# Properties

 Label 525.2.bf.b Level 525 Weight 2 Character orbit 525.bf Analytic conductor 4.192 Analytic rank 0 Dimension 8 CM discriminant -3 Inner twists 16

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.bf (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{24} - \zeta_{24}^{5} ) q^{3} + 2 \zeta_{24}^{2} q^{4} + ( -2 \zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{7} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{24} - \zeta_{24}^{5} ) q^{3} + 2 \zeta_{24}^{2} q^{4} + ( -2 \zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{7} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{9} + ( 4 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{12} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{13} + 4 \zeta_{24}^{4} q^{16} + ( 7 \zeta_{24}^{2} - 7 \zeta_{24}^{6} ) q^{19} + ( -5 + 4 \zeta_{24}^{4} ) q^{21} + ( 3 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{27} + ( -6 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{28} + ( -7 + 7 \zeta_{24}^{4} ) q^{31} + 6 q^{36} + ( -14 \zeta_{24}^{3} + 7 \zeta_{24}^{7} ) q^{37} + 3 \zeta_{24}^{2} q^{39} + ( -7 \zeta_{24} - 7 \zeta_{24}^{5} ) q^{43} + ( 4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{48} + ( 3 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{49} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{52} + ( 7 \zeta_{24}^{3} - 14 \zeta_{24}^{7} ) q^{57} -14 \zeta_{24}^{4} q^{61} + ( -6 \zeta_{24} + 9 \zeta_{24}^{5} ) q^{63} + 8 \zeta_{24}^{6} q^{64} + ( 7 \zeta_{24}^{3} + 7 \zeta_{24}^{7} ) q^{67} + ( 18 \zeta_{24} - 9 \zeta_{24}^{5} ) q^{73} + 14 q^{76} + ( -13 \zeta_{24}^{2} + 13 \zeta_{24}^{6} ) q^{79} + ( 9 - 9 \zeta_{24}^{4} ) q^{81} + ( -10 \zeta_{24}^{2} + 8 \zeta_{24}^{6} ) q^{84} + ( -4 - \zeta_{24}^{4} ) q^{91} + ( -7 \zeta_{24} + 14 \zeta_{24}^{5} ) q^{93} + ( -8 \zeta_{24}^{3} + 16 \zeta_{24}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 16q^{16} - 24q^{21} - 28q^{31} + 48q^{36} - 56q^{61} + 112q^{76} + 36q^{81} - 36q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-\zeta_{24}^{6}$$ $$-1$$ $$-\zeta_{24}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i
0 −0.448288 1.67303i −1.73205 1.00000i 0 0 −1.48356 2.19067i 0 −2.59808 + 1.50000i 0
32.2 0 0.448288 + 1.67303i −1.73205 1.00000i 0 0 1.48356 + 2.19067i 0 −2.59808 + 1.50000i 0
107.1 0 −1.67303 0.448288i 1.73205 1.00000i 0 0 2.19067 + 1.48356i 0 2.59808 + 1.50000i 0
107.2 0 1.67303 + 0.448288i 1.73205 1.00000i 0 0 −2.19067 1.48356i 0 2.59808 + 1.50000i 0
368.1 0 −1.67303 + 0.448288i 1.73205 + 1.00000i 0 0 2.19067 1.48356i 0 2.59808 1.50000i 0
368.2 0 1.67303 0.448288i 1.73205 + 1.00000i 0 0 −2.19067 + 1.48356i 0 2.59808 1.50000i 0
443.1 0 −0.448288 + 1.67303i −1.73205 + 1.00000i 0 0 −1.48356 + 2.19067i 0 −2.59808 1.50000i 0
443.2 0 0.448288 1.67303i −1.73205 + 1.00000i 0 0 1.48356 2.19067i 0 −2.59808 1.50000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 443.2 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
5.c odd 4 2 inner
7.c even 3 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
21.h odd 6 1 inner
35.j even 6 1 inner
35.l odd 12 2 inner
105.o odd 6 1 inner
105.x even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bf.b 8
3.b odd 2 1 CM 525.2.bf.b 8
5.b even 2 1 inner 525.2.bf.b 8
5.c odd 4 2 inner 525.2.bf.b 8
7.c even 3 1 inner 525.2.bf.b 8
15.d odd 2 1 inner 525.2.bf.b 8
15.e even 4 2 inner 525.2.bf.b 8
21.h odd 6 1 inner 525.2.bf.b 8
35.j even 6 1 inner 525.2.bf.b 8
35.l odd 12 2 inner 525.2.bf.b 8
105.o odd 6 1 inner 525.2.bf.b 8
105.x even 12 2 inner 525.2.bf.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bf.b 8 1.a even 1 1 trivial
525.2.bf.b 8 3.b odd 2 1 CM
525.2.bf.b 8 5.b even 2 1 inner
525.2.bf.b 8 5.c odd 4 2 inner
525.2.bf.b 8 7.c even 3 1 inner
525.2.bf.b 8 15.d odd 2 1 inner
525.2.bf.b 8 15.e even 4 2 inner
525.2.bf.b 8 21.h odd 6 1 inner
525.2.bf.b 8 35.j even 6 1 inner
525.2.bf.b 8 35.l odd 12 2 inner
525.2.bf.b 8 105.o odd 6 1 inner
525.2.bf.b 8 105.x even 12 2 inner

## Hecke kernels

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

 $$T_{2}$$ $$T_{13}^{4} + 9$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$
$3$ $$1 - 9 T^{4} + 81 T^{8}$$
$5$ 
$7$ $$1 + 71 T^{4} + 2401 T^{8}$$
$11$ $$( 1 + 11 T^{2} + 121 T^{4} )^{4}$$
$13$ $$( 1 + 191 T^{4} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 289 T^{4} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 26 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 529 T^{4} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 29 T^{2} )^{8}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{4}( 1 + 11 T + 31 T^{2} )^{4}$$
$37$ $$( 1 - 2062 T^{4} + 1874161 T^{8} )( 1 - 529 T^{4} + 1874161 T^{8} )$$
$41$ $$( 1 - 41 T^{2} )^{8}$$
$43$ $$( 1 + 23 T^{4} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 - 2209 T^{4} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 2809 T^{4} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 59 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 + T + 61 T^{2} )^{4}( 1 + 13 T + 61 T^{2} )^{4}$$
$67$ $$( 1 + 2903 T^{4} + 20151121 T^{8} )( 1 + 5906 T^{4} + 20151121 T^{8} )$$
$71$ $$( 1 - 71 T^{2} )^{8}$$
$73$ $$( 1 - 8542 T^{4} + 28398241 T^{8} )( 1 + 9791 T^{4} + 28398241 T^{8} )$$
$79$ $$( 1 - 142 T^{2} + 6241 T^{4} )^{2}( 1 + 131 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 6889 T^{4} )^{4}$$
$89$ $$( 1 - 89 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 - 18814 T^{4} + 88529281 T^{8} )^{2}$$