# Properties

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

# 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 32) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{5} + 3 \beta_{2} q^{9}+O(q^{10})$$ q - b1 * q^5 + 3*b2 * q^9 $$q - \beta_1 q^{5} + 3 \beta_{2} q^{9} - 3 \beta_{3} q^{13} - 2 q^{17} - \beta_{2} q^{25} - 5 \beta_{3} q^{29} - \beta_1 q^{37} - 10 \beta_{2} q^{41} - 3 \beta_{3} q^{45} + 7 q^{49} - 7 \beta_1 q^{53} - 5 \beta_{3} q^{61} - 12 q^{65} + 6 \beta_{2} q^{73} - 9 q^{81} + 2 \beta_1 q^{85} + 10 \beta_{2} q^{89} + 18 q^{97}+O(q^{100})$$ q - b1 * q^5 + 3*b2 * q^9 - 3*b3 * q^13 - 2 * q^17 - b2 * q^25 - 5*b3 * q^29 - b1 * q^37 - 10*b2 * q^41 - 3*b3 * q^45 + 7 * q^49 - 7*b1 * q^53 - 5*b3 * q^61 - 12 * q^65 + 6*b2 * q^73 - 9 * q^81 + 2*b1 * q^85 + 10*b2 * q^89 + 18 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 8 q^{17} + 28 q^{49} - 48 q^{65} - 36 q^{81} + 72 q^{97}+O(q^{100})$$ 4 * q - 8 * q^17 + 28 * q^49 - 48 * q^65 - 36 * q^81 + 72 * q^97

