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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 143.7
Character \(\chi\) \(=\) 567.143
Dual form 567.2.ba.a.341.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.905074 + 1.07862i) q^{2} +(0.00302336 + 0.0171463i) q^{4} +(-1.61655 + 1.35644i) q^{5} +(1.08812 - 2.41164i) q^{7} +(-2.46004 - 1.42030i) q^{8} -2.97133i q^{10} +(2.38051 - 2.83698i) q^{11} +(2.27684 - 6.25557i) q^{13} +(1.61642 + 3.35639i) q^{14} +(3.72576 - 1.35607i) q^{16} +3.23775 q^{17} +5.32762i q^{19} +(-0.0281454 - 0.0236168i) q^{20} +(0.905502 + 5.13536i) q^{22} +(0.239559 - 0.658182i) q^{23} +(-0.0949575 + 0.538530i) q^{25} +(4.68670 + 8.11761i) q^{26} +(0.0446405 + 0.0113661i) q^{28} +(-1.50083 - 4.12351i) q^{29} +(-1.47986 + 0.260940i) q^{31} +(0.0336848 - 0.0925482i) q^{32} +(-2.93040 + 3.49232i) q^{34} +(1.51224 + 5.37450i) q^{35} +(-3.24831 + 5.62624i) q^{37} +(-5.74650 - 4.82189i) q^{38} +(5.90332 - 1.04091i) q^{40} +(10.4849 + 3.81619i) q^{41} +(1.29121 - 7.32280i) q^{43} +(0.0558409 + 0.0322398i) q^{44} +(0.493113 + 0.854097i) q^{46} +(1.04114 - 5.90462i) q^{47} +(-4.63197 - 5.24832i) q^{49} +(-0.494929 - 0.589833i) q^{50} +(0.114144 + 0.0201266i) q^{52} +(0.766350 + 0.442452i) q^{53} +7.81514i q^{55} +(-6.10208 + 4.38724i) q^{56} +(5.80608 + 2.11324i) q^{58} +(-5.85868 - 2.13238i) q^{59} +(7.32422 + 1.29146i) q^{61} +(1.05793 - 1.83239i) q^{62} +(4.03421 + 6.98746i) q^{64} +(4.80471 + 13.2008i) q^{65} +(8.05809 - 6.76154i) q^{67} +(0.00978888 + 0.0555155i) q^{68} +(-7.16576 - 3.23318i) q^{70} +(3.83248 - 2.21268i) q^{71} +(-2.65123 + 1.53069i) q^{73} +(-3.12864 - 8.59586i) q^{74} +(-0.0913491 + 0.0161073i) q^{76} +(-4.25147 - 8.82791i) q^{77} +(-7.29250 - 6.11913i) q^{79} +(-4.18344 + 7.24593i) q^{80} +(-13.6058 + 7.85533i) q^{82} +(11.4377 - 4.16299i) q^{83} +(-5.23397 + 4.39182i) q^{85} +(6.72991 + 8.02040i) q^{86} +(-9.88551 + 3.59803i) q^{88} +4.35832 q^{89} +(-12.6087 - 12.2977i) q^{91} +(0.0120097 + 0.00211763i) q^{92} +(5.42656 + 6.46712i) q^{94} +(-7.22662 - 8.61234i) q^{95} +(6.85869 + 1.20937i) q^{97} +(9.85324 - 0.246039i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34}+ \cdots - 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.905074 + 1.07862i −0.639984 + 0.762703i −0.984368 − 0.176126i \(-0.943643\pi\)
0.344384 + 0.938829i \(0.388088\pi\)
\(3\) 0 0
\(4\) 0.00302336 + 0.0171463i 0.00151168 + 0.00857316i
\(5\) −1.61655 + 1.35644i −0.722942 + 0.606620i −0.928197 − 0.372088i \(-0.878642\pi\)
0.205256 + 0.978708i \(0.434197\pi\)
\(6\) 0 0
\(7\) 1.08812 − 2.41164i 0.411273 − 0.911512i
\(8\) −2.46004 − 1.42030i −0.869754 − 0.502153i
\(9\) 0 0
\(10\) − 2.97133i − 0.939616i
