# Properties

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

# 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: $$2^{4}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( 2 - 4 \zeta_{12}^{2} ) q^{7} -2 q^{9} +O(q^{10})$$ $$q - q^{3} + ( 2 - 4 \zeta_{12}^{2} ) q^{7} -2 q^{9} + 3 \zeta_{12}^{3} q^{11} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{13} + 3 \zeta_{12}^{3} q^{17} -\zeta_{12}^{3} q^{19} + ( -2 + 4 \zeta_{12}^{2} ) q^{21} + 5 q^{27} + ( -6 + 12 \zeta_{12}^{2} ) q^{29} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{31} -3 \zeta_{12}^{3} q^{33} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{37} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{39} -9 q^{41} + 4 q^{43} + ( 6 - 12 \zeta_{12}^{2} ) q^{47} -5 q^{49} -3 \zeta_{12}^{3} q^{51} + \zeta_{12}^{3} q^{57} + 12 \zeta_{12}^{3} q^{59} + ( -2 + 4 \zeta_{12}^{2} ) q^{61} + ( -4 + 8 \zeta_{12}^{2} ) q^{63} + 11 q^{67} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{71} -7 \zeta_{12}^{3} q^{73} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{79} + q^{81} -15 q^{83} + ( 6 - 12 \zeta_{12}^{2} ) q^{87} + 3 q^{89} + 12 \zeta_{12}^{3} q^{91} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{93} + 14 \zeta_{12}^{3} q^{97} -6 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 8q^{9} + O(q^{10})$$ $$4q - 4q^{3} - 8q^{9} + 20q^{27} - 36q^{41} + 16q^{43} - 20q^{49} + 44q^{67} + 4q^{81} - 60q^{83} + 12q^{89} + O(q^{100})$$

## 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 −1.00000 0 0 0 3.46410i 0 −2.00000 0
1249.2 0 −1.00000 0 0 0 3.46410i 0 −2.00000 0
1249.3 0 −1.00000 0 0 0 3.46410i 0 −2.00000 0
1249.4 0 −1.00000 0 0 0 3.46410i 0 −2.00000 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
20.d odd 2 1 inner
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.c 4
4.b odd 2 1 1600.2.f.g 4
5.b even 2 1 1600.2.f.g 4
5.c odd 4 1 1600.2.d.e 4
5.c odd 4 1 1600.2.d.f yes 4
8.b even 2 1 1600.2.f.g 4
8.d odd 2 1 inner 1600.2.f.c 4
20.d odd 2 1 inner 1600.2.f.c 4
20.e even 4 1 1600.2.d.e 4
20.e even 4 1 1600.2.d.f yes 4
40.e odd 2 1 1600.2.f.g 4
40.f even 2 1 inner 1600.2.f.c 4
40.i odd 4 1 1600.2.d.e 4
40.i odd 4 1 1600.2.d.f yes 4
40.k even 4 1 1600.2.d.e 4
40.k even 4 1 1600.2.d.f yes 4
80.i odd 4 1 6400.2.a.cf 2
80.i odd 4 1 6400.2.a.cg 2
80.j even 4 1 6400.2.a.cf 2
80.j even 4 1 6400.2.a.cg 2
80.s even 4 1 6400.2.a.ba 2
80.s even 4 1 6400.2.a.bb 2
80.t odd 4 1 6400.2.a.ba 2
80.t odd 4 1 6400.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.d.e 4 5.c odd 4 1
1600.2.d.e 4 20.e even 4 1
1600.2.d.e 4 40.i odd 4 1
1600.2.d.e 4 40.k even 4 1
1600.2.d.f yes 4 5.c odd 4 1
1600.2.d.f yes 4 20.e even 4 1
1600.2.d.f yes 4 40.i odd 4 1
1600.2.d.f yes 4 40.k even 4 1
1600.2.f.c 4 1.a even 1 1 trivial
1600.2.f.c 4 8.d odd 2 1 inner
1600.2.f.c 4 20.d odd 2 1 inner
1600.2.f.c 4 40.f even 2 1 inner
1600.2.f.g 4 4.b odd 2 1
1600.2.f.g 4 5.b even 2 1
1600.2.f.g 4 8.b even 2 1
1600.2.f.g 4 40.e odd 2 1
6400.2.a.ba 2 80.s even 4 1
6400.2.a.ba 2 80.t odd 4 1
6400.2.a.bb 2 80.s even 4 1
6400.2.a.bb 2 80.t odd 4 1
6400.2.a.cf 2 80.i odd 4 1
6400.2.a.cf 2 80.j even 4 1
6400.2.a.cg 2 80.i odd 4 1
6400.2.a.cg 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 $$S_{2}^{\mathrm{new}}(1600, [\chi])$$:

 $$T_{3} + 1$$ $$T_{31}^{2} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$( 12 + T^{2} )^{2}$$
$11$ $$( 9 + T^{2} )^{2}$$
$13$ $$( -12 + T^{2} )^{2}$$
$17$ $$( 9 + T^{2} )^{2}$$
$19$ $$( 1 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$( 108 + T^{2} )^{2}$$
$31$ $$( -48 + T^{2} )^{2}$$
$37$ $$( -108 + T^{2} )^{2}$$
$41$ $$( 9 + T )^{4}$$
$43$ $$( -4 + T )^{4}$$
$47$ $$( 108 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$( 144 + T^{2} )^{2}$$
$61$ $$( 12 + T^{2} )^{2}$$
$67$ $$( -11 + T )^{4}$$
$71$ $$( -108 + T^{2} )^{2}$$
$73$ $$( 49 + T^{2} )^{2}$$
$79$ $$( -108 + T^{2} )^{2}$$
$83$ $$( 15 + T )^{4}$$
$89$ $$( -3 + T )^{4}$$
$97$ $$( 196 + T^{2} )^{2}$$