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Properties

Label 135.2.q.a.68.13
Level $135$
Weight $2$
Character 135.68
Analytic conductor $1.078$
Analytic rank $0$
Dimension $192$
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] = [135,2,Mod(2,135)] 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("135.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([2, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.q (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: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 68.13
Character \(\chi\) \(=\) 135.68
Dual form 135.2.q.a.2.13

$q$-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+(0.866263 - 1.23715i) q^{2} +(-0.938839 + 1.45553i) q^{3} +(-0.0960923 - 0.264011i) q^{4} +(0.525423 + 2.17346i) q^{5} +(0.987435 + 2.42236i) q^{6} +(1.45360 - 0.677826i) q^{7} +(2.50778 + 0.671958i) q^{8} +(-1.23716 - 2.73303i) q^{9} +(3.14405 + 1.23276i) q^{10} +(-1.78545 - 2.12782i) q^{11} +(0.474493 + 0.107999i) q^{12} +(-0.251952 + 0.176418i) q^{13} +(0.420628 - 2.38550i) q^{14} +(-3.65683 - 1.27576i) q^{15} +(3.43416 - 2.88160i) q^{16} +(-7.02145 + 1.88139i) q^{17} +(-4.45287 - 0.836964i) q^{18} +(5.07203 - 2.92834i) q^{19} +(0.523329 - 0.347571i) q^{20} +(-0.378100 + 2.75214i) q^{21} +(-4.17910 + 0.365624i) q^{22} +(-0.0543910 + 0.116642i) q^{23} +(-3.33246 + 3.01930i) q^{24} +(-4.44786 + 2.28397i) q^{25} +0.464527i q^{26} +(5.13951 + 0.765143i) q^{27} +(-0.318634 - 0.318634i) q^{28} +(-1.51659 - 8.60103i) q^{29} +(-4.74609 + 3.41892i) q^{30} +(4.22026 - 1.53605i) q^{31} +(-0.137538 - 1.57207i) q^{32} +(4.77337 - 0.601106i) q^{33} +(-3.75485 + 10.3164i) q^{34} +(2.23699 + 2.80320i) q^{35} +(-0.602668 + 0.589247i) q^{36} +(-0.558369 - 2.08386i) q^{37} +(0.770913 - 8.81158i) q^{38} +(-0.0202409 - 0.532353i) q^{39} +(-0.142827 + 5.80362i) q^{40} +(-4.62442 - 0.815410i) q^{41} +(3.07728 + 2.85184i) q^{42} +(-4.90494 - 0.429127i) q^{43} +(-0.390200 + 0.675847i) q^{44} +(5.29009 - 4.12492i) q^{45} +(0.0971868 + 0.168333i) q^{46} +(3.58654 + 7.69135i) q^{47} +(0.970146 + 7.70389i) q^{48} +(-2.84600 + 3.39173i) q^{49} +(-1.02740 + 7.48120i) q^{50} +(3.85358 - 11.9863i) q^{51} +(0.0707870 + 0.0495656i) q^{52} +(-0.483003 + 0.483003i) q^{53} +(5.39876 - 5.69554i) q^{54} +(3.68661 - 4.99861i) q^{55} +(4.10079 - 0.723079i) q^{56} +(-0.499526 + 10.1318i) q^{57} +(-11.9546 - 5.57450i) q^{58} +(-7.43127 - 6.23558i) q^{59} +(0.0145787 + 1.08804i) q^{60} +(2.50049 + 0.910105i) q^{61} +(1.75553 - 6.55173i) q^{62} +(-3.65086 - 3.13415i) q^{63} +(5.70071 + 3.29131i) q^{64} +(-0.515819 - 0.454912i) q^{65} +(3.39133 - 6.42609i) q^{66} +(0.619257 + 0.884390i) q^{67} +(1.17142 + 1.67295i) q^{68} +(-0.118712 - 0.188676i) q^{69} +(5.40580 - 0.339179i) q^{70} +(9.03942 + 5.21891i) q^{71} +(-1.26605 - 7.68515i) q^{72} +(1.78199 - 6.65047i) q^{73} +(-3.06175 - 1.11439i) q^{74} +(0.851425 - 8.61830i) q^{75} +(-1.26050 - 1.05768i) q^{76} +(-4.03763 - 1.88278i) q^{77} +(-0.676135 - 0.436116i) q^{78} +(-2.04188 + 0.360038i) q^{79} +(8.06743 + 5.94995i) q^{80} +(-5.93887 + 6.76239i) q^{81} +(-5.01475 + 5.01475i) q^{82} +(10.7962 + 7.55956i) q^{83} +(0.762929 - 0.164636i) q^{84} +(-7.77836 - 14.2723i) q^{85} +(-4.77986 + 5.69642i) q^{86} +(13.9429 + 5.86754i) q^{87} +(-3.04772 - 6.53585i) q^{88} +(5.58181 + 9.66798i) q^{89} +(-0.520536 - 10.1179i) q^{90} +(-0.246656 + 0.427222i) q^{91} +(0.0360214 + 0.00315146i) q^{92} +(-1.72638 + 7.58484i) q^{93} +(12.6222 + 2.22564i) q^{94} +(9.02959 + 9.48524i) q^{95} +(2.41732 + 1.27573i) q^{96} +(-1.03291 + 11.8062i) q^{97} +(1.73070 + 6.45906i) q^{98} +(-3.60649 + 7.51214i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 36 q^{6} - 12 q^{7} - 18 q^{8} - 6 q^{10} - 36 q^{11} - 12 q^{12} - 12 q^{13} - 12 q^{15} - 24 q^{16} - 18 q^{17} - 54 q^{18} + 36 q^{20} - 24 q^{21} - 12 q^{22}+ \cdots + 324 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866263 − 1.23715i 0.612540 − 0.874798i −0.386328 − 0.922361i \(-0.626257\pi\)
0.998868 + 0.0475633i \(0.0151456\pi\)
\(3\) −0.938839 + 1.45553i −0.542039 + 0.840353i
\(4\) −0.0960923 − 0.264011i −0.0480461 − 0.132006i
\(5\) 0.525423 + 2.17346i 0.234976 + 0.972001i
\(6\) 0.987435 + 2.42236i 0.403119 + 0.988925i
\(7\) 1.45360 − 0.677826i 0.549410 − 0.256194i −0.128032 − 0.991770i \(-0.540866\pi\)
0.677442 + 0.735576i \(0.263088\pi\)
\(8\) 2.50778 + 0.671958i 0.886634 + 0.237573i
\(9\) −1.23716 − 2.73303i −0.412387 − 0.911009i
\(10\) 3.14405 + 1.23276i 0.994237 + 0.389833i
\(11\) −1.78545 − 2.12782i −0.538334 − 0.641561i 0.426479 − 0.904497i \(-0.359754\pi\)
−0.964813 + 0.262936i \(0.915309\pi\)
\(12\) 0.474493 + 0.107999i 0.136974 + 0.0311765i
\(13\) −0.251952 + 0.176418i −0.0698788 + 0.0489296i −0.607994 − 0.793942i \(-0.708025\pi\)
0.538115 + 0.842871i \(0.319137\pi\)
\(14\) 0.420628 − 2.38550i 0.112418 − 0.637552i
\(15\) −3.65683 − 1.27576i −0.944191 − 0.329400i
\(16\) 3.43416 − 2.88160i 0.858539 − 0.720400i
\(17\) −7.02145 + 1.88139i −1.70295 + 0.456304i −0.973680 − 0.227921i \(-0.926807\pi\)
−0.729271 + 0.684225i \(0.760141\pi\)
\(18\) −4.45287 − 0.836964i −1.04955 − 0.197274i
\(19\) 5.07203 − 2.92834i 1.16360 − 0.671807i 0.211439 − 0.977391i \(-0.432185\pi\)
0.952165 + 0.305584i \(0.0988518\pi\)
\(20\) 0.523329 − 0.347571i 0.117020 − 0.0777191i
\(21\) −0.378100 + 2.75214i −0.0825083 + 0.600566i
\(22\) −4.17910 + 0.365624i −0.890988 + 0.0779514i
\(23\) −0.0543910 + 0.116642i −0.0113413 + 0.0243215i −0.911897 − 0.410419i \(-0.865382\pi\)
0.900556 + 0.434741i \(0.143160\pi\)
\(24\) −3.33246 + 3.01930i −0.680236 + 0.616312i
\(25\) −4.44786 + 2.28397i −0.889572 + 0.456795i
\(26\) 0.464527i 0.0911012i
\(27\) 5.13951 + 0.765143i 0.989099 + 0.147252i
