# Properties

 Label 525.4.a.h Level $525$ Weight $4$ Character orbit 525.a Self dual yes Analytic conductor $30.976$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.9760027530$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{2} + 3q^{3} + q^{4} + 9q^{6} - 7q^{7} - 21q^{8} + 9q^{9} + O(q^{10})$$ $$q + 3q^{2} + 3q^{3} + q^{4} + 9q^{6} - 7q^{7} - 21q^{8} + 9q^{9} - 6q^{11} + 3q^{12} - 41q^{13} - 21q^{14} - 71q^{16} - 27q^{17} + 27q^{18} - 4q^{19} - 21q^{21} - 18q^{22} - 75q^{23} - 63q^{24} - 123q^{26} + 27q^{27} - 7q^{28} - 123q^{29} - 205q^{31} - 45q^{32} - 18q^{33} - 81q^{34} + 9q^{36} + 262q^{37} - 12q^{38} - 123q^{39} + 57q^{41} - 63q^{42} - 407q^{43} - 6q^{44} - 225q^{46} + 60q^{47} - 213q^{48} + 49q^{49} - 81q^{51} - 41q^{52} - 327q^{53} + 81q^{54} + 147q^{56} - 12q^{57} - 369q^{58} + 33q^{59} - 427q^{61} - 615q^{62} - 63q^{63} + 433q^{64} - 54q^{66} + 628q^{67} - 27q^{68} - 225q^{69} + 300q^{71} - 189q^{72} - 98q^{73} + 786q^{74} - 4q^{76} + 42q^{77} - 369q^{78} + 686q^{79} + 81q^{81} + 171q^{82} - 1401q^{83} - 21q^{84} - 1221q^{86} - 369q^{87} + 126q^{88} + 714q^{89} + 287q^{91} - 75q^{92} - 615q^{93} + 180q^{94} - 135q^{96} - 494q^{97} + 147q^{98} - 54q^{99} + O(q^{100})$$

## 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
3.00000 3.00000 1.00000 0 9.00000 −7.00000 −21.0000 9.00000 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 525.4.a.h yes 1
3.b odd 2 1 1575.4.a.a 1
5.b even 2 1 525.4.a.c 1
5.c odd 4 2 525.4.d.d 2
15.d odd 2 1 1575.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.c 1 5.b even 2 1
525.4.a.h yes 1 1.a even 1 1 trivial
525.4.d.d 2 5.c odd 4 2
1575.4.a.a 1 3.b odd 2 1
1575.4.a.j 1 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(525))$$:

 $$T_{2} - 3$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 8 T^{2}$$
$3$ $$1 - 3 T$$
$5$ 1
$7$ $$1 + 7 T$$
$11$ $$1 + 6 T + 1331 T^{2}$$
$13$ $$1 + 41 T + 2197 T^{2}$$
$17$ $$1 + 27 T + 4913 T^{2}$$
$19$ $$1 + 4 T + 6859 T^{2}$$
$23$ $$1 + 75 T + 12167 T^{2}$$
$29$ $$1 + 123 T + 24389 T^{2}$$
$31$ $$1 + 205 T + 29791 T^{2}$$
$37$ $$1 - 262 T + 50653 T^{2}$$
$41$ $$1 - 57 T + 68921 T^{2}$$
$43$ $$1 + 407 T + 79507 T^{2}$$
$47$ $$1 - 60 T + 103823 T^{2}$$
$53$ $$1 + 327 T + 148877 T^{2}$$
$59$ $$1 - 33 T + 205379 T^{2}$$
$61$ $$1 + 427 T + 226981 T^{2}$$
$67$ $$1 - 628 T + 300763 T^{2}$$
$71$ $$1 - 300 T + 357911 T^{2}$$
$73$ $$1 + 98 T + 389017 T^{2}$$
$79$ $$1 - 686 T + 493039 T^{2}$$
$83$ $$1 + 1401 T + 571787 T^{2}$$
$89$ $$1 - 714 T + 704969 T^{2}$$
$97$ $$1 + 494 T + 912673 T^{2}$$