# Properties

 Label 1600.2.d.g Level $1600$ Weight $2$ Character orbit 1600.d Analytic conductor $12.776$ Analytic rank $0$ Dimension $4$ CM discriminant -40 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 320) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{7} + 3 q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{7} + 3 q^{9} -\beta_{1} q^{11} + \beta_{2} q^{13} -3 \beta_{1} q^{19} + \beta_{3} q^{23} -\beta_{2} q^{37} + 2 q^{41} + 3 \beta_{3} q^{47} + 13 q^{49} -3 \beta_{2} q^{53} -7 \beta_{1} q^{59} -3 \beta_{3} q^{63} + 2 \beta_{2} q^{77} + 9 q^{81} + 14 q^{89} -10 \beta_{1} q^{91} -3 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} + 8q^{41} + 52q^{49} + 36q^{81} + 56q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} + 4 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 8 \nu$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{2} + 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 6$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 2 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
801.1
 0.618034i − 0.618034i − 1.61803i 1.61803i
0 0 0 0 0 −4.47214 0 3.00000 0
801.2 0 0 0 0 0 −4.47214 0 3.00000 0
801.3 0 0 0 0 0 4.47214 0 3.00000 0
801.4 0 0 0 0 0 4.47214 0 3.00000 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
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
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.d.g 4
4.b odd 2 1 inner 1600.2.d.g 4
5.b even 2 1 inner 1600.2.d.g 4
5.c odd 4 2 320.2.f.a 4
8.b even 2 1 inner 1600.2.d.g 4
8.d odd 2 1 inner 1600.2.d.g 4
15.e even 4 2 2880.2.d.e 4
16.e even 4 1 6400.2.a.bi 2
16.e even 4 1 6400.2.a.bj 2
16.f odd 4 1 6400.2.a.bi 2
16.f odd 4 1 6400.2.a.bj 2
20.d odd 2 1 inner 1600.2.d.g 4
20.e even 4 2 320.2.f.a 4
40.e odd 2 1 CM 1600.2.d.g 4
40.f even 2 1 inner 1600.2.d.g 4
40.i odd 4 2 320.2.f.a 4
40.k even 4 2 320.2.f.a 4
60.l odd 4 2 2880.2.d.e 4
80.i odd 4 1 1280.2.c.b 2
80.i odd 4 1 1280.2.c.c 2
80.j even 4 1 1280.2.c.b 2
80.j even 4 1 1280.2.c.c 2
80.k odd 4 1 6400.2.a.bi 2
80.k odd 4 1 6400.2.a.bj 2
80.q even 4 1 6400.2.a.bi 2
80.q even 4 1 6400.2.a.bj 2
80.s even 4 1 1280.2.c.b 2
80.s even 4 1 1280.2.c.c 2
80.t odd 4 1 1280.2.c.b 2
80.t odd 4 1 1280.2.c.c 2
120.q odd 4 2 2880.2.d.e 4
120.w even 4 2 2880.2.d.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.f.a 4 5.c odd 4 2
320.2.f.a 4 20.e even 4 2
320.2.f.a 4 40.i odd 4 2
320.2.f.a 4 40.k even 4 2
1280.2.c.b 2 80.i odd 4 1
1280.2.c.b 2 80.j even 4 1
1280.2.c.b 2 80.s even 4 1
1280.2.c.b 2 80.t odd 4 1
1280.2.c.c 2 80.i odd 4 1
1280.2.c.c 2 80.j even 4 1
1280.2.c.c 2 80.s even 4 1
1280.2.c.c 2 80.t odd 4 1
1600.2.d.g 4 1.a even 1 1 trivial
1600.2.d.g 4 4.b odd 2 1 inner
1600.2.d.g 4 5.b even 2 1 inner
1600.2.d.g 4 8.b even 2 1 inner
1600.2.d.g 4 8.d odd 2 1 inner
1600.2.d.g 4 20.d odd 2 1 inner
1600.2.d.g 4 40.e odd 2 1 CM
1600.2.d.g 4 40.f even 2 1 inner
2880.2.d.e 4 15.e even 4 2
2880.2.d.e 4 60.l odd 4 2
2880.2.d.e 4 120.q odd 4 2
2880.2.d.e 4 120.w even 4 2
6400.2.a.bi 2 16.e even 4 1
6400.2.a.bi 2 16.f odd 4 1
6400.2.a.bi 2 80.k odd 4 1
6400.2.a.bi 2 80.q even 4 1
6400.2.a.bj 2 16.e even 4 1
6400.2.a.bj 2 16.f odd 4 1
6400.2.a.bj 2 80.k odd 4 1
6400.2.a.bj 2 80.q even 4 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{7}^{2} - 20$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -20 + T^{2} )^{2}$$
$11$ $$( 4 + T^{2} )^{2}$$
$13$ $$( 20 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( 36 + T^{2} )^{2}$$
$23$ $$( -20 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$( 20 + T^{2} )^{2}$$
$41$ $$( -2 + T )^{4}$$
$43$ $$T^{4}$$
$47$ $$( -180 + T^{2} )^{2}$$
$53$ $$( 180 + T^{2} )^{2}$$
$59$ $$( 196 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( -14 + T )^{4}$$
$97$ $$T^{4}$$