\(28\) −0.318634 − 0.318634i −0.0602161 − 0.0602161i
\(29\) −1.51659 − 8.60103i −0.281625 − 1.59717i −0.717099 − 0.696971i \(-0.754531\pi\)
0.435475 − 0.900201i \(-0.356581\pi\)
\(30\) −4.74609 + 3.41892i −0.866513 + 0.624206i
\(31\) 4.22026 − 1.53605i 0.757981 − 0.275883i 0.0660211 − 0.997818i \(-0.478970\pi\)
0.691960 + 0.721936i \(0.256747\pi\)
\(32\) −0.137538 − 1.57207i −0.0243135 − 0.277905i
\(33\) 4.77337 − 0.601106i 0.830936 − 0.104639i
\(34\) −3.75485 + 10.3164i −0.643952 + 1.76924i
\(35\) 2.23699 + 2.80320i 0.378119 + 0.473828i
\(36\) −0.602668 + 0.589247i −0.100445 + 0.0982079i
\(37\) −0.558369 − 2.08386i −0.0917953 − 0.342585i 0.904719 − 0.426009i \(-0.140081\pi\)
−0.996514 + 0.0834245i \(0.973414\pi\)
\(38\) 0.770913 − 8.81158i 0.125059 − 1.42943i
\(39\) −0.0202409 − 0.532353i −0.00324114 − 0.0852446i
\(40\) −0.142827 + 5.80362i −0.0225830 + 0.917633i
\(41\) −4.62442 − 0.815410i −0.722213 − 0.127346i −0.199553 − 0.979887i \(-0.563949\pi\)
−0.522660 + 0.852541i \(0.675060\pi\)
\(42\) 3.07728 + 2.85184i 0.474834 + 0.440049i
\(43\) −4.90494 − 0.429127i −0.747997 − 0.0654412i −0.293218 − 0.956046i \(-0.594726\pi\)
−0.454779 + 0.890604i \(0.650282\pi\)
\(44\) −0.390200 + 0.675847i −0.0588249 + 0.101888i
\(45\) 5.29009 − 4.12492i 0.788600 − 0.614906i
\(46\) 0.0971868 + 0.168333i 0.0143294 + 0.0248193i
\(47\) 3.58654 + 7.69135i 0.523150 + 1.12190i 0.973424 + 0.229009i \(0.0735484\pi\)
−0.450274 + 0.892890i \(0.648674\pi\)
\(48\) 0.970146 + 7.70389i 0.140028 + 1.11196i
\(49\) −2.84600 + 3.39173i −0.406571 + 0.484533i
\(50\) −1.02740 + 7.48120i −0.145296 + 1.05800i
\(51\) 3.85358 − 11.9863i 0.539609 − 1.67841i
\(52\) 0.0707870 + 0.0495656i 0.00981640 + 0.00687352i
\(53\) −0.483003 + 0.483003i −0.0663456 + 0.0663456i −0.739501 − 0.673155i \(-0.764938\pi\)
0.673155 + 0.739501i \(0.264938\pi\)
\(54\) 5.39876 − 5.69554i 0.734679 − 0.775064i
\(55\) 3.68661 − 4.99861i 0.497103 − 0.674013i
\(56\) 4.10079 − 0.723079i 0.547991 − 0.0966255i
\(57\) −0.499526 + 10.1318i −0.0661638 + 1.34198i
\(58\) −11.9546 − 5.57450i −1.56971 − 0.731968i
\(59\) −7.43127 − 6.23558i −0.967469 − 0.811803i 0.0146830 − 0.999892i \(-0.495326\pi\)
−0.982152 + 0.188089i \(0.939771\pi\)
\(60\) 0.0145787 + 1.08804i 0.00188211 + 0.140465i
\(61\) 2.50049 + 0.910105i 0.320155 + 0.116527i 0.497099 − 0.867694i \(-0.334399\pi\)
−0.176943 + 0.984221i \(0.556621\pi\)
\(62\) 1.75553 − 6.55173i 0.222952 − 0.832070i
\(63\) −3.65086 − 3.13415i −0.459965 − 0.394866i
\(64\) 5.70071 + 3.29131i 0.712589 + 0.411413i
\(65\) −0.515819 − 0.454912i −0.0639795 − 0.0564249i
\(66\) 3.39133 − 6.42609i 0.417444 − 0.790997i
\(67\) 0.619257 + 0.884390i 0.0756543 + 0.108045i 0.855196 − 0.518305i \(-0.173437\pi\)
−0.779542 + 0.626350i \(0.784548\pi\)
\(68\) 1.17142 + 1.67295i 0.142055 + 0.202876i
\(69\) −0.118712 − 0.188676i −0.0142912 − 0.0227139i
