# Properties

 Label 1850.2.a.y Level $1850$ Weight $2$ Character orbit 1850.a Self dual yes Analytic conductor $14.772$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.7723243739$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1524.1 Defining polynomial: $$x^{3} - x^{2} - 7 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} -\beta_{1} q^{3} + q^{4} + \beta_{1} q^{6} + ( -1 - \beta_{1} ) q^{7} - q^{8} + ( 2 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q - q^{2} -\beta_{1} q^{3} + q^{4} + \beta_{1} q^{6} + ( -1 - \beta_{1} ) q^{7} - q^{8} + ( 2 + \beta_{1} + \beta_{2} ) q^{9} + ( -2 - \beta_{2} ) q^{11} -\beta_{1} q^{12} + ( -3 + \beta_{1} ) q^{13} + ( 1 + \beta_{1} ) q^{14} + q^{16} + ( 2 + \beta_{1} ) q^{17} + ( -2 - \beta_{1} - \beta_{2} ) q^{18} + ( -1 + \beta_{1} - \beta_{2} ) q^{19} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{21} + ( 2 + \beta_{2} ) q^{22} + ( -2 + 2 \beta_{1} ) q^{23} + \beta_{1} q^{24} + ( 3 - \beta_{1} ) q^{26} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{27} + ( -1 - \beta_{1} ) q^{28} + ( -4 - 2 \beta_{1} ) q^{29} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{31} - q^{32} + ( -1 + 4 \beta_{1} ) q^{33} + ( -2 - \beta_{1} ) q^{34} + ( 2 + \beta_{1} + \beta_{2} ) q^{36} - q^{37} + ( 1 - \beta_{1} + \beta_{2} ) q^{38} + ( -5 + 2 \beta_{1} - \beta_{2} ) q^{39} + ( 7 + 2 \beta_{2} ) q^{41} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{42} + ( -6 + 3 \beta_{1} + \beta_{2} ) q^{43} + ( -2 - \beta_{2} ) q^{44} + ( 2 - 2 \beta_{1} ) q^{46} -\beta_{1} q^{48} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{49} + ( -5 - 3 \beta_{1} - \beta_{2} ) q^{51} + ( -3 + \beta_{1} ) q^{52} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{54} + ( 1 + \beta_{1} ) q^{56} + ( -6 + 2 \beta_{1} - \beta_{2} ) q^{57} + ( 4 + 2 \beta_{1} ) q^{58} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{59} + ( 5 - 3 \beta_{1} ) q^{61} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{62} + ( -6 - 6 \beta_{1} - 2 \beta_{2} ) q^{63} + q^{64} + ( 1 - 4 \beta_{1} ) q^{66} + ( -\beta_{1} + 2 \beta_{2} ) q^{67} + ( 2 + \beta_{1} ) q^{68} + ( -10 - 2 \beta_{2} ) q^{69} + ( 3 + \beta_{1} + 2 \beta_{2} ) q^{71} + ( -2 - \beta_{1} - \beta_{2} ) q^{72} + ( -5 + \beta_{1} - \beta_{2} ) q^{73} + q^{74} + ( -1 + \beta_{1} - \beta_{2} ) q^{76} + ( 1 + 4 \beta_{1} + \beta_{2} ) q^{77} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{78} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{79} + ( 3 + 5 \beta_{1} - \beta_{2} ) q^{81} + ( -7 - 2 \beta_{2} ) q^{82} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{83} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{84} + ( 6 - 3 \beta_{1} - \beta_{2} ) q^{86} + ( 10 + 6 \beta_{1} + 2 \beta_{2} ) q^{87} + ( 2 + \beta_{2} ) q^{88} + ( 2 - 2 \beta_{1} + 3 \beta_{2} ) q^{89} + ( -2 + \beta_{1} - \beta_{2} ) q^{91} + ( -2 + 2 \beta_{1} ) q^{92} + ( -11 - \beta_{1} - 2 \beta_{2} ) q^{93} + \beta_{1} q^{96} + ( -4 + 2 \beta_{2} ) q^{97} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{98} + ( -14 - 3 \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} - q^{3} + 3q^{4} + q^{6} - 4q^{7} - 3q^{8} + 6q^{9} + O(q^{10})$$ $$3q - 3q^{2} - q^{3} + 3q^{4} + q^{6} - 4q^{7} - 3q^{8} + 6q^{9} - 5q^{11} - q^{12} - 8q^{13} + 4q^{14} + 3q^{16} + 7q^{17} - 6q^{18} - q^{19} + 16q^{21} + 5q^{22} - 4q^{23} + q^{24} + 8q^{26} - 13q^{27} - 4q^{28} - 14q^{29} + 6q^{31} - 3q^{32} + q^{33} - 7q^{34} + 6q^{36} - 3q^{37} + q^{38} - 12q^{39} + 19q^{41} - 16q^{42} - 16q^{43} - 5q^{44} + 4q^{46} - q^{48} - q^{49} - 17q^{51} - 8q^{52} + 6q^{53} + 13q^{54} + 4q^{56} - 15q^{57} + 14q^{58} - 8q^{59} + 12q^{61} - 6q^{62} - 22q^{63} + 3q^{64} - q^{66} - 3q^{67} + 7q^{68} - 28q^{69} + 8q^{71} - 6q^{72} - 13q^{73} + 3q^{74} - q^{76} + 6q^{77} + 12q^{78} + 2q^{79} + 15q^{81} - 19q^{82} + 19q^{83} + 16q^{84} + 16q^{86} + 34q^{87} + 5q^{88} + q^{89} - 4q^{91} - 4q^{92} - 32q^{93} + q^{96} - 14q^{97} + q^{98} - 44q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 5$$

