# Properties

 Label 1110.2.i.l 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: $$\Q(\sqrt{-3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 10 \nu$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{3} + 10 \beta_{2}$$$$)/3$$

## Character values

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

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

## 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
 −1.93649 − 1.11803i 1.93649 + 1.11803i −1.93649 + 1.11803i 1.93649 − 1.11803i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i −1.00000 −2.43649 4.22013i −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 1.43649 + 2.48808i −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 −2.43649 + 4.22013i −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 1.43649 2.48808i −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.l 4
37.c even 3 1 inner 1110.2.i.l 4

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

 $$T_{7}^{4} + 2 T_{7}^{3} + 18 T_{7}^{2} - 28 T_{7} + 196$$ $$T_{11} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$196 - 28 T + 18 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$( 5 + T )^{4}$$
$13$ $$( 1 - T + T^{2} )^{2}$$
$17$ $$1 + 8 T + 63 T^{2} + 8 T^{3} + T^{4}$$
$19$ $$3600 + 60 T^{2} + T^{4}$$
$23$ $$( 6 + T )^{4}$$
$29$ $$( -15 + T^{2} )^{2}$$
$31$ $$( -60 + T^{2} )^{2}$$
$37$ $$( 37 + 11 T + T^{2} )^{2}$$
$41$ $$36 + 36 T + 42 T^{2} - 6 T^{3} + T^{4}$$
$43$ $$( -56 + 4 T + T^{2} )^{2}$$
$47$ $$( -51 + 6 T + T^{2} )^{2}$$
$53$ $$36 + 36 T + 42 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$( 25 + 5 T + T^{2} )^{2}$$
$61$ $$100 - 100 T + 90 T^{2} - 10 T^{3} + T^{4}$$
$67$ $$7225 + 1700 T + 315 T^{2} + 20 T^{3} + T^{4}$$
$71$ $$15876 - 756 T + 162 T^{2} + 6 T^{3} + T^{4}$$
$73$ $$( -12 + T )^{4}$$
$79$ $$57600 + 240 T^{2} + T^{4}$$
$83$ $$196 + 28 T + 18 T^{2} - 2 T^{3} + T^{4}$$
$89$ $$1936 - 352 T + 108 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$( 10 + T )^{4}$$