Newspace parameters
| Level: | \( N \) | \(=\) | \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3042.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(24.2904922949\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{10})\) |
|
|
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| Defining polynomial: |
\( x^{4} + 25 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 234) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 5\nu ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{3} + 5\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times\).
| \(n\) | \(677\) | \(847\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1351.1 |
|
− | 1.00000i | 0 | −1.00000 | − | 4.16228i | 0 | − | 3.16228i | 1.00000i | 0 | −4.16228 | |||||||||||||||||||||||||||
| 1351.2 | − | 1.00000i | 0 | −1.00000 | 2.16228i | 0 | 3.16228i | 1.00000i | 0 | 2.16228 | ||||||||||||||||||||||||||||||
| 1351.3 | 1.00000i | 0 | −1.00000 | − | 2.16228i | 0 | − | 3.16228i | − | 1.00000i | 0 | 2.16228 | ||||||||||||||||||||||||||||
| 1351.4 | 1.00000i | 0 | −1.00000 | 4.16228i | 0 | 3.16228i | − | 1.00000i | 0 | −4.16228 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3042.2.b.j | 4 | |
| 3.b | odd | 2 | 1 | 3042.2.b.k | 4 | ||
| 13.b | even | 2 | 1 | inner | 3042.2.b.j | 4 | |
| 13.d | odd | 4 | 1 | 3042.2.a.r | 2 | ||
| 13.d | odd | 4 | 1 | 3042.2.a.x | 2 | ||
| 13.f | odd | 12 | 2 | 234.2.h.e | yes | 4 | |
| 39.d | odd | 2 | 1 | 3042.2.b.k | 4 | ||
| 39.f | even | 4 | 1 | 3042.2.a.q | 2 | ||
| 39.f | even | 4 | 1 | 3042.2.a.w | 2 | ||
| 39.k | even | 12 | 2 | 234.2.h.d | ✓ | 4 | |
| 52.l | even | 12 | 2 | 1872.2.t.n | 4 | ||
| 156.v | odd | 12 | 2 | 1872.2.t.p | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 234.2.h.d | ✓ | 4 | 39.k | even | 12 | 2 | |
| 234.2.h.e | yes | 4 | 13.f | odd | 12 | 2 | |
| 1872.2.t.n | 4 | 52.l | even | 12 | 2 | ||
| 1872.2.t.p | 4 | 156.v | odd | 12 | 2 | ||
| 3042.2.a.q | 2 | 39.f | even | 4 | 1 | ||
| 3042.2.a.r | 2 | 13.d | odd | 4 | 1 | ||
| 3042.2.a.w | 2 | 39.f | even | 4 | 1 | ||
| 3042.2.a.x | 2 | 13.d | odd | 4 | 1 | ||
| 3042.2.b.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 3042.2.b.j | 4 | 13.b | even | 2 | 1 | inner | |
| 3042.2.b.k | 4 | 3.b | odd | 2 | 1 | ||
| 3042.2.b.k | 4 | 39.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3042, [\chi])\):
|
\( T_{5}^{4} + 22T_{5}^{2} + 81 \)
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\( T_{7}^{2} + 10 \)
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\( T_{17} + 3 \)
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\( T_{23}^{2} - 8T_{23} + 6 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 22T^{2} + 81 \)
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| $7$ |
\( (T^{2} + 10)^{2} \)
|
| $11$ |
\( T^{4} + 28T^{2} + 36 \)
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| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T + 3)^{4} \)
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| $19$ |
\( T^{4} + 28T^{2} + 36 \)
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| $23$ |
\( (T^{2} - 8 T + 6)^{2} \)
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| $29$ |
\( (T^{2} - 10 T + 15)^{2} \)
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| $31$ |
\( (T^{2} + 40)^{2} \)
|
| $37$ |
\( T^{4} + 118T^{2} + 1521 \)
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| $41$ |
\( (T^{2} + 9)^{2} \)
|
| $43$ |
\( (T^{2} + 12 T + 26)^{2} \)
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| $47$ |
\( T^{4} + 148T^{2} + 2916 \)
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| $53$ |
\( (T^{2} - 6 T - 81)^{2} \)
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| $59$ |
\( T^{4} + 112T^{2} + 576 \)
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| $61$ |
\( (T^{2} + 6 T - 1)^{2} \)
|
| $67$ |
\( T^{4} + 92T^{2} + 676 \)
|
| $71$ |
\( T^{4} + 52T^{2} + 36 \)
|
| $73$ |
\( (T^{2} + 1)^{2} \)
|
| $79$ |
\( (T + 4)^{4} \)
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| $83$ |
\( T^{4} + 252T^{2} + 2916 \)
|
| $89$ |
\( (T^{2} + 144)^{2} \)
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| $97$ |
\( (T^{2} + 16)^{2} \)
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