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Properties

Label 567.2.be.a.503.8
Level $567$
Weight $2$
Character 567.503
Analytic conductor $4.528$
Analytic rank $0$
Dimension $132$
Inner twists $4$

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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(62,567)] 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("567.62"); 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, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.be (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 503.8
Character \(\chi\) \(=\) 567.503
Dual form 567.2.be.a.62.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.471430 + 1.29524i) q^{2} +(0.0766803 + 0.0643424i) q^{4} +(0.459149 - 2.60396i) q^{5} +(-1.47152 - 2.19878i) q^{7} +(-2.50689 + 1.44736i) q^{8} +(3.15631 + 1.82230i) q^{10} +(0.681802 - 0.120220i) q^{11} +(0.171823 + 0.472079i) q^{13} +(3.54167 - 0.869398i) q^{14} +(-0.658089 - 3.73221i) q^{16} +(2.42375 - 4.19806i) q^{17} +(-1.03438 + 0.597199i) q^{19} +(0.202753 - 0.170130i) q^{20} +(-0.165708 + 0.939774i) q^{22} +(4.74912 - 5.65978i) q^{23} +(-1.87135 - 0.681115i) q^{25} -0.692459 q^{26} +(0.0286387 - 0.263284i) q^{28} +(2.19806 - 6.03911i) q^{29} +(5.17613 - 6.16868i) q^{31} +(-0.557110 - 0.0982336i) q^{32} +(4.29488 + 5.11844i) q^{34} +(-6.40120 + 2.82220i) q^{35} +(-3.71414 + 6.43307i) q^{37} +(-0.285881 - 1.62131i) q^{38} +(2.61782 + 7.19241i) q^{40} +(-3.25278 + 1.18391i) q^{41} +(-1.81376 - 10.2863i) q^{43} +(0.0600160 + 0.0346503i) q^{44} +(5.09191 + 8.81945i) q^{46} +(2.93133 - 2.45968i) q^{47} +(-2.66929 + 6.47108i) q^{49} +(1.76442 - 2.10275i) q^{50} +(-0.0171993 + 0.0472546i) q^{52} +10.3394i q^{53} -1.83059i q^{55} +(6.87135 + 3.38231i) q^{56} +(6.78588 + 5.69403i) q^{58} +(-1.13100 + 6.41423i) q^{59} +(2.36159 + 2.81444i) q^{61} +(5.54975 + 9.61245i) q^{62} +(4.17966 - 7.23938i) q^{64} +(1.30817 - 0.230665i) q^{65} +(-3.24000 + 1.17926i) q^{67} +(0.455968 - 0.165959i) q^{68} +(-0.637725 - 9.62158i) q^{70} +(-0.286623 - 0.165482i) q^{71} +(3.13921 - 1.81243i) q^{73} +(-6.58144 - 7.84345i) q^{74} +(-0.117742 - 0.0207610i) q^{76} +(-1.26762 - 1.32223i) q^{77} +(9.71572 + 3.53623i) q^{79} -10.0207 q^{80} -4.77127i q^{82} +(-12.8972 - 4.69419i) q^{83} +(-9.81874 - 8.23890i) q^{85} +(14.1784 + 2.50003i) q^{86} +(-1.53520 + 1.28819i) q^{88} +(4.84538 + 8.39244i) q^{89} +(0.785159 - 1.07247i) q^{91} +(0.728328 - 0.128424i) q^{92} +(1.80397 + 4.95636i) q^{94} +(1.08015 + 2.96769i) q^{95} +(-3.19371 + 0.563137i) q^{97} +(-7.12324 - 6.50804i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 12 q^{2} - 12 q^{4} - 6 q^{7} + 18 q^{8} + 18 q^{11} - 3 q^{14} - 24 q^{16} - 12 q^{22} - 12 q^{23} - 12 q^{25} - 12 q^{28} + 48 q^{29} + 6 q^{32} + 36 q^{35} - 6 q^{37} - 12 q^{43} + 18 q^{44}+ \cdots - 126 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.471430 + 1.29524i −0.333351 + 0.915875i 0.653882 + 0.756596i \(0.273139\pi\)
