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Properties

Label 729.2.e.t.163.1
Level $729$
Weight $2$
Character 729.163
Analytic conductor $5.821$
Analytic rank $0$
Dimension $12$
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] = [729,2,Mod(82,729)] 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("729.82"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,3,0,-3,6,0,6,6,0,-6,-15,0,-3,-21,0,9,-9,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 105x^{8} + 266x^{6} + 306x^{4} + 132x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 163.1
Root \(0.0878222i\) of defining polynomial
Character \(\chi\) \(=\) 729.163
Dual form 729.2.e.t.568.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.162544 + 0.0591613i) q^{2} +(-1.50917 + 1.26634i) q^{4} +(-0.648847 - 3.67980i) q^{5} +(2.32226 + 1.94861i) q^{7} +(0.343364 - 0.594724i) q^{8} +(0.323168 + 0.559743i) q^{10} +(-0.432678 + 2.45384i) q^{11} +(0.718995 + 0.261693i) q^{13} +(-0.492752 - 0.179347i) q^{14} +(0.663574 - 3.76332i) q^{16} +(-2.31139 - 4.00345i) q^{17} +(0.305922 - 0.529872i) q^{19} +(5.63910 + 4.73177i) q^{20} +(-0.0748430 - 0.424456i) q^{22} +(4.99796 - 4.19379i) q^{23} +(-8.42143 + 3.06515i) q^{25} -0.132351 q^{26} -5.97229 q^{28} +(6.15583 - 2.24054i) q^{29} +(5.01792 - 4.21053i) q^{31} +(0.353281 + 2.00355i) q^{32} +(0.612552 + 0.513992i) q^{34} +(5.66369 - 9.80980i) q^{35} +(-2.47984 - 4.29522i) q^{37} +(-0.0183779 + 0.104226i) q^{38} +(-2.41125 - 0.877625i) q^{40} +(-4.94301 - 1.79911i) q^{41} +(0.967320 - 5.48594i) q^{43} +(-2.45442 - 4.25118i) q^{44} +(-0.564280 + 0.977362i) q^{46} +(0.848483 + 0.711962i) q^{47} +(0.380286 + 2.15671i) q^{49} +(1.18752 - 0.996445i) q^{50} +(-1.41648 + 0.515556i) q^{52} +8.84310 q^{53} +9.31038 q^{55} +(1.95627 - 0.712023i) q^{56} +(-0.868041 + 0.728373i) q^{58} +(-2.05804 - 11.6717i) q^{59} +(6.27161 + 5.26250i) q^{61} +(-0.566533 + 0.981264i) q^{62} +(3.64541 + 6.31404i) q^{64} +(0.496458 - 2.81555i) q^{65} +(1.13923 + 0.414644i) q^{67} +(8.55801 + 3.11486i) q^{68} +(-0.340240 + 1.92960i) q^{70} +(2.45973 + 4.26038i) q^{71} +(-2.14972 + 3.72343i) q^{73} +(0.657195 + 0.551452i) q^{74} +(0.209312 + 1.18707i) q^{76} +(-5.78637 + 4.85534i) q^{77} +(11.0833 - 4.03399i) q^{79} -14.2788 q^{80} +0.909895 q^{82} +(-8.47234 + 3.08368i) q^{83} +(-13.2321 + 11.1031i) q^{85} +(0.167323 + 0.948936i) q^{86} +(1.31079 + 1.09989i) q^{88} +(-3.76943 + 6.52884i) q^{89} +(1.15976 + 2.00876i) q^{91} +(-2.23199 + 12.6583i) q^{92} +(-0.180037 - 0.0655279i) q^{94} +(-2.14832 - 0.781924i) q^{95} +(-0.164680 + 0.933947i) q^{97} +(-0.189407 - 0.328062i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{4} + 6 q^{5} + 6 q^{7} + 6 q^{8} - 6 q^{10} - 15 q^{11} - 3 q^{13} - 21 q^{14} + 9 q^{16} - 9 q^{17} - 12 q^{19} - 3 q^{20} + 33 q^{22} + 15 q^{23} - 12 q^{25} - 48 q^{26} + 6 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.162544 + 0.0591613i −0.114936 + 0.0418333i −0.398848 − 0.917017i \(-0.630590\pi\)
0.283912 + 0.958850i \(0.408368\pi\)
\(3\) 0 0
\(4\) −1.50917 + 1.26634i −0.754584 + 0.633171i
\(5\) −0.648847 − 3.67980i −0.290173 − 1.64565i −0.686198 − 0.727415i \(-0.740722\pi\)
