# Properties

 Label 370.2.e.d 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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$-\zeta_{12}$$ $$1$$

## 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
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.00000 −1.73205 3.00000i 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 1.73205 + 3.00000i 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 −1.73205 + 3.00000i 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 1.73205 3.00000i 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.d 4
37.c even 3 1 inner 370.2.e.d 4

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

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{7}^{4} + 12 T_{7}^{2} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$144 + 12 T^{2} + T^{4}$$
$11$ $$( -8 + 4 T + T^{2} )^{2}$$
$13$ $$121 - 22 T + 15 T^{2} + 2 T^{3} + T^{4}$$
$17$ $$64 - 32 T + 24 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$( 4 - 2 T + T^{2} )^{2}$$
$23$ $$( -8 + 4 T + T^{2} )^{2}$$
$29$ $$( 2 + T )^{4}$$
$31$ $$( 13 - 10 T + T^{2} )^{2}$$
$37$ $$1369 + 74 T - 33 T^{2} + 2 T^{3} + T^{4}$$
$41$ $$529 + 230 T + 123 T^{2} - 10 T^{3} + T^{4}$$
$43$ $$( -39 + 6 T + T^{2} )^{2}$$
$47$ $$( -12 + T^{2} )^{2}$$
$53$ $$4761 - 1242 T + 255 T^{2} - 18 T^{3} + T^{4}$$
$59$ $$64 + 32 T + 24 T^{2} - 4 T^{3} + T^{4}$$
$61$ $$1936 - 176 T + 60 T^{2} + 4 T^{3} + T^{4}$$
$67$ $$256 + 256 T + 240 T^{2} + 16 T^{3} + T^{4}$$
$71$ $$256 + 256 T + 240 T^{2} + 16 T^{3} + T^{4}$$
$73$ $$( 24 - 12 T + T^{2} )^{2}$$
$79$ $$256 - 256 T + 240 T^{2} - 16 T^{3} + T^{4}$$
$83$ $$30976 + 1408 T + 240 T^{2} - 8 T^{3} + T^{4}$$
$89$ $$( 4 - 2 T + T^{2} )^{2}$$
$97$ $$( 2 + T )^{4}$$