# Properties

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

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## 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} - 4q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} - 4q^{7} + q^{8} + q^{9} - q^{11} + q^{12} - 4q^{14} + q^{16} - 8q^{17} + q^{18} + q^{19} - 4q^{21} - q^{22} - 3q^{23} + q^{24} + q^{27} - 4q^{28} - q^{29} + q^{31} + q^{32} - q^{33} - 8q^{34} + q^{36} - 2q^{37} + q^{38} - 10q^{41} - 4q^{42} - 8q^{43} - q^{44} - 3q^{46} + q^{48} + 9q^{49} - 8q^{51} + 3q^{53} + q^{54} - 4q^{56} + q^{57} - q^{58} + 4q^{59} + 5q^{61} + q^{62} - 4q^{63} + q^{64} - q^{66} - 5q^{67} - 8q^{68} - 3q^{69} - 6q^{71} + q^{72} - 13q^{73} - 2q^{74} + q^{76} + 4q^{77} - 5q^{79} + q^{81} - 10q^{82} + 11q^{83} - 4q^{84} - 8q^{86} - q^{87} - q^{88} + 3q^{89} - 3q^{92} + q^{93} + q^{96} + 10q^{97} + 9q^{98} - q^{99} + O(q^{100})$$

## 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 −4.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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.w yes 1
3.b odd 2 1 8550.2.a.c 1
5.b even 2 1 2850.2.a.f 1
5.c odd 4 2 2850.2.d.o 2
15.d odd 2 1 8550.2.a.bk 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.f 1 5.b even 2 1
2850.2.a.w yes 1 1.a even 1 1 trivial
2850.2.d.o 2 5.c odd 4 2
8550.2.a.c 1 3.b odd 2 1
8550.2.a.bk 1 15.d odd 2 1

## Hecke kernels

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

 $$T_{7} + 4$$ $$T_{11} + 1$$ $$T_{13}$$ $$T_{23} + 3$$

## Hecke characteristic polynomials

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