# Properties

 Label 2574.2.a.i Level $2574$ Weight $2$ Character orbit 2574.a Self dual yes Analytic conductor $20.553$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2574 = 2 \cdot 3^{2} \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2574.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.5534934803$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 858) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} - 3q^{7} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} + q^{5} - 3q^{7} - q^{8} - q^{10} - q^{11} - q^{13} + 3q^{14} + q^{16} + 4q^{17} - 2q^{19} + q^{20} + q^{22} + q^{23} - 4q^{25} + q^{26} - 3q^{28} + 9q^{29} - 4q^{31} - q^{32} - 4q^{34} - 3q^{35} - 6q^{37} + 2q^{38} - q^{40} - q^{41} + 11q^{43} - q^{44} - q^{46} + 2q^{49} + 4q^{50} - q^{52} + 10q^{53} - q^{55} + 3q^{56} - 9q^{58} + 3q^{59} + 5q^{61} + 4q^{62} + q^{64} - q^{65} + 3q^{67} + 4q^{68} + 3q^{70} - 10q^{71} + 9q^{73} + 6q^{74} - 2q^{76} + 3q^{77} + 10q^{79} + q^{80} + q^{82} + 6q^{83} + 4q^{85} - 11q^{86} + q^{88} + 8q^{89} + 3q^{91} + q^{92} - 2q^{95} + 2q^{97} - 2q^{98} + O(q^{100})$$

## 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 0 1.00000 1.00000 0 −3.00000 −1.00000 0 −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
$$2$$ $$1$$
$$3$$ $$-1$$
$$11$$ $$1$$
$$13$$ $$1$$

## 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 2574.2.a.i 1
3.b odd 2 1 858.2.a.g 1
12.b even 2 1 6864.2.a.u 1
33.d even 2 1 9438.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
858.2.a.g 1 3.b odd 2 1
2574.2.a.i 1 1.a even 1 1 trivial
6864.2.a.u 1 12.b even 2 1
9438.2.a.d 1 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2574))$$:

 $$T_{5} - 1$$ $$T_{7} + 3$$ $$T_{17} - 4$$ $$T_{23} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$3 + T$$
$11$ $$1 + T$$
$13$ $$1 + T$$
$17$ $$-4 + T$$
$19$ $$2 + T$$
$23$ $$-1 + T$$
$29$ $$-9 + T$$
$31$ $$4 + T$$
$37$ $$6 + T$$
$41$ $$1 + T$$
$43$ $$-11 + T$$
$47$ $$T$$
$53$ $$-10 + T$$
$59$ $$-3 + T$$
$61$ $$-5 + T$$
$67$ $$-3 + T$$
$71$ $$10 + T$$
$73$ $$-9 + T$$
$79$ $$-10 + T$$
$83$ $$-6 + T$$
$89$ $$-8 + T$$
$97$ $$-2 + T$$