Newspace parameters
| Level: | \( N \) | \(=\) | \( 1690 = 2 \cdot 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1690.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(13.4947179416\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{3}) \) |
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| Defining polynomial: |
\( x^{2} - 3 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 130) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). 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 |
|
1.00000 | −0.732051 | 1.00000 | −1.00000 | −0.732051 | −2.46410 | 1.00000 | −2.46410 | −1.00000 | ||||||||||||||||||||||||
| 1.2 | 1.00000 | 2.73205 | 1.00000 | −1.00000 | 2.73205 | 4.46410 | 1.00000 | 4.46410 | −1.00000 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 1690.2.a.o | 2 | |
| 5.b | even | 2 | 1 | 8450.2.a.z | 2 | ||
| 13.b | even | 2 | 1 | 1690.2.a.l | 2 | ||
| 13.c | even | 3 | 2 | 1690.2.e.k | 4 | ||
| 13.d | odd | 4 | 2 | 1690.2.d.h | 4 | ||
| 13.e | even | 6 | 2 | 130.2.e.d | ✓ | 4 | |
| 13.f | odd | 12 | 2 | 1690.2.l.c | 4 | ||
| 13.f | odd | 12 | 2 | 1690.2.l.d | 4 | ||
| 39.h | odd | 6 | 2 | 1170.2.i.n | 4 | ||
| 52.i | odd | 6 | 2 | 1040.2.q.p | 4 | ||
| 65.d | even | 2 | 1 | 8450.2.a.bg | 2 | ||
| 65.l | even | 6 | 2 | 650.2.e.f | 4 | ||
| 65.r | odd | 12 | 2 | 650.2.o.a | 4 | ||
| 65.r | odd | 12 | 2 | 650.2.o.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.2.e.d | ✓ | 4 | 13.e | even | 6 | 2 | |
| 650.2.e.f | 4 | 65.l | even | 6 | 2 | ||
| 650.2.o.a | 4 | 65.r | odd | 12 | 2 | ||
| 650.2.o.e | 4 | 65.r | odd | 12 | 2 | ||
| 1040.2.q.p | 4 | 52.i | odd | 6 | 2 | ||
| 1170.2.i.n | 4 | 39.h | odd | 6 | 2 | ||
| 1690.2.a.l | 2 | 13.b | even | 2 | 1 | ||
| 1690.2.a.o | 2 | 1.a | even | 1 | 1 | trivial | |
| 1690.2.d.h | 4 | 13.d | odd | 4 | 2 | ||
| 1690.2.e.k | 4 | 13.c | even | 3 | 2 | ||
| 1690.2.l.c | 4 | 13.f | odd | 12 | 2 | ||
| 1690.2.l.d | 4 | 13.f | odd | 12 | 2 | ||
| 8450.2.a.z | 2 | 5.b | even | 2 | 1 | ||
| 8450.2.a.bg | 2 | 65.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1690))\):
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\( T_{3}^{2} - 2T_{3} - 2 \)
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\( T_{7}^{2} - 2T_{7} - 11 \)
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\( T_{11}^{2} - 3 \)
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\( T_{31}^{2} + 10T_{31} + 22 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{2} \)
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| $3$ |
\( T^{2} - 2T - 2 \)
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| $5$ |
\( (T + 1)^{2} \)
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| $7$ |
\( T^{2} - 2T - 11 \)
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| $11$ |
\( T^{2} - 3 \)
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| $13$ |
\( T^{2} \)
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| $17$ |
\( T^{2} - 6T + 6 \)
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| $19$ |
\( T^{2} + 4T - 23 \)
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| $23$ |
\( T^{2} - 12 \)
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| $29$ |
\( T^{2} - 12T + 24 \)
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| $31$ |
\( T^{2} + 10T + 22 \)
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| $37$ |
\( T^{2} - 8T - 11 \)
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| $41$ |
\( T^{2} - 48 \)
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| $43$ |
\( T^{2} - 4T - 44 \)
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| $47$ |
\( T^{2} - 6T - 3 \)
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| $53$ |
\( T^{2} - 3 \)
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| $59$ |
\( T^{2} + 12T + 24 \)
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| $61$ |
\( T^{2} - 10T + 22 \)
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| $67$ |
\( T^{2} + 4T - 104 \)
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| $71$ |
\( T^{2} + 24T + 132 \)
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| $73$ |
\( T^{2} - 2T - 2 \)
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| $79$ |
\( T^{2} - 10T + 22 \)
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| $83$ |
\( T^{2} + 6T - 66 \)
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| $89$ |
\( (T - 9)^{2} \)
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| $97$ |
\( T^{2} - 14T + 46 \)
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