# Properties

 Label 1148.2.i.a 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} -4 q^{13} -3 q^{15} + 7 \zeta_{6} q^{19} + ( -3 + 2 \zeta_{6} ) q^{21} -6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} -5 q^{27} + 6 q^{29} + ( 10 - 10 \zeta_{6} ) q^{31} -3 \zeta_{6} q^{33} + ( -3 + 9 \zeta_{6} ) q^{35} -2 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{39} - q^{41} -4 q^{43} + ( -6 + 6 \zeta_{6} ) q^{45} -12 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( 6 - 6 \zeta_{6} ) q^{53} -9 q^{55} -7 q^{57} + ( -6 + 6 \zeta_{6} ) q^{59} + 13 \zeta_{6} q^{61} + ( -2 + 6 \zeta_{6} ) q^{63} -12 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{67} + 6 q^{69} -9 q^{71} + ( -14 + 14 \zeta_{6} ) q^{73} -4 \zeta_{6} q^{75} + ( -9 + 6 \zeta_{6} ) q^{77} + \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} + ( -6 + 6 \zeta_{6} ) q^{87} + 12 \zeta_{6} q^{89} + ( -8 - 4 \zeta_{6} ) q^{91} + 10 \zeta_{6} q^{93} + ( -21 + 21 \zeta_{6} ) q^{95} + 2 q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + 3q^{5} + 5q^{7} + 2q^{9} + O(q^{10})$$ $$2q - q^{3} + 3q^{5} + 5q^{7} + 2q^{9} - 3q^{11} - 8q^{13} - 6q^{15} + 7q^{19} - 4q^{21} - 6q^{23} - 4q^{25} - 10q^{27} + 12q^{29} + 10q^{31} - 3q^{33} + 3q^{35} - 2q^{37} + 4q^{39} - 2q^{41} - 8q^{43} - 6q^{45} - 12q^{47} + 11q^{49} + 6q^{53} - 18q^{55} - 14q^{57} - 6q^{59} + 13q^{61} + 2q^{63} - 12q^{65} + 4q^{67} + 12q^{69} - 18q^{71} - 14q^{73} - 4q^{75} - 12q^{77} + q^{79} - q^{81} + 24q^{83} - 6q^{87} + 12q^{89} - 20q^{91} + 10q^{93} - 21q^{95} + 4q^{97} - 12q^{99} + O(q^{100})$$

## 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 1.50000 + 2.59808i 0 2.50000 + 0.866025i 0 1.00000 + 1.73205i 0
821.1 0 −0.500000 0.866025i 0 1.50000 2.59808i 0 2.50000 0.866025i 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.a 2
7.c even 3 1 inner 1148.2.i.a 2
7.c even 3 1 8036.2.a.d 1
7.d odd 6 1 8036.2.a.c 1

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

## Hecke kernels

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

 $$T_{3}^{2} + T_{3} + 1$$ $$T_{11}^{2} + 3 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$49 - 7 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$100 - 10 T + T^{2}$$
$37$ $$4 + 2 T + T^{2}$$
$41$ $$( 1 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$144 + 12 T + T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$36 + 6 T + T^{2}$$
$61$ $$169 - 13 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$( 9 + T )^{2}$$
$73$ $$196 + 14 T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$144 - 12 T + T^{2}$$
$97$ $$( -2 + T )^{2}$$