Newspace parameters
| Level: | \( N \) | \(=\) | \( 850 = 2 \cdot 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 850.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.78728417181\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{17})\) |
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| Defining polynomial: |
\( x^{4} + 9x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 170) |
| 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
\( x^{4} + 9x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 4 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 5 \)
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| \(\nu^{3}\) | \(=\) |
\( 4\beta_{2} - 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/850\mathbb{Z}\right)^\times\).
| \(n\) | \(477\) | \(751\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 749.1 |
|
− | 1.00000i | − | 2.56155i | −1.00000 | 0 | −2.56155 | − | 5.12311i | 1.00000i | −3.56155 | 0 | |||||||||||||||||||||||||||
| 749.2 | − | 1.00000i | 1.56155i | −1.00000 | 0 | 1.56155 | 3.12311i | 1.00000i | 0.561553 | 0 | ||||||||||||||||||||||||||||||
| 749.3 | 1.00000i | − | 1.56155i | −1.00000 | 0 | 1.56155 | − | 3.12311i | − | 1.00000i | 0.561553 | 0 | ||||||||||||||||||||||||||||
| 749.4 | 1.00000i | 2.56155i | −1.00000 | 0 | −2.56155 | 5.12311i | − | 1.00000i | −3.56155 | 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 | 850.2.c.i | 4 | |
| 5.b | even | 2 | 1 | inner | 850.2.c.i | 4 | |
| 5.c | odd | 4 | 1 | 170.2.a.f | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 850.2.a.n | 2 | ||
| 15.e | even | 4 | 1 | 1530.2.a.r | 2 | ||
| 15.e | even | 4 | 1 | 7650.2.a.de | 2 | ||
| 20.e | even | 4 | 1 | 1360.2.a.m | 2 | ||
| 20.e | even | 4 | 1 | 6800.2.a.be | 2 | ||
| 35.f | even | 4 | 1 | 8330.2.a.bq | 2 | ||
| 40.i | odd | 4 | 1 | 5440.2.a.bj | 2 | ||
| 40.k | even | 4 | 1 | 5440.2.a.bd | 2 | ||
| 85.f | odd | 4 | 1 | 2890.2.b.i | 4 | ||
| 85.g | odd | 4 | 1 | 2890.2.a.u | 2 | ||
| 85.i | odd | 4 | 1 | 2890.2.b.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 170.2.a.f | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 850.2.a.n | 2 | 5.c | odd | 4 | 1 | ||
| 850.2.c.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 850.2.c.i | 4 | 5.b | even | 2 | 1 | inner | |
| 1360.2.a.m | 2 | 20.e | even | 4 | 1 | ||
| 1530.2.a.r | 2 | 15.e | even | 4 | 1 | ||
| 2890.2.a.u | 2 | 85.g | odd | 4 | 1 | ||
| 2890.2.b.i | 4 | 85.f | odd | 4 | 1 | ||
| 2890.2.b.i | 4 | 85.i | odd | 4 | 1 | ||
| 5440.2.a.bd | 2 | 40.k | even | 4 | 1 | ||
| 5440.2.a.bj | 2 | 40.i | odd | 4 | 1 | ||
| 6800.2.a.be | 2 | 20.e | even | 4 | 1 | ||
| 7650.2.a.de | 2 | 15.e | even | 4 | 1 | ||
| 8330.2.a.bq | 2 | 35.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(850, [\chi])\):
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\( T_{3}^{4} + 9T_{3}^{2} + 16 \)
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\( T_{7}^{4} + 36T_{7}^{2} + 256 \)
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\( T_{11} + 4 \)
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\( T_{13}^{4} + 21T_{13}^{2} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
|
| $3$ |
\( T^{4} + 9T^{2} + 16 \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + 36T^{2} + 256 \)
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| $11$ |
\( (T + 4)^{4} \)
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| $13$ |
\( T^{4} + 21T^{2} + 4 \)
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| $17$ |
\( (T^{2} + 1)^{2} \)
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| $19$ |
\( (T^{2} - T - 4)^{2} \)
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| $23$ |
\( T^{4} + 36T^{2} + 256 \)
|
| $29$ |
\( (T^{2} + T - 38)^{2} \)
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| $31$ |
\( (T^{2} + 9 T + 16)^{2} \)
|
| $37$ |
\( T^{4} + 52T^{2} + 64 \)
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| $41$ |
\( (T^{2} + 8 T - 52)^{2} \)
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| $43$ |
\( T^{4} + 52T^{2} + 64 \)
|
| $47$ |
\( T^{4} + 49T^{2} + 256 \)
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| $53$ |
\( T^{4} + 13T^{2} + 4 \)
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| $59$ |
\( (T^{2} - 13 T + 4)^{2} \)
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| $61$ |
\( (T^{2} - 19 T + 86)^{2} \)
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| $67$ |
\( T^{4} + 84T^{2} + 64 \)
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| $71$ |
\( (T^{2} + 21 T + 72)^{2} \)
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| $73$ |
\( T^{4} + 237T^{2} + 8836 \)
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| $79$ |
\( T^{4} \)
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| $83$ |
\( T^{4} + 144T^{2} + 4096 \)
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| $89$ |
\( (T^{2} - 7 T - 26)^{2} \)
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| $97$ |
\( T^{4} + 117T^{2} + 324 \)
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