Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1089,2,Mod(364,1089)] 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("1089.364"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1089, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1089.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,2,9,-12,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.69570878012\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 364.8
Character \(\chi\) \(=\) 1089.364
Dual form 1089.2.e.p.727.8

$q$-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+(-0.288906 - 0.500400i) q^{2} +(-1.33220 + 1.10691i) q^{3} +(0.833067 - 1.44291i) q^{4} +(-1.50221 + 2.60191i) q^{5} +(0.938776 + 0.346842i) q^{6} +(0.582160 + 1.00833i) q^{7} -2.11834 q^{8} +(0.549521 - 2.94924i) q^{9} +1.73599 q^{10} +(0.487356 + 2.84438i) q^{12} +(1.97309 - 3.41748i) q^{13} +(0.336379 - 0.582626i) q^{14} +(-0.878818 - 5.12908i) q^{15} +(-1.05413 - 1.82581i) q^{16} -0.314191 q^{17} +(-1.63456 + 0.577073i) q^{18} +6.36839 q^{19} +(2.50289 + 4.33513i) q^{20} +(-1.89168 - 0.698904i) q^{21} +(0.0427501 - 0.0740453i) q^{23} +(2.82205 - 2.34480i) q^{24} +(-2.01329 - 3.48713i) q^{25} -2.28015 q^{26} +(2.53246 + 4.53725i) q^{27} +1.93991 q^{28} +(3.84991 + 6.66824i) q^{29} +(-2.31269 + 1.92158i) q^{30} +(-3.26442 + 5.65414i) q^{31} +(-2.72743 + 4.72404i) q^{32} +(0.0907715 + 0.157221i) q^{34} -3.49812 q^{35} +(-3.79771 - 3.24983i) q^{36} -6.24309 q^{37} +(-1.83987 - 3.18674i) q^{38} +(1.15428 + 6.73680i) q^{39} +(3.18219 - 5.51172i) q^{40} +(-2.90454 + 5.03080i) q^{41} +(0.196787 + 1.14851i) q^{42} +(3.39229 + 5.87562i) q^{43} +(6.84817 + 5.86020i) q^{45} -0.0494030 q^{46} +(0.163868 + 0.283827i) q^{47} +(3.42532 + 1.26552i) q^{48} +(2.82218 - 4.88816i) q^{49} +(-1.16331 + 2.01490i) q^{50} +(0.418565 - 0.347779i) q^{51} +(-3.28742 - 5.69398i) q^{52} -2.42269 q^{53} +(1.53880 - 2.57808i) q^{54} +(-1.23321 - 2.13598i) q^{56} +(-8.48398 + 7.04920i) q^{57} +(2.22452 - 3.85299i) q^{58} +(1.14802 - 1.98842i) q^{59} +(-8.13293 - 3.00481i) q^{60} +(6.84360 + 11.8535i) q^{61} +3.77244 q^{62} +(3.29372 - 1.16283i) q^{63} -1.06465 q^{64} +(5.92799 + 10.2676i) q^{65} +(-5.83989 + 10.1150i) q^{67} +(-0.261742 + 0.453350i) q^{68} +(0.0250094 + 0.145964i) q^{69} +(1.01063 + 1.75046i) q^{70} +8.71765 q^{71} +(-1.16407 + 6.24749i) q^{72} +2.94656 q^{73} +(1.80367 + 3.12404i) q^{74} +(6.54203 + 2.41703i) q^{75} +(5.30529 - 9.18904i) q^{76} +(3.03761 - 2.52391i) q^{78} +(4.44360 + 7.69654i) q^{79} +6.33413 q^{80} +(-8.39605 - 3.24134i) q^{81} +3.35655 q^{82} +(-2.47889 - 4.29357i) q^{83} +(-2.58435 + 