# Properties

 Label 2576.2.f.d Level $2576$ Weight $2$ Character orbit 2576.f Analytic conductor $20.569$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2576 = 2^{4} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2576.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.5694635607$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 322) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -\beta_{3} q^{7} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -\beta_{3} q^{7} + q^{9} + \beta_{2} q^{11} + 3 \beta_{1} q^{13} -2 \beta_{3} q^{19} + \beta_{2} q^{21} + ( 3 - \beta_{2} ) q^{23} -5 q^{25} -4 \beta_{1} q^{27} + 6 q^{29} -6 \beta_{1} q^{31} + 2 \beta_{3} q^{33} -3 \beta_{2} q^{37} + 6 q^{39} -4 \beta_{1} q^{41} -3 \beta_{2} q^{43} -2 \beta_{1} q^{47} + 7 q^{49} -\beta_{2} q^{53} + 2 \beta_{2} q^{57} + 7 \beta_{1} q^{59} -4 \beta_{3} q^{61} -\beta_{3} q^{63} + 3 \beta_{2} q^{67} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{69} -6 q^{71} -6 \beta_{1} q^{73} + 5 \beta_{1} q^{75} -7 \beta_{1} q^{77} -5 q^{81} + 6 \beta_{3} q^{83} -6 \beta_{1} q^{87} -6 \beta_{3} q^{89} -3 \beta_{2} q^{91} -12 q^{93} + 2 \beta_{3} q^{97} + \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{9} + O(q^{10})$$ $$4q + 4q^{9} + 12q^{23} - 20q^{25} + 24q^{29} + 24q^{39} + 28q^{49} - 24q^{71} - 20q^{81} - 48q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 11 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} + 11 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2576\mathbb{Z}\right)^\times$$.

 $$n$$ $$645$$ $$1473$$ $$1569$$ $$2255$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
321.1
 1.16372i − 2.57794i − 1.16372i 2.57794i
0 1.41421i 0 0 0 −2.64575 0 1.00000 0
321.2 0 1.41421i 0 0 0 2.64575 0 1.00000 0
321.3 0 1.41421i 0 0 0 −2.64575 0 1.00000 0
321.4 0 1.41421i 0 0 0 2.64575 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2576.2.f.d 4
4.b odd 2 1 322.2.c.a 4
7.b odd 2 1 inner 2576.2.f.d 4
12.b even 2 1 2898.2.g.f 4
23.b odd 2 1 inner 2576.2.f.d 4
28.d even 2 1 322.2.c.a 4
84.h odd 2 1 2898.2.g.f 4
92.b even 2 1 322.2.c.a 4
161.c even 2 1 inner 2576.2.f.d 4
276.h odd 2 1 2898.2.g.f 4
644.h odd 2 1 322.2.c.a 4
1932.b even 2 1 2898.2.g.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.c.a 4 4.b odd 2 1
322.2.c.a 4 28.d even 2 1
322.2.c.a 4 92.b even 2 1
322.2.c.a 4 644.h odd 2 1
2576.2.f.d 4 1.a even 1 1 trivial
2576.2.f.d 4 7.b odd 2 1 inner
2576.2.f.d 4 23.b odd 2 1 inner
2576.2.f.d 4 161.c even 2 1 inner
2898.2.g.f 4 12.b even 2 1
2898.2.g.f 4 84.h odd 2 1
2898.2.g.f 4 276.h odd 2 1
2898.2.g.f 4 1932.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 $$S_{2}^{\mathrm{new}}(2576, [\chi])$$:

 $$T_{3}^{2} + 2$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 2 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( -7 + T^{2} )^{2}$$
$11$ $$( 14 + T^{2} )^{2}$$
$13$ $$( 18 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( -28 + T^{2} )^{2}$$
$23$ $$( 23 - 6 T + T^{2} )^{2}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$( 72 + T^{2} )^{2}$$
$37$ $$( 126 + T^{2} )^{2}$$
$41$ $$( 32 + T^{2} )^{2}$$
$43$ $$( 126 + T^{2} )^{2}$$
$47$ $$( 8 + T^{2} )^{2}$$
$53$ $$( 14 + T^{2} )^{2}$$
$59$ $$( 98 + T^{2} )^{2}$$
$61$ $$( -112 + T^{2} )^{2}$$
$67$ $$( 126 + T^{2} )^{2}$$
$71$ $$( 6 + T )^{4}$$
$73$ $$( 72 + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$( -252 + T^{2} )^{2}$$
$89$ $$( -252 + T^{2} )^{2}$$
$97$ $$( -28 + T^{2} )^{2}$$