# Properties

 Label 1050.2.i.c Level $1050$ Weight $2$ Character orbit 1050.i Analytic conductor $8.384$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + q^{6} + ( 2 + \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + q^{6} + ( 2 + \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} + ( -6 + 6 \zeta_{6} ) q^{11} -\zeta_{6} q^{12} -4 q^{13} + ( 1 - 3 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( 3 - 3 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} + 4 \zeta_{6} q^{19} + ( -3 + 2 \zeta_{6} ) q^{21} + 6 q^{22} -3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{24} + 4 \zeta_{6} q^{26} + q^{27} + ( -3 + 2 \zeta_{6} ) q^{28} -6 q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} -6 \zeta_{6} q^{33} -3 q^{34} + q^{36} -8 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} + ( 4 - 4 \zeta_{6} ) q^{39} -3 q^{41} + ( 2 + \zeta_{6} ) q^{42} + 8 q^{43} -6 \zeta_{6} q^{44} + ( -3 + 3 \zeta_{6} ) q^{46} -9 \zeta_{6} q^{47} + q^{48} + ( 3 + 5 \zeta_{6} ) q^{49} + 3 \zeta_{6} q^{51} + ( 4 - 4 \zeta_{6} ) q^{52} + ( -12 + 12 \zeta_{6} ) q^{53} -\zeta_{6} q^{54} + ( 2 + \zeta_{6} ) q^{56} -4 q^{57} + 6 \zeta_{6} q^{58} + ( -6 + 6 \zeta_{6} ) q^{59} -2 \zeta_{6} q^{61} + 5 q^{62} + ( 1 - 3 \zeta_{6} ) q^{63} + q^{64} + ( -6 + 6 \zeta_{6} ) q^{66} + ( -8 + 8 \zeta_{6} ) q^{67} + 3 \zeta_{6} q^{68} + 3 q^{69} -9 q^{71} -\zeta_{6} q^{72} + ( -14 + 14 \zeta_{6} ) q^{73} + ( -8 + 8 \zeta_{6} ) q^{74} -4 q^{76} + ( -18 + 12 \zeta_{6} ) q^{77} -4 q^{78} + 7 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 3 \zeta_{6} q^{82} -6 q^{83} + ( 1 - 3 \zeta_{6} ) q^{84} -8 \zeta_{6} q^{86} + ( 6 - 6 \zeta_{6} ) q^{87} + ( -6 + 6 \zeta_{6} ) q^{88} -3 \zeta_{6} q^{89} + ( -8 - 4 \zeta_{6} ) q^{91} + 3 q^{92} -5 \zeta_{6} q^{93} + ( -9 + 9 \zeta_{6} ) q^{94} -\zeta_{6} q^{96} + 17 q^{97} + ( 5 - 8 \zeta_{6} ) q^{98} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{3} - q^{4} + 2q^{6} + 5q^{7} + 2q^{8} - q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} - q^{4} + 2q^{6} + 5q^{7} + 2q^{8} - q^{9} - 6q^{11} - q^{12} - 8q^{13} - q^{14} - q^{16} + 3q^{17} - q^{18} + 4q^{19} - 4q^{21} + 12q^{22} - 3q^{23} - q^{24} + 4q^{26} + 2q^{27} - 4q^{28} - 12q^{29} - 5q^{31} - q^{32} - 6q^{33} - 6q^{34} + 2q^{36} - 8q^{37} + 4q^{38} + 4q^{39} - 6q^{41} + 5q^{42} + 16q^{43} - 6q^{44} - 3q^{46} - 9q^{47} + 2q^{48} + 11q^{49} + 3q^{51} + 4q^{52} - 12q^{53} - q^{54} + 5q^{56} - 8q^{57} + 6q^{58} - 6q^{59} - 2q^{61} + 10q^{62} - q^{63} + 2q^{64} - 6q^{66} - 8q^{67} + 3q^{68} + 6q^{69} - 18q^{71} - q^{72} - 14q^{73} - 8q^{74} - 8q^{76} - 24q^{77} - 8q^{78} + 7q^{79} - q^{81} + 3q^{82} - 12q^{83} - q^{84} - 8q^{86} + 6q^{87} - 6q^{88} - 3q^{89} - 20q^{91} + 6q^{92} - 5q^{93} - 9q^{94} - q^{96} + 34q^{97} + 2q^{98} + 12q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 2.50000 + 0.866025i 1.00000 −0.500000 0.866025i 0
751.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 1.00000 2.50000 0.866025i 1.00000 −0.500000 + 0.866025i 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 1050.2.i.c 2
5.b even 2 1 1050.2.i.q yes 2
5.c odd 4 2 1050.2.o.h 4
7.c even 3 1 inner 1050.2.i.c 2
7.c even 3 1 7350.2.a.cy 1
7.d odd 6 1 7350.2.a.cg 1
35.i odd 6 1 7350.2.a.bm 1
35.j even 6 1 1050.2.i.q yes 2
35.j even 6 1 7350.2.a.s 1
35.l odd 12 2 1050.2.o.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.i.c 2 1.a even 1 1 trivial
1050.2.i.c 2 7.c even 3 1 inner
1050.2.i.q yes 2 5.b even 2 1
1050.2.i.q yes 2 35.j even 6 1
1050.2.o.h 4 5.c odd 4 2
1050.2.o.h 4 35.l odd 12 2
7350.2.a.s 1 35.j even 6 1
7350.2.a.bm 1 35.i odd 6 1
7350.2.a.cg 1 7.d odd 6 1
7350.2.a.cy 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 $$S_{2}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{2} + 6 T_{11} + 36$$ $$T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$36 + 6 T + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$( 3 + T )^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$81 + 9 T + T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$36 + 6 T + T^{2}$$
$61$ $$4 + 2 T + T^{2}$$
$67$ $$64 + 8 T + T^{2}$$
$71$ $$( 9 + T )^{2}$$
$73$ $$196 + 14 T + T^{2}$$
$79$ $$49 - 7 T + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$9 + 3 T + T^{2}$$
$97$ $$( -17 + T )^{2}$$