# Properties

 Label 2800.2.a.bl Level $2800$ Weight $2$ Character orbit 2800.a Self dual yes Analytic conductor $22.358$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −2.44949 2.44949
0 −2.44949 0 0 0 −1.00000 0 3.00000 0
1.2 0 2.44949 0 0 0 −1.00000 0 3.00000 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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.bl 2
4.b odd 2 1 350.2.a.h 2
5.b even 2 1 2800.2.a.bm 2
5.c odd 4 2 560.2.g.e 4
12.b even 2 1 3150.2.a.bs 2
15.e even 4 2 5040.2.t.t 4
20.d odd 2 1 350.2.a.g 2
20.e even 4 2 70.2.c.a 4
28.d even 2 1 2450.2.a.bq 2
40.i odd 4 2 2240.2.g.i 4
40.k even 4 2 2240.2.g.j 4
60.h even 2 1 3150.2.a.bt 2
60.l odd 4 2 630.2.g.g 4
140.c even 2 1 2450.2.a.bl 2
140.j odd 4 2 490.2.c.e 4
140.w even 12 4 490.2.i.c 8
140.x odd 12 4 490.2.i.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 20.e even 4 2
350.2.a.g 2 20.d odd 2 1
350.2.a.h 2 4.b odd 2 1
490.2.c.e 4 140.j odd 4 2
490.2.i.c 8 140.w even 12 4
490.2.i.f 8 140.x odd 12 4
560.2.g.e 4 5.c odd 4 2
630.2.g.g 4 60.l odd 4 2
2240.2.g.i 4 40.i odd 4 2
2240.2.g.j 4 40.k even 4 2
2450.2.a.bl 2 140.c even 2 1
2450.2.a.bq 2 28.d even 2 1
2800.2.a.bl 2 1.a even 1 1 trivial
2800.2.a.bm 2 5.b even 2 1
3150.2.a.bs 2 12.b even 2 1
3150.2.a.bt 2 60.h even 2 1
5040.2.t.t 4 15.e even 4 2

## Hecke kernels

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

 $$T_{3}^{2} - 6$$ $$T_{11}^{2} - 24$$ $$T_{13}^{2} + 4 T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-6 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-24 + T^{2}$$
$13$ $$-2 + 4 T + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$10 + 8 T + T^{2}$$
$23$ $$-20 + 4 T + T^{2}$$
$29$ $$-20 - 4 T + T^{2}$$
$31$ $$-8 + 8 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$12 + 12 T + T^{2}$$
$43$ $$-8 - 8 T + T^{2}$$
$47$ $$-8 - 8 T + T^{2}$$
$53$ $$12 - 12 T + T^{2}$$
$59$ $$10 - 8 T + T^{2}$$
$61$ $$30 - 12 T + T^{2}$$
$67$ $$( 8 + T )^{2}$$
$71$ $$12 - 12 T + T^{2}$$
$73$ $$-20 - 4 T + T^{2}$$
$79$ $$-20 + 4 T + T^{2}$$
$83$ $$-6 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$-60 + 12 T + T^{2}$$