Properties

 Label 2800.2.g.a Level $2800$ Weight $2$ Character orbit 2800.g Analytic conductor $22.358$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 350) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} -i q^{7} -6 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} -i q^{7} -6 q^{9} + 5 q^{11} -6 i q^{13} + i q^{17} -3 q^{19} + 3 q^{21} -9 i q^{27} + 6 q^{29} + 4 q^{31} + 15 i q^{33} -8 i q^{37} + 18 q^{39} + 11 q^{41} + 8 i q^{43} + 2 i q^{47} - q^{49} -3 q^{51} + 4 i q^{53} -9 i q^{57} + 4 q^{59} -2 q^{61} + 6 i q^{63} + 9 i q^{67} + 10 q^{71} -7 i q^{73} -5 i q^{77} -2 q^{79} + 9 q^{81} -11 i q^{83} + 18 i q^{87} + 11 q^{89} -6 q^{91} + 12 i q^{93} + 10 i q^{97} -30 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 12q^{9} + O(q^{10})$$ $$2q - 12q^{9} + 10q^{11} - 6q^{19} + 6q^{21} + 12q^{29} + 8q^{31} + 36q^{39} + 22q^{41} - 2q^{49} - 6q^{51} + 8q^{59} - 4q^{61} + 20q^{71} - 4q^{79} + 18q^{81} + 22q^{89} - 12q^{91} - 60q^{99} + O(q^{100})$$

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
 − 1.00000i 1.00000i
0 3.00000i 0 0 0 1.00000i 0 −6.00000 0
449.2 0 3.00000i 0 0 0 1.00000i 0 −6.00000 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
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.a 2
4.b odd 2 1 350.2.c.a 2
5.b even 2 1 inner 2800.2.g.a 2
5.c odd 4 1 2800.2.a.b 1
5.c odd 4 1 2800.2.a.bg 1
12.b even 2 1 3150.2.g.v 2
20.d odd 2 1 350.2.c.a 2
20.e even 4 1 350.2.a.c 1
20.e even 4 1 350.2.a.d yes 1
28.d even 2 1 2450.2.c.r 2
60.h even 2 1 3150.2.g.v 2
60.l odd 4 1 3150.2.a.j 1
60.l odd 4 1 3150.2.a.bq 1
140.c even 2 1 2450.2.c.r 2
140.j odd 4 1 2450.2.a.a 1
140.j odd 4 1 2450.2.a.bg 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.c 1 20.e even 4 1
350.2.a.d yes 1 20.e even 4 1
350.2.c.a 2 4.b odd 2 1
350.2.c.a 2 20.d odd 2 1
2450.2.a.a 1 140.j odd 4 1
2450.2.a.bg 1 140.j odd 4 1
2450.2.c.r 2 28.d even 2 1
2450.2.c.r 2 140.c even 2 1
2800.2.a.b 1 5.c odd 4 1
2800.2.a.bg 1 5.c odd 4 1
2800.2.g.a 2 1.a even 1 1 trivial
2800.2.g.a 2 5.b even 2 1 inner
3150.2.a.j 1 60.l odd 4 1
3150.2.a.bq 1 60.l odd 4 1
3150.2.g.v 2 12.b even 2 1
3150.2.g.v 2 60.h even 2 1

Hecke kernels

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

 $$T_{3}^{2} + 9$$ $$T_{11} - 5$$ $$T_{13}^{2} + 36$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$1 + T^{2}$$
$19$ $$( 3 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( -11 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$4 + T^{2}$$
$53$ $$16 + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$81 + T^{2}$$
$71$ $$( -10 + T )^{2}$$
$73$ $$49 + T^{2}$$
$79$ $$( 2 + T )^{2}$$
$83$ $$121 + T^{2}$$
$89$ $$( -11 + T )^{2}$$
$97$ $$100 + T^{2}$$