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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 467.3
Character \(\chi\) \(=\) 567.467
Dual form 567.2.bd.a.17.3

$q$-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 \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.22112 − 0.391643i −1.57057 − 0.276934i −0.680496 − 0.732752i \(-0.738235\pi\)
−0.890073 + 0.455819i \(0.849347\pi\)
\(3\) 0 0
\(4\) 2.90060 + 1.05573i 1.45030 + 0.527866i
\(5\) 3.20807 + 1.16764i 1.43469 + 0.522186i 0.938273 − 0.345896i \(-0.112425\pi\)
0.496421 + 0.868082i \(0.334647\pi\)
\(6\) 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 −0.999007 − 0.0445598i \(-0.985811\pi\)
0.736641 − 0.676284i \(-0.236411\pi\)
\(12\) 0 0
\(13\) 2.11236 − 5.80365i 0.585862 − 1.60964i −0.192131 − 0.981369i \(-0.561540\pi\)
0.777993 − 0.628273i \(-0.216238\pi\)
\(14\) 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 −0.683150 − 0.730278i \(-0.739390\pi\)
0.974014 + 0.226486i \(0.0727237\pi\)
\(18\) 0 0
\(19\) −1.07808 + 0.622427i −0.247328 + 0.142795i −0.618540 − 0.785753i \(-0.712276\pi\)
0.371212 + 0.928548i \(0.378942\pi\)
\(20\) 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 0.0621072 − 0.998069i \(-0.480218\pi\)
0.399722 + 0.916637i \(0.369107\pi\)
\(24\) 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 0.859577 − 0.511006i \(-0.170727\pi\)
−0.986943 + 0.161072i \(0.948505\pi\)
\(30\) 0 0
\(31\) −0.538767 + 1.48025i −0.0967653 + 0.265861i −0.978626 − 0.205650i \(-0.934069\pi\)
0.881860 + 0.471511i \(0.156291\pi\)
\(32\) 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 −0.115633 − 0.993292i \(-0.536890\pi\)
−0.727056 + 0.686578i \(0.759112\pi\)
\(42\) 0 0
\(43\) −0.110284 + 0.625454i −0.0168182 + 0.0953809i −0.992061 − 0.125754i \(-0.959865\pi\)
0.975243 + 0.221135i \(0.0709761\pi\)
\(44\) − 7.85333i − 1.18393i
\(45\) 0 0
\(46\) −5.07240 −0.747885
\(47\) 11.3454 − 4.12940i 1.65490 − 0.602334i 0.665351 − 0.746531i \(-0.268282\pi\)
0.989549 + 0.144197i \(0.0460598\pi\)
\(48\) 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 −0.186833 − 0.982392i \(-0.559822\pi\)
0.757360 + 0.652998i \(0.226489\pi\)
\(54\) 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 −0.769884 − 0.638183i \(-0.779686\pi\)
0.494799 + 0.869007i \(0.335242\pi\)
\(60\) 0 0
\(61\) 2.52148 + 6.92772i 0.322843 + 0.887004i 0.989871 + 0.141971i \(0.0453440\pi\)
−0.667028 + 0.745033i \(0.732434\pi\)
\(62\) 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 −0.943672 − 0.330882i \(-0.892654\pi\)
0.773594 − 0.633682i \(-0.218457\pi\)
\(68\) 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 −0.740659 − 0.671881i \(-0.765487\pi\)
0.211536 + 0.977370i \(0.432153\pi\)
\(72\) 0 0
\(73\) − 13.4098i − 1.56949i −0.619817 − 0.784747i \(-0.712793\pi\)
0.619817 − 0.784747i \(-0.287207\pi\)
\(74\) −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 −0.973910 − 0.226934i \(-0.927130\pi\)
0.992792 + 0.119849i \(0.0382410\pi\)
\(80\) −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 −0.469439 − 0.882965i \(-0.655544\pi\)
0.207948 + 0.978140i \(0.433322\pi\)
\(84\) 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 0.999491 + 0.0319144i \(0.0101604\pi\)
−0.472107 + 0.881541i \(0.656506\pi\)
\(90\) 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 −0.244503 − 0.969649i \(-0.578625\pi\)
−0.561397 + 0.827547i \(0.689736\pi\)
\(98\) −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.467.3 132
3.2 odd 2 189.2.bd.a.47.20 yes 132
7.3 odd 6 567.2.ba.a.143.20 132
21.17 even 6 189.2.ba.a.101.3 ✓ 132
27.4 even 9 189.2.ba.a.131.3 yes 132
27.23 odd 18 567.2.ba.a.341.20 132
189.31 odd 18 189.2.bd.a.185.20 yes 132
189.185 even 18 inner 567.2.bd.a.17.3 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.3 ✓ 132 21.17 even 6
189.2.ba.a.131.3 yes 132 27.4 even 9
189.2.bd.a.47.20 yes 132 3.2 odd 2
189.2.bd.a.185.20 yes 132 189.31 odd 18
567.2.ba.a.143.20 132 7.3 odd 6
567.2.ba.a.341.20 132 27.23 odd 18
567.2.bd.a.17.3 132 189.185 even 18 inner
567.2.bd.a.467.3 132 1.1 even 1 trivial