# Properties

 Label 2850.2.a.bl 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.

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

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
 −1.48119 2.17009 0.311108
−1.00000 1.00000 1.00000 0 −1.00000 −3.35026 −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 4.42864 −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.bl 3
3.b odd 2 1 8550.2.a.cq 3
5.b even 2 1 2850.2.a.bm 3
5.c odd 4 2 570.2.d.c 6
15.d odd 2 1 8550.2.a.ce 3
15.e even 4 2 1710.2.d.f 6

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$-16 - 16 T + T^{3}$$
$11$ $$16 + 8 T - 8 T^{2} + T^{3}$$
$13$ $$-32 - 16 T + 4 T^{2} + T^{3}$$
$17$ $$-8 - 20 T + 2 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$16 + 8 T - 8 T^{2} + T^{3}$$
$29$ $$8 + 20 T - 10 T^{2} + T^{3}$$
$31$ $$16 - 16 T + T^{3}$$
$37$ $$-32 - 16 T + 4 T^{2} + T^{3}$$
$41$ $$400 - 80 T - 4 T^{2} + T^{3}$$
$43$ $$760 - 124 T - 6 T^{2} + T^{3}$$
$47$ $$-16 + 8 T + 8 T^{2} + T^{3}$$
$53$ $$( -6 + T )^{3}$$
$59$ $$8 - 4 T - 10 T^{2} + T^{3}$$
$61$ $$152 + 12 T - 14 T^{2} + T^{3}$$
$67$ $$-320 + 48 T + 20 T^{2} + T^{3}$$
$71$ $$320 + 48 T - 20 T^{2} + T^{3}$$
$73$ $$-256 - 64 T + 8 T^{2} + T^{3}$$
$79$ $$-16 - 16 T + T^{3}$$
$83$ $$160 - 176 T - 4 T^{2} + T^{3}$$
$89$ $$-400 + 176 T - 24 T^{2} + T^{3}$$
$97$ $$296 + 148 T + 22 T^{2} + T^{3}$$
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