# Properties

 Label 4020.2.g.a Level 4020 Weight 2 Character orbit 4020.g Analytic conductor 32.100 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4020.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.0998616126$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + ( 1 - 2 i ) q^{5} - q^{9} +O(q^{10})$$ $$q + i q^{3} + ( 1 - 2 i ) q^{5} - q^{9} + 2 i q^{13} + ( 2 + i ) q^{15} -2 i q^{17} -8 i q^{23} + ( -3 - 4 i ) q^{25} -i q^{27} -6 q^{29} -2 q^{31} -2 q^{39} -6 q^{41} + 4 i q^{43} + ( -1 + 2 i ) q^{45} + 7 q^{49} + 2 q^{51} -4 i q^{53} -2 q^{59} -10 q^{61} + ( 4 + 2 i ) q^{65} -i q^{67} + 8 q^{69} -10 q^{71} -8 i q^{73} + ( 4 - 3 i ) q^{75} + 14 q^{79} + q^{81} -12 i q^{83} + ( -4 - 2 i ) q^{85} -6 i q^{87} + 2 q^{89} -2 i q^{93} + 14 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{5} - 2q^{9} + 4q^{15} - 6q^{25} - 12q^{29} - 4q^{31} - 4q^{39} - 12q^{41} - 2q^{45} + 14q^{49} + 4q^{51} - 4q^{59} - 20q^{61} + 8q^{65} + 16q^{69} - 20q^{71} + 8q^{75} + 28q^{79} + 2q^{81} - 8q^{85} + 4q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$$.

 $$n$$ $$1141$$ $$2011$$ $$2681$$ $$3217$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1609.1
 − 1.00000i 1.00000i
0 1.00000i 0 1.00000 + 2.00000i 0 0 0 −1.00000 0
1609.2 0 1.00000i 0 1.00000 2.00000i 0 0 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 4020.2.g.a 2
5.b even 2 1 inner 4020.2.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.g.a 2 1.a even 1 1 trivial
4020.2.g.a 2 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}$$ acting on $$S_{2}^{\mathrm{new}}(4020, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + T^{2}$$
$5$ $$1 - 2 T + 5 T^{2}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$1 - 22 T^{2} + 169 T^{4}$$
$17$ $$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$1 + 18 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 2 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 37 T^{2} )^{2}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$1 - 70 T^{2} + 1849 T^{4}$$
$47$ $$( 1 - 47 T^{2} )^{2}$$
$53$ $$( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} )$$
$59$ $$( 1 + 2 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 10 T + 61 T^{2} )^{2}$$
$67$ $$1 + T^{2}$$
$71$ $$( 1 + 10 T + 71 T^{2} )^{2}$$
$73$ $$1 - 82 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 14 T + 79 T^{2} )^{2}$$
$83$ $$1 - 22 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 2 T + 89 T^{2} )^{2}$$
$97$ $$1 + 2 T^{2} + 9409 T^{4}$$