# Properties

 Label 1050.2.o.c Level $1050$ Weight $2$ Character orbit 1050.o Analytic conductor $8.384$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} + \zeta_{12} q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} - q^{6} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} + \zeta_{12} q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} - q^{6} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + ( -3 + 3 \zeta_{12}^{2} ) q^{11} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{12} -5 \zeta_{12}^{3} q^{13} + ( 2 + \zeta_{12}^{2} ) q^{14} -\zeta_{12}^{2} q^{16} -\zeta_{12} q^{18} + 5 \zeta_{12}^{2} q^{19} + ( 1 - 3 \zeta_{12}^{2} ) q^{21} -3 \zeta_{12}^{3} q^{22} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{23} + ( -1 + \zeta_{12}^{2} ) q^{24} + 5 \zeta_{12}^{2} q^{26} + \zeta_{12}^{3} q^{27} + ( -3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{28} + ( 10 - 10 \zeta_{12}^{2} ) q^{31} + \zeta_{12} q^{32} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{33} + q^{36} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{37} -5 \zeta_{12} q^{38} + ( 5 - 5 \zeta_{12}^{2} ) q^{39} + 9 q^{41} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{42} -8 \zeta_{12}^{3} q^{43} + 3 \zeta_{12}^{2} q^{44} + ( -9 + 9 \zeta_{12}^{2} ) q^{46} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{47} -\zeta_{12}^{3} q^{48} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} -5 \zeta_{12} q^{52} -3 \zeta_{12} q^{53} -\zeta_{12}^{2} q^{54} + ( 3 - 2 \zeta_{12}^{2} ) q^{56} + 5 \zeta_{12}^{3} q^{57} + ( 12 - 12 \zeta_{12}^{2} ) q^{59} -8 \zeta_{12}^{2} q^{61} + 10 \zeta_{12}^{3} q^{62} + ( \zeta_{12} - 3 \zeta_{12}^{3} ) q^{63} - q^{64} + ( 3 - 3 \zeta_{12}^{2} ) q^{66} -8 \zeta_{12} q^{67} + 9 q^{69} -6 q^{71} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{72} + 2 \zeta_{12} q^{73} + ( 1 - \zeta_{12}^{2} ) q^{74} + 5 q^{76} + ( 9 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} + 5 \zeta_{12}^{3} q^{78} + 8 \zeta_{12}^{2} q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{82} + ( -2 - \zeta_{12}^{2} ) q^{84} + 8 \zeta_{12}^{2} q^{86} -3 \zeta_{12} q^{88} + 6 \zeta_{12}^{2} q^{89} + ( -15 + 10 \zeta_{12}^{2} ) q^{91} -9 \zeta_{12}^{3} q^{92} + ( 10 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{93} + ( -3 + 3 \zeta_{12}^{2} ) q^{94} + \zeta_{12}^{2} q^{96} + 8 \zeta_{12}^{3} q^{97} + ( -3 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{98} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 4q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{4} - 4q^{6} + 2q^{9} - 6q^{11} + 10q^{14} - 2q^{16} + 10q^{19} - 2q^{21} - 2q^{24} + 10q^{26} + 20q^{31} + 4q^{36} + 10q^{39} + 36q^{41} + 6q^{44} - 18q^{46} - 4q^{49} - 2q^{54} + 8q^{56} + 24q^{59} - 16q^{61} - 4q^{64} + 6q^{66} + 36q^{69} - 24q^{71} + 2q^{74} + 20q^{76} + 16q^{79} - 2q^{81} - 10q^{84} + 16q^{86} + 12q^{89} - 40q^{91} - 6q^{94} + 2q^{96} - 12q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
499.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 0 −1.00000 −1.73205 + 2.00000i 1.00000i 0.500000 0.866025i 0
499.2 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 −1.00000 1.73205 2.00000i 1.00000i 0.500000 0.866025i 0
949.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 0 −1.00000 −1.73205 2.00000i 1.00000i 0.500000 + 0.866025i 0
949.2 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0 −1.00000 1.73205 + 2.00000i 1.00000i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.o.c 4
5.b even 2 1 inner 1050.2.o.c 4
5.c odd 4 1 210.2.i.c 2
5.c odd 4 1 1050.2.i.i 2
7.c even 3 1 inner 1050.2.o.c 4
15.e even 4 1 630.2.k.a 2
20.e even 4 1 1680.2.bg.n 2
35.f even 4 1 1470.2.i.p 2
35.j even 6 1 inner 1050.2.o.c 4
35.k even 12 1 1470.2.a.e 1
35.k even 12 1 1470.2.i.p 2
35.k even 12 1 7350.2.a.cx 1
35.l odd 12 1 210.2.i.c 2
35.l odd 12 1 1050.2.i.i 2
35.l odd 12 1 1470.2.a.f 1
35.l odd 12 1 7350.2.a.cd 1
105.w odd 12 1 4410.2.a.w 1
105.x even 12 1 630.2.k.a 2
105.x even 12 1 4410.2.a.bh 1
140.w even 12 1 1680.2.bg.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.c 2 5.c odd 4 1
210.2.i.c 2 35.l odd 12 1
630.2.k.a 2 15.e even 4 1
630.2.k.a 2 105.x even 12 1
1050.2.i.i 2 5.c odd 4 1
1050.2.i.i 2 35.l odd 12 1
1050.2.o.c 4 1.a even 1 1 trivial
1050.2.o.c 4 5.b even 2 1 inner
1050.2.o.c 4 7.c even 3 1 inner
1050.2.o.c 4 35.j even 6 1 inner
1470.2.a.e 1 35.k even 12 1
1470.2.a.f 1 35.l odd 12 1
1470.2.i.p 2 35.f even 4 1
1470.2.i.p 2 35.k even 12 1
1680.2.bg.n 2 20.e even 4 1
1680.2.bg.n 2 140.w even 12 1
4410.2.a.w 1 105.w odd 12 1
4410.2.a.bh 1 105.x even 12 1
7350.2.a.cd 1 35.l odd 12 1
7350.2.a.cx 1 35.k even 12 1

## Hecke kernels

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

 $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{13}^{2} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$49 + 2 T^{2} + T^{4}$$
$11$ $$( 9 + 3 T + T^{2} )^{2}$$
$13$ $$( 25 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( 25 - 5 T + T^{2} )^{2}$$
$23$ $$6561 - 81 T^{2} + T^{4}$$
$29$ $$T^{4}$$
$31$ $$( 100 - 10 T + T^{2} )^{2}$$
$37$ $$1 - T^{2} + T^{4}$$
$41$ $$( -9 + T )^{4}$$
$43$ $$( 64 + T^{2} )^{2}$$
$47$ $$81 - 9 T^{2} + T^{4}$$
$53$ $$81 - 9 T^{2} + T^{4}$$
$59$ $$( 144 - 12 T + T^{2} )^{2}$$
$61$ $$( 64 + 8 T + T^{2} )^{2}$$
$67$ $$4096 - 64 T^{2} + T^{4}$$
$71$ $$( 6 + T )^{4}$$
$73$ $$16 - 4 T^{2} + T^{4}$$
$79$ $$( 64 - 8 T + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$( 36 - 6 T + T^{2} )^{2}$$
$97$ $$( 64 + T^{2} )^{2}$$