# Properties

 Label 1600.2.o.j Level $1600$ Weight $2$ Character orbit 1600.o Analytic conductor $12.776$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.o (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{24}^{3} q^{3} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7} + \zeta_{24}^{6} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{24}^{3} q^{3} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{7} + \zeta_{24}^{6} q^{9} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{11} + ( 4 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{13} + ( 4 \zeta_{24}^{3} - 8 \zeta_{24}^{7} ) q^{17} + ( -2 + 4 \zeta_{24}^{4} ) q^{19} + ( -4 + 8 \zeta_{24}^{4} ) q^{21} + ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{23} + ( 4 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{27} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{29} -8 \zeta_{24}^{6} q^{31} + ( 4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{33} -8 \zeta_{24}^{3} q^{37} + 8 q^{39} + 6 q^{41} -2 \zeta_{24}^{3} q^{43} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{47} + 5 \zeta_{24}^{6} q^{49} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{51} + ( -12 \zeta_{24} + 12 \zeta_{24}^{5} ) q^{53} + ( -4 \zeta_{24}^{3} + 8 \zeta_{24}^{7} ) q^{57} + ( -6 + 12 \zeta_{24}^{4} ) q^{59} + ( -8 + 16 \zeta_{24}^{4} ) q^{61} + ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{63} + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{67} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{69} + ( -4 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{73} + 12 \zeta_{24}^{3} q^{77} -8 q^{79} + 11 q^{81} -6 \zeta_{24}^{3} q^{83} + ( -8 \zeta_{24} - 8 \zeta_{24}^{5} ) q^{87} + 6 \zeta_{24}^{6} q^{89} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{91} + ( 16 \zeta_{24} - 16 \zeta_{24}^{5} ) q^{93} + ( -4 \zeta_{24}^{3} + 8 \zeta_{24}^{7} ) q^{97} + ( -2 + 4 \zeta_{24}^{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 64q^{39} + 48q^{41} - 64q^{79} + 88q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$-\zeta_{24}^{3}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
543.1
 −0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 − 0.258819i
0 −1.41421 1.41421i 0 0 0 −2.44949 2.44949i 0 1.00000i 0
543.2 0 −1.41421 1.41421i 0 0 0 2.44949 + 2.44949i 0 1.00000i 0
543.3 0 1.41421 + 1.41421i 0 0 0 −2.44949 2.44949i 0 1.00000i 0
543.4 0 1.41421 + 1.41421i 0 0 0 2.44949 + 2.44949i 0 1.00000i 0
607.1 0 −1.41421 + 1.41421i 0 0 0 −2.44949 + 2.44949i 0 1.00000i 0
607.2 0 −1.41421 + 1.41421i 0 0 0 2.44949 2.44949i 0 1.00000i 0
607.3 0 1.41421 1.41421i 0 0 0 −2.44949 + 2.44949i 0 1.00000i 0
607.4 0 1.41421 1.41421i 0 0 0 2.44949 2.44949i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 607.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
20.e even 4 2 inner
40.f even 2 1 inner
40.k even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.o.j yes 8
4.b odd 2 1 1600.2.o.g 8
5.b even 2 1 inner 1600.2.o.j yes 8
5.c odd 4 2 1600.2.o.g 8
8.b even 2 1 inner 1600.2.o.j yes 8
8.d odd 2 1 1600.2.o.g 8
20.d odd 2 1 1600.2.o.g 8
20.e even 4 2 inner 1600.2.o.j yes 8
40.e odd 2 1 1600.2.o.g 8
40.f even 2 1 inner 1600.2.o.j yes 8
40.i odd 4 2 1600.2.o.g 8
40.k even 4 2 inner 1600.2.o.j yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.o.g 8 4.b odd 2 1
1600.2.o.g 8 5.c odd 4 2
1600.2.o.g 8 8.d odd 2 1
1600.2.o.g 8 20.d odd 2 1
1600.2.o.g 8 40.e odd 2 1
1600.2.o.g 8 40.i odd 4 2
1600.2.o.j yes 8 1.a even 1 1 trivial
1600.2.o.j yes 8 5.b even 2 1 inner
1600.2.o.j yes 8 8.b even 2 1 inner
1600.2.o.j yes 8 20.e even 4 2 inner
1600.2.o.j yes 8 40.f even 2 1 inner
1600.2.o.j yes 8 40.k even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1600, [\chi])$$:

 $$T_{3}^{4} + 16$$ $$T_{7}^{4} + 144$$ $$T_{79} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 16 + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 144 + T^{4} )^{2}$$
$11$ $$( -12 + T^{2} )^{4}$$
$13$ $$( 256 + T^{4} )^{2}$$
$17$ $$( 2304 + T^{4} )^{2}$$
$19$ $$( 12 + T^{2} )^{4}$$
$23$ $$( 144 + T^{4} )^{2}$$
$29$ $$( -48 + T^{2} )^{4}$$
$31$ $$( 64 + T^{2} )^{4}$$
$37$ $$( 4096 + T^{4} )^{2}$$
$41$ $$( -6 + T )^{8}$$
$43$ $$( 16 + T^{4} )^{2}$$
$47$ $$( 144 + T^{4} )^{2}$$
$53$ $$( 20736 + T^{4} )^{2}$$
$59$ $$( 108 + T^{2} )^{4}$$
$61$ $$( 192 + T^{2} )^{4}$$
$67$ $$( 16 + T^{4} )^{2}$$
$71$ $$T^{8}$$
$73$ $$( 2304 + T^{4} )^{2}$$
$79$ $$( 8 + T )^{8}$$
$83$ $$( 1296 + T^{4} )^{2}$$
$89$ $$( 36 + T^{2} )^{4}$$
$97$ $$( 2304 + T^{4} )^{2}$$