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Properties

Label 567.2.ba.a.143.19
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 143.19
Character \(\chi\) \(=\) 567.143
Dual form 567.2.ba.a.341.19

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.22264 - 1.45709i) q^{2} +(-0.280959 - 1.59339i) q^{4} +(0.691544 - 0.580274i) q^{5} +(-0.448602 + 2.60744i) q^{7} +(0.629295 + 0.363324i) q^{8} -1.71711i q^{10} +(2.24705 - 2.67793i) q^{11} +(1.35464 - 3.72184i) q^{13} +(3.25080 + 3.84163i) q^{14} +(4.33960 - 1.57949i) q^{16} +0.467865 q^{17} -3.62874i q^{19} +(-1.11890 - 0.938869i) q^{20} +(-1.15465 - 6.54832i) q^{22} +(1.19589 - 3.28569i) q^{23} +(-0.726726 + 4.12147i) q^{25} +(-3.76682 - 6.52432i) q^{26} +(4.28072 - 0.0177830i) q^{28} +(1.79273 + 4.92550i) q^{29} +(-6.86071 + 1.20973i) q^{31} +(2.50728 - 6.88869i) q^{32} +(0.572032 - 0.681722i) q^{34} +(1.20280 + 2.06347i) q^{35} +(-4.92689 + 8.53362i) q^{37} +(-5.28741 - 4.43666i) q^{38} +(0.646012 - 0.113909i) q^{40} +(2.71326 + 0.987547i) q^{41} +(-1.54504 + 8.76238i) q^{43} +(-4.89833 - 2.82805i) q^{44} +(-3.32540 - 5.75976i) q^{46} +(0.579339 - 3.28560i) q^{47} +(-6.59751 - 2.33941i) q^{49} +(5.11683 + 6.09800i) q^{50} +(-6.31096 - 1.11279i) q^{52} +(-8.75304 - 5.05357i) q^{53} -3.15581i q^{55} +(-1.22965 + 1.47786i) q^{56} +(9.36878 + 3.40996i) q^{58} +(3.04590 + 1.10862i) q^{59} +(0.0601824 + 0.0106118i) q^{61} +(-6.62553 + 11.4757i) q^{62} +(-2.35384 - 4.07697i) q^{64} +(-1.22290 - 3.35988i) q^{65} +(7.85348 - 6.58985i) q^{67} +(-0.131451 - 0.745493i) q^{68} +(4.47727 + 0.770299i) q^{70} +(2.19935 - 1.26979i) q^{71} +(-7.24870 + 4.18504i) q^{73} +(6.41043 + 17.6125i) q^{74} +(-5.78202 + 1.01953i) q^{76} +(5.97452 + 7.06039i) q^{77} +(7.23876 + 6.07404i) q^{79} +(2.08449 - 3.61044i) q^{80} +(4.75630 - 2.74605i) q^{82} +(-10.6153 + 3.86367i) q^{83} +(0.323549 - 0.271490i) q^{85} +(10.8786 + 12.9646i) q^{86} +(2.38702 - 0.868803i) q^{88} +3.03736 q^{89} +(9.09679 + 5.20177i) q^{91} +(-5.57140 - 0.982388i) q^{92} +(-4.07909 - 4.86127i) q^{94} +(-2.10566 - 2.50943i) q^{95} +(-5.02929 - 0.886800i) q^{97} +(-11.4751 + 6.75291i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34}+ \cdots - 27 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22264 − 1.45709i 0.864540 − 1.03032i −0.134682 − 0.990889i \(-0.543001\pi\)
0.999222 − 0.0394303i \(-0.0125543\pi\)
\(3\) 0 0
\(4\) −0.280959 − 1.59339i −0.140479 − 0.796697i
\(5\) 0.691544 − 0.580274i 0.309268 − 0.259506i −0.474922 − 0.880028i \(-0.657524\pi\)
0.784189 + 0.620522i \(0.213079\pi\)
\(6\) 0 0
\(7\) −0.448602 + 2.60744i −0.169556 + 0.985521i
\(8\) 0.629295 + 0.363324i 0.222489 + 0.128454i
\(9\) 0 0
\(10\) − 1.71711i − 0.542998i
