# Properties

 Label 1050.4.a.g Level $1050$ Weight $4$ Character orbit 1050.a Self dual yes Analytic conductor $61.952$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$61.9520055060$$ 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

 $$f(q)$$ $$=$$ $$q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{6} - 7q^{7} - 8q^{8} + 9q^{9} + O(q^{10})$$ $$q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{6} - 7q^{7} - 8q^{8} + 9q^{9} - 72q^{11} + 12q^{12} + 34q^{13} + 14q^{14} + 16q^{16} - 6q^{17} - 18q^{18} + 92q^{19} - 21q^{21} + 144q^{22} + 180q^{23} - 24q^{24} - 68q^{26} + 27q^{27} - 28q^{28} - 114q^{29} + 56q^{31} - 32q^{32} - 216q^{33} + 12q^{34} + 36q^{36} + 34q^{37} - 184q^{38} + 102q^{39} + 6q^{41} + 42q^{42} - 164q^{43} - 288q^{44} - 360q^{46} - 168q^{47} + 48q^{48} + 49q^{49} - 18q^{51} + 136q^{52} - 654q^{53} - 54q^{54} + 56q^{56} + 276q^{57} + 228q^{58} - 492q^{59} - 250q^{61} - 112q^{62} - 63q^{63} + 64q^{64} + 432q^{66} + 124q^{67} - 24q^{68} + 540q^{69} + 36q^{71} - 72q^{72} - 1010q^{73} - 68q^{74} + 368q^{76} + 504q^{77} - 204q^{78} + 56q^{79} + 81q^{81} - 12q^{82} - 228q^{83} - 84q^{84} + 328q^{86} - 342q^{87} + 576q^{88} + 390q^{89} - 238q^{91} + 720q^{92} + 168q^{93} + 336q^{94} - 96q^{96} + 70q^{97} - 98q^{98} - 648q^{99} + O(q^{100})$$

## 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
−2.00000 3.00000 4.00000 0 −6.00000 −7.00000 −8.00000 9.00000 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 1050.4.a.g 1
5.b even 2 1 42.4.a.a 1
5.c odd 4 2 1050.4.g.a 2
15.d odd 2 1 126.4.a.a 1
20.d odd 2 1 336.4.a.l 1
35.c odd 2 1 294.4.a.i 1
35.i odd 6 2 294.4.e.b 2
35.j even 6 2 294.4.e.c 2
40.e odd 2 1 1344.4.a.a 1
40.f even 2 1 1344.4.a.o 1
60.h even 2 1 1008.4.a.b 1
105.g even 2 1 882.4.a.g 1
105.o odd 6 2 882.4.g.w 2
105.p even 6 2 882.4.g.o 2
140.c even 2 1 2352.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 5.b even 2 1
126.4.a.a 1 15.d odd 2 1
294.4.a.i 1 35.c odd 2 1
294.4.e.b 2 35.i odd 6 2
294.4.e.c 2 35.j even 6 2
336.4.a.l 1 20.d odd 2 1
882.4.a.g 1 105.g even 2 1
882.4.g.o 2 105.p even 6 2
882.4.g.w 2 105.o odd 6 2
1008.4.a.b 1 60.h even 2 1
1050.4.a.g 1 1.a even 1 1 trivial
1050.4.g.a 2 5.c odd 4 2
1344.4.a.a 1 40.e odd 2 1
1344.4.a.o 1 40.f even 2 1
2352.4.a.a 1 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1050))$$:

 $$T_{11} + 72$$ $$T_{13} - 34$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$-3 + T$$
$5$ $$T$$
$7$ $$7 + T$$
$11$ $$72 + T$$
$13$ $$-34 + T$$
$17$ $$6 + T$$
$19$ $$-92 + T$$
$23$ $$-180 + T$$
$29$ $$114 + T$$
$31$ $$-56 + T$$
$37$ $$-34 + T$$
$41$ $$-6 + T$$
$43$ $$164 + T$$
$47$ $$168 + T$$
$53$ $$654 + T$$
$59$ $$492 + T$$
$61$ $$250 + T$$
$67$ $$-124 + T$$
$71$ $$-36 + T$$
$73$ $$1010 + T$$
$79$ $$-56 + T$$
$83$ $$228 + T$$
$89$ $$-390 + T$$
$97$ $$-70 + T$$