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Properties

Label 2888.1.s.a.1029.1
Level $2888$
Weight $1$
Character 2888.1029
Analytic conductor $1.441$
Analytic rank $0$
Dimension $6$
Projective image $D_{3}$
CM discriminant -152
Inner twists $12$

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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] = [2888,1,Mod(333,2888)] 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("2888.333"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 9, 17])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2888.s (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,3,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.44129975648\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.152.1

Embedding invariants

Embedding label 1029.1
Root \(-0.173648 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 2888.1029
Dual form 2888.1.s.a.2293.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.766044 + 0.642788i) q^{2} +(0.939693 - 0.342020i) q^{3} +(0.173648 + 0.984808i) q^{4} +(0.939693 + 0.342020i) q^{6} +(0.500000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{8} +(0.500000 + 0.866025i) q^{12} +(0.939693 + 0.342020i) q^{13} +(-0.173648 + 0.984808i) q^{14} +(-0.939693 + 0.342020i) q^{16} +(-0.766044 - 0.642788i) q^{17} +(0.766044 + 0.642788i) q^{21} +(-0.173648 - 0.984808i) q^{23} +(-0.173648 + 0.984808i) q^{24} +(-0.939693 - 0.342020i) q^{25} +(0.500000 + 0.866025i) q^{26} +(-0.500000 + 0.866025i) q^{27} +(-0.766044 + 0.642788i) q^{28} +(-0.766044 + 0.642788i) q^{29} +(-0.939693 - 0.342020i) q^{32} +(-0.173648 - 0.984808i) q^{34} +2.00000 q^{37} +1.00000 q^{39} +(0.173648 + 0.984808i) q^{42} +(0.500000 - 0.866025i) q^{46} +(1.53209 - 1.28558i) q^{47} +(-0.766044 + 0.642788i) q^{48} +(-0.500000 - 0.866025i) q^{50} +(-0.939693 - 0.342020i) q^{51} +(-0.173648 + 0.984808i) q^{52} +(-0.173648 - 0.984808i) q^{53} +(-0.939693 + 0.342020i) q^{54} -1.00000 q^{56} -1.00000 q^{58} +(-0.766044 - 0.642788i) q^{59} +(-0.500000 - 0.866025i) q^{64} +(-0.766044 + 0.642788i) q^{67} +(0.500000 - 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{69} +(0.939693 - 0.342020i) q^{73} +(1.53209 + 1.28558i) q^{74} -1.00000 q^{75} +(0.766044 + 0.642788i) q^{78} +(-0.173648 + 0.984808i) q^{81} +(-0.500000 + 0.866025i) q^{84} +(-0.500000 + 0.866025i) q^{87} +(0.173648 + 0.984808i) q^{91} +(0.939693 - 0.342020i) q^{92} +2.00000 q^{94} -1.00000 q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{7} - 3 q^{8} + 3 q^{12} + 3 q^{26} - 3 q^{27} + 12 q^{37} + 6 q^{39} + 3 q^{46} - 3 q^{50} - 6 q^{56} - 6 q^{58} - 3 q^{64} + 3 q^{68} - 3 q^{69} - 6 q^{75} - 3 q^{84} - 3 q^{87} + 12 q^{94}+ \cdots - 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times\).

