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Properties

Label 729.2.e.k.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 \(1.37340i\) of defining polynomial
Character \(\chi\) \(=\) 729.163
Dual form 729.2.e.k.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)
 
\(f(q)\) \(=\) \(q+(-2.54193 + 0.925187i) q^{2} +(4.07335 - 3.41794i) q^{4} +(-0.290407 - 1.64698i) q^{5} +(0.383475 + 0.321774i) q^{7} +(-4.48686 + 7.77147i) q^{8} +(2.26195 + 3.91782i) q^{10} +(-0.333008 + 1.88858i) q^{11} +(-2.92473 - 1.06452i) q^{13} +(-1.27247 - 0.463140i) q^{14} +(2.36851 - 13.4325i) q^{16} +(-1.33234 - 2.30767i) q^{17} +(-2.89832 + 5.02003i) q^{19} +(-6.81220 - 5.71612i) q^{20} +(-0.900809 - 5.10874i) q^{22} +(-3.55894 + 2.98631i) q^{23} +(2.07026 - 0.753515i) q^{25} +8.41934 q^{26} +2.66183 q^{28} +(2.45736 - 0.894407i) q^{29} +(-3.53499 + 2.96621i) q^{31} +(3.29045 + 18.6611i) q^{32} +(5.52173 + 4.63328i) q^{34} +(0.418591 - 0.725020i) q^{35} +(2.42934 + 4.20773i) q^{37} +(2.72285 - 15.4421i) q^{38} +(14.1024 + 5.13287i) q^{40} +(10.8517 + 3.94970i) q^{41} +(-1.56359 + 8.86754i) q^{43} +(5.09861 + 8.83106i) q^{44} +(6.28369 - 10.8837i) q^{46} +(5.23380 + 4.39168i) q^{47} +(-1.17202 - 6.64687i) q^{49} +(-4.56532 + 3.83076i) q^{50} +(-15.5519 + 5.66043i) q^{52} -5.43322 q^{53} +3.20716 q^{55} +(-4.22125 + 1.53641i) q^{56} +(-5.41895 + 4.54704i) q^{58} +(-0.380517 - 2.15802i) q^{59} +(5.24000 + 4.39688i) q^{61} +(6.24140 - 10.8104i) q^{62} +(-11.9893 - 20.7661i) q^{64} +(-0.903872 + 5.12611i) q^{65} +(-11.7307 - 4.26964i) q^{67} +(-13.3146 - 4.84610i) q^{68} +(-0.393249 + 2.23022i) q^{70} +(1.41784 + 2.45578i) q^{71} +(-4.96749 + 8.60394i) q^{73} +(-10.0681 - 8.44817i) q^{74} +(5.35234 + 30.3546i) q^{76} +(-0.735397 + 0.617071i) q^{77} +(-4.99091 + 1.81654i) q^{79} -22.8109 q^{80} -31.2385 q^{82} +(2.56362 - 0.933082i) q^{83} +(-3.41377 + 2.86449i) q^{85} +(-4.22960 - 23.9873i) q^{86} +(-13.1829 - 11.0618i) q^{88} +(-5.60945 + 9.71585i) q^{89} +(-0.779029 - 1.34932i) q^{91} +(-4.28977 + 24.3285i) q^{92} +(-17.3671 - 6.32110i) q^{94} +(9.10957 + 3.31561i) q^{95} +(-1.19629 + 6.78448i) q^{97} +(9.12879 + 15.8115i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 6 q^{7} - 6 q^{8} - 6 q^{10} + 15 q^{11} - 3 q^{13} + 21 q^{14} + 9 q^{16} + 9 q^{17} - 12 q^{19} + 3 q^{20} + 33 q^{22} - 15 q^{23} - 12 q^{25} + 48 q^{26} + 6 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) \) 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 \(0.794532\pi\)
−0.998615 + 0.0526096i \(0.983246\pi\)
\(3\) 0 0
\(4\) 4.07335 − 3.41794i 2.03667 − 1.70897i
\(5\) −0.290407 − 1.64698i −0.129874 − 0.736551i −0.978293 − 0.207226i \(-0.933556\pi\)
