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Properties

Label 729.2.e.t.163.2
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 163.2
Root \(-1.37340i\) of defining polynomial
Character \(\chi\) \(=\) 729.163
Dual form 729.2.e.t.568.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.54193 - 0.925187i) q^{2} +(4.07335 - 3.41794i) q^{4} +(0.290407 + 1.64698i) q^{5} +(0.383475 + 0.321774i) q^{7} +(4.48686 - 7.77147i) q^{8} +(2.26195 + 3.91782i) q^{10} +(0.333008 - 1.88858i) q^{11} +(-2.92473 - 1.06452i) q^{13} +(1.27247 + 0.463140i) q^{14} +(2.36851 - 13.4325i) q^{16} +(1.33234 + 2.30767i) q^{17} +(-2.89832 + 5.02003i) q^{19} +(6.81220 + 5.71612i) q^{20} +(-0.900809 - 5.10874i) q^{22} +(3.55894 - 2.98631i) q^{23} +(2.07026 - 0.753515i) q^{25} -8.41934 q^{26} +2.66183 q^{28} +(-2.45736 + 0.894407i) q^{29} +(-3.53499 + 2.96621i) q^{31} +(-3.29045 - 18.6611i) q^{32} +(5.52173 + 4.63328i) q^{34} +(-0.418591 + 0.725020i) q^{35} +(2.42934 + 4.20773i) q^{37} +(-2.72285 + 15.4421i) q^{38} +(14.1024 + 5.13287i) q^{40} +(-10.8517 - 3.94970i) q^{41} +(-1.56359 + 8.86754i) q^{43} +(-5.09861 - 8.83106i) q^{44} +(6.28369 - 10.8837i) q^{46} +(-5.23380 - 4.39168i) q^{47} +(-1.17202 - 6.64687i) q^{49} +(4.56532 - 3.83076i) q^{50} +(-15.5519 + 5.66043i) q^{52} +5.43322 q^{53} +3.20716 q^{55} +(4.22125 - 1.53641i) q^{56} +(-5.41895 + 4.54704i) q^{58} +(0.380517 + 2.15802i) q^{59} +(5.24000 + 4.39688i) q^{61} +(-6.24140 + 10.8104i) q^{62} +(-11.9893 - 20.7661i) q^{64} +(0.903872 - 5.12611i) q^{65} +(-11.7307 - 4.26964i) q^{67} +(13.3146 + 4.84610i) q^{68} +(-0.393249 + 2.23022i) q^{70} +(-1.41784 - 2.45578i) q^{71} +(-4.96749 + 8.60394i) q^{73} +(10.0681 + 8.44817i) q^{74} +(5.35234 + 30.3546i) q^{76} +(0.735397 - 0.617071i) q^{77} +(-4.99091 + 1.81654i) q^{79} +22.8109 q^{80} -31.2385 q^{82} +(-2.56362 + 0.933082i) q^{83} +(-3.41377 + 2.86449i) q^{85} +(4.22960 + 23.9873i) q^{86} +(-13.1829 - 11.0618i) q^{88} +(5.60945 - 9.71585i) q^{89} +(-0.779029 - 1.34932i) q^{91} +(4.28977 - 24.3285i) q^{92} +(-17.3671 - 6.32110i) q^{94} +(-9.10957 - 3.31561i) q^{95} +(-1.19629 + 6.78448i) q^{97} +(-9.12879 - 15.8115i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{4} + 6 q^{5} + 6 q^{7} + 6 q^{8} - 6 q^{10} - 15 q^{11} - 3 q^{13} - 21 q^{14} + 9 q^{16} - 9 q^{17} - 12 q^{19} - 3 q^{20} + 33 q^{22} + 15 q^{23} - 12 q^{25} - 48 q^{26} + 6 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.54193 − 0.925187i 1.79742 − 0.654206i 0.798800 − 0.601596i \(-0.205468\pi\)
0.998615 − 0.0526096i \(-0.0167539\pi\)
\(3\) 0 0
\(4\) 4.07335 − 3.41794i 2.03667 − 1.70897i
\(5\) 0.290407 + 1.64698i 0.129874 + 0.736551i 0.978293 + 0.207226i \(0.0664436\pi\)
−0.848419 + 0.529325i \(0.822445\pi\)
\(6\) 0 0
\(7\) 0.383475 + 0.321774i 0.144940 + 0.121619i 0.712375 − 0.701799i \(-0.247620\pi\)
−0.567435 + 0.823418i \(0.692064\pi\)
\(8\) 4.48686 − 7.77147i 1.58634 − 2.74763i
\(9\) 0 0
