Newspace parameters
| Level: | \( N \) | \(=\) | \( 575 = 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 575.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(92.2206963925\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.7925.1 |
| Defining polynomial: |
\( x^{3} - x^{2} - 13x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 23) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 13x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} - \beta _1 + 17 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.31473 | 27.6631 | −3.75366 | 0 | −147.022 | 11.7843 | 190.021 | 522.249 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0.164504 | 1.43100 | −31.9729 | 0 | 0.235406 | 43.8366 | −10.5238 | −240.952 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 9.15022 | −9.09413 | 51.7266 | 0 | −83.2134 | 226.379 | 180.503 | −160.297 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(23\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 575.6.a.b | 3 | |
| 5.b | even | 2 | 1 | 23.6.a.a | ✓ | 3 | |
| 15.d | odd | 2 | 1 | 207.6.a.b | 3 | ||
| 20.d | odd | 2 | 1 | 368.6.a.e | 3 | ||
| 115.c | odd | 2 | 1 | 529.6.a.a | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.6.a.a | ✓ | 3 | 5.b | even | 2 | 1 | |
| 207.6.a.b | 3 | 15.d | odd | 2 | 1 | ||
| 368.6.a.e | 3 | 20.d | odd | 2 | 1 | ||
| 529.6.a.a | 3 | 115.c | odd | 2 | 1 | ||
| 575.6.a.b | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 4T_{2}^{2} - 48T_{2} + 8 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(575))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - 4 T^{2} - 48 T + 8 \)
$3$
\( T^{3} - 20 T^{2} - 225 T + 360 \)
$5$
\( T^{3} \)
$7$
\( T^{3} - 282 T^{2} + 13108 T - 116944 \)
$11$
\( T^{3} - 136 T^{2} - 43688 T + 840152 \)
$13$
\( T^{3} - 1116 T^{2} + \cdots + 255615102 \)
$17$
\( T^{3} - 896 T^{2} + \cdots - 220718408 \)
$19$
\( T^{3} - 1654 T^{2} + \cdots + 460771768 \)
$23$
\( (T - 529)^{3} \)
$29$
\( T^{3} + 844 T^{2} + \cdots - 33789223458 \)
$31$
\( T^{3} + 3020 T^{2} + \cdots - 117638912880 \)
$37$
\( T^{3} + 8938 T^{2} + \cdots - 1048082031344 \)
$41$
\( T^{3} + 12792 T^{2} + \cdots - 1564944049486 \)
$43$
\( T^{3} - 16730 T^{2} + \cdots + 95315904000 \)
$47$
\( T^{3} + 22500 T^{2} + \cdots - 916008439440 \)
$53$
\( T^{3} + 17108 T^{2} + \cdots - 7849670295504 \)
$59$
\( T^{3} - 54176 T^{2} + \cdots - 1725012447168 \)
$61$
\( T^{3} + 71324 T^{2} + \cdots + 11439907465152 \)
$67$
\( T^{3} - 62960 T^{2} + \cdots + 11971711378840 \)
$71$
\( T^{3} - 98400 T^{2} + \cdots - 24837760695040 \)
$73$
\( T^{3} - 81772 T^{2} + \cdots - 7199078503954 \)
$79$
\( T^{3} + \cdots + 235690469012368 \)
$83$
\( T^{3} + \cdots + 297029282761704 \)
$89$
\( T^{3} + \cdots + 125799322340896 \)
$97$
\( T^{3} + \cdots - 693755159518744 \)
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