# Properties

 Label 1110.2.a.s Level $1110$ Weight $2$ Character orbit 1110.a Self dual yes Analytic conductor $8.863$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.54764.1 Defining polynomial: $$x^{4} - x^{3} - 9 x^{2} + 3 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 9 x^{2} + 3 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 7 \nu - 2$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 11 \nu - 2$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} + 7 \nu - 6$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{2} + 11 \beta_{1} + 4$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.655762 3.36007 −0.339102 −2.67673
1.00000 1.00000 1.00000 1.00000 1.00000 −3.46569 1.00000 1.00000 1.00000
1.2 1.00000 1.00000 1.00000 1.00000 1.00000 0.487359 1.00000 1.00000 1.00000
1.3 1.00000 1.00000 1.00000 1.00000 1.00000 1.84556 1.00000 1.00000 1.00000
1.4 1.00000 1.00000 1.00000 1.00000 1.00000 5.13277 1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$37$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.a.s 4
3.b odd 2 1 3330.2.a.bj 4
4.b odd 2 1 8880.2.a.cg 4
5.b even 2 1 5550.2.a.cj 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.s 4 1.a even 1 1 trivial
3330.2.a.bj 4 3.b odd 2 1
5550.2.a.cj 4 5.b even 2 1
8880.2.a.cg 4 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1110))$$:

 $$T_{7}^{4} - 4 T_{7}^{3} - 13 T_{7}^{2} + 40 T_{7} - 16$$ $$T_{11}^{4} - 2 T_{11}^{3} - 23 T_{11}^{2} + 68 T_{11} - 40$$ $$T_{13}^{4} + 3 T_{13}^{3} - 38 T_{13}^{2} - 44 T_{13} + 328$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$-16 + 40 T - 13 T^{2} - 4 T^{3} + T^{4}$$
$11$ $$-40 + 68 T - 23 T^{2} - 2 T^{3} + T^{4}$$
$13$ $$328 - 44 T - 38 T^{2} + 3 T^{3} + T^{4}$$
$17$ $$-764 + 412 T - 47 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$-80 + 208 T - 50 T^{2} - 3 T^{3} + T^{4}$$
$23$ $$1280 - 64 T - 76 T^{2} + T^{3} + T^{4}$$
$29$ $$1208 - 204 T - 82 T^{2} + 3 T^{3} + T^{4}$$
$31$ $$32 - 38 T^{2} - 3 T^{3} + T^{4}$$
$37$ $$( 1 + T )^{4}$$
$41$ $$-80 - 84 T + 4 T^{2} + 11 T^{3} + T^{4}$$
$43$ $$1280 - 128 T - 76 T^{2} + 5 T^{3} + T^{4}$$
$47$ $$1280 - 96 T - 88 T^{2} + T^{4}$$
$53$ $$-2524 - 244 T + 121 T^{2} + 22 T^{3} + T^{4}$$
$59$ $$-256 - 320 T - 52 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$1096 + 196 T - 130 T^{2} + 5 T^{3} + T^{4}$$
$67$ $$128 + 32 T - 72 T^{2} - 6 T^{3} + T^{4}$$
$71$ $$512 - 1536 T - 56 T^{2} + 18 T^{3} + T^{4}$$
$73$ $$-2336 - 1132 T - 128 T^{2} + 5 T^{3} + T^{4}$$
$79$ $$( -4 + T )^{4}$$
$83$ $$5344 + 704 T - 234 T^{2} + T^{3} + T^{4}$$
$89$ $$-3208 - 1956 T - 234 T^{2} + 5 T^{3} + T^{4}$$
$97$ $$9344 + 340 T - 236 T^{2} - 7 T^{3} + T^{4}$$