Properties

 Label 1050.2.g.i Level 1050 Weight 2 Character orbit 1050.g 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.g (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -i q^{2} + i q^{3} - q^{4} + q^{6} -i q^{7} + i q^{8} - q^{9} +O(q^{10})$$ $$q -i q^{2} + i q^{3} - q^{4} + q^{6} -i q^{7} + i q^{8} - q^{9} + 2 q^{11} -i q^{12} -i q^{13} - q^{14} + q^{16} -3 i q^{17} + i q^{18} + q^{21} -2 i q^{22} -i q^{23} - q^{24} - q^{26} -i q^{27} + i q^{28} + 5 q^{29} + 7 q^{31} -i q^{32} + 2 i q^{33} -3 q^{34} + q^{36} + 2 i q^{37} + q^{39} + 7 q^{41} -i q^{42} -11 i q^{43} -2 q^{44} - q^{46} -8 i q^{47} + i q^{48} - q^{49} + 3 q^{51} + i q^{52} -i q^{53} - q^{54} + q^{56} -5 i q^{58} + 5 q^{59} -3 q^{61} -7 i q^{62} + i q^{63} - q^{64} + 2 q^{66} + 12 i q^{67} + 3 i q^{68} + q^{69} + 12 q^{71} -i q^{72} -6 i q^{73} + 2 q^{74} -2 i q^{77} -i q^{78} -10 q^{79} + q^{81} -7 i q^{82} -11 i q^{83} - q^{84} -11 q^{86} + 5 i q^{87} + 2 i q^{88} + 10 q^{89} - q^{91} + i q^{92} + 7 i q^{93} -8 q^{94} + q^{96} + 2 i q^{97} + i q^{98} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + 4q^{11} - 2q^{14} + 2q^{16} + 2q^{21} - 2q^{24} - 2q^{26} + 10q^{29} + 14q^{31} - 6q^{34} + 2q^{36} + 2q^{39} + 14q^{41} - 4q^{44} - 2q^{46} - 2q^{49} + 6q^{51} - 2q^{54} + 2q^{56} + 10q^{59} - 6q^{61} - 2q^{64} + 4q^{66} + 2q^{69} + 24q^{71} + 4q^{74} - 20q^{79} + 2q^{81} - 2q^{84} - 22q^{86} + 20q^{89} - 2q^{91} - 16q^{94} + 2q^{96} - 4q^{99} + O(q^{100})$$

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.i 2
3.b odd 2 1 3150.2.g.h 2
5.b even 2 1 inner 1050.2.g.i 2
5.c odd 4 1 1050.2.a.b 1
5.c odd 4 1 1050.2.a.r yes 1
15.d odd 2 1 3150.2.g.h 2
15.e even 4 1 3150.2.a.n 1
15.e even 4 1 3150.2.a.y 1
20.e even 4 1 8400.2.a.d 1
20.e even 4 1 8400.2.a.ck 1
35.f even 4 1 7350.2.a.bi 1
35.f even 4 1 7350.2.a.cb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.a.b 1 5.c odd 4 1
1050.2.a.r yes 1 5.c odd 4 1
1050.2.g.i 2 1.a even 1 1 trivial
1050.2.g.i 2 5.b even 2 1 inner
3150.2.a.n 1 15.e even 4 1
3150.2.a.y 1 15.e even 4 1
3150.2.g.h 2 3.b odd 2 1
3150.2.g.h 2 15.d odd 2 1
7350.2.a.bi 1 35.f even 4 1
7350.2.a.cb 1 35.f even 4 1
8400.2.a.d 1 20.e even 4 1
8400.2.a.ck 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 $$S_{2}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11} - 2$$ $$T_{13}^{2} + 1$$ $$T_{17}^{2} + 9$$ $$T_{19}$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ 
$7$ $$1 + T^{2}$$
$11$ $$( 1 - 2 T + 11 T^{2} )^{2}$$
$13$ $$1 - 25 T^{2} + 169 T^{4}$$
$17$ $$1 - 25 T^{2} + 289 T^{4}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$1 - 45 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 5 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 - 7 T + 41 T^{2} )^{2}$$
$43$ $$1 + 35 T^{2} + 1849 T^{4}$$
$47$ $$1 - 30 T^{2} + 2209 T^{4}$$
$53$ $$1 - 105 T^{2} + 2809 T^{4}$$
$59$ $$( 1 - 5 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 3 T + 61 T^{2} )^{2}$$
$67$ $$1 + 10 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 12 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} )$$
$79$ $$( 1 + 10 T + 79 T^{2} )^{2}$$
$83$ $$1 - 45 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 10 T + 89 T^{2} )^{2}$$
$97$ $$1 - 190 T^{2} + 9409 T^{4}$$