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: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 8x^{5} - 2x^{4} + 16x^{3} + 5x^{2} - 6x - 2 \)
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| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 8x^{5} - 2x^{4} + 16x^{3} + 5x^{2} - 6x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 2\nu + 3 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 8\nu^{4} - \nu^{3} + 15\nu^{2} + \nu - 4 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 7\nu^{4} + 4\nu^{3} + 13\nu^{2} - 3\nu - 5 \)
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| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 7\nu^{4} + 5\nu^{3} + 12\nu^{2} - 7\nu - 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta_{2} + 4\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta_{3} + 6\beta_{2} + 2\beta _1 + 7 \)
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| \(\nu^{5}\) | \(=\) |
\( 6\beta_{6} - 7\beta_{5} + \beta_{4} + \beta_{3} + 9\beta_{2} + 18\beta _1 + 2 \)
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| \(\nu^{6}\) | \(=\) |
\( 9\beta_{6} - 9\beta_{5} + \beta_{4} + 8\beta_{3} + 34\beta_{2} + 19\beta _1 + 30 \)
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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 |
|
−1.92480 | 2.17582 | 1.70485 | 0 | −4.18801 | −1.00000 | 0.568104 | 1.73417 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.71085 | 0.260291 | 0.927009 | 0 | −0.445320 | −1.00000 | 1.83573 | −2.93225 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.687115 | −1.04121 | −1.52787 | 0 | 0.715431 | −1.00000 | 2.42406 | −1.91588 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.340436 | 2.66843 | −1.88410 | 0 | −0.908430 | −1.00000 | 1.32229 | 4.12053 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.700339 | −1.90743 | −1.50953 | 0 | −1.33585 | −1.00000 | −2.45786 | 0.638282 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.51372 | 1.55652 | 0.291349 | 0 | 2.35614 | −1.00000 | −2.58642 | −0.577244 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.44914 | −1.71242 | 3.99829 | 0 | −4.19396 | −1.00000 | 4.89410 | −0.0676101 | 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.y | 7 | |
| 5.b | even | 2 | 1 | 2275.2.a.w | 7 | ||
| 5.c | odd | 4 | 2 | 455.2.c.b | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 455.2.c.b | ✓ | 14 | 5.c | odd | 4 | 2 | |
| 2275.2.a.w | 7 | 5.b | even | 2 | 1 | ||
| 2275.2.a.y | 7 | 1.a | even | 1 | 1 | trivial | |
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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\( T_{2}^{7} - 8T_{2}^{5} - 2T_{2}^{4} + 16T_{2}^{3} + 5T_{2}^{2} - 6T_{2} - 2 \)
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\( T_{3}^{7} - 2T_{3}^{6} - 9T_{3}^{5} + 14T_{3}^{4} + 27T_{3}^{3} - 26T_{3}^{2} - 26T_{3} + 8 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 8 T^{5} + \cdots - 2 \)
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| $3$ |
\( T^{7} - 2 T^{6} + \cdots + 8 \)
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| $5$ |
\( T^{7} \)
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| $7$ |
\( (T + 1)^{7} \)
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| $11$ |
\( T^{7} + 6 T^{6} + \cdots - 512 \)
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| $13$ |
\( (T - 1)^{7} \)
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| $17$ |
\( T^{7} + 2 T^{6} + \cdots - 4 \)
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| $19$ |
\( T^{7} + 10 T^{6} + \cdots + 15782 \)
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| $23$ |
\( T^{7} + 10 T^{6} + \cdots + 12832 \)
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| $29$ |
\( T^{7} + 26 T^{6} + \cdots + 302 \)
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| $31$ |
\( T^{7} + 6 T^{6} + \cdots - 43898 \)
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| $37$ |
\( T^{7} - 4 T^{6} + \cdots - 29668 \)
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| $41$ |
\( T^{7} + 28 T^{6} + \cdots - 46208 \)
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| $43$ |
\( T^{7} + 2 T^{6} + \cdots - 22432 \)
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| $47$ |
\( T^{7} - 20 T^{6} + \cdots - 2048 \)
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| $53$ |
\( T^{7} + 8 T^{6} + \cdots + 95552 \)
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| $59$ |
\( T^{7} + 28 T^{6} + \cdots + 54848 \)
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| $61$ |
\( T^{7} + 26 T^{6} + \cdots - 1967744 \)
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| $67$ |
\( T^{7} + 2 T^{6} + \cdots - 621536 \)
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| $71$ |
\( T^{7} + 14 T^{6} + \cdots + 794624 \)
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| $73$ |
\( T^{7} - 2 T^{6} + \cdots + 14816 \)
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| $79$ |
\( T^{7} - 14 T^{6} + \cdots - 436112 \)
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| $83$ |
\( T^{7} - 28 T^{6} + \cdots - 29530048 \)
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| $89$ |
\( T^{7} + 30 T^{6} + \cdots + 32188 \)
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| $97$ |
\( T^{7} - 18 T^{6} + \cdots + 1139296 \)
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