# Properties

 Label 1050.2.g.f Level 1050 Weight 2 Character orbit 1050.g Analytic conductor 8.384 Analytic rank 1 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.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 1.00000i 1.00000i −1.00000 0
799.2 1.00000i 1.00000i −1.00000 0 1.00000 1.00000i 1.00000i −1.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.g.f 2
3.b odd 2 1 3150.2.g.t 2
5.b even 2 1 inner 1050.2.g.f 2
5.c odd 4 1 210.2.a.a 1
5.c odd 4 1 1050.2.a.q 1
15.d odd 2 1 3150.2.g.t 2
15.e even 4 1 630.2.a.i 1
15.e even 4 1 3150.2.a.t 1
20.e even 4 1 1680.2.a.o 1
20.e even 4 1 8400.2.a.m 1
35.f even 4 1 1470.2.a.g 1
35.f even 4 1 7350.2.a.bo 1
35.k even 12 2 1470.2.i.n 2
35.l odd 12 2 1470.2.i.t 2
40.i odd 4 1 6720.2.a.cg 1
40.k even 4 1 6720.2.a.z 1
60.l odd 4 1 5040.2.a.bg 1
105.k odd 4 1 4410.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.a 1 5.c odd 4 1
630.2.a.i 1 15.e even 4 1
1050.2.a.q 1 5.c odd 4 1
1050.2.g.f 2 1.a even 1 1 trivial
1050.2.g.f 2 5.b even 2 1 inner
1470.2.a.g 1 35.f even 4 1
1470.2.i.n 2 35.k even 12 2
1470.2.i.t 2 35.l odd 12 2
1680.2.a.o 1 20.e even 4 1
3150.2.a.t 1 15.e even 4 1
3150.2.g.t 2 3.b odd 2 1
3150.2.g.t 2 15.d odd 2 1
4410.2.a.bc 1 105.k odd 4 1
5040.2.a.bg 1 60.l odd 4 1
6720.2.a.z 1 40.k even 4 1
6720.2.a.cg 1 40.i odd 4 1
7350.2.a.bo 1 35.f even 4 1
8400.2.a.m 1 20.e even 4 1

## Hecke kernels

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

 $$T_{11} + 4$$ $$T_{13}^{2} + 4$$ $$T_{17}^{2} + 36$$ $$T_{19}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ 1
$7$ $$1 + T^{2}$$
$11$ $$( 1 + 4 T + 11 T^{2} )^{2}$$
$13$ $$1 - 22 T^{2} + 169 T^{4}$$
$17$ $$1 + 2 T^{2} + 289 T^{4}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$1 + 18 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 10 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 8 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$1 - 22 T^{2} + 1849 T^{4}$$
$47$ $$1 - 78 T^{2} + 2209 T^{4}$$
$53$ $$1 - 6 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 6 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 67 T^{2} )^{2}$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} )$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}$$
$83$ $$1 - 150 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 14 T + 89 T^{2} )^{2}$$
$97$ $$1 - 190 T^{2} + 9409 T^{4}$$