Newspace parameters
| Level: | \( N \) | \(=\) | \( 7744 = 2^{6} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7744.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(61.8361513253\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{33}) \) |
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| Defining polynomial: |
\( x^{2} - x - 8 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 484) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $N(\mathrm{U}(1))$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). 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 | −3.37228 | 0 | 1.37228 | 0 | 0 | 0 | 8.37228 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 2.37228 | 0 | −4.37228 | 0 | 0 | 0 | 2.62772 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(11\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7744.2.a.bx | 2 | |
| 4.b | odd | 2 | 1 | 7744.2.a.cn | 2 | ||
| 8.b | even | 2 | 1 | 484.2.a.d | ✓ | 2 | |
| 8.d | odd | 2 | 1 | 1936.2.a.s | 2 | ||
| 11.b | odd | 2 | 1 | CM | 7744.2.a.bx | 2 | |
| 24.h | odd | 2 | 1 | 4356.2.a.n | 2 | ||
| 44.c | even | 2 | 1 | 7744.2.a.cn | 2 | ||
| 88.b | odd | 2 | 1 | 484.2.a.d | ✓ | 2 | |
| 88.g | even | 2 | 1 | 1936.2.a.s | 2 | ||
| 88.o | even | 10 | 4 | 484.2.e.g | 8 | ||
| 88.p | odd | 10 | 4 | 484.2.e.g | 8 | ||
| 264.m | even | 2 | 1 | 4356.2.a.n | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 484.2.a.d | ✓ | 2 | 8.b | even | 2 | 1 | |
| 484.2.a.d | ✓ | 2 | 88.b | odd | 2 | 1 | |
| 484.2.e.g | 8 | 88.o | even | 10 | 4 | ||
| 484.2.e.g | 8 | 88.p | odd | 10 | 4 | ||
| 1936.2.a.s | 2 | 8.d | odd | 2 | 1 | ||
| 1936.2.a.s | 2 | 88.g | even | 2 | 1 | ||
| 4356.2.a.n | 2 | 24.h | odd | 2 | 1 | ||
| 4356.2.a.n | 2 | 264.m | even | 2 | 1 | ||
| 7744.2.a.bx | 2 | 1.a | even | 1 | 1 | trivial | |
| 7744.2.a.bx | 2 | 11.b | odd | 2 | 1 | CM | |
| 7744.2.a.cn | 2 | 4.b | odd | 2 | 1 | ||
| 7744.2.a.cn | 2 | 44.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7744))\):
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\( T_{3}^{2} + T_{3} - 8 \)
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\( T_{5}^{2} + 3T_{5} - 6 \)
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\( T_{7} \)
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\( T_{13} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
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| $3$ |
\( T^{2} + T - 8 \)
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| $5$ |
\( T^{2} + 3T - 6 \)
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| $7$ |
\( T^{2} \)
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| $11$ |
\( T^{2} \)
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| $13$ |
\( T^{2} \)
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| $17$ |
\( T^{2} \)
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| $19$ |
\( T^{2} \)
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| $23$ |
\( T^{2} - 9T + 12 \)
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| $29$ |
\( T^{2} \)
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| $31$ |
\( T^{2} - 5T - 68 \)
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| $37$ |
\( T^{2} - 7T - 62 \)
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| $41$ |
\( T^{2} \)
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| $43$ |
\( T^{2} \)
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| $47$ |
\( (T + 12)^{2} \)
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| $53$ |
\( (T + 6)^{2} \)
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| $59$ |
\( T^{2} + 15T + 48 \)
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| $61$ |
\( T^{2} \)
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| $67$ |
\( T^{2} - 13T - 32 \)
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| $71$ |
\( T^{2} - 3T - 204 \)
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| $73$ |
\( T^{2} \)
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| $79$ |
\( T^{2} \)
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| $83$ |
\( T^{2} \)
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| $89$ |
\( T^{2} - 9T - 186 \)
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| $97$ |
\( T^{2} + 17T - 2 \)
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