Newspace parameters
| Level: | \( N \) | \(=\) | \( 5100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5100.k (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(40.7237050309\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 1020) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{8}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{8}^{3} + 2\zeta_{8} \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\zeta_{8}^{3} + 2\zeta_{8} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 4 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5100\mathbb{Z}\right)^\times\).
| \(n\) | \(2551\) | \(3301\) | \(3401\) | \(3877\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4249.1 |
|
0 | 1.00000 | 0 | 0 | 0 | −3.82843 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 4249.2 | 0 | 1.00000 | 0 | 0 | 0 | −3.82843 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 4249.3 | 0 | 1.00000 | 0 | 0 | 0 | 1.82843 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 4249.4 | 0 | 1.00000 | 0 | 0 | 0 | 1.82843 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 85.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5100.2.k.i | 4 | |
| 5.b | even | 2 | 1 | 5100.2.k.h | 4 | ||
| 5.c | odd | 4 | 1 | 1020.2.e.d | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 5100.2.e.f | 4 | ||
| 15.e | even | 4 | 1 | 3060.2.e.g | 4 | ||
| 17.b | even | 2 | 1 | 5100.2.k.h | 4 | ||
| 20.e | even | 4 | 1 | 4080.2.h.n | 4 | ||
| 85.c | even | 2 | 1 | inner | 5100.2.k.i | 4 | |
| 85.g | odd | 4 | 1 | 1020.2.e.d | ✓ | 4 | |
| 85.g | odd | 4 | 1 | 5100.2.e.f | 4 | ||
| 255.o | even | 4 | 1 | 3060.2.e.g | 4 | ||
| 340.r | even | 4 | 1 | 4080.2.h.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1020.2.e.d | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 1020.2.e.d | ✓ | 4 | 85.g | odd | 4 | 1 | |
| 3060.2.e.g | 4 | 15.e | even | 4 | 1 | ||
| 3060.2.e.g | 4 | 255.o | even | 4 | 1 | ||
| 4080.2.h.n | 4 | 20.e | even | 4 | 1 | ||
| 4080.2.h.n | 4 | 340.r | even | 4 | 1 | ||
| 5100.2.e.f | 4 | 5.c | odd | 4 | 1 | ||
| 5100.2.e.f | 4 | 85.g | odd | 4 | 1 | ||
| 5100.2.k.h | 4 | 5.b | even | 2 | 1 | ||
| 5100.2.k.h | 4 | 17.b | even | 2 | 1 | ||
| 5100.2.k.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 5100.2.k.i | 4 | 85.c | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5100, [\chi])\):
|
\( T_{7}^{2} + 2T_{7} - 7 \)
|
|
\( T_{23}^{2} - 72 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 1)^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 2 T - 7)^{2} \)
|
| $11$ |
\( T^{4} + 34T^{2} + 1 \)
|
| $13$ |
\( T^{4} + 24T^{2} + 16 \)
|
| $17$ |
\( T^{4} + 2T^{2} + 289 \)
|
| $19$ |
\( (T + 3)^{4} \)
|
| $23$ |
\( (T^{2} - 72)^{2} \)
|
| $29$ |
\( T^{4} + 114T^{2} + 1681 \)
|
| $31$ |
\( T^{4} + 48T^{2} + 64 \)
|
| $37$ |
\( (T^{2} + 2 T - 71)^{2} \)
|
| $41$ |
\( T^{4} + 34T^{2} + 1 \)
|
| $43$ |
\( (T^{2} + 64)^{2} \)
|
| $47$ |
\( T^{4} + 82T^{2} + 529 \)
|
| $53$ |
\( T^{4} + 274 T^{2} + 14161 \)
|
| $59$ |
\( (T^{2} + 16 T + 56)^{2} \)
|
| $61$ |
\( (T^{2} + 72)^{2} \)
|
| $67$ |
\( T^{4} + 216T^{2} + 1296 \)
|
| $71$ |
\( T^{4} + 96T^{2} + 256 \)
|
| $73$ |
\( (T^{2} + 26 T + 161)^{2} \)
|
| $79$ |
\( (T^{2} + 4)^{2} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( (T^{2} - 200)^{2} \)
|
| $97$ |
\( (T + 14)^{4} \)
|
| show more | |
| show less |