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Properties

Label 2888.1.f.b
Level $2888$
Weight $1$
Character orbit 2888.f
Self dual yes
Analytic conductor $1.441$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -8
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2888.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.44129975648\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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: $S_3$
Artin field: Galois closure of 3.1.2888.1

$q$-expansion

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

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(1445\) \(2167\) \(2529\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
723.1
0
1.00000 −1.00000 1.00000 0 −1.00000 0 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.f.b 1
8.d odd 2 1 CM 2888.1.f.b 1
19.b odd 2 1 2888.1.f.a 1
19.c even 3 2 152.1.k.a ✓ 2
19.d odd 6 2 2888.1.k.a 2
19.e even 9 6 2888.1.u.c 6
19.f odd 18 6 2888.1.u.d 6
57.h odd 6 2 1368.1.bz.a 2
76.g odd 6 2 608.1.o.a 2
95.i even 6 2 3800.1.bd.c 2
95.m odd 12 4 3800.1.bn.b 4
152.b even 2 1 2888.1.f.a 1
152.k odd 6 2 152.1.k.a ✓ 2
152.o even 6 2 2888.1.k.a 2
152.p even 6 2 608.1.o.a 2
152.u odd 18 6 2888.1.u.c 6
152.v even 18 6 2888.1.u.d 6
456.u even 6 2 1368.1.bz.a 2
760.bm odd 6 2 3800.1.bd.c 2
760.bw even 12 4 3800.1.bn.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.k.a ✓ 2 19.c even 3 2
152.1.k.a ✓ 2 152.k odd 6 2
608.1.o.a 2 76.g odd 6 2
608.1.o.a 2 152.p even 6 2
1368.1.bz.a 2 57.h odd 6 2
1368.1.bz.a 2 456.u even 6 2
2888.1.f.a 1 19.b odd 2 1
2888.1.f.a 1 152.b even 2 1
2888.1.f.b 1 1.a even 1 1 trivial
2888.1.f.b 1 8.d odd 2 1 CM
2888.1.k.a 2 19.d odd 6 2
2888.1.k.a 2 152.o even 6 2
2888.1.u.c 6 19.e even 9 6
2888.1.u.c 6 152.u odd 18 6
2888.1.u.d 6 19.f odd 18 6
2888.1.u.d 6 152.v even 18 6
3800.1.bd.c 2 95.i even 6 2
3800.1.bd.c 2 760.bm odd 6 2
3800.1.bn.b 4 95.m odd 12 4
3800.1.bn.b 4 760.bw even 12 4

Hecke kernels

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

Hecke characteristic polynomials

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