# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} - q^{4} -3 q^{6} + ( -3 + 2 \zeta_{6} ) q^{7} + ( 1 - 2 \zeta_{6} ) q^{8} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} - q^{4} -3 q^{6} + ( -3 + 2 \zeta_{6} ) q^{7} + ( 1 - 2 \zeta_{6} ) q^{8} -3 q^{9} + ( 2 - 4 \zeta_{6} ) q^{11} + ( -1 + 2 \zeta_{6} ) q^{12} + ( 1 + 4 \zeta_{6} ) q^{14} -5 q^{16} -6 q^{17} + ( -3 + 6 \zeta_{6} ) q^{18} + ( -2 + 4 \zeta_{6} ) q^{19} + ( 1 + 4 \zeta_{6} ) q^{21} -6 q^{22} + ( 2 - 4 \zeta_{6} ) q^{23} -3 q^{24} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 3 - 2 \zeta_{6} ) q^{28} + ( 4 - 8 \zeta_{6} ) q^{29} + ( 2 - 4 \zeta_{6} ) q^{31} + ( -3 + 6 \zeta_{6} ) q^{32} -6 q^{33} + ( -6 + 12 \zeta_{6} ) q^{34} + 3 q^{36} + 2 q^{37} + 6 q^{38} + 6 q^{41} + ( 9 - 6 \zeta_{6} ) q^{42} + 8 q^{43} + ( -2 + 4 \zeta_{6} ) q^{44} -6 q^{46} + 12 q^{47} + ( -5 + 10 \zeta_{6} ) q^{48} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -6 + 12 \zeta_{6} ) q^{51} + 9 q^{54} + ( 1 + 4 \zeta_{6} ) q^{56} + 6 q^{57} -12 q^{58} -12 q^{59} + ( 4 - 8 \zeta_{6} ) q^{61} -6 q^{62} + ( 9 - 6 \zeta_{6} ) q^{63} - q^{64} + ( -6 + 12 \zeta_{6} ) q^{66} -8 q^{67} + 6 q^{68} -6 q^{69} + ( -2 + 4 \zeta_{6} ) q^{71} + ( -3 + 6 \zeta_{6} ) q^{72} + ( 4 - 8 \zeta_{6} ) q^{73} + ( 2 - 4 \zeta_{6} ) q^{74} + ( 2 - 4 \zeta_{6} ) q^{76} + ( 2 + 8 \zeta_{6} ) q^{77} + 8 q^{79} + 9 q^{81} + ( 6 - 12 \zeta_{6} ) q^{82} + ( -1 - 4 \zeta_{6} ) q^{84} + ( 8 - 16 \zeta_{6} ) q^{86} -12 q^{87} -6 q^{88} + 6 q^{89} + ( -2 + 4 \zeta_{6} ) q^{92} -6 q^{93} + ( 12 - 24 \zeta_{6} ) q^{94} + 9 q^{96} + ( 4 - 8 \zeta_{6} ) q^{97} + ( -11 - 2 \zeta_{6} ) q^{98} + ( -6 + 12 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 6q^{6} - 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 6q^{6} - 4q^{7} - 6q^{9} + 6q^{14} - 10q^{16} - 12q^{17} + 6q^{21} - 12q^{22} - 6q^{24} + 4q^{28} - 12q^{33} + 6q^{36} + 4q^{37} + 12q^{38} + 12q^{41} + 12q^{42} + 16q^{43} - 12q^{46} + 24q^{47} + 2q^{49} + 18q^{54} + 6q^{56} + 12q^{57} - 24q^{58} - 24q^{59} - 12q^{62} + 12q^{63} - 2q^{64} - 16q^{67} + 12q^{68} - 12q^{69} + 12q^{77} + 16q^{79} + 18q^{81} - 6q^{84} - 24q^{87} - 12q^{88} + 12q^{89} - 12q^{93} + 18q^{96} - 24q^{98} + O(q^{100})$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i 1.73205i −1.00000 0 −3.00000 −2.00000 + 1.73205i 1.73205i −3.00000 0
251.2 1.73205i 1.73205i −1.00000 0 −3.00000 −2.00000 1.73205i 1.73205i −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
21.c 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.a 2
3.b odd 2 1 525.2.b.b 2
5.b even 2 1 105.2.b.a 2
5.c odd 4 2 525.2.g.b 4
7.b odd 2 1 525.2.b.b 2
15.d odd 2 1 105.2.b.b yes 2
15.e even 4 2 525.2.g.c 4
20.d odd 2 1 1680.2.f.b 2
21.c even 2 1 inner 525.2.b.a 2
35.c odd 2 1 105.2.b.b yes 2
35.f even 4 2 525.2.g.c 4
35.i odd 6 1 735.2.s.a 2
35.i odd 6 1 735.2.s.f 2
35.j even 6 1 735.2.s.b 2
35.j even 6 1 735.2.s.d 2
60.h even 2 1 1680.2.f.c 2
105.g even 2 1 105.2.b.a 2
105.k odd 4 2 525.2.g.b 4
105.o odd 6 1 735.2.s.a 2
105.o odd 6 1 735.2.s.f 2
105.p even 6 1 735.2.s.b 2
105.p even 6 1 735.2.s.d 2
140.c even 2 1 1680.2.f.c 2
420.o odd 2 1 1680.2.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.a 2 5.b even 2 1
105.2.b.a 2 105.g even 2 1
105.2.b.b yes 2 15.d odd 2 1
105.2.b.b yes 2 35.c odd 2 1
525.2.b.a 2 1.a even 1 1 trivial
525.2.b.a 2 21.c even 2 1 inner
525.2.b.b 2 3.b odd 2 1
525.2.b.b 2 7.b odd 2 1
525.2.g.b 4 5.c odd 4 2
525.2.g.b 4 105.k odd 4 2
525.2.g.c 4 15.e even 4 2
525.2.g.c 4 35.f even 4 2
735.2.s.a 2 35.i odd 6 1
735.2.s.a 2 105.o odd 6 1
735.2.s.b 2 35.j even 6 1
735.2.s.b 2 105.p even 6 1
735.2.s.d 2 35.j even 6 1
735.2.s.d 2 105.p even 6 1
735.2.s.f 2 35.i odd 6 1
735.2.s.f 2 105.o odd 6 1
1680.2.f.b 2 20.d odd 2 1
1680.2.f.b 2 420.o odd 2 1
1680.2.f.c 2 60.h even 2 1
1680.2.f.c 2 140.c even 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 3$$ $$T_{11}^{2} + 12$$ $$T_{17} + 6$$ $$T_{37} - 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + 4 T^{4}$$
$3$ $$1 + 3 T^{2}$$
$5$ 
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$1 - 10 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 13 T^{2} )^{2}$$
$17$ $$( 1 + 6 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$23$ $$1 - 34 T^{2} + 529 T^{4}$$
$29$ $$1 - 10 T^{2} + 841 T^{4}$$
$31$ $$1 - 50 T^{2} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 6 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{2}$$
$47$ $$( 1 - 12 T + 47 T^{2} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{2}$$
$59$ $$( 1 + 12 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$
$67$ $$( 1 + 8 T + 67 T^{2} )^{2}$$
$71$ $$1 - 130 T^{2} + 5041 T^{4}$$
$73$ $$1 - 98 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 - 6 T + 89 T^{2} )^{2}$$
$97$ $$1 - 146 T^{2} + 9409 T^{4}$$