Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [162,6,Mod(1,162)] 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("162.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 162.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-12,0,48,-54,0,132] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.9821788097\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.125628.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 63x + 159 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 18)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-8.54724\) of defining polynomial
Character \(\chi\) \(=\) 162.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +16.0000 q^{4} -78.4839 q^{5} +221.131 q^{7} -64.0000 q^{8} +313.936 q^{10} -230.289 q^{11} -771.586 q^{13} -884.524 q^{14} +256.000 q^{16} +769.880 q^{17} -383.367 q^{19} -1255.74 q^{20} +921.155 q^{22} -386.356 q^{23} +3034.73 q^{25} +3086.34 q^{26} +3538.10 q^{28} +789.767 q^{29} +3219.93 q^{31} -1024.00 q^{32} -3079.52 q^{34} -17355.2 q^{35} +2465.33 q^{37} +1533.47 q^{38} +5022.97 q^{40} +9243.46 q^{41} +10631.5 q^{43} -3684.62 q^{44} +1545.42 q^{46} -1952.06 q^{47} +32091.9 q^{49} -12138.9 q^{50} -12345.4 q^{52} +32589.2 q^{53} +18074.0 q^{55} -14152.4 q^{56} -3159.07 q^{58} +23832.1 q^{59} +37608.8 q^{61} -12879.7 q^{62} +4096.00 q^{64} +60557.1 q^{65} -23051.1 q^{67} +12318.1 q^{68} +69421.0 q^{70} +66050.4 q^{71} +65130.0 q^{73} -9861.33 q^{74} -6133.87 q^{76} -50924.0 q^{77} -70867.4 q^{79} -20091.9 q^{80} -36973.8 q^{82} -55287.0 q^{83} -60423.2 q^{85} -42526.0 q^{86} +14738.5 q^{88} -10598.6 q^{89} -170622. q^{91} -6181.69 q^{92} +7808.26 q^{94} +30088.1 q^{95} -82819.8 q^{97} -128368. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 12 q^{2} + 48 q^{4} - 54 q^{5} + 132 q^{7} - 192 q^{8} + 216 q^{10} - 315 q^{11} + 744 q^{13} - 528 q^{14} + 768 q^{16} - 1449 q^{17} + 1131 q^{19} - 864 q^{20} + 1260 q^{22} - 3168 q^{23} + 2883 q^{25}+ \cdots - 211428 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) −78.4839 −1.40396 −0.701982 − 0.712195i \(-0.747701\pi\)
−0.701982 + 0.712195i \(0.747701\pi\)
\(6\) 0 0
\(7\) 221.131 1.70571 0.852854 − 0.522150i \(-0.174870\pi\)
0.852854 + 0.522150i \(0.174870\pi\)
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) 313.936 0.992752
\(11\) −230.289 −0.573840 −0.286920 − 0.957955i \(-0.592631\pi\)
−0.286920 + 0.957955i \(0.592631\pi\)
\(12\) 0 0
\(13\) −771.586 −1.26627 −0.633134 − 0.774042i \(-0.718232\pi\)
−0.633134 + 0.774042i \(0.718232\pi\)
\(14\) −884.524 −1.20612
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 769.880 0.646101 0.323051 − 0.946382i \(-0.395292\pi\)
0.323051 + 0.946382i \(0.395292\pi\)
\(18\) 0 0
\(19\) −383.367 −0.243630 −0.121815 − 0.992553i \(-0.538871\pi\)
−0.121815 + 0.992553i \(0.538871\pi\)
\(20\) −1255.74 −0.701982
\(21\) 0 0
\(22\) 921.155 0.405766
\(23\) −386.356 −0.152289 −0.0761444 − 0.997097i \(-0.524261\pi\)
−0.0761444 + 0.997097i \(0.524261\pi\)
\(24\) 0 0
\(25\) 3034.73 0.971114
\(26\) 3086.34 0.895387
\(27\) 0 0
\(28\) 3538.10 0.852854
\(29\) 789.767 0.174383 0.0871914 − 0.996192i \(-0.472211\pi\)
0.0871914 + 0.996192i \(0.472211\pi\)
\(30\) 0 0