\(11\) 2.38051 − 2.83698i 0.717751 − 0.855382i −0.276659 − 0.960968i \(-0.589227\pi\)
0.994410 + 0.105586i \(0.0336717\pi\)
\(12\) 0 0
\(13\) 2.27684 − 6.25557i 0.631482 − 1.73498i −0.0454806 − 0.998965i \(-0.514482\pi\)
0.676963 − 0.736017i \(-0.263296\pi\)
\(14\) 1.61642 + 3.35639i 0.432005 + 0.897032i
\(15\) 0 0
\(16\) 3.72576 − 1.35607i 0.931441 − 0.339017i
\(17\) 3.23775 0.785270 0.392635 − 0.919694i \(-0.371564\pi\)
0.392635 + 0.919694i \(0.371564\pi\)
\(18\) 0 0
\(19\) 5.32762i 1.22224i 0.791538 + 0.611120i \(0.209281\pi\)
−0.791538 + 0.611120i \(0.790719\pi\)
\(20\) −0.0281454 − 0.0236168i −0.00629351 − 0.00528088i
\(21\) 0 0
\(22\) 0.905502 + 5.13536i 0.193054 + 1.09486i
\(23\) 0.239559 − 0.658182i 0.0499514 − 0.137240i −0.912208 − 0.409727i \(-0.865624\pi\)
0.962160 + 0.272487i \(0.0878461\pi\)
\(24\) 0 0
\(25\) −0.0949575 + 0.538530i −0.0189915 + 0.107706i
\(26\) 4.68670 + 8.11761i 0.919138 + 1.59199i
\(27\) 0 0
\(28\) 0.0446405 + 0.0113661i 0.00843625 + 0.00214799i
\(29\) −1.50083 − 4.12351i −0.278698 − 0.765716i −0.997511 − 0.0705128i \(-0.977536\pi\)
0.718813 − 0.695204i \(-0.244686\pi\)
\(30\) 0 0
\(31\) −1.47986 + 0.260940i −0.265791 + 0.0468662i −0.304955 − 0.952367i \(-0.598642\pi\)
0.0391641 + 0.999233i \(0.487530\pi\)
\(32\) 0.0336848 − 0.0925482i 0.00595468 − 0.0163604i
\(33\) 0 0
\(34\) −2.93040 + 3.49232i −0.502560 + 0.598927i
\(35\) 1.51224 + 5.37450i 0.255616 + 0.908456i
\(36\) 0 0
\(37\) −3.24831 + 5.62624i −0.534019 + 0.924947i 0.465191 + 0.885210i \(0.345985\pi\)
−0.999210 + 0.0397374i \(0.987348\pi\)
\(38\) −5.74650 − 4.82189i −0.932206 − 0.782213i
\(39\) 0 0
\(40\) 5.90332 − 1.04091i 0.933397 − 0.164583i
\(41\) 10.4849 + 3.81619i 1.63746 + 0.595988i 0.986593 − 0.163200i \(-0.0521817\pi\)
0.650871 + 0.759188i \(0.274404\pi\)
\(42\) 0 0
\(43\) 1.29121 − 7.32280i 0.196907 − 1.11672i −0.712769 − 0.701399i \(-0.752559\pi\)
0.909676 − 0.415318i \(-0.136330\pi\)
\(44\) 0.0558409 + 0.0322398i 0.00841834 + 0.00486033i
\(45\) 0 0
\(46\) 0.493113 + 0.854097i 0.0727056 + 0.125930i
\(47\) 1.04114 − 5.90462i 0.151866 − 0.861278i −0.809728 − 0.586805i \(-0.800386\pi\)
0.961595 − 0.274473i \(-0.0885033\pi\)
\(48\) 0 0
\(49\) −4.63197 − 5.24832i −0.661710 − 0.749760i
\(50\) −0.494929 − 0.589833i −0.0699935 − 0.0834150i
\(51\) 0 0
\(52\) 0.114144 + 0.0201266i 0.0158289 + 0.00279106i
\(53\) 0.766350 + 0.442452i 0.105266 + 0.0607755i 0.551709 − 0.834037i \(-0.313976\pi\)
−0.446443 + 0.894812i \(0.647309\pi\)
\(54\) 0 0
\(55\) 7.81514i 1.05379i
\(56\) −6.10208 + 4.38724i −0.815424 + 0.586270i
\(57\) 0 0
\(58\) 5.80608 + 2.11324i 0.762376 + 0.277482i
\(59\) −5.85868 − 2.13238i −0.762735 − 0.277613i −0.0687807 − 0.997632i \(-0.521911\pi\)
