# Properties

 Label 1110.2.bb.b 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} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{12} + 5 \zeta_{12} q^{13} + q^{14} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{15} -\zeta_{12}^{2} q^{16} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + \zeta_{12} q^{18} + ( 7 - 7 \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}^{3} q^{23} + ( 1 - \zeta_{12}^{2} ) q^{24} + ( 4 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{25} + 5 q^{26} + \zeta_{12}^{3} q^{27} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{28} + 3 q^{29} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{30} + 2 q^{31} -\zeta_{12} q^{32} + ( 2 - 2 \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} -7 \zeta_{12}^{3} q^{38} + 5 \zeta_{12}^{2} q^{39} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{40} + \zeta_{12} q^{42} -2 \zeta_{12}^{3} q^{43} + ( -1 - 2 \zeta_{12}^{3} ) q^{45} + 4 \zeta_{12}^{2} q^{46} + 4 \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} + 2 q^{51} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{52} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{53} + \zeta_{12}^{2} q^{54} + ( 1 - \zeta_{12}^{2} ) q^{56} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{57} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{58} + ( -2 + \zeta_{12}^{3} ) q^{60} + ( 4 - 4 \zeta_{12}^{2} ) q^{61} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{62} + \zeta_{12}^{3} q^{63} - q^{64} + ( -5 \zeta_{12} - 10 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{65} -4 \zeta_{12} q^{67} -2 \zeta_{12}^{3} q^{68} + ( -4 + 4 \zeta_{12}^{2} ) q^{69} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{70} + ( 13 - 13 \zeta_{12}^{2} ) q^{71} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{72} + 8 \zeta_{12}^{3} q^{73} + ( 7 - 3 \zeta_{12}^{2} ) q^{74} + ( 4 + 3 \zeta_{12}^{3} ) q^{75} -7 \zeta_{12}^{2} q^{76} + 5 \zeta_{12} q^{78} + ( -10 + 10 \zeta_{12}^{2} ) q^{79} + ( 1 + 2 \zeta_{12}^{3} ) q^{80} + ( -1 + \zeta_{12}^{2} ) q^{81} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{83} + q^{84} + ( -4 + 2 \zeta_{12}^{3} ) q^{85} -2 \zeta_{12}^{2} q^{86} + 3 \zeta_{12} q^{87} -12 \zeta_{12}^{2} q^{89} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{90} + 5 \zeta_{12}^{2} q^{91} + 4 \zeta_{12} q^{92} + 2 \zeta_{12} q^{93} + 4 \zeta_{12}^{2} q^{94} + ( -14 \zeta_{12} + 7 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{95} -\zeta_{12}^{2} q^{96} + 8 \zeta_{12}^{3} q^{97} -6 \zeta_{12} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 2q^{5} + 4q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{4} - 2q^{5} + 4q^{6} + 2q^{9} - 8q^{10} + 4q^{14} - 4q^{15} - 2q^{16} + 14q^{19} + 2q^{20} + 2q^{21} + 2q^{24} + 6q^{25} + 20q^{26} + 12q^{29} - 2q^{30} + 8q^{31} + 4q^{34} - 4q^{35} + 4q^{36} + 10q^{39} - 4q^{40} - 4q^{45} + 8q^{46} - 12q^{49} + 8q^{50} + 8q^{51} + 2q^{54} + 2q^{56} - 8q^{60} + 8q^{61} - 4q^{64} - 20q^{65} - 8q^{69} - 2q^{70} + 26q^{71} + 22q^{74} + 16q^{75} - 14q^{76} - 20q^{79} + 4q^{80} - 2q^{81} + 4q^{84} - 16q^{85} - 4q^{86} - 24q^{89} - 4q^{90} + 10q^{91} + 8q^{94} + 14q^{95} - 2q^{96} + O(q^{100})$$

## 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.b 4
5.b even 2 1 inner 1110.2.bb.b 4
37.c even 3 1 inner 1110.2.bb.b 4
185.n even 6 1 inner 1110.2.bb.b 4

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$25 + 10 T - T^{2} + 2 T^{3} + T^{4}$$
$7$ $$1 - T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$625 - 25 T^{2} + T^{4}$$
$17$ $$16 - 4 T^{2} + T^{4}$$
$19$ $$( 49 - 7 T + T^{2} )^{2}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$( -3 + T )^{4}$$
$31$ $$( -2 + T )^{4}$$
$37$ $$1369 - 73 T^{2} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 4 + T^{2} )^{2}$$
$47$ $$( 16 + T^{2} )^{2}$$
$53$ $$4096 - 64 T^{2} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 16 - 4 T + T^{2} )^{2}$$
$67$ $$256 - 16 T^{2} + T^{4}$$
$71$ $$( 169 - 13 T + T^{2} )^{2}$$
$73$ $$( 64 + T^{2} )^{2}$$
$79$ $$( 100 + 10 T + T^{2} )^{2}$$
$83$ $$6561 - 81 T^{2} + T^{4}$$
$89$ $$( 144 + 12 T + T^{2} )^{2}$$
$97$ $$( 64 + T^{2} )^{2}$$