# Properties

 Label 525.2.a.e Level $525$ Weight $2$ Character orbit 525.a Self dual yes Analytic conductor $4.192$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.19214610612$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{2} - q^{3} + 3 \beta q^{4} + ( 1 + \beta ) q^{6} + q^{7} + ( -1 - 4 \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{2} - q^{3} + 3 \beta q^{4} + ( 1 + \beta ) q^{6} + q^{7} + ( -1 - 4 \beta ) q^{8} + q^{9} + ( 1 - 4 \beta ) q^{11} -3 \beta q^{12} + ( -4 + 2 \beta ) q^{13} + ( -1 - \beta ) q^{14} + ( 5 + 3 \beta ) q^{16} + ( -2 + 6 \beta ) q^{17} + ( -1 - \beta ) q^{18} -2 \beta q^{19} - q^{21} + ( 3 + 7 \beta ) q^{22} -5 q^{23} + ( 1 + 4 \beta ) q^{24} + 2 q^{26} - q^{27} + 3 \beta q^{28} + ( -5 + 6 \beta ) q^{29} + ( -2 + 4 \beta ) q^{31} + ( -6 - 3 \beta ) q^{32} + ( -1 + 4 \beta ) q^{33} + ( -4 - 10 \beta ) q^{34} + 3 \beta q^{36} + ( 1 - 4 \beta ) q^{37} + ( 2 + 4 \beta ) q^{38} + ( 4 - 2 \beta ) q^{39} -8 q^{41} + ( 1 + \beta ) q^{42} + ( -5 - 2 \beta ) q^{43} + ( -12 - 9 \beta ) q^{44} + ( 5 + 5 \beta ) q^{46} + ( -4 - 2 \beta ) q^{47} + ( -5 - 3 \beta ) q^{48} + q^{49} + ( 2 - 6 \beta ) q^{51} + ( 6 - 6 \beta ) q^{52} + ( 6 - 4 \beta ) q^{53} + ( 1 + \beta ) q^{54} + ( -1 - 4 \beta ) q^{56} + 2 \beta q^{57} + ( -1 - 7 \beta ) q^{58} + ( -4 + 2 \beta ) q^{59} + ( 4 - 12 \beta ) q^{61} + ( -2 - 6 \beta ) q^{62} + q^{63} + ( -1 + 6 \beta ) q^{64} + ( -3 - 7 \beta ) q^{66} + ( -7 + 6 \beta ) q^{67} + ( 18 + 12 \beta ) q^{68} + 5 q^{69} + ( -7 + 4 \beta ) q^{71} + ( -1 - 4 \beta ) q^{72} + ( 2 - 2 \beta ) q^{73} + ( 3 + 7 \beta ) q^{74} + ( -6 - 6 \beta ) q^{76} + ( 1 - 4 \beta ) q^{77} -2 q^{78} + ( 5 - 2 \beta ) q^{79} + q^{81} + ( 8 + 8 \beta ) q^{82} + ( -6 - 4 \beta ) q^{83} -3 \beta q^{84} + ( 7 + 9 \beta ) q^{86} + ( 5 - 6 \beta ) q^{87} + ( 15 + 16 \beta ) q^{88} + ( 4 - 6 \beta ) q^{89} + ( -4 + 2 \beta ) q^{91} -15 \beta q^{92} + ( 2 - 4 \beta ) q^{93} + ( 6 + 8 \beta ) q^{94} + ( 6 + 3 \beta ) q^{96} + ( -6 - 4 \beta ) q^{97} + ( -1 - \beta ) q^{98} + ( 1 - 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} - 2q^{3} + 3q^{4} + 3q^{6} + 2q^{7} - 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 3q^{2} - 2q^{3} + 3q^{4} + 3q^{6} + 2q^{7} - 6q^{8} + 2q^{9} - 2q^{11} - 3q^{12} - 6q^{13} - 3q^{14} + 13q^{16} + 2q^{17} - 3q^{18} - 2q^{19} - 2q^{21} + 13q^{22} - 10q^{23} + 6q^{24} + 4q^{26} - 2q^{27} + 3q^{28} - 4q^{29} - 15q^{32} + 2q^{33} - 18q^{34} + 3q^{36} - 2q^{37} + 8q^{38} + 6q^{39} - 16q^{41} + 3q^{42} - 12q^{43} - 33q^{44} + 15q^{46} - 10q^{47} - 13q^{48} + 2q^{49} - 2q^{51} + 6q^{52} + 8q^{53} + 3q^{54} - 6q^{56} + 2q^{57} - 9q^{58} - 6q^{59} - 4q^{61} - 10q^{62} + 2q^{63} + 4q^{64} - 13q^{66} - 8q^{67} + 48q^{68} + 10q^{69} - 10q^{71} - 6q^{72} + 2q^{73} + 13q^{74} - 18q^{76} - 2q^{77} - 4q^{78} + 8q^{79} + 2q^{81} + 24q^{82} - 16q^{83} - 3q^{84} + 23q^{86} + 4q^{87} + 46q^{88} + 2q^{89} - 6q^{91} - 15q^{92} + 20q^{94} + 15q^{96} - 16q^{97} - 3q^{98} - 2q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
−2.61803 −1.00000 4.85410 0 2.61803 1.00000 −7.47214 1.00000 0
1.2 −0.381966 −1.00000 −1.85410 0 0.381966 1.00000 1.47214 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.a.e 2
3.b odd 2 1 1575.2.a.v 2
4.b odd 2 1 8400.2.a.da 2
5.b even 2 1 525.2.a.i yes 2
5.c odd 4 2 525.2.d.e 4
7.b odd 2 1 3675.2.a.r 2
15.d odd 2 1 1575.2.a.l 2
15.e even 4 2 1575.2.d.f 4
20.d odd 2 1 8400.2.a.cy 2
35.c odd 2 1 3675.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.a.e 2 1.a even 1 1 trivial
525.2.a.i yes 2 5.b even 2 1
525.2.d.e 4 5.c odd 4 2
1575.2.a.l 2 15.d odd 2 1
1575.2.a.v 2 3.b odd 2 1
1575.2.d.f 4 15.e even 4 2
3675.2.a.r 2 7.b odd 2 1
3675.2.a.bh 2 35.c odd 2 1
8400.2.a.cy 2 20.d odd 2 1
8400.2.a.da 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(525))$$:

 $$T_{2}^{2} + 3 T_{2} + 1$$ $$T_{11}^{2} + 2 T_{11} - 19$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-19 + 2 T + T^{2}$$
$13$ $$4 + 6 T + T^{2}$$
$17$ $$-44 - 2 T + T^{2}$$
$19$ $$-4 + 2 T + T^{2}$$
$23$ $$( 5 + T )^{2}$$
$29$ $$-41 + 4 T + T^{2}$$
$31$ $$-20 + T^{2}$$
$37$ $$-19 + 2 T + T^{2}$$
$41$ $$( 8 + T )^{2}$$
$43$ $$31 + 12 T + T^{2}$$
$47$ $$20 + 10 T + T^{2}$$
$53$ $$-4 - 8 T + T^{2}$$
$59$ $$4 + 6 T + T^{2}$$
$61$ $$-176 + 4 T + T^{2}$$
$67$ $$-29 + 8 T + T^{2}$$
$71$ $$5 + 10 T + T^{2}$$
$73$ $$-4 - 2 T + T^{2}$$
$79$ $$11 - 8 T + T^{2}$$
$83$ $$44 + 16 T + T^{2}$$
$89$ $$-44 - 2 T + T^{2}$$
$97$ $$44 + 16 T + T^{2}$$