# Properties

 Label 2520.2.bi.e 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: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 3) q^{7} +O(q^{10})$$ q - z * q^5 + (-2*z + 3) * q^7 $$q - \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 3) q^{7} + (\zeta_{6} - 1) q^{11} - 3 q^{13} + (2 \zeta_{6} - 2) q^{17} + 5 \zeta_{6} q^{19} + 7 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 6 q^{29} + (4 \zeta_{6} - 4) q^{31} + ( - \zeta_{6} - 2) q^{35} + 5 \zeta_{6} q^{37} + 5 q^{41} + 6 q^{43} - 9 \zeta_{6} q^{47} + ( - 8 \zeta_{6} + 5) q^{49} + ( - 11 \zeta_{6} + 11) q^{53} + q^{55} + ( - 8 \zeta_{6} + 8) q^{59} + 12 \zeta_{6} q^{61} + 3 \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{67} + 4 q^{71} + (12 \zeta_{6} - 12) q^{73} + (3 \zeta_{6} - 1) q^{77} - 14 \zeta_{6} q^{79} + 4 q^{83} + 2 q^{85} + 6 \zeta_{6} q^{89} + (6 \zeta_{6} - 9) q^{91} + ( - 5 \zeta_{6} + 5) q^{95} + 6 q^{97} +O(q^{100})$$ q - z * q^5 + (-2*z + 3) * q^7 + (z - 1) * q^11 - 3 * q^13 + (2*z - 2) * q^17 + 5*z * q^19 + 7*z * q^23 + (z - 1) * q^25 + 6 * q^29 + (4*z - 4) * q^31 + (-z - 2) * q^35 + 5*z * q^37 + 5 * q^41 + 6 * q^43 - 9*z * q^47 + (-8*z + 5) * q^49 + (-11*z + 11) * q^53 + q^55 + (-8*z + 8) * q^59 + 12*z * q^61 + 3*z * q^65 + (-4*z + 4) * q^67 + 4 * q^71 + (12*z - 12) * q^73 + (3*z - 1) * q^77 - 14*z * q^79 + 4 * q^83 + 2 * q^85 + 6*z * q^89 + (6*z - 9) * q^91 + (-5*z + 5) * q^95 + 6 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q - q^5 + 4 * q^7 $$2 q - q^{5} + 4 q^{7} - q^{11} - 6 q^{13} - 2 q^{17} + 5 q^{19} + 7 q^{23} - q^{25} + 12 q^{29} - 4 q^{31} - 5 q^{35} + 5 q^{37} + 10 q^{41} + 12 q^{43} - 9 q^{47} + 2 q^{49} + 11 q^{53} + 2 q^{55} + 8 q^{59} + 12 q^{61} + 3 q^{65} + 4 q^{67} + 8 q^{71} - 12 q^{73} + q^{77} - 14 q^{79} + 8 q^{83} + 4 q^{85} + 6 q^{89} - 12 q^{91} + 5 q^{95} + 12 q^{97}+O(q^{100})$$ 2 * q - q^5 + 4 * q^7 - q^11 - 6 * q^13 - 2 * q^17 + 5 * q^19 + 7 * q^23 - q^25 + 12 * q^29 - 4 * q^31 - 5 * q^35 + 5 * q^37 + 10 * q^41 + 12 * q^43 - 9 * q^47 + 2 * q^49 + 11 * q^53 + 2 * q^55 + 8 * q^59 + 12 * q^61 + 3 * q^65 + 4 * q^67 + 8 * q^71 - 12 * q^73 + q^77 - 14 * q^79 + 8 * q^83 + 4 * q^85 + 6 * q^89 - 12 * q^91 + 5 * q^95 + 12 * q^97

## 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 2.00000 1.73205i 0 0 0
1801.1 0 0 0 −0.500000 + 0.866025i 0 2.00000 + 1.73205i 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.e 2
3.b odd 2 1 280.2.q.c 2
7.c even 3 1 inner 2520.2.bi.e 2
12.b even 2 1 560.2.q.c 2
15.d odd 2 1 1400.2.q.a 2
15.e even 4 2 1400.2.bh.e 4
21.c even 2 1 1960.2.q.c 2
21.g even 6 1 1960.2.a.m 1
21.g even 6 1 1960.2.q.c 2
21.h odd 6 1 280.2.q.c 2
21.h odd 6 1 1960.2.a.a 1
84.j odd 6 1 3920.2.a.i 1
84.n even 6 1 560.2.q.c 2
84.n even 6 1 3920.2.a.bf 1
105.o odd 6 1 1400.2.q.a 2
105.o odd 6 1 9800.2.a.bi 1
105.p even 6 1 9800.2.a.g 1
105.x even 12 2 1400.2.bh.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.c 2 3.b odd 2 1
280.2.q.c 2 21.h odd 6 1
560.2.q.c 2 12.b even 2 1
560.2.q.c 2 84.n even 6 1
1400.2.q.a 2 15.d odd 2 1
1400.2.q.a 2 105.o odd 6 1
1400.2.bh.e 4 15.e even 4 2
1400.2.bh.e 4 105.x even 12 2
1960.2.a.a 1 21.h odd 6 1
1960.2.a.m 1 21.g even 6 1
1960.2.q.c 2 21.c even 2 1
1960.2.q.c 2 21.g even 6 1
2520.2.bi.e 2 1.a even 1 1 trivial
2520.2.bi.e 2 7.c even 3 1 inner
3920.2.a.i 1 84.j odd 6 1
3920.2.a.bf 1 84.n even 6 1
9800.2.a.g 1 105.p even 6 1
9800.2.a.bi 1 105.o odd 6 1

## Hecke kernels

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

 $$T_{11}^{2} + T_{11} + 1$$ T11^2 + T11 + 1 $$T_{13} + 3$$ T13 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} - 4T + 7$$
$11$ $$T^{2} + T + 1$$
$13$ $$(T + 3)^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} - 5T + 25$$
$23$ $$T^{2} - 7T + 49$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$T^{2} - 5T + 25$$
$41$ $$(T - 5)^{2}$$
$43$ $$(T - 6)^{2}$$
$47$ $$T^{2} + 9T + 81$$
$53$ $$T^{2} - 11T + 121$$
$59$ $$T^{2} - 8T + 64$$
$61$ $$T^{2} - 12T + 144$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T - 4)^{2}$$
$73$ $$T^{2} + 12T + 144$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T - 6)^{2}$$