2.14730i) q^{84} +(0.471981 - 0.817496i) q^{85} +(1.96011 - 3.39500i) q^{86} +(-12.5100 - 4.62195i) q^{87} +2.12862 q^{89} +(0.953966 - 5.11987i) q^{90} +4.59461 q^{91} +(-0.0712273 - 0.123369i) q^{92} +(-1.90973 - 11.1459i) q^{93} +(0.0946848 - 0.163999i) q^{94} +(-9.56668 + 16.5700i) q^{95} +(-1.59559 - 9.31238i) q^{96} +(0.0444196 + 0.0769370i) q^{97} -3.26138 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 2 q^{2} + 9 q^{3} - 12 q^{4} + q^{5} - q^{6} - q^{7} - 12 q^{8} - q^{9} - 4 q^{10} - 8 q^{12} - 3 q^{13} - 5 q^{15} + 8 q^{16} - 40 q^{17} + 17 q^{18} - 6 q^{19} + 5 q^{20} - 8 q^{21} + 10 q^{23}+ \cdots - 164 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.288906 − 0.500400i −0.204287 − 0.353836i 0.745618 − 0.666374i \(-0.232154\pi\)
−0.949905 + 0.312537i \(0.898821\pi\)
\(3\) −1.33220 + 1.10691i −0.769147 + 0.639072i
\(4\) 0.833067 − 1.44291i 0.416533 − 0.721457i
\(5\) −1.50221 + 2.60191i −0.671810 + 1.16361i 0.305580 + 0.952166i \(0.401150\pi\)
−0.977390 + 0.211443i \(0.932184\pi\)
\(6\) 0.938776 + 0.346842i 0.383254 + 0.141598i
\(7\) 0.582160 + 1.00833i 0.220036 + 0.381113i 0.954819 − 0.297189i \(-0.0960492\pi\)
−0.734783 + 0.678303i \(0.762716\pi\)
\(8\) −2.11834 −0.748945
\(9\) 0.549521 − 2.94924i 0.183174 − 0.983081i
\(10\) 1.73599 0.548970
\(11\) 0 0
\(12\) 0.487356 + 2.84438i 0.140688 + 0.821101i
\(13\) 1.97309 − 3.41748i 0.547236 − 0.947840i −0.451227 − 0.892409i \(-0.649014\pi\)
0.998463 − 0.0554305i \(-0.0176531\pi\)
\(14\) 0.336379 − 0.582626i 0.0899011 − 0.155713i
\(15\) −0.878818 − 5.12908i −0.226910 − 1.32432i
\(16\) −1.05413 − 1.82581i −0.263533 − 0.456453i
\(17\) −0.314191 −0.0762024 −0.0381012 − 0.999274i \(-0.512131\pi\)
−0.0381012 + 0.999274i \(0.512131\pi\)
\(18\) −1.63456 + 0.577073i −0.385270 + 0.136018i
\(19\) 6.36839 1.46101 0.730504 − 0.682908i \(-0.239285\pi\)
0.730504 + 0.682908i \(0.239285\pi\)
\(20\) 2.50289 + 4.33513i 0.559663 + 0.969365i
\(21\) −1.89168 − 0.698904i −0.412799 − 0.152513i
\(22\) 0 0
\(23\) 0.0427501 − 0.0740453i 0.00891401 − 0.0154395i −0.861534 − 0.507700i \(-0.830496\pi\)
0.870448 + 0.492260i \(0.163829\pi\)
\(24\) 2.82205 − 2.34480i 0.576049 − 0.478630i
\(25\) −2.01329 − 3.48713i −0.402659 − 0.697425i
\(26\) −2.28015 −0.447173
\(27\) 2.53246 + 4.53725i 0.487372 + 0.873195i
\(28\) 1.93991 0.366609
\(29\) 3.84991 + 6.66824i 0.714910 + 1.23826i 0.962994 + 0.269522i \(0.0868658\pi\)
−0.248084 + 0.968739i \(0.579801\pi\)
\(30\) −2.31269 + 1.92158i −0.422238 + 0.350831i
\(31\) −3.26442 + 5.65414i −0.586307 + 1.01551i 0.408404 + 0.912801i \(0.366085\pi\)
−0.994711 + 0.102712i \(0.967248\pi\)
