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Properties

Label 729.2.e.s.325.1
Level $729$
Weight $2$
Character 729.325
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 325.1
Root \(1.22778i\) of defining polynomial
Character \(\chi\) \(=\) 729.325
Dual form 729.2.e.s.406.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.20913 + 1.01458i) q^{2} +(0.0853237 - 0.483895i) q^{4} +(-1.57728 + 0.574083i) q^{5} +(0.482617 + 2.73706i) q^{7} +(-1.19062 - 2.06222i) q^{8} +(1.32468 - 2.29442i) q^{10} +(3.90087 + 1.41980i) q^{11} +(5.26736 + 4.41984i) q^{13} +(-3.36051 - 2.81980i) q^{14} +(4.45535 + 1.62162i) q^{16} +(-0.488276 + 0.845718i) q^{17} +(-1.34264 - 2.32553i) q^{19} +(0.143217 + 0.812221i) q^{20} +(-6.15715 + 2.24102i) q^{22} +(0.280124 - 1.58866i) q^{23} +(-1.67198 + 1.40296i) q^{25} -10.8532 q^{26} +1.36563 q^{28} +(-6.30292 + 5.28878i) q^{29} +(0.181301 - 1.02821i) q^{31} +(-2.55707 + 0.930697i) q^{32} +(-0.267660 - 1.51798i) q^{34} +(-2.33252 - 4.04005i) q^{35} +(0.654172 - 1.13306i) q^{37} +(3.98286 + 1.44964i) q^{38} +(3.06183 + 2.56918i) q^{40} +(-3.71391 - 3.11634i) q^{41} +(-9.24679 - 3.36556i) q^{43} +(1.01987 - 1.76647i) q^{44} +(1.27312 + 2.20510i) q^{46} +(2.17020 + 12.3078i) q^{47} +(-0.680721 + 0.247762i) q^{49} +(0.598226 - 3.39271i) q^{50} +(2.58817 - 2.17173i) q^{52} +7.34280 q^{53} -6.96786 q^{55} +(5.06980 - 4.25406i) q^{56} +(2.25515 - 12.7896i) q^{58} +(-8.50598 + 3.09592i) q^{59} +(-0.223267 - 1.26621i) q^{61} +(0.823982 + 1.42718i) q^{62} +(-2.59373 + 4.49247i) q^{64} +(-10.8455 - 3.94742i) q^{65} +(-3.55927 - 2.98658i) q^{67} +(0.367577 + 0.308434i) q^{68} +(6.91926 + 2.51841i) q^{70} +(-2.81187 + 4.87030i) q^{71} +(2.28072 + 3.95033i) q^{73} +(0.358600 + 2.03372i) q^{74} +(-1.23987 + 0.451276i) q^{76} +(-2.00345 + 11.3621i) q^{77} +(3.56732 - 2.99333i) q^{79} -7.95828 q^{80} +7.65237 q^{82} +(4.41578 - 3.70528i) q^{83} +(0.284635 - 1.61425i) q^{85} +(14.5952 - 5.31221i) q^{86} +(-1.71652 - 9.73490i) q^{88} +(-2.27221 - 3.93558i) q^{89} +(-9.55523 + 16.5502i) q^{91} +(-0.744844 - 0.271101i) q^{92} +(-15.1113 - 12.6799i) q^{94} +(3.45278 + 2.89722i) q^{95} +(-8.05828 - 2.93297i) q^{97} +(0.571704 - 0.990221i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{4} - 12 q^{5} - 3 q^{7} + 6 q^{8} - 6 q^{10} + 3 q^{11} + 6 q^{13} + 6 q^{14} + 27 q^{16} - 9 q^{17} - 12 q^{19} - 39 q^{20} - 39 q^{22} - 21 q^{23} + 6 q^{25} - 48 q^{26} + 6 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.20913 + 1.01458i −0.854982 + 0.717415i −0.960881 − 0.276961i \(-0.910673\pi\)
0.105899 + 0.994377i \(0.466228\pi\)
\(3\) 0 0
\(4\) 0.0853237 − 0.483895i 0.0426619 − 0.241948i
\(5\) −1.57728 + 0.574083i −0.705382 + 0.256738i −0.669707 − 0.742626i \(-0.733580\pi\)
−0.0356747 + 0.999363i \(0.511358\pi\)
\(6\) 0 0
\(7\) 0.482617 + 2.73706i 0.182412 + 1.03451i 0.929235 + 0.369488i \(0.120467\pi\)
−0.746823 + 0.665023i \(0.768422\pi\)
\(8\) −1.19062 − 2.06222i −0.420948 − 0.729104i
