Newspace parameters
| Level: | \( N \) | \(=\) | \( 2275 = 5^{2} \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2275.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(18.1659664598\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.169.1 |
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|
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| Defining polynomial: |
\( x^{3} - x^{2} - 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
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 |
|
−2.37720 | −2.65109 | 3.65109 | 0 | 6.30219 | 1.00000 | −3.92498 | 4.02830 | 0 | |||||||||||||||||||||||||||
| 1.2 | −1.27389 | 1.37720 | −0.377203 | 0 | −1.75441 | 1.00000 | 3.02830 | −1.10331 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 1.65109 | 0.273891 | 0.726109 | 0 | 0.452219 | 1.00000 | −2.10331 | −2.92498 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 2275.2.a.l | ✓ | 3 |
| 5.b | even | 2 | 1 | 2275.2.a.n | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2275.2.a.l | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 2275.2.a.n | yes | 3 | 5.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2275))\):
|
\( T_{2}^{3} + 2T_{2}^{2} - 3T_{2} - 5 \)
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\( T_{3}^{3} + T_{3}^{2} - 4T_{3} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 2 T^{2} + \cdots - 5 \)
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| $3$ |
\( T^{3} + T^{2} - 4T + 1 \)
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| $5$ |
\( T^{3} \)
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| $7$ |
\( (T - 1)^{3} \)
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| $11$ |
\( T^{3} + 5 T^{2} + \cdots - 5 \)
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| $13$ |
\( (T - 1)^{3} \)
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| $17$ |
\( T^{3} - 3 T^{2} + \cdots - 1 \)
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| $19$ |
\( T^{3} - 39T + 65 \)
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| $23$ |
\( T^{3} + 9 T^{2} + \cdots - 25 \)
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| $29$ |
\( T^{3} - 13T - 13 \)
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| $31$ |
\( T^{3} + 5 T^{2} + \cdots - 5 \)
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| $37$ |
\( T^{3} + 25 T^{2} + \cdots + 415 \)
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| $41$ |
\( T^{3} + 10 T^{2} + \cdots + 25 \)
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| $43$ |
\( T^{3} - 6 T^{2} + \cdots - 8 \)
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| $47$ |
\( T^{3} + 3 T^{2} + \cdots - 467 \)
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| $53$ |
\( T^{3} - T^{2} + \cdots + 25 \)
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| $59$ |
\( T^{3} + 10 T^{2} + \cdots + 25 \)
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| $61$ |
\( T^{3} + 19 T^{2} + \cdots + 125 \)
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| $67$ |
\( T^{3} - 8 T^{2} + \cdots + 125 \)
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| $71$ |
\( T^{3} - T^{2} + \cdots - 53 \)
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| $73$ |
\( T^{3} + 14 T^{2} + \cdots + 1 \)
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| $79$ |
\( T^{3} - 11 T^{2} + \cdots - 109 \)
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| $83$ |
\( T^{3} - 9 T^{2} + \cdots + 181 \)
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| $89$ |
\( (T - 5)^{3} \)
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| $97$ |
\( T^{3} - 15 T^{2} + \cdots + 265 \)
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