# Properties

 Label 1148.1.o.c Level $1148$ Weight $1$ Character orbit 1148.o Analytic conductor $0.573$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.258309184.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{4} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{2} q^{4} + \zeta_{12}^{4} q^{5} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{6} + \zeta_{12} q^{7} + q^{8} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{4} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{2} q^{4} + \zeta_{12}^{4} q^{5} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{6} + \zeta_{12} q^{7} + q^{8} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{9} -\zeta_{12}^{2} q^{10} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{11} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{12} + \zeta_{12}^{5} q^{14} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{15} + \zeta_{12}^{4} q^{16} + ( -1 - \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{18} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{19} + q^{20} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{21} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{22} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{24} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{27} -\zeta_{12}^{3} q^{28} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{30} -\zeta_{12}^{2} q^{32} + ( 1 - \zeta_{12}^{2} - 2 \zeta_{12}^{4} ) q^{33} + \zeta_{12}^{5} q^{35} + ( 1 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{36} -2 \zeta_{12}^{4} q^{37} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{38} + \zeta_{12}^{4} q^{40} + q^{41} + ( -1 - \zeta_{12}^{2} ) q^{42} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{44} + ( -1 - \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{45} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{48} + \zeta_{12}^{2} q^{49} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{54} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{55} + \zeta_{12} q^{56} + ( -2 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{57} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{60} -\zeta_{12}^{4} q^{61} + ( -\zeta_{12} + \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{63} + q^{64} + ( 1 + 2 \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{66} -\zeta_{12}^{3} q^{70} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{71} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{72} + 2 \zeta_{12}^{2} q^{73} + 2 \zeta_{12}^{2} q^{74} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{76} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{77} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{79} -\zeta_{12}^{2} q^{80} -\zeta_{12}^{2} q^{81} + \zeta_{12}^{4} q^{82} + ( 1 - \zeta_{12}^{4} ) q^{84} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{88} + ( 1 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{90} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{95} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{96} - q^{98} + ( 2 \zeta_{12} - 2 \zeta_{12}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} - 2q^{5} + 4q^{8} - 4q^{9} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} - 2q^{5} + 4q^{8} - 4q^{9} - 2q^{10} - 2q^{16} - 4q^{18} + 4q^{20} - 2q^{32} + 6q^{33} + 8q^{36} + 4q^{37} - 2q^{40} + 4q^{41} - 6q^{42} - 4q^{45} + 2q^{49} - 12q^{57} + 2q^{61} + 4q^{64} + 6q^{66} - 4q^{72} + 4q^{73} + 4q^{74} - 2q^{80} - 2q^{81} - 2q^{82} + 6q^{84} + 8q^{90} - 4q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$493$$ $$575$$ $$785$$ $$\chi(n)$$ $$\zeta_{12}^{4}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.500000 + 0.866025i −0.866025 1.50000i −0.500000 0.866025i −0.500000 + 0.866025i 1.73205 −0.866025 0.500000i 1.00000 −1.00000 + 1.73205i −0.500000 0.866025i
163.2 −0.500000 + 0.866025i 0.866025 + 1.50000i −0.500000 0.866025i −0.500000 + 0.866025i −1.73205 0.866025 + 0.500000i 1.00000 −1.00000 + 1.73205i −0.500000 0.866025i
655.1 −0.500000 0.866025i −0.866025 + 1.50000i −0.500000 + 0.866025i −0.500000 0.866025i 1.73205 −0.866025 + 0.500000i 1.00000 −1.00000 1.73205i −0.500000 + 0.866025i
655.2 −0.500000 0.866025i 0.866025 1.50000i −0.500000 + 0.866025i −0.500000 0.866025i −1.73205 0.866025 0.500000i 1.00000 −1.00000 1.73205i −0.500000 + 0.866025i
 $$n$$: 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
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.c 4
4.b odd 2 1 inner 1148.1.o.c 4
7.c even 3 1 inner 1148.1.o.c 4
28.g odd 6 1 inner 1148.1.o.c 4
41.b even 2 1 inner 1148.1.o.c 4
164.d odd 2 1 CM 1148.1.o.c 4
287.j even 6 1 inner 1148.1.o.c 4
1148.o odd 6 1 inner 1148.1.o.c 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 3 T_{3}^{2} + 9$$ acting on $$S_{1}^{\mathrm{new}}(1148, [\chi])$$.

## Hecke characteristic polynomials

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