# Properties

 Label 1148.1.o.a Level $1148$ Weight $1$ Character orbit 1148.o Analytic conductor $0.573$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -164 Inner twists $4$

# 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: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.8036.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + \zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} + \zeta_{6} q^{5} + q^{6} + \zeta_{6} q^{7} + q^{8} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + \zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} + \zeta_{6} q^{5} + q^{6} + \zeta_{6} q^{7} + q^{8} -\zeta_{6}^{2} q^{10} + \zeta_{6}^{2} q^{11} -\zeta_{6} q^{12} -\zeta_{6}^{2} q^{14} - q^{15} -\zeta_{6} q^{16} -\zeta_{6} q^{19} - q^{20} - q^{21} + q^{22} + \zeta_{6}^{2} q^{24} - q^{27} - q^{28} + \zeta_{6} q^{30} + \zeta_{6}^{2} q^{32} -\zeta_{6} q^{33} + \zeta_{6}^{2} q^{35} -2 \zeta_{6} q^{37} + \zeta_{6}^{2} q^{38} + \zeta_{6} q^{40} + q^{41} + \zeta_{6} q^{42} -\zeta_{6} q^{44} + 2 \zeta_{6} q^{47} + q^{48} + \zeta_{6}^{2} q^{49} + \zeta_{6} q^{54} - q^{55} + \zeta_{6} q^{56} + q^{57} -\zeta_{6}^{2} q^{60} + \zeta_{6} q^{61} + q^{64} + \zeta_{6}^{2} q^{66} -2 \zeta_{6}^{2} q^{67} + q^{70} + q^{71} + 2 \zeta_{6}^{2} q^{73} + 2 \zeta_{6}^{2} q^{74} + q^{76} - q^{77} -\zeta_{6} q^{79} -\zeta_{6}^{2} q^{80} -\zeta_{6}^{2} q^{81} -\zeta_{6} q^{82} -\zeta_{6}^{2} q^{84} + \zeta_{6}^{2} q^{88} -2 \zeta_{6}^{2} q^{94} -\zeta_{6}^{2} q^{95} -\zeta_{6} q^{96} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{3} - q^{4} + q^{5} + 2q^{6} + q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{3} - q^{4} + q^{5} + 2q^{6} + q^{7} + 2q^{8} + q^{10} - q^{11} - q^{12} + q^{14} - 2q^{15} - q^{16} - q^{19} - 2q^{20} - 2q^{21} + 2q^{22} - q^{24} - 2q^{27} - 2q^{28} + q^{30} - q^{32} - q^{33} - q^{35} - 2q^{37} - q^{38} + q^{40} + 2q^{41} + q^{42} - q^{44} + 2q^{47} + 2q^{48} - q^{49} + q^{54} - 2q^{55} + q^{56} + 2q^{57} + q^{60} + q^{61} + 2q^{64} - q^{66} + 2q^{67} + 2q^{70} + 2q^{71} - 2q^{73} - 2q^{74} + 2q^{76} - 2q^{77} - q^{79} + q^{80} + q^{81} - q^{82} + q^{84} - q^{88} + 2q^{94} + q^{95} - q^{96} + 2q^{98} + O(q^{100})$$

## Character values

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

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

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i 1.00000 0.500000 0.866025i 1.00000 0 0.500000 + 0.866025i
655.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 0.500000 + 0.866025i 1.00000 0 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 $$\Q(\sqrt{-41})$$
7.c even 3 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.a 2
4.b odd 2 1 1148.1.o.b yes 2
7.c even 3 1 inner 1148.1.o.a 2
28.g odd 6 1 1148.1.o.b yes 2
41.b even 2 1 1148.1.o.b yes 2
164.d odd 2 1 CM 1148.1.o.a 2
287.j even 6 1 1148.1.o.b yes 2
1148.o odd 6 1 inner 1148.1.o.a 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

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