# Properties

 Label 2520.2.bi.f Level $2520$ Weight $2$ Character orbit 2520.bi Analytic conductor $20.122$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2520.bi (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.1223013094$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 840) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -\zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} + ( 2 - 2 \zeta_{6} ) q^{11} + 5 q^{13} + ( 4 - 4 \zeta_{6} ) q^{17} -3 \zeta_{6} q^{19} -4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -4 q^{29} + ( 1 - \zeta_{6} ) q^{31} + ( 1 - 3 \zeta_{6} ) q^{35} + \zeta_{6} q^{37} -6 q^{41} - q^{43} + 10 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} -2 q^{55} + 6 \zeta_{6} q^{61} -5 \zeta_{6} q^{65} + ( 15 - 15 \zeta_{6} ) q^{67} + 6 q^{71} + ( 3 - 3 \zeta_{6} ) q^{73} + ( 6 - 4 \zeta_{6} ) q^{77} -9 \zeta_{6} q^{79} -6 q^{83} -4 q^{85} + ( 10 + 5 \zeta_{6} ) q^{91} + ( -3 + 3 \zeta_{6} ) q^{95} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{5} + 5q^{7} + O(q^{10})$$ $$2q - q^{5} + 5q^{7} + 2q^{11} + 10q^{13} + 4q^{17} - 3q^{19} - 4q^{23} - q^{25} - 8q^{29} + q^{31} - q^{35} + q^{37} - 12q^{41} - 2q^{43} + 10q^{47} + 11q^{49} - 4q^{55} + 6q^{61} - 5q^{65} + 15q^{67} + 12q^{71} + 3q^{73} + 8q^{77} - 9q^{79} - 12q^{83} - 8q^{85} + 25q^{91} - 3q^{95} + 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$1081$$ $$1261$$ $$2017$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −0.500000 0.866025i 0 2.50000 + 0.866025i 0 0 0
1801.1 0 0 0 −0.500000 + 0.866025i 0 2.50000 0.866025i 0 0 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 2520.2.bi.f 2
3.b odd 2 1 840.2.bg.f 2
7.c even 3 1 inner 2520.2.bi.f 2
12.b even 2 1 1680.2.bg.f 2
21.g even 6 1 5880.2.a.bg 1
21.h odd 6 1 840.2.bg.f 2
21.h odd 6 1 5880.2.a.e 1
84.n even 6 1 1680.2.bg.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bg.f 2 3.b odd 2 1
840.2.bg.f 2 21.h odd 6 1
1680.2.bg.f 2 12.b even 2 1
1680.2.bg.f 2 84.n even 6 1
2520.2.bi.f 2 1.a even 1 1 trivial
2520.2.bi.f 2 7.c even 3 1 inner
5880.2.a.e 1 21.h odd 6 1
5880.2.a.bg 1 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2520, [\chi])$$:

 $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{13} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$( -5 + T )^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$9 + 3 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$( 4 + T )^{2}$$
$31$ $$1 - T + T^{2}$$
$37$ $$1 - T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( 1 + T )^{2}$$
$47$ $$100 - 10 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$36 - 6 T + T^{2}$$
$67$ $$225 - 15 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$9 - 3 T + T^{2}$$
$79$ $$81 + 9 T + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$( -2 + T )^{2}$$