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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(17,567)] 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("567.17"); 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([11, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.bd (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 17.16
Character \(\chi\) \(=\) 567.17
Dual form 567.2.bd.a.467.16

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.38665 - 0.244504i) q^{2} +(-0.0163608 + 0.00595485i) q^{4} +(-1.98299 + 0.721749i) q^{5} +(-2.18605 - 1.49037i) q^{7} +(-2.46004 + 1.42030i) q^{8} +(-2.57325 + 1.48566i) q^{10} +(1.26664 - 3.48007i) q^{11} +(-2.27684 - 6.25557i) q^{13} +(-3.39569 - 1.53213i) q^{14} +(-3.03727 + 2.54857i) q^{16} +(1.61887 + 2.80397i) q^{17} +(-4.61385 - 2.66381i) q^{19} +(0.0281454 - 0.0236168i) q^{20} +(0.905502 - 5.13536i) q^{22} +(-0.689782 - 0.121627i) q^{23} +(-0.418902 + 0.351501i) q^{25} +(-4.68670 - 8.11761i) q^{26} +(0.0446405 + 0.0113661i) q^{28} +(-1.50083 + 4.12351i) q^{29} +(-0.513951 - 1.41207i) q^{31} +(0.0633067 - 0.0754460i) q^{32} +(2.93040 + 3.49232i) q^{34} +(5.41058 + 1.37761i) q^{35} +6.49662 q^{37} +(-7.04913 - 2.56567i) q^{38} +(3.85312 - 4.59197i) q^{40} +(-10.4849 + 3.81619i) q^{41} +(1.29121 + 7.32280i) q^{43} +0.0644796i q^{44} -0.986227 q^{46} +(5.63412 + 2.05065i) q^{47} +(2.55760 + 6.51603i) q^{49} +(-0.494929 + 0.589833i) q^{50} +(0.0745020 + 0.0887880i) q^{52} +(-0.766350 - 0.442452i) q^{53} +7.81514i q^{55} +(7.49453 + 0.561517i) q^{56} +(-1.07292 + 6.08484i) q^{58} +(-4.77604 - 4.00757i) q^{59} +(2.54368 - 6.98869i) q^{61} +(-1.05793 - 1.83239i) q^{62} +(4.03421 - 6.98746i) q^{64} +(9.02990 + 10.7614i) q^{65} +(1.82662 - 10.3593i) q^{67} +(-0.0431834 - 0.0362352i) q^{68} +(7.83942 + 0.587358i) q^{70} +(3.83248 + 2.21268i) q^{71} -3.06138i q^{73} +(9.00855 - 1.58845i) q^{74} +(0.0913491 + 0.0161073i) q^{76} +(-7.95554 + 5.71984i) q^{77} +(-1.65307 - 9.37505i) q^{79} +(4.18344 - 7.24593i) q^{80} +(-13.6058 + 7.85533i) q^{82} +(-11.4377 - 4.16299i) q^{83} +(-5.23397 - 4.39182i) q^{85} +(3.58091 + 9.83847i) q^{86} +(1.82677 + 10.3601i) q^{88} +(2.17916 - 3.77442i) q^{89} +(-4.34583 + 17.0683i) q^{91} +(0.0120097 - 0.00211763i) q^{92} +(8.31397 + 1.46598i) q^{94} +(11.0718 + 1.95226i) q^{95} +(-6.85869 + 1.20937i) q^{97} +(5.13970 + 8.41014i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} - 9 q^{10} - 9 q^{11} + 42 q^{14} - 15 q^{16} + 9 q^{17} - 9 q^{19} + 18 q^{20} - 12 q^{22} - 30 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31}+ \cdots + 180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.38665 − 0.244504i 0.980512 − 0.172891i 0.339654 − 0.940550i \(-0.389690\pi\)
0.640858 + 0.767660i \(0.278579\pi\)
