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Properties

Label 162.4.c.j.109.1
Level $162$
Weight $4$
Character 162.109
Analytic conductor $9.558$
Analytic rank $0$
Dimension $4$
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] = [162,4,Mod(55,162)] 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("162.55"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,0,-8,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.55830942093\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 162.109
Dual form 162.4.c.j.55.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(0.401924 - 0.696152i) q^{5} +(1.19615 + 2.07180i) q^{7} -8.00000 q^{8} +1.60770 q^{10} +(29.7846 + 51.5885i) q^{11} +(-38.8731 + 67.3301i) q^{13} +(-2.39230 + 4.14359i) q^{14} +(-8.00000 - 13.8564i) q^{16} -2.84232 q^{17} -118.315 q^{19} +(1.60770 + 2.78461i) q^{20} +(-59.5692 + 103.177i) q^{22} +(55.3923 - 95.9423i) q^{23} +(62.1769 + 107.694i) q^{25} -155.492 q^{26} -9.56922 q^{28} +(62.8135 + 108.796i) q^{29} +(-31.5307 + 54.6128i) q^{31} +(16.0000 - 27.7128i) q^{32} +(-2.84232 - 4.92305i) q^{34} +1.92305 q^{35} -227.392 q^{37} +(-118.315 - 204.928i) q^{38} +(-3.21539 + 5.56922i) q^{40} +(162.315 - 281.138i) q^{41} +(136.412 + 236.272i) q^{43} -238.277 q^{44} +221.569 q^{46} +(-2.46926 - 4.27688i) q^{47} +(168.638 - 292.090i) q^{49} +(-124.354 + 215.387i) q^{50} +(-155.492 - 269.321i) q^{52} +598.908 q^{53} +47.8846 q^{55} +(-9.56922 - 16.5744i) q^{56} +(-125.627 + 217.592i) q^{58} +(-335.138 + 580.477i) q^{59} +(-232.265 - 402.295i) q^{61} -126.123 q^{62} +64.0000 q^{64} +(31.2480 + 54.1232i) q^{65} +(384.535 - 666.033i) q^{67} +(5.68465 - 9.84610i) q^{68} +(1.92305 + 3.33082i) q^{70} +611.569 q^{71} +923.831 q^{73} +(-227.392 - 393.855i) q^{74} +(236.631 - 409.856i) q^{76} +(-71.2539 + 123.415i) q^{77} +(-19.9038 - 34.4744i) q^{79} -12.8616 q^{80} +649.261 q^{82} +(221.885 + 384.315i) q^{83} +(-1.14240 + 1.97869i) q^{85} +(-272.823 + 472.543i) q^{86} +(-238.277 - 412.708i) q^{88} +78.4576 q^{89} -185.992 q^{91} +(221.569 + 383.769i) q^{92} +(4.93851 - 8.55376i) q^{94} +(-47.5538 + 82.3655i) q^{95} +(45.5077 + 78.8217i) q^{97} +674.554 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} + 12 q^{5} - 16 q^{7} - 32 q^{8} + 48 q^{10} + 36 q^{11} - 10 q^{13} + 32 q^{14} - 32 q^{16} - 240 q^{17} - 16 q^{19} + 48 q^{20} - 72 q^{22} + 180 q^{23} + 124 q^{25} - 40 q^{26}+ \cdots + 1368 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 + 1.73205i 0.353553 + 0.612372i
\(3\) 0 0
\(4\) −2.00000 + 3.46410i −0.250000 + 0.433013i
\(5\) 0.401924 − 0.696152i 0.0359492 − 0.0622658i −0.847491 − 0.530810i \(-0.821888\pi\)
