Newspace parameters
| Level: | \( N \) | \(=\) | \( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3300.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(26.3506326670\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{13})\) |
|
|
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| Defining polynomial: |
\( x^{4} + 7x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 660) |
| 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
\( x^{4} + 7x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 4\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 10\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} + 7 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{2} + 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3300\mathbb{Z}\right)^\times\).
| \(n\) | \(1201\) | \(1651\) | \(2201\) | \(2377\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1849.1 |
|
0 | − | 1.00000i | 0 | 0 | 0 | − | 2.60555i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||
| 1849.2 | 0 | − | 1.00000i | 0 | 0 | 0 | 4.60555i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1849.3 | 0 | 1.00000i | 0 | 0 | 0 | − | 4.60555i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1849.4 | 0 | 1.00000i | 0 | 0 | 0 | 2.60555i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3300.2.c.k | 4 | |
| 3.b | odd | 2 | 1 | 9900.2.c.v | 4 | ||
| 5.b | even | 2 | 1 | inner | 3300.2.c.k | 4 | |
| 5.c | odd | 4 | 1 | 660.2.a.f | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 3300.2.a.t | 2 | ||
| 15.d | odd | 2 | 1 | 9900.2.c.v | 4 | ||
| 15.e | even | 4 | 1 | 1980.2.a.j | 2 | ||
| 15.e | even | 4 | 1 | 9900.2.a.bn | 2 | ||
| 20.e | even | 4 | 1 | 2640.2.a.y | 2 | ||
| 55.e | even | 4 | 1 | 7260.2.a.z | 2 | ||
| 60.l | odd | 4 | 1 | 7920.2.a.bn | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 660.2.a.f | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 1980.2.a.j | 2 | 15.e | even | 4 | 1 | ||
| 2640.2.a.y | 2 | 20.e | even | 4 | 1 | ||
| 3300.2.a.t | 2 | 5.c | odd | 4 | 1 | ||
| 3300.2.c.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 3300.2.c.k | 4 | 5.b | even | 2 | 1 | inner | |
| 7260.2.a.z | 2 | 55.e | even | 4 | 1 | ||
| 7920.2.a.bn | 2 | 60.l | odd | 4 | 1 | ||
| 9900.2.a.bn | 2 | 15.e | even | 4 | 1 | ||
| 9900.2.c.v | 4 | 3.b | odd | 2 | 1 | ||
| 9900.2.c.v | 4 | 15.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 \(S_{2}^{\mathrm{new}}(3300, [\chi])\):
|
\( T_{7}^{4} + 28T_{7}^{2} + 144 \)
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\( T_{13}^{4} + 44T_{13}^{2} + 16 \)
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\( T_{17}^{4} + 28T_{17}^{2} + 144 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( (T^{2} + 1)^{2} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + 28T^{2} + 144 \)
|
| $11$ |
\( (T + 1)^{4} \)
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| $13$ |
\( T^{4} + 44T^{2} + 16 \)
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| $17$ |
\( T^{4} + 28T^{2} + 144 \)
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| $19$ |
\( (T + 2)^{4} \)
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| $23$ |
\( T^{4} \)
|
| $29$ |
\( (T^{2} + 4 T - 48)^{2} \)
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| $31$ |
\( (T^{2} + 4 T - 48)^{2} \)
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| $37$ |
\( (T^{2} + 52)^{2} \)
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| $41$ |
\( (T^{2} + 4 T - 48)^{2} \)
|
| $43$ |
\( T^{4} + 28T^{2} + 144 \)
|
| $47$ |
\( T^{4} + 112T^{2} + 2304 \)
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| $53$ |
\( (T^{2} + 36)^{2} \)
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| $59$ |
\( T^{4} \)
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| $61$ |
\( (T^{2} - 8 T - 36)^{2} \)
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| $67$ |
\( (T^{2} + 208)^{2} \)
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| $71$ |
\( (T + 12)^{4} \)
|
| $73$ |
\( T^{4} + 28T^{2} + 144 \)
|
| $79$ |
\( (T + 14)^{4} \)
|
| $83$ |
\( T^{4} + 364 T^{2} + 24336 \)
|
| $89$ |
\( (T^{2} + 4 T - 204)^{2} \)
|
| $97$ |
\( (T^{2} + 100)^{2} \)
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