Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,8,Mod(1,588)] 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("588.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,108,0,-196] 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: yes
Analytic conductor: \(183.682394985\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 655x^{2} - 5704x - 84 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-16.7957\) of defining polynomial
Character \(\chi\) \(=\) 588.1

$q$-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)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} -40.1550 q^{5} +729.000 q^{9} +4188.19 q^{11} -508.908 q^{13} -1084.19 q^{15} -10613.0 q^{17} +17994.8 q^{19} -66655.5 q^{23} -76512.6 q^{25} +19683.0 q^{27} +34469.9 q^{29} -46149.1 q^{31} +113081. q^{33} -423576. q^{37} -13740.5 q^{39} -500533. q^{41} +569541. q^{43} -29273.0 q^{45} +507298. q^{47} -286551. q^{51} +987663. q^{53} -168177. q^{55} +485860. q^{57} -579095. q^{59} +189294. q^{61} +20435.2 q^{65} -1.75802e6 q^{67} -1.79970e6 q^{69} +5436.62 q^{71} +1.72904e6 q^{73} -2.06584e6 q^{75} +1.68419e6 q^{79} +531441. q^{81} +5.83451e6 q^{83} +426166. q^{85} +930687. q^{87} -1.28692e7 q^{89} -1.24602e6 q^{93} -722581. q^{95} +6.50023e6 q^{97} +3.05319e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 108 q^{3} - 196 q^{5} + 2916 q^{9} - 406 q^{11} - 1974 q^{13} - 5292 q^{15} - 7436 q^{17} + 15874 q^{19} + 6788 q^{23} - 69898 q^{25} + 78732 q^{27} - 94544 q^{29} - 55890 q^{31} - 10962 q^{33} - 93742 q^{37}+ \cdots - 295974 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 27.0000 0.577350
\(4\) 0 0
\(5\) −40.1550 −0.143663 −0.0718315 − 0.997417i \(-0.522884\pi\)
−0.0718315 + 0.997417i \(0.522884\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 4188.19 0.948750 0.474375 − 0.880323i \(-0.342674\pi\)
0.474375 + 0.880323i \(0.342674\pi\)
\(12\) 0 0
\(13\) −508.908 −0.0642447 −0.0321224 − 0.999484i \(-0.510227\pi\)
−0.0321224 + 0.999484i \(0.510227\pi\)
\(14\) 0 0
\(15\) −1084.19 −0.0829438
\(16\) 0 0
\(17\) −10613.0 −0.523923 −0.261962 − 0.965078i \(-0.584369\pi\)
−0.261962 + 0.965078i \(0.584369\pi\)
\(18\) 0 0
\(19\) 17994.8 0.601879 0.300940 − 0.953643i \(-0.402700\pi\)
0.300940 + 0.953643i \(0.402700\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −66655.5 −1.14232 −0.571161 − 0.820838i \(-0.693507\pi\)
−0.571161 + 0.820838i \(0.693507\pi\)
\(24\) 0 0
\(25\) −76512.6 −0.979361
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) 34469.9 0.262450 0.131225 − 0.991353i \(-0.458109\pi\)
0.131225 + 0.991353i \(0.458109\pi\)
\(30\) 0 0
\(31\) −46149.1 −0.278226 −0.139113 − 0.990277i \(-0.544425\pi\)
−0.139113 + 0.990277i \(0.544425\pi\)
\(32\) 0 0
\(33\) 113081. 0.547761
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −423576. −1.37476 −0.687378 − 0.726300i \(-0.741238\pi\)
−0.687378 + 0.726300i \(0.741238\pi\)
\(38\) 0 0
\(39\) −13740.5 −0.0370917
\(40\) 0 0
\(41\) −500533. −1.13420 −0.567100 − 0.823649i \(-0.691935\pi\)
−0.567100 + 0.823649i \(0.691935\pi\)
\(42\) 0 0
\(43\) 569541. 1.09241 0.546204 − 0.837652i \(-0.316072\pi\)
0.546204 + 0.837652i \(0.316072\pi\)
\(44\) 0 0
\(45\) −29273.0 −0.0478876
\(46\) 0 0
\(47\) 507298. 0.712723 0.356362 − 0.934348i \(-0.384017\pi\)
0.356362 + 0.934348i \(0.384017\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −286551. −0.302487
\(52\) 0 0
\(53\) 987663. 0.911262 0.455631 − 0.890169i \(-0.349414\pi\)
0.455631 + 0.890169i \(0.349414\pi\)
\(54\) 0 0
\(55\) −168177. −0.136300
\(56\) 0 0
\(57\) 485860. 0.347495
\(58\) 0 0
\(59\) −579095. −0.367086 −0.183543 − 0.983012i \(-0.558757\pi\)
−0.183543 + 0.983012i \(0.558757\pi\)
\(60\) 0 0
\(61\) 189294. 0.106778 0.0533892 − 0.998574i \(-0.482998\pi\)
0.0533892 + 0.998574i \(0.482998\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 20435.2 0.00922959
\(66\) 0 0
\(67\) −1.75802e6 −0.714103 −0.357052 − 0.934085i \(-0.616218\pi\)
−0.357052 + 0.934085i \(0.616218\pi\)
\(68\) 0 0
\(69\) −1.79970e6 −0.659520
\(70\) 0 0
\(71\) 5436.62 0.00180271 0.000901353 − 1.00000i \(-0.499713\pi\)
0.000901353 1.00000i \(0.499713\pi\)
\(72\) 0 0
\(73\) 1.72904e6 0.520206 0.260103 − 0.965581i \(-0.416243\pi\)
0.260103 + 0.965581i \(0.416243\pi\)
\(74\) 0 0
\(75\) −2.06584e6 −0.565434
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.68419e6 0.384322 0.192161 − 0.981363i \(-0.438450\pi\)
0.192161 + 0.981363i \(0.438450\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 5.83451e6 1.12003 0.560017 − 0.828481i \(-0.310795\pi\)
0.560017 + 0.828481i \(0.310795\pi\)
\(84\) 0 0
\(85\) 426166. 0.0752683
\(86\) 0 0
\(87\) 930687. 0.151526
\(88\) 0 0
\(89\) −1.28692e7 −1.93503 −0.967513 − 0.252823i \(-0.918641\pi\)
−0.967513 + 0.252823i \(0.918641\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.24602e6 −0.160634
\(94\) 0 0
\(95\) −722581. −0.0864678
\(96\) 0 0
\(97\) 6.50023e6 0.723149 0.361575 − 0.932343i \(-0.382239\pi\)
0.361575 + 0.932343i \(0.382239\pi\)
\(98\) 0 0
\(99\) 3.05319e6 0.316250
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 588.8.a.j.1.2 4
7.2 even 3 588.8.i.o.361.3 8
7.3 odd 6 84.8.i.a.37.2 yes 8
7.4 even 3 588.8.i.o.373.3 8
7.5 odd 6 84.8.i.a.25.2 ✓ 8
7.6 odd 2 588.8.a.i.1.3 4
21.5 even 6 252.8.k.b.109.3 8
21.17 even 6 252.8.k.b.37.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.i.a.25.2 ✓ 8 7.5 odd 6
84.8.i.a.37.2 yes 8 7.3 odd 6
252.8.k.b.37.3 8 21.17 even 6
252.8.k.b.109.3 8 21.5 even 6
588.8.a.i.1.3 4 7.6 odd 2
588.8.a.j.1.2 4 1.1 even 1 trivial
588.8.i.o.361.3 8 7.2 even 3
588.8.i.o.373.3 8 7.4 even 3