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Properties

Label 729.2.e.j.649.1
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,-6,0,6,6,0,-3,6,0,-6,12,0,-3,15,0,-36,-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 649.1
Root \(-3.10658i\) of defining polynomial
Character \(\chi\) \(=\) 729.649
Dual form 729.2.e.j.82.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.369007 + 2.09274i) q^{2} +(-2.36403 - 0.860436i) q^{4} +(-1.58643 + 1.33117i) q^{5} +(-4.55626 + 1.65834i) q^{7} +(0.547989 - 0.949144i) q^{8} +(-2.20040 - 3.81121i) q^{10} +(3.17869 + 2.66724i) q^{11} +(-0.211159 - 1.19754i) q^{13} +(-1.78920 - 10.1470i) q^{14} +(-2.07024 - 1.73714i) q^{16} +(-1.18182 - 2.04697i) q^{17} +(0.919003 - 1.59176i) q^{19} +(4.89576 - 1.78191i) q^{20} +(-6.75481 + 5.66796i) q^{22} +(4.04403 + 1.47191i) q^{23} +(-0.123500 + 0.700401i) q^{25} +2.58407 q^{26} +12.1980 q^{28} +(0.517788 - 2.93652i) q^{29} +(-1.38472 - 0.503996i) q^{31} +(6.07846 - 5.10043i) q^{32} +(4.71989 - 1.71790i) q^{34} +(5.02066 - 8.69603i) q^{35} +(-4.48554 - 7.76918i) q^{37} +(2.99203 + 2.51061i) q^{38} +(0.394130 + 2.23522i) q^{40} +(0.392536 + 2.22618i) q^{41} +(-4.20164 - 3.52560i) q^{43} +(-5.21953 - 9.04050i) q^{44} +(-4.57260 + 7.91998i) q^{46} +(-6.74994 + 2.45678i) q^{47} +(12.6471 - 10.6122i) q^{49} +(-1.42019 - 0.516906i) q^{50} +(-0.531222 + 3.01271i) q^{52} -6.32803 q^{53} -8.59334 q^{55} +(-0.922773 + 5.23330i) q^{56} +(5.95432 + 2.16719i) q^{58} +(0.200900 - 0.168575i) q^{59} +(4.18690 - 1.52391i) q^{61} +(1.56571 - 2.71188i) q^{62} +(5.72840 + 9.92188i) q^{64} +(1.92913 + 1.61873i) q^{65} +(0.717359 + 4.06834i) q^{67} +(1.03257 + 5.85598i) q^{68} +(16.3459 + 13.7159i) q^{70} +(-1.54276 - 2.67213i) q^{71} +(-6.38003 + 11.0505i) q^{73} +(17.9141 - 6.52021i) q^{74} +(-3.54216 + 2.97222i) q^{76} +(-18.9062 - 6.88128i) q^{77} +(0.790517 - 4.48325i) q^{79} +5.59674 q^{80} -4.80368 q^{82} +(-1.46786 + 8.32464i) q^{83} +(4.59975 + 1.67417i) q^{85} +(8.92862 - 7.49200i) q^{86} +(4.27348 - 1.55542i) q^{88} +(-8.48158 + 14.6905i) q^{89} +(2.94803 + 5.10614i) q^{91} +(-8.29373 - 6.95926i) q^{92} +(-2.65063 - 15.0325i) q^{94} +(0.660975 + 3.74857i) q^{95} +(-3.91431 - 3.28450i) q^{97} +(17.5417 + 30.3831i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8} - 6 q^{10} + 12 q^{11} - 3 q^{13} + 15 q^{14} - 36 q^{16} - 9 q^{17} - 12 q^{19} + 42 q^{20} + 6 q^{22} + 6 q^{23} + 6 q^{25} - 48 q^{26} + 6 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.369007 + 2.09274i −0.260928 + 1.47979i 0.519458 + 0.854496i \(0.326134\pi\)
−0.780386 + 0.625298i \(0.784977\pi\)
\(3\) 0 0
\(4\) −2.36403 − 0.860436i −1.18201 − 0.430218i
\(5\) −1.58643 + 1.33117i −0.709474 + 0.595319i −0.924452 − 0.381300i \(-0.875477\pi\)
