# Properties

 Label 1024.2.e.i Level $1024$ Weight $2$ Character orbit 1024.e Analytic conductor $8.177$ Analytic rank $0$ Dimension $4$ CM no 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: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 128) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{8} q^{3} + 2 \zeta_{8}^{3} q^{5} + 4 \zeta_{8}^{2} q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{8} q^{3} + 2 \zeta_{8}^{3} q^{5} + 4 \zeta_{8}^{2} q^{7} + \zeta_{8}^{2} q^{9} + 2 \zeta_{8}^{3} q^{11} -2 \zeta_{8} q^{13} -4 q^{15} + 2 q^{17} -2 \zeta_{8} q^{19} + 8 \zeta_{8}^{3} q^{21} + 4 \zeta_{8}^{2} q^{23} + \zeta_{8}^{2} q^{25} -4 \zeta_{8}^{3} q^{27} -6 \zeta_{8} q^{29} -4 q^{33} -8 \zeta_{8} q^{35} + 10 \zeta_{8}^{3} q^{37} -4 \zeta_{8}^{2} q^{39} -6 \zeta_{8}^{2} q^{41} -6 \zeta_{8}^{3} q^{43} -2 \zeta_{8} q^{45} + 8 q^{47} -9 q^{49} + 4 \zeta_{8} q^{51} + 6 \zeta_{8}^{3} q^{53} -4 \zeta_{8}^{2} q^{55} -4 \zeta_{8}^{2} q^{57} + 14 \zeta_{8}^{3} q^{59} + 2 \zeta_{8} q^{61} -4 q^{63} + 4 q^{65} + 10 \zeta_{8} q^{67} + 8 \zeta_{8}^{3} q^{69} -12 \zeta_{8}^{2} q^{71} + 14 \zeta_{8}^{2} q^{73} + 2 \zeta_{8}^{3} q^{75} -8 \zeta_{8} q^{77} + 8 q^{79} + 11 q^{81} + 6 \zeta_{8} q^{83} + 4 \zeta_{8}^{3} q^{85} -12 \zeta_{8}^{2} q^{87} + 2 \zeta_{8}^{2} q^{89} -8 \zeta_{8}^{3} q^{91} + 4 q^{95} -2 q^{97} -2 \zeta_{8} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 16q^{15} + 8q^{17} - 16q^{33} + 32q^{47} - 36q^{49} - 16q^{63} + 16q^{65} + 32q^{79} + 44q^{81} + 16q^{95} - 8q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$5$$ $$1023$$ $$\chi(n)$$ $$-\zeta_{8}^{2}$$ $$1$$

