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Properties

Label 729.2.a.a.1.3
Level $729$
Weight $2$
Character 729.1
Self dual yes
Analytic conductor $5.821$
Analytic rank $1$
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,-3,0,3] 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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1397493.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.68091\) 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)
 
\(f(q)\) \(=\) \(q-0.801527 q^{2} -1.35755 q^{4} -2.74984 q^{5} +2.37683 q^{7} +2.69117 q^{8} +2.20407 q^{10} -0.250159 q^{11} +2.61198 q^{13} -1.90510 q^{14} +0.558064 q^{16} -0.293377 q^{17} -2.78475 q^{19} +3.73306 q^{20} +0.200509 q^{22} -6.68984 q^{23} +2.56163 q^{25} -2.09357 q^{26} -3.22668 q^{28} -0.355057 q^{29} +2.76547 q^{31} -5.82964 q^{32} +0.235149 q^{34} -6.53592 q^{35} -6.99238 q^{37} +2.23205 q^{38} -7.40029 q^{40} -9.71761 q^{41} +0.260706 q^{43} +0.339604 q^{44} +5.36209 q^{46} -11.4256 q^{47} -1.35066 q^{49} -2.05321 q^{50} -3.54591 q^{52} +5.43137 q^{53} +0.687897 q^{55} +6.39646 q^{56} +0.284588 q^{58} +5.97693 q^{59} -11.8468 q^{61} -2.21660 q^{62} +3.55649 q^{64} -7.18254 q^{65} +1.81030 q^{67} +0.398275 q^{68} +5.23871 q^{70} -0.370510 q^{71} +5.02679 q^{73} +5.60458 q^{74} +3.78044 q^{76} -0.594586 q^{77} +0.802822 q^{79} -1.53459 q^{80} +7.78892 q^{82} -2.75565 q^{83} +0.806740 q^{85} -0.208963 q^{86} -0.673220 q^{88} +10.4507 q^{89} +6.20825 q^{91} +9.08183 q^{92} +9.15789 q^{94} +7.65761 q^{95} -14.8346 q^{97} +1.08259 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8} + 3 q^{10} - 12 q^{11} - 6 q^{14} - 3 q^{16} - 9 q^{17} + 3 q^{19} - 6 q^{20} + 6 q^{22} - 15 q^{23} - 6 q^{25} - 15 q^{26} - 6 q^{28} - 12 q^{29} - 12 q^{35}+ \cdots + 45 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.801527 −0.566765 −0.283383 − 0.959007i \(-0.591457\pi\)
−0.283383 + 0.959007i \(0.591457\pi\)
\(3\) 0 0
\(4\) −1.35755 −0.678777
\(5\) −2.74984 −1.22977 −0.614883 − 0.788618i \(-0.710797\pi\)
−0.614883 + 0.788618i \(0.710797\pi\)
\(6\) 0 0
\(7\) 2.37683 0.898359 0.449179 − 0.893442i \(-0.351716\pi\)
0.449179 + 0.893442i \(0.351716\pi\)
\(8\) 2.69117 0.951472
\(9\) 0 0
\(10\) 2.20407 0.696989
\(11\) −0.250159 −0.0754257 −0.0377129 − 0.999289i \(-0.512007\pi\)
−0.0377129 + 0.999289i \(0.512007\pi\)
\(12\) 0 0
\(13\) 2.61198 0.724434 0.362217 − 0.932094i \(-0.382020\pi\)
0.362217 + 0.932094i \(0.382020\pi\)
\(14\) −1.90510 −0.509158
