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: | \(6\) |
| Coefficient field: | 6.6.45853772.1 |
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| Defining polynomial: |
\( x^{6} - x^{5} - 10x^{4} + 10x^{3} + 23x^{2} - 14x - 18 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 10x^{4} + 10x^{3} + 23x^{2} - 14x - 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} + 2\nu + 9 \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - 9\nu^{3} + \nu^{2} + 14\nu + 3 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 9\nu^{2} - 8\nu - 12 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 10\nu^{2} + 8\nu - 9 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + \nu^{4} - 9\nu^{3} - 8\nu^{2} + 16\nu + 16 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_{3} - \beta_{2} ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{2} + \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} - 2\beta_{3} - 4\beta_{2} - 1 \)
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| \(\nu^{4}\) | \(=\) |
\( -8\beta_{5} - \beta_{4} + \beta_{3} + 9\beta_{2} + 9\beta _1 + 23 \)
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| \(\nu^{5}\) | \(=\) |
\( 10\beta_{5} + 20\beta_{4} - 11\beta_{3} - 29\beta_{2} - \beta _1 - 16 \)
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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 |
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−2.67080 | 3.23496 | 5.13320 | 0 | −8.63994 | −1.00000 | −8.36815 | 7.46495 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.28471 | −0.432805 | 3.21989 | 0 | 0.988833 | −1.00000 | −2.78708 | −2.81268 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.35280 | −2.76555 | −0.169936 | 0 | 3.74123 | −1.00000 | 2.93549 | 4.64827 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.332820 | 0.594815 | −1.88923 | 0 | −0.197967 | −1.00000 | 1.29442 | −2.64619 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.62413 | 1.57457 | 0.637810 | 0 | 2.55731 | −1.00000 | −2.21238 | −0.520730 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.01700 | −2.20599 | 2.06828 | 0 | −4.44947 | −1.00000 | 0.137712 | 1.86638 | 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.s | 6 | |
| 5.b | even | 2 | 1 | 455.2.a.e | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 4095.2.a.bl | 6 | ||
| 20.d | odd | 2 | 1 | 7280.2.a.ce | 6 | ||
| 35.c | odd | 2 | 1 | 3185.2.a.r | 6 | ||
| 65.d | even | 2 | 1 | 5915.2.a.u | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 455.2.a.e | ✓ | 6 | 5.b | even | 2 | 1 | |
| 2275.2.a.s | 6 | 1.a | even | 1 | 1 | trivial | |
| 3185.2.a.r | 6 | 35.c | odd | 2 | 1 | ||
| 4095.2.a.bl | 6 | 15.d | odd | 2 | 1 | ||
| 5915.2.a.u | 6 | 65.d | even | 2 | 1 | ||
| 7280.2.a.ce | 6 | 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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\( T_{2}^{6} + 3T_{2}^{5} - 6T_{2}^{4} - 20T_{2}^{3} + 6T_{2}^{2} + 31T_{2} + 9 \)
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\( T_{3}^{6} - 13T_{3}^{4} - 2T_{3}^{3} + 35T_{3}^{2} - 4T_{3} - 8 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 3 T^{5} + \cdots + 9 \)
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| $3$ |
\( T^{6} - 13 T^{4} + \cdots - 8 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( (T + 1)^{6} \)
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| $11$ |
\( T^{6} + 2 T^{5} + \cdots - 1152 \)
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| $13$ |
\( (T + 1)^{6} \)
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| $17$ |
\( T^{6} + 8 T^{5} + \cdots + 9948 \)
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| $19$ |
\( T^{6} - 6 T^{5} + \cdots + 8 \)
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| $23$ |
\( T^{6} + 4 T^{5} + \cdots - 1152 \)
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| $29$ |
\( T^{6} + 14 T^{5} + \cdots - 5004 \)
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| $31$ |
\( T^{6} + 6 T^{5} + \cdots - 24536 \)
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| $37$ |
\( T^{6} + 20 T^{5} + \cdots - 4564 \)
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| $41$ |
\( T^{6} + 4 T^{5} + \cdots + 89748 \)
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| $43$ |
\( T^{6} + 10 T^{5} + \cdots - 105856 \)
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| $47$ |
\( T^{6} - 6 T^{5} + \cdots + 36864 \)
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| $53$ |
\( T^{6} + 2 T^{5} + \cdots + 4032 \)
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| $59$ |
\( T^{6} + 18 T^{5} + \cdots - 8856 \)
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| $61$ |
\( T^{6} - 92 T^{4} + \cdots - 64 \)
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| $67$ |
\( T^{6} + 20 T^{5} + \cdots + 15472 \)
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| $71$ |
\( T^{6} + 4 T^{5} + \cdots - 169344 \)
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| $73$ |
\( T^{6} + 6 T^{5} + \cdots - 10432 \)
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| $79$ |
\( T^{6} + 4 T^{5} + \cdots - 1545312 \)
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| $83$ |
\( T^{6} - 6 T^{5} + \cdots + 2304 \)
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| $89$ |
\( T^{6} + 12 T^{5} + \cdots - 692316 \)
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| $97$ |
\( T^{6} + 26 T^{5} + \cdots + 152512 \)
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