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 68.11
Character \(\chi\) \(=\) 135.68
Dual form 135.2.q.a.2.11

$q$-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+(0.343150 - 0.490069i) q^{2} +(0.150104 - 1.72553i) q^{3} +(0.561625 + 1.54305i) q^{4} +(2.04155 + 0.912186i) q^{5} +(-0.794122 - 0.665679i) q^{6} +(-0.136472 + 0.0636379i) q^{7} +(2.10468 + 0.563947i) q^{8} +(-2.95494 - 0.518020i) q^{9} +(1.14759 - 0.687482i) q^{10} +(-3.54970 - 4.23037i) q^{11} +(2.74689 - 0.737484i) q^{12} +(-1.54940 + 1.08490i) q^{13} +(-0.0156434 + 0.0887180i) q^{14} +(1.88045 - 3.38584i) q^{15} +(-1.51722 + 1.27310i) q^{16} +(4.40907 - 1.18141i) q^{17} +(-1.26785 + 1.27036i) q^{18} +(-6.55178 + 3.78267i) q^{19} +(-0.260967 + 3.66252i) q^{20} +(0.0893244 + 0.245039i) q^{21} +(-3.29125 + 0.287947i) q^{22} +(0.412075 - 0.883698i) q^{23} +(1.28903 - 3.54705i) q^{24} +(3.33583 + 3.72454i) q^{25} +1.13159i q^{26} +(-1.33741 + 5.02109i) q^{27} +(-0.174843 - 0.174843i) q^{28} +(0.486461 + 2.75886i) q^{29} +(-1.01402 - 2.08340i) q^{30} +(1.70974 - 0.622294i) q^{31} +(0.483084 + 5.52167i) q^{32} +(-7.83247 + 5.49013i) q^{33} +(0.934001 - 2.56615i) q^{34} +(-0.336664 + 0.00543199i) q^{35} +(-0.860234 - 4.85055i) q^{36} +(-1.53079 - 5.71298i) q^{37} +(-0.394473 + 4.50885i) q^{38} +(1.63946 + 2.83639i) q^{39} +(3.78238 + 3.07118i) q^{40} +(-0.682276 - 0.120304i) q^{41} +(0.150738 + 0.0403102i) q^{42} +(1.73456 + 0.151755i) q^{43} +(4.53407 - 7.85325i) q^{44} +(-5.56011 - 3.75302i) q^{45} +(-0.291669 - 0.505186i) q^{46} +(-4.70696 - 10.0941i) q^{47} +(1.96903 + 2.80911i) q^{48} +(-4.48494 + 5.34494i) q^{49} +(2.96997 - 0.356711i) q^{50} +(-1.37674 - 7.78533i) q^{51} +(-2.54423 - 1.78149i) q^{52} +(-0.204978 + 0.204978i) q^{53} +(2.00175 + 2.37841i) q^{54} +(-3.38800 - 11.8745i) q^{55} +(-0.323118 + 0.0569745i) q^{56} +(5.54368 + 11.8731i) q^{57} +(1.51896 + 0.708303i) q^{58} +(-0.796564 - 0.668396i) q^{59} +(6.28063 + 1.00007i) q^{60} +(9.12620 + 3.32167i) q^{61} +(0.281730 - 1.05143i) q^{62} +(0.436232 - 0.117351i) q^{63} +(-0.558711 - 0.322572i) q^{64} +(-4.15280 + 0.801535i) q^{65} +(0.00283170 + 5.72239i) q^{66} +(6.27537 + 8.96216i) q^{67} +(4.29921 + 6.13991i) q^{68} +(-1.46300 - 0.843697i) q^{69} +(-0.112864 + 0.166852i) q^{70} +(-2.53024 - 1.46084i) q^{71} +(-5.92706 - 2.75670i) q^{72} +(2.66706 - 9.95358i) q^{73} +(-3.32505 - 1.21022i) q^{74} +(6.92755 - 5.19702i) q^{75} +(-9.51649 - 7.98529i) q^{76} +(0.753647 + 0.351431i) q^{77} +(1.95261 + 0.169857i) q^{78} +(11.9179 - 2.10144i) q^{79} +(-4.25878 + 1.21510i) q^{80} +(8.46331 + 3.06144i) q^{81} +(-0.293080 + 0.293080i) q^{82} +(5.85978 + 4.10306i) q^{83} +(-0.327942 + 