0.883440 + 0.468544i \(0.155221\pi\)
\(6\) 0 0
\(7\) 1.19615 + 2.07180i 0.0645862 + 0.111867i 0.896510 − 0.443023i \(-0.146094\pi\)
−0.831924 + 0.554889i \(0.812761\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 1.60770 0.0508398
\(11\) 29.7846 + 51.5885i 0.816400 + 1.41405i 0.908318 + 0.418279i \(0.137367\pi\)
−0.0919186 + 0.995767i \(0.529300\pi\)
\(12\) 0 0
\(13\) −38.8731 + 67.3301i −0.829342 + 1.43646i 0.0692128 + 0.997602i \(0.477951\pi\)
−0.898555 + 0.438861i \(0.855382\pi\)
\(14\) −2.39230 + 4.14359i −0.0456693 + 0.0791016i
\(15\) 0 0
\(16\) −8.00000 − 13.8564i −0.125000 − 0.216506i
\(17\) −2.84232 −0.0405509 −0.0202754 − 0.999794i \(-0.506454\pi\)
−0.0202754 + 0.999794i \(0.506454\pi\)
\(18\) 0 0
\(19\) −118.315 −1.42860 −0.714300 − 0.699840i \(-0.753255\pi\)
−0.714300 + 0.699840i \(0.753255\pi\)
\(20\) 1.60770 + 2.78461i 0.0179746 + 0.0311329i
\(21\) 0 0
\(22\) −59.5692 + 103.177i −0.577282 + 0.999881i
\(23\) 55.3923 − 95.9423i 0.502178 − 0.869798i −0.497819 − 0.867281i \(-0.665866\pi\)
0.999997 − 0.00251677i \(-0.000801112\pi\)
\(24\) 0 0
\(25\) 62.1769 + 107.694i 0.497415 + 0.861549i
\(26\) −155.492 −1.17287
\(27\) 0 0
\(28\) −9.56922 −0.0645862
\(29\) 62.8135 + 108.796i 0.402213 + 0.696653i 0.993993 − 0.109447i \(-0.0349079\pi\)
−0.591780 + 0.806100i \(0.701575\pi\)
\(30\) 0 0
\(31\) −31.5307 + 54.6128i −0.182680 + 0.316412i −0.942792 − 0.333380i \(-0.891811\pi\)
0.760112 + 0.649792i \(0.225144\pi\)
\(32\) 16.0000 − 27.7128i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −2.84232 − 4.92305i −0.0143369 − 0.0248322i
\(35\) 1.92305 0.00928727
\(36\) 0 0
\(37\) −227.392 −1.01035 −0.505177 − 0.863016i \(-0.668573\pi\)
−0.505177 + 0.863016i \(0.668573\pi\)
\(38\) −118.315 − 204.928i −0.505086 − 0.874835i
\(39\) 0 0
\(40\) −3.21539 + 5.56922i −0.0127099 + 0.0220143i
\(41\) 162.315 − 281.138i 0.618278 − 1.07089i −0.371522 − 0.928424i \(-0.621164\pi\)
0.989800 − 0.142465i \(-0.0455028\pi\)
\(42\) 0 0
\(43\) 136.412 + 236.272i 0.483781 + 0.837933i 0.999826 − 0.0186284i \(-0.00592994\pi\)
−0.516046 + 0.856561i \(0.672597\pi\)
\(44\) −238.277 −0.816400
\(45\) 0 0
\(46\) 221.569 0.710187
\(47\) −2.46926 − 4.27688i −0.00766336 − 0.0132733i 0.862168 − 0.506622i \(-0.169106\pi\)
−0.869832 + 0.493349i \(0.835773\pi\)
\(48\) 0 0
\(49\) 168.638 − 292.090i 0.491657 − 0.851575i
\(50\) −124.354 + 215.387i −0.351726 + 0.609207i
\(51\) 0 0
\(52\) −155.492 − 269.321i −0.414671 − 0.718231i
\(53\) 598.908 1.55219 0.776097 − 0.630614i \(-0.217197\pi\)
0.776097 + 0.630614i \(0.217197\pi\)
\(54\) 0 0
\(55\) 47.8846 0.117396
