# Properties

 Label 1050.4.a.d 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

 $$f(q)$$ $$=$$ $$q - 2q^{2} - 3q^{3} + 4q^{4} + 6q^{6} + 7q^{7} - 8q^{8} + 9q^{9} + O(q^{10})$$ $$q - 2q^{2} - 3q^{3} + 4q^{4} + 6q^{6} + 7q^{7} - 8q^{8} + 9q^{9} - 8q^{11} - 12q^{12} + 42q^{13} - 14q^{14} + 16q^{16} + 2q^{17} - 18q^{18} - 124q^{19} - 21q^{21} + 16q^{22} - 76q^{23} + 24q^{24} - 84q^{26} - 27q^{27} + 28q^{28} + 254q^{29} - 72q^{31} - 32q^{32} + 24q^{33} - 4q^{34} + 36q^{36} - 398q^{37} + 248q^{38} - 126q^{39} + 462q^{41} + 42q^{42} - 212q^{43} - 32q^{44} + 152q^{46} + 264q^{47} - 48q^{48} + 49q^{49} - 6q^{51} + 168q^{52} + 162q^{53} + 54q^{54} - 56q^{56} + 372q^{57} - 508q^{58} - 772q^{59} + 30q^{61} + 144q^{62} + 63q^{63} + 64q^{64} - 48q^{66} + 764q^{67} + 8q^{68} + 228q^{69} - 236q^{71} - 72q^{72} - 418q^{73} + 796q^{74} - 496q^{76} - 56q^{77} + 252q^{78} + 552q^{79} + 81q^{81} - 924q^{82} - 1036q^{83} - 84q^{84} + 424q^{86} - 762q^{87} + 64q^{88} + 30q^{89} + 294q^{91} - 304q^{92} + 216q^{93} - 528q^{94} + 96q^{96} + 1190q^{97} - 98q^{98} - 72q^{99} + O(q^{100})$$

## 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.d 1
5.b even 2 1 42.4.a.b 1
5.c odd 4 2 1050.4.g.n 2
15.d odd 2 1 126.4.a.c 1
20.d odd 2 1 336.4.a.d 1
35.c odd 2 1 294.4.a.h 1
35.i odd 6 2 294.4.e.d 2
35.j even 6 2 294.4.e.a 2
40.e odd 2 1 1344.4.a.t 1
40.f even 2 1 1344.4.a.f 1
60.h even 2 1 1008.4.a.j 1
105.g even 2 1 882.4.a.d 1
105.o odd 6 2 882.4.g.s 2
105.p even 6 2 882.4.g.r 2
140.c even 2 1 2352.4.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.b 1 5.b even 2 1
126.4.a.c 1 15.d odd 2 1
294.4.a.h 1 35.c odd 2 1
294.4.e.a 2 35.j even 6 2
294.4.e.d 2 35.i odd 6 2
336.4.a.d 1 20.d odd 2 1
882.4.a.d 1 105.g even 2 1
882.4.g.r 2 105.p even 6 2
882.4.g.s 2 105.o odd 6 2
1008.4.a.j 1 60.h even 2 1
1050.4.a.d 1 1.a even 1 1 trivial
1050.4.g.n 2 5.c odd 4 2
1344.4.a.f 1 40.f even 2 1
1344.4.a.t 1 40.e odd 2 1
2352.4.a.ba 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1050))$$:

 $$T_{11} + 8$$ $$T_{13} - 42$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$3 + T$$
$5$ $$T$$
$7$ $$-7 + T$$
$11$ $$8 + T$$
$13$ $$-42 + T$$
$17$ $$-2 + T$$
$19$ $$124 + T$$
$23$ $$76 + T$$
$29$ $$-254 + T$$
$31$ $$72 + T$$
$37$ $$398 + T$$
$41$ $$-462 + T$$
$43$ $$212 + T$$
$47$ $$-264 + T$$
$53$ $$-162 + T$$
$59$ $$772 + T$$
$61$ $$-30 + T$$
$67$ $$-764 + T$$
$71$ $$236 + T$$
$73$ $$418 + T$$
$79$ $$-552 + T$$
$83$ $$1036 + T$$
$89$ $$-30 + T$$
$97$ $$-1190 + T$$