Newspace parameters
Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 504.s (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(29.7369626429\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 168) |
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 \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).
\(n\) | \(73\) | \(127\) | \(253\) | \(281\) |
\(\chi(n)\) | \(-1 + \zeta_{6}\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
289.1 |
|
0 | 0 | 0 | 3.50000 | − | 6.06218i | 0 | −17.5000 | − | 6.06218i | 0 | 0 | 0 | ||||||||||||||||||||
361.1 | 0 | 0 | 0 | 3.50000 | + | 6.06218i | 0 | −17.5000 | + | 6.06218i | 0 | 0 | 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 | 504.4.s.e | 2 | |
3.b | odd | 2 | 1 | 168.4.q.a | ✓ | 2 | |
7.c | even | 3 | 1 | inner | 504.4.s.e | 2 | |
12.b | even | 2 | 1 | 336.4.q.a | 2 | ||
21.g | even | 6 | 1 | 1176.4.a.i | 1 | ||
21.h | odd | 6 | 1 | 168.4.q.a | ✓ | 2 | |
21.h | odd | 6 | 1 | 1176.4.a.f | 1 | ||
84.j | odd | 6 | 1 | 2352.4.a.c | 1 | ||
84.n | even | 6 | 1 | 336.4.q.a | 2 | ||
84.n | even | 6 | 1 | 2352.4.a.bg | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
168.4.q.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
168.4.q.a | ✓ | 2 | 21.h | odd | 6 | 1 | |
336.4.q.a | 2 | 12.b | even | 2 | 1 | ||
336.4.q.a | 2 | 84.n | even | 6 | 1 | ||
504.4.s.e | 2 | 1.a | even | 1 | 1 | trivial | |
504.4.s.e | 2 | 7.c | even | 3 | 1 | inner | |
1176.4.a.f | 1 | 21.h | odd | 6 | 1 | ||
1176.4.a.i | 1 | 21.g | even | 6 | 1 | ||
2352.4.a.c | 1 | 84.j | odd | 6 | 1 | ||
2352.4.a.bg | 1 | 84.n | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 7T_{5} + 49 \)
acting on \(S_{4}^{\mathrm{new}}(504, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} - 7T + 49 \)
$7$
\( T^{2} + 35T + 343 \)
$11$
\( T^{2} - 7T + 49 \)
$13$
\( (T + 52)^{2} \)
$17$
\( T^{2} - 72T + 5184 \)
$19$
\( T^{2} + 20T + 400 \)
$23$
\( T^{2} + 48T + 2304 \)
$29$
\( (T - 243)^{2} \)
$31$
\( T^{2} + 95T + 9025 \)
$37$
\( T^{2} + 352T + 123904 \)
$41$
\( (T - 296)^{2} \)
$43$
\( (T - 158)^{2} \)
$47$
\( T^{2} + 142T + 20164 \)
$53$
\( T^{2} + 375T + 140625 \)
$59$
\( T^{2} - 279T + 77841 \)
$61$
\( T^{2} + 246T + 60516 \)
$67$
\( T^{2} - 730T + 532900 \)
$71$
\( (T + 338)^{2} \)
$73$
\( T^{2} - 542T + 293764 \)
$79$
\( T^{2} - 305T + 93025 \)
$83$
\( (T + 1123)^{2} \)
$89$
\( T^{2} + 426T + 181476 \)
$97$
\( (T + 369)^{2} \)
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