# Properties

 Label 525.2.b.d Level 525 Weight 2 Character orbit 525.b Analytic conductor 4.192 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + 2 q^{4} + ( 3 - \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + 2 q^{4} + ( 3 - \zeta_{6} ) q^{7} -3 q^{9} + ( 2 - 4 \zeta_{6} ) q^{12} + ( 3 - 6 \zeta_{6} ) q^{13} + 4 q^{16} + ( -5 + 10 \zeta_{6} ) q^{19} + ( 1 - 5 \zeta_{6} ) q^{21} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 6 - 2 \zeta_{6} ) q^{28} + ( 5 - 10 \zeta_{6} ) q^{31} -6 q^{36} -10 q^{37} -9 q^{39} + 5 q^{43} + ( 4 - 8 \zeta_{6} ) q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 6 - 12 \zeta_{6} ) q^{52} + 15 q^{57} + ( -5 + 10 \zeta_{6} ) q^{61} + ( -9 + 3 \zeta_{6} ) q^{63} + 8 q^{64} -5 q^{67} + ( -8 + 16 \zeta_{6} ) q^{73} + ( -10 + 20 \zeta_{6} ) q^{76} -4 q^{79} + 9 q^{81} + ( 2 - 10 \zeta_{6} ) q^{84} + ( 3 - 15 \zeta_{6} ) q^{91} -15 q^{93} + ( -11 + 22 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{4} + 5q^{7} - 6q^{9} + O(q^{10})$$ $$2q + 4q^{4} + 5q^{7} - 6q^{9} + 8q^{16} - 3q^{21} + 10q^{28} - 12q^{36} - 20q^{37} - 18q^{39} + 10q^{43} + 11q^{49} + 30q^{57} - 15q^{63} + 16q^{64} - 10q^{67} - 8q^{79} + 18q^{81} - 6q^{84} - 9q^{91} - 30q^{93} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 2.00000 0 0 2.50000 0.866025i 0 −3.00000 0
251.2 0 1.73205i 2.00000 0 0 2.50000 + 0.866025i 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.b.d yes 2
3.b odd 2 1 CM 525.2.b.d yes 2
5.b even 2 1 525.2.b.c 2
5.c odd 4 2 525.2.g.a 4
7.b odd 2 1 inner 525.2.b.d yes 2
15.d odd 2 1 525.2.b.c 2
15.e even 4 2 525.2.g.a 4
21.c even 2 1 inner 525.2.b.d yes 2
35.c odd 2 1 525.2.b.c 2
35.f even 4 2 525.2.g.a 4
105.g even 2 1 525.2.b.c 2
105.k odd 4 2 525.2.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.b.c 2 5.b even 2 1
525.2.b.c 2 15.d odd 2 1
525.2.b.c 2 35.c odd 2 1
525.2.b.c 2 105.g even 2 1
525.2.b.d yes 2 1.a even 1 1 trivial
525.2.b.d yes 2 3.b odd 2 1 CM
525.2.b.d yes 2 7.b odd 2 1 inner
525.2.b.d yes 2 21.c even 2 1 inner
525.2.g.a 4 5.c odd 4 2
525.2.g.a 4 15.e even 4 2
525.2.g.a 4 35.f even 4 2
525.2.g.a 4 105.k odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}$$ $$T_{11}$$ $$T_{17}$$ $$T_{37} + 10$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} )^{2}$$
$3$ $$1 + 3 T^{2}$$
$5$ 1
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$( 1 - T + 19 T^{2} )( 1 + T + 19 T^{2} )$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 - 29 T^{2} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$( 1 + 10 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 5 T + 43 T^{2} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$( 1 + 5 T + 67 T^{2} )^{2}$$
$71$ $$( 1 - 71 T^{2} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$
$79$ $$( 1 + 4 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} )$$