## 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
 3.13264 0.140435 −2.27307
−1.00000 −3.13264 1.00000 0 3.13264 −4.13264 −1.00000 6.81342 0
1.2 −1.00000 −0.140435 1.00000 0 0.140435 −1.14044 −1.00000 −2.98028 0
1.3 −1.00000 2.27307 1.00000 0 −2.27307 1.27307 −1.00000 2.16686 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
$$2$$ $$1$$
$$5$$ $$1$$
$$37$$ $$1$$

## 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 1850.2.a.y 3
5.b even 2 1 1850.2.a.bc yes 3
5.c odd 4 2 1850.2.b.p 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.y 3 1.a even 1 1 trivial
1850.2.a.bc yes 3 5.b even 2 1
1850.2.b.p 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1850))$$:

 $$T_{3}^{3} + T_{3}^{2} - 7 T_{3} - 1$$ $$T_{7}^{3} + 4 T_{7}^{2} - 2 T_{7} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$-1 - 7 T + T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-6 - 2 T + 4 T^{2} + T^{3}$$
$11$ $$-51 - 9 T + 5 T^{2} + T^{3}$$
$13$ $$-2 + 14 T + 8 T^{2} + T^{3}$$
$17$ $$3 + 9 T - 7 T^{2} + T^{3}$$
$19$ $$11 - 25 T + T^{2} + T^{3}$$
$23$ $$-48 - 24 T + 4 T^{2} + T^{3}$$
$29$ $$-24 + 36 T + 14 T^{2} + T^{3}$$
$31$ $$214 - 36 T - 6 T^{2} + T^{3}$$
$37$ $$( 1 + T )^{3}$$
$41$ $$399 + 51 T - 19 T^{2} + T^{3}$$
$43$ $$-564 + 4 T + 16 T^{2} + T^{3}$$
$47$ $$T^{3}$$
$53$ $$216 - 84 T - 6 T^{2} + T^{3}$$
$59$ $$84 - 60 T + 8 T^{2} + T^{3}$$
$61$ $$238 - 18 T - 12 T^{2} + T^{3}$$
$67$ $$17 - 75 T + 3 T^{2} + T^{3}$$
$71$ $$378 - 54 T - 8 T^{2} + T^{3}$$
$73$ $$-9 + 31 T + 13 T^{2} + T^{3}$$
$79$ $$-88 - 100 T - 2 T^{2} + T^{3}$$
$83$ $$-63 + 75 T - 19 T^{2} + T^{3}$$
$89$ $$147 - 189 T - T^{2} + T^{3}$$
$97$ $$-8 - 4 T + 14 T^{2} + T^{3}$$