Newspace parameters
| Level: | \( N \) | \(=\) | \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7623.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(60.8699614608\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.568.1 |
|
|
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| Defining polynomial: |
\( x^{3} - x^{2} - 6x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 847) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 6x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 4 \)
|
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 |
|
−2.76156 | 0 | 5.62620 | 2.62620 | 0 | 1.00000 | −10.0140 | 0 | −7.25240 | |||||||||||||||||||||||||||
| 1.2 | −1.36333 | 0 | −0.141336 | −3.14134 | 0 | 1.00000 | 2.91934 | 0 | 4.28267 | ||||||||||||||||||||||||||||
| 1.3 | 2.12489 | 0 | 2.51514 | −0.484862 | 0 | 1.00000 | 1.09461 | 0 | −1.03028 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(11\) | \( +1 \) |
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 | 7623.2.a.bz | 3 | |
| 3.b | odd | 2 | 1 | 847.2.a.j | yes | 3 | |
| 11.b | odd | 2 | 1 | 7623.2.a.ce | 3 | ||
| 21.c | even | 2 | 1 | 5929.2.a.y | 3 | ||
| 33.d | even | 2 | 1 | 847.2.a.i | ✓ | 3 | |
| 33.f | even | 10 | 4 | 847.2.f.u | 12 | ||
| 33.h | odd | 10 | 4 | 847.2.f.t | 12 | ||
| 231.h | odd | 2 | 1 | 5929.2.a.t | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 847.2.a.i | ✓ | 3 | 33.d | even | 2 | 1 | |
| 847.2.a.j | yes | 3 | 3.b | odd | 2 | 1 | |
| 847.2.f.t | 12 | 33.h | odd | 10 | 4 | ||
| 847.2.f.u | 12 | 33.f | even | 10 | 4 | ||
| 5929.2.a.t | 3 | 231.h | odd | 2 | 1 | ||
| 5929.2.a.y | 3 | 21.c | even | 2 | 1 | ||
| 7623.2.a.bz | 3 | 1.a | even | 1 | 1 | trivial | |
| 7623.2.a.ce | 3 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):
|
\( T_{2}^{3} + 2T_{2}^{2} - 5T_{2} - 8 \)
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\( T_{5}^{3} + T_{5}^{2} - 8T_{5} - 4 \)
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\( T_{13}^{3} - 8T_{13}^{2} + 2T_{13} + 64 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 2 T^{2} + \cdots - 8 \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} + T^{2} - 8T - 4 \)
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| $7$ |
\( (T - 1)^{3} \)
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| $11$ |
\( T^{3} \)
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| $13$ |
\( T^{3} - 8 T^{2} + \cdots + 64 \)
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| $17$ |
\( T^{3} + 8 T^{2} + \cdots - 64 \)
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| $19$ |
\( T^{3} - 40T - 64 \)
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| $23$ |
\( T^{3} + 7 T^{2} + \cdots - 8 \)
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| $29$ |
\( T^{3} - 40T - 64 \)
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| $31$ |
\( T^{3} + 13 T^{2} + \cdots + 58 \)
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| $37$ |
\( T^{3} + 17 T^{2} + \cdots + 16 \)
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| $41$ |
\( T^{3} + 16 T^{2} + \cdots + 80 \)
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| $43$ |
\( T^{3} - 4 T^{2} + \cdots + 32 \)
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| $47$ |
\( T^{3} + 4 T^{2} + \cdots + 8 \)
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| $53$ |
\( T^{3} - 10T^{2} + 64 \)
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| $59$ |
\( T^{3} + T^{2} - 6T + 2 \)
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| $61$ |
\( T^{3} + 16 T^{2} + \cdots + 80 \)
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| $67$ |
\( T^{3} + 3 T^{2} + \cdots - 424 \)
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| $71$ |
\( T^{3} + 5 T^{2} + \cdots - 1480 \)
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| $73$ |
\( T^{3} - 16 T^{2} + \cdots - 80 \)
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| $79$ |
\( T^{3} - 28 T^{2} + \cdots - 512 \)
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| $83$ |
\( T^{3} + 8 T^{2} + \cdots + 64 \)
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| $89$ |
\( T^{3} - 21 T^{2} + \cdots - 100 \)
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| $97$ |
\( T^{3} + 11 T^{2} + \cdots - 452 \)
|
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