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Properties

Label 729.2.c.d.244.5
Level $729$
Weight $2$
Character 729.244
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(244,729)] 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("729.244"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,3,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == 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: \(6\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 244.5
Root \(1.13697i\) of defining polynomial
Character \(\chi\) \(=\) 729.244
Dual form 729.2.c.d.487.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.06251 + 1.84033i) q^{2} +(-1.25787 + 2.17870i) q^{4} +(1.03547 - 1.79349i) q^{5} +(-2.42434 - 4.19907i) q^{7} -1.09598 q^{8} +4.40081 q^{10} +(-2.07474 - 3.59356i) q^{11} +(0.608008 - 1.05310i) q^{13} +(5.15178 - 8.92315i) q^{14} +(1.35126 + 2.34044i) q^{16} +2.36364 q^{17} -1.83801 q^{19} +(2.60498 + 4.51196i) q^{20} +(4.40889 - 7.63642i) q^{22} +(2.15178 - 3.72700i) q^{23} +(0.355603 + 0.615922i) q^{25} +2.58407 q^{26} +12.1980 q^{28} +(-1.49091 - 2.58233i) q^{29} +(-0.736793 + 1.27616i) q^{31} +(-3.96743 + 6.87180i) q^{32} +(2.51140 + 4.34987i) q^{34} -10.0413 q^{35} +8.97108 q^{37} +(-1.95291 - 3.38254i) q^{38} +(-1.13485 + 1.96562i) q^{40} +(-1.13026 + 1.95767i) q^{41} +(2.74243 + 4.75003i) q^{43} +10.4391 q^{44} +9.14521 q^{46} +(-3.59157 - 6.22077i) q^{47} +(-8.25481 + 14.2978i) q^{49} +(-0.755667 + 1.30885i) q^{50} +(1.52959 + 2.64933i) q^{52} -6.32803 q^{53} -8.59334 q^{55} +(2.65702 + 4.60209i) q^{56} +(3.16823 - 5.48753i) q^{58} +(-0.131128 + 0.227121i) q^{59} +(2.22780 + 3.85867i) q^{61} -3.13141 q^{62} -11.4568 q^{64} +(-1.25915 - 2.18091i) q^{65} +(-2.06555 + 3.57764i) q^{67} +(-2.97316 + 5.14966i) q^{68} +(-10.6690 - 18.4793i) q^{70} +3.08551 q^{71} +12.7601 q^{73} +(9.53190 + 16.5097i) q^{74} +(2.31198 - 4.00447i) q^{76} +(-10.0598 + 17.4240i) q^{77} +(-2.27620 - 3.94250i) q^{79} +5.59674 q^{80} -4.80368 q^{82} +(4.22653 + 7.32056i) q^{83} +(2.44748 - 4.23915i) q^{85} +(-5.82774 + 10.0939i) q^{86} +(2.27387 + 3.93846i) q^{88} +16.9632 q^{89} -5.89606 q^{91} +(5.41335 + 9.37619i) q^{92} +(7.63218 - 13.2193i) q^{94} +(-1.90320 + 3.29644i) q^{95} +(2.55489 + 4.42519i) q^{97} -35.0834 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 9 q^{4} - 3 q^{5} - 6 q^{7} - 12 q^{8} + 12 q^{10} - 6 q^{11} - 6 q^{13} + 24 q^{14} - 15 q^{16} + 18 q^{17} + 24 q^{19} - 21 q^{20} - 3 q^{22} - 12 q^{23} - 9 q^{25} - 48 q^{26} + 6 q^{28}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.06251 + 1.84033i 0.751311 + 1.30131i 0.947187 + 0.320680i \(0.103912\pi\)
−0.195876 + 0.980629i \(0.562755\pi\)
\(3\) 0 0
\(4\) −1.25787 + 2.17870i −0.628937 + 1.08935i
\(5\) 1.03547 − 1.79349i 0.463076 − 0.802072i −0.536036 − 0.844195i \(-0.680079\pi\)
