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Properties

Label 585.2.bt.b.571.17
Level $585$
Weight $2$
Character 585.571
Analytic conductor $4.671$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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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] = [585,2,Mod(376,585)] 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("585.376"); 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([4, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bt (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [108] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(54\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 571.17
Character \(\chi\) \(=\) 585.571
Dual form 585.2.bt.b.376.17

$q$-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+(-1.09899 + 0.634500i) q^{2} +(1.58392 + 0.700848i) q^{3} +(-0.194820 + 0.337439i) q^{4} +(-0.866025 - 0.500000i) q^{5} +(-2.18540 + 0.234776i) q^{6} +(4.16228 - 2.40310i) q^{7} -3.03245i q^{8} +(2.01762 + 2.22018i) q^{9} +1.26900 q^{10} +(-3.25705 + 1.88046i) q^{11} +(-0.545074 + 0.397937i) q^{12} +(3.09393 + 1.85137i) q^{13} +(-3.04953 + 5.28194i) q^{14} +(-1.02129 - 1.39891i) q^{15} +(1.53445 + 2.65774i) q^{16} +2.36135 q^{17} +(-3.62604 - 1.15976i) q^{18} -4.60952i q^{19} +(0.337439 - 0.194820i) q^{20} +(8.27694 - 0.889188i) q^{21} +(2.38630 - 4.13319i) q^{22} +(4.07745 - 7.06236i) q^{23} +(2.12529 - 4.80317i) q^{24} +(0.500000 + 0.866025i) q^{25} +(-4.57488 - 0.0715345i) q^{26} +(1.63975 + 4.93064i) q^{27} +1.87269i q^{28} +(2.72644 + 4.72233i) q^{29} +(2.01000 + 0.889376i) q^{30} +(-1.31066 - 0.756711i) q^{31} +(1.87969 + 1.08524i) q^{32} +(-6.47683 + 0.695803i) q^{33} +(-2.59509 + 1.49827i) q^{34} -4.80619 q^{35} +(-1.14225 + 0.248288i) q^{36} +9.70426i q^{37} +(2.92474 + 5.06579i) q^{38} +(3.60302 + 5.10081i) q^{39} +(-1.51623 + 2.62618i) q^{40} +(3.95889 + 2.28566i) q^{41} +(-8.53205 + 6.22892i) q^{42} +(-0.981245 - 1.69957i) q^{43} -1.46541i q^{44} +(-0.637223 - 2.93154i) q^{45} +10.3486i q^{46} +(-8.48359 + 4.89800i) q^{47} +(0.567773 + 5.28508i) q^{48} +(8.04974 - 13.9426i) q^{49} +(-1.09899 - 0.634500i) q^{50} +(3.74019 + 1.65495i) q^{51} +(-1.22749 + 0.683327i) q^{52} -8.05166 q^{53} +(-4.93055 - 4.37828i) q^{54} +3.76092 q^{55} +(-7.28728 - 12.6219i) q^{56} +(3.23057 - 7.30112i) q^{57} +(-5.99263 - 3.45985i) q^{58} +(-0.839155 - 0.484487i) q^{59} +(0.671017 - 0.0720870i) q^{60} +(4.76134 + 8.24689i) q^{61} +1.92053 q^{62} +(13.7332 + 4.39248i) q^{63} -8.89213 q^{64} +(-1.75374 - 3.15030i) q^{65} +(6.67646 - 4.87422i) q^{66} +(-1.76799 - 1.02075i) q^{67} +(-0.460039 + 0.796811i) q^{68} +(11.4080 - 8.32855i) q^{69} +(5.28194 - 3.04953i) q^{70} +7.78951i q^{71} +(6.73259 - 6.11835i) q^{72} +1.99976i q^{73} +(-6.15735 - 10.6648i) q^{74} +(0.185009 + 1.72214i) q^{75} +(1.55543 + 0.898028i) q^{76} +(-9.03784 + 15.6540i) q^{77} +(-7.19613 - 3.31960i) q^{78} +(-3.61917 - 6.26859i) q^{79} -3.06890i q^{80} +(-0.858397 + 8.95897i) q^{81} -5.80101 q^{82} +(-3.04623 + 1.75874i) q^{83} +(-1.31247 + 2.96619i) q^{84} +(-2.04499 - 1.18067i) q^{85} +(2.15675 + 1.24520i) q^{86} +(1.00883 + 9.39062i) q^{87} +(5.70240 + 9.87685i) q^{88} -17.2510i q^{89} +(2.56036 + 2.81741i) q^{90} +(17.3269 + 0.270929i) q^{91} +(1.58874 + 2.75178i) q^{92} +(-1.54565 - 2.11715i) q^{93} +(6.21556 - 10.7657i) q^{94} +(-2.30476 + 3.99196i) q^{95} +(2.21669 + 3.03631i) q^{96} +(-2.99976 + 1.73191i) q^{97} +20.4302i q^{98} +(-10.7465 - 3.43718i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q - 6 q^{3} + 52 q^{4} - 14 q^{9} - 8 q^{10} - 36 q^{12} - 4 q^{13} - 8 q^{14} - 64 q^{16} + 36 q^{17} + 24 q^{22} - 22 q^{23} + 54 q^{25} + 40 q^{26} + 48 q^{27} - 16 q^{29} + 20 q^{30} - 40 q^{35}+ \cdots - 56 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.09899 + 0.634500i −0.777100 + 0.448659i −0.835402 − 0.549640i \(-0.814765\pi\)
0.0583014 + 0.998299i \(0.481432\pi\)
\(3\) 1.58392 + 0.700848i 0.914478 + 0.404635i
\(4\) −0.194820 + 0.337439i −0.0974102 + 0.168719i
\(5\) −0.866025 − 0.500000i −0.387298 − 0.223607i
\(6\) −2.18540 + 0.234776i −0.892184 + 0.0958469i
\(7\) 4.16228 − 2.40310i 1.57320 − 0.908285i 0.577422 − 0.816446i \(-0.304059\pi\)
0.995774 − 0.0918391i \(-0.0292745\pi\)
\(8\) − 3.03245i − 1.07213i
\(9\) 2.01762 + 2.22018i 0.672541 + 0.740060i
\(10\) 1.26900 0.401293
\(11\) −3.25705 + 1.88046i −0.982037 + 0.566980i −0.902885 − 0.429883i \(-0.858555\pi\)
−0.0791527 + 0.996863i \(0.525221\pi\)
\(12\) −0.545074 + 0.397937i −0.157349 + 0.114875i
\(13\) 3.09393 + 1.85137i 0.858102 + 0.513479i
\(14\) −3.04953 + 5.28194i −0.815020 + 1.41166i
\(15\) −1.02129 − 1.39891i −0.263697 − 0.361198i
\(16\) 1.53445 + 2.65774i 0.383612 + 0.664436i
\(17\) 2.36135 0.572711 0.286356 − 0.958123i \(-0.407556\pi\)
0.286356 + 0.958123i \(0.407556\pi\)
\(18\) −3.62604 − 1.15976i −0.854666 − 0.273359i
\(19\) − 4.60952i − 1.05750i −0.848779 − 0.528748i \(-0.822662\pi\)
0.848779 − 0.528748i \(-0.177338\pi\)
\(20\) 0.337439 − 0.194820i 0.0754536 − 0.0435632i
\(21\) 8.27694 − 0.889188i 1.80618 − 0.194037i
\(22\) 2.38630 − 4.13319i 0.508761 − 0.881200i
