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Properties

Label 567.2.bd.a.17.4
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 17.4
Character \(\chi\) \(=\) 567.17
Dual form 567.2.bd.a.467.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.87320 + 0.330296i) q^{2} +(1.52040 - 0.553380i) q^{4} +(0.848304 - 0.308757i) q^{5} +(2.48993 + 0.894564i) q^{7} +(0.629295 - 0.363324i) q^{8} +(-1.48706 + 0.858555i) q^{10} +(1.19563 - 3.28497i) q^{11} +(-1.35464 - 3.72184i) q^{13} +(-4.95961 - 0.853285i) q^{14} +(-3.53767 + 2.96846i) q^{16} +(0.233932 + 0.405183i) q^{17} +(3.14258 + 1.81437i) q^{19} +(1.11890 - 0.938869i) q^{20} +(-1.15465 + 6.54832i) q^{22} +(-3.44344 - 0.607171i) q^{23} +(-3.20593 + 2.69010i) q^{25} +(3.76682 + 6.52432i) q^{26} +(4.28072 - 0.0177830i) q^{28} +(1.79273 - 4.92550i) q^{29} +(-2.38270 - 6.54641i) q^{31} +(4.71214 - 5.61571i) q^{32} +(-0.572032 - 0.681722i) q^{34} +(2.38842 - 0.00992201i) q^{35} +9.85377 q^{37} +(-6.48596 - 2.36070i) q^{38} +(0.421655 - 0.502508i) q^{40} +(-2.71326 + 0.987547i) q^{41} +(-1.54504 - 8.76238i) q^{43} -5.65611i q^{44} +6.65080 q^{46} +(3.13508 + 1.14108i) q^{47} +(5.39951 + 4.45481i) q^{49} +(5.11683 - 6.09800i) q^{50} +(-4.11919 - 4.90906i) q^{52} +(8.75304 + 5.05357i) q^{53} -3.15581i q^{55} +(1.89192 - 0.341706i) q^{56} +(-1.73128 + 9.81858i) q^{58} +(2.48304 + 2.08352i) q^{59} +(0.0209011 - 0.0574254i) q^{61} +(6.62553 + 11.4757i) q^{62} +(-2.35384 + 4.07697i) q^{64} +(-2.29829 - 2.73900i) q^{65} +(1.78024 - 10.0962i) q^{67} +(0.579891 + 0.486586i) q^{68} +(-4.47071 + 0.807471i) q^{70} +(2.19935 + 1.26979i) q^{71} -8.37008i q^{73} +(-18.4581 + 3.25466i) q^{74} +(5.78202 + 1.01953i) q^{76} +(5.91566 - 7.10978i) q^{77} +(1.64089 + 9.30597i) q^{79} +(-2.08449 + 3.61044i) q^{80} +(4.75630 - 2.74605i) q^{82} +(10.6153 + 3.86367i) q^{83} +(0.323549 + 0.271490i) q^{85} +(5.78836 + 15.9034i) q^{86} +(-0.441103 - 2.50162i) q^{88} +(1.51868 - 2.63043i) q^{89} +(-0.0435318 - 10.4789i) q^{91} +(-5.57140 + 0.982388i) q^{92} +(-6.24952 - 1.10196i) q^{94} +(3.22606 + 0.568842i) q^{95} +(5.02929 - 0.886800i) q^{97} +(-11.5858 - 6.56131i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} - 9 q^{10} - 9 q^{11} + 42 q^{14} - 15 q^{16} + 9 q^{17} - 9 q^{19} + 18 q^{20} - 12 q^{22} - 30 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31}+ \cdots + 180 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.87320 + 0.330296i −1.32455 + 0.233554i −0.790794 − 0.612082i \(-0.790332\pi\)
−0.533759 + 0.845637i \(0.679221\pi\)
\(3\) 0 0
\(4\) 1.52040 − 0.553380i 0.760200 − 0.276690i
\(5\) 0.848304 − 0.308757i 0.379373 − 0.138080i −0.145293 − 0.989389i \(-0.546412\pi\)