\(11\) 2.24705 − 2.67793i 0.677512 − 0.807427i −0.312274 − 0.949992i \(-0.601091\pi\)
0.989785 + 0.142565i \(0.0455350\pi\)
\(12\) 0 0
\(13\) 1.35464 − 3.72184i 0.375709 − 1.03225i −0.597407 − 0.801938i \(-0.703802\pi\)
0.973116 − 0.230315i \(-0.0739755\pi\)
\(14\) 3.25080 + 3.84163i 0.868813 + 1.02672i
\(15\) 0 0
\(16\) 4.33960 − 1.57949i 1.08490 − 0.394871i
\(17\) 0.467865 0.113474 0.0567369 − 0.998389i \(-0.481930\pi\)
0.0567369 + 0.998389i \(0.481930\pi\)
\(18\) 0 0
\(19\) − 3.62874i − 0.832490i −0.909252 − 0.416245i \(-0.863346\pi\)
0.909252 − 0.416245i \(-0.136654\pi\)
\(20\) −1.11890 − 0.938869i −0.250194 − 0.209938i
\(21\) 0 0
\(22\) −1.15465 − 6.54832i −0.246171 − 1.39611i
\(23\) 1.19589 − 3.28569i 0.249361 − 0.685114i −0.750349 − 0.661042i \(-0.770115\pi\)
0.999710 − 0.0240721i \(-0.00766312\pi\)
\(24\) 0 0
\(25\) −0.726726 + 4.12147i −0.145345 + 0.824294i
\(26\) −3.76682 − 6.52432i −0.738734 − 1.27953i
\(27\) 0 0
\(28\) 4.28072 − 0.0177830i 0.808981 − 0.00336068i
\(29\) 1.79273 + 4.92550i 0.332902 + 0.914642i 0.987353 + 0.158537i \(0.0506777\pi\)
−0.654451 + 0.756105i \(0.727100\pi\)
\(30\) 0 0
\(31\) −6.86071 + 1.20973i −1.23222 + 0.217274i −0.751579 − 0.659644i \(-0.770707\pi\)
−0.480641 + 0.876917i \(0.659596\pi\)
\(32\) 2.50728 − 6.88869i 0.443229 − 1.21776i
\(33\) 0 0
\(34\) 0.572032 − 0.681722i 0.0981027 − 0.116914i
\(35\) 1.20280 + 2.06347i 0.203311 + 0.348790i
\(36\) 0 0
\(37\) −4.92689 + 8.53362i −0.809975 + 1.40292i 0.102905 + 0.994691i \(0.467186\pi\)
−0.912881 + 0.408227i \(0.866147\pi\)
\(38\) −5.28741 − 4.43666i −0.857731 − 0.719721i
\(39\) 0 0
\(40\) 0.646012 − 0.113909i 0.102144 − 0.0180107i
\(41\) 2.71326 + 0.987547i 0.423741 + 0.154229i 0.545084 − 0.838382i \(-0.316498\pi\)
−0.121343 + 0.992611i \(0.538720\pi\)
\(42\) 0 0
\(43\) −1.54504 + 8.76238i −0.235617 + 1.33625i 0.605693 + 0.795698i \(0.292896\pi\)
−0.841310 + 0.540553i \(0.818215\pi\)
\(44\) −4.89833 − 2.82805i −0.738452 − 0.426345i
\(45\) 0 0
\(46\) −3.32540 − 5.75976i −0.490303 − 0.849230i
\(47\) 0.579339 − 3.28560i 0.0845053 − 0.479253i −0.912957 − 0.408056i \(-0.866207\pi\)
0.997462 − 0.0711975i \(-0.0226821\pi\)
\(48\) 0 0
\(49\) −6.59751 − 2.33941i −0.942502 − 0.334201i
\(50\) 5.11683 + 6.09800i 0.723629 + 0.862387i
\(51\) 0 0
\(52\) −6.31096 − 1.11279i −0.875173 − 0.154317i
\(53\) −8.75304 − 5.05357i −1.20232 − 0.694161i −0.241251 − 0.970463i \(-0.577558\pi\)
−0.961071 + 0.276302i \(0.910891\pi\)
\(54\) 0 0
\(55\) − 3.15581i − 0.425530i
\(56\) −1.22965 + 1.47786i −0.164319 + 0.197488i
\(57\) 0 0
\(58\) 9.36878 + 3.40996i 1.23018 + 0.447749i
\(59\) 3.04590 + 1.10862i 0.396543 + 0.144330i 0.532592 − 0.846372i \(-0.321218\pi\)
