# Properties

 Label 2800.2.g.s Level $2800$ Weight $2$ Character orbit 2800.g Analytic conductor $22.358$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.3581125660$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 175) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{3} -\beta_{1} q^{7} + ( -3 + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{3} -\beta_{1} q^{7} + ( -3 + 2 \beta_{3} ) q^{9} + ( -2 - \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{2} ) q^{13} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{17} + 2 \beta_{3} q^{19} + ( -1 + \beta_{3} ) q^{21} + ( 4 \beta_{1} - \beta_{2} ) q^{23} + ( 10 \beta_{1} - 2 \beta_{2} ) q^{27} -5 q^{29} + ( 3 + 3 \beta_{3} ) q^{31} + ( -3 \beta_{1} - \beta_{2} ) q^{33} + 3 \beta_{1} q^{37} -4 q^{39} + ( 7 + \beta_{3} ) q^{41} + ( 4 \beta_{1} + \beta_{2} ) q^{43} + 2 \beta_{1} q^{47} - q^{49} + 8 q^{51} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 10 \beta_{1} - 2 \beta_{2} ) q^{57} + ( 5 - 3 \beta_{3} ) q^{59} + ( -3 + 3 \beta_{3} ) q^{61} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{63} + ( 2 \beta_{1} - \beta_{2} ) q^{67} + ( 9 - 5 \beta_{3} ) q^{69} + ( -2 + 3 \beta_{3} ) q^{71} + ( 11 \beta_{1} + \beta_{2} ) q^{73} + ( 2 \beta_{1} + \beta_{2} ) q^{77} + 5 \beta_{3} q^{79} + ( 11 - 6 \beta_{3} ) q^{81} + ( -\beta_{1} + 3 \beta_{2} ) q^{83} + ( 5 \beta_{1} - 5 \beta_{2} ) q^{87} + ( -15 + \beta_{3} ) q^{89} + ( 1 + \beta_{3} ) q^{91} + 12 \beta_{1} q^{93} + ( 3 \beta_{1} - \beta_{2} ) q^{97} + ( -4 - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{9} + O(q^{10})$$ $$4q - 12q^{9} - 8q^{11} - 4q^{21} - 20q^{29} + 12q^{31} - 16q^{39} + 28q^{41} - 4q^{49} + 32q^{51} + 20q^{59} - 12q^{61} + 36q^{69} - 8q^{71} + 44q^{81} - 60q^{89} + 4q^{91} - 16q^{99} + O(q^{100})$$

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 $$\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$801$$ $$2101$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 0.618034i 1.61803i
0 3.23607i 0 0 0 1.00000i 0 −7.47214 0
449.2 0 1.23607i 0 0 0 1.00000i 0 1.47214 0
449.3 0 1.23607i 0 0 0 1.00000i 0 1.47214 0
449.4 0 3.23607i 0 0 0 1.00000i 0 −7.47214 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
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{4} + 12 T_{3}^{2} + 16$$ $$T_{11}^{2} + 4 T_{11} - 1$$ $$T_{13}^{4} + 12 T_{13}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 12 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -1 + 4 T + T^{2} )^{2}$$
$13$ $$16 + 12 T^{2} + T^{4}$$
$17$ $$256 + 48 T^{2} + T^{4}$$
$19$ $$( -20 + T^{2} )^{2}$$
$23$ $$121 + 42 T^{2} + T^{4}$$
$29$ $$( 5 + T )^{4}$$
$31$ $$( -36 - 6 T + T^{2} )^{2}$$
$37$ $$( 9 + T^{2} )^{2}$$
$41$ $$( 44 - 14 T + T^{2} )^{2}$$
$43$ $$121 + 42 T^{2} + T^{4}$$
$47$ $$( 4 + T^{2} )^{2}$$
$53$ $$16 + 72 T^{2} + T^{4}$$
$59$ $$( -20 - 10 T + T^{2} )^{2}$$
$61$ $$( -36 + 6 T + T^{2} )^{2}$$
$67$ $$1 + 18 T^{2} + T^{4}$$
$71$ $$( -41 + 4 T + T^{2} )^{2}$$
$73$ $$13456 + 252 T^{2} + T^{4}$$
$79$ $$( -125 + T^{2} )^{2}$$
$83$ $$1936 + 92 T^{2} + T^{4}$$
$89$ $$( 220 + 30 T + T^{2} )^{2}$$
$97$ $$16 + 28 T^{2} + T^{4}$$