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Properties

Label 729.2.e.l.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 \(-1.37340i\) of defining polynomial
Character \(\chi\) \(=\) 729.649
Dual form 729.2.e.l.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)
 
\(f(q)\) \(=\) \(q+(0.469730 - 2.66397i) q^{2} +(-4.99670 - 1.81865i) q^{4} +(-1.28112 + 1.07499i) q^{5} +(-0.470402 + 0.171212i) q^{7} +(-4.48686 + 7.77147i) q^{8} +(2.26195 + 3.91782i) q^{10} +(-1.46906 - 1.23269i) q^{11} +(0.540469 + 3.06515i) q^{13} +(0.235142 + 1.33356i) q^{14} +(10.4486 + 8.76745i) q^{16} +(-1.33234 - 2.30767i) q^{17} +(-2.89832 + 5.02003i) q^{19} +(8.35640 - 3.04148i) q^{20} +(-3.97390 + 3.33449i) q^{22} +(4.36569 + 1.58898i) q^{23} +(-0.382569 + 2.16966i) q^{25} +8.41934 q^{26} +2.66183 q^{28} +(-0.454102 + 2.57534i) q^{29} +(4.33631 + 1.57829i) q^{31} +(14.5157 - 12.1801i) q^{32} +(-6.77341 + 2.46532i) q^{34} +(0.418591 - 0.725020i) q^{35} +(2.42934 + 4.20773i) q^{37} +(12.0118 + 10.0791i) q^{38} +(-2.60603 - 14.7795i) q^{40} +(-2.00532 - 11.3727i) q^{41} +(-6.89772 - 5.78788i) q^{43} +(5.09861 + 8.83106i) q^{44} +(6.28369 - 10.8837i) q^{46} +(-6.42020 + 2.33676i) q^{47} +(-5.17035 + 4.33844i) q^{49} +(5.60020 + 2.03831i) q^{50} +(2.87388 - 16.2986i) q^{52} -5.43322 q^{53} +3.20716 q^{55} +(0.780056 - 4.42392i) q^{56} +(6.64732 + 2.41943i) q^{58} +(-1.67864 + 1.40855i) q^{59} +(-6.42781 + 2.33953i) q^{61} +(6.24140 - 10.8104i) q^{62} +(-11.9893 - 20.7661i) q^{64} +(-3.98741 - 3.34583i) q^{65} +(2.16775 + 12.2939i) q^{67} +(2.46043 + 13.9538i) q^{68} +(-1.73481 - 1.45568i) q^{70} +(1.41784 + 2.45578i) q^{71} +(-4.96749 + 8.60394i) q^{73} +(12.3504 - 4.49518i) q^{74} +(23.6117 - 19.8126i) q^{76} +(0.902098 + 0.328337i) q^{77} +(0.922282 - 5.23052i) q^{79} -22.8109 q^{80} -31.2385 q^{82} +(-0.473738 + 2.68670i) q^{83} +(4.18761 + 1.52416i) q^{85} +(-18.6588 + 15.6566i) q^{86} +(16.1712 - 5.88584i) q^{88} +(-5.60945 + 9.71585i) q^{89} +(-0.779029 - 1.34932i) q^{91} +(-18.9242 - 15.8793i) q^{92} +(3.20930 + 18.2009i) q^{94} +(-1.68338 - 9.54693i) q^{95} +(-5.27739 - 4.42826i) q^{97} +(9.12879 + 15.8115i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 3 q^{4} + 12 q^{5} - 3 q^{7} - 6 q^{8} - 6 q^{10} - 3 q^{11} + 6 q^{13} - 6 q^{14} + 27 q^{16} + 9 q^{17} - 12 q^{19} + 39 q^{20} - 39 q^{22} + 21 q^{23} + 6 q^{25} + 48 q^{26} + 6 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.469730 − 2.66397i 0.332149 − 1.88371i −0.121597 − 0.992580i \(-0.538802\pi\)
0.453746 − 0.891131i \(-0.350087\pi\)
\(3\) 0 0
\(4\) −4.99670 − 1.81865i −2.49835 − 0.909325i
\(5\) −1.28112 + 1.07499i −0.572935 + 0.480749i −0.882618 − 0.470090i \(-0.844221\pi\)
0.309684 + 0.950840i \(0.399777\pi\)
\(6\) 0 0
\(7\) −0.470402 + 0.171212i −0.177795 + 0.0647122i −0.429384 − 0.903122i \(-0.641269\pi\)
0.251589 + 0.967834i \(0.419047\pi\)
\(8\) −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 \(-0.371136\pi\)
−0.836807 + 0.547498i \(0.815581\pi\)
\(12\) 0 0
\(13\) 0.540469 + 3.06515i 0.149899 + 0.850120i 0.963302 + 0.268419i \(0.0865013\pi\)
−0.813403 + 0.581700i \(0.802388\pi\)
\(14\) 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 \(-0.271405\pi\)
−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 \(0.935132\pi\)
\(20\) 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 \(-0.227939\pi\)
