# Properties

 Label 1148.1.bc.a Level $1148$ Weight $1$ Character orbit 1148.bc Analytic conductor $0.573$ Analytic rank $0$ Dimension $8$ Projective image $A_{5}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1148 = 2^{2} \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1148.bc (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.572926634503$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$A_{5}$$ Projective field: Galois closure of 5.1.553849156.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{20}^{5} q^{3} -\zeta_{20} q^{5} -\zeta_{20}^{2} q^{7} +O(q^{10})$$ $$q + \zeta_{20}^{5} q^{3} -\zeta_{20} q^{5} -\zeta_{20}^{2} q^{7} + ( \zeta_{20}^{2} + \zeta_{20}^{6} ) q^{11} + \zeta_{20}^{3} q^{13} -\zeta_{20}^{6} q^{15} + ( -\zeta_{20} + \zeta_{20}^{7} ) q^{17} + \zeta_{20}^{7} q^{19} -\zeta_{20}^{7} q^{21} + ( -\zeta_{20}^{2} + \zeta_{20}^{4} ) q^{23} + \zeta_{20}^{5} q^{27} + ( -1 + \zeta_{20}^{2} ) q^{29} + ( \zeta_{20}^{3} - \zeta_{20}^{5} ) q^{31} + ( -\zeta_{20} + \zeta_{20}^{7} ) q^{33} + \zeta_{20}^{3} q^{35} -\zeta_{20}^{6} q^{37} + \zeta_{20}^{8} q^{39} -\zeta_{20}^{7} q^{41} + \zeta_{20}^{8} q^{43} + ( \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{47} + \zeta_{20}^{4} q^{49} + ( -\zeta_{20}^{2} - \zeta_{20}^{6} ) q^{51} + ( -\zeta_{20}^{3} - \zeta_{20}^{7} ) q^{55} -\zeta_{20}^{2} q^{57} + ( \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{59} + ( \zeta_{20} - \zeta_{20}^{3} ) q^{61} -\zeta_{20}^{4} q^{65} + ( -\zeta_{20}^{7} + \zeta_{20}^{9} ) q^{69} + ( -\zeta_{20}^{3} - \zeta_{20}^{7} ) q^{73} + ( -\zeta_{20}^{4} - \zeta_{20}^{8} ) q^{77} + q^{79} - q^{81} -\zeta_{20}^{5} q^{83} + ( \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{85} + ( -\zeta_{20}^{5} + \zeta_{20}^{7} ) q^{87} + ( \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{89} -\zeta_{20}^{5} q^{91} + ( 1 + \zeta_{20}^{8} ) q^{93} -\zeta_{20}^{8} q^{95} + \zeta_{20} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{7} + O(q^{10})$$ $$8q - 2q^{7} + 4q^{11} - 2q^{15} - 4q^{23} - 6q^{29} - 2q^{37} - 2q^{39} - 2q^{43} - 2q^{49} - 4q^{51} - 2q^{57} + 2q^{65} + 4q^{77} + 8q^{79} - 8q^{81} + 4q^{85} + 6q^{93} + 2q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times$$.

 $$n$$ $$493$$ $$575$$ $$785$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{20}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
461.1
 0.951057 − 0.309017i −0.951057 + 0.309017i −0.587785 + 0.809017i 0.587785 − 0.809017i 0.587785 + 0.809017i −0.587785 − 0.809017i −0.951057 − 0.309017i 0.951057 + 0.309017i
0 1.00000i 0 −0.951057 + 0.309017i 0 −0.809017 + 0.587785i 0 0 0
461.2 0 1.00000i 0 0.951057 0.309017i 0 −0.809017 + 0.587785i 0 0 0
713.1 0 1.00000i 0 0.587785 0.809017i 0 0.309017 + 0.951057i 0 0 0
713.2 0 1.00000i 0 −0.587785 + 0.809017i 0 0.309017 + 0.951057i 0 0 0
797.1 0 1.00000i 0 −0.587785 0.809017i 0 0.309017 0.951057i 0 0 0
797.2 0 1.00000i 0 0.587785 + 0.809017i 0 0.309017 0.951057i 0 0 0
1021.1 0 1.00000i 0 0.951057 + 0.309017i 0 −0.809017 0.587785i 0 0 0
1021.2 0 1.00000i 0 −0.951057 0.309017i 0 −0.809017 0.587785i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1021.2 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.b odd 2 1 inner
41.d even 5 1 inner
287.o odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.1.bc.a 8
7.b odd 2 1 inner 1148.1.bc.a 8
41.d even 5 1 inner 1148.1.bc.a 8
287.o odd 10 1 inner 1148.1.bc.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.1.bc.a 8 1.a even 1 1 trivial
1148.1.bc.a 8 7.b odd 2 1 inner
1148.1.bc.a 8 41.d even 5 1 inner
1148.1.bc.a 8 287.o odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1148, [\chi])$$.

## Hecke characteristic polynomials

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