# Properties

 Label 1050.2.i.h 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.

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

## 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 0.500000 2.59808i 1.00000 −0.500000 0.866025i 0
751.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 0.500000 + 2.59808i 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.h 2
5.b even 2 1 1050.2.i.m yes 2
5.c odd 4 2 1050.2.o.e 4
7.c even 3 1 inner 1050.2.i.h 2
7.c even 3 1 7350.2.a.by 1
7.d odd 6 1 7350.2.a.cq 1
35.i odd 6 1 7350.2.a.n 1
35.j even 6 1 1050.2.i.m yes 2
35.j even 6 1 7350.2.a.bb 1
35.l odd 12 2 1050.2.o.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.i.h 2 1.a even 1 1 trivial
1050.2.i.h 2 7.c even 3 1 inner
1050.2.i.m yes 2 5.b even 2 1
1050.2.i.m yes 2 35.j even 6 1
1050.2.o.e 4 5.c odd 4 2
1050.2.o.e 4 35.l odd 12 2
7350.2.a.n 1 35.i odd 6 1
7350.2.a.bb 1 35.j even 6 1
7350.2.a.by 1 7.c even 3 1
7350.2.a.cq 1 7.d odd 6 1

## Hecke kernels

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

 $$T_{11}$$ $$T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ 1
$7$ $$1 - T + 7 T^{2}$$
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 4 T + 13 T^{2} )^{2}$$
$17$ $$1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4}$$
$19$ $$1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4}$$
$23$ $$1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4}$$
$37$ $$( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} )$$
$41$ $$( 1 + 9 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 10 T + 43 T^{2} )^{2}$$
$47$ $$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$
$53$ $$1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4}$$
$59$ $$1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4}$$
$61$ $$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$
$67$ $$1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 3 T + 71 T^{2} )^{2}$$
$73$ $$1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4}$$
$79$ $$1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 7 T + 97 T^{2} )^{2}$$