Newspace parameters
Level: | \( N \) | \(=\) | \( 1444 = 2^{2} \cdot 19^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1444.h (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.720649878242\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\zeta_{6})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{2} - x + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 76) |
Projective image: | \(D_{3}\) |
Projective field: | Galois closure of 3.1.76.1 |
Artin image: | $C_3\times S_3$ |
Artin field: | Galois closure of 6.0.39617584.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times\).
\(n\) | \(723\) | \(1085\) |
\(\chi(n)\) | \(1\) | \(\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
69.1 |
|
0 | 0 | 0 | 0.500000 | + | 0.866025i | 0 | −1.00000 | 0 | −0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||
293.1 | 0 | 0 | 0 | 0.500000 | − | 0.866025i | 0 | −1.00000 | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-19}) \) |
19.c | even | 3 | 1 | inner |
19.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1444.1.h.a | 2 | |
19.b | odd | 2 | 1 | CM | 1444.1.h.a | 2 | |
19.c | even | 3 | 1 | 76.1.c.a | ✓ | 1 | |
19.c | even | 3 | 1 | inner | 1444.1.h.a | 2 | |
19.d | odd | 6 | 1 | 76.1.c.a | ✓ | 1 | |
19.d | odd | 6 | 1 | inner | 1444.1.h.a | 2 | |
19.e | even | 9 | 6 | 1444.1.j.a | 6 | ||
19.f | odd | 18 | 6 | 1444.1.j.a | 6 | ||
57.f | even | 6 | 1 | 684.1.h.a | 1 | ||
57.h | odd | 6 | 1 | 684.1.h.a | 1 | ||
76.f | even | 6 | 1 | 304.1.e.a | 1 | ||
76.g | odd | 6 | 1 | 304.1.e.a | 1 | ||
95.h | odd | 6 | 1 | 1900.1.e.a | 1 | ||
95.i | even | 6 | 1 | 1900.1.e.a | 1 | ||
95.l | even | 12 | 2 | 1900.1.g.a | 2 | ||
95.m | odd | 12 | 2 | 1900.1.g.a | 2 | ||
133.g | even | 3 | 1 | 3724.1.bc.c | 2 | ||
133.h | even | 3 | 1 | 3724.1.bc.c | 2 | ||
133.i | even | 6 | 1 | 3724.1.bc.b | 2 | ||
133.j | odd | 6 | 1 | 3724.1.bc.c | 2 | ||
133.k | odd | 6 | 1 | 3724.1.bc.b | 2 | ||
133.m | odd | 6 | 1 | 3724.1.e.c | 1 | ||
133.n | odd | 6 | 1 | 3724.1.bc.c | 2 | ||
133.p | even | 6 | 1 | 3724.1.e.c | 1 | ||
133.s | even | 6 | 1 | 3724.1.bc.b | 2 | ||
133.t | odd | 6 | 1 | 3724.1.bc.b | 2 | ||
152.k | odd | 6 | 1 | 1216.1.e.b | 1 | ||
152.l | odd | 6 | 1 | 1216.1.e.a | 1 | ||
152.o | even | 6 | 1 | 1216.1.e.b | 1 | ||
152.p | even | 6 | 1 | 1216.1.e.a | 1 | ||
228.m | even | 6 | 1 | 2736.1.o.b | 1 | ||
228.n | odd | 6 | 1 | 2736.1.o.b | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
76.1.c.a | ✓ | 1 | 19.c | even | 3 | 1 | |
76.1.c.a | ✓ | 1 | 19.d | odd | 6 | 1 | |
304.1.e.a | 1 | 76.f | even | 6 | 1 | ||
304.1.e.a | 1 | 76.g | odd | 6 | 1 | ||
684.1.h.a | 1 | 57.f | even | 6 | 1 | ||
684.1.h.a | 1 | 57.h | odd | 6 | 1 | ||
1216.1.e.a | 1 | 152.l | odd | 6 | 1 | ||
1216.1.e.a | 1 | 152.p | even | 6 | 1 | ||
1216.1.e.b | 1 | 152.k | odd | 6 | 1 | ||
1216.1.e.b | 1 | 152.o | even | 6 | 1 | ||
1444.1.h.a | 2 | 1.a | even | 1 | 1 | trivial | |
1444.1.h.a | 2 | 19.b | odd | 2 | 1 | CM | |
1444.1.h.a | 2 | 19.c | even | 3 | 1 | inner | |
1444.1.h.a | 2 | 19.d | odd | 6 | 1 | inner | |
1444.1.j.a | 6 | 19.e | even | 9 | 6 | ||
1444.1.j.a | 6 | 19.f | odd | 18 | 6 | ||
1900.1.e.a | 1 | 95.h | odd | 6 | 1 | ||
1900.1.e.a | 1 | 95.i | even | 6 | 1 | ||
1900.1.g.a | 2 | 95.l | even | 12 | 2 | ||
1900.1.g.a | 2 | 95.m | odd | 12 | 2 | ||
2736.1.o.b | 1 | 228.m | even | 6 | 1 | ||
2736.1.o.b | 1 | 228.n | odd | 6 | 1 | ||
3724.1.e.c | 1 | 133.m | odd | 6 | 1 | ||
3724.1.e.c | 1 | 133.p | even | 6 | 1 | ||
3724.1.bc.b | 2 | 133.i | even | 6 | 1 | ||
3724.1.bc.b | 2 | 133.k | odd | 6 | 1 | ||
3724.1.bc.b | 2 | 133.s | even | 6 | 1 | ||
3724.1.bc.b | 2 | 133.t | odd | 6 | 1 | ||
3724.1.bc.c | 2 | 133.g | even | 3 | 1 | ||
3724.1.bc.c | 2 | 133.h | even | 3 | 1 | ||
3724.1.bc.c | 2 | 133.j | odd | 6 | 1 | ||
3724.1.bc.c | 2 | 133.n | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{1}^{\mathrm{new}}(1444, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} - T + 1 \)
$7$
\( (T + 1)^{2} \)
$11$
\( (T + 1)^{2} \)
$13$
\( T^{2} \)
$17$
\( T^{2} - T + 1 \)
$19$
\( T^{2} \)
$23$
\( T^{2} + 2T + 4 \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} - T + 1 \)
$47$
\( T^{2} - T + 1 \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} - T + 1 \)
$67$
\( T^{2} \)
$71$
\( T^{2} \)
$73$
\( T^{2} - T + 1 \)
$79$
\( T^{2} \)
$83$
\( (T - 2)^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} \)
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