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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 17.12
Character \(\chi\) \(=\) 567.17
Dual form 567.2.bd.a.467.12

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.0147002 - 0.00259205i) q^{2} +(-1.87918 + 0.683964i) q^{4} +(-1.54651 + 0.562885i) q^{5} +(2.21011 - 1.45445i) q^{7} +(-0.0517058 + 0.0298524i) q^{8} +(-0.0212751 + 0.0122832i) q^{10} +(-1.57083 + 4.31582i) q^{11} +(-1.52751 - 4.19679i) q^{13} +(0.0287191 - 0.0271094i) q^{14} +(3.06315 - 2.57029i) q^{16} +(-1.88705 - 3.26847i) q^{17} +(-1.27160 - 0.734160i) q^{19} +(2.52118 - 2.11552i) q^{20} +(-0.0119048 + 0.0675152i) q^{22} +(-7.31736 - 1.29025i) q^{23} +(-1.75536 + 1.47292i) q^{25} +(-0.0333330 - 0.0577345i) q^{26} +(-3.15839 + 4.24480i) q^{28} +(2.83246 - 7.78213i) q^{29} +(-2.87381 - 7.89573i) q^{31} +(0.115122 - 0.137197i) q^{32} +(-0.0362121 - 0.0431559i) q^{34} +(-2.59927 + 3.49336i) q^{35} -0.794639 q^{37} +(-0.0205958 - 0.00749627i) q^{38} +(0.0631602 - 0.0752714i) q^{40} +(-3.97658 + 1.44736i) q^{41} +(-0.303215 - 1.71962i) q^{43} -9.18457i q^{44} -0.110911 q^{46} +(1.54917 + 0.563852i) q^{47} +(2.76916 - 6.42898i) q^{49} +(-0.0219863 + 0.0262023i) q^{50} +(5.74091 + 6.84175i) q^{52} +(-3.66888 - 2.11823i) q^{53} -7.55866i q^{55} +(-0.0708567 + 0.141180i) q^{56} +(0.0214662 - 0.121741i) q^{58} +(8.85828 + 7.43298i) q^{59} +(1.02900 - 2.82714i) q^{61} +(-0.0627118 - 0.108620i) q^{62} +(-3.99733 + 6.92357i) q^{64} +(4.72462 + 5.63058i) q^{65} +(-1.88292 + 10.6785i) q^{67} +(5.78162 + 4.85135i) q^{68} +(-0.0291550 + 0.0580906i) q^{70} +(-6.66808 - 3.84982i) q^{71} +15.8281i q^{73} +(-0.0116814 + 0.00205974i) q^{74} +(2.89171 + 0.509886i) q^{76} +(2.80543 + 11.8231i) q^{77} +(0.276799 + 1.56981i) q^{79} +(-3.29043 + 5.69919i) q^{80} +(-0.0547050 + 0.0315840i) q^{82} +(-4.61667 - 1.68033i) q^{83} +(4.75812 + 3.99254i) q^{85} +(-0.00891467 - 0.0244929i) q^{86} +(-0.0476164 - 0.270046i) q^{88} +(3.59252 - 6.22243i) q^{89} +(-9.47998 - 7.05369i) q^{91} +(14.6331 - 2.58021i) q^{92} +(0.0242347 + 0.00427323i) q^{94} +(2.37980 + 0.419623i) q^{95} +(-1.56544 + 0.276029i) q^{97} +(0.0240431 - 0.101685i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} - 9 q^{10} - 9 q^{11} + 42 q^{14} - 15 q^{16} + 9 q^{17} - 9 q^{19} + 18 q^{20} - 12 q^{22} - 30 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31}+ \cdots + 180 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0147002 − 0.00259205i 0.0103946 − 0.00183285i −0.168448 − 0.985710i \(-0.553876\pi\)
0.178843 + 0.983878i \(0.442765\pi\)
\(3\) 0 0
\(4\) −1.87918 + 0.683964i −0.939588 + 0.341982i
\(5\) −1.54651 + 0.562885i −0.691622 + 0.251730i −0.663829 − 0.747884i \(-0.731070\pi\)
−0.0277922 + 0.999614i \(0.508848\pi\)
\(6\) 0 0
\(7\) 2.21011 − 1.45445i 0.835342 − 0.549730i
