# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 320) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{7} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{7} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{9} + ( 2 - 4 \zeta_{12}^{2} ) q^{11} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{13} + ( 2 - 4 \zeta_{12}^{2} ) q^{17} + 2 \zeta_{12}^{3} q^{19} + ( -4 + 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{21} + ( 3 - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{23} -4 q^{27} + ( 6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{31} + ( -2 + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{33} + 6 q^{37} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{39} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{41} + ( -5 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{43} + ( 3 - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{47} + ( -5 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{49} + ( -2 + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{51} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{53} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{57} -6 \zeta_{12}^{3} q^{59} + ( -4 + 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{61} + ( 7 - 14 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{63} + ( 5 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{67} -6 \zeta_{12}^{3} q^{69} + ( -6 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{71} + ( 6 - 12 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{73} + ( -6 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} -12 q^{79} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{81} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{83} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} + ( -6 + 12 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{91} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{93} + ( -6 + 12 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{97} + ( 2 - 4 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} + 4q^{9} - 16q^{27} + 24q^{31} + 24q^{37} + 24q^{39} + 24q^{41} - 20q^{43} - 20q^{49} + 20q^{67} - 24q^{71} - 24q^{77} - 48q^{79} + 4q^{81} - 12q^{83} + 24q^{89} + O(q^{100})$$

## Character values

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

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
0 −2.73205 0 0 0 4.73205i 0 4.46410 0
1249.2 0 −2.73205 0 0 0 4.73205i 0 4.46410 0
1249.3 0 0.732051 0 0 0 1.26795i 0 −2.46410 0
1249.4 0 0.732051 0 0 0 1.26795i 0 −2.46410 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
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.f.e 4
4.b odd 2 1 1600.2.f.h 4
5.b even 2 1 1600.2.f.i 4
5.c odd 4 1 320.2.d.a 4
5.c odd 4 1 1600.2.d.h 4
8.b even 2 1 1600.2.f.i 4
8.d odd 2 1 1600.2.f.d 4
15.e even 4 1 2880.2.k.e 4
20.d odd 2 1 1600.2.f.d 4
20.e even 4 1 320.2.d.b yes 4
20.e even 4 1 1600.2.d.b 4
40.e odd 2 1 1600.2.f.h 4
40.f even 2 1 inner 1600.2.f.e 4
40.i odd 4 1 320.2.d.a 4
40.i odd 4 1 1600.2.d.h 4
40.k even 4 1 320.2.d.b yes 4
40.k even 4 1 1600.2.d.b 4
60.l odd 4 1 2880.2.k.l 4
80.i odd 4 1 1280.2.a.p 2
80.i odd 4 1 6400.2.a.cd 2
80.j even 4 1 1280.2.a.m 2
80.j even 4 1 6400.2.a.ck 2
80.s even 4 1 1280.2.a.c 2
80.s even 4 1 6400.2.a.bf 2
80.t odd 4 1 1280.2.a.b 2
80.t odd 4 1 6400.2.a.y 2
120.q odd 4 1 2880.2.k.l 4
120.w even 4 1 2880.2.k.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.d.a 4 5.c odd 4 1
320.2.d.a 4 40.i odd 4 1
320.2.d.b yes 4 20.e even 4 1
320.2.d.b yes 4 40.k even 4 1
1280.2.a.b 2 80.t odd 4 1
1280.2.a.c 2 80.s even 4 1
1280.2.a.m 2 80.j even 4 1
1280.2.a.p 2 80.i odd 4 1
1600.2.d.b 4 20.e even 4 1
1600.2.d.b 4 40.k even 4 1
1600.2.d.h 4 5.c odd 4 1
1600.2.d.h 4 40.i odd 4 1
1600.2.f.d 4 8.d odd 2 1
1600.2.f.d 4 20.d odd 2 1
1600.2.f.e 4 1.a even 1 1 trivial
1600.2.f.e 4 40.f even 2 1 inner
1600.2.f.h 4 4.b odd 2 1
1600.2.f.h 4 40.e odd 2 1
1600.2.f.i 4 5.b even 2 1
1600.2.f.i 4 8.b even 2 1
2880.2.k.e 4 15.e even 4 1
2880.2.k.e 4 120.w even 4 1
2880.2.k.l 4 60.l odd 4 1
2880.2.k.l 4 120.q odd 4 1
6400.2.a.y 2 80.t odd 4 1
6400.2.a.bf 2 80.s even 4 1
6400.2.a.cd 2 80.i odd 4 1
6400.2.a.ck 2 80.j even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 2 T_{3} - 2$$ $$T_{31}^{2} - 12 T_{31} + 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -2 + 2 T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$36 + 24 T^{2} + T^{4}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$( -12 + T^{2} )^{2}$$
$17$ $$( 12 + T^{2} )^{2}$$
$19$ $$( 4 + T^{2} )^{2}$$
$23$ $$324 + 72 T^{2} + T^{4}$$
$29$ $$T^{4}$$
$31$ $$( 24 - 12 T + T^{2} )^{2}$$
$37$ $$( -6 + T )^{4}$$
$41$ $$( 24 - 12 T + T^{2} )^{2}$$
$43$ $$( -2 + 10 T + T^{2} )^{2}$$
$47$ $$324 + 72 T^{2} + T^{4}$$
$53$ $$( -108 + T^{2} )^{2}$$
$59$ $$( 36 + T^{2} )^{2}$$
$61$ $$144 + 168 T^{2} + T^{4}$$
$67$ $$( -2 - 10 T + T^{2} )^{2}$$
$71$ $$( -72 + 12 T + T^{2} )^{2}$$
$73$ $$8464 + 248 T^{2} + T^{4}$$
$79$ $$( 12 + T )^{4}$$
$83$ $$( 6 + 6 T + T^{2} )^{2}$$
$89$ $$( -12 - 12 T + T^{2} )^{2}$$
$97$ $$8464 + 248 T^{2} + T^{4}$$