0.214977 + 0.976619i \(0.431032\pi\)
\(6\) 0 0
\(7\) −4.55626 + 1.65834i −1.72211 + 0.626795i −0.998019 − 0.0629144i \(-0.979960\pi\)
−0.724086 + 0.689709i \(0.757738\pi\)
\(8\) 0.547989 − 0.949144i 0.193743 − 0.335573i
\(9\) 0 0
\(10\) −2.20040 − 3.81121i −0.695829 − 1.20521i
\(11\) 3.17869 + 2.66724i 0.958412 + 0.804203i 0.980694 − 0.195549i \(-0.0626487\pi\)
−0.0222820 + 0.999752i \(0.507093\pi\)
\(12\) 0 0
\(13\) −0.211159 − 1.19754i −0.0585649 − 0.332138i 0.941422 − 0.337230i \(-0.109490\pi\)
−0.999987 + 0.00509231i \(0.998379\pi\)
\(14\) −1.78920 − 10.1470i −0.478183 − 2.71191i
\(15\) 0 0
\(16\) −2.07024 − 1.73714i −0.517561 − 0.434285i
\(17\) −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 \(0.0990488\pi\)
\(20\) 4.89576 − 1.78191i 1.09473 − 0.398448i
\(21\) 0 0
\(22\) −6.75481 + 5.66796i −1.44013 + 1.20841i
\(23\) 4.04403 + 1.47191i 0.843239 + 0.306914i 0.727281 − 0.686340i \(-0.240784\pi\)
0.115958 + 0.993254i \(0.463006\pi\)
\(24\) 0 0
\(25\) −0.123500 + 0.700401i −0.0246999 + 0.140080i
\(26\) 2.58407 0.506777
\(27\) 0 0
\(28\) 12.1980 2.30521
\(29\) 0.517788 − 2.93652i 0.0961507 − 0.545298i −0.898238 − 0.439510i \(-0.855152\pi\)
0.994389 − 0.105788i \(-0.0337366\pi\)
\(30\) 0 0
\(31\) −1.38472 − 0.503996i −0.248703 − 0.0905204i 0.214661 − 0.976689i \(-0.431135\pi\)
−0.463364 + 0.886168i \(0.653358\pi\)
\(32\) 6.07846 − 5.10043i 1.07453 − 0.901638i
\(33\) 0 0
\(34\) 4.71989 − 1.71790i 0.809454 − 0.294617i
\(35\) 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 \(-0.430622\pi\)
\(38\) 2.99203 + 2.51061i 0.485371 + 0.407275i
\(39\) 0 0
\(40\) 0.394130 + 2.23522i 0.0623174 + 0.353420i
\(41\) 0.392536 + 2.22618i 0.0613038 + 0.347671i 0.999996 + 0.00291413i \(0.000927599\pi\)
−0.938692 + 0.344757i \(0.887961\pi\)
\(42\) 0 0
\(43\) −4.20164 − 3.52560i −0.640745 − 0.537649i 0.263502 − 0.964659i \(-0.415122\pi\)
−0.904247 + 0.427010i \(0.859567\pi\)
\(44\) −5.21953 − 9.04050i −0.786874 − 1.36291i
\(45\) 0 0
\(46\) −4.57260 + 7.91998i −0.674194 + 1.16774i
\(47\) −6.74994 + 2.45678i −0.984579 + 0.358358i −0.783619 − 0.621242i \(-0.786628\pi\)
−0.200960 + 0.979599i \(0.564406\pi\)
\(48\) 0 0
\(49\) 12.6471 − 10.6122i 1.80673 − 1.51603i
\(50\) −1.42019 − 0.516906i −0.200845 − 0.0731016i
\(51\) 0 0
\(52\) −0.531222 + 3.01271i −0.0736673 + 0.417788i
\(53\) −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
\(56\) −0.922773 + 5.23330i −0.123311 + 0.699330i
\(57\) 0 0
\(58\) 5.95432 + 2.16719i 0.781840 + 0.284567i
\(59\) 0.200900 − 0.168575i 0.0261550 − 0.0219466i −0.629616 − 0.776906i \(-0.716788\pi\)
0.655771 + 0.754959i \(0.272343\pi\)
\(60\) 0 0
\(61\) 4.18690 − 1.52391i 0.536078 − 0.195116i −0.0597724 − 0.998212i \(-0.519037\pi\)