\(32\) −2.72743 + 4.72404i −0.482146 + 0.835101i
\(33\) 0 0
\(34\) 0.0907715 + 0.157221i 0.0155672 + 0.0269632i
\(35\) −3.49812 −0.591290
\(36\) −3.79771 − 3.24983i −0.632952 − 0.541638i
\(37\) −6.24309 −1.02636 −0.513179 − 0.858282i \(-0.671532\pi\)
−0.513179 + 0.858282i \(0.671532\pi\)
\(38\) −1.83987 − 3.18674i −0.298466 − 0.516958i
\(39\) 1.15428 + 6.73680i 0.184833 + 1.07875i
\(40\) 3.18219 − 5.51172i 0.503149 − 0.871480i
\(41\) −2.90454 + 5.03080i −0.453612 + 0.785679i −0.998607 − 0.0527599i \(-0.983198\pi\)
0.544995 + 0.838439i \(0.316532\pi\)
\(42\) 0.196787 + 1.14851i 0.0303649 + 0.177220i
\(43\) 3.39229 + 5.87562i 0.517320 + 0.896024i 0.999798 + 0.0201157i \(0.00640347\pi\)
−0.482478 + 0.875908i \(0.660263\pi\)
\(44\) 0 0
\(45\) 6.84817 + 5.86020i 1.02086 + 0.873587i
\(46\) −0.0494030 −0.00728408
\(47\) 0.163868 + 0.283827i 0.0239026 + 0.0414005i 0.877729 − 0.479157i \(-0.159058\pi\)
−0.853827 + 0.520557i \(0.825724\pi\)
\(48\) 3.42532 + 1.26552i 0.494402 + 0.182663i
\(49\) 2.82218 − 4.88816i 0.403168 − 0.698308i
\(50\) −1.16331 + 2.01490i −0.164516 + 0.284950i
\(51\) 0.418565 − 0.347779i 0.0586108 − 0.0486988i
\(52\) −3.28742 − 5.69398i −0.455884 − 0.789614i
\(53\) −2.42269 −0.332782 −0.166391 − 0.986060i \(-0.553211\pi\)
−0.166391 + 0.986060i \(0.553211\pi\)
\(54\) 1.53880 − 2.57808i 0.209404 − 0.350832i
\(55\) 0 0
\(56\) −1.23321 − 2.13598i −0.164795 − 0.285433i
\(57\) −8.48398 + 7.04920i −1.12373 + 0.933690i
\(58\) 2.22452 − 3.85299i 0.292094 − 0.505922i
\(59\) 1.14802 − 1.98842i 0.149459 − 0.258870i −0.781569 − 0.623819i \(-0.785580\pi\)
0.931028 + 0.364949i \(0.118913\pi\)
\(60\) −8.13293 − 3.00481i −1.04996 − 0.387919i
\(61\) 6.84360 + 11.8535i 0.876233 + 1.51768i 0.855443 + 0.517897i \(0.173285\pi\)
0.0207900 + 0.999784i \(0.493382\pi\)
\(62\) 3.77244 0.479100
\(63\) 3.29372 − 1.16283i 0.414970 − 0.146503i
\(64\) −1.06465 −0.133081
\(65\) 5.92799 + 10.2676i 0.735277 + 1.27354i
\(66\) 0 0
\(67\) −5.83989 + 10.1150i −0.713456 + 1.23574i 0.250096 + 0.968221i \(0.419538\pi\)
−0.963552 + 0.267521i \(0.913795\pi\)
\(68\) −0.261742 + 0.453350i −0.0317408 + 0.0549767i
\(69\) 0.0250094 + 0.145964i 0.00301078 + 0.0175719i
\(70\) 1.01063 + 1.75046i 0.120793 + 0.209220i
\(71\) 8.71765 1.03459 0.517297 − 0.855806i \(-0.326938\pi\)
0.517297 + 0.855806i \(0.326938\pi\)
\(72\) −1.16407 + 6.24749i −0.137187 + 0.736273i
\(73\) 2.94656 0.344869 0.172434 − 0.985021i \(-0.444837\pi\)
0.172434 + 0.985021i \(0.444837\pi\)
\(74\) 1.80367 + 3.12404i 0.209672 + 0.363162i
\(75\) 6.54203 + 2.41703i 0.755409 + 0.279095i
\(76\) 5.30529 − 9.18904i 0.608559 − 1.05405i
