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: | 6.6.134584629.1 |
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| Defining polynomial: |
\( x^{6} - x^{5} - 8x^{4} + 4x^{3} + 14x^{2} - 3x - 6 \)
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| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 8x^{4} + 4x^{3} + 14x^{2} - 3x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 7\nu^{2} + 3\nu + 7 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 3\nu^{2} + 7\nu - 1 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 6\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 8\beta_{2} + 10\beta _1 + 15 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 8\beta_{3} + 12\beta_{2} + 42\beta _1 + 14 \)
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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 |
|
−2.79352 | −1.41058 | 5.80378 | 0 | 3.94050 | −1.00000 | −10.6259 | −1.01025 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.37422 | −2.54671 | −0.111522 | 0 | 3.49974 | −1.00000 | 2.90169 | 3.48574 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.872890 | 2.66664 | −1.23806 | 0 | −2.32768 | −1.00000 | 2.82647 | 4.11095 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.705969 | −2.40741 | −1.50161 | 0 | −1.69956 | −1.00000 | −2.47203 | 2.79564 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.17366 | 2.10923 | −0.622525 | 0 | 2.47552 | −1.00000 | −3.07795 | 1.44887 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.16100 | −0.411161 | 2.66994 | 0 | −0.888521 | −1.00000 | 1.44775 | −2.83095 | 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.t | ✓ | 6 |
| 5.b | even | 2 | 1 | 2275.2.a.u | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2275.2.a.t | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2275.2.a.u | yes | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2275))\):
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\( T_{2}^{6} + T_{2}^{5} - 8T_{2}^{4} - 4T_{2}^{3} + 14T_{2}^{2} + 3T_{2} - 6 \)
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\( T_{3}^{6} + 2T_{3}^{5} - 11T_{3}^{4} - 23T_{3}^{3} + 25T_{3}^{2} + 62T_{3} + 20 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 8 T^{4} + \cdots - 6 \)
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| $3$ |
\( T^{6} + 2 T^{5} + \cdots + 20 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( (T + 1)^{6} \)
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| $11$ |
\( T^{6} + 9 T^{5} + \cdots + 48 \)
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| $13$ |
\( (T + 1)^{6} \)
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| $17$ |
\( T^{6} + 9 T^{5} + \cdots - 24 \)
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| $19$ |
\( T^{6} - 69 T^{4} + \cdots - 40 \)
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| $23$ |
\( T^{6} + 4 T^{5} + \cdots + 10587 \)
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| $29$ |
\( T^{6} - 3 T^{5} + \cdots - 75 \)
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| $31$ |
\( T^{6} + 3 T^{5} + \cdots - 3032 \)
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| $37$ |
\( T^{6} - 9 T^{5} + \cdots - 62 \)
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| $41$ |
\( T^{6} + 10 T^{5} + \cdots - 192 \)
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| $43$ |
\( T^{6} + 7 T^{5} + \cdots - 134488 \)
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| $47$ |
\( T^{6} - 3 T^{5} + \cdots - 2736 \)
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| $53$ |
\( T^{6} + 34 T^{5} + \cdots - 154404 \)
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| $59$ |
\( T^{6} - 18 T^{5} + \cdots - 24 \)
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| $61$ |
\( T^{6} + 6 T^{5} + \cdots - 111340 \)
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| $67$ |
\( T^{6} - 30 T^{5} + \cdots + 674 \)
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| $71$ |
\( T^{6} + 25 T^{5} + \cdots - 14502 \)
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| $73$ |
\( T^{6} - 8 T^{5} + \cdots + 221248 \)
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| $79$ |
\( T^{6} + 14 T^{5} + \cdots + 2713 \)
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| $83$ |
\( T^{6} + 3 T^{5} + \cdots - 1440 \)
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| $89$ |
\( T^{6} - 9 T^{5} + \cdots - 214944 \)
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| $97$ |
\( T^{6} + 33 T^{5} + \cdots - 497512 \)
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