# Properties

 Label 2850.2.a.bn Level $2850$ Weight $2$ Character orbit 2850.a Self dual yes Analytic conductor $22.757$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 5$$$$)/2$$

## 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
 0.311108 2.17009 −1.48119
1.00000 1.00000 1.00000 0 1.00000 −4.42864 1.00000 1.00000 0
1.2 1.00000 1.00000 1.00000 0 1.00000 1.07838 1.00000 1.00000 0
1.3 1.00000 1.00000 1.00000 0 1.00000 3.35026 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.bn 3
3.b odd 2 1 8550.2.a.cf 3
5.b even 2 1 2850.2.a.bk 3
5.c odd 4 2 570.2.d.d 6
15.d odd 2 1 8550.2.a.cr 3
15.e even 4 2 1710.2.d.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.d.d 6 5.c odd 4 2
1710.2.d.e 6 15.e even 4 2
2850.2.a.bk 3 5.b even 2 1
2850.2.a.bn 3 1.a even 1 1 trivial
8550.2.a.cf 3 3.b odd 2 1
8550.2.a.cr 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2850))$$:

 $$T_{7}^{3} - 16 T_{7} + 16$$ $$T_{11}^{3} + 4 T_{11}^{2} - 16 T_{11} - 32$$ $$T_{13}^{3} - 6 T_{13}^{2} - 4 T_{13} + 8$$ $$T_{23}^{3} - 6 T_{23}^{2} - 4 T_{23} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$16 - 16 T + T^{3}$$
$11$ $$-32 - 16 T + 4 T^{2} + T^{3}$$
$13$ $$8 - 4 T - 6 T^{2} + T^{3}$$
$17$ $$8 + 20 T - 10 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$8 - 4 T - 6 T^{2} + T^{3}$$
$29$ $$40 + 12 T - 10 T^{2} + T^{3}$$
$31$ $$152 + 28 T - 14 T^{2} + T^{3}$$
$37$ $$-232 - 84 T - 2 T^{2} + T^{3}$$
$41$ $$-16 - 16 T + T^{3}$$
$43$ $$-8 - 4 T + 10 T^{2} + T^{3}$$
$47$ $$-184 - 52 T + 2 T^{2} + T^{3}$$
$53$ $$152 + 12 T - 14 T^{2} + T^{3}$$
$59$ $$-40 + 52 T - 14 T^{2} + T^{3}$$
$61$ $$( -2 + T )^{3}$$
$67$ $$-128 - 32 T + 8 T^{2} + T^{3}$$
$71$ $$-64 - 80 T + 4 T^{2} + T^{3}$$
$73$ $$-256 - 64 T + 8 T^{2} + T^{3}$$
$79$ $$-200 + 124 T - 22 T^{2} + T^{3}$$
$83$ $$16 - 120 T + T^{3}$$
$89$ $$-80 - 48 T - 4 T^{2} + T^{3}$$
$97$ $$-216 - 180 T + 6 T^{2} + T^{3}$$
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