# Properties

 Label 525.2.g.a Level 525 Weight 2 Character orbit 525.g Analytic conductor 4.192 Analytic rank 0 Dimension 4 CM discriminant -3 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5^{2}$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} -2 q^{4} + ( \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} -2 q^{4} + ( \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 q^{9} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{12} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{13} + 4 q^{16} + ( -5 + 10 \zeta_{12}^{2} ) q^{19} + ( -4 + 5 \zeta_{12}^{2} ) q^{21} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( -2 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{28} + ( -5 + 10 \zeta_{12}^{2} ) q^{31} -6 q^{36} + 10 \zeta_{12}^{3} q^{37} + 9 q^{39} + 5 \zeta_{12}^{3} q^{43} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{48} + ( -3 - 5 \zeta_{12}^{2} ) q^{49} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{52} -15 \zeta_{12}^{3} q^{57} + ( 5 - 10 \zeta_{12}^{2} ) q^{61} + ( 3 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{63} -8 q^{64} + 5 \zeta_{12}^{3} q^{67} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{73} + ( 10 - 20 \zeta_{12}^{2} ) q^{76} + 4 q^{79} + 9 q^{81} + ( 8 - 10 \zeta_{12}^{2} ) q^{84} + ( -12 + 15 \zeta_{12}^{2} ) q^{91} -15 \zeta_{12}^{3} q^{93} + ( -22 \zeta_{12} + 11 \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} + 12q^{9} + O(q^{10})$$ $$4q - 8q^{4} + 12q^{9} + 16q^{16} - 6q^{21} - 24q^{36} + 36q^{39} - 22q^{49} - 32q^{64} + 16q^{79} + 36q^{81} + 12q^{84} - 18q^{91} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
524.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i
0 −1.73205 −2.00000 0 0 0.866025 2.50000i 0 3.00000 0
524.2 0 −1.73205 −2.00000 0 0 0.866025 + 2.50000i 0 3.00000 0
524.3 0 1.73205 −2.00000 0 0 −0.866025 2.50000i 0 3.00000 0
524.4 0 1.73205 −2.00000 0 0 −0.866025 + 2.50000i 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 $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g 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.g.a 4
3.b odd 2 1 CM 525.2.g.a 4
5.b even 2 1 inner 525.2.g.a 4
5.c odd 4 1 525.2.b.c 2
5.c odd 4 1 525.2.b.d yes 2
7.b odd 2 1 inner 525.2.g.a 4
15.d odd 2 1 inner 525.2.g.a 4
15.e even 4 1 525.2.b.c 2
15.e even 4 1 525.2.b.d yes 2
21.c even 2 1 inner 525.2.g.a 4
35.c odd 2 1 inner 525.2.g.a 4
35.f even 4 1 525.2.b.c 2
35.f even 4 1 525.2.b.d yes 2
105.g even 2 1 inner 525.2.g.a 4
105.k odd 4 1 525.2.b.c 2
105.k odd 4 1 525.2.b.d yes 2

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

## Hecke kernels

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

 $$T_{2}$$ $$T_{41}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{4}$$
$3$ $$( 1 - 3 T^{2} )^{2}$$
$5$ 
$7$ $$1 + 11 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 11 T^{2} )^{4}$$
$13$ $$( 1 - T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 17 T^{2} )^{4}$$
$19$ $$( 1 - T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2}$$
$23$ $$( 1 + 23 T^{2} )^{4}$$
$29$ $$( 1 - 29 T^{2} )^{4}$$
$31$ $$( 1 - 7 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 26 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{4}$$
$43$ $$( 1 - 61 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 47 T^{2} )^{4}$$
$53$ $$( 1 + 53 T^{2} )^{4}$$
$59$ $$( 1 + 59 T^{2} )^{4}$$
$61$ $$( 1 - 13 T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 109 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 71 T^{2} )^{4}$$
$73$ $$( 1 - 46 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 4 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 83 T^{2} )^{4}$$
$89$ $$( 1 + 89 T^{2} )^{4}$$
$97$ $$( 1 - 169 T^{2} + 9409 T^{4} )^{2}$$