↑

Properties

Label 729.2.e.s.82.1
Level $729$
Weight $2$
Character 729.82
Analytic conductor $5.821$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

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 82.1
Root \(-1.37340i\) of defining polynomial
Character \(\chi\) \(=\) 729.82
Dual form 729.2.e.s.649.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.469730 - 2.66397i) q^{2} +(-4.99670 + 1.81865i) q^{4} +(1.28112 + 1.07499i) q^{5} +(-0.470402 - 0.171212i) q^{7} +(4.48686 + 7.77147i) q^{8} +(2.26195 - 3.91782i) q^{10} +(1.46906 - 1.23269i) q^{11} +(0.540469 - 3.06515i) q^{13} +(-0.235142 + 1.33356i) q^{14} +(10.4486 - 8.76745i) q^{16} +(1.33234 - 2.30767i) q^{17} +(-2.89832 - 5.02003i) q^{19} +(-8.35640 - 3.04148i) q^{20} +(-3.97390 - 3.33449i) q^{22} +(-4.36569 + 1.58898i) q^{23} +(-0.382569 - 2.16966i) q^{25} -8.41934 q^{26} +2.66183 q^{28} +(0.454102 + 2.57534i) q^{29} +(4.33631 - 1.57829i) q^{31} +(-14.5157 - 12.1801i) q^{32} +(-6.77341 - 2.46532i) q^{34} +(-0.418591 - 0.725020i) q^{35} +(2.42934 - 4.20773i) q^{37} +(-12.0118 + 10.0791i) q^{38} +(-2.60603 + 14.7795i) q^{40} +(2.00532 - 11.3727i) q^{41} +(-6.89772 + 5.78788i) q^{43} +(-5.09861 + 8.83106i) q^{44} +(6.28369 + 10.8837i) q^{46} +(6.42020 + 2.33676i) q^{47} +(-5.17035 - 4.33844i) q^{49} +(-5.60020 + 2.03831i) q^{50} +(2.87388 + 16.2986i) q^{52} +5.43322 q^{53} +3.20716 q^{55} +(-0.780056 - 4.42392i) q^{56} +(6.64732 - 2.41943i) q^{58} +(1.67864 + 1.40855i) q^{59} +(-6.42781 - 2.33953i) q^{61} +(-6.24140 - 10.8104i) q^{62} +(-11.9893 + 20.7661i) q^{64} +(3.98741 - 3.34583i) q^{65} +(2.16775 - 12.2939i) q^{67} +(-2.46043 + 13.9538i) q^{68} +(-1.73481 + 1.45568i) q^{70} +(-1.41784 + 2.45578i) q^{71} +(-4.96749 - 8.60394i) q^{73} +(-12.3504 - 4.49518i) q^{74} +(23.6117 + 19.8126i) q^{76} +(-0.902098 + 0.328337i) q^{77} +(0.922282 + 5.23052i) q^{79} +22.8109 q^{80} -31.2385 q^{82} +(0.473738 + 2.68670i) q^{83} +(4.18761 - 1.52416i) q^{85} +(18.6588 + 15.6566i) q^{86} +(16.1712 + 5.88584i) q^{88} +(5.60945 + 9.71585i) q^{89} +(-0.779029 + 1.34932i) q^{91} +(18.9242 - 15.8793i) q^{92} +(3.20930 - 18.2009i) q^{94} +(1.68338 - 9.54693i) q^{95} +(-5.27739 + 4.42826i) q^{97} +(-9.12879 + 15.8115i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{4} - 12 q^{5} - 3 q^{7} + 6 q^{8} - 6 q^{10} + 3 q^{11} + 6 q^{13} + 6 q^{14} + 27 q^{16} - 9 q^{17} - 12 q^{19} - 39 q^{20} - 39 q^{22} - 21 q^{23} + 6 q^{25} - 48 q^{26} + 6 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{4}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.469730 − 2.66397i −0.332149 − 1.88371i −0.453746 − 0.891131i \(-0.649913\pi\)
0.121597 − 0.992580i \(-0.461198\pi\)
\(3\) 0 0
\(4\) −4.99670 + 1.81865i −2.49835 + 0.909325i
\(5\) 1.28112 + 1.07499i 0.572935 + 0.480749i 0.882618 − 0.470090i \(-0.155779\pi\)
−0.309684 + 0.950840i \(0.600223\pi\)
\(6\) 0 0
\(7\) −0.470402 − 0.171212i −0.177795 − 0.0647122i 0.251589 − 0.967834i \(-0.419047\pi\)
