# Properties

 Label 840.2.j.a Level $840$ Weight $2$ Character orbit 840.j Analytic conductor $6.707$ Analytic rank $1$ 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.j (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

 $$f(q)$$ $$=$$ $$q + ( -1 - i ) q^{2} - q^{3} + 2 i q^{4} + ( -2 + i ) q^{5} + ( 1 + i ) q^{6} -i q^{7} + ( 2 - 2 i ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 - i ) q^{2} - q^{3} + 2 i q^{4} + ( -2 + i ) q^{5} + ( 1 + i ) q^{6} -i q^{7} + ( 2 - 2 i ) q^{8} + q^{9} + ( 3 + i ) q^{10} + 4 i q^{11} -2 i q^{12} + 2 q^{13} + ( -1 + i ) q^{14} + ( 2 - i ) q^{15} -4 q^{16} + 2 i q^{17} + ( -1 - i ) q^{18} -8 i q^{19} + ( -2 - 4 i ) q^{20} + i q^{21} + ( 4 - 4 i ) q^{22} + 4 i q^{23} + ( -2 + 2 i ) q^{24} + ( 3 - 4 i ) q^{25} + ( -2 - 2 i ) q^{26} - q^{27} + 2 q^{28} + 2 i q^{29} + ( -3 - i ) q^{30} -4 q^{31} + ( 4 + 4 i ) q^{32} -4 i q^{33} + ( 2 - 2 i ) q^{34} + ( 1 + 2 i ) q^{35} + 2 i q^{36} -6 q^{37} + ( -8 + 8 i ) q^{38} -2 q^{39} + ( -2 + 6 i ) q^{40} + ( 1 - i ) q^{42} -10 q^{43} -8 q^{44} + ( -2 + i ) q^{45} + ( 4 - 4 i ) q^{46} -6 i q^{47} + 4 q^{48} - q^{49} + ( -7 + i ) q^{50} -2 i q^{51} + 4 i q^{52} -14 q^{53} + ( 1 + i ) q^{54} + ( -4 - 8 i ) q^{55} + ( -2 - 2 i ) q^{56} + 8 i q^{57} + ( 2 - 2 i ) q^{58} -6 i q^{59} + ( 2 + 4 i ) q^{60} + 2 i q^{61} + ( 4 + 4 i ) q^{62} -i q^{63} -8 i q^{64} + ( -4 + 2 i ) q^{65} + ( -4 + 4 i ) q^{66} + 10 q^{67} -4 q^{68} -4 i q^{69} + ( 1 - 3 i ) q^{70} -12 q^{71} + ( 2 - 2 i ) q^{72} + 6 i q^{73} + ( 6 + 6 i ) q^{74} + ( -3 + 4 i ) q^{75} + 16 q^{76} + 4 q^{77} + ( 2 + 2 i ) q^{78} -14 q^{79} + ( 8 - 4 i ) q^{80} + q^{81} -12 q^{83} -2 q^{84} + ( -2 - 4 i ) q^{85} + ( 10 + 10 i ) q^{86} -2 i q^{87} + ( 8 + 8 i ) q^{88} + 8 q^{89} + ( 3 + i ) q^{90} -2 i q^{91} -8 q^{92} + 4 q^{93} + ( -6 + 6 i ) q^{94} + ( 8 + 16 i ) q^{95} + ( -4 - 4 i ) q^{96} + 6 i q^{97} + ( 1 + i ) q^{98} + 4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} - 4 q^{5} + 2 q^{6} + 4 q^{8} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} - 2 q^{3} - 4 q^{5} + 2 q^{6} + 4 q^{8} + 2 q^{9} + 6 q^{10} + 4 q^{13} - 2 q^{14} + 4 q^{15} - 8 q^{16} - 2 q^{18} - 4 q^{20} + 8 q^{22} - 4 q^{24} + 6 q^{25} - 4 q^{26} - 2 q^{27} + 4 q^{28} - 6 q^{30} - 8 q^{31} + 8 q^{32} + 4 q^{34} + 2 q^{35} - 12 q^{37} - 16 q^{38} - 4 q^{39} - 4 q^{40} + 2 q^{42} - 20 q^{43} - 16 q^{44} - 4 q^{45} + 8 q^{46} + 8 q^{48} - 2 q^{49} - 14 q^{50} - 28 q^{53} + 2 q^{54} - 8 q^{55} - 4 q^{56} + 4 q^{58} + 4 q^{60} + 8 q^{62} - 8 q^{65} - 8 q^{66} + 20 q^{67} - 8 q^{68} + 2 q^{70} - 24 q^{71} + 4 q^{72} + 12 q^{74} - 6 q^{75} + 32 q^{76} + 8 q^{77} + 4 q^{78} - 28 q^{79} + 16 q^{80} + 2 q^{81} - 24 q^{83} - 4 q^{84} - 4 q^{85} + 20 q^{86} + 16 q^{88} + 16 q^{89} + 6 q^{90} - 16 q^{92} + 8 q^{93} - 12 q^{94} + 16 q^{95} - 8 q^{96} + 2 q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
589.1
 1.00000i − 1.00000i
−1.00000 1.00000i −1.00000 2.00000i −2.00000 + 1.00000i 1.00000 + 1.00000i 1.00000i 2.00000 2.00000i 1.00000 3.00000 + 1.00000i
589.2 −1.00000 + 1.00000i −1.00000 2.00000i −2.00000 1.00000i 1.00000 1.00000i 1.00000i 2.00000 + 2.00000i 1.00000 3.00000 1.00000i
 $$n$$: 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
40.f 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.j.a 2
4.b odd 2 1 3360.2.j.c 2
5.b even 2 1 840.2.j.d yes 2
8.b even 2 1 840.2.j.d yes 2
8.d odd 2 1 3360.2.j.b 2
20.d odd 2 1 3360.2.j.b 2
40.e odd 2 1 3360.2.j.c 2
40.f even 2 1 inner 840.2.j.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.j.a 2 1.a even 1 1 trivial
840.2.j.a 2 40.f even 2 1 inner
840.2.j.d yes 2 5.b even 2 1
840.2.j.d yes 2 8.b even 2 1
3360.2.j.b 2 8.d odd 2 1
3360.2.j.b 2 20.d odd 2 1
3360.2.j.c 2 4.b odd 2 1
3360.2.j.c 2 40.e odd 2 1

## Hecke kernels

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

 $$T_{11}^{2} + 16$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + 2 T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$5 + 4 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$64 + T^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$4 + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$( 14 + T )^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$4 + T^{2}$$
$67$ $$( -10 + T )^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( 14 + T )^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$( -8 + T )^{2}$$
$97$ $$36 + T^{2}$$