# Properties

 Label 525.2.a.k Level $525$ Weight $2$ Character orbit 525.a Self dual yes Analytic conductor $4.192$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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## 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: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} - q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} -\beta_{1} q^{6} - q^{7} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} - q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} -\beta_{1} q^{6} - q^{7} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{8} + q^{9} + 2 q^{11} + ( -1 - \beta_{1} - \beta_{2} ) q^{12} + ( -2 + \beta_{1} - \beta_{2} ) q^{13} -\beta_{1} q^{14} + ( 3 + 4 \beta_{1} ) q^{16} + ( -\beta_{1} + \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( 2 + \beta_{1} - \beta_{2} ) q^{19} + q^{21} + 2 \beta_{1} q^{22} + ( 2 - \beta_{1} - \beta_{2} ) q^{23} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{24} + ( 4 - 3 \beta_{1} + \beta_{2} ) q^{26} - q^{27} + ( -1 - \beta_{1} - \beta_{2} ) q^{28} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{31} + ( 8 + 3 \beta_{1} + 2 \beta_{2} ) q^{32} -2 q^{33} + ( -4 + \beta_{1} - \beta_{2} ) q^{34} + ( 1 + \beta_{1} + \beta_{2} ) q^{36} -4 \beta_{1} q^{37} + ( 4 + \beta_{1} + \beta_{2} ) q^{38} + ( 2 - \beta_{1} + \beta_{2} ) q^{39} + ( 3 \beta_{1} - \beta_{2} ) q^{41} + \beta_{1} q^{42} + 4 \beta_{2} q^{43} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{44} + ( -2 - \beta_{1} - \beta_{2} ) q^{46} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( -3 - 4 \beta_{1} ) q^{48} + q^{49} + ( \beta_{1} - \beta_{2} ) q^{51} + ( -6 + \beta_{1} - \beta_{2} ) q^{52} + ( 6 - 3 \beta_{1} - \beta_{2} ) q^{53} -\beta_{1} q^{54} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{56} + ( -2 - \beta_{1} + \beta_{2} ) q^{57} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{58} + ( 4 + 4 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -8 - 3 \beta_{1} - 3 \beta_{2} ) q^{62} - q^{63} + ( 1 + 7 \beta_{1} + 3 \beta_{2} ) q^{64} -2 \beta_{1} q^{66} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 4 - 3 \beta_{1} - \beta_{2} ) q^{68} + ( -2 + \beta_{1} + \beta_{2} ) q^{69} + 2 q^{71} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{72} + ( -6 - \beta_{1} + \beta_{2} ) q^{73} + ( -12 - 4 \beta_{1} - 4 \beta_{2} ) q^{74} + ( -2 + 5 \beta_{1} + 3 \beta_{2} ) q^{76} -2 q^{77} + ( -4 + 3 \beta_{1} - \beta_{2} ) q^{78} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{79} + q^{81} + ( 10 + \beta_{1} + 3 \beta_{2} ) q^{82} + ( 4 - 4 \beta_{1} ) q^{83} + ( 1 + \beta_{1} + \beta_{2} ) q^{84} + ( -4 + 8 \beta_{1} ) q^{86} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{87} + ( 4 + 4 \beta_{1} + 2 \beta_{2} ) q^{88} + ( 4 + \beta_{1} + \beta_{2} ) q^{89} + ( 2 - \beta_{1} + \beta_{2} ) q^{91} + ( -6 - 3 \beta_{1} + \beta_{2} ) q^{92} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{93} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{94} + ( -8 - 3 \beta_{1} - 2 \beta_{2} ) q^{96} + ( -10 + 5 \beta_{1} + 3 \beta_{2} ) q^{97} + \beta_{1} q^{98} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} - 3 q^{7} + 9 q^{8} + 3 q^{9} + O(q^{10})$$ $$3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} - 3 q^{7} + 9 q^{8} + 3 q^{9} + 6 q^{11} - 5 q^{12} - 6 q^{13} - q^{14} + 13 q^{16} + q^{18} + 6 q^{19} + 3 q^{21} + 2 q^{22} + 4 q^{23} - 9 q^{24} + 10 q^{26} - 3 q^{27} - 5 q^{28} + 2 q^{29} + 2 q^{31} + 29 q^{32} - 6 q^{33} - 12 q^{34} + 5 q^{36} - 4 q^{37} + 14 q^{38} + 6 q^{39} + 2 q^{41} + q^{42} + 4 q^{43} + 10 q^{44} - 8 q^{46} + 8 q^{47} - 13 q^{48} + 3 q^{49} - 18 q^{52} + 14 q^{53} - q^{54} - 9 q^{56} - 6 q^{57} - 18 q^{58} + 16 q^{59} - 6 q^{61} - 30 q^{62} - 3 q^{63} + 13 q^{64} - 2 q^{66} + 8 q^{67} + 8 q^{68} - 4 q^{69} + 6 q^{71} + 9 q^{72} - 18 q^{73} - 44 q^{74} + 2 q^{76} - 6 q^{77} - 10 q^{78} + 12 q^{79} + 3 q^{81} + 34 q^{82} + 8 q^{83} + 5 q^{84} - 4 q^{86} - 2 q^{87} + 18 q^{88} + 14 q^{89} + 6 q^{91} - 20 q^{92} - 2 q^{93} - 16 q^{94} - 29 q^{96} - 22 q^{97} + q^{98} + 6 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} + 2$$

