Newspace parameters
| Level: | \( N \) | \(=\) | \( 348 = 2^{2} \cdot 3 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 348.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(9.48231319974\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-5}, \sqrt{-29})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 17x^{2} + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{29}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} + 17x^{2} + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 11\nu ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{2} - 11\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/348\mathbb{Z}\right)^\times\).
| \(n\) | \(175\) | \(205\) | \(233\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 233.1 |
|
0 | 2.00000 | − | 2.23607i | 0 | 4.47214i | 0 | −11.5208 | 0 | −1.00000 | − | 8.94427i | 0 | ||||||||||||||||||||||||||
| 233.2 | 0 | 2.00000 | − | 2.23607i | 0 | 4.47214i | 0 | 0.520797 | 0 | −1.00000 | − | 8.94427i | 0 | |||||||||||||||||||||||||||
| 233.3 | 0 | 2.00000 | + | 2.23607i | 0 | − | 4.47214i | 0 | −11.5208 | 0 | −1.00000 | + | 8.94427i | 0 | ||||||||||||||||||||||||||
| 233.4 | 0 | 2.00000 | + | 2.23607i | 0 | − | 4.47214i | 0 | 0.520797 | 0 | −1.00000 | + | 8.94427i | 0 | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 348.3.d.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 348.3.d.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 348.3.d.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 348.3.d.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 20 \)
acting on \(S_{3}^{\mathrm{new}}(348, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 4 T + 9)^{2} \)
|
| $5$ |
\( (T^{2} + 20)^{2} \)
|
| $7$ |
\( (T^{2} + 11 T - 6)^{2} \)
|
| $11$ |
\( T^{4} + 385 T^{2} + 28900 \)
|
| $13$ |
\( (T^{2} + 9 T - 306)^{2} \)
|
| $17$ |
\( T^{4} + 425 T^{2} + 22500 \)
|
| $19$ |
\( (T^{2} - 12 T - 544)^{2} \)
|
| $23$ |
\( T^{4} + 1460 T^{2} + 518400 \)
|
| $29$ |
\( (T^{2} + 29)^{2} \)
|
| $31$ |
\( (T^{2} - 78 T + 1376)^{2} \)
|
| $37$ |
\( (T + 24)^{4} \)
|
| $41$ |
\( T^{4} + 5960 T^{2} + 7952400 \)
|
| $43$ |
\( (T^{2} + 54 T - 576)^{2} \)
|
| $47$ |
\( T^{4} + 2185 T^{2} + 532900 \)
|
| $53$ |
\( T^{4} + 4340 T^{2} + 518400 \)
|
| $59$ |
\( T^{4} + 8360 T^{2} + 2624400 \)
|
| $61$ |
\( (T^{2} + 62 T + 816)^{2} \)
|
| $67$ |
\( (T^{2} - 59 T - 906)^{2} \)
|
| $71$ |
\( (T^{2} + 3920)^{2} \)
|
| $73$ |
\( (T - 14)^{4} \)
|
| $79$ |
\( (T + 122)^{4} \)
|
| $83$ |
\( T^{4} + 20340 T^{2} + 8294400 \)
|
| $89$ |
\( T^{4} + 10625 T^{2} + 14062500 \)
|
| $97$ |
\( (T^{2} - 174 T + 6264)^{2} \)
|
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