# Properties

 Label 420.2.k.a Level $420$ Weight $2$ Character orbit 420.k 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.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + ( -2 - i ) q^{5} + i q^{7} - q^{9} +O(q^{10})$$ $$q + i q^{3} + ( -2 - i ) q^{5} + i q^{7} - q^{9} -4 q^{11} + 6 i q^{13} + ( 1 - 2 i ) q^{15} + 2 i q^{17} -6 q^{19} - q^{21} -2 i q^{23} + ( 3 + 4 i ) q^{25} -i q^{27} -6 q^{29} -2 q^{31} -4 i q^{33} + ( 1 - 2 i ) q^{35} -4 i q^{37} -6 q^{39} + 8 q^{41} + 4 i q^{43} + ( 2 + i ) q^{45} + 4 i q^{47} - q^{49} -2 q^{51} -6 i q^{53} + ( 8 + 4 i ) q^{55} -6 i q^{57} -4 q^{59} + 14 q^{61} -i q^{63} + ( 6 - 12 i ) q^{65} + 4 i q^{67} + 2 q^{69} + 10 i q^{73} + ( -4 + 3 i ) q^{75} -4 i q^{77} + q^{81} + 16 i q^{83} + ( 2 - 4 i ) q^{85} -6 i q^{87} -8 q^{89} -6 q^{91} -2 i q^{93} + ( 12 + 6 i ) q^{95} + 10 i q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5} - 2 q^{9} + O(q^{10})$$ $$2 q - 4 q^{5} - 2 q^{9} - 8 q^{11} + 2 q^{15} - 12 q^{19} - 2 q^{21} + 6 q^{25} - 12 q^{29} - 4 q^{31} + 2 q^{35} - 12 q^{39} + 16 q^{41} + 4 q^{45} - 2 q^{49} - 4 q^{51} + 16 q^{55} - 8 q^{59} + 28 q^{61} + 12 q^{65} + 4 q^{69} - 8 q^{75} + 2 q^{81} + 4 q^{85} - 16 q^{89} - 12 q^{91} + 24 q^{95} + 8 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$211$$ $$241$$ $$281$$ $$337$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 −2.00000 + 1.00000i 0 1.00000i 0 −1.00000 0
169.2 0 1.00000i 0 −2.00000 1.00000i 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 420.2.k.a 2
3.b odd 2 1 1260.2.k.d 2
4.b odd 2 1 1680.2.t.a 2
5.b even 2 1 inner 420.2.k.a 2
5.c odd 4 1 2100.2.a.e 1
5.c odd 4 1 2100.2.a.j 1
7.b odd 2 1 2940.2.k.d 2
7.c even 3 2 2940.2.bb.h 4
7.d odd 6 2 2940.2.bb.c 4
12.b even 2 1 5040.2.t.o 2
15.d odd 2 1 1260.2.k.d 2
15.e even 4 1 6300.2.a.n 1
15.e even 4 1 6300.2.a.bc 1
20.d odd 2 1 1680.2.t.a 2
20.e even 4 1 8400.2.a.bh 1
20.e even 4 1 8400.2.a.cd 1
35.c odd 2 1 2940.2.k.d 2
35.i odd 6 2 2940.2.bb.c 4
35.j even 6 2 2940.2.bb.h 4
60.h even 2 1 5040.2.t.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.k.a 2 1.a even 1 1 trivial
420.2.k.a 2 5.b even 2 1 inner
1260.2.k.d 2 3.b odd 2 1
1260.2.k.d 2 15.d odd 2 1
1680.2.t.a 2 4.b odd 2 1
1680.2.t.a 2 20.d odd 2 1
2100.2.a.e 1 5.c odd 4 1
2100.2.a.j 1 5.c odd 4 1
2940.2.k.d 2 7.b odd 2 1
2940.2.k.d 2 35.c odd 2 1
2940.2.bb.c 4 7.d odd 6 2
2940.2.bb.c 4 35.i odd 6 2
2940.2.bb.h 4 7.c even 3 2
2940.2.bb.h 4 35.j even 6 2
5040.2.t.o 2 12.b even 2 1
5040.2.t.o 2 60.h even 2 1
6300.2.a.n 1 15.e even 4 1
6300.2.a.bc 1 15.e even 4 1
8400.2.a.bh 1 20.e even 4 1
8400.2.a.cd 1 20.e even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$5 + 4 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$16 + T^{2}$$
$41$ $$( -8 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( -14 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$256 + T^{2}$$
$89$ $$( 8 + T )^{2}$$
$97$ $$100 + T^{2}$$