## 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
 0.311108 −1.48119 2.17009
−1.90321 −1.00000 1.62222 0 1.90321 −1.00000 0.719004 1.00000 0
1.2 0.193937 −1.00000 −1.96239 0 −0.193937 −1.00000 −0.768452 1.00000 0
1.3 2.70928 −1.00000 5.34017 0 −2.70928 −1.00000 9.04945 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.k 3
3.b odd 2 1 1575.2.a.w 3
4.b odd 2 1 8400.2.a.dj 3
5.b even 2 1 525.2.a.j 3
5.c odd 4 2 105.2.d.b 6
7.b odd 2 1 3675.2.a.bj 3
15.d odd 2 1 1575.2.a.x 3
15.e even 4 2 315.2.d.e 6
20.d odd 2 1 8400.2.a.dg 3
20.e even 4 2 1680.2.t.k 6
35.c odd 2 1 3675.2.a.bi 3
35.f even 4 2 735.2.d.b 6
35.k even 12 4 735.2.q.f 12
35.l odd 12 4 735.2.q.e 12
60.l odd 4 2 5040.2.t.v 6
105.k odd 4 2 2205.2.d.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b 6 5.c odd 4 2
315.2.d.e 6 15.e even 4 2
525.2.a.j 3 5.b even 2 1
525.2.a.k 3 1.a even 1 1 trivial
735.2.d.b 6 35.f even 4 2
735.2.q.e 12 35.l odd 12 4
735.2.q.f 12 35.k even 12 4
1575.2.a.w 3 3.b odd 2 1
1575.2.a.x 3 15.d odd 2 1
1680.2.t.k 6 20.e even 4 2
2205.2.d.l 6 105.k odd 4 2
3675.2.a.bi 3 35.c odd 2 1
3675.2.a.bj 3 7.b odd 2 1
5040.2.t.v 6 60.l odd 4 2
8400.2.a.dg 3 20.d odd 2 1
8400.2.a.dj 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(525))$$:

 $$T_{2}^{3} - T_{2}^{2} - 5 T_{2} + 1$$ $$T_{11} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T - T^{2} + T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$( -2 + T )^{3}$$
$13$ $$-8 - 4 T + 6 T^{2} + T^{3}$$
$17$ $$-16 - 16 T + T^{3}$$
$19$ $$40 - 4 T - 6 T^{2} + T^{3}$$
$23$ $$16 - 8 T - 4 T^{2} + T^{3}$$
$29$ $$40 - 52 T - 2 T^{2} + T^{3}$$
$31$ $$184 - 52 T - 2 T^{2} + T^{3}$$
$37$ $$-64 - 80 T + 4 T^{2} + T^{3}$$
$41$ $$200 - 60 T - 2 T^{2} + T^{3}$$
$43$ $$832 - 144 T - 4 T^{2} + T^{3}$$
$47$ $$128 - 32 T - 8 T^{2} + T^{3}$$
$53$ $$296 + 12 T - 14 T^{2} + T^{3}$$
$59$ $$1280 - 64 T - 16 T^{2} + T^{3}$$
$61$ $$-248 - 52 T + 6 T^{2} + T^{3}$$
$67$ $$128 - 32 T - 8 T^{2} + T^{3}$$
$71$ $$( -2 + T )^{3}$$
$73$ $$104 + 92 T + 18 T^{2} + T^{3}$$
$79$ $$320 - 16 T - 12 T^{2} + T^{3}$$
$83$ $$256 - 64 T - 8 T^{2} + T^{3}$$
$89$ $$-40 + 52 T - 14 T^{2} + T^{3}$$
$97$ $$-1864 - 36 T + 22 T^{2} + T^{3}$$
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