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Properties

Label 567.2.bd.a.17.3
Level $567$
Weight $2$
Character 567.17
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(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 17.3
Character \(\chi\) \(=\) 567.17
Dual form 567.2.bd.a.467.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.22112 + 0.391643i) q^{2} +(2.90060 - 1.05573i) q^{4} +(3.20807 - 1.16764i) q^{5} +(-2.62457 - 0.334082i) q^{7} +(-2.12267 + 1.22552i) q^{8} +(-6.66821 + 3.84989i) q^{10} +(-0.870169 + 2.39077i) q^{11} +(2.11236 + 5.80365i) q^{13} +(5.96033 - 0.285861i) q^{14} +(-0.494450 + 0.414893i) q^{16} +(1.19927 + 2.07719i) q^{17} +(-1.07808 - 0.622427i) q^{19} +(8.07261 - 6.77373i) q^{20} +(0.996420 - 5.65098i) q^{22} +(2.21485 + 0.390538i) q^{23} +(5.09811 - 4.27783i) q^{25} +(-6.96475 - 12.0633i) q^{26} +(-7.96554 + 1.80181i) q^{28} +(-0.685884 + 1.88445i) q^{29} +(-0.538767 - 1.48025i) q^{31} +(4.08675 - 4.87040i) q^{32} +(-3.47723 - 4.14400i) q^{34} +(-8.80991 + 1.99281i) q^{35} +10.0245 q^{37} +(2.63830 + 0.960264i) q^{38} +(-5.37870 + 6.41008i) q^{40} +(-5.39584 + 1.96393i) q^{41} +(-0.110284 - 0.625454i) q^{43} +7.85333i q^{44} -5.07240 q^{46} +(11.3454 + 4.12940i) q^{47} +(6.77678 + 1.75365i) q^{49} +(-9.64814 + 11.4982i) q^{50} +(12.2542 + 14.6040i) q^{52} +(4.15350 + 2.39802i) q^{53} +8.68580i q^{55} +(5.98053 - 2.50733i) q^{56} +(0.785398 - 4.45421i) q^{58} +(-2.11297 - 1.77299i) q^{59} +(2.52148 - 6.92772i) q^{61} +(1.77639 + 3.07681i) q^{62} +(-6.52423 + 11.3003i) q^{64} +(13.5532 + 16.1520i) q^{65} +(-1.39215 + 7.89530i) q^{67} +(5.67155 + 4.75899i) q^{68} +(18.7874 - 7.87660i) q^{70} +(-4.45847 - 2.57410i) q^{71} +13.4098i q^{73} +(-22.2655 + 3.92601i) q^{74} +(-3.78418 - 0.667254i) q^{76} +(3.08253 - 5.98404i) q^{77} +(0.167826 + 0.951791i) q^{79} +(-1.10178 + 1.90835i) q^{80} +(11.2156 - 6.47536i) q^{82} +(-2.38230 - 0.867085i) q^{83} +(6.27275 + 5.26346i) q^{85} +(0.489910 + 1.34602i) q^{86} +(-1.08286 - 6.14122i) q^{88} +(4.97533 - 8.61753i) q^{89} +(-3.60514 - 15.9378i) q^{91} +(6.83671 - 1.20550i) q^{92} +(-26.8168 - 4.72852i) q^{94} +(-4.18532 - 0.737985i) q^{95} +(-7.93719 + 1.39954i) q^{97} +(-15.7388 - 1.24098i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} - 9 q^{10} - 9 q^{11} + 42 q^{14} - 15 q^{16} + 9 q^{17} - 9 q^{19} + 18 q^{20} - 12 q^{22} - 30 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31}+ \cdots + 180 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.22112 + 0.391643i −1.57057 + 0.276934i −0.890073 − 0.455819i \(-0.849347\pi\)
−0.680496 + 0.732752i \(0.738235\pi\)
\(3\) 0 0
