# Properties

 Label 370.2.e.e Level $370$ Weight $2$ Character orbit 370.e 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.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{10})$$ Defining polynomial: $$x^{4} + 10 x^{2} + 100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 10 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/10$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$10 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$10 \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 −1.58114 − 2.73861i 1.58114 + 2.73861i −1.58114 + 2.73861i 1.58114 − 2.73861i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 −2.58114 4.47066i −1.00000 1.00000 1.73205i −1.00000
121.2 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 0.581139 + 1.00656i −1.00000 1.00000 1.73205i −1.00000
211.1 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −2.58114 + 4.47066i −1.00000 1.00000 + 1.73205i −1.00000
211.2 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0.581139 1.00656i −1.00000 1.00000 + 1.73205i −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.e 4
37.c even 3 1 inner 370.2.e.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.e.e 4 1.a even 1 1 trivial
370.2.e.e 4 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 $$S_{2}^{\mathrm{new}}(370, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{7}^{4} + 4 T_{7}^{3} + 22 T_{7}^{2} - 24 T_{7} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$36 - 24 T + 22 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$( -6 - 4 T + T^{2} )^{2}$$
$13$ $$81 + 18 T + 13 T^{2} - 2 T^{3} + T^{4}$$
$17$ $$36 + 24 T + 22 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$1296 - 144 T + 52 T^{2} + 4 T^{3} + T^{4}$$
$23$ $$( -10 + T^{2} )^{2}$$
$29$ $$( -36 - 4 T + T^{2} )^{2}$$
$31$ $$( -9 - 2 T + T^{2} )^{2}$$
$37$ $$1369 + 666 T + 145 T^{2} + 18 T^{3} + T^{4}$$
$41$ $$1521 + 78 T + 43 T^{2} - 2 T^{3} + T^{4}$$
$43$ $$( -39 - 2 T + T^{2} )^{2}$$
$47$ $$( -6 - 4 T + T^{2} )^{2}$$
$53$ $$12321 + 2442 T + 373 T^{2} + 22 T^{3} + T^{4}$$
$59$ $$36 - 24 T + 22 T^{2} + 4 T^{3} + T^{4}$$
$61$ $$1296 + 144 T + 52 T^{2} - 4 T^{3} + T^{4}$$
$67$ $$1296 + 144 T + 52 T^{2} - 4 T^{3} + T^{4}$$
$71$ $$1600 + 40 T^{2} + T^{4}$$
$73$ $$( -54 + 12 T + T^{2} )^{2}$$
$79$ $$( 4 - 2 T + T^{2} )^{2}$$
$83$ $$( 16 - 4 T + T^{2} )^{2}$$
$89$ $$20736 + 1152 T + 208 T^{2} - 8 T^{3} + T^{4}$$
$97$ $$( -4 + 12 T + T^{2} )^{2}$$