# Properties

 Label 1148.1.o.d Level $1148$ Weight $1$ Character orbit 1148.o Analytic conductor $0.573$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ CM discriminant -164 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1148 = 2^{2} \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1148.o (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.572926634503$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{8} q^{2} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{3} -\zeta_{24}^{4} q^{4} + ( \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{5} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} ) q^{6} -\zeta_{24}^{11} q^{7} - q^{8} + ( -\zeta_{24}^{6} + \zeta_{24}^{8} - \zeta_{24}^{10} ) q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{8} q^{2} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{3} -\zeta_{24}^{4} q^{4} + ( \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{5} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} ) q^{6} -\zeta_{24}^{11} q^{7} - q^{8} + ( -\zeta_{24}^{6} + \zeta_{24}^{8} - \zeta_{24}^{10} ) q^{9} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{10} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} ) q^{11} + ( -\zeta_{24} + \zeta_{24}^{3} ) q^{12} -\zeta_{24}^{7} q^{14} + ( \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} - \zeta_{24}^{9} ) q^{15} + \zeta_{24}^{8} q^{16} + ( -\zeta_{24}^{2} + \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{18} + ( -\zeta_{24}^{7} - \zeta_{24}^{9} ) q^{19} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{20} + ( -\zeta_{24}^{8} + \zeta_{24}^{10} ) q^{21} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{22} + ( \zeta_{24}^{9} - \zeta_{24}^{11} ) q^{24} + ( -1 - \zeta_{24}^{4} - \zeta_{24}^{8} ) q^{25} + ( -\zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{27} -\zeta_{24}^{3} q^{28} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{30} + \zeta_{24}^{4} q^{32} + ( -1 + \zeta_{24}^{4} ) q^{33} + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{35} + ( 1 - \zeta_{24}^{2} + \zeta_{24}^{10} ) q^{36} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} ) q^{38} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{40} + q^{41} + ( -\zeta_{24}^{4} + \zeta_{24}^{6} ) q^{42} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{44} + ( 1 - \zeta_{24}^{2} + 2 \zeta_{24}^{4} - \zeta_{24}^{6} + \zeta_{24}^{8} ) q^{45} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{47} + ( \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{48} -\zeta_{24}^{10} q^{49} + ( -1 - \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{50} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{54} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{9} - \zeta_{24}^{11} ) q^{55} + \zeta_{24}^{11} q^{56} + ( -\zeta_{24}^{4} + \zeta_{24}^{8} ) q^{57} + ( -\zeta_{24} - \zeta_{24}^{7} + \zeta_{24}^{9} - \zeta_{24}^{11} ) q^{60} + \zeta_{24}^{8} q^{61} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} - \zeta_{24}^{9} ) q^{63} + q^{64} + ( 1 + \zeta_{24}^{8} ) q^{66} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{67} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{70} + ( \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{71} + ( \zeta_{24}^{6} - \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{72} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} + \zeta_{24}^{9} - \zeta_{24}^{11} ) q^{75} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{76} + ( -\zeta_{24}^{2} - \zeta_{24}^{4} ) q^{77} + ( -\zeta_{24} + \zeta_{24}^{3} ) q^{79} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{80} + ( -1 + \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{6} - \zeta_{24}^{8} ) q^{81} -\zeta_{24}^{8} q^{82} + ( -1 + \zeta_{24}^{2} ) q^{84} + ( \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{88} + ( 2 - \zeta_{24}^{2} + \zeta_{24}^{4} - \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{90} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{94} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{95} + ( \zeta_{24} - \zeta_{24}^{3} ) q^{96} -\zeta_{24}^{6} q^{98} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{2} - 4q^{4} - 8q^{8} - 4q^{9} + O(q^{10})$$ $$8q + 4q^{2} - 4q^{4} - 8q^{8} - 4q^{9} - 4q^{16} + 4q^{18} + 4q^{21} - 8q^{25} + 4q^{32} - 4q^{33} + 8q^{36} + 8q^{41} - 4q^{42} + 12q^{45} - 16q^{50} - 8q^{57} - 4q^{61} + 8q^{64} + 4q^{66} + 4q^{72} - 4q^{77} - 8q^{81} + 4q^{82} - 8q^{84} + 24q^{90} + O(q^{100})$$

## Character values

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

 $$n$$ $$493$$ $$575$$ $$785$$ $$\chi(n)$$ $$\zeta_{24}^{8}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 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
0.500000 0.866025i −0.965926 1.67303i −0.500000 0.866025i 0.866025 1.50000i −1.93185 0.258819 + 0.965926i −1.00000 −1.36603 + 2.36603i −0.866025 1.50000i
163.2 0.500000 0.866025i −0.258819 0.448288i −0.500000 0.866025i −0.866025 + 1.50000i −0.517638 0.965926 0.258819i −1.00000 0.366025 0.633975i 0.866025 + 1.50000i
163.3 0.500000 0.866025i 0.258819 + 0.448288i −0.500000 0.866025i −0.866025 + 1.50000i 0.517638 −0.965926 + 0.258819i −1.00000 0.366025 0.633975i 0.866025 + 1.50000i
163.4 0.500000 0.866025i 0.965926 + 1.67303i −0.500000 0.866025i 0.866025 1.50000i 1.93185 −0.258819 0.965926i −1.00000 −1.36603 + 2.36603i −0.866025 1.50000i
655.1 0.500000 + 0.866025i −0.965926 + 1.67303i −0.500000 + 0.866025i 0.866025 + 1.50000i −1.93185 0.258819 0.965926i −1.00000 −1.36603 2.36603i −0.866025 + 1.50000i
655.2 0.500000 + 0.866025i −0.258819 + 0.448288i −0.500000 + 0.866025i −0.866025 1.50000i −0.517638 0.965926 + 0.258819i −1.00000 0.366025 + 0.633975i 0.866025 1.50000i
655.3 0.500000 + 0.866025i 0.258819 0.448288i −0.500000 + 0.866025i −0.866025 1.50000i 0.517638 −0.965926 0.258819i −1.00000 0.366025 + 0.633975i 0.866025 1.50000i
655.4 0.500000 + 0.866025i 0.965926 1.67303i −0.500000 + 0.866025i 0.866025 + 1.50000i 1.93185 −0.258819 + 0.965926i −1.00000 −1.36603 2.36603i −0.866025 + 1.50000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 655.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
164.d odd 2 1 CM by $$\Q(\sqrt{-41})$$
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner
41.b even 2 1 inner
287.j even 6 1 inner
1148.o odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.1.o.d 8
4.b odd 2 1 inner 1148.1.o.d 8
7.c even 3 1 inner 1148.1.o.d 8
28.g odd 6 1 inner 1148.1.o.d 8
41.b even 2 1 inner 1148.1.o.d 8
164.d odd 2 1 CM 1148.1.o.d 8
287.j even 6 1 inner 1148.1.o.d 8
1148.o odd 6 1 inner 1148.1.o.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.1.o.d 8 1.a even 1 1 trivial
1148.1.o.d 8 4.b odd 2 1 inner
1148.1.o.d 8 7.c even 3 1 inner
1148.1.o.d 8 28.g odd 6 1 inner
1148.1.o.d 8 41.b even 2 1 inner
1148.1.o.d 8 164.d odd 2 1 CM
1148.1.o.d 8 287.j even 6 1 inner
1148.1.o.d 8 1148.o odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 4 T_{3}^{6} + 15 T_{3}^{4} + 4 T_{3}^{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1148, [\chi])$$.

## Hecke characteristic polynomials

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