# Properties

 Label 525.2.a.i Level 525 Weight 2 Character orbit 525.a Self dual yes Analytic conductor 4.192 Analytic rank 0 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ 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
 −0.618034 1.61803
0.381966 1.00000 −1.85410 0 0.381966 −1.00000 −1.47214 1.00000 0
1.2 2.61803 1.00000 4.85410 0 2.61803 −1.00000 7.47214 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.i yes 2
3.b odd 2 1 1575.2.a.l 2
4.b odd 2 1 8400.2.a.cy 2
5.b even 2 1 525.2.a.e 2
5.c odd 4 2 525.2.d.e 4
7.b odd 2 1 3675.2.a.bh 2
15.d odd 2 1 1575.2.a.v 2
15.e even 4 2 1575.2.d.f 4
20.d odd 2 1 8400.2.a.da 2
35.c odd 2 1 3675.2.a.r 2

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

## Atkin-Lehner signs

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

## 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

$p$ $F_p(T)$
$2$ $$1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4}$$
$3$ $$( 1 - T )^{2}$$
$5$ 
$7$ $$( 1 + T )^{2}$$
$11$ $$1 + 2 T + 3 T^{2} + 22 T^{3} + 121 T^{4}$$
$13$ $$1 - 6 T + 30 T^{2} - 78 T^{3} + 169 T^{4}$$
$17$ $$1 + 2 T - 10 T^{2} + 34 T^{3} + 289 T^{4}$$
$19$ $$1 + 2 T + 34 T^{2} + 38 T^{3} + 361 T^{4}$$
$23$ $$( 1 - 5 T + 23 T^{2} )^{2}$$
$29$ $$1 + 4 T + 17 T^{2} + 116 T^{3} + 841 T^{4}$$
$31$ $$1 + 42 T^{2} + 961 T^{4}$$
$37$ $$1 - 2 T + 55 T^{2} - 74 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 8 T + 41 T^{2} )^{2}$$
$43$ $$1 - 12 T + 117 T^{2} - 516 T^{3} + 1849 T^{4}$$
$47$ $$1 - 10 T + 114 T^{2} - 470 T^{3} + 2209 T^{4}$$
$53$ $$1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4}$$
$59$ $$1 + 6 T + 122 T^{2} + 354 T^{3} + 3481 T^{4}$$
$61$ $$1 + 4 T - 54 T^{2} + 244 T^{3} + 3721 T^{4}$$
$67$ $$1 - 8 T + 105 T^{2} - 536 T^{3} + 4489 T^{4}$$
$71$ $$1 + 10 T + 147 T^{2} + 710 T^{3} + 5041 T^{4}$$
$73$ $$1 + 2 T + 142 T^{2} + 146 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T + 169 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$1 - 16 T + 210 T^{2} - 1328 T^{3} + 6889 T^{4}$$
$89$ $$1 - 2 T + 134 T^{2} - 178 T^{3} + 7921 T^{4}$$
$97$ $$1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4}$$