825i
\(13\) 1.29130 − 0.469995i 0.358143 − 0.130353i −0.156681 − 0.987649i \(-0.550080\pi\)
0.514824 + 0.857296i \(0.327857\pi\)
\(14\) 0.333590 + 0.192598i 0.0891558 + 0.0514741i
\(15\) 0.413170 + 1.21828i 0.106680 + 0.314559i
\(16\) 2.90637 − 2.43873i 0.726592 − 0.609683i
\(17\) 3.67129 + 1.33624i 0.890419 + 0.324086i 0.746407 − 0.665490i \(-0.231777\pi\)
0.144012 + 0.989576i \(0.454000\pi\)
\(18\) 0.0861268 − 0.488450i 0.0203003 − 0.115129i
\(19\) 2.47154 + 0.435800i 0.567011 + 0.0999793i 0.449803 − 0.893128i \(-0.351494\pi\)
0.117208 + 0.993107i \(0.462605\pi\)
\(20\) −3.30346 − 2.89870i −0.738676 − 0.648168i
\(21\) −0.913531 + 0.766543i −0.199349 + 0.167273i
\(22\) −0.0678984 + 0.0569735i −0.0144760 + 0.0121468i
\(23\) −1.93058 + 3.34387i −0.402555 + 0.697245i −0.994033 − 0.109075i \(-0.965211\pi\)
0.591479 + 0.806320i \(0.298544\pi\)
\(24\) −0.145000 − 0.398383i −0.0295979 − 0.0813196i
\(25\) −2.68886 + 4.21545i −0.537773 + 0.843090i
\(26\) 0.127682 + 0.221151i 0.0250405 + 0.0433714i
\(27\) 2.82450 + 1.63073i 0.543576 + 0.313834i
\(28\) 1.39342 − 3.82840i 0.263332 − 0.723499i
\(29\) −6.06811 + 3.50343i −1.12682 + 0.650570i −0.943133 − 0.332415i \(-0.892137\pi\)
−0.183687 + 0.982985i \(0.558803\pi\)
\(30\) −0.209622 + 0.114926i −0.0382716 + 0.0209825i
\(31\) − 1.94048i − 0.348520i −0.984700 − 0.174260i \(-0.944247\pi\)
0.984700 − 0.174260i \(-0.0557534\pi\)
\(32\) 1.66910 + 1.40054i 0.295057 + 0.247582i
\(33\) −0.0938520 − 0.257856i −0.0163375 − 0.0448870i
\(34\) −0.126072 + 0.714992i −0.0216212 + 0.122620i
\(35\) −4.54570 − 0.905520i −0.768363 − 0.153061i
\(36\) −5.24586 −0.874310
\(37\) 5.95714 + 1.22985i 0.979347 + 0.202187i
\(38\) 0.466374i 0.0756558i
\(39\) −0.778569 + 0.137283i −0.124671 + 0.0219828i
\(40\) 0.855214 − 1.40846i 0.135221 − 0.222697i
\(41\) −5.79284 + 2.10842i −0.904690 + 0.329280i −0.752131 − 0.659014i \(-0.770974\pi\)
−0.152559 + 0.988294i \(0.548752\pi\)
\(42\) −0.169762 − 0.142447i −0.0261948 − 0.0219801i
\(43\) −10.6440 −1.62320 −0.811601 − 0.584213i \(-0.801403\pi\)
−0.811601 + 0.584213i \(0.801403\pi\)
\(44\) 0.718138 + 0.602589i 0.108263 + 0.0908438i
\(45\) 0.906234 + 5.89890i 0.135093 + 0.879355i
\(46\) −0.674251 − 0.245407i −0.0994129 − 0.0361833i
\(47\) 8.85447 + 5.11213i 1.29156 + 0.745681i 0.978930 − 0.204196i \(-0.0654579\pi\)
0.312626 + 0.949876i \(0.398791\pi\)
\(48\) −1.89030 + 1.09137i −0.272841 + 0.157525i
\(49\) 0.469429 + 2.66227i 0.0670614 + 0.380324i
\(50\) −0.858226 − 0.356053i −0.121371 − 0.0503535i
\(51\) −1.94656 − 1.12385i −0.272573 − 0.157370i
\(52\) 2.06901 − 1.73610i 0.286919 − 0.240754i
\(53\) −3.57349 − 4.25872i −0.490857 − 0.584980i 0.462578 − 0.886578i \(-0.346924\pi\)
−0.953435 + 0.301598i \(0.902480\pi\)
\(54\) −0.207291 + 0.569527i −0.0282087 + 0.0775028i
\(55\) 0.553543 − 0.911634i 0.0746397 − 0.122925i
\(56\) 1.50428 + 0.265245i 0.201018 + 0.0354449i
\(57\) −1.35677 − 0.493824i −0.179709 − 0.0654086i
\(58\) −0.836966 − 0.997457i −0.109899 − 0.130973i
\(59\) −1.93057 − 2.30076i −0.251339 − 0.299534i 0.625592 − 0.780150i \(-0.284857\pi\)
−0.876931 + 0.480616i \(0.840413\pi\)
\(60\) 1.58206 + 1.97234i 0.204243 + 0.254628i