0.848419 − 0.529325i \(-0.177555\pi\)
\(6\) 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 \(0.148914\pi\)
−0.992956 + 0.118482i \(0.962197\pi\)
\(12\) 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 0.657995 − 0.753022i \(-0.271405\pi\)
−0.981134 + 0.193329i \(0.938071\pi\)
\(18\) 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 −0.933398 − 0.358842i \(-0.883172\pi\)
0.191308 + 0.981530i \(0.438727\pi\)
\(24\) 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 −0.103625 − 0.994616i \(-0.533044\pi\)
0.559946 + 0.828529i \(0.310822\pi\)
\(30\) 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 0.995211 − 0.0977502i \(-0.0311646\pi\)
0.699543 + 0.714590i \(0.253387\pi\)
\(42\) 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 0.939017 − 0.343872i \(-0.111738\pi\)
−0.175589 + 0.984464i \(0.556183\pi\)
\(48\) 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 \(-0.621724\pi\)
−0.373155 + 0.927769i \(0.621724\pi\)
\(54\) 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 0.949968 − 0.312348i \(-0.101115\pi\)
−0.999507 + 0.0313973i \(0.990004\pi\)
\(60\) 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 0.937811 − 0.347147i \(-0.112850\pi\)
−0.769544 + 0.638594i \(0.779516\pi\)
\(72\) 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 −0.197468 − 0.980309i \(-0.563272\pi\)
0.478862 + 0.877890i \(0.341050\pi\)
\(84\) 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 \(0.369356\pi\)
−0.993603 + 0.112928i \(0.963977\pi\)
\(90\) 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.k.163.1 12
3.2 odd 2 729.2.e.t.163.2 12
9.2 odd 6 729.2.e.j.406.1 12
9.4 even 3 729.2.e.l.649.2 12
9.5 odd 6 729.2.e.s.649.1 12
9.7 even 3 729.2.e.u.406.2 12
27.2 odd 18 729.2.a.b.1.1 ✓ 6
27.4 even 9 inner 729.2.e.k.568.1 12
27.5 odd 18 729.2.e.s.82.1 12
27.7 even 9 729.2.c.a.487.1 12
27.11 odd 18 729.2.c.d.244.6 12
27.13 even 9 729.2.e.u.325.2 12
27.14 odd 18 729.2.e.j.325.1 12
27.16 even 9 729.2.c.a.244.1 12
27.20 odd 18 729.2.c.d.487.6 12
27.22 even 9 729.2.e.l.82.2 12
27.23 odd 18 729.2.e.t.568.2 12
27.25 even 9 729.2.a.e.1.6 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.1 ✓ 6 27.2 odd 18
729.2.a.e.1.6 yes 6 27.25 even 9
729.2.c.a.244.1 12 27.16 even 9
729.2.c.a.487.1 12 27.7 even 9
729.2.c.d.244.6 12 27.11 odd 18
729.2.c.d.487.6 12 27.20 odd 18
729.2.e.j.325.1 12 27.14 odd 18
729.2.e.j.406.1 12 9.2 odd 6
729.2.e.k.163.1 12 1.1 even 1 trivial
729.2.e.k.568.1 12 27.4 even 9 inner
729.2.e.l.82.2 12 27.22 even 9
729.2.e.l.649.2 12 9.4 even 3
729.2.e.s.82.1 12 27.5 odd 18
729.2.e.s.649.1 12 9.5 odd 6
729.2.e.t.163.2 12 3.2 odd 2
729.2.e.t.568.2 12 27.23 odd 18
729.2.e.u.325.2 12 27.13 even 9
729.2.e.u.406.2 12 9.7 even 3