# Properties

 Label 1110.2.x.a Level $1110$ Weight $2$ Character orbit 1110.x Analytic conductor $8.863$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.x (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.86339462436$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times$$.

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
751.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i −1.23205 + 2.13397i 1.00000i −0.500000 0.866025i 1.00000
751.2 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i 2.23205 3.86603i 1.00000i −0.500000 0.866025i 1.00000
841.1 −0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i −1.23205 2.13397i 1.00000i −0.500000 + 0.866025i 1.00000
841.2 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i 2.23205 + 3.86603i 1.00000i −0.500000 + 0.866025i 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 1110.2.x.a 4
37.e even 6 1 inner 1110.2.x.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.x.a 4 1.a even 1 1 trivial
1110.2.x.a 4 37.e even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 2 T_{7}^{3} + 15 T_{7}^{2} + 22 T_{7} + 121$$ acting on $$S_{2}^{\mathrm{new}}(1110, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$1 - T^{2} + T^{4}$$
$7$ $$121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$( -2 + T )^{4}$$
$13$ $$1 - T^{2} + T^{4}$$
$17$ $$64 + 96 T + 56 T^{2} + 12 T^{3} + T^{4}$$
$19$ $$81 - 9 T^{2} + T^{4}$$
$23$ $$36 + 24 T^{2} + T^{4}$$
$29$ $$1 + 14 T^{2} + T^{4}$$
$31$ $$64 + 32 T^{2} + T^{4}$$
$37$ $$1369 + 47 T^{2} + T^{4}$$
$41$ $$676 - 52 T + 30 T^{2} + 2 T^{3} + T^{4}$$
$43$ $$324 + 72 T^{2} + T^{4}$$
$47$ $$( 6 - 6 T + T^{2} )^{2}$$
$53$ $$4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4}$$
$59$ $$676 + 468 T + 134 T^{2} + 18 T^{3} + T^{4}$$
$61$ $$13924 + 708 T - 106 T^{2} - 6 T^{3} + T^{4}$$
$67$ $$144 + 12 T^{2} + T^{4}$$
$71$ $$81 - 108 T + 135 T^{2} - 12 T^{3} + T^{4}$$
$73$ $$( 94 - 22 T + T^{2} )^{2}$$
$79$ $$4 - 36 T + 110 T^{2} - 18 T^{3} + T^{4}$$
$83$ $$1521 + 234 T + 75 T^{2} - 6 T^{3} + T^{4}$$
$89$ $$576 + 288 T + 24 T^{2} - 12 T^{3} + T^{4}$$
$97$ $$19044 + 312 T^{2} + T^{4}$$