\(10\) 2.26195 + 3.91782i 0.715293 + 1.23892i
\(11\) 0.333008 − 1.88858i 0.100406 − 0.569429i −0.892550 − 0.450948i \(-0.851086\pi\)
0.992956 − 0.118482i \(-0.0378027\pi\)
\(12\) 0 0
\(13\) −2.92473 − 1.06452i −0.811175 − 0.295243i −0.0970658 − 0.995278i \(-0.530946\pi\)
−0.714109 + 0.700034i \(0.753168\pi\)
\(14\) 1.27247 + 0.463140i 0.340081 + 0.123779i
\(15\) 0 0
\(16\) 2.36851 − 13.4325i 0.592129 − 3.35813i
\(17\) 1.33234 + 2.30767i 0.323139 + 0.559693i 0.981134 − 0.193329i \(-0.0619285\pi\)
−0.657995 + 0.753022i \(0.728595\pi\)
\(18\) 0 0
\(19\) −2.89832 + 5.02003i −0.664920 + 1.15167i 0.314387 + 0.949295i \(0.398201\pi\)
−0.979307 + 0.202380i \(0.935132\pi\)
\(20\) 6.81220 + 5.71612i 1.52325 + 1.27816i
\(21\) 0 0
\(22\) −0.900809 − 5.10874i −0.192053 − 1.08919i
\(23\) 3.55894 − 2.98631i 0.742091 − 0.622688i −0.191308 − 0.981530i \(-0.561273\pi\)
0.933398 + 0.358842i \(0.116828\pi\)
\(24\) 0 0
\(25\) 2.07026 − 0.753515i 0.414053 − 0.150703i
\(26\) −8.41934 −1.65117
\(27\) 0 0
\(28\) 2.66183 0.503039
\(29\) −2.45736 + 0.894407i −0.456321 + 0.166087i −0.559946 − 0.828529i \(-0.689178\pi\)
0.103625 + 0.994616i \(0.466956\pi\)
\(30\) 0 0
\(31\) −3.53499 + 2.96621i −0.634903 + 0.532747i −0.902448 − 0.430798i \(-0.858232\pi\)
0.267545 + 0.963545i \(0.413788\pi\)
\(32\) −3.29045 − 18.6611i −0.581674 − 3.29884i
\(33\) 0 0
\(34\) 5.52173 + 4.63328i 0.946969 + 0.794602i
\(35\) −0.418591 + 0.725020i −0.0707547 + 0.122551i
\(36\) 0 0
\(37\) 2.42934 + 4.20773i 0.399381 + 0.691747i 0.993650 − 0.112519i \(-0.0358919\pi\)
−0.594269 + 0.804266i \(0.702559\pi\)
\(38\) −2.72285 + 15.4421i −0.441705 + 2.50503i
\(39\) 0 0
\(40\) 14.1024 + 5.13287i 2.22979 + 0.811578i
\(41\) −10.8517 − 3.94970i −1.69475 − 0.616840i −0.699543 − 0.714590i \(-0.746613\pi\)
−0.995211 + 0.0977502i \(0.968835\pi\)
\(42\) 0 0
\(43\) −1.56359 + 8.86754i −0.238445 + 1.35229i 0.596792 + 0.802396i \(0.296442\pi\)
−0.835236 + 0.549891i \(0.814669\pi\)
\(44\) −5.09861 − 8.83106i −0.768645 − 1.33133i
\(45\) 0 0
\(46\) 6.28369 − 10.8837i 0.926479 − 1.60471i
\(47\) −5.23380 − 4.39168i −0.763428 − 0.640592i 0.175589 − 0.984464i \(-0.443817\pi\)
−0.939017 + 0.343872i \(0.888262\pi\)
\(48\) 0 0
\(49\) −1.17202 − 6.64687i −0.167432 − 0.949553i
\(50\) 4.56532 − 3.83076i 0.645634 − 0.541752i
\(51\) 0 0
\(52\) −15.5519 + 5.66043i −2.15666 + 0.784960i
\(53\) 5.43322 0.746309 0.373155 − 0.927769i \(-0.378276\pi\)
0.373155 + 0.927769i \(0.378276\pi\)
\(54\) 0 0
\(55\) 3.20716 0.432454
\(56\) 4.22125 − 1.53641i 0.564089 − 0.205311i
\(57\) 0 0
\(58\) −5.41895 + 4.54704i −0.711543 + 0.597055i
\(59\) 0.380517 + 2.15802i 0.0495392 + 0.280950i 0.999507 − 0.0313973i \(-0.00999570\pi\)
−0.949968 + 0.312348i \(0.898885\pi\)
\(60\) 0 0
\(61\) 5.24000 + 4.39688i 0.670914 + 0.562963i 0.913336 − 0.407208i \(-0.133497\pi\)
−0.242422 + 0.970171i \(0.577942\pi\)
\(62\) −6.24140 + 10.8104i −0.792659 + 1.37293i
\(63\) 0 0
\(64\) −11.9893 − 20.7661i −1.49866 − 2.59576i
