Newspace parameters
| Level: | \( N \) | \(=\) | \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2300.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(18.3655924649\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.53121.1 |
|
|
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| Defining polynomial: |
\( x^{4} - 7x^{2} - x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 7x^{2} - x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu - 1 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta _1 + 1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −2.50653 | 0 | 0 | 0 | −0.797915 | 0 | 3.28271 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.631352 | 0 | 0 | 0 | −3.16780 | 0 | −2.60139 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.474520 | 0 | 0 | 0 | 4.21479 | 0 | −2.77483 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.66337 | 0 | 0 | 0 | 0.750930 | 0 | 4.09352 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(23\) | \( +1 \) |
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 | 2300.2.a.m | yes | 4 |
| 4.b | odd | 2 | 1 | 9200.2.a.cm | 4 | ||
| 5.b | even | 2 | 1 | 2300.2.a.l | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 2300.2.c.j | 8 | ||
| 20.d | odd | 2 | 1 | 9200.2.a.co | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2300.2.a.l | ✓ | 4 | 5.b | even | 2 | 1 | |
| 2300.2.a.m | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 2300.2.c.j | 8 | 5.c | odd | 4 | 2 | ||
| 9200.2.a.cm | 4 | 4.b | odd | 2 | 1 | ||
| 9200.2.a.co | 4 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2300))\):
|
\( T_{3}^{4} - 7T_{3}^{2} - T_{3} + 2 \)
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\( T_{7}^{4} - T_{7}^{3} - 14T_{7}^{2} + 8 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} - 7T^{2} - T + 2 \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - T^{3} - 14 T^{2} + 8 \)
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| $11$ |
\( T^{4} - T^{3} + \cdots + 120 \)
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| $13$ |
\( T^{4} + T^{3} + \cdots - 17 \)
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| $17$ |
\( T^{4} - 8 T^{3} + \cdots - 48 \)
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| $19$ |
\( T^{4} - 7 T^{3} + \cdots + 72 \)
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| $23$ |
\( (T + 1)^{4} \)
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| $29$ |
\( T^{4} + 5 T^{3} + \cdots - 93 \)
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| $31$ |
\( T^{4} - 14 T^{3} + \cdots - 10 \)
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| $37$ |
\( T^{4} - 4 T^{3} + \cdots + 416 \)
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| $41$ |
\( T^{4} - 3 T^{3} + \cdots + 381 \)
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| $43$ |
\( T^{4} + 7 T^{3} + \cdots + 72 \)
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| $47$ |
\( T^{4} - 12 T^{3} + \cdots - 1242 \)
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| $53$ |
\( T^{4} + 10 T^{3} + \cdots + 288 \)
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| $59$ |
\( T^{4} - 11 T^{3} + \cdots + 12 \)
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| $61$ |
\( T^{4} - 22 T^{3} + \cdots - 416 \)
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| $67$ |
\( T^{4} - 12 T^{3} + \cdots + 992 \)
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| $71$ |
\( T^{4} - 18 T^{3} + \cdots - 1800 \)
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| $73$ |
\( T^{4} + 9 T^{3} + \cdots - 139 \)
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| $79$ |
\( T^{4} - 25 T^{3} + \cdots + 40 \)
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| $83$ |
\( T^{4} + 7 T^{3} + \cdots + 72 \)
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| $89$ |
\( T^{4} - 236 T^{2} + \cdots - 3840 \)
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| $97$ |
\( T^{4} - 8 T^{3} + \cdots - 1040 \)
|
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