\(77\) 0 0
\(78\) 3.03761 − 2.52391i 0.343942 − 0.285776i
\(79\) 4.44360 + 7.69654i 0.499944 + 0.865928i 1.00000 6.50093e-5i \(-2.06931e-5\pi\)
−0.500056 + 0.865993i \(0.666687\pi\)
\(80\) 6.33413 0.708178
\(81\) −8.39605 − 3.24134i −0.932895 − 0.360149i
\(82\) 3.35655 0.370669
\(83\) −2.47889 − 4.29357i −0.272094 − 0.471280i 0.697304 − 0.716775i \(-0.254383\pi\)
−0.969398 + 0.245495i \(0.921049\pi\)
\(84\) −2.58435 + 2.14730i −0.281976 + 0.234290i
\(85\) 0.471981 − 0.817496i 0.0511936 − 0.0886699i
\(86\) 1.96011 − 3.39500i 0.211364 − 0.366093i
\(87\) −12.5100 − 4.62195i −1.34121 − 0.495525i
\(88\) 0 0
\(89\) 2.12862 0.225634 0.112817 − 0.993616i \(-0.464013\pi\)
0.112817 + 0.993616i \(0.464013\pi\)
\(90\) 0.953966 − 5.11987i 0.100557 − 0.539681i
\(91\) 4.59461 0.481646
\(92\) −0.0712273 − 0.123369i −0.00742596 − 0.0128621i
\(93\) −1.90973 − 11.1459i −0.198030 − 1.15577i
\(94\) 0.0946848 − 0.163999i 0.00976599 − 0.0169152i
\(95\) −9.56668 + 16.5700i −0.981521 + 1.70004i
\(96\) −1.59559 − 9.31238i −0.162849 − 0.950441i
\(97\) 0.0444196 + 0.0769370i 0.00451013 + 0.00781177i 0.868272 − 0.496089i \(-0.165231\pi\)
−0.863762 + 0.503901i \(0.831898\pi\)
\(98\) −3.26138 −0.329449
\(99\) 0 0
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 1089.2.e.p.364.8 36
9.4 even 3 9801.2.a.cm.1.11 18
9.5 odd 6 9801.2.a.cp.1.8 18
9.7 even 3 inner 1089.2.e.p.727.8 36
11.5 even 5 99.2.m.b.58.6 yes 72
11.9 even 5 99.2.m.b.4.4 ✓ 72
11.10 odd 2 1089.2.e.o.364.11 36
33.5 odd 10 297.2.n.b.91.4 72
33.20 odd 10 297.2.n.b.37.6 72
99.5 odd 30 891.2.f.e.487.6 36
99.16 even 15 99.2.m.b.25.4 yes 72
99.20 odd 30 297.2.n.b.235.4 72
99.31 even 15 891.2.f.f.730.4 36
99.32 even 6 9801.2.a.cn.1.11 18
99.38 odd 30 297.2.n.b.289.6 72
99.43 odd 6 1089.2.e.o.727.11 36
99.49 even 15 891.2.f.f.487.4 36
99.76 odd 6 9801.2.a.co.1.8 18
99.86 odd 30 891.2.f.e.730.6 36
99.97 even 15 99.2.m.b.70.6 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.m.b.4.4 ✓ 72 11.9 even 5
99.2.m.b.25.4 yes 72 99.16 even 15
99.2.m.b.58.6 yes 72 11.5 even 5
99.2.m.b.70.6 yes 72 99.97 even 15
297.2.n.b.37.6 72 33.20 odd 10
297.2.n.b.91.4 72 33.5 odd 10
297.2.n.b.235.4 72 99.20 odd 30
297.2.n.b.289.6 72 99.38 odd 30
891.2.f.e.487.6 36 99.5 odd 30
891.2.f.e.730.6 36 99.86 odd 30
891.2.f.f.487.4 36 99.49 even 15
891.2.f.f.730.4 36 99.31 even 15
1089.2.e.o.364.11 36 11.10 odd 2
1089.2.e.o.727.11 36 99.43 odd 6
1089.2.e.p.364.8 36 1.1 even 1 trivial
1089.2.e.p.727.8 36 9.7 even 3 inner
9801.2.a.cm.1.11 18 9.4 even 3
9801.2.a.cn.1.11 18 99.32 even 6
9801.2.a.co.1.8 18 99.76 odd 6
9801.2.a.cp.1.8 18 9.5 odd 6