0.999112 + 0.0421233i \(0.0134122\pi\)
\(6\) 0 0
\(7\) −2.42434 − 4.19907i −0.916313 − 1.58710i −0.804967 − 0.593319i \(-0.797817\pi\)
−0.111346 − 0.993782i \(-0.535516\pi\)
\(8\) −1.09598 −0.387487
\(9\) 0 0
\(10\) 4.40081 1.39166
\(11\) −2.07474 − 3.59356i −0.625559 − 1.08350i −0.988432 − 0.151662i \(-0.951538\pi\)
0.362874 − 0.931838i \(-0.381796\pi\)
\(12\) 0 0
\(13\) 0.608008 − 1.05310i 0.168631 − 0.292077i −0.769308 − 0.638878i \(-0.779399\pi\)
0.937939 + 0.346801i \(0.112732\pi\)
\(14\) 5.15178 − 8.92315i 1.37687 − 2.38481i
\(15\) 0 0
\(16\) 1.35126 + 2.34044i 0.337814 + 0.585111i
\(17\) 2.36364 0.573266 0.286633 − 0.958040i \(-0.407464\pi\)
0.286633 + 0.958040i \(0.407464\pi\)
\(18\) 0 0
\(19\) −1.83801 −0.421668 −0.210834 − 0.977522i \(-0.567618\pi\)
−0.210834 + 0.977522i \(0.567618\pi\)
\(20\) 2.60498 + 4.51196i 0.582491 + 1.00890i
\(21\) 0 0
\(22\) 4.40889 − 7.63642i 0.939979 − 1.62809i
\(23\) 2.15178 − 3.72700i 0.448678 − 0.777133i −0.549622 − 0.835413i \(-0.685228\pi\)
0.998300 + 0.0582801i \(0.0185617\pi\)
\(24\) 0 0
\(25\) 0.355603 + 0.615922i 0.0711206 + 0.123184i
\(26\) 2.58407 0.506777
\(27\) 0 0
\(28\) 12.1980 2.30521
\(29\) −1.49091 − 2.58233i −0.276855 − 0.479527i 0.693746 − 0.720219i \(-0.255959\pi\)
−0.970601 + 0.240692i \(0.922626\pi\)
\(30\) 0 0
\(31\) −0.736793 + 1.27616i −0.132332 + 0.229206i −0.924575 − 0.381000i \(-0.875580\pi\)
0.792243 + 0.610206i \(0.208913\pi\)
\(32\) −3.96743 + 6.87180i −0.701350 + 1.21477i
\(33\) 0 0
\(34\) 2.51140 + 4.34987i 0.430701 + 0.745997i
\(35\) −10.0413 −1.69729
\(36\) 0 0
\(37\) 8.97108 1.47484 0.737418 − 0.675436i \(-0.236045\pi\)
0.737418 + 0.675436i \(0.236045\pi\)
\(38\) −1.95291 − 3.38254i −0.316804 − 0.548720i
\(39\) 0 0
\(40\) −1.13485 + 1.96562i −0.179436 + 0.310792i
\(41\) −1.13026 + 1.95767i −0.176517 + 0.305737i −0.940685 − 0.339280i \(-0.889816\pi\)
0.764168 + 0.645017i \(0.223150\pi\)
\(42\) 0 0
\(43\) 2.74243 + 4.75003i 0.418216 + 0.724372i 0.995760 − 0.0919876i \(-0.0293220\pi\)
−0.577544 + 0.816360i \(0.695989\pi\)
\(44\) 10.4391 1.57375
\(45\) 0 0
\(46\) 9.14521 1.34839
\(47\) −3.59157 − 6.22077i −0.523884 − 0.907393i −0.999613 − 0.0278017i \(-0.991149\pi\)
0.475730 − 0.879591i \(-0.342184\pi\)
\(48\) 0 0
\(49\) −8.25481 + 14.2978i −1.17926 + 2.04254i
\(50\) −0.755667 + 1.30885i −0.106867 + 0.185100i
\(51\) 0 0
\(52\) 1.52959 + 2.64933i 0.212116 + 0.367396i
\(53\) −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
\(56\) 2.65702 + 4.60209i 0.355059 + 0.614980i
\(57\) 0 0
\(58\) 3.16823 − 5.48753i 0.416009 − 0.720548i
\(59\) −0.131128 + 0.227121i −0.0170715 + 0.0295686i −0.874435 − 0.485143i \(-0.838768\pi\)
0.857363 + 0.514711i \(0.172101\pi\)
\(60\) 0 0
\(61\) 2.22780 + 3.85867i 0.285241 + 0.494052i 0.972668 − 0.232202i \(-0.0745931\pi\)
−0.687427 + 0.726254i \(0.741260\pi\)
\(62\) −3.13141 −0.397690
