Newspace parameters
| Level: | \( N \) | \(=\) | \( 1352 = 2^{3} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1352.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(10.7957743533\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.13968.1 |
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 7x^{2} + 8x + 4 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 104) |
| 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
\( x^{4} - 2x^{3} - 7x^{2} + 8x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 4 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 6\nu + 2 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 2\beta_{2} + 7\beta _1 + 2 \)
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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 |
|
0 | −3.27743 | 0 | −2.61023 | 0 | −1.12182 | 0 | 7.74153 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.38651 | 0 | −3.44247 | 0 | 3.17452 | 0 | −1.07759 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.386509 | 0 | 3.17452 | 0 | −3.44247 | 0 | −2.85061 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.27743 | 0 | −1.12182 | 0 | −2.61023 | 0 | 2.18667 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 1352.2.a.k | 4 | |
| 4.b | odd | 2 | 1 | 2704.2.a.bd | 4 | ||
| 13.b | even | 2 | 1 | 1352.2.a.l | 4 | ||
| 13.c | even | 3 | 2 | 1352.2.i.k | 8 | ||
| 13.d | odd | 4 | 2 | 1352.2.f.f | 8 | ||
| 13.e | even | 6 | 2 | 1352.2.i.l | 8 | ||
| 13.f | odd | 12 | 2 | 104.2.o.a | ✓ | 8 | |
| 13.f | odd | 12 | 2 | 1352.2.o.f | 8 | ||
| 39.k | even | 12 | 2 | 936.2.bi.b | 8 | ||
| 52.b | odd | 2 | 1 | 2704.2.a.be | 4 | ||
| 52.f | even | 4 | 2 | 2704.2.f.q | 8 | ||
| 52.l | even | 12 | 2 | 208.2.w.c | 8 | ||
| 104.u | even | 12 | 2 | 832.2.w.i | 8 | ||
| 104.x | odd | 12 | 2 | 832.2.w.g | 8 | ||
| 156.v | odd | 12 | 2 | 1872.2.by.n | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.2.o.a | ✓ | 8 | 13.f | odd | 12 | 2 | |
| 208.2.w.c | 8 | 52.l | even | 12 | 2 | ||
| 832.2.w.g | 8 | 104.x | odd | 12 | 2 | ||
| 832.2.w.i | 8 | 104.u | even | 12 | 2 | ||
| 936.2.bi.b | 8 | 39.k | even | 12 | 2 | ||
| 1352.2.a.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 1352.2.a.l | 4 | 13.b | even | 2 | 1 | ||
| 1352.2.f.f | 8 | 13.d | odd | 4 | 2 | ||
| 1352.2.i.k | 8 | 13.c | even | 3 | 2 | ||
| 1352.2.i.l | 8 | 13.e | even | 6 | 2 | ||
| 1352.2.o.f | 8 | 13.f | odd | 12 | 2 | ||
| 1872.2.by.n | 8 | 156.v | odd | 12 | 2 | ||
| 2704.2.a.bd | 4 | 4.b | odd | 2 | 1 | ||
| 2704.2.a.be | 4 | 52.b | odd | 2 | 1 | ||
| 2704.2.f.q | 8 | 52.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1352))\):
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\( T_{3}^{4} + 2T_{3}^{3} - 7T_{3}^{2} - 8T_{3} + 4 \)
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\( T_{5}^{4} + 4T_{5}^{3} - 7T_{5}^{2} - 40T_{5} - 32 \)
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\( T_{7}^{4} + 4T_{7}^{3} - 7T_{7}^{2} - 40T_{7} - 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
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| $5$ |
\( T^{4} + 4 T^{3} + \cdots - 32 \)
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| $7$ |
\( T^{4} + 4 T^{3} + \cdots - 32 \)
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| $11$ |
\( T^{4} + 4 T^{3} + \cdots - 32 \)
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| $13$ |
\( T^{4} \)
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| $17$ |
\( T^{4} - 34T^{2} + 97 \)
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| $19$ |
\( T^{4} + 16 T^{3} + \cdots - 284 \)
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| $23$ |
\( T^{4} - 2 T^{3} + \cdots + 4 \)
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| $29$ |
\( T^{4} - 8 T^{3} + \cdots + 61 \)
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| $31$ |
\( T^{4} + 4 T^{3} + \cdots + 16 \)
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| $37$ |
\( T^{4} + 12 T^{3} + \cdots - 299 \)
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| $41$ |
\( (T^{2} - 8 T + 13)^{2} \)
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| $43$ |
\( T^{4} - 6 T^{3} + \cdots + 832 \)
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| $47$ |
\( T^{4} + 20 T^{3} + \cdots - 368 \)
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| $53$ |
\( T^{4} - 10 T^{3} + \cdots - 1724 \)
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| $59$ |
\( T^{4} + 4 T^{3} + \cdots + 208 \)
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| $61$ |
\( T^{4} - 4 T^{3} + \cdots - 947 \)
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| $67$ |
\( T^{4} + 8 T^{3} + \cdots - 236 \)
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| $71$ |
\( T^{4} + 16 T^{3} + \cdots - 1196 \)
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| $73$ |
\( T^{4} + 24 T^{3} + \cdots + 144 \)
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| $79$ |
\( T^{4} - 8 T^{3} + \cdots + 3328 \)
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| $83$ |
\( T^{4} + 24 T^{3} + \cdots + 256 \)
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| $89$ |
\( T^{4} + 16 T^{3} + \cdots + 13648 \)
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| $97$ |
\( T^{4} + 32 T^{3} + \cdots - 7472 \)
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