# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{73})$$ Defining polynomial: $$x^{4} + 37 x^{2} + 324$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 37 x^{2} + 324$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 19 \nu$$$$)/18$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 19$$ $$\nu^{3}$$ $$=$$ $$18 \beta_{2} - 19 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 − 4.77200i 3.77200i − 3.77200i 4.77200i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
961.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
961.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
961.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 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.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.h.d 4
3.b odd 2 1 3330.2.h.j 4
37.b even 2 1 inner 1110.2.h.d 4
111.d odd 2 1 3330.2.h.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.h.d 4 1.a even 1 1 trivial
1110.2.h.d 4 37.b even 2 1 inner
3330.2.h.j 4 3.b odd 2 1
3330.2.h.j 4 111.d odd 2 1

## Hecke kernels

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

 $$T_{7} + 1$$ $$T_{13}^{4} + 37 T_{13}^{2} + 324$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$( 1 + T )^{4}$$
$13$ $$324 + 37 T^{2} + T^{4}$$
$17$ $$( 1 + T^{2} )^{2}$$
$19$ $$36 + 61 T^{2} + T^{4}$$
$23$ $$36 + 61 T^{2} + T^{4}$$
$29$ $$36 + 61 T^{2} + T^{4}$$
$31$ $$36 + 61 T^{2} + T^{4}$$
$37$ $$1369 - 74 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$41$ $$( -6 - 7 T + T^{2} )^{2}$$
$43$ $$144 + 97 T^{2} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$( -69 + 4 T + T^{2} )^{2}$$
$59$ $$5184 + 148 T^{2} + T^{4}$$
$61$ $$2916 + 181 T^{2} + T^{4}$$
$67$ $$( -72 - 2 T + T^{2} )^{2}$$
$71$ $$( 6 + T )^{4}$$
$73$ $$( 2 - 9 T + T^{2} )^{2}$$
$79$ $$4096 + 164 T^{2} + T^{4}$$
$83$ $$( -12 - 5 T + T^{2} )^{2}$$
$89$ $$20736 + 369 T^{2} + T^{4}$$
$97$ $$576 + 121 T^{2} + T^{4}$$