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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 80.15
Character \(\chi\) \(=\) 567.80
Dual form 567.2.p.e.404.15

$q$-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 \(-0.298205\pi\)
0.993917 − 0.110134i \(-0.0351280\pi\)
\(3\) 0 0
\(4\) 2.35494 − 4.07887i 1.17747 − 2.03943i
\(5\) −1.85700 − 3.21642i −0.830477 − 1.43843i −0.897661 − 0.440687i \(-0.854735\pi\)
0.0671841 − 0.997741i \(-0.478599\pi\)
\(6\) 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 0.870246 − 0.492617i \(-0.163960\pi\)
0.00850447 + 0.999964i \(0.497293\pi\)
\(12\) 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 −0.720665 − 0.693284i \(-0.756163\pi\)
0.960734 + 0.277472i \(0.0894966\pi\)
\(18\) 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 −0.141662 − 0.989915i \(-0.545245\pi\)
0.786461 + 0.617640i \(0.211911\pi\)
\(24\) 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 \(-0.931280\pi\)
0.976786 − 0.214218i \(-0.0687203\pi\)
\(30\) 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 \(-0.218612\pi\)
0.773286 + 0.634058i \(0.218612\pi\)
\(42\) 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 0.979999 − 0.199005i \(-0.0637709\pi\)
−0.662342 + 0.749201i \(0.730438\pi\)
\(48\) 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 0.114881 − 0.993379i \(-0.463351\pi\)
−0.802851 + 0.596179i \(0.796685\pi\)
\(54\) 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 −0.899219 − 0.437498i \(-0.855865\pi\)
0.828494 + 0.559998i \(0.189198\pi\)
\(60\) 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 \(0.258523\pi\)
−0.687922 + 0.725784i \(0.741477\pi\)
\(72\) 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 \(-0.162473\pi\)
0.872538 + 0.488547i \(0.162473\pi\)
\(84\) 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 0.902733 − 0.430201i \(-0.141557\pi\)
−0.823932 + 0.566689i \(0.808224\pi\)
\(90\) 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.80.15 yes 32
3.2 odd 2 inner 567.2.p.e.80.2 ✓ 32
7.5 odd 6 inner 567.2.p.e.404.2 yes 32
9.2 odd 6 567.2.s.g.458.15 32
9.4 even 3 567.2.i.g.269.15 32
9.5 odd 6 567.2.i.g.269.2 32
9.7 even 3 567.2.s.g.458.2 32
21.5 even 6 inner 567.2.p.e.404.15 yes 32
63.5 even 6 567.2.s.g.26.2 32
63.40 odd 6 567.2.s.g.26.15 32
63.47 even 6 567.2.i.g.215.2 32
63.61 odd 6 567.2.i.g.215.15 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.i.g.215.2 32 63.47 even 6
567.2.i.g.215.15 32 63.61 odd 6
567.2.i.g.269.2 32 9.5 odd 6
567.2.i.g.269.15 32 9.4 even 3
567.2.p.e.80.2 ✓ 32 3.2 odd 2 inner
567.2.p.e.80.15 yes 32 1.1 even 1 trivial
567.2.p.e.404.2 yes 32 7.5 odd 6 inner
567.2.p.e.404.15 yes 32 21.5 even 6 inner
567.2.s.g.26.2 32 63.5 even 6
567.2.s.g.26.15 32 63.40 odd 6
567.2.s.g.458.2 32 9.7 even 3
567.2.s.g.458.15 32 9.2 odd 6