\(n\) \(1445\) \(2167\) \(2529\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{13}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(3\) 0.939693 − 0.342020i 0.939693 − 0.342020i 0.173648 − 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(4\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(5\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(6\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(7\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(13\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 − 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(14\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(15\) 0 0
\(16\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(17\) −0.766044 − 0.642788i −0.766044 − 0.642788i 0.173648 − 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(22\) 0 0
\(23\) −0.173648 − 0.984808i −0.173648 − 0.984808i −0.939693 − 0.342020i \(-0.888889\pi\)
0.766044 − 0.642788i \(-0.222222\pi\)
\(24\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(25\) −0.939693 − 0.342020i −0.939693 − 0.342020i
\(26\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(27\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(28\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(29\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 − 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) −0.939693 − 0.342020i −0.939693 − 0.342020i
\(33\) 0 0
\(34\) −0.173648 − 0.984808i −0.173648 − 0.984808i
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(38\) 0 0
\(39\) 1.00000 1.00000
\(40\) 0 0
\(41\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(42\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(43\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(47\) 1.53209 − 1.28558i 1.53209 − 1.28558i 0.766044 − 0.642788i \(-0.222222\pi\)
0.766044 − 0.642788i \(-0.222222\pi\)
\(48\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(49\) 0 0
\(50\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(51\) −0.939693 − 0.342020i −0.939693 − 0.342020i
\(52\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(53\) −0.173648 − 0.984808i −0.173648 − 0.984808i −0.939693 − 0.342020i \(-0.888889\pi\)
0.766044 − 0.642788i \(-0.222222\pi\)
\(54\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(55\) 0 0
\(56\) −1.00000 −1.00000
\(57\) 0 0
\(58\) −1.00000 −1.00000
\(59\) −0.766044 − 0.642788i −0.766044 − 0.642788i 0.173648 − 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(60\) 0 0
\(61\) 0 0 −0.173648 − 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 − 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(68\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(69\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(70\) 0 0
\(71\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(72\) 0 0
\(73\) 0.939693 − 0.342020i 0.939693 − 0.342020i 0.173648 − 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(74\) 1.53209 + 1.28558i 1.53209 + 1.28558i
\(75\) −1.00000 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(79\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(80\) 0 0
\(81\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(82\) 0 0
\(83\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(84\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(85\) 0 0
\(86\) 0 0
\(87\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(88\) 0 0
\(89\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(90\) 0 0
\(91\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(92\) 0.939693 − 0.342020i 0.939693 − 0.342020i
\(93\) 0 0
\(94\) 2.00000 2.00000
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(98\) 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 2888.1.s.a.1029.1 6
8.5 even 2 2888.1.s.b.1029.1 6
19.2 odd 18 2888.1.l.b.69.1 2
