# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} ) q^{3} - q^{5} + ( -1 + \beta_{1} + \beta_{2} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} ) q^{3} - q^{5} + ( -1 + \beta_{1} + \beta_{2} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{3} ) q^{13} + ( -1 - \beta_{1} ) q^{15} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{17} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{19} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{21} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{23} + q^{25} + ( 3 + 2 \beta_{2} + \beta_{3} ) q^{27} -2 \beta_{2} q^{29} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{31} + ( 2 - \beta_{1} + 3 \beta_{3} ) q^{33} + ( 1 - \beta_{1} - \beta_{2} ) q^{35} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{39} + ( -6 + 2 \beta_{1} - 2 \beta_{3} ) q^{41} + ( -6 - \beta_{1} + \beta_{3} ) q^{43} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{45} + ( -2 - \beta_{1} + \beta_{3} ) q^{47} + ( -3 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{49} + ( 6 + 2 \beta_{2} - 2 \beta_{3} ) q^{51} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{53} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{55} + ( -2 + \beta_{1} + 4 \beta_{2} + 5 \beta_{3} ) q^{57} + ( 4 + 4 \beta_{1} - 4 \beta_{3} ) q^{59} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{61} + ( -3 + \beta_{2} + 5 \beta_{3} ) q^{63} + ( -\beta_{1} - \beta_{3} ) q^{65} + ( 6 + 3 \beta_{1} - 3 \beta_{3} ) q^{67} + ( 6 - 3 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{69} + ( 5 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} ) q^{71} + ( \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{73} + ( 1 + \beta_{1} ) q^{75} + ( -2 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{77} + ( 8 + 2 \beta_{1} - 2 \beta_{3} ) q^{79} + ( 1 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{81} + ( -6 - \beta_{1} + \beta_{3} ) q^{83} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{85} + ( -2 \beta_{2} - 4 \beta_{3} ) q^{87} + ( -10 - 2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{91} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{93} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{95} + ( -5 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} ) q^{97} + ( -4 + 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + O(q^{10})$$ $$4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} - 2 q^{15} + 4 q^{25} + 14 q^{27} + 16 q^{33} + 6 q^{35} - 12 q^{39} - 32 q^{41} - 20 q^{43} - 4 q^{47} + 20 q^{51} - 2 q^{63} + 12 q^{67} + 28 q^{69} + 2 q^{75} - 8 q^{77} + 24 q^{79} + 4 q^{81} - 20 q^{83} - 8 q^{87} - 32 q^{89} - 4 q^{91} + 8 q^{93} - 24 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 − 1.61803i 1.61803i − 0.618034i 0.618034i
0 −0.618034 1.61803i 0 −1.00000 0 −2.61803 + 0.381966i 0 −2.23607 + 2.00000i 0
41.2 0 −0.618034 + 1.61803i 0 −1.00000 0 −2.61803 0.381966i 0 −2.23607 2.00000i 0
41.3 0 1.61803 0.618034i 0 −1.00000 0 −0.381966 2.61803i 0 2.23607 2.00000i 0
41.4 0 1.61803 + 0.618034i 0 −1.00000 0 −0.381966 + 2.61803i 0 2.23607 + 2.00000i 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
21.c 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.d.d yes 4
3.b odd 2 1 420.2.d.c 4
4.b odd 2 1 1680.2.f.f 4
5.b even 2 1 2100.2.d.g 4
5.c odd 4 1 2100.2.f.b 4
5.c odd 4 1 2100.2.f.h 4
7.b odd 2 1 420.2.d.c 4
12.b even 2 1 1680.2.f.j 4
15.d odd 2 1 2100.2.d.h 4
15.e even 4 1 2100.2.f.a 4
15.e even 4 1 2100.2.f.g 4
21.c even 2 1 inner 420.2.d.d yes 4
28.d even 2 1 1680.2.f.j 4
35.c odd 2 1 2100.2.d.h 4
35.f even 4 1 2100.2.f.a 4
35.f even 4 1 2100.2.f.g 4
84.h odd 2 1 1680.2.f.f 4
105.g even 2 1 2100.2.d.g 4
105.k odd 4 1 2100.2.f.b 4
105.k odd 4 1 2100.2.f.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.d.c 4 3.b odd 2 1
420.2.d.c 4 7.b odd 2 1
420.2.d.d yes 4 1.a even 1 1 trivial
420.2.d.d yes 4 21.c even 2 1 inner
1680.2.f.f 4 4.b odd 2 1
1680.2.f.f 4 84.h odd 2 1
1680.2.f.j 4 12.b even 2 1
1680.2.f.j 4 28.d even 2 1
2100.2.d.g 4 5.b even 2 1
2100.2.d.g 4 105.g even 2 1
2100.2.d.h 4 15.d odd 2 1
2100.2.d.h 4 35.c odd 2 1
2100.2.f.a 4 15.e even 4 1
2100.2.f.a 4 35.f even 4 1
2100.2.f.b 4 5.c odd 4 1
2100.2.f.b 4 105.k odd 4 1
2100.2.f.g 4 15.e even 4 1
2100.2.f.g 4 35.f even 4 1
2100.2.f.h 4 5.c odd 4 1
2100.2.f.h 4 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(420, [\chi])$$:

 $$T_{11}^{4} + 28 T_{11}^{2} + 16$$ $$T_{17}^{2} - 20$$ $$T_{41}^{2} + 16 T_{41} + 44$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 - 6 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$49 + 42 T + 18 T^{2} + 6 T^{3} + T^{4}$$
$11$ $$16 + 28 T^{2} + T^{4}$$
$13$ $$16 + 12 T^{2} + T^{4}$$
$17$ $$( -20 + T^{2} )^{2}$$
$19$ $$400 + 60 T^{2} + T^{4}$$
$23$ $$1936 + 92 T^{2} + T^{4}$$
$29$ $$( 16 + T^{2} )^{2}$$
$31$ $$16 + 12 T^{2} + T^{4}$$
$37$ $$( -20 + T^{2} )^{2}$$
$41$ $$( 44 + 16 T + T^{2} )^{2}$$
$43$ $$( 20 + 10 T + T^{2} )^{2}$$
$47$ $$( -4 + 2 T + T^{2} )^{2}$$
$53$ $$1936 + 108 T^{2} + T^{4}$$
$59$ $$( -80 + T^{2} )^{2}$$
$61$ $$4096 + 192 T^{2} + T^{4}$$
$67$ $$( -36 - 6 T + T^{2} )^{2}$$
$71$ $$15376 + 252 T^{2} + T^{4}$$
$73$ $$5776 + 172 T^{2} + T^{4}$$
$79$ $$( 16 - 12 T + T^{2} )^{2}$$
$83$ $$( 20 + 10 T + T^{2} )^{2}$$
$89$ $$( 44 + 16 T + T^{2} )^{2}$$
$97$ $$13456 + 268 T^{2} + T^{4}$$