# Properties

 Label 1600.2.d.d Level $1600$ Weight $2$ Character orbit 1600.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( -4 - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( -4 - \beta_{3} ) q^{9} + ( -\beta_{1} - \beta_{2} ) q^{11} + ( 3 - \beta_{3} ) q^{17} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{19} + ( -5 \beta_{1} - 3 \beta_{2} ) q^{27} + ( 9 + 2 \beta_{3} ) q^{33} + ( 3 - 2 \beta_{3} ) q^{41} -5 \beta_{1} q^{43} -7 q^{49} + ( -5 \beta_{1} + \beta_{2} ) q^{51} + ( 17 + \beta_{3} ) q^{57} + 3 \beta_{1} q^{59} + ( -5 \beta_{1} + 3 \beta_{2} ) q^{67} + ( -1 - 3 \beta_{3} ) q^{73} + ( 19 + 5 \beta_{3} ) q^{81} + ( 5 \beta_{1} - \beta_{2} ) q^{83} + ( -9 + \beta_{3} ) q^{89} + 10 q^{97} + ( 7 \beta_{1} + 10 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 16q^{9} + O(q^{10})$$ $$4q - 16q^{9} + 12q^{17} + 36q^{33} + 12q^{41} - 28q^{49} + 68q^{57} - 4q^{73} + 76q^{81} - 36q^{89} + 40q^{97} + O(q^{100})$$

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

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

## Character values

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

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

## 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
 1.22474 − 1.22474i −1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 + 1.22474i
0 3.44949i 0 0 0 0 0 −8.89898 0
801.2 0 1.44949i 0 0 0 0 0 0.898979 0
801.3 0 1.44949i 0 0 0 0 0 0.898979 0
801.4 0 3.44949i 0 0 0 0 0 −8.89898 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 $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b 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.d yes 4
4.b odd 2 1 inner 1600.2.d.d yes 4
5.b even 2 1 1600.2.d.c 4
5.c odd 4 1 1600.2.f.f 4
5.c odd 4 1 1600.2.f.j 4
8.b even 2 1 inner 1600.2.d.d yes 4
8.d odd 2 1 CM 1600.2.d.d yes 4
16.e even 4 1 6400.2.a.bc 2
16.e even 4 1 6400.2.a.ci 2
16.f odd 4 1 6400.2.a.bc 2
16.f odd 4 1 6400.2.a.ci 2
20.d odd 2 1 1600.2.d.c 4
20.e even 4 1 1600.2.f.f 4
20.e even 4 1 1600.2.f.j 4
40.e odd 2 1 1600.2.d.c 4
40.f even 2 1 1600.2.d.c 4
40.i odd 4 1 1600.2.f.f 4
40.i odd 4 1 1600.2.f.j 4
40.k even 4 1 1600.2.f.f 4
40.k even 4 1 1600.2.f.j 4
80.k odd 4 1 6400.2.a.bd 2
80.k odd 4 1 6400.2.a.ch 2
80.q even 4 1 6400.2.a.bd 2
80.q even 4 1 6400.2.a.ch 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.d.c 4 5.b even 2 1
1600.2.d.c 4 20.d odd 2 1
1600.2.d.c 4 40.e odd 2 1
1600.2.d.c 4 40.f even 2 1
1600.2.d.d yes 4 1.a even 1 1 trivial
1600.2.d.d yes 4 4.b odd 2 1 inner
1600.2.d.d yes 4 8.b even 2 1 inner
1600.2.d.d yes 4 8.d odd 2 1 CM
1600.2.f.f 4 5.c odd 4 1
1600.2.f.f 4 20.e even 4 1
1600.2.f.f 4 40.i odd 4 1
1600.2.f.f 4 40.k even 4 1
1600.2.f.j 4 5.c odd 4 1
1600.2.f.j 4 20.e even 4 1
1600.2.f.j 4 40.i odd 4 1
1600.2.f.j 4 40.k even 4 1
6400.2.a.bc 2 16.e even 4 1
6400.2.a.bc 2 16.f odd 4 1
6400.2.a.bd 2 80.k odd 4 1
6400.2.a.bd 2 80.q even 4 1
6400.2.a.ch 2 80.k odd 4 1
6400.2.a.ch 2 80.q even 4 1
6400.2.a.ci 2 16.e even 4 1
6400.2.a.ci 2 16.f odd 4 1

## Hecke kernels

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

 $$T_{3}^{4} + 14 T_{3}^{2} + 25$$ $$T_{7}$$ $$T_{17}^{2} - 6 T_{17} - 15$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$25 + 14 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$9 + 30 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -15 - 6 T + T^{2} )^{2}$$
$19$ $$2809 + 110 T^{2} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( -87 - 6 T + T^{2} )^{2}$$
$43$ $$( 100 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$( 36 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$25 + 206 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -215 + 2 T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$5625 + 174 T^{2} + T^{4}$$
$89$ $$( 57 + 18 T + T^{2} )^{2}$$
$97$ $$( -10 + T )^{4}$$