Properties

Label 2888.1.u.c
Level $2888$
Weight $1$
Character orbit 2888.u
Analytic conductor $1.441$
Analytic rank $0$
Dimension $6$
Projective image $D_{3}$
CM discriminant -8
Inner twists $12$

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Newspace parameters

Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2888.u (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.44129975648\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
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 3.1.2888.1
Artin image: $C_9\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{54} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{18}^{5} q^{2} -\zeta_{18}^{2} q^{3} -\zeta_{18} q^{4} + \zeta_{18}^{7} q^{6} + \zeta_{18}^{6} q^{8} +O(q^{10})\) \( q -\zeta_{18}^{5} q^{2} -\zeta_{18}^{2} q^{3} -\zeta_{18} q^{4} + \zeta_{18}^{7} q^{6} + \zeta_{18}^{6} q^{8} -\zeta_{18}^{6} q^{11} + \zeta_{18}^{3} q^{12} + \zeta_{18}^{2} q^{16} -2 \zeta_{18}^{5} q^{17} -\zeta_{18}^{2} q^{22} -\zeta_{18}^{8} q^{24} -\zeta_{18}^{7} q^{25} + \zeta_{18}^{6} q^{27} -\zeta_{18}^{7} q^{32} + \zeta_{18}^{8} q^{33} -2 \zeta_{18} q^{34} -\zeta_{18}^{2} q^{41} + 2 \zeta_{18}^{8} q^{43} + \zeta_{18}^{7} q^{44} -\zeta_{18}^{4} q^{48} + \zeta_{18}^{6} q^{49} -\zeta_{18}^{3} q^{50} + 2 \zeta_{18}^{7} q^{51} + \zeta_{18}^{2} q^{54} + \zeta_{18}^{5} q^{59} -\zeta_{18}^{3} q^{64} + \zeta_{18}^{4} q^{66} -\zeta_{18}^{4} q^{67} + 2 \zeta_{18}^{6} q^{68} -\zeta_{18}^{2} q^{73} - q^{75} -\zeta_{18}^{8} q^{81} + \zeta_{18}^{7} q^{82} + \zeta_{18}^{3} q^{83} + 2 \zeta_{18}^{4} q^{86} + \zeta_{18}^{3} q^{88} -2 \zeta_{18}^{7} q^{89} - q^{96} + \zeta_{18}^{5} q^{97} + \zeta_{18}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{8} + O(q^{10}) \) \( 6 q - 3 q^{8} + 3 q^{11} + 3 q^{12} - 3 q^{27} - 3 q^{49} - 3 q^{50} - 3 q^{64} - 6 q^{68} - 6 q^{75} + 3 q^{83} + 3 q^{88} - 6 q^{96} + O(q^{100}) \)

