# Properties

 Label 1600.2.o.l Level $1600$ Weight $2$ Character orbit 1600.o Analytic conductor $12.776$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 320) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( \beta_{1} - \beta_{4} ) q^{7} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( \beta_{1} - \beta_{4} ) q^{7} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{9} + ( -\beta_{6} + \beta_{7} ) q^{11} + ( \beta_{3} + \beta_{7} ) q^{13} + ( -3 + 3 \beta_{1} ) q^{17} + ( -\beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{19} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{6} - \beta_{7} ) q^{21} + ( -2 + 2 \beta_{1} - \beta_{5} ) q^{23} + ( 4 \beta_{3} + 2 \beta_{7} ) q^{27} + ( -\beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{29} + ( 3 \beta_{1} + \beta_{4} - \beta_{5} ) q^{31} + ( 2 + 2 \beta_{1} ) q^{33} + ( 3 \beta_{2} - 3 \beta_{6} ) q^{37} + ( -4 + \beta_{1} - \beta_{4} - \beta_{5} ) q^{39} + ( -2 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{41} + ( -\beta_{2} + 4 \beta_{6} ) q^{43} + ( 2 + \beta_{1} + \beta_{4} ) q^{47} + ( 4 \beta_{1} - \beta_{4} + \beta_{5} ) q^{49} + ( 3 \beta_{2} + 3 \beta_{3} ) q^{51} + ( -\beta_{3} + \beta_{7} ) q^{53} + ( -4 + 4 \beta_{1} - 2 \beta_{5} ) q^{57} + ( -\beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{59} + ( -\beta_{2} + \beta_{3} - 2 \beta_{6} - 2 \beta_{7} ) q^{61} + ( 10 - 10 \beta_{1} + \beta_{5} ) q^{63} + ( \beta_{3} + 4 \beta_{7} ) q^{67} + ( 4 \beta_{2} + 4 \beta_{3} - \beta_{6} + \beta_{7} ) q^{69} + ( 9 \beta_{1} - \beta_{4} + \beta_{5} ) q^{71} + ( 5 + 3 \beta_{1} + 2 \beta_{4} ) q^{73} + ( 2 \beta_{2} - 6 \beta_{6} ) q^{77} + 12 q^{79} + ( -11 + \beta_{1} - \beta_{4} - \beta_{5} ) q^{81} + ( -\beta_{2} - 2 \beta_{6} ) q^{83} + ( 8 + 6 \beta_{1} + 2 \beta_{4} ) q^{87} + ( -6 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{89} + ( -\beta_{2} - \beta_{3} - 2 \beta_{6} + 2 \beta_{7} ) q^{91} + ( 8 \beta_{3} + 2 \beta_{7} ) q^{93} + ( 5 - 5 \beta_{1} + 2 \beta_{5} ) q^{97} + ( -2 \beta_{2} + 2 \beta_{3} + 3 \beta_{6} + 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{7} + O(q^{10})$$ $$8q + 4q^{7} - 24q^{17} - 12q^{23} + 16q^{33} - 24q^{39} - 24q^{41} + 12q^{47} - 24q^{57} + 76q^{63} + 32q^{73} + 96q^{79} - 80q^{81} + 56q^{87} + 32q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{7} + 5 \nu^{5} - 5 \nu^{3} + 16 \nu$$$$)/40$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{7} + 2 \nu^{6} - 15 \nu^{5} - 10 \nu^{4} - 25 \nu^{3} - 30 \nu^{2} - 20 \nu - 16$$$$)/40$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{7} - 2 \nu^{6} - 15 \nu^{5} + 10 \nu^{4} - 25 \nu^{3} + 30 \nu^{2} - 20 \nu + 16$$$$)/40$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 5 \nu^{6} - 5 \nu^{5} - 15 \nu^{4} - 15 \nu^{3} - 5 \nu^{2} - 42 \nu - 40$$$$)/20$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + 3 \nu^{5} - 6 \nu^{4} + 5 \nu^{3} - 2 \nu^{2} + 20 \nu - 16$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{7} + 7 \nu^{6} - 5 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} + 15 \nu^{2} - 30 \nu + 44$$$$)/20$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 7 \nu^{6} + 5 \nu^{5} + 5 \nu^{4} + 15 \nu^{3} + 15 \nu^{2} + 30 \nu + 44$$$$)/20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_{1} - 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} - 10 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_{1} - 4$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{7} + 5 \beta_{6} + \beta_{5} - \beta_{4} - 6 \beta_{3} - 6 \beta_{2} + 11 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$5 \beta_{7} + 5 \beta_{6} - 5 \beta_{3} + 5 \beta_{2} - 18$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$10 \beta_{7} - 10 \beta_{6} - 7 \beta_{5} + 7 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 13 \beta_{1}$$$$)/4$$

