# Properties

 Label 1600.2.c.o Level $1600$ Weight $2$ Character orbit 1600.c 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 800) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + 2 \beta_{2} q^{7} -2 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + 2 \beta_{2} q^{7} -2 q^{9} + \beta_{3} q^{11} -4 \beta_{1} q^{13} + 7 \beta_{1} q^{17} + 3 \beta_{3} q^{19} -10 q^{21} + 2 \beta_{2} q^{23} + \beta_{2} q^{27} -2 \beta_{3} q^{31} + 5 \beta_{1} q^{33} + 2 \beta_{1} q^{37} + 4 \beta_{3} q^{39} + 5 q^{41} -4 \beta_{2} q^{47} -13 q^{49} -7 \beta_{3} q^{51} -6 \beta_{1} q^{53} + 15 \beta_{1} q^{57} -4 \beta_{3} q^{59} -10 q^{61} -4 \beta_{2} q^{63} + \beta_{2} q^{67} -10 q^{69} + 4 \beta_{3} q^{71} -9 \beta_{1} q^{73} + 10 \beta_{1} q^{77} -2 \beta_{3} q^{79} -11 q^{81} -5 \beta_{2} q^{83} + 5 q^{89} + 8 \beta_{3} q^{91} -10 \beta_{1} q^{93} -2 \beta_{1} q^{97} -2 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{9} + O(q^{10})$$ $$4q - 8q^{9} - 40q^{21} + 20q^{41} - 52q^{49} - 40q^{61} - 40q^{69} - 44q^{81} + 20q^{89} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.61803i − 0.618034i 1.61803i 0.618034i
0 2.23607i 0 0 0 4.47214i 0 −2.00000 0
449.2 0 2.23607i 0 0 0 4.47214i 0 −2.00000 0
449.3 0 2.23607i 0 0 0 4.47214i 0 −2.00000 0
449.4 0 2.23607i 0 0 0 4.47214i 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.c.o 4
4.b odd 2 1 inner 1600.2.c.o 4
5.b even 2 1 inner 1600.2.c.o 4
5.c odd 4 1 1600.2.a.ba 2
5.c odd 4 1 1600.2.a.bb 2
8.b even 2 1 800.2.c.g 4
8.d odd 2 1 800.2.c.g 4
20.d odd 2 1 inner 1600.2.c.o 4
20.e even 4 1 1600.2.a.ba 2
20.e even 4 1 1600.2.a.bb 2
24.f even 2 1 7200.2.f.bg 4
24.h odd 2 1 7200.2.f.bg 4
40.e odd 2 1 800.2.c.g 4
40.f even 2 1 800.2.c.g 4
40.i odd 4 1 800.2.a.k 2
40.i odd 4 1 800.2.a.l yes 2
40.k even 4 1 800.2.a.k 2
40.k even 4 1 800.2.a.l yes 2
120.i odd 2 1 7200.2.f.bg 4
120.m even 2 1 7200.2.f.bg 4
120.q odd 4 1 7200.2.a.cf 2
120.q odd 4 1 7200.2.a.cn 2
120.w even 4 1 7200.2.a.cf 2
120.w even 4 1 7200.2.a.cn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.a.k 2 40.i odd 4 1
800.2.a.k 2 40.k even 4 1
800.2.a.l yes 2 40.i odd 4 1
800.2.a.l yes 2 40.k even 4 1
800.2.c.g 4 8.b even 2 1
800.2.c.g 4 8.d odd 2 1
800.2.c.g 4 40.e odd 2 1
800.2.c.g 4 40.f even 2 1
1600.2.a.ba 2 5.c odd 4 1
1600.2.a.ba 2 20.e even 4 1
1600.2.a.bb 2 5.c odd 4 1
1600.2.a.bb 2 20.e even 4 1
1600.2.c.o 4 1.a even 1 1 trivial
1600.2.c.o 4 4.b odd 2 1 inner
1600.2.c.o 4 5.b even 2 1 inner
1600.2.c.o 4 20.d odd 2 1 inner
7200.2.a.cf 2 120.q odd 4 1
7200.2.a.cf 2 120.w even 4 1
7200.2.a.cn 2 120.q odd 4 1
7200.2.a.cn 2 120.w even 4 1
7200.2.f.bg 4 24.f even 2 1
7200.2.f.bg 4 24.h odd 2 1
7200.2.f.bg 4 120.i odd 2 1
7200.2.f.bg 4 120.m even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 5$$ $$T_{7}^{2} + 20$$ $$T_{11}^{2} - 5$$ $$T_{19}^{2} - 45$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 5 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 20 + T^{2} )^{2}$$
$11$ $$( -5 + T^{2} )^{2}$$
$13$ $$( 16 + T^{2} )^{2}$$
$17$ $$( 49 + T^{2} )^{2}$$
$19$ $$( -45 + T^{2} )^{2}$$
$23$ $$( 20 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( -20 + T^{2} )^{2}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( -5 + T )^{4}$$
$43$ $$T^{4}$$
$47$ $$( 80 + T^{2} )^{2}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( -80 + T^{2} )^{2}$$
$61$ $$( 10 + T )^{4}$$
$67$ $$( 5 + T^{2} )^{2}$$
$71$ $$( -80 + T^{2} )^{2}$$
$73$ $$( 81 + T^{2} )^{2}$$
$79$ $$( -20 + T^{2} )^{2}$$
$83$ $$( 125 + T^{2} )^{2}$$
$89$ $$( -5 + T )^{4}$$
$97$ $$( 4 + T^{2} )^{2}$$