0.155933 + 0.987768i \(0.450162\pi\)
\(24\) 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 0.913176 + 0.407566i \(0.133622\pi\)
−0.997500 + 0.0706626i \(0.977489\pi\)
\(30\) 0 0
\(31\) 4.33631 + 1.57829i 0.778824 + 0.283469i 0.700682 − 0.713473i \(-0.252879\pi\)
0.0781418 + 0.996942i \(0.475101\pi\)
\(32\) 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 0.993650 − 0.112519i \(-0.0358919\pi\)
−0.594269 + 0.804266i \(0.702559\pi\)
\(38\) 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 −0.582260 − 0.813003i \(-0.697831\pi\)
0.269082 − 0.963117i \(-0.413280\pi\)
\(42\) 0 0
\(43\) −6.89772 − 5.78788i −1.05189 − 0.882643i −0.0586014 − 0.998281i \(-0.518664\pi\)
−0.993291 + 0.115639i \(0.963109\pi\)
\(44\) 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 \(-0.777150\pi\)
−0.171707 + 0.985148i \(0.554928\pi\)
\(48\) 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 \(-0.621724\pi\)
−0.373155 + 0.927769i \(0.621724\pi\)
\(54\) 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 \(-0.767782\pi\)
0.526944 + 0.849900i \(0.323338\pi\)
\(60\) 0 0
\(61\) −6.42781 + 2.33953i −0.822997 + 0.299547i −0.718982 − 0.695029i \(-0.755391\pi\)
−0.104016 + 0.994576i \(0.533169\pi\)
\(62\) 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 0.769509 + 0.638635i \(0.220501\pi\)
−0.504676 + 0.863309i \(0.668388\pi\)
\(68\) 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 \(-0.112850\pi\)
−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 \(0.969172\pi\)
\(74\) 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 −0.887942 − 0.459956i \(-0.847865\pi\)
0.991706 − 0.128524i \(-0.0410238\pi\)
\(80\) −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 \(-0.992284\pi\)
0.947707 + 0.319143i \(0.103395\pi\)
\(84\) 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 0.399003 + 0.916950i \(0.369356\pi\)
−0.993603 + 0.112928i \(0.963977\pi\)
\(90\) 0 0
\(91\) −0.779029 − 1.34932i −0.0816644 − 0.141447i
\(92\) −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 0.334274 − 0.942476i \(-0.391509\pi\)
−0.870112 + 0.492855i \(0.835953\pi\)
\(98\) 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.l.649.2 12
3.2 odd 2 729.2.e.s.649.1 12
9.2 odd 6 729.2.e.t.163.2 12
9.4 even 3 729.2.e.u.406.2 12
9.5 odd 6 729.2.e.j.406.1 12
9.7 even 3 729.2.e.k.163.1 12
27.2 odd 18 729.2.c.d.487.6 12
27.4 even 9 729.2.e.u.325.2 12
27.5 odd 18 729.2.e.t.568.2 12
27.7 even 9 729.2.c.a.244.1 12
27.11 odd 18 729.2.a.b.1.1 ✓ 6
27.13 even 9 inner 729.2.e.l.82.2 12
27.14 odd 18 729.2.e.s.82.1 12
27.16 even 9 729.2.a.e.1.6 yes 6
27.20 odd 18 729.2.c.d.244.6 12
27.22 even 9 729.2.e.k.568.1 12
27.23 odd 18 729.2.e.j.325.1 12
27.25 even 9 729.2.c.a.487.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.1 ✓ 6 27.11 odd 18
729.2.a.e.1.6 yes 6 27.16 even 9
729.2.c.a.244.1 12 27.7 even 9
729.2.c.a.487.1 12 27.25 even 9
729.2.c.d.244.6 12 27.20 odd 18
729.2.c.d.487.6 12 27.2 odd 18
729.2.e.j.325.1 12 27.23 odd 18
729.2.e.j.406.1 12 9.5 odd 6
729.2.e.k.163.1 12 9.7 even 3
729.2.e.k.568.1 12 27.22 even 9
729.2.e.l.82.2 12 27.13 even 9 inner
729.2.e.l.649.2 12 1.1 even 1 trivial
729.2.e.s.82.1 12 27.14 odd 18
729.2.e.s.649.1 12 3.2 odd 2
729.2.e.t.163.2 12 9.2 odd 6
729.2.e.t.568.2 12 27.5 odd 18
729.2.e.u.325.2 12 27.4 even 9
729.2.e.u.406.2 12 9.4 even 3