Properties

Label 2673.1.q.a.703.1
Level $2673$
Weight $1$
Character 2673.703
Analytic conductor $1.334$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -11
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] = [2673,1,Mod(109,2673)] 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("2673.109"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2673, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([14, 9])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2673 = 3^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2673.q (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.33400077877\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 297)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.459450093735369.1

Embedding invariants

Embedding label 703.1
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 2673.703
Dual form 2673.1.q.a.1000.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.939693 - 0.342020i) q^{4} +(-1.17365 + 0.984808i) q^{5} +(-0.766044 - 0.642788i) q^{11} +(0.766044 + 0.642788i) q^{16} +(1.43969 - 0.524005i) q^{20} +(-0.939693 - 0.342020i) q^{23} +(0.233956 - 1.32683i) q^{25} +(1.76604 + 0.642788i) q^{31} +(-0.766044 - 1.32683i) q^{37} +(0.500000 + 0.866025i) q^{44} +(0.326352 - 0.118782i) q^{47} +(0.766044 - 0.642788i) q^{49} +1.87939 q^{53} +1.53209 q^{55} +(1.43969 - 1.20805i) q^{59} +(-0.500000 - 0.866025i) q^{64} +(-0.326352 - 1.85083i) q^{67} +(0.173648 + 0.300767i) q^{71} -1.53209 q^{80} +(-0.500000 + 0.866025i) q^{89} +(0.766044 + 0.642788i) q^{92} +(0.266044 + 0.223238i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{5} + 3 q^{20} + 6 q^{25} + 6 q^{31} + 3 q^{44} + 3 q^{47} + 3 q^{59} - 3 q^{64} - 3 q^{67} - 3 q^{89} - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2673\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(2432\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{9}\right)\)