0.396025 − 0.918240i \(-0.370389\pi\)
\(6\) 0 0
\(7\) 2.32226 + 1.94861i 0.877733 + 0.736505i 0.965712 − 0.259617i \(-0.0835964\pi\)
−0.0879791 + 0.996122i \(0.528041\pi\)
\(8\) 0.343364 − 0.594724i 0.121398 − 0.210267i
\(9\) 0 0
\(10\) 0.323168 + 0.559743i 0.102195 + 0.177006i
\(11\) −0.432678 + 2.45384i −0.130457 + 0.739861i 0.847458 + 0.530862i \(0.178132\pi\)
−0.977916 + 0.208999i \(0.932979\pi\)
\(12\) 0 0
\(13\) 0.718995 + 0.261693i 0.199413 + 0.0725805i 0.439796 − 0.898098i \(-0.355051\pi\)
−0.240383 + 0.970678i \(0.577273\pi\)
\(14\) −0.492752 − 0.179347i −0.131694 − 0.0479326i
\(15\) 0 0
\(16\) 0.663574 − 3.76332i 0.165894 − 0.940829i
\(17\) −2.31139 − 4.00345i −0.560595 − 0.970979i −0.997445 − 0.0714442i \(-0.977239\pi\)
0.436850 − 0.899534i \(-0.356094\pi\)
\(18\) 0 0
\(19\) 0.305922 − 0.529872i 0.0701833 − 0.121561i −0.828798 − 0.559548i \(-0.810975\pi\)
0.898982 + 0.437987i \(0.144308\pi\)
\(20\) 5.63910 + 4.73177i 1.26094 + 1.05806i
\(21\) 0 0
\(22\) −0.0748430 − 0.424456i −0.0159566 − 0.0904942i
\(23\) 4.99796 − 4.19379i 1.04215 − 0.874465i 0.0499011 − 0.998754i \(-0.484109\pi\)
0.992246 + 0.124289i \(0.0396650\pi\)
\(24\) 0 0
\(25\) −8.42143 + 3.06515i −1.68429 + 0.613030i
\(26\) −0.132351 −0.0259561
\(27\) 0 0
\(28\) −5.97229 −1.12866
\(29\) 6.15583 − 2.24054i 1.14311 − 0.416057i 0.300073 − 0.953916i \(-0.402989\pi\)
0.843035 + 0.537859i \(0.180767\pi\)
\(30\) 0 0
\(31\) 5.01792 − 4.21053i 0.901245 − 0.756234i −0.0691887 − 0.997604i \(-0.522041\pi\)
0.970433 + 0.241370i \(0.0775966\pi\)
\(32\) 0.353281 + 2.00355i 0.0624518 + 0.354182i
\(33\) 0 0
\(34\) 0.612552 + 0.513992i 0.105052 + 0.0881490i
\(35\) 5.66369 − 9.80980i 0.957338 − 1.65816i
\(36\) 0 0
\(37\) −2.47984 − 4.29522i −0.407684 − 0.706129i 0.586946 − 0.809626i \(-0.300330\pi\)
−0.994630 + 0.103497i \(0.966997\pi\)
\(38\) −0.0183779 + 0.104226i −0.00298129 + 0.0169078i
\(39\) 0 0
\(40\) −2.41125 − 0.877625i −0.381253 − 0.138765i
\(41\) −4.94301 − 1.79911i −0.771968 − 0.280973i −0.0741488 − 0.997247i \(-0.523624\pi\)
−0.697819 + 0.716274i \(0.745846\pi\)
\(42\) 0 0
\(43\) 0.967320 − 5.48594i 0.147515 − 0.836599i −0.817798 − 0.575505i \(-0.804806\pi\)
0.965313 − 0.261094i \(-0.0840832\pi\)
\(44\) −2.45442 − 4.25118i −0.370018 − 0.640889i
\(45\) 0 0
\(46\) −0.564280 + 0.977362i −0.0831986 + 0.144104i
\(47\) 0.848483 + 0.711962i 0.123764 + 0.103850i 0.702569 − 0.711616i \(-0.252036\pi\)
−0.578805 + 0.815466i \(0.696481\pi\)
\(48\) 0 0
\(49\) 0.380286 + 2.15671i 0.0543265 + 0.308101i
\(50\) 1.18752 − 0.996445i 0.167940 − 0.140919i
\(51\) 0 0
\(52\) −1.41648 + 0.515556i −0.196430 + 0.0714947i
\(53\) 8.84310 1.21469 0.607346 − 0.794437i \(-0.292234\pi\)
0.607346 + 0.794437i \(0.292234\pi\)
\(54\) 0 0
\(55\) 9.31038 1.25541
\(56\) 1.95627 − 0.712023i 0.261417 − 0.0951480i
\(57\) 0 0
\(58\) −0.868041 + 0.728373i −0.113979 + 0.0956400i
\(59\) −2.05804 − 11.6717i −0.267933 − 1.51953i −0.760550 − 0.649280i \(-0.775071\pi\)
0.492616 − 0.870247i \(-0.336041\pi\)
\(60\) 0 0
\(61\) 6.27161 + 5.26250i 0.802997 + 0.673795i 0.948925 − 0.315500i \(-0.102172\pi\)