0.524666 + 0.851308i \(0.324190\pi\)
\(6\) 0 0
\(7\) 2.48993 + 0.894564i 0.941105 + 0.338114i
\(8\) 0.629295 − 0.363324i 0.222489 − 0.128454i
\(9\) 0 0
\(10\) −1.48706 + 0.858555i −0.470250 + 0.271499i
\(11\) 1.19563 − 3.28497i 0.360497 − 0.990456i −0.618358 − 0.785897i \(-0.712202\pi\)
0.978854 − 0.204559i \(-0.0655761\pi\)
\(12\) 0 0
\(13\) −1.35464 − 3.72184i −0.375709 − 1.03225i −0.973116 − 0.230315i \(-0.926024\pi\)
0.597407 − 0.801938i \(-0.296198\pi\)
\(14\) −4.95961 − 0.853285i −1.32551 − 0.228050i
\(15\) 0 0
\(16\) −3.53767 + 2.96846i −0.884419 + 0.742115i
\(17\) 0.233932 + 0.405183i 0.0567369 + 0.0982712i 0.892999 − 0.450059i \(-0.148597\pi\)
−0.836262 + 0.548330i \(0.815264\pi\)
\(18\) 0 0
\(19\) 3.14258 + 1.81437i 0.720958 + 0.416245i 0.815105 − 0.579313i \(-0.196679\pi\)
−0.0941474 + 0.995558i \(0.530013\pi\)
\(20\) 1.11890 − 0.938869i 0.250194 − 0.209938i
\(21\) 0 0
\(22\) −1.15465 + 6.54832i −0.246171 + 1.39611i
\(23\) −3.44344 − 0.607171i −0.718006 − 0.126604i −0.197304 − 0.980342i \(-0.563219\pi\)
−0.520702 + 0.853738i \(0.674330\pi\)
\(24\) 0 0
\(25\) −3.20593 + 2.69010i −0.641187 + 0.538020i
\(26\) 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 −0.654451 − 0.756105i \(-0.727100\pi\)
0.987353 − 0.158537i \(-0.0506777\pi\)
\(30\) 0 0
\(31\) −2.38270 − 6.54641i −0.427945 − 1.17577i −0.947057 − 0.321064i \(-0.895959\pi\)
0.519112 − 0.854706i \(-0.326263\pi\)
\(32\) 4.71214 − 5.61571i 0.832997 − 0.992727i
\(33\) 0 0
\(34\) −0.572032 − 0.681722i −0.0981027 − 0.116914i
\(35\) 2.38842 − 0.00992201i 0.403717 − 0.00167713i
\(36\) 0 0
\(37\) 9.85377 1.61995 0.809975 − 0.586464i \(-0.199481\pi\)
0.809975 + 0.586464i \(0.199481\pi\)
\(38\) −6.48596 − 2.36070i −1.05216 − 0.382956i
\(39\) 0 0
\(40\) 0.421655 − 0.502508i 0.0666695 − 0.0794536i
\(41\) −2.71326 + 0.987547i −0.423741 + 0.154229i −0.545084 − 0.838382i \(-0.683502\pi\)
0.121343 + 0.992611i \(0.461280\pi\)
\(42\) 0 0
\(43\) −1.54504 − 8.76238i −0.235617 − 1.33625i −0.841310 − 0.540553i \(-0.818215\pi\)
0.605693 − 0.795698i \(-0.292896\pi\)
\(44\) − 5.65611i − 0.852690i
\(45\) 0 0
\(46\) 6.65080 0.980606
\(47\) 3.13508 + 1.14108i 0.457298 + 0.166443i 0.560390 − 0.828229i \(-0.310651\pi\)
−0.103092 + 0.994672i \(0.532874\pi\)
\(48\) 0 0
\(49\) 5.39951 + 4.45481i 0.771358 + 0.636401i
\(50\) 5.11683 − 6.09800i 0.723629 − 0.862387i
\(51\) 0 0
\(52\) −4.11919 − 4.90906i −0.571228 − 0.680764i
\(53\) 8.75304 + 5.05357i 1.20232 + 0.694161i 0.961071 − 0.276302i \(-0.0891090\pi\)
0.241251 + 0.970463i \(0.422442\pi\)
\(54\) 0 0
\(55\) − 3.15581i − 0.425530i
\(56\) 1.89192 − 0.341706i 0.252818 − 0.0456624i
\(57\) 0 0
\(58\) −1.73128 + 9.81858i −0.227328 + 1.28924i
