# Properties

 Label 1050.2.g.a 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: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} - q^{4} - q^{6} -i q^{7} -i q^{8} - q^{9} +O(q^{10})$$ $$q + i q^{2} + i q^{3} - q^{4} - q^{6} -i q^{7} -i q^{8} - q^{9} -4 q^{11} -i q^{12} -6 i q^{13} + q^{14} + q^{16} + 2 i q^{17} -i q^{18} + 4 q^{19} + q^{21} -4 i q^{22} -8 i q^{23} + q^{24} + 6 q^{26} -i q^{27} + i q^{28} + 2 q^{29} + i q^{32} -4 i q^{33} -2 q^{34} + q^{36} -10 i q^{37} + 4 i q^{38} + 6 q^{39} -6 q^{41} + i q^{42} + 4 i q^{43} + 4 q^{44} + 8 q^{46} + i q^{48} - q^{49} -2 q^{51} + 6 i q^{52} -6 i q^{53} + q^{54} - q^{56} + 4 i q^{57} + 2 i q^{58} -4 q^{59} + 6 q^{61} + i q^{63} - q^{64} + 4 q^{66} + 4 i q^{67} -2 i q^{68} + 8 q^{69} + 8 q^{71} + i q^{72} -10 i q^{73} + 10 q^{74} -4 q^{76} + 4 i q^{77} + 6 i q^{78} + q^{81} -6 i q^{82} + 4 i q^{83} - q^{84} -4 q^{86} + 2 i q^{87} + 4 i q^{88} + 6 q^{89} -6 q^{91} + 8 i q^{92} - q^{96} -14 i q^{97} -i q^{98} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 2 q^{21} + 2 q^{24} + 12 q^{26} + 4 q^{29} - 4 q^{34} + 2 q^{36} + 12 q^{39} - 12 q^{41} + 8 q^{44} + 16 q^{46} - 2 q^{49} - 4 q^{51} + 2 q^{54} - 2 q^{56} - 8 q^{59} + 12 q^{61} - 2 q^{64} + 8 q^{66} + 16 q^{69} + 16 q^{71} + 20 q^{74} - 8 q^{76} + 2 q^{81} - 2 q^{84} - 8 q^{86} + 12 q^{89} - 12 q^{91} - 2 q^{96} + 8 q^{99} + O(q^{100})$$

## 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.a 2
3.b odd 2 1 3150.2.g.r 2
5.b even 2 1 inner 1050.2.g.a 2
5.c odd 4 1 42.2.a.a 1
5.c odd 4 1 1050.2.a.i 1
15.d odd 2 1 3150.2.g.r 2
15.e even 4 1 126.2.a.a 1
15.e even 4 1 3150.2.a.bo 1
20.e even 4 1 336.2.a.d 1
20.e even 4 1 8400.2.a.k 1
35.f even 4 1 294.2.a.g 1
35.f even 4 1 7350.2.a.f 1
35.k even 12 2 294.2.e.a 2
35.l odd 12 2 294.2.e.c 2
40.i odd 4 1 1344.2.a.q 1
40.k even 4 1 1344.2.a.i 1
45.k odd 12 2 1134.2.f.g 2
45.l even 12 2 1134.2.f.j 2
55.e even 4 1 5082.2.a.d 1
60.l odd 4 1 1008.2.a.j 1
65.h odd 4 1 7098.2.a.f 1
80.i odd 4 1 5376.2.c.bc 2
80.j even 4 1 5376.2.c.e 2
80.s even 4 1 5376.2.c.e 2
80.t odd 4 1 5376.2.c.bc 2
105.k odd 4 1 882.2.a.b 1
105.w odd 12 2 882.2.g.j 2
105.x even 12 2 882.2.g.h 2
120.q odd 4 1 4032.2.a.m 1
120.w even 4 1 4032.2.a.e 1
140.j odd 4 1 2352.2.a.l 1
140.w even 12 2 2352.2.q.i 2
140.x odd 12 2 2352.2.q.n 2
280.s even 4 1 9408.2.a.n 1
280.y odd 4 1 9408.2.a.bw 1
420.w even 4 1 7056.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 5.c odd 4 1
126.2.a.a 1 15.e even 4 1
294.2.a.g 1 35.f even 4 1
294.2.e.a 2 35.k even 12 2
294.2.e.c 2 35.l odd 12 2
336.2.a.d 1 20.e even 4 1
882.2.a.b 1 105.k odd 4 1
882.2.g.h 2 105.x even 12 2
882.2.g.j 2 105.w odd 12 2
1008.2.a.j 1 60.l odd 4 1
1050.2.a.i 1 5.c odd 4 1
1050.2.g.a 2 1.a even 1 1 trivial
1050.2.g.a 2 5.b even 2 1 inner
1134.2.f.g 2 45.k odd 12 2
1134.2.f.j 2 45.l even 12 2
1344.2.a.i 1 40.k even 4 1
1344.2.a.q 1 40.i odd 4 1
2352.2.a.l 1 140.j odd 4 1
2352.2.q.i 2 140.w even 12 2
2352.2.q.n 2 140.x odd 12 2
3150.2.a.bo 1 15.e even 4 1
3150.2.g.r 2 3.b odd 2 1
3150.2.g.r 2 15.d odd 2 1
4032.2.a.e 1 120.w even 4 1
4032.2.a.m 1 120.q odd 4 1
5082.2.a.d 1 55.e even 4 1
5376.2.c.e 2 80.j even 4 1
5376.2.c.e 2 80.s even 4 1
5376.2.c.bc 2 80.i odd 4 1
5376.2.c.bc 2 80.t odd 4 1
7056.2.a.k 1 420.w even 4 1
7098.2.a.f 1 65.h odd 4 1
7350.2.a.f 1 35.f even 4 1
8400.2.a.k 1 20.e even 4 1
9408.2.a.n 1 280.s even 4 1
9408.2.a.bw 1 280.y odd 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$196 + T^{2}$$