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Properties

Label 135.2.q.a.38.16
Level $135$
Weight $2$
Character 135.38
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 38.16
Character \(\chi\) \(=\) 135.38
Dual form 135.2.q.a.32.16

$q$-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+(0.237511 + 2.71476i) q^{2} +(0.914466 + 1.47097i) q^{3} +(-5.34392 + 0.942278i) q^{4} +(0.428650 - 2.19460i) q^{5} +(-3.77614 + 2.83193i) q^{6} +(0.907744 + 1.29639i) q^{7} +(-2.41667 - 9.01913i) q^{8} +(-1.32750 + 2.69030i) q^{9} +(6.05963 + 0.642443i) q^{10} +(0.162954 + 0.447712i) q^{11} +(-6.27290 - 6.99907i) q^{12} +(2.76700 + 0.242081i) q^{13} +(-3.30380 + 2.77222i) q^{14} +(3.62017 - 1.37635i) q^{15} +(13.7126 - 4.99098i) q^{16} +(0.898665 - 3.35386i) q^{17} +(-7.61884 - 2.96488i) q^{18} +(1.97319 + 1.13922i) q^{19} +(-0.222753 + 12.1317i) q^{20} +(-1.07685 + 2.52077i) q^{21} +(-1.17673 + 0.548719i) q^{22} +(-0.329603 - 0.230790i) q^{23} +(11.0569 - 11.8025i) q^{24} +(-4.63252 - 1.88143i) q^{25} +7.56924i q^{26} +(-5.17131 + 0.507474i) q^{27} +(-6.07248 - 6.07248i) q^{28} +(-4.29568 - 3.60450i) q^{29} +(4.59631 + 9.50102i) q^{30} +(0.908475 + 5.15222i) q^{31} +(8.91403 + 19.1162i) q^{32} +(-0.509556 + 0.649118i) q^{33} +(9.31839 + 1.64308i) q^{34} +(3.23417 - 1.43643i) q^{35} +(4.55906 - 15.6276i) q^{36} +(1.44483 + 0.387140i) q^{37} +(-2.62407 + 5.62733i) q^{38} +(2.17423 + 4.29154i) q^{39} +(-20.8293 + 1.43756i) q^{40} +(-4.99389 - 5.95149i) q^{41} +(-7.09907 - 2.32469i) q^{42} +(-1.07467 - 0.501128i) q^{43} +(-1.29268 - 2.23899i) q^{44} +(5.33510 + 4.06654i) q^{45} +(0.548257 - 0.949609i) q^{46} +(9.92230 - 6.94767i) q^{47} +(19.8813 + 15.6068i) q^{48} +(1.53751 - 4.22426i) q^{49} +(4.00736 - 13.0231i) q^{50} +(5.75523 - 1.74508i) q^{51} +(-15.0147 + 1.31362i) q^{52} +(0.274270 - 0.274270i) q^{53} +(-2.60592 - 13.9184i) q^{54} +(1.05240 - 0.165706i) q^{55} +(9.49862 - 11.3200i) q^{56} +(0.128655 + 3.94428i) q^{57} +(8.76510 - 12.5179i) q^{58} +(3.33385 + 1.21342i) q^{59} +(-18.0490 + 10.7663i) q^{60} +(-1.59716 + 9.05792i) q^{61} +(-13.7713 + 3.69001i) q^{62} +(-4.69272 + 0.721141i) q^{63} +(-24.5036 + 14.1471i) q^{64} +(1.71734 - 5.96868i) q^{65} +(-1.88323 - 1.22915i) q^{66} +(1.15101 - 13.1561i) q^{67} +(-1.64212 + 18.7696i) q^{68} +(0.0380751 - 0.695886i) q^{69} +(4.66773 + 8.43883i) q^{70} +(-13.8198 + 7.97886i) q^{71} +(27.4723 + 5.47136i) q^{72} +(-7.73957 + 2.07381i) q^{73} +(-0.707832 + 4.01431i) q^{74} +(-1.46875 - 8.53480i) q^{75} +(-11.6180 - 4.22862i) q^{76} +(-0.432491 + 0.617661i) q^{77} +(-11.1341 + 6.92182i) q^{78} +(2.85683 - 3.40464i) q^{79} +(-5.07529 - 32.2331i) q^{80} +(-5.47547 - 