Newspace parameters
Level: | \( N \) | \(=\) | \( 23 \) |
Weight: | \( k \) | \(=\) | \( 11 \) |
Character orbit: | \([\chi]\) | \(=\) | 23.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(14.6132168115\) |
Analytic rank: | \(0\) |
Dimension: | \(3\) |
Coefficient field: | 3.3.621.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{3} - 6x - 3 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \( x^{3} - 6x - 3 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - \nu - 4 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + \beta _1 + 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).
\(n\) | \(5\) |
\(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
22.1 |
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−62.2986 | −122.026 | 2857.12 | 0 | 7602.03 | 0 | −114201. | −44158.7 | 0 | |||||||||||||||||||||||||||
22.2 | 18.4544 | −346.393 | −683.435 | 0 | −6392.47 | 0 | −31509.7 | 60938.9 | 0 | ||||||||||||||||||||||||||||
22.3 | 43.8442 | 468.418 | 898.316 | 0 | 20537.4 | 0 | −5510.51 | 160367. | 0 | ||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 23.11.b.a | ✓ | 3 |
3.b | odd | 2 | 1 | 207.11.d.a | 3 | ||
23.b | odd | 2 | 1 | CM | 23.11.b.a | ✓ | 3 |
69.c | even | 2 | 1 | 207.11.d.a | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
23.11.b.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
23.11.b.a | ✓ | 3 | 23.b | odd | 2 | 1 | CM |
207.11.d.a | 3 | 3.b | odd | 2 | 1 | ||
207.11.d.a | 3 | 69.c | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 3072T_{2} + 50407 \)
acting on \(S_{11}^{\mathrm{new}}(23, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - 3072T + 50407 \)
$3$
\( T^{3} - 177147 T - 19799482 \)
$5$
\( T^{3} \)
$7$
\( T^{3} \)
$11$
\( T^{3} \)
$13$
\( T^{3} - 413575475547 T - 96\!\cdots\!82 \)
$17$
\( T^{3} \)
$19$
\( T^{3} \)
$23$
\( (T + 6436343)^{3} \)
$29$
\( T^{3} + \cdots + 45\!\cdots\!74 \)
$31$
\( T^{3} + \cdots - 31\!\cdots\!74 \)
$37$
\( T^{3} \)
$41$
\( T^{3} + \cdots - 31\!\cdots\!74 \)
$43$
\( T^{3} \)
$47$
\( T^{3} + \cdots + 17\!\cdots\!82 \)
$53$
\( T^{3} \)
$59$
\( (T - 1281228026)^{3} \)
$61$
\( T^{3} \)
$67$
\( T^{3} \)
$71$
\( T^{3} + \cdots + 32\!\cdots\!26 \)
$73$
\( T^{3} + \cdots + 50\!\cdots\!18 \)
$79$
\( T^{3} \)
$83$
\( T^{3} \)
$89$
\( T^{3} \)
$97$
\( T^{3} \)
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