↑

Properties

Label 486.2.g.b.73.5
Level $486$
Weight $2$
Character 486.73
Analytic conductor $3.881$
Analytic rank $0$
Dimension $90$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [486,2,Mod(19,486)] 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("486.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(486, base_ring=CyclotomicField(54)) chi = DirichletCharacter(H, H._module([52])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 486 = 2 \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 486.g (of order \(27\), degree \(18\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [90] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.88072953823\)
Analytic rank: \(0\)
Dimension: \(90\)
Relative dimension: \(5\) over \(\Q(\zeta_{27})\)
Twist minimal: no (minimal twist has level 162)
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label 73.5
Character \(\chi\) \(=\) 486.73
Dual form 486.2.g.b.253.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.893633 + 0.448799i) q^{2} +(0.597159 - 0.802123i) q^{4} +(1.15296 - 3.85116i) q^{5} +(-0.663579 + 1.53835i) q^{7} +(-0.173648 + 0.984808i) q^{8} +(0.698073 + 3.95897i) q^{10} +(-0.0131573 - 0.00311835i) q^{11} +(3.84080 - 2.52614i) q^{13} +(-0.0974140 - 1.67253i) q^{14} +(-0.286803 - 0.957990i) q^{16} +(-4.83704 - 1.76054i) q^{17} +(2.66092 - 0.968497i) q^{19} +(-2.40060 - 3.22457i) q^{20} +(0.0131573 - 0.00311835i) q^{22} +(0.253001 + 0.586523i) q^{23} +(-9.32465 - 6.13292i) q^{25} +(-2.29854 + 3.98119i) q^{26} +(0.837684 + 1.45091i) q^{28} +(0.262835 - 4.51270i) q^{29} +(-9.48738 - 1.10892i) q^{31} +(0.686242 + 0.727374i) q^{32} +(5.11267 - 0.597585i) q^{34} +(5.15935 + 4.32920i) q^{35} +(6.03321 - 5.06246i) q^{37} +(-1.94323 + 2.05970i) q^{38} +(3.59244 + 1.80419i) q^{40} +(-6.30010 - 3.16403i) q^{41} +(4.34864 - 4.60929i) q^{43} +(-0.0103583 + 0.00869166i) q^{44} +(-0.489321 - 0.410589i) q^{46} +(11.3730 - 1.32931i) q^{47} +(2.87751 + 3.04998i) q^{49} +(11.0853 + 1.29568i) q^{50} +(0.267296 - 4.58930i) q^{52} +(2.96943 + 5.14321i) q^{53} +(-0.0271791 + 0.0470757i) q^{55} +(-1.39975 - 0.920630i) q^{56} +(1.79042 + 4.15065i) q^{58} +(2.42002 - 0.573555i) q^{59} +(1.39556 + 1.87456i) q^{61} +(8.97591 - 3.26697i) q^{62} +(-0.939693 - 0.342020i) q^{64} +(-5.30025 - 17.7041i) q^{65} +(0.232730 + 3.99581i) q^{67} +(-4.30065 + 2.82858i) q^{68} +(-6.55350 - 1.55321i) q^{70} +(1.01165 + 5.73737i) q^{71} +(1.45371 - 8.24441i) q^{73} +(-3.11944 + 7.23168i) q^{74} +(0.812139 - 2.71273i) q^{76} +(0.0135280 - 0.0181713i) q^{77} +(-4.39144 + 2.20546i) q^{79} -4.02004 q^{80} +7.04998 q^{82} +(-13.5542 + 6.80719i) q^{83} +(-12.3570 + 16.5984i) q^{85} +(-1.81744 + 6.07068i) q^{86} +(0.00535572 - 0.0124160i) q^{88} +(-1.34899 + 7.65049i) q^{89} +(1.33740 + 7.58479i) q^{91} +(0.621545 + 0.147309i) q^{92} +(-9.56667 + 6.29210i) q^{94} +(-0.661893 - 11.3643i) q^{95} +(0.923670 + 3.08527i) q^{97} +(-3.94027 - 1.43414i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 90 q - 18 q^{13} + 9 q^{20} - 27 q^{23} - 18 q^{25} + 27 q^{26} - 18 q^{28} + 27 q^{29} + 54 q^{31} + 27 q^{35} + 18 q^{38} + 9 q^{41} - 36 q^{43} + 18 q^{46} + 27 q^{47} + 36 q^{52} + 27 q^{53} - 54 q^{55}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/486\mathbb{Z}\right)^\times\).

