# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.u (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$-1$$ $$\zeta_{8}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−0.707107 0.707107i −1.41421 1.00000i 1.00000i −0.707107 + 0.707107i 0.292893 + 1.70711i 2.00000 0.707107 0.707107i 1.00000 + 2.82843i 1.00000
191.2 0.707107 + 0.707107i 1.41421 1.00000i 1.00000i 0.707107 0.707107i 1.70711 + 0.292893i 2.00000 −0.707107 + 0.707107i 1.00000 2.82843i 1.00000
401.1 −0.707107 + 0.707107i −1.41421 + 1.00000i 1.00000i −0.707107 0.707107i 0.292893 1.70711i 2.00000 0.707107 + 0.707107i 1.00000 2.82843i 1.00000
401.2 0.707107 0.707107i 1.41421 + 1.00000i 1.00000i 0.707107 + 0.707107i 1.70711 0.292893i 2.00000 −0.707107 0.707107i 1.00000 + 2.82843i 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
3.b odd 2 1 inner
37.d odd 4 1 inner
111.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.u.d 4
3.b odd 2 1 inner 1110.2.u.d 4
37.d odd 4 1 inner 1110.2.u.d 4
111.g even 4 1 inner 1110.2.u.d 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} - 2$$ acting on $$S_{2}^{\mathrm{new}}(1110, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ $$9 - 2 T^{2} + T^{4}$$
$5$ $$1 + T^{4}$$
$7$ $$( -2 + T )^{4}$$
$11$ $$T^{4}$$
$13$ $$( 2 - 2 T + T^{2} )^{2}$$
$17$ $$1296 + T^{4}$$
$19$ $$( 18 + 6 T + T^{2} )^{2}$$
$23$ $$256 + T^{4}$$
$29$ $$1296 + T^{4}$$
$31$ $$( 98 + 14 T + T^{2} )^{2}$$
$37$ $$( 37 + 2 T + T^{2} )^{2}$$
$41$ $$( -32 + T^{2} )^{2}$$
$43$ $$( 98 - 14 T + T^{2} )^{2}$$
$47$ $$( 8 + T^{2} )^{2}$$
$53$ $$( 32 + T^{2} )^{2}$$
$59$ $$4096 + T^{4}$$
$61$ $$( 162 + 18 T + T^{2} )^{2}$$
$67$ $$( 16 + T^{2} )^{2}$$
$71$ $$( 8 + T^{2} )^{2}$$
$73$ $$( 36 + T^{2} )^{2}$$
$79$ $$( 18 - 6 T + T^{2} )^{2}$$
$83$ $$( 8 + T^{2} )^{2}$$
$89$ $$16 + T^{4}$$
$97$ $$( 98 - 14 T + T^{2} )^{2}$$