# Properties

 Label 525.4.a.g 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: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{2} + 3q^{3} + q^{4} + 9q^{6} - 7q^{7} - 21q^{8} + 9q^{9} + O(q^{10})$$ $$q + 3q^{2} + 3q^{3} + q^{4} + 9q^{6} - 7q^{7} - 21q^{8} + 9q^{9} - 36q^{11} + 3q^{12} + 34q^{13} - 21q^{14} - 71q^{16} - 42q^{17} + 27q^{18} - 124q^{19} - 21q^{21} - 108q^{22} - 63q^{24} + 102q^{26} + 27q^{27} - 7q^{28} + 102q^{29} - 160q^{31} - 45q^{32} - 108q^{33} - 126q^{34} + 9q^{36} - 398q^{37} - 372q^{38} + 102q^{39} - 318q^{41} - 63q^{42} + 268q^{43} - 36q^{44} - 240q^{47} - 213q^{48} + 49q^{49} - 126q^{51} + 34q^{52} + 498q^{53} + 81q^{54} + 147q^{56} - 372q^{57} + 306q^{58} - 132q^{59} + 398q^{61} - 480q^{62} - 63q^{63} + 433q^{64} - 324q^{66} - 92q^{67} - 42q^{68} - 720q^{71} - 189q^{72} + 502q^{73} - 1194q^{74} - 124q^{76} + 252q^{77} + 306q^{78} - 1024q^{79} + 81q^{81} - 954q^{82} + 204q^{83} - 21q^{84} + 804q^{86} + 306q^{87} + 756q^{88} + 354q^{89} - 238q^{91} - 480q^{93} - 720q^{94} - 135q^{96} + 286q^{97} + 147q^{98} - 324q^{99} + O(q^{100})$$

## 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

## 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.g 1
3.b odd 2 1 1575.4.a.b 1
5.b even 2 1 21.4.a.a 1
5.c odd 4 2 525.4.d.c 2
15.d odd 2 1 63.4.a.c 1
20.d odd 2 1 336.4.a.f 1
35.c odd 2 1 147.4.a.c 1
35.i odd 6 2 147.4.e.g 2
35.j even 6 2 147.4.e.i 2
40.e odd 2 1 1344.4.a.n 1
40.f even 2 1 1344.4.a.ba 1
60.h even 2 1 1008.4.a.v 1
105.g even 2 1 441.4.a.j 1
105.o odd 6 2 441.4.e.b 2
105.p even 6 2 441.4.e.d 2
140.c even 2 1 2352.4.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.a 1 5.b even 2 1
63.4.a.c 1 15.d odd 2 1
147.4.a.c 1 35.c odd 2 1
147.4.e.g 2 35.i odd 6 2
147.4.e.i 2 35.j even 6 2
336.4.a.f 1 20.d odd 2 1
441.4.a.j 1 105.g even 2 1
441.4.e.b 2 105.o odd 6 2
441.4.e.d 2 105.p even 6 2
525.4.a.g 1 1.a even 1 1 trivial
525.4.d.c 2 5.c odd 4 2
1008.4.a.v 1 60.h even 2 1
1344.4.a.n 1 40.e odd 2 1
1344.4.a.ba 1 40.f even 2 1
1575.4.a.b 1 3.b odd 2 1
2352.4.a.r 1 140.c even 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(525))$$:

 $$T_{2} - 3$$ $$T_{11} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 8 T^{2}$$
$3$ $$1 - 3 T$$
$5$ 1
$7$ $$1 + 7 T$$
$11$ $$1 + 36 T + 1331 T^{2}$$
$13$ $$1 - 34 T + 2197 T^{2}$$
$17$ $$1 + 42 T + 4913 T^{2}$$
$19$ $$1 + 124 T + 6859 T^{2}$$
$23$ $$1 + 12167 T^{2}$$
$29$ $$1 - 102 T + 24389 T^{2}$$
$31$ $$1 + 160 T + 29791 T^{2}$$
$37$ $$1 + 398 T + 50653 T^{2}$$
$41$ $$1 + 318 T + 68921 T^{2}$$
$43$ $$1 - 268 T + 79507 T^{2}$$
$47$ $$1 + 240 T + 103823 T^{2}$$
$53$ $$1 - 498 T + 148877 T^{2}$$
$59$ $$1 + 132 T + 205379 T^{2}$$
$61$ $$1 - 398 T + 226981 T^{2}$$
$67$ $$1 + 92 T + 300763 T^{2}$$
$71$ $$1 + 720 T + 357911 T^{2}$$
$73$ $$1 - 502 T + 389017 T^{2}$$
$79$ $$1 + 1024 T + 493039 T^{2}$$
$83$ $$1 - 204 T + 571787 T^{2}$$
$89$ $$1 - 354 T + 704969 T^{2}$$
$97$ $$1 - 286 T + 912673 T^{2}$$