Properties

 Label 1110.2.i.k Level $1110$ Weight $2$ Character orbit 1110.i 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.i (of order $$3$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{41})$$ Defining polynomial: $$x^{4} - x^{3} + 11 x^{2} + 10 x + 100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 11 x^{2} + 10 x + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 11 \nu^{2} - 11 \nu + 100$$$$)/110$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 21$$$$)/11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 10 \beta_{2} + \beta_{1} - 11$$ $$\nu^{3}$$ $$=$$ $$11 \beta_{3} - 21$$

Character values

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

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 −1.35078 − 2.33962i 1.85078 + 3.20565i −1.35078 + 2.33962i 1.85078 − 3.20565i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.00000 1.50000 + 2.59808i −1.00000 −0.500000 + 0.866025i −1.00000
121.2 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.00000 1.50000 + 2.59808i −1.00000 −0.500000 + 0.866025i −1.00000
211.1 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.00000 1.50000 2.59808i −1.00000 −0.500000 0.866025i −1.00000
211.2 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.00000 1.50000 2.59808i −1.00000 −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.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.i.k 4
37.c even 3 1 inner 1110.2.i.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.k 4 1.a even 1 1 trivial
1110.2.i.k 4 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1110, [\chi])$$:

 $$T_{7}^{2} - 3 T_{7} + 9$$ $$T_{11} - 2$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$( 9 - 3 T + T^{2} )^{2}$$
$11$ $$( -2 + T )^{4}$$
$13$ $$1681 + 41 T^{2} + T^{4}$$
$17$ $$( 4 - 2 T + T^{2} )^{2}$$
$19$ $$16 - 20 T + 29 T^{2} + 5 T^{3} + T^{4}$$
$23$ $$( -40 + 2 T + T^{2} )^{2}$$
$29$ $$( -8 - 3 T + T^{2} )^{2}$$
$31$ $$( -8 - 3 T + T^{2} )^{2}$$
$37$ $$1369 + 185 T - 12 T^{2} + 5 T^{3} + T^{4}$$
$41$ $$256 - 160 T + 116 T^{2} + 10 T^{3} + T^{4}$$
$43$ $$( -10 - T + T^{2} )^{2}$$
$47$ $$( 8 + T )^{4}$$
$53$ $$1024 + 192 T + 68 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$( 4 + 2 T + T^{2} )^{2}$$
$61$ $$1024 + 192 T + 68 T^{2} - 6 T^{3} + T^{4}$$
$67$ $$2116 - 690 T + 179 T^{2} - 15 T^{3} + T^{4}$$
$71$ $$100 - 10 T + 11 T^{2} + T^{3} + T^{4}$$
$73$ $$( 62 - 17 T + T^{2} )^{2}$$
$79$ $$8100 - 270 T + 99 T^{2} + 3 T^{3} + T^{4}$$
$83$ $$100 - 90 T + 71 T^{2} - 9 T^{3} + T^{4}$$
$89$ $$256 - 160 T + 116 T^{2} + 10 T^{3} + T^{4}$$
$97$ $$( 122 + 23 T + T^{2} )^{2}$$