\(31\) 3219.93 0.601787 0.300893 − 0.953658i \(-0.402715\pi\)
0.300893 + 0.953658i \(0.402715\pi\)
\(32\) −1024.00 −0.176777
\(33\) 0 0
\(34\) −3079.52 −0.456863
\(35\) −17355.2 −2.39475
\(36\) 0 0
\(37\) 2465.33 0.296054 0.148027 − 0.988983i \(-0.452708\pi\)
0.148027 + 0.988983i \(0.452708\pi\)
\(38\) 1533.47 0.172272
\(39\) 0 0
\(40\) 5022.97 0.496376
\(41\) 9243.46 0.858766 0.429383 − 0.903122i \(-0.358731\pi\)
0.429383 + 0.903122i \(0.358731\pi\)
\(42\) 0 0
\(43\) 10631.5 0.876847 0.438424 − 0.898769i \(-0.355537\pi\)
0.438424 + 0.898769i \(0.355537\pi\)
\(44\) −3684.62 −0.286920
\(45\) 0 0
\(46\) 1545.42 0.107684
\(47\) −1952.06 −0.128899 −0.0644495 − 0.997921i \(-0.520529\pi\)
−0.0644495 + 0.997921i \(0.520529\pi\)
\(48\) 0 0
\(49\) 32091.9 1.90944
\(50\) −12138.9 −0.686681
\(51\) 0 0
\(52\) −12345.4 −0.633134
\(53\) 32589.2 1.59362 0.796809 − 0.604231i \(-0.206520\pi\)
0.796809 + 0.604231i \(0.206520\pi\)
\(54\) 0 0
\(55\) 18074.0 0.805651
\(56\) −14152.4 −0.603059
\(57\) 0 0
\(58\) −3159.07 −0.123307
\(59\) 23832.1 0.891317 0.445659 − 0.895203i \(-0.352970\pi\)
0.445659 + 0.895203i \(0.352970\pi\)
\(60\) 0 0
\(61\) 37608.8 1.29409 0.647046 − 0.762451i \(-0.276004\pi\)
0.647046 + 0.762451i \(0.276004\pi\)
\(62\) −12879.7 −0.425528
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 60557.1 1.77779
\(66\) 0 0
\(67\) −23051.1 −0.627343 −0.313672 − 0.949531i \(-0.601559\pi\)
−0.313672 + 0.949531i \(0.601559\pi\)
\(68\) 12318.1 0.323051
\(69\) 0 0
\(70\) 69421.0 1.69335
\(71\) 66050.4 1.55500 0.777498 − 0.628885i \(-0.216489\pi\)
0.777498 + 0.628885i \(0.216489\pi\)
\(72\) 0 0
\(73\) 65130.0 1.43045 0.715227 − 0.698893i \(-0.246324\pi\)
0.715227 + 0.698893i \(0.246324\pi\)
\(74\) −9861.33 −0.209342
\(75\) 0 0
\(76\) −6133.87 −0.121815
\(77\) −50924.0 −0.978804
\(78\) 0 0
\(79\) −70867.4 −1.27755 −0.638776 − 0.769392i \(-0.720559\pi\)
−0.638776 + 0.769392i \(0.720559\pi\)
\(80\) −20091.9 −0.350991
\(81\) 0 0
\(82\) −36973.8 −0.607239
\(83\) −55287.0 −0.880903 −0.440451 − 0.897776i \(-0.645182\pi\)
−0.440451 + 0.897776i \(0.645182\pi\)
\(84\) 0 0
\(85\) −60423.2 −0.907103
\(86\) −42526.0 −0.620024
\(87\) 0 0
\(88\) 14738.5 0.202883
\(89\) −10598.6 −0.141831 −0.0709156 − 0.997482i \(-0.522592\pi\)
−0.0709156 + 0.997482i \(0.522592\pi\)
\(90\) 0 0
\(91\) −170622. −2.15988
\(92\) −6181.69 −0.0761444
\(93\) 0 0
\(94\) 7808.26 0.0911453
\(95\) 30088.1 0.342047
\(96\) 0 0
\(97\) −82819.8 −0.893727 −0.446864 − 0.894602i \(-0.647459\pi\)
−0.446864 + 0.894602i \(0.647459\pi\)
\(98\) −128368. −1.35018
\(99\) 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 162.6.a.i.1.1 3
3.2 odd 2 162.6.a.j.1.3 3
9.2 odd 6 18.6.c.b.13.1 yes 6
9.4 even 3 54.6.c.b.19.3 6
9.5 odd 6 18.6.c.b.7.1 ✓ 6
9.7 even 3 54.6.c.b.37.3 6
36.7 odd 6 432.6.i.b.145.3 6
36.11 even 6 144.6.i.b.49.3 6
36.23 even 6 144.6.i.b.97.3 6
36.31 odd 6 432.6.i.b.289.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.1 ✓ 6 9.5 odd 6
18.6.c.b.13.1 yes 6 9.2 odd 6
54.6.c.b.19.3 6 9.4 even 3
54.6.c.b.37.3 6 9.7 even 3
144.6.i.b.49.3 6 36.11 even 6
144.6.i.b.97.3 6 36.23 even 6
162.6.a.i.1.1 3 1.1 even 1 trivial
162.6.a.j.1.3 3 3.2 odd 2
432.6.i.b.145.3 6 36.7 odd 6
432.6.i.b.289.3 6 36.31 odd 6