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Properties

Label 729.2.e.s.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,-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 163.2
Root \(-3.10658i\) of defining polynomial
Character \(\chi\) \(=\) 729.163
Dual form 729.2.e.s.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)
 
\(f(q)\) \(=\) \(q+(1.99687 - 0.726803i) q^{2} +(1.92717 - 1.61709i) q^{4} +(-0.359615 - 2.03948i) q^{5} +(3.71430 + 3.11667i) q^{7} +(0.547989 - 0.949144i) q^{8} +(-2.20040 - 3.81121i) q^{10} +(0.720551 - 4.08645i) q^{11} +(1.14268 + 0.415902i) q^{13} +(9.68219 + 3.52403i) q^{14} +(-0.469286 + 2.66145i) q^{16} +(-1.18182 - 2.04697i) q^{17} +(0.919003 - 1.59176i) q^{19} +(-3.99106 - 3.34890i) q^{20} +(-1.53119 - 8.68382i) q^{22} +(-3.29673 + 2.76628i) q^{23} +(0.668315 - 0.243247i) q^{25} +2.58407 q^{26} +12.1980 q^{28} +(-2.80199 + 1.01984i) q^{29} +(1.12883 - 0.947203i) q^{31} +(1.37788 + 7.81432i) q^{32} +(-3.84769 - 3.22859i) q^{34} +(5.02066 - 8.69603i) q^{35} +(-4.48554 - 7.76918i) q^{37} +(0.678238 - 3.84648i) q^{38} +(-2.13282 - 0.776284i) q^{40} +(-2.12420 - 0.773145i) q^{41} +(-0.952435 + 5.40153i) q^{43} +(-5.21953 - 9.04050i) q^{44} +(-4.57260 + 7.91998i) q^{46} +(5.50260 + 4.61723i) q^{47} +(2.86687 + 16.2588i) q^{49} +(1.15775 - 0.971466i) q^{50} +(2.87470 - 1.04630i) q^{52} -6.32803 q^{53} -8.59334 q^{55} +(4.99356 - 1.81751i) q^{56} +(-4.85400 + 4.07299i) q^{58} +(0.0455404 + 0.258272i) q^{59} +(-3.41319 - 2.86401i) q^{61} +(1.56571 - 2.71188i) q^{62} +(5.72840 + 9.92188i) q^{64} +(0.437297 - 2.48004i) q^{65} +(-3.88197 - 1.41292i) q^{67} +(-5.58771 - 2.03376i) q^{68} +(3.70532 - 21.0139i) q^{70} +(-1.54276 - 2.67213i) q^{71} +(-6.38003 + 11.0505i) q^{73} +(-14.6037 - 12.2540i) q^{74} +(-0.802942 - 4.55371i) q^{76} +(15.4124 - 12.9326i) q^{77} +(-4.27786 + 1.55701i) q^{79} +5.59674 q^{80} -4.80368 q^{82} +(7.94328 - 2.89112i) q^{83} +(-3.74975 + 3.14642i) q^{85} +(2.02395 + 11.4784i) q^{86} +(-3.48378 - 2.92323i) q^{88} +(-8.48158 + 14.6905i) q^{89} +(2.94803 + 5.10614i) q^{91} +(-1.88004 + 10.6622i) q^{92} +(14.3438 + 5.22072i) q^{94} +(-3.57685 - 1.30187i) q^{95} +(-0.887302 + 5.03214i) q^{97} +(17.5417 + 30.3831i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{4} - 12 q^{5} - 3 q^{7} + 6 q^{8} - 6 q^{10} + 3 q^{11} + 6 q^{13} + 6 q^{14} + 27 q^{16} - 9 q^{17} - 12 q^{19} - 39 q^{20} - 39 q^{22} - 21 q^{23} + 6 q^{25} - 48 q^{26} + 6 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.99687 − 0.726803i 1.41200 − 0.513927i 0.480286 − 0.877112i \(-0.340533\pi\)
0.931717 + 0.363185i \(0.118311\pi\)
\(3\) 0 0
\(4\) 1.92717 − 1.61709i 0.963587 − 0.808546i
\(5\) −0.359615 − 2.03948i −0.160825 − 0.912082i −0.953266 − 0.302134i \(-0.902301\pi\)
0.792441 − 0.609949i \(-0.208810\pi\)
\(6\) 0 0
\(7\) 3.71430 + 3.11667i 1.40387 + 1.17799i 0.959349 + 0.282222i \(0.0910716\pi\)
0.444524 + 0.895767i \(0.353373\pi\)
\(8\) 0.547989 − 0.949144i 0.193743 − 0.335573i
