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.
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 |
|
−0.500000 | + | 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 | |||||||||||||||||||||||||||
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
\( T_{3}^{6} + 3T_{3}^{4} + 2T_{3}^{3} + 9T_{3}^{2} + 3T_{3} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2888, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + T + 1)^{3} \)
$3$
\( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \)
$5$
\( T^{6} \)
$7$
\( T^{6} \)
$11$
\( (T^{3} - 3 T + 1)^{2} \)
$13$
\( T^{6} \)
$17$
\( (T^{2} - T + 1)^{3} \)
$19$
\( T^{6} \)
$23$
\( T^{6} \)
$29$
\( T^{6} \)
$31$
\( T^{6} \)
$37$
\( T^{6} \)
$41$
\( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \)
$43$
\( (T^{2} - T + 1)^{3} \)
$47$
\( T^{6} \)
$53$
\( T^{6} \)
$59$
\( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \)
$61$
\( T^{6} \)
$67$
\( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \)
$71$
\( T^{6} \)
$73$
\( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \)
$79$
\( T^{6} \)
$83$
\( (T^{3} - 3 T + 1)^{2} \)
$89$
\( (T^{2} - T + 1)^{3} \)
$97$
\( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \)
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