Newspace parameters
| Level: | \( N \) | \(=\) | \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 770.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(6.14848095564\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.316.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 4x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 6 ) / 2 \)
|
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 | −2.24914 | 1.00000 | 1.00000 | −2.24914 | −1.00000 | 1.00000 | 2.05863 | 1.00000 | |||||||||||||||||||||||||||
| 1.2 | 1.00000 | 1.14637 | 1.00000 | 1.00000 | 1.14637 | −1.00000 | 1.00000 | −1.68585 | 1.00000 | ||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 3.10278 | 1.00000 | 1.00000 | 3.10278 | −1.00000 | 1.00000 | 6.62721 | 1.00000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 770.2.a.m | ✓ | 3 |
| 3.b | odd | 2 | 1 | 6930.2.a.ce | 3 | ||
| 4.b | odd | 2 | 1 | 6160.2.a.bf | 3 | ||
| 5.b | even | 2 | 1 | 3850.2.a.bt | 3 | ||
| 5.c | odd | 4 | 2 | 3850.2.c.ba | 6 | ||
| 7.b | odd | 2 | 1 | 5390.2.a.ca | 3 | ||
| 11.b | odd | 2 | 1 | 8470.2.a.ci | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 770.2.a.m | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 3850.2.a.bt | 3 | 5.b | even | 2 | 1 | ||
| 3850.2.c.ba | 6 | 5.c | odd | 4 | 2 | ||
| 5390.2.a.ca | 3 | 7.b | odd | 2 | 1 | ||
| 6160.2.a.bf | 3 | 4.b | odd | 2 | 1 | ||
| 6930.2.a.ce | 3 | 3.b | odd | 2 | 1 | ||
| 8470.2.a.ci | 3 | 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(770))\):
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\( T_{3}^{3} - 2T_{3}^{2} - 6T_{3} + 8 \)
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\( T_{13}^{3} - 2T_{13}^{2} - 16T_{13} + 16 \)
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\( T_{17}^{3} - 2T_{17}^{2} - 28T_{17} - 8 \)
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\( T_{19}^{3} + 6T_{19}^{2} - 46T_{19} - 232 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{3} \)
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| $3$ |
\( T^{3} - 2 T^{2} + \cdots + 8 \)
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| $5$ |
\( (T - 1)^{3} \)
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| $7$ |
\( (T + 1)^{3} \)
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| $11$ |
\( (T - 1)^{3} \)
|
| $13$ |
\( T^{3} - 2 T^{2} + \cdots + 16 \)
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| $17$ |
\( T^{3} - 2 T^{2} + \cdots - 8 \)
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| $19$ |
\( T^{3} + 6 T^{2} + \cdots - 232 \)
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| $23$ |
\( T^{3} - 10 T^{2} + \cdots - 16 \)
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| $29$ |
\( T^{3} - 66T - 92 \)
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| $31$ |
\( T^{3} + 12 T^{2} + \cdots - 32 \)
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| $37$ |
\( T^{3} - 4 T^{2} + \cdots + 4 \)
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| $41$ |
\( T^{3} - 4 T^{2} + \cdots - 164 \)
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| $43$ |
\( T^{3} - 8 T^{2} + \cdots + 848 \)
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| $47$ |
\( T^{3} - 112T - 128 \)
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| $53$ |
\( T^{3} - 12 T^{2} + \cdots + 292 \)
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| $59$ |
\( T^{3} + 4 T^{2} + \cdots - 128 \)
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| $61$ |
\( T^{3} + 6 T^{2} + \cdots - 344 \)
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| $67$ |
\( T^{3} - 4 T^{2} + \cdots - 64 \)
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| $71$ |
\( T^{3} + 28 T^{2} + \cdots + 64 \)
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| $73$ |
\( T^{3} + 14 T^{2} + \cdots - 8 \)
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| $79$ |
\( T^{3} - 18 T^{2} + \cdots + 256 \)
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| $83$ |
\( T^{3} + 8 T^{2} + \cdots - 32 \)
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| $89$ |
\( T^{3} - 18 T^{2} + \cdots + 1208 \)
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| $97$ |
\( T^{3} + 4 T^{2} + \cdots + 316 \)
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