# Properties

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

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 20q^{4} + 2q^{5} - 40q^{6} + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 20q^{4} + 2q^{5} - 40q^{6} + 20q^{9} + 24q^{14} - 20q^{16} - 14q^{19} - 2q^{20} - 12q^{21} - 20q^{24} - 2q^{25} - 24q^{26} + 12q^{29} - 2q^{30} - 40q^{31} + 10q^{34} - 2q^{35} + 40q^{36} + 12q^{39} + 20q^{41} + 4q^{45} + 2q^{46} + 28q^{49} + 12q^{50} - 20q^{51} - 20q^{54} + 32q^{55} + 12q^{56} - 40q^{59} + 12q^{61} - 40q^{64} + 2q^{65} + 2q^{69} - 22q^{70} + 12q^{71} - 30q^{74} - 24q^{75} + 14q^{76} - 24q^{79} - 4q^{80} - 20q^{81} - 24q^{84} + 40q^{85} - 4q^{86} - 34q^{89} - 4q^{91} - 8q^{94} + 6q^{95} + 20q^{96} + O(q^{100})$$

## 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 2.22909 + 0.176566i −1.00000 −3.02191 + 1.74470i 1.00000i 0.500000 0.866025i −1.84216 1.26745i
1009.2 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.221132 + 2.22511i −1.00000 −0.666623 + 0.384875i 1.00000i 0.500000 0.866025i 1.30406 1.81643i
1009.3 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −1.62132 1.53991i −1.00000 −0.149067 + 0.0860637i 1.00000i 0.500000 0.866025i 0.634148 + 2.14426i
1009.4 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −1.88087 + 1.20927i −1.00000 −0.304101 + 0.175573i 1.00000i 0.500000 0.866025i 2.23352 0.106818i
1009.5 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 0.649619 + 2.13962i −1.00000 0.381600 0.220317i 1.00000i 0.500000 0.866025i 0.507226 2.17778i
1009.6 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.329099 2.21172i −1.00000 0.903939 0.521889i 1.00000i 0.500000 0.866025i −0.820851 + 2.07995i
1009.7 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 1.61819 1.54320i −1.00000 1.82925 1.05612i 1.00000i 0.500000 0.866025i −2.17299 + 0.527353i
1009.8 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 1.45774 1.69558i −1.00000 −3.96572 + 2.28961i 1.00000i 0.500000 0.866025i −2.11023 + 0.739551i
1009.9 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −2.23585 + 0.0310134i −1.00000 −4.17929 + 2.41291i 1.00000i 0.500000 0.866025i 1.95181 + 1.09107i
1009.10 −0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 0.833642 + 2.07486i −1.00000 3.97576 2.29541i 1.00000i 0.500000 0.866025i 0.315474 2.21370i
1009.11 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 1.38006 + 1.75939i −1.00000 −3.97576 + 2.29541i 1.00000i 0.500000 0.866025i 0.315474 + 2.21370i
1009.12 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 1.14478 1.92080i −1.00000 4.17929 2.41291i 1.00000i 0.500000 0.866025i 1.95181 1.09107i
1009.13 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −2.19729 + 0.414645i −1.00000 3.96572 2.28961i 1.00000i 0.500000 0.866025i −2.11023 0.739551i
1009.14 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −2.14554 + 0.629796i −1.00000 −1.82925 + 1.05612i 1.00000i 0.500000 0.866025i −2.17299 0.527353i
1009.15 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.75085 1.39087i −1.00000 −0.903939 + 0.521889i 1.00000i 0.500000 0.866025i −0.820851 2.07995i
1009.16 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 1.52816 + 1.63240i −1.00000 −0.381600 + 0.220317i 1.00000i 0.500000 0.866025i 0.507226 + 2.17778i
1009.17 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 1.98769 1.02425i −1.00000 0.304101 0.175573i 1.00000i 0.500000 0.866025i 2.23352 + 0.106818i
1009.18 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −0.522942 2.17406i −1.00000 0.149067 0.0860637i 1.00000i 0.500000 0.866025i 0.634148 2.14426i
1009.19 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 2.03757 + 0.921047i −1.00000 0.666623 0.384875i 1.00000i 0.500000 0.866025i 1.30406 + 1.81643i
1009.20 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −0.961633 + 2.01873i −1.00000 3.02191 1.74470i 1.00000i 0.500000 0.866025i −1.84216 + 1.26745i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1099.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.e 40
5.b even 2 1 inner 1110.2.bb.e 40
37.c even 3 1 inner 1110.2.bb.e 40
185.n even 6 1 inner 1110.2.bb.e 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.bb.e 40 1.a even 1 1 trivial
1110.2.bb.e 40 5.b even 2 1 inner
1110.2.bb.e 40 37.c even 3 1 inner
1110.2.bb.e 40 185.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 $$S_{2}^{\mathrm{new}}(1110, [\chi])$$:

 $$11\!\cdots\!36$$$$T_{7}^{18} +$$$$12\!\cdots\!84$$$$T_{7}^{16} - 775494567232 T_{7}^{14} + 351706556496 T_{7}^{12} - 89236701824 T_{7}^{10} + 16141091520 T_{7}^{8} - 1687392256 T_{7}^{6} + 122601728 T_{7}^{4} - 3280896 T_{7}^{2} + 65536$$">$$T_{7}^{40} - \cdots$$ $$T_{11}^{10} - \cdots$$