# Properties

 Label 2850.2.a.bg Level $2850$ Weight $2$ Character orbit 2850.a Self dual yes Analytic conductor $22.757$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{10})$$ Defining polynomial: $$x^{2} - 10$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} + \beta q^{7} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} + \beta q^{7} + q^{8} + q^{9} + ( 1 + \beta ) q^{11} - q^{12} + ( 2 + \beta ) q^{13} + \beta q^{14} + q^{16} + ( 4 - \beta ) q^{17} + q^{18} - q^{19} -\beta q^{21} + ( 1 + \beta ) q^{22} + ( 1 - 2 \beta ) q^{23} - q^{24} + ( 2 + \beta ) q^{26} - q^{27} + \beta q^{28} + ( -7 + \beta ) q^{29} + ( -1 + 2 \beta ) q^{31} + q^{32} + ( -1 - \beta ) q^{33} + ( 4 - \beta ) q^{34} + q^{36} + 10 q^{37} - q^{38} + ( -2 - \beta ) q^{39} -2 \beta q^{41} -\beta q^{42} + ( -6 - \beta ) q^{43} + ( 1 + \beta ) q^{44} + ( 1 - 2 \beta ) q^{46} + 6 q^{47} - q^{48} + 3 q^{49} + ( -4 + \beta ) q^{51} + ( 2 + \beta ) q^{52} + ( 5 - \beta ) q^{53} - q^{54} + \beta q^{56} + q^{57} + ( -7 + \beta ) q^{58} + ( 2 - 3 \beta ) q^{59} + ( -1 - \beta ) q^{61} + ( -1 + 2 \beta ) q^{62} + \beta q^{63} + q^{64} + ( -1 - \beta ) q^{66} + ( 3 - 3 \beta ) q^{67} + ( 4 - \beta ) q^{68} + ( -1 + 2 \beta ) q^{69} + ( -2 + \beta ) q^{71} + q^{72} - q^{73} + 10 q^{74} - q^{76} + ( 10 + \beta ) q^{77} + ( -2 - \beta ) q^{78} + ( 1 - 2 \beta ) q^{79} + q^{81} -2 \beta q^{82} + ( -3 + 3 \beta ) q^{83} -\beta q^{84} + ( -6 - \beta ) q^{86} + ( 7 - \beta ) q^{87} + ( 1 + \beta ) q^{88} + ( -1 - 2 \beta ) q^{89} + ( 10 + 2 \beta ) q^{91} + ( 1 - 2 \beta ) q^{92} + ( 1 - 2 \beta ) q^{93} + 6 q^{94} - q^{96} + ( 10 - 3 \beta ) q^{97} + 3 q^{98} + ( 1 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 2q^{8} + 2q^{9} + 2q^{11} - 2q^{12} + 4q^{13} + 2q^{16} + 8q^{17} + 2q^{18} - 2q^{19} + 2q^{22} + 2q^{23} - 2q^{24} + 4q^{26} - 2q^{27} - 14q^{29} - 2q^{31} + 2q^{32} - 2q^{33} + 8q^{34} + 2q^{36} + 20q^{37} - 2q^{38} - 4q^{39} - 12q^{43} + 2q^{44} + 2q^{46} + 12q^{47} - 2q^{48} + 6q^{49} - 8q^{51} + 4q^{52} + 10q^{53} - 2q^{54} + 2q^{57} - 14q^{58} + 4q^{59} - 2q^{61} - 2q^{62} + 2q^{64} - 2q^{66} + 6q^{67} + 8q^{68} - 2q^{69} - 4q^{71} + 2q^{72} - 2q^{73} + 20q^{74} - 2q^{76} + 20q^{77} - 4q^{78} + 2q^{79} + 2q^{81} - 6q^{83} - 12q^{86} + 14q^{87} + 2q^{88} - 2q^{89} + 20q^{91} + 2q^{92} + 2q^{93} + 12q^{94} - 2q^{96} + 20q^{97} + 6q^{98} + 2q^{99} + O(q^{100})$$

## 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
 −3.16228 3.16228
1.00000 −1.00000 1.00000 0 −1.00000 −3.16228 1.00000 1.00000 0
1.2 1.00000 −1.00000 1.00000 0 −1.00000 3.16228 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.bg yes 2
3.b odd 2 1 8550.2.a.bq 2
5.b even 2 1 2850.2.a.bf 2
5.c odd 4 2 2850.2.d.v 4
15.d odd 2 1 8550.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bf 2 5.b even 2 1
2850.2.a.bg yes 2 1.a even 1 1 trivial
2850.2.d.v 4 5.c odd 4 2
8550.2.a.bq 2 3.b odd 2 1
8550.2.a.ca 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2850))$$:

 $$T_{7}^{2} - 10$$ $$T_{11}^{2} - 2 T_{11} - 9$$ $$T_{13}^{2} - 4 T_{13} - 6$$ $$T_{23}^{2} - 2 T_{23} - 39$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-10 + T^{2}$$
$11$ $$-9 - 2 T + T^{2}$$
$13$ $$-6 - 4 T + T^{2}$$
$17$ $$6 - 8 T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$-39 - 2 T + T^{2}$$
$29$ $$39 + 14 T + T^{2}$$
$31$ $$-39 + 2 T + T^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$-40 + T^{2}$$
$43$ $$26 + 12 T + T^{2}$$
$47$ $$( -6 + T )^{2}$$
$53$ $$15 - 10 T + T^{2}$$
$59$ $$-86 - 4 T + T^{2}$$
$61$ $$-9 + 2 T + T^{2}$$
$67$ $$-81 - 6 T + T^{2}$$
$71$ $$-6 + 4 T + T^{2}$$
$73$ $$( 1 + T )^{2}$$
$79$ $$-39 - 2 T + T^{2}$$
$83$ $$-81 + 6 T + T^{2}$$
$89$ $$-39 + 2 T + T^{2}$$
$97$ $$10 - 20 T + T^{2}$$
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