# Properties

 Label 420.2.f.b Level $420$ Weight $2$ Character orbit 420.f Analytic conductor $3.354$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} -\beta_{2} q^{5} + ( \beta_{5} + \beta_{6} ) q^{7} + ( 1 - \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{6} q^{3} -\beta_{2} q^{5} + ( \beta_{5} + \beta_{6} ) q^{7} + ( 1 - \beta_{4} ) q^{9} -\beta_{4} q^{11} + ( \beta_{3} - \beta_{6} ) q^{13} + ( -\beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{15} + ( -2 \beta_{3} - 2 \beta_{6} ) q^{17} + ( -2 \beta_{1} - \beta_{4} ) q^{19} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{21} + ( \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{23} + 5 q^{25} + ( 3 \beta_{3} + 2 \beta_{6} ) q^{27} -2 \beta_{4} q^{29} + ( 2 \beta_{1} + \beta_{4} ) q^{31} + ( 3 \beta_{3} + \beta_{6} ) q^{33} + ( -3 \beta_{3} - 3 \beta_{6} - \beta_{7} ) q^{35} + ( -2 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} ) q^{37} + ( -4 + \beta_{4} ) q^{39} -2 \beta_{2} q^{41} + ( \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{43} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{45} + ( -3 \beta_{3} - 3 \beta_{6} ) q^{47} + ( -3 - 2 \beta_{1} - \beta_{4} ) q^{49} + ( 4 + 2 \beta_{4} ) q^{51} + ( \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{53} + ( 2 \beta_{1} + \beta_{4} ) q^{55} + ( \beta_{3} + 4 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{57} + 4 \beta_{2} q^{59} + ( \beta_{3} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{63} + ( \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{65} + ( -\beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{67} + ( -2 \beta_{1} + 4 \beta_{2} - \beta_{4} ) q^{69} + 5 \beta_{4} q^{71} + ( -3 \beta_{3} + 3 \beta_{6} ) q^{73} + 5 \beta_{6} q^{75} + ( \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{77} -12 q^{79} + ( -7 - 2 \beta_{4} ) q^{81} + ( -5 \beta_{3} - 5 \beta_{6} ) q^{83} + ( 2 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} ) q^{85} + ( 6 \beta_{3} + 2 \beta_{6} ) q^{87} + 2 \beta_{2} q^{89} + ( -4 + 2 \beta_{1} + \beta_{4} ) q^{91} + ( -\beta_{3} - 4 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{93} + 5 \beta_{4} q^{95} + ( -3 \beta_{3} + 3 \beta_{6} ) q^{97} + ( -8 - \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{9} + O(q^{10})$$ $$8 q + 8 q^{9} + 16 q^{21} + 40 q^{25} - 32 q^{39} - 24 q^{49} + 32 q^{51} - 96 q^{79} - 56 q^{81} - 32 q^{91} - 64 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{7} + 16 \nu^{3} + 6 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{4} + 7$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{7} - \nu^{6} - \nu^{5} + 13 \nu^{3} - 8 \nu^{2} - 5 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$4 \nu^{7} + 2 \nu^{5} + 26 \nu^{3} + 10 \nu$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{6} - 10 \nu^{2}$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{7} - \nu^{6} + \nu^{5} - 13 \nu^{3} - 8 \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$-4 \nu^{7} + \nu^{6} + \nu^{5} - 29 \nu^{3} + 8 \nu^{2} + 11 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{3} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} + \beta_{5} - \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} + \beta_{6} - \beta_{4} - 3 \beta_{3} + 2 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{2} - 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{7} + 3 \beta_{6} + 3 \beta_{4} - 8 \beta_{3} - 5 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$5 \beta_{6} - 8 \beta_{5} + 5 \beta_{3}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$13 \beta_{7} - 8 \beta_{6} + 8 \beta_{4} + 21 \beta_{3} - 13 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 −0.437016 − 0.437016i 1.14412 + 1.14412i −0.437016 + 0.437016i 1.14412 − 1.14412i 0.437016 + 0.437016i −1.14412 − 1.14412i 0.437016 − 0.437016i −1.14412 + 1.14412i