\(9\) 0 0
\(10\) 1.32468 − 2.29442i 0.418901 − 0.725558i
\(11\) 3.90087 + 1.41980i 1.17616 + 0.428086i 0.854843 − 0.518886i \(-0.173653\pi\)
0.321314 + 0.946973i \(0.395875\pi\)
\(12\) 0 0
\(13\) 5.26736 + 4.41984i 1.46090 + 1.22584i 0.924110 + 0.382128i \(0.124809\pi\)
0.536792 + 0.843715i \(0.319636\pi\)
\(14\) −3.36051 − 2.81980i −0.898133 − 0.753623i
\(15\) 0 0
\(16\) 4.45535 + 1.62162i 1.11384 + 0.405404i
\(17\) −0.488276 + 0.845718i −0.118424 + 0.205117i −0.919143 − 0.393923i \(-0.871118\pi\)
0.800719 + 0.599040i \(0.204451\pi\)
\(18\) 0 0
\(19\) −1.34264 − 2.32553i −0.308024 − 0.533513i 0.669906 − 0.742446i \(-0.266334\pi\)
−0.977930 + 0.208933i \(0.933001\pi\)
\(20\) 0.143217 + 0.812221i 0.0320242 + 0.181618i
\(21\) 0 0
\(22\) −6.15715 + 2.24102i −1.31271 + 0.477787i
\(23\) 0.280124 − 1.58866i 0.0584099 − 0.331259i −0.941575 − 0.336804i \(-0.890654\pi\)
0.999985 + 0.00554518i \(0.00176509\pi\)
\(24\) 0 0
\(25\) −1.67198 + 1.40296i −0.334396 + 0.280591i
\(26\) −10.8532 −2.12848
\(27\) 0 0
\(28\) 1.36563 0.258079
\(29\) −6.30292 + 5.28878i −1.17042 + 0.982101i −0.999995 − 0.00326885i \(-0.998959\pi\)
−0.170428 + 0.985370i \(0.554515\pi\)
\(30\) 0 0
\(31\) 0.181301 − 1.02821i 0.0325626 − 0.184672i −0.964188 − 0.265219i \(-0.914556\pi\)
0.996751 + 0.0805475i \(0.0256669\pi\)
\(32\) −2.55707 + 0.930697i −0.452030 + 0.164526i
\(33\) 0 0
\(34\) −0.267660 − 1.51798i −0.0459033 − 0.260331i
\(35\) −2.33252 − 4.04005i −0.394268 − 0.682893i
\(36\) 0 0
\(37\) 0.654172 − 1.13306i 0.107545 − 0.186274i −0.807230 − 0.590237i \(-0.799034\pi\)
0.914775 + 0.403963i \(0.132368\pi\)
\(38\) 3.98286 + 1.44964i 0.646105 + 0.235163i
\(39\) 0 0
\(40\) 3.06183 + 2.56918i 0.484118 + 0.406223i
\(41\) −3.71391 − 3.11634i −0.580016 − 0.486691i 0.304936 − 0.952373i \(-0.401365\pi\)
−0.884952 + 0.465682i \(0.845809\pi\)
\(42\) 0 0
\(43\) −9.24679 − 3.36556i −1.41012 − 0.513243i −0.478957 − 0.877839i \(-0.658985\pi\)
−0.931166 + 0.364596i \(0.881207\pi\)
\(44\) 1.01987 − 1.76647i 0.153751 − 0.266305i
\(45\) 0 0
\(46\) 1.27312 + 2.20510i 0.187711 + 0.325125i
\(47\) 2.17020 + 12.3078i 0.316557 + 1.79528i 0.563355 + 0.826215i \(0.309510\pi\)
−0.246798 + 0.969067i \(0.579379\pi\)
\(48\) 0 0
\(49\) −0.680721 + 0.247762i −0.0972458 + 0.0353946i
\(50\) 0.598226 − 3.39271i 0.0846019 − 0.479801i
\(51\) 0 0
\(52\) 2.58817 − 2.17173i 0.358914 − 0.301165i
\(53\) 7.34280 1.00861 0.504305 − 0.863525i \(-0.331749\pi\)
0.504305 + 0.863525i \(0.331749\pi\)
\(54\) 0 0
\(55\) −6.96786 −0.939546
\(56\) 5.06980 − 4.25406i 0.677480 − 0.568473i
\(57\) 0 0
\(58\) 2.25515 − 12.7896i 0.296116 − 1.67936i
\(59\) −8.50598 + 3.09592i −1.10738 + 0.403055i −0.830033 − 0.557714i \(-0.811679\pi\)
−0.277351 + 0.960769i \(0.589456\pi\)
\(60\) 0 0
\(61\) −0.223267 − 1.26621i −0.0285864 − 0.162121i 0.967173 − 0.254120i \(-0.0817857\pi\)
−0.995759 + 0.0919982i \(0.970675\pi\)
\(62\) 0.823982 + 1.42718i 0.104646 + 0.181252i
\(63\) 0 0
\(64\) −2.59373 + 4.49247i −0.324216 + 0.561558i
