# Properties

 Label 2520.2.bi.d 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{5} + (3 \zeta_{6} - 1) q^{7} +O(q^{10})$$ q - z * q^5 + (3*z - 1) * q^7 $$q - \zeta_{6} q^{5} + (3 \zeta_{6} - 1) q^{7} + (2 \zeta_{6} - 2) q^{11} + 4 q^{13} - 6 \zeta_{6} q^{19} + 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 3 q^{29} + ( - 2 \zeta_{6} + 3) q^{35} + 12 \zeta_{6} q^{37} + 7 q^{41} - 9 q^{43} + (3 \zeta_{6} - 8) q^{49} + (6 \zeta_{6} - 6) q^{53} + 2 q^{55} + (10 \zeta_{6} - 10) q^{59} - 5 \zeta_{6} q^{61} - 4 \zeta_{6} q^{65} + (11 \zeta_{6} - 11) q^{67} + 10 q^{71} + ( - 8 \zeta_{6} + 8) q^{73} + ( - 2 \zeta_{6} - 4) q^{77} - 6 \zeta_{6} q^{79} + 3 q^{83} + 17 \zeta_{6} q^{89} + (12 \zeta_{6} - 4) q^{91} + (6 \zeta_{6} - 6) q^{95} - 2 q^{97} +O(q^{100})$$ q - z * q^5 + (3*z - 1) * q^7 + (2*z - 2) * q^11 + 4 * q^13 - 6*z * q^19 + 3*z * q^23 + (z - 1) * q^25 + 3 * q^29 + (-2*z + 3) * q^35 + 12*z * q^37 + 7 * q^41 - 9 * q^43 + (3*z - 8) * q^49 + (6*z - 6) * q^53 + 2 * q^55 + (10*z - 10) * q^59 - 5*z * q^61 - 4*z * q^65 + (11*z - 11) * q^67 + 10 * q^71 + (-8*z + 8) * q^73 + (-2*z - 4) * q^77 - 6*z * q^79 + 3 * q^83 + 17*z * q^89 + (12*z - 4) * q^91 + (6*z - 6) * q^95 - 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} + q^{7}+O(q^{10})$$ 2 * q - q^5 + q^7 $$2 q - q^{5} + q^{7} - 2 q^{11} + 8 q^{13} - 6 q^{19} + 3 q^{23} - q^{25} + 6 q^{29} + 4 q^{35} + 12 q^{37} + 14 q^{41} - 18 q^{43} - 13 q^{49} - 6 q^{53} + 4 q^{55} - 10 q^{59} - 5 q^{61} - 4 q^{65} - 11 q^{67} + 20 q^{71} + 8 q^{73} - 10 q^{77} - 6 q^{79} + 6 q^{83} + 17 q^{89} + 4 q^{91} - 6 q^{95} - 4 q^{97}+O(q^{100})$$ 2 * q - q^5 + q^7 - 2 * q^11 + 8 * q^13 - 6 * q^19 + 3 * q^23 - q^25 + 6 * q^29 + 4 * q^35 + 12 * q^37 + 14 * q^41 - 18 * q^43 - 13 * q^49 - 6 * q^53 + 4 * q^55 - 10 * q^59 - 5 * q^61 - 4 * q^65 - 11 * q^67 + 20 * q^71 + 8 * q^73 - 10 * q^77 - 6 * q^79 + 6 * q^83 + 17 * q^89 + 4 * q^91 - 6 * q^95 - 4 * 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 0.500000 + 2.59808i 0 0 0
1801.1 0 0 0 −0.500000 + 0.866025i 0 0.500000 2.59808i 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.d 2
3.b odd 2 1 280.2.q.b 2
7.c even 3 1 inner 2520.2.bi.d 2
12.b even 2 1 560.2.q.e 2
15.d odd 2 1 1400.2.q.c 2
15.e even 4 2 1400.2.bh.c 4
21.c even 2 1 1960.2.q.d 2
21.g even 6 1 1960.2.a.l 1
21.g even 6 1 1960.2.q.d 2
21.h odd 6 1 280.2.q.b 2
21.h odd 6 1 1960.2.a.c 1
84.j odd 6 1 3920.2.a.q 1
84.n even 6 1 560.2.q.e 2
84.n even 6 1 3920.2.a.v 1
105.o odd 6 1 1400.2.q.c 2
105.o odd 6 1 9800.2.a.z 1
105.p even 6 1 9800.2.a.o 1
105.x even 12 2 1400.2.bh.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.b 2 3.b odd 2 1
280.2.q.b 2 21.h odd 6 1
560.2.q.e 2 12.b even 2 1
560.2.q.e 2 84.n even 6 1
1400.2.q.c 2 15.d odd 2 1
1400.2.q.c 2 105.o odd 6 1
1400.2.bh.c 4 15.e even 4 2
1400.2.bh.c 4 105.x even 12 2
1960.2.a.c 1 21.h odd 6 1
1960.2.a.l 1 21.g even 6 1
1960.2.q.d 2 21.c even 2 1
1960.2.q.d 2 21.g even 6 1
2520.2.bi.d 2 1.a even 1 1 trivial
2520.2.bi.d 2 7.c even 3 1 inner
3920.2.a.q 1 84.j odd 6 1
3920.2.a.v 1 84.n even 6 1
9800.2.a.o 1 105.p even 6 1
9800.2.a.z 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 $$S_{2}^{\mathrm{new}}(2520, [\chi])$$:

 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} - T + 7$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 6T + 36$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 12T + 144$$
$41$ $$(T - 7)^{2}$$
$43$ $$(T + 9)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} + 10T + 100$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} - 8T + 64$$
$79$ $$T^{2} + 6T + 36$$
$83$ $$(T - 3)^{2}$$
$89$ $$T^{2} - 17T + 289$$
$97$ $$(T + 2)^{2}$$