7.14278i) q^{81} +(14.9708 - 14.9708i) q^{82} +(-2.90211 + 0.253902i) q^{83} +(3.37935 - 14.4855i) q^{84} +(-6.97517 - 3.40984i) q^{85} +(1.10520 - 3.03651i) q^{86} +(1.37386 - 9.61501i) q^{87} +(3.64417 - 2.55168i) q^{88} +(-3.50126 + 6.06435i) q^{89} +(-9.77254 + 15.4494i) q^{90} +(2.19789 + 3.80686i) q^{91} +(1.97884 + 0.922748i) q^{92} +(-6.74799 + 6.04787i) q^{93} +(21.2179 + 25.2866i) q^{94} +(3.34594 - 3.84203i) q^{95} +(-19.9678 + 30.5934i) q^{96} +(-3.19969 + 6.86175i) q^{97} +(11.8331 + 3.17066i) q^{98} +(-1.42080 - 0.155944i) 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 \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.237511 + 2.71476i 0.167946 + 1.91963i 0.350732 + 0.936476i \(0.385933\pi\)
−0.182786 + 0.983153i \(0.558512\pi\)
\(3\) 0.914466 + 1.47097i 0.527967 + 0.849265i
\(4\) −5.34392 + 0.942278i −2.67196 + 0.471139i
\(5\) 0.428650 − 2.19460i 0.191698 − 0.981454i
\(6\) −3.77614 + 2.83193i −1.54160 + 1.15613i
\(7\) 0.907744 + 1.29639i 0.343095 + 0.489990i 0.953204 − 0.302327i \(-0.0977634\pi\)
−0.610109 + 0.792317i \(0.708874\pi\)
\(8\) −2.41667 − 9.01913i −0.854422 − 3.18874i
\(9\) −1.32750 + 2.69030i −0.442501 + 0.896768i
\(10\) 6.05963 + 0.642443i 1.91622 + 0.203158i
\(11\) 0.162954 + 0.447712i 0.0491325 + 0.134990i 0.961832 − 0.273642i \(-0.0882282\pi\)
−0.912699 + 0.408632i \(0.866006\pi\)
\(12\) −6.27290 − 6.99907i −1.81083 − 2.02046i
\(13\) 2.76700 + 0.242081i 0.767427 + 0.0671411i 0.464144 − 0.885760i \(-0.346362\pi\)
0.303282 + 0.952901i \(0.401917\pi\)
\(14\) −3.30380 + 2.77222i −0.882978 + 0.740907i
\(15\) 3.62017 − 1.37635i 0.934725 − 0.355373i
\(16\) 13.7126 − 4.99098i 3.42815 − 1.24775i
\(17\) 0.898665 − 3.35386i 0.217958 − 0.813431i −0.767146 − 0.641473i \(-0.778324\pi\)
0.985104 − 0.171959i \(-0.0550095\pi\)
\(18\) −7.61884 − 2.96488i −1.79578 − 0.698830i
\(19\) 1.97319 + 1.13922i 0.452681 + 0.261356i 0.708962 − 0.705247i \(-0.249164\pi\)
−0.256281 + 0.966602i \(0.582497\pi\)
\(20\) −0.222753 + 12.1317i −0.0498090 + 2.71272i
\(21\) −1.07685 + 2.52077i −0.234989 + 0.550077i
\(22\) −1.17673 + 0.548719i −0.250880 + 0.116987i
\(23\) −0.329603 − 0.230790i −0.0687269 − 0.0481231i 0.538708 − 0.842493i \(-0.318913\pi\)
−0.607435 + 0.794369i \(0.707801\pi\)
\(24\) 11.0569 − 11.8025i 2.25698 − 2.40918i
\(25\) −4.63252 − 1.88143i −0.926504 − 0.376286i
\(26\) 7.56924i 1.48445i
\(27\) −5.17131 + 0.507474i −0.995220 + 0.0976633i
\(28\) −6.07248 − 6.07248i −1.14759 − 1.14759i
\(29\) −4.29568 − 3.60450i −0.797687 − 0.669339i 0.149948 − 0.988694i \(-0.452089\pi\)
−0.947635 + 0.319355i \(0.896534\pi\)
\(30\) 4.59631 + 9.50102i 0.839167 + 1.73464i
\(31\) 0.908475 + 5.15222i 0.163167 + 0.925366i 0.950934 + 0.309393i \(0.100126\pi\)
