# Properties

 Label 1050.2.a.i Level $1050$ Weight $2$ Character orbit 1050.a Self dual yes Analytic conductor $8.384$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.38429221223$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 1.00000 1.00000 0 −1.00000 1.00000 −1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.a.i 1
3.b odd 2 1 3150.2.a.bo 1
4.b odd 2 1 8400.2.a.k 1
5.b even 2 1 42.2.a.a 1
5.c odd 4 2 1050.2.g.a 2
7.b odd 2 1 7350.2.a.f 1
15.d odd 2 1 126.2.a.a 1
15.e even 4 2 3150.2.g.r 2
20.d odd 2 1 336.2.a.d 1
35.c odd 2 1 294.2.a.g 1
35.i odd 6 2 294.2.e.a 2
35.j even 6 2 294.2.e.c 2
40.e odd 2 1 1344.2.a.i 1
40.f even 2 1 1344.2.a.q 1
45.h odd 6 2 1134.2.f.j 2
45.j even 6 2 1134.2.f.g 2
55.d odd 2 1 5082.2.a.d 1
60.h even 2 1 1008.2.a.j 1
65.d even 2 1 7098.2.a.f 1
80.k odd 4 2 5376.2.c.e 2
80.q even 4 2 5376.2.c.bc 2
105.g even 2 1 882.2.a.b 1
105.o odd 6 2 882.2.g.h 2
105.p even 6 2 882.2.g.j 2
120.i odd 2 1 4032.2.a.e 1
120.m even 2 1 4032.2.a.m 1
140.c even 2 1 2352.2.a.l 1
140.p odd 6 2 2352.2.q.i 2
140.s even 6 2 2352.2.q.n 2
280.c odd 2 1 9408.2.a.n 1
280.n even 2 1 9408.2.a.bw 1
420.o odd 2 1 7056.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 5.b even 2 1
126.2.a.a 1 15.d odd 2 1
294.2.a.g 1 35.c odd 2 1
294.2.e.a 2 35.i odd 6 2
294.2.e.c 2 35.j even 6 2
336.2.a.d 1 20.d odd 2 1
882.2.a.b 1 105.g even 2 1
882.2.g.h 2 105.o odd 6 2
882.2.g.j 2 105.p even 6 2
1008.2.a.j 1 60.h even 2 1
1050.2.a.i 1 1.a even 1 1 trivial
1050.2.g.a 2 5.c odd 4 2
1134.2.f.g 2 45.j even 6 2
1134.2.f.j 2 45.h odd 6 2
1344.2.a.i 1 40.e odd 2 1
1344.2.a.q 1 40.f even 2 1
2352.2.a.l 1 140.c even 2 1
2352.2.q.i 2 140.p odd 6 2
2352.2.q.n 2 140.s even 6 2
3150.2.a.bo 1 3.b odd 2 1
3150.2.g.r 2 15.e even 4 2
4032.2.a.e 1 120.i odd 2 1
4032.2.a.m 1 120.m even 2 1
5082.2.a.d 1 55.d odd 2 1
5376.2.c.e 2 80.k odd 4 2
5376.2.c.bc 2 80.q even 4 2
7056.2.a.k 1 420.o odd 2 1
7098.2.a.f 1 65.d even 2 1
7350.2.a.f 1 7.b odd 2 1
8400.2.a.k 1 4.b odd 2 1
9408.2.a.n 1 280.c odd 2 1
9408.2.a.bw 1 280.n even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1050))$$:

 $$T_{11} + 4$$ $$T_{13} + 6$$ $$T_{17} + 2$$ $$T_{19} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$4 + T$$
$13$ $$6 + T$$
$17$ $$2 + T$$
$19$ $$4 + T$$
$23$ $$8 + T$$
$29$ $$2 + T$$
$31$ $$T$$
$37$ $$-10 + T$$
$41$ $$6 + T$$
$43$ $$-4 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$-4 + T$$
$61$ $$-6 + T$$
$67$ $$4 + T$$
$71$ $$-8 + T$$
$73$ $$10 + T$$
$79$ $$T$$
$83$ $$-4 + T$$
$89$ $$6 + T$$
$97$ $$-14 + T$$