# Properties

 Label 1110.2.d.f Level $1110$ Weight $2$ Character orbit 1110.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## Character values

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

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
889.1
 − 1.61803i 0.618034i − 0.618034i 1.61803i
1.00000i 1.00000i −1.00000 2.23607i −1.00000 2.00000i 1.00000i −1.00000 −2.23607
889.2 1.00000i 1.00000i −1.00000 2.23607i −1.00000 2.00000i 1.00000i −1.00000 2.23607
889.3 1.00000i 1.00000i −1.00000 2.23607i −1.00000 2.00000i 1.00000i −1.00000 2.23607
889.4 1.00000i 1.00000i −1.00000 2.23607i −1.00000 2.00000i 1.00000i −1.00000 −2.23607
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.d.f 4
3.b odd 2 1 3330.2.d.j 4
5.b even 2 1 inner 1110.2.d.f 4
5.c odd 4 1 5550.2.a.bu 2
5.c odd 4 1 5550.2.a.bz 2
15.d odd 2 1 3330.2.d.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.f 4 1.a even 1 1 trivial
1110.2.d.f 4 5.b even 2 1 inner
3330.2.d.j 4 3.b odd 2 1
3330.2.d.j 4 15.d odd 2 1
5550.2.a.bu 2 5.c odd 4 1
5550.2.a.bz 2 5.c odd 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$( 4 + T^{2} )^{2}$$
$11$ $$( 4 - 6 T + T^{2} )^{2}$$
$13$ $$( 20 + T^{2} )^{2}$$
$17$ $$( 20 + T^{2} )^{2}$$
$19$ $$( 20 - 10 T + T^{2} )^{2}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$( 4 + T )^{4}$$
$31$ $$( -16 - 4 T + T^{2} )^{2}$$
$37$ $$( 1 + T^{2} )^{2}$$
$41$ $$( -20 + T^{2} )^{2}$$
$43$ $$256 + 112 T^{2} + T^{4}$$
$47$ $$1296 + 108 T^{2} + T^{4}$$
$53$ $$1936 + 168 T^{2} + T^{4}$$
$59$ $$( -76 - 4 T + T^{2} )^{2}$$
$61$ $$( 4 - 6 T + T^{2} )^{2}$$
$67$ $$256 + 112 T^{2} + T^{4}$$
$71$ $$( -16 - 4 T + T^{2} )^{2}$$
$73$ $$256 + 112 T^{2} + T^{4}$$
$79$ $$( -16 + 4 T + T^{2} )^{2}$$
$83$ $$30976 + 368 T^{2} + T^{4}$$
$89$ $$( 124 - 24 T + T^{2} )^{2}$$
$97$ $$13456 + 268 T^{2} + T^{4}$$