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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{33}) \) |
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|
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| Defining polynomial: |
\( x^{2} - x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.
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 | −3.37228 | 1.00000 | 1.00000 | −3.37228 | −3.37228 | 1.00000 | 8.37228 | 1.00000 | ||||||||||||||||||||||||
| 1.2 | 1.00000 | 2.37228 | 1.00000 | 1.00000 | 2.37228 | 2.37228 | 1.00000 | 2.62772 | 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.r | 2 | |
| 4.b | odd | 2 | 1 | 9680.2.a.bt | 2 | ||
| 5.b | even | 2 | 1 | 6050.2.a.cb | 2 | ||
| 11.b | odd | 2 | 1 | 110.2.a.d | ✓ | 2 | |
| 33.d | even | 2 | 1 | 990.2.a.m | 2 | ||
| 44.c | even | 2 | 1 | 880.2.a.n | 2 | ||
| 55.d | odd | 2 | 1 | 550.2.a.n | 2 | ||
| 55.e | even | 4 | 2 | 550.2.b.f | 4 | ||
| 77.b | even | 2 | 1 | 5390.2.a.bp | 2 | ||
| 88.b | odd | 2 | 1 | 3520.2.a.bq | 2 | ||
| 88.g | even | 2 | 1 | 3520.2.a.bj | 2 | ||
| 132.d | odd | 2 | 1 | 7920.2.a.bq | 2 | ||
| 165.d | even | 2 | 1 | 4950.2.a.bw | 2 | ||
| 165.l | odd | 4 | 2 | 4950.2.c.bc | 4 | ||
| 220.g | even | 2 | 1 | 4400.2.a.bl | 2 | ||
| 220.i | odd | 4 | 2 | 4400.2.b.p | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.2.a.d | ✓ | 2 | 11.b | odd | 2 | 1 | |
| 550.2.a.n | 2 | 55.d | odd | 2 | 1 | ||
| 550.2.b.f | 4 | 55.e | even | 4 | 2 | ||
| 880.2.a.n | 2 | 44.c | even | 2 | 1 | ||
| 990.2.a.m | 2 | 33.d | even | 2 | 1 | ||
| 1210.2.a.r | 2 | 1.a | even | 1 | 1 | trivial | |
| 3520.2.a.bj | 2 | 88.g | even | 2 | 1 | ||
| 3520.2.a.bq | 2 | 88.b | odd | 2 | 1 | ||
| 4400.2.a.bl | 2 | 220.g | even | 2 | 1 | ||
| 4400.2.b.p | 4 | 220.i | odd | 4 | 2 | ||
| 4950.2.a.bw | 2 | 165.d | even | 2 | 1 | ||
| 4950.2.c.bc | 4 | 165.l | odd | 4 | 2 | ||
| 5390.2.a.bp | 2 | 77.b | even | 2 | 1 | ||
| 6050.2.a.cb | 2 | 5.b | even | 2 | 1 | ||
| 7920.2.a.bq | 2 | 132.d | odd | 2 | 1 | ||
| 9680.2.a.bt | 2 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1210))\):
|
\( T_{3}^{2} + T_{3} - 8 \)
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\( T_{7}^{2} + T_{7} - 8 \)
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\( T_{13} + 2 \)
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\( T_{17}^{2} - 3T_{17} - 6 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{2} \)
|
| $3$ |
\( T^{2} + T - 8 \)
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| $5$ |
\( (T - 1)^{2} \)
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| $7$ |
\( T^{2} + T - 8 \)
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| $11$ |
\( T^{2} \)
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| $13$ |
\( (T + 2)^{2} \)
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| $17$ |
\( T^{2} - 3T - 6 \)
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| $19$ |
\( T^{2} + 7T + 4 \)
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| $23$ |
\( T^{2} + 6T - 24 \)
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| $29$ |
\( T^{2} - 3T - 6 \)
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| $31$ |
\( T^{2} - T - 8 \)
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| $37$ |
\( T^{2} - 13T + 34 \)
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| $41$ |
\( T^{2} - 132 \)
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| $43$ |
\( (T - 4)^{2} \)
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| $47$ |
\( T^{2} + 6T - 24 \)
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| $53$ |
\( T^{2} - 9T - 54 \)
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| $59$ |
\( T^{2} - 6T - 24 \)
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| $61$ |
\( T^{2} - 5T - 2 \)
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| $67$ |
\( (T - 8)^{2} \)
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| $71$ |
\( T^{2} - 3T - 72 \)
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| $73$ |
\( T^{2} - 8T - 116 \)
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| $79$ |
\( T^{2} - 14T + 16 \)
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| $83$ |
\( T^{2} + 6T - 24 \)
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| $89$ |
\( T^{2} - 3T - 6 \)
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| $97$ |
\( T^{2} + 14T + 16 \)
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