−0.987234 + 0.159279i \(0.949083\pi\)
\(3\) 0 0
\(4\) 0.0766803 + 0.0643424i 0.0383402 + 0.0321712i
\(5\) 0.459149 − 2.60396i 0.205338 − 1.16453i −0.691570 − 0.722310i \(-0.743081\pi\)
0.896907 − 0.442218i \(-0.145808\pi\)
\(6\) 0 0
\(7\) −1.47152 − 2.19878i −0.556180 − 0.831062i
\(8\) −2.50689 + 1.44736i −0.886321 + 0.511718i
\(9\) 0 0
\(10\) 3.15631 + 1.82230i 0.998113 + 0.576261i
\(11\) 0.681802 − 0.120220i 0.205571 − 0.0362477i −0.0699143 − 0.997553i \(-0.522273\pi\)
0.275485 + 0.961305i \(0.411161\pi\)
\(12\) 0 0
\(13\) 0.171823 + 0.472079i 0.0476550 + 0.130931i 0.961237 − 0.275724i \(-0.0889174\pi\)
−0.913582 + 0.406655i \(0.866695\pi\)
\(14\) 3.54167 − 0.869398i 0.946552 − 0.232356i
\(15\) 0 0
\(16\) −0.658089 − 3.73221i −0.164522 − 0.933052i
\(17\) 2.42375 − 4.19806i 0.587846 − 1.01818i −0.406668 − 0.913576i \(-0.633309\pi\)
0.994514 − 0.104604i \(-0.0333574\pi\)
\(18\) 0 0
\(19\) −1.03438 + 0.597199i −0.237303 + 0.137007i −0.613936 − 0.789355i \(-0.710415\pi\)
0.376634 + 0.926362i \(0.377082\pi\)
\(20\) 0.202753 − 0.170130i 0.0453370 − 0.0380422i
\(21\) 0 0
\(22\) −0.165708 + 0.939774i −0.0353290 + 0.200361i
\(23\) 4.74912 − 5.65978i 0.990259 − 1.18014i 0.00662300 − 0.999978i \(-0.497892\pi\)
0.983636 − 0.180167i \(-0.0576637\pi\)
\(24\) 0 0
\(25\) −1.87135 − 0.681115i −0.374270 − 0.136223i
\(26\) −0.692459 −0.135802
\(27\) 0 0
\(28\) 0.0286387 − 0.263284i 0.00541221 − 0.0497560i
\(29\) 2.19806 − 6.03911i 0.408169 − 1.12143i −0.549984 − 0.835175i \(-0.685366\pi\)
0.958152 − 0.286259i \(-0.0924117\pi\)
\(30\) 0 0
\(31\) 5.17613 − 6.16868i 0.929661 − 1.10793i −0.0642712 − 0.997932i \(-0.520472\pi\)
0.993932 − 0.109994i \(-0.0350833\pi\)
\(32\) −0.557110 − 0.0982336i −0.0984841 − 0.0173654i
\(33\) 0 0
\(34\) 4.29488 + 5.11844i 0.736566 + 0.877806i
\(35\) −6.40120 + 2.82220i −1.08200 + 0.477039i
\(36\) 0 0
\(37\) −3.71414 + 6.43307i −0.610600 + 1.05759i 0.380539 + 0.924765i \(0.375738\pi\)
−0.991139 + 0.132826i \(0.957595\pi\)
\(38\) −0.285881 − 1.62131i −0.0463759 − 0.263011i
\(39\) 0 0
\(40\) 2.61782 + 7.19241i 0.413914 + 1.13722i
\(41\) −3.25278 + 1.18391i −0.507998 + 0.184896i −0.583288 − 0.812265i \(-0.698234\pi\)
0.0752897 + 0.997162i \(0.476012\pi\)
\(42\) 0 0
\(43\) −1.81376 − 10.2863i −0.276596 − 1.56865i −0.733846 − 0.679315i \(-0.762277\pi\)
0.457250 − 0.889338i \(-0.348834\pi\)
\(44\) 0.0600160 + 0.0346503i 0.00904776 + 0.00522373i
\(45\) 0 0
\(46\) 5.09191 + 8.81945i 0.750761 + 1.30036i
\(47\) 2.93133 − 2.45968i 0.427579 − 0.358781i −0.403458 − 0.914998i \(-0.632192\pi\)
0.831037 + 0.556217i \(0.187748\pi\)
\(48\) 0 0
\(49\) −2.66929 + 6.47108i −0.381327 + 0.924440i
\(50\) 1.76442 − 2.10275i 0.249527 − 0.297374i
\(51\) 0 0
\(52\) −0.0171993 + 0.0472546i −0.00238511 + 0.00655304i
\(53\) 10.3394i 1.42022i 0.704090 + 0.710111i \(0.251355\pi\)
−0.704090 + 0.710111i \(0.748645\pi\)
\(54\) 0 0
\(55\) − 1.83059i − 0.246836i