−0.787767 + 0.615973i \(0.788763\pi\)
\(32\) 8.91403 + 19.1162i 1.57579 + 3.37930i
\(33\) −0.509556 + 0.649118i −0.0887022 + 0.112997i
\(34\) 9.31839 + 1.64308i 1.59809 + 0.281787i
\(35\) 3.23417 − 1.43643i 0.546674 − 0.242802i
\(36\) 4.55906 − 15.6276i 0.759844 − 2.60461i
\(37\) 1.44483 + 0.387140i 0.237528 + 0.0636454i 0.375619 − 0.926774i \(-0.377430\pi\)
−0.138091 + 0.990420i \(0.544097\pi\)
\(38\) −2.62407 + 5.62733i −0.425680 + 0.912873i
\(39\) 2.17423 + 4.29154i 0.348156 + 0.687197i
\(40\) −20.8293 + 1.43756i −3.29340 + 0.227299i
\(41\) −4.99389 − 5.95149i −0.779915 − 0.929467i 0.219015 − 0.975722i \(-0.429716\pi\)
−0.998930 + 0.0462550i \(0.985271\pi\)
\(42\) −7.09907 − 2.32469i −1.09541 − 0.358708i
\(43\) −1.07467 − 0.501128i −0.163886 − 0.0764213i 0.338944 − 0.940806i \(-0.389930\pi\)
−0.502830 + 0.864385i \(0.667708\pi\)
\(44\) −1.29268 − 2.23899i −0.194879 − 0.337541i
\(45\) 5.33510 + 4.06654i 0.795310 + 0.606203i
\(46\) 0.548257 − 0.949609i 0.0808361 − 0.140012i
\(47\) 9.92230 − 6.94767i 1.44732 − 1.01342i 0.454894 − 0.890546i \(-0.349677\pi\)
0.992422 − 0.122876i \(-0.0392117\pi\)
\(48\) 19.8813 + 15.6068i 2.86962 + 2.25264i
\(49\) 1.53751 − 4.22426i 0.219644 − 0.603466i
\(50\) 4.00736 − 13.0231i 0.566727 − 1.84174i
\(51\) 5.75523 − 1.74508i 0.805893 − 0.244361i
\(52\) −15.0147 + 1.31362i −2.08217 + 0.182166i
\(53\) 0.274270 − 0.274270i 0.0376738 − 0.0376738i −0.688019 − 0.725693i \(-0.741519\pi\)
0.725693 + 0.688019i \(0.241519\pi\)
\(54\) −2.60592 − 13.9184i −0.354620 − 1.89405i
\(55\) 1.05240 − 0.165706i 0.141905 − 0.0223439i
\(56\) 9.49862 − 11.3200i 1.26931 − 1.51270i
\(57\) 0.128655 + 3.94428i 0.0170407 + 0.522433i
\(58\) 8.76510 − 12.5179i 1.15091 − 1.64368i
\(59\) 3.33385 + 1.21342i 0.434031 + 0.157974i 0.549790 − 0.835303i \(-0.314708\pi\)
−0.115759 + 0.993277i \(0.536930\pi\)
\(60\) −18.0490 + 10.7663i −2.33012 + 1.38993i
\(61\) −1.59716 + 9.05792i −0.204495 + 1.15975i 0.693738 + 0.720228i \(0.255963\pi\)
−0.898233 + 0.439520i \(0.855149\pi\)
\(62\) −13.7713 + 3.69001i −1.74896 + 0.468631i
\(63\) −4.69272 + 0.721141i −0.591228 + 0.0908552i
\(64\) −24.5036 + 14.1471i −3.06294 + 1.76839i
\(65\) 1.71734 − 5.96868i 0.213010 − 0.740323i
\(66\) −1.88323 − 1.22915i −0.231809 − 0.151298i
\(67\) 1.15101 − 13.1561i 0.140618 − 1.60727i −0.518394 − 0.855142i \(-0.673470\pi\)
0.659012 − 0.752132i \(-0.270975\pi\)
\(68\) −1.64212 + 18.7696i −0.199137 + 2.27614i
\(69\) 0.0380751 − 0.695886i 0.00458370 − 0.0837748i
\(70\) 4.66773 + 8.43883i 0.557901 + 1.00863i
\(71\) −13.8198 + 7.97886i −1.64011 + 0.946916i −0.659313 + 0.751869i \(0.729153\pi\)
−0.980794 + 0.195047i \(0.937514\pi\)