\(63\) 0 0
\(64\) −11.4568 −1.43210
\(65\) −1.25915 − 2.18091i −0.156178 − 0.270508i
\(66\) 0 0
\(67\) −2.06555 + 3.57764i −0.252347 + 0.437078i −0.964172 − 0.265279i \(-0.914536\pi\)
0.711824 + 0.702358i \(0.247869\pi\)
\(68\) −2.97316 + 5.14966i −0.360548 + 0.624488i
\(69\) 0 0
\(70\) −10.6690 − 18.4793i −1.27519 − 2.20870i
\(71\) 3.08551 0.366183 0.183091 − 0.983096i \(-0.441390\pi\)
0.183091 + 0.983096i \(0.441390\pi\)
\(72\) 0 0
\(73\) 12.7601 1.49345 0.746726 − 0.665132i \(-0.231625\pi\)
0.746726 + 0.665132i \(0.231625\pi\)
\(74\) 9.53190 + 16.5097i 1.10806 + 1.91922i
\(75\) 0 0
\(76\) 2.31198 − 4.00447i 0.265202 − 0.459344i
\(77\) −10.0598 + 17.4240i −1.14642 + 1.98565i
\(78\) 0 0
\(79\) −2.27620 − 3.94250i −0.256093 − 0.443566i 0.709099 − 0.705109i \(-0.249102\pi\)
−0.965192 + 0.261543i \(0.915769\pi\)
\(80\) 5.59674 0.625734
\(81\) 0 0
\(82\) −4.80368 −0.530478
\(83\) 4.22653 + 7.32056i 0.463922 + 0.803536i 0.999152 − 0.0411700i \(-0.0131085\pi\)
−0.535230 + 0.844706i \(0.679775\pi\)
\(84\) 0 0
\(85\) 2.44748 − 4.23915i 0.265466 − 0.459801i
\(86\) −5.82774 + 10.0939i −0.628421 + 1.08846i
\(87\) 0 0
\(88\) 2.27387 + 3.93846i 0.242396 + 0.419842i
\(89\) 16.9632 1.79809 0.899046 − 0.437854i \(-0.144261\pi\)
0.899046 + 0.437854i \(0.144261\pi\)
\(90\) 0 0
\(91\) −5.89606 −0.618075
\(92\) 5.41335 + 9.37619i 0.564380 + 0.977535i
\(93\) 0 0
\(94\) 7.63218 − 13.2193i 0.787199 − 1.36347i
\(95\) −1.90320 + 3.29644i −0.195264 + 0.338208i
\(96\) 0 0
\(97\) 2.55489 + 4.42519i 0.259409 + 0.449310i 0.966084 − 0.258229i \(-0.0831388\pi\)
−0.706674 + 0.707539i \(0.749805\pi\)
\(98\) −35.0834 −3.54396
\(99\) 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.c.d.244.5 12
3.2 odd 2 729.2.c.a.244.2 12
9.2 odd 6 729.2.c.a.487.2 12
9.4 even 3 729.2.a.b.1.2 ✓ 6
9.5 odd 6 729.2.a.e.1.5 yes 6
9.7 even 3 inner 729.2.c.d.487.5 12
27.2 odd 18 729.2.e.u.82.2 12
27.4 even 9 729.2.e.j.649.1 12
27.5 odd 18 729.2.e.l.163.1 12
27.7 even 9 729.2.e.s.568.2 12
27.11 odd 18 729.2.e.k.325.2 12
27.13 even 9 729.2.e.t.406.1 12
27.14 odd 18 729.2.e.k.406.2 12
27.16 even 9 729.2.e.t.325.1 12
27.20 odd 18 729.2.e.l.568.1 12
27.22 even 9 729.2.e.s.163.2 12
27.23 odd 18 729.2.e.u.649.2 12
27.25 even 9 729.2.e.j.82.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.2 ✓ 6 9.4 even 3
729.2.a.e.1.5 yes 6 9.5 odd 6
729.2.c.a.244.2 12 3.2 odd 2
729.2.c.a.487.2 12 9.2 odd 6
729.2.c.d.244.5 12 1.1 even 1 trivial
729.2.c.d.487.5 12 9.7 even 3 inner
729.2.e.j.82.1 12 27.25 even 9
729.2.e.j.649.1 12 27.4 even 9
729.2.e.k.325.2 12 27.11 odd 18
729.2.e.k.406.2 12 27.14 odd 18
729.2.e.l.163.1 12 27.5 odd 18
729.2.e.l.568.1 12 27.20 odd 18
729.2.e.s.163.2 12 27.22 even 9
729.2.e.s.568.2 12 27.7 even 9
729.2.e.t.325.1 12 27.16 even 9
729.2.e.t.406.1 12 27.13 even 9
729.2.e.u.82.2 12 27.2 odd 18
729.2.e.u.649.2 12 27.23 odd 18