\(70\) 5.40580 − 0.339179i 0.646117 − 0.0405397i
\(71\) 9.03942 + 5.21891i 1.07278 + 0.619371i 0.928940 − 0.370230i \(-0.120721\pi\)
0.143842 + 0.989601i \(0.454054\pi\)
\(72\) −1.26605 − 7.68515i −0.149205 − 0.905703i
\(73\) 1.78199 − 6.65047i 0.208566 − 0.778378i −0.779767 − 0.626070i \(-0.784663\pi\)
0.988333 − 0.152309i \(-0.0486707\pi\)
\(74\) −3.06175 − 1.11439i −0.355921 − 0.129545i
\(75\) 0.851425 − 8.61830i 0.0983141 − 0.995155i
\(76\) −1.26050 − 1.05768i −0.144589 − 0.121325i
\(77\) −4.03763 − 1.88278i −0.460130 − 0.214562i
\(78\) −0.676135 − 0.436116i −0.0765572 − 0.0493804i
\(79\) −2.04188 + 0.360038i −0.229729 + 0.0405074i −0.287327 − 0.957832i \(-0.592767\pi\)
0.0575985 + 0.998340i \(0.481656\pi\)
\(80\) 8.06743 + 5.94995i 0.901966 + 0.665224i
\(81\) −5.93887 + 6.76239i −0.659874 + 0.751376i
\(82\) −5.01475 + 5.01475i −0.553787 + 0.553787i
\(83\) 10.7962 + 7.55956i 1.18503 + 0.829770i 0.988573 − 0.150742i \(-0.0481662\pi\)
0.196461 + 0.980512i \(0.437055\pi\)
\(84\) 0.762929 − 0.164636i 0.0832423 − 0.0179633i
\(85\) −7.77836 − 14.2723i −0.843681 − 1.54805i
\(86\) −4.77986 + 5.69642i −0.515426 + 0.614261i
\(87\) 13.9429 + 5.86754i 1.49484 + 0.629066i
\(88\) −3.04772 − 6.53585i −0.324888 − 0.696724i
\(89\) 5.58181 + 9.66798i 0.591671 + 1.02480i 0.994007 + 0.109312i \(0.0348648\pi\)
−0.402337 + 0.915492i \(0.631802\pi\)
\(90\) −0.520536 − 10.1179i −0.0548693 − 1.06652i
\(91\) −0.246656 + 0.427222i −0.0258566 + 0.0447850i
\(92\) 0.0360214 + 0.00315146i 0.00375549 + 0.000328562i
\(93\) −1.72638 + 7.58484i −0.179017 + 0.786511i
\(94\) 12.6222 + 2.22564i 1.30189 + 0.229558i
\(95\) 9.02959 + 9.48524i 0.926416 + 0.973165i
\(96\) 2.41732 + 1.27573i 0.246717 + 0.130203i
\(97\) −1.03291 + 11.8062i −0.104876 + 1.19874i 0.743275 + 0.668986i \(0.233271\pi\)
−0.848151 + 0.529754i \(0.822284\pi\)
\(98\) 1.73070 + 6.45906i 0.174827 + 0.652464i
\(99\) −3.60649 + 7.51214i −0.362466 + 0.754999i
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 135.2.q.a.68.13 yes 192
3.2 odd 2 405.2.r.a.233.4 192
5.2 odd 4 inner 135.2.q.a.122.13 yes 192
5.3 odd 4 675.2.ba.b.257.4 192
5.4 even 2 675.2.ba.b.68.4 192
15.2 even 4 405.2.r.a.152.4 192
27.2 odd 18 inner 135.2.q.a.83.13 yes 192
27.25 even 9 405.2.r.a.8.4 192
135.2 even 36 inner 135.2.q.a.2.13 ✓ 192
135.29 odd 18 675.2.ba.b.218.4 192
135.52 odd 36 405.2.r.a.332.4 192
135.83 even 36 675.2.ba.b.407.4 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.q.a.2.13 ✓ 192 135.2 even 36 inner
135.2.q.a.68.13 yes 192 1.1 even 1 trivial
135.2.q.a.83.13 yes 192 27.2 odd 18 inner
135.2.q.a.122.13 yes 192 5.2 odd 4 inner
405.2.r.a.8.4 192 27.25 even 9
405.2.r.a.152.4 192 15.2 even 4
405.2.r.a.233.4 192 3.2 odd 2
405.2.r.a.332.4 192 135.52 odd 36
675.2.ba.b.68.4 192 5.4 even 2
675.2.ba.b.218.4 192 135.29 odd 18
675.2.ba.b.257.4 192 5.3 odd 4
675.2.ba.b.407.4 192 135.83 even 36