# Properties

 Label 525.2.i.b Level $525$ Weight $2$ Character orbit 525.i 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + q^{6} + ( 1 - 3 \zeta_{6} ) q^{7} -3 q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + q^{6} + ( 1 - 3 \zeta_{6} ) q^{7} -3 q^{8} -\zeta_{6} q^{9} + \zeta_{6} q^{12} -3 q^{13} + ( -3 + 2 \zeta_{6} ) q^{14} + \zeta_{6} q^{16} + ( -2 + 2 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} -\zeta_{6} q^{19} + ( 2 + \zeta_{6} ) q^{21} -2 \zeta_{6} q^{23} + ( 3 - 3 \zeta_{6} ) q^{24} + 3 \zeta_{6} q^{26} + q^{27} + ( -2 - \zeta_{6} ) q^{28} -8 q^{29} + ( 8 - 8 \zeta_{6} ) q^{31} + ( -5 + 5 \zeta_{6} ) q^{32} + 2 q^{34} - q^{36} -7 \zeta_{6} q^{37} + ( -1 + \zeta_{6} ) q^{38} + ( 3 - 3 \zeta_{6} ) q^{39} + ( 1 - 3 \zeta_{6} ) q^{42} -8 q^{43} + ( -2 + 2 \zeta_{6} ) q^{46} + 10 \zeta_{6} q^{47} - q^{48} + ( -8 + 3 \zeta_{6} ) q^{49} -2 \zeta_{6} q^{51} + ( -3 + 3 \zeta_{6} ) q^{52} + ( 14 - 14 \zeta_{6} ) q^{53} -\zeta_{6} q^{54} + ( -3 + 9 \zeta_{6} ) q^{56} + q^{57} + 8 \zeta_{6} q^{58} + ( -10 + 10 \zeta_{6} ) q^{59} -7 \zeta_{6} q^{61} -8 q^{62} + ( -3 + 2 \zeta_{6} ) q^{63} + 7 q^{64} + ( 5 - 5 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} + 2 q^{69} -12 q^{71} + 3 \zeta_{6} q^{72} + ( 11 - 11 \zeta_{6} ) q^{73} + ( -7 + 7 \zeta_{6} ) q^{74} - q^{76} -3 q^{78} + 7 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 14 q^{83} + ( 3 - 2 \zeta_{6} ) q^{84} + 8 \zeta_{6} q^{86} + ( 8 - 8 \zeta_{6} ) q^{87} + 6 \zeta_{6} q^{89} + ( -3 + 9 \zeta_{6} ) q^{91} -2 q^{92} + 8 \zeta_{6} q^{93} + ( 10 - 10 \zeta_{6} ) q^{94} -5 \zeta_{6} q^{96} + 9 q^{97} + ( 3 + 5 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{3} + q^{4} + 2q^{6} - q^{7} - 6q^{8} - q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} + q^{4} + 2q^{6} - q^{7} - 6q^{8} - q^{9} + q^{12} - 6q^{13} - 4q^{14} + q^{16} - 2q^{17} - q^{18} - q^{19} + 5q^{21} - 2q^{23} + 3q^{24} + 3q^{26} + 2q^{27} - 5q^{28} - 16q^{29} + 8q^{31} - 5q^{32} + 4q^{34} - 2q^{36} - 7q^{37} - q^{38} + 3q^{39} - q^{42} - 16q^{43} - 2q^{46} + 10q^{47} - 2q^{48} - 13q^{49} - 2q^{51} - 3q^{52} + 14q^{53} - q^{54} + 3q^{56} + 2q^{57} + 8q^{58} - 10q^{59} - 7q^{61} - 16q^{62} - 4q^{63} + 14q^{64} + 5q^{67} + 2q^{68} + 4q^{69} - 24q^{71} + 3q^{72} + 11q^{73} - 7q^{74} - 2q^{76} - 6q^{78} + 7q^{79} - q^{81} + 28q^{83} + 4q^{84} + 8q^{86} + 8q^{87} + 6q^{89} + 3q^{91} - 4q^{92} + 8q^{93} + 10q^{94} - 5q^{96} + 18q^{97} + 11q^{98} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i 0 1.00000 −0.500000 2.59808i −3.00000 −0.500000 0.866025i 0
226.1 −0.500000 + 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i 0 1.00000 −0.500000 + 2.59808i −3.00000 −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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.i.b 2
5.b even 2 1 525.2.i.d yes 2
5.c odd 4 2 525.2.r.c 4
7.c even 3 1 inner 525.2.i.b 2
7.c even 3 1 3675.2.a.m 1
7.d odd 6 1 3675.2.a.k 1
35.i odd 6 1 3675.2.a.g 1
35.j even 6 1 525.2.i.d yes 2
35.j even 6 1 3675.2.a.e 1
35.l odd 12 2 525.2.r.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.i.b 2 1.a even 1 1 trivial
525.2.i.b 2 7.c even 3 1 inner
525.2.i.d yes 2 5.b even 2 1
525.2.i.d yes 2 35.j even 6 1
525.2.r.c 4 5.c odd 4 2
525.2.r.c 4 35.l odd 12 2
3675.2.a.e 1 35.j even 6 1
3675.2.a.g 1 35.i odd 6 1
3675.2.a.k 1 7.d odd 6 1
3675.2.a.m 1 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( 3 + T )^{2}$$
$17$ $$4 + 2 T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$4 + 2 T + T^{2}$$
$29$ $$( 8 + T )^{2}$$
$31$ $$64 - 8 T + T^{2}$$
$37$ $$49 + 7 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$100 - 10 T + T^{2}$$
$53$ $$196 - 14 T + T^{2}$$
$59$ $$100 + 10 T + T^{2}$$
$61$ $$49 + 7 T + T^{2}$$
$67$ $$25 - 5 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$121 - 11 T + T^{2}$$
$79$ $$49 - 7 T + T^{2}$$
$83$ $$( -14 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( -9 + T )^{2}$$