↑

Properties

Label 585.2.j.f.451.1
Level $585$
Weight $2$
Character 585.451
Analytic conductor $4.671$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [585,2,Mod(406,585)] 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("585.406"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1714608.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 451.1
Root \(0.500000 + 2.23871i\) of defining polynomial
Character \(\chi\) \(=\) 585.451
Dual form 585.2.j.f.406.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.13090 + 1.95878i) q^{2} +(-1.55787 - 2.69832i) q^{4} -1.00000 q^{5} +(0.630901 + 1.09275i) q^{7} +2.52360 q^{8} +(1.13090 - 1.95878i) q^{10} +(2.26180 - 3.91756i) q^{11} +(-3.45058 + 1.04571i) q^{13} -2.85395 q^{14} +(0.261802 - 0.453455i) q^{16} +(-2.24665 - 3.89131i) q^{17} +(-2.55787 - 4.43037i) q^{19} +(1.55787 + 2.69832i) q^{20} +(5.11575 + 8.86074i) q^{22} +(1.11575 - 1.93253i) q^{23} +1.00000 q^{25} +(1.85395 - 7.94151i) q^{26} +(1.96573 - 3.40474i) q^{28} +(-0.688776 + 1.19299i) q^{29} +8.87085 q^{31} +(3.11575 + 5.39664i) q^{32} +10.1630 q^{34} +(-0.630901 - 1.09275i) q^{35} +(0.115749 - 0.200484i) q^{37} +11.5708 q^{38} -2.52360 q^{40} +(0.573026 - 0.992511i) q^{41} +(-3.18878 - 5.52312i) q^{43} -14.0944 q^{44} +(2.52360 + 4.37101i) q^{46} +10.7854 q^{47} +(2.70393 - 4.68334i) q^{49} +(-1.13090 + 1.95878i) q^{50} +(8.19723 + 7.68167i) q^{52} -4.52360 q^{53} +(-2.26180 + 3.91756i) q^{55} +(1.59214 + 2.75768i) q^{56} +(-1.55787 - 2.69832i) q^{58} +(-0.426974 - 0.739540i) q^{59} +(-2.31968 - 4.01780i) q^{61} +(-10.0321 + 17.3760i) q^{62} -13.0472 q^{64} +(3.45058 - 1.04571i) q^{65} +(-6.56633 + 11.3732i) q^{67} +(-7.00000 + 12.1244i) q^{68} +2.85395 q^{70} +(-4.80453 - 8.32168i) q^{71} +13.7854 q^{73} +(0.261802 + 0.453455i) q^{74} +(-7.96970 + 13.8039i) q^{76} +5.70789 q^{77} -8.87085 q^{79} +(-0.261802 + 0.453455i) q^{80} +(1.29607 + 2.24486i) q^{82} +8.23150 q^{83} +(2.24665 + 3.89131i) q^{85} +14.4248 q^{86} +(5.70789 - 9.88636i) q^{88} +(3.31122 - 5.73521i) q^{89} +(-3.31968 - 3.11089i) q^{91} -6.95279 q^{92} +(-12.1972 + 21.1262i) q^{94} +(2.55787 + 4.43037i) q^{95} +(-5.33483 - 9.24019i) q^{97} +(6.11575 + 10.5928i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 6 q^{5} - 3 q^{7} - 12 q^{8} + 3 q^{13} - 24 q^{14} - 12 q^{16} - 12 q^{19} + 6 q^{20} + 24 q^{22} + 6 q^{25} + 18 q^{26} - 12 q^{28} + 6 q^{29} + 6 q^{31} + 12 q^{32} + 3 q^{35} - 6 q^{37}+ \cdots + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.13090 + 1.95878i −0.799668 + 1.38507i 0.120165 + 0.992754i \(0.461658\pi\)
−0.919832 + 0.392311i \(0.871676\pi\)
\(3\) 0 0
\(4\) −1.55787 − 2.69832i −0.778937 − 1.34916i
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.630901 + 1.09275i 0.238458 + 0.413022i 0.960272 − 0.279066i \(-0.0900247\pi\)
