Newspace parameters
| Level: | \( N \) | \(=\) | \( 2175 = 3 \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2175.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(17.3674624396\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.469.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 5x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 435) |
| 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
\( x^{3} - x^{2} - 5x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
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 |
|
−2.39138 | 1.00000 | 3.71871 | 0 | −2.39138 | 1.32733 | −4.11009 | 1.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | −0.772866 | 1.00000 | −1.40268 | 0 | −0.772866 | −2.17554 | 2.62981 | 1.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 2.16425 | 1.00000 | 2.68397 | 0 | 2.16425 | 4.84822 | 1.48028 | 1.00000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 2175.2.a.u | 3 | |
| 3.b | odd | 2 | 1 | 6525.2.a.bf | 3 | ||
| 5.b | even | 2 | 1 | 435.2.a.i | ✓ | 3 | |
| 5.c | odd | 4 | 2 | 2175.2.c.m | 6 | ||
| 15.d | odd | 2 | 1 | 1305.2.a.q | 3 | ||
| 20.d | odd | 2 | 1 | 6960.2.a.cl | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 435.2.a.i | ✓ | 3 | 5.b | even | 2 | 1 | |
| 1305.2.a.q | 3 | 15.d | odd | 2 | 1 | ||
| 2175.2.a.u | 3 | 1.a | even | 1 | 1 | trivial | |
| 2175.2.c.m | 6 | 5.c | odd | 4 | 2 | ||
| 6525.2.a.bf | 3 | 3.b | odd | 2 | 1 | ||
| 6960.2.a.cl | 3 | 20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2175))\):
|
\( T_{2}^{3} + T_{2}^{2} - 5T_{2} - 4 \)
|
|
\( T_{7}^{3} - 4T_{7}^{2} - 7T_{7} + 14 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} - 5T - 4 \)
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| $3$ |
\( (T - 1)^{3} \)
|
| $5$ |
\( T^{3} \)
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| $7$ |
\( T^{3} - 4 T^{2} + \cdots + 14 \)
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| $11$ |
\( (T - 3)^{3} \)
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| $13$ |
\( T^{3} + 6T^{2} - T - 2 \)
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| $17$ |
\( T^{3} + 2 T^{2} + \cdots - 8 \)
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| $19$ |
\( T^{3} + 4 T^{2} + \cdots - 88 \)
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| $23$ |
\( T^{3} + T^{2} + \cdots - 112 \)
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| $29$ |
\( (T + 1)^{3} \)
|
| $31$ |
\( T^{3} \)
|
| $37$ |
\( T^{3} + 13T^{2} - 256 \)
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| $41$ |
\( T^{3} - 13 T^{2} + \cdots - 28 \)
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| $43$ |
\( T^{3} - 13 T^{2} + \cdots + 308 \)
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| $47$ |
\( T^{3} + 2 T^{2} + \cdots - 266 \)
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| $53$ |
\( T^{3} - 3 T^{2} + \cdots - 316 \)
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| $59$ |
\( T^{3} - 22 T^{2} + \cdots - 256 \)
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| $61$ |
\( T^{3} - 10 T^{2} + \cdots + 112 \)
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| $67$ |
\( T^{3} - 28 T^{2} + \cdots - 194 \)
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| $71$ |
\( T^{3} - 192T + 488 \)
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| $73$ |
\( T^{3} + 3 T^{2} + \cdots - 1168 \)
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| $79$ |
\( T^{3} + 2 T^{2} + \cdots - 224 \)
|
| $83$ |
\( T^{3} - 15 T^{2} + \cdots - 44 \)
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| $89$ |
\( T^{3} - 30 T^{2} + \cdots + 602 \)
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| $97$ |
\( T^{3} - T^{2} + \cdots - 76 \)
|
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