# Properties

 Label 1024.2.e.h 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: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 512) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{3} - \beta_1 q^{5} + ( - \beta_{3} - \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10})$$ q + (b2 - 1) * q^3 - b1 * q^5 + (-b3 - b1) * q^7 + b2 * q^9 $$q + (\beta_{2} - 1) q^{3} - \beta_1 q^{5} + ( - \beta_{3} - \beta_1) q^{7} + \beta_{2} q^{9} + (3 \beta_{2} + 3) q^{11} + 3 \beta_{3} q^{13} + ( - \beta_{3} + \beta_1) q^{15} + ( - 3 \beta_{2} + 3) q^{19} + 2 \beta_1 q^{21} + ( - 3 \beta_{3} - 3 \beta_1) q^{23} - \beta_{2} q^{25} + ( - 4 \beta_{2} - 4) q^{27} - \beta_{3} q^{29} + ( - 2 \beta_{3} + 2 \beta_1) q^{31} - 6 q^{33} + (4 \beta_{2} - 4) q^{35} - 3 \beta_1 q^{37} + ( - 3 \beta_{3} - 3 \beta_1) q^{39} - 6 \beta_{2} q^{41} + ( - 3 \beta_{2} - 3) q^{43} - \beta_{3} q^{45} - q^{49} - \beta_1 q^{53} + ( - 3 \beta_{3} - 3 \beta_1) q^{55} + 6 \beta_{2} q^{57} + ( - \beta_{2} - 1) q^{59} - 3 \beta_{3} q^{61} + ( - \beta_{3} + \beta_1) q^{63} + 12 q^{65} + ( - 9 \beta_{2} + 9) q^{67} + 6 \beta_1 q^{69} + ( - 3 \beta_{3} - 3 \beta_1) q^{71} + 12 \beta_{2} q^{73} + (\beta_{2} + 1) q^{75} - 6 \beta_{3} q^{77} + ( - 2 \beta_{3} + 2 \beta_1) q^{79} + 5 q^{81} + (3 \beta_{2} - 3) q^{83} + (\beta_{3} + \beta_1) q^{87} - 12 \beta_{2} q^{89} + (12 \beta_{2} + 12) q^{91} + 4 \beta_{3} q^{93} + (3 \beta_{3} - 3 \beta_1) q^{95} - 8 q^{97} + (3 \beta_{2} - 3) q^{99}+O(q^{100})$$ q + (b2 - 1) * q^3 - b1 * q^5 + (-b3 - b1) * q^7 + b2 * q^9 + (3*b2 + 3) * q^11 + 3*b3 * q^13 + (-b3 + b1) * q^15 + (-3*b2 + 3) * q^19 + 2*b1 * q^21 + (-3*b3 - 3*b1) * q^23 - b2 * q^25 + (-4*b2 - 4) * q^27 - b3 * q^29 + (-2*b3 + 2*b1) * q^31 - 6 * q^33 + (4*b2 - 4) * q^35 - 3*b1 * q^37 + (-3*b3 - 3*b1) * q^39 - 6*b2 * q^41 + (-3*b2 - 3) * q^43 - b3 * q^45 - q^49 - b1 * q^53 + (-3*b3 - 3*b1) * q^55 + 6*b2 * q^57 + (-b2 - 1) * q^59 - 3*b3 * q^61 + (-b3 + b1) * q^63 + 12 * q^65 + (-9*b2 + 9) * q^67 + 6*b1 * q^69 + (-3*b3 - 3*b1) * q^71 + 12*b2 * q^73 + (b2 + 1) * q^75 - 6*b3 * q^77 + (-2*b3 + 2*b1) * q^79 + 5 * q^81 + (3*b2 - 3) * q^83 + (b3 + b1) * q^87 - 12*b2 * q^89 + (12*b2 + 12) * q^91 + 4*b3 * q^93 + (3*b3 - 3*b1) * q^95 - 8 * q^97 + (3*b2 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3}+O(q^{10})$$ 4 * q - 4 * q^3 $$4 q - 4 q^{3} + 12 q^{11} + 12 q^{19} - 16 q^{27} - 24 q^{33} - 16 q^{35} - 12 q^{43} - 4 q^{49} - 4 q^{59} + 48 q^{65} + 36 q^{67} + 4 q^{75} + 20 q^{81} - 12 q^{83} + 48 q^{91} - 32 q^{97} - 12 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 + 12 * q^11 + 12 * q^19 - 16 * q^27 - 24 * q^33 - 16 * q^35 - 12 * q^43 - 4 * q^49 - 4 * q^59 + 48 * q^65 + 36 * q^67 + 4 * q^75 + 20 * q^81 - 12 * q^83 + 48 * q^91 - 32 * q^97 - 12 * 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 −1.00000 1.00000i 0 −1.41421 + 1.41421i 0 2.82843i 0 1.00000i 0
257.2 0 −1.00000 1.00000i 0 1.41421 1.41421i 0 2.82843i 0 1.00000i 0
769.1 0 −1.00000 + 1.00000i 0 −1.41421 1.41421i 0 2.82843i 0 1.00000i 0
769.2 0 −1.00000 + 1.00000i 0 1.41421 + 1.41421i 0 2.82843i 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.d odd 2 1 inner
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.h 4
4.b odd 2 1 1024.2.e.n 4
8.b even 2 1 1024.2.e.n 4
8.d odd 2 1 inner 1024.2.e.h 4
16.e even 4 1 inner 1024.2.e.h 4
16.e even 4 1 1024.2.e.n 4
16.f odd 4 1 inner 1024.2.e.h 4
16.f odd 4 1 1024.2.e.n 4
32.g even 8 1 512.2.a.b 2
32.g even 8 1 512.2.a.e yes 2
32.g even 8 2 512.2.b.e 4
32.h odd 8 1 512.2.a.b 2
32.h odd 8 1 512.2.a.e yes 2
32.h odd 8 2 512.2.b.e 4
96.o even 8 1 4608.2.a.c 2
96.o even 8 1 4608.2.a.p 2
96.o even 8 2 4608.2.d.j 4
96.p odd 8 1 4608.2.a.c 2
96.p odd 8 1 4608.2.a.p 2
96.p odd 8 2 4608.2.d.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.b 2 32.g even 8 1
512.2.a.b 2 32.h odd 8 1
512.2.a.e yes 2 32.g even 8 1
512.2.a.e yes 2 32.h odd 8 1
512.2.b.e 4 32.g even 8 2
512.2.b.e 4 32.h odd 8 2
1024.2.e.h 4 1.a even 1 1 trivial
1024.2.e.h 4 8.d odd 2 1 inner
1024.2.e.h 4 16.e even 4 1 inner
1024.2.e.h 4 16.f odd 4 1 inner
1024.2.e.n 4 4.b odd 2 1
1024.2.e.n 4 8.b even 2 1
1024.2.e.n 4 16.e even 4 1
1024.2.e.n 4 16.f odd 4 1
4608.2.a.c 2 96.o even 8 1
4608.2.a.c 2 96.p odd 8 1
4608.2.a.p 2 96.o even 8 1
4608.2.a.p 2 96.p odd 8 1
4608.2.d.j 4 96.o even 8 2
4608.2.d.j 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 $$S_{2}^{\mathrm{new}}(1024, [\chi])$$:

