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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(143,567)] 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("567.143"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([7, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.ba (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 341.20
Character \(\chi\) \(=\) 567.341
Dual form 567.2.ba.a.143.20

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.44973 + 1.72772i) q^{2} +(-0.536009 + 3.03986i) q^{4} +(2.61524 + 2.19445i) q^{5} +(-0.126746 + 2.64271i) q^{7} +(-2.12267 + 1.22552i) q^{8} +7.69979i q^{10} +(-1.63538 - 1.94897i) q^{11} +(-2.11236 - 5.80365i) q^{13} +(-4.74963 + 3.61225i) q^{14} +(0.606533 + 0.220760i) q^{16} +2.39853 q^{17} -1.24485i q^{19} +(-8.07261 + 6.77373i) q^{20} +(0.996420 - 5.65098i) q^{22} +(-0.769210 - 2.11339i) q^{23} +(1.15565 + 6.55401i) q^{25} +(6.96475 - 12.0633i) q^{26} +(-7.96554 - 1.80181i) q^{28} +(-0.685884 + 1.88445i) q^{29} +(-1.55132 - 0.273539i) q^{31} +(2.17451 + 5.97442i) q^{32} +(3.47723 + 4.14400i) q^{34} +(-6.13078 + 6.63320i) q^{35} +(-5.01223 - 8.68144i) q^{37} +(2.15077 - 1.80471i) q^{38} +(-8.24065 - 1.45305i) q^{40} +(5.39584 - 1.96393i) q^{41} +(-0.110284 - 0.625454i) q^{43} +(6.80118 - 3.92666i) q^{44} +(2.53620 - 4.39283i) q^{46} +(2.09655 + 11.8901i) q^{47} +(-6.96787 - 0.669906i) q^{49} +(-9.64814 + 11.4982i) q^{50} +(18.7745 - 3.31045i) q^{52} +(-4.15350 + 2.39802i) q^{53} -8.68580i q^{55} +(-2.96967 - 5.76493i) q^{56} +(-4.25016 + 1.54693i) q^{58} +(-2.59194 + 0.943390i) q^{59} +(7.26032 - 1.28019i) q^{61} +(-1.77639 - 3.07681i) q^{62} +(-6.52423 + 11.3003i) q^{64} +(7.21149 - 19.8134i) q^{65} +(-6.14145 - 5.15329i) q^{67} +(-1.28564 + 7.29120i) q^{68} +(-20.3483 - 0.975916i) q^{70} +(-4.45847 - 2.57410i) q^{71} +(11.6132 + 6.70488i) q^{73} +(7.73274 - 21.2455i) q^{74} +(3.78418 + 0.667254i) q^{76} +(5.35785 - 4.07482i) q^{77} +(0.740362 - 0.621237i) q^{79} +(1.10178 + 1.90835i) q^{80} +(11.2156 + 6.47536i) q^{82} +(2.38230 + 0.867085i) q^{83} +(6.27275 + 5.26346i) q^{85} +(0.920729 - 1.09728i) q^{86} +(5.85988 + 2.13282i) q^{88} +9.95066 q^{89} +(15.6051 - 4.84676i) q^{91} +(6.83671 - 1.20550i) q^{92} +(-17.5034 + 20.8598i) q^{94} +(2.73177 - 3.25560i) q^{95} +(7.93719 - 1.39954i) q^{97} +(-8.94414 - 13.0097i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34}+ \cdots - 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.44973 + 1.72772i 1.02512 + 1.22169i 0.974830 + 0.222950i \(0.0715688\pi\)
0.0502858 + 0.998735i \(0.483987\pi\)
\(3\) 0 0
\(4\) −0.536009 + 3.03986i −0.268005 + 1.51993i
\(5\) 2.61524 + 2.19445i 1.16957 + 0.981388i 0.999991 − 0.00412812i \(-0.00131402\pi\)
0.169581 + 0.985516i \(0.445758\pi\)
