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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(80,567)] 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("567.80"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,16,0,0,-8,0,0,0,0] 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: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 80.2
Character \(\chi\) \(=\) 567.80
Dual form 567.2.p.e.404.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.24330 + 1.29517i) q^{2} +(2.35494 - 4.07887i) q^{4} +(1.85700 + 3.21642i) q^{5} +(-2.20314 + 1.46498i) q^{7} +7.01949i q^{8} +(-8.33163 - 4.81027i) q^{10} +(-2.91449 - 1.68268i) q^{11} +0.152939i q^{13} +(3.04491 - 6.13984i) q^{14} +(-4.38157 - 7.58910i) q^{16} +(-0.989830 + 1.71444i) q^{17} +(0.839544 - 0.484711i) q^{19} +17.4925 q^{20} +8.71743 q^{22} +(-3.09235 + 1.78537i) q^{23} +(-4.39691 + 7.61568i) q^{25} +(-0.198082 - 0.343088i) q^{26} +(0.787200 + 12.4363i) q^{28} +2.30720i q^{29} +(-3.68784 - 2.12918i) q^{31} +(7.50025 + 4.33027i) q^{32} -5.12799i q^{34} +(-8.80323 - 4.36577i) q^{35} +(1.45972 + 2.52831i) q^{37} +(-1.25557 + 2.17471i) q^{38} +(-22.5776 + 13.0352i) q^{40} -9.90289 q^{41} +9.71743 q^{43} +(-13.7269 + 7.92520i) q^{44} +(4.62471 - 8.01023i) q^{46} +(-2.17774 - 3.77196i) q^{47} +(2.70767 - 6.45512i) q^{49} -22.7790i q^{50} +(0.623818 + 0.360162i) q^{52} +(5.00850 + 2.89166i) q^{53} -12.4990i q^{55} +(-10.2834 - 15.4649i) q^{56} +(-2.98822 - 5.17574i) q^{58} +(0.543251 - 0.940938i) q^{59} +(-9.55242 + 5.51510i) q^{61} +11.0306 q^{62} -4.90749 q^{64} +(-0.491917 + 0.284008i) q^{65} +(-2.12714 + 3.68431i) q^{67} +(4.66197 + 8.07477i) q^{68} +(25.4027 - 1.60796i) q^{70} -12.2311i q^{71} +(-9.75901 - 5.63436i) q^{73} +(-6.54919 - 3.78118i) q^{74} -4.56585i q^{76} +(8.88612 - 0.562480i) q^{77} +(-1.31193 - 2.27232i) q^{79} +(16.2732 - 28.1860i) q^{80} +(22.2152 - 12.8259i) q^{82} -15.8984 q^{83} -7.35246 q^{85} +(-21.7991 + 12.5857i) q^{86} +(11.8116 - 20.4582i) q^{88} +(-0.743409 - 1.28762i) q^{89} +(-0.224053 - 0.336946i) q^{91} +16.8177i q^{92} +(9.77066 + 5.64110i) q^{94} +(3.11807 + 1.80022i) q^{95} +13.2916i q^{97} +(2.28635 + 17.9877i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{4} - 8 q^{7} - 28 q^{16} + 24 q^{22} - 16 q^{25} - 16 q^{28} - 48 q^{31} - 4 q^{37} + 56 q^{43} + 12 q^{46} - 4 q^{49} + 48 q^{52} + 36 q^{58} + 12 q^{61} - 80 q^{64} - 20 q^{67} + 120 q^{70}+ \cdots + 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.24330 + 1.29517i −1.58625 + 0.915824i −0.592337 + 0.805690i \(0.701795\pi\)
−0.993917 + 0.110134i \(0.964872\pi\)
\(3\) 0 0
\(4\) 2.35494 − 4.07887i 1.17747 − 2.03943i
\(5\) 1.85700 + 3.21642i 0.830477 + 1.43843i 0.897661 + 0.440687i \(0.145265\pi\)
