Newspace parameters
| Level: | \( N \) | \(=\) | \( 2890 = 2 \cdot 5 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2890.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(23.0767661842\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{19})\) |
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|
|
| Defining polynomial: |
\( x^{4} - 9x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| 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
\( x^{4} - 9x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 4\nu ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 14\nu ) / 5 \)
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| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 9 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 9 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} + 7\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2890\mathbb{Z}\right)^\times\).
| \(n\) | \(581\) | \(1157\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2311.1 |
|
1.00000 | − | 1.00000i | 1.00000 | − | 1.00000i | − | 1.00000i | − | 4.35890i | 1.00000 | 2.00000 | − | 1.00000i | |||||||||||||||||||||||||
| 2311.2 | 1.00000 | − | 1.00000i | 1.00000 | − | 1.00000i | − | 1.00000i | 4.35890i | 1.00000 | 2.00000 | − | 1.00000i | |||||||||||||||||||||||||||
| 2311.3 | 1.00000 | 1.00000i | 1.00000 | 1.00000i | 1.00000i | − | 4.35890i | 1.00000 | 2.00000 | 1.00000i | ||||||||||||||||||||||||||||||
| 2311.4 | 1.00000 | 1.00000i | 1.00000 | 1.00000i | 1.00000i | 4.35890i | 1.00000 | 2.00000 | 1.00000i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2890.2.b.k | 4 | |
| 17.b | even | 2 | 1 | inner | 2890.2.b.k | 4 | |
| 17.c | even | 4 | 1 | 2890.2.a.r | ✓ | 2 | |
| 17.c | even | 4 | 1 | 2890.2.a.s | yes | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2890.2.a.r | ✓ | 2 | 17.c | even | 4 | 1 | |
| 2890.2.a.s | yes | 2 | 17.c | even | 4 | 1 | |
| 2890.2.b.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 2890.2.b.k | 4 | 17.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2890, [\chi])\):
|
\( T_{3}^{2} + 1 \)
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\( T_{7}^{2} + 19 \)
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\( T_{11}^{4} + 40T_{11}^{2} + 324 \)
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\( T_{13}^{2} - 2T_{13} - 18 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
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| $3$ |
\( (T^{2} + 1)^{2} \)
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| $5$ |
\( (T^{2} + 1)^{2} \)
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| $7$ |
\( (T^{2} + 19)^{2} \)
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| $11$ |
\( T^{4} + 40T^{2} + 324 \)
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| $13$ |
\( (T^{2} - 2 T - 18)^{2} \)
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| $17$ |
\( T^{4} \)
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| $19$ |
\( (T^{2} - 6 T - 10)^{2} \)
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| $23$ |
\( T^{4} + 70T^{2} + 9 \)
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| $29$ |
\( T^{4} \)
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| $31$ |
\( T^{4} + 56T^{2} + 100 \)
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| $37$ |
\( T^{4} + 40T^{2} + 324 \)
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| $41$ |
\( T^{4} + 154T^{2} + 5625 \)
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| $43$ |
\( (T - 10)^{4} \)
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| $47$ |
\( (T^{2} - 4 T - 72)^{2} \)
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| $53$ |
\( (T^{2} - 4 T - 72)^{2} \)
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| $59$ |
\( (T - 6)^{4} \)
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| $61$ |
\( (T^{2} + 100)^{2} \)
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| $67$ |
\( (T + 7)^{4} \)
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| $71$ |
\( T^{4} + 88T^{2} + 36 \)
|
| $73$ |
\( T^{4} + 56T^{2} + 100 \)
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| $79$ |
\( T^{4} + 200T^{2} + 3844 \)
|
| $83$ |
\( (T^{2} + 14 T - 27)^{2} \)
|
| $89$ |
\( (T + 9)^{4} \)
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| $97$ |
\( T^{4} + 200T^{2} + 3844 \)
|
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