Newspace parameters
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(84.2015054018\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 21) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(176\) | \(451\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
274.1 |
|
− | 5.00000i | 9.00000i | 7.00000 | 0 | 45.0000 | 49.0000i | − | 195.000i | −81.0000 | 0 | ||||||||||||||||||||||
274.2 | 5.00000i | − | 9.00000i | 7.00000 | 0 | 45.0000 | − | 49.0000i | 195.000i | −81.0000 | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.6.d.c | 2 | |
5.b | even | 2 | 1 | inner | 525.6.d.c | 2 | |
5.c | odd | 4 | 1 | 21.6.a.c | ✓ | 1 | |
5.c | odd | 4 | 1 | 525.6.a.b | 1 | ||
15.e | even | 4 | 1 | 63.6.a.b | 1 | ||
20.e | even | 4 | 1 | 336.6.a.i | 1 | ||
35.f | even | 4 | 1 | 147.6.a.f | 1 | ||
35.k | even | 12 | 2 | 147.6.e.d | 2 | ||
35.l | odd | 12 | 2 | 147.6.e.c | 2 | ||
60.l | odd | 4 | 1 | 1008.6.a.a | 1 | ||
105.k | odd | 4 | 1 | 441.6.a.c | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
21.6.a.c | ✓ | 1 | 5.c | odd | 4 | 1 | |
63.6.a.b | 1 | 15.e | even | 4 | 1 | ||
147.6.a.f | 1 | 35.f | even | 4 | 1 | ||
147.6.e.c | 2 | 35.l | odd | 12 | 2 | ||
147.6.e.d | 2 | 35.k | even | 12 | 2 | ||
336.6.a.i | 1 | 20.e | even | 4 | 1 | ||
441.6.a.c | 1 | 105.k | odd | 4 | 1 | ||
525.6.a.b | 1 | 5.c | odd | 4 | 1 | ||
525.6.d.c | 2 | 1.a | even | 1 | 1 | trivial | |
525.6.d.c | 2 | 5.b | even | 2 | 1 | inner | |
1008.6.a.a | 1 | 60.l | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 25 \)
acting on \(S_{6}^{\mathrm{new}}(525, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 25 \)
$3$
\( T^{2} + 81 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 2401 \)
$11$
\( (T - 52)^{2} \)
$13$
\( T^{2} + 592900 \)
$17$
\( T^{2} + 4088484 \)
$19$
\( (T + 1732)^{2} \)
$23$
\( T^{2} + 331776 \)
$29$
\( (T + 5518)^{2} \)
$31$
\( (T - 6336)^{2} \)
$37$
\( T^{2} + 53846244 \)
$41$
\( (T + 3262)^{2} \)
$43$
\( T^{2} + 29376400 \)
$47$
\( T^{2} + 746496 \)
$53$
\( T^{2} + 17489124 \)
$59$
\( (T - 11220)^{2} \)
$61$
\( (T + 45602)^{2} \)
$67$
\( T^{2} + 1948816 \)
$71$
\( (T - 18720)^{2} \)
$73$
\( T^{2} + 2149435044 \)
$79$
\( (T + 97424)^{2} \)
$83$
\( T^{2} + 6597987984 \)
$89$
\( (T - 3182)^{2} \)
$97$
\( T^{2} + 24147396 \)
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