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Properties

Label 243.1.d.a.80.1
Level $243$
Weight $1$
Character 243.80
Analytic conductor $0.121$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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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] = [243,1,Mod(80,243)] 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("243.80"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 243.d (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(0.121272798070\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of \(\Q(\sqrt[3]{3})\)
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.177147.1

Embedding invariants

Embedding label 80.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 243.80
Dual form 243.1.d.a.161.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{7} +(0.500000 + 0.866025i) q^{13} +(-0.500000 + 0.866025i) q^{16} -1.00000 q^{19} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{28} +(0.500000 + 0.866025i) q^{31} -1.00000 q^{37} +(0.500000 - 0.866025i) q^{43} +(0.500000 - 0.866025i) q^{52} +(-1.00000 + 1.73205i) q^{61} +1.00000 q^{64} +(-1.00000 - 1.73205i) q^{67} +2.00000 q^{73} +(0.500000 + 0.866025i) q^{76} +(0.500000 - 0.866025i) q^{79} +1.00000 q^{91} +(0.500000 - 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{4} + q^{7} + q^{13} - q^{16} - 2 q^{19} - q^{25} - 2 q^{28} + q^{31} - 2 q^{37} + q^{43} + q^{52} - 2 q^{61} + 2 q^{64} - 2 q^{67} + 4 q^{73} + q^{76} + q^{79} + 2 q^{91} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(5\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) −1.00000 −1.00000
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(53\) 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 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 − 1.73205i −1.00000 − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 − 0.866025i \(-0.666667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 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
\(91\) 1.00000 1.00000
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(98\) 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 243.1.d.a.80.1 2
3.2 odd 2 CM 243.1.d.a.80.1 2
4.3 odd 2 3888.1.q.b.2753.1 2
9.2 odd 6 243.1.b.a.242.1 ✓ 1
9.4 even 3 inner 243.1.d.a.161.1 2
9.5 odd 6 inner 243.1.d.a.161.1 2
9.7 even 3 243.1.b.a.242.1 ✓ 1
12.11 even 2 3888.1.q.b.2753.1 2
27.2 odd 18 729.1.f.a.404.1 6
27.4 even 9 729.1.f.a.566.1 6
27.5 odd 18 729.1.f.a.323.1 6
27.7 even 9 729.1.f.a.647.1 6
27.11 odd 18 729.1.f.a.161.1 6
27.13 even 9 729.1.f.a.80.1 6
27.14 odd 18 729.1.f.a.80.1 6
27.16 even 9 729.1.f.a.161.1 6
27.20 odd 18 729.1.f.a.647.1 6
27.22 even 9 729.1.f.a.323.1 6
27.23 odd 18 729.1.f.a.566.1 6
27.25 even 9 729.1.f.a.404.1 6
36.7 odd 6 3888.1.e.b.1457.1 1
36.11 even 6 3888.1.e.b.1457.1 1
36.23 even 6 3888.1.q.b.161.1 2
36.31 odd 6 3888.1.q.b.161.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
243.1.b.a.242.1 ✓ 1 9.2 odd 6
243.1.b.a.242.1 ✓ 1 9.7 even 3
243.1.d.a.80.1 2 1.1 even 1 trivial
243.1.d.a.80.1 2 3.2 odd 2 CM
243.1.d.a.161.1 2 9.4 even 3 inner
243.1.d.a.161.1 2 9.5 odd 6 inner
729.1.f.a.80.1 6 27.13 even 9
729.1.f.a.80.1 6 27.14 odd 18
729.1.f.a.161.1 6 27.11 odd 18
729.1.f.a.161.1 6 27.16 even 9
729.1.f.a.323.1 6 27.5 odd 18
729.1.f.a.323.1 6 27.22 even 9
729.1.f.a.404.1 6 27.2 odd 18
729.1.f.a.404.1 6 27.25 even 9
729.1.f.a.566.1 6 27.4 even 9
729.1.f.a.566.1 6 27.23 odd 18
729.1.f.a.647.1 6 27.7 even 9
729.1.f.a.647.1 6 27.20 odd 18
3888.1.e.b.1457.1 1 36.7 odd 6
3888.1.e.b.1457.1 1 36.11 even 6
3888.1.q.b.161.1 2 36.23 even 6
3888.1.q.b.161.1 2 36.31 odd 6
3888.1.q.b.2753.1 2 4.3 odd 2
3888.1.q.b.2753.1 2 12.11 even 2