\(65\) −10.8455 − 3.94742i −1.34521 − 0.489618i
\(66\) 0 0
\(67\) −3.55927 − 2.98658i −0.434834 − 0.364869i 0.398938 − 0.916978i \(-0.369379\pi\)
−0.833772 + 0.552109i \(0.813823\pi\)
\(68\) 0.367577 + 0.308434i 0.0445753 + 0.0374031i
\(69\) 0 0
\(70\) 6.91926 + 2.51841i 0.827010 + 0.301007i
\(71\) −2.81187 + 4.87030i −0.333707 + 0.577998i −0.983236 − 0.182339i \(-0.941633\pi\)
0.649528 + 0.760337i \(0.274966\pi\)
\(72\) 0 0
\(73\) 2.28072 + 3.95033i 0.266938 + 0.462351i 0.968070 − 0.250681i \(-0.0806547\pi\)
−0.701131 + 0.713032i \(0.747321\pi\)
\(74\) 0.358600 + 2.03372i 0.0416864 + 0.236416i
\(75\) 0 0
\(76\) −1.23987 + 0.451276i −0.142223 + 0.0517649i
\(77\) −2.00345 + 11.3621i −0.228314 + 1.29484i
\(78\) 0 0
\(79\) 3.56732 − 2.99333i 0.401354 − 0.336776i −0.419662 − 0.907680i \(-0.637852\pi\)
0.821017 + 0.570904i \(0.193407\pi\)
\(80\) −7.95828 −0.889763
\(81\) 0 0
\(82\) 7.65237 0.845063
\(83\) 4.41578 − 3.70528i 0.484695 − 0.406707i −0.367426 − 0.930053i \(-0.619761\pi\)
0.852121 + 0.523346i \(0.175316\pi\)
\(84\) 0 0
\(85\) 0.284635 − 1.61425i 0.0308730 − 0.175090i
\(86\) 14.5952 − 5.31221i 1.57384 − 0.572830i
\(87\) 0 0
\(88\) −1.71652 − 9.73490i −0.182982 − 1.03774i
\(89\) −2.27221 − 3.93558i −0.240854 − 0.417171i 0.720104 − 0.693866i \(-0.244094\pi\)
−0.960958 + 0.276695i \(0.910761\pi\)
\(90\) 0 0
\(91\) −9.55523 + 16.5502i −1.00166 + 1.73493i
\(92\) −0.744844 − 0.271101i −0.0776554 − 0.0282643i
\(93\) 0 0
\(94\) −15.1113 − 12.6799i −1.55861 − 1.30783i
\(95\) 3.45278 + 2.89722i 0.354247 + 0.297249i
\(96\) 0 0
\(97\) −8.05828 − 2.93297i −0.818194 − 0.297798i −0.101190 − 0.994867i \(-0.532265\pi\)
−0.717004 + 0.697069i \(0.754487\pi\)
\(98\) 0.571704 − 0.990221i 0.0577508 − 0.100027i
\(99\) 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.325.1 12
3.2 odd 2 729.2.e.l.325.2 12
9.2 odd 6 729.2.e.u.568.1 12
9.4 even 3 729.2.e.t.82.1 12
9.5 odd 6 729.2.e.k.82.2 12
9.7 even 3 729.2.e.j.568.2 12
27.2 odd 18 729.2.e.u.163.1 12
27.4 even 9 729.2.a.b.1.3 ✓ 6
27.5 odd 18 729.2.c.a.244.3 12
27.7 even 9 729.2.e.t.649.1 12
27.11 odd 18 729.2.e.l.406.2 12
27.13 even 9 729.2.c.d.487.4 12
27.14 odd 18 729.2.c.a.487.3 12
27.16 even 9 inner 729.2.e.s.406.1 12
27.20 odd 18 729.2.e.k.649.2 12
27.22 even 9 729.2.c.d.244.4 12
27.23 odd 18 729.2.a.e.1.4 yes 6
27.25 even 9 729.2.e.j.163.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.3 ✓ 6 27.4 even 9
729.2.a.e.1.4 yes 6 27.23 odd 18
729.2.c.a.244.3 12 27.5 odd 18
729.2.c.a.487.3 12 27.14 odd 18
729.2.c.d.244.4 12 27.22 even 9
729.2.c.d.487.4 12 27.13 even 9
729.2.e.j.163.2 12 27.25 even 9
729.2.e.j.568.2 12 9.7 even 3
729.2.e.k.82.2 12 9.5 odd 6
729.2.e.k.649.2 12 27.20 odd 18
729.2.e.l.325.2 12 3.2 odd 2
729.2.e.l.406.2 12 27.11 odd 18
729.2.e.s.325.1 12 1.1 even 1 trivial
729.2.e.s.406.1 12 27.16 even 9 inner
729.2.e.t.82.1 12 9.4 even 3
729.2.e.t.649.1 12 27.7 even 9
729.2.e.u.163.1 12 27.2 odd 18
729.2.e.u.568.1 12 9.2 odd 6