19.3 odd 18 2888.1.l.b.293.1 2
19.4 even 9 inner 2888.1.s.a.333.1 6
19.5 even 9 152.1.g.b.37.1 yes 1
19.6 even 9 inner 2888.1.s.a.2293.1 6
19.7 even 3 inner 2888.1.s.a.1021.1 6
19.8 odd 6 2888.1.s.b.477.1 6
19.9 even 9 inner 2888.1.s.a.2789.1 6
19.10 odd 18 2888.1.s.b.2789.1 6
19.11 even 3 inner 2888.1.s.a.477.1 6
19.12 odd 6 2888.1.s.b.1021.1 6
19.13 odd 18 2888.1.s.b.2293.1 6
19.14 odd 18 152.1.g.a.37.1 ✓ 1
19.15 odd 18 2888.1.s.b.333.1 6
19.16 even 9 2888.1.l.a.293.1 2
19.17 even 9 2888.1.l.a.69.1 2
19.18 odd 2 2888.1.s.b.1029.1 6
57.5 odd 18 1368.1.i.a.37.1 1
57.14 even 18 1368.1.i.b.37.1 1
76.43 odd 18 608.1.g.b.113.1 1
76.71 even 18 608.1.g.a.113.1 1
95.14 odd 18 3800.1.o.b.1101.1 1
95.24 even 18 3800.1.o.a.1101.1 1
95.33 even 36 3800.1.b.a.949.2 2
95.43 odd 36 3800.1.b.b.949.1 2
95.52 even 36 3800.1.b.a.949.1 2
95.62 odd 36 3800.1.b.b.949.2 2
152.5 even 18 152.1.g.a.37.1 ✓ 1
152.13 odd 18 inner 2888.1.s.a.2293.1 6
152.21 odd 18 2888.1.l.a.69.1 2
152.29 odd 18 inner 2888.1.s.a.2789.1 6
152.37 odd 2 CM 2888.1.s.a.1029.1 6
152.43 odd 18 608.1.g.a.113.1 1
152.45 even 6 2888.1.s.b.1021.1 6
152.53 odd 18 inner 2888.1.s.a.333.1 6
152.61 even 18 2888.1.s.b.333.1 6
152.69 odd 6 inner 2888.1.s.a.1021.1 6
152.85 even 18 2888.1.s.b.2789.1 6
152.93 even 18 2888.1.l.b.69.1 2
152.101 even 18 2888.1.s.b.2293.1 6
152.109 odd 18 152.1.g.b.37.1 yes 1
152.117 odd 18 2888.1.l.a.293.1 2
152.125 even 6 2888.1.s.b.477.1 6
152.141 odd 6 inner 2888.1.s.a.477.1 6
152.147 even 18 608.1.g.b.113.1 1
152.149 even 18 2888.1.l.b.293.1 2
456.5 odd 18 1368.1.i.b.37.1 1
456.413 even 18 1368.1.i.a.37.1 1
760.109 odd 18 3800.1.o.a.1101.1 1
760.157 odd 36 3800.1.b.a.949.1 2
760.309 even 18 3800.1.o.b.1101.1 1
760.413 even 36 3800.1.b.b.949.1 2
760.613 odd 36 3800.1.b.a.949.2 2
760.717 even 36 3800.1.b.b.949.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.1.g.a.37.1 ✓ 1 19.14 odd 18
152.1.g.a.37.1 ✓ 1 152.5 even 18
152.1.g.b.37.1 yes 1 19.5 even 9
152.1.g.b.37.1 yes 1 152.109 odd 18
608.1.g.a.113.1 1 76.71 even 18
608.1.g.a.113.1 1 152.43 odd 18
608.1.g.b.113.1 1 76.43 odd 18
608.1.g.b.113.1 1 152.147 even 18
1368.1.i.a.37.1 1 57.5 odd 18
1368.1.i.a.37.1 1 456.413 even 18
1368.1.i.b.37.1 1 57.14 even 18
1368.1.i.b.37.1 1 456.5 odd 18
2888.1.l.a.69.1 2 19.17 even 9
2888.1.l.a.69.1 2 152.21 odd 18
2888.1.l.a.293.1 2 19.16 even 9
2888.1.l.a.293.1 2 152.117 odd 18
2888.1.l.b.69.1 2 19.2 odd 18
2888.1.l.b.69.1 2 152.93 even 18
2888.1.l.b.293.1 2 19.3 odd 18
2888.1.l.b.293.1 2 152.149 even 18
2888.1.s.a.333.1 6 19.4 even 9 inner
2888.1.s.a.333.1 6 152.53 odd 18 inner
2888.1.s.a.477.1 6 19.11 even 3 inner
2888.1.s.a.477.1 6 152.141 odd 6 inner
2888.1.s.a.1021.1 6 19.7 even 3 inner
2888.1.s.a.1021.1 6 152.69 odd 6 inner
2888.1.s.a.1029.1 6 1.1 even 1 trivial
2888.1.s.a.1029.1 6 152.37 odd 2 CM
2888.1.s.a.2293.1 6 19.6 even 9 inner
2888.1.s.a.2293.1 6 152.13 odd 18 inner
2888.1.s.a.2789.1 6 19.9 even 9 inner
2888.1.s.a.2789.1 6 152.29 odd 18 inner
2888.1.s.b.333.1 6 19.15 odd 18
2888.1.s.b.333.1 6 152.61 even 18
2888.1.s.b.477.1 6 19.8 odd 6
2888.1.s.b.477.1 6 152.125 even 6
2888.1.s.b.1021.1 6 19.12 odd 6
2888.1.s.b.1021.1 6 152.45 even 6
2888.1.s.b.1029.1 6 8.5 even 2
2888.1.s.b.1029.1 6 19.18 odd 2
2888.1.s.b.2293.1 6 19.13 odd 18
2888.1.s.b.2293.1 6 152.101 even 18
2888.1.s.b.2789.1 6 19.10 odd 18
2888.1.s.b.2789.1 6 152.85 even 18
3800.1.b.a.949.1 2 95.52 even 36
3800.1.b.a.949.1 2 760.157 odd 36
3800.1.b.a.949.2 2 95.33 even 36
3800.1.b.a.949.2 2 760.613 odd 36
3800.1.b.b.949.1 2 95.43 odd 36
3800.1.b.b.949.1 2 760.413 even 36
3800.1.b.b.949.2 2 95.62 odd 36
3800.1.b.b.949.2 2 760.717 even 36
3800.1.o.a.1101.1 1 95.24 even 18
3800.1.o.a.1101.1 1 760.109 odd 18
3800.1.o.b.1101.1 1 95.14 odd 18
3800.1.o.b.1101.1 1 760.309 even 18