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Properties

Label 729.2.e.s.649.2
Level $729$
Weight $2$
Character 729.649
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,-12,0,-3,6,0,-6,3,0,6,6,0,27,-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 649.2
Root \(-0.0878222i\) of defining polynomial
Character \(\chi\) \(=\) 729.649
Dual form 729.2.e.s.82.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.0300370 - 0.170348i) q^{2} +(1.85127 + 0.673807i) q^{4} +(-2.86237 + 2.40182i) q^{5} +(-2.84868 + 1.03683i) q^{7} +(0.343364 - 0.594724i) q^{8} +(0.323168 + 0.559743i) q^{10} +(-1.90875 - 1.60163i) q^{11} +(-0.132865 - 0.753515i) q^{13} +(0.0910570 + 0.516410i) q^{14} +(2.92734 + 2.45633i) q^{16} +(-2.31139 - 4.00345i) q^{17} +(0.305922 - 0.529872i) q^{19} +(-6.91738 + 2.51772i) q^{20} +(-0.330168 + 0.277044i) q^{22} +(-6.13091 - 2.23147i) q^{23} +(1.55622 - 8.82575i) q^{25} -0.132351 q^{26} -5.97229 q^{28} +(-1.13755 + 6.45137i) q^{29} +(-6.15539 - 2.24038i) q^{31} +(1.55849 - 1.30773i) q^{32} +(-0.751407 + 0.273490i) q^{34} +(5.66369 - 9.80980i) q^{35} +(-2.47984 - 4.29522i) q^{37} +(-0.0810738 - 0.0680290i) q^{38} +(0.445582 + 2.52702i) q^{40} +(0.913431 + 5.18032i) q^{41} +(4.26731 + 3.58070i) q^{43} +(-2.45442 - 4.25118i) q^{44} +(-0.564280 + 0.977362i) q^{46} +(-1.04082 + 0.378827i) q^{47} +(1.67762 - 1.40769i) q^{49} +(-1.45670 - 0.530197i) q^{50} +(0.261754 - 1.48448i) q^{52} +8.84310 q^{53} +9.31038 q^{55} +(-0.361503 + 2.05019i) q^{56} +(1.06481 + 0.387559i) q^{58} +(-9.07897 + 7.61816i) q^{59} +(-7.69327 + 2.80012i) q^{61} +(-0.566533 + 0.981264i) q^{62} +(3.64541 + 6.31404i) q^{64} +(2.19011 + 1.83772i) q^{65} +(-0.210520 - 1.19392i) q^{67} +(-1.58146 - 8.96889i) q^{68} +(-1.50096 - 1.25946i) q^{70} +(2.45973 + 4.26038i) q^{71} +(-2.14972 + 3.72343i) q^{73} +(-0.806169 + 0.293421i) q^{74} +(0.923375 - 0.774804i) q^{76} +(7.09803 + 2.58347i) q^{77} +(-2.04811 + 11.6154i) q^{79} -14.2788 q^{80} +0.909895 q^{82} +(1.56562 - 8.87910i) q^{83} +(16.2316 + 5.90782i) q^{85} +(0.738141 - 0.619374i) q^{86} +(-1.60792 + 0.585237i) q^{88} +(-3.76943 + 6.52884i) q^{89} +(1.15976 + 2.00876i) q^{91} +(-9.84638 - 8.26210i) q^{92} +(0.0332694 + 0.188680i) q^{94} +(0.396993 + 2.25146i) q^{95} +(-0.726481 - 0.609590i) q^{97} +(-0.189407 - 0.328062i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{4} - 12 q^{5} - 3 q^{7} + 6 q^{8} - 6 q^{10} + 3 q^{11} + 6 q^{13} + 6 q^{14} + 27 q^{16} - 9 q^{17} - 12 q^{19} - 39 q^{20} - 39 q^{22} - 21 q^{23} + 6 q^{25} - 48 q^{26} + 6 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{5}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0300370 − 0.170348i 0.0212393 − 0.120454i −0.972345 − 0.233550i \(-0.924966\pi\)
0.993584 + 0.113096i \(0.0360768\pi\)
\(3\) 0 0
