# Properties

 Label 1148.2.i.b Level $1148$ Weight $2$ Character orbit 1148.i Analytic conductor $9.167$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1148 = 2^{2} \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1148.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.16682615204$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( 5 - 5 \zeta_{6} ) q^{11} -2 q^{13} + q^{15} + ( 3 - 3 \zeta_{6} ) q^{17} + 3 \zeta_{6} q^{19} + ( -3 + \zeta_{6} ) q^{21} + \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + 5 q^{27} -10 q^{29} + ( 11 - 11 \zeta_{6} ) q^{31} -5 \zeta_{6} q^{33} + ( 2 - 3 \zeta_{6} ) q^{35} -7 \zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{39} - q^{41} + 8 q^{43} + ( -2 + 2 \zeta_{6} ) q^{45} + 7 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} -3 \zeta_{6} q^{51} + ( 11 - 11 \zeta_{6} ) q^{53} + 5 q^{55} + 3 q^{57} + ( 7 - 7 \zeta_{6} ) q^{59} + \zeta_{6} q^{61} + ( 4 - 6 \zeta_{6} ) q^{63} -2 \zeta_{6} q^{65} + ( -7 + 7 \zeta_{6} ) q^{67} + q^{69} -12 q^{71} + ( 5 - 5 \zeta_{6} ) q^{73} -4 \zeta_{6} q^{75} + ( -15 + 5 \zeta_{6} ) q^{77} -9 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 3 q^{85} + ( -10 + 10 \zeta_{6} ) q^{87} + 3 \zeta_{6} q^{89} + ( 2 + 4 \zeta_{6} ) q^{91} -11 \zeta_{6} q^{93} + ( -3 + 3 \zeta_{6} ) q^{95} + 2 q^{97} + 10 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9} + 5 q^{11} - 4 q^{13} + 2 q^{15} + 3 q^{17} + 3 q^{19} - 5 q^{21} + q^{23} + 4 q^{25} + 10 q^{27} - 20 q^{29} + 11 q^{31} - 5 q^{33} + q^{35} - 7 q^{37} - 2 q^{39} - 2 q^{41} + 16 q^{43} - 2 q^{45} + 7 q^{47} + 2 q^{49} - 3 q^{51} + 11 q^{53} + 10 q^{55} + 6 q^{57} + 7 q^{59} + q^{61} + 2 q^{63} - 2 q^{65} - 7 q^{67} + 2 q^{69} - 24 q^{71} + 5 q^{73} - 4 q^{75} - 25 q^{77} - 9 q^{79} - q^{81} + 6 q^{85} - 10 q^{87} + 3 q^{89} + 8 q^{91} - 11 q^{93} - 3 q^{95} + 4 q^{97} + 20 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 −2.00000 1.73205i 0 1.00000 + 1.73205i 0
821.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 −2.00000 + 1.73205i 0 1.00000 1.73205i 0
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.i.b 2
7.c even 3 1 inner 1148.2.i.b 2
7.c even 3 1 8036.2.a.a 1
7.d odd 6 1 8036.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.i.b 2 1.a even 1 1 trivial
1148.2.i.b 2 7.c even 3 1 inner
8036.2.a.a 1 7.c even 3 1
8036.2.a.e 1 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1148, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{11}^{2} - 5 T_{11} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$9 - 3 T + T^{2}$$
$23$ $$1 - T + T^{2}$$
$29$ $$( 10 + T )^{2}$$
$31$ $$121 - 11 T + T^{2}$$
$37$ $$49 + 7 T + T^{2}$$
$41$ $$( 1 + T )^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$49 - 7 T + T^{2}$$
$53$ $$121 - 11 T + T^{2}$$
$59$ $$49 - 7 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$49 + 7 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$25 - 5 T + T^{2}$$
$79$ $$81 + 9 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$9 - 3 T + T^{2}$$
$97$ $$( -2 + T )^{2}$$