\(15\) 0 0
\(16\) 0.558064 0.139516
\(17\) −0.293377 −0.0711543 −0.0355772 − 0.999367i \(-0.511327\pi\)
−0.0355772 + 0.999367i \(0.511327\pi\)
\(18\) 0 0
\(19\) −2.78475 −0.638864 −0.319432 − 0.947609i \(-0.603492\pi\)
−0.319432 + 0.947609i \(0.603492\pi\)
\(20\) 3.73306 0.834737
\(21\) 0 0
\(22\) 0.200509 0.0427487
\(23\) −6.68984 −1.39493 −0.697464 − 0.716620i \(-0.745688\pi\)
−0.697464 + 0.716620i \(0.745688\pi\)
\(24\) 0 0
\(25\) 2.56163 0.512325
\(26\) −2.09357 −0.410584
\(27\) 0 0
\(28\) −3.22668 −0.609786
\(29\) −0.355057 −0.0659324 −0.0329662 − 0.999456i \(-0.510495\pi\)
−0.0329662 + 0.999456i \(0.510495\pi\)
\(30\) 0 0
\(31\) 2.76547 0.496692 0.248346 − 0.968671i \(-0.420113\pi\)
0.248346 + 0.968671i \(0.420113\pi\)
\(32\) −5.82964 −1.03055
\(33\) 0 0
\(34\) 0.235149 0.0403278
\(35\) −6.53592 −1.10477
\(36\) 0 0
\(37\) −6.99238 −1.14954 −0.574770 − 0.818315i \(-0.694909\pi\)
−0.574770 + 0.818315i \(0.694909\pi\)
\(38\) 2.23205 0.362086
\(39\) 0 0
\(40\) −7.40029 −1.17009
\(41\) −9.71761 −1.51764 −0.758818 − 0.651303i \(-0.774223\pi\)
−0.758818 + 0.651303i \(0.774223\pi\)
\(42\) 0 0
\(43\) 0.260706 0.0397574 0.0198787 − 0.999802i \(-0.493672\pi\)
0.0198787 + 0.999802i \(0.493672\pi\)
\(44\) 0.339604 0.0511973
\(45\) 0 0
\(46\) 5.36209 0.790597
\(47\) −11.4256 −1.66659 −0.833295 − 0.552829i \(-0.813548\pi\)
−0.833295 + 0.552829i \(0.813548\pi\)
\(48\) 0 0
\(49\) −1.35066 −0.192952
\(50\) −2.05321 −0.290368
\(51\) 0 0
\(52\) −3.54591 −0.491729
\(53\) 5.43137 0.746056 0.373028 − 0.927820i \(-0.378320\pi\)
0.373028 + 0.927820i \(0.378320\pi\)
\(54\) 0 0
\(55\) 0.687897 0.0927560
\(56\) 6.39646 0.854764
\(57\) 0 0
\(58\) 0.284588 0.0373682
\(59\) 5.97693 0.778130 0.389065 − 0.921210i \(-0.372798\pi\)
0.389065 + 0.921210i \(0.372798\pi\)
\(60\) 0 0
\(61\) −11.8468 −1.51682 −0.758411 − 0.651776i \(-0.774024\pi\)
−0.758411 + 0.651776i \(0.774024\pi\)
\(62\) −2.21660 −0.281508
\(63\) 0 0
\(64\) 3.55649 0.444561
\(65\) −7.18254 −0.890884
\(66\) 0 0
\(67\) 1.81030 0.221164 0.110582 − 0.993867i \(-0.464729\pi\)
0.110582 + 0.993867i \(0.464729\pi\)
\(68\) 0.398275 0.0482979
\(69\) 0 0
\(70\) 5.23871 0.626146
\(71\) −0.370510 −0.0439714 −0.0219857 − 0.999758i \(-0.506999\pi\)
−0.0219857 + 0.999758i \(0.506999\pi\)
\(72\) 0 0
\(73\) 5.02679 0.588341 0.294171 − 0.955753i \(-0.404957\pi\)
0.294171 + 0.955753i \(0.404957\pi\)
\(74\) 5.60458 0.651519
\(75\) 0 0
\(76\) 3.78044 0.433647
\(77\) −0.594586 −0.0677594