\(4\) 1.85127 + 0.673807i 0.925635 + 0.336903i
\(5\) −2.86237 + 2.40182i −1.28009 + 1.07412i −0.286861 + 0.957972i \(0.592612\pi\)
−0.993231 + 0.116153i \(0.962944\pi\)
\(6\) 0 0
\(7\) −2.84868 + 1.03683i −1.07670 + 0.391886i −0.818678 − 0.574253i \(-0.805292\pi\)
−0.258021 + 0.966139i \(0.583070\pi\)
\(8\) 0.343364 − 0.594724i 0.121398 − 0.210267i
\(9\) 0 0
\(10\) 0.323168 + 0.559743i 0.102195 + 0.177006i
\(11\) −1.90875 − 1.60163i −0.575510 − 0.482910i 0.307959 − 0.951400i \(-0.400354\pi\)
−0.883469 + 0.468490i \(0.844798\pi\)
\(12\) 0 0
\(13\) −0.132865 − 0.753515i −0.0368501 − 0.208987i 0.960823 − 0.277162i \(-0.0893937\pi\)
−0.997673 + 0.0681742i \(0.978283\pi\)
\(14\) 0.0910570 + 0.516410i 0.0243360 + 0.138016i
\(15\) 0 0
\(16\) 2.92734 + 2.45633i 0.731835 + 0.614083i
\(17\) −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 \(0.144308\pi\)
\(20\) −6.91738 + 2.51772i −1.54677 + 0.562980i
\(21\) 0 0
\(22\) −0.330168 + 0.277044i −0.0703920 + 0.0590659i
\(23\) −6.13091 − 2.23147i −1.27838 − 0.465293i −0.388486 − 0.921455i \(-0.627002\pi\)
−0.889897 + 0.456161i \(0.849224\pi\)
\(24\) 0 0
\(25\) 1.55622 − 8.82575i 0.311244 − 1.76515i
\(26\) −0.132351 −0.0259561
\(27\) 0 0
\(28\) −5.97229 −1.12866
\(29\) −1.13755 + 6.45137i −0.211238 + 1.19799i 0.676079 + 0.736829i \(0.263678\pi\)
−0.887317 + 0.461160i \(0.847433\pi\)
\(30\) 0 0
\(31\) −6.15539 − 2.24038i −1.10554 − 0.402384i −0.276184 − 0.961105i \(-0.589070\pi\)
−0.829356 + 0.558721i \(0.811292\pi\)
\(32\) 1.55849 − 1.30773i 0.275504 − 0.231176i
\(33\) 0 0
\(34\) −0.751407 + 0.273490i −0.128865 + 0.0469031i
\(35\) 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 \(0.966997\pi\)
\(38\) −0.0810738 − 0.0680290i −0.0131519 − 0.0110358i
\(39\) 0 0
\(40\) 0.445582 + 2.52702i 0.0704527 + 0.399557i
\(41\) 0.913431 + 5.18032i 0.142654 + 0.809031i 0.969221 + 0.246192i \(0.0791795\pi\)
−0.826567 + 0.562838i \(0.809709\pi\)
\(42\) 0 0
\(43\) 4.26731 + 3.58070i 0.650758 + 0.546051i 0.907301 − 0.420482i \(-0.138139\pi\)
−0.256543 + 0.966533i \(0.582583\pi\)
\(44\) −2.45442 − 4.25118i −0.370018 − 0.640889i
\(45\) 0 0
\(46\) −0.564280 + 0.977362i −0.0831986 + 0.144104i
\(47\) −1.04082 + 0.378827i −0.151819 + 0.0552576i −0.416812 − 0.908993i \(-0.636853\pi\)
0.264993 + 0.964250i \(0.414630\pi\)
\(48\) 0 0
\(49\) 1.67762 − 1.40769i 0.239660 − 0.201099i
\(50\) −1.45670 − 0.530197i −0.206009 − 0.0749812i
\(51\) 0 0
\(52\) 0.261754 − 1.48448i 0.0362988 − 0.205861i
\(53\) 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
\(56\) −0.361503 + 2.05019i −0.0483079 + 0.273968i
\(57\) 0 0
\(58\) 1.06481 + 0.387559i 0.139816 + 0.0508890i
\(59\) −9.07897 + 7.61816i −1.18198 + 0.991800i −0.182018 + 0.983295i \(0.558263\pi\)
−0.999964 + 0.00850504i \(0.997293\pi\)
\(60\) 0 0
\(61\) −7.69327 + 2.80012i −0.985022 + 0.358519i −0.783791 − 0.621025i \(-0.786716\pi\)
