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Properties

Label 444.1.bf.a.197.1
Level $444$
Weight $1$
Character 444.197
Analytic conductor $0.222$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
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] = [444,1,Mod(53,444)] 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("444.53"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 9, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 444.bf (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: \(0.221584865609\)
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 9.1.18208693489126464.1

Embedding invariants

Embedding label 197.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 444.197
Dual form 444.1.bf.a.293.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.766044 + 0.642788i) q^{3} +(-0.326352 + 0.118782i) q^{7} +(0.173648 + 0.984808i) q^{9} +(0.0603074 - 0.342020i) q^{13} +(-0.766044 - 0.642788i) q^{19} +(-0.326352 - 0.118782i) q^{21} +(0.766044 - 0.642788i) q^{25} +(-0.500000 + 0.866025i) q^{27} -1.87939 q^{31} +(-0.500000 - 0.866025i) q^{37} +(0.266044 - 0.223238i) q^{39} +1.53209 q^{43} +(-0.673648 + 0.565258i) q^{49} +(-0.173648 - 0.984808i) q^{57} +(0.347296 - 1.96962i) q^{61} +(-0.173648 - 0.300767i) q^{63} +(-1.43969 + 0.524005i) q^{67} +1.53209 q^{73} +1.00000 q^{75} +(-1.43969 + 0.524005i) q^{79} +(-0.939693 + 0.342020i) q^{81} +(0.0209445 + 0.118782i) q^{91} +(-1.43969 - 1.20805i) q^{93} +(-0.766044 + 1.32683i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{7} + 6 q^{13} - 3 q^{21} - 3 q^{27} - 3 q^{37} - 3 q^{39} - 3 q^{49} - 3 q^{67} + 6 q^{75} - 3 q^{79} - 3 q^{91} - 3 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{7}{9}\right)\)

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
\(3\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(4\) 0 0
\(5\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(6\) 0 0
\(7\) −0.326352 + 0.118782i −0.326352 + 0.118782i −0.500000 − 0.866025i \(-0.666667\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(8\) 0 0
\(9\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(10\) 0 0
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) 0.0603074 − 0.342020i 0.0603074 − 0.342020i −0.939693 − 0.342020i \(-0.888889\pi\)
1.00000 \(0\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.173648 − 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(18\) 0 0
\(19\) −0.766044 − 0.642788i −0.766044 − 0.642788i 0.173648 − 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(20\) 0 0
\(21\) −0.326352 − 0.118782i −0.326352 − 0.118782i
\(22\) 0 0
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 0.766044 − 0.642788i 0.766044 − 0.642788i
\(26\) 0 0
\(27\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(28\) 0 0
\(29\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) −1.87939 −1.87939 −0.939693 − 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(38\) 0 0
\(39\) 0.266044 − 0.223238i 0.266044 − 0.223238i
\(40\) 0 0
\(41\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(42\) 0 0
\(43\) 1.53209 1.53209 0.766044 − 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(44\) 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
\(49\) −0.673648 + 0.565258i −0.673648 + 0.565258i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.173648 − 0.984808i −0.173648 − 0.984808i
\(58\) 0 0
\(59\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(60\) 0 0
\(61\) 0.347296 − 1.96962i 0.347296 − 1.96962i 0.173648 − 0.984808i \(-0.444444\pi\)
0.173648 − 0.984808i \(-0.444444\pi\)
\(62\) 0 0
\(63\) −0.173648 − 0.300767i −0.173648 − 0.300767i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.43969 + 0.524005i −1.43969 + 0.524005i −0.939693 − 0.342020i \(-0.888889\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(72\) 0 0
\(73\) 1.53209 1.53209 0.766044 − 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(74\) 0 0
\(75\) 1.00000 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.43969 + 0.524005i −1.43969 + 0.524005i −0.939693 − 0.342020i \(-0.888889\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(82\) 0 0
\(83\) 0 0 −0.173648 − 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(90\) 0 0
\(91\) 0.0209445 + 0.118782i 0.0209445 + 0.118782i
\(92\) 0 0
\(93\) −1.43969 − 1.20805i −1.43969 − 1.20805i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.766044 + 1.32683i −0.766044 + 1.32683i 0.173648 + 0.984808i \(0.444444\pi\)
−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 444.1.bf.a.197.1 ✓ 6
3.2 odd 2 CM 444.1.bf.a.197.1 ✓ 6
4.3 odd 2 1776.1.dr.a.641.1 6
12.11 even 2 1776.1.dr.a.641.1 6
37.34 even 9 inner 444.1.bf.a.293.1 yes 6
111.71 odd 18 inner 444.1.bf.a.293.1 yes 6
148.71 odd 18 1776.1.dr.a.737.1 6
444.71 even 18 1776.1.dr.a.737.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.1.bf.a.197.1 ✓ 6 1.1 even 1 trivial
444.1.bf.a.197.1 ✓ 6 3.2 odd 2 CM
444.1.bf.a.293.1 yes 6 37.34 even 9 inner
444.1.bf.a.293.1 yes 6 111.71 odd 18 inner
1776.1.dr.a.641.1 6 4.3 odd 2
1776.1.dr.a.641.1 6 12.11 even 2
1776.1.dr.a.737.1 6 148.71 odd 18
1776.1.dr.a.737.1 6 444.71 even 18