# Properties

 Label 370.2.e.a Level $370$ Weight $2$ Character orbit 370.e Analytic conductor $2.954$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i −1.00000 1.73205i −0.500000 0.866025i −0.500000 0.866025i −2.00000 0 −1.00000 −0.500000 + 0.866025i −1.00000
211.1 0.500000 + 0.866025i −1.00000 + 1.73205i −0.500000 + 0.866025i −0.500000 + 0.866025i −2.00000 0 −1.00000 −0.500000 0.866025i −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.e.a 2
37.c even 3 1 inner 370.2.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.e.a 2 1.a even 1 1 trivial
370.2.e.a 2 37.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$:

 $$T_{3}^{2} + 2 T_{3} + 4$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$4 + 2 T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 5 + T )^{2}$$
$13$ $$1 + T + T^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$25 - 5 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$37 + T + T^{2}$$
$41$ $$100 + 10 T + T^{2}$$
$43$ $$( -6 + T )^{2}$$
$47$ $$( -9 + T )^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$1 + T + T^{2}$$
$61$ $$4 + 2 T + T^{2}$$
$67$ $$64 + 8 T + T^{2}$$
$71$ $$196 + 14 T + T^{2}$$
$73$ $$( -12 + T )^{2}$$
$79$ $$64 - 8 T + T^{2}$$
$83$ $$4 + 2 T + T^{2}$$
$89$ $$1 + T + T^{2}$$
$97$ $$( -10 + T )^{2}$$