0.275452i) q^{84} +(10.0790 + 1.60999i) q^{85} +(0.669585 - 0.797981i) q^{86} +(4.83353 - 0.425289i) q^{87} +(-5.08528 - 10.9054i) q^{88} +(-6.74192 - 11.6773i) q^{89} +(-3.74719 + 1.43699i) q^{90} +(0.142409 - 0.246659i) q^{91} +(1.59502 + 0.139546i) q^{92} +(-0.817150 - 3.04362i) q^{93} +(-6.56201 - 1.15706i) q^{94} +(-16.8263 + 1.74606i) q^{95} +(9.60035 - 0.00475069i) q^{96} +(0.311872 - 3.56471i) q^{97} +(1.08038 + 4.03205i) q^{98} +(8.29773 + 14.3393i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.343150 − 0.490069i 0.242644 − 0.346531i −0.679296 − 0.733864i \(-0.737715\pi\)
0.921940 + 0.387333i \(0.126604\pi\)
\(3\) 0.150104 − 1.72553i 0.0866628 − 0.996238i
\(4\) 0.561625 + 1.54305i 0.280812 + 0.771525i
\(5\) 2.04155 + 0.912186i 0.913008 + 0.407942i
\(6\) −0.794122 − 0.665679i −0.324199 − 0.271762i
\(7\) −0.136472 + 0.0636379i −0.0515816 + 0.0240529i −0.448238 − 0.893914i \(-0.647948\pi\)
0.396656 + 0.917967i \(0.370170\pi\)
\(8\) 2.10468 + 0.563947i 0.744117 + 0.199385i
\(9\) −2.95494 − 0.518020i −0.984979 − 0.172673i
\(10\) 1.14759 − 0.687482i 0.362900 − 0.217401i
\(11\) −3.54970 − 4.23037i −1.07027 − 1.27550i −0.959515 − 0.281657i \(-0.909116\pi\)
−0.110760 − 0.993847i \(-0.535328\pi\)
\(12\) 2.74689 − 0.737484i 0.792959 − 0.212893i
\(13\) −1.54940 + 1.08490i −0.429725 + 0.300897i −0.768356 − 0.640022i \(-0.778925\pi\)
0.338631 + 0.940919i \(0.390036\pi\)
\(14\) −0.0156434 + 0.0887180i −0.00418087 + 0.0237109i
\(15\) 1.88045 − 3.38584i 0.485531 − 0.874219i
\(16\) −1.51722 + 1.27310i −0.379305 + 0.318275i
\(17\) 4.40907 − 1.18141i 1.06936 − 0.286533i 0.319127 − 0.947712i \(-0.396610\pi\)
0.750228 + 0.661179i \(0.229944\pi\)
\(18\) −1.26785 + 1.27036i −0.298836 + 0.299428i
\(19\) −6.55178 + 3.78267i −1.50308 + 0.867804i −0.503087 + 0.864236i \(0.667803\pi\)
−0.999994 + 0.00356839i \(0.998864\pi\)
\(20\) −0.260967 + 3.66252i −0.0583539 + 0.818964i
\(21\) 0.0893244 + 0.245039i 0.0194922 + 0.0534720i
\(22\) −3.29125 + 0.287947i −0.701697 + 0.0613906i
\(23\) 0.412075 − 0.883698i 0.0859236 − 0.184264i −0.858651 − 0.512561i \(-0.828697\pi\)
0.944575 + 0.328297i \(0.106475\pi\)
\(24\) 1.28903 − 3.54705i 0.263122 − 0.724038i
\(25\) 3.33583 + 3.72454i 0.667166 + 0.744909i
\(26\) 1.13159i 0.221924i
\(27\) −1.33741 + 5.02109i −0.257385 + 0.966309i
\(28\) −0.174843 − 0.174843i −0.0330421 − 0.0330421i
\(29\) 0.486461 + 2.75886i 0.0903336 + 0.512307i 0.996078 + 0.0884824i \(0.0282017\pi\)
−0.905744 + 0.423825i \(0.860687\pi\)
\(30\) −1.01402 − 2.08340i −0.185133 − 0.380375i
\(31\) 1.70974 − 0.622294i 0.307078 − 0.111767i −0.183885 − 0.982948i \(-0.558867\pi\)
0.490963 + 0.871181i \(0.336645\pi\)