−0.693954 + 0.720019i \(0.744133\pi\)
\(60\) 0 0
\(61\) 7.32422 + 1.29146i 0.937771 + 0.165354i 0.621588 − 0.783344i \(-0.286488\pi\)
0.316183 + 0.948698i \(0.397599\pi\)
\(62\) 1.05793 − 1.83239i 0.134357 − 0.232713i
\(63\) 0 0
\(64\) 4.03421 + 6.98746i 0.504277 + 0.873433i
\(65\) 4.80471 + 13.2008i 0.595951 + 1.63736i
\(66\) 0 0
\(67\) 8.05809 − 6.76154i 0.984453 − 0.826054i −0.000302478 − 1.00000i \(-0.500096\pi\)
0.984755 + 0.173946i \(0.0556518\pi\)
\(68\) 0.00978888 + 0.0555155i 0.00118708 + 0.00673224i
\(69\) 0 0
\(70\) −7.16576 − 3.23318i −0.856472 − 0.386438i
\(71\) 3.83248 − 2.21268i 0.454832 − 0.262597i −0.255037 − 0.966931i \(-0.582088\pi\)
0.709869 + 0.704334i \(0.248754\pi\)
\(72\) 0 0
\(73\) −2.65123 + 1.53069i −0.310303 + 0.179154i −0.647062 − 0.762437i \(-0.724003\pi\)
0.336759 + 0.941591i \(0.390669\pi\)
\(74\) −3.12864 − 8.59586i −0.363697 − 0.999249i
\(75\) 0 0
\(76\) −0.0913491 + 0.0161073i −0.0104785 + 0.00184763i
\(77\) −4.25147 − 8.82791i −0.484500 − 1.00603i
\(78\) 0 0
\(79\) −7.29250 − 6.11913i −0.820470 − 0.688456i 0.132612 − 0.991168i \(-0.457664\pi\)
−0.953082 + 0.302712i \(0.902108\pi\)
\(80\) −4.18344 + 7.24593i −0.467723 + 0.810120i
\(81\) 0 0
\(82\) −13.6058 + 7.85533i −1.50251 + 0.867476i
\(83\) 11.4377 − 4.16299i 1.25545 − 0.456947i 0.373212 − 0.927746i \(-0.378256\pi\)
0.882241 + 0.470799i \(0.156034\pi\)
\(84\) 0 0
\(85\) −5.23397 + 4.39182i −0.567704 + 0.476360i
\(86\) 6.72991 + 8.02040i 0.725705 + 0.864862i
\(87\) 0 0
\(88\) −9.88551 + 3.59803i −1.05380 + 0.383552i
\(89\) 4.35832 0.461982 0.230991 − 0.972956i \(-0.425803\pi\)
0.230991 + 0.972956i \(0.425803\pi\)
\(90\) 0 0
\(91\) −12.6087 − 12.2977i −1.32175 − 1.28915i
\(92\) 0.0120097 + 0.00211763i 0.00125210 + 0.000220778i
\(93\) 0 0
\(94\) 5.42656 + 6.46712i 0.559707 + 0.667033i
\(95\) −7.22662 − 8.61234i −0.741435 − 0.883608i
\(96\) 0 0
\(97\) 6.85869 + 1.20937i 0.696394 + 0.122793i 0.510629 − 0.859801i \(-0.329413\pi\)
0.185765 + 0.982594i \(0.440524\pi\)
\(98\) 9.85324 − 0.246039i 0.995328 − 0.0248537i
\(99\) 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.143.7 132
3.2 odd 2 189.2.ba.a.101.16 ✓ 132
7.5 odd 6 567.2.bd.a.467.16 132
21.5 even 6 189.2.bd.a.47.7 yes 132
27.4 even 9 189.2.bd.a.185.7 yes 132
27.23 odd 18 567.2.bd.a.17.16 132
189.131 even 18 inner 567.2.ba.a.341.7 132
189.166 odd 18 189.2.ba.a.131.16 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.16 ✓ 132 3.2 odd 2
189.2.ba.a.131.16 yes 132 189.166 odd 18
189.2.bd.a.47.7 yes 132 21.5 even 6
189.2.bd.a.185.7 yes 132 27.4 even 9
567.2.ba.a.143.7 132 1.1 even 1 trivial
567.2.ba.a.341.7 132 189.131 even 18 inner
567.2.bd.a.17.16 132 27.23 odd 18
567.2.bd.a.467.16 132 7.5 odd 6