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Properties

Label 2888.1.k.b
Level $2888$
Weight $1$
Character orbit 2888.k
Analytic conductor $1.441$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.44129975648\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.69564674215936.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{18}^{3} q^{2} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{3} + \zeta_{18}^{6} q^{4} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{6} + q^{8} + ( -\zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{6} ) q^{9} +O(q^{10})\) \( q -\zeta_{18}^{3} q^{2} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{3} + \zeta_{18}^{6} q^{4} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{6} + q^{8} + ( -\zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{6} ) q^{9} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{11} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{12} -\zeta_{18}^{3} q^{16} + \zeta_{18}^{3} q^{17} + ( 1 + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{18} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{22} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{24} + \zeta_{18}^{6} q^{25} + ( \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{6} - \zeta_{18}^{7} ) q^{27} + \zeta_{18}^{6} q^{32} + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{6} ) q^{33} -\zeta_{18}^{6} q^{34} + ( -\zeta_{18}^{3} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{36} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{41} + \zeta_{18}^{3} q^{43} + ( -\zeta_{18}^{5} - \zeta_{18}^{7} ) q^{44} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{48} + q^{49} + q^{50} + ( -\zeta_{18}^{4} - \zeta_{18}^{8} ) q^{51} + ( 1 - \zeta_{18} - \zeta_{18}^{5} + \zeta_{18}^{6} ) q^{54} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{59} + q^{64} + ( 1 - \zeta_{18}^{3} - \zeta_{18}^{5} - \zeta_{18}^{7} ) q^{66} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{67} - q^{68} + ( -\zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{6} ) q^{72} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{73} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{75} + ( \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{81} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{82} + ( \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{83} -\zeta_{18}^{6} q^{86} + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{88} -\zeta_{18}^{6} q^{89} + ( \zeta_{18}^{2} - \zeta_{18}^{7} ) q^{96} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{97} -\zeta_{18}^{3} q^{98} + ( 1 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - \zeta_{18}^{5} - \zeta_{18}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} + 6 q^{8} - 3 q^{9} + O(q^{10}) \) \( 6 q - 3 q^{2} - 3 q^{4} + 6 q^{8} - 3 q^{9} - 3 q^{16} + 3 q^{17} + 6 q^{18} - 3 q^{25} - 6 q^{27} - 3 q^{32} + 3 q^{33} + 3 q^{34} - 3 q^{36} + 3 q^{43} + 6 q^{49} + 6 q^{50} + 3 q^{54} + 6 q^{64} + 3 q^{66} - 6 q^{68} - 3 q^{72} - 3 q^{81} + 3 q^{86} + 3 q^{89} - 3 q^{98} + 3 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(1445\) \(2167\) \(2529\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{18}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2595.1
0.939693 − 0.342020i
−0.766044 − 0.642788i
−0.173648 + 0.984808i
0.939693 + 0.342020i
−0.766044 + 0.642788i
−0.173648 − 0.984808i
−0.500000 + 0.866025i −0.766044 + 1.32683i −0.500000 − 0.866025i 0 −0.766044 − 1.32683i 0 1.00000 −0.673648 − 1.16679i 0
2595.2 −0.500000 + 0.866025i −0.173648 + 0.300767i −0.500000 − 0.866025i 0 −0.173648 − 0.300767i 0 1.00000 0.439693 + 0.761570i 0
