Newspace parameters
| Level: | \( N \) | \(=\) | \( 1183 = 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1183.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(9.44630255912\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 91) |
| 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
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 2\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).
| \(n\) | \(339\) | \(1016\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
− | 2.61803i | −2.61803 | −4.85410 | − | 2.61803i | 6.85410i | 1.00000i | 7.47214i | 3.85410 | −6.85410 | ||||||||||||||||||||||||||||
| 337.2 | − | 0.381966i | −0.381966 | 1.85410 | − | 0.381966i | 0.145898i | 1.00000i | − | 1.47214i | −2.85410 | −0.145898 | ||||||||||||||||||||||||||||
| 337.3 | 0.381966i | −0.381966 | 1.85410 | 0.381966i | − | 0.145898i | − | 1.00000i | 1.47214i | −2.85410 | −0.145898 | |||||||||||||||||||||||||||||
| 337.4 | 2.61803i | −2.61803 | −4.85410 | 2.61803i | − | 6.85410i | − | 1.00000i | − | 7.47214i | 3.85410 | −6.85410 | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1183.2.c.c | 4 | |
| 13.b | even | 2 | 1 | inner | 1183.2.c.c | 4 | |
| 13.d | odd | 4 | 1 | 1183.2.a.c | 2 | ||
| 13.d | odd | 4 | 1 | 1183.2.a.g | 2 | ||
| 13.f | odd | 12 | 2 | 91.2.f.a | ✓ | 4 | |
| 39.k | even | 12 | 2 | 819.2.o.c | 4 | ||
| 52.l | even | 12 | 2 | 1456.2.s.h | 4 | ||
| 91.i | even | 4 | 1 | 8281.2.a.n | 2 | ||
| 91.i | even | 4 | 1 | 8281.2.a.bb | 2 | ||
| 91.w | even | 12 | 2 | 637.2.g.c | 4 | ||
| 91.x | odd | 12 | 2 | 637.2.h.g | 4 | ||
| 91.ba | even | 12 | 2 | 637.2.h.f | 4 | ||
| 91.bc | even | 12 | 2 | 637.2.f.c | 4 | ||
| 91.bd | odd | 12 | 2 | 637.2.g.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.2.f.a | ✓ | 4 | 13.f | odd | 12 | 2 | |
| 637.2.f.c | 4 | 91.bc | even | 12 | 2 | ||
| 637.2.g.b | 4 | 91.bd | odd | 12 | 2 | ||
| 637.2.g.c | 4 | 91.w | even | 12 | 2 | ||
| 637.2.h.f | 4 | 91.ba | even | 12 | 2 | ||
| 637.2.h.g | 4 | 91.x | odd | 12 | 2 | ||
| 819.2.o.c | 4 | 39.k | even | 12 | 2 | ||
| 1183.2.a.c | 2 | 13.d | odd | 4 | 1 | ||
| 1183.2.a.g | 2 | 13.d | odd | 4 | 1 | ||
| 1183.2.c.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 1183.2.c.c | 4 | 13.b | even | 2 | 1 | inner | |
| 1456.2.s.h | 4 | 52.l | even | 12 | 2 | ||
| 8281.2.a.n | 2 | 91.i | even | 4 | 1 | ||
| 8281.2.a.bb | 2 | 91.i | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 7T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 7T^{2} + 1 \)
|
| $3$ |
\( (T^{2} + 3 T + 1)^{2} \)
|
| $5$ |
\( T^{4} + 7T^{2} + 1 \)
|
| $7$ |
\( (T^{2} + 1)^{2} \)
|
| $11$ |
\( T^{4} + 27T^{2} + 81 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} + 6 T - 11)^{2} \)
|
| $19$ |
\( T^{4} + 27T^{2} + 81 \)
|
| $23$ |
\( (T^{2} - 20)^{2} \)
|
| $29$ |
\( (T^{2} - 3 T - 29)^{2} \)
|
| $31$ |
\( T^{4} + 98T^{2} + 1681 \)
|
| $37$ |
\( (T^{2} + 16)^{2} \)
|
| $41$ |
\( T^{4} + 28T^{2} + 16 \)
|
| $43$ |
\( (T^{2} + 5 T - 95)^{2} \)
|
| $47$ |
\( (T^{2} + 5)^{2} \)
|
| $53$ |
\( (T^{2} - 12 T + 31)^{2} \)
|
| $59$ |
\( (T^{2} + 5)^{2} \)
|
| $61$ |
\( (T + 6)^{4} \)
|
| $67$ |
\( T^{4} + 162T^{2} + 81 \)
|
| $71$ |
\( T^{4} + 268 T^{2} + 13456 \)
|
| $73$ |
\( (T^{2} + 4)^{2} \)
|
| $79$ |
\( (T - 4)^{4} \)
|
| $83$ |
\( (T^{2} + 45)^{2} \)
|
| $89$ |
\( T^{4} + 283T^{2} + 6241 \)
|
| $97$ |
\( T^{4} + 503 T^{2} + 52441 \)
|
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