# Properties

 Label 1600.4.a.cg Level $1600$ Weight $4$ Character orbit 1600.a Self dual yes Analytic conductor $94.403$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$94.4030560092$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} -7 \beta q^{7} -7 q^{9} +O(q^{10})$$ $$q -\beta q^{3} -7 \beta q^{7} -7 q^{9} -2 \beta q^{11} -62 q^{13} + 46 q^{17} + 24 \beta q^{19} + 140 q^{21} + 43 \beta q^{23} + 34 \beta q^{27} + 90 q^{29} + 34 \beta q^{31} + 40 q^{33} -214 q^{37} + 62 \beta q^{39} -10 q^{41} + 15 \beta q^{43} + 89 \beta q^{47} + 637 q^{49} -46 \beta q^{51} -678 q^{53} -480 q^{57} -92 \beta q^{59} -250 q^{61} + 49 \beta q^{63} -11 \beta q^{67} -860 q^{69} + 82 \beta q^{71} -522 q^{73} + 280 q^{77} -196 \beta q^{79} -491 q^{81} -85 \beta q^{83} -90 \beta q^{87} + 970 q^{89} + 434 \beta q^{91} -680 q^{93} + 934 q^{97} + 14 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 14q^{9} + O(q^{10})$$ $$2q - 14q^{9} - 124q^{13} + 92q^{17} + 280q^{21} + 180q^{29} + 80q^{33} - 428q^{37} - 20q^{41} + 1274q^{49} - 1356q^{53} - 960q^{57} - 500q^{61} - 1720q^{69} - 1044q^{73} + 560q^{77} - 982q^{81} + 1940q^{89} - 1360q^{93} + 1868q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
0 −4.47214 0 0 0 −31.3050 0 −7.00000 0
1.2 0 4.47214 0 0 0 31.3050 0 −7.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.4.a.cg 2
4.b odd 2 1 inner 1600.4.a.cg 2
5.b even 2 1 320.4.a.q 2
8.b even 2 1 800.4.a.o 2
8.d odd 2 1 800.4.a.o 2
20.d odd 2 1 320.4.a.q 2
40.e odd 2 1 160.4.a.d 2
40.f even 2 1 160.4.a.d 2
40.i odd 4 2 800.4.c.l 4
40.k even 4 2 800.4.c.l 4
80.k odd 4 2 1280.4.d.v 4
80.q even 4 2 1280.4.d.v 4
120.i odd 2 1 1440.4.a.bb 2
120.m even 2 1 1440.4.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.d 2 40.e odd 2 1
160.4.a.d 2 40.f even 2 1
320.4.a.q 2 5.b even 2 1
320.4.a.q 2 20.d odd 2 1
800.4.a.o 2 8.b even 2 1
800.4.a.o 2 8.d odd 2 1
800.4.c.l 4 40.i odd 4 2
800.4.c.l 4 40.k even 4 2
1280.4.d.v 4 80.k odd 4 2
1280.4.d.v 4 80.q even 4 2
1440.4.a.bb 2 120.i odd 2 1
1440.4.a.bb 2 120.m even 2 1
1600.4.a.cg 2 1.a even 1 1 trivial
1600.4.a.cg 2 4.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1600))$$:

 $$T_{3}^{2} - 20$$ $$T_{7}^{2} - 980$$ $$T_{11}^{2} - 80$$ $$T_{13} + 62$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-20 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-980 + T^{2}$$
$11$ $$-80 + T^{2}$$
$13$ $$( 62 + T )^{2}$$
$17$ $$( -46 + T )^{2}$$
$19$ $$-11520 + T^{2}$$
$23$ $$-36980 + T^{2}$$
$29$ $$( -90 + T )^{2}$$
$31$ $$-23120 + T^{2}$$
$37$ $$( 214 + T )^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$-4500 + T^{2}$$
$47$ $$-158420 + T^{2}$$
$53$ $$( 678 + T )^{2}$$
$59$ $$-169280 + T^{2}$$
$61$ $$( 250 + T )^{2}$$
$67$ $$-2420 + T^{2}$$
$71$ $$-134480 + T^{2}$$
$73$ $$( 522 + T )^{2}$$
$79$ $$-768320 + T^{2}$$
$83$ $$-144500 + T^{2}$$
$89$ $$( -970 + T )^{2}$$
$97$ $$( -934 + T )^{2}$$