# Properties

 Label 525.2.t.d Level 525 Weight 2 Character orbit 525.t 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.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} + ( 3 - 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} + ( 3 - 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( -4 + 2 \zeta_{6} ) q^{12} + ( -4 + 8 \zeta_{6} ) q^{13} -4 \zeta_{6} q^{16} + ( 4 - 2 \zeta_{6} ) q^{19} + ( 5 - \zeta_{6} ) q^{21} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -2 + 6 \zeta_{6} ) q^{28} + ( -5 - 5 \zeta_{6} ) q^{31} -6 q^{36} + 11 \zeta_{6} q^{37} + ( -12 + 12 \zeta_{6} ) q^{39} -5 q^{43} + ( 4 - 8 \zeta_{6} ) q^{48} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -8 - 8 \zeta_{6} ) q^{52} + 6 q^{57} + ( 18 - 9 \zeta_{6} ) q^{61} + ( 6 + 3 \zeta_{6} ) q^{63} + 8 q^{64} + ( 16 - 16 \zeta_{6} ) q^{67} + ( -1 - \zeta_{6} ) q^{73} + ( -4 + 8 \zeta_{6} ) q^{76} -17 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -8 + 10 \zeta_{6} ) q^{84} + ( 4 + 16 \zeta_{6} ) q^{91} -15 \zeta_{6} q^{93} + ( 3 - 6 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - 2q^{4} + 4q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} - 2q^{4} + 4q^{7} + 3q^{9} - 6q^{12} - 4q^{16} + 6q^{19} + 9q^{21} + 2q^{28} - 15q^{31} - 12q^{36} + 11q^{37} - 12q^{39} - 10q^{43} + 2q^{49} - 24q^{52} + 12q^{57} + 27q^{61} + 15q^{63} + 16q^{64} + 16q^{67} - 3q^{73} - 17q^{79} - 9q^{81} - 6q^{84} + 24q^{91} - 15q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.50000 0.866025i −1.00000 1.73205i 0 0 2.00000 + 1.73205i 0 1.50000 2.59808i 0
101.1 0 1.50000 + 0.866025i −1.00000 + 1.73205i 0 0 2.00000 1.73205i 0 1.50000 + 2.59808i 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.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.t.d yes 2
3.b odd 2 1 CM 525.2.t.d yes 2
5.b even 2 1 525.2.t.b 2
5.c odd 4 2 525.2.q.c 4
7.d odd 6 1 inner 525.2.t.d yes 2
15.d odd 2 1 525.2.t.b 2
15.e even 4 2 525.2.q.c 4
21.g even 6 1 inner 525.2.t.d yes 2
35.i odd 6 1 525.2.t.b 2
35.k even 12 2 525.2.q.c 4
105.p even 6 1 525.2.t.b 2
105.w odd 12 2 525.2.q.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.q.c 4 5.c odd 4 2
525.2.q.c 4 15.e even 4 2
525.2.q.c 4 35.k even 12 2
525.2.q.c 4 105.w odd 12 2
525.2.t.b 2 5.b even 2 1
525.2.t.b 2 15.d odd 2 1
525.2.t.b 2 35.i odd 6 1
525.2.t.b 2 105.p even 6 1
525.2.t.d yes 2 1.a even 1 1 trivial
525.2.t.d yes 2 3.b odd 2 1 CM
525.2.t.d yes 2 7.d odd 6 1 inner
525.2.t.d yes 2 21.g even 6 1 inner

## Hecke kernels

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

 $$T_{2}$$ $$T_{13}^{2} + 48$$ $$T_{37}^{2} - 11 T_{37} + 121$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2} + 4 T^{4}$$
$3$ $$1 - 3 T + 3 T^{2}$$
$5$ 
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 7 T + 19 T^{2} )( 1 + T + 19 T^{2} )$$
$23$ $$1 + 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 29 T^{2} )^{2}$$
$31$ $$( 1 + 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$
$37$ $$( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} )$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 + 5 T + 43 T^{2} )^{2}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 + 53 T^{2} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 14 T + 61 T^{2} )( 1 - 13 T + 61 T^{2} )$$
$67$ $$( 1 - 11 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} )$$
$71$ $$( 1 - 71 T^{2} )^{2}$$
$73$ $$( 1 - 7 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$
$79$ $$( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 - 89 T^{2} + 7921 T^{4}$$
$97$ $$( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} )$$