Newspace parameters
| Level: | \( N \) | \(=\) | \( 8381 = 17^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8381.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(66.9226219340\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
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|
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| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 29) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−2.41421 | −2.41421 | 3.82843 | 1.00000 | 5.82843 | 2.82843 | −4.41421 | 2.82843 | −2.41421 | ||||||||||||||||||||||||
| 1.2 | 0.414214 | 0.414214 | −1.82843 | 1.00000 | 0.171573 | −2.82843 | −1.58579 | −2.82843 | 0.414214 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 8381.2.a.e | 2 | |
| 17.b | even | 2 | 1 | 29.2.a.a | ✓ | 2 | |
| 51.c | odd | 2 | 1 | 261.2.a.d | 2 | ||
| 68.d | odd | 2 | 1 | 464.2.a.h | 2 | ||
| 85.c | even | 2 | 1 | 725.2.a.b | 2 | ||
| 85.g | odd | 4 | 2 | 725.2.b.b | 4 | ||
| 119.d | odd | 2 | 1 | 1421.2.a.j | 2 | ||
| 136.e | odd | 2 | 1 | 1856.2.a.w | 2 | ||
| 136.h | even | 2 | 1 | 1856.2.a.r | 2 | ||
| 187.b | odd | 2 | 1 | 3509.2.a.j | 2 | ||
| 204.h | even | 2 | 1 | 4176.2.a.bq | 2 | ||
| 221.b | even | 2 | 1 | 4901.2.a.g | 2 | ||
| 255.h | odd | 2 | 1 | 6525.2.a.o | 2 | ||
| 493.c | even | 2 | 1 | 841.2.a.d | 2 | ||
| 493.h | odd | 4 | 2 | 841.2.b.a | 4 | ||
| 493.p | even | 14 | 6 | 841.2.d.j | 12 | ||
| 493.q | even | 14 | 6 | 841.2.d.f | 12 | ||
| 493.y | odd | 28 | 12 | 841.2.e.k | 24 | ||
| 1479.h | odd | 2 | 1 | 7569.2.a.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.2.a.a | ✓ | 2 | 17.b | even | 2 | 1 | |
| 261.2.a.d | 2 | 51.c | odd | 2 | 1 | ||
| 464.2.a.h | 2 | 68.d | odd | 2 | 1 | ||
| 725.2.a.b | 2 | 85.c | even | 2 | 1 | ||
| 725.2.b.b | 4 | 85.g | odd | 4 | 2 | ||
| 841.2.a.d | 2 | 493.c | even | 2 | 1 | ||
| 841.2.b.a | 4 | 493.h | odd | 4 | 2 | ||
| 841.2.d.f | 12 | 493.q | even | 14 | 6 | ||
| 841.2.d.j | 12 | 493.p | even | 14 | 6 | ||
| 841.2.e.k | 24 | 493.y | odd | 28 | 12 | ||
| 1421.2.a.j | 2 | 119.d | odd | 2 | 1 | ||
| 1856.2.a.r | 2 | 136.h | even | 2 | 1 | ||
| 1856.2.a.w | 2 | 136.e | odd | 2 | 1 | ||
| 3509.2.a.j | 2 | 187.b | odd | 2 | 1 | ||
| 4176.2.a.bq | 2 | 204.h | even | 2 | 1 | ||
| 4901.2.a.g | 2 | 221.b | even | 2 | 1 | ||
| 6525.2.a.o | 2 | 255.h | odd | 2 | 1 | ||
| 7569.2.a.c | 2 | 1479.h | odd | 2 | 1 | ||
| 8381.2.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8381))\):
|
\( T_{2}^{2} + 2T_{2} - 1 \)
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\( T_{3}^{2} + 2T_{3} - 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 2T - 1 \)
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| $3$ |
\( T^{2} + 2T - 1 \)
|
| $5$ |
\( (T - 1)^{2} \)
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| $7$ |
\( T^{2} - 8 \)
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| $11$ |
\( T^{2} + 2T - 1 \)
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| $13$ |
\( T^{2} + 2T - 7 \)
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| $17$ |
\( T^{2} \)
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| $19$ |
\( (T - 6)^{2} \)
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| $23$ |
\( T^{2} - 4T - 28 \)
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| $29$ |
\( (T + 1)^{2} \)
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| $31$ |
\( T^{2} + 6T - 41 \)
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| $37$ |
\( (T - 4)^{2} \)
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| $41$ |
\( T^{2} + 8T - 56 \)
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| $43$ |
\( T^{2} - 10T + 23 \)
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| $47$ |
\( T^{2} - 2T - 17 \)
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| $53$ |
\( T^{2} - 2T - 71 \)
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| $59$ |
\( T^{2} - 4T - 28 \)
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| $61$ |
\( T^{2} - 4T - 4 \)
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| $67$ |
\( T^{2} - 32 \)
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| $71$ |
\( T^{2} - 12T + 28 \)
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| $73$ |
\( (T + 4)^{2} \)
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| $79$ |
\( T^{2} - 2T - 1 \)
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| $83$ |
\( T^{2} - 4T - 28 \)
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| $89$ |
\( T^{2} + 8T - 56 \)
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| $97$ |
\( T^{2} - 8T - 56 \)
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