 $$T_{3}^{2} + 2T_{3} + 2$$ T3^2 + 2*T3 + 2 $$T_{5}^{4} + 16$$ T5^4 + 16 $$T_{47}$$ T47

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 2 T + 2)^{2}$$
$5$ $$T^{4} + 16$$
$7$ $$(T^{2} + 8)^{2}$$
$11$ $$(T^{2} - 6 T + 18)^{2}$$
$13$ $$T^{4} + 1296$$
$17$ $$T^{4}$$
$19$ $$(T^{2} - 6 T + 18)^{2}$$
$23$ $$(T^{2} + 72)^{2}$$
$29$ $$T^{4} + 16$$
$31$ $$(T^{2} - 32)^{2}$$
$37$ $$T^{4} + 1296$$
$41$ $$(T^{2} + 36)^{2}$$
$43$ $$(T^{2} + 6 T + 18)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 16$$
$59$ $$(T^{2} + 2 T + 2)^{2}$$
$61$ $$T^{4} + 1296$$
$67$ $$(T^{2} - 18 T + 162)^{2}$$
$71$ $$(T^{2} + 72)^{2}$$
$73$ $$(T^{2} + 144)^{2}$$
$79$ $$(T^{2} - 32)^{2}$$
$83$ $$(T^{2} + 6 T + 18)^{2}$$
$89$ $$(T^{2} + 144)^{2}$$
$97$ $$(T + 8)^{4}$$