Newspace parameters
| Level: | \( N \) | \(=\) | \( 1725 = 3 \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1725.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(13.7741943487\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
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| Defining polynomial: |
\( x^{4} + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 345) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{8}^{2} \)
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| \(\beta_{2}\) | \(=\) |
\( \zeta_{8}^{3} + \zeta_{8} \)
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| \(\beta_{3}\) | \(=\) |
\( -\zeta_{8}^{3} + \zeta_{8} \)
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| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
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| \(\zeta_{8}^{2}\) | \(=\) |
\( \beta_1 \)
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| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 2 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times\).
| \(n\) | \(277\) | \(1151\) | \(1201\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1174.1 |
|
− | 1.41421i | − | 1.00000i | 0 | 0 | −1.41421 | 3.82843i | − | 2.82843i | −1.00000 | 0 | |||||||||||||||||||||||||||
| 1174.2 | − | 1.41421i | 1.00000i | 0 | 0 | 1.41421 | 1.82843i | − | 2.82843i | −1.00000 | 0 | |||||||||||||||||||||||||||||
| 1174.3 | 1.41421i | − | 1.00000i | 0 | 0 | 1.41421 | − | 1.82843i | 2.82843i | −1.00000 | 0 | |||||||||||||||||||||||||||||
| 1174.4 | 1.41421i | 1.00000i | 0 | 0 | −1.41421 | − | 3.82843i | 2.82843i | −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 | 1725.2.b.s | 4 | |
| 5.b | even | 2 | 1 | inner | 1725.2.b.s | 4 | |
| 5.c | odd | 4 | 1 | 345.2.a.h | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 1725.2.a.z | 2 | ||
| 15.e | even | 4 | 1 | 1035.2.a.j | 2 | ||
| 15.e | even | 4 | 1 | 5175.2.a.bj | 2 | ||
| 20.e | even | 4 | 1 | 5520.2.a.bm | 2 | ||
| 115.e | even | 4 | 1 | 7935.2.a.q | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 345.2.a.h | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 1035.2.a.j | 2 | 15.e | even | 4 | 1 | ||
| 1725.2.a.z | 2 | 5.c | odd | 4 | 1 | ||
| 1725.2.b.s | 4 | 1.a | even | 1 | 1 | trivial | |
| 1725.2.b.s | 4 | 5.b | even | 2 | 1 | inner | |
| 5175.2.a.bj | 2 | 15.e | even | 4 | 1 | ||
| 5520.2.a.bm | 2 | 20.e | even | 4 | 1 | ||
| 7935.2.a.q | 2 | 115.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1725, [\chi])\):
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\( T_{2}^{2} + 2 \)
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\( T_{7}^{4} + 18T_{7}^{2} + 49 \)
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\( T_{11}^{2} + 8T_{11} + 14 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{2} \)
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| $3$ |
\( (T^{2} + 1)^{2} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + 18T^{2} + 49 \)
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| $11$ |
\( (T^{2} + 8 T + 14)^{2} \)
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| $13$ |
\( T^{4} + 12T^{2} + 4 \)
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| $17$ |
\( T^{4} + 102T^{2} + 2401 \)
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| $19$ |
\( (T^{2} - 4 T - 14)^{2} \)
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| $23$ |
\( (T^{2} + 1)^{2} \)
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| $29$ |
\( (T^{2} - 10 T + 23)^{2} \)
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| $31$ |
\( (T^{2} + 14 T + 41)^{2} \)
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| $37$ |
\( (T^{2} + 9)^{2} \)
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| $41$ |
\( (T^{2} + 18 T + 79)^{2} \)
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| $43$ |
\( (T^{2} + 36)^{2} \)
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| $47$ |
\( T^{4} + 204T^{2} + 9604 \)
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| $53$ |
\( T^{4} + 54T^{2} + 81 \)
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| $59$ |
\( (T^{2} + 6 T - 89)^{2} \)
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| $61$ |
\( (T^{2} - 8 T + 14)^{2} \)
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| $67$ |
\( T^{4} + 162T^{2} + 3969 \)
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| $71$ |
\( (T^{2} + 2 T - 97)^{2} \)
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| $73$ |
\( T^{4} + 108T^{2} + 324 \)
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| $79$ |
\( (T^{2} - 8 T - 112)^{2} \)
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| $83$ |
\( T^{4} + 198T^{2} + 1 \)
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| $89$ |
\( (T^{2} - 8 T - 184)^{2} \)
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| $97$ |
\( T^{4} + 48T^{2} + 64 \)
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