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Properties

Label 729.2.a.b.1.6
Level $729$
Weight $2$
Character 729.1
Self dual yes
Analytic conductor $5.821$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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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(1,729)] 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("729.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-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: yes
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.7459857.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.578404\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.45779 q^{2} +4.04073 q^{4} +3.08026 q^{5} -2.65867 q^{7} +5.01568 q^{8} +7.57064 q^{10} +3.43434 q^{11} -3.34396 q^{13} -6.53444 q^{14} +4.24603 q^{16} -2.57282 q^{17} -2.09676 q^{19} +12.4465 q^{20} +8.44089 q^{22} -0.534444 q^{23} +4.48802 q^{25} -8.21874 q^{26} -10.7430 q^{28} -2.53089 q^{29} +7.71470 q^{31} +0.404491 q^{32} -6.32344 q^{34} -8.18939 q^{35} -10.2957 q^{37} -5.15340 q^{38} +15.4496 q^{40} +4.88501 q^{41} +2.74149 q^{43} +13.8772 q^{44} -1.31355 q^{46} +5.65800 q^{47} +0.0685109 q^{49} +11.0306 q^{50} -13.5120 q^{52} -6.42657 q^{53} +10.5787 q^{55} -13.3350 q^{56} -6.22040 q^{58} -1.65495 q^{59} +14.3722 q^{61} +18.9611 q^{62} -7.49791 q^{64} -10.3003 q^{65} -5.87898 q^{67} -10.3961 q^{68} -20.1278 q^{70} -14.8163 q^{71} +1.88140 q^{73} -25.3046 q^{74} -8.47245 q^{76} -9.13077 q^{77} +17.1935 q^{79} +13.0789 q^{80} +12.0063 q^{82} -3.96878 q^{83} -7.92496 q^{85} +6.73801 q^{86} +17.2256 q^{88} -5.09880 q^{89} +8.89047 q^{91} -2.15954 q^{92} +13.9062 q^{94} -6.45858 q^{95} -10.6319 q^{97} +0.168385 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 24 q^{14} + 15 q^{16} + 9 q^{17} + 12 q^{19} + 21 q^{20} + 3 q^{22} + 12 q^{23} + 9 q^{25} - 24 q^{26} + 3 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.45779 1.73792 0.868960 − 0.494883i \(-0.164789\pi\)
0.868960 + 0.494883i \(0.164789\pi\)
\(3\) 0 0
\(4\) 4.04073 2.02036
\(5\) 3.08026 1.37754 0.688768 − 0.724982i \(-0.258152\pi\)
0.688768 + 0.724982i \(0.258152\pi\)
\(6\) 0 0
\(7\) −2.65867 −1.00488 −0.502441 − 0.864612i \(-0.667565\pi\)
−0.502441 + 0.864612i \(0.667565\pi\)
\(8\) 5.01568 1.77331
\(9\) 0 0
\(10\) 7.57064 2.39405
\(11\) 3.43434 1.03549 0.517746 − 0.855534i \(-0.326771\pi\)
0.517746 + 0.855534i \(0.326771\pi\)
\(12\) 0 0
\(13\) −3.34396 −0.927447 −0.463723 − 0.885980i \(-0.653487\pi\)
−0.463723 + 0.885980i \(0.653487\pi\)
\(14\) −6.53444 −1.74640
\(15\) 0 0
\(16\) 4.24603 1.06151
\(17\) −2.57282 −0.624000 −0.312000 − 0.950082i \(-0.600999\pi\)
−0.312000 + 0.950082i \(0.600999\pi\)
\(18\) 0 0
\(19\) −2.09676 −0.481030 −0.240515 − 0.970645i \(-0.577316\pi\)
−0.240515 + 0.970645i \(0.577316\pi\)
\(20\) 12.4465 2.78312
\(21\) 0 0
\(22\) 8.44089 1.79960
\(23\) −0.534444 −0.111439 −0.0557196 − 0.998446i \(-0.517745\pi\)
−0.0557196 + 0.998446i \(0.517745\pi\)
\(24\) 0 0
\(25\) 4.48802 0.897604
\(26\) −8.21874 −1.61183
\(27\) 0 0
\(28\) −10.7430 −2.03023
\(29\) −2.53089 −0.469975 −0.234988 − 0.971998i \(-0.575505\pi\)
−0.234988 + 0.971998i \(0.575505\pi\)
\(30\) 0 0
\(31\) 7.71470 1.38560 0.692801 − 0.721129i \(-0.256377\pi\)
0.692801 + 0.721129i \(0.256377\pi\)
\(32\) 0.404491 0.0715045
\(33\) 0 0
\(34\) −6.32344 −1.08446
\(35\) −8.18939 −1.38426
\(36\) 0 0
\(37\) −10.2957 −1.69260 −0.846298 − 0.532709i \(-0.821174\pi\)
