# Properties

 Label 840.2.t.a Level $840$ Weight $2$ Character orbit 840.t Analytic conductor $6.707$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$840 = 2^{3} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 840.t (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.70743376979$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -i q^{3} + ( -1 - 2 i ) q^{5} -i q^{7} - q^{9} +O(q^{10})$$ $$q -i q^{3} + ( -1 - 2 i ) q^{5} -i q^{7} - q^{9} + 2 q^{11} -2 i q^{13} + ( -2 + i ) q^{15} -2 q^{19} - q^{21} -8 i q^{23} + ( -3 + 4 i ) q^{25} + i q^{27} -2 q^{29} -6 q^{31} -2 i q^{33} + ( -2 + i ) q^{35} + 8 i q^{37} -2 q^{39} -10 q^{41} + ( 1 + 2 i ) q^{45} -12 i q^{47} - q^{49} + 2 i q^{53} + ( -2 - 4 i ) q^{55} + 2 i q^{57} + 2 q^{61} + i q^{63} + ( -4 + 2 i ) q^{65} + 4 i q^{67} -8 q^{69} + 14 q^{71} + 2 i q^{73} + ( 4 + 3 i ) q^{75} -2 i q^{77} -4 q^{79} + q^{81} -16 i q^{83} + 2 i q^{87} + 6 q^{89} -2 q^{91} + 6 i q^{93} + ( 2 + 4 i ) q^{95} + 2 i q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{5} - 2 q^{9} + 4 q^{11} - 4 q^{15} - 4 q^{19} - 2 q^{21} - 6 q^{25} - 4 q^{29} - 12 q^{31} - 4 q^{35} - 4 q^{39} - 20 q^{41} + 2 q^{45} - 2 q^{49} - 4 q^{55} + 4 q^{61} - 8 q^{65} - 16 q^{69} + 28 q^{71} + 8 q^{75} - 8 q^{79} + 2 q^{81} + 12 q^{89} - 4 q^{91} + 4 q^{95} - 4 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$281$$ $$337$$ $$421$$ $$631$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 1.00000i − 1.00000i
0 1.00000i 0 −1.00000 2.00000i 0 1.00000i 0 −1.00000 0
169.2 0 1.00000i 0 −1.00000 + 2.00000i 0 1.00000i 0 −1.00000 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 840.2.t.a 2
3.b odd 2 1 2520.2.t.c 2
4.b odd 2 1 1680.2.t.c 2
5.b even 2 1 inner 840.2.t.a 2
5.c odd 4 1 4200.2.a.k 1
5.c odd 4 1 4200.2.a.x 1
12.b even 2 1 5040.2.t.n 2
15.d odd 2 1 2520.2.t.c 2
20.d odd 2 1 1680.2.t.c 2
20.e even 4 1 8400.2.a.u 1
20.e even 4 1 8400.2.a.br 1
60.h even 2 1 5040.2.t.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.a 2 1.a even 1 1 trivial
840.2.t.a 2 5.b even 2 1 inner
1680.2.t.c 2 4.b odd 2 1
1680.2.t.c 2 20.d odd 2 1
2520.2.t.c 2 3.b odd 2 1
2520.2.t.c 2 15.d odd 2 1
4200.2.a.k 1 5.c odd 4 1
4200.2.a.x 1 5.c odd 4 1
5040.2.t.n 2 12.b even 2 1
5040.2.t.n 2 60.h even 2 1
8400.2.a.u 1 20.e even 4 1
8400.2.a.br 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(840, [\chi])$$:

 $$T_{11} - 2$$ $$T_{19} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$( 6 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$144 + T^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -14 + T )^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$256 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$4 + T^{2}$$