# Properties

 Label 370.2.n.c Level $370$ Weight $2$ Character orbit 370.n 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.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.

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

## Character values

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

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

## 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 0 0.500000 + 0.866025i 1.86603 1.23205i 0 2.36603 1.36603i 1.00000i −1.50000 + 2.59808i −2.23205 + 0.133975i
269.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.133975 2.23205i 0 0.633975 0.366025i 1.00000i −1.50000 + 2.59808i 1.23205 1.86603i
359.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.86603 + 1.23205i 0 2.36603 + 1.36603i 1.00000i −1.50000 2.59808i −2.23205 0.133975i
359.2 0.866025 0.500000i 0 0.500000 0.866025i 0.133975 + 2.23205i 0 0.633975 + 0.366025i 1.00000i −1.50000 2.59808i 1.23205 + 1.86603i
 $$n$$: 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.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.c yes 4
5.b even 2 1 370.2.n.a 4
37.c even 3 1 370.2.n.a 4
185.n even 6 1 inner 370.2.n.c yes 4

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

 $$T_{3}$$ $$T_{7}^{4} - 6 T_{7}^{3} + 14 T_{7}^{2} - 12 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 - 20 T + 11 T^{2} - 4 T^{3} + T^{4}$$
$7$ $$4 - 12 T + 14 T^{2} - 6 T^{3} + T^{4}$$
$11$ $$( 6 - 6 T + T^{2} )^{2}$$
$13$ $$64 - 96 T + 56 T^{2} - 12 T^{3} + T^{4}$$
$17$ $$( 3 + 3 T + T^{2} )^{2}$$
$19$ $$36 - 36 T + 30 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$64 + 32 T^{2} + T^{4}$$
$29$ $$( -3 + T^{2} )^{2}$$
$31$ $$( -74 + 2 T + T^{2} )^{2}$$
$37$ $$1369 + 47 T^{2} + T^{4}$$
$41$ $$2209 + 94 T + 51 T^{2} - 2 T^{3} + T^{4}$$
$43$ $$2116 + 104 T^{2} + T^{4}$$
$47$ $$484 + 56 T^{2} + T^{4}$$
$53$ $$10816 + 3744 T + 536 T^{2} + 36 T^{3} + T^{4}$$
$59$ $$576 + 288 T + 120 T^{2} + 12 T^{3} + T^{4}$$
$61$ $$1089 + 396 T + 111 T^{2} + 12 T^{3} + T^{4}$$
$67$ $$2116 - 276 T - 34 T^{2} + 6 T^{3} + T^{4}$$
$71$ $$19044 - 828 T + 174 T^{2} + 6 T^{3} + T^{4}$$
$73$ $$35344 + 392 T^{2} + T^{4}$$
$79$ $$676 + 364 T + 222 T^{2} - 14 T^{3} + T^{4}$$
$83$ $$576 + 288 T + 24 T^{2} - 12 T^{3} + T^{4}$$
$89$ $$9 - 18 T + 39 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$529 + 62 T^{2} + T^{4}$$