Newspace parameters
| Level: | \( N \) | \(=\) | \( 1568 = 2^{5} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1568.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(12.5205430369\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
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| Defining polynomial: |
\( x^{2} - x - 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 224) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). 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 |
|
0 | −3.23607 | 0 | 1.23607 | 0 | 0 | 0 | 7.47214 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 1.23607 | 0 | −3.23607 | 0 | 0 | 0 | −1.47214 | 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 | 1568.2.a.k | 2 | |
| 4.b | odd | 2 | 1 | 1568.2.a.v | 2 | ||
| 7.b | odd | 2 | 1 | 224.2.a.d | yes | 2 | |
| 7.c | even | 3 | 2 | 1568.2.i.w | 4 | ||
| 7.d | odd | 6 | 2 | 1568.2.i.m | 4 | ||
| 8.b | even | 2 | 1 | 3136.2.a.by | 2 | ||
| 8.d | odd | 2 | 1 | 3136.2.a.bf | 2 | ||
| 21.c | even | 2 | 1 | 2016.2.a.o | 2 | ||
| 28.d | even | 2 | 1 | 224.2.a.c | ✓ | 2 | |
| 28.f | even | 6 | 2 | 1568.2.i.v | 4 | ||
| 28.g | odd | 6 | 2 | 1568.2.i.n | 4 | ||
| 35.c | odd | 2 | 1 | 5600.2.a.z | 2 | ||
| 56.e | even | 2 | 1 | 448.2.a.j | 2 | ||
| 56.h | odd | 2 | 1 | 448.2.a.i | 2 | ||
| 84.h | odd | 2 | 1 | 2016.2.a.r | 2 | ||
| 112.j | even | 4 | 2 | 1792.2.b.k | 4 | ||
| 112.l | odd | 4 | 2 | 1792.2.b.m | 4 | ||
| 140.c | even | 2 | 1 | 5600.2.a.bk | 2 | ||
| 168.e | odd | 2 | 1 | 4032.2.a.bw | 2 | ||
| 168.i | even | 2 | 1 | 4032.2.a.bv | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.2.a.c | ✓ | 2 | 28.d | even | 2 | 1 | |
| 224.2.a.d | yes | 2 | 7.b | odd | 2 | 1 | |
| 448.2.a.i | 2 | 56.h | odd | 2 | 1 | ||
| 448.2.a.j | 2 | 56.e | even | 2 | 1 | ||
| 1568.2.a.k | 2 | 1.a | even | 1 | 1 | trivial | |
| 1568.2.a.v | 2 | 4.b | odd | 2 | 1 | ||
| 1568.2.i.m | 4 | 7.d | odd | 6 | 2 | ||
| 1568.2.i.n | 4 | 28.g | odd | 6 | 2 | ||
| 1568.2.i.v | 4 | 28.f | even | 6 | 2 | ||
| 1568.2.i.w | 4 | 7.c | even | 3 | 2 | ||
| 1792.2.b.k | 4 | 112.j | even | 4 | 2 | ||
| 1792.2.b.m | 4 | 112.l | odd | 4 | 2 | ||
| 2016.2.a.o | 2 | 21.c | even | 2 | 1 | ||
| 2016.2.a.r | 2 | 84.h | odd | 2 | 1 | ||
| 3136.2.a.bf | 2 | 8.d | odd | 2 | 1 | ||
| 3136.2.a.by | 2 | 8.b | even | 2 | 1 | ||
| 4032.2.a.bv | 2 | 168.i | even | 2 | 1 | ||
| 4032.2.a.bw | 2 | 168.e | odd | 2 | 1 | ||
| 5600.2.a.z | 2 | 35.c | odd | 2 | 1 | ||
| 5600.2.a.bk | 2 | 140.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1568))\):
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\( T_{3}^{2} + 2T_{3} - 4 \)
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\( T_{5}^{2} + 2T_{5} - 4 \)
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\( T_{11}^{2} - 4T_{11} - 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
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| $3$ |
\( T^{2} + 2T - 4 \)
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| $5$ |
\( T^{2} + 2T - 4 \)
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| $7$ |
\( T^{2} \)
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| $11$ |
\( T^{2} - 4T - 16 \)
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| $13$ |
\( T^{2} + 6T + 4 \)
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| $17$ |
\( T^{2} - 20 \)
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| $19$ |
\( T^{2} - 2T - 4 \)
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| $23$ |
\( (T + 4)^{2} \)
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| $29$ |
\( T^{2} - 20 \)
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| $31$ |
\( T^{2} - 4T - 16 \)
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| $37$ |
\( T^{2} - 20 \)
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| $41$ |
\( T^{2} - 8T - 4 \)
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| $43$ |
\( T^{2} + 4T - 16 \)
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| $47$ |
\( T^{2} - 12T + 16 \)
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| $53$ |
\( (T + 10)^{2} \)
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| $59$ |
\( T^{2} + 14T + 44 \)
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| $61$ |
\( T^{2} + 18T + 76 \)
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| $67$ |
\( (T + 4)^{2} \)
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| $71$ |
\( T^{2} + 8T - 64 \)
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| $73$ |
\( T^{2} + 12T - 44 \)
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| $79$ |
\( T^{2} + 8T - 64 \)
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| $83$ |
\( T^{2} + 14T + 44 \)
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| $89$ |
\( (T - 6)^{2} \)
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| $97$ |
\( T^{2} + 16T + 44 \)
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