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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,6,Mod(361,588)] 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("588.361"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 588.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-36,0,0,0,0,0,-324,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 91x^{6} - 2x^{5} + 5907x^{4} - 304x^{3} + 167650x^{2} + 161744x + 3378244 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.4
Root \(3.18047 - 5.50873i\) of defining polynomial
Character \(\chi\) \(=\) 588.361
Dual form 588.6.i.p.373.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.50000 + 7.79423i) q^{3} +(38.9057 + 67.3866i) q^{5} +(-40.5000 - 70.1481i) q^{9} +(-29.1331 + 50.4600i) q^{11} +792.502 q^{13} -700.302 q^{15} +(232.320 - 402.391i) q^{17} +(492.278 + 852.651i) q^{19} +(317.971 + 550.742i) q^{23} +(-1464.80 + 2537.11i) q^{25} +729.000 q^{27} +6396.56 q^{29} +(4214.31 - 7299.40i) q^{31} +(-262.198 - 454.140i) q^{33} +(702.479 + 1216.73i) q^{37} +(-3566.26 + 6176.94i) q^{39} +19074.6 q^{41} +6023.14 q^{43} +(3151.36 - 5458.32i) q^{45} +(-940.471 - 1628.94i) q^{47} +(2090.88 + 3621.52i) q^{51} +(-2627.70 + 4551.30i) q^{53} -4533.77 q^{55} -8861.01 q^{57} +(-1556.38 + 2695.73i) q^{59} +(11047.3 + 19134.4i) q^{61} +(30832.8 + 53404.0i) q^{65} +(27355.9 - 47381.8i) q^{67} -5723.48 q^{69} -15211.7 q^{71} +(25038.3 - 43367.6i) q^{73} +(-13183.2 - 22834.0i) q^{75} +(-571.083 - 989.144i) q^{79} +(-3280.50 + 5681.99i) q^{81} -122870. q^{83} +36154.3 q^{85} +(-28784.5 + 49856.2i) q^{87} +(-35246.3 - 61048.4i) q^{89} +(37928.8 + 65694.6i) q^{93} +(-38304.8 + 66345.9i) q^{95} -132975. q^{97} +4719.56 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 36 q^{3} - 324 q^{9} + 1872 q^{17} + 1728 q^{19} + 3648 q^{23} + 3996 q^{25} + 5832 q^{27} - 2496 q^{29} + 3888 q^{31} + 12032 q^{37} + 18144 q^{41} - 4256 q^{43} + 19872 q^{47} + 16848 q^{51} + 22248 q^{53}+ \cdots - 641088 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −4.50000 + 7.79423i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) 38.9057 + 67.3866i 0.695966 + 1.20545i 0.969854 + 0.243687i \(0.0783569\pi\)
−0.273888 + 0.961762i \(0.588310\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −40.5000 − 70.1481i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) −29.1331 + 50.4600i −0.0725947 + 0.125738i −0.900038 − 0.435812i \(-0.856461\pi\)
0.827443 + 0.561550i \(0.189795\pi\)
\(12\) 0 0
\(13\) 792.502 1.30059 0.650297 − 0.759680i \(-0.274644\pi\)
0.650297 + 0.759680i \(0.274644\pi\)
\(14\) 0 0
\(15\) −700.302 −0.803632
\(16\) 0 0
\(17\) 232.320 − 402.391i 0.194969 − 0.337696i −0.751922 − 0.659253i \(-0.770873\pi\)
0.946890 + 0.321557i \(0.104206\pi\)
\(18\) 0 0
\(19\) 492.278 + 852.651i 0.312843 + 0.541860i 0.978977 − 0.203972i \(-0.0653853\pi\)
−0.666134 + 0.745832i \(0.732052\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 317.971 + 550.742i 0.125334 + 0.217084i 0.921863 − 0.387515i \(-0.126667\pi\)
−0.796530 + 0.604600i \(0.793333\pi\)
\(24\) 0 0
\(25\) −1464.80 + 2537.11i −0.468737 + 0.811876i
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) 6396.56 1.41238 0.706189 − 0.708023i \(-0.250413\pi\)
0.706189 + 0.708023i \(0.250413\pi\)
\(30\) 0 0
