Newspace parameters
| Level: | \( N \) | \(=\) | \( 51 = 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 51.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(0.407237050309\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{17}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} - x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). 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 |
|
−2.56155 | −1.00000 | 4.56155 | 3.56155 | 2.56155 | 0 | −6.56155 | 1.00000 | −9.12311 | ||||||||||||||||||||||||
| 1.2 | 1.56155 | −1.00000 | 0.438447 | −0.561553 | −1.56155 | 0 | −2.43845 | 1.00000 | −0.876894 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(17\) | \( -1 \) |
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 | 51.2.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 153.2.a.e | 2 | ||
| 4.b | odd | 2 | 1 | 816.2.a.m | 2 | ||
| 5.b | even | 2 | 1 | 1275.2.a.n | 2 | ||
| 5.c | odd | 4 | 2 | 1275.2.b.d | 4 | ||
| 7.b | odd | 2 | 1 | 2499.2.a.o | 2 | ||
| 8.b | even | 2 | 1 | 3264.2.a.bl | 2 | ||
| 8.d | odd | 2 | 1 | 3264.2.a.bg | 2 | ||
| 11.b | odd | 2 | 1 | 6171.2.a.p | 2 | ||
| 12.b | even | 2 | 1 | 2448.2.a.v | 2 | ||
| 13.b | even | 2 | 1 | 8619.2.a.q | 2 | ||
| 15.d | odd | 2 | 1 | 3825.2.a.s | 2 | ||
| 17.b | even | 2 | 1 | 867.2.a.f | 2 | ||
| 17.c | even | 4 | 2 | 867.2.d.c | 4 | ||
| 17.d | even | 8 | 4 | 867.2.e.f | 8 | ||
| 17.e | odd | 16 | 8 | 867.2.h.j | 16 | ||
| 21.c | even | 2 | 1 | 7497.2.a.v | 2 | ||
| 24.f | even | 2 | 1 | 9792.2.a.cz | 2 | ||
| 24.h | odd | 2 | 1 | 9792.2.a.cy | 2 | ||
| 51.c | odd | 2 | 1 | 2601.2.a.t | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.2.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 153.2.a.e | 2 | 3.b | odd | 2 | 1 | ||
| 816.2.a.m | 2 | 4.b | odd | 2 | 1 | ||
| 867.2.a.f | 2 | 17.b | even | 2 | 1 | ||
| 867.2.d.c | 4 | 17.c | even | 4 | 2 | ||
| 867.2.e.f | 8 | 17.d | even | 8 | 4 | ||
| 867.2.h.j | 16 | 17.e | odd | 16 | 8 | ||
| 1275.2.a.n | 2 | 5.b | even | 2 | 1 | ||
| 1275.2.b.d | 4 | 5.c | odd | 4 | 2 | ||
| 2448.2.a.v | 2 | 12.b | even | 2 | 1 | ||
| 2499.2.a.o | 2 | 7.b | odd | 2 | 1 | ||
| 2601.2.a.t | 2 | 51.c | odd | 2 | 1 | ||
| 3264.2.a.bg | 2 | 8.d | odd | 2 | 1 | ||
| 3264.2.a.bl | 2 | 8.b | even | 2 | 1 | ||
| 3825.2.a.s | 2 | 15.d | odd | 2 | 1 | ||
| 6171.2.a.p | 2 | 11.b | odd | 2 | 1 | ||
| 7497.2.a.v | 2 | 21.c | even | 2 | 1 | ||
| 8619.2.a.q | 2 | 13.b | even | 2 | 1 | ||
| 9792.2.a.cy | 2 | 24.h | odd | 2 | 1 | ||
| 9792.2.a.cz | 2 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(51))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + T - 4 \)
|
| $3$ |
\( (T + 1)^{2} \)
|
| $5$ |
\( T^{2} - 3T - 2 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + T - 4 \)
|
| $13$ |
\( T^{2} - 5T + 2 \)
|
| $17$ |
\( (T - 1)^{2} \)
|
| $19$ |
\( T^{2} - 3T - 36 \)
|
| $23$ |
\( T^{2} + 9T + 16 \)
|
| $29$ |
\( T^{2} - 68 \)
|
| $31$ |
\( T^{2} + 2T - 16 \)
|
| $37$ |
\( T^{2} + 2T - 16 \)
|
| $41$ |
\( T^{2} + 3T - 2 \)
|
| $43$ |
\( T^{2} + 3T - 36 \)
|
| $47$ |
\( T^{2} + 14T + 32 \)
|
| $53$ |
\( T^{2} - 8T - 52 \)
|
| $59$ |
\( T^{2} - 6T - 8 \)
|
| $61$ |
\( T^{2} - 10T + 8 \)
|
| $67$ |
\( (T - 4)^{2} \)
|
| $71$ |
\( T^{2} - 4T - 64 \)
|
| $73$ |
\( T^{2} + 8T - 52 \)
|
| $79$ |
\( T^{2} - 6T - 144 \)
|
| $83$ |
\( T^{2} + 10T + 8 \)
|
| $89$ |
\( T^{2} - 6T - 8 \)
|
| $97$ |
\( T^{2} + 14T + 32 \)
|
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