Properties

 Label 1260.1.cf.a Level $1260$ Weight $1$ Character orbit 1260.cf Analytic conductor $0.629$ Analytic rank $0$ Dimension $8$ Projective image $S_{4}$ CM/RM no Inner twists $8$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1260.cf (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.628821915918$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$S_{4}$$ Projective field: Galois closure of 4.2.132300.3

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{7} q^{5} -\zeta_{24}^{10} q^{7} +O(q^{10})$$ $$q -\zeta_{24}^{7} q^{5} -\zeta_{24}^{10} q^{7} -\zeta_{24}^{6} q^{13} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{17} + \zeta_{24}^{4} q^{19} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{23} -\zeta_{24}^{2} q^{25} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{29} + \zeta_{24}^{8} q^{31} -\zeta_{24}^{5} q^{35} -\zeta_{24}^{10} q^{37} + \zeta_{24}^{6} q^{43} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{47} -\zeta_{24}^{8} q^{49} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{59} -\zeta_{24} q^{65} + \zeta_{24}^{2} q^{67} -\zeta_{24}^{2} q^{73} -\zeta_{24}^{4} q^{79} + ( 1 + \zeta_{24}^{6} ) q^{85} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{89} -\zeta_{24}^{4} q^{91} -\zeta_{24}^{11} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 4 q^{19} - 4 q^{31} + 4 q^{49} - 4 q^{79} + 8 q^{85} - 4 q^{91} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$757$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-\zeta_{24}^{4}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
809.1
 −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 0.258819 + 0.965926i
0 0 0 −0.965926 0.258819i 0 −0.866025 + 0.500000i 0 0 0
809.2 0 0 0 −0.258819 + 0.965926i 0 0.866025 0.500000i 0 0 0
809.3 0 0 0 0.258819 0.965926i 0 0.866025 0.500000i 0 0 0
809.4 0 0 0 0.965926 + 0.258819i 0 −0.866025 + 0.500000i 0 0 0
989.1 0 0 0 −0.965926 + 0.258819i 0 −0.866025 0.500000i 0 0 0
989.2 0 0 0 −0.258819 0.965926i 0 0.866025 + 0.500000i 0 0 0
989.3 0 0 0 0.258819 + 0.965926i 0 0.866025 + 0.500000i 0 0 0
989.4 0 0 0 0.965926 0.258819i 0 −0.866025 0.500000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 989.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.1.cf.a 8
3.b odd 2 1 inner 1260.1.cf.a 8
5.b even 2 1 inner 1260.1.cf.a 8
7.c even 3 1 inner 1260.1.cf.a 8
15.d odd 2 1 inner 1260.1.cf.a 8
21.h odd 6 1 inner 1260.1.cf.a 8
35.j even 6 1 inner 1260.1.cf.a 8
105.o odd 6 1 inner 1260.1.cf.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.cf.a 8 1.a even 1 1 trivial
1260.1.cf.a 8 3.b odd 2 1 inner
1260.1.cf.a 8 5.b even 2 1 inner
1260.1.cf.a 8 7.c even 3 1 inner
1260.1.cf.a 8 15.d odd 2 1 inner
1260.1.cf.a 8 21.h odd 6 1 inner
1260.1.cf.a 8 35.j even 6 1 inner
1260.1.cf.a 8 105.o odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1260, [\chi])$$.

Hecke characteristic polynomials

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