# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
1.00000 −1.00000 1.00000 0 −1.00000 −0.732051 1.00000 1.00000 0
1.2 1.00000 −1.00000 1.00000 0 −1.00000 2.73205 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.bh yes 2
3.b odd 2 1 8550.2.a.bt 2
5.b even 2 1 2850.2.a.be 2
5.c odd 4 2 2850.2.d.u 4
15.d odd 2 1 8550.2.a.bz 2

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

 $$T_{7}^{2} - 2 T_{7} - 2$$ $$T_{11}^{2} - 27$$ $$T_{13}^{2} - 6 T_{13} + 6$$ $$T_{23}^{2} - 10 T_{23} + 13$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-2 - 2 T + T^{2}$$
$11$ $$-27 + T^{2}$$
$13$ $$6 - 6 T + T^{2}$$
$17$ $$-2 + 2 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$13 - 10 T + T^{2}$$
$29$ $$-11 + 8 T + T^{2}$$
$31$ $$-47 - 2 T + T^{2}$$
$37$ $$-44 + 4 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$46 - 14 T + T^{2}$$
$47$ $$-12 + T^{2}$$
$53$ $$-59 - 8 T + T^{2}$$
$59$ $$-18 + 6 T + T^{2}$$
$61$ $$97 - 20 T + T^{2}$$
$67$ $$-27 + T^{2}$$
$71$ $$-2 + 10 T + T^{2}$$
$73$ $$109 - 22 T + T^{2}$$
$79$ $$-47 + 2 T + T^{2}$$
$83$ $$-11 - 16 T + T^{2}$$
$89$ $$-39 + 6 T + T^{2}$$
$97$ $$-122 - 10 T + T^{2}$$