# Properties

 Label 1050.2.i.o 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: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} - q^{6} + ( 3 - 2 \zeta_{6} ) q^{7} - q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} - q^{6} + ( 3 - 2 \zeta_{6} ) q^{7} - q^{8} -\zeta_{6} q^{9} + ( 5 - 5 \zeta_{6} ) q^{11} -\zeta_{6} q^{12} + q^{13} + ( 2 + \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( 1 - \zeta_{6} ) q^{18} -7 \zeta_{6} q^{19} + ( -1 + 3 \zeta_{6} ) q^{21} + 5 q^{22} -3 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} + \zeta_{6} q^{26} + q^{27} + ( -1 + 3 \zeta_{6} ) q^{28} + ( 6 - 6 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} + 5 \zeta_{6} q^{33} + 2 q^{34} + q^{36} + 5 \zeta_{6} q^{37} + ( 7 - 7 \zeta_{6} ) q^{38} + ( -1 + \zeta_{6} ) q^{39} -9 q^{41} + ( -3 + 2 \zeta_{6} ) q^{42} + 10 q^{43} + 5 \zeta_{6} q^{44} + ( 3 - 3 \zeta_{6} ) q^{46} + 13 \zeta_{6} q^{47} + q^{48} + ( 5 - 8 \zeta_{6} ) q^{49} + 2 \zeta_{6} q^{51} + ( -1 + \zeta_{6} ) q^{52} + ( 1 - \zeta_{6} ) q^{53} + \zeta_{6} q^{54} + ( -3 + 2 \zeta_{6} ) q^{56} + 7 q^{57} + ( -4 + 4 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + 6 q^{62} + ( -2 - \zeta_{6} ) q^{63} + q^{64} + ( -5 + 5 \zeta_{6} ) q^{66} + ( -6 + 6 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} + 3 q^{69} -2 q^{71} + \zeta_{6} q^{72} + ( -4 + 4 \zeta_{6} ) q^{73} + ( -5 + 5 \zeta_{6} ) q^{74} + 7 q^{76} + ( 5 - 15 \zeta_{6} ) q^{77} - q^{78} + 14 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -9 \zeta_{6} q^{82} -10 q^{83} + ( -2 - \zeta_{6} ) q^{84} + 10 \zeta_{6} q^{86} + ( -5 + 5 \zeta_{6} ) q^{88} -10 \zeta_{6} q^{89} + ( 3 - 2 \zeta_{6} ) q^{91} + 3 q^{92} + 6 \zeta_{6} q^{93} + ( -13 + 13 \zeta_{6} ) q^{94} + \zeta_{6} q^{96} + 8 q^{97} + ( 8 - 3 \zeta_{6} ) q^{98} -5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{3} - q^{4} - 2q^{6} + 4q^{7} - 2q^{8} - q^{9} + O(q^{10})$$ $$2q + q^{2} - q^{3} - q^{4} - 2q^{6} + 4q^{7} - 2q^{8} - q^{9} + 5q^{11} - q^{12} + 2q^{13} + 5q^{14} - q^{16} + 2q^{17} + q^{18} - 7q^{19} + q^{21} + 10q^{22} - 3q^{23} + q^{24} + q^{26} + 2q^{27} + q^{28} + 6q^{31} + q^{32} + 5q^{33} + 4q^{34} + 2q^{36} + 5q^{37} + 7q^{38} - q^{39} - 18q^{41} - 4q^{42} + 20q^{43} + 5q^{44} + 3q^{46} + 13q^{47} + 2q^{48} + 2q^{49} + 2q^{51} - q^{52} + q^{53} + q^{54} - 4q^{56} + 14q^{57} - 4q^{59} + 2q^{61} + 12q^{62} - 5q^{63} + 2q^{64} - 5q^{66} - 6q^{67} + 2q^{68} + 6q^{69} - 4q^{71} + q^{72} - 4q^{73} - 5q^{74} + 14q^{76} - 5q^{77} - 2q^{78} + 14q^{79} - q^{81} - 9q^{82} - 20q^{83} - 5q^{84} + 10q^{86} - 5q^{88} - 10q^{89} + 4q^{91} + 6q^{92} + 6q^{93} - 13q^{94} + q^{96} + 16q^{97} + 13q^{98} - 10q^{99} + O(q^{100})$$

## 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 2.00000 1.73205i −1.00000 −0.500000 0.866025i 0
751.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 −1.00000 2.00000 + 1.73205i −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.o 2
5.b even 2 1 1050.2.i.f 2
5.c odd 4 2 210.2.n.a 4
7.c even 3 1 inner 1050.2.i.o 2
7.c even 3 1 7350.2.a.t 1
7.d odd 6 1 7350.2.a.b 1
15.e even 4 2 630.2.u.c 4
20.e even 4 2 1680.2.di.a 4
35.f even 4 2 1470.2.n.i 4
35.i odd 6 1 7350.2.a.ch 1
35.j even 6 1 1050.2.i.f 2
35.j even 6 1 7350.2.a.bn 1
35.k even 12 2 1470.2.g.a 2
35.k even 12 2 1470.2.n.i 4
35.l odd 12 2 210.2.n.a 4
35.l odd 12 2 1470.2.g.f 2
105.x even 12 2 630.2.u.c 4
140.w even 12 2 1680.2.di.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a 4 5.c odd 4 2
210.2.n.a 4 35.l odd 12 2
630.2.u.c 4 15.e even 4 2
630.2.u.c 4 105.x even 12 2
1050.2.i.f 2 5.b even 2 1
1050.2.i.f 2 35.j even 6 1
1050.2.i.o 2 1.a even 1 1 trivial
1050.2.i.o 2 7.c even 3 1 inner
1470.2.g.a 2 35.k even 12 2
1470.2.g.f 2 35.l odd 12 2
1470.2.n.i 4 35.f even 4 2
1470.2.n.i 4 35.k even 12 2
1680.2.di.a 4 20.e even 4 2
1680.2.di.a 4 140.w even 12 2
7350.2.a.b 1 7.d odd 6 1
7350.2.a.t 1 7.c even 3 1
7350.2.a.bn 1 35.j even 6 1
7350.2.a.ch 1 35.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{2} - 5 T_{11} + 25$$ $$T_{13} - 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ 1
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4}$$
$13$ $$( 1 - T + 13 T^{2} )^{2}$$
$17$ $$1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4}$$
$19$ $$( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$23$ $$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4}$$
$37$ $$1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 9 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 10 T + 43 T^{2} )^{2}$$
$47$ $$1 - 13 T + 122 T^{2} - 611 T^{3} + 2209 T^{4}$$
$53$ $$1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4}$$
$59$ $$1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4}$$
$61$ $$1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4}$$
$67$ $$1 + 6 T - 31 T^{2} + 402 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 2 T + 71 T^{2} )^{2}$$
$73$ $$1 + 4 T - 57 T^{2} + 292 T^{3} + 5329 T^{4}$$
$79$ $$1 - 14 T + 117 T^{2} - 1106 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 10 T + 83 T^{2} )^{2}$$
$89$ $$1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 8 T + 97 T^{2} )^{2}$$