# Properties

 Label 1024.2.b.f Level $1024$ Weight $2$ Character orbit 1024.b Analytic conductor $8.177$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1024.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.17668116698$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 512) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{3} - \beta q^{5} + 4 q^{7} - 5 q^{9} +O(q^{10})$$ q + 2*b * q^3 - b * q^5 + 4 * q^7 - 5 * q^9 $$q + 2 \beta q^{3} - \beta q^{5} + 4 q^{7} - 5 q^{9} - 2 \beta q^{11} + 3 \beta q^{13} + 4 q^{15} + 2 \beta q^{19} + 8 \beta q^{21} + 4 q^{23} + 3 q^{25} - 4 \beta q^{27} + 3 \beta q^{29} + 8 q^{31} + 8 q^{33} - 4 \beta q^{35} + \beta q^{37} - 12 q^{39} - 8 q^{41} + 2 \beta q^{43} + 5 \beta q^{45} - 8 q^{47} + 9 q^{49} - \beta q^{53} - 4 q^{55} - 8 q^{57} + 6 \beta q^{59} - 3 \beta q^{61} - 20 q^{63} + 6 q^{65} - 2 \beta q^{67} + 8 \beta q^{69} + 12 q^{71} + 2 q^{73} + 6 \beta q^{75} - 8 \beta q^{77} + q^{81} - 10 \beta q^{83} - 12 q^{87} - 14 q^{89} + 12 \beta q^{91} + 16 \beta q^{93} + 4 q^{95} - 16 q^{97} + 10 \beta q^{99} +O(q^{100})$$ q + 2*b * q^3 - b * q^5 + 4 * q^7 - 5 * q^9 - 2*b * q^11 + 3*b * q^13 + 4 * q^15 + 2*b * q^19 + 8*b * q^21 + 4 * q^23 + 3 * q^25 - 4*b * q^27 + 3*b * q^29 + 8 * q^31 + 8 * q^33 - 4*b * q^35 + b * q^37 - 12 * q^39 - 8 * q^41 + 2*b * q^43 + 5*b * q^45 - 8 * q^47 + 9 * q^49 - b * q^53 - 4 * q^55 - 8 * q^57 + 6*b * q^59 - 3*b * q^61 - 20 * q^63 + 6 * q^65 - 2*b * q^67 + 8*b * q^69 + 12 * q^71 + 2 * q^73 + 6*b * q^75 - 8*b * q^77 + q^81 - 10*b * q^83 - 12 * q^87 - 14 * q^89 + 12*b * q^91 + 16*b * q^93 + 4 * q^95 - 16 * q^97 + 10*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{7} - 10 q^{9}+O(q^{10})$$ 2 * q + 8 * q^7 - 10 * q^9 $$2 q + 8 q^{7} - 10 q^{9} + 8 q^{15} + 8 q^{23} + 6 q^{25} + 16 q^{31} + 16 q^{33} - 24 q^{39} - 16 q^{41} - 16 q^{47} + 18 q^{49} - 8 q^{55} - 16 q^{57} - 40 q^{63} + 12 q^{65} + 24 q^{71} + 4 q^{73} + 2 q^{81} - 24 q^{87} - 28 q^{89} + 8 q^{95} - 32 q^{97}+O(q^{100})$$ 2 * q + 8 * q^7 - 10 * q^9 + 8 * q^15 + 8 * q^23 + 6 * q^25 + 16 * q^31 + 16 * q^33 - 24 * q^39 - 16 * q^41 - 16 * q^47 + 18 * q^49 - 8 * q^55 - 16 * q^57 - 40 * q^63 + 12 * q^65 + 24 * q^71 + 4 * q^73 + 2 * q^81 - 24 * q^87 - 28 * q^89 + 8 * q^95 - 32 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$1023$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
513.1
 − 1.41421i 1.41421i
0 2.82843i 0 1.41421i 0 4.00000 0 −5.00000 0
513.2 0 2.82843i 0 1.41421i 0 4.00000 0 −5.00000 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
8.b even 2 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.a 2 32.h odd 8 2
512.2.e.b yes 2 32.g even 8 2
512.2.e.g yes 2 32.g even 8 2
512.2.e.h yes 2 32.h odd 8 2
1024.2.a.a 2 16.e even 4 2
1024.2.a.f 2 16.f odd 4 2
1024.2.b.a 2 4.b odd 2 1
1024.2.b.a 2 8.d odd 2 1
1024.2.b.f 2 1.a even 1 1 trivial
1024.2.b.f 2 8.b even 2 1 inner
4608.2.k.f 2 96.o even 8 2
4608.2.k.j 2 96.p odd 8 2
4608.2.k.o 2 96.p odd 8 2
4608.2.k.s 2 96.o even 8 2
9216.2.a.b 2 48.i odd 4 2
9216.2.a.u 2 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1024, [\chi])$$:

 $$T_{3}^{2} + 8$$ T3^2 + 8 $$T_{5}^{2} + 2$$ T5^2 + 2 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 8$$
$5$ $$T^{2} + 2$$
$7$ $$(T - 4)^{2}$$
$11$ $$T^{2} + 8$$
$13$ $$T^{2} + 18$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 8$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} + 18$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 2$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2} + 8$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 2$$
$59$ $$T^{2} + 72$$
$61$ $$T^{2} + 18$$
$67$ $$T^{2} + 8$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 200$$
$89$ $$(T + 14)^{2}$$
$97$ $$(T + 16)^{2}$$