# Properties

 Label 525.2.b.j Level 525 Weight 2 Character orbit 525.b Analytic conductor 4.192 Analytic rank 0 Dimension 8 CM no Inner twists 8

# 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: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{24}^{4} ) q^{2} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{3} - q^{4} + ( \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{6} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{7} + ( 1 - 2 \zeta_{24}^{4} ) q^{8} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{24}^{4} ) q^{2} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{3} - q^{4} + ( \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{6} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{7} + ( 1 - 2 \zeta_{24}^{4} ) q^{8} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{9} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{11} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{12} -4 \zeta_{24}^{6} q^{13} + ( 3 \zeta_{24} - 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{14} -5 q^{16} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{17} + ( 1 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{18} + ( -1 - 2 \zeta_{24} - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{21} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{22} + ( -2 + 4 \zeta_{24}^{4} ) q^{23} + ( \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{24} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{26} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 5 \zeta_{24}^{6} ) q^{27} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{28} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{29} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{31} + ( -3 + 6 \zeta_{24}^{4} ) q^{32} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{33} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{34} + ( -1 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{36} + ( -4 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{39} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{41} + ( -1 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{42} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{43} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{44} + 6 q^{46} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{47} + ( 5 \zeta_{24} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{48} + ( 5 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{49} + ( 4 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{51} + 4 \zeta_{24}^{6} q^{52} + ( -\zeta_{24} - 10 \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} + 5 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{54} + ( 3 \zeta_{24} - 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{56} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{58} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{59} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{61} + ( 12 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} ) q^{62} + ( -4 + 3 \zeta_{24} + 3 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{63} - q^{64} + ( 2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{66} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{67} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{68} + ( -2 \zeta_{24} + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{69} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{71} + ( 1 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{72} + 8 \zeta_{24}^{6} q^{73} + ( 4 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{77} + ( -4 + 4 \zeta_{24} + 4 \zeta_{24}^{3} + 8 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{78} -8 q^{79} + ( -7 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{81} -6 \zeta_{24}^{6} q^{82} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{83} + ( 1 + 2 \zeta_{24} + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{84} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{86} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{6} ) q^{87} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{88} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{89} + ( -4 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{91} + ( 2 - 4 \zeta_{24}^{4} ) q^{92} + ( 8 + 4 \zeta_{24} + 4 \zeta_{24}^{3} - 16 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{93} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{94} + ( -3 \zeta_{24} + 6 \zeta_{24}^{2} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{96} -8 \zeta_{24}^{6} q^{97} + ( 5 - 6 \zeta_{24} - 6 \zeta_{24}^{3} - 10 \zeta_{24}^{4} + 6 \zeta_{24}^{5} ) q^{98} + ( 8 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} + 8q^{9} + O(q^{10})$$ $$8q - 8q^{4} + 8q^{9} - 40q^{16} - 8q^{21} - 8q^{36} - 32q^{39} + 48q^{46} + 40q^{49} + 32q^{51} - 8q^{64} - 64q^{79} - 56q^{81} + 8q^{84} - 32q^{91} + 64q^{99} + O(q^{100})$$

