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Properties

Label 27.5.b.a.26.1
Level $27$
Weight $5$
Character 27.26
Self dual yes
Analytic conductor $2.791$
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] = [27,5,Mod(26,27)] 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("27.26"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 27.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,16] 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: \(2.79098900326\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 26.1
Character \(\chi\) \(=\) 27.26

$q$-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)
 
\(f(q)\) \(=\) \(q+16.0000 q^{4} +71.0000 q^{7} -337.000 q^{13} +256.000 q^{16} -601.000 q^{19} +625.000 q^{25} +1136.00 q^{28} +194.000 q^{31} -529.000 q^{37} -3214.00 q^{43} +2640.00 q^{49} -5392.00 q^{52} +7199.00 q^{61} +4096.00 q^{64} +2903.00 q^{67} -1249.00 q^{73} -9616.00 q^{76} +4679.00 q^{79} -23927.0 q^{91} +9071.00 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 16.0000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 71.0000 1.44898 0.724490 − 0.689286i \(-0.242075\pi\)
0.724490 + 0.689286i \(0.242075\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −337.000 −1.99408 −0.997041 − 0.0768662i \(-0.975509\pi\)
−0.997041 + 0.0768662i \(0.975509\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −601.000 −1.66482 −0.832410 − 0.554160i \(-0.813039\pi\)
−0.832410 + 0.554160i \(0.813039\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 1136.00 1.44898
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 194.000 0.201873 0.100937 − 0.994893i \(-0.467816\pi\)
0.100937 + 0.994893i \(0.467816\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −529.000 −0.386413 −0.193207 − 0.981158i \(-0.561889\pi\)
−0.193207 + 0.981158i \(0.561889\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −3214.00 −1.73824 −0.869118 − 0.494604i \(-0.835313\pi\)
−0.869118 + 0.494604i \(0.835313\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2640.00 1.09954
\(50\) 0 0
\(51\) 0 0
\(52\) −5392.00 −1.99408
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 7199.00 1.93469 0.967347 − 0.253454i \(-0.0815666\pi\)
0.967347 + 0.253454i \(0.0815666\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4096.00 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 2903.00 0.646692 0.323346 − 0.946281i \(-0.395192\pi\)
0.323346 + 0.946281i \(0.395192\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1249.00 −0.234378 −0.117189 − 0.993110i \(-0.537388\pi\)
−0.117189 + 0.993110i \(0.537388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −9616.00 −1.66482
\(77\) 0 0
\(78\) 0 0
\(79\) 4679.00 0.749720 0.374860 − 0.927082i \(-0.377691\pi\)
0.374860 + 0.927082i \(0.377691\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −23927.0 −2.88939
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9071.00 0.964077 0.482038 − 0.876150i \(-0.339897\pi\)
0.482038 + 0.876150i \(0.339897\pi\)
\(98\) 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 27.5.b.a.26.1 ✓ 1
3.2 odd 2 CM 27.5.b.a.26.1 ✓ 1
4.3 odd 2 432.5.e.a.161.1 1
5.2 odd 4 675.5.d.c.674.2 2
5.3 odd 4 675.5.d.c.674.1 2
5.4 even 2 675.5.c.b.26.1 1
9.2 odd 6 81.5.d.a.53.1 2
9.4 even 3 81.5.d.a.26.1 2
9.5 odd 6 81.5.d.a.26.1 2
9.7 even 3 81.5.d.a.53.1 2
12.11 even 2 432.5.e.a.161.1 1
15.2 even 4 675.5.d.c.674.2 2
15.8 even 4 675.5.d.c.674.1 2
15.14 odd 2 675.5.c.b.26.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.b.a.26.1 ✓ 1 1.1 even 1 trivial
27.5.b.a.26.1 ✓ 1 3.2 odd 2 CM
81.5.d.a.26.1 2 9.4 even 3
81.5.d.a.26.1 2 9.5 odd 6
81.5.d.a.53.1 2 9.2 odd 6
81.5.d.a.53.1 2 9.7 even 3
432.5.e.a.161.1 1 4.3 odd 2
432.5.e.a.161.1 1 12.11 even 2
675.5.c.b.26.1 1 5.4 even 2
675.5.c.b.26.1 1 15.14 odd 2
675.5.d.c.674.1 2 5.3 odd 4
675.5.d.c.674.1 2 15.8 even 4
675.5.d.c.674.2 2 5.2 odd 4
675.5.d.c.674.2 2 15.2 even 4