# Properties

 Label 2576.2.a.c Level $2576$ Weight $2$ Character orbit 2576.a Self dual yes Analytic conductor $20.569$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2576 = 2^{4} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2576.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.5694635607$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 322) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{3} - q^{7} + q^{9} + O(q^{10})$$ $$q - 2q^{3} - q^{7} + q^{9} - 4q^{11} + 6q^{17} + 6q^{19} + 2q^{21} + q^{23} - 5q^{25} + 4q^{27} + 10q^{29} - 4q^{31} + 8q^{33} - 2q^{37} - 10q^{41} + 4q^{43} - 12q^{47} + q^{49} - 12q^{51} - 6q^{53} - 12q^{57} + 2q^{59} - q^{63} - 2q^{69} + 8q^{71} - 6q^{73} + 10q^{75} + 4q^{77} + 8q^{79} - 11q^{81} + 14q^{83} - 20q^{87} - 14q^{89} + 8q^{93} - 2q^{97} - 4q^{99} + O(q^{100})$$

## 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
0 −2.00000 0 0 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
$$2$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 2576.2.a.c 1
4.b odd 2 1 322.2.a.b 1
12.b even 2 1 2898.2.a.p 1
20.d odd 2 1 8050.2.a.n 1
28.d even 2 1 2254.2.a.a 1
92.b even 2 1 7406.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.b 1 4.b odd 2 1
2254.2.a.a 1 28.d even 2 1
2576.2.a.c 1 1.a even 1 1 trivial
2898.2.a.p 1 12.b even 2 1
7406.2.a.e 1 92.b even 2 1
8050.2.a.n 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2576))$$:

 $$T_{3} + 2$$ $$T_{5}$$ $$T_{11} + 4$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$2 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$4 + T$$
$13$ $$T$$
$17$ $$-6 + T$$
$19$ $$-6 + T$$
$23$ $$-1 + T$$
$29$ $$-10 + T$$
$31$ $$4 + T$$
$37$ $$2 + T$$
$41$ $$10 + T$$
$43$ $$-4 + T$$
$47$ $$12 + T$$
$53$ $$6 + T$$
$59$ $$-2 + T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$-8 + T$$
$73$ $$6 + T$$
$79$ $$-8 + T$$
$83$ $$-14 + T$$
$89$ $$14 + T$$
$97$ $$2 + T$$