# Properties

 Label 420.2.q.a Level $420$ Weight $2$ Character orbit 420.q Analytic conductor $3.354$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 420.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{11} + q^{13} - q^{15} + ( 4 - 4 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + ( 3 - 2 \zeta_{6} ) q^{21} -4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} - q^{27} + ( 5 - 5 \zeta_{6} ) q^{31} -2 \zeta_{6} q^{33} + ( 1 - 3 \zeta_{6} ) q^{35} + 5 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{39} + 2 q^{41} -9 q^{43} + ( -1 + \zeta_{6} ) q^{45} + 2 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{51} + ( -12 + 12 \zeta_{6} ) q^{53} -2 q^{55} + q^{57} + ( 8 - 8 \zeta_{6} ) q^{59} + 14 \zeta_{6} q^{61} + ( 1 - 3 \zeta_{6} ) q^{63} -\zeta_{6} q^{65} + ( -9 + 9 \zeta_{6} ) q^{67} -4 q^{69} + 2 q^{71} + ( -1 + \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + ( 6 - 4 \zeta_{6} ) q^{77} + 3 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -18 q^{83} -4 q^{85} + 4 \zeta_{6} q^{89} + ( 2 + \zeta_{6} ) q^{91} -5 \zeta_{6} q^{93} + ( 1 - \zeta_{6} ) q^{95} + 10 q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{5} + 5 q^{7} - q^{9} + O(q^{10})$$ $$2 q + q^{3} - q^{5} + 5 q^{7} - q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} + 4 q^{17} + q^{19} + 4 q^{21} - 4 q^{23} - q^{25} - 2 q^{27} + 5 q^{31} - 2 q^{33} - q^{35} + 5 q^{37} + q^{39} + 4 q^{41} - 18 q^{43} - q^{45} + 2 q^{47} + 11 q^{49} - 4 q^{51} - 12 q^{53} - 4 q^{55} + 2 q^{57} + 8 q^{59} + 14 q^{61} - q^{63} - q^{65} - 9 q^{67} - 8 q^{69} + 4 q^{71} - q^{73} + q^{75} + 8 q^{77} + 3 q^{79} - q^{81} - 36 q^{83} - 8 q^{85} + 4 q^{89} + 5 q^{91} - 5 q^{93} + q^{95} + 20 q^{97} - 4 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$241$$ $$281$$ $$337$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 2.50000 0.866025i 0 −0.500000 + 0.866025i 0
361.1 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 2.50000 + 0.866025i 0 −0.500000 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.q.a 2
3.b odd 2 1 1260.2.s.d 2
4.b odd 2 1 1680.2.bg.a 2
5.b even 2 1 2100.2.q.a 2
5.c odd 4 2 2100.2.bc.c 4
7.b odd 2 1 2940.2.q.h 2
7.c even 3 1 inner 420.2.q.a 2
7.c even 3 1 2940.2.a.d 1
7.d odd 6 1 2940.2.a.h 1
7.d odd 6 1 2940.2.q.h 2
21.g even 6 1 8820.2.a.y 1
21.h odd 6 1 1260.2.s.d 2
21.h odd 6 1 8820.2.a.j 1
28.g odd 6 1 1680.2.bg.a 2
35.j even 6 1 2100.2.q.a 2
35.l odd 12 2 2100.2.bc.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.a 2 1.a even 1 1 trivial
420.2.q.a 2 7.c even 3 1 inner
1260.2.s.d 2 3.b odd 2 1
1260.2.s.d 2 21.h odd 6 1
1680.2.bg.a 2 4.b odd 2 1
1680.2.bg.a 2 28.g odd 6 1
2100.2.q.a 2 5.b even 2 1
2100.2.q.a 2 35.j even 6 1
2100.2.bc.c 4 5.c odd 4 2
2100.2.bc.c 4 35.l odd 12 2
2940.2.a.d 1 7.c even 3 1
2940.2.a.h 1 7.d odd 6 1
2940.2.q.h 2 7.b odd 2 1
2940.2.q.h 2 7.d odd 6 1
8820.2.a.j 1 21.h odd 6 1
8820.2.a.y 1 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} - 2 T_{11} + 4$$ acting on $$S_{2}^{\mathrm{new}}(420, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$25 - 5 T + T^{2}$$
$37$ $$25 - 5 T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$( 9 + T )^{2}$$
$47$ $$4 - 2 T + T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$64 - 8 T + T^{2}$$
$61$ $$196 - 14 T + T^{2}$$
$67$ $$81 + 9 T + T^{2}$$
$71$ $$( -2 + T )^{2}$$
$73$ $$1 + T + T^{2}$$
$79$ $$9 - 3 T + T^{2}$$
$83$ $$( 18 + T )^{2}$$
$89$ $$16 - 4 T + T^{2}$$
$97$ $$( -10 + T )^{2}$$