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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [135,2,Mod(4,135)] 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("135.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(18)) 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.p (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: \(1.07798042729\)
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 124.7
Character \(\chi\) \(=\) 135.124
Dual form 135.2.p.a.49.7

$q$-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+(-0.195467 + 0.232949i) q^{2} +(-1.57417 + 0.722492i) q^{3} +(0.331239 + 1.87855i) q^{4} +(-2.07799 - 0.825805i) q^{5} +(0.139394 - 0.507924i) q^{6} +(-3.36181 - 0.592777i) q^{7} +(-1.02906 - 0.594126i) q^{8} +(1.95601 - 2.27465i) q^{9} +(0.598549 - 0.322648i) q^{10} +(-2.75854 + 1.00403i) q^{11} +(-1.87866 - 2.71783i) q^{12} +(3.53380 + 4.21142i) q^{13} +(0.795210 - 0.667260i) q^{14} +(3.86774 - 0.201377i) q^{15} +(-3.24543 + 1.18124i) q^{16} +(-1.83496 + 1.05942i) q^{17} +(0.147541 + 0.900269i) q^{18} +(-2.20534 + 3.81976i) q^{19} +(0.863003 - 4.17714i) q^{20} +(5.72033 - 1.49575i) q^{21} +(0.305317 - 0.838853i) q^{22} +(4.56507 - 0.804944i) q^{23} +(2.04916 + 0.191769i) q^{24} +(3.63609 + 3.43203i) q^{25} -1.67179 q^{26} +(-1.43567 + 4.99388i) q^{27} -6.51167i q^{28} +(0.308828 + 0.259138i) q^{29} +(-0.709106 + 0.940348i) q^{30} +(-0.573021 - 3.24976i) q^{31} +(1.17202 - 3.22009i) q^{32} +(3.61701 - 3.57353i) q^{33} +(0.111885 - 0.634533i) q^{34} +(6.49629 + 4.00798i) q^{35} +(4.92094 + 2.92101i) q^{36} +(3.52072 - 2.03269i) q^{37} +(-0.458737 - 1.26037i) q^{38} +(-8.60551 - 4.07634i) q^{39} +(1.64774 + 2.08439i) q^{40} +(-3.73852 + 3.13699i) q^{41} +(-0.769703 + 1.62491i) q^{42} +(0.0361245 + 0.0992511i) q^{43} +(-2.79985 - 4.84948i) q^{44} +(-5.94299 + 3.11142i) q^{45} +(-0.704810 + 1.22077i) q^{46} +(-11.3497 - 2.00126i) q^{47} +(4.25542 - 4.20427i) q^{48} +(4.37252 + 1.59147i) q^{49} +(-1.51022 + 0.176174i) q^{50} +(2.12312 - 2.99344i) q^{51} +(-6.74082 + 8.03340i) q^{52} +11.5814i q^{53} +(-0.882691 - 1.31058i) q^{54} +(6.56135 + 0.191657i) q^{55} +(3.10730 + 2.60734i) q^{56} +(0.711828 - 7.60629i) q^{57} +(-0.120732 + 0.0212882i) q^{58} +(5.15923 + 1.87781i) q^{59} +(1.65944 + 7.19904i) q^{60} +(1.18992 - 6.74834i) q^{61} +(0.869034 + 0.501737i) q^{62} +(-7.92409 + 6.48745i) q^{63} +(-2.93269 - 5.07957i) q^{64} +(-3.86539 - 11.6695i) q^{65} +(0.125444 + 1.54108i) q^{66} +(-2.45463 - 2.92531i) q^{67} +(-2.59797 - 3.09614i) q^{68} +(-6.60462 + 4.56534i) q^{69} +(-2.20346 + 0.729873i) q^{70} +(6.75694 + 11.7034i) q^{71} +(-3.36427 + 1.17862i) q^{72} +(-6.69232 - 3.86382i) q^{73} +(-0.214673 + 1.21747i) q^{74} +(-8.20344 - 2.77554i) q^{75} +(-7.90610 - 2.87758i) q^{76} +(9.86885 - 1.74014i) q^{77} +(2.63167 - 1.20785i) q^{78} +(-1.80591 - 1.51534i) q^{79} +(7.71945 + 0.225486i) q^{80} +(-1.34805 - 8.89847i) q^{81} -1.48406i q^{82} +(-4.37992 + 5.21978i) q^{83} +(4.70463 + 10.2505i) q^{84} +(4.68790 - 0.686136i) q^{85} +(-0.0301816 - 0.0109852i) q^{86} +(-0.673373 - 0.184800i) q^{87} +(3.43521 + 0.605720i) q^{88} +(0.870723 - 1.50814i) q^{89} +(0.436858 - 1.99259i) q^{90} +(-9.38352 - 16.2527i) q^{91} +(3.02425 + 8.30907i) q^{92} +(3.24996 + 4.70167i) q^{93} +(2.68468 - 2.25271i) q^{94} +(7.73705 - 6.11625i) q^{95} +(0.481539 + 5.91574i) q^{96} +(-0.152119 - 0.417944i) q^{97} +(-1.22541 + 0.707493i) q^{98} +(-3.11193 + 8.23860i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 12 q^{4} - 9 q^{5} - 6 q^{6} - 18 q^{9} - 3 q^{10} - 6 q^{11} - 18 q^{14} - 21 q^{15} - 24 q^{16} - 6 q^{19} - 57 q^{20} + 24 q^{21} - 30 q^{24} + 3 q^{25} + 48 q^{26} - 30 q^{29} - 51 q^{30} - 30 q^{31}+ \cdots + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(56\) \(82\)
\(\chi(n)\) \(e\left(\frac{2}{9}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.195467 + 0.232949i −0.138216 + 0.164720i −0.830712 − 0.556702i \(-0.812067\pi\)
0.692496 + 0.721422i \(0.256511\pi\)
\(3\) −1.57417 + 0.722492i −0.908846 + 0.417131i
\(4\) 0.331239 + 1.87855i 0.165619 + 0.939274i
\(5\) −2.07799 − 0.825805i −0.929306 − 0.369311i
\(6\) 0.139394 − 0.507924i 0.0569076 − 0.207359i
\(7\) −3.36181 − 0.592777i −1.27064 − 0.224049i −0.502641 − 0.864495i \(-0.667638\pi\)
−0.768003 + 0.640446i \(0.778749\pi\)
\(8\) −1.02906 − 0.594126i −0.363826 − 0.210055i
\(9\) 1.95601 − 2.27465i 0.652003 − 0.758216i
\(10\) 0.598549 − 0.322648i 0.189278 − 0.102030i
\(11\) −2.75854 + 1.00403i −0.831732 + 0.302726i −0.722569 − 0.691298i \(-0.757039\pi\)
−0.109162 + 0.994024i \(0.534817\pi\)
\(12\) −1.87866 − 2.71783i −0.542323 − 0.784571i
\(13\) 3.53380 + 4.21142i 0.980100 + 1.16804i 0.985778 + 0.168053i \(0.0537481\pi\)
−0.00567819 + 0.999984i \(0.501807\pi\)
\(14\) 0.795210 − 0.667260i 0.212529 − 0.178333i
\(15\) 3.86774 − 0.201377i 0.998647 − 0.0519954i
\(16\) −3.24543 + 1.18124i −0.811358 + 0.295310i
\(17\) −1.83496 + 1.05942i −0.445044 + 0.256946i −0.705735 − 0.708476i \(-0.749383\pi\)
0.260691 + 0.965422i \(0.416050\pi\)
\(18\) 0.147541 + 0.900269i 0.0347757 + 0.212195i
\(19\) −2.20534 + 3.81976i −0.505940 + 0.876313i 0.494037 + 0.869441i \(0.335521\pi\)
−0.999976 + 0.00687212i \(0.997813\pi\)
\(20\) 0.863003 − 4.17714i 0.192973 − 0.934038i
\(21\) 5.72033 − 1.49575i 1.24828 − 0.326399i
\(22\) 0.305317 − 0.838853i 0.0650939 − 0.178844i
\(23\) 4.56507 − 0.804944i 0.951882 − 0.167843i 0.323918 − 0.946085i \(-0.395000\pi\)
0.627964 + 0.778243i \(0.283889\pi\)
\(24\) 2.04916 + 0.191769i 0.418283 + 0.0391446i
\(25\) 3.63609 + 3.43203i 0.727219 + 0.686406i
\(26\) −1.67179 −0.327864
\(27\) −1.43567 + 4.99388i −0.276295 + 0.961073i