−0.136049 + 0.990702i \(0.543440\pi\)
\(60\) 0 0
\(61\) 0.0601824 + 0.0106118i 0.00770557 + 0.00135870i 0.177500 − 0.984121i \(-0.443199\pi\)
−0.169794 + 0.985480i \(0.554310\pi\)
\(62\) −6.62553 + 11.4757i −0.841443 + 1.45742i
\(63\) 0 0
\(64\) −2.35384 − 4.07697i −0.294230 − 0.509621i
\(65\) −1.22290 − 3.35988i −0.151681 − 0.416741i
\(66\) 0 0
\(67\) 7.85348 − 6.58985i 0.959455 − 0.805078i −0.0214094 − 0.999771i \(-0.506815\pi\)
0.980864 + 0.194693i \(0.0623709\pi\)
\(68\) −0.131451 − 0.745493i −0.0159407 − 0.0904043i
\(69\) 0 0
\(70\) 4.47727 + 0.770299i 0.535136 + 0.0920684i
\(71\) 2.19935 − 1.26979i 0.261014 − 0.150697i −0.363783 − 0.931484i \(-0.618515\pi\)
0.624797 + 0.780787i \(0.285182\pi\)
\(72\) 0 0
\(73\) −7.24870 + 4.18504i −0.848396 + 0.489822i −0.860109 − 0.510110i \(-0.829605\pi\)
0.0117134 + 0.999931i \(0.496271\pi\)
\(74\) 6.41043 + 17.6125i 0.745197 + 2.04741i
\(75\) 0 0
\(76\) −5.78202 + 1.01953i −0.663243 + 0.116948i
\(77\) 5.97452 + 7.06039i 0.680860 + 0.804606i
\(78\) 0 0
\(79\) 7.23876 + 6.07404i 0.814424 + 0.683383i 0.951659 − 0.307156i \(-0.0993773\pi\)
−0.137236 + 0.990538i \(0.543822\pi\)
\(80\) 2.08449 − 3.61044i 0.233053 − 0.403659i
\(81\) 0 0
\(82\) 4.75630 − 2.74605i 0.525246 − 0.303251i
\(83\) −10.6153 + 3.86367i −1.16518 + 0.424092i −0.850947 − 0.525252i \(-0.823971\pi\)
−0.314237 + 0.949344i \(0.601749\pi\)
\(84\) 0 0
\(85\) 0.323549 − 0.271490i 0.0350938 − 0.0294472i
\(86\) 10.8786 + 12.9646i 1.17306 + 1.39800i
\(87\) 0 0
\(88\) 2.38702 − 0.868803i 0.254457 − 0.0926147i
\(89\) 3.03736 0.321959 0.160980 − 0.986958i \(-0.448535\pi\)
0.160980 + 0.986958i \(0.448535\pi\)
\(90\) 0 0
\(91\) 9.09679 + 5.20177i 0.953603 + 0.545294i
\(92\) −5.57140 − 0.982388i −0.580858 − 0.102421i
\(93\) 0 0
\(94\) −4.07909 − 4.86127i −0.420726 − 0.501401i
\(95\) −2.10566 − 2.50943i −0.216036 − 0.257462i
\(96\) 0 0
\(97\) −5.02929 − 0.886800i −0.510647 − 0.0900409i −0.0876131 − 0.996155i \(-0.527924\pi\)
−0.423034 + 0.906114i \(0.639035\pi\)
\(98\) −11.4751 + 6.75291i −1.15916 + 0.682147i
\(99\) 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.143.19 132
3.2 odd 2 189.2.ba.a.101.4 ✓ 132
7.5 odd 6 567.2.bd.a.467.4 132
21.5 even 6 189.2.bd.a.47.19 yes 132
27.4 even 9 189.2.bd.a.185.19 yes 132
27.23 odd 18 567.2.bd.a.17.4 132
189.131 even 18 inner 567.2.ba.a.341.19 132
189.166 odd 18 189.2.ba.a.131.4 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.4 ✓ 132 3.2 odd 2
189.2.ba.a.131.4 yes 132 189.166 odd 18
189.2.bd.a.47.19 yes 132 21.5 even 6
189.2.bd.a.185.19 yes 132 27.4 even 9
567.2.ba.a.143.19 132 1.1 even 1 trivial
567.2.ba.a.341.19 132 189.131 even 18 inner
567.2.bd.a.17.4 132 27.23 odd 18
567.2.bd.a.467.4 132 7.5 odd 6