−0.0671841 + 0.997741i \(0.521401\pi\)
\(6\) 0 0
\(7\) −2.20314 + 1.46498i −0.832709 + 0.553710i
\(8\) 7.01949i 2.48176i
\(9\) 0 0
\(10\) −8.33163 − 4.81027i −2.63469 − 1.52114i
\(11\) −2.91449 − 1.68268i −0.878751 − 0.507347i −0.00850447 − 0.999964i \(-0.502707\pi\)
−0.870246 + 0.492617i \(0.836040\pi\)
\(12\) 0 0
\(13\) 0.152939i 0.0424177i 0.999775 + 0.0212088i \(0.00675149\pi\)
−0.999775 + 0.0212088i \(0.993249\pi\)
\(14\) 3.04491 − 6.13984i 0.813788 − 1.64094i
\(15\) 0 0
\(16\) −4.38157 − 7.58910i −1.09539 − 1.89728i
\(17\) −0.989830 + 1.71444i −0.240069 + 0.415812i −0.960734 − 0.277472i \(-0.910503\pi\)
0.720665 + 0.693284i \(0.243837\pi\)
\(18\) 0 0
\(19\) 0.839544 − 0.484711i 0.192605 − 0.111200i −0.400597 − 0.916254i \(-0.631197\pi\)
0.593201 + 0.805054i \(0.297864\pi\)
\(20\) 17.4925 3.91144
\(21\) 0 0
\(22\) 8.71743 1.85856
\(23\) −3.09235 + 1.78537i −0.644799 + 0.372275i −0.786461 − 0.617640i \(-0.788089\pi\)
0.141662 + 0.989915i \(0.454755\pi\)
\(24\) 0 0
\(25\) −4.39691 + 7.61568i −0.879383 + 1.52314i
\(26\) −0.198082 − 0.343088i −0.0388471 − 0.0672852i
\(27\) 0 0
\(28\) 0.787200 + 12.4363i 0.148767 + 2.35023i
\(29\) 2.30720i 0.428436i 0.976786 + 0.214218i \(0.0687203\pi\)
−0.976786 + 0.214218i \(0.931280\pi\)
\(30\) 0 0
\(31\) −3.68784 − 2.12918i −0.662356 − 0.382412i 0.130818 − 0.991406i \(-0.458240\pi\)
−0.793174 + 0.608995i \(0.791573\pi\)
\(32\) 7.50025 + 4.33027i 1.32587 + 0.765491i
\(33\) 0 0
\(34\) − 5.12799i − 0.879444i
\(35\) −8.80323 − 4.36577i −1.48802 − 0.737949i
\(36\) 0 0
\(37\) 1.45972 + 2.52831i 0.239977 + 0.415652i 0.960707 − 0.277563i \(-0.0895268\pi\)
−0.720731 + 0.693215i \(0.756194\pi\)
\(38\) −1.25557 + 2.17471i −0.203680 + 0.352784i
\(39\) 0 0
\(40\) −22.5776 + 13.0352i −3.56984 + 2.06105i
\(41\) −9.90289 −1.54657 −0.773286 − 0.634058i \(-0.781388\pi\)
−0.773286 + 0.634058i \(0.781388\pi\)
\(42\) 0 0
\(43\) 9.71743 1.48189 0.740947 − 0.671563i \(-0.234377\pi\)
0.740947 + 0.671563i \(0.234377\pi\)
\(44\) −13.7269 + 7.92520i −2.06940 + 1.19477i
\(45\) 0 0
\(46\) 4.62471 − 8.01023i 0.681876 − 1.18104i
\(47\) −2.17774 − 3.77196i −0.317656 − 0.550197i 0.662342 − 0.749201i \(-0.269562\pi\)
−0.979999 + 0.199005i \(0.936229\pi\)
\(48\) 0 0
\(49\) 2.70767 − 6.45512i 0.386810 − 0.922159i
\(50\) − 22.7790i − 3.22144i
\(51\) 0 0
\(52\) 0.623818 + 0.360162i 0.0865080 + 0.0499454i
\(53\) 5.00850 + 2.89166i 0.687970 + 0.397200i 0.802851 − 0.596179i \(-0.203315\pi\)
−0.114881 + 0.993379i \(0.536649\pi\)
\(54\) 0 0
\(55\) − 12.4990i − 1.68536i
\(56\) −10.2834 − 15.4649i −1.37418 − 2.06659i
\(57\) 0 0
\(58\) −2.98822 − 5.17574i −0.392372 − 0.679608i
\(59\) 0.543251 − 0.940938i 0.0707252 − 0.122500i −0.828494 − 0.559998i \(-0.810802\pi\)
0.899219 + 0.437498i \(0.144135\pi\)
