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Properties

Label 567.2.ba.a.143.2
Level $567$
Weight $2$
Character 567.143
Analytic conductor $4.528$
Analytic rank $0$
Dimension $132$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

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 143.2
Character \(\chi\) \(=\) 567.143
Dual form 567.2.ba.a.341.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.61592 + 1.92578i) q^{2} +(-0.750127 - 4.25418i) q^{4} +(2.10349 - 1.76504i) q^{5} +(-0.122508 - 2.64291i) q^{7} +(5.05050 + 2.91591i) q^{8} +6.90302i q^{10} +(-2.69072 + 3.20668i) q^{11} +(-0.0398496 + 0.109486i) q^{13} +(5.28762 + 4.03481i) q^{14} +(-5.65800 + 2.05934i) q^{16} -5.14689 q^{17} -5.21844i q^{19} +(-9.08669 - 7.62463i) q^{20} +(-1.82736 - 10.3635i) q^{22} +(2.56763 - 7.05452i) q^{23} +(0.441075 - 2.50146i) q^{25} +(-0.146452 - 0.253662i) q^{26} +(-11.1515 + 2.50369i) q^{28} +(-1.21597 - 3.34086i) q^{29} +(-0.305335 + 0.0538389i) q^{31} +(1.18784 - 3.26355i) q^{32} +(8.31695 - 9.91175i) q^{34} +(-4.92254 - 5.34312i) q^{35} +(-0.959970 + 1.66272i) q^{37} +(10.0496 + 8.43258i) q^{38} +(15.7704 - 2.78075i) q^{40} +(-6.29347 - 2.29063i) q^{41} +(0.411100 - 2.33146i) q^{43} +(15.6602 + 9.04140i) q^{44} +(9.43633 + 16.3442i) q^{46} +(0.876823 - 4.97271i) q^{47} +(-6.96998 + 0.647559i) q^{49} +(4.10452 + 4.89157i) q^{50} +(0.495665 + 0.0873991i) q^{52} +(3.76047 + 2.17111i) q^{53} +11.4945i q^{55} +(7.08776 - 13.7053i) q^{56} +(8.39867 + 3.05687i) q^{58} +(-9.84174 - 3.58210i) q^{59} +(-1.13841 - 0.200732i) q^{61} +(0.389715 - 0.675007i) q^{62} +(-1.65570 - 2.86775i) q^{64} +(0.109424 + 0.300639i) q^{65} +(8.49204 - 7.12567i) q^{67} +(3.86082 + 21.8958i) q^{68} +(18.2441 - 0.845678i) q^{70} +(5.47124 - 3.15882i) q^{71} +(-2.23588 + 1.29089i) q^{73} +(-1.65079 - 4.53550i) q^{74} +(-22.2002 + 3.91449i) q^{76} +(8.80461 + 6.71850i) q^{77} +(12.0963 + 10.1500i) q^{79} +(-8.26674 + 14.3184i) q^{80} +(14.5810 - 8.41833i) q^{82} +(10.6778 - 3.88640i) q^{83} +(-10.8264 + 9.08446i) q^{85} +(3.82557 + 4.55914i) q^{86} +(-22.9399 + 8.34943i) q^{88} +0.913664 q^{89} +(0.294244 + 0.0919061i) q^{91} +(-31.9372 - 5.63140i) q^{92} +(8.15946 + 9.72406i) q^{94} +(-9.21076 - 10.9770i) q^{95} +(-5.88114 - 1.03700i) q^{97} +(10.0159 - 14.4690i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34}+ \cdots - 27 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61592 + 1.92578i −1.14263 + 1.36173i −0.220248 + 0.975444i \(0.570687\pi\)
−0.922379 + 0.386286i \(0.873758\pi\)
\(3\) 0 0
\(4\) −0.750127 − 4.25418i −0.375063 − 2.12709i
\(5\) 2.10349 − 1.76504i 0.940711 − 0.789350i −0.0369981 − 0.999315i \(-0.511780\pi\)
0.977709 + 0.209965i \(0.0673351\pi\)
\(6\) 0 0