\(9\) 0 0
\(10\) −2.20040 − 3.81121i −0.695829 − 1.20521i
\(11\) 0.720551 − 4.08645i 0.217254 − 1.23211i −0.659697 − 0.751532i \(-0.729315\pi\)
0.876951 − 0.480579i \(-0.159573\pi\)
\(12\) 0 0
\(13\) 1.14268 + 0.415902i 0.316923 + 0.115350i 0.495583 − 0.868560i \(-0.334954\pi\)
−0.178661 + 0.983911i \(0.557176\pi\)
\(14\) 9.68219 + 3.52403i 2.58767 + 0.941836i
\(15\) 0 0
\(16\) −0.469286 + 2.66145i −0.117322 + 0.665363i
\(17\) −1.18182 − 2.04697i −0.286633 − 0.496463i 0.686371 − 0.727252i \(-0.259203\pi\)
−0.973004 + 0.230789i \(0.925869\pi\)
\(18\) 0 0
\(19\) 0.919003 − 1.59176i 0.210834 − 0.365175i −0.741142 − 0.671348i \(-0.765715\pi\)
0.951976 + 0.306174i \(0.0990488\pi\)
\(20\) −3.99106 − 3.34890i −0.892429 − 0.748837i
\(21\) 0 0
\(22\) −1.53119 − 8.68382i −0.326451 − 1.85140i
\(23\) −3.29673 + 2.76628i −0.687415 + 0.576809i −0.918162 − 0.396204i \(-0.870327\pi\)
0.230748 + 0.973014i \(0.425883\pi\)
\(24\) 0 0
\(25\) 0.668315 − 0.243247i 0.133663 − 0.0486494i
\(26\) 2.58407 0.506777
\(27\) 0 0
\(28\) 12.1980 2.30521
\(29\) −2.80199 + 1.01984i −0.520317 + 0.189380i −0.588810 − 0.808272i \(-0.700403\pi\)
0.0684925 + 0.997652i \(0.478181\pi\)
\(30\) 0 0
\(31\) 1.12883 − 0.947203i 0.202744 − 0.170123i −0.535762 − 0.844369i \(-0.679976\pi\)
0.738507 + 0.674246i \(0.235531\pi\)
\(32\) 1.37788 + 7.81432i 0.243576 + 1.38139i
\(33\) 0 0
\(34\) −3.84769 − 3.22859i −0.659873 − 0.553699i
\(35\) 5.02066 − 8.69603i 0.848646 − 1.46990i
\(36\) 0 0
\(37\) −4.48554 − 7.76918i −0.737418 − 1.27725i −0.953654 − 0.300905i \(-0.902711\pi\)
0.216236 − 0.976341i \(-0.430622\pi\)
\(38\) 0.678238 − 3.84648i 0.110025 − 0.623981i
\(39\) 0 0
\(40\) −2.13282 − 0.776284i −0.337229 − 0.122741i
\(41\) −2.12420 − 0.773145i −0.331744 − 0.120745i 0.170778 − 0.985310i \(-0.445372\pi\)
−0.502522 + 0.864565i \(0.667594\pi\)
\(42\) 0 0
\(43\) −0.952435 + 5.40153i −0.145245 + 0.823726i 0.821925 + 0.569596i \(0.192900\pi\)
−0.967170 + 0.254130i \(0.918211\pi\)
\(44\) −5.21953 − 9.04050i −0.786874 − 1.36291i
\(45\) 0 0
\(46\) −4.57260 + 7.91998i −0.674194 + 1.16774i
\(47\) 5.50260 + 4.61723i 0.802636 + 0.673492i 0.948838 − 0.315763i \(-0.102260\pi\)
−0.146202 + 0.989255i \(0.546705\pi\)
\(48\) 0 0
\(49\) 2.86687 + 16.2588i 0.409552 + 2.32269i
\(50\) 1.15775 − 0.971466i 0.163730 − 0.137386i
\(51\) 0 0
\(52\) 2.87470 − 1.04630i 0.398649 − 0.145096i
\(53\) −6.32803 −0.869222 −0.434611 − 0.900618i \(-0.643114\pi\)
−0.434611 + 0.900618i \(0.643114\pi\)
\(54\) 0 0
\(55\) −8.59334 −1.15873
\(56\) 4.99356 − 1.81751i 0.667293 − 0.242875i
\(57\) 0 0
\(58\) −4.85400 + 4.07299i −0.637362 + 0.534810i
\(59\) 0.0455404 + 0.258272i 0.00592886 + 0.0336242i 0.987629 − 0.156811i \(-0.0501213\pi\)
−0.981700 + 0.190435i \(0.939010\pi\)
\(60\) 0 0
\(61\) −3.41319 − 2.86401i −0.437014 − 0.366699i 0.397576 − 0.917569i \(-0.369851\pi\)
−0.834591 + 0.550870i \(0.814296\pi\)
\(62\) 1.56571 − 2.71188i 0.198845 − 0.344410i
\(63\) 0 0