## 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 −1.41421 1.41421i 0 1.41421 1.41421i 0 4.00000i 0 1.00000i 0
257.2 0 1.41421 + 1.41421i 0 −1.41421 + 1.41421i 0 4.00000i 0 1.00000i 0
769.1 0 −1.41421 + 1.41421i 0 1.41421 + 1.41421i 0 4.00000i 0 1.00000i 0
769.2 0 1.41421 1.41421i 0 −1.41421 1.41421i 0 4.00000i 0 1.00000i 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
16.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.e.i 4
4.b odd 2 1 1024.2.e.m 4
8.b even 2 1 inner 1024.2.e.i 4
8.d odd 2 1 1024.2.e.m 4
16.e even 4 2 inner 1024.2.e.i 4
16.f odd 4 2 1024.2.e.m 4
32.g even 8 1 128.2.a.a 1
32.g even 8 1 128.2.a.d yes 1
32.g even 8 2 256.2.b.c 2
32.h odd 8 1 128.2.a.b yes 1
32.h odd 8 1 128.2.a.c yes 1
32.h odd 8 2 256.2.b.a 2
96.o even 8 1 1152.2.a.h 1
96.o even 8 1 1152.2.a.r 1
96.o even 8 2 2304.2.d.b 2
96.p odd 8 1 1152.2.a.c 1
96.p odd 8 1 1152.2.a.m 1
96.p odd 8 2 2304.2.d.r 2
160.u even 8 1 3200.2.c.f 2
160.u even 8 1 3200.2.c.k 2
160.v odd 8 1 3200.2.c.e 2
160.v odd 8 1 3200.2.c.l 2
160.y odd 8 1 3200.2.a.e 1
160.y odd 8 1 3200.2.a.u 1
160.z even 8 1 3200.2.a.h 1
160.z even 8 1 3200.2.a.x 1
160.ba even 8 1 3200.2.c.f 2
160.ba even 8 1 3200.2.c.k 2
160.bb odd 8 1 3200.2.c.e 2
160.bb odd 8 1 3200.2.c.l 2
224.v odd 8 1 6272.2.a.a 1
224.v odd 8 1 6272.2.a.h 1
224.x even 8 1 6272.2.a.b 1
224.x even 8 1 6272.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 32.g even 8 1
128.2.a.b yes 1 32.h odd 8 1
128.2.a.c yes 1 32.h odd 8 1
128.2.a.d yes 1 32.g even 8 1
256.2.b.a 2 32.h odd 8 2
256.2.b.c 2 32.g even 8 2
1024.2.e.i 4 1.a even 1 1 trivial
1024.2.e.i 4 8.b even 2 1 inner
1024.2.e.i 4 16.e even 4 2 inner
1024.2.e.m 4 4.b odd 2 1
1024.2.e.m 4 8.d odd 2 1
1024.2.e.m 4 16.f odd 4 2
1152.2.a.c 1 96.p odd 8 1
1152.2.a.h 1 96.o even 8 1
1152.2.a.m 1 96.p odd 8 1
1152.2.a.r 1 96.o even 8 1
2304.2.d.b 2 96.o even 8 2
2304.2.d.r 2 96.p odd 8 2
3200.2.a.e 1 160.y odd 8 1
3200.2.a.h 1 160.z even 8 1
3200.2.a.u 1 160.y odd 8 1
3200.2.a.x 1 160.z even 8 1
3200.2.c.e 2 160.v odd 8 1
3200.2.c.e 2 160.bb odd 8 1
3200.2.c.f 2 160.u even 8 1
3200.2.c.f 2 160.ba even 8 1
3200.2.c.k 2 160.u even 8 1
3200.2.c.k 2 160.ba even 8 1
3200.2.c.l 2 160.v odd 8 1
3200.2.c.l 2 160.bb odd 8 1
6272.2.a.a 1 224.v odd 8 1
6272.2.a.b 1 224.x even 8 1
6272.2.a.g 1 224.x even 8 1
6272.2.a.h 1 224.v odd 8 1

## Hecke kernels

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

 $$T_{3}^{4} + 16$$ $$T_{5}^{4} + 16$$ $$T_{47} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + T^{4}$$
$5$ $$16 + T^{4}$$
$7$ $$( 16 + T^{2} )^{2}$$
$11$ $$16 + T^{4}$$
$13$ $$16 + T^{4}$$
$17$ $$( -2 + T )^{4}$$
$19$ $$16 + T^{4}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$1296 + T^{4}$$
$31$ $$T^{4}$$
$37$ $$10000 + T^{4}$$
$41$ $$( 36 + T^{2} )^{2}$$
$43$ $$1296 + T^{4}$$
$47$ $$( -8 + T )^{4}$$
$53$ $$1296 + T^{4}$$
$59$ $$38416 + T^{4}$$
$61$ $$16 + T^{4}$$
$67$ $$10000 + T^{4}$$
$71$ $$( 144 + T^{2} )^{2}$$
$73$ $$( 196 + T^{2} )^{2}$$
$79$ $$( -8 + T )^{4}$$
$83$ $$1296 + T^{4}$$
$89$ $$( 4 + T^{2} )^{2}$$
$97$ $$( 2 + T )^{4}$$