Newspace parameters
| Level: | \( N \) | \(=\) | \( 7168 = 2^{10} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7168.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(57.2367681689\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{10 +4 \sqrt{2}})\) |
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| Defining polynomial: |
\( x^{4} - 6x^{2} - 4x + 2 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1792) |
| 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
\( x^{4} - 6x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 3 \)
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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 | −2.68554 | 0 | 0.526602 | 0 | 1.00000 | 0 | 4.21215 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.334904 | 0 | −4.22274 | 0 | 1.00000 | 0 | −2.88784 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.27133 | 0 | −1.11239 | 0 | 1.00000 | 0 | −1.38372 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.74912 | 0 | 0.808530 | 0 | 1.00000 | 0 | 0.0594122 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
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 | 7168.2.a.t | 4 | |
| 4.b | odd | 2 | 1 | 7168.2.a.s | 4 | ||
| 8.b | even | 2 | 1 | 7168.2.a.x | 4 | ||
| 8.d | odd | 2 | 1 | 7168.2.a.w | 4 | ||
| 32.g | even | 8 | 2 | 1792.2.m.a | ✓ | 8 | |
| 32.g | even | 8 | 2 | 1792.2.m.d | yes | 8 | |
| 32.h | odd | 8 | 2 | 1792.2.m.b | yes | 8 | |
| 32.h | odd | 8 | 2 | 1792.2.m.c | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1792.2.m.a | ✓ | 8 | 32.g | even | 8 | 2 | |
| 1792.2.m.b | yes | 8 | 32.h | odd | 8 | 2 | |
| 1792.2.m.c | yes | 8 | 32.h | odd | 8 | 2 | |
| 1792.2.m.d | yes | 8 | 32.g | even | 8 | 2 | |
| 7168.2.a.s | 4 | 4.b | odd | 2 | 1 | ||
| 7168.2.a.t | 4 | 1.a | even | 1 | 1 | trivial | |
| 7168.2.a.w | 4 | 8.d | odd | 2 | 1 | ||
| 7168.2.a.x | 4 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7168))\):
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\( T_{3}^{4} - 6T_{3}^{2} + 4T_{3} + 2 \)
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\( T_{5}^{4} + 4T_{5}^{3} - 2T_{5}^{2} - 4T_{5} + 2 \)
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\( T_{11}^{4} - 32T_{11}^{2} + 64T_{11} + 16 \)
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\( T_{13}^{4} - 8T_{13}^{3} + 10T_{13}^{2} + 36T_{13} - 62 \)
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\( T_{17}^{4} + 4T_{17}^{3} - 4T_{17}^{2} - 16T_{17} + 8 \)
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\( T_{23}^{4} + 8T_{23}^{3} + 4T_{23}^{2} - 48T_{23} + 4 \)
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\( T_{31}^{4} - 4T_{31}^{3} - 44T_{31}^{2} - 32T_{31} + 8 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} - 6 T^{2} + \cdots + 2 \)
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| $5$ |
\( T^{4} + 4 T^{3} + \cdots + 2 \)
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| $7$ |
\( (T - 1)^{4} \)
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| $11$ |
\( T^{4} - 32 T^{2} + \cdots + 16 \)
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| $13$ |
\( T^{4} - 8 T^{3} + \cdots - 62 \)
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| $17$ |
\( T^{4} + 4 T^{3} + \cdots + 8 \)
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| $19$ |
\( T^{4} - 4 T^{3} + \cdots + 178 \)
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| $23$ |
\( T^{4} + 8 T^{3} + \cdots + 4 \)
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| $29$ |
\( T^{4} + 16 T^{3} + \cdots + 4 \)
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| $31$ |
\( T^{4} - 4 T^{3} + \cdots + 8 \)
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| $37$ |
\( T^{4} + 16 T^{3} + \cdots - 508 \)
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| $41$ |
\( T^{4} - 12 T^{3} + \cdots - 248 \)
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| $43$ |
\( T^{4} + 16 T^{3} + \cdots - 992 \)
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| $47$ |
\( T^{4} + 20 T^{3} + \cdots + 392 \)
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| $53$ |
\( T^{4} + 8 T^{3} + \cdots - 284 \)
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| $59$ |
\( T^{4} - 4 T^{3} + \cdots + 4594 \)
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| $61$ |
\( T^{4} + 20 T^{3} + \cdots - 6398 \)
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| $67$ |
\( T^{4} - 16 T^{3} + \cdots - 4544 \)
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| $71$ |
\( T^{4} - 16 T^{3} + \cdots - 2416 \)
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| $73$ |
\( T^{4} - 8 T^{3} + \cdots + 1424 \)
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| $79$ |
\( T^{4} - 288 T^{2} + \cdots + 19088 \)
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| $83$ |
\( T^{4} + 8 T^{3} + \cdots + 98 \)
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| $89$ |
\( T^{4} - 8 T^{3} + \cdots - 4976 \)
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| $97$ |
\( T^{4} - 36 T^{3} + \cdots - 3064 \)
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