\(3\) 0 0
\(4\) −0.0163608 + 0.00595485i −0.00818041 + 0.00297743i
\(5\) −1.98299 + 0.721749i −0.886819 + 0.322776i −0.744958 − 0.667111i \(-0.767531\pi\)
−0.141861 + 0.989887i \(0.545309\pi\)
\(6\) 0 0
\(7\) −2.18605 − 1.49037i −0.826248 − 0.563307i
\(8\) −2.46004 + 1.42030i −0.869754 + 0.502153i
\(9\) 0 0
\(10\) −2.57325 + 1.48566i −0.813732 + 0.469808i
\(11\) 1.26664 − 3.48007i 0.381907 − 1.04928i −0.588645 − 0.808392i \(-0.700338\pi\)
0.970552 − 0.240890i \(-0.0774394\pi\)
\(12\) 0 0
\(13\) −2.27684 − 6.25557i −0.631482 − 1.73498i −0.676963 − 0.736017i \(-0.736704\pi\)
0.0454806 − 0.998965i \(-0.485518\pi\)
\(14\) −3.39569 − 1.53213i −0.907536 − 0.409478i
\(15\) 0 0
\(16\) −3.03727 + 2.54857i −0.759318 + 0.637143i
\(17\) 1.61887 + 2.80397i 0.392635 + 0.680063i 0.992796 − 0.119816i \(-0.0382303\pi\)
−0.600161 + 0.799879i \(0.704897\pi\)
\(18\) 0 0
\(19\) −4.61385 − 2.66381i −1.05849 − 0.611120i −0.133476 − 0.991052i \(-0.542614\pi\)
−0.925014 + 0.379932i \(0.875947\pi\)
\(20\) 0.0281454 − 0.0236168i 0.00629351 − 0.00528088i
\(21\) 0 0
\(22\) 0.905502 − 5.13536i 0.193054 − 1.09486i
\(23\) −0.689782 − 0.121627i −0.143829 − 0.0253610i 0.101270 − 0.994859i \(-0.467709\pi\)
−0.245099 + 0.969498i \(0.578821\pi\)
\(24\) 0 0
\(25\) −0.418902 + 0.351501i −0.0837805 + 0.0703002i
\(26\) −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 0.718813 + 0.695204i \(0.244686\pi\)
−0.997511 + 0.0705128i \(0.977536\pi\)
\(30\) 0 0
\(31\) −0.513951 − 1.41207i −0.0923084 − 0.253615i 0.884943 − 0.465699i \(-0.154197\pi\)
−0.977251 + 0.212084i \(0.931975\pi\)
\(32\) 0.0633067 − 0.0754460i 0.0111911 − 0.0133371i
\(33\) 0 0
\(34\) 2.93040 + 3.49232i 0.502560 + 0.598927i
\(35\) 5.41058 + 1.37761i 0.914554 + 0.232859i
\(36\) 0 0
\(37\) 6.49662 1.06804 0.534019 − 0.845473i \(-0.320681\pi\)
0.534019 + 0.845473i \(0.320681\pi\)
\(38\) −7.04913 − 2.56567i −1.14352 − 0.416207i
\(39\) 0 0
\(40\) 3.85312 − 4.59197i 0.609232 − 0.726054i
\(41\) −10.4849 + 3.81619i −1.63746 + 0.595988i −0.986593 − 0.163200i \(-0.947818\pi\)
−0.650871 + 0.759188i \(0.725596\pi\)
\(42\) 0 0
\(43\) 1.29121 + 7.32280i 0.196907 + 1.11672i 0.909676 + 0.415318i \(0.136330\pi\)
−0.712769 + 0.701399i \(0.752559\pi\)
\(44\) 0.0644796i 0.00972066i
\(45\) 0 0
\(46\) −0.986227 −0.145411
\(47\) 5.63412 + 2.05065i 0.821821 + 0.299119i 0.718498 − 0.695529i \(-0.244830\pi\)
0.103324 + 0.994648i \(0.467052\pi\)
\(48\) 0 0
\(49\) 2.55760 + 6.51603i 0.365371 + 0.930862i
\(50\) −0.494929 + 0.589833i −0.0699935 + 0.0834150i
\(51\) 0 0
\(52\) 0.0745020 + 0.0887880i 0.0103316 + 0.0123127i
\(53\) −0.766350 − 0.442452i −0.105266 − 0.0607755i 0.446443 − 0.894812i \(-0.352691\pi\)
−0.551709 + 0.834037i \(0.686024\pi\)
\(54\) 0 0
\(55\) 7.81514i 1.05379i
