Newspace parameters
| Level: | \( N \) | \(=\) | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1210.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(9.66189864457\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.5225.1 |
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| Defining polynomial: |
\( x^{4} - x^{3} - 8x^{2} + x + 11 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 110) |
| 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
\( x^{4} - x^{3} - 8x^{2} + x + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 4\nu + 3 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 4\nu^{2} + 2\nu - 11 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 4\beta_{2} + 6\beta _1 + 5 \)
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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 |
|
−1.00000 | −2.10522 | 1.00000 | 1.00000 | 2.10522 | −2.18609 | −1.00000 | 1.43195 | −1.00000 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −0.264977 | 1.00000 | 1.00000 | 0.264977 | −4.31175 | −1.00000 | −2.92979 | −1.00000 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 2.48718 | 1.00000 | 1.00000 | −2.48718 | −0.431946 | −1.00000 | 3.18609 | −1.00000 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 2.88301 | 1.00000 | 1.00000 | −2.88301 | 3.92979 | −1.00000 | 5.31175 | −1.00000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
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 | 1210.2.a.u | 4 | |
| 4.b | odd | 2 | 1 | 9680.2.a.cj | 4 | ||
| 5.b | even | 2 | 1 | 6050.2.a.dh | 4 | ||
| 11.b | odd | 2 | 1 | 1210.2.a.v | 4 | ||
| 11.c | even | 5 | 2 | 110.2.g.c | ✓ | 8 | |
| 33.h | odd | 10 | 2 | 990.2.n.j | 8 | ||
| 44.c | even | 2 | 1 | 9680.2.a.ci | 4 | ||
| 44.h | odd | 10 | 2 | 880.2.bo.g | 8 | ||
| 55.d | odd | 2 | 1 | 6050.2.a.cy | 4 | ||
| 55.j | even | 10 | 2 | 550.2.h.l | 8 | ||
| 55.k | odd | 20 | 4 | 550.2.ba.f | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.2.g.c | ✓ | 8 | 11.c | even | 5 | 2 | |
| 550.2.h.l | 8 | 55.j | even | 10 | 2 | ||
| 550.2.ba.f | 16 | 55.k | odd | 20 | 4 | ||
| 880.2.bo.g | 8 | 44.h | odd | 10 | 2 | ||
| 990.2.n.j | 8 | 33.h | odd | 10 | 2 | ||
| 1210.2.a.u | 4 | 1.a | even | 1 | 1 | trivial | |
| 1210.2.a.v | 4 | 11.b | odd | 2 | 1 | ||
| 6050.2.a.cy | 4 | 55.d | odd | 2 | 1 | ||
| 6050.2.a.dh | 4 | 5.b | even | 2 | 1 | ||
| 9680.2.a.ci | 4 | 44.c | even | 2 | 1 | ||
| 9680.2.a.cj | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1210))\):
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\( T_{3}^{4} - 3T_{3}^{3} - 5T_{3}^{2} + 14T_{3} + 4 \)
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\( T_{7}^{4} + 3T_{7}^{3} - 15T_{7}^{2} - 44T_{7} - 16 \)
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\( T_{13}^{4} + 7T_{13}^{3} - 21T_{13}^{2} - 132T_{13} + 176 \)
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\( T_{17}^{4} + T_{17}^{3} - 33T_{17}^{2} + 74T_{17} - 44 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
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| $3$ |
\( T^{4} - 3 T^{3} + \cdots + 4 \)
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| $5$ |
\( (T - 1)^{4} \)
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| $7$ |
\( T^{4} + 3 T^{3} + \cdots - 16 \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} + 7 T^{3} + \cdots + 176 \)
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| $17$ |
\( T^{4} + T^{3} + \cdots - 44 \)
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| $19$ |
\( T^{4} - 4 T^{3} + \cdots - 149 \)
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| $23$ |
\( T^{4} - 13 T^{3} + \cdots - 484 \)
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| $29$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
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| $31$ |
\( T^{4} - 12 T^{3} + \cdots - 1136 \)
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| $37$ |
\( T^{4} + 9 T^{3} + \cdots - 44 \)
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| $41$ |
\( T^{4} - 59 T^{2} + \cdots - 271 \)
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| $43$ |
\( T^{4} - 19 T^{3} + \cdots - 44 \)
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| $47$ |
\( T^{4} - 3 T^{3} + \cdots - 244 \)
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| $53$ |
\( T^{4} - 3 T^{3} + \cdots + 1516 \)
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| $59$ |
\( T^{4} - 2 T^{3} + \cdots + 5651 \)
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| $61$ |
\( T^{4} + 10 T^{3} + \cdots - 2816 \)
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| $67$ |
\( T^{4} + 13 T^{3} + \cdots - 604 \)
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| $71$ |
\( T^{4} - 16 T^{3} + \cdots - 5296 \)
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| $73$ |
\( T^{4} + T^{3} + \cdots + 4076 \)
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| $79$ |
\( T^{4} - 2 T^{3} + \cdots - 64 \)
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| $83$ |
\( T^{4} - 27 T^{3} + \cdots - 1324 \)
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| $89$ |
\( T^{4} - 16 T^{3} + \cdots + 1 \)
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| $97$ |
\( T^{4} + 13 T^{3} + \cdots - 2384 \)
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