# Properties

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

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## 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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

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

## 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
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 −1.36603 2.36603i −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 0.366025 + 0.633975i −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 −1.36603 + 2.36603i −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 0.366025 0.633975i −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.n 4
37.c even 3 1 inner 1110.2.i.n 4

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

 $$T_{7}^{4} + 2 T_{7}^{3} + 6 T_{7}^{2} - 4 T_{7} + 4$$ $$T_{11}^{2} - 6 T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$( -3 - 6 T + T^{2} )^{2}$$
$13$ $$9 - 18 T + 39 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$1089 + 396 T + 111 T^{2} + 12 T^{3} + T^{4}$$
$19$ $$144 + 12 T^{2} + T^{4}$$
$23$ $$( -44 - 4 T + T^{2} )^{2}$$
$29$ $$( -27 + T^{2} )^{2}$$
$31$ $$( -12 + T^{2} )^{2}$$
$37$ $$1369 - 74 T - 33 T^{2} - 2 T^{3} + T^{4}$$
$41$ $$6084 + 1404 T + 246 T^{2} + 18 T^{3} + T^{4}$$
$43$ $$( -104 - 4 T + T^{2} )^{2}$$
$47$ $$( -11 - 2 T + T^{2} )^{2}$$
$53$ $$484 - 220 T + 78 T^{2} - 10 T^{3} + T^{4}$$
$59$ $$9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4}$$
$61$ $$4356 + 396 T + 102 T^{2} - 6 T^{3} + T^{4}$$
$67$ $$1089 + 396 T + 111 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$484 - 220 T + 78 T^{2} - 10 T^{3} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 16 + 4 T + T^{2} )^{2}$$
$83$ $$2500 - 500 T + 150 T^{2} + 10 T^{3} + T^{4}$$
$89$ $$144 + 12 T^{2} + T^{4}$$
$97$ $$( -44 - 4 T + T^{2} )^{2}$$
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