# Properties

 Label 1024.2.a.h Level $1024$ Weight $2$ Character orbit 1024.a Self dual yes Analytic conductor $8.177$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1024.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.17668116698$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{16})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 512) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -2 - \beta_{2} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + ( 1 + 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -2 - \beta_{2} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + ( 1 + 2 \beta_{2} ) q^{9} + ( -\beta_{1} - 2 \beta_{3} ) q^{11} + ( -2 + \beta_{2} ) q^{13} + ( -3 \beta_{1} - \beta_{3} ) q^{15} -2 \beta_{2} q^{17} + ( -\beta_{1} + 2 \beta_{3} ) q^{19} -4 q^{21} + ( -\beta_{1} - 3 \beta_{3} ) q^{23} + ( 1 + 4 \beta_{2} ) q^{25} + 2 \beta_{3} q^{27} + ( -6 + \beta_{2} ) q^{29} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{31} + ( -4 - 6 \beta_{2} ) q^{33} + 2 \beta_{1} q^{35} + ( -2 + 5 \beta_{2} ) q^{37} + ( -\beta_{1} + \beta_{3} ) q^{39} -4 q^{41} -\beta_{1} q^{43} + ( -6 - 5 \beta_{2} ) q^{45} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{47} + ( 1 - 4 \beta_{2} ) q^{49} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{51} + ( -6 - \beta_{2} ) q^{53} + ( 5 \beta_{1} + 3 \beta_{3} ) q^{55} + ( -4 + 2 \beta_{2} ) q^{57} + \beta_{1} q^{59} + ( -6 - 5 \beta_{2} ) q^{61} + ( -\beta_{1} - 3 \beta_{3} ) q^{63} + 2 q^{65} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{67} + ( -4 - 8 \beta_{2} ) q^{69} + ( 5 \beta_{1} - \beta_{3} ) q^{71} + ( 2 + 6 \beta_{2} ) q^{73} + ( 5 \beta_{1} + 4 \beta_{3} ) q^{75} + ( -4 + 8 \beta_{2} ) q^{77} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{79} + ( -3 - 2 \beta_{2} ) q^{81} + ( 3 \beta_{1} - 4 \beta_{3} ) q^{83} + ( 4 + 4 \beta_{2} ) q^{85} + ( -5 \beta_{1} + \beta_{3} ) q^{87} + ( 2 - 2 \beta_{2} ) q^{89} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{91} + ( 8 + 8 \beta_{2} ) q^{93} + ( \beta_{1} - \beta_{3} ) q^{95} + ( 8 + 2 \beta_{2} ) q^{97} -7 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{5} + 4q^{9} + O(q^{10})$$ $$4q - 8q^{5} + 4q^{9} - 8q^{13} - 16q^{21} + 4q^{25} - 24q^{29} - 16q^{33} - 8q^{37} - 16q^{41} - 24q^{45} + 4q^{49} - 24q^{53} - 16q^{57} - 24q^{61} + 8q^{65} - 16q^{69} + 8q^{73} - 16q^{77} - 12q^{81} + 16q^{85} + 8q^{89} + 32q^{93} + 32q^{97} + O(q^{100})$$

## 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
 −1.84776 0.765367 −0.765367 1.84776
0 −2.61313 0 −3.41421 0 1.53073 0 3.82843 0
1.2 0 −1.08239 0 −0.585786 0 3.69552 0 −1.82843 0
1.3 0 1.08239 0 −0.585786 0 −3.69552 0 −1.82843 0
1.4 0 2.61313 0 −3.41421 0 −1.53073 0 3.82843 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
$$2$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.a.h 4
3.b odd 2 1 9216.2.a.bp 4
4.b odd 2 1 inner 1024.2.a.h 4
8.b even 2 1 1024.2.a.i 4
8.d odd 2 1 1024.2.a.i 4
12.b even 2 1 9216.2.a.bp 4
16.e even 4 2 1024.2.b.g 8
16.f odd 4 2 1024.2.b.g 8
24.f even 2 1 9216.2.a.w 4
24.h odd 2 1 9216.2.a.w 4
32.g even 8 2 512.2.e.i 8
32.g even 8 2 512.2.e.j yes 8
32.h odd 8 2 512.2.e.i 8
32.h odd 8 2 512.2.e.j yes 8
96.o even 8 2 4608.2.k.bd 8
96.o even 8 2 4608.2.k.bi 8
96.p odd 8 2 4608.2.k.bd 8
96.p odd 8 2 4608.2.k.bi 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.i 8 32.g even 8 2
512.2.e.i 8 32.h odd 8 2
512.2.e.j yes 8 32.g even 8 2
512.2.e.j yes 8 32.h odd 8 2
1024.2.a.h 4 1.a even 1 1 trivial
1024.2.a.h 4 4.b odd 2 1 inner
1024.2.a.i 4 8.b even 2 1
1024.2.a.i 4 8.d odd 2 1
1024.2.b.g 8 16.e even 4 2
1024.2.b.g 8 16.f odd 4 2
4608.2.k.bd 8 96.o even 8 2
4608.2.k.bd 8 96.p odd 8 2
4608.2.k.bi 8 96.o even 8 2
4608.2.k.bi 8 96.p odd 8 2
9216.2.a.w 4 24.f even 2 1
9216.2.a.w 4 24.h odd 2 1
9216.2.a.bp 4 3.b odd 2 1
9216.2.a.bp 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1024))$$:

 $$T_{3}^{4} - 8 T_{3}^{2} + 8$$ $$T_{5}^{2} + 4 T_{5} + 2$$ $$T_{7}^{4} - 16 T_{7}^{2} + 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$8 - 8 T^{2} + T^{4}$$
$5$ $$( 2 + 4 T + T^{2} )^{2}$$
$7$ $$32 - 16 T^{2} + T^{4}$$
$11$ $$392 - 40 T^{2} + T^{4}$$
$13$ $$( 2 + 4 T + T^{2} )^{2}$$
$17$ $$( -8 + T^{2} )^{2}$$
$19$ $$8 - 40 T^{2} + T^{4}$$
$23$ $$1568 - 80 T^{2} + T^{4}$$
$29$ $$( 34 + 12 T + T^{2} )^{2}$$
$31$ $$512 - 64 T^{2} + T^{4}$$
$37$ $$( -46 + 4 T + T^{2} )^{2}$$
$41$ $$( 4 + T )^{4}$$
$43$ $$8 - 8 T^{2} + T^{4}$$
$47$ $$512 - 64 T^{2} + T^{4}$$
$53$ $$( 34 + 12 T + T^{2} )^{2}$$
$59$ $$8 - 8 T^{2} + T^{4}$$
$61$ $$( -14 + 12 T + T^{2} )^{2}$$
$67$ $$392 - 104 T^{2} + T^{4}$$
$71$ $$9248 - 208 T^{2} + T^{4}$$
$73$ $$( -68 - 4 T + T^{2} )^{2}$$
$79$ $$8192 - 256 T^{2} + T^{4}$$
$83$ $$2312 - 200 T^{2} + T^{4}$$
$89$ $$( -4 - 4 T + T^{2} )^{2}$$
$97$ $$( 56 - 16 T + T^{2} )^{2}$$