\(32\) 0.483084 + 5.52167i 0.0853979 + 0.976103i
\(33\) −7.83247 + 5.49013i −1.36346 + 0.955710i
\(34\) 0.934001 − 2.56615i 0.160180 − 0.440090i
\(35\) −0.336664 + 0.00543199i −0.0569065 + 0.000918174i
\(36\) −0.860234 − 4.85055i −0.143372 − 0.808425i
\(37\) −1.53079 − 5.71298i −0.251660 − 0.939209i −0.969918 − 0.243432i \(-0.921727\pi\)
0.718258 − 0.695777i \(-0.244940\pi\)
\(38\) −0.394473 + 4.50885i −0.0639920 + 0.731431i
\(39\) 1.63946 + 2.83639i 0.262524 + 0.454185i
\(40\) 3.78238 + 3.07118i 0.598047 + 0.485597i
\(41\) −0.682276 − 0.120304i −0.106554 − 0.0187883i 0.120117 − 0.992760i \(-0.461673\pi\)
−0.226671 + 0.973971i \(0.572784\pi\)
\(42\) 0.150738 + 0.0403102i 0.0232594 + 0.00621999i
\(43\) 1.73456 + 0.151755i 0.264518 + 0.0231424i 0.218643 − 0.975805i \(-0.429837\pi\)
0.0458750 + 0.998947i \(0.485392\pi\)
\(44\) 4.53407 − 7.85325i 0.683537 − 1.18392i
\(45\) −5.56011 − 3.75302i −0.828853 − 0.559467i
\(46\) −0.291669 − 0.505186i −0.0430043 − 0.0744856i
\(47\) −4.70696 − 10.0941i −0.686581 − 1.47238i −0.871892 − 0.489698i \(-0.837107\pi\)
0.185311 − 0.982680i \(-0.440671\pi\)
\(48\) 1.96903 + 2.80911i 0.284206 + 0.405460i
\(49\) −4.48494 + 5.34494i −0.640705 + 0.763563i
\(50\) 2.96997 − 0.356711i 0.420018 − 0.0504466i
\(51\) −1.37674 − 7.78533i −0.192782 − 1.09016i
\(52\) −2.54423 − 1.78149i −0.352822 − 0.247049i
\(53\) −0.204978 + 0.204978i −0.0281559 + 0.0281559i −0.721045 − 0.692889i \(-0.756338\pi\)
0.692889 + 0.721045i \(0.256338\pi\)
\(54\) 2.00175 + 2.37841i 0.272403 + 0.323661i
\(55\) −3.38800 − 11.8745i −0.456838 − 1.60116i
\(56\) −0.323118 + 0.0569745i −0.0431785 + 0.00761353i
\(57\) 5.54368 + 11.8731i 0.734278 + 1.57263i
\(58\) 1.51896 + 0.708303i 0.199449 + 0.0930047i
\(59\) −0.796564 − 0.668396i −0.103704 − 0.0870178i 0.589462 − 0.807796i \(-0.299340\pi\)
−0.693165 + 0.720779i \(0.743784\pi\)
\(60\) 6.28063 + 1.00007i 0.810826 + 0.129108i
\(61\) 9.12620 + 3.32167i 1.16849 + 0.425296i 0.852123 − 0.523341i \(-0.175315\pi\)
0.316367 + 0.948637i \(0.397537\pi\)
\(62\) 0.281730 − 1.05143i 0.0357797 − 0.133532i
\(63\) 0.436232 − 0.117351i 0.0549601 − 0.0147848i
\(64\) −0.558711 − 0.322572i −0.0698389 − 0.0403215i
\(65\) −4.15280 + 0.801535i −0.515091 + 0.0994182i
\(66\) 0.00283170 + 5.72239i 0.000348558 + 0.704378i
\(67\) 6.27537 + 8.96216i 0.766659 + 1.09490i 0.992679 + 0.120780i \(0.0385394\pi\)
−0.226021 + 0.974122i \(0.572572\pi\)
\(68\) 4.29921 + 6.13991i 0.521356 + 0.744573i
\(69\) −1.46300 − 0.843697i −0.176124 − 0.101569i
\(70\) −0.112864 + 0.166852i −0.0134898 + 0.0199427i
\(71\) −2.53024 − 1.46084i −0.300285 − 0.173369i 0.342286 − 0.939596i \(-0.388799\pi\)
−0.642571 + 0.766226i \(0.722132\pi\)
