# Properties

 Label 1024.2.e.o Level $1024$ Weight $2$ Character orbit 1024.e Analytic conductor $8.177$ Analytic rank $0$ Dimension $4$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1024.e (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.17668116698$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 512) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{3} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{9}+O(q^{10})$$ q + (b2 + b1 + 1) * q^3 + (2*b3 + 3*b2 + 2*b1) * q^9 $$q + (\beta_{2} + \beta_1 + 1) q^{3} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{9} + ( - \beta_{3} + 3 \beta_{2} - 3) q^{11} + ( - 2 \beta_{3} + 2 \beta_1) q^{17} + (\beta_{2} - 3 \beta_1 + 1) q^{19} + 5 \beta_{2} q^{25} + (4 \beta_{3} + 8 \beta_{2} - 8) q^{27} + (2 \beta_{3} - 2 \beta_1 - 2) q^{33} - 6 \beta_{2} q^{41} + ( - 3 \beta_{3} + 5 \beta_{2} - 5) q^{43} + 7 q^{49} + (8 \beta_{2} + 4 \beta_1 + 8) q^{51} + ( - 2 \beta_{3} - 10 \beta_{2} - 2 \beta_1) q^{57} + ( - 5 \beta_{3} + 3 \beta_{2} - 3) q^{59} + (7 \beta_{2} + 3 \beta_1 + 7) q^{67} + ( - 6 \beta_{3} - 6 \beta_1) q^{73} + (5 \beta_{3} + 5 \beta_{2} - 5) q^{75} + (6 \beta_{3} - 6 \beta_1 - 23) q^{81} + ( - 9 \beta_{2} - \beta_1 - 9) q^{83} + ( - 2 \beta_{3} - 2 \beta_1) q^{89} + ( - 6 \beta_{3} + 6 \beta_1) q^{97} + ( - \beta_{2} - 9 \beta_1 - 1) q^{99}+O(q^{100})$$ q + (b2 + b1 + 1) * q^3 + (2*b3 + 3*b2 + 2*b1) * q^9 + (-b3 + 3*b2 - 3) * q^11 + (-2*b3 + 2*b1) * q^17 + (b2 - 3*b1 + 1) * q^19 + 5*b2 * q^25 + (4*b3 + 8*b2 - 8) * q^27 + (2*b3 - 2*b1 - 2) * q^33 - 6*b2 * q^41 + (-3*b3 + 5*b2 - 5) * q^43 + 7 * q^49 + (8*b2 + 4*b1 + 8) * q^51 + (-2*b3 - 10*b2 - 2*b1) * q^57 + (-5*b3 + 3*b2 - 3) * q^59 + (7*b2 + 3*b1 + 7) * q^67 + (-6*b3 - 6*b1) * q^73 + (5*b3 + 5*b2 - 5) * q^75 + (6*b3 - 6*b1 - 23) * q^81 + (-9*b2 - b1 - 9) * q^83 + (-2*b3 - 2*b1) * q^89 + (-6*b3 + 6*b1) * q^97 + (-b2 - 9*b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3}+O(q^{10})$$ 4 * q + 4 * q^3 $$4 q + 4 q^{3} - 12 q^{11} + 4 q^{19} - 32 q^{27} - 8 q^{33} - 20 q^{43} + 28 q^{49} + 32 q^{51} - 12 q^{59} + 28 q^{67} - 20 q^{75} - 92 q^{81} - 36 q^{83} - 4 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 - 12 * q^11 + 4 * q^19 - 32 * q^27 - 8 * q^33 - 20 * q^43 + 28 * q^49 + 32 * q^51 - 12 * q^59 + 28 * q^67 - 20 * q^75 - 92 * q^81 - 36 * q^83 - 4 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}$$ 2*v $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$2\zeta_{8}^{3}$$ 2*v^3
 $$\zeta_{8}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{8}^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## Character values

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

 $$n$$ $$5$$ $$1023$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 −0.414214 0.414214i 0 0 0 0 0 2.65685i 0
257.2 0 2.41421 + 2.41421i 0 0 0 0 0 8.65685i 0
769.1 0 −0.414214 + 0.414214i 0 0 0 0 0 2.65685i 0
769.2 0 2.41421 2.41421i 0 0 0 0 0 8.65685i 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.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
16.e even 4 1 inner
16.f odd 4 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{4} - 4T_{3}^{3} + 8T_{3}^{2} + 8T_{3} + 4$$ T3^4 - 4*T3^3 + 8*T3^2 + 8*T3 + 4 $$T_{5}$$ T5 $$T_{47}$$ T47

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 4 T^{3} + 8 T^{2} + 8 T + 4$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 196$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 32)^{2}$$
$19$ $$T^{4} - 4 T^{3} + 8 T^{2} + 136 T + 1156$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 36)^{2}$$
$43$ $$T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 196$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 6724$$
$61$ $$T^{4}$$
$67$ $$T^{4} - 28 T^{3} + 392 T^{2} + \cdots + 3844$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 288)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 36 T^{3} + 648 T^{2} + \cdots + 24964$$
$89$ $$(T^{2} + 32)^{2}$$
$97$ $$(T^{2} - 288)^{2}$$