\(23\) 4.07745 − 7.06236i 0.850208 − 1.47260i −0.0308128 − 0.999525i \(-0.509810\pi\)
0.881021 − 0.473078i \(-0.156857\pi\)
\(24\) 2.12529 − 4.80317i 0.433823 − 0.980443i
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) −4.57488 − 0.0715345i −0.897208 − 0.0140291i
\(27\) 1.63975 + 4.93064i 0.315570 + 0.948902i
\(28\) 1.87269i 0.353905i
\(29\) 2.72644 + 4.72233i 0.506287 + 0.876914i 0.999974 + 0.00727453i \(0.00231558\pi\)
−0.493687 + 0.869640i \(0.664351\pi\)
\(30\) 2.01000 + 0.889376i 0.366974 + 0.162377i
\(31\) −1.31066 − 0.756711i −0.235402 − 0.135909i 0.377660 − 0.925944i \(-0.376729\pi\)
−0.613062 + 0.790035i \(0.710062\pi\)
\(32\) 1.87969 + 1.08524i 0.332285 + 0.191845i
\(33\) −6.47683 + 0.695803i −1.12747 + 0.121124i
\(34\) −2.59509 + 1.49827i −0.445054 + 0.256952i
\(35\) −4.80619 −0.812395
\(36\) −1.14225 + 0.248288i −0.190375 + 0.0413813i
\(37\) 9.70426i 1.59537i 0.603074 + 0.797685i \(0.293942\pi\)
−0.603074 + 0.797685i \(0.706058\pi\)
\(38\) 2.92474 + 5.06579i 0.474455 + 0.821780i
\(39\) 3.60302 + 5.10081i 0.576944 + 0.816783i
\(40\) −1.51623 + 2.62618i −0.239736 + 0.415236i
\(41\) 3.95889 + 2.28566i 0.618274 + 0.356961i 0.776197 − 0.630491i \(-0.217146\pi\)
−0.157923 + 0.987451i \(0.550480\pi\)
\(42\) −8.53205 + 6.22892i −1.31652 + 0.961144i
\(43\) −0.981245 − 1.69957i −0.149638 − 0.259181i 0.781455 − 0.623961i \(-0.214478\pi\)
−0.931094 + 0.364780i \(0.881144\pi\)
\(44\) − 1.46541i − 0.220918i
\(45\) −0.637223 − 2.93154i −0.0949916 − 0.437009i
\(46\) 10.3486i 1.52581i
\(47\) −8.48359 + 4.89800i −1.23746 + 0.714447i −0.968574 − 0.248726i \(-0.919988\pi\)
−0.268884 + 0.963173i \(0.586655\pi\)
\(48\) 0.567773 + 5.28508i 0.0819510 + 0.762835i
\(49\) 8.04974 − 13.9426i 1.14996 − 1.99179i
\(50\) −1.09899 − 0.634500i −0.155420 − 0.0897318i
\(51\) 3.74019 + 1.65495i 0.523732 + 0.231739i
\(52\) −1.22749 + 0.683327i −0.170222 + 0.0947605i
\(53\) −8.05166 −1.10598 −0.552990 − 0.833188i \(-0.686513\pi\)
−0.552990 + 0.833188i \(0.686513\pi\)
\(54\) −4.93055 − 4.37828i −0.670963 − 0.595809i
\(55\) 3.76092 0.507122
\(56\) −7.28728 − 12.6219i −0.973803 − 1.68668i
\(57\) 3.23057 − 7.30112i 0.427900 − 0.967056i
\(58\) −5.99263 − 3.45985i −0.786871 − 0.454300i
\(59\) −0.839155 − 0.484487i −0.109249 − 0.0630748i 0.444380 − 0.895838i \(-0.353424\pi\)
−0.553629 + 0.832764i \(0.686757\pi\)
\(60\) 0.671017 − 0.0720870i 0.0866279 − 0.00930639i
\(61\) 4.76134 + 8.24689i 0.609628 + 1.05591i 0.991302 + 0.131609i \(0.0420144\pi\)
−0.381674 + 0.924297i \(0.624652\pi\)
\(62\) 1.92053 0.243908
\(63\) 13.7332 + 4.39248i 1.73022 + 0.553400i
