Newspace parameters
| Level: | \( N \) | \(=\) | \( 103 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 103.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.80654672291\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.33153125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 10x^{3} + 20x - 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 10x^{3} + 20x - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 6\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} - \nu + 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 8\beta_{2} + \beta _1 + 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/103\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 102.1 |
|
−3.93313 | 0 | 11.4695 | 0 | 0 | −8.24529 | −29.3787 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 102.2 | −1.90809 | 0 | −0.359188 | 0 | 0 | 0.0201584 | 8.31773 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 102.3 | −0.522718 | 0 | −3.72677 | 0 | 0 | 13.3210 | 4.03892 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 102.4 | 2.75387 | 0 | 3.58378 | 0 | 0 | 8.21268 | −1.14620 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 102.5 | 3.61007 | 0 | 9.03264 | 0 | 0 | −13.3085 | 18.1682 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 103.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-103}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 103.3.b.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 927.3.d.a | 5 | ||
| 103.b | odd | 2 | 1 | CM | 103.3.b.a | ✓ | 5 |
| 309.c | even | 2 | 1 | 927.3.d.a | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 103.3.b.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 103.3.b.a | ✓ | 5 | 103.b | odd | 2 | 1 | CM |
| 927.3.d.a | 5 | 3.b | odd | 2 | 1 | ||
| 927.3.d.a | 5 | 309.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 20T_{2}^{3} + 80T_{2} + 39 \)
acting on \(S_{3}^{\mathrm{new}}(103, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 20 T^{3} + \cdots + 39 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - 245 T^{3} + \cdots - 242 \)
|
| $11$ |
\( T^{5} \)
|
| $13$ |
\( T^{5} - 845 T^{3} + \cdots - 740938 \)
|
| $17$ |
\( T^{5} - 1445 T^{3} + \cdots + 1616478 \)
|
| $19$ |
\( T^{5} - 1805 T^{3} + \cdots - 3880586 \)
|
| $23$ |
\( T^{5} - 2645 T^{3} + \cdots - 12852498 \)
|
| $29$ |
\( T^{5} - 4205 T^{3} + \cdots + 40931094 \)
|
| $31$ |
\( T^{5} \)
|
| $37$ |
\( T^{5} \)
|
| $41$ |
\( T^{5} - 8405 T^{3} + \cdots + 73002798 \)
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| $43$ |
\( T^{5} \)
|
| $47$ |
\( T^{5} \)
|
| $53$ |
\( T^{5} \)
|
| $59$ |
\( T^{5} - 17405 T^{3} + \cdots - 666692346 \)
|
| $61$ |
\( T^{5} + \cdots + 1536274646 \)
|
| $67$ |
\( T^{5} \)
|
| $71$ |
\( T^{5} \)
|
| $73$ |
\( T^{5} \)
|
| $79$ |
\( T^{5} + \cdots - 6124300066 \)
|
| $83$ |
\( T^{5} + \cdots - 4314495114 \)
|
| $89$ |
\( T^{5} \)
|
| $97$ |
\( T^{5} + \cdots - 12924099586 \)
|
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