## 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
 0.965926 + 0.258819i −0.258819 + 0.965926i −0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 + 0.258819i
1.73205i −1.41421 1.00000i −1.00000 0 −1.73205 + 2.44949i 2.44949 1.00000i 1.73205i 1.00000 + 2.82843i 0
251.2 1.73205i −1.41421 + 1.00000i −1.00000 0 1.73205 + 2.44949i −2.44949 + 1.00000i 1.73205i 1.00000 2.82843i 0
251.3 1.73205i 1.41421 1.00000i −1.00000 0 −1.73205 2.44949i −2.44949 1.00000i 1.73205i 1.00000 2.82843i 0
251.4 1.73205i 1.41421 + 1.00000i −1.00000 0 1.73205 2.44949i 2.44949 + 1.00000i 1.73205i 1.00000 + 2.82843i 0
251.5 1.73205i −1.41421 1.00000i −1.00000 0 1.73205 2.44949i −2.44949 1.00000i 1.73205i 1.00000 + 2.82843i 0
251.6 1.73205i −1.41421 + 1.00000i −1.00000 0 −1.73205 2.44949i 2.44949 + 1.00000i 1.73205i 1.00000 2.82843i 0
251.7 1.73205i 1.41421 1.00000i −1.00000 0 1.73205 + 2.44949i 2.44949 1.00000i 1.73205i 1.00000 2.82843i 0
251.8 1.73205i 1.41421 + 1.00000i −1.00000 0 −1.73205 + 2.44949i −2.44949 + 1.00000i 1.73205i 1.00000 + 2.82843i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.8 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 inner
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.b.j 8
3.b odd 2 1 inner 525.2.b.j 8
5.b even 2 1 inner 525.2.b.j 8
5.c odd 4 1 105.2.g.a 4
5.c odd 4 1 105.2.g.c yes 4
7.b odd 2 1 inner 525.2.b.j 8
15.d odd 2 1 inner 525.2.b.j 8
15.e even 4 1 105.2.g.a 4
15.e even 4 1 105.2.g.c yes 4
20.e even 4 1 1680.2.k.a 4
20.e even 4 1 1680.2.k.c 4
21.c even 2 1 inner 525.2.b.j 8
35.c odd 2 1 inner 525.2.b.j 8
35.f even 4 1 105.2.g.a 4
35.f even 4 1 105.2.g.c yes 4
35.k even 12 2 735.2.p.a 8
35.k even 12 2 735.2.p.c 8
35.l odd 12 2 735.2.p.a 8
35.l odd 12 2 735.2.p.c 8
60.l odd 4 1 1680.2.k.a 4
60.l odd 4 1 1680.2.k.c 4
105.g even 2 1 inner 525.2.b.j 8
105.k odd 4 1 105.2.g.a 4
105.k odd 4 1 105.2.g.c yes 4
105.w odd 12 2 735.2.p.a 8
105.w odd 12 2 735.2.p.c 8
105.x even 12 2 735.2.p.a 8
105.x even 12 2 735.2.p.c 8
140.j odd 4 1 1680.2.k.a 4
140.j odd 4 1 1680.2.k.c 4
420.w even 4 1 1680.2.k.a 4
420.w even 4 1 1680.2.k.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.a 4 5.c odd 4 1
105.2.g.a 4 15.e even 4 1
105.2.g.a 4 35.f even 4 1
105.2.g.a 4 105.k odd 4 1
105.2.g.c yes 4 5.c odd 4 1
105.2.g.c yes 4 15.e even 4 1
105.2.g.c yes 4 35.f even 4 1
105.2.g.c yes 4 105.k odd 4 1
525.2.b.j 8 1.a even 1 1 trivial
525.2.b.j 8 3.b odd 2 1 inner
525.2.b.j 8 5.b even 2 1 inner
525.2.b.j 8 7.b odd 2 1 inner
525.2.b.j 8 15.d odd 2 1 inner
525.2.b.j 8 21.c even 2 1 inner
525.2.b.j 8 35.c odd 2 1 inner
525.2.b.j 8 105.g even 2 1 inner
735.2.p.a 8 35.k even 12 2
735.2.p.a 8 35.l odd 12 2
735.2.p.a 8 105.w odd 12 2
735.2.p.a 8 105.x even 12 2
735.2.p.c 8 35.k even 12 2
735.2.p.c 8 35.l odd 12 2
735.2.p.c 8 105.w odd 12 2
735.2.p.c 8 105.x even 12 2
1680.2.k.a 4 20.e even 4 1
1680.2.k.a 4 60.l odd 4 1
1680.2.k.a 4 140.j odd 4 1
1680.2.k.a 4 420.w even 4 1
1680.2.k.c 4 20.e even 4 1
1680.2.k.c 4 60.l odd 4 1
1680.2.k.c 4 140.j odd 4 1
1680.2.k.c 4 420.w even 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 3$$ $$T_{11}^{2} + 8$$ $$T_{17}^{2} - 8$$ $$T_{37}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + 4 T^{4} )^{4}$$
$3$ $$( 1 - 2 T^{2} + 9 T^{4} )^{2}$$
$5$ 
$7$ $$( 1 - 10 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 6 T + 11 T^{2} )^{4}( 1 + 6 T + 11 T^{2} )^{4}$$
$13$ $$( 1 - 6 T + 13 T^{2} )^{4}( 1 + 6 T + 13 T^{2} )^{4}$$
$17$ $$( 1 + 26 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 - 19 T^{2} )^{8}$$
$23$ $$( 1 - 34 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 26 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 + 34 T^{2} + 961 T^{4} )^{4}$$
$37$ $$( 1 + 37 T^{2} )^{8}$$
$41$ $$( 1 + 70 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 62 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 86 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 - 53 T^{2} )^{8}$$
$59$ $$( 1 + 70 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 - 26 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 110 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 134 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 82 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{8}$$
$83$ $$( 1 + 158 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 + 70 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 - 18 T + 97 T^{2} )^{4}( 1 + 18 T + 97 T^{2} )^{4}$$