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Properties

Label 2888.1.l.a.293.1
Level $2888$
Weight $1$
Character 2888.293
Analytic conductor $1.441$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -152
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] = [2888,1,Mod(69,2888)] 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("2888.69"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 5])) 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.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.44129975648\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.1267762688.3

Embedding invariants

Embedding label 293.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2888.293
Dual form 2888.1.l.a.69.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{12} +(0.500000 + 0.866025i) q^{13} +(0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.500000 - 0.866025i) q^{17} +(-0.500000 + 0.866025i) q^{21} +(0.500000 + 0.866025i) q^{23} +(0.500000 - 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{26} +1.00000 q^{27} +(0.500000 + 0.866025i) q^{28} +(0.500000 + 0.866025i) q^{29} +(-0.500000 - 0.866025i) q^{32} +(0.500000 + 0.866025i) q^{34} +2.00000 q^{37} +1.00000 q^{39} +(-0.500000 - 0.866025i) q^{42} -1.00000 q^{46} +(-1.00000 - 1.73205i) q^{47} +(0.500000 + 0.866025i) q^{48} +1.00000 q^{50} +(-0.500000 - 0.866025i) q^{51} +(0.500000 - 0.866025i) q^{52} +(0.500000 + 0.866025i) q^{53} +(-0.500000 + 0.866025i) q^{54} -1.00000 q^{56} -1.00000 q^{58} +(0.500000 - 0.866025i) q^{59} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{67} -1.00000 q^{68} +1.00000 q^{69} +(0.500000 - 0.866025i) q^{73} +(-1.00000 + 1.73205i) q^{74} -1.00000 q^{75} +(-0.500000 + 0.866025i) q^{78} +(0.500000 - 0.866025i) q^{81} +1.00000 q^{84} +1.00000 q^{87} +(-0.500000 - 0.866025i) q^{91} +(0.500000 - 0.866025i) q^{92} +2.00000 q^{94} -1.00000 q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} + q^{6} - 2 q^{7} + 2 q^{8} - 2 q^{12} + q^{13} + q^{14} - q^{16} + q^{17} - q^{21} + q^{23} + q^{24} - q^{25} - 2 q^{26} + 2 q^{27} + q^{28} + q^{29} - q^{32}+ \cdots - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1445\) \(2167\) \(2529\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(3\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(4\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(5\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(7\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 1.00000 1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −1.00000 −1.00000
\(13\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(22\) 0 0
\(23\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(25\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(26\) −1.00000 −1.00000
\(27\) 1.00000 1.00000
\(28\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(29\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(33\) 0 0
\(34\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(35\) 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 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(43\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −1.00000
\(47\) −1.00000 − 1.73205i −1.00000 − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 − 0.866025i \(-0.666667\pi\)
\(48\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(49\) 0 0
\(50\) 1.00000 1.00000
\(51\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(52\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(53\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(55\) 0 0
\(56\) −1.00000 −1.00000
\(57\) 0 0
\(58\) −1.00000 −1.00000
\(59\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(60\) 0 0
\(61\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) −1.00000 −1.00000
\(69\) 1.00000 1.00000
\(70\) 0 0
\(71\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(72\) 0 0
\(73\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(74\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(75\) −1.00000 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(79\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 1.00000 1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000 1.00000
