# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.n (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.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i −2.59808 + 1.50000i 0.500000 + 0.866025i 1.86603 + 1.23205i 3.00000 1.73205 1.00000i 1.00000i 3.00000 5.19615i −1.00000 2.00000i
269.2 0.866025 + 0.500000i 2.59808 1.50000i 0.500000 + 0.866025i 0.133975 + 2.23205i 3.00000 −1.73205 + 1.00000i 1.00000i 3.00000 5.19615i −1.00000 + 2.00000i
359.1 −0.866025 + 0.500000i −2.59808 1.50000i 0.500000 0.866025i 1.86603 1.23205i 3.00000 1.73205 + 1.00000i 1.00000i 3.00000 + 5.19615i −1.00000 + 2.00000i
359.2 0.866025 0.500000i 2.59808 + 1.50000i 0.500000 0.866025i 0.133975 2.23205i 3.00000 −1.73205 1.00000i 1.00000i 3.00000 + 5.19615i −1.00000 2.00000i
 $$n$$: 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
5.b even 2 1 inner
37.c even 3 1 inner
185.n 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.n.d 4
5.b even 2 1 inner 370.2.n.d 4
37.c even 3 1 inner 370.2.n.d 4
185.n even 6 1 inner 370.2.n.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.n.d 4 1.a even 1 1 trivial
370.2.n.d 4 5.b even 2 1 inner
370.2.n.d 4 37.c even 3 1 inner
370.2.n.d 4 185.n 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 $$S_{2}^{\mathrm{new}}(370, [\chi])$$:

 $$T_{3}^{4} - 9 T_{3}^{2} + 81$$ $$T_{7}^{4} - 4 T_{7}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$81 - 9 T^{2} + T^{4}$$
$5$ $$25 - 20 T + 11 T^{2} - 4 T^{3} + T^{4}$$
$7$ $$16 - 4 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$1 - T^{2} + T^{4}$$
$17$ $$1296 - 36 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$( 7 + T )^{4}$$
$37$ $$1369 + 47 T^{2} + T^{4}$$
$41$ $$( 1 - T + T^{2} )^{2}$$
$43$ $$( 121 + T^{2} )^{2}$$
$47$ $$( 100 + T^{2} )^{2}$$
$53$ $$625 - 25 T^{2} + T^{4}$$
$59$ $$( 36 + 6 T + T^{2} )^{2}$$
$61$ $$( 36 - 6 T + T^{2} )^{2}$$
$67$ $$256 - 16 T^{2} + T^{4}$$
$71$ $$( 144 - 12 T + T^{2} )^{2}$$
$73$ $$( 256 + T^{2} )^{2}$$
$79$ $$( 64 + 8 T + T^{2} )^{2}$$
$83$ $$20736 - 144 T^{2} + T^{4}$$
$89$ $$( 324 - 18 T + T^{2} )^{2}$$
$97$ $$( 4 + T^{2} )^{2}$$