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: | \(2\) |
Coefficient field: | \(\Q(\zeta_{6})\) |
Defining polynomial: |
\( x^{2} - x + 1 \)
|
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: | $C_6\times S_3$ |
Artin field: | Galois closure of 12.0.4452139149819904.6 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times\).
\(n\) | \(1445\) | \(2167\) | \(2529\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-\zeta_{6}\) |
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 |
|
0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0 | −1.00000 | 0 | 0 | ||||||||||||||||
2819.1 | 0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0 | −1.00000 | 0 | 0 | |||||||||||||||||
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.a | 2 | |
8.d | odd | 2 | 1 | CM | 2888.1.k.a | 2 | |
19.b | odd | 2 | 1 | 152.1.k.a | ✓ | 2 | |
19.c | even | 3 | 1 | 2888.1.f.a | 1 | ||
19.c | even | 3 | 1 | inner | 2888.1.k.a | 2 | |
19.d | odd | 6 | 1 | 152.1.k.a | ✓ | 2 | |
19.d | odd | 6 | 1 | 2888.1.f.b | 1 | ||
19.e | even | 9 | 6 | 2888.1.u.d | 6 | ||
19.f | odd | 18 | 6 | 2888.1.u.c | 6 | ||
57.d | even | 2 | 1 | 1368.1.bz.a | 2 | ||
57.f | even | 6 | 1 | 1368.1.bz.a | 2 | ||
76.d | even | 2 | 1 | 608.1.o.a | 2 | ||
76.f | even | 6 | 1 | 608.1.o.a | 2 | ||
95.d | odd | 2 | 1 | 3800.1.bd.c | 2 | ||
95.g | even | 4 | 2 | 3800.1.bn.b | 4 | ||
95.h | odd | 6 | 1 | 3800.1.bd.c | 2 | ||
95.l | even | 12 | 2 | 3800.1.bn.b | 4 | ||
152.b | even | 2 | 1 | 152.1.k.a | ✓ | 2 | |
152.g | odd | 2 | 1 | 608.1.o.a | 2 | ||
152.k | odd | 6 | 1 | 2888.1.f.a | 1 | ||
152.k | odd | 6 | 1 | inner | 2888.1.k.a | 2 | |
152.l | odd | 6 | 1 | 608.1.o.a | 2 | ||
152.o | even | 6 | 1 | 152.1.k.a | ✓ | 2 | |
152.o | even | 6 | 1 | 2888.1.f.b | 1 | ||
152.u | odd | 18 | 6 | 2888.1.u.d | 6 | ||
152.v | even | 18 | 6 | 2888.1.u.c | 6 | ||
456.l | odd | 2 | 1 | 1368.1.bz.a | 2 | ||
456.s | odd | 6 | 1 | 1368.1.bz.a | 2 | ||
760.p | even | 2 | 1 | 3800.1.bd.c | 2 | ||
760.y | odd | 4 | 2 | 3800.1.bn.b | 4 | ||
760.bf | even | 6 | 1 | 3800.1.bd.c | 2 | ||
760.bu | odd | 12 | 2 | 3800.1.bn.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
152.1.k.a | ✓ | 2 | 19.b | odd | 2 | 1 | |
152.1.k.a | ✓ | 2 | 19.d | odd | 6 | 1 | |
152.1.k.a | ✓ | 2 | 152.b | even | 2 | 1 | |
152.1.k.a | ✓ | 2 | 152.o | even | 6 | 1 | |
608.1.o.a | 2 | 76.d | even | 2 | 1 | ||
608.1.o.a | 2 | 76.f | even | 6 | 1 | ||
608.1.o.a | 2 | 152.g | odd | 2 | 1 | ||
608.1.o.a | 2 | 152.l | odd | 6 | 1 | ||
1368.1.bz.a | 2 | 57.d | even | 2 | 1 | ||
1368.1.bz.a | 2 | 57.f | even | 6 | 1 | ||
1368.1.bz.a | 2 | 456.l | odd | 2 | 1 | ||
1368.1.bz.a | 2 | 456.s | odd | 6 | 1 | ||
2888.1.f.a | 1 | 19.c | even | 3 | 1 | ||
2888.1.f.a | 1 | 152.k | odd | 6 | 1 | ||
2888.1.f.b | 1 | 19.d | odd | 6 | 1 | ||
2888.1.f.b | 1 | 152.o | even | 6 | 1 | ||
2888.1.k.a | 2 | 1.a | even | 1 | 1 | trivial | |
2888.1.k.a | 2 | 8.d | odd | 2 | 1 | CM | |
2888.1.k.a | 2 | 19.c | even | 3 | 1 | inner | |
2888.1.k.a | 2 | 152.k | odd | 6 | 1 | inner | |
2888.1.u.c | 6 | 19.f | odd | 18 | 6 | ||
2888.1.u.c | 6 | 152.v | even | 18 | 6 | ||
2888.1.u.d | 6 | 19.e | even | 9 | 6 | ||
2888.1.u.d | 6 | 152.u | odd | 18 | 6 | ||
3800.1.bd.c | 2 | 95.d | odd | 2 | 1 | ||
3800.1.bd.c | 2 | 95.h | odd | 6 | 1 | ||
3800.1.bd.c | 2 | 760.p | even | 2 | 1 | ||
3800.1.bd.c | 2 | 760.bf | even | 6 | 1 | ||
3800.1.bn.b | 4 | 95.g | even | 4 | 2 | ||
3800.1.bn.b | 4 | 95.l | even | 12 | 2 | ||
3800.1.bn.b | 4 | 760.y | odd | 4 | 2 | ||
3800.1.bn.b | 4 | 760.bu | odd | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + T_{3} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2888, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - T + 1 \)
$3$
\( T^{2} + T + 1 \)
$5$
\( T^{2} \)
$7$
\( T^{2} \)
$11$
\( (T + 1)^{2} \)
$13$
\( T^{2} \)
$17$
\( T^{2} + 2T + 4 \)
$19$
\( T^{2} \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} \)
$41$
\( T^{2} + T + 1 \)
$43$
\( T^{2} + 2T + 4 \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} + T + 1 \)
$61$
\( T^{2} \)
$67$
\( T^{2} + T + 1 \)
$71$
\( T^{2} \)
$73$
\( T^{2} - T + 1 \)
$79$
\( T^{2} \)
$83$
\( (T + 1)^{2} \)
$89$
\( T^{2} - 2T + 4 \)
$97$
\( T^{2} + T + 1 \)
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