\(28\) − 6.51167i − 1.23059i
\(29\) 0.308828 + 0.259138i 0.0573480 + 0.0481207i 0.671011 − 0.741447i \(-0.265860\pi\)
−0.613663 + 0.789568i \(0.710305\pi\)
\(30\) −0.709106 + 0.940348i −0.129465 + 0.171683i
\(31\) −0.573021 − 3.24976i −0.102918 − 0.583674i −0.992032 − 0.125989i \(-0.959790\pi\)
0.889114 − 0.457686i \(-0.151321\pi\)
\(32\) 1.17202 − 3.22009i 0.207186 − 0.569238i
\(33\) 3.61701 − 3.57353i 0.629640 − 0.622072i
\(34\) 0.111885 − 0.634533i 0.0191882 − 0.108821i
\(35\) 6.49629 + 4.00798i 1.09807 + 0.677473i
\(36\) 4.92094 + 2.92101i 0.820157 + 0.486834i
\(37\) 3.52072 − 2.03269i 0.578803 − 0.334172i −0.181854 − 0.983325i \(-0.558210\pi\)
0.760658 + 0.649153i \(0.224877\pi\)
\(38\) −0.458737 − 1.26037i −0.0744169 − 0.204459i
\(39\) −8.60551 − 4.07634i −1.37798 − 0.652736i
\(40\) 1.64774 + 2.08439i 0.260530 + 0.329570i
\(41\) −3.73852 + 3.13699i −0.583859 + 0.489916i −0.886212 − 0.463280i \(-0.846672\pi\)
0.302353 + 0.953196i \(0.402228\pi\)
\(42\) −0.769703 + 1.62491i −0.118768 + 0.250729i
\(43\) 0.0361245 + 0.0992511i 0.00550893 + 0.0151357i 0.942417 − 0.334441i \(-0.108548\pi\)
−0.936908 + 0.349577i \(0.886325\pi\)
\(44\) −2.79985 − 4.84948i −0.422093 − 0.731087i
\(45\) −5.94299 + 3.11142i −0.885928 + 0.463823i
\(46\) −0.704810 + 1.22077i −0.103919 + 0.179992i
\(47\) −11.3497 − 2.00126i −1.65552 − 0.291913i −0.733684 − 0.679490i \(-0.762201\pi\)
−0.921837 + 0.387577i \(0.873312\pi\)
\(48\) 4.25542 − 4.20427i 0.614217 − 0.606834i
\(49\) 4.37252 + 1.59147i 0.624646 + 0.227352i
\(50\) −1.51022 + 0.176174i −0.213578 + 0.0249148i
\(51\) 2.12312 − 2.99344i 0.297296 − 0.419166i
\(52\) −6.74082 + 8.03340i −0.934784 + 1.11403i
\(53\) 11.5814i 1.59083i 0.606068 + 0.795413i \(0.292746\pi\)
−0.606068 + 0.795413i \(0.707254\pi\)
\(54\) −0.882691 − 1.31058i −0.120119 − 0.178347i
\(55\) 6.56135 + 0.191657i 0.884733 + 0.0258431i
\(56\) 3.10730 + 2.60734i 0.415231 + 0.348420i
\(57\) 0.711828 − 7.60629i 0.0942839 − 1.00748i
\(58\) −0.120732 + 0.0212882i −0.0158528 + 0.00279528i
\(59\) 5.15923 + 1.87781i 0.671674 + 0.244469i 0.655269 − 0.755396i \(-0.272555\pi\)
0.0164056 + 0.999865i \(0.494778\pi\)
\(60\) 1.65944 + 7.19904i 0.214233 + 0.929392i
\(61\) 1.18992 − 6.74834i 0.152353 − 0.864037i −0.808813 − 0.588066i \(-0.799890\pi\)
0.961166 − 0.275971i \(-0.0889993\pi\)
\(62\) 0.869034 + 0.501737i 0.110367 + 0.0637207i
\(63\) −7.92409 + 6.48745i −0.998341 + 0.817342i
\(64\) −2.93269 − 5.07957i −0.366586 − 0.634946i
\(65\) −3.86539 − 11.6695i −0.479443 − 1.44743i
\(66\) 0.125444 + 1.54108i 0.0154411 + 0.189694i
\(67\) −2.45463 − 2.92531i −0.299881 − 0.357384i 0.594971 − 0.803747i \(-0.297163\pi\)
−0.894852 + 0.446363i \(0.852719\pi\)
\(68\) −2.59797 − 3.09614i −0.315051 − 0.375463i
\(69\) −6.60462 + 4.56534i −0.795102 + 0.549603i