\(31\) 4214.31 − 7299.40i 0.787630 − 1.36422i −0.139785 − 0.990182i \(-0.544641\pi\)
0.927415 − 0.374034i \(-0.122026\pi\)
\(32\) 0 0
\(33\) −262.198 − 454.140i −0.0419126 − 0.0725947i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 702.479 + 1216.73i 0.0843585 + 0.146113i 0.905118 − 0.425161i \(-0.139783\pi\)
−0.820759 + 0.571274i \(0.806449\pi\)
\(38\) 0 0
\(39\) −3566.26 + 6176.94i −0.375449 + 0.650297i
\(40\) 0 0
\(41\) 19074.6 1.77213 0.886066 − 0.463559i \(-0.153428\pi\)
0.886066 + 0.463559i \(0.153428\pi\)
\(42\) 0 0
\(43\) 6023.14 0.496766 0.248383 − 0.968662i \(-0.420101\pi\)
0.248383 + 0.968662i \(0.420101\pi\)
\(44\) 0 0
\(45\) 3151.36 − 5458.32i 0.231989 − 0.401816i
\(46\) 0 0
\(47\) −940.471 − 1628.94i −0.0621013 − 0.107563i 0.833303 − 0.552816i \(-0.186447\pi\)
−0.895404 + 0.445254i \(0.853114\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2090.88 + 3621.52i 0.112565 + 0.194969i
\(52\) 0 0
\(53\) −2627.70 + 4551.30i −0.128495 + 0.222560i −0.923094 − 0.384575i \(-0.874348\pi\)
0.794599 + 0.607135i \(0.207681\pi\)
\(54\) 0 0
\(55\) −4533.77 −0.202094
\(56\) 0 0
\(57\) −8861.01 −0.361240
\(58\) 0 0
\(59\) −1556.38 + 2695.73i −0.0582083 + 0.100820i −0.893661 − 0.448742i \(-0.851872\pi\)
0.835453 + 0.549562i \(0.185205\pi\)
\(60\) 0 0
\(61\) 11047.3 + 19134.4i 0.380128 + 0.658401i 0.991080 − 0.133267i \(-0.0425466\pi\)
−0.610952 + 0.791667i \(0.709213\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 30832.8 + 53404.0i 0.905169 + 1.56780i
\(66\) 0 0
\(67\) 27355.9 − 47381.8i 0.744498 − 1.28951i −0.205931 − 0.978567i \(-0.566022\pi\)
0.950429 − 0.310942i \(-0.100645\pi\)
\(68\) 0 0
\(69\) −5723.48 −0.144723
\(70\) 0 0
\(71\) −15211.7 −0.358122 −0.179061 − 0.983838i \(-0.557306\pi\)
−0.179061 + 0.983838i \(0.557306\pi\)
\(72\) 0 0
\(73\) 25038.3 − 43367.6i 0.549917 − 0.952484i −0.448362 − 0.893852i \(-0.647993\pi\)
0.998280 − 0.0586326i \(-0.0186741\pi\)
\(74\) 0 0
\(75\) −13183.2 − 22834.0i −0.270625 − 0.468737i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −571.083 − 989.144i −0.0102951 − 0.0178317i 0.860832 − 0.508889i \(-0.169944\pi\)
−0.871127 + 0.491058i \(0.836610\pi\)
\(80\) 0 0
\(81\) −3280.50 + 5681.99i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −122870. −1.95772 −0.978859 − 0.204537i \(-0.934431\pi\)
−0.978859 + 0.204537i \(0.934431\pi\)
\(84\) 0 0
\(85\) 36154.3 0.542766
\(86\) 0 0
\(87\) −28784.5 + 49856.2i −0.407719 + 0.706189i
\(88\) 0 0
\(89\) −35246.3 − 61048.4i −0.471670 − 0.816957i 0.527804 − 0.849366i \(-0.323015\pi\)
−0.999475 + 0.0324090i \(0.989682\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 37928.8 + 65694.6i 0.454738 + 0.787630i
\(94\) 0 0
\(95\) −38304.8 + 66345.9i −0.435456 + 0.754232i
\(96\) 0 0
\(97\) −132975. −1.43497 −0.717484 − 0.696575i \(-0.754706\pi\)
−0.717484 + 0.696575i \(0.754706\pi\)
\(98\) 0 0
\(99\) 4719.56 0.0483965
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 588.6.i.p.361.4 8
7.2 even 3 inner 588.6.i.p.373.4 8
7.3 odd 6 588.6.a.m.1.4 ✓ 4
7.4 even 3 588.6.a.o.1.1 yes 4
7.5 odd 6 588.6.i.q.373.1 8
7.6 odd 2 588.6.i.q.361.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.6.a.m.1.4 ✓ 4 7.3 odd 6
588.6.a.o.1.1 yes 4 7.4 even 3
588.6.i.p.361.4 8 1.1 even 1 trivial
588.6.i.p.373.4 8 7.2 even 3 inner
588.6.i.q.361.1 8 7.6 odd 2
588.6.i.q.373.1 8 7.5 odd 6