Newspace parameters
| Level: | \( N \) | \(=\) | \( 464 = 2^{4} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 464.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(0.231566165862\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{3}) \) |
| Defining polynomial: |
\( x^{2} - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{6}\) |
| Projective field: | Galois closure of 6.0.53824.1 |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).
| \(n\) | \(117\) | \(175\) | \(321\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 463.1 |
|
0 | −1.73205 | 0 | −1.00000 | 0 | 0 | 0 | 2.00000 | 0 | ||||||||||||||||||||||||
| 463.2 | 0 | 1.73205 | 0 | −1.00000 | 0 | 0 | 0 | 2.00000 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 116.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-29}) \) |
| 4.b | odd | 2 | 1 | inner |
| 29.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 464.1.h.b | ✓ | 2 |
| 4.b | odd | 2 | 1 | inner | 464.1.h.b | ✓ | 2 |
| 8.b | even | 2 | 1 | 1856.1.h.d | 2 | ||
| 8.d | odd | 2 | 1 | 1856.1.h.d | 2 | ||
| 29.b | even | 2 | 1 | inner | 464.1.h.b | ✓ | 2 |
| 116.d | odd | 2 | 1 | CM | 464.1.h.b | ✓ | 2 |
| 232.b | odd | 2 | 1 | 1856.1.h.d | 2 | ||
| 232.g | even | 2 | 1 | 1856.1.h.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 464.1.h.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 464.1.h.b | ✓ | 2 | 4.b | odd | 2 | 1 | inner |
| 464.1.h.b | ✓ | 2 | 29.b | even | 2 | 1 | inner |
| 464.1.h.b | ✓ | 2 | 116.d | odd | 2 | 1 | CM |
| 1856.1.h.d | 2 | 8.b | even | 2 | 1 | ||
| 1856.1.h.d | 2 | 8.d | odd | 2 | 1 | ||
| 1856.1.h.d | 2 | 232.b | odd | 2 | 1 | ||
| 1856.1.h.d | 2 | 232.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 3 \)
acting on \(S_{1}^{\mathrm{new}}(464, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 3 \)
$5$
\( (T + 1)^{2} \)
$7$
\( T^{2} \)
$11$
\( T^{2} - 3 \)
$13$
\( (T - 1)^{2} \)
$17$
\( T^{2} \)
$19$
\( T^{2} \)
$23$
\( T^{2} \)
$29$
\( (T + 1)^{2} \)
$31$
\( T^{2} - 3 \)
$37$
\( T^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} - 3 \)
$47$
\( T^{2} - 3 \)
$53$
\( (T - 1)^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} \)
$67$
\( T^{2} \)
$71$
\( T^{2} \)
$73$
\( T^{2} \)
$79$
\( T^{2} - 3 \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} \)
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