# Properties

 Label 1110.2.o.a Level $1110$ Weight $2$ Character orbit 1110.o Analytic conductor $8.863$ Analytic rank $0$ Dimension $36$ 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.o (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 36q^{2} + 36q^{4} + 4q^{5} + 4q^{7} - 36q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 36q^{2} + 36q^{4} + 4q^{5} + 4q^{7} - 36q^{8} - 4q^{10} - 8q^{13} - 4q^{14} + 36q^{16} - 4q^{19} + 4q^{20} + 4q^{25} + 8q^{26} + 4q^{28} - 36q^{29} - 4q^{31} - 36q^{32} + 4q^{33} + 28q^{35} + 28q^{37} + 4q^{38} - 4q^{39} - 4q^{40} - 32q^{43} + 16q^{47} - 4q^{50} - 8q^{52} - 8q^{53} - 8q^{55} - 4q^{56} - 8q^{57} + 36q^{58} + 4q^{59} - 4q^{61} + 4q^{62} - 4q^{63} + 36q^{64} - 52q^{65} - 4q^{66} - 16q^{67} + 8q^{69} - 28q^{70} - 8q^{71} + 4q^{73} - 28q^{74} + 16q^{75} - 4q^{76} - 8q^{77} + 4q^{78} + 12q^{79} + 4q^{80} - 36q^{81} + 8q^{83} - 8q^{85} + 32q^{86} + 24q^{89} + 56q^{91} - 8q^{93} - 16q^{94} + 20q^{95} + 8q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
253.1 −1.00000 −0.707107 + 0.707107i 1.00000 −2.23554 + 0.0484605i 0.707107 0.707107i 2.77693 2.77693i −1.00000 1.00000i 2.23554 0.0484605i
253.2 −1.00000 −0.707107 + 0.707107i 1.00000 −2.22064 0.262217i 0.707107 0.707107i −2.69527 + 2.69527i −1.00000 1.00000i 2.22064 + 0.262217i
253.3 −1.00000 −0.707107 + 0.707107i 1.00000 0.0687789 2.23501i 0.707107 0.707107i 2.13605 2.13605i −1.00000 1.00000i −0.0687789 + 2.23501i
253.4 −1.00000 −0.707107 + 0.707107i 1.00000 −0.530977 + 2.17211i 0.707107 0.707107i −1.93388 + 1.93388i −1.00000 1.00000i 0.530977 2.17211i
253.5 −1.00000 −0.707107 + 0.707107i 1.00000 1.63524 + 1.52512i 0.707107 0.707107i 1.84299 1.84299i −1.00000 1.00000i −1.63524 1.52512i
253.6 −1.00000 −0.707107 + 0.707107i 1.00000 1.69741 1.45560i 0.707107 0.707107i −0.764053 + 0.764053i −1.00000 1.00000i −1.69741 + 1.45560i
253.7 −1.00000 −0.707107 + 0.707107i 1.00000 −0.866398 + 2.06140i 0.707107 0.707107i −0.662100 + 0.662100i −1.00000 1.00000i 0.866398 2.06140i
253.8 −1.00000 −0.707107 + 0.707107i 1.00000 2.23596 + 0.0221544i 0.707107 0.707107i −0.281427 + 0.281427i −1.00000 1.00000i −2.23596 0.0221544i
253.9 −1.00000 −0.707107 + 0.707107i 1.00000 1.21617 1.87641i 0.707107 0.707107i −2.24767 + 2.24767i −1.00000 1.00000i −1.21617 + 1.87641i
253.10 −1.00000 0.707107 0.707107i 1.00000 1.91070 + 1.16157i −0.707107 + 0.707107i 2.90687 2.90687i −1.00000 1.00000i −1.91070 1.16157i
253.11 −1.00000 0.707107 0.707107i 1.00000 −0.891783 2.05054i −0.707107 + 0.707107i −1.56460 + 1.56460i −1.00000 1.00000i 0.891783 + 2.05054i
253.12 −1.00000 0.707107 0.707107i 1.00000 0.407420 + 2.19864i −0.707107 + 0.707107i 1.64494 1.64494i −1.00000 1.00000i −0.407420 2.19864i
253.13 −1.00000 0.707107 0.707107i 1.00000 −1.65797 1.50038i −0.707107 + 0.707107i −1.06172 + 1.06172i −1.00000 1.00000i 1.65797 + 1.50038i
253.14 −1.00000 0.707107 0.707107i 1.00000 −2.02734 + 0.943349i −0.707107 + 0.707107i −0.516238 + 0.516238i −1.00000 1.00000i 2.02734 0.943349i
253.15 −1.00000 0.707107 0.707107i 1.00000 1.94044 1.11117i −0.707107 + 0.707107i 0.636201 0.636201i −1.00000 1.00000i −1.94044 + 1.11117i
253.16 −1.00000 0.707107 0.707107i 1.00000 0.914315 2.04060i −0.707107 + 0.707107i 0.867388 0.867388i −1.00000 1.00000i −0.914315 + 2.04060i
253.17 −1.00000 0.707107 0.707107i 1.00000 2.05225 + 0.887844i −0.707107 + 0.707107i −1.84414 + 1.84414i −1.00000 1.00000i −2.05225 0.887844i
253.18 −1.00000 0.707107 0.707107i 1.00000 −1.64803 + 1.51129i −0.707107 + 0.707107i 2.75974 2.75974i −1.00000 1.00000i 1.64803 1.51129i
487.1 −1.00000 −0.707107 0.707107i 1.00000 −2.23554 0.0484605i 0.707107 + 0.707107i 2.77693 + 2.77693i −1.00000 1.00000i 2.23554 + 0.0484605i
487.2 −1.00000 −0.707107 0.707107i 1.00000 −2.22064 + 0.262217i 0.707107 + 0.707107i −2.69527 2.69527i −1.00000 1.00000i 2.22064 0.262217i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 487.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.k 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.o.a yes 36
5.c odd 4 1 1110.2.l.a 36
37.d odd 4 1 1110.2.l.a 36
185.k even 4 1 inner 1110.2.o.a yes 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.l.a 36 5.c odd 4 1
1110.2.l.a 36 37.d odd 4 1
1110.2.o.a yes 36 1.a even 1 1 trivial
1110.2.o.a yes 36 185.k even 4 1 inner