−0.846298 + 0.532709i \(0.821174\pi\)
\(38\) −5.15340 −0.835992
\(39\) 0 0
\(40\) 15.4496 2.44280
\(41\) 4.88501 0.762910 0.381455 − 0.924387i \(-0.375423\pi\)
0.381455 + 0.924387i \(0.375423\pi\)
\(42\) 0 0
\(43\) 2.74149 0.418073 0.209037 − 0.977908i \(-0.432967\pi\)
0.209037 + 0.977908i \(0.432967\pi\)
\(44\) 13.8772 2.09207
\(45\) 0 0
\(46\) −1.31355 −0.193673
\(47\) 5.65800 0.825304 0.412652 − 0.910889i \(-0.364603\pi\)
0.412652 + 0.910889i \(0.364603\pi\)
\(48\) 0 0
\(49\) 0.0685109 0.00978728
\(50\) 11.0306 1.55996
\(51\) 0 0
\(52\) −13.5120 −1.87378
\(53\) −6.42657 −0.882758 −0.441379 − 0.897321i \(-0.645511\pi\)
−0.441379 + 0.897321i \(0.645511\pi\)
\(54\) 0 0
\(55\) 10.5787 1.42643
\(56\) −13.3350 −1.78197
\(57\) 0 0
\(58\) −6.22040 −0.816779
\(59\) −1.65495 −0.215456 −0.107728 − 0.994180i \(-0.534358\pi\)
−0.107728 + 0.994180i \(0.534358\pi\)
\(60\) 0 0
\(61\) 14.3722 1.84017 0.920086 − 0.391716i \(-0.128118\pi\)
0.920086 + 0.391716i \(0.128118\pi\)
\(62\) 18.9611 2.40806
\(63\) 0 0
\(64\) −7.49791 −0.937239
\(65\) −10.3003 −1.27759
\(66\) 0 0
\(67\) −5.87898 −0.718232 −0.359116 − 0.933293i \(-0.616922\pi\)
−0.359116 + 0.933293i \(0.616922\pi\)
\(68\) −10.3961 −1.26071
\(69\) 0 0
\(70\) −20.1278 −2.40573
\(71\) −14.8163 −1.75837 −0.879184 − 0.476483i \(-0.841911\pi\)
−0.879184 + 0.476483i \(0.841911\pi\)
\(72\) 0 0
\(73\) 1.88140 0.220201 0.110101 − 0.993920i \(-0.464883\pi\)
0.110101 + 0.993920i \(0.464883\pi\)
\(74\) −25.3046 −2.94160
\(75\) 0 0
\(76\) −8.47245 −0.971857
\(77\) −9.13077 −1.04055
\(78\) 0 0
\(79\) 17.1935 1.93442 0.967209 − 0.253981i \(-0.0817400\pi\)
0.967209 + 0.253981i \(0.0817400\pi\)
\(80\) 13.0789 1.46227
\(81\) 0 0
\(82\) 12.0063 1.32588
\(83\) −3.96878 −0.435631 −0.217815 − 0.975990i \(-0.569893\pi\)
−0.217815 + 0.975990i \(0.569893\pi\)
\(84\) 0 0
\(85\) −7.92496 −0.859582
\(86\) 6.73801 0.726578
\(87\) 0 0
\(88\) 17.2256 1.83625
\(89\) −5.09880 −0.540471 −0.270236 − 0.962794i \(-0.587102\pi\)
−0.270236 + 0.962794i \(0.587102\pi\)
\(90\) 0 0
\(91\) 8.89047 0.931974
\(92\) −2.15954 −0.225148
\(93\) 0 0
\(94\) 13.9062 1.43431
\(95\) −6.45858 −0.662637
\(96\) 0 0
\(97\) −10.6319 −1.07950 −0.539752 − 0.841824i \(-0.681482\pi\)
−0.539752 + 0.841824i \(0.681482\pi\)
\(98\) 0.168385 0.0170095
\(99\) 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.a.b.1.6 ✓ 6
3.2 odd 2 729.2.a.e.1.1 yes 6
9.2 odd 6 729.2.c.a.244.6 12
9.4 even 3 729.2.c.d.487.1 12
9.5 odd 6 729.2.c.a.487.6 12
9.7 even 3 729.2.c.d.244.1 12
27.2 odd 18 729.2.e.u.568.2 12
27.4 even 9 729.2.e.s.406.2 12
27.5 odd 18 729.2.e.k.649.1 12
27.7 even 9 729.2.e.s.325.2 12
27.11 odd 18 729.2.e.k.82.1 12
27.13 even 9 729.2.e.j.163.1 12
27.14 odd 18 729.2.e.u.163.2 12
27.16 even 9 729.2.e.t.82.2 12
27.20 odd 18 729.2.e.l.325.1 12
27.22 even 9 729.2.e.t.649.2 12
27.23 odd 18 729.2.e.l.406.1 12
27.25 even 9 729.2.e.j.568.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.6 ✓ 6 1.1 even 1 trivial
729.2.a.e.1.1 yes 6 3.2 odd 2
729.2.c.a.244.6 12 9.2 odd 6
729.2.c.a.487.6 12 9.5 odd 6
729.2.c.d.244.1 12 9.7 even 3
729.2.c.d.487.1 12 9.4 even 3
729.2.e.j.163.1 12 27.13 even 9
729.2.e.j.568.1 12 27.25 even 9
729.2.e.k.82.1 12 27.11 odd 18
729.2.e.k.649.1 12 27.5 odd 18
729.2.e.l.325.1 12 27.20 odd 18
729.2.e.l.406.1 12 27.23 odd 18
729.2.e.s.325.2 12 27.7 even 9
729.2.e.s.406.2 12 27.4 even 9
729.2.e.t.82.2 12 27.16 even 9
729.2.e.t.649.2 12 27.22 even 9
729.2.e.u.163.2 12 27.14 odd 18
729.2.e.u.568.2 12 27.2 odd 18