Properties

 Label 2800.2.g.v 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: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1400) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{7} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{7} + ( -2 + \beta_{3} ) q^{9} + ( 1 + 2 \beta_{3} ) q^{11} + 2 \beta_{2} q^{13} -\beta_{1} q^{17} + ( 1 + \beta_{3} ) q^{19} + ( 1 - \beta_{3} ) q^{21} + ( \beta_{1} - 3 \beta_{2} ) q^{23} + ( \beta_{1} + 4 \beta_{2} ) q^{27} + ( -6 + \beta_{3} ) q^{29} + ( 4 + 2 \beta_{3} ) q^{31} + ( \beta_{1} + 8 \beta_{2} ) q^{33} + ( -5 \beta_{1} - \beta_{2} ) q^{37} + ( 2 - 2 \beta_{3} ) q^{39} + ( -5 + \beta_{3} ) q^{41} + ( -\beta_{1} - 5 \beta_{2} ) q^{43} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{47} - q^{49} + ( 5 - \beta_{3} ) q^{51} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( \beta_{1} + 4 \beta_{2} ) q^{57} + ( -4 + 6 \beta_{3} ) q^{59} + ( 6 + 2 \beta_{3} ) q^{61} + ( \beta_{1} - \beta_{2} ) q^{63} + ( -2 \beta_{1} + 11 \beta_{2} ) q^{67} + ( -8 + 4 \beta_{3} ) q^{69} + ( 6 - 3 \beta_{3} ) q^{71} + ( -\beta_{1} + 8 \beta_{2} ) q^{73} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{77} + ( 2 - 3 \beta_{3} ) q^{79} -7 q^{81} + ( -3 \beta_{1} + 6 \beta_{2} ) q^{83} + ( -6 \beta_{1} + 4 \beta_{2} ) q^{87} + ( 3 - 5 \beta_{3} ) q^{89} -2 q^{91} + ( 4 \beta_{1} + 8 \beta_{2} ) q^{93} -6 \beta_{2} q^{97} + ( 6 - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{9} + O(q^{10})$$ $$4q - 6q^{9} + 8q^{11} + 6q^{19} + 2q^{21} - 22q^{29} + 20q^{31} + 4q^{39} - 18q^{41} - 4q^{49} + 18q^{51} - 4q^{59} + 28q^{61} - 24q^{69} + 18q^{71} + 2q^{79} - 28q^{81} + 2q^{89} - 8q^{91} + 22q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 5 \beta_{1}$$

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
 − 2.56155i − 1.56155i 1.56155i 2.56155i
0 2.56155i 0 0 0 1.00000i 0 −3.56155 0
449.2 0 1.56155i 0 0 0 1.00000i 0 0.561553 0
449.3 0 1.56155i 0 0 0 1.00000i 0 0.561553 0
449.4 0 2.56155i 0 0 0 1.00000i 0 −3.56155 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.v 4
4.b odd 2 1 1400.2.g.j 4
5.b even 2 1 inner 2800.2.g.v 4
5.c odd 4 1 2800.2.a.bj 2
5.c odd 4 1 2800.2.a.bo 2
20.d odd 2 1 1400.2.g.j 4
20.e even 4 1 1400.2.a.o 2
20.e even 4 1 1400.2.a.q yes 2
140.j odd 4 1 9800.2.a.bt 2
140.j odd 4 1 9800.2.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.a.o 2 20.e even 4 1
1400.2.a.q yes 2 20.e even 4 1
1400.2.g.j 4 4.b odd 2 1
1400.2.g.j 4 20.d odd 2 1
2800.2.a.bj 2 5.c odd 4 1
2800.2.a.bo 2 5.c odd 4 1
2800.2.g.v 4 1.a even 1 1 trivial
2800.2.g.v 4 5.b even 2 1 inner
9800.2.a.bt 2 140.j odd 4 1
9800.2.a.bx 2 140.j odd 4 1

Hecke kernels

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

 $$T_{3}^{4} + 9 T_{3}^{2} + 16$$ $$T_{11}^{2} - 4 T_{11} - 13$$ $$T_{13}^{2} + 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 9 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -13 - 4 T + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$16 + 9 T^{2} + T^{4}$$
$19$ $$( -2 - 3 T + T^{2} )^{2}$$
$23$ $$64 + 33 T^{2} + T^{4}$$
$29$ $$( 26 + 11 T + T^{2} )^{2}$$
$31$ $$( 8 - 10 T + T^{2} )^{2}$$
$37$ $$10816 + 217 T^{2} + T^{4}$$
$41$ $$( 16 + 9 T + T^{2} )^{2}$$
$43$ $$256 + 49 T^{2} + T^{4}$$
$47$ $$( 68 + T^{2} )^{2}$$
$53$ $$64 + 52 T^{2} + T^{4}$$
$59$ $$( -152 + 2 T + T^{2} )^{2}$$
$61$ $$( 32 - 14 T + T^{2} )^{2}$$
$67$ $$16129 + 322 T^{2} + T^{4}$$
$71$ $$( -18 - 9 T + T^{2} )^{2}$$
$73$ $$4624 + 153 T^{2} + T^{4}$$
$79$ $$( -38 - T + T^{2} )^{2}$$
$83$ $$324 + 189 T^{2} + T^{4}$$
$89$ $$( -106 - T + T^{2} )^{2}$$
$97$ $$( 36 + T^{2} )^{2}$$