\(n\) \(245\)
\(\chi(n)\) \(e\left(\frac{23}{27}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.893633 + 0.448799i −0.631894 + 0.317349i
\(3\) 0 0
\(4\) 0.597159 − 0.802123i 0.298579 − 0.401062i
\(5\) 1.15296 − 3.85116i 0.515620 − 1.72229i −0.161920 − 0.986804i \(-0.551769\pi\)
0.677540 − 0.735486i \(-0.263046\pi\)
\(6\) 0 0
\(7\) −0.663579 + 1.53835i −0.250809 + 0.581441i −0.996382 − 0.0849824i \(-0.972917\pi\)
0.745573 + 0.666424i \(0.232176\pi\)
\(8\) −0.173648 + 0.984808i −0.0613939 + 0.348182i
\(9\) 0 0
\(10\) 0.698073 + 3.95897i 0.220750 + 1.25194i
\(11\) −0.0131573 − 0.00311835i −0.00396709 − 0.000940217i 0.228632 − 0.973513i \(-0.426575\pi\)
−0.232599 + 0.972573i \(0.574723\pi\)
\(12\) 0 0
\(13\) 3.84080 − 2.52614i 1.06525 − 0.700624i 0.109254 − 0.994014i \(-0.465154\pi\)
0.955993 + 0.293390i \(0.0947834\pi\)
\(14\) −0.0974140 − 1.67253i −0.0260350 − 0.447003i
\(15\) 0 0
\(16\) −0.286803 − 0.957990i −0.0717008 − 0.239497i
\(17\) −4.83704 − 1.76054i −1.17315 − 0.426993i −0.319375 − 0.947628i \(-0.603473\pi\)
−0.853779 + 0.520635i \(0.825695\pi\)
\(18\) 0 0
\(19\) 2.66092 − 0.968497i 0.610457 − 0.222188i −0.0182459 − 0.999834i \(-0.505808\pi\)
0.628703 + 0.777645i \(0.283586\pi\)
\(20\) −2.40060 − 3.22457i −0.536791 − 0.721035i
\(21\) 0 0
\(22\) 0.0131573 − 0.00311835i 0.00280516 − 0.000664834i
\(23\) 0.253001 + 0.586523i 0.0527544 + 0.122298i 0.942555 − 0.334051i \(-0.108416\pi\)
−0.889801 + 0.456350i \(0.849157\pi\)
\(24\) 0 0
\(25\) −9.32465 − 6.13292i −1.86493 − 1.22658i
\(26\) −2.29854 + 3.98119i −0.450781 + 0.780775i
\(27\) 0 0
\(28\) 0.837684 + 1.45091i 0.158307 + 0.274196i
\(29\) 0.262835 − 4.51270i 0.0488072 − 0.837987i −0.881722 − 0.471769i \(-0.843616\pi\)
0.930529 − 0.366218i \(-0.119347\pi\)
\(30\) 0 0
\(31\) −9.48738 − 1.10892i −1.70398 − 0.199167i −0.792321 − 0.610105i \(-0.791127\pi\)
−0.911663 + 0.410938i \(0.865201\pi\)
\(32\) 0.686242 + 0.727374i 0.121312 + 0.128583i
\(33\) 0 0
\(34\) 5.11267 − 0.597585i 0.876815 − 0.102485i
\(35\) 5.15935 + 4.32920i 0.872088 + 0.731769i
\(36\) 0 0
\(37\) 6.03321 − 5.06246i 0.991853 − 0.832264i 0.00601844 − 0.999982i \(-0.498084\pi\)
0.985835 + 0.167718i \(0.0536398\pi\)
\(38\) −1.94323 + 2.05970i −0.315233 + 0.334127i
\(39\) 0 0
\(40\) 3.59244 + 1.80419i 0.568015 + 0.285268i
\(41\) −6.30010 − 3.16403i −0.983910 − 0.494138i −0.117285 − 0.993098i \(-0.537419\pi\)
−0.866625 + 0.498960i \(0.833715\pi\)
\(42\) 0 0
\(43\) 4.34864 − 4.60929i 0.663162 − 0.702911i −0.305248 − 0.952273i \(-0.598739\pi\)
0.968410 + 0.249362i \(0.0802209\pi\)
\(44\) −0.0103583 + 0.00869166i −0.00156158 + 0.00131032i
\(45\) 0 0
\(46\) −0.489321 − 0.410589i −0.0721465 − 0.0605381i
\(47\) 11.3730 − 1.32931i 1.65892 − 0.193900i 0.765598 − 0.643319i \(-0.222443\pi\)
0.893321 + 0.449419i \(0.148369\pi\)
\(48\) 0 0
\(49\) 2.87751 + 3.04998i 0.411073 + 0.435712i