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 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(3\) 0 0
\(4\) −0.939693 − 0.342020i −0.939693 − 0.342020i
\(5\) −1.17365 + 0.984808i −1.17365 + 0.984808i −0.173648 + 0.984808i \(0.555556\pi\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.766044 − 0.642788i −0.766044 − 0.642788i
\(12\) 0 0
\(13\) 0 0 −0.173648 − 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(17\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 1.43969 − 0.524005i 1.43969 − 0.524005i
\(21\) 0 0
\(22\) 0 0
\(23\) −0.939693 − 0.342020i −0.939693 − 0.342020i −0.173648 − 0.984808i \(-0.555556\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(24\) 0 0
\(25\) 0.233956 − 1.32683i 0.233956 − 1.32683i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(30\) 0 0
\(31\) 1.76604 + 0.642788i 1.76604 + 0.642788i 1.00000 \(0\)
0.766044 + 0.642788i \(0.222222\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.766044 − 1.32683i −0.766044 − 1.32683i −0.939693 − 0.342020i \(-0.888889\pi\)
0.173648 − 0.984808i \(-0.444444\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.173648 − 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(42\) 0 0
\(43\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(44\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(45\) 0 0
\(46\) 0 0
\(47\) 0.326352 − 0.118782i 0.326352 − 0.118782i −0.173648 − 0.984808i \(-0.555556\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 0 0
\(49\) 0.766044 − 0.642788i 0.766044 − 0.642788i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.87939 1.87939 0.939693 − 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(54\) 0 0
\(55\) 1.53209 1.53209
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.43969 − 1.20805i 1.43969 − 1.20805i 0.500000 − 0.866025i \(-0.333333\pi\)
0.939693 − 0.342020i \(-0.111111\pi\)
\(60\) 0 0
\(61\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.326352 − 1.85083i −0.326352 − 1.85083i −0.500000 − 0.866025i \(-0.666667\pi\)
0.173648 − 0.984808i \(-0.444444\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.173648 + 0.300767i 0.173648 + 0.300767i 0.939693 − 0.342020i \(-0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(72\) 0 0
\(73\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(80\) −1.53209 −1.53209
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.266044 + 0.223238i 0.266044 + 0.223238i 0.766044 − 0.642788i \(-0.222222\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 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 2673.1.q.a.703.1 6
3.2 odd 2 2673.1.q.d.703.1 6
9.2 odd 6 297.1.q.a.142.1 ✓ 6
9.4 even 3 2673.1.q.c.1594.1 6
9.5 odd 6 2673.1.q.b.1594.1 6
9.7 even 3 891.1.q.a.505.1 6
11.10 odd 2 CM 2673.1.q.a.703.1 6
27.4 even 9 2673.1.q.c.109.1 6
27.5 odd 18 297.1.q.a.274.1 yes 6
27.13 even 9 inner 2673.1.q.a.1000.1 6
27.14 odd 18 2673.1.q.d.1000.1 6
27.22 even 9 891.1.q.a.307.1 6
27.23 odd 18 2673.1.q.b.109.1 6
33.32 even 2 2673.1.q.d.703.1 6
99.2 even 30 3267.1.bf.a.1492.1 24
99.20 odd 30 3267.1.bf.a.1492.1 24
99.29 even 30 3267.1.bf.a.1546.1 24
99.32 even 6 2673.1.q.b.1594.1 6
99.38 odd 30 3267.1.bf.a.844.1 24
99.43 odd 6 891.1.q.a.505.1 6
99.47 odd 30 3267.1.bf.a.2653.1 24
99.65 even 6 297.1.q.a.142.1 ✓ 6
99.74 even 30 3267.1.bf.a.2653.1 24
99.76 odd 6 2673.1.q.c.1594.1 6
99.83 even 30 3267.1.bf.a.844.1 24
99.92 odd 30 3267.1.bf.a.1546.1 24
297.5 odd 90 3267.1.bf.a.1570.1 24
297.32 even 18 297.1.q.a.274.1 yes 6
297.59 odd 90 3267.1.bf.a.2272.1 24
297.76 odd 18 891.1.q.a.307.1 6
297.86 odd 90 3267.1.bf.a.2218.1 24
297.113 odd 90 3267.1.bf.a.112.1 24
297.131 even 18 2673.1.q.b.109.1 6
297.140 even 90 3267.1.bf.a.112.1 24
297.167 even 90 3267.1.bf.a.2218.1 24
297.175 odd 18 inner 2673.1.q.a.1000.1 6
297.194 even 90 3267.1.bf.a.2272.1 24
297.230 even 18 2673.1.q.d.1000.1 6
297.248 even 90 3267.1.bf.a.1570.1 24
297.274 odd 18 2673.1.q.c.109.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
297.1.q.a.142.1 ✓ 6 9.2 odd 6
297.1.q.a.142.1 ✓ 6 99.65 even 6
297.1.q.a.274.1 yes 6 27.5 odd 18
297.1.q.a.274.1 yes 6 297.32 even 18
891.1.q.a.307.1 6 27.22 even 9
891.1.q.a.307.1 6 297.76 odd 18
891.1.q.a.505.1 6 9.7 even 3
891.1.q.a.505.1 6 99.43 odd 6
2673.1.q.a.703.1 6 1.1 even 1 trivial
2673.1.q.a.703.1 6 11.10 odd 2 CM
2673.1.q.a.1000.1 6 27.13 even 9 inner
2673.1.q.a.1000.1 6 297.175 odd 18 inner
2673.1.q.b.109.1 6 27.23 odd 18
2673.1.q.b.109.1 6 297.131 even 18
2673.1.q.b.1594.1 6 9.5 odd 6
2673.1.q.b.1594.1 6 99.32 even 6
2673.1.q.c.109.1 6 27.4 even 9
2673.1.q.c.109.1 6 297.274 odd 18
2673.1.q.c.1594.1 6 9.4 even 3
2673.1.q.c.1594.1 6 99.76 odd 6
2673.1.q.d.703.1 6 3.2 odd 2
2673.1.q.d.703.1 6 33.32 even 2
2673.1.q.d.1000.1 6 27.14 odd 18
2673.1.q.d.1000.1 6 297.230 even 18
3267.1.bf.a.112.1 24 297.113 odd 90
3267.1.bf.a.112.1 24 297.140 even 90
3267.1.bf.a.844.1 24 99.38 odd 30
3267.1.bf.a.844.1 24 99.83 even 30
3267.1.bf.a.1492.1 24 99.2 even 30
3267.1.bf.a.1492.1 24 99.20 odd 30
3267.1.bf.a.1546.1 24 99.29 even 30
3267.1.bf.a.1546.1 24 99.92 odd 30
3267.1.bf.a.1570.1 24 297.5 odd 90
3267.1.bf.a.1570.1 24 297.248 even 90
3267.1.bf.a.2218.1 24 297.86 odd 90
3267.1.bf.a.2218.1 24 297.167 even 90
3267.1.bf.a.2272.1 24 297.59 odd 90
3267.1.bf.a.2272.1 24 297.194 even 90
3267.1.bf.a.2653.1 24 99.47 odd 30
3267.1.bf.a.2653.1 24 99.74 even 30