\(6\) 0 0
\(7\) −0.126746 + 2.64271i −0.0479054 + 0.998852i
\(8\) −2.12267 + 1.22552i −0.750477 + 0.433288i
\(9\) 0 0
\(10\) 7.69979i 2.43489i
\(11\) −1.63538 − 1.94897i −0.493086 − 0.587637i 0.460913 − 0.887445i \(-0.347522\pi\)
−0.954000 + 0.299808i \(0.903077\pi\)
\(12\) 0 0
\(13\) −2.11236 − 5.80365i −0.585862 − 1.60964i −0.777993 − 0.628273i \(-0.783762\pi\)
0.192131 − 0.981369i \(-0.438460\pi\)
\(14\) −4.74963 + 3.61225i −1.26939 + 0.965413i
\(15\) 0 0
\(16\) 0.606533 + 0.220760i 0.151633 + 0.0551900i
\(17\) 2.39853 0.581730 0.290865 − 0.956764i \(-0.406057\pi\)
0.290865 + 0.956764i \(0.406057\pi\)
\(18\) 0 0
\(19\) − 1.24485i − 0.285589i −0.989752 − 0.142795i \(-0.954391\pi\)
0.989752 − 0.142795i \(-0.0456089\pi\)
\(20\) −8.07261 + 6.77373i −1.80509 + 1.51465i
\(21\) 0 0
\(22\) 0.996420 − 5.65098i 0.212437 − 1.20479i
\(23\) −0.769210 − 2.11339i −0.160391 − 0.440672i 0.833300 − 0.552821i \(-0.186449\pi\)
−0.993691 + 0.112149i \(0.964227\pi\)
\(24\) 0 0
\(25\) 1.15565 + 6.55401i 0.231130 + 1.31080i
\(26\) 6.96475 − 12.0633i 1.36590 − 2.36581i
\(27\) 0 0
\(28\) −7.96554 − 1.80181i −1.50535 − 0.340510i
\(29\) −0.685884 + 1.88445i −0.127366 + 0.349934i −0.986943 − 0.161072i \(-0.948505\pi\)
0.859577 + 0.511006i \(0.170727\pi\)
\(30\) 0 0
\(31\) −1.55132 − 0.273539i −0.278625 − 0.0491291i 0.0325896 − 0.999469i \(-0.489625\pi\)
−0.311214 + 0.950340i \(0.600736\pi\)
\(32\) 2.17451 + 5.97442i 0.384403 + 1.05614i
\(33\) 0 0
\(34\) 3.47723 + 4.14400i 0.596340 + 0.710691i
\(35\) −6.13078 + 6.63320i −1.03629 + 1.12122i
\(36\) 0 0
\(37\) −5.01223 − 8.68144i −0.824006 − 1.42722i −0.902677 − 0.430320i \(-0.858401\pi\)
0.0786706 − 0.996901i \(-0.474932\pi\)
\(38\) 2.15077 − 1.80471i 0.348900 − 0.292762i
\(39\) 0 0
\(40\) −8.24065 − 1.45305i −1.30296 − 0.229747i
\(41\) 5.39584 − 1.96393i 0.842689 − 0.306714i 0.115633 − 0.993292i \(-0.463110\pi\)
0.727056 + 0.686578i \(0.240888\pi\)
\(42\) 0 0
\(43\) −0.110284 − 0.625454i −0.0168182 − 0.0953809i 0.975243 − 0.221135i \(-0.0709761\pi\)
−0.992061 + 0.125754i \(0.959865\pi\)
\(44\) 6.80118 − 3.92666i 1.02532 − 0.591967i
\(45\) 0 0
\(46\) 2.53620 − 4.39283i 0.373943 − 0.647687i
\(47\) 2.09655 + 11.8901i 0.305813 + 1.73435i 0.619653 + 0.784876i \(0.287274\pi\)
−0.313839 + 0.949476i \(0.601615\pi\)
\(48\) 0 0
\(49\) −6.96787 − 0.669906i −0.995410 − 0.0957009i
\(50\) −9.64814 + 11.4982i −1.36445 + 1.62609i
\(51\) 0 0
\(52\) 18.7745 − 3.31045i 2.60356 − 0.459077i
\(53\) −4.15350 + 2.39802i −0.570527 + 0.329394i −0.757360 − 0.652998i \(-0.773511\pi\)
0.186833 + 0.982392i \(0.440178\pi\)
\(54\) 0 0
\(55\) − 8.68580i − 1.17119i
\(56\) −2.96967 − 5.76493i −0.396838 − 0.770372i
\(57\) 0 0
\(58\) −4.25016 + 1.54693i −0.558074 + 0.203122i
