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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.12197.1 |
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|
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| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} + 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 455) |
| 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
\( x^{4} - x^{3} - 5x^{2} + 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 5\beta _1 + 1 \)
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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 |
|
−2.04515 | −2.55619 | 2.18265 | 0 | 5.22780 | 1.00000 | −0.373543 | 3.53411 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −0.241765 | 2.89448 | −1.94155 | 0 | −0.699784 | 1.00000 | 0.952930 | 5.37801 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0.819751 | −1.40013 | −1.32801 | 0 | −1.14776 | 1.00000 | −2.72814 | −1.03963 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 2.46717 | 1.06184 | 4.08691 | 0 | 2.61974 | 1.00000 | 5.14875 | −1.87249 | 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.p | 4 | |
| 5.b | even | 2 | 1 | 455.2.a.c | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 4095.2.a.bf | 4 | ||
| 20.d | odd | 2 | 1 | 7280.2.a.bw | 4 | ||
| 35.c | odd | 2 | 1 | 3185.2.a.l | 4 | ||
| 65.d | even | 2 | 1 | 5915.2.a.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 455.2.a.c | ✓ | 4 | 5.b | even | 2 | 1 | |
| 2275.2.a.p | 4 | 1.a | even | 1 | 1 | trivial | |
| 3185.2.a.l | 4 | 35.c | odd | 2 | 1 | ||
| 4095.2.a.bf | 4 | 15.d | odd | 2 | 1 | ||
| 5915.2.a.n | 4 | 65.d | even | 2 | 1 | ||
| 7280.2.a.bw | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2275))\):
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\( T_{2}^{4} - T_{2}^{3} - 5T_{2}^{2} + 3T_{2} + 1 \)
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\( T_{3}^{4} - 9T_{3}^{2} - 2T_{3} + 11 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} - 5 T^{2} + \cdots + 1 \)
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| $3$ |
\( T^{4} - 9 T^{2} + \cdots + 11 \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( (T - 1)^{4} \)
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| $11$ |
\( T^{4} - 2 T^{3} + \cdots - 80 \)
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| $13$ |
\( (T + 1)^{4} \)
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| $17$ |
\( T^{4} - 41 T^{2} + \cdots + 163 \)
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| $19$ |
\( T^{4} - 18 T^{3} + \cdots + 121 \)
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| $23$ |
\( T^{4} - 12 T^{3} + \cdots - 880 \)
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| $29$ |
\( T^{4} + 2 T^{3} + \cdots + 61 \)
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| $31$ |
\( T^{4} - 18 T^{3} + \cdots + 85 \)
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| $37$ |
\( T^{4} + 4 T^{3} + \cdots + 895 \)
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| $41$ |
\( T^{4} - 12 T^{3} + \cdots + 405 \)
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| $43$ |
\( T^{4} + 2 T^{3} + \cdots + 16 \)
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| $47$ |
\( T^{4} - 18 T^{3} + \cdots - 304 \)
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| $53$ |
\( T^{4} + 2 T^{3} + \cdots + 2704 \)
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| $59$ |
\( T^{4} - 30 T^{3} + \cdots + 2221 \)
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| $61$ |
\( T^{4} - 120 T^{2} + \cdots + 80 \)
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| $67$ |
\( T^{4} + 12 T^{3} + \cdots - 977 \)
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| $71$ |
\( T^{4} + 8 T^{3} + \cdots + 1168 \)
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| $73$ |
\( T^{4} + 34 T^{3} + \cdots - 1648 \)
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| $79$ |
\( T^{4} - 12 T^{3} + \cdots + 17 \)
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| $83$ |
\( T^{4} + 2 T^{3} + \cdots + 10768 \)
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| $89$ |
\( T^{4} - 24 T^{3} + \cdots - 6215 \)
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| $97$ |
\( T^{4} + 14 T^{3} + \cdots - 10000 \)
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