0.595850 + 0.803096i \(0.296815\pi\)
\(62\) 1.56571 − 2.71188i 0.198845 − 0.344410i
\(63\) 0 0
\(64\) 5.72840 + 9.92188i 0.716050 + 1.24024i
\(65\) 1.92913 + 1.61873i 0.239279 + 0.200779i
\(66\) 0 0
\(67\) 0.717359 + 4.06834i 0.0876393 + 0.497027i 0.996756 + 0.0804853i \(0.0256470\pi\)
−0.909116 + 0.416542i \(0.863242\pi\)
\(68\) 1.03257 + 5.85598i 0.125217 + 0.710142i
\(69\) 0 0
\(70\) 16.3459 + 13.7159i 1.95371 + 1.63936i
\(71\) −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 \(0.898291\pi\)
\(74\) 17.9141 − 6.52021i 2.08247 − 0.757958i
\(75\) 0 0
\(76\) −3.54216 + 2.97222i −0.406313 + 0.340937i
\(77\) −18.9062 − 6.88128i −2.15456 − 0.784195i
\(78\) 0 0
\(79\) 0.790517 − 4.48325i 0.0889401 − 0.504404i −0.907497 − 0.420060i \(-0.862009\pi\)
0.996437 − 0.0843449i \(-0.0268797\pi\)
\(80\) 5.59674 0.625734
\(81\) 0 0
\(82\) −4.80368 −0.530478
\(83\) −1.46786 + 8.32464i −0.161118 + 0.913748i 0.791859 + 0.610705i \(0.209114\pi\)
−0.952977 + 0.303043i \(0.901997\pi\)
\(84\) 0 0
\(85\) 4.59975 + 1.67417i 0.498913 + 0.181590i
\(86\) 8.92862 − 7.49200i 0.962797 − 0.807883i
\(87\) 0 0
\(88\) 4.27348 − 1.55542i 0.455555 − 0.165808i
\(89\) −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
\(92\) −8.29373 − 6.95926i −0.864681 − 0.725553i
\(93\) 0 0
\(94\) −2.65063 − 15.0325i −0.273391 − 1.55048i
\(95\) 0.660975 + 3.74857i 0.0678146 + 0.384596i
\(96\) 0 0
\(97\) −3.91431 − 3.28450i −0.397438 − 0.333490i 0.422064 − 0.906566i \(-0.361306\pi\)
−0.819502 + 0.573076i \(0.805750\pi\)
\(98\) 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.j.649.1 12
3.2 odd 2 729.2.e.u.649.2 12
9.2 odd 6 729.2.e.l.163.1 12
9.4 even 3 729.2.e.t.406.1 12
9.5 odd 6 729.2.e.k.406.2 12
9.7 even 3 729.2.e.s.163.2 12
27.2 odd 18 729.2.c.a.487.2 12
27.4 even 9 729.2.e.t.325.1 12
27.5 odd 18 729.2.e.l.568.1 12
27.7 even 9 729.2.c.d.244.5 12
27.11 odd 18 729.2.a.e.1.5 yes 6
27.13 even 9 inner 729.2.e.j.82.1 12
27.14 odd 18 729.2.e.u.82.2 12
27.16 even 9 729.2.a.b.1.2 ✓ 6
27.20 odd 18 729.2.c.a.244.2 12
27.22 even 9 729.2.e.s.568.2 12
27.23 odd 18 729.2.e.k.325.2 12
27.25 even 9 729.2.c.d.487.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.2 ✓ 6 27.16 even 9
729.2.a.e.1.5 yes 6 27.11 odd 18
729.2.c.a.244.2 12 27.20 odd 18
729.2.c.a.487.2 12 27.2 odd 18
729.2.c.d.244.5 12 27.7 even 9
729.2.c.d.487.5 12 27.25 even 9
729.2.e.j.82.1 12 27.13 even 9 inner
729.2.e.j.649.1 12 1.1 even 1 trivial
729.2.e.k.325.2 12 27.23 odd 18
729.2.e.k.406.2 12 9.5 odd 6
729.2.e.l.163.1 12 9.2 odd 6
729.2.e.l.568.1 12 27.5 odd 18
729.2.e.s.163.2 12 9.7 even 3
729.2.e.s.568.2 12 27.22 even 9
729.2.e.t.325.1 12 27.4 even 9
729.2.e.t.406.1 12 9.4 even 3
729.2.e.u.82.2 12 27.14 odd 18
729.2.e.u.649.2 12 3.2 odd 2