\(4\) 2.90060 − 1.05573i 1.45030 − 0.527866i
\(5\) 3.20807 − 1.16764i 1.43469 − 0.522186i 0.496421 − 0.868082i \(-0.334647\pi\)
0.938273 + 0.345896i \(0.112425\pi\)
\(6\) 0 0
\(7\) −2.62457 − 0.334082i −0.991996 − 0.126271i
\(8\) −2.12267 + 1.22552i −0.750477 + 0.433288i
\(9\) 0 0
\(10\) −6.66821 + 3.84989i −2.10867 + 1.21744i
\(11\) −0.870169 + 2.39077i −0.262366 + 0.720844i 0.736641 + 0.676284i \(0.236411\pi\)
−0.999007 + 0.0445598i \(0.985811\pi\)
\(12\) 0 0
\(13\) 2.11236 + 5.80365i 0.585862 + 1.60964i 0.777993 + 0.628273i \(0.216238\pi\)
−0.192131 + 0.981369i \(0.561540\pi\)
\(14\) 5.96033 − 0.285861i 1.59297 − 0.0763994i
\(15\) 0 0
\(16\) −0.494450 + 0.414893i −0.123613 + 0.103723i
\(17\) 1.19927 + 2.07719i 0.290865 + 0.503793i 0.974014 − 0.226486i \(-0.0727237\pi\)
−0.683150 + 0.730278i \(0.739390\pi\)
\(18\) 0 0
\(19\) −1.07808 − 0.622427i −0.247328 − 0.142795i 0.371212 − 0.928548i \(-0.378942\pi\)
−0.618540 + 0.785753i \(0.712276\pi\)
\(20\) 8.07261 − 6.77373i 1.80509 − 1.51465i
\(21\) 0 0
\(22\) 0.996420 − 5.65098i 0.212437 − 1.20479i
\(23\) 2.21485 + 0.390538i 0.461829 + 0.0814329i 0.399722 − 0.916637i \(-0.369107\pi\)
0.0621072 + 0.998069i \(0.480218\pi\)
\(24\) 0 0
\(25\) 5.09811 − 4.27783i 1.01962 − 0.855565i
\(26\) −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
\(31\) −0.538767 − 1.48025i −0.0967653 − 0.265861i 0.881860 − 0.471511i \(-0.156291\pi\)
−0.978626 + 0.205650i \(0.934069\pi\)
\(32\) 4.08675 − 4.87040i 0.722442 − 0.860972i
\(33\) 0 0
\(34\) −3.47723 − 4.14400i −0.596340 − 0.710691i
\(35\) −8.80991 + 1.99281i −1.48915 + 0.336846i
\(36\) 0 0
\(37\) 10.0245 1.64801 0.824006 − 0.566581i \(-0.191734\pi\)
0.824006 + 0.566581i \(0.191734\pi\)
\(38\) 2.63830 + 0.960264i 0.427989 + 0.155775i
\(39\) 0 0
\(40\) −5.37870 + 6.41008i −0.850447 + 1.01352i
\(41\) −5.39584 + 1.96393i −0.842689 + 0.306714i −0.727056 − 0.686578i \(-0.759112\pi\)
−0.115633 + 0.993292i \(0.536890\pi\)
\(42\) 0 0
\(43\) −0.110284 − 0.625454i −0.0168182 − 0.0953809i 0.975243 − 0.221135i \(-0.0709761\pi\)
−0.992061 + 0.125754i \(0.959865\pi\)
\(44\) 7.85333i 1.18393i
\(45\) 0 0
\(46\) −5.07240 −0.747885
\(47\) 11.3454 + 4.12940i 1.65490 + 0.602334i 0.989549 − 0.144197i \(-0.0460598\pi\)
0.665351 + 0.746531i \(0.268282\pi\)
\(48\) 0 0
\(49\) 6.77678 + 1.75365i 0.968111 + 0.250521i
\(50\) −9.64814 + 11.4982i −1.36445 + 1.62609i
\(51\) 0 0
\(52\) 12.2542 + 14.6040i 1.69935 + 2.02521i
\(53\) 4.15350 + 2.39802i 0.570527 + 0.329394i 0.757360 − 0.652998i \(-0.226489\pi\)
−0.186833 + 0.982392i \(0.559822\pi\)
\(54\) 0 0
\(55\) 8.68580i 1.17119i
\(56\) 5.98053 − 2.50733i 0.799181 − 0.335056i
\(57\) 0 0
