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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [270,2,Mod(23,270)] 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("270.23"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([22, 27])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.r (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: \(2.15596085457\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(18\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 113.13
Character \(\chi\) \(=\) 270.113
Dual form 270.2.r.a.227.13

$q$-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+(0.996195 + 0.0871557i) q^{2} +(-0.831700 + 1.51930i) q^{3} +(0.984808 + 0.173648i) q^{4} +(2.19457 - 0.428806i) q^{5} +(-0.960951 + 1.44103i) q^{6} +(0.926438 + 0.648699i) q^{7} +(0.965926 + 0.258819i) q^{8} +(-1.61655 - 2.52720i) q^{9} +(2.22359 - 0.235905i) q^{10} +(-1.09129 + 2.99831i) q^{11} +(-1.08289 + 1.35180i) q^{12} +(-0.104097 - 1.18984i) q^{13} +(0.866375 + 0.726975i) q^{14} +(-1.17373 + 3.69085i) q^{15} +(0.939693 + 0.342020i) q^{16} +(3.07725 - 0.824548i) q^{17} +(-1.39014 - 2.65848i) q^{18} +(-4.45771 + 2.57366i) q^{19} +(2.23569 - 0.0412092i) q^{20} +(-1.75609 + 0.868015i) q^{21} +(-1.34846 + 2.89179i) q^{22} +(-0.618138 - 0.882792i) q^{23} +(-1.19658 + 1.25227i) q^{24} +(4.63225 - 1.88209i) q^{25} -1.19438i q^{26} +(5.18407 - 0.354154i) q^{27} +(0.799718 + 0.799718i) q^{28} +(-4.18252 + 3.50955i) q^{29} +(-1.49095 + 3.57450i) q^{30} +(1.18252 - 6.70642i) q^{31} +(0.906308 + 0.422618i) q^{32} +(-3.64770 - 4.15170i) q^{33} +(3.13741 - 0.553210i) q^{34} +(2.31130 + 1.02635i) q^{35} +(-1.15315 - 2.76952i) q^{36} +(-2.00428 - 7.48009i) q^{37} +(-4.66505 + 2.17535i) q^{38} +(1.89430 + 0.831432i) q^{39} +(2.23077 + 0.153801i) q^{40} +(-3.43229 + 4.09044i) q^{41} +(-1.82506 + 0.711659i) q^{42} +(-2.90752 - 6.23520i) q^{43} +(-1.59537 + 2.76326i) q^{44} +(-4.63131 - 4.85293i) q^{45} +(-0.538845 - 0.933307i) q^{46} +(2.77457 - 3.96249i) q^{47} +(-1.30117 + 1.14322i) q^{48} +(-1.95666 - 5.37589i) q^{49} +(4.77866 - 1.47120i) q^{50} +(-1.30661 + 5.36105i) q^{51} +(0.104097 - 1.18984i) q^{52} +(3.59451 - 3.59451i) q^{53} +(5.19521 + 0.0990153i) q^{54} +(-1.10923 + 7.04794i) q^{55} +(0.726975 + 0.866375i) q^{56} +(-0.202689 - 8.91311i) q^{57} +(-4.47248 + 3.13167i) q^{58} +(-5.86932 + 2.13626i) q^{59} +(-1.79681 + 3.43096i) q^{60} +(-2.36034 - 13.3861i) q^{61} +(1.76253 - 6.57784i) q^{62} +(0.141760 - 3.38995i) q^{63} +(0.866025 + 0.500000i) q^{64} +(-0.738658 - 2.56654i) q^{65} +(-3.27198 - 4.45382i) q^{66} +(4.47588 - 0.391589i) q^{67} +(3.17368 - 0.277661i) q^{68} +(1.85533 - 0.204919i) q^{69} +(2.21305 + 1.22389i) q^{70} +(5.95249 + 3.43667i) q^{71} +(-0.907380 - 2.85949i) q^{72} +(-2.81344 + 10.4999i) q^{73} +(-1.34472 - 7.62631i) q^{74} +(-0.993182 + 8.60312i) q^{75} +(-4.83689 + 1.76049i) q^{76} +(-2.95602 + 2.06982i) q^{77} +(1.81463 + 0.993367i) q^{78} +(1.98062 + 2.36041i) q^{79} +(2.20888 + 0.347640i) q^{80} +(-3.77352 + 8.17071i) q^{81} +(-3.77573 + 3.77573i) q^{82} +(-1.48805 + 17.0085i) q^{83} +(-1.88014 + 0.549887i) q^{84} +(6.39967 - 3.12907i) q^{85} +(-2.35302 - 6.46488i) q^{86} +(-1.85347 - 9.27341i) q^{87} +(-1.83013 + 2.61369i) q^{88} +(7.57232 + 13.1156i) q^{89} +(-4.19073 - 5.23811i) q^{90} +(0.675406 - 1.16984i) q^{91} +(-0.455452 - 0.976719i) q^{92} +(9.20557 + 7.37433i) q^{93} +(3.10936 - 3.70559i) q^{94} +(-8.67914 + 7.55956i) q^{95} +(-1.39586 + 1.02546i) q^{96} +(-0.363881 + 0.169680i) q^{97} +(-1.48068 - 5.52597i) q^{98} +(9.34147 - 2.08899i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 216 q + 12 q^{6} + 24 q^{11} - 24 q^{20} + 24 q^{23} + 36 q^{25} + 12 q^{27} - 72 q^{30} - 12 q^{33} - 108 q^{35} - 12 q^{36} - 36 q^{38} - 72 q^{41} - 48 q^{42} - 60 q^{45} - 48 q^{47} - 12 q^{48} - 48 q^{50}+ \cdots - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.996195 + 0.0871557i 0.704416 + 0.0616284i
\(3\) −0.831700 + 1.51930i −0.480182 + 0.877169i
\(4\) 0.984808 + 0.173648i 0.492404 + 0.0868241i
\(5\) 2.19457 − 0.428806i 0.981440 − 0.191768i
\(6\) −0.960951 + 1.44103i −0.392306 + 0.588299i
\(7\) 0.926438 + 0.648699i 0.350161 + 0.245185i 0.735403 − 0.677631i \(-0.236993\pi\)
−0.385242 + 0.922816i \(0.625882\pi\)
\(8\) 0.965926 + 0.258819i 0.341506 + 0.0915064i
\(9\) −1.61655 − 2.52720i −0.538851 − 0.842401i
\(10\) 2.22359 − 0.235905i 0.703161 − 0.0745998i
\(11\) −1.09129 + 2.99831i −0.329038 + 0.904024i 0.659318 + 0.751864i \(0.270845\pi\)
−0.988356 + 0.152160i \(0.951377\pi\)
\(12\) −1.08289 + 1.35180i −0.312603 + 0.390230i
\(13\) −0.104097 − 1.18984i −0.0288714 − 0.330001i −0.996871 − 0.0790487i \(-0.974812\pi\)
0.967999 − 0.250953i \(-0.0807438\pi\)
\(14\) 0.866375 + 0.726975i 0.231548 + 0.194292i
\(15\) −1.17373 + 3.69085i −0.303057 + 0.952972i
\(16\) 0.939693 + 0.342020i 0.234923 + 0.0855050i
\(17\) 3.07725 − 0.824548i 0.746344 − 0.199982i 0.134448 − 0.990921i \(-0.457074\pi\)
0.611896 + 0.790938i \(0.290407\pi\)
\(18\) −1.39014 − 2.65848i −0.327659 − 0.626610i
\(19\) −4.45771 + 2.57366i −1.02267 + 0.590438i −0.914876 − 0.403736i \(-0.867712\pi\)
−0.107792 + 0.994173i \(0.534378\pi\)
\(20\) 2.23569 − 0.0412092i 0.499915 − 0.00921465i
\(21\) −1.75609 + 0.868015i −0.383210 + 0.189417i
\(22\) −1.34846 + 2.89179i −0.287493 + 0.616531i
\(23\) −0.618138 − 0.882792i −0.128891 − 0.184075i 0.749520 − 0.661982i \(-0.230284\pi\)
−0.878410 + 0.477907i \(0.841396\pi\)
\(24\) −1.19658 + 1.25227i −0.244252 + 0.255619i
\(25\) 4.63225 − 1.88209i 0.926450 − 0.376418i