\(61\) 4.47819 + 12.3037i 0.573373 + 1.57533i 0.799137 + 0.601149i \(0.205290\pi\)
−0.225764 + 0.974182i \(0.572488\pi\)
\(62\) 0.355122 − 0.0626176i 0.0451005 − 0.00795244i
\(63\) −4.79123 + 2.76622i −0.603639 + 0.348511i
\(64\) 3.59155 − 6.22074i 0.448943 − 0.777593i
\(65\) −2.30965 − 2.02665i −0.286476 − 0.251375i
\(66\) 0.0441611 − 0.0254964i 0.00543585 − 0.00313839i
\(67\) 1.43708 − 1.71265i 0.175567 − 0.209233i −0.671084 − 0.741382i \(-0.734171\pi\)
0.846651 + 0.532149i \(0.178615\pi\)
\(68\) 7.67889 0.931203
\(69\) 1.42787 − 1.70167i 0.171896 − 0.204857i
\(70\) 0.0190312 − 0.861116i 0.00227466 − 0.102923i
\(71\) 1.33200 − 7.55415i 0.158079 − 0.896513i −0.797837 − 0.602873i \(-0.794022\pi\)
0.955916 − 0.293639i \(-0.0948665\pi\)
\(72\) −0.341533 − 1.93693i −0.0402501 − 0.228269i
\(73\) − 0.472707i − 0.0553262i −0.999617 − 0.0276631i \(-0.991193\pi\)
0.999617 − 0.0276631i \(-0.00880657\pi\)
\(74\) −0.0328408 + 1.12989i −0.00381766 + 0.131347i
\(75\) 1.94457 − 2.11973i 0.224539 − 0.244766i
\(76\) 4.85774 − 0.856550i 0.557221 − 0.0982530i
\(77\) 0.973655 + 0.171682i 0.110958 + 0.0195649i
\(78\) −0.0502475 − 0.138054i −0.00568941 − 0.0156315i
\(79\) 5.91818 − 7.05302i 0.665848 − 0.793526i −0.322365 − 0.946616i \(-0.604478\pi\)
0.988212 + 0.153089i \(0.0489222\pi\)
\(80\) −7.90595 − 3.07701i −0.883912 − 0.344020i
\(81\) 4.69638 + 3.94073i 0.521820 + 0.437859i
\(82\) −0.572787 − 0.992096i −0.0632537 − 0.109559i
\(83\) 2.68018 − 7.36374i 0.294188 − 0.808275i −0.701254 − 0.712911i \(-0.747376\pi\)
0.995442 − 0.0953641i \(-0.0304015\pi\)
\(84\) −1.17194 + 2.02986i −0.127869 + 0.221476i
\(85\) −1.32654 − 8.63480i −0.143884 − 0.936576i
\(86\) −0.343474 − 1.94794i −0.0370378 − 0.210052i
\(87\) 3.78802 − 1.37873i 0.406119 − 0.147815i
\(88\) −0.175740 + 0.304390i −0.0187339 + 0.0324481i
\(89\) 5.21281 + 6.21239i 0.552557 + 0.658512i 0.967954 − 0.251128i \(-0.0808016\pi\)
−0.415397 + 0.909640i \(0.636357\pi\)
\(90\) −1.05030 + 0.356200i −0.110711 + 0.0375467i
\(91\) 0.974225 − 2.67666i 0.102127 − 0.280590i
\(92\) −1.31782 + 7.47370i −0.137392 + 0.779188i
\(93\) −0.193858 + 1.09942i −0.0201021 + 0.114005i
\(94\) −0.649831 + 1.78540i −0.0670250 + 0.184150i
\(95\) −1.80236 − 5.31448i −0.184918 − 0.545255i
\(96\) −0.805746 − 0.960251i −0.0822362 − 0.0980052i
\(97\) −6.94919 + 12.0364i −0.705584 + 1.22211i 0.260897 + 0.965367i \(0.415982\pi\)
−0.966480 + 0.256740i \(0.917352\pi\)
\(98\) −0.472066 + 0.171818i −0.0476859 + 0.0173562i
\(99\) −0.221060 − 1.25369i −0.0222173 − 0.126001i
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.v.a.139.9 yes 96
5.2 odd 4 925.2.bb.e.176.8 96
5.3 odd 4 925.2.bb.e.176.9 96
5.4 even 2 inner 185.2.v.a.139.8 yes 96
37.4 even 18 inner 185.2.v.a.4.8 ✓ 96
185.4 even 18 inner 185.2.v.a.4.9 yes 96
185.78 odd 36 925.2.bb.e.226.9 96
185.152 odd 36 925.2.bb.e.226.8 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.v.a.4.8 ✓ 96 37.4 even 18 inner
185.2.v.a.4.9 yes 96 185.4 even 18 inner
185.2.v.a.139.8 yes 96 5.4 even 2 inner
185.2.v.a.139.9 yes 96 1.1 even 1 trivial
925.2.bb.e.176.8 96 5.2 odd 4
925.2.bb.e.176.9 96 5.3 odd 4
925.2.bb.e.226.8 96 185.152 odd 36
925.2.bb.e.226.9 96 185.78 odd 36