Newspace parameters
| Level: | \( N \) | \(=\) | \( 416 = 2^{5} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 416.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(24.5447945624\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{47})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 47x^{2} + 2209 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} + 47x^{2} + 2209 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 47 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 47 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 47\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 47\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(287\) | \(353\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 289.1 |
|
0 | −3.42783 | − | 5.93717i | 0 | 12.0000 | 0 | −3.42783 | + | 5.93717i | 0 | −10.0000 | + | 17.3205i | 0 | ||||||||||||||||||||||||
| 289.2 | 0 | 3.42783 | + | 5.93717i | 0 | 12.0000 | 0 | 3.42783 | − | 5.93717i | 0 | −10.0000 | + | 17.3205i | 0 | |||||||||||||||||||||||||
| 321.1 | 0 | −3.42783 | + | 5.93717i | 0 | 12.0000 | 0 | −3.42783 | − | 5.93717i | 0 | −10.0000 | − | 17.3205i | 0 | |||||||||||||||||||||||||
| 321.2 | 0 | 3.42783 | − | 5.93717i | 0 | 12.0000 | 0 | 3.42783 | + | 5.93717i | 0 | −10.0000 | − | 17.3205i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 13.c | even | 3 | 1 | inner |
| 52.j | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 416.4.i.c | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 416.4.i.c | ✓ | 4 |
| 13.c | even | 3 | 1 | inner | 416.4.i.c | ✓ | 4 |
| 52.j | odd | 6 | 1 | inner | 416.4.i.c | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 416.4.i.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 416.4.i.c | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 416.4.i.c | ✓ | 4 | 13.c | even | 3 | 1 | inner |
| 416.4.i.c | ✓ | 4 | 52.j | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 47T_{3}^{2} + 2209 \)
acting on \(S_{4}^{\mathrm{new}}(416, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 47T^{2} + 2209 \)
|
| $5$ |
\( (T - 12)^{4} \)
|
| $7$ |
\( T^{4} + 47T^{2} + 2209 \)
|
| $11$ |
\( T^{4} + 47T^{2} + 2209 \)
|
| $13$ |
\( (T^{2} + 26 T + 2197)^{2} \)
|
| $17$ |
\( (T^{2} + 81 T + 6561)^{2} \)
|
| $19$ |
\( T^{4} + 7943 T^{2} + 63091249 \)
|
| $23$ |
\( T^{4} + 24863 T^{2} + 618168769 \)
|
| $29$ |
\( (T^{2} - 41 T + 1681)^{2} \)
|
| $31$ |
\( (T^{2} - 27072)^{2} \)
|
| $37$ |
\( (T^{2} - 39 T + 1521)^{2} \)
|
| $41$ |
\( (T^{2} + 9 T + 81)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 1173953169 \)
|
| $47$ |
\( (T^{2} - 54332)^{2} \)
|
| $53$ |
\( (T + 164)^{4} \)
|
| $59$ |
\( T^{4} + \cdots + 4140021649 \)
|
| $61$ |
\( (T^{2} + 795 T + 632025)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 95090206689 \)
|
| $71$ |
\( T^{4} + \cdots + 5110391169 \)
|
| $73$ |
\( (T - 512)^{4} \)
|
| $79$ |
\( (T^{2} - 1173308)^{2} \)
|
| $83$ |
\( (T^{2} - 18800)^{2} \)
|
| $89$ |
\( (T^{2} + 1103 T + 1216609)^{2} \)
|
| $97$ |
\( (T^{2} - 1145 T + 1311025)^{2} \)
|
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