# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.bb (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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1009.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.23205 + 1.86603i 1.00000 0.866025 0.500000i 1.00000i 0.500000 0.866025i 2.00000 1.00000i
1009.2 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 2.23205 0.133975i 1.00000 −0.866025 + 0.500000i 1.00000i 0.500000 0.866025i 2.00000 + 1.00000i
1099.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i −1.23205 1.86603i 1.00000 0.866025 + 0.500000i 1.00000i 0.500000 + 0.866025i 2.00000 + 1.00000i
1099.2 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 2.23205 + 0.133975i 1.00000 −0.866025 0.500000i 1.00000i 0.500000 + 0.866025i 2.00000 1.00000i
 $$n$$: 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 1110.2.bb.c 4
5.b even 2 1 inner 1110.2.bb.c 4
37.c even 3 1 inner 1110.2.bb.c 4
185.n even 6 1 inner 1110.2.bb.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.bb.c 4 1.a even 1 1 trivial
1110.2.bb.c 4 5.b even 2 1 inner
1110.2.bb.c 4 37.c even 3 1 inner
1110.2.bb.c 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 $$S_{2}^{\mathrm{new}}(1110, [\chi])$$:

 $$T_{7}^{4} - T_{7}^{2} + 1$$ $$T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$25 - 10 T - T^{2} - 2 T^{3} + T^{4}$$
$7$ $$1 - T^{2} + T^{4}$$
$11$ $$( -4 + T )^{4}$$
$13$ $$81 - 9 T^{2} + T^{4}$$
$17$ $$1296 - 36 T^{2} + T^{4}$$
$19$ $$( 1 + T + T^{2} )^{2}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$( -5 + T )^{4}$$
$31$ $$( 6 + T )^{4}$$
$37$ $$1369 - 73 T^{2} + T^{4}$$
$41$ $$( 64 - 8 T + T^{2} )^{2}$$
$43$ $$( 36 + T^{2} )^{2}$$
$47$ $$( 144 + T^{2} )^{2}$$
$53$ $$256 - 16 T^{2} + T^{4}$$
$59$ $$( 16 + 4 T + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$4096 - 64 T^{2} + T^{4}$$
$71$ $$( 9 - 3 T + T^{2} )^{2}$$
$73$ $$( 256 + T^{2} )^{2}$$
$79$ $$( 100 - 10 T + T^{2} )^{2}$$
$83$ $$83521 - 289 T^{2} + T^{4}$$
$89$ $$( 16 + 4 T + T^{2} )^{2}$$
$97$ $$( 144 + T^{2} )^{2}$$