\(56\) 6.87135 + 3.38231i 0.918223 + 0.451980i
\(57\) 0 0
\(58\) 6.78588 + 5.69403i 0.891030 + 0.747663i
\(59\) −1.13100 + 6.41423i −0.147244 + 0.835061i 0.818294 + 0.574799i \(0.194920\pi\)
−0.965538 + 0.260262i \(0.916191\pi\)
\(60\) 0 0
\(61\) 2.36159 + 2.81444i 0.302371 + 0.360352i 0.895739 − 0.444579i \(-0.146647\pi\)
−0.593369 + 0.804931i \(0.702202\pi\)
\(62\) 5.54975 + 9.61245i 0.704819 + 1.22078i
\(63\) 0 0
\(64\) 4.17966 − 7.23938i 0.522457 − 0.904922i
\(65\) 1.30817 − 0.230665i 0.162258 − 0.0286105i
\(66\) 0 0
\(67\) −3.24000 + 1.17926i −0.395829 + 0.144070i −0.532263 − 0.846579i \(-0.678658\pi\)
0.136434 + 0.990649i \(0.456436\pi\)
\(68\) 0.455968 − 0.165959i 0.0552942 − 0.0201254i
\(69\) 0 0
\(70\) −0.637725 − 9.62158i −0.0762227 − 1.15000i
\(71\) −0.286623 − 0.165482i −0.0340159 − 0.0196391i 0.482896 − 0.875678i \(-0.339585\pi\)
−0.516912 + 0.856039i \(0.672918\pi\)
\(72\) 0 0
\(73\) 3.13921 − 1.81243i 0.367417 − 0.212128i −0.304912 − 0.952380i \(-0.598627\pi\)
0.672329 + 0.740252i \(0.265294\pi\)
\(74\) −6.58144 − 7.84345i −0.765077 − 0.911783i
\(75\) 0 0
\(76\) −0.117742 − 0.0207610i −0.0135059 − 0.00238145i
\(77\) −1.26762 − 1.32223i −0.144459 − 0.150682i
\(78\) 0 0
\(79\) 9.71572 + 3.53623i 1.09310 + 0.397857i 0.824770 − 0.565469i \(-0.191305\pi\)
0.268334 + 0.963326i \(0.413527\pi\)
\(80\) −10.0207 −1.12035
\(81\) 0 0
\(82\) − 4.77127i − 0.526898i
\(83\) −12.8972 − 4.69419i −1.41565 − 0.515254i −0.482866 − 0.875694i \(-0.660404\pi\)
−0.932782 + 0.360440i \(0.882627\pi\)
\(84\) 0 0
\(85\) −9.81874 − 8.23890i −1.06499 − 0.893634i
\(86\) 14.1784 + 2.50003i 1.52889 + 0.269585i
\(87\) 0 0
\(88\) −1.53520 + 1.28819i −0.163653 + 0.137321i
\(89\) 4.84538 + 8.39244i 0.513609 + 0.889597i 0.999875 + 0.0157861i \(0.00502510\pi\)
−0.486266 + 0.873811i \(0.661642\pi\)
\(90\) 0 0
\(91\) 0.785159 − 1.07247i 0.0823070 − 0.112426i
\(92\) 0.728328 − 0.128424i 0.0759334 − 0.0133891i
\(93\) 0 0
\(94\) 1.80397 + 4.95636i 0.186065 + 0.511209i
\(95\) 1.08015 + 2.96769i 0.110821 + 0.304478i
\(96\) 0 0
\(97\) −3.19371 + 0.563137i −0.324272 + 0.0571779i −0.333415 − 0.942780i \(-0.608201\pi\)
0.00914255 + 0.999958i \(0.497090\pi\)
\(98\) −7.12324 − 6.50804i −0.719556 − 0.657411i
\(99\) 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.be.a.503.8 132
3.2 odd 2 189.2.be.a.104.16 yes 132
7.6 odd 2 inner 567.2.be.a.503.7 132
21.20 even 2 189.2.be.a.104.15 yes 132
27.7 even 9 189.2.be.a.20.15 ✓ 132
27.20 odd 18 inner 567.2.be.a.62.7 132
189.20 even 18 inner 567.2.be.a.62.8 132
189.34 odd 18 189.2.be.a.20.16 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.be.a.20.15 ✓ 132 27.7 even 9
189.2.be.a.20.16 yes 132 189.34 odd 18
189.2.be.a.104.15 yes 132 21.20 even 2
189.2.be.a.104.16 yes 132 3.2 odd 2
567.2.be.a.62.7 132 27.20 odd 18 inner
567.2.be.a.62.8 132 189.20 even 18 inner
567.2.be.a.503.7 132 7.6 odd 2 inner
567.2.be.a.503.8 132 1.1 even 1 trivial