# Properties

 Label 370.2.a.g Level $370$ Weight $2$ Character orbit 370.a Self dual yes Analytic conductor $2.954$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.892.1 Defining polynomial: $$x^{3} - x^{2} - 8 x + 10$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} -\beta_{2} q^{3} + q^{4} - q^{5} -\beta_{2} q^{6} -\beta_{1} q^{7} + q^{8} + ( 3 + 2 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + q^{2} -\beta_{2} q^{3} + q^{4} - q^{5} -\beta_{2} q^{6} -\beta_{1} q^{7} + q^{8} + ( 3 + 2 \beta_{1} ) q^{9} - q^{10} + ( 4 - \beta_{1} + \beta_{2} ) q^{11} -\beta_{2} q^{12} + 2 \beta_{2} q^{13} -\beta_{1} q^{14} + \beta_{2} q^{15} + q^{16} + ( \beta_{1} - \beta_{2} ) q^{17} + ( 3 + 2 \beta_{1} ) q^{18} -\beta_{2} q^{19} - q^{20} + ( 2 + 2 \beta_{2} ) q^{21} + ( 4 - \beta_{1} + \beta_{2} ) q^{22} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{23} -\beta_{2} q^{24} + q^{25} + 2 \beta_{2} q^{26} + ( -4 - 4 \beta_{2} ) q^{27} -\beta_{1} q^{28} + ( -2 + \beta_{1} + \beta_{2} ) q^{29} + \beta_{2} q^{30} + 3 \beta_{1} q^{31} + q^{32} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{33} + ( \beta_{1} - \beta_{2} ) q^{34} + \beta_{1} q^{35} + ( 3 + 2 \beta_{1} ) q^{36} - q^{37} -\beta_{2} q^{38} + ( -12 - 4 \beta_{1} ) q^{39} - q^{40} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{41} + ( 2 + 2 \beta_{2} ) q^{42} + ( -4 + \beta_{1} + \beta_{2} ) q^{43} + ( 4 - \beta_{1} + \beta_{2} ) q^{44} + ( -3 - 2 \beta_{1} ) q^{45} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 2 \beta_{1} - \beta_{2} ) q^{47} -\beta_{2} q^{48} + ( -1 - \beta_{1} + \beta_{2} ) q^{49} + q^{50} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{51} + 2 \beta_{2} q^{52} + ( 6 + 3 \beta_{1} - \beta_{2} ) q^{53} + ( -4 - 4 \beta_{2} ) q^{54} + ( -4 + \beta_{1} - \beta_{2} ) q^{55} -\beta_{1} q^{56} + ( 6 + 2 \beta_{1} ) q^{57} + ( -2 + \beta_{1} + \beta_{2} ) q^{58} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{59} + \beta_{2} q^{60} + ( 2 + \beta_{1} + \beta_{2} ) q^{61} + 3 \beta_{1} q^{62} + ( -12 - \beta_{1} - 2 \beta_{2} ) q^{63} + q^{64} -2 \beta_{2} q^{65} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{66} + ( 2 \beta_{1} - \beta_{2} ) q^{67} + ( \beta_{1} - \beta_{2} ) q^{68} + ( -8 - 4 \beta_{1} + 4 \beta_{2} ) q^{69} + \beta_{1} q^{70} + ( 8 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 3 + 2 \beta_{1} ) q^{72} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{73} - q^{74} -\beta_{2} q^{75} -\beta_{2} q^{76} + ( 4 - 5 \beta_{1} - \beta_{2} ) q^{77} + ( -12 - 4 \beta_{1} ) q^{78} + ( -8 - 2 \beta_{1} - \beta_{2} ) q^{79} - q^{80} + ( 15 + 2 \beta_{1} + 4 \beta_{2} ) q^{81} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{82} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{83} + ( 2 + 2 \beta_{2} ) q^{84} + ( -\beta_{1} + \beta_{2} ) q^{85} + ( -4 + \beta_{1} + \beta_{2} ) q^{86} + ( -8 - 2 \beta_{1} ) q^{87} + ( 4 - \beta_{1} + \beta_{2} ) q^{88} + 6 q^{89} + ( -3 - 2 \beta_{1} ) q^{90} + ( -4 - 4 \beta_{2} ) q^{91} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{92} + ( -6 - 6 \beta_{2} ) q^{93} + ( 2 \beta_{1} - \beta_{2} ) q^{94} + \beta_{2} q^{95} -\beta_{2} q^{96} + ( -6 - 3 \beta_{1} + \beta_{2} ) q^{97} + ( -1 - \beta_{1} + \beta_{2} ) q^{98} + ( 4 + 7 \beta_{1} + 5 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 3q^{4} - 3q^{5} - q^{7} + 3q^{8} + 11q^{9} + O(q^{10})$$ $$3q + 3q^{2} + 3q^{4} - 3q^{5} - q^{7} + 3q^{8} + 11q^{9} - 3q^{10} + 11q^{11} - q^{14} + 3q^{16} + q^{17} + 11q^{18} - 3q^{20} + 6q^{21} + 11q^{22} - 2q^{23} + 3q^{25} - 12q^{27} - q^{28} - 5q^{29} + 3q^{31} + 3q^{32} - 14q^{33} + q^{34} + q^{35} + 11q^{36} - 3q^{37} - 40q^{39} - 3q^{40} - 9q^{41} + 6q^{42} - 11q^{43} + 11q^{44} - 11q^{45} - 2q^{46} + 2q^{47} - 4q^{49} + 3q^{50} + 14q^{51} + 21q^{53} - 12q^{54} - 11q^{55} - q^{56} + 20q^{57} - 5q^{58} + 14q^{59} + 7q^{61} + 3q^{62} - 37q^{63} + 3q^{64} - 14q^{66} + 2q^{67} + q^{68} - 28q^{69} + q^{70} + 22q^{71} + 11q^{72} - 16q^{73} - 3q^{74} + 7q^{77} - 40q^{78} - 26q^{79} - 3q^{80} + 47q^{81} - 9q^{82} + 2q^{83} + 6q^{84} - q^{85} - 11q^{86} - 26q^{87} + 11q^{88} + 18q^{89} - 11q^{90} - 12q^{91} - 2q^{92} - 18q^{93} + 2q^{94} - 21q^{97} - 4q^{98} + 19q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 8 x + 10$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta_{1} + 6$$

