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Properties

Label 3267.1.q.a.1211.1
Level $3267$
Weight $1$
Character 3267.1211
Analytic conductor $1.630$
Analytic rank $0$
Dimension $6$
Projective image $D_{18}$
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] = [3267,1,Mod(122,3267)] 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("3267.122"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3267, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([17, 0])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3267.q (of order \(18\), degree \(6\), 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: \(1.63044539627\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

Embedding invariants

Embedding label 1211.1
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 3267.1211
Dual form 3267.1.q.a.1937.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.939693 - 0.342020i) q^{3} +(0.766044 + 0.642788i) q^{4} +(-1.26604 - 0.223238i) q^{5} +(0.766044 - 0.642788i) q^{9} +(0.939693 + 0.342020i) q^{12} +(-1.26604 + 0.223238i) q^{15} +(0.173648 + 0.984808i) q^{16} +(-0.826352 - 0.984808i) q^{20} +(1.11334 - 1.32683i) q^{23} +(0.613341 + 0.223238i) q^{25} +(0.500000 - 0.866025i) q^{27} +(1.43969 + 1.20805i) q^{31} +1.00000 q^{36} +(0.766044 - 1.32683i) q^{37} +(-1.11334 + 0.642788i) q^{45} +(1.26604 + 1.50881i) q^{47} +(0.500000 + 0.866025i) q^{48} +(-0.173648 + 0.984808i) q^{49} -0.684040i q^{53} +(0.673648 + 0.118782i) q^{59} +(-1.11334 - 0.642788i) q^{60} +(-0.500000 + 0.866025i) q^{64} +(-1.76604 + 0.642788i) q^{67} +(0.592396 - 1.62760i) q^{69} +(-1.70574 - 0.984808i) q^{71} +0.652704 q^{75} -1.28558i q^{80} +(0.173648 - 0.984808i) q^{81} +(-1.50000 + 0.866025i) q^{89} +(1.70574 - 0.300767i) q^{92} +(1.76604 + 0.642788i) q^{93} +(0.0603074 + 0.342020i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 3 q^{15} - 6 q^{20} - 3 q^{25} + 3 q^{27} + 3 q^{31} + 6 q^{36} + 3 q^{47} + 3 q^{48} + 3 q^{59} - 3 q^{64} - 6 q^{67} + 6 q^{75} - 9 q^{89} + 6 q^{93} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(3026\)
\(\chi(n)\) \(1\) \(e\left(\frac{11}{18}\right)\)