\(78\) 0 0
\(79\) 0.802822 0.0903245 0.0451622 − 0.998980i \(-0.485620\pi\)
0.0451622 + 0.998980i \(0.485620\pi\)
\(80\) −1.53459 −0.171572
\(81\) 0 0
\(82\) 7.78892 0.860143
\(83\) −2.75565 −0.302472 −0.151236 − 0.988498i \(-0.548325\pi\)
−0.151236 + 0.988498i \(0.548325\pi\)
\(84\) 0 0
\(85\) 0.806740 0.0875032
\(86\) −0.208963 −0.0225331
\(87\) 0 0
\(88\) −0.673220 −0.0717655
\(89\) 10.4507 1.10777 0.553884 − 0.832594i \(-0.313145\pi\)
0.553884 + 0.832594i \(0.313145\pi\)
\(90\) 0 0
\(91\) 6.20825 0.650801
\(92\) 9.08183 0.946846
\(93\) 0 0
\(94\) 9.15789 0.944565
\(95\) 7.65761 0.785654
\(96\) 0 0
\(97\) −14.8346 −1.50623 −0.753113 − 0.657891i \(-0.771449\pi\)
−0.753113 + 0.657891i \(0.771449\pi\)
\(98\) 1.08259 0.109358
\(99\) 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.a.1.3 6
3.2 odd 2 729.2.a.d.1.4 6
9.2 odd 6 729.2.c.b.244.3 12
9.4 even 3 729.2.c.e.487.4 12
9.5 odd 6 729.2.c.b.487.3 12
9.7 even 3 729.2.c.e.244.4 12
27.2 odd 18 81.2.e.a.64.1 12
27.4 even 9 243.2.e.d.136.1 12
27.5 odd 18 243.2.e.b.217.2 12
27.7 even 9 243.2.e.d.109.1 12
27.11 odd 18 243.2.e.b.28.2 12
27.13 even 9 27.2.e.a.7.2 yes 12
27.14 odd 18 81.2.e.a.19.1 12
27.16 even 9 243.2.e.c.28.1 12
27.20 odd 18 243.2.e.a.109.2 12
27.22 even 9 243.2.e.c.217.1 12
27.23 odd 18 243.2.e.a.136.2 12
27.25 even 9 27.2.e.a.4.2 ✓ 12
108.67 odd 18 432.2.u.c.385.2 12
108.79 odd 18 432.2.u.c.193.2 12
135.13 odd 36 675.2.u.b.574.2 24
135.52 odd 36 675.2.u.b.274.2 24
135.67 odd 36 675.2.u.b.574.3 24
135.79 even 18 675.2.l.c.301.1 12
135.94 even 18 675.2.l.c.601.1 12
135.133 odd 36 675.2.u.b.274.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.4.2 ✓ 12 27.25 even 9
27.2.e.a.7.2 yes 12 27.13 even 9
81.2.e.a.19.1 12 27.14 odd 18
81.2.e.a.64.1 12 27.2 odd 18
243.2.e.a.109.2 12 27.20 odd 18
243.2.e.a.136.2 12 27.23 odd 18
243.2.e.b.28.2 12 27.11 odd 18
243.2.e.b.217.2 12 27.5 odd 18
243.2.e.c.28.1 12 27.16 even 9
243.2.e.c.217.1 12 27.22 even 9
243.2.e.d.109.1 12 27.7 even 9
243.2.e.d.136.1 12 27.4 even 9
432.2.u.c.193.2 12 108.79 odd 18
432.2.u.c.385.2 12 108.67 odd 18
675.2.l.c.301.1 12 135.79 even 18
675.2.l.c.601.1 12 135.94 even 18
675.2.u.b.274.2 24 135.52 odd 36
675.2.u.b.274.3 24 135.133 odd 36
675.2.u.b.574.2 24 135.13 odd 36
675.2.u.b.574.3 24 135.67 odd 36
729.2.a.a.1.3 6 1.1 even 1 trivial
729.2.a.d.1.4 6 3.2 odd 2
729.2.c.b.244.3 12 9.2 odd 6
729.2.c.b.487.3 12 9.5 odd 6
729.2.c.e.244.4 12 9.7 even 3
729.2.c.e.487.4 12 9.4 even 3