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Properties

Label 27.2.a.a.1.1
Level $27$
Weight $2$
Character 27.1
Self dual yes
Analytic conductor $0.216$
Analytic rank $0$
Dimension $1$
CM discriminant -3
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] = [27,2,Mod(1,27)] 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("27.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 27.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.215596085457\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 27.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} -1.00000 q^{7} +5.00000 q^{13} +4.00000 q^{16} -7.00000 q^{19} -5.00000 q^{25} +2.00000 q^{28} -4.00000 q^{31} +11.0000 q^{37} +8.00000 q^{43} -6.00000 q^{49} -10.0000 q^{52} -1.00000 q^{61} -8.00000 q^{64} +5.00000 q^{67} -7.00000 q^{73} +14.0000 q^{76} +17.0000 q^{79} -5.00000 q^{91} -19.0000 q^{97} +O(q^{100})\)

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 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 11.0000 1.80839 0.904194 − 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) −10.0000 −1.38675
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 14.0000 1.60591
\(77\) 0 0
\(78\) 0 0
\(79\) 17.0000 1.91265 0.956325 − 0.292306i \(-0.0944227\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −5.00000 −0.524142
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −19.0000 −1.92916 −0.964579 − 0.263795i \(-0.915026\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 0 0
\(99\) 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 27.2.a.a.1.1 ✓ 1
3.2 odd 2 CM 27.2.a.a.1.1 ✓ 1
4.3 odd 2 432.2.a.e.1.1 1
5.2 odd 4 675.2.b.f.649.1 2
5.3 odd 4 675.2.b.f.649.2 2
5.4 even 2 675.2.a.e.1.1 1
7.6 odd 2 1323.2.a.i.1.1 1
8.3 odd 2 1728.2.a.o.1.1 1
8.5 even 2 1728.2.a.n.1.1 1
9.2 odd 6 81.2.c.a.28.1 2
9.4 even 3 81.2.c.a.55.1 2
9.5 odd 6 81.2.c.a.55.1 2
9.7 even 3 81.2.c.a.28.1 2
11.10 odd 2 3267.2.a.f.1.1 1
12.11 even 2 432.2.a.e.1.1 1
13.12 even 2 4563.2.a.e.1.1 1
15.2 even 4 675.2.b.f.649.1 2
15.8 even 4 675.2.b.f.649.2 2
15.14 odd 2 675.2.a.e.1.1 1
17.16 even 2 7803.2.a.k.1.1 1
19.18 odd 2 9747.2.a.f.1.1 1
21.20 even 2 1323.2.a.i.1.1 1
24.5 odd 2 1728.2.a.n.1.1 1
24.11 even 2 1728.2.a.o.1.1 1
27.2 odd 18 729.2.e.f.568.1 6
27.4 even 9 729.2.e.f.406.1 6
27.5 odd 18 729.2.e.f.649.1 6
27.7 even 9 729.2.e.f.325.1 6
27.11 odd 18 729.2.e.f.82.1 6
27.13 even 9 729.2.e.f.163.1 6
27.14 odd 18 729.2.e.f.163.1 6
27.16 even 9 729.2.e.f.82.1 6
27.20 odd 18 729.2.e.f.325.1 6
27.22 even 9 729.2.e.f.649.1 6
27.23 odd 18 729.2.e.f.406.1 6
27.25 even 9 729.2.e.f.568.1 6
33.32 even 2 3267.2.a.f.1.1 1
36.7 odd 6 1296.2.i.i.433.1 2
36.11 even 6 1296.2.i.i.433.1 2
36.23 even 6 1296.2.i.i.865.1 2
36.31 odd 6 1296.2.i.i.865.1 2
39.38 odd 2 4563.2.a.e.1.1 1
51.50 odd 2 7803.2.a.k.1.1 1
57.56 even 2 9747.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
27.2.a.a.1.1 ✓ 1 3.2 odd 2 CM
81.2.c.a.28.1 2 9.2 odd 6
81.2.c.a.28.1 2 9.7 even 3
81.2.c.a.55.1 2 9.4 even 3
81.2.c.a.55.1 2 9.5 odd 6
432.2.a.e.1.1 1 4.3 odd 2
432.2.a.e.1.1 1 12.11 even 2
675.2.a.e.1.1 1 5.4 even 2
675.2.a.e.1.1 1 15.14 odd 2
675.2.b.f.649.1 2 5.2 odd 4
675.2.b.f.649.1 2 15.2 even 4
675.2.b.f.649.2 2 5.3 odd 4
675.2.b.f.649.2 2 15.8 even 4
729.2.e.f.82.1 6 27.11 odd 18
729.2.e.f.82.1 6 27.16 even 9
729.2.e.f.163.1 6 27.13 even 9
729.2.e.f.163.1 6 27.14 odd 18
729.2.e.f.325.1 6 27.7 even 9
729.2.e.f.325.1 6 27.20 odd 18
729.2.e.f.406.1 6 27.4 even 9
729.2.e.f.406.1 6 27.23 odd 18
729.2.e.f.568.1 6 27.2 odd 18
729.2.e.f.568.1 6 27.25 even 9
729.2.e.f.649.1 6 27.5 odd 18
729.2.e.f.649.1 6 27.22 even 9
1296.2.i.i.433.1 2 36.7 odd 6
1296.2.i.i.433.1 2 36.11 even 6
1296.2.i.i.865.1 2 36.23 even 6
1296.2.i.i.865.1 2 36.31 odd 6
1323.2.a.i.1.1 1 7.6 odd 2
1323.2.a.i.1.1 1 21.20 even 2
1728.2.a.n.1.1 1 8.5 even 2
1728.2.a.n.1.1 1 24.5 odd 2
1728.2.a.o.1.1 1 8.3 odd 2
1728.2.a.o.1.1 1 24.11 even 2
3267.2.a.f.1.1 1 11.10 odd 2
3267.2.a.f.1.1 1 33.32 even 2
4563.2.a.e.1.1 1 13.12 even 2
4563.2.a.e.1.1 1 39.38 odd 2
7803.2.a.k.1.1 1 17.16 even 2
7803.2.a.k.1.1 1 51.50 odd 2
9747.2.a.f.1.1 1 19.18 odd 2
9747.2.a.f.1.1 1 57.56 even 2