\(65\) 0.903872 − 5.12611i 0.112111 − 0.635816i
\(66\) 0 0
\(67\) −11.7307 − 4.26964i −1.43314 − 0.521620i −0.495309 − 0.868717i \(-0.664945\pi\)
−0.937829 + 0.347097i \(0.887167\pi\)
\(68\) 13.3146 + 4.84610i 1.61463 + 0.587677i
\(69\) 0 0
\(70\) −0.393249 + 2.23022i −0.0470022 + 0.266563i
\(71\) −1.41784 − 2.45578i −0.168267 − 0.291447i 0.769544 − 0.638594i \(-0.220484\pi\)
−0.937811 + 0.347147i \(0.887150\pi\)
\(72\) 0 0
\(73\) −4.96749 + 8.60394i −0.581400 + 1.00701i 0.413913 + 0.910316i \(0.364162\pi\)
−0.995314 + 0.0966986i \(0.969172\pi\)
\(74\) 10.0681 + 8.44817i 1.17040 + 0.982080i
\(75\) 0 0
\(76\) 5.35234 + 30.3546i 0.613955 + 3.48191i
\(77\) 0.735397 − 0.617071i 0.0838063 − 0.0703218i
\(78\) 0 0
\(79\) −4.99091 + 1.81654i −0.561521 + 0.204377i −0.607158 − 0.794581i \(-0.707690\pi\)
0.0456370 + 0.998958i \(0.485468\pi\)
\(80\) 22.8109 2.55033
\(81\) 0 0
\(82\) −31.2385 −3.44972
\(83\) −2.56362 + 0.933082i −0.281394 + 0.102419i −0.478862 − 0.877890i \(-0.658950\pi\)
0.197468 + 0.980309i \(0.436728\pi\)
\(84\) 0 0
\(85\) −3.41377 + 2.86449i −0.370275 + 0.310698i
\(86\) 4.22960 + 23.9873i 0.456090 + 2.58661i
\(87\) 0 0
\(88\) −13.1829 − 11.0618i −1.40530 − 1.17919i
\(89\) 5.60945 − 9.71585i 0.594600 − 1.02988i −0.399003 − 0.916950i \(-0.630644\pi\)
0.993603 − 0.112928i \(-0.0360230\pi\)
\(90\) 0 0
\(91\) −0.779029 − 1.34932i −0.0816644 − 0.141447i
\(92\) 4.28977 − 24.3285i 0.447240 − 2.53642i
\(93\) 0 0
\(94\) −17.3671 − 6.32110i −1.79128 − 0.651971i
\(95\) −9.10957 − 3.31561i −0.934623 − 0.340175i
\(96\) 0 0
\(97\) −1.19629 + 6.78448i −0.121465 + 0.688860i 0.861881 + 0.507111i \(0.169287\pi\)
−0.983345 + 0.181748i \(0.941824\pi\)
\(98\) −9.12879 − 15.8115i −0.922147 − 1.59721i
\(99\) 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.163.2 12
3.2 odd 2 729.2.e.k.163.1 12
9.2 odd 6 729.2.e.u.406.2 12
9.4 even 3 729.2.e.s.649.1 12
9.5 odd 6 729.2.e.l.649.2 12
9.7 even 3 729.2.e.j.406.1 12
27.2 odd 18 729.2.a.e.1.6 yes 6
27.4 even 9 inner 729.2.e.t.568.2 12
27.5 odd 18 729.2.e.l.82.2 12
27.7 even 9 729.2.c.d.487.6 12
27.11 odd 18 729.2.c.a.244.1 12
27.13 even 9 729.2.e.j.325.1 12
27.14 odd 18 729.2.e.u.325.2 12
27.16 even 9 729.2.c.d.244.6 12
27.20 odd 18 729.2.c.a.487.1 12
27.22 even 9 729.2.e.s.82.1 12
27.23 odd 18 729.2.e.k.568.1 12
27.25 even 9 729.2.a.b.1.1 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.1 ✓ 6 27.25 even 9
729.2.a.e.1.6 yes 6 27.2 odd 18
729.2.c.a.244.1 12 27.11 odd 18
729.2.c.a.487.1 12 27.20 odd 18
729.2.c.d.244.6 12 27.16 even 9
729.2.c.d.487.6 12 27.7 even 9
729.2.e.j.325.1 12 27.13 even 9
729.2.e.j.406.1 12 9.7 even 3
729.2.e.k.163.1 12 3.2 odd 2
729.2.e.k.568.1 12 27.23 odd 18
729.2.e.l.82.2 12 27.5 odd 18
729.2.e.l.649.2 12 9.5 odd 6
729.2.e.s.82.1 12 27.22 even 9
729.2.e.s.649.1 12 9.4 even 3
729.2.e.t.163.2 12 1.1 even 1 trivial
729.2.e.t.568.2 12 27.4 even 9 inner
729.2.e.u.325.2 12 27.14 odd 18
729.2.e.u.406.2 12 9.2 odd 6