−0.145928 + 0.989295i \(0.546617\pi\)
\(62\) −0.566533 + 0.981264i −0.0719498 + 0.124621i
\(63\) 0 0
\(64\) 3.64541 + 6.31404i 0.455677 + 0.789255i
\(65\) 0.496458 − 2.81555i 0.0615781 − 0.349227i
\(66\) 0 0
\(67\) 1.13923 + 0.414644i 0.139179 + 0.0506568i 0.410670 − 0.911784i \(-0.365295\pi\)
−0.271492 + 0.962441i \(0.587517\pi\)
\(68\) 8.55801 + 3.11486i 1.03781 + 0.377733i
\(69\) 0 0
\(70\) −0.340240 + 1.92960i −0.0406665 + 0.230631i
\(71\) 2.45973 + 4.26038i 0.291916 + 0.505614i 0.974263 − 0.225415i \(-0.0723738\pi\)
−0.682346 + 0.731029i \(0.739040\pi\)
\(72\) 0 0
\(73\) −2.14972 + 3.72343i −0.251606 + 0.435795i −0.963968 − 0.266017i \(-0.914292\pi\)
0.712362 + 0.701812i \(0.247625\pi\)
\(74\) 0.657195 + 0.551452i 0.0763973 + 0.0641050i
\(75\) 0 0
\(76\) 0.209312 + 1.18707i 0.0240098 + 0.136166i
\(77\) −5.78637 + 4.85534i −0.659418 + 0.553318i
\(78\) 0 0
\(79\) 11.0833 − 4.03399i 1.24697 − 0.453860i 0.367593 − 0.929987i \(-0.380182\pi\)
0.879376 + 0.476127i \(0.157960\pi\)
\(80\) −14.2788 −1.59642
\(81\) 0 0
\(82\) 0.909895 0.100481
\(83\) −8.47234 + 3.08368i −0.929960 + 0.338478i −0.762194 − 0.647349i \(-0.775878\pi\)
−0.167766 + 0.985827i \(0.553655\pi\)
\(84\) 0 0
\(85\) −13.2321 + 11.1031i −1.43523 + 1.20430i
\(86\) 0.167323 + 0.948936i 0.0180429 + 0.102326i
\(87\) 0 0
\(88\) 1.31079 + 1.09989i 0.139731 + 0.117248i
\(89\) −3.76943 + 6.52884i −0.399558 + 0.692055i −0.993671 − 0.112326i \(-0.964170\pi\)
0.594113 + 0.804382i \(0.297503\pi\)
\(90\) 0 0
\(91\) 1.15976 + 2.00876i 0.121576 + 0.210575i
\(92\) −2.23199 + 12.6583i −0.232701 + 1.31972i
\(93\) 0 0
\(94\) −0.180037 − 0.0655279i −0.0185694 − 0.00675869i
\(95\) −2.14832 − 0.781924i −0.220413 − 0.0802237i
\(96\) 0 0
\(97\) −0.164680 + 0.933947i −0.0167207 + 0.0948279i −0.992026 − 0.126033i \(-0.959776\pi\)
0.975305 + 0.220861i \(0.0708866\pi\)
\(98\) −0.189407 − 0.328062i −0.0191330 − 0.0331393i
\(99\) 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 729.2.e.t.163.1 12
3.2 odd 2 729.2.e.k.163.2 12
9.2 odd 6 729.2.e.u.406.1 12
9.4 even 3 729.2.e.s.649.2 12
9.5 odd 6 729.2.e.l.649.1 12
9.7 even 3 729.2.e.j.406.2 12
27.2 odd 18 729.2.a.e.1.3 yes 6
27.4 even 9 inner 729.2.e.t.568.1 12
27.5 odd 18 729.2.e.l.82.1 12
27.7 even 9 729.2.c.d.487.3 12
27.11 odd 18 729.2.c.a.244.4 12
27.13 even 9 729.2.e.j.325.2 12
27.14 odd 18 729.2.e.u.325.1 12
27.16 even 9 729.2.c.d.244.3 12
27.20 odd 18 729.2.c.a.487.4 12
27.22 even 9 729.2.e.s.82.2 12
27.23 odd 18 729.2.e.k.568.2 12
27.25 even 9 729.2.a.b.1.4 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.4 ✓ 6 27.25 even 9
729.2.a.e.1.3 yes 6 27.2 odd 18
729.2.c.a.244.4 12 27.11 odd 18
729.2.c.a.487.4 12 27.20 odd 18
729.2.c.d.244.3 12 27.16 even 9
729.2.c.d.487.3 12 27.7 even 9
729.2.e.j.325.2 12 27.13 even 9
729.2.e.j.406.2 12 9.7 even 3
729.2.e.k.163.2 12 3.2 odd 2
729.2.e.k.568.2 12 27.23 odd 18
729.2.e.l.82.1 12 27.5 odd 18
729.2.e.l.649.1 12 9.5 odd 6
729.2.e.s.82.2 12 27.22 even 9
729.2.e.s.649.2 12 9.4 even 3
729.2.e.t.163.1 12 1.1 even 1 trivial
729.2.e.t.568.1 12 27.4 even 9 inner
729.2.e.u.325.1 12 27.14 odd 18
729.2.e.u.406.1 12 9.2 odd 6