Newspace parameters
| Level: | \( N \) | \(=\) | \( 507 = 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 507.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(29.9139683729\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{14})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 49 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 39) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{2} ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} + 7\nu ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 14\nu ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} - \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{3} + 14\beta_{2} - 7\beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).
| \(n\) | \(170\) | \(340\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
− | 4.74166i | −3.00000 | −14.4833 | − | 4.51669i | 14.2250i | − | 7.48331i | 30.7417i | 9.00000 | −21.4166 | |||||||||||||||||||||||||||
| 337.2 | − | 2.74166i | −3.00000 | 0.483315 | 19.4833i | 8.22497i | − | 7.48331i | − | 23.2583i | 9.00000 | 53.4166 | ||||||||||||||||||||||||||||
| 337.3 | 2.74166i | −3.00000 | 0.483315 | − | 19.4833i | − | 8.22497i | 7.48331i | 23.2583i | 9.00000 | 53.4166 | |||||||||||||||||||||||||||||
| 337.4 | 4.74166i | −3.00000 | −14.4833 | 4.51669i | − | 14.2250i | 7.48331i | − | 30.7417i | 9.00000 | −21.4166 | |||||||||||||||||||||||||||||
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 | 507.4.b.f | 4 | |
| 13.b | even | 2 | 1 | inner | 507.4.b.f | 4 | |
| 13.d | odd | 4 | 1 | 39.4.a.b | ✓ | 2 | |
| 13.d | odd | 4 | 1 | 507.4.a.f | 2 | ||
| 39.f | even | 4 | 1 | 117.4.a.c | 2 | ||
| 39.f | even | 4 | 1 | 1521.4.a.s | 2 | ||
| 52.f | even | 4 | 1 | 624.4.a.r | 2 | ||
| 65.g | odd | 4 | 1 | 975.4.a.j | 2 | ||
| 91.i | even | 4 | 1 | 1911.4.a.h | 2 | ||
| 104.j | odd | 4 | 1 | 2496.4.a.bc | 2 | ||
| 104.m | even | 4 | 1 | 2496.4.a.s | 2 | ||
| 156.l | odd | 4 | 1 | 1872.4.a.t | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.4.a.b | ✓ | 2 | 13.d | odd | 4 | 1 | |
| 117.4.a.c | 2 | 39.f | even | 4 | 1 | ||
| 507.4.a.f | 2 | 13.d | odd | 4 | 1 | ||
| 507.4.b.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 507.4.b.f | 4 | 13.b | even | 2 | 1 | inner | |
| 624.4.a.r | 2 | 52.f | even | 4 | 1 | ||
| 975.4.a.j | 2 | 65.g | odd | 4 | 1 | ||
| 1521.4.a.s | 2 | 39.f | even | 4 | 1 | ||
| 1872.4.a.t | 2 | 156.l | odd | 4 | 1 | ||
| 1911.4.a.h | 2 | 91.i | even | 4 | 1 | ||
| 2496.4.a.s | 2 | 104.m | even | 4 | 1 | ||
| 2496.4.a.bc | 2 | 104.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(507, [\chi])\):
|
\( T_{2}^{4} + 30T_{2}^{2} + 169 \)
|
|
\( T_{5}^{4} + 400T_{5}^{2} + 7744 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 30T^{2} + 169 \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( T^{4} + 400T^{2} + 7744 \)
|
| $7$ |
\( (T^{2} + 56)^{2} \)
|
| $11$ |
\( T^{4} + 5000 T^{2} + 2347024 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} + 164 T + 6500)^{2} \)
|
| $19$ |
\( T^{4} + 1264 T^{2} + 270400 \)
|
| $23$ |
\( (T^{2} + 8 T - 32240)^{2} \)
|
| $29$ |
\( (T^{2} - 404 T + 32740)^{2} \)
|
| $31$ |
\( T^{4} + 19728 T^{2} + 82156096 \)
|
| $37$ |
\( T^{4} + 26952 T^{2} + 71842576 \)
|
| $41$ |
\( T^{4} + \cdots + 12928599616 \)
|
| $43$ |
\( (T^{2} - 616 T + 57008)^{2} \)
|
| $47$ |
\( T^{4} + 81160 T^{2} + 141800464 \)
|
| $53$ |
\( (T^{2} + 164 T - 194876)^{2} \)
|
| $59$ |
\( T^{4} + 54600 T^{2} + 306250000 \)
|
| $61$ |
\( (T^{2} - 628 T - 160348)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 121734001216 \)
|
| $71$ |
\( T^{4} + \cdots + 2807728144 \)
|
| $73$ |
\( T^{4} + \cdots + 14795316496 \)
|
| $79$ |
\( (T^{2} + 432 T - 61760)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 180023701264 \)
|
| $89$ |
\( T^{4} + \cdots + 75625000000 \)
|
| $97$ |
\( T^{4} + \cdots + 368259640336 \)
|
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