\(72\) −5.92706 − 2.75670i −0.698511 − 0.324880i
\(73\) 2.66706 − 9.95358i 0.312155 − 1.16498i −0.614454 − 0.788953i \(-0.710624\pi\)
0.926609 − 0.376026i \(-0.122710\pi\)
\(74\) −3.32505 − 1.21022i −0.386529 − 0.140685i
\(75\) 6.92755 − 5.19702i 0.799925 − 0.600101i
\(76\) −9.51649 − 7.98529i −1.09162 − 0.915975i
\(77\) 0.753647 + 0.351431i 0.0858860 + 0.0400493i
\(78\) 1.95261 + 0.169857i 0.221089 + 0.0192325i
\(79\) 11.9179 − 2.10144i 1.34086 − 0.236430i 0.543236 − 0.839580i \(-0.317199\pi\)
0.797628 + 0.603150i \(0.206088\pi\)
\(80\) −4.25878 + 1.21510i −0.476146 + 0.135853i
\(81\) 8.46331 + 3.06144i 0.940368 + 0.340159i
\(82\) −0.293080 + 0.293080i −0.0323653 + 0.0323653i
\(83\) 5.85978 + 4.10306i 0.643194 + 0.450370i 0.849121 − 0.528198i \(-0.177132\pi\)
−0.205927 + 0.978567i \(0.566021\pi\)
\(84\) −0.327942 + 0.275452i −0.0357814 + 0.0300543i
\(85\) 10.0790 + 1.60999i 1.09322 + 0.174628i
\(86\) 0.669585 − 0.797981i 0.0722033 − 0.0860485i
\(87\) 4.83353 − 0.425289i 0.518208 − 0.0455958i
\(88\) −5.08528 − 10.9054i −0.542092 − 1.16252i
\(89\) −6.74192 − 11.6773i −0.714642 − 1.23780i −0.963097 − 0.269153i \(-0.913256\pi\)
0.248455 − 0.968643i \(-0.420077\pi\)
\(90\) −3.74719 + 1.43699i −0.394988 + 0.151472i
\(91\) 0.142409 − 0.246659i 0.0149285 − 0.0258569i
\(92\) 1.59502 + 0.139546i 0.166293 + 0.0145487i
\(93\) −0.817150 − 3.04362i −0.0847345 − 0.315609i
\(94\) −6.56201 − 1.15706i −0.676819 − 0.119341i
\(95\) −16.8263 + 1.74606i −1.72634 + 0.179142i
\(96\) 9.60035 − 0.00475069i 0.979831 − 0.000484865i
\(97\) 0.311872 − 3.56471i 0.0316658 − 0.361941i −0.963786 − 0.266676i \(-0.914075\pi\)
0.995452 − 0.0952651i \(-0.0303699\pi\)
\(98\) 1.08038 + 4.03205i 0.109135 + 0.407298i
\(99\) 8.29773 + 14.3393i 0.833953 + 1.44115i
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.68.11 yes 192
3.2 odd 2 405.2.r.a.233.6 192
5.2 odd 4 inner 135.2.q.a.122.11 yes 192
5.3 odd 4 675.2.ba.b.257.6 192
5.4 even 2 675.2.ba.b.68.6 192
15.2 even 4 405.2.r.a.152.6 192
27.2 odd 18 inner 135.2.q.a.83.11 yes 192
27.25 even 9 405.2.r.a.8.6 192
135.2 even 36 inner 135.2.q.a.2.11 ✓ 192
135.29 odd 18 675.2.ba.b.218.6 192
135.52 odd 36 405.2.r.a.332.6 192
135.83 even 36 675.2.ba.b.407.6 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.q.a.2.11 ✓ 192 135.2 even 36 inner
135.2.q.a.68.11 yes 192 1.1 even 1 trivial
135.2.q.a.83.11 yes 192 27.2 odd 18 inner
135.2.q.a.122.11 yes 192 5.2 odd 4 inner
405.2.r.a.8.6 192 27.25 even 9
405.2.r.a.152.6 192 15.2 even 4
405.2.r.a.233.6 192 3.2 odd 2
405.2.r.a.332.6 192 135.52 odd 36
675.2.ba.b.68.6 192 5.4 even 2
675.2.ba.b.218.6 192 135.29 odd 18
675.2.ba.b.257.6 192 5.3 odd 4
675.2.ba.b.407.6 192 135.83 even 36