Character values

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

\(n\) \(1445\) \(2167\) \(2529\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{18}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−0.766044 0.642788i
−0.173648 + 0.984808i
−0.173648 0.984808i
−0.766044 + 0.642788i
0.939693 + 0.342020i
0.939693 0.342020i
−0.939693 0.342020i −0.173648 0.984808i 0.766044 + 0.642788i 0 −0.173648 + 0.984808i 0 −0.500000 0.866025i 0 0
595.1 0.766044 0.642788i 0.939693 + 0.342020i 0.173648 0.984808i 0 0.939693 0.342020i 0 −0.500000 0.866025i 0 0
1859.1 0.766044 + 0.642788i 0.939693 0.342020i 0.173648 + 0.984808i 0 0.939693 + 0.342020i 0 −0.500000 + 0.866025i 0 0
1867.1 −0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 0.642788i 0 −0.173648 0.984808i 0 −0.500000 + 0.866025i 0 0
2411.1 0.173648 0.984808i −0.766044 0.642788i −0.939693 0.342020i 0 −0.766044 + 0.642788i 0 −0.500000 + 0.866025i 0 0
2555.1 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i 0 −0.766044 0.642788i 0 −0.500000 0.866025i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2555.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
19.c even 3 2 inner
19.e even 9 3 inner
152.k odd 6 2 inner
152.u odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.u.c 6
8.d odd 2 1 CM 2888.1.u.c 6
19.b odd 2 1 2888.1.u.d 6
19.c even 3 2 inner 2888.1.u.c 6
19.d odd 6 2 2888.1.u.d 6
19.e even 9 2 152.1.k.a 2
19.e even 9 1 2888.1.f.b 1
19.e even 9 3 inner 2888.1.u.c 6
19.f odd 18 1 2888.1.f.a 1
19.f odd 18 2 2888.1.k.a 2
19.f odd 18 3 2888.1.u.d 6
57.l odd 18 2 1368.1.bz.a 2
76.l odd 18 2 608.1.o.a 2
95.p even 18 2 3800.1.bd.c 2
95.q odd 36 4 3800.1.bn.b 4
152.b even 2 1 2888.1.u.d 6
152.k odd 6 2 inner 2888.1.u.c 6
152.o even 6 2 2888.1.u.d 6
152.t even 18 2 608.1.o.a 2
152.u odd 18 2 152.1.k.a 2
152.u odd 18 1 2888.1.f.b 1
152.u odd 18 3 inner 2888.1.u.c 6
152.v even 18 1 2888.1.f.a 1
152.v even 18 2 2888.1.k.a 2
152.v even 18 3 2888.1.u.d 6
456.bu even 18 2 1368.1.bz.a 2
760.bz odd 18 2 3800.1.bd.c 2
760.cp even 36 4 3800.1.bn.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.k.a 2 19.e even 9 2
152.1.k.a 2 152.u odd 18 2
608.1.o.a 2 76.l odd 18 2
608.1.o.a 2 152.t even 18 2
1368.1.bz.a 2 57.l odd 18 2
1368.1.bz.a 2 456.bu even 18 2
2888.1.f.a 1 19.f odd 18 1
2888.1.f.a 1 152.v even 18 1
2888.1.f.b 1 19.e even 9 1
2888.1.f.b 1 152.u odd 18 1
2888.1.k.a 2 19.f odd 18 2
2888.1.k.a 2 152.v even 18 2
2888.1.u.c 6 1.a even 1 1 trivial
2888.1.u.c 6 8.d odd 2 1 CM
2888.1.u.c 6 19.c even 3 2 inner
2888.1.u.c 6 19.e even 9 3 inner
2888.1.u.c 6 152.k odd 6 2 inner
2888.1.u.c 6 152.u odd 18 3 inner
2888.1.u.d 6 19.b odd 2 1
2888.1.u.d 6 19.d odd 6 2
2888.1.u.d 6 19.f odd 18 3
2888.1.u.d 6 152.b even 2 1
2888.1.u.d 6 152.o even 6 2
2888.1.u.d 6 152.v even 18 3
3800.1.bd.c 2 95.p even 18 2
3800.1.bd.c 2 760.bz odd 18 2
3800.1.bn.b 4 95.q odd 36 4
3800.1.bn.b 4 760.cp even 36 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - T_{3}^{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2888, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{3} + T^{6} \)
$3$ \( 1 - T^{3} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( T^{6} \)
$11$ \( ( 1 - T + T^{2} )^{3} \)
$13$ \( T^{6} \)
$17$ \( 64 + 8 T^{3} + T^{6} \)
$19$ \( T^{6} \)
$23$ \( T^{6} \)
$29$ \( T^{6} \)
$31$ \( T^{6} \)
$37$ \( T^{6} \)
$41$ \( 1 - T^{3} + T^{6} \)
$43$ \( 64 + 8 T^{3} + T^{6} \)
$47$ \( T^{6} \)
$53$ \( T^{6} \)
$59$ \( 1 - T^{3} + T^{6} \)
$61$ \( T^{6} \)
$67$ \( 1 - T^{3} + T^{6} \)
$71$ \( T^{6} \)
$73$ \( 1 - T^{3} + T^{6} \)
$79$ \( T^{6} \)
$83$ \( ( 1 - T + T^{2} )^{3} \)
$89$ \( 64 + 8 T^{3} + T^{6} \)
$97$ \( 1 - T^{3} + T^{6} \)
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