# Properties

 Label 2535.2.a.h 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

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{4} - q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 - 2 * q^4 - q^5 - q^7 + q^9 $$q + q^{3} - 2 q^{4} - q^{5} - q^{7} + q^{9} + 6 q^{11} - 2 q^{12} - q^{15} + 4 q^{16} - 4 q^{19} + 2 q^{20} - q^{21} - 6 q^{23} + q^{25} + q^{27} + 2 q^{28} - 6 q^{29} + 5 q^{31} + 6 q^{33} + q^{35} - 2 q^{36} + 2 q^{37} + 11 q^{43} - 12 q^{44} - q^{45} + 6 q^{47} + 4 q^{48} - 6 q^{49} - 6 q^{55} - 4 q^{57} + 6 q^{59} + 2 q^{60} - q^{61} - q^{63} - 8 q^{64} + 11 q^{67} - 6 q^{69} - 6 q^{71} + 5 q^{73} + q^{75} + 8 q^{76} - 6 q^{77} + 11 q^{79} - 4 q^{80} + q^{81} + 12 q^{83} + 2 q^{84} - 6 q^{87} + 12 q^{89} + 12 q^{92} + 5 q^{93} + 4 q^{95} + 17 q^{97} + 6 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^4 - q^5 - q^7 + q^9 + 6 * q^11 - 2 * q^12 - q^15 + 4 * q^16 - 4 * q^19 + 2 * q^20 - q^21 - 6 * q^23 + q^25 + q^27 + 2 * q^28 - 6 * q^29 + 5 * q^31 + 6 * q^33 + q^35 - 2 * q^36 + 2 * q^37 + 11 * q^43 - 12 * q^44 - q^45 + 6 * q^47 + 4 * q^48 - 6 * q^49 - 6 * q^55 - 4 * q^57 + 6 * q^59 + 2 * q^60 - q^61 - q^63 - 8 * q^64 + 11 * q^67 - 6 * q^69 - 6 * q^71 + 5 * q^73 + q^75 + 8 * q^76 - 6 * q^77 + 11 * q^79 - 4 * q^80 + q^81 + 12 * q^83 + 2 * q^84 - 6 * q^87 + 12 * q^89 + 12 * q^92 + 5 * q^93 + 4 * q^95 + 17 * q^97 + 6 * q^99

## 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
0 1.00000 −2.00000 −1.00000 0 −1.00000 0 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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.h 1
3.b odd 2 1 7605.2.a.l 1
13.b even 2 1 2535.2.a.i 1
13.c even 3 2 195.2.i.b 2
39.d odd 2 1 7605.2.a.k 1
39.i odd 6 2 585.2.j.a 2
65.n even 6 2 975.2.i.d 2
65.q odd 12 4 975.2.bb.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.b 2 13.c even 3 2
585.2.j.a 2 39.i odd 6 2
975.2.i.d 2 65.n even 6 2
975.2.bb.b 4 65.q odd 12 4
2535.2.a.h 1 1.a even 1 1 trivial
2535.2.a.i 1 13.b even 2 1
7605.2.a.k 1 39.d odd 2 1
7605.2.a.l 1 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2535))$$:

 $$T_{2}$$ T2 $$T_{7} + 1$$ T7 + 1 $$T_{11} - 6$$ T11 - 6

## Hecke characteristic polynomials

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