Newspace parameters
Level: | \( N \) | \(=\) | \( 1444 = 2^{2} \cdot 19^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1444.e (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(11.5303980519\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
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) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
429.1 |
|
0 | 1.00000 | − | 1.73205i | 0 | 0.500000 | − | 0.866025i | 0 | −3.00000 | 0 | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||
653.1 | 0 | 1.00000 | + | 1.73205i | 0 | 0.500000 | + | 0.866025i | 0 | −3.00000 | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1444.2.e.c | 2 | |
19.b | odd | 2 | 1 | 1444.2.e.a | 2 | ||
19.c | even | 3 | 1 | 1444.2.a.a | 1 | ||
19.c | even | 3 | 1 | inner | 1444.2.e.c | 2 | |
19.d | odd | 6 | 1 | 76.2.a.a | ✓ | 1 | |
19.d | odd | 6 | 1 | 1444.2.e.a | 2 | ||
57.f | even | 6 | 1 | 684.2.a.b | 1 | ||
76.f | even | 6 | 1 | 304.2.a.a | 1 | ||
76.g | odd | 6 | 1 | 5776.2.a.p | 1 | ||
95.h | odd | 6 | 1 | 1900.2.a.b | 1 | ||
95.l | even | 12 | 2 | 1900.2.c.b | 2 | ||
133.p | even | 6 | 1 | 3724.2.a.a | 1 | ||
152.l | odd | 6 | 1 | 1216.2.a.c | 1 | ||
152.o | even | 6 | 1 | 1216.2.a.q | 1 | ||
209.g | even | 6 | 1 | 9196.2.a.f | 1 | ||
228.n | odd | 6 | 1 | 2736.2.a.q | 1 | ||
380.s | even | 6 | 1 | 7600.2.a.p | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
76.2.a.a | ✓ | 1 | 19.d | odd | 6 | 1 | |
304.2.a.a | 1 | 76.f | even | 6 | 1 | ||
684.2.a.b | 1 | 57.f | even | 6 | 1 | ||
1216.2.a.c | 1 | 152.l | odd | 6 | 1 | ||
1216.2.a.q | 1 | 152.o | even | 6 | 1 | ||
1444.2.a.a | 1 | 19.c | even | 3 | 1 | ||
1444.2.e.a | 2 | 19.b | odd | 2 | 1 | ||
1444.2.e.a | 2 | 19.d | odd | 6 | 1 | ||
1444.2.e.c | 2 | 1.a | even | 1 | 1 | trivial | |
1444.2.e.c | 2 | 19.c | even | 3 | 1 | inner | |
1900.2.a.b | 1 | 95.h | odd | 6 | 1 | ||
1900.2.c.b | 2 | 95.l | even | 12 | 2 | ||
2736.2.a.q | 1 | 228.n | odd | 6 | 1 | ||
3724.2.a.a | 1 | 133.p | even | 6 | 1 | ||
5776.2.a.p | 1 | 76.g | odd | 6 | 1 | ||
7600.2.a.p | 1 | 380.s | even | 6 | 1 | ||
9196.2.a.f | 1 | 209.g | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 2T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(1444, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 2T + 4 \)
$5$
\( T^{2} - T + 1 \)
$7$
\( (T + 3)^{2} \)
$11$
\( (T - 5)^{2} \)
$13$
\( T^{2} + 4T + 16 \)
$17$
\( T^{2} - 3T + 9 \)
$19$
\( T^{2} \)
$23$
\( T^{2} + 8T + 64 \)
$29$
\( T^{2} + 2T + 4 \)
$31$
\( (T + 4)^{2} \)
$37$
\( (T + 10)^{2} \)
$41$
\( T^{2} - 10T + 100 \)
$43$
\( T^{2} + T + 1 \)
$47$
\( T^{2} - T + 1 \)
$53$
\( T^{2} + 4T + 16 \)
$59$
\( T^{2} - 6T + 36 \)
$61$
\( T^{2} - 13T + 169 \)
$67$
\( T^{2} + 12T + 144 \)
$71$
\( T^{2} - 2T + 4 \)
$73$
\( T^{2} + 9T + 81 \)
$79$
\( T^{2} - 8T + 64 \)
$83$
\( (T + 12)^{2} \)
$89$
\( T^{2} - 12T + 144 \)
$97$
\( T^{2} + 8T + 64 \)
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