−0.721814 + 0.692087i \(0.756691\pi\)
\(8\) 2.52360 0.892229
\(9\) 0 0
\(10\) 1.13090 − 1.95878i 0.357622 − 0.619420i
\(11\) 2.26180 − 3.91756i 0.681959 − 1.18119i −0.292423 − 0.956289i \(-0.594462\pi\)
0.974382 − 0.224899i \(-0.0722051\pi\)
\(12\) 0 0
\(13\) −3.45058 + 1.04571i −0.957018 + 0.290028i
\(14\) −2.85395 −0.762749
\(15\) 0 0
\(16\) 0.261802 − 0.453455i 0.0654506 − 0.113364i
\(17\) −2.24665 − 3.89131i −0.544893 − 0.943782i −0.998614 − 0.0526381i \(-0.983237\pi\)
0.453721 − 0.891144i \(-0.350096\pi\)
\(18\) 0 0
\(19\) −2.55787 − 4.43037i −0.586817 − 1.01640i −0.994646 − 0.103338i \(-0.967048\pi\)
0.407830 − 0.913058i \(-0.366286\pi\)
\(20\) 1.55787 + 2.69832i 0.348351 + 0.603362i
\(21\) 0 0
\(22\) 5.11575 + 8.86074i 1.09068 + 1.88912i
\(23\) 1.11575 − 1.93253i 0.232650 − 0.402961i −0.725937 − 0.687761i \(-0.758594\pi\)
0.958587 + 0.284800i \(0.0919271\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.85395 − 7.94151i 0.363589 − 1.55746i
\(27\) 0 0
\(28\) 1.96573 − 3.40474i 0.371488 − 0.643436i
\(29\) −0.688776 + 1.19299i −0.127902 + 0.221534i −0.922864 − 0.385127i \(-0.874158\pi\)
0.794961 + 0.606660i \(0.207491\pi\)
\(30\) 0 0
\(31\) 8.87085 1.59325 0.796626 − 0.604472i \(-0.206616\pi\)
0.796626 + 0.604472i \(0.206616\pi\)
\(32\) 3.11575 + 5.39664i 0.550792 + 0.954000i
\(33\) 0 0
\(34\) 10.1630 1.74293
\(35\) −0.630901 − 1.09275i −0.106642 − 0.184709i
\(36\) 0 0
\(37\) 0.115749 − 0.200484i 0.0190291 − 0.0329593i −0.856354 − 0.516389i \(-0.827276\pi\)
0.875383 + 0.483430i \(0.160609\pi\)
\(38\) 11.5708 1.87703
\(39\) 0 0
\(40\) −2.52360 −0.399017
\(41\) 0.573026 − 0.992511i 0.0894917 − 0.155004i −0.817805 − 0.575496i \(-0.804809\pi\)
0.907296 + 0.420492i \(0.138142\pi\)
\(42\) 0 0
\(43\) −3.18878 − 5.52312i −0.486284 − 0.842268i 0.513592 − 0.858035i \(-0.328315\pi\)
−0.999876 + 0.0157664i \(0.994981\pi\)
\(44\) −14.0944 −2.12481
\(45\) 0 0
\(46\) 2.52360 + 4.37101i 0.372085 + 0.644470i
\(47\) 10.7854 1.57321 0.786607 − 0.617454i \(-0.211836\pi\)
0.786607 + 0.617454i \(0.211836\pi\)
\(48\) 0 0
\(49\) 2.70393 − 4.68334i 0.386275 − 0.669049i
\(50\) −1.13090 + 1.95878i −0.159934 + 0.277013i
\(51\) 0 0
\(52\) 8.19723 + 7.68167i 1.13675 + 1.06526i
\(53\) −4.52360 −0.621365 −0.310682 − 0.950514i \(-0.600558\pi\)
−0.310682 + 0.950514i \(0.600558\pi\)
\(54\) 0 0
\(55\) −2.26180 + 3.91756i −0.304981 + 0.528243i
\(56\) 1.59214 + 2.75768i 0.212759 + 0.368510i
\(57\) 0 0
\(58\) −1.55787 − 2.69832i −0.204559 − 0.354307i
\(59\) −0.426974 − 0.739540i −0.0555872 − 0.0962799i 0.836893 − 0.547367i \(-0.184370\pi\)
−0.892480 + 0.451087i \(0.851036\pi\)
\(60\) 0 0
\(61\) −2.31968 − 4.01780i −0.297004 − 0.514426i 0.678445 − 0.734651i \(-0.262654\pi\)
