# Properties

 Label 1110.2.o.b Level $1110$ Weight $2$ Character orbit 1110.o Analytic conductor $8.863$ Analytic rank $0$ Dimension $40$ 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: $$40$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 40q^{2} + 40q^{4} - 4q^{7} + 40q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 40q^{2} + 40q^{4} - 4q^{7} + 40q^{8} + 8q^{13} - 4q^{14} + 40q^{16} - 4q^{19} - 8q^{25} + 8q^{26} - 4q^{28} + 28q^{31} + 40q^{32} - 4q^{33} + 12q^{35} + 8q^{37} - 4q^{38} - 4q^{39} - 16q^{47} - 8q^{50} + 16q^{51} + 8q^{52} + 20q^{53} - 8q^{55} - 4q^{56} - 8q^{57} + 4q^{59} - 8q^{61} + 28q^{62} + 4q^{63} + 40q^{64} + 4q^{65} - 4q^{66} - 16q^{67} + 8q^{69} + 12q^{70} + 40q^{71} - 8q^{73} + 8q^{74} + 16q^{75} - 4q^{76} + 24q^{77} - 4q^{78} + 12q^{79} - 40q^{81} - 8q^{83} + 8q^{85} - 12q^{89} - 24q^{91} - 8q^{93} - 16q^{94} + 28q^{95} - 8q^{99} + O(q^{100})$$

## 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 −1.40994 1.73553i −0.707107 + 0.707107i −3.05055 + 3.05055i 1.00000 1.00000i −1.40994 1.73553i
253.2 1.00000 −0.707107 + 0.707107i 1.00000 −1.17802 + 1.90060i −0.707107 + 0.707107i −2.67041 + 2.67041i 1.00000 1.00000i −1.17802 + 1.90060i
253.3 1.00000 −0.707107 + 0.707107i 1.00000 1.26431 1.84432i −0.707107 + 0.707107i 1.64144 1.64144i 1.00000 1.00000i 1.26431 1.84432i
253.4 1.00000 −0.707107 + 0.707107i 1.00000 −0.825074 + 2.07828i −0.707107 + 0.707107i 1.45633 1.45633i 1.00000 1.00000i −0.825074 + 2.07828i
253.5 1.00000 −0.707107 + 0.707107i 1.00000 −2.23211 + 0.132937i −0.707107 + 0.707107i 1.47899 1.47899i 1.00000 1.00000i −2.23211 + 0.132937i
253.6 1.00000 −0.707107 + 0.707107i 1.00000 −1.19936 1.88721i −0.707107 + 0.707107i −1.50593 + 1.50593i 1.00000 1.00000i −1.19936 1.88721i
253.7 1.00000 −0.707107 + 0.707107i 1.00000 2.22571 0.215025i −0.707107 + 0.707107i −1.35684 + 1.35684i 1.00000 1.00000i 2.22571 0.215025i
253.8 1.00000 −0.707107 + 0.707107i 1.00000 0.906760 2.04396i −0.707107 + 0.707107i 1.03373 1.03373i 1.00000 1.00000i 0.906760 2.04396i
253.9 1.00000 −0.707107 + 0.707107i 1.00000 1.62651 + 1.53443i −0.707107 + 0.707107i −2.68468 + 2.68468i 1.00000 1.00000i 1.62651 + 1.53443i
253.10 1.00000 −0.707107 + 0.707107i 1.00000 0.821222 + 2.07981i −0.707107 + 0.707107i 1.82950 1.82950i 1.00000 1.00000i 0.821222 + 2.07981i
253.11 1.00000 0.707107 0.707107i 1.00000 0.837144 + 2.07345i 0.707107 0.707107i −2.93723 + 2.93723i 1.00000 1.00000i 0.837144 + 2.07345i
253.12 1.00000 0.707107 0.707107i 1.00000 −0.161376 + 2.23024i 0.707107 0.707107i 1.80704 1.80704i 1.00000 1.00000i −0.161376 + 2.23024i
253.13 1.00000 0.707107 0.707107i 1.00000 −2.22514 + 0.220830i 0.707107 0.707107i −1.18593 + 1.18593i 1.00000 1.00000i −2.22514 + 0.220830i
253.14 1.00000 0.707107 0.707107i 1.00000 2.21223 + 0.325656i 0.707107 0.707107i 0.597471 0.597471i 1.00000 1.00000i 2.21223 + 0.325656i
253.15 1.00000 0.707107 0.707107i 1.00000 −0.314477 2.21384i 0.707107 0.707107i −0.376687 + 0.376687i 1.00000 1.00000i −0.314477 2.21384i
253.16 1.00000 0.707107 0.707107i 1.00000 0.620781 2.14817i 0.707107 0.707107i −0.434986 + 0.434986i 1.00000 1.00000i 0.620781 2.14817i
253.17 1.00000 0.707107 0.707107i 1.00000 2.00267 + 0.994632i 0.707107 0.707107i 0.593998 0.593998i 1.00000 1.00000i 2.00267 + 0.994632i
253.18 1.00000 0.707107 0.707107i 1.00000 −2.22216 0.248996i 0.707107 0.707107i −2.88943 + 2.88943i 1.00000 1.00000i −2.22216 0.248996i
253.19 1.00000 0.707107 0.707107i 1.00000 −2.18349 + 0.482069i 0.707107 0.707107i 3.34184 3.34184i 1.00000 1.00000i −2.18349 + 0.482069i
253.20 1.00000 0.707107 0.707107i 1.00000 1.43381 1.71586i 0.707107 0.707107i 3.31235 3.31235i 1.00000 1.00000i 1.43381 1.71586i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 487.20 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.b yes 40
5.c odd 4 1 1110.2.l.b 40
37.d odd 4 1 1110.2.l.b 40
185.k even 4 1 inner 1110.2.o.b yes 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.l.b 40 5.c odd 4 1
1110.2.l.b 40 37.d odd 4 1
1110.2.o.b yes 40 1.a even 1 1 trivial
1110.2.o.b yes 40 185.k even 4 1 inner