\(26\) − 1.19438i − 0.234237i
\(27\) 5.18407 − 0.354154i 0.997675 − 0.0681569i
\(28\) 0.799718 + 0.799718i 0.151132 + 0.151132i
\(29\) −4.18252 + 3.50955i −0.776675 + 0.651708i −0.942409 − 0.334463i \(-0.891445\pi\)
0.165734 + 0.986170i \(0.447001\pi\)
\(30\) −1.49095 + 3.57450i −0.272208 + 0.652612i
\(31\) 1.18252 − 6.70642i 0.212387 − 1.20451i −0.672996 − 0.739646i \(-0.734993\pi\)
0.885383 − 0.464862i \(-0.153896\pi\)
\(32\) 0.906308 + 0.422618i 0.160214 + 0.0747091i
\(33\) −3.64770 − 4.15170i −0.634984 − 0.722718i
\(34\) 3.13741 − 0.553210i 0.538061 − 0.0948747i
\(35\) 2.31130 + 1.02635i 0.390680 + 0.173485i
\(36\) −1.15315 − 2.76952i −0.192191 − 0.461587i
\(37\) −2.00428 − 7.48009i −0.329502 − 1.22972i −0.909708 − 0.415249i \(-0.863695\pi\)
0.580206 − 0.814470i \(-0.302972\pi\)
\(38\) −4.66505 + 2.17535i −0.756771 + 0.352888i
\(39\) 1.89430 + 0.831432i 0.303330 + 0.133136i
\(40\) 2.23077 + 0.153801i 0.352716 + 0.0243180i
\(41\) −3.43229 + 4.09044i −0.536033 + 0.638820i −0.964294 − 0.264836i \(-0.914682\pi\)
0.428260 + 0.903655i \(0.359127\pi\)
\(42\) −1.82506 + 0.711659i −0.281612 + 0.109811i
\(43\) −2.90752 − 6.23520i −0.443393 − 0.950859i −0.993293 − 0.115628i \(-0.963112\pi\)
0.549900 − 0.835231i \(-0.314666\pi\)
\(44\) −1.59537 + 2.76326i −0.240510 + 0.416576i
\(45\) −4.63131 − 4.85293i −0.690395 − 0.723432i
\(46\) −0.538845 − 0.933307i −0.0794484 − 0.137609i
\(47\) 2.77457 − 3.96249i 0.404712 − 0.577989i −0.564125 − 0.825690i \(-0.690786\pi\)
0.968836 + 0.247701i \(0.0796751\pi\)
\(48\) −1.30117 + 1.14322i −0.187808 + 0.165009i
\(49\) −1.95666 − 5.37589i −0.279523 − 0.767984i
\(50\) 4.77866 − 1.47120i 0.675804 − 0.208059i
\(51\) −1.30661 + 5.36105i −0.182963 + 0.750697i
\(52\) 0.104097 − 1.18984i 0.0144357 − 0.165001i
\(53\) 3.59451 − 3.59451i 0.493743 − 0.493743i −0.415740 − 0.909483i \(-0.636477\pi\)
0.909483 + 0.415740i \(0.136477\pi\)
\(54\) 5.19521 + 0.0990153i 0.706978 + 0.0134743i
\(55\) −1.10923 + 7.04794i −0.149568 + 0.950344i
\(56\) 0.726975 + 0.866375i 0.0971461 + 0.115774i
\(57\) −0.202689 − 8.91311i −0.0268468 − 1.18057i
\(58\) −4.47248 + 3.13167i −0.587266 + 0.411208i
\(59\) −5.86932 + 2.13626i −0.764120 + 0.278117i −0.694535 − 0.719459i \(-0.744390\pi\)
−0.0695853 + 0.997576i \(0.522168\pi\)
\(60\) −1.79681 + 3.43096i −0.231967 + 0.442935i
\(61\) −2.36034 − 13.3861i −0.302210 − 1.71392i −0.636354 − 0.771397i \(-0.719558\pi\)
0.334144 − 0.942522i \(-0.391553\pi\)
\(62\) 1.76253 − 6.57784i 0.223841 − 0.835386i
\(63\) 0.141760 − 3.38995i 0.0178600 − 0.427094i
\(64\) 0.866025 + 0.500000i 0.108253 + 0.0625000i
\(65\) −0.738658 − 2.56654i −0.0916192 − 0.318340i
\(66\) −3.27198 − 4.45382i −0.402753 − 0.548227i
\(67\) 4.47588 − 0.391589i 0.546816 − 0.0478402i 0.189599 − 0.981862i \(-0.439281\pi\)