\(8\) −0.0517058 + 0.0298524i −0.0182808 + 0.0105544i
\(9\) 0 0
\(10\) −0.0212751 + 0.0122832i −0.00672777 + 0.00388428i
\(11\) −1.57083 + 4.31582i −0.473623 + 1.30127i 0.441199 + 0.897409i \(0.354553\pi\)
−0.914821 + 0.403859i \(0.867669\pi\)
\(12\) 0 0
\(13\) −1.52751 − 4.19679i −0.423654 − 1.16398i −0.949600 − 0.313463i \(-0.898511\pi\)
0.525946 − 0.850518i \(-0.323711\pi\)
\(14\) 0.0287191 − 0.0271094i 0.00767551 − 0.00724530i
\(15\) 0 0
\(16\) 3.06315 − 2.57029i 0.765788 − 0.642573i
\(17\) −1.88705 − 3.26847i −0.457677 − 0.792720i 0.541160 − 0.840919i \(-0.317985\pi\)
−0.998838 + 0.0481990i \(0.984652\pi\)
\(18\) 0 0
\(19\) −1.27160 − 0.734160i −0.291726 − 0.168428i 0.346994 − 0.937867i \(-0.387202\pi\)
−0.638720 + 0.769439i \(0.720536\pi\)
\(20\) 2.52118 − 2.11552i 0.563752 − 0.473044i
\(21\) 0 0
\(22\) −0.0119048 + 0.0675152i −0.00253810 + 0.0143943i
\(23\) −7.31736 − 1.29025i −1.52578 − 0.269035i −0.653076 − 0.757293i \(-0.726522\pi\)
−0.872700 + 0.488257i \(0.837633\pi\)
\(24\) 0 0
\(25\) −1.75536 + 1.47292i −0.351072 + 0.294584i
\(26\) −0.0333330 − 0.0577345i −0.00653714 − 0.0113227i
\(27\) 0 0
\(28\) −3.15839 + 4.24480i −0.596880 + 0.802192i
\(29\) 2.83246 − 7.78213i 0.525975 − 1.44511i −0.337795 − 0.941220i \(-0.609681\pi\)
0.863771 − 0.503885i \(-0.168097\pi\)
\(30\) 0 0
\(31\) −2.87381 − 7.89573i −0.516152 − 1.41812i −0.874728 − 0.484615i \(-0.838960\pi\)
0.358576 − 0.933501i \(-0.383262\pi\)
\(32\) 0.115122 − 0.137197i 0.0203508 − 0.0242532i
\(33\) 0 0
\(34\) −0.0362121 − 0.0431559i −0.00621033 − 0.00740118i
\(35\) −2.59927 + 3.49336i −0.439358 + 0.590485i
\(36\) 0 0
\(37\) −0.794639 −0.130638 −0.0653190 − 0.997864i \(-0.520806\pi\)
−0.0653190 + 0.997864i \(0.520806\pi\)
\(38\) −0.0205958 − 0.00749627i −0.00334109 − 0.00121606i
\(39\) 0 0
\(40\) 0.0631602 − 0.0752714i 0.00998651 − 0.0119015i
\(41\) −3.97658 + 1.44736i −0.621037 + 0.226039i −0.633326 − 0.773885i \(-0.718311\pi\)
0.0122884 + 0.999924i \(0.496088\pi\)
\(42\) 0 0
\(43\) −0.303215 − 1.71962i −0.0462399 − 0.262239i 0.952920 − 0.303222i \(-0.0980623\pi\)
−0.999160 + 0.0409825i \(0.986951\pi\)
\(44\) − 9.18457i − 1.38463i
\(45\) 0 0
\(46\) −0.110911 −0.0163530
\(47\) 1.54917 + 0.563852i 0.225970 + 0.0822463i 0.452524 − 0.891752i \(-0.350524\pi\)
−0.226554 + 0.973999i \(0.572746\pi\)
\(48\) 0 0
\(49\) 2.76916 − 6.42898i 0.395594 − 0.918425i
\(50\) −0.0219863 + 0.0262023i −0.00310933 + 0.00370556i
\(51\) 0 0
\(52\) 5.74091 + 6.84175i 0.796121 + 0.948780i
\(53\) −3.66888 − 2.11823i −0.503959 − 0.290961i 0.226388 − 0.974037i \(-0.427308\pi\)
−0.730347 + 0.683076i \(0.760642\pi\)
\(54\) 0 0
\(55\) − 7.55866i − 1.01921i
\(56\) −0.0708567 + 0.141180i −0.00946863 + 0.0188660i
\(57\) 0 0
\(58\) 0.0214662 − 0.121741i 0.00281865 − 0.0159854i
