Newspace parameters
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(84.2015054018\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 21) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
−5.00000 | −9.00000 | −7.00000 | 0 | 45.0000 | 49.0000 | 195.000 | 81.0000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(5\) | \(1\) |
\(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.6.a.b | 1 | |
5.b | even | 2 | 1 | 21.6.a.c | ✓ | 1 | |
5.c | odd | 4 | 2 | 525.6.d.c | 2 | ||
15.d | odd | 2 | 1 | 63.6.a.b | 1 | ||
20.d | odd | 2 | 1 | 336.6.a.i | 1 | ||
35.c | odd | 2 | 1 | 147.6.a.f | 1 | ||
35.i | odd | 6 | 2 | 147.6.e.d | 2 | ||
35.j | even | 6 | 2 | 147.6.e.c | 2 | ||
60.h | even | 2 | 1 | 1008.6.a.a | 1 | ||
105.g | even | 2 | 1 | 441.6.a.c | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
21.6.a.c | ✓ | 1 | 5.b | even | 2 | 1 | |
63.6.a.b | 1 | 15.d | odd | 2 | 1 | ||
147.6.a.f | 1 | 35.c | odd | 2 | 1 | ||
147.6.e.c | 2 | 35.j | even | 6 | 2 | ||
147.6.e.d | 2 | 35.i | odd | 6 | 2 | ||
336.6.a.i | 1 | 20.d | odd | 2 | 1 | ||
441.6.a.c | 1 | 105.g | even | 2 | 1 | ||
525.6.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
525.6.d.c | 2 | 5.c | odd | 4 | 2 | ||
1008.6.a.a | 1 | 60.h | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 5 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(525))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 5 \)
$3$
\( T + 9 \)
$5$
\( T \)
$7$
\( T - 49 \)
$11$
\( T - 52 \)
$13$
\( T - 770 \)
$17$
\( T - 2022 \)
$19$
\( T - 1732 \)
$23$
\( T - 576 \)
$29$
\( T - 5518 \)
$31$
\( T - 6336 \)
$37$
\( T - 7338 \)
$41$
\( T + 3262 \)
$43$
\( T + 5420 \)
$47$
\( T + 864 \)
$53$
\( T + 4182 \)
$59$
\( T + 11220 \)
$61$
\( T + 45602 \)
$67$
\( T + 1396 \)
$71$
\( T - 18720 \)
$73$
\( T + 46362 \)
$79$
\( T - 97424 \)
$83$
\( T - 81228 \)
$89$
\( T + 3182 \)
$97$
\( T + 4914 \)
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