Properties

Label 297.1.q.a.142.1
Level $297$
Weight $1$
Character 297.142
Analytic conductor $0.148$
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] = [297,1,Mod(43,297)] 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("297.43"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(297, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([4, 9])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 297.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: \(0.148222308752\)
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: \(D_{9}\)
Projective field: Galois closure of 9.1.459450093735369.1

Embedding invariants

Embedding label 142.1
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 297.142
Dual form 297.1.q.a.274.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.939693 - 0.342020i) q^{3} +(0.766044 - 0.642788i) q^{4} +(0.266044 + 1.50881i) q^{5} +(0.766044 + 0.642788i) q^{9} +(0.173648 - 0.984808i) q^{11} +(-0.939693 + 0.342020i) q^{12} +(0.266044 - 1.50881i) q^{15} +(0.173648 - 0.984808i) q^{16} +(1.17365 + 0.984808i) q^{20} +(-0.766044 + 0.642788i) q^{23} +(-1.26604 + 0.460802i) q^{25} +(-0.500000 - 0.866025i) q^{27} +(-1.43969 + 1.20805i) q^{31} +(-0.500000 + 0.866025i) q^{33} +1.00000 q^{36} +(-0.766044 - 1.32683i) q^{37} +(-0.500000 - 0.866025i) q^{44} +(-0.766044 + 1.32683i) q^{45} +(0.266044 + 0.223238i) q^{47} +(-0.500000 + 0.866025i) q^{48} +(0.173648 + 0.984808i) q^{49} -1.87939 q^{53} +1.53209 q^{55} +(-0.326352 - 1.85083i) q^{59} +(-0.766044 - 1.32683i) q^{60} +(-0.500000 - 0.866025i) q^{64} +(1.76604 + 0.642788i) q^{67} +(0.939693 - 0.342020i) q^{69} +(-0.173648 - 0.300767i) q^{71} +1.34730 q^{75} +1.53209 q^{80} +(0.173648 + 0.984808i) q^{81} +(0.500000 - 0.866025i) q^{89} +(-0.173648 + 0.984808i) q^{92} +(1.76604 - 0.642788i) q^{93} +(0.0603074 - 0.342020i) q^{97} +(0.766044 - 0.642788i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 3 q^{15} + 6 q^{20} - 3 q^{25} - 3 q^{27} - 3 q^{31} - 3 q^{33} + 6 q^{36} - 3 q^{44} - 3 q^{47} - 3 q^{48} - 3 q^{59} - 3 q^{64} + 6 q^{67} + 6 q^{75} + 3 q^{89} + 6 q^{93} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/297\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(244\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\) \(-1\)