\(64\) −8.89213 −1.11152
\(65\) −1.75374 − 3.15030i −0.217524 − 0.390747i
\(66\) 6.67646 − 4.87422i 0.821815 − 0.599976i
\(67\) −1.76799 − 1.02075i −0.215994 − 0.124704i 0.388100 − 0.921617i \(-0.373132\pi\)
−0.604094 + 0.796913i \(0.706465\pi\)
\(68\) −0.460039 + 0.796811i −0.0557879 + 0.0966275i
\(69\) 11.4080 − 8.32855i 1.37336 − 1.00264i
\(70\) 5.28194 − 3.04953i 0.631312 − 0.364488i
\(71\) 7.78951i 0.924445i 0.886764 + 0.462223i \(0.152948\pi\)
−0.886764 + 0.462223i \(0.847052\pi\)
\(72\) 6.73259 − 6.11835i 0.793443 − 0.721054i
\(73\) 1.99976i 0.234055i 0.993129 + 0.117027i \(0.0373365\pi\)
−0.993129 + 0.117027i \(0.962663\pi\)
\(74\) −6.15735 − 10.6648i −0.715777 − 1.23976i
\(75\) 0.185009 + 1.72214i 0.0213630 + 0.198856i
\(76\) 1.55543 + 0.898028i 0.178420 + 0.103011i
\(77\) −9.03784 + 15.6540i −1.02996 + 1.78394i
\(78\) −7.19613 − 3.31960i −0.814801 − 0.375871i
\(79\) −3.61917 − 6.26859i −0.407189 − 0.705271i 0.587385 − 0.809308i \(-0.300157\pi\)
−0.994573 + 0.104036i \(0.966824\pi\)
\(80\) − 3.06890i − 0.343113i
\(81\) −0.858397 + 8.95897i −0.0953774 + 0.995441i
\(82\) −5.80101 −0.640615
\(83\) −3.04623 + 1.75874i −0.334367 + 0.193047i −0.657778 − 0.753211i \(-0.728504\pi\)
0.323411 + 0.946259i \(0.395170\pi\)
\(84\) −1.31247 + 2.96619i −0.143202 + 0.323638i
\(85\) −2.04499 − 1.18067i −0.221810 − 0.128062i
\(86\) 2.15675 + 1.24520i 0.232568 + 0.134273i
\(87\) 1.00883 + 9.39062i 0.108158 + 1.00678i
\(88\) 5.70240 + 9.87685i 0.607878 + 1.05288i
\(89\) − 17.2510i − 1.82860i −0.405037 − 0.914300i \(-0.632742\pi\)
0.405037 − 0.914300i \(-0.367258\pi\)
\(90\) 2.56036 + 2.81741i 0.269886 + 0.296981i
\(91\) 17.3269 + 0.270929i 1.81635 + 0.0284011i
\(92\) 1.58874 + 2.75178i 0.165638 + 0.286893i
\(93\) −1.54565 − 2.11715i −0.160276 − 0.219538i
\(94\) 6.21556 − 10.7657i 0.641086 − 1.11039i
\(95\) −2.30476 + 3.99196i −0.236463 + 0.409566i
\(96\) 2.21669 + 3.03631i 0.226240 + 0.309892i
\(97\) −2.99976 + 1.73191i −0.304580 + 0.175849i −0.644498 − 0.764606i \(-0.722934\pi\)
0.339919 + 0.940455i \(0.389600\pi\)
\(98\) 20.4302i 2.06377i
\(99\) −10.7465 − 3.43718i −1.08006 − 0.345450i
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.bt.b.571.17 yes 108
9.7 even 3 inner 585.2.bt.b.376.38 yes 108
13.12 even 2 inner 585.2.bt.b.571.38 yes 108
117.25 even 6 inner 585.2.bt.b.376.17 ✓ 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.bt.b.376.17 ✓ 108 117.25 even 6 inner
585.2.bt.b.376.38 yes 108 9.7 even 3 inner
585.2.bt.b.571.17 yes 108 1.1 even 1 trivial
585.2.bt.b.571.38 yes 108 13.12 even 2 inner