# Properties

 Label 2520.1.h.a Level $2520$ Weight $1$ Character orbit 2520.h Self dual yes Analytic conductor $1.258$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -24, -280, 105 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2520.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.25764383184$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-6}, \sqrt{-70})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.60480.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + q^{10} + q^{14} + q^{16} - q^{20} + q^{25} - q^{28} - q^{32} + q^{35} + q^{40} + q^{49} - q^{50} + 2q^{53} + q^{56} + 2q^{59} + q^{64} - q^{70} + 2q^{73} - 2q^{79} - q^{80} - 2q^{97} - q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$281$$ $$631$$ $$1081$$ $$1261$$ $$2017$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1189.1
 0
−1.00000 0 1.00000 −1.00000 0 −1.00000 −1.00000 0 1.00000
 $$n$$: 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
105.g even 2 1 RM by $$\Q(\sqrt{105})$$
280.c odd 2 1 CM by $$\Q(\sqrt{-70})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.1.h.a 1
3.b odd 2 1 2520.1.h.d yes 1
5.b even 2 1 2520.1.h.c yes 1
7.b odd 2 1 2520.1.h.b yes 1
8.b even 2 1 2520.1.h.d yes 1
15.d odd 2 1 2520.1.h.b yes 1
21.c even 2 1 2520.1.h.c yes 1
24.h odd 2 1 CM 2520.1.h.a 1
35.c odd 2 1 2520.1.h.d yes 1
40.f even 2 1 2520.1.h.b yes 1
56.h odd 2 1 2520.1.h.c yes 1
105.g even 2 1 RM 2520.1.h.a 1
120.i odd 2 1 2520.1.h.c yes 1
168.i even 2 1 2520.1.h.b yes 1
280.c odd 2 1 CM 2520.1.h.a 1
840.u even 2 1 2520.1.h.d yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.1.h.a 1 1.a even 1 1 trivial
2520.1.h.a 1 24.h odd 2 1 CM
2520.1.h.a 1 105.g even 2 1 RM
2520.1.h.a 1 280.c odd 2 1 CM
2520.1.h.b yes 1 7.b odd 2 1
2520.1.h.b yes 1 15.d odd 2 1
2520.1.h.b yes 1 40.f even 2 1
2520.1.h.b yes 1 168.i even 2 1
2520.1.h.c yes 1 5.b even 2 1
2520.1.h.c yes 1 21.c even 2 1
2520.1.h.c yes 1 56.h odd 2 1
2520.1.h.c yes 1 120.i odd 2 1
2520.1.h.d yes 1 3.b odd 2 1
2520.1.h.d yes 1 8.b even 2 1
2520.1.h.d yes 1 35.c odd 2 1
2520.1.h.d yes 1 840.u even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2520, [\chi])$$:

 $$T_{13}$$ $$T_{53} - 2$$ $$T_{59} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$1 + T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$-2 + T$$
$59$ $$-2 + T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$-2 + T$$
$79$ $$2 + T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$2 + T$$