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 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(3\) −0.939693 − 0.342020i −0.939693 − 0.342020i
\(4\) 0.766044 − 0.642788i 0.766044 − 0.642788i
\(5\) 0.266044 + 1.50881i 0.266044 + 1.50881i 0.766044 + 0.642788i \(0.222222\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(8\) 0 0
\(9\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(10\) 0 0
\(11\) 0.173648 − 0.984808i 0.173648 − 0.984808i
\(12\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(13\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(14\) 0 0
\(15\) 0.266044 − 1.50881i 0.266044 − 1.50881i
\(16\) 0.173648 − 0.984808i 0.173648 − 0.984808i
\(17\) 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 \(0.666667\pi\)
\(20\) 1.17365 + 0.984808i 1.17365 + 0.984808i
\(21\) 0 0
\(22\) 0 0
\(23\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 − 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(24\) 0 0
\(25\) −1.26604 + 0.460802i −1.26604 + 0.460802i
\(26\) 0 0
\(27\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(28\) 0 0
\(29\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(30\) 0 0
\(31\) −1.43969 + 1.20805i −1.43969 + 1.20805i −0.500000 + 0.866025i \(0.666667\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(32\) 0 0
\(33\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) −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 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(42\) 0 0
\(43\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(44\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(45\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(46\) 0 0
\(47\) 0.266044 + 0.223238i 0.266044 + 0.223238i 0.766044 − 0.642788i \(-0.222222\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) −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
\(53\) −1.87939 −1.87939 −0.939693 − 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(54\) 0 0
\(55\) 1.53209 1.53209
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.326352 − 1.85083i −0.326352 − 1.85083i −0.500000 − 0.866025i \(-0.666667\pi\)
0.173648 − 0.984808i \(-0.444444\pi\)
\(60\) −0.766044 − 1.32683i −0.766044 − 1.32683i
\(61\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(62\) 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 1.00000 \(0\)
0.766044 + 0.642788i \(0.222222\pi\)
\(68\) 0 0
\(69\) 0.939693 − 0.342020i 0.939693 − 0.342020i
\(70\) 0 0
\(71\) −0.173648 − 0.300767i −0.173648 − 0.300767i 0.766044 − 0.642788i \(-0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(72\) 0 0
\(73\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 0 0
\(75\) 1.34730 1.34730
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(80\) 1.53209 1.53209
\(81\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(82\) 0 0
\(83\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(84\) 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 \(-0.666667\pi\)
1.00000 \(0\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(93\) 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 −0.939693 − 0.342020i \(-0.888889\pi\)
1.00000 \(0\)
\(98\) 0 0
\(99\) 0.766044 − 0.642788i 0.766044 − 0.642788i
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 297.1.q.a.142.1 ✓ 6
3.2 odd 2 891.1.q.a.505.1 6
9.2 odd 6 2673.1.q.c.1594.1 6
9.4 even 3 2673.1.q.d.703.1 6
9.5 odd 6 2673.1.q.a.703.1 6
9.7 even 3 2673.1.q.b.1594.1 6
11.2 odd 10 3267.1.bf.a.1492.1 24
11.3 even 5 3267.1.bf.a.2653.1 24
11.4 even 5 3267.1.bf.a.1546.1 24
11.5 even 5 3267.1.bf.a.844.1 24
11.6 odd 10 3267.1.bf.a.844.1 24
11.7 odd 10 3267.1.bf.a.1546.1 24
11.8 odd 10 3267.1.bf.a.2653.1 24
11.9 even 5 3267.1.bf.a.1492.1 24
11.10 odd 2 CM 297.1.q.a.142.1 ✓ 6
27.4 even 9 inner 297.1.q.a.274.1 yes 6
27.5 odd 18 2673.1.q.a.1000.1 6
27.13 even 9 2673.1.q.b.109.1 6
27.14 odd 18 2673.1.q.c.109.1 6
27.22 even 9 2673.1.q.d.1000.1 6
27.23 odd 18 891.1.q.a.307.1 6
33.32 even 2 891.1.q.a.505.1 6
99.32 even 6 2673.1.q.a.703.1 6
99.43 odd 6 2673.1.q.b.1594.1 6
99.65 even 6 2673.1.q.c.1594.1 6
99.76 odd 6 2673.1.q.d.703.1 6
297.4 even 45 3267.1.bf.a.2272.1 24
297.31 even 45 3267.1.bf.a.2218.1 24
297.32 even 18 2673.1.q.a.1000.1 6
297.58 even 45 3267.1.bf.a.112.1 24
297.76 odd 18 2673.1.q.d.1000.1 6
297.85 odd 90 3267.1.bf.a.112.1 24
297.112 odd 90 3267.1.bf.a.2218.1 24
297.131 even 18 891.1.q.a.307.1 6
297.139 odd 90 3267.1.bf.a.2272.1 24
297.175 odd 18 2673.1.q.b.109.1 6
297.193 odd 90 3267.1.bf.a.1570.1 24
297.230 even 18 2673.1.q.c.109.1 6
297.247 even 45 3267.1.bf.a.1570.1 24
297.274 odd 18 inner 297.1.q.a.274.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
297.1.q.a.142.1 ✓ 6 1.1 even 1 trivial
297.1.q.a.142.1 ✓ 6 11.10 odd 2 CM
297.1.q.a.274.1 yes 6 27.4 even 9 inner
297.1.q.a.274.1 yes 6 297.274 odd 18 inner
891.1.q.a.307.1 6 27.23 odd 18
891.1.q.a.307.1 6 297.131 even 18
891.1.q.a.505.1 6 3.2 odd 2
891.1.q.a.505.1 6 33.32 even 2
2673.1.q.a.703.1 6 9.5 odd 6
2673.1.q.a.703.1 6 99.32 even 6
2673.1.q.a.1000.1 6 27.5 odd 18
2673.1.q.a.1000.1 6 297.32 even 18
2673.1.q.b.109.1 6 27.13 even 9
2673.1.q.b.109.1 6 297.175 odd 18
2673.1.q.b.1594.1 6 9.7 even 3
2673.1.q.b.1594.1 6 99.43 odd 6
2673.1.q.c.109.1 6 27.14 odd 18
2673.1.q.c.109.1 6 297.230 even 18
2673.1.q.c.1594.1 6 9.2 odd 6
2673.1.q.c.1594.1 6 99.65 even 6
2673.1.q.d.703.1 6 9.4 even 3
2673.1.q.d.703.1 6 99.76 odd 6
2673.1.q.d.1000.1 6 27.22 even 9
2673.1.q.d.1000.1 6 297.76 odd 18
3267.1.bf.a.112.1 24 297.58 even 45
3267.1.bf.a.112.1 24 297.85 odd 90
3267.1.bf.a.844.1 24 11.5 even 5
3267.1.bf.a.844.1 24 11.6 odd 10
3267.1.bf.a.1492.1 24 11.2 odd 10
3267.1.bf.a.1492.1 24 11.9 even 5
3267.1.bf.a.1546.1 24 11.4 even 5
3267.1.bf.a.1546.1 24 11.7 odd 10
3267.1.bf.a.1570.1 24 297.193 odd 90
3267.1.bf.a.1570.1 24 297.247 even 45
3267.1.bf.a.2218.1 24 297.31 even 45
3267.1.bf.a.2218.1 24 297.112 odd 90
3267.1.bf.a.2272.1 24 297.4 even 45
3267.1.bf.a.2272.1 24 297.139 odd 90
3267.1.bf.a.2653.1 24 11.3 even 5
3267.1.bf.a.2653.1 24 11.8 odd 10