Newspace parameters
| Level: | \( N \) | \(=\) | \( 2952 = 2^{3} \cdot 3^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2952.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.47323991729\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
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| Defining polynomial: |
\( x^{2} - 2 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{4}\) |
| Projective field: | Galois closure of 4.2.121032.1 |
| Artin image: | $D_8$ |
| Artin field: | Galois closure of 8.0.205797003264.3 |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2952\mathbb{Z}\right)^\times\).
| \(n\) | \(1441\) | \(1477\) | \(2215\) | \(2297\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 163.1 |
|
1.00000 | 0 | 1.00000 | 0 | 0 | −1.41421 | 1.00000 | 0 | 0 | ||||||||||||||||||||||||
| 163.2 | 1.00000 | 0 | 1.00000 | 0 | 0 | 1.41421 | 1.00000 | 0 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 328.c | odd | 2 | 1 | CM by \(\Q(\sqrt{-82}) \) |
| 8.d | odd | 2 | 1 | inner |
| 41.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2952.1.h.c | yes | 2 |
| 3.b | odd | 2 | 1 | 2952.1.h.b | ✓ | 2 | |
| 8.d | odd | 2 | 1 | inner | 2952.1.h.c | yes | 2 |
| 24.f | even | 2 | 1 | 2952.1.h.b | ✓ | 2 | |
| 41.b | even | 2 | 1 | inner | 2952.1.h.c | yes | 2 |
| 123.b | odd | 2 | 1 | 2952.1.h.b | ✓ | 2 | |
| 328.c | odd | 2 | 1 | CM | 2952.1.h.c | yes | 2 |
| 984.p | even | 2 | 1 | 2952.1.h.b | ✓ | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2952.1.h.b | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 2952.1.h.b | ✓ | 2 | 24.f | even | 2 | 1 | |
| 2952.1.h.b | ✓ | 2 | 123.b | odd | 2 | 1 | |
| 2952.1.h.b | ✓ | 2 | 984.p | even | 2 | 1 | |
| 2952.1.h.c | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 2952.1.h.c | yes | 2 | 8.d | odd | 2 | 1 | inner |
| 2952.1.h.c | yes | 2 | 41.b | even | 2 | 1 | inner |
| 2952.1.h.c | yes | 2 | 328.c | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2952, [\chi])\):
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\( T_{7}^{2} - 2 \)
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\( T_{59} + 2 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{2} \)
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| $3$ |
\( T^{2} \)
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| $5$ |
\( T^{2} \)
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| $7$ |
\( T^{2} - 2 \)
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| $11$ |
\( T^{2} \)
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| $13$ |
\( T^{2} - 2 \)
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| $17$ |
\( T^{2} \)
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| $19$ |
\( T^{2} \)
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| $23$ |
\( T^{2} \)
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| $29$ |
\( T^{2} - 2 \)
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| $31$ |
\( T^{2} \)
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| $37$ |
\( T^{2} \)
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| $41$ |
\( (T + 1)^{2} \)
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| $43$ |
\( T^{2} \)
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| $47$ |
\( T^{2} - 2 \)
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| $53$ |
\( T^{2} - 2 \)
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| $59$ |
\( (T + 2)^{2} \)
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| $61$ |
\( T^{2} \)
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| $67$ |
\( T^{2} \)
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| $71$ |
\( T^{2} - 2 \)
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| $73$ |
\( T^{2} \)
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| $79$ |
\( T^{2} - 2 \)
|
| $83$ |
\( T^{2} \)
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| $89$ |
\( T^{2} \)
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| $97$ |
\( T^{2} \)
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