# Properties

 Label 525.4.a.b Level $525$ Weight $4$ Character orbit 525.a Self dual yes Analytic conductor $30.976$ Analytic rank $0$ 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q - 4q^{2} + 3q^{3} + 8q^{4} - 12q^{6} + 7q^{7} + 9q^{9} + O(q^{10})$$ $$q - 4q^{2} + 3q^{3} + 8q^{4} - 12q^{6} + 7q^{7} + 9q^{9} + 62q^{11} + 24q^{12} + 62q^{13} - 28q^{14} - 64q^{16} - 84q^{17} - 36q^{18} + 100q^{19} + 21q^{21} - 248q^{22} + 42q^{23} - 248q^{26} + 27q^{27} + 56q^{28} - 10q^{29} - 48q^{31} + 256q^{32} + 186q^{33} + 336q^{34} + 72q^{36} + 246q^{37} - 400q^{38} + 186q^{39} - 248q^{41} - 84q^{42} - 68q^{43} + 496q^{44} - 168q^{46} - 324q^{47} - 192q^{48} + 49q^{49} - 252q^{51} + 496q^{52} - 258q^{53} - 108q^{54} + 300q^{57} + 40q^{58} + 120q^{59} + 622q^{61} + 192q^{62} + 63q^{63} - 512q^{64} - 744q^{66} - 904q^{67} - 672q^{68} + 126q^{69} - 678q^{71} + 642q^{73} - 984q^{74} + 800q^{76} + 434q^{77} - 744q^{78} + 740q^{79} + 81q^{81} + 992q^{82} - 468q^{83} + 168q^{84} + 272q^{86} - 30q^{87} + 200q^{89} + 434q^{91} + 336q^{92} - 144q^{93} + 1296q^{94} + 768q^{96} + 1266q^{97} - 196q^{98} + 558q^{99} + O(q^{100})$$

## 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
−4.00000 3.00000 8.00000 0 −12.0000 7.00000 0 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.b 1
3.b odd 2 1 1575.4.a.k 1
5.b even 2 1 21.4.a.b 1
5.c odd 4 2 525.4.d.b 2
15.d odd 2 1 63.4.a.a 1
20.d odd 2 1 336.4.a.h 1
35.c odd 2 1 147.4.a.g 1
35.i odd 6 2 147.4.e.b 2
35.j even 6 2 147.4.e.c 2
40.e odd 2 1 1344.4.a.i 1
40.f even 2 1 1344.4.a.w 1
60.h even 2 1 1008.4.a.m 1
105.g even 2 1 441.4.a.b 1
105.o odd 6 2 441.4.e.m 2
105.p even 6 2 441.4.e.n 2
140.c even 2 1 2352.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 5.b even 2 1
63.4.a.a 1 15.d odd 2 1
147.4.a.g 1 35.c odd 2 1
147.4.e.b 2 35.i odd 6 2
147.4.e.c 2 35.j even 6 2
336.4.a.h 1 20.d odd 2 1
441.4.a.b 1 105.g even 2 1
441.4.e.m 2 105.o odd 6 2
441.4.e.n 2 105.p even 6 2
525.4.a.b 1 1.a even 1 1 trivial
525.4.d.b 2 5.c odd 4 2
1008.4.a.m 1 60.h even 2 1
1344.4.a.i 1 40.e odd 2 1
1344.4.a.w 1 40.f even 2 1
1575.4.a.k 1 3.b odd 2 1
2352.4.a.l 1 140.c even 2 1

## Hecke kernels

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

 $$T_{2} + 4$$ $$T_{11} - 62$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T$$
$3$ $$-3 + T$$
$5$ $$T$$
$7$ $$-7 + T$$
$11$ $$-62 + T$$
$13$ $$-62 + T$$
$17$ $$84 + T$$
$19$ $$-100 + T$$
$23$ $$-42 + T$$
$29$ $$10 + T$$
$31$ $$48 + T$$
$37$ $$-246 + T$$
$41$ $$248 + T$$
$43$ $$68 + T$$
$47$ $$324 + T$$
$53$ $$258 + T$$
$59$ $$-120 + T$$
$61$ $$-622 + T$$
$67$ $$904 + T$$
$71$ $$678 + T$$
$73$ $$-642 + T$$
$79$ $$-740 + T$$
$83$ $$468 + T$$
$89$ $$-200 + T$$
$97$ $$-1266 + T$$