# Properties

 Label 1110.2.ba.a Level $1110$ Weight $2$ Character orbit 1110.ba 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.ba (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 18q^{2} - 18q^{4} + 2q^{5} + 36q^{8} + 18q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 18q^{2} - 18q^{4} + 2q^{5} + 36q^{8} + 18q^{9} + 2q^{10} + 4q^{11} - 14q^{13} - 2q^{15} - 18q^{16} + 18q^{18} + 6q^{19} - 4q^{20} - 2q^{22} - 20q^{23} + 4q^{25} + 28q^{26} - 2q^{30} - 18q^{32} - 6q^{33} - 40q^{35} - 36q^{36} + 20q^{37} + 6q^{39} + 2q^{40} + 10q^{41} - 2q^{44} - 2q^{45} + 10q^{46} + 10q^{49} - 2q^{50} - 14q^{52} - 12q^{53} + 56q^{55} + 8q^{57} + 30q^{58} + 18q^{59} + 4q^{60} - 6q^{61} - 12q^{62} + 36q^{64} + 40q^{65} + 36q^{67} + 12q^{69} + 20q^{70} - 24q^{71} + 18q^{72} - 34q^{74} + 8q^{75} - 6q^{76} - 24q^{77} - 6q^{78} + 2q^{80} - 18q^{81} - 20q^{82} + 36q^{83} + 26q^{85} - 10q^{87} + 4q^{88} + 4q^{90} - 36q^{91} + 10q^{92} + 12q^{93} + 12q^{94} - 30q^{95} + 52q^{97} + 10q^{98} + 2q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
529.1 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −1.83366 + 1.27972i 1.00000i 2.57051 + 1.48408i 1.00000 0.500000 + 0.866025i 2.02510 + 0.948138i
529.2 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −0.648845 2.13986i 1.00000i 1.49882 + 0.865342i 1.00000 0.500000 + 0.866025i −1.52875 + 1.63185i
529.3 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −0.491597 + 2.18136i 1.00000i −0.916644 0.529225i 1.00000 0.500000 + 0.866025i 2.13491 0.664945i
529.4 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i 2.18544 + 0.473136i 1.00000i 0.827955 + 0.478020i 1.00000 0.500000 + 0.866025i −0.682971 2.12921i
529.5 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i 1.51023 1.64901i 1.00000i 0.0701099 + 0.0404780i 1.00000 0.500000 + 0.866025i −2.18319 0.483391i
529.6 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −2.21345 0.317246i 1.00000i −1.17143 0.676327i 1.00000 0.500000 + 0.866025i 0.831981 + 2.07553i
529.7 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −0.00696233 + 2.23606i 1.00000i 3.60632 + 2.08211i 1.00000 0.500000 + 0.866025i 1.93996 1.11200i
529.8 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i 1.03441 1.98242i 1.00000i −3.17295 1.83190i 1.00000 0.500000 + 0.866025i −2.23403 + 0.0953865i
529.9 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i 1.83047 + 1.28429i 1.00000i −3.31268 1.91258i 1.00000 0.500000 + 0.866025i 0.196990 2.22737i
529.10 −0.500000 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 1.05954 + 1.96911i 1.00000i −3.29356 1.90154i 1.00000 0.500000 + 0.866025i 1.17553 1.90214i
529.11 −0.500000 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 2.17865 0.503482i 1.00000i −3.20005 1.84755i 1.00000 0.500000 + 0.866025i −1.52535 1.63502i
529.12 −0.500000 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −2.12688 + 0.690199i 1.00000i 3.89483 + 2.24868i 1.00000 0.500000 + 0.866025i 1.66117 + 1.49683i
529.13 −0.500000 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 0.668714 + 2.13373i 1.00000i 2.13280 + 1.23137i 1.00000 0.500000 + 0.866025i 1.51351 1.64599i
529.14 −0.500000 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 2.16244 + 0.569075i 1.00000i 1.99912 + 1.15419i 1.00000 0.500000 + 0.866025i −0.588388 2.15727i
529.15 −0.500000 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −2.18834 + 0.459547i 1.00000i −1.07219 0.619032i 1.00000 0.500000 + 0.866025i 1.49215 + 1.66538i
529.16 −0.500000 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 0.603880 2.15298i 1.00000i 0.998396 + 0.576424i 1.00000 0.500000 + 0.866025i −2.16648 + 0.553515i
529.17 −0.500000 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −0.581519 2.15913i 1.00000i 1.29297 + 0.746494i 1.00000 0.500000 + 0.866025i −1.57910 + 1.58317i
529.18 −0.500000 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −2.14251 0.640045i 1.00000i −2.75229 1.58904i 1.00000 0.500000 + 0.866025i 0.516959 + 2.17549i
619.1 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −1.83366 1.27972i 1.00000i 2.57051 1.48408i 1.00000 0.500000 0.866025i 2.02510 0.948138i
619.2 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −0.648845 + 2.13986i 1.00000i 1.49882 0.865342i 1.00000 0.500000 0.866025i −1.52875 1.63185i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 619.18 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.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.ba.a 36
5.b even 2 1 1110.2.ba.b yes 36
37.e even 6 1 1110.2.ba.b yes 36
185.l even 6 1 inner 1110.2.ba.a 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.ba.a 36 1.a even 1 1 trivial
1110.2.ba.a 36 185.l even 6 1 inner
1110.2.ba.b yes 36 5.b even 2 1
1110.2.ba.b yes 36 37.e even 6 1