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Properties

Label 162.4.c.j.55.1
Level $162$
Weight $4$
Character 162.55
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 55.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 162.55
Dual form 162.4.c.j.109.1

$q$-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 \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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 0.883440 − 0.468544i \(-0.155221\pi\)
−0.847491 + 0.530810i \(0.821888\pi\)
\(6\) 0 0
\(7\) 1.19615 − 2.07180i 0.0645862 − 0.111867i −0.831924 − 0.554889i \(-0.812761\pi\)
0.896510 + 0.443023i \(0.146094\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 1.60770 0.0508398
\(11\) 29.7846 − 51.5885i 0.816400 − 1.41405i −0.0919186 − 0.995767i \(-0.529300\pi\)
0.908318 − 0.418279i \(-0.137367\pi\)
\(12\) 0 0
\(13\) −38.8731 − 67.3301i −0.829342 − 1.43646i −0.898555 − 0.438861i \(-0.855382\pi\)
0.0692128 − 0.997602i \(-0.477951\pi\)
\(14\) −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 0.999997 + 0.00251677i \(0.000801112\pi\)
−0.497819 + 0.867281i \(0.665866\pi\)
\(24\) 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 −0.591780 − 0.806100i \(-0.701575\pi\)
0.993993 + 0.109447i \(0.0349079\pi\)
\(30\) 0 0
\(31\) −31.5307 − 54.6128i −0.182680 − 0.316412i 0.760112 − 0.649792i \(-0.225144\pi\)
−0.942792 + 0.333380i \(0.891811\pi\)
\(32\) 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 0.989800 + 0.142465i \(0.0455028\pi\)
−0.371522 + 0.928424i \(0.621164\pi\)
\(42\) 0 0
\(43\) 136.412 − 236.272i 0.483781 − 0.837933i −0.516046 − 0.856561i \(-0.672597\pi\)
0.999826 + 0.0186284i \(0.00592994\pi\)
\(44\) −238.277 −0.816400
\(45\) 0 0
\(46\) 221.569 0.710187
\(47\) −2.46926 + 4.27688i −0.00766336 + 0.0132733i −0.869832 − 0.493349i \(-0.835773\pi\)
0.862168 + 0.506622i \(0.169106\pi\)
\(48\) 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 −0.952714 − 0.303867i \(-0.901722\pi\)
0.213201 − 0.977008i \(-0.431611\pi\)
\(60\) 0 0
\(61\) −232.265 + 402.295i −0.487517 + 0.844404i −0.999897 − 0.0143547i \(-0.995431\pi\)
0.512380 + 0.858759i \(0.328764\pi\)
\(62\) −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 0.968056 + 0.250734i \(0.0806718\pi\)
−0.266886 + 0.963728i \(0.585995\pi\)
\(68\) 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 −0.879851 − 0.475251i \(-0.842357\pi\)
0.851504 + 0.524348i \(0.175691\pi\)
\(80\) −12.8616 −0.0179746
\(81\) 0 0
\(82\) 649.261 0.874377
\(83\) 221.885 − 384.315i 0.293434 − 0.508242i −0.681186 − 0.732111i \(-0.738535\pi\)
0.974619 + 0.223869i \(0.0718687\pi\)
\(84\) 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 −0.841225 − 0.540686i \(-0.818165\pi\)
0.888860 + 0.458179i \(0.151498\pi\)
\(98\) 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.55.1 4
3.2 odd 2 162.4.c.i.55.2 4
9.2 odd 6 162.4.a.h.1.1 yes 2
9.4 even 3 inner 162.4.c.j.109.1 4
9.5 odd 6 162.4.c.i.109.2 4
9.7 even 3 162.4.a.e.1.2 ✓ 2
36.7 odd 6 1296.4.a.j.1.2 2
36.11 even 6 1296.4.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.4.a.e.1.2 ✓ 2 9.7 even 3
162.4.a.h.1.1 yes 2 9.2 odd 6
162.4.c.i.55.2 4 3.2 odd 2
162.4.c.i.109.2 4 9.5 odd 6
162.4.c.j.55.1 4 1.1 even 1 trivial
162.4.c.j.109.1 4 9.4 even 3 inner
1296.4.a.j.1.2 2 36.7 odd 6
1296.4.a.s.1.1 2 36.11 even 6