Newspace parameters
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.c (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.826026740080\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
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|
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| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 + \zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
−1.00000 | + | 1.73205i | 2.50000 | + | 4.33013i | −2.00000 | − | 3.46410i | 4.50000 | − | 7.79423i | −10.0000 | −14.0000 | − | 12.1244i | 8.00000 | 1.00000 | − | 1.73205i | 9.00000 | + | 15.5885i | ||||||||||
| 11.1 | −1.00000 | − | 1.73205i | 2.50000 | − | 4.33013i | −2.00000 | + | 3.46410i | 4.50000 | + | 7.79423i | −10.0000 | −14.0000 | + | 12.1244i | 8.00000 | 1.00000 | + | 1.73205i | 9.00000 | − | 15.5885i | |||||||||||
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 | 14.4.c.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 126.4.g.d | 2 | ||
| 4.b | odd | 2 | 1 | 112.4.i.a | 2 | ||
| 5.b | even | 2 | 1 | 350.4.e.e | 2 | ||
| 5.c | odd | 4 | 2 | 350.4.j.b | 4 | ||
| 7.b | odd | 2 | 1 | 98.4.c.a | 2 | ||
| 7.c | even | 3 | 1 | inner | 14.4.c.a | ✓ | 2 |
| 7.c | even | 3 | 1 | 98.4.a.d | 1 | ||
| 7.d | odd | 6 | 1 | 98.4.a.f | 1 | ||
| 7.d | odd | 6 | 1 | 98.4.c.a | 2 | ||
| 8.b | even | 2 | 1 | 448.4.i.b | 2 | ||
| 8.d | odd | 2 | 1 | 448.4.i.e | 2 | ||
| 21.c | even | 2 | 1 | 882.4.g.u | 2 | ||
| 21.g | even | 6 | 1 | 882.4.a.c | 1 | ||
| 21.g | even | 6 | 1 | 882.4.g.u | 2 | ||
| 21.h | odd | 6 | 1 | 126.4.g.d | 2 | ||
| 21.h | odd | 6 | 1 | 882.4.a.f | 1 | ||
| 28.f | even | 6 | 1 | 784.4.a.c | 1 | ||
| 28.g | odd | 6 | 1 | 112.4.i.a | 2 | ||
| 28.g | odd | 6 | 1 | 784.4.a.p | 1 | ||
| 35.i | odd | 6 | 1 | 2450.4.a.d | 1 | ||
| 35.j | even | 6 | 1 | 350.4.e.e | 2 | ||
| 35.j | even | 6 | 1 | 2450.4.a.q | 1 | ||
| 35.l | odd | 12 | 2 | 350.4.j.b | 4 | ||
| 56.k | odd | 6 | 1 | 448.4.i.e | 2 | ||
| 56.p | even | 6 | 1 | 448.4.i.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.4.c.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 14.4.c.a | ✓ | 2 | 7.c | even | 3 | 1 | inner |
| 98.4.a.d | 1 | 7.c | even | 3 | 1 | ||
| 98.4.a.f | 1 | 7.d | odd | 6 | 1 | ||
| 98.4.c.a | 2 | 7.b | odd | 2 | 1 | ||
| 98.4.c.a | 2 | 7.d | odd | 6 | 1 | ||
| 112.4.i.a | 2 | 4.b | odd | 2 | 1 | ||
| 112.4.i.a | 2 | 28.g | odd | 6 | 1 | ||
| 126.4.g.d | 2 | 3.b | odd | 2 | 1 | ||
| 126.4.g.d | 2 | 21.h | odd | 6 | 1 | ||
| 350.4.e.e | 2 | 5.b | even | 2 | 1 | ||
| 350.4.e.e | 2 | 35.j | even | 6 | 1 | ||
| 350.4.j.b | 4 | 5.c | odd | 4 | 2 | ||
| 350.4.j.b | 4 | 35.l | odd | 12 | 2 | ||
| 448.4.i.b | 2 | 8.b | even | 2 | 1 | ||
| 448.4.i.b | 2 | 56.p | even | 6 | 1 | ||
| 448.4.i.e | 2 | 8.d | odd | 2 | 1 | ||
| 448.4.i.e | 2 | 56.k | odd | 6 | 1 | ||
| 784.4.a.c | 1 | 28.f | even | 6 | 1 | ||
| 784.4.a.p | 1 | 28.g | odd | 6 | 1 | ||
| 882.4.a.c | 1 | 21.g | even | 6 | 1 | ||
| 882.4.a.f | 1 | 21.h | odd | 6 | 1 | ||
| 882.4.g.u | 2 | 21.c | even | 2 | 1 | ||
| 882.4.g.u | 2 | 21.g | even | 6 | 1 | ||
| 2450.4.a.d | 1 | 35.i | odd | 6 | 1 | ||
| 2450.4.a.q | 1 | 35.j | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 5T_{3} + 25 \)
acting on \(S_{4}^{\mathrm{new}}(14, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 2T + 4 \)
|
| $3$ |
\( T^{2} - 5T + 25 \)
|
| $5$ |
\( T^{2} - 9T + 81 \)
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| $7$ |
\( T^{2} + 28T + 343 \)
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| $11$ |
\( T^{2} - 57T + 3249 \)
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| $13$ |
\( (T + 70)^{2} \)
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| $17$ |
\( T^{2} + 51T + 2601 \)
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| $19$ |
\( T^{2} + 5T + 25 \)
|
| $23$ |
\( T^{2} + 69T + 4761 \)
|
| $29$ |
\( (T - 114)^{2} \)
|
| $31$ |
\( T^{2} + 23T + 529 \)
|
| $37$ |
\( T^{2} - 253T + 64009 \)
|
| $41$ |
\( (T + 42)^{2} \)
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| $43$ |
\( (T + 124)^{2} \)
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| $47$ |
\( T^{2} + 201T + 40401 \)
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| $53$ |
\( T^{2} - 393T + 154449 \)
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| $59$ |
\( T^{2} + 219T + 47961 \)
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| $61$ |
\( T^{2} - 709T + 502681 \)
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| $67$ |
\( T^{2} + 419T + 175561 \)
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| $71$ |
\( (T + 96)^{2} \)
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| $73$ |
\( T^{2} - 313T + 97969 \)
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| $79$ |
\( T^{2} + 461T + 212521 \)
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| $83$ |
\( (T + 588)^{2} \)
|
| $89$ |
\( T^{2} - 1017 T + 1034289 \)
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| $97$ |
\( (T + 1834)^{2} \)
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