\(58\) 0.785398 − 4.45421i 0.103128 − 0.584867i
\(59\) −2.11297 − 1.77299i −0.275085 − 0.230824i 0.494799 − 0.869007i \(-0.335242\pi\)
−0.769884 + 0.638183i \(0.779686\pi\)
\(60\) 0 0
\(61\) 2.52148 − 6.92772i 0.322843 − 0.887004i −0.667028 − 0.745033i \(-0.732434\pi\)
0.989871 − 0.141971i \(-0.0453440\pi\)
\(62\) 1.77639 + 3.07681i 0.225602 + 0.390755i
\(63\) 0 0
\(64\) −6.52423 + 11.3003i −0.815529 + 1.41254i
\(65\) 13.5532 + 16.1520i 1.68106 + 2.00341i
\(66\) 0 0
\(67\) −1.39215 + 7.89530i −0.170079 + 0.964564i 0.773594 + 0.633682i \(0.218457\pi\)
−0.943672 + 0.330882i \(0.892654\pi\)
\(68\) 5.67155 + 4.75899i 0.687776 + 0.577113i
\(69\) 0 0
\(70\) 18.7874 − 7.87660i 2.24552 − 0.941434i
\(71\) −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
\(73\) 13.4098i 1.56949i 0.619817 + 0.784747i \(0.287207\pi\)
−0.619817 + 0.784747i \(0.712793\pi\)
\(74\) −22.2655 + 3.92601i −2.58832 + 0.456390i
\(75\) 0 0
\(76\) −3.78418 − 0.667254i −0.434076 − 0.0765392i
\(77\) 3.08253 − 5.98404i 0.351287 − 0.681945i
\(78\) 0 0
\(79\) 0.167826 + 0.951791i 0.0188819 + 0.107085i 0.992792 − 0.119849i \(-0.0382410\pi\)
−0.973910 + 0.226934i \(0.927130\pi\)
\(80\) −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 0.207948 − 0.978140i \(-0.433322\pi\)
−0.469439 + 0.882965i \(0.655544\pi\)
\(84\) 0 0
\(85\) 6.27275 + 5.26346i 0.680375 + 0.570903i
\(86\) 0.489910 + 1.34602i 0.0528283 + 0.145145i
\(87\) 0 0
\(88\) −1.08286 − 6.14122i −0.115434 − 0.654656i
\(89\) 4.97533 − 8.61753i 0.527384 − 0.913456i −0.472107 − 0.881541i \(-0.656506\pi\)
0.999491 − 0.0319144i \(-0.0101604\pi\)
\(90\) 0 0
\(91\) −3.60514 − 15.9378i −0.377921 − 1.67074i
\(92\) 6.83671 − 1.20550i 0.712776 − 0.125682i
\(93\) 0 0
\(94\) −26.8168 − 4.72852i −2.76594 − 0.487710i
\(95\) −4.18532 − 0.737985i −0.429405 − 0.0757156i
\(96\) 0 0
\(97\) −7.93719 + 1.39954i −0.805899 + 0.142102i −0.561397 − 0.827547i \(-0.689736\pi\)
−0.244503 + 0.969649i \(0.578625\pi\)
\(98\) −15.7388 − 1.24098i −1.58986 − 0.125358i
\(99\) 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.17.3 132
3.2 odd 2 189.2.bd.a.185.20 yes 132
7.5 odd 6 567.2.ba.a.341.20 132
21.5 even 6 189.2.ba.a.131.3 yes 132
27.7 even 9 189.2.ba.a.101.3 ✓ 132
27.20 odd 18 567.2.ba.a.143.20 132
189.47 even 18 inner 567.2.bd.a.467.3 132
189.61 odd 18 189.2.bd.a.47.20 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.3 ✓ 132 27.7 even 9
189.2.ba.a.131.3 yes 132 21.5 even 6
189.2.bd.a.47.20 yes 132 189.61 odd 18
189.2.bd.a.185.20 yes 132 3.2 odd 2
567.2.ba.a.143.20 132 27.20 odd 18
567.2.ba.a.341.20 132 7.5 odd 6
567.2.bd.a.17.3 132 1.1 even 1 trivial
567.2.bd.a.467.3 132 189.47 even 18 inner