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 0 −1.41421 + 1.41421i 0 0 0 3.00000i 0
257.2 0 0 0 1.41421 1.41421i 0 0 0 3.00000i 0
769.1 0 0 0 −1.41421 1.41421i 0 0 0 3.00000i 0
769.2 0 0 0 1.41421 + 1.41421i 0 0 0 3.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
8.b even 2 1 inner
8.d odd 2 1 inner
16.e even 4 2 inner
16.f odd 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.j 4
4.b odd 2 1 CM 1024.2.e.j 4
8.b even 2 1 inner 1024.2.e.j 4
8.d odd 2 1 inner 1024.2.e.j 4
16.e even 4 2 inner 1024.2.e.j 4
16.f odd 4 2 inner 1024.2.e.j 4
32.g even 8 1 32.2.a.a 1
32.g even 8 1 64.2.a.a 1
32.g even 8 2 256.2.b.b 2
32.h odd 8 1 32.2.a.a 1
32.h odd 8 1 64.2.a.a 1
32.h odd 8 2 256.2.b.b 2
96.o even 8 1 288.2.a.d 1
96.o even 8 1 576.2.a.c 1
96.o even 8 2 2304.2.d.j 2
96.p odd 8 1 288.2.a.d 1
96.p odd 8 1 576.2.a.c 1
96.p odd 8 2 2304.2.d.j 2
160.u even 8 1 800.2.c.e 2
160.u even 8 1 1600.2.c.l 2
160.v odd 8 1 800.2.c.e 2
160.v odd 8 1 1600.2.c.l 2
160.y odd 8 1 800.2.a.d 1
160.y odd 8 1 1600.2.a.n 1
160.z even 8 1 800.2.a.d 1
160.z even 8 1 1600.2.a.n 1
160.ba even 8 1 800.2.c.e 2
160.ba even 8 1 1600.2.c.l 2
160.bb odd 8 1 800.2.c.e 2
160.bb odd 8 1 1600.2.c.l 2
224.v odd 8 1 1568.2.a.e 1
224.v odd 8 1 3136.2.a.m 1
224.x even 8 1 1568.2.a.e 1
224.x even 8 1 3136.2.a.m 1
224.bc odd 24 2 1568.2.i.f 2
224.bd even 24 2 1568.2.i.g 2
224.be even 24 2 1568.2.i.f 2
224.bf odd 24 2 1568.2.i.g 2
288.bc even 24 2 2592.2.i.t 2
288.bd odd 24 2 2592.2.i.t 2
288.be odd 24 2 2592.2.i.e 2
288.bf even 24 2 2592.2.i.e 2
352.o odd 8 1 3872.2.a.f 1
352.o odd 8 1 7744.2.a.v 1
352.q even 8 1 3872.2.a.f 1
352.q even 8 1 7744.2.a.v 1
416.be odd 8 1 5408.2.a.g 1
416.bg even 8 1 5408.2.a.g 1
480.bq odd 8 1 7200.2.f.m 2
480.br even 8 1 7200.2.f.m 2
480.bs even 8 1 7200.2.a.v 1
480.bu odd 8 1 7200.2.a.v 1
480.ca odd 8 1 7200.2.f.m 2
480.cb even 8 1 7200.2.f.m 2
544.bc even 8 1 9248.2.a.f 1
544.bj odd 8 1 9248.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 32.g even 8 1
32.2.a.a 1 32.h odd 8 1
64.2.a.a 1 32.g even 8 1
64.2.a.a 1 32.h odd 8 1
256.2.b.b 2 32.g even 8 2
256.2.b.b 2 32.h odd 8 2
288.2.a.d 1 96.o even 8 1
288.2.a.d 1 96.p odd 8 1
576.2.a.c 1 96.o even 8 1
576.2.a.c 1 96.p odd 8 1
800.2.a.d 1 160.y odd 8 1
800.2.a.d 1 160.z even 8 1
800.2.c.e 2 160.u even 8 1
800.2.c.e 2 160.v odd 8 1
800.2.c.e 2 160.ba even 8 1
800.2.c.e 2 160.bb odd 8 1
1024.2.e.j 4 1.a even 1 1 trivial
1024.2.e.j 4 4.b odd 2 1 CM
1024.2.e.j 4 8.b even 2 1 inner
1024.2.e.j 4 8.d odd 2 1 inner
1024.2.e.j 4 16.e even 4 2 inner
1024.2.e.j 4 16.f odd 4 2 inner
1568.2.a.e 1 224.v odd 8 1
1568.2.a.e 1 224.x even 8 1
1568.2.i.f 2 224.bc odd 24 2
1568.2.i.f 2 224.be even 24 2
1568.2.i.g 2 224.bd even 24 2
1568.2.i.g 2 224.bf odd 24 2
1600.2.a.n 1 160.y odd 8 1
1600.2.a.n 1 160.z even 8 1
1600.2.c.l 2 160.u even 8 1
1600.2.c.l 2 160.v odd 8 1
1600.2.c.l 2 160.ba even 8 1
1600.2.c.l 2 160.bb odd 8 1
2304.2.d.j 2 96.o even 8 2
2304.2.d.j 2 96.p odd 8 2
2592.2.i.e 2 288.be odd 24 2
2592.2.i.e 2 288.bf even 24 2
2592.2.i.t 2 288.bc even 24 2
2592.2.i.t 2 288.bd odd 24 2
3136.2.a.m 1 224.v odd 8 1
3136.2.a.m 1 224.x even 8 1
3872.2.a.f 1 352.o odd 8 1
3872.2.a.f 1 352.q even 8 1
5408.2.a.g 1 416.be odd 8 1
5408.2.a.g 1 416.bg even 8 1
7200.2.a.v 1 480.bs even 8 1
7200.2.a.v 1 480.bu odd 8 1
7200.2.f.m 2 480.bq odd 8 1
7200.2.f.m 2 480.br even 8 1
7200.2.f.m 2 480.ca odd 8 1
7200.2.f.m 2 480.cb even 8 1
7744.2.a.v 1 352.o odd 8 1
7744.2.a.v 1 352.q even 8 1
9248.2.a.f 1 544.bc even 8 1
9248.2.a.f 1 544.bj odd 8 1

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{5}^{4} + 16$$ T5^4 + 16 $$T_{47}$$ T47

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 16$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 1296$$
$17$ $$(T + 2)^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} + 10000$$
$31$ $$T^{4}$$
$37$ $$T^{4} + 16$$
$41$ $$(T^{2} + 100)^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 38416$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 10000$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 36)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} + 100)^{2}$$
$97$ $$(T - 18)^{4}$$