Newspace parameters
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.d (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.44717948317\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−1.41421 | − | 2.44949i | −12.9853 | − | 7.49706i | −4.00000 | + | 6.92820i | 21.9853 | − | 12.6932i | 42.4098i | −7.00000 | − | 48.4974i | 22.6274 | 71.9117 | + | 124.555i | −62.1838 | − | 35.9018i | ||||||||||||||||
| 3.2 | 1.41421 | + | 2.44949i | 3.98528 | + | 2.30090i | −4.00000 | + | 6.92820i | 5.01472 | − | 2.89525i | 13.0159i | −7.00000 | − | 48.4974i | −22.6274 | −29.9117 | − | 51.8086i | 14.1838 | + | 8.18900i | |||||||||||||||||
| 5.1 | −1.41421 | + | 2.44949i | −12.9853 | + | 7.49706i | −4.00000 | − | 6.92820i | 21.9853 | + | 12.6932i | − | 42.4098i | −7.00000 | + | 48.4974i | 22.6274 | 71.9117 | − | 124.555i | −62.1838 | + | 35.9018i | ||||||||||||||||
| 5.2 | 1.41421 | − | 2.44949i | 3.98528 | − | 2.30090i | −4.00000 | − | 6.92820i | 5.01472 | + | 2.89525i | − | 13.0159i | −7.00000 | + | 48.4974i | −22.6274 | −29.9117 | + | 51.8086i | 14.1838 | − | 8.18900i | ||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 14.5.d.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 126.5.n.a | 4 | ||
| 4.b | odd | 2 | 1 | 112.5.s.b | 4 | ||
| 5.b | even | 2 | 1 | 350.5.k.a | 4 | ||
| 5.c | odd | 4 | 2 | 350.5.i.a | 8 | ||
| 7.b | odd | 2 | 1 | 98.5.d.a | 4 | ||
| 7.c | even | 3 | 1 | 98.5.b.b | 4 | ||
| 7.c | even | 3 | 1 | 98.5.d.a | 4 | ||
| 7.d | odd | 6 | 1 | inner | 14.5.d.a | ✓ | 4 |
| 7.d | odd | 6 | 1 | 98.5.b.b | 4 | ||
| 21.g | even | 6 | 1 | 126.5.n.a | 4 | ||
| 21.g | even | 6 | 1 | 882.5.c.b | 4 | ||
| 21.h | odd | 6 | 1 | 882.5.c.b | 4 | ||
| 28.f | even | 6 | 1 | 112.5.s.b | 4 | ||
| 28.f | even | 6 | 1 | 784.5.c.b | 4 | ||
| 28.g | odd | 6 | 1 | 784.5.c.b | 4 | ||
| 35.i | odd | 6 | 1 | 350.5.k.a | 4 | ||
| 35.k | even | 12 | 2 | 350.5.i.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.5.d.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 14.5.d.a | ✓ | 4 | 7.d | odd | 6 | 1 | inner |
| 98.5.b.b | 4 | 7.c | even | 3 | 1 | ||
| 98.5.b.b | 4 | 7.d | odd | 6 | 1 | ||
| 98.5.d.a | 4 | 7.b | odd | 2 | 1 | ||
| 98.5.d.a | 4 | 7.c | even | 3 | 1 | ||
| 112.5.s.b | 4 | 4.b | odd | 2 | 1 | ||
| 112.5.s.b | 4 | 28.f | even | 6 | 1 | ||
| 126.5.n.a | 4 | 3.b | odd | 2 | 1 | ||
| 126.5.n.a | 4 | 21.g | even | 6 | 1 | ||
| 350.5.i.a | 8 | 5.c | odd | 4 | 2 | ||
| 350.5.i.a | 8 | 35.k | even | 12 | 2 | ||
| 350.5.k.a | 4 | 5.b | even | 2 | 1 | ||
| 350.5.k.a | 4 | 35.i | odd | 6 | 1 | ||
| 784.5.c.b | 4 | 28.f | even | 6 | 1 | ||
| 784.5.c.b | 4 | 28.g | odd | 6 | 1 | ||
| 882.5.c.b | 4 | 21.g | even | 6 | 1 | ||
| 882.5.c.b | 4 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(14, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 8T^{2} + 64 \)
|
| $3$ |
\( T^{4} + 18 T^{3} + \cdots + 4761 \)
|
| $5$ |
\( T^{4} - 54 T^{3} + \cdots + 21609 \)
|
| $7$ |
\( (T^{2} + 14 T + 2401)^{2} \)
|
| $11$ |
\( T^{4} + 54 T^{3} + \cdots + 194481 \)
|
| $13$ |
\( T^{4} + 62592 T^{2} + 955551744 \)
|
| $17$ |
\( T^{4} + \cdots + 2084013801 \)
|
| $19$ |
\( T^{4} + \cdots + 37762205625 \)
|
| $23$ |
\( T^{4} + \cdots + 161403866001 \)
|
| $29$ |
\( (T^{2} - 1620 T + 651492)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 566643101049 \)
|
| $37$ |
\( T^{4} + \cdots + 284591307841 \)
|
| $41$ |
\( T^{4} + \cdots + 1046087110656 \)
|
| $43$ |
\( (T^{2} - 1172 T - 3015836)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 5548271897529 \)
|
| $53$ |
\( T^{4} + \cdots + 8389590597441 \)
|
| $59$ |
\( T^{4} + \cdots + 149611524833241 \)
|
| $61$ |
\( T^{4} + \cdots + 66399241936329 \)
|
| $67$ |
\( T^{4} + \cdots + 256392053989009 \)
|
| $71$ |
\( (T^{2} - 9396 T + 20407716)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 29440640401929 \)
|
| $79$ |
\( T^{4} + \cdots + 54769812839569 \)
|
| $83$ |
\( T^{4} + \cdots + 314640617324544 \)
|
| $89$ |
\( T^{4} + \cdots + 28434692391561 \)
|
| $97$ |
\( T^{4} + \cdots + 12\!\cdots\!36 \)
|
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