\(88\) 0 0
\(89\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(92\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(93\) 0 0
\(94\) 2.00000 2.00000
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−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 2888.1.l.a.293.1 2
8.5 even 2 2888.1.l.b.293.1 2
19.2 odd 18 2888.1.s.b.2293.1 6
19.3 odd 18 2888.1.s.b.2789.1 6
19.4 even 9 2888.1.s.a.1021.1 6
19.5 even 9 2888.1.s.a.333.1 6
19.6 even 9 2888.1.s.a.1029.1 6
19.7 even 3 inner 2888.1.l.a.69.1 2
19.8 odd 6 152.1.g.a.37.1 ✓ 1
19.9 even 9 2888.1.s.a.477.1 6
19.10 odd 18 2888.1.s.b.477.1 6
19.11 even 3 152.1.g.b.37.1 yes 1
19.12 odd 6 2888.1.l.b.69.1 2
19.13 odd 18 2888.1.s.b.1029.1 6
19.14 odd 18 2888.1.s.b.333.1 6
19.15 odd 18 2888.1.s.b.1021.1 6
19.16 even 9 2888.1.s.a.2789.1 6
19.17 even 9 2888.1.s.a.2293.1 6
19.18 odd 2 2888.1.l.b.293.1 2
57.8 even 6 1368.1.i.b.37.1 1
57.11 odd 6 1368.1.i.a.37.1 1
76.11 odd 6 608.1.g.b.113.1 1
76.27 even 6 608.1.g.a.113.1 1
95.8 even 12 3800.1.b.a.949.2 2
95.27 even 12 3800.1.b.a.949.1 2
95.49 even 6 3800.1.o.a.1101.1 1
95.68 odd 12 3800.1.b.b.949.1 2
95.84 odd 6 3800.1.o.b.1101.1 1
95.87 odd 12 3800.1.b.b.949.2 2
152.5 even 18 2888.1.s.b.333.1 6
152.11 odd 6 608.1.g.a.113.1 1
152.13 odd 18 2888.1.s.a.1029.1 6
152.21 odd 18 2888.1.s.a.2293.1 6
152.27 even 6 608.1.g.b.113.1 1
152.29 odd 18 2888.1.s.a.477.1 6
152.37 odd 2 CM 2888.1.l.a.293.1 2
152.45 even 6 2888.1.l.b.69.1 2
152.53 odd 18 2888.1.s.a.1021.1 6
152.61 even 18 2888.1.s.b.1021.1 6
152.69 odd 6 inner 2888.1.l.a.69.1 2
152.85 even 18 2888.1.s.b.477.1 6
152.93 even 18 2888.1.s.b.2293.1 6
152.101 even 18 2888.1.s.b.1029.1 6
152.109 odd 18 2888.1.s.a.333.1 6
152.117 odd 18 2888.1.s.a.2789.1 6
152.125 even 6 152.1.g.a.37.1 ✓ 1
152.141 odd 6 152.1.g.b.37.1 yes 1
152.149 even 18 2888.1.s.b.2789.1 6
456.125 odd 6 1368.1.i.b.37.1 1
456.293 even 6 1368.1.i.a.37.1 1
760.277 odd 12 3800.1.b.a.949.1 2
760.293 even 12 3800.1.b.b.949.1 2
760.429 even 6 3800.1.o.b.1101.1 1
760.597 even 12 3800.1.b.b.949.2 2
760.733 odd 12 3800.1.b.a.949.2 2
760.749 odd 6 3800.1.o.a.1101.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.1.g.a.37.1 ✓ 1 19.8 odd 6
152.1.g.a.37.1 ✓ 1 152.125 even 6
152.1.g.b.37.1 yes 1 19.11 even 3
152.1.g.b.37.1 yes 1 152.141 odd 6
608.1.g.a.113.1 1 76.27 even 6
608.1.g.a.113.1 1 152.11 odd 6
608.1.g.b.113.1 1 76.11 odd 6
608.1.g.b.113.1 1 152.27 even 6
1368.1.i.a.37.1 1 57.11 odd 6
1368.1.i.a.37.1 1 456.293 even 6
1368.1.i.b.37.1 1 57.8 even 6
1368.1.i.b.37.1 1 456.125 odd 6
2888.1.l.a.69.1 2 19.7 even 3 inner
2888.1.l.a.69.1 2 152.69 odd 6 inner
2888.1.l.a.293.1 2 1.1 even 1 trivial
2888.1.l.a.293.1 2 152.37 odd 2 CM
2888.1.l.b.69.1 2 19.12 odd 6
2888.1.l.b.69.1 2 152.45 even 6
2888.1.l.b.293.1 2 8.5 even 2
2888.1.l.b.293.1 2 19.18 odd 2
2888.1.s.a.333.1 6 19.5 even 9
2888.1.s.a.333.1 6 152.109 odd 18
2888.1.s.a.477.1 6 19.9 even 9
2888.1.s.a.477.1 6 152.29 odd 18
2888.1.s.a.1021.1 6 19.4 even 9
2888.1.s.a.1021.1 6 152.53 odd 18
2888.1.s.a.1029.1 6 19.6 even 9
2888.1.s.a.1029.1 6 152.13 odd 18
2888.1.s.a.2293.1 6 19.17 even 9
2888.1.s.a.2293.1 6 152.21 odd 18
2888.1.s.a.2789.1 6 19.16 even 9
2888.1.s.a.2789.1 6 152.117 odd 18
2888.1.s.b.333.1 6 19.14 odd 18
2888.1.s.b.333.1 6 152.5 even 18
2888.1.s.b.477.1 6 19.10 odd 18
2888.1.s.b.477.1 6 152.85 even 18
2888.1.s.b.1021.1 6 19.15 odd 18
2888.1.s.b.1021.1 6 152.61 even 18
2888.1.s.b.1029.1 6 19.13 odd 18
2888.1.s.b.1029.1 6 152.101 even 18
2888.1.s.b.2293.1 6 19.2 odd 18
2888.1.s.b.2293.1 6 152.93 even 18
2888.1.s.b.2789.1 6 19.3 odd 18
2888.1.s.b.2789.1 6 152.149 even 18
3800.1.b.a.949.1 2 95.27 even 12
3800.1.b.a.949.1 2 760.277 odd 12
3800.1.b.a.949.2 2 95.8 even 12
3800.1.b.a.949.2 2 760.733 odd 12
3800.1.b.b.949.1 2 95.68 odd 12
3800.1.b.b.949.1 2 760.293 even 12
3800.1.b.b.949.2 2 95.87 odd 12
3800.1.b.b.949.2 2 760.597 even 12
3800.1.o.a.1101.1 1 95.49 even 6
3800.1.o.a.1101.1 1 760.749 odd 6
3800.1.o.b.1101.1 1 95.84 odd 6
3800.1.o.b.1101.1 1 760.429 even 6