## 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
 2.59774 −2.91729 1.31955
1.00000 −3.34596 1.00000 −1.00000 −3.34596 −2.59774 1.00000 8.19547 −1.00000
1.2 1.00000 0.406728 1.00000 −1.00000 0.406728 2.91729 1.00000 −2.83457 −1.00000
1.3 1.00000 2.93923 1.00000 −1.00000 2.93923 −1.31955 1.00000 5.63910 −1.00000
 $$n$$: 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 370.2.a.g 3
3.b odd 2 1 3330.2.a.bg 3
4.b odd 2 1 2960.2.a.u 3
5.b even 2 1 1850.2.a.z 3
5.c odd 4 2 1850.2.b.o 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.g 3 1.a even 1 1 trivial
1850.2.a.z 3 5.b even 2 1
1850.2.b.o 6 5.c odd 4 2
2960.2.a.u 3 4.b odd 2 1
3330.2.a.bg 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(370))$$:

 $$T_{3}^{3} - 10 T_{3} + 4$$ $$T_{7}^{3} + T_{7}^{2} - 8 T_{7} - 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$4 - 10 T + T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$-10 - 8 T + T^{2} + T^{3}$$
$11$ $$8 + 28 T - 11 T^{2} + T^{3}$$
$13$ $$-32 - 40 T + T^{3}$$
$17$ $$-8 - 12 T - T^{2} + T^{3}$$
$19$ $$4 - 10 T + T^{3}$$
$23$ $$64 - 48 T + 2 T^{2} + T^{3}$$
$29$ $$-76 - 16 T + 5 T^{2} + T^{3}$$
$31$ $$270 - 72 T - 3 T^{2} + T^{3}$$
$37$ $$( 1 + T )^{3}$$
$41$ $$-364 - 40 T + 9 T^{2} + T^{3}$$
$43$ $$-80 + 16 T + 11 T^{2} + T^{3}$$
$47$ $$56 - 30 T - 2 T^{2} + T^{3}$$
$53$ $$316 + 80 T - 21 T^{2} + T^{3}$$
$59$ $$80 + 34 T - 14 T^{2} + T^{3}$$
$61$ $$4 - 8 T - 7 T^{2} + T^{3}$$
$67$ $$56 - 30 T - 2 T^{2} + T^{3}$$
$71$ $$640 + 64 T - 22 T^{2} + T^{3}$$
$73$ $$-208 + 36 T + 16 T^{2} + T^{3}$$
$79$ $$224 + 170 T + 26 T^{2} + T^{3}$$
$83$ $$-664 - 158 T - 2 T^{2} + T^{3}$$
$89$ $$( -6 + T )^{3}$$
$97$ $$-316 + 80 T + 21 T^{2} + T^{3}$$