# Properties

 Label 370.2.m.b Level $370$ Weight $2$ Character orbit 370.m Analytic conductor $2.954$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.m (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 2 \beta_{1} + 3$$

## Character values

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

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
159.1
 1.68614 + 0.396143i −1.18614 − 1.26217i 1.68614 − 0.396143i −1.18614 + 1.26217i
0.500000 + 0.866025i −0.686141 0.396143i −0.500000 + 0.866025i 0.686141 2.12819i 0.792287i 3.00000 + 1.73205i −1.00000 −1.18614 2.05446i 2.18614 0.469882i
159.2 0.500000 + 0.866025i 2.18614 + 1.26217i −0.500000 + 0.866025i −2.18614 0.469882i 2.52434i 3.00000 + 1.73205i −1.00000 1.68614 + 2.92048i −0.686141 2.12819i
249.1 0.500000 0.866025i −0.686141 + 0.396143i −0.500000 0.866025i 0.686141 + 2.12819i 0.792287i 3.00000 1.73205i −1.00000 −1.18614 + 2.05446i 2.18614 + 0.469882i
249.2 0.500000 0.866025i 2.18614 1.26217i −0.500000 0.866025i −2.18614 + 0.469882i 2.52434i 3.00000 1.73205i −1.00000 1.68614 2.92048i −0.686141 + 2.12819i
 $$n$$: 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
185.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.m.b yes 4
5.b even 2 1 370.2.m.a 4
37.e even 6 1 370.2.m.a 4
185.l even 6 1 inner 370.2.m.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.m.a 4 5.b even 2 1
370.2.m.a 4 37.e even 6 1
370.2.m.b yes 4 1.a even 1 1 trivial
370.2.m.b yes 4 185.l even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 3 T_{3}^{3} + T_{3}^{2} + 6 T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$4 + 6 T + T^{2} - 3 T^{3} + T^{4}$$
$5$ $$25 + 15 T + 4 T^{2} + 3 T^{3} + T^{4}$$
$7$ $$( 12 - 6 T + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$4 + 10 T + 27 T^{2} - 5 T^{3} + T^{4}$$
$17$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$( -24 - 6 T + T^{2} )^{2}$$
$29$ $$1156 + 79 T^{2} + T^{4}$$
$31$ $$144 + 123 T^{2} + T^{4}$$
$37$ $$( 37 + T + T^{2} )^{2}$$
$41$ $$1089 + 33 T^{2} + T^{4}$$
$43$ $$( -74 - T + T^{2} )^{2}$$
$47$ $$64 + 28 T^{2} + T^{4}$$
$53$ $$16 + 36 T + 31 T^{2} + 9 T^{3} + T^{4}$$
$59$ $$4096 + 1920 T + 364 T^{2} + 30 T^{3} + T^{4}$$
$61$ $$324 - 162 T + 9 T^{2} + 9 T^{3} + T^{4}$$
$67$ $$9216 - 576 T - 84 T^{2} + 6 T^{3} + T^{4}$$
$71$ $$576 + 144 T + 60 T^{2} - 6 T^{3} + T^{4}$$
$73$ $$( 48 + T^{2} )^{2}$$
$79$ $$9216 + 576 T - 84 T^{2} - 6 T^{3} + T^{4}$$
$83$ $$16 - 96 T + 196 T^{2} - 24 T^{3} + T^{4}$$
$89$ $$15376 + 4836 T + 631 T^{2} + 39 T^{3} + T^{4}$$
$97$ $$( -62 - 7 T + T^{2} )^{2}$$