\(72\) 27.4723 + 5.47136i 3.23765 + 0.644806i
\(73\) −7.73957 + 2.07381i −0.905849 + 0.242721i −0.681526 − 0.731794i \(-0.738683\pi\)
−0.224322 + 0.974515i \(0.572017\pi\)
\(74\) −0.707832 + 4.01431i −0.0822838 + 0.466655i
\(75\) −1.46875 − 8.53480i −0.169597 − 0.985513i
\(76\) −11.6180 − 4.22862i −1.33268 − 0.485056i
\(77\) −0.432491 + 0.617661i −0.0492869 + 0.0703890i
\(78\) −11.1341 + 6.92182i −1.26069 + 0.783741i
\(79\) 2.85683 − 3.40464i 0.321418 − 0.383051i −0.581006 − 0.813899i \(-0.697341\pi\)
0.902425 + 0.430848i \(0.141785\pi\)
\(80\) −5.07529 − 32.2331i −0.567434 − 3.60377i
\(81\) −5.47547 − 7.14278i −0.608385 − 0.793642i
\(82\) 14.9708 − 14.9708i 1.65325 − 1.65325i
\(83\) −2.90211 + 0.253902i −0.318548 + 0.0278694i −0.245309 − 0.969445i \(-0.578889\pi\)
−0.0732395 + 0.997314i \(0.523334\pi\)
\(84\) 3.37935 − 14.4855i 0.368718 − 1.58050i
\(85\) −6.97517 − 3.40984i −0.756563 − 0.369849i
\(86\) 1.10520 − 3.03651i 0.119177 − 0.327435i
\(87\) 1.37386 − 9.61501i 0.147293 − 1.03084i
\(88\) 3.64417 − 2.55168i 0.388470 − 0.272010i
\(89\) −3.50126 + 6.06435i −0.371132 + 0.642820i −0.989740 − 0.142880i \(-0.954364\pi\)
0.618608 + 0.785700i \(0.287697\pi\)
\(90\) −9.77254 + 15.4494i −1.03012 + 1.62851i
\(91\) 2.19789 + 3.80686i 0.230402 + 0.399068i
\(92\) 1.97884 + 0.922748i 0.206308 + 0.0962031i
\(93\) −6.74799 + 6.04787i −0.699734 + 0.627135i
\(94\) 21.2179 + 25.2866i 2.18846 + 2.60811i
\(95\) 3.34594 − 3.84203i 0.343287 − 0.394184i
\(96\) −19.9678 + 30.5934i −2.03795 + 3.12242i
\(97\) −3.19969 + 6.86175i −0.324879 + 0.696705i −0.999126 − 0.0418007i \(-0.986691\pi\)
0.674247 + 0.738506i \(0.264468\pi\)
\(98\) 11.8331 + 3.17066i 1.19532 + 0.320285i
\(99\) −1.42080 − 0.155944i −0.142796 − 0.0156730i
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.38.16 yes 192
3.2 odd 2 405.2.r.a.278.1 192
5.2 odd 4 inner 135.2.q.a.92.1 yes 192
5.3 odd 4 675.2.ba.b.632.16 192
5.4 even 2 675.2.ba.b.443.1 192
15.2 even 4 405.2.r.a.197.16 192
27.5 odd 18 inner 135.2.q.a.113.1 yes 192
27.22 even 9 405.2.r.a.368.16 192
135.22 odd 36 405.2.r.a.287.1 192
135.32 even 36 inner 135.2.q.a.32.16 ✓ 192
135.59 odd 18 675.2.ba.b.518.16 192
135.113 even 36 675.2.ba.b.32.1 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.q.a.32.16 ✓ 192 135.32 even 36 inner
135.2.q.a.38.16 yes 192 1.1 even 1 trivial
135.2.q.a.92.1 yes 192 5.2 odd 4 inner
135.2.q.a.113.1 yes 192 27.5 odd 18 inner
405.2.r.a.197.16 192 15.2 even 4
405.2.r.a.278.1 192 3.2 odd 2
405.2.r.a.287.1 192 135.22 odd 36
405.2.r.a.368.16 192 27.22 even 9
675.2.ba.b.32.1 192 135.113 even 36
675.2.ba.b.443.1 192 5.4 even 2
675.2.ba.b.518.16 192 135.59 odd 18
675.2.ba.b.632.16 192 5.3 odd 4