# Properties

 Label 1600.2.f.f Level $1600$ Weight $2$ Character orbit 1600.f Analytic conductor $12.776$ Analytic rank $0$ Dimension $4$ CM discriminant -8 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: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{3} ) q^{3} + ( 4 - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{3} ) q^{3} + ( 4 - 2 \beta_{3} ) q^{9} + ( 3 \beta_{1} - \beta_{2} ) q^{11} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{17} + ( \beta_{1} + 3 \beta_{2} ) q^{19} + ( -13 + 3 \beta_{3} ) q^{27} + ( -9 \beta_{1} + 4 \beta_{2} ) q^{33} + ( 3 + 4 \beta_{3} ) q^{41} + 10 q^{43} + 7 q^{49} + ( 9 \beta_{1} + \beta_{2} ) q^{51} + ( 17 \beta_{1} - 2 \beta_{2} ) q^{57} + 6 \beta_{1} q^{59} + ( -7 - 3 \beta_{3} ) q^{67} + ( \beta_{1} - 6 \beta_{2} ) q^{73} + ( 19 - 10 \beta_{3} ) q^{81} + ( -9 - \beta_{3} ) q^{83} + ( 9 + 2 \beta_{3} ) q^{89} + 10 \beta_{1} q^{97} + ( 24 \beta_{1} - 10 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} + 16q^{9} + O(q^{10})$$ $$4q - 4q^{3} + 16q^{9} - 52q^{27} + 12q^{41} + 40q^{43} + 28q^{49} - 28q^{67} + 76q^{81} - 36q^{83} + 36q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 3 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 3 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{3} + 3 \beta_{2}$$$$)/2$$

## 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
 −1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 − 1.22474i 1.22474 + 1.22474i
0 −3.44949 0 0 0 0 0 8.89898 0
1249.2 0 −3.44949 0 0 0 0 0 8.89898 0
1249.3 0 1.44949 0 0 0 0 0 −0.898979 0
1249.4 0 1.44949 0 0 0 0 0 −0.898979 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 CM by $$\Q(\sqrt{-2})$$
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.f 4
4.b odd 2 1 1600.2.f.j 4
5.b even 2 1 1600.2.f.j 4
5.c odd 4 1 1600.2.d.c 4
5.c odd 4 1 1600.2.d.d yes 4
8.b even 2 1 1600.2.f.j 4
8.d odd 2 1 CM 1600.2.f.f 4
20.d odd 2 1 inner 1600.2.f.f 4
20.e even 4 1 1600.2.d.c 4
20.e even 4 1 1600.2.d.d yes 4
40.e odd 2 1 1600.2.f.j 4
40.f even 2 1 inner 1600.2.f.f 4
40.i odd 4 1 1600.2.d.c 4
40.i odd 4 1 1600.2.d.d yes 4
40.k even 4 1 1600.2.d.c 4
40.k even 4 1 1600.2.d.d yes 4
80.i odd 4 1 6400.2.a.ch 2
80.i odd 4 1 6400.2.a.ci 2
80.j even 4 1 6400.2.a.ch 2
80.j even 4 1 6400.2.a.ci 2
80.s even 4 1 6400.2.a.bc 2
80.s even 4 1 6400.2.a.bd 2
80.t odd 4 1 6400.2.a.bc 2
80.t odd 4 1 6400.2.a.bd 2

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

 $$T_{3}^{2} + 2 T_{3} - 5$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -5 + 2 T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$9 + 30 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$225 + 66 T^{2} + T^{4}$$
$19$ $$2809 + 110 T^{2} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( -87 - 6 T + T^{2} )^{2}$$
$43$ $$( -10 + T )^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$( 36 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( -5 + 14 T + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$46225 + 434 T^{2} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 75 + 18 T + T^{2} )^{2}$$
$89$ $$( 57 - 18 T + T^{2} )^{2}$$
$97$ $$( 100 + T^{2} )^{2}$$