−0.201231 + 0.979544i \(0.564494\pi\)
\(62\) −0.566533 + 0.981264i −0.0719498 + 0.124621i
\(63\) 0 0
\(64\) 3.64541 + 6.31404i 0.455677 + 0.789255i
\(65\) 2.19011 + 1.83772i 0.271650 + 0.227941i
\(66\) 0 0
\(67\) −0.210520 − 1.19392i −0.0257192 − 0.145861i 0.969244 − 0.246102i \(-0.0791497\pi\)
−0.994963 + 0.100241i \(0.968039\pi\)
\(68\) −1.58146 − 8.96889i −0.191780 − 1.08764i
\(69\) 0 0
\(70\) −1.50096 − 1.25946i −0.179399 − 0.150534i
\(71\) 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 \(0.247625\pi\)
\(74\) −0.806169 + 0.293421i −0.0937152 + 0.0341095i
\(75\) 0 0
\(76\) 0.923375 − 0.774804i 0.105918 − 0.0888761i
\(77\) 7.09803 + 2.58347i 0.808896 + 0.294414i
\(78\) 0 0
\(79\) −2.04811 + 11.6154i −0.230431 + 1.30684i 0.621596 + 0.783338i \(0.286485\pi\)
−0.852027 + 0.523499i \(0.824627\pi\)
\(80\) −14.2788 −1.59642
\(81\) 0 0
\(82\) 0.909895 0.100481
\(83\) 1.56562 − 8.87910i 0.171850 − 0.974608i −0.769868 − 0.638203i \(-0.779678\pi\)
0.941718 − 0.336404i \(-0.109211\pi\)
\(84\) 0 0
\(85\) 16.2316 + 5.90782i 1.76057 + 0.640793i
\(86\) 0.738141 − 0.619374i 0.0795958 − 0.0667888i
\(87\) 0 0
\(88\) −1.60792 + 0.585237i −0.171405 + 0.0623865i
\(89\) −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
\(92\) −9.84638 − 8.26210i −1.02656 − 0.861383i
\(93\) 0 0
\(94\) 0.0332694 + 0.188680i 0.00343148 + 0.0194609i
\(95\) 0.396993 + 2.25146i 0.0407306 + 0.230995i
\(96\) 0 0
\(97\) −0.726481 − 0.609590i −0.0737630 − 0.0618945i 0.605161 − 0.796103i \(-0.293109\pi\)
−0.678924 + 0.734209i \(0.737553\pi\)
\(98\) −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.s.649.2 12
3.2 odd 2 729.2.e.l.649.1 12
9.2 odd 6 729.2.e.k.163.2 12
9.4 even 3 729.2.e.j.406.2 12
9.5 odd 6 729.2.e.u.406.1 12
9.7 even 3 729.2.e.t.163.1 12
27.2 odd 18 729.2.c.a.487.4 12
27.4 even 9 729.2.e.j.325.2 12
27.5 odd 18 729.2.e.k.568.2 12
27.7 even 9 729.2.c.d.244.3 12
27.11 odd 18 729.2.a.e.1.3 yes 6
27.13 even 9 inner 729.2.e.s.82.2 12
27.14 odd 18 729.2.e.l.82.1 12
27.16 even 9 729.2.a.b.1.4 ✓ 6
27.20 odd 18 729.2.c.a.244.4 12
27.22 even 9 729.2.e.t.568.1 12
27.23 odd 18 729.2.e.u.325.1 12
27.25 even 9 729.2.c.d.487.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.4 ✓ 6 27.16 even 9
729.2.a.e.1.3 yes 6 27.11 odd 18
729.2.c.a.244.4 12 27.20 odd 18
729.2.c.a.487.4 12 27.2 odd 18
729.2.c.d.244.3 12 27.7 even 9
729.2.c.d.487.3 12 27.25 even 9
729.2.e.j.325.2 12 27.4 even 9
729.2.e.j.406.2 12 9.4 even 3
729.2.e.k.163.2 12 9.2 odd 6
729.2.e.k.568.2 12 27.5 odd 18
729.2.e.l.82.1 12 27.14 odd 18
729.2.e.l.649.1 12 3.2 odd 2
729.2.e.s.82.2 12 27.13 even 9 inner
729.2.e.s.649.2 12 1.1 even 1 trivial
729.2.e.t.163.1 12 9.7 even 3
729.2.e.t.568.1 12 27.22 even 9
729.2.e.u.325.1 12 27.23 odd 18
729.2.e.u.406.1 12 9.5 odd 6