Newspace parameters
| Level: | \( N \) | \(=\) | \( 312 = 2^{3} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 312.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.49133254306\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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 \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).
| \(n\) | \(79\) | \(145\) | \(157\) | \(209\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
0 | 1.00000 | 0 | − | 2.00000i | 0 | − | 2.00000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||
| 25.2 | 0 | 1.00000 | 0 | 2.00000i | 0 | 2.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||
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 | 312.2.c.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 936.2.c.a | 2 | ||
| 4.b | odd | 2 | 1 | 624.2.c.b | 2 | ||
| 8.b | even | 2 | 1 | 2496.2.c.g | 2 | ||
| 8.d | odd | 2 | 1 | 2496.2.c.n | 2 | ||
| 12.b | even | 2 | 1 | 1872.2.c.a | 2 | ||
| 13.b | even | 2 | 1 | inner | 312.2.c.b | ✓ | 2 |
| 13.d | odd | 4 | 1 | 4056.2.a.n | 1 | ||
| 13.d | odd | 4 | 1 | 4056.2.a.p | 1 | ||
| 39.d | odd | 2 | 1 | 936.2.c.a | 2 | ||
| 52.b | odd | 2 | 1 | 624.2.c.b | 2 | ||
| 52.f | even | 4 | 1 | 8112.2.a.e | 1 | ||
| 52.f | even | 4 | 1 | 8112.2.a.k | 1 | ||
| 104.e | even | 2 | 1 | 2496.2.c.g | 2 | ||
| 104.h | odd | 2 | 1 | 2496.2.c.n | 2 | ||
| 156.h | even | 2 | 1 | 1872.2.c.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 312.2.c.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 312.2.c.b | ✓ | 2 | 13.b | even | 2 | 1 | inner |
| 624.2.c.b | 2 | 4.b | odd | 2 | 1 | ||
| 624.2.c.b | 2 | 52.b | odd | 2 | 1 | ||
| 936.2.c.a | 2 | 3.b | odd | 2 | 1 | ||
| 936.2.c.a | 2 | 39.d | odd | 2 | 1 | ||
| 1872.2.c.a | 2 | 12.b | even | 2 | 1 | ||
| 1872.2.c.a | 2 | 156.h | even | 2 | 1 | ||
| 2496.2.c.g | 2 | 8.b | even | 2 | 1 | ||
| 2496.2.c.g | 2 | 104.e | even | 2 | 1 | ||
| 2496.2.c.n | 2 | 8.d | odd | 2 | 1 | ||
| 2496.2.c.n | 2 | 104.h | odd | 2 | 1 | ||
| 4056.2.a.n | 1 | 13.d | odd | 4 | 1 | ||
| 4056.2.a.p | 1 | 13.d | odd | 4 | 1 | ||
| 8112.2.a.e | 1 | 52.f | even | 4 | 1 | ||
| 8112.2.a.k | 1 | 52.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T - 1)^{2} \)
|
| $5$ |
\( T^{2} + 4 \)
|
| $7$ |
\( T^{2} + 4 \)
|
| $11$ |
\( T^{2} + 16 \)
|
| $13$ |
\( T^{2} + 6T + 13 \)
|
| $17$ |
\( (T - 6)^{2} \)
|
| $19$ |
\( T^{2} + 4 \)
|
| $23$ |
\( (T + 4)^{2} \)
|
| $29$ |
\( (T - 6)^{2} \)
|
| $31$ |
\( T^{2} + 4 \)
|
| $37$ |
\( T^{2} + 64 \)
|
| $41$ |
\( T^{2} + 36 \)
|
| $43$ |
\( (T + 4)^{2} \)
|
| $47$ |
\( T^{2} + 64 \)
|
| $53$ |
\( (T - 2)^{2} \)
|
| $59$ |
\( T^{2} + 64 \)
|
| $61$ |
\( (T + 14)^{2} \)
|
| $67$ |
\( T^{2} + 196 \)
|
| $71$ |
\( T^{2} + 144 \)
|
| $73$ |
\( T^{2} + 16 \)
|
| $79$ |
\( T^{2} \)
|
| $83$ |
\( T^{2} \)
|
| $89$ |
\( T^{2} + 36 \)
|
| $97$ |
\( T^{2} + 144 \)
|
| show more | |
| show less |