# Properties

 Label 2520.2.bi.l Level $2520$ Weight $2$ Character orbit 2520.bi Analytic conductor $20.122$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + \beta_{1} q^{11} + ( -1 + 3 \beta_{3} ) q^{13} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{17} + ( 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{19} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{25} + ( 2 - 5 \beta_{3} ) q^{29} + ( 5 + 5 \beta_{2} ) q^{31} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{35} + ( \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{37} + ( 4 - 3 \beta_{3} ) q^{41} + ( -1 - 7 \beta_{3} ) q^{43} + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{47} + ( 5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{49} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{53} -\beta_{3} q^{55} + ( -10 - 3 \beta_{1} - 10 \beta_{2} ) q^{59} + ( 2 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} ) q^{61} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{65} + ( 1 + 7 \beta_{1} + \beta_{2} ) q^{67} + ( -4 - 5 \beta_{3} ) q^{71} + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 2 + 4 \beta_{2} + \beta_{3} ) q^{77} + ( 2 \beta_{1} - 9 \beta_{2} + 2 \beta_{3} ) q^{79} + ( -2 - 3 \beta_{3} ) q^{83} + ( -4 + \beta_{3} ) q^{85} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{89} + ( -12 - 4 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{91} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{95} -6 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} - 2q^{7} + O(q^{10})$$ $$4q + 2q^{5} - 2q^{7} - 4q^{13} - 8q^{17} + 6q^{19} - 2q^{25} + 8q^{29} + 10q^{31} + 2q^{35} + 18q^{37} + 16q^{41} - 4q^{43} - 4q^{47} + 10q^{49} - 8q^{53} - 20q^{59} - 24q^{61} - 2q^{65} + 2q^{67} - 16q^{71} + 6q^{73} + 18q^{79} - 8q^{83} - 16q^{85} - 4q^{89} - 34q^{91} - 6q^{95} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$281$$ $$631$$ $$1081$$ $$1261$$ $$2017$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{2}$$ $$1$$ $$1$$

## 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
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0 0 0 0.500000 + 0.866025i 0 −2.62132 + 0.358719i 0 0 0
361.2 0 0 0 0.500000 + 0.866025i 0 1.62132 2.09077i 0 0 0
1801.1 0 0 0 0.500000 0.866025i 0 −2.62132 0.358719i 0 0 0
1801.2 0 0 0 0.500000 0.866025i 0 1.62132 + 2.09077i 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.l 4
3.b odd 2 1 840.2.bg.g 4
7.c even 3 1 inner 2520.2.bi.l 4
12.b even 2 1 1680.2.bg.r 4
21.g even 6 1 5880.2.a.bm 2
21.h odd 6 1 840.2.bg.g 4
21.h odd 6 1 5880.2.a.bs 2
84.n even 6 1 1680.2.bg.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bg.g 4 3.b odd 2 1
840.2.bg.g 4 21.h odd 6 1
1680.2.bg.r 4 12.b even 2 1
1680.2.bg.r 4 84.n even 6 1
2520.2.bi.l 4 1.a even 1 1 trivial
2520.2.bi.l 4 7.c even 3 1 inner
5880.2.a.bm 2 21.g even 6 1
5880.2.a.bs 2 21.h odd 6 1

## Hecke kernels

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

 $$T_{11}^{4} + 2 T_{11}^{2} + 4$$ $$T_{13}^{2} + 2 T_{13} - 17$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$4 + 2 T^{2} + T^{4}$$
$13$ $$( -17 + 2 T + T^{2} )^{2}$$
$17$ $$196 + 112 T + 50 T^{2} + 8 T^{3} + T^{4}$$
$19$ $$1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$2500 + 50 T^{2} + T^{4}$$
$29$ $$( -46 - 4 T + T^{2} )^{2}$$
$31$ $$( 25 - 5 T + T^{2} )^{2}$$
$37$ $$6241 - 1422 T + 245 T^{2} - 18 T^{3} + T^{4}$$
$41$ $$( -2 - 8 T + T^{2} )^{2}$$
$43$ $$( -97 + 2 T + T^{2} )^{2}$$
$47$ $$784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4}$$
$53$ $$64 + 64 T + 56 T^{2} + 8 T^{3} + T^{4}$$
$59$ $$6724 + 1640 T + 318 T^{2} + 20 T^{3} + T^{4}$$
$61$ $$18496 + 3264 T + 440 T^{2} + 24 T^{3} + T^{4}$$
$67$ $$9409 + 194 T + 101 T^{2} - 2 T^{3} + T^{4}$$
$71$ $$( -34 + 8 T + T^{2} )^{2}$$
$73$ $$81 + 54 T + 45 T^{2} - 6 T^{3} + T^{4}$$
$79$ $$5329 - 1314 T + 251 T^{2} - 18 T^{3} + T^{4}$$
$83$ $$( -14 + 4 T + T^{2} )^{2}$$
$89$ $$196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4}$$
$97$ $$( -72 + T^{2} )^{2}$$