Newspace parameters
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 441.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.52140272914\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{7}) \) |
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| Defining polynomial: |
\( x^{2} - 7 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $N(\mathrm{U}(1))$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). 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 |
|
−2.64575 | 0 | 5.00000 | 0 | 0 | 0 | −7.93725 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 2.64575 | 0 | 5.00000 | 0 | 0 | 0 | 7.93725 | 0 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 3.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 441.2.a.h | ✓ | 2 |
| 3.b | odd | 2 | 1 | inner | 441.2.a.h | ✓ | 2 |
| 4.b | odd | 2 | 1 | 7056.2.a.co | 2 | ||
| 7.b | odd | 2 | 1 | CM | 441.2.a.h | ✓ | 2 |
| 7.c | even | 3 | 2 | 441.2.e.h | 4 | ||
| 7.d | odd | 6 | 2 | 441.2.e.h | 4 | ||
| 12.b | even | 2 | 1 | 7056.2.a.co | 2 | ||
| 21.c | even | 2 | 1 | inner | 441.2.a.h | ✓ | 2 |
| 21.g | even | 6 | 2 | 441.2.e.h | 4 | ||
| 21.h | odd | 6 | 2 | 441.2.e.h | 4 | ||
| 28.d | even | 2 | 1 | 7056.2.a.co | 2 | ||
| 84.h | odd | 2 | 1 | 7056.2.a.co | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 441.2.a.h | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 441.2.a.h | ✓ | 2 | 3.b | odd | 2 | 1 | inner |
| 441.2.a.h | ✓ | 2 | 7.b | odd | 2 | 1 | CM |
| 441.2.a.h | ✓ | 2 | 21.c | even | 2 | 1 | inner |
| 441.2.e.h | 4 | 7.c | even | 3 | 2 | ||
| 441.2.e.h | 4 | 7.d | odd | 6 | 2 | ||
| 441.2.e.h | 4 | 21.g | even | 6 | 2 | ||
| 441.2.e.h | 4 | 21.h | odd | 6 | 2 | ||
| 7056.2.a.co | 2 | 4.b | odd | 2 | 1 | ||
| 7056.2.a.co | 2 | 12.b | even | 2 | 1 | ||
| 7056.2.a.co | 2 | 28.d | even | 2 | 1 | ||
| 7056.2.a.co | 2 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(441))\):
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\( T_{2}^{2} - 7 \)
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\( T_{5} \)
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\( T_{13} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 7 \)
|
| $3$ |
\( T^{2} \)
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| $5$ |
\( T^{2} \)
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| $7$ |
\( T^{2} \)
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| $11$ |
\( T^{2} - 28 \)
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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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| $23$ |
\( T^{2} - 28 \)
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| $29$ |
\( T^{2} - 112 \)
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| $31$ |
\( T^{2} \)
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| $37$ |
\( (T - 6)^{2} \)
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| $41$ |
\( T^{2} \)
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| $43$ |
\( (T - 12)^{2} \)
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| $47$ |
\( T^{2} \)
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| $53$ |
\( T^{2} - 112 \)
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| $59$ |
\( T^{2} \)
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| $61$ |
\( T^{2} \)
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| $67$ |
\( (T - 4)^{2} \)
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| $71$ |
\( T^{2} - 28 \)
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| $73$ |
\( T^{2} \)
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| $79$ |
\( (T - 8)^{2} \)
|
| $83$ |
\( T^{2} \)
|
| $89$ |
\( T^{2} \)
|
| $97$ |
\( T^{2} \)
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