Newspace parameters
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.i (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(17.9629878191\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{109})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 28x^{2} + 27x + 729 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 28) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 28x^{2} + 27x + 729 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 28\nu^{2} - 28\nu + 729 ) / 756 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 28\nu^{2} + 1540\nu - 729 ) / 756 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 41 ) / 14 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 55\beta _1 - 55 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 14\beta_{3} - 41 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(85\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | 1.77985 | + | 3.08278i | 0 | 20.8209 | − | 36.0629i | 0 | −28.0000 | + | 126.582i | 0 | 115.164 | − | 199.470i | 0 | ||||||||||||||||||||||
| 65.2 | 0 | 12.2202 | + | 21.1659i | 0 | −41.8209 | + | 72.4360i | 0 | −28.0000 | − | 126.582i | 0 | −177.164 | + | 306.858i | 0 | |||||||||||||||||||||||
| 81.1 | 0 | 1.77985 | − | 3.08278i | 0 | 20.8209 | + | 36.0629i | 0 | −28.0000 | − | 126.582i | 0 | 115.164 | + | 199.470i | 0 | |||||||||||||||||||||||
| 81.2 | 0 | 12.2202 | − | 21.1659i | 0 | −41.8209 | − | 72.4360i | 0 | −28.0000 | + | 126.582i | 0 | −177.164 | − | 306.858i | 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 | 112.6.i.e | 4 | |
| 4.b | odd | 2 | 1 | 28.6.e.b | ✓ | 4 | |
| 7.c | even | 3 | 1 | inner | 112.6.i.e | 4 | |
| 7.c | even | 3 | 1 | 784.6.a.o | 2 | ||
| 7.d | odd | 6 | 1 | 784.6.a.bd | 2 | ||
| 12.b | even | 2 | 1 | 252.6.k.d | 4 | ||
| 28.d | even | 2 | 1 | 196.6.e.k | 4 | ||
| 28.f | even | 6 | 1 | 196.6.a.h | 2 | ||
| 28.f | even | 6 | 1 | 196.6.e.k | 4 | ||
| 28.g | odd | 6 | 1 | 28.6.e.b | ✓ | 4 | |
| 28.g | odd | 6 | 1 | 196.6.a.j | 2 | ||
| 84.n | even | 6 | 1 | 252.6.k.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.6.e.b | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 28.6.e.b | ✓ | 4 | 28.g | odd | 6 | 1 | |
| 112.6.i.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 112.6.i.e | 4 | 7.c | even | 3 | 1 | inner | |
| 196.6.a.h | 2 | 28.f | even | 6 | 1 | ||
| 196.6.a.j | 2 | 28.g | odd | 6 | 1 | ||
| 196.6.e.k | 4 | 28.d | even | 2 | 1 | ||
| 196.6.e.k | 4 | 28.f | even | 6 | 1 | ||
| 252.6.k.d | 4 | 12.b | even | 2 | 1 | ||
| 252.6.k.d | 4 | 84.n | even | 6 | 1 | ||
| 784.6.a.o | 2 | 7.c | even | 3 | 1 | ||
| 784.6.a.bd | 2 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 28T_{3}^{3} + 697T_{3}^{2} - 2436T_{3} + 7569 \)
acting on \(S_{6}^{\mathrm{new}}(112, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 28 T^{3} + \cdots + 7569 \)
|
| $5$ |
\( T^{4} + 42 T^{3} + \cdots + 12131289 \)
|
| $7$ |
\( (T^{2} + 56 T + 16807)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 3700410561 \)
|
| $13$ |
\( (T^{2} + 644 T - 147452)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 2678994081 \)
|
| $19$ |
\( T^{4} + \cdots + 11959250986225 \)
|
| $23$ |
\( T^{4} + \cdots + 5547141326169 \)
|
| $29$ |
\( (T^{2} - 5532 T - 4654908)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 370117150388041 \)
|
| $37$ |
\( T^{4} + \cdots + 11\!\cdots\!69 \)
|
| $41$ |
\( (T^{2} - 4116 T - 68342940)^{2} \)
|
| $43$ |
\( (T + 6716)^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 27\!\cdots\!69 \)
|
| $53$ |
\( T^{4} + \cdots + 48\!\cdots\!09 \)
|
| $59$ |
\( T^{4} + \cdots + 323868145417569 \)
|
| $61$ |
\( T^{4} + \cdots + 31\!\cdots\!09 \)
|
| $67$ |
\( T^{4} + \cdots + 14\!\cdots\!41 \)
|
| $71$ |
\( (T^{2} + 71568 T - 491520960)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 99\!\cdots\!61 \)
|
| $79$ |
\( T^{4} + \cdots + 17\!\cdots\!89 \)
|
| $83$ |
\( (T^{2} + 6216 T - 1440901872)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 43\!\cdots\!25 \)
|
| $97$ |
\( (T^{2} - 8260 T - 5263077500)^{2} \)
|
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