Newspace parameters
| Level: | \( N \) | \(=\) | \( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6800.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(54.2982733745\) |
| Analytic rank: | \(1\) |
| 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 68) |
| 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 |
|
0 | −0.732051 | 0 | 0 | 0 | 0.732051 | 0 | −2.46410 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 2.73205 | 0 | 0 | 0 | −2.73205 | 0 | 4.46410 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(17\) | \( -1 \) |
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 | 6800.2.a.bh | 2 | |
| 4.b | odd | 2 | 1 | 1700.2.a.d | 2 | ||
| 5.b | even | 2 | 1 | 272.2.a.e | 2 | ||
| 15.d | odd | 2 | 1 | 2448.2.a.y | 2 | ||
| 20.d | odd | 2 | 1 | 68.2.a.a | ✓ | 2 | |
| 20.e | even | 4 | 2 | 1700.2.e.c | 4 | ||
| 40.e | odd | 2 | 1 | 1088.2.a.p | 2 | ||
| 40.f | even | 2 | 1 | 1088.2.a.t | 2 | ||
| 60.h | even | 2 | 1 | 612.2.a.e | 2 | ||
| 85.c | even | 2 | 1 | 4624.2.a.x | 2 | ||
| 120.i | odd | 2 | 1 | 9792.2.a.cs | 2 | ||
| 120.m | even | 2 | 1 | 9792.2.a.cr | 2 | ||
| 140.c | even | 2 | 1 | 3332.2.a.h | 2 | ||
| 220.g | even | 2 | 1 | 8228.2.a.k | 2 | ||
| 340.d | odd | 2 | 1 | 1156.2.a.a | 2 | ||
| 340.n | odd | 4 | 2 | 1156.2.b.c | 4 | ||
| 340.ba | odd | 8 | 4 | 1156.2.e.d | 8 | ||
| 340.bg | even | 16 | 8 | 1156.2.h.f | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 68.2.a.a | ✓ | 2 | 20.d | odd | 2 | 1 | |
| 272.2.a.e | 2 | 5.b | even | 2 | 1 | ||
| 612.2.a.e | 2 | 60.h | even | 2 | 1 | ||
| 1088.2.a.p | 2 | 40.e | odd | 2 | 1 | ||
| 1088.2.a.t | 2 | 40.f | even | 2 | 1 | ||
| 1156.2.a.a | 2 | 340.d | odd | 2 | 1 | ||
| 1156.2.b.c | 4 | 340.n | odd | 4 | 2 | ||
| 1156.2.e.d | 8 | 340.ba | odd | 8 | 4 | ||
| 1156.2.h.f | 16 | 340.bg | even | 16 | 8 | ||
| 1700.2.a.d | 2 | 4.b | odd | 2 | 1 | ||
| 1700.2.e.c | 4 | 20.e | even | 4 | 2 | ||
| 2448.2.a.y | 2 | 15.d | odd | 2 | 1 | ||
| 3332.2.a.h | 2 | 140.c | even | 2 | 1 | ||
| 4624.2.a.x | 2 | 85.c | even | 2 | 1 | ||
| 6800.2.a.bh | 2 | 1.a | even | 1 | 1 | trivial | |
| 8228.2.a.k | 2 | 220.g | even | 2 | 1 | ||
| 9792.2.a.cr | 2 | 120.m | even | 2 | 1 | ||
| 9792.2.a.cs | 2 | 120.i | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6800))\):
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\( T_{3}^{2} - 2T_{3} - 2 \)
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\( T_{7}^{2} + 2T_{7} - 2 \)
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\( T_{11}^{2} - 6T_{11} + 6 \)
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\( T_{13}^{2} + 4T_{13} - 8 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
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| $3$ |
\( T^{2} - 2T - 2 \)
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| $5$ |
\( T^{2} \)
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| $7$ |
\( T^{2} + 2T - 2 \)
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| $11$ |
\( T^{2} - 6T + 6 \)
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| $13$ |
\( T^{2} + 4T - 8 \)
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| $17$ |
\( (T - 1)^{2} \)
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| $19$ |
\( T^{2} + 4T - 8 \)
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| $23$ |
\( T^{2} + 6T + 6 \)
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| $29$ |
\( T^{2} - 12 \)
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| $31$ |
\( T^{2} - 2T - 26 \)
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| $37$ |
\( T^{2} + 16T + 52 \)
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| $41$ |
\( (T + 6)^{2} \)
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| $43$ |
\( T^{2} - 4T - 104 \)
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| $47$ |
\( T^{2} - 48 \)
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| $53$ |
\( T^{2} + 12T - 12 \)
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| $59$ |
\( T^{2} + 12T + 24 \)
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| $61$ |
\( T^{2} + 8T + 4 \)
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| $67$ |
\( T^{2} - 16T + 16 \)
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| $71$ |
\( T^{2} - 6T - 18 \)
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| $73$ |
\( (T + 2)^{2} \)
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| $79$ |
\( T^{2} - 14T + 22 \)
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| $83$ |
\( T^{2} + 12T + 24 \)
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| $89$ |
\( T^{2} - 12T + 24 \)
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| $97$ |
\( T^{2} + 4T - 44 \)
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