# Properties

 Label 2800.2.a.bp Level $2800$ Weight $2$ Character orbit 2800.a Self dual yes Analytic conductor $22.358$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.3581125660$$ Analytic rank: $$0$$ 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: $$2$$ Twist minimal: no (minimal twist has level 175) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} - q^{7} + ( 3 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} - q^{7} + ( 3 + 2 \beta ) q^{9} + ( -2 + \beta ) q^{11} + ( -1 + \beta ) q^{13} + ( -2 + 2 \beta ) q^{17} + 2 \beta q^{19} + ( -1 - \beta ) q^{21} + ( -4 - \beta ) q^{23} + ( 10 + 2 \beta ) q^{27} + 5 q^{29} + ( 3 - 3 \beta ) q^{31} + ( 3 - \beta ) q^{33} + 3 q^{37} + 4 q^{39} + ( 7 - \beta ) q^{41} + ( -4 + \beta ) q^{43} + 2 q^{47} + q^{49} + 8 q^{51} + ( 4 + 2 \beta ) q^{53} + ( 10 + 2 \beta ) q^{57} + ( -5 - 3 \beta ) q^{59} + ( -3 - 3 \beta ) q^{61} + ( -3 - 2 \beta ) q^{63} + ( 2 + \beta ) q^{67} + ( -9 - 5 \beta ) q^{69} + ( -2 - 3 \beta ) q^{71} + ( -11 + \beta ) q^{73} + ( 2 - \beta ) q^{77} + 5 \beta q^{79} + ( 11 + 6 \beta ) q^{81} + ( 1 + 3 \beta ) q^{83} + ( 5 + 5 \beta ) q^{87} + ( 15 + \beta ) q^{89} + ( 1 - \beta ) q^{91} -12 q^{93} + ( 3 + \beta ) q^{97} + ( 4 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{7} + 6q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{7} + 6q^{9} - 4q^{11} - 2q^{13} - 4q^{17} - 2q^{21} - 8q^{23} + 20q^{27} + 10q^{29} + 6q^{31} + 6q^{33} + 6q^{37} + 8q^{39} + 14q^{41} - 8q^{43} + 4q^{47} + 2q^{49} + 16q^{51} + 8q^{53} + 20q^{57} - 10q^{59} - 6q^{61} - 6q^{63} + 4q^{67} - 18q^{69} - 4q^{71} - 22q^{73} + 4q^{77} + 22q^{81} + 2q^{83} + 10q^{87} + 30q^{89} + 2q^{91} - 24q^{93} + 6q^{97} + 8q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
0 −1.23607 0 0 0 −1.00000 0 −1.47214 0
1.2 0 3.23607 0 0 0 −1.00000 0 7.47214 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.2.a.bp 2
4.b odd 2 1 175.2.a.e yes 2
5.b even 2 1 2800.2.a.bh 2
5.c odd 4 2 2800.2.g.s 4
12.b even 2 1 1575.2.a.n 2
20.d odd 2 1 175.2.a.d 2
20.e even 4 2 175.2.b.c 4
28.d even 2 1 1225.2.a.u 2
60.h even 2 1 1575.2.a.s 2
60.l odd 4 2 1575.2.d.k 4
140.c even 2 1 1225.2.a.n 2
140.j odd 4 2 1225.2.b.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.a.d 2 20.d odd 2 1
175.2.a.e yes 2 4.b odd 2 1
175.2.b.c 4 20.e even 4 2
1225.2.a.n 2 140.c even 2 1
1225.2.a.u 2 28.d even 2 1
1225.2.b.k 4 140.j odd 4 2
1575.2.a.n 2 12.b even 2 1
1575.2.a.s 2 60.h even 2 1
1575.2.d.k 4 60.l odd 4 2
2800.2.a.bh 2 5.b even 2 1
2800.2.a.bp 2 1.a even 1 1 trivial
2800.2.g.s 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2800))$$:

 $$T_{3}^{2} - 2 T_{3} - 4$$ $$T_{11}^{2} + 4 T_{11} - 1$$ $$T_{13}^{2} + 2 T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-1 + 4 T + T^{2}$$
$13$ $$-4 + 2 T + T^{2}$$
$17$ $$-16 + 4 T + T^{2}$$
$19$ $$-20 + T^{2}$$
$23$ $$11 + 8 T + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$-36 - 6 T + T^{2}$$
$37$ $$( -3 + T )^{2}$$
$41$ $$44 - 14 T + T^{2}$$
$43$ $$11 + 8 T + T^{2}$$
$47$ $$( -2 + T )^{2}$$
$53$ $$-4 - 8 T + T^{2}$$
$59$ $$-20 + 10 T + T^{2}$$
$61$ $$-36 + 6 T + T^{2}$$
$67$ $$-1 - 4 T + T^{2}$$
$71$ $$-41 + 4 T + T^{2}$$
$73$ $$116 + 22 T + T^{2}$$
$79$ $$-125 + T^{2}$$
$83$ $$-44 - 2 T + T^{2}$$
$89$ $$220 - 30 T + T^{2}$$
$97$ $$4 - 6 T + T^{2}$$