# Properties

 Label 2535.2.a.k Level $2535$ Weight $2$ Character orbit 2535.a Self dual yes Analytic conductor $20.242$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2535 = 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2535.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.2420769124$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - 3q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - 3q^{8} + q^{9} - q^{10} - 4q^{11} - q^{12} - q^{15} - q^{16} + 2q^{17} + q^{18} + 4q^{19} + q^{20} - 4q^{22} + 8q^{23} - 3q^{24} + q^{25} + q^{27} - 2q^{29} - q^{30} + 8q^{31} + 5q^{32} - 4q^{33} + 2q^{34} - q^{36} - 6q^{37} + 4q^{38} + 3q^{40} + 6q^{41} - 4q^{43} + 4q^{44} - q^{45} + 8q^{46} + 8q^{47} - q^{48} - 7q^{49} + q^{50} + 2q^{51} + 6q^{53} + q^{54} + 4q^{55} + 4q^{57} - 2q^{58} + 12q^{59} + q^{60} - 2q^{61} + 8q^{62} + 7q^{64} - 4q^{66} + 4q^{67} - 2q^{68} + 8q^{69} - 3q^{72} + 6q^{73} - 6q^{74} + q^{75} - 4q^{76} + 16q^{79} + q^{80} + q^{81} + 6q^{82} + 4q^{83} - 2q^{85} - 4q^{86} - 2q^{87} + 12q^{88} - 10q^{89} - q^{90} - 8q^{92} + 8q^{93} + 8q^{94} - 4q^{95} + 5q^{96} - 18q^{97} - 7q^{98} - 4q^{99} + O(q^{100})$$

## 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
1.00000 1.00000 −1.00000 −1.00000 1.00000 0 −3.00000 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 2535.2.a.k 1
3.b odd 2 1 7605.2.a.h 1
13.b even 2 1 195.2.a.a 1
39.d odd 2 1 585.2.a.g 1
52.b odd 2 1 3120.2.a.k 1
65.d even 2 1 975.2.a.i 1
65.h odd 4 2 975.2.c.e 2
91.b odd 2 1 9555.2.a.b 1
156.h even 2 1 9360.2.a.o 1
195.e odd 2 1 2925.2.a.d 1
195.s even 4 2 2925.2.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.a 1 13.b even 2 1
585.2.a.g 1 39.d odd 2 1
975.2.a.i 1 65.d even 2 1
975.2.c.e 2 65.h odd 4 2
2535.2.a.k 1 1.a even 1 1 trivial
2925.2.a.d 1 195.e odd 2 1
2925.2.c.f 2 195.s even 4 2
3120.2.a.k 1 52.b odd 2 1
7605.2.a.h 1 3.b odd 2 1
9360.2.a.o 1 156.h even 2 1
9555.2.a.b 1 91.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2535))$$:

 $$T_{2} - 1$$ $$T_{7}$$ $$T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$4 + T$$
$13$ $$T$$
$17$ $$-2 + T$$
$19$ $$-4 + T$$
$23$ $$-8 + T$$
$29$ $$2 + T$$
$31$ $$-8 + T$$
$37$ $$6 + T$$
$41$ $$-6 + T$$
$43$ $$4 + T$$
$47$ $$-8 + T$$
$53$ $$-6 + T$$
$59$ $$-12 + T$$
$61$ $$2 + T$$
$67$ $$-4 + T$$
$71$ $$T$$
$73$ $$-6 + T$$
$79$ $$-16 + T$$
$83$ $$-4 + T$$
$89$ $$10 + T$$
$97$ $$18 + T$$