# Properties

 Label 1110.2.bb.d Level $1110$ Weight $2$ Character orbit 1110.bb Analytic conductor $8.863$ Analytic rank $0$ Dimension $28$ 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.bb (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q + 14q^{4} + 2q^{5} + 28q^{6} + 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + 14q^{4} + 2q^{5} + 28q^{6} + 14q^{9} + 4q^{10} - 12q^{11} + 8q^{14} + 2q^{15} - 14q^{16} - 20q^{19} - 2q^{20} + 4q^{21} + 14q^{24} - 8q^{25} - 20q^{26} - 36q^{29} + 2q^{30} + 24q^{31} + 38q^{34} - 2q^{35} + 28q^{36} - 10q^{39} + 2q^{40} - 6q^{44} + 4q^{45} + 8q^{46} + 50q^{49} - 4q^{50} + 76q^{51} + 14q^{54} - 28q^{55} + 4q^{56} - 26q^{59} + 4q^{60} - 28q^{61} - 28q^{64} + 60q^{65} - 12q^{66} - 8q^{69} - 10q^{70} - 64q^{71} + 24q^{74} - 8q^{75} + 20q^{76} + 32q^{79} - 4q^{80} - 14q^{81} + 8q^{84} + 16q^{85} - 8q^{86} + 76q^{89} + 2q^{90} - 8q^{91} - 38q^{94} - 70q^{95} - 14q^{96} - 6q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1009.1 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −0.00835632 + 2.23605i 1.00000 3.70065 2.13657i 1.00000i 0.500000 0.866025i 1.12526 1.93230i
1009.2 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −0.589266 2.15703i 1.00000 −3.59896 + 2.07786i 1.00000i 0.500000 0.866025i −0.568195 + 2.16267i
1009.3 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −2.16486 + 0.559786i 1.00000 −2.76595 + 1.59692i 1.00000i 0.500000 0.866025i 2.15472 + 0.597644i
1009.4 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −2.14608 0.627973i 1.00000 2.93485 1.69443i 1.00000i 0.500000 0.866025i 1.54457 + 1.61688i
1009.5 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 2.09232 + 0.788796i 1.00000 0.268117 0.154798i 1.00000i 0.500000 0.866025i −1.41760 1.72928i
1009.6 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 1.64155 1.51833i 1.00000 1.09384 0.631530i 1.00000i 0.500000 0.866025i −2.18079 + 0.494135i
1009.7 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.808671 + 2.08472i 1.00000 −3.36459 + 1.94255i 1.00000i 0.500000 0.866025i 0.342030 2.20975i
1009.8 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 1.40108 + 1.74269i 1.00000 3.36459 1.94255i 1.00000i 0.500000 0.866025i 0.342030 + 2.20975i
1009.9 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −2.13568 + 0.662460i 1.00000 −1.09384 + 0.631530i 1.00000i 0.500000 0.866025i −2.18079 0.494135i
1009.10 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.363043 + 2.20640i 1.00000 −0.268117 + 0.154798i 1.00000i 0.500000 0.866025i −1.41760 + 1.72928i
1009.11 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 0.529198 2.17254i 1.00000 −2.93485 + 1.69443i 1.00000i 0.500000 0.866025i 1.54457 1.61688i
1009.12 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 1.56722 1.59494i 1.00000 2.76595 1.59692i 1.00000i 0.500000 0.866025i 2.15472 0.597644i
1009.13 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −1.57341 1.58883i 1.00000 3.59896 2.07786i 1.00000i 0.500000 0.866025i −0.568195 2.16267i
1009.14 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 1.94066 + 1.11079i 1.00000 −3.70065 + 2.13657i 1.00000i 0.500000 0.866025i 1.12526 + 1.93230i
1099.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i −0.00835632 2.23605i 1.00000 3.70065 + 2.13657i 1.00000i 0.500000 + 0.866025i 1.12526 + 1.93230i
1099.2 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i −0.589266 + 2.15703i 1.00000 −3.59896 2.07786i 1.00000i 0.500000 + 0.866025i −0.568195 2.16267i
1099.3 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i −2.16486 0.559786i 1.00000 −2.76595 1.59692i 1.00000i 0.500000 + 0.866025i 2.15472 0.597644i
1099.4 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i −2.14608 + 0.627973i 1.00000 2.93485 + 1.69443i 1.00000i 0.500000 + 0.866025i 1.54457 1.61688i
1099.5 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i 2.09232 0.788796i 1.00000 0.268117 + 0.154798i 1.00000i 0.500000 + 0.866025i −1.41760 + 1.72928i
1099.6 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i 1.64155 + 1.51833i 1.00000 1.09384 + 0.631530i 1.00000i 0.500000 + 0.866025i −2.18079 0.494135i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1099.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n 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.bb.d 28
5.b even 2 1 inner 1110.2.bb.d 28
37.c even 3 1 inner 1110.2.bb.d 28
185.n even 6 1 inner 1110.2.bb.d 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.bb.d 28 1.a even 1 1 trivial
1110.2.bb.d 28 5.b even 2 1 inner
1110.2.bb.d 28 37.c even 3 1 inner
1110.2.bb.d 28 185.n even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1110, [\chi])$$:

 $$T_{7}^{28} - \cdots$$ $$T_{11}^{7} + 3 T_{11}^{6} - 61 T_{11}^{5} - 129 T_{11}^{4} + 1256 T_{11}^{3} + 1502 T_{11}^{2} - 8680 T_{11} - 2680$$