Newspace parameters
| Level: | \( N \) | \(=\) | \( 1183 = 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1183.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(9.44630255912\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 91) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).
| \(n\) | \(339\) | \(1016\) |
| \(\chi(n)\) | \(-\zeta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 170.1 |
|
0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | −0.500000 | − | 2.59808i | 3.00000 | 1.50000 | − | 2.59808i | 0 | ||||||||||||||||
| 508.1 | 0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | −0.500000 | + | 2.59808i | 3.00000 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1183.2.e.b | 2 | |
| 7.c | even | 3 | 1 | inner | 1183.2.e.b | 2 | |
| 7.c | even | 3 | 1 | 8281.2.a.f | 1 | ||
| 7.d | odd | 6 | 1 | 8281.2.a.e | 1 | ||
| 13.b | even | 2 | 1 | 91.2.e.a | ✓ | 2 | |
| 39.d | odd | 2 | 1 | 819.2.j.b | 2 | ||
| 52.b | odd | 2 | 1 | 1456.2.r.g | 2 | ||
| 91.b | odd | 2 | 1 | 637.2.e.a | 2 | ||
| 91.r | even | 6 | 1 | 91.2.e.a | ✓ | 2 | |
| 91.r | even | 6 | 1 | 637.2.a.c | 1 | ||
| 91.s | odd | 6 | 1 | 637.2.a.d | 1 | ||
| 91.s | odd | 6 | 1 | 637.2.e.a | 2 | ||
| 273.w | odd | 6 | 1 | 819.2.j.b | 2 | ||
| 273.w | odd | 6 | 1 | 5733.2.a.c | 1 | ||
| 273.ba | even | 6 | 1 | 5733.2.a.d | 1 | ||
| 364.bl | odd | 6 | 1 | 1456.2.r.g | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.2.e.a | ✓ | 2 | 13.b | even | 2 | 1 | |
| 91.2.e.a | ✓ | 2 | 91.r | even | 6 | 1 | |
| 637.2.a.c | 1 | 91.r | even | 6 | 1 | ||
| 637.2.a.d | 1 | 91.s | odd | 6 | 1 | ||
| 637.2.e.a | 2 | 91.b | odd | 2 | 1 | ||
| 637.2.e.a | 2 | 91.s | odd | 6 | 1 | ||
| 819.2.j.b | 2 | 39.d | odd | 2 | 1 | ||
| 819.2.j.b | 2 | 273.w | odd | 6 | 1 | ||
| 1183.2.e.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 1183.2.e.b | 2 | 7.c | even | 3 | 1 | inner | |
| 1456.2.r.g | 2 | 52.b | odd | 2 | 1 | ||
| 1456.2.r.g | 2 | 364.bl | odd | 6 | 1 | ||
| 5733.2.a.c | 1 | 273.w | odd | 6 | 1 | ||
| 5733.2.a.d | 1 | 273.ba | even | 6 | 1 | ||
| 8281.2.a.e | 1 | 7.d | odd | 6 | 1 | ||
| 8281.2.a.f | 1 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\):
|
\( T_{2}^{2} - T_{2} + 1 \)
|
|
\( T_{3} \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - T + 1 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + T + 7 \)
|
| $11$ |
\( T^{2} + 3T + 9 \)
|
| $13$ |
\( T^{2} \)
|
| $17$ |
\( T^{2} + 7T + 49 \)
|
| $19$ |
\( T^{2} + 7T + 49 \)
|
| $23$ |
\( T^{2} - 6T + 36 \)
|
| $29$ |
\( (T + 5)^{2} \)
|
| $31$ |
\( T^{2} \)
|
| $37$ |
\( T^{2} - 8T + 64 \)
|
| $41$ |
\( T^{2} \)
|
| $43$ |
\( (T - 2)^{2} \)
|
| $47$ |
\( T^{2} - 7T + 49 \)
|
| $53$ |
\( T^{2} - 3T + 9 \)
|
| $59$ |
\( T^{2} + 7T + 49 \)
|
| $61$ |
\( T^{2} - 7T + 49 \)
|
| $67$ |
\( T^{2} + 3T + 9 \)
|
| $71$ |
\( (T - 5)^{2} \)
|
| $73$ |
\( T^{2} - 14T + 196 \)
|
| $79$ |
\( T^{2} - 6T + 36 \)
|
| $83$ |
\( T^{2} \)
|
| $89$ |
\( T^{2} \)
|
| $97$ |
\( (T - 14)^{2} \)
|
| show more | |
| show less |