2595.3 −0.500000 + 0.866025i 0.939693 − 1.62760i −0.500000 − 0.866025i 0 0.939693 + 1.62760i 0 1.00000 −1.26604 − 2.19285i 0
2819.1 −0.500000 − 0.866025i −0.766044 − 1.32683i −0.500000 + 0.866025i 0 −0.766044 + 1.32683i 0 1.00000 −0.673648 + 1.16679i 0
2819.2 −0.500000 − 0.866025i −0.173648 − 0.300767i −0.500000 + 0.866025i 0 −0.173648 + 0.300767i 0 1.00000 0.439693 − 0.761570i 0
2819.3 −0.500000 − 0.866025i 0.939693 + 1.62760i −0.500000 + 0.866025i 0 0.939693 − 1.62760i 0 1.00000 −1.26604 + 2.19285i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2819.3
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 1 inner
152.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.k.b 6
8.d odd 2 1 CM 2888.1.k.b 6
19.b odd 2 1 2888.1.k.c 6
19.c even 3 1 2888.1.f.d 3
19.c even 3 1 inner 2888.1.k.b 6
19.d odd 6 1 2888.1.f.c 3
19.d odd 6 1 2888.1.k.c 6
19.e even 9 2 152.1.u.a ✓ 6
19.e even 9 2 2888.1.u.b 6
19.e even 9 2 2888.1.u.g 6
19.f odd 18 2 2888.1.u.a 6
19.f odd 18 2 2888.1.u.e 6
19.f odd 18 2 2888.1.u.f 6
57.l odd 18 2 1368.1.eh.a 6
76.l odd 18 2 608.1.bg.a 6
95.p even 18 2 3800.1.cv.c 6
95.q odd 36 4 3800.1.cq.b 12
152.b even 2 1 2888.1.k.c 6
152.k odd 6 1 2888.1.f.d 3
152.k odd 6 1 inner 2888.1.k.b 6
152.o even 6 1 2888.1.f.c 3
152.o even 6 1 2888.1.k.c 6
152.t even 18 2 608.1.bg.a 6
152.u odd 18 2 152.1.u.a ✓ 6
152.u odd 18 2 2888.1.u.b 6
152.u odd 18 2 2888.1.u.g 6
152.v even 18 2 2888.1.u.a 6
152.v even 18 2 2888.1.u.e 6
152.v even 18 2 2888.1.u.f 6
456.bu even 18 2 1368.1.eh.a 6
760.bz odd 18 2 3800.1.cv.c 6
760.cp even 36 4 3800.1.cq.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a ✓ 6 19.e even 9 2
152.1.u.a ✓ 6 152.u odd 18 2
608.1.bg.a 6 76.l odd 18 2
608.1.bg.a 6 152.t even 18 2
1368.1.eh.a 6 57.l odd 18 2
1368.1.eh.a 6 456.bu even 18 2
2888.1.f.c 3 19.d odd 6 1
2888.1.f.c 3 152.o even 6 1
2888.1.f.d 3 19.c even 3 1
2888.1.f.d 3 152.k odd 6 1
2888.1.k.b 6 1.a even 1 1 trivial
2888.1.k.b 6 8.d odd 2 1 CM
2888.1.k.b 6 19.c even 3 1 inner
2888.1.k.b 6 152.k odd 6 1 inner
2888.1.k.c 6 19.b odd 2 1
2888.1.k.c 6 19.d odd 6 1
2888.1.k.c 6 152.b even 2 1
2888.1.k.c 6 152.o even 6 1
2888.1.u.a 6 19.f odd 18 2
2888.1.u.a 6 152.v even 18 2
2888.1.u.b 6 19.e even 9 2
2888.1.u.b 6 152.u odd 18 2
2888.1.u.e 6 19.f odd 18 2
2888.1.u.e 6 152.v even 18 2
2888.1.u.f 6 19.f odd 18 2
2888.1.u.f 6 152.v even 18 2
2888.1.u.g 6 19.e even 9 2
2888.1.u.g 6 152.u odd 18 2
3800.1.cq.b 12 95.q odd 36 4
3800.1.cq.b 12 760.cp even 36 4
3800.1.cv.c 6 95.p even 18 2
3800.1.cv.c 6 760.bz odd 18 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 3 T_{3}^{4} + 2 T_{3}^{3} + 9 T_{3}^{2} + 3 T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2888, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{3} \)
$3$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( T^{6} \)
$11$ \( ( 1 - 3 T + T^{3} )^{2} \)
$13$ \( T^{6} \)
$17$ \( ( 1 - T + T^{2} )^{3} \)
$19$ \( T^{6} \)
$23$ \( T^{6} \)
$29$ \( T^{6} \)
$31$ \( T^{6} \)
$37$ \( T^{6} \)
$41$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} \)
$43$ \( ( 1 - T + T^{2} )^{3} \)
$47$ \( T^{6} \)
$53$ \( T^{6} \)
$59$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} \)
$61$ \( T^{6} \)
$67$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} \)
$71$ \( T^{6} \)
$73$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} \)
$79$ \( T^{6} \)
$83$ \( ( 1 - 3 T + T^{3} )^{2} \)
$89$ \( ( 1 - T + T^{2} )^{3} \)
$97$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} \)
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