\(59\) 8.85828 + 7.43298i 1.15325 + 0.967691i 0.999791 − 0.0204636i \(-0.00651421\pi\)
0.153459 + 0.988155i \(0.450959\pi\)
\(60\) 0 0
\(61\) 1.02900 − 2.82714i 0.131749 − 0.361978i −0.856224 − 0.516605i \(-0.827195\pi\)
0.987973 + 0.154627i \(0.0494176\pi\)
\(62\) −0.0627118 − 0.108620i −0.00796441 − 0.0137948i
\(63\) 0 0
\(64\) −3.99733 + 6.92357i −0.499666 + 0.865447i
\(65\) 4.72462 + 5.63058i 0.586017 + 0.698388i
\(66\) 0 0
\(67\) −1.88292 + 10.6785i −0.230035 + 1.30459i 0.622788 + 0.782391i \(0.286000\pi\)
−0.852823 + 0.522201i \(0.825111\pi\)
\(68\) 5.78162 + 4.85135i 0.701124 + 0.588313i
\(69\) 0 0
\(70\) −0.0291550 + 0.0580906i −0.00348469 + 0.00694316i
\(71\) −6.66808 − 3.84982i −0.791355 − 0.456889i 0.0490844 − 0.998795i \(-0.484370\pi\)
−0.840439 + 0.541906i \(0.817703\pi\)
\(72\) 0 0
\(73\) 15.8281i 1.85254i 0.376858 + 0.926271i \(0.377005\pi\)
−0.376858 + 0.926271i \(0.622995\pi\)
\(74\) −0.0116814 + 0.00205974i −0.00135793 + 0.000239440i
\(75\) 0 0
\(76\) 2.89171 + 0.509886i 0.331701 + 0.0584879i
\(77\) 2.80543 + 11.8231i 0.319709 + 1.34737i
\(78\) 0 0
\(79\) 0.276799 + 1.56981i 0.0311423 + 0.176617i 0.996412 − 0.0846402i \(-0.0269741\pi\)
−0.965269 + 0.261257i \(0.915863\pi\)
\(80\) −3.29043 + 5.69919i −0.367881 + 0.637189i
\(81\) 0 0
\(82\) −0.0547050 + 0.0315840i −0.00604116 + 0.00348787i
\(83\) −4.61667 − 1.68033i −0.506745 − 0.184440i 0.0759804 − 0.997109i \(-0.475791\pi\)
−0.582725 + 0.812669i \(0.698014\pi\)
\(84\) 0 0
\(85\) 4.75812 + 3.99254i 0.516091 + 0.433051i
\(86\) −0.00891467 − 0.0244929i −0.000961293 − 0.00264113i
\(87\) 0 0
\(88\) −0.0476164 − 0.270046i −0.00507592 − 0.0287870i
\(89\) 3.59252 − 6.22243i 0.380807 − 0.659577i −0.610371 − 0.792116i \(-0.708980\pi\)
0.991178 + 0.132539i \(0.0423130\pi\)
\(90\) 0 0
\(91\) −9.47998 − 7.05369i −0.993772 − 0.739427i
\(92\) 14.6331 − 2.58021i 1.52561 − 0.269005i
\(93\) 0 0
\(94\) 0.0242347 + 0.00427323i 0.00249962 + 0.000440750i
\(95\) 2.37980 + 0.419623i 0.244162 + 0.0430524i
\(96\) 0 0
\(97\) −1.56544 + 0.276029i −0.158946 + 0.0280265i −0.252555 − 0.967583i \(-0.581271\pi\)
0.0936086 + 0.995609i \(0.470160\pi\)
\(98\) 0.0240431 − 0.101685i 0.00242872 − 0.0102718i
\(99\) 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.17.12 132
3.2 odd 2 189.2.bd.a.185.11 yes 132
7.5 odd 6 567.2.ba.a.341.11 132
21.5 even 6 189.2.ba.a.131.12 yes 132
27.7 even 9 189.2.ba.a.101.12 ✓ 132
27.20 odd 18 567.2.ba.a.143.11 132
189.47 even 18 inner 567.2.bd.a.467.12 132
189.61 odd 18 189.2.bd.a.47.11 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.12 ✓ 132 27.7 even 9
189.2.ba.a.131.12 yes 132 21.5 even 6
189.2.bd.a.47.11 yes 132 189.61 odd 18
189.2.bd.a.185.11 yes 132 3.2 odd 2
567.2.ba.a.143.11 132 27.20 odd 18
567.2.ba.a.341.11 132 7.5 odd 6
567.2.bd.a.17.12 132 1.1 even 1 trivial
567.2.bd.a.467.12 132 189.47 even 18 inner