# Properties

 Label 370.2.l.a Level $370$ Weight $2$ Character orbit 370.l 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.l (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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0.866025 1.50000i 0.500000 0.866025i 0.866025 + 0.500000i 1.73205i −0.366025 + 0.633975i 1.00000i 0 −1.00000
11.2 0.866025 0.500000i −0.866025 + 1.50000i 0.500000 0.866025i −0.866025 0.500000i 1.73205i 1.36603 2.36603i 1.00000i 0 −1.00000
101.1 −0.866025 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i 0.866025 0.500000i 1.73205i −0.366025 0.633975i 1.00000i 0 −1.00000
101.2 0.866025 + 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.73205i 1.36603 + 2.36603i 1.00000i 0 −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.e 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.l.a 4
37.e even 6 1 inner 370.2.l.a 4

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

## Hecke kernels

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

 $$T_{3}^{4} + 3 T_{3}^{2} + 9$$ $$T_{7}^{4} - 2 T_{7}^{3} + 6 T_{7}^{2} + 4 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$1 - T^{2} + T^{4}$$
$7$ $$4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$( 6 - 6 T + T^{2} )^{2}$$
$13$ $$529 - 414 T + 131 T^{2} - 18 T^{3} + T^{4}$$
$17$ $$4 - 12 T + 14 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$16 - 48 T + 44 T^{2} + 12 T^{3} + T^{4}$$
$23$ $$36 + 24 T^{2} + T^{4}$$
$29$ $$1936 + 104 T^{2} + T^{4}$$
$31$ $$2209 + 98 T^{2} + T^{4}$$
$37$ $$( 37 + T + T^{2} )^{2}$$
$41$ $$121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4}$$
$43$ $$169 + 74 T^{2} + T^{4}$$
$47$ $$( 22 + 10 T + T^{2} )^{2}$$
$53$ $$9801 - 594 T + 135 T^{2} + 6 T^{3} + T^{4}$$
$59$ $$4356 + 1980 T + 366 T^{2} + 30 T^{3} + T^{4}$$
$61$ $$16 - 48 T + 44 T^{2} + 12 T^{3} + T^{4}$$
$67$ $$( 64 + 8 T + T^{2} )^{2}$$
$71$ $$10816 - 416 T + 120 T^{2} + 4 T^{3} + T^{4}$$
$73$ $$( -66 - 6 T + T^{2} )^{2}$$
$79$ $$20736 - 144 T^{2} + T^{4}$$
$83$ $$17424 + 3168 T + 444 T^{2} + 24 T^{3} + T^{4}$$
$89$ $$1936 + 1056 T + 236 T^{2} + 24 T^{3} + T^{4}$$
$97$ $$8464 + 248 T^{2} + T^{4}$$