# Properties

 Label 1110.2.h.e 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: $$\Q(i, \sqrt{33})$$ Defining polynomial: $$x^{4} + 17 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 17 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 9 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 9$$ $$\nu^{3}$$ $$=$$ $$8 \beta_{2} - 9 \beta_{1}$$

## 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
 − 2.37228i 3.37228i 2.37228i − 3.37228i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i −2.37228 1.00000i 1.00000 −1.00000
961.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.37228 1.00000i 1.00000 −1.00000
961.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −2.37228 1.00000i 1.00000 −1.00000
961.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.37228 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.e 4
3.b odd 2 1 3330.2.h.k 4
37.b even 2 1 inner 1110.2.h.e 4
111.d odd 2 1 3330.2.h.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.h.e 4 1.a even 1 1 trivial
1110.2.h.e 4 37.b even 2 1 inner
3330.2.h.k 4 3.b odd 2 1
3330.2.h.k 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 $$S_{2}^{\mathrm{new}}(1110, [\chi])$$:

 $$T_{7}^{2} - T_{7} - 8$$ $$T_{13}^{4} + 68 T_{13}^{2} + 1024$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( -8 - T + T^{2} )^{2}$$
$11$ $$( 4 - 7 T + T^{2} )^{2}$$
$13$ $$1024 + 68 T^{2} + T^{4}$$
$17$ $$4 + 29 T^{2} + T^{4}$$
$19$ $$( 36 + T^{2} )^{2}$$
$23$ $$576 + 84 T^{2} + T^{4}$$
$29$ $$1156 + 101 T^{2} + T^{4}$$
$31$ $$484 + 77 T^{2} + T^{4}$$
$37$ $$1369 - 58 T^{2} + T^{4}$$
$41$ $$( -74 + T + T^{2} )^{2}$$
$43$ $$36 + 21 T^{2} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$( 22 + 11 T + T^{2} )^{2}$$
$59$ $$( 64 + T^{2} )^{2}$$
$61$ $$4096 + 161 T^{2} + T^{4}$$
$67$ $$( -24 - 6 T + T^{2} )^{2}$$
$71$ $$( -24 + 6 T + T^{2} )^{2}$$
$73$ $$( -14 + T )^{4}$$
$79$ $$1024 + 68 T^{2} + T^{4}$$
$83$ $$( -96 - 12 T + T^{2} )^{2}$$
$89$ $$576 + 84 T^{2} + T^{4}$$
$97$ $$5184 + 153 T^{2} + T^{4}$$