0 −1.41421 1.00000i 0 −2.23607 0 −1.41421 2.23607i 0 1.00000 + 2.82843i 0
209.2 0 −1.41421 1.00000i 0 2.23607 0 −1.41421 + 2.23607i 0 1.00000 + 2.82843i 0
209.3 0 −1.41421 + 1.00000i 0 −2.23607 0 −1.41421 + 2.23607i 0 1.00000 2.82843i 0
209.4 0 −1.41421 + 1.00000i 0 2.23607 0 −1.41421 2.23607i 0 1.00000 2.82843i 0
209.5 0 1.41421 1.00000i 0 −2.23607 0 1.41421 2.23607i 0 1.00000 2.82843i 0
209.6 0 1.41421 1.00000i 0 2.23607 0 1.41421 + 2.23607i 0 1.00000 2.82843i 0
209.7 0 1.41421 + 1.00000i 0 −2.23607 0 1.41421 + 2.23607i 0 1.00000 + 2.82843i 0
209.8 0 1.41421 + 1.00000i 0 2.23607 0 1.41421 2.23607i 0 1.00000 + 2.82843i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.8 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
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g 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.f.b 8
3.b odd 2 1 inner 420.2.f.b 8
4.b odd 2 1 1680.2.k.g 8
5.b even 2 1 inner 420.2.f.b 8
5.c odd 4 1 2100.2.d.f 4
5.c odd 4 1 2100.2.d.i 4
7.b odd 2 1 inner 420.2.f.b 8
12.b even 2 1 1680.2.k.g 8
15.d odd 2 1 inner 420.2.f.b 8
15.e even 4 1 2100.2.d.f 4
15.e even 4 1 2100.2.d.i 4
20.d odd 2 1 1680.2.k.g 8
21.c even 2 1 inner 420.2.f.b 8
28.d even 2 1 1680.2.k.g 8
35.c odd 2 1 inner 420.2.f.b 8
35.f even 4 1 2100.2.d.f 4
35.f even 4 1 2100.2.d.i 4
60.h even 2 1 1680.2.k.g 8
84.h odd 2 1 1680.2.k.g 8
105.g even 2 1 inner 420.2.f.b 8
105.k odd 4 1 2100.2.d.f 4
105.k odd 4 1 2100.2.d.i 4
140.c even 2 1 1680.2.k.g 8
420.o odd 2 1 1680.2.k.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.f.b 8 1.a even 1 1 trivial
420.2.f.b 8 3.b odd 2 1 inner
420.2.f.b 8 5.b even 2 1 inner
420.2.f.b 8 7.b odd 2 1 inner
420.2.f.b 8 15.d odd 2 1 inner
420.2.f.b 8 21.c even 2 1 inner
420.2.f.b 8 35.c odd 2 1 inner
420.2.f.b 8 105.g even 2 1 inner
1680.2.k.g 8 4.b odd 2 1
1680.2.k.g 8 12.b even 2 1
1680.2.k.g 8 20.d odd 2 1
1680.2.k.g 8 28.d even 2 1
1680.2.k.g 8 60.h even 2 1
1680.2.k.g 8 84.h odd 2 1
1680.2.k.g 8 140.c even 2 1
1680.2.k.g 8 420.o odd 2 1
2100.2.d.f 4 5.c odd 4 1
2100.2.d.f 4 15.e even 4 1
2100.2.d.f 4 35.f even 4 1
2100.2.d.f 4 105.k odd 4 1
2100.2.d.i 4 5.c odd 4 1
2100.2.d.i 4 15.e even 4 1
2100.2.d.i 4 35.f even 4 1
2100.2.d.i 4 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(420, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 9 - 2 T^{2} + T^{4} )^{2}$$
$5$ $$( -5 + T^{2} )^{4}$$
$7$ $$( 49 + 6 T^{2} + T^{4} )^{2}$$
$11$ $$( 8 + T^{2} )^{4}$$
$13$ $$( -8 + T^{2} )^{4}$$
$17$ $$( 16 + T^{2} )^{4}$$
$19$ $$( 40 + T^{2} )^{4}$$
$23$ $$( -40 + T^{2} )^{4}$$
$29$ $$( 32 + T^{2} )^{4}$$
$31$ $$( 40 + T^{2} )^{4}$$
$37$ $$( 80 + T^{2} )^{4}$$
$41$ $$( -20 + T^{2} )^{4}$$
$43$ $$( 20 + T^{2} )^{4}$$
$47$ $$( 36 + T^{2} )^{4}$$
$53$ $$( -40 + T^{2} )^{4}$$
$59$ $$( -80 + T^{2} )^{4}$$
$61$ $$T^{8}$$
$67$ $$( 20 + T^{2} )^{4}$$
$71$ $$( 200 + T^{2} )^{4}$$
$73$ $$( -72 + T^{2} )^{4}$$
$79$ $$( 12 + T )^{8}$$
$83$ $$( 100 + T^{2} )^{4}$$
$89$ $$( -20 + T^{2} )^{4}$$
$97$ $$( -72 + T^{2} )^{4}$$