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Properties

Label 486.2.g.b.253.5
Level $486$
Weight $2$
Character 486.253
Analytic conductor $3.881$
Analytic rank $0$
Dimension $90$
Inner twists $2$

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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 253.5
Character \(\chi\) \(=\) 486.253
Dual form 486.2.g.b.73.5

$q$-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 \(\left(\mathbb{Z}/486\mathbb{Z}\right)^\times\).

\(n\) \(245\)
\(\chi(n)\) \(e\left(\frac{4}{27}\right)\)

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 0.677540 + 0.735486i \(0.263046\pi\)
−0.161920 + 0.986804i \(0.551769\pi\)
\(6\) 0 0
\(7\) −0.663579 − 1.53835i −0.250809 − 0.581441i 0.745573 − 0.666424i \(-0.232176\pi\)
−0.996382 + 0.0849824i \(0.972917\pi\)
\(8\) −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 −0.232599 − 0.972573i \(-0.574723\pi\)
0.228632 + 0.973513i \(0.426575\pi\)
\(12\) 0 0
\(13\) 3.84080 + 2.52614i 1.06525 + 0.700624i 0.955993 − 0.293390i \(-0.0947834\pi\)
0.109254 + 0.994014i \(0.465154\pi\)
\(14\) −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 −0.853779 − 0.520635i \(-0.825695\pi\)
−0.319375 + 0.947628i \(0.603473\pi\)
\(18\) 0 0
\(19\) 2.66092 + 0.968497i 0.610457 + 0.222188i 0.628703 − 0.777645i \(-0.283586\pi\)
−0.0182459 + 0.999834i \(0.505808\pi\)
\(20\) −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 −0.889801 − 0.456350i \(-0.849157\pi\)
0.942555 + 0.334051i \(0.108416\pi\)
\(24\) 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 0.930529 + 0.366218i \(0.119347\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(30\) 0 0
\(31\) −9.48738 + 1.10892i −1.70398 + 0.199167i −0.911663 − 0.410938i \(-0.865201\pi\)
−0.792321 + 0.610105i \(0.791127\pi\)
\(32\) 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 0.985835 − 0.167718i \(-0.0536398\pi\)
0.00601844 + 0.999982i \(0.498084\pi\)
\(38\) −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 −0.866625 − 0.498960i \(-0.833715\pi\)
−0.117285 + 0.993098i \(0.537419\pi\)
\(42\) 0 0
\(43\) 4.34864 + 4.60929i 0.663162 + 0.702911i 0.968410 − 0.249362i \(-0.0802209\pi\)
−0.305248 + 0.952273i \(0.598739\pi\)
\(44\) −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 0.893321 − 0.449419i \(-0.148369\pi\)
0.765598 + 0.643319i \(0.222443\pi\)
\(48\) 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 −0.586769 − 0.809754i \(-0.699601\pi\)
0.994652 + 0.103280i \(0.0329339\pi\)
\(54\) 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 0.385103 − 0.922873i \(-0.374166\pi\)
−0.0700438 + 0.997544i \(0.522314\pi\)
\(60\) 0 0
\(61\) 1.39556 − 1.87456i 0.178683 − 0.240013i −0.703754 − 0.710444i \(-0.748494\pi\)
0.882437 + 0.470431i \(0.155902\pi\)
\(62\) 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 −0.953793 − 0.300463i \(-0.902859\pi\)
0.982226 − 0.187703i \(-0.0601042\pi\)
\(68\) −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 −0.864058 − 0.503392i \(-0.832085\pi\)
0.984119 − 0.177509i \(-0.0568038\pi\)
\(72\) 0 0
\(73\) 1.45371 + 8.24441i 0.170144 + 0.964935i 0.943601 + 0.331086i \(0.107415\pi\)
−0.773457 + 0.633849i \(0.781474\pi\)
\(74\) −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 0.184272 − 0.982875i \(-0.441007\pi\)
−0.678348 + 0.734741i \(0.737304\pi\)
\(80\) −4.02004 −0.449454
\(81\) 0 0
\(82\) 7.04998 0.778541
\(83\) −13.5542 − 6.80719i −1.48777 − 0.747186i −0.495191 − 0.868784i \(-0.664902\pi\)
−0.992579 + 0.121598i \(0.961198\pi\)
\(84\) 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 −0.968957 − 0.247229i \(-0.920480\pi\)
0.825965 − 0.563722i \(-0.190631\pi\)
\(90\) 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 −0.898206 − 0.439575i \(-0.855129\pi\)
0.991990 + 0.126313i \(0.0403143\pi\)
\(98\) −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.253.5 90
3.2 odd 2 162.2.g.b.13.2 ✓ 90
81.25 even 27 inner 486.2.g.b.73.5 90
81.56 odd 54 162.2.g.b.25.2 yes 90
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.b.13.2 ✓ 90 3.2 odd 2
162.2.g.b.25.2 yes 90 81.56 odd 54
486.2.g.b.73.5 90 81.25 even 27 inner
486.2.g.b.253.5 90 1.1 even 1 trivial