## Character values

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

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$-\beta_{1}$$ $$-1$$ $$-1$$

## 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
 1.09445 − 0.895644i 0.228425 + 1.39564i −0.228425 + 1.39564i −1.09445 − 0.895644i 1.09445 + 0.895644i 0.228425 − 1.39564i −0.228425 − 1.39564i −1.09445 + 0.895644i
0 −2.18890 2.18890i 0 0 0 −1.79129 1.79129i 0 6.58258i 0
543.2 0 −0.456850 0.456850i 0 0 0 2.79129 + 2.79129i 0 2.58258i 0
543.3 0 0.456850 + 0.456850i 0 0 0 2.79129 + 2.79129i 0 2.58258i 0
543.4 0 2.18890 + 2.18890i 0 0 0 −1.79129 1.79129i 0 6.58258i 0
607.1 0 −2.18890 + 2.18890i 0 0 0 −1.79129 + 1.79129i 0 6.58258i 0
607.2 0 −0.456850 + 0.456850i 0 0 0 2.79129 2.79129i 0 2.58258i 0
607.3 0 0.456850 0.456850i 0 0 0 2.79129 2.79129i 0 2.58258i 0
607.4 0 2.18890 2.18890i 0 0 0 −1.79129 + 1.79129i 0 6.58258i 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
8.b even 2 1 inner
20.e even 4 1 inner
40.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.o.l 8
4.b odd 2 1 1600.2.o.e 8
5.b even 2 1 320.2.o.e 8
5.c odd 4 1 320.2.o.f yes 8
5.c odd 4 1 1600.2.o.e 8
8.b even 2 1 inner 1600.2.o.l 8
8.d odd 2 1 1600.2.o.e 8
20.d odd 2 1 320.2.o.f yes 8
20.e even 4 1 320.2.o.e 8
20.e even 4 1 inner 1600.2.o.l 8
40.e odd 2 1 320.2.o.f yes 8
40.f even 2 1 320.2.o.e 8
40.i odd 4 1 320.2.o.f yes 8
40.i odd 4 1 1600.2.o.e 8
40.k even 4 1 320.2.o.e 8
40.k even 4 1 inner 1600.2.o.l 8
80.i odd 4 1 1280.2.n.n 8
80.j even 4 1 1280.2.n.p 8
80.k odd 4 2 1280.2.n.n 8
80.q even 4 2 1280.2.n.p 8
80.s even 4 1 1280.2.n.p 8
80.t odd 4 1 1280.2.n.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.o.e 8 5.b even 2 1
320.2.o.e 8 20.e even 4 1
320.2.o.e 8 40.f even 2 1
320.2.o.e 8 40.k even 4 1
320.2.o.f yes 8 5.c odd 4 1
320.2.o.f yes 8 20.d odd 2 1
320.2.o.f yes 8 40.e odd 2 1
320.2.o.f yes 8 40.i odd 4 1
1280.2.n.n 8 80.i odd 4 1
1280.2.n.n 8 80.k odd 4 2
1280.2.n.n 8 80.t odd 4 1
1280.2.n.p 8 80.j even 4 1
1280.2.n.p 8 80.q even 4 2
1280.2.n.p 8 80.s even 4 1
1600.2.o.e 8 4.b odd 2 1
1600.2.o.e 8 5.c odd 4 1
1600.2.o.e 8 8.d odd 2 1
1600.2.o.e 8 40.i odd 4 1
1600.2.o.l 8 1.a even 1 1 trivial
1600.2.o.l 8 8.b even 2 1 inner
1600.2.o.l 8 20.e even 4 1 inner
1600.2.o.l 8 40.k even 4 1 inner

## Hecke kernels

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

 $$T_{3}^{8} + 92 T_{3}^{4} + 16$$ $$T_{7}^{4} - 2 T_{7}^{3} + 2 T_{7}^{2} + 20 T_{7} + 100$$ $$T_{79} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$16 + 92 T^{4} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 100 + 20 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$11$ $$( 16 - 20 T^{2} + T^{4} )^{2}$$
$13$ $$( 36 + T^{4} )^{2}$$
$17$ $$( 18 + 6 T + T^{2} )^{4}$$
$19$ $$( 12 + T^{2} )^{4}$$
$23$ $$( 36 - 36 T + 18 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$29$ $$( -28 + T^{2} )^{4}$$
$31$ $$( 144 + 60 T^{2} + T^{4} )^{2}$$
$37$ $$( 2916 + T^{4} )^{2}$$
$41$ $$( -12 + 6 T + T^{2} )^{4}$$
$43$ $$1296 + 18972 T^{4} + T^{8}$$
$47$ $$( 36 + 36 T + 18 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$53$ $$( 196 + T^{4} )^{2}$$
$59$ $$( 28 + T^{2} )^{4}$$
$61$ $$( 3600 + 132 T^{2} + T^{4} )^{2}$$
$67$ $$1296 + 18972 T^{4} + T^{8}$$
$71$ $$( 3600 + 204 T^{2} + T^{4} )^{2}$$
$73$ $$( 100 + 160 T + 128 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$79$ $$( -12 + T )^{8}$$
$83$ $$810000 + 2556 T^{4} + T^{8}$$
$89$ $$( 2304 + 240 T^{2} + T^{4} )^{2}$$
$97$ $$( 100 + 160 T + 128 T^{2} - 16 T^{3} + T^{4} )^{2}$$