\(70\) −2.20346 + 0.729873i −0.263364 + 0.0872365i
\(71\) 6.75694 + 11.7034i 0.801901 + 1.38893i 0.918363 + 0.395738i \(0.129511\pi\)
−0.116462 + 0.993195i \(0.537155\pi\)
\(72\) −3.36427 + 1.17862i −0.396483 + 0.138902i
\(73\) −6.69232 − 3.86382i −0.783277 − 0.452225i 0.0543132 − 0.998524i \(-0.482703\pi\)
−0.837591 + 0.546299i \(0.816036\pi\)
\(74\) −0.214673 + 1.21747i −0.0249552 + 0.141528i
\(75\) −8.20344 − 2.77554i −0.947251 − 0.320492i
\(76\) −7.90610 − 2.87758i −0.906891 − 0.330082i
\(77\) 9.86885 − 1.74014i 1.12466 − 0.198308i
\(78\) 2.63167 − 1.20785i 0.297978 − 0.136762i
\(79\) −1.80591 − 1.51534i −0.203181 − 0.170489i 0.535520 − 0.844523i \(-0.320116\pi\)
−0.738701 + 0.674034i \(0.764560\pi\)
\(80\) 7.71945 + 0.225486i 0.863061 + 0.0252100i
\(81\) −1.34805 − 8.89847i −0.149784 − 0.988719i
\(82\) − 1.48406i − 0.163887i
\(83\) −4.37992 + 5.21978i −0.480758 + 0.572946i −0.950842 − 0.309676i \(-0.899779\pi\)
0.470084 + 0.882622i \(0.344224\pi\)
\(84\) 4.70463 + 10.2505i 0.513317 + 1.11842i
\(85\) 4.68790 − 0.686136i 0.508475 − 0.0744219i
\(86\) −0.0301816 − 0.0109852i −0.00325456 − 0.00118456i
\(87\) −0.673373 − 0.184800i −0.0721931 − 0.0198127i
\(88\) 3.43521 + 0.605720i 0.366195 + 0.0645700i
\(89\) 0.870723 − 1.50814i 0.0922964 − 0.159862i −0.816181 − 0.577797i \(-0.803913\pi\)
0.908477 + 0.417935i \(0.137246\pi\)
\(90\) 0.436858 − 1.99259i 0.0460489 − 0.210037i
\(91\) −9.38352 − 16.2527i −0.983660 − 1.70375i
\(92\) 3.02425 + 8.30907i 0.315300 + 0.866280i
\(93\) 3.24996 + 4.70167i 0.337005 + 0.487540i
\(94\) 2.68468 − 2.25271i 0.276904 − 0.232350i
\(95\) 7.73705 − 6.11625i 0.793805 − 0.627514i
\(96\) 0.481539 + 5.91574i 0.0491469 + 0.603773i
\(97\) −0.152119 − 0.417944i −0.0154454 − 0.0424358i 0.931730 − 0.363151i \(-0.118299\pi\)
−0.947176 + 0.320715i \(0.896077\pi\)
\(98\) −1.22541 + 0.707493i −0.123785 + 0.0714676i
\(99\) −3.11193 + 8.23860i −0.312760 + 0.828010i
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.p.a.124.7 yes 96
3.2 odd 2 405.2.p.a.289.10 96
5.2 odd 4 675.2.l.h.151.7 96
5.3 odd 4 675.2.l.h.151.10 96
5.4 even 2 inner 135.2.p.a.124.10 yes 96
15.14 odd 2 405.2.p.a.289.7 96
27.5 odd 18 405.2.p.a.199.7 96
27.22 even 9 inner 135.2.p.a.49.10 yes 96
135.22 odd 36 675.2.l.h.76.7 96
135.49 even 18 inner 135.2.p.a.49.7 ✓ 96
135.59 odd 18 405.2.p.a.199.10 96
135.103 odd 36 675.2.l.h.76.10 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.p.a.49.7 ✓ 96 135.49 even 18 inner
135.2.p.a.49.10 yes 96 27.22 even 9 inner
135.2.p.a.124.7 yes 96 1.1 even 1 trivial
135.2.p.a.124.10 yes 96 5.4 even 2 inner
405.2.p.a.199.7 96 27.5 odd 18
405.2.p.a.199.10 96 135.59 odd 18
405.2.p.a.289.7 96 15.14 odd 2
405.2.p.a.289.10 96 3.2 odd 2
675.2.l.h.76.7 96 135.22 odd 36
675.2.l.h.76.10 96 135.103 odd 36
675.2.l.h.151.7 96 5.2 odd 4
675.2.l.h.151.10 96 5.3 odd 4