\(50\) 11.0853 + 1.29568i 1.56769 + 0.183237i
\(51\) 0 0
\(52\) 0.267296 − 4.58930i 0.0370673 − 0.636421i
\(53\) 2.96943 + 5.14321i 0.407883 + 0.706474i 0.994652 − 0.103280i \(-0.0329339\pi\)
−0.586769 + 0.809754i \(0.699601\pi\)
\(54\) 0 0
\(55\) −0.0271791 + 0.0470757i −0.00366484 + 0.00634768i
\(56\) −1.39975 − 0.920630i −0.187049 − 0.123024i
\(57\) 0 0
\(58\) 1.79042 + 4.15065i 0.235093 + 0.545007i
\(59\) 2.42002 − 0.573555i 0.315060 − 0.0746705i −0.0700438 − 0.997544i \(-0.522314\pi\)
0.385103 + 0.922873i \(0.374166\pi\)
\(60\) 0 0
\(61\) 1.39556 + 1.87456i 0.178683 + 0.240013i 0.882437 − 0.470431i \(-0.155902\pi\)
−0.703754 + 0.710444i \(0.748494\pi\)
\(62\) 8.97591 − 3.26697i 1.13994 − 0.414905i
\(63\) 0 0
\(64\) −0.939693 − 0.342020i −0.117462 − 0.0427525i
\(65\) −5.30025 − 17.7041i −0.657415 − 2.19592i
\(66\) 0 0
\(67\) 0.232730 + 3.99581i 0.0284324 + 0.488166i 0.982226 + 0.187703i \(0.0601042\pi\)
−0.953793 + 0.300463i \(0.902859\pi\)
\(68\) −4.30065 + 2.82858i −0.521530 + 0.343016i
\(69\) 0 0
\(70\) −6.55350 − 1.55321i −0.783293 − 0.185644i
\(71\) 1.01165 + 5.73737i 0.120061 + 0.680901i 0.984119 + 0.177509i \(0.0568038\pi\)
−0.864058 + 0.503392i \(0.832085\pi\)
\(72\) 0 0
\(73\) 1.45371 − 8.24441i 0.170144 − 0.964935i −0.773457 − 0.633849i \(-0.781474\pi\)
0.943601 − 0.331086i \(-0.107415\pi\)
\(74\) −3.11944 + 7.23168i −0.362628 + 0.840666i
\(75\) 0 0
\(76\) 0.812139 − 2.71273i 0.0931587 − 0.311172i
\(77\) 0.0135280 − 0.0181713i 0.00154166 − 0.00207081i
\(78\) 0 0
\(79\) −4.39144 + 2.20546i −0.494076 + 0.248134i −0.678348 − 0.734741i \(-0.737304\pi\)
0.184272 + 0.982875i \(0.441007\pi\)
\(80\) −4.02004 −0.449454
\(81\) 0 0
\(82\) 7.04998 0.778541
\(83\) −13.5542 + 6.80719i −1.48777 + 0.747186i −0.992579 − 0.121598i \(-0.961198\pi\)
−0.495191 + 0.868784i \(0.664902\pi\)
\(84\) 0 0
\(85\) −12.3570 + 16.5984i −1.34031 + 1.80035i
\(86\) −1.81744 + 6.07068i −0.195980 + 0.654619i
\(87\) 0 0
\(88\) 0.00535572 − 0.0124160i 0.000570922 − 0.00132355i
\(89\) −1.34899 + 7.65049i −0.142992 + 0.810950i 0.825965 + 0.563722i \(0.190631\pi\)
−0.968957 + 0.247229i \(0.920480\pi\)
\(90\) 0 0
\(91\) 1.33740 + 7.58479i 0.140198 + 0.795102i
\(92\) 0.621545 + 0.147309i 0.0648006 + 0.0153580i
\(93\) 0 0
\(94\) −9.56667 + 6.29210i −0.986726 + 0.648980i
\(95\) −0.661893 − 11.3643i −0.0679088 − 1.16595i
\(96\) 0 0
\(97\) 0.923670 + 3.08527i 0.0937845 + 0.313262i 0.991990 − 0.126313i \(-0.0403143\pi\)
−0.898206 + 0.439575i \(0.855129\pi\)
\(98\) −3.94027 − 1.43414i −0.398027 − 0.144870i
\(99\) 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 486.2.g.b.73.5 90
3.2 odd 2 162.2.g.b.25.2 yes 90
81.13 even 27 inner 486.2.g.b.253.5 90
81.68 odd 54 162.2.g.b.13.2 ✓ 90
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.b.13.2 ✓ 90 81.68 odd 54
162.2.g.b.25.2 yes 90 3.2 odd 2
486.2.g.b.73.5 90 1.1 even 1 trivial
486.2.g.b.253.5 90 81.13 even 27 inner