\(59\) 2.48304 + 2.08352i 0.323265 + 0.271251i 0.789949 − 0.613173i \(-0.210107\pi\)
−0.466684 + 0.884424i \(0.654551\pi\)
\(60\) 0 0
\(61\) 0.0209011 − 0.0574254i 0.00267611 − 0.00735257i −0.938347 − 0.345694i \(-0.887644\pi\)
0.941023 + 0.338341i \(0.109866\pi\)
\(62\) 6.62553 + 11.4757i 0.841443 + 1.45742i
\(63\) 0 0
\(64\) −2.35384 + 4.07697i −0.294230 + 0.509621i
\(65\) −2.29829 − 2.73900i −0.285068 − 0.339731i
\(66\) 0 0
\(67\) 1.78024 − 10.0962i 0.217491 − 1.23345i −0.659041 − 0.752107i \(-0.729038\pi\)
0.876532 − 0.481344i \(-0.159851\pi\)
\(68\) 0.579891 + 0.486586i 0.0703221 + 0.0590072i
\(69\) 0 0
\(70\) −4.47071 + 0.807471i −0.534352 + 0.0965113i
\(71\) 2.19935 + 1.26979i 0.261014 + 0.150697i 0.624797 − 0.780787i \(-0.285182\pi\)
−0.363783 + 0.931484i \(0.618515\pi\)
\(72\) 0 0
\(73\) − 8.37008i − 0.979643i −0.871823 − 0.489822i \(-0.837062\pi\)
0.871823 − 0.489822i \(-0.162938\pi\)
\(74\) −18.4581 + 3.25466i −2.14571 + 0.378346i
\(75\) 0 0
\(76\) 5.78202 + 1.01953i 0.663243 + 0.116948i
\(77\) 5.91566 − 7.10978i 0.674152 − 0.810235i
\(78\) 0 0
\(79\) 1.64089 + 9.30597i 0.184615 + 1.04700i 0.926449 + 0.376420i \(0.122845\pi\)
−0.741834 + 0.670583i \(0.766044\pi\)
\(80\) −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 \(-0.176029\pi\)
0.314237 + 0.949344i \(0.398251\pi\)
\(84\) 0 0
\(85\) 0.323549 + 0.271490i 0.0350938 + 0.0294472i
\(86\) 5.78836 + 15.9034i 0.624175 + 1.71491i
\(87\) 0 0
\(88\) −0.441103 − 2.50162i −0.0470217 − 0.266673i
\(89\) 1.51868 − 2.63043i 0.160980 − 0.278825i −0.774241 − 0.632891i \(-0.781868\pi\)
0.935220 + 0.354066i \(0.115201\pi\)
\(90\) 0 0
\(91\) −0.0435318 − 10.4789i −0.00456337 − 1.09849i
\(92\) −5.57140 + 0.982388i −0.580858 + 0.102421i
\(93\) 0 0
\(94\) −6.24952 − 1.10196i −0.644589 − 0.113658i
\(95\) 3.22606 + 0.568842i 0.330987 + 0.0583620i
\(96\) 0 0
\(97\) 5.02929 − 0.886800i 0.510647 − 0.0900409i 0.0876131 − 0.996155i \(-0.472076\pi\)
0.423034 + 0.906114i \(0.360965\pi\)
\(98\) −11.5858 − 6.56131i −1.17034 − 0.662792i
\(99\) 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.17.4 132
3.2 odd 2 189.2.bd.a.185.19 yes 132
7.5 odd 6 567.2.ba.a.341.19 132
21.5 even 6 189.2.ba.a.131.4 yes 132
27.7 even 9 189.2.ba.a.101.4 ✓ 132
27.20 odd 18 567.2.ba.a.143.19 132
189.47 even 18 inner 567.2.bd.a.467.4 132
189.61 odd 18 189.2.bd.a.47.19 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.4 ✓ 132 27.7 even 9
189.2.ba.a.131.4 yes 132 21.5 even 6
189.2.bd.a.47.19 yes 132 189.61 odd 18
189.2.bd.a.185.19 yes 132 3.2 odd 2
567.2.ba.a.143.19 132 27.20 odd 18
567.2.ba.a.341.19 132 7.5 odd 6
567.2.bd.a.17.4 132 1.1 even 1 trivial
567.2.bd.a.467.4 132 189.47 even 18 inner