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 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(3\) 0.939693 − 0.342020i 0.939693 − 0.342020i
\(4\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(5\) −1.26604 − 0.223238i −1.26604 − 0.223238i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(6\) 0 0
\(7\) 0 0 −0.642788 − 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(8\) 0 0
\(9\) 0.766044 − 0.642788i 0.766044 − 0.642788i
\(10\) 0 0
\(11\) 0 0
\(12\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(13\) 0 0 −0.342020 − 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(14\) 0 0
\(15\) −1.26604 + 0.223238i −1.26604 + 0.223238i
\(16\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(17\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) −0.826352 − 0.984808i −0.826352 − 0.984808i
\(21\) 0 0
\(22\) 0 0
\(23\) 1.11334 − 1.32683i 1.11334 − 1.32683i 0.173648 − 0.984808i \(-0.444444\pi\)
0.939693 − 0.342020i \(-0.111111\pi\)
\(24\) 0 0
\(25\) 0.613341 + 0.223238i 0.613341 + 0.223238i
\(26\) 0 0
\(27\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(28\) 0 0
\(29\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(30\) 0 0
\(31\) 1.43969 + 1.20805i 1.43969 + 1.20805i 0.939693 + 0.342020i \(0.111111\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 0.766044 − 1.32683i 0.766044 − 1.32683i −0.173648 − 0.984808i \(-0.555556\pi\)
0.939693 − 0.342020i \(-0.111111\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(42\) 0 0
\(43\) 0 0 0.984808 − 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(44\) 0 0
\(45\) −1.11334 + 0.642788i −1.11334 + 0.642788i
\(46\) 0 0
\(47\) 1.26604 + 1.50881i 1.26604 + 1.50881i 0.766044 + 0.642788i \(0.222222\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(49\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 0.684040i − 0.684040i −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 − 0.342020i \(-0.111111\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.673648 + 0.118782i 0.673648 + 0.118782i 0.500000 − 0.866025i \(-0.333333\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(60\) −1.11334 − 0.642788i −1.11334 − 0.642788i
\(61\) 0 0 −0.642788 − 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) −1.76604 + 0.642788i −1.76604 + 0.642788i −0.766044 + 0.642788i \(0.777778\pi\)
−1.00000 \(1.00000\pi\)
\(68\) 0 0
\(69\) 0.592396 − 1.62760i 0.592396 − 1.62760i
\(70\) 0 0
\(71\) −1.70574 − 0.984808i −1.70574 − 0.984808i −0.939693 − 0.342020i \(-0.888889\pi\)
−0.766044 − 0.642788i \(-0.777778\pi\)
\(72\) 0 0
\(73\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) 0 0
\(75\) 0.652704 0.652704
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.342020 − 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(80\) − 1.28558i − 1.28558i
\(81\) 0.173648 − 0.984808i 0.173648 − 0.984808i
\(82\) 0 0
\(83\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.70574 − 0.300767i 1.70574 − 0.300767i
\(93\) 1.76604 + 0.642788i 1.76604 + 0.642788i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.0603074 + 0.342020i 0.0603074 + 0.342020i 1.00000 \(0\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(98\) 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 3267.1.q.a.1211.1 ✓ 6
11.2 odd 10 3267.1.be.a.1049.1 24
11.3 even 5 3267.1.be.a.3155.1 24
11.4 even 5 3267.1.be.a.995.1 24
11.5 even 5 3267.1.be.a.1697.1 24
11.6 odd 10 3267.1.be.a.1697.1 24
11.7 odd 10 3267.1.be.a.995.1 24
11.8 odd 10 3267.1.be.a.3155.1 24
11.9 even 5 3267.1.be.a.1049.1 24
11.10 odd 2 CM 3267.1.q.a.1211.1 ✓ 6
27.20 odd 18 inner 3267.1.q.a.1937.1 yes 6
297.20 odd 90 3267.1.be.a.1775.1 24
297.47 odd 90 3267.1.be.a.614.1 24
297.74 even 90 3267.1.be.a.614.1 24
297.101 even 90 3267.1.be.a.1775.1 24
297.128 even 90 3267.1.be.a.1721.1 24
297.182 even 90 3267.1.be.a.2423.1 24
297.236 odd 90 3267.1.be.a.2423.1 24
297.263 even 18 inner 3267.1.q.a.1937.1 yes 6
297.290 odd 90 3267.1.be.a.1721.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3267.1.q.a.1211.1 ✓ 6 1.1 even 1 trivial
3267.1.q.a.1211.1 ✓ 6 11.10 odd 2 CM
3267.1.q.a.1937.1 yes 6 27.20 odd 18 inner
3267.1.q.a.1937.1 yes 6 297.263 even 18 inner
3267.1.be.a.614.1 24 297.47 odd 90
3267.1.be.a.614.1 24 297.74 even 90
3267.1.be.a.995.1 24 11.4 even 5
3267.1.be.a.995.1 24 11.7 odd 10
3267.1.be.a.1049.1 24 11.2 odd 10
3267.1.be.a.1049.1 24 11.9 even 5
3267.1.be.a.1697.1 24 11.5 even 5
3267.1.be.a.1697.1 24 11.6 odd 10
3267.1.be.a.1721.1 24 297.128 even 90
3267.1.be.a.1721.1 24 297.290 odd 90
3267.1.be.a.1775.1 24 297.20 odd 90
3267.1.be.a.1775.1 24 297.101 even 90
3267.1.be.a.2423.1 24 297.182 even 90
3267.1.be.a.2423.1 24 297.236 odd 90
3267.1.be.a.3155.1 24 11.3 even 5
3267.1.be.a.3155.1 24 11.8 odd 10