\(7\) −0.122508 − 2.64291i −0.0463038 − 0.998927i
\(8\) 5.05050 + 2.91591i 1.78562 + 1.03093i
\(9\) 0 0
\(10\) 6.90302i 2.18293i
\(11\) −2.69072 + 3.20668i −0.811283 + 0.966850i −0.999884 − 0.0152039i \(-0.995160\pi\)
0.188601 + 0.982054i \(0.439605\pi\)
\(12\) 0 0
\(13\) −0.0398496 + 0.109486i −0.0110523 + 0.0303659i −0.945096 − 0.326793i \(-0.894032\pi\)
0.934044 + 0.357158i \(0.116254\pi\)
\(14\) 5.28762 + 4.03481i 1.41318 + 1.07835i
\(15\) 0 0
\(16\) −5.65800 + 2.05934i −1.41450 + 0.514836i
\(17\) −5.14689 −1.24830 −0.624152 − 0.781303i \(-0.714555\pi\)
−0.624152 + 0.781303i \(0.714555\pi\)
\(18\) 0 0
\(19\) − 5.21844i − 1.19719i −0.801051 − 0.598597i \(-0.795725\pi\)
0.801051 − 0.598597i \(-0.204275\pi\)
\(20\) −9.08669 − 7.62463i −2.03184 − 1.70492i
\(21\) 0 0
\(22\) −1.82736 − 10.3635i −0.389594 − 2.20950i
\(23\) 2.56763 − 7.05452i 0.535389 − 1.47097i −0.317186 − 0.948363i \(-0.602738\pi\)
0.852575 − 0.522605i \(-0.175040\pi\)
\(24\) 0 0
\(25\) 0.441075 − 2.50146i 0.0882150 − 0.500292i
\(26\) −0.146452 − 0.253662i −0.0287215 − 0.0497472i
\(27\) 0 0
\(28\) −11.1515 + 2.50369i −2.10744 + 0.473153i
\(29\) −1.21597 − 3.34086i −0.225801 − 0.620383i 0.774119 − 0.633040i \(-0.218193\pi\)
−0.999920 + 0.0126573i \(0.995971\pi\)
\(30\) 0 0
\(31\) −0.305335 + 0.0538389i −0.0548398 + 0.00966974i −0.201001 − 0.979591i \(-0.564419\pi\)
0.146161 + 0.989261i \(0.453308\pi\)
\(32\) 1.18784 − 3.26355i 0.209982 − 0.576920i
\(33\) 0 0
\(34\) 8.31695 − 9.91175i 1.42634 − 1.69985i
\(35\) −4.92254 − 5.34312i −0.832062 − 0.903152i
\(36\) 0 0
\(37\) −0.959970 + 1.66272i −0.157818 + 0.273349i −0.934082 − 0.357060i \(-0.883779\pi\)
0.776264 + 0.630409i \(0.217113\pi\)
\(38\) 10.0496 + 8.43258i 1.63025 + 1.36795i
\(39\) 0 0
\(40\) 15.7704 − 2.78075i 2.49352 − 0.439674i
\(41\) −6.29347 − 2.29063i −0.982874 − 0.357737i −0.199917 − 0.979813i \(-0.564067\pi\)
−0.782957 + 0.622076i \(0.786290\pi\)
\(42\) 0 0
\(43\) 0.411100 − 2.33146i 0.0626921 − 0.355545i −0.937284 − 0.348568i \(-0.886668\pi\)
0.999976 − 0.00697673i \(-0.00222078\pi\)
\(44\) 15.6602 + 9.04140i 2.36086 + 1.36304i
\(45\) 0 0
\(46\) 9.43633 + 16.3442i 1.39131 + 2.40982i
\(47\) 0.876823 − 4.97271i 0.127898 − 0.725345i −0.851647 − 0.524117i \(-0.824396\pi\)
0.979544 − 0.201228i \(-0.0644933\pi\)
\(48\) 0 0
\(49\) −6.96998 + 0.647559i −0.995712 + 0.0925084i
\(50\) 4.10452 + 4.89157i 0.580466 + 0.691773i
\(51\) 0 0
\(52\) 0.495665 + 0.0873991i 0.0687364 + 0.0121201i
\(53\) 3.76047 + 2.17111i 0.516541 + 0.298225i 0.735518 − 0.677505i \(-0.236939\pi\)
−0.218977 + 0.975730i \(0.570272\pi\)
\(54\) 0 0
\(55\) 11.4945i 1.54991i
\(56\) 7.08776 − 13.7053i 0.947142 − 1.83144i
\(57\) 0 0
\(58\) 8.39867 + 3.05687i 1.10280 + 0.401386i