\(56\) −9.56922 − 16.5744i −0.0228347 − 0.0395508i
\(57\) 0 0
\(58\) −125.627 + 217.592i −0.284407 + 0.492608i
\(59\) −335.138 + 580.477i −0.739514 + 1.28088i 0.213201 + 0.977008i \(0.431611\pi\)
−0.952714 + 0.303867i \(0.901722\pi\)
\(60\) 0 0
\(61\) −232.265 − 402.295i −0.487517 − 0.844404i 0.512380 − 0.858759i \(-0.328764\pi\)
−0.999897 + 0.0143547i \(0.995431\pi\)
\(62\) −126.123 −0.258349
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 31.2480 + 54.1232i 0.0596283 + 0.103279i
\(66\) 0 0
\(67\) 384.535 − 666.033i 0.701170 − 1.21446i −0.266886 − 0.963728i \(-0.585995\pi\)
0.968056 − 0.250734i \(-0.0806718\pi\)
\(68\) 5.68465 − 9.84610i 0.0101377 − 0.0175590i
\(69\) 0 0
\(70\) 1.92305 + 3.33082i 0.00328355 + 0.00568727i
\(71\) 611.569 1.02225 0.511126 − 0.859506i \(-0.329228\pi\)
0.511126 + 0.859506i \(0.329228\pi\)
\(72\) 0 0
\(73\) 923.831 1.48118 0.740590 − 0.671957i \(-0.234546\pi\)
0.740590 + 0.671957i \(0.234546\pi\)
\(74\) −227.392 − 393.855i −0.357214 − 0.618712i
\(75\) 0 0
\(76\) 236.631 − 409.856i 0.357150 − 0.618602i
\(77\) −71.2539 + 123.415i −0.105456 + 0.182656i
\(78\) 0 0
\(79\) −19.9038 − 34.4744i −0.0283462 − 0.0490971i 0.851504 − 0.524348i \(-0.175691\pi\)
−0.879851 + 0.475251i \(0.842357\pi\)
\(80\) −12.8616 −0.0179746
\(81\) 0 0
\(82\) 649.261 0.874377
\(83\) 221.885 + 384.315i 0.293434 + 0.508242i 0.974619 − 0.223869i \(-0.0718687\pi\)
−0.681186 + 0.732111i \(0.738535\pi\)
\(84\) 0 0
\(85\) −1.14240 + 1.97869i −0.00145777 + 0.00252493i
\(86\) −272.823 + 472.543i −0.342085 + 0.592508i
\(87\) 0 0
\(88\) −238.277 − 412.708i −0.288641 − 0.499941i
\(89\) 78.4576 0.0934437 0.0467218 − 0.998908i \(-0.485123\pi\)
0.0467218 + 0.998908i \(0.485123\pi\)
\(90\) 0 0
\(91\) −185.992 −0.214256
\(92\) 221.569 + 383.769i 0.251089 + 0.434899i
\(93\) 0 0
\(94\) 4.93851 − 8.55376i 0.00541882 − 0.00938566i
\(95\) −47.5538 + 82.3655i −0.0513570 + 0.0889529i
\(96\) 0 0
\(97\) 45.5077 + 78.8217i 0.0476352 + 0.0825065i 0.888860 − 0.458179i \(-0.151498\pi\)
−0.841225 + 0.540686i \(0.818165\pi\)
\(98\) 674.554 0.695308
\(99\) 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 162.4.c.j.109.1 4
3.2 odd 2 162.4.c.i.109.2 4
9.2 odd 6 162.4.c.i.55.2 4
9.4 even 3 162.4.a.e.1.2 ✓ 2
9.5 odd 6 162.4.a.h.1.1 yes 2
9.7 even 3 inner 162.4.c.j.55.1 4
36.23 even 6 1296.4.a.s.1.1 2
36.31 odd 6 1296.4.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.4.a.e.1.2 ✓ 2 9.4 even 3
162.4.a.h.1.1 yes 2 9.5 odd 6
162.4.c.i.55.2 4 9.2 odd 6
162.4.c.i.109.2 4 3.2 odd 2
162.4.c.j.55.1 4 9.7 even 3 inner
162.4.c.j.109.1 4 1.1 even 1 trivial
1296.4.a.j.1.2 2 36.31 odd 6
1296.4.a.s.1.1 2 36.23 even 6