\(59\) −2.59194 + 0.943390i −0.337442 + 0.122819i −0.505183 − 0.863012i \(-0.668575\pi\)
0.167741 + 0.985831i \(0.446353\pi\)
\(60\) 0 0
\(61\) 7.26032 − 1.28019i 0.929589 − 0.163912i 0.311703 − 0.950179i \(-0.399100\pi\)
0.617886 + 0.786268i \(0.287989\pi\)
\(62\) −1.77639 − 3.07681i −0.225602 − 0.390755i
\(63\) 0 0
\(64\) −6.52423 + 11.3003i −0.815529 + 1.41254i
\(65\) 7.21149 − 19.8134i 0.894476 − 2.45755i
\(66\) 0 0
\(67\) −6.14145 − 5.15329i −0.750298 − 0.629574i 0.185284 − 0.982685i \(-0.440680\pi\)
−0.935582 + 0.353111i \(0.885124\pi\)
\(68\) −1.28564 + 7.29120i −0.155906 + 0.884188i
\(69\) 0 0
\(70\) −20.3483 − 0.975916i −2.43209 − 0.116644i
\(71\) −4.45847 − 2.57410i −0.529123 − 0.305489i 0.211536 − 0.977370i \(-0.432153\pi\)
−0.740659 + 0.671881i \(0.765487\pi\)
\(72\) 0 0
\(73\) 11.6132 + 6.70488i 1.35922 + 0.784747i 0.989519 − 0.144403i \(-0.0461263\pi\)
0.369702 + 0.929150i \(0.379460\pi\)
\(74\) 7.73274 − 21.2455i 0.898912 − 2.46974i
\(75\) 0 0
\(76\) 3.78418 + 0.667254i 0.434076 + 0.0765392i
\(77\) 5.35785 − 4.07482i 0.610584 − 0.464369i
\(78\) 0 0
\(79\) 0.740362 − 0.621237i 0.0832972 − 0.0698946i −0.600188 − 0.799859i \(-0.704908\pi\)
0.683485 + 0.729964i \(0.260463\pi\)
\(80\) 1.10178 + 1.90835i 0.123183 + 0.213360i
\(81\) 0 0
\(82\) 11.2156 + 6.47536i 1.23856 + 0.715084i
\(83\) 2.38230 + 0.867085i 0.261491 + 0.0951749i 0.469439 − 0.882965i \(-0.344456\pi\)
−0.207948 + 0.978140i \(0.566678\pi\)
\(84\) 0 0
\(85\) 6.27275 + 5.26346i 0.680375 + 0.570903i
\(86\) 0.920729 − 1.09728i 0.0992848 − 0.118323i
\(87\) 0 0
\(88\) 5.85988 + 2.13282i 0.624666 + 0.227360i
\(89\) 9.95066 1.05477 0.527384 − 0.849627i \(-0.323173\pi\)
0.527384 + 0.849627i \(0.323173\pi\)
\(90\) 0 0
\(91\) 15.6051 − 4.84676i 1.63586 − 0.508079i
\(92\) 6.83671 − 1.20550i 0.712776 − 0.125682i
\(93\) 0 0
\(94\) −17.5034 + 20.8598i −1.80534 + 2.15152i
\(95\) 2.73177 − 3.25560i 0.280274 − 0.334017i
\(96\) 0 0
\(97\) 7.93719 − 1.39954i 0.805899 − 0.142102i 0.244503 − 0.969649i \(-0.421375\pi\)
0.561397 + 0.827547i \(0.310264\pi\)
\(98\) −8.94414 − 13.0097i −0.903494 − 1.31418i
\(99\) 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.ba.a.341.20 132
3.2 odd 2 189.2.ba.a.131.3 yes 132
7.3 odd 6 567.2.bd.a.17.3 132
21.17 even 6 189.2.bd.a.185.20 yes 132
27.7 even 9 189.2.bd.a.47.20 yes 132
27.20 odd 18 567.2.bd.a.467.3 132
189.101 even 18 inner 567.2.ba.a.143.20 132
189.115 odd 18 189.2.ba.a.101.3 ✓ 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.3 ✓ 132 189.115 odd 18
189.2.ba.a.131.3 yes 132 3.2 odd 2
189.2.bd.a.47.20 yes 132 27.7 even 9
189.2.bd.a.185.20 yes 132 21.17 even 6
567.2.ba.a.143.20 132 189.101 even 18 inner
567.2.ba.a.341.20 132 1.1 even 1 trivial
567.2.bd.a.17.3 132 7.3 odd 6
567.2.bd.a.467.3 132 27.20 odd 18