# Properties

 Label 2520.1.eq.a Level $2520$ Weight $1$ Character orbit 2520.eq Analytic conductor $1.258$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ CM discriminant -40 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2520.eq (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.25764383184$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{10} q^{2} -\zeta_{24}^{8} q^{4} -\zeta_{24}^{4} q^{5} + \zeta_{24}^{9} q^{7} -\zeta_{24}^{6} q^{8} +O(q^{10})$$ $$q -\zeta_{24}^{10} q^{2} -\zeta_{24}^{8} q^{4} -\zeta_{24}^{4} q^{5} + \zeta_{24}^{9} q^{7} -\zeta_{24}^{6} q^{8} -\zeta_{24}^{2} q^{10} + ( -\zeta_{24} - \zeta_{24}^{3} ) q^{11} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{13} + \zeta_{24}^{7} q^{14} -\zeta_{24}^{4} q^{16} -\zeta_{24}^{10} q^{19} - q^{20} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{22} + ( -1 + \zeta_{24}^{8} ) q^{23} + \zeta_{24}^{8} q^{25} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{26} + \zeta_{24}^{5} q^{28} -\zeta_{24}^{2} q^{32} + \zeta_{24} q^{35} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{37} -\zeta_{24}^{8} q^{38} + \zeta_{24}^{10} q^{40} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{41} + ( \zeta_{24}^{9} + \zeta_{24}^{11} ) q^{44} + ( \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{46} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{47} -\zeta_{24}^{6} q^{49} + \zeta_{24}^{6} q^{50} + ( -\zeta_{24}^{7} + \zeta_{24}^{9} ) q^{52} + \zeta_{24}^{2} q^{53} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{55} + \zeta_{24}^{3} q^{56} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{59} - q^{64} + ( -\zeta_{24}^{3} + \zeta_{24}^{5} ) q^{65} -\zeta_{24}^{11} q^{70} + ( -\zeta_{24}^{7} + \zeta_{24}^{9} ) q^{74} -\zeta_{24}^{6} q^{76} + ( 1 - \zeta_{24}^{10} ) q^{77} + \zeta_{24}^{8} q^{80} + ( -\zeta_{24}^{9} - \zeta_{24}^{11} ) q^{82} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{88} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{89} + ( \zeta_{24}^{8} - \zeta_{24}^{10} ) q^{91} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{92} + ( -1 - \zeta_{24}^{4} ) q^{94} -\zeta_{24}^{2} q^{95} -\zeta_{24}^{4} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} - 4q^{5} + O(q^{10})$$ $$8q + 4q^{4} - 4q^{5} - 4q^{16} - 8q^{20} - 12q^{23} - 4q^{25} + 4q^{38} - 8q^{64} + 8q^{77} - 4q^{80} - 4q^{91} - 12q^{94} - 4q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$281$$ $$631$$ $$1081$$ $$1261$$ $$2017$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{24}^{4}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
899.1
 −0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i
−0.866025 + 0.500000i 0 0.500000 0.866025i −0.500000 0.866025i 0 −0.707107 0.707107i 1.00000i 0 0.866025 + 0.500000i
899.2 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.500000 0.866025i 0 0.707107 + 0.707107i 1.00000i 0 0.866025 + 0.500000i
899.3 0.866025 0.500000i 0 0.500000 0.866025i −0.500000 0.866025i 0 −0.707107 + 0.707107i 1.00000i 0 −0.866025 0.500000i
899.4 0.866025 0.500000i 0 0.500000 0.866025i −0.500000 0.866025i 0 0.707107 0.707107i 1.00000i 0 −0.866025 0.500000i
1979.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.707107 + 0.707107i 1.00000i 0 0.866025 0.500000i
1979.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.707107 0.707107i 1.00000i 0 0.866025 0.500000i
1979.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.707107 0.707107i 1.00000i 0 −0.866025 + 0.500000i
1979.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.707107 + 0.707107i 1.00000i 0 −0.866025 + 0.500000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1979.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
15.d odd 2 1 inner
21.g even 6 1 inner
24.f even 2 1 inner
35.i odd 6 1 inner
56.m even 6 1 inner
840.ct odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.1.eq.a 8
3.b odd 2 1 2520.1.eq.b yes 8
5.b even 2 1 2520.1.eq.b yes 8
7.d odd 6 1 2520.1.eq.b yes 8
8.d odd 2 1 2520.1.eq.b yes 8
15.d odd 2 1 inner 2520.1.eq.a 8
21.g even 6 1 inner 2520.1.eq.a 8
24.f even 2 1 inner 2520.1.eq.a 8
35.i odd 6 1 inner 2520.1.eq.a 8
40.e odd 2 1 CM 2520.1.eq.a 8
56.m even 6 1 inner 2520.1.eq.a 8
105.p even 6 1 2520.1.eq.b yes 8
120.m even 2 1 2520.1.eq.b yes 8
168.be odd 6 1 2520.1.eq.b yes 8
280.ba even 6 1 2520.1.eq.b yes 8
840.ct odd 6 1 inner 2520.1.eq.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.1.eq.a 8 1.a even 1 1 trivial
2520.1.eq.a 8 15.d odd 2 1 inner
2520.1.eq.a 8 21.g even 6 1 inner
2520.1.eq.a 8 24.f even 2 1 inner
2520.1.eq.a 8 35.i odd 6 1 inner
2520.1.eq.a 8 40.e odd 2 1 CM
2520.1.eq.a 8 56.m even 6 1 inner
2520.1.eq.a 8 840.ct odd 6 1 inner
2520.1.eq.b yes 8 3.b odd 2 1
2520.1.eq.b yes 8 5.b even 2 1
2520.1.eq.b yes 8 7.d odd 6 1
2520.1.eq.b yes 8 8.d odd 2 1
2520.1.eq.b yes 8 105.p even 6 1
2520.1.eq.b yes 8 120.m even 2 1
2520.1.eq.b yes 8 168.be odd 6 1
2520.1.eq.b yes 8 280.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23}^{2} + 3 T_{23} + 3$$ acting on $$S_{1}^{\mathrm{new}}(2520, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 1 + T + T^{2} )^{4}$$
$7$ $$( 1 + T^{4} )^{2}$$
$11$ $$1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8}$$
$13$ $$( 1 + 4 T^{2} + T^{4} )^{2}$$
$17$ $$T^{8}$$
$19$ $$( 1 - T^{2} + T^{4} )^{2}$$
$23$ $$( 3 + 3 T + T^{2} )^{4}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8}$$
$41$ $$( 1 - 4 T^{2} + T^{4} )^{2}$$
$43$ $$T^{8}$$
$47$ $$( 9 + 3 T^{2} + T^{4} )^{2}$$
$53$ $$( 1 - T^{2} + T^{4} )^{2}$$
$59$ $$( 4 + 2 T^{2} + T^{4} )^{2}$$
$61$ $$T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$( 4 + 2 T^{2} + T^{4} )^{2}$$
$97$ $$T^{8}$$