# Properties

 Label 1050.2.o.k 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: yes 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.

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

## 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 0.866025 2.50000i 1.00000i 0.500000 0.866025i 0
499.2 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 0 1.00000 −0.866025 + 2.50000i 1.00000i 0.500000 0.866025i 0
949.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0 1.00000 0.866025 + 2.50000i 1.00000i 0.500000 + 0.866025i 0
949.2 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 0 1.00000 −0.866025 2.50000i 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.k 4
5.b even 2 1 inner 1050.2.o.k 4
5.c odd 4 1 1050.2.i.a 2
5.c odd 4 1 1050.2.i.t yes 2
7.c even 3 1 inner 1050.2.o.k 4
35.j even 6 1 inner 1050.2.o.k 4
35.k even 12 1 7350.2.a.y 1
35.k even 12 1 7350.2.a.bp 1
35.l odd 12 1 1050.2.i.a 2
35.l odd 12 1 1050.2.i.t yes 2
35.l odd 12 1 7350.2.a.c 1
35.l odd 12 1 7350.2.a.cl 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.i.a 2 5.c odd 4 1
1050.2.i.a 2 35.l odd 12 1
1050.2.i.t yes 2 5.c odd 4 1
1050.2.i.t yes 2 35.l odd 12 1
1050.2.o.k 4 1.a even 1 1 trivial
1050.2.o.k 4 5.b even 2 1 inner
1050.2.o.k 4 7.c even 3 1 inner
1050.2.o.k 4 35.j even 6 1 inner
7350.2.a.c 1 35.l odd 12 1
7350.2.a.y 1 35.k even 12 1
7350.2.a.bp 1 35.k even 12 1
7350.2.a.cl 1 35.l odd 12 1

## Hecke kernels

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

 $$T_{11}^{2} - 4 T_{11} + 16$$ $$T_{13}^{2} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ 1
$7$ $$1 + 11 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 25 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$
$23$ $$1 + 42 T^{2} + 1235 T^{4} + 22218 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 4 T + 29 T^{2} )^{4}$$
$31$ $$( 1 - 31 T^{2} + 961 T^{4} )^{2}$$
$37$ $$1 + 65 T^{2} + 2856 T^{4} + 88985 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 - 12 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 22 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 102 T^{2} + 7595 T^{4} + 286518 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2}$$
$67$ $$1 + 125 T^{2} + 11136 T^{4} + 561125 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 16 T + 71 T^{2} )^{4}$$
$73$ $$1 + 25 T^{2} - 4704 T^{4} + 133225 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 17 T + 79 T^{2} )^{2}( 1 + 4 T + 79 T^{2} )^{2}$$
$83$ $$( 1 - 130 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 95 T^{2} + 9409 T^{4} )^{2}$$