\(56\) 7.49453 + 0.561517i 1.00150 + 0.0750358i
\(57\) 0 0
\(58\) −1.07292 + 6.08484i −0.140881 + 0.798978i
\(59\) −4.77604 − 4.00757i −0.621787 − 0.521742i 0.276577 − 0.960992i \(-0.410800\pi\)
−0.898365 + 0.439250i \(0.855244\pi\)
\(60\) 0 0
\(61\) 2.54368 − 6.98869i 0.325684 − 0.894811i −0.663506 − 0.748171i \(-0.730932\pi\)
0.989190 − 0.146639i \(-0.0468456\pi\)
\(62\) −1.05793 − 1.83239i −0.134357 − 0.232713i
\(63\) 0 0
\(64\) 4.03421 − 6.98746i 0.504277 − 0.873433i
\(65\) 9.02990 + 10.7614i 1.12002 + 1.33479i
\(66\) 0 0
\(67\) 1.82662 − 10.3593i 0.223157 − 1.26559i −0.643019 − 0.765850i \(-0.722319\pi\)
0.866177 − 0.499738i \(-0.166570\pi\)
\(68\) −0.0431834 − 0.0362352i −0.00523675 − 0.00439416i
\(69\) 0 0
\(70\) 7.83942 + 0.587358i 0.936990 + 0.0702027i
\(71\) 3.83248 + 2.21268i 0.454832 + 0.262597i 0.709869 − 0.704334i \(-0.248754\pi\)
−0.255037 + 0.966931i \(0.582088\pi\)
\(72\) 0 0
\(73\) − 3.06138i − 0.358307i −0.983821 − 0.179154i \(-0.942664\pi\)
0.983821 − 0.179154i \(-0.0573359\pi\)
\(74\) 9.00855 − 1.58845i 1.04722 − 0.184654i
\(75\) 0 0
\(76\) 0.0913491 + 0.0161073i 0.0104785 + 0.00184763i
\(77\) −7.95554 + 5.71984i −0.906618 + 0.651836i
\(78\) 0 0
\(79\) −1.65307 − 9.37505i −0.185985 − 1.05478i −0.924683 − 0.380739i \(-0.875670\pi\)
0.738697 − 0.674037i \(-0.235441\pi\)
\(80\) 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 \(-0.621744\pi\)
−0.882241 + 0.470799i \(0.843966\pi\)
\(84\) 0 0
\(85\) −5.23397 − 4.39182i −0.567704 − 0.476360i
\(86\) 3.58091 + 9.83847i 0.386140 + 1.06091i
\(87\) 0 0
\(88\) 1.82677 + 10.3601i 0.194734 + 1.10439i
\(89\) 2.17916 − 3.77442i 0.230991 − 0.400088i −0.727109 − 0.686522i \(-0.759137\pi\)
0.958100 + 0.286434i \(0.0924699\pi\)
\(90\) 0 0
\(91\) −4.34583 + 17.0683i −0.455567 + 1.78924i
\(92\) 0.0120097 − 0.00211763i 0.00125210 − 0.000220778i
\(93\) 0 0
\(94\) 8.31397 + 1.46598i 0.857520 + 0.151204i
\(95\) 11.0718 + 1.95226i 1.13594 + 0.200298i
\(96\) 0 0
\(97\) −6.85869 + 1.20937i −0.696394 + 0.122793i −0.510629 − 0.859801i \(-0.670587\pi\)
−0.185765 + 0.982594i \(0.559476\pi\)
\(98\) 5.13970 + 8.41014i 0.519188 + 0.849552i
\(99\) 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.bd.a.17.16 132
3.2 odd 2 189.2.bd.a.185.7 yes 132
7.5 odd 6 567.2.ba.a.341.7 132
21.5 even 6 189.2.ba.a.131.16 yes 132
27.7 even 9 189.2.ba.a.101.16 ✓ 132
27.20 odd 18 567.2.ba.a.143.7 132
189.47 even 18 inner 567.2.bd.a.467.16 132
189.61 odd 18 189.2.bd.a.47.7 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.16 ✓ 132 27.7 even 9
189.2.ba.a.131.16 yes 132 21.5 even 6
189.2.bd.a.47.7 yes 132 189.61 odd 18
189.2.bd.a.185.7 yes 132 3.2 odd 2
567.2.ba.a.143.7 132 27.20 odd 18
567.2.ba.a.341.7 132 7.5 odd 6
567.2.bd.a.17.16 132 1.1 even 1 trivial
567.2.bd.a.467.16 132 189.47 even 18 inner