0.357217 + 0.934021i \(0.383726\pi\)
\(68\) 3.17368 − 0.277661i 0.384866 − 0.0336714i
\(69\) 1.85533 − 0.204919i 0.223356 − 0.0246694i
\(70\) 2.21305 + 1.22389i 0.264510 + 0.146283i
\(71\) 5.95249 + 3.43667i 0.706431 + 0.407858i 0.809738 − 0.586792i \(-0.199609\pi\)
−0.103307 + 0.994649i \(0.532943\pi\)
\(72\) −0.907380 − 2.85949i −0.106936 − 0.336994i
\(73\) −2.81344 + 10.4999i −0.329289 + 1.22892i 0.580641 + 0.814160i \(0.302802\pi\)
−0.909930 + 0.414763i \(0.863865\pi\)
\(74\) −1.34472 − 7.62631i −0.156321 − 0.886540i
\(75\) −0.993182 + 8.60312i −0.114683 + 0.993402i
\(76\) −4.83689 + 1.76049i −0.554830 + 0.201942i
\(77\) −2.95602 + 2.06982i −0.336869 + 0.235878i
\(78\) 1.81463 + 0.993367i 0.205466 + 0.112477i
\(79\) 1.98062 + 2.36041i 0.222837 + 0.265567i 0.865867 − 0.500274i \(-0.166767\pi\)
−0.643030 + 0.765841i \(0.722323\pi\)
\(80\) 2.20888 + 0.347640i 0.246960 + 0.0388673i
\(81\) −3.77352 + 8.17071i −0.419280 + 0.907857i
\(82\) −3.77573 + 3.77573i −0.416960 + 0.416960i
\(83\) −1.48805 + 17.0085i −0.163334 + 1.86692i 0.266264 + 0.963900i \(0.414211\pi\)
−0.429599 + 0.903020i \(0.641345\pi\)
\(84\) −1.88014 + 0.549887i −0.205140 + 0.0599976i
\(85\) 6.39967 − 3.12907i 0.694142 − 0.339395i
\(86\) −2.35302 − 6.46488i −0.253733 − 0.697126i
\(87\) −1.85347 − 9.27341i −0.198712 − 0.994213i
\(88\) −1.83013 + 2.61369i −0.195092 + 0.278621i
\(89\) 7.57232 + 13.1156i 0.802664 + 1.39025i 0.917857 + 0.396912i \(0.129918\pi\)
−0.115193 + 0.993343i \(0.536749\pi\)
\(90\) −4.19073 − 5.23811i −0.441741 − 0.552145i
\(91\) 0.675406 − 1.16984i 0.0708018 − 0.122632i
\(92\) −0.455452 − 0.976719i −0.0474841 − 0.101830i
\(93\) 9.20557 + 7.37433i 0.954573 + 0.764683i
\(94\) 3.10936 − 3.70559i 0.320706 − 0.382203i
\(95\) −8.67914 + 7.55956i −0.890461 + 0.775594i
\(96\) −1.39586 + 1.02546i −0.142464 + 0.104661i
\(97\) −0.363881 + 0.169680i −0.0369465 + 0.0172284i −0.441004 − 0.897505i \(-0.645377\pi\)
0.404057 + 0.914734i \(0.367600\pi\)
\(98\) −1.48068 − 5.52597i −0.149571 − 0.558207i
\(99\) 9.34147 − 2.08899i 0.938853 − 0.209952i
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 270.2.r.a.113.13 ✓ 216
3.2 odd 2 810.2.s.a.773.1 216
5.2 odd 4 inner 270.2.r.a.167.8 yes 216
15.2 even 4 810.2.s.a.287.17 216
27.11 odd 18 inner 270.2.r.a.173.8 yes 216
27.16 even 9 810.2.s.a.683.17 216
135.92 even 36 inner 270.2.r.a.227.13 yes 216
135.97 odd 36 810.2.s.a.197.1 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.2.r.a.113.13 ✓ 216 1.1 even 1 trivial
270.2.r.a.167.8 yes 216 5.2 odd 4 inner
270.2.r.a.173.8 yes 216 27.11 odd 18 inner
270.2.r.a.227.13 yes 216 135.92 even 36 inner
810.2.s.a.197.1 216 135.97 odd 36
810.2.s.a.287.17 216 15.2 even 4
810.2.s.a.683.17 216 27.16 even 9
810.2.s.a.773.1 216 3.2 odd 2