\(60\) 0 0
\(61\) −9.55242 + 5.51510i −1.22306 + 0.706136i −0.965570 − 0.260145i \(-0.916230\pi\)
−0.257493 + 0.966280i \(0.582896\pi\)
\(62\) 11.0306 1.40089
\(63\) 0 0
\(64\) −4.90749 −0.613436
\(65\) −0.491917 + 0.284008i −0.0610147 + 0.0352269i
\(66\) 0 0
\(67\) −2.12714 + 3.68431i −0.259871 + 0.450111i −0.966207 − 0.257766i \(-0.917014\pi\)
0.706336 + 0.707877i \(0.250347\pi\)
\(68\) 4.66197 + 8.07477i 0.565347 + 0.979210i
\(69\) 0 0
\(70\) 25.4027 − 1.60796i 3.03621 − 0.192188i
\(71\) − 12.2311i − 1.45157i −0.687922 − 0.725784i \(-0.741477\pi\)
0.687922 − 0.725784i \(-0.258523\pi\)
\(72\) 0 0
\(73\) −9.75901 − 5.63436i −1.14221 − 0.659452i −0.195229 − 0.980758i \(-0.562545\pi\)
−0.946976 + 0.321305i \(0.895878\pi\)
\(74\) −6.54919 − 3.78118i −0.761328 − 0.439553i
\(75\) 0 0
\(76\) − 4.56585i − 0.523739i
\(77\) 8.88612 − 0.562480i 1.01267 − 0.0641006i
\(78\) 0 0
\(79\) −1.31193 − 2.27232i −0.147603 − 0.255657i 0.782738 − 0.622352i \(-0.213823\pi\)
−0.930341 + 0.366695i \(0.880489\pi\)
\(80\) 16.2732 − 28.1860i 1.81940 − 3.15129i
\(81\) 0 0
\(82\) 22.2152 − 12.8259i 2.45325 − 1.41639i
\(83\) −15.8984 −1.74508 −0.872538 − 0.488547i \(-0.837527\pi\)
−0.872538 + 0.488547i \(0.837527\pi\)
\(84\) 0 0
\(85\) −7.35246 −0.797487
\(86\) −21.7991 + 12.5857i −2.35066 + 1.35715i
\(87\) 0 0
\(88\) 11.8116 − 20.4582i 1.25912 − 2.18085i
\(89\) −0.743409 − 1.28762i −0.0788012 − 0.136488i 0.823932 − 0.566689i \(-0.191776\pi\)
−0.902733 + 0.430201i \(0.858443\pi\)
\(90\) 0 0
\(91\) −0.224053 − 0.336946i −0.0234871 − 0.0353216i
\(92\) 16.8177i 1.75337i
\(93\) 0 0
\(94\) 9.77066 + 5.64110i 1.00777 + 0.581835i
\(95\) 3.11807 + 1.80022i 0.319907 + 0.184698i
\(96\) 0 0
\(97\) 13.2916i 1.34956i 0.738020 + 0.674779i \(0.235761\pi\)
−0.738020 + 0.674779i \(0.764239\pi\)
\(98\) 2.28635 + 17.9877i 0.230957 + 1.81703i
\(99\) 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 567.2.p.e.80.2 ✓ 32
3.2 odd 2 inner 567.2.p.e.80.15 yes 32
7.5 odd 6 inner 567.2.p.e.404.15 yes 32
9.2 odd 6 567.2.s.g.458.2 32
9.4 even 3 567.2.i.g.269.2 32
9.5 odd 6 567.2.i.g.269.15 32
9.7 even 3 567.2.s.g.458.15 32
21.5 even 6 inner 567.2.p.e.404.2 yes 32
63.5 even 6 567.2.s.g.26.15 32
63.40 odd 6 567.2.s.g.26.2 32
63.47 even 6 567.2.i.g.215.15 32
63.61 odd 6 567.2.i.g.215.2 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.i.g.215.2 32 63.61 odd 6
567.2.i.g.215.15 32 63.47 even 6
567.2.i.g.269.2 32 9.4 even 3
567.2.i.g.269.15 32 9.5 odd 6
567.2.p.e.80.2 ✓ 32 1.1 even 1 trivial
567.2.p.e.80.15 yes 32 3.2 odd 2 inner
567.2.p.e.404.2 yes 32 21.5 even 6 inner
567.2.p.e.404.15 yes 32 7.5 odd 6 inner
567.2.s.g.26.2 32 63.40 odd 6
567.2.s.g.26.15 32 63.5 even 6
567.2.s.g.458.2 32 9.2 odd 6
567.2.s.g.458.15 32 9.7 even 3