Newspace parameters
| Level: | \( N \) | \(=\) | \( 1369 = 37^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1369.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(10.9315200367\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
|
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| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 37) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1369\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1368.1 |
|
0 | −1.00000 | 2.00000 | 0 | 0 | −1.00000 | 0 | −2.00000 | 0 | ||||||||||||||||||||||||
| 1368.2 | 0 | −1.00000 | 2.00000 | 0 | 0 | −1.00000 | 0 | −2.00000 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1369.2.b.a | 2 | |
| 37.b | even | 2 | 1 | inner | 1369.2.b.a | 2 | |
| 37.d | odd | 4 | 1 | 37.2.a.b | ✓ | 1 | |
| 37.d | odd | 4 | 1 | 1369.2.a.c | 1 | ||
| 111.g | even | 4 | 1 | 333.2.a.b | 1 | ||
| 148.g | even | 4 | 1 | 592.2.a.a | 1 | ||
| 185.f | even | 4 | 1 | 925.2.b.e | 2 | ||
| 185.j | odd | 4 | 1 | 925.2.a.b | 1 | ||
| 185.k | even | 4 | 1 | 925.2.b.e | 2 | ||
| 259.j | even | 4 | 1 | 1813.2.a.b | 1 | ||
| 296.j | even | 4 | 1 | 2368.2.a.m | 1 | ||
| 296.m | odd | 4 | 1 | 2368.2.a.d | 1 | ||
| 407.f | even | 4 | 1 | 4477.2.a.a | 1 | ||
| 444.j | odd | 4 | 1 | 5328.2.a.k | 1 | ||
| 481.j | odd | 4 | 1 | 6253.2.a.b | 1 | ||
| 555.m | even | 4 | 1 | 8325.2.a.p | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 37.2.a.b | ✓ | 1 | 37.d | odd | 4 | 1 | |
| 333.2.a.b | 1 | 111.g | even | 4 | 1 | ||
| 592.2.a.a | 1 | 148.g | even | 4 | 1 | ||
| 925.2.a.b | 1 | 185.j | odd | 4 | 1 | ||
| 925.2.b.e | 2 | 185.f | even | 4 | 1 | ||
| 925.2.b.e | 2 | 185.k | even | 4 | 1 | ||
| 1369.2.a.c | 1 | 37.d | odd | 4 | 1 | ||
| 1369.2.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 1369.2.b.a | 2 | 37.b | even | 2 | 1 | inner | |
| 1813.2.a.b | 1 | 259.j | even | 4 | 1 | ||
| 2368.2.a.d | 1 | 296.m | odd | 4 | 1 | ||
| 2368.2.a.m | 1 | 296.j | even | 4 | 1 | ||
| 4477.2.a.a | 1 | 407.f | even | 4 | 1 | ||
| 5328.2.a.k | 1 | 444.j | odd | 4 | 1 | ||
| 6253.2.a.b | 1 | 481.j | odd | 4 | 1 | ||
| 8325.2.a.p | 1 | 555.m | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1369, [\chi])\):
|
\( T_{2} \)
|
|
\( T_{3} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T + 1)^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( (T + 1)^{2} \)
|
| $11$ |
\( (T + 3)^{2} \)
|
| $13$ |
\( T^{2} + 16 \)
|
| $17$ |
\( T^{2} + 36 \)
|
| $19$ |
\( T^{2} + 4 \)
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| $23$ |
\( T^{2} + 36 \)
|
| $29$ |
\( T^{2} + 36 \)
|
| $31$ |
\( T^{2} + 16 \)
|
| $37$ |
\( T^{2} \)
|
| $41$ |
\( (T - 9)^{2} \)
|
| $43$ |
\( T^{2} + 64 \)
|
| $47$ |
\( (T - 3)^{2} \)
|
| $53$ |
\( (T + 3)^{2} \)
|
| $59$ |
\( T^{2} + 144 \)
|
| $61$ |
\( T^{2} + 64 \)
|
| $67$ |
\( (T - 4)^{2} \)
|
| $71$ |
\( (T + 15)^{2} \)
|
| $73$ |
\( (T + 11)^{2} \)
|
| $79$ |
\( T^{2} + 100 \)
|
| $83$ |
\( (T - 9)^{2} \)
|
| $89$ |
\( T^{2} + 36 \)
|
| $97$ |
\( T^{2} + 64 \)
|
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