Newspace parameters
| Level: | \( N \) | \(=\) | \( 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 23.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.68882785570\) |
| Analytic rank: | \(1\) |
| 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: | yes |
| 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 |
|
−9.15022 | 9.09413 | 51.7266 | 3.86330 | −83.2134 | −226.379 | −180.503 | −160.297 | −35.3501 | |||||||||||||||||||||||||||
| 1.2 | −0.164504 | −1.43100 | −31.9729 | −37.9865 | 0.235406 | −43.8366 | 10.5238 | −240.952 | 6.24894 | ||||||||||||||||||||||||||||
| 1.3 | 5.31473 | −27.6631 | −3.75366 | −23.8768 | −147.022 | −11.7843 | −190.021 | 522.249 | −126.899 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(23\) | \(1\) |
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 | 23.6.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 207.6.a.b | 3 | ||
| 4.b | odd | 2 | 1 | 368.6.a.e | 3 | ||
| 5.b | even | 2 | 1 | 575.6.a.b | 3 | ||
| 23.b | 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 | 1.a | even | 1 | 1 | trivial |
| 207.6.a.b | 3 | 3.b | odd | 2 | 1 | ||
| 368.6.a.e | 3 | 4.b | odd | 2 | 1 | ||
| 529.6.a.a | 3 | 23.b | odd | 2 | 1 | ||
| 575.6.a.b | 3 | 5.b | even | 2 | 1 | ||
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(23))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} + 4 T^{2} - 48 T - 8 \)
$3$
\( T^{3} + 20 T^{2} - 225 T - 360 \)
$5$
\( T^{3} + 58 T^{2} + 668 T - 3504 \)
$7$
\( T^{3} + 282 T^{2} + 13108 T + 116944 \)
$11$
\( T^{3} - 136 T^{2} - 43688 T + 840152 \)
$13$
\( T^{3} + 1116 T^{2} + \cdots - 255615102 \)
$17$
\( T^{3} + 896 T^{2} + \cdots + 220718408 \)
$19$
\( T^{3} - 1654 T^{2} + \cdots + 460771768 \)
$23$
\( (T + 529)^{3} \)
$29$
\( T^{3} + 844 T^{2} + \cdots - 33789223458 \)
$31$
\( T^{3} + 3020 T^{2} + \cdots - 117638912880 \)
$37$
\( T^{3} - 8938 T^{2} + \cdots + 1048082031344 \)
$41$
\( T^{3} + 12792 T^{2} + \cdots - 1564944049486 \)
$43$
\( T^{3} + 16730 T^{2} + \cdots - 95315904000 \)
$47$
\( T^{3} - 22500 T^{2} + \cdots + 916008439440 \)
$53$
\( T^{3} - 17108 T^{2} + \cdots + 7849670295504 \)
$59$
\( T^{3} - 54176 T^{2} + \cdots - 1725012447168 \)
$61$
\( T^{3} + 71324 T^{2} + \cdots + 11439907465152 \)
$67$
\( T^{3} + 62960 T^{2} + \cdots - 11971711378840 \)
$71$
\( T^{3} - 98400 T^{2} + \cdots - 24837760695040 \)
$73$
\( T^{3} + 81772 T^{2} + \cdots + 7199078503954 \)
$79$
\( T^{3} + \cdots + 235690469012368 \)
$83$
\( T^{3} + \cdots - 297029282761704 \)
$89$
\( T^{3} + \cdots + 125799322340896 \)
$97$
\( T^{3} + \cdots + 693755159518744 \)
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