\(64\) 5.72840 + 9.92188i 0.716050 + 1.24024i
\(65\) 0.437297 − 2.48004i 0.0542401 − 0.307611i
\(66\) 0 0
\(67\) −3.88197 − 1.41292i −0.474258 − 0.172616i 0.0938223 − 0.995589i \(-0.470091\pi\)
−0.568080 + 0.822973i \(0.692314\pi\)
\(68\) −5.58771 − 2.03376i −0.677609 − 0.246630i
\(69\) 0 0
\(70\) 3.70532 − 21.0139i 0.442870 − 2.51164i
\(71\) −1.54276 − 2.67213i −0.183091 − 0.317124i 0.759840 − 0.650110i \(-0.225277\pi\)
−0.942932 + 0.332986i \(0.891944\pi\)
\(72\) 0 0
\(73\) −6.38003 + 11.0505i −0.746726 + 1.29337i 0.202658 + 0.979250i \(0.435042\pi\)
−0.949384 + 0.314118i \(0.898291\pi\)
\(74\) −14.6037 − 12.2540i −1.69765 − 1.42450i
\(75\) 0 0
\(76\) −0.802942 − 4.55371i −0.0921038 − 0.522346i
\(77\) 15.4124 − 12.9326i 1.75641 − 1.47380i
\(78\) 0 0
\(79\) −4.27786 + 1.55701i −0.481297 + 0.175178i −0.571263 − 0.820767i \(-0.693546\pi\)
0.0899659 + 0.995945i \(0.471324\pi\)
\(80\) 5.59674 0.625734
\(81\) 0 0
\(82\) −4.80368 −0.530478
\(83\) 7.94328 − 2.89112i 0.871888 − 0.317341i 0.132956 − 0.991122i \(-0.457553\pi\)
0.738931 + 0.673781i \(0.235331\pi\)
\(84\) 0 0
\(85\) −3.74975 + 3.14642i −0.406718 + 0.341277i
\(86\) 2.02395 + 11.4784i 0.218248 + 1.23775i
\(87\) 0 0
\(88\) −3.48378 − 2.92323i −0.371372 − 0.311618i
\(89\) −8.48158 + 14.6905i −0.899046 + 1.55719i −0.0703304 + 0.997524i \(0.522405\pi\)
−0.828716 + 0.559670i \(0.810928\pi\)
\(90\) 0 0
\(91\) 2.94803 + 5.10614i 0.309038 + 0.535269i
\(92\) −1.88004 + 10.6622i −0.196007 + 1.11161i
\(93\) 0 0
\(94\) 14.3438 + 5.22072i 1.47945 + 0.538476i
\(95\) −3.57685 − 1.30187i −0.366977 − 0.133569i
\(96\) 0 0
\(97\) −0.887302 + 5.03214i −0.0900919 + 0.510937i 0.906049 + 0.423172i \(0.139083\pi\)
−0.996141 + 0.0877646i \(0.972028\pi\)
\(98\) 17.5417 + 30.3831i 1.77198 + 3.06916i
\(99\) 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.163.2 12
3.2 odd 2 729.2.e.l.163.1 12
9.2 odd 6 729.2.e.k.406.2 12
9.4 even 3 729.2.e.j.649.1 12
9.5 odd 6 729.2.e.u.649.2 12
9.7 even 3 729.2.e.t.406.1 12
27.2 odd 18 729.2.a.e.1.5 yes 6
27.4 even 9 inner 729.2.e.s.568.2 12
27.5 odd 18 729.2.e.u.82.2 12
27.7 even 9 729.2.c.d.487.5 12
27.11 odd 18 729.2.c.a.244.2 12
27.13 even 9 729.2.e.t.325.1 12
27.14 odd 18 729.2.e.k.325.2 12
27.16 even 9 729.2.c.d.244.5 12
27.20 odd 18 729.2.c.a.487.2 12
27.22 even 9 729.2.e.j.82.1 12
27.23 odd 18 729.2.e.l.568.1 12
27.25 even 9 729.2.a.b.1.2 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.2 ✓ 6 27.25 even 9
729.2.a.e.1.5 yes 6 27.2 odd 18
729.2.c.a.244.2 12 27.11 odd 18
729.2.c.a.487.2 12 27.20 odd 18
729.2.c.d.244.5 12 27.16 even 9
729.2.c.d.487.5 12 27.7 even 9
729.2.e.j.82.1 12 27.22 even 9
729.2.e.j.649.1 12 9.4 even 3
729.2.e.k.325.2 12 27.14 odd 18
729.2.e.k.406.2 12 9.2 odd 6
729.2.e.l.163.1 12 3.2 odd 2
729.2.e.l.568.1 12 27.23 odd 18
729.2.e.s.163.2 12 1.1 even 1 trivial
729.2.e.s.568.2 12 27.4 even 9 inner
729.2.e.t.325.1 12 27.13 even 9
729.2.e.t.406.1 12 9.7 even 3
729.2.e.u.82.2 12 27.5 odd 18
729.2.e.u.649.2 12 9.5 odd 6