Newspace parameters
Level: | \( N \) | \(=\) | \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 468.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.73699881460\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 156) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).
\(n\) | \(145\) | \(209\) | \(235\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
181.1 |
|
0 | 0 | 0 | − | 3.46410i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||
181.2 | 0 | 0 | 0 | 3.46410i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 468.2.b.a | 2 | |
3.b | odd | 2 | 1 | 156.2.b.a | ✓ | 2 | |
4.b | odd | 2 | 1 | 1872.2.c.c | 2 | ||
12.b | even | 2 | 1 | 624.2.c.f | 2 | ||
13.b | even | 2 | 1 | inner | 468.2.b.a | 2 | |
13.d | odd | 4 | 2 | 6084.2.a.v | 2 | ||
15.d | odd | 2 | 1 | 3900.2.c.c | 2 | ||
15.e | even | 4 | 2 | 3900.2.j.h | 4 | ||
21.c | even | 2 | 1 | 7644.2.e.g | 2 | ||
24.f | even | 2 | 1 | 2496.2.c.e | 2 | ||
24.h | odd | 2 | 1 | 2496.2.c.l | 2 | ||
39.d | odd | 2 | 1 | 156.2.b.a | ✓ | 2 | |
39.f | even | 4 | 2 | 2028.2.a.g | 2 | ||
39.h | odd | 6 | 1 | 2028.2.q.b | 2 | ||
39.h | odd | 6 | 1 | 2028.2.q.c | 2 | ||
39.i | odd | 6 | 1 | 2028.2.q.b | 2 | ||
39.i | odd | 6 | 1 | 2028.2.q.c | 2 | ||
39.k | even | 12 | 4 | 2028.2.i.i | 4 | ||
52.b | odd | 2 | 1 | 1872.2.c.c | 2 | ||
156.h | even | 2 | 1 | 624.2.c.f | 2 | ||
156.l | odd | 4 | 2 | 8112.2.a.bs | 2 | ||
195.e | odd | 2 | 1 | 3900.2.c.c | 2 | ||
195.s | even | 4 | 2 | 3900.2.j.h | 4 | ||
273.g | even | 2 | 1 | 7644.2.e.g | 2 | ||
312.b | odd | 2 | 1 | 2496.2.c.l | 2 | ||
312.h | even | 2 | 1 | 2496.2.c.e | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
156.2.b.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
156.2.b.a | ✓ | 2 | 39.d | odd | 2 | 1 | |
468.2.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
468.2.b.a | 2 | 13.b | even | 2 | 1 | inner | |
624.2.c.f | 2 | 12.b | even | 2 | 1 | ||
624.2.c.f | 2 | 156.h | even | 2 | 1 | ||
1872.2.c.c | 2 | 4.b | odd | 2 | 1 | ||
1872.2.c.c | 2 | 52.b | odd | 2 | 1 | ||
2028.2.a.g | 2 | 39.f | even | 4 | 2 | ||
2028.2.i.i | 4 | 39.k | even | 12 | 4 | ||
2028.2.q.b | 2 | 39.h | odd | 6 | 1 | ||
2028.2.q.b | 2 | 39.i | odd | 6 | 1 | ||
2028.2.q.c | 2 | 39.h | odd | 6 | 1 | ||
2028.2.q.c | 2 | 39.i | odd | 6 | 1 | ||
2496.2.c.e | 2 | 24.f | even | 2 | 1 | ||
2496.2.c.e | 2 | 312.h | even | 2 | 1 | ||
2496.2.c.l | 2 | 24.h | odd | 2 | 1 | ||
2496.2.c.l | 2 | 312.b | odd | 2 | 1 | ||
3900.2.c.c | 2 | 15.d | odd | 2 | 1 | ||
3900.2.c.c | 2 | 195.e | odd | 2 | 1 | ||
3900.2.j.h | 4 | 15.e | even | 4 | 2 | ||
3900.2.j.h | 4 | 195.s | even | 4 | 2 | ||
6084.2.a.v | 2 | 13.d | odd | 4 | 2 | ||
7644.2.e.g | 2 | 21.c | even | 2 | 1 | ||
7644.2.e.g | 2 | 273.g | even | 2 | 1 | ||
8112.2.a.bs | 2 | 156.l | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 12 \)
acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} + 12 \)
$7$
\( T^{2} \)
$11$
\( T^{2} + 12 \)
$13$
\( T^{2} + 2T + 13 \)
$17$
\( (T + 6)^{2} \)
$19$
\( T^{2} + 48 \)
$23$
\( T^{2} \)
$29$
\( (T - 6)^{2} \)
$31$
\( T^{2} + 48 \)
$37$
\( T^{2} \)
$41$
\( T^{2} + 12 \)
$43$
\( (T - 8)^{2} \)
$47$
\( T^{2} + 12 \)
$53$
\( (T + 6)^{2} \)
$59$
\( T^{2} + 12 \)
$61$
\( (T - 10)^{2} \)
$67$
\( T^{2} + 192 \)
$71$
\( T^{2} + 108 \)
$73$
\( T^{2} + 48 \)
$79$
\( (T + 8)^{2} \)
$83$
\( T^{2} + 12 \)
$89$
\( T^{2} + 300 \)
$97$
\( T^{2} + 48 \)
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