# Properties

 Label 1600.2.n.u Level $1600$ Weight $2$ Character orbit 1600.n 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.n (of order $$4$$, degree $$2$$, not 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 400) 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24} - \zeta_{24}^{5} ) q^{3} -4 \zeta_{24}^{3} q^{7} + 2 \zeta_{24}^{6} q^{9} +O(q^{10})$$ $$q + ( \zeta_{24} - \zeta_{24}^{5} ) q^{3} -4 \zeta_{24}^{3} q^{7} + 2 \zeta_{24}^{6} q^{9} + ( -3 + 6 \zeta_{24}^{4} ) q^{11} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{13} + ( 3 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{17} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{19} -4 q^{21} + ( -6 \zeta_{24} + 6 \zeta_{24}^{5} ) q^{23} + 5 \zeta_{24}^{3} q^{27} + 6 \zeta_{24}^{6} q^{29} + ( 2 - 4 \zeta_{24}^{4} ) q^{31} + ( 3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{33} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{39} -3 q^{41} + ( -4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{43} + 6 \zeta_{24}^{3} q^{47} + 9 \zeta_{24}^{6} q^{49} + ( 3 - 6 \zeta_{24}^{4} ) q^{51} + ( 6 \zeta_{24} + 6 \zeta_{24}^{5} ) q^{53} + ( -\zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{57} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{59} + 4 q^{61} + ( 8 \zeta_{24} - 8 \zeta_{24}^{5} ) q^{63} -7 \zeta_{24}^{3} q^{67} + 6 \zeta_{24}^{6} q^{69} + ( -6 + 12 \zeta_{24}^{4} ) q^{71} + ( 9 \zeta_{24} + 9 \zeta_{24}^{5} ) q^{73} + ( 12 \zeta_{24}^{3} - 24 \zeta_{24}^{7} ) q^{77} - q^{81} + ( 3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{83} + 6 \zeta_{24}^{3} q^{87} + 3 \zeta_{24}^{6} q^{89} + ( 8 - 16 \zeta_{24}^{4} ) q^{91} + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{93} + ( 4 \zeta_{24}^{3} - 8 \zeta_{24}^{7} ) q^{97} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 32q^{21} - 24q^{41} + 32q^{61} - 8q^{81} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1343.1
 0.258819 + 0.965926i −0.965926 − 0.258819i −0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 + 0.258819i 0.258819 − 0.965926i 0.965926 − 0.258819i −0.258819 + 0.965926i
0 −0.707107 + 0.707107i 0 0 0 2.82843 + 2.82843i 0 2.00000i 0
1343.2 0 −0.707107 + 0.707107i 0 0 0 2.82843 + 2.82843i 0 2.00000i 0
1343.3 0 0.707107 0.707107i 0 0 0 −2.82843 2.82843i 0 2.00000i 0
1343.4 0 0.707107 0.707107i 0 0 0 −2.82843 2.82843i 0 2.00000i 0
1407.1 0 −0.707107 0.707107i 0 0 0 2.82843 2.82843i 0 2.00000i 0
1407.2 0 −0.707107 0.707107i 0 0 0 2.82843 2.82843i 0 2.00000i 0
1407.3 0 0.707107 + 0.707107i 0 0 0 −2.82843 + 2.82843i 0 2.00000i 0
1407.4 0 0.707107 + 0.707107i 0 0 0 −2.82843 + 2.82843i 0 2.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1407.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e 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.n.u 8
4.b odd 2 1 inner 1600.2.n.u 8
5.b even 2 1 inner 1600.2.n.u 8
5.c odd 4 2 inner 1600.2.n.u 8
8.b even 2 1 400.2.n.d 8
8.d odd 2 1 400.2.n.d 8
20.d odd 2 1 inner 1600.2.n.u 8
20.e even 4 2 inner 1600.2.n.u 8
24.f even 2 1 3600.2.x.m 8
24.h odd 2 1 3600.2.x.m 8
40.e odd 2 1 400.2.n.d 8
40.f even 2 1 400.2.n.d 8
40.i odd 4 2 400.2.n.d 8
40.k even 4 2 400.2.n.d 8
120.i odd 2 1 3600.2.x.m 8
120.m even 2 1 3600.2.x.m 8
120.q odd 4 2 3600.2.x.m 8
120.w even 4 2 3600.2.x.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.n.d 8 8.b even 2 1
400.2.n.d 8 8.d odd 2 1
400.2.n.d 8 40.e odd 2 1
400.2.n.d 8 40.f even 2 1
400.2.n.d 8 40.i odd 4 2
400.2.n.d 8 40.k even 4 2
1600.2.n.u 8 1.a even 1 1 trivial
1600.2.n.u 8 4.b odd 2 1 inner
1600.2.n.u 8 5.b even 2 1 inner
1600.2.n.u 8 5.c odd 4 2 inner
1600.2.n.u 8 20.d odd 2 1 inner
1600.2.n.u 8 20.e even 4 2 inner
3600.2.x.m 8 24.f even 2 1
3600.2.x.m 8 24.h odd 2 1
3600.2.x.m 8 120.i odd 2 1
3600.2.x.m 8 120.m even 2 1
3600.2.x.m 8 120.q odd 4 2
3600.2.x.m 8 120.w even 4 2

## Hecke kernels

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

 $$T_{3}^{4} + 1$$ $$T_{7}^{4} + 256$$ $$T_{11}^{2} + 27$$ $$T_{13}^{4} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 256 + T^{4} )^{2}$$
$11$ $$( 27 + T^{2} )^{4}$$
$13$ $$( 144 + T^{4} )^{2}$$
$17$ $$( 729 + T^{4} )^{2}$$
$19$ $$( -3 + T^{2} )^{4}$$
$23$ $$( 1296 + T^{4} )^{2}$$
$29$ $$( 36 + T^{2} )^{4}$$
$31$ $$( 12 + T^{2} )^{4}$$
$37$ $$T^{8}$$
$41$ $$( 3 + T )^{8}$$
$43$ $$( 256 + T^{4} )^{2}$$
$47$ $$( 1296 + T^{4} )^{2}$$
$53$ $$( 11664 + T^{4} )^{2}$$
$59$ $$( -108 + T^{2} )^{4}$$
$61$ $$( -4 + T )^{8}$$
$67$ $$( 2401 + T^{4} )^{2}$$
$71$ $$( 108 + T^{2} )^{4}$$
$73$ $$( 59049 + T^{4} )^{2}$$
$79$ $$T^{8}$$
$83$ $$( 81 + T^{4} )^{2}$$
$89$ $$( 9 + T^{2} )^{4}$$
$97$ $$( 2304 + T^{4} )^{2}$$