# Properties

 Label 420.2.q.d Level $420$ Weight $2$ Character orbit 420.q Analytic conductor $3.354$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} ) q^{3} -\beta_{1} q^{5} + ( \beta_{1} + \beta_{3} ) q^{7} + \beta_{1} q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} ) q^{3} -\beta_{1} q^{5} + ( \beta_{1} + \beta_{3} ) q^{7} + \beta_{1} q^{9} + ( \beta_{2} + \beta_{3} ) q^{11} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{13} - q^{15} + ( -\beta_{2} - \beta_{3} ) q^{17} -7 \beta_{1} q^{19} + ( 1 - \beta_{2} ) q^{21} + ( -2 \beta_{2} + \beta_{3} ) q^{23} + ( -1 - \beta_{1} ) q^{25} + q^{27} + ( -6 - \beta_{2} + 2 \beta_{3} ) q^{29} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -2 \beta_{2} + \beta_{3} ) q^{33} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{35} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{37} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( \beta_{2} - 2 \beta_{3} ) q^{41} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{43} + ( 1 + \beta_{1} ) q^{45} -6 \beta_{1} q^{47} + ( 5 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{49} + ( 2 \beta_{2} - \beta_{3} ) q^{51} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -\beta_{2} + 2 \beta_{3} ) q^{55} -7 q^{57} + ( 6 + 6 \beta_{1} + \beta_{2} + \beta_{3} ) q^{59} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{63} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{65} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{67} + ( \beta_{2} - 2 \beta_{3} ) q^{69} + ( 3 \beta_{2} - 6 \beta_{3} ) q^{71} + ( -5 - 5 \beta_{1} - \beta_{2} - \beta_{3} ) q^{73} + \beta_{1} q^{75} + ( 6 + 12 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{77} + 11 \beta_{1} q^{79} + ( -1 - \beta_{1} ) q^{81} + ( -6 + \beta_{2} - 2 \beta_{3} ) q^{83} + ( \beta_{2} - 2 \beta_{3} ) q^{85} + ( 6 + 6 \beta_{1} - \beta_{2} - \beta_{3} ) q^{87} + ( -6 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{89} + ( 12 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{91} + ( -\beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{93} + ( -7 - 7 \beta_{1} ) q^{95} + ( 8 + 2 \beta_{2} - 4 \beta_{3} ) q^{97} + ( \beta_{2} - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 2 q^{5} - 2 q^{7} - 2 q^{9} + O(q^{10})$$ $$4 q - 2 q^{3} + 2 q^{5} - 2 q^{7} - 2 q^{9} - 4 q^{13} - 4 q^{15} + 14 q^{19} + 4 q^{21} - 2 q^{25} + 4 q^{27} - 24 q^{29} + 2 q^{31} + 2 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{43} + 2 q^{45} + 12 q^{47} + 10 q^{49} - 28 q^{57} + 12 q^{59} + 8 q^{61} - 2 q^{63} - 2 q^{65} + 2 q^{67} - 10 q^{73} - 2 q^{75} - 22 q^{79} - 2 q^{81} - 24 q^{83} + 12 q^{87} + 12 q^{89} + 38 q^{91} + 2 q^{93} - 14 q^{95} + 32 q^{97} + O(q^{100})$$

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

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

## Character values

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

 $$n$$ $$211$$ $$241$$ $$281$$ $$337$$ $$\chi(n)$$ $$1$$ $$\beta_{1}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 −0.500000 0.866025i 0 0.500000 0.866025i 0 −2.62132 0.358719i 0 −0.500000 + 0.866025i 0
121.2 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 1.62132 + 2.09077i 0 −0.500000 + 0.866025i 0
361.1 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 −2.62132 + 0.358719i 0 −0.500000 0.866025i 0
361.2 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 1.62132 2.09077i 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.d 4
3.b odd 2 1 1260.2.s.e 4
4.b odd 2 1 1680.2.bg.t 4
5.b even 2 1 2100.2.q.k 4
5.c odd 4 2 2100.2.bc.f 8
7.b odd 2 1 2940.2.q.q 4
7.c even 3 1 inner 420.2.q.d 4
7.c even 3 1 2940.2.a.r 2
7.d odd 6 1 2940.2.a.p 2
7.d odd 6 1 2940.2.q.q 4
21.g even 6 1 8820.2.a.bf 2
21.h odd 6 1 1260.2.s.e 4
21.h odd 6 1 8820.2.a.bk 2
28.g odd 6 1 1680.2.bg.t 4
35.j even 6 1 2100.2.q.k 4
35.l odd 12 2 2100.2.bc.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d 4 1.a even 1 1 trivial
420.2.q.d 4 7.c even 3 1 inner
1260.2.s.e 4 3.b odd 2 1
1260.2.s.e 4 21.h odd 6 1
1680.2.bg.t 4 4.b odd 2 1
1680.2.bg.t 4 28.g odd 6 1
2100.2.q.k 4 5.b even 2 1
2100.2.q.k 4 35.j even 6 1
2100.2.bc.f 8 5.c odd 4 2
2100.2.bc.f 8 35.l odd 12 2
2940.2.a.p 2 7.d odd 6 1
2940.2.a.r 2 7.c even 3 1
2940.2.q.q 4 7.b odd 2 1
2940.2.q.q 4 7.d odd 6 1
8820.2.a.bf 2 21.g even 6 1
8820.2.a.bk 2 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 18 T_{11}^{2} + 324$$ acting on $$S_{2}^{\mathrm{new}}(420, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$324 + 18 T^{2} + T^{4}$$
$13$ $$( -17 + 2 T + T^{2} )^{2}$$
$17$ $$324 + 18 T^{2} + T^{4}$$
$19$ $$( 49 - 7 T + T^{2} )^{2}$$
$23$ $$324 + 18 T^{2} + T^{4}$$
$29$ $$( 18 + 12 T + T^{2} )^{2}$$
$31$ $$5041 + 142 T + 75 T^{2} - 2 T^{3} + T^{4}$$
$37$ $$289 + 34 T + 21 T^{2} - 2 T^{3} + T^{4}$$
$41$ $$( -18 + T^{2} )^{2}$$
$43$ $$( -17 + 2 T + T^{2} )^{2}$$
$47$ $$( 36 - 6 T + T^{2} )^{2}$$
$53$ $$5184 + 72 T^{2} + T^{4}$$
$59$ $$324 - 216 T + 126 T^{2} - 12 T^{3} + T^{4}$$
$61$ $$3136 + 448 T + 120 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$289 + 34 T + 21 T^{2} - 2 T^{3} + T^{4}$$
$71$ $$( -162 + T^{2} )^{2}$$
$73$ $$49 + 70 T + 93 T^{2} + 10 T^{3} + T^{4}$$
$79$ $$( 121 + 11 T + T^{2} )^{2}$$
$83$ $$( 18 + 12 T + T^{2} )^{2}$$
$89$ $$324 - 216 T + 126 T^{2} - 12 T^{3} + T^{4}$$
$97$ $$( -8 - 16 T + T^{2} )^{2}$$