# Properties

 Label 420.2.l.a Level $420$ Weight $2$ Character orbit 420.l Analytic conductor $3.354$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 420.l (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{2} + ( -1 + \zeta_{8} + \zeta_{8}^{3} ) q^{3} -2 q^{4} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( 2 + \zeta_{8} + \zeta_{8}^{3} ) q^{6} + q^{7} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{2} + ( -1 + \zeta_{8} + \zeta_{8}^{3} ) q^{3} -2 q^{4} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( 2 + \zeta_{8} + \zeta_{8}^{3} ) q^{6} + q^{7} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} + ( 1 - 3 \zeta_{8}^{2} ) q^{10} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{11} + ( 2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{12} + 6 \zeta_{8}^{2} q^{13} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{14} + ( -1 - 2 \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{15} + 4 q^{16} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{17} + ( -4 + \zeta_{8} + \zeta_{8}^{3} ) q^{18} + 6 \zeta_{8}^{2} q^{19} + ( -4 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{20} + ( -1 + \zeta_{8} + \zeta_{8}^{3} ) q^{21} + 6 \zeta_{8}^{2} q^{22} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{23} + ( -4 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{24} + ( 4 + 3 \zeta_{8}^{2} ) q^{25} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{26} + ( 5 + \zeta_{8} + \zeta_{8}^{3} ) q^{27} -2 q^{28} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{29} + ( -1 + 4 \zeta_{8} + 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{30} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} + ( 3 \zeta_{8} - 6 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{33} -6 \zeta_{8}^{2} q^{34} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{35} + ( 2 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{36} + 6 \zeta_{8}^{2} q^{37} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{38} + ( -6 \zeta_{8} - 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{39} + ( -2 + 6 \zeta_{8}^{2} ) q^{40} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{41} + ( 2 + \zeta_{8} + \zeta_{8}^{3} ) q^{42} + 8 q^{43} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{44} + ( 2 - 2 \zeta_{8} - 6 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{45} + 2 q^{46} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{47} + ( -4 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{48} + q^{49} + ( -\zeta_{8} - 7 \zeta_{8}^{3} ) q^{50} + ( -3 \zeta_{8} + 6 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{51} -12 \zeta_{8}^{2} q^{52} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{53} + ( 2 - 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{54} + ( -9 - 3 \zeta_{8}^{2} ) q^{55} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{56} + ( -6 \zeta_{8} - 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{57} -4 q^{58} + ( 2 + 4 \zeta_{8} - 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{60} -10 q^{61} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{63} -8 q^{64} + ( 6 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{65} + ( -6 \zeta_{8} - 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{66} -4 q^{67} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{68} + ( -2 - \zeta_{8} - \zeta_{8}^{3} ) q^{69} + ( 1 - 3 \zeta_{8}^{2} ) q^{70} + ( 9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{71} + ( 8 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{72} -6 \zeta_{8}^{2} q^{73} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{74} + ( -4 + \zeta_{8} - 3 \zeta_{8}^{2} + 7 \zeta_{8}^{3} ) q^{75} -12 \zeta_{8}^{2} q^{76} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{77} + ( -6 \zeta_{8} + 12 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{78} + ( 8 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{80} + ( -7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} + 2 q^{82} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{83} + ( 2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{84} + ( 9 + 3 \zeta_{8}^{2} ) q^{85} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{86} + ( 4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{87} -12 \zeta_{8}^{2} q^{88} + ( -5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{89} + ( -1 - 8 \zeta_{8} + 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{90} + 6 \zeta_{8}^{2} q^{91} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{92} -4 q^{94} + ( 6 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{95} + ( 8 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{96} -6 \zeta_{8}^{2} q^{97} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{98} + ( 3 \zeta_{8} + 12 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 8 q^{4} + 8 q^{6} + 4 q^{7} - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} - 8 q^{4} + 8 q^{6} + 4 q^{7} - 4 q^{9} + 4 q^{10} + 8 q^{12} - 4 q^{15} + 16 q^{16} - 16 q^{18} - 4 q^{21} - 16 q^{24} + 16 q^{25} + 20 q^{27} - 8 q^{28} - 4 q^{30} + 8 q^{36} - 8 q^{40} + 8 q^{42} + 32 q^{43} + 8 q^{45} + 8 q^{46} - 16 q^{48} + 4 q^{49} + 8 q^{54} - 36 q^{55} - 16 q^{58} + 8 q^{60} - 40 q^{61} - 4 q^{63} - 32 q^{64} - 16 q^{67} - 8 q^{69} + 4 q^{70} + 32 q^{72} - 16 q^{75} - 28 q^{81} + 8 q^{82} + 8 q^{84} + 36 q^{85} + 16 q^{87} - 4 q^{90} - 16 q^{94} + 32 q^{96} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i
1.41421i −1.00000 + 1.41421i −2.00000 −2.12132 + 0.707107i 2.00000 + 1.41421i 1.00000 2.82843i −1.00000 2.82843i 1.00000 + 3.00000i
239.2 1.41421i −1.00000 + 1.41421i −2.00000 2.12132 + 0.707107i 2.00000 + 1.41421i 1.00000 2.82843i −1.00000 2.82843i 1.00000 3.00000i
239.3 1.41421i −1.00000 1.41421i −2.00000 −2.12132 0.707107i 2.00000 1.41421i 1.00000 2.82843i −1.00000 + 2.82843i 1.00000 3.00000i
239.4 1.41421i −1.00000 1.41421i −2.00000 2.12132 0.707107i 2.00000 1.41421i 1.00000 2.82843i −1.00000 + 2.82843i 1.00000 + 3.00000i
 $$n$$: 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
3.b odd 2 1 inner
20.d odd 2 1 inner
60.h 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.l.a 4
3.b odd 2 1 inner 420.2.l.a 4
4.b odd 2 1 420.2.l.b yes 4
5.b even 2 1 420.2.l.b yes 4
12.b even 2 1 420.2.l.b yes 4
15.d odd 2 1 420.2.l.b yes 4
20.d odd 2 1 inner 420.2.l.a 4
60.h even 2 1 inner 420.2.l.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.l.a 4 1.a even 1 1 trivial
420.2.l.a 4 3.b odd 2 1 inner
420.2.l.a 4 20.d odd 2 1 inner
420.2.l.a 4 60.h even 2 1 inner
420.2.l.b yes 4 4.b odd 2 1
420.2.l.b yes 4 5.b even 2 1
420.2.l.b yes 4 12.b even 2 1
420.2.l.b yes 4 15.d odd 2 1

## Hecke kernels

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

 $$T_{11}^{2} - 18$$ $$T_{17}^{2} - 18$$ $$T_{43} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$( 3 + 2 T + T^{2} )^{2}$$
$5$ $$25 - 8 T^{2} + T^{4}$$
$7$ $$( -1 + T )^{4}$$
$11$ $$( -18 + T^{2} )^{2}$$
$13$ $$( 36 + T^{2} )^{2}$$
$17$ $$( -18 + T^{2} )^{2}$$
$19$ $$( 36 + T^{2} )^{2}$$
$23$ $$( 2 + T^{2} )^{2}$$
$29$ $$( 8 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 36 + T^{2} )^{2}$$
$41$ $$( 2 + T^{2} )^{2}$$
$43$ $$( -8 + T )^{4}$$
$47$ $$( 8 + T^{2} )^{2}$$
$53$ $$( -72 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( 10 + T )^{4}$$
$67$ $$( 4 + T )^{4}$$
$71$ $$( -162 + T^{2} )^{2}$$
$73$ $$( 36 + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$( 8 + T^{2} )^{2}$$
$89$ $$( 50 + T^{2} )^{2}$$
$97$ $$( 36 + T^{2} )^{2}$$