−0.975449 + 0.220225i \(0.929321\pi\)
\(62\) −10.0321 + 17.3760i −1.27407 + 2.20676i
\(63\) 0 0
\(64\) −13.0472 −1.63090
\(65\) 3.45058 − 1.04571i 0.427992 − 0.129704i
\(66\) 0 0
\(67\) −6.56633 + 11.3732i −0.802205 + 1.38946i 0.115958 + 0.993254i \(0.463006\pi\)
−0.918162 + 0.396205i \(0.870327\pi\)
\(68\) −7.00000 + 12.1244i −0.848875 + 1.47029i
\(69\) 0 0
\(70\) 2.85395 0.341112
\(71\) −4.80453 − 8.32168i −0.570192 − 0.987602i −0.996546 − 0.0830453i \(-0.973535\pi\)
0.426354 − 0.904557i \(-0.359798\pi\)
\(72\) 0 0
\(73\) 13.7854 1.61346 0.806730 − 0.590920i \(-0.201235\pi\)
0.806730 + 0.590920i \(0.201235\pi\)
\(74\) 0.261802 + 0.453455i 0.0304339 + 0.0527130i
\(75\) 0 0
\(76\) −7.96970 + 13.8039i −0.914187 + 1.58342i
\(77\) 5.70789 0.650475
\(78\) 0 0
\(79\) −8.87085 −0.998049 −0.499024 − 0.866588i \(-0.666308\pi\)
−0.499024 + 0.866588i \(0.666308\pi\)
\(80\) −0.261802 + 0.453455i −0.0292704 + 0.0506978i
\(81\) 0 0
\(82\) 1.29607 + 2.24486i 0.143127 + 0.247904i
\(83\) 8.23150 0.903524 0.451762 − 0.892138i \(-0.350796\pi\)
0.451762 + 0.892138i \(0.350796\pi\)
\(84\) 0 0
\(85\) 2.24665 + 3.89131i 0.243683 + 0.422072i
\(86\) 14.4248 1.55546
\(87\) 0 0
\(88\) 5.70789 − 9.88636i 0.608464 − 1.05389i
\(89\) 3.31122 − 5.73521i 0.350989 − 0.607931i −0.635434 − 0.772155i \(-0.719179\pi\)
0.986423 + 0.164224i \(0.0525121\pi\)
\(90\) 0 0
\(91\) −3.31968 − 3.11089i −0.347997 − 0.326110i
\(92\) −6.95279 −0.724879
\(93\) 0 0
\(94\) −12.1972 + 21.1262i −1.25805 + 2.17900i
\(95\) 2.55787 + 4.43037i 0.262432 + 0.454546i
\(96\) 0 0
\(97\) −5.33483 − 9.24019i −0.541670 − 0.938200i −0.998808 − 0.0488041i \(-0.984459\pi\)
0.457139 − 0.889395i \(-0.348874\pi\)
\(98\) 6.11575 + 10.5928i 0.617784 + 1.07003i
\(99\) 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 585.2.j.f.451.1 6
3.2 odd 2 195.2.i.d.61.3 yes 6
13.3 even 3 inner 585.2.j.f.406.1 6
13.4 even 6 7605.2.a.bw.1.1 3
13.9 even 3 7605.2.a.bv.1.3 3
15.2 even 4 975.2.bb.k.724.5 12
15.8 even 4 975.2.bb.k.724.2 12
15.14 odd 2 975.2.i.l.451.1 6
39.17 odd 6 2535.2.a.ba.1.3 3
39.29 odd 6 195.2.i.d.16.3 ✓ 6
39.35 odd 6 2535.2.a.bb.1.1 3
195.29 odd 6 975.2.i.l.601.1 6
195.68 even 12 975.2.bb.k.874.5 12
195.107 even 12 975.2.bb.k.874.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.i.d.16.3 ✓ 6 39.29 odd 6
195.2.i.d.61.3 yes 6 3.2 odd 2
585.2.j.f.406.1 6 13.3 even 3 inner
585.2.j.f.451.1 6 1.1 even 1 trivial
975.2.i.l.451.1 6 15.14 odd 2
975.2.i.l.601.1 6 195.29 odd 6
975.2.bb.k.724.2 12 15.8 even 4
975.2.bb.k.724.5 12 15.2 even 4
975.2.bb.k.874.2 12 195.107 even 12
975.2.bb.k.874.5 12 195.68 even 12
2535.2.a.ba.1.3 3 39.17 odd 6
2535.2.a.bb.1.1 3 39.35 odd 6
7605.2.a.bv.1.3 3 13.9 even 3
7605.2.a.bw.1.1 3 13.4 even 6