−0.429384 + 0.903122i \(0.641269\pi\)
\(8\) 4.48686 + 7.77147i 1.58634 + 2.74763i
\(9\) 0 0
\(10\) 2.26195 − 3.91782i 0.715293 − 1.23892i
\(11\) 1.46906 − 1.23269i 0.442937 − 0.371669i −0.393870 − 0.919166i \(-0.628864\pi\)
0.836807 + 0.547498i \(0.184419\pi\)
\(12\) 0 0
\(13\) 0.540469 − 3.06515i 0.149899 − 0.850120i −0.813403 − 0.581700i \(-0.802388\pi\)
0.963302 − 0.268419i \(-0.0865013\pi\)
\(14\) −0.235142 + 1.33356i −0.0628445 + 0.356409i
\(15\) 0 0
\(16\) 10.4486 − 8.76745i 2.61216 − 2.19186i
\(17\) 1.33234 − 2.30767i 0.323139 − 0.559693i −0.657995 − 0.753022i \(-0.728595\pi\)
0.981134 + 0.193329i \(0.0619285\pi\)
\(18\) 0 0
\(19\) −2.89832 − 5.02003i −0.664920 − 1.15167i −0.979307 − 0.202380i \(-0.935132\pi\)
0.314387 − 0.949295i \(-0.398201\pi\)
\(20\) −8.35640 − 3.04148i −1.86855 − 0.680096i
\(21\) 0 0
\(22\) −3.97390 − 3.33449i −0.847237 − 0.710917i
\(23\) −4.36569 + 1.58898i −0.910309 + 0.331325i −0.754376 − 0.656442i \(-0.772061\pi\)
−0.155933 + 0.987768i \(0.549838\pi\)
\(24\) 0 0
\(25\) −0.382569 − 2.16966i −0.0765139 − 0.433932i
\(26\) −8.41934 −1.65117
\(27\) 0 0
\(28\) 2.66183 0.503039
\(29\) 0.454102 + 2.57534i 0.0843247 + 0.478229i 0.997500 + 0.0706626i \(0.0225114\pi\)
−0.913176 + 0.407566i \(0.866378\pi\)
\(30\) 0 0
\(31\) 4.33631 − 1.57829i 0.778824 − 0.283469i 0.0781418 − 0.996942i \(-0.475101\pi\)
0.700682 + 0.713473i \(0.252879\pi\)
\(32\) −14.5157 − 12.1801i −2.56604 − 2.15316i
\(33\) 0 0
\(34\) −6.77341 − 2.46532i −1.16163 − 0.422799i
\(35\) −0.418591 − 0.725020i −0.0707547 − 0.122551i
\(36\) 0 0
\(37\) 2.42934 − 4.20773i 0.399381 − 0.691747i −0.594269 − 0.804266i \(-0.702559\pi\)
0.993650 + 0.112519i \(0.0358919\pi\)
\(38\) −12.0118 + 10.0791i −1.94857 + 1.63504i
\(39\) 0 0
\(40\) −2.60603 + 14.7795i −0.412049 + 2.33685i
\(41\) 2.00532 − 11.3727i 0.313178 − 1.77612i −0.269082 − 0.963117i \(-0.586720\pi\)
0.582260 − 0.813003i \(-0.302169\pi\)
\(42\) 0 0
\(43\) −6.89772 + 5.78788i −1.05189 + 0.882643i −0.993291 − 0.115639i \(-0.963109\pi\)
−0.0586014 + 0.998281i \(0.518664\pi\)
\(44\) −5.09861 + 8.83106i −0.768645 + 1.33133i
\(45\) 0 0
\(46\) 6.28369 + 10.8837i 0.926479 + 1.60471i
\(47\) 6.42020 + 2.33676i 0.936483 + 0.340852i 0.764776 − 0.644296i \(-0.222850\pi\)
0.171707 + 0.985148i \(0.445072\pi\)
\(48\) 0 0
\(49\) −5.17035 − 4.33844i −0.738621 − 0.619777i
\(50\) −5.60020 + 2.03831i −0.791988 + 0.288260i
\(51\) 0 0
\(52\) 2.87388 + 16.2986i 0.398535 + 2.26020i
\(53\) 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
\(56\) −0.780056 − 4.42392i −0.104239 − 0.591171i
\(57\) 0 0
\(58\) 6.64732 − 2.41943i 0.872836 − 0.317686i
\(59\) 1.67864 + 1.40855i 0.218541 + 0.183377i 0.745485 − 0.666522i \(-0.232218\pi\)
−0.526944 + 0.849900i \(0.676662\pi\)
\(60\) 0 0
\(61\) −6.42781 − 2.33953i −0.822997 − 0.299547i −0.104016 − 0.994576i \(-0.533169\pi\)
−0.718982 + 0.695029i \(0.755391\pi\)
\(62\) −6.24140 − 10.8104i −0.792659 − 1.37293i
