Newspace parameters
| Level: | \( N \) | \(=\) | \( 645 = 3 \cdot 5 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 645.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.15035093037\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.148.1 |
|
|
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| Defining polynomial: |
\( x^{3} - x^{2} - 3x + 1 \)
|
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
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 |
|
−2.21432 | −1.00000 | 2.90321 | 1.00000 | 2.21432 | −2.68889 | −2.00000 | 1.00000 | −2.21432 | |||||||||||||||||||||||||||
| 1.2 | 0.539189 | −1.00000 | −1.70928 | 1.00000 | −0.539189 | −0.829914 | −2.00000 | 1.00000 | 0.539189 | ||||||||||||||||||||||||||||
| 1.3 | 1.67513 | −1.00000 | 0.806063 | 1.00000 | −1.67513 | −4.48119 | −2.00000 | 1.00000 | 1.67513 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(43\) | \( -1 \) |
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 | 645.2.a.i | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1935.2.a.p | 3 | ||
| 5.b | even | 2 | 1 | 3225.2.a.r | 3 | ||
| 15.d | odd | 2 | 1 | 9675.2.a.bs | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 645.2.a.i | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1935.2.a.p | 3 | 3.b | odd | 2 | 1 | ||
| 3225.2.a.r | 3 | 5.b | even | 2 | 1 | ||
| 9675.2.a.bs | 3 | 15.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(645))\):
|
\( T_{2}^{3} - 4T_{2} + 2 \)
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\( T_{7}^{3} + 8T_{7}^{2} + 18T_{7} + 10 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 4T + 2 \)
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| $3$ |
\( (T + 1)^{3} \)
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| $5$ |
\( (T - 1)^{3} \)
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| $7$ |
\( T^{3} + 8 T^{2} + \cdots + 10 \)
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| $11$ |
\( T^{3} + T^{2} + \cdots + 29 \)
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| $13$ |
\( T^{3} + 3T^{2} - T - 1 \)
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| $17$ |
\( T^{3} + 3 T^{2} + \cdots + 25 \)
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| $19$ |
\( T^{3} + 6 T^{2} + \cdots - 2 \)
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| $23$ |
\( T^{3} + 9T^{2} - T - 5 \)
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| $29$ |
\( T^{3} + 2 T^{2} + \cdots - 200 \)
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| $31$ |
\( T^{3} + 7 T^{2} + \cdots - 475 \)
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| $37$ |
\( T^{3} + 6 T^{2} + \cdots - 40 \)
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| $41$ |
\( T^{3} - 3 T^{2} + \cdots + 155 \)
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| $43$ |
\( (T - 1)^{3} \)
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| $47$ |
\( T^{3} + 12 T^{2} + \cdots - 100 \)
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| $53$ |
\( T^{3} - 3 T^{2} + \cdots - 305 \)
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| $59$ |
\( (T + 4)^{3} \)
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| $61$ |
\( T^{3} - 6 T^{2} + \cdots + 248 \)
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| $67$ |
\( T^{3} + 27 T^{2} + \cdots + 361 \)
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| $71$ |
\( T^{3} - 12 T^{2} + \cdots + 50 \)
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| $73$ |
\( T^{3} + 28 T^{2} + \cdots + 320 \)
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| $79$ |
\( T^{3} + 8 T^{2} + \cdots - 740 \)
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| $83$ |
\( T^{3} + 33 T^{2} + \cdots + 1075 \)
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| $89$ |
\( T^{3} - 6 T^{2} + \cdots + 54 \)
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| $97$ |
\( T^{3} + 15 T^{2} + \cdots - 281 \)
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