\(59\) −9.84174 − 3.58210i −1.28128 − 0.466350i −0.390428 − 0.920634i \(-0.627673\pi\)
−0.890857 + 0.454284i \(0.849895\pi\)
\(60\) 0 0
\(61\) −1.13841 − 0.200732i −0.145758 − 0.0257011i 0.100293 − 0.994958i \(-0.468022\pi\)
−0.246051 + 0.969257i \(0.579133\pi\)
\(62\) 0.389715 − 0.675007i 0.0494939 − 0.0857260i
\(63\) 0 0
\(64\) −1.65570 − 2.86775i −0.206962 − 0.358469i
\(65\) 0.109424 + 0.300639i 0.0135723 + 0.0372897i
\(66\) 0 0
\(67\) 8.49204 − 7.12567i 1.03747 − 0.870539i 0.0457477 − 0.998953i \(-0.485433\pi\)
0.991721 + 0.128414i \(0.0409885\pi\)
\(68\) 3.86082 + 21.8958i 0.468193 + 2.65525i
\(69\) 0 0
\(70\) 18.2441 − 0.845678i 2.18058 − 0.101078i
\(71\) 5.47124 − 3.15882i 0.649317 − 0.374883i −0.138878 − 0.990310i \(-0.544350\pi\)
0.788194 + 0.615426i \(0.211016\pi\)
\(72\) 0 0
\(73\) −2.23588 + 1.29089i −0.261690 + 0.151087i −0.625105 − 0.780541i \(-0.714944\pi\)
0.363415 + 0.931627i \(0.381611\pi\)
\(74\) −1.65079 − 4.53550i −0.191900 − 0.527241i
\(75\) 0 0
\(76\) −22.2002 + 3.91449i −2.54654 + 0.449023i
\(77\) 8.80461 + 6.71850i 1.00338 + 0.765644i
\(78\) 0 0
\(79\) 12.0963 + 10.1500i 1.36094 + 1.14197i 0.975690 + 0.219155i \(0.0703299\pi\)
0.385252 + 0.922811i \(0.374115\pi\)
\(80\) −8.26674 + 14.3184i −0.924250 + 1.60085i
\(81\) 0 0
\(82\) 14.5810 − 8.41833i 1.61020 − 0.929649i
\(83\) 10.6778 − 3.88640i 1.17204 − 0.426588i 0.318657 − 0.947870i \(-0.396768\pi\)
0.853384 + 0.521282i \(0.174546\pi\)
\(84\) 0 0
\(85\) −10.8264 + 9.08446i −1.17429 + 0.985348i
\(86\) 3.82557 + 4.55914i 0.412522 + 0.491624i
\(87\) 0 0
\(88\) −22.9399 + 8.34943i −2.44540 + 0.890052i
\(89\) 0.913664 0.0968482 0.0484241 − 0.998827i \(-0.484580\pi\)
0.0484241 + 0.998827i \(0.484580\pi\)
\(90\) 0 0
\(91\) 0.294244 + 0.0919061i 0.0308451 + 0.00963438i
\(92\) −31.9372 − 5.63140i −3.32969 − 0.587114i
\(93\) 0 0
\(94\) 8.15946 + 9.72406i 0.841584 + 1.00296i
\(95\) −9.21076 − 10.9770i −0.945005 − 1.12621i
\(96\) 0 0
\(97\) −5.88114 − 1.03700i −0.597140 − 0.105292i −0.133094 − 0.991103i \(-0.542491\pi\)
−0.464045 + 0.885812i \(0.653602\pi\)
\(98\) 10.0159 − 14.4690i 1.01176 − 1.46159i
\(99\) 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.143.2 132
3.2 odd 2 189.2.ba.a.101.21 ✓ 132
7.5 odd 6 567.2.bd.a.467.21 132
21.5 even 6 189.2.bd.a.47.2 yes 132
27.4 even 9 189.2.bd.a.185.2 yes 132
27.23 odd 18 567.2.bd.a.17.21 132
189.131 even 18 inner 567.2.ba.a.341.2 132
189.166 odd 18 189.2.ba.a.131.21 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.21 ✓ 132 3.2 odd 2
189.2.ba.a.131.21 yes 132 189.166 odd 18
189.2.bd.a.47.2 yes 132 21.5 even 6
189.2.bd.a.185.2 yes 132 27.4 even 9
567.2.ba.a.143.2 132 1.1 even 1 trivial
567.2.ba.a.341.2 132 189.131 even 18 inner
567.2.bd.a.17.21 132 27.23 odd 18
567.2.bd.a.467.21 132 7.5 odd 6