# Properties

 Label 2576.2.a.l 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 644) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 - q^7 - 2 * q^9 $$q + q^{3} - q^{7} - 2 q^{9} + 2 q^{11} - 3 q^{13} - q^{21} + q^{23} - 5 q^{25} - 5 q^{27} + q^{29} + 5 q^{31} + 2 q^{33} - 8 q^{37} - 3 q^{39} - 7 q^{41} + 4 q^{43} - 3 q^{47} + q^{49} - 12 q^{53} - 4 q^{59} - 6 q^{61} + 2 q^{63} + 12 q^{67} + q^{69} - 13 q^{71} + 3 q^{73} - 5 q^{75} - 2 q^{77} - 4 q^{79} + q^{81} - 16 q^{83} + q^{87} + 4 q^{89} + 3 q^{91} + 5 q^{93} + 10 q^{97} - 4 q^{99}+O(q^{100})$$ q + q^3 - q^7 - 2 * q^9 + 2 * q^11 - 3 * q^13 - q^21 + q^23 - 5 * q^25 - 5 * q^27 + q^29 + 5 * q^31 + 2 * q^33 - 8 * q^37 - 3 * q^39 - 7 * q^41 + 4 * q^43 - 3 * q^47 + q^49 - 12 * q^53 - 4 * q^59 - 6 * q^61 + 2 * q^63 + 12 * q^67 + q^69 - 13 * q^71 + 3 * q^73 - 5 * q^75 - 2 * q^77 - 4 * q^79 + q^81 - 16 * q^83 + q^87 + 4 * q^89 + 3 * q^91 + 5 * q^93 + 10 * q^97 - 4 * 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 0 0 0 −1.00000 0 −2.00000 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.l 1
4.b odd 2 1 644.2.a.a 1
12.b even 2 1 5796.2.a.e 1
28.d even 2 1 4508.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.a.a 1 4.b odd 2 1
2576.2.a.l 1 1.a even 1 1 trivial
4508.2.a.c 1 28.d even 2 1
5796.2.a.e 1 12.b even 2 1

## Hecke kernels

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

 $$T_{3} - 1$$ T3 - 1 $$T_{5}$$ T5 $$T_{11} - 2$$ T11 - 2 $$T_{13} + 3$$ T13 + 3

## Hecke characteristic polynomials

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