# Properties

 Label 525.2.r.b Level 525 Weight 2 Character orbit 525.r Analytic conductor 4.192 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{3} + ( -2 + 2 \zeta_{12}^{2} ) q^{4} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{3} + ( -2 + 2 \zeta_{12}^{2} ) q^{4} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{12} + \zeta_{12}^{3} q^{13} -4 \zeta_{12}^{2} q^{16} -6 \zeta_{12} q^{17} + 5 \zeta_{12}^{2} q^{19} + ( -3 + 2 \zeta_{12}^{2} ) q^{21} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{23} + \zeta_{12}^{3} q^{27} + ( -4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} + 6 q^{29} + ( -5 + 5 \zeta_{12}^{2} ) q^{31} -2 q^{36} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{37} + ( -1 + \zeta_{12}^{2} ) q^{39} + 12 q^{41} + \zeta_{12}^{3} q^{43} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{47} -4 \zeta_{12}^{3} q^{48} + ( -3 - 5 \zeta_{12}^{2} ) q^{49} -6 \zeta_{12}^{2} q^{51} -2 \zeta_{12} q^{52} + 5 \zeta_{12}^{3} q^{57} + ( -6 + 6 \zeta_{12}^{2} ) q^{59} -2 \zeta_{12}^{2} q^{61} + ( -3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{63} + 8 q^{64} + 7 \zeta_{12} q^{67} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{68} -6 q^{69} + 12 q^{71} + 11 \zeta_{12} q^{73} -10 q^{76} -13 \zeta_{12}^{2} q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} + 12 \zeta_{12}^{3} q^{83} + ( 2 - 6 \zeta_{12}^{2} ) q^{84} + 6 \zeta_{12} q^{87} + 6 \zeta_{12}^{2} q^{89} + ( -2 - \zeta_{12}^{2} ) q^{91} -12 \zeta_{12}^{3} q^{92} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{93} -10 \zeta_{12}^{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 2q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 2q^{9} - 8q^{16} + 10q^{19} - 8q^{21} + 24q^{29} - 10q^{31} - 8q^{36} - 2q^{39} + 48q^{41} - 22q^{49} - 12q^{51} - 12q^{59} - 4q^{61} + 32q^{64} - 24q^{69} + 48q^{71} - 40q^{76} - 26q^{79} - 2q^{81} - 4q^{84} + 12q^{89} - 10q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
424.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 −0.866025 0.500000i −1.00000 + 1.73205i 0 0 0.866025 2.50000i 0 0.500000 + 0.866025i 0
424.2 0 0.866025 + 0.500000i −1.00000 + 1.73205i 0 0 −0.866025 + 2.50000i 0 0.500000 + 0.866025i 0
499.1 0 −0.866025 + 0.500000i −1.00000 1.73205i 0 0 0.866025 + 2.50000i 0 0.500000 0.866025i 0
499.2 0 0.866025 0.500000i −1.00000 1.73205i 0 0 −0.866025 2.50000i 0 0.500000 0.866025i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j 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.r.b 4
5.b even 2 1 inner 525.2.r.b 4
5.c odd 4 1 105.2.i.a 2
5.c odd 4 1 525.2.i.c 2
7.c even 3 1 inner 525.2.r.b 4
15.e even 4 1 315.2.j.b 2
20.e even 4 1 1680.2.bg.m 2
35.f even 4 1 735.2.i.c 2
35.j even 6 1 inner 525.2.r.b 4
35.k even 12 1 735.2.a.d 1
35.k even 12 1 735.2.i.c 2
35.k even 12 1 3675.2.a.i 1
35.l odd 12 1 105.2.i.a 2
35.l odd 12 1 525.2.i.c 2
35.l odd 12 1 735.2.a.e 1
35.l odd 12 1 3675.2.a.h 1
105.w odd 12 1 2205.2.a.d 1
105.x even 12 1 315.2.j.b 2
105.x even 12 1 2205.2.a.f 1
140.w even 12 1 1680.2.bg.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.a 2 5.c odd 4 1
105.2.i.a 2 35.l odd 12 1
315.2.j.b 2 15.e even 4 1
315.2.j.b 2 105.x even 12 1
525.2.i.c 2 5.c odd 4 1
525.2.i.c 2 35.l odd 12 1
525.2.r.b 4 1.a even 1 1 trivial
525.2.r.b 4 5.b even 2 1 inner
525.2.r.b 4 7.c even 3 1 inner
525.2.r.b 4 35.j even 6 1 inner
735.2.a.d 1 35.k even 12 1
735.2.a.e 1 35.l odd 12 1
735.2.i.c 2 35.f even 4 1
735.2.i.c 2 35.k even 12 1
1680.2.bg.m 2 20.e even 4 1
1680.2.bg.m 2 140.w even 12 1
2205.2.a.d 1 105.w odd 12 1
2205.2.a.f 1 105.x even 12 1
3675.2.a.h 1 35.l odd 12 1
3675.2.a.i 1 35.k even 12 1

## Hecke kernels

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

 $$T_{2}$$ $$T_{11}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} + 4 T^{4} )^{2}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ 
$7$ $$1 + 11 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 11 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 25 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{4}$$
$31$ $$( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 - 12 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 85 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 + 53 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 12 T + 71 T^{2} )^{4}$$
$73$ $$1 + 25 T^{2} - 4704 T^{4} + 133225 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 4 T + 79 T^{2} )^{2}( 1 + 17 T + 79 T^{2} )^{2}$$
$83$ $$( 1 - 22 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 94 T^{2} + 9409 T^{4} )^{2}$$