Properties

 Label 2850.2.a.j Level $2850$ Weight $2$ Character orbit 2850.a Self dual yes Analytic conductor $22.757$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$22.7573645761$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} - 4q^{11} + q^{12} - 2q^{13} + q^{16} + 6q^{17} - q^{18} - q^{19} + 4q^{22} + 4q^{23} - q^{24} + 2q^{26} + q^{27} - 2q^{29} + 4q^{31} - q^{32} - 4q^{33} - 6q^{34} + q^{36} - 10q^{37} + q^{38} - 2q^{39} + 10q^{41} - 4q^{43} - 4q^{44} - 4q^{46} + 4q^{47} + q^{48} - 7q^{49} + 6q^{51} - 2q^{52} + 10q^{53} - q^{54} - q^{57} + 2q^{58} + 12q^{59} + 14q^{61} - 4q^{62} + q^{64} + 4q^{66} + 12q^{67} + 6q^{68} + 4q^{69} + 8q^{71} - q^{72} + 6q^{73} + 10q^{74} - q^{76} + 2q^{78} - 4q^{79} + q^{81} - 10q^{82} - 12q^{83} + 4q^{86} - 2q^{87} + 4q^{88} - 6q^{89} + 4q^{92} + 4q^{93} - 4q^{94} - q^{96} - 10q^{97} + 7q^{98} - 4q^{99} + O(q^{100})$$

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 0 −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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$1$$

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 2850.2.a.j 1
3.b odd 2 1 8550.2.a.ba 1
5.b even 2 1 114.2.a.b 1
5.c odd 4 2 2850.2.d.b 2
15.d odd 2 1 342.2.a.b 1
20.d odd 2 1 912.2.a.k 1
35.c odd 2 1 5586.2.a.y 1
40.e odd 2 1 3648.2.a.c 1
40.f even 2 1 3648.2.a.x 1
60.h even 2 1 2736.2.a.d 1
95.d odd 2 1 2166.2.a.d 1
285.b even 2 1 6498.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.b 1 5.b even 2 1
342.2.a.b 1 15.d odd 2 1
912.2.a.k 1 20.d odd 2 1
2166.2.a.d 1 95.d odd 2 1
2736.2.a.d 1 60.h even 2 1
2850.2.a.j 1 1.a even 1 1 trivial
2850.2.d.b 2 5.c odd 4 2
3648.2.a.c 1 40.e odd 2 1
3648.2.a.x 1 40.f even 2 1
5586.2.a.y 1 35.c odd 2 1
6498.2.a.p 1 285.b even 2 1
8550.2.a.ba 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2850))$$:

 $$T_{7}$$ $$T_{11} + 4$$ $$T_{13} + 2$$ $$T_{23} - 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$4 + T$$
$13$ $$2 + T$$
$17$ $$-6 + T$$
$19$ $$1 + T$$
$23$ $$-4 + T$$
$29$ $$2 + T$$
$31$ $$-4 + T$$
$37$ $$10 + T$$
$41$ $$-10 + T$$
$43$ $$4 + T$$
$47$ $$-4 + T$$
$53$ $$-10 + T$$
$59$ $$-12 + T$$
$61$ $$-14 + T$$
$67$ $$-12 + T$$
$71$ $$-8 + T$$
$73$ $$-6 + T$$
$79$ $$4 + T$$
$83$ $$12 + T$$
$89$ $$6 + T$$
$97$ $$10 + T$$