\(63\) 0 0
\(64\) −11.9893 + 20.7661i −1.49866 + 2.59576i
\(65\) 3.98741 − 3.34583i 0.494577 − 0.414999i
\(66\) 0 0
\(67\) 2.16775 − 12.2939i 0.264833 − 1.50194i −0.504676 − 0.863309i \(-0.668388\pi\)
0.769509 − 0.638635i \(-0.220501\pi\)
\(68\) −2.46043 + 13.9538i −0.298371 + 1.69215i
\(69\) 0 0
\(70\) −1.73481 + 1.45568i −0.207349 + 0.173986i
\(71\) −1.41784 + 2.45578i −0.168267 + 0.291447i −0.937811 − 0.347147i \(-0.887150\pi\)
0.769544 + 0.638594i \(0.220484\pi\)
\(72\) 0 0
\(73\) −4.96749 − 8.60394i −0.581400 − 1.00701i −0.995314 − 0.0966986i \(-0.969172\pi\)
0.413913 − 0.910316i \(-0.364162\pi\)
\(74\) −12.3504 − 4.49518i −1.43571 − 0.522554i
\(75\) 0 0
\(76\) 23.6117 + 19.8126i 2.70845 + 2.27266i
\(77\) −0.902098 + 0.328337i −0.102804 + 0.0374175i
\(78\) 0 0
\(79\) 0.922282 + 5.23052i 0.103765 + 0.588480i 0.991706 + 0.128524i \(0.0410238\pi\)
−0.887942 + 0.459956i \(0.847865\pi\)
\(80\) 22.8109 2.55033
\(81\) 0 0
\(82\) −31.2385 −3.44972
\(83\) 0.473738 + 2.68670i 0.0519995 + 0.294904i 0.999706 − 0.0242387i \(-0.00771617\pi\)
−0.947707 + 0.319143i \(0.896605\pi\)
\(84\) 0 0
\(85\) 4.18761 − 1.52416i 0.454210 − 0.165319i
\(86\) 18.6588 + 15.6566i 2.01203 + 1.68829i
\(87\) 0 0
\(88\) 16.1712 + 5.88584i 1.72386 + 0.627433i
\(89\) 5.60945 + 9.71585i 0.594600 + 1.02988i 0.993603 + 0.112928i \(0.0360230\pi\)
−0.399003 + 0.916950i \(0.630644\pi\)
\(90\) 0 0
\(91\) −0.779029 + 1.34932i −0.0816644 + 0.141447i
\(92\) 18.9242 − 15.8793i 1.97299 − 1.65553i
\(93\) 0 0
\(94\) 3.20930 − 18.2009i 0.331014 − 1.87728i
\(95\) 1.68338 − 9.54693i 0.172711 − 0.979494i
\(96\) 0 0
\(97\) −5.27739 + 4.42826i −0.535838 + 0.449621i −0.870112 − 0.492855i \(-0.835953\pi\)
0.334274 + 0.942476i \(0.391509\pi\)
\(98\) −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.s.82.1 12
3.2 odd 2 729.2.e.l.82.2 12
9.2 odd 6 729.2.e.u.325.2 12
9.4 even 3 729.2.e.t.568.2 12
9.5 odd 6 729.2.e.k.568.1 12
9.7 even 3 729.2.e.j.325.1 12
27.2 odd 18 729.2.e.l.649.2 12
27.4 even 9 729.2.c.d.487.6 12
27.5 odd 18 729.2.a.e.1.6 yes 6
27.7 even 9 729.2.e.j.406.1 12
27.11 odd 18 729.2.e.k.163.1 12
27.13 even 9 729.2.c.d.244.6 12
27.14 odd 18 729.2.c.a.244.1 12
27.16 even 9 729.2.e.t.163.2 12
27.20 odd 18 729.2.e.u.406.2 12
27.22 even 9 729.2.a.b.1.1 ✓ 6
27.23 odd 18 729.2.c.a.487.1 12
27.25 even 9 inner 729.2.e.s.649.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.1 ✓ 6 27.22 even 9
729.2.a.e.1.6 yes 6 27.5 odd 18
729.2.c.a.244.1 12 27.14 odd 18
729.2.c.a.487.1 12 27.23 odd 18
729.2.c.d.244.6 12 27.13 even 9
729.2.c.d.487.6 12 27.4 even 9
729.2.e.j.325.1 12 9.7 even 3
729.2.e.j.406.1 12 27.7 even 9
729.2.e.k.163.1 12 27.11 odd 18
729.2.e.k.568.1 12 9.5 odd 6
729.2.e.l.82.2 12 3.2 odd 2
729.2.e.l.649.2 12 27.2 odd 18
729.2.e.s.82.1 12 1.1 even 1 trivial
729.2.e.s.649.1 12 27.25 even 9 inner
729.2.e.t.163.2 12 27.16 even 9
729.2.e.t.568.2 12 9.4 even 3
729.2.e.u.325.2 12 9.2 odd 6
729.2.e.u.406.2 12 27.20 odd 18