# Properties

 Label 1110.2.u.f Level $1110$ Weight $2$ Character orbit 1110.u 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.u (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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 24q^{7} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 24q^{7} - 8q^{9} - 40q^{10} + 16q^{13} - 40q^{16} + 8q^{18} + 16q^{19} - 16q^{22} - 8q^{31} + 24q^{33} + 24q^{34} + 32q^{37} + 16q^{39} + 12q^{42} + 32q^{43} + 8q^{45} - 56q^{46} - 32q^{49} - 44q^{51} - 16q^{52} - 24q^{54} + 16q^{55} + 8q^{57} - 72q^{61} + 24q^{63} - 28q^{66} + 16q^{69} - 24q^{70} + 8q^{72} - 16q^{76} - 48q^{79} + 120q^{81} - 24q^{82} - 24q^{84} + 20q^{87} + 16q^{88} + 8q^{90} - 92q^{93} - 8q^{94} - 40q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1 −0.707107 0.707107i 0.169604 1.72373i 1.00000i 0.707107 0.707107i −1.33879 + 1.09893i 1.82689 0.707107 0.707107i −2.94247 0.584702i −1.00000
191.2 −0.707107 0.707107i −0.392697 1.68695i 1.00000i 0.707107 0.707107i −0.915172 + 1.47053i −0.873718 0.707107 0.707107i −2.69158 + 1.32492i −1.00000
191.3 −0.707107 0.707107i 0.125486 + 1.72750i 1.00000i 0.707107 0.707107i 1.13279 1.31026i 0.386861 0.707107 0.707107i −2.96851 + 0.433555i −1.00000
191.4 −0.707107 0.707107i −0.912090 1.47244i 1.00000i 0.707107 0.707107i −0.396230 + 1.68612i −2.79858 0.707107 0.707107i −1.33618 + 2.68600i −1.00000
191.5 −0.707107 0.707107i 1.33218 + 1.10694i 1.00000i 0.707107 0.707107i −0.159268 1.72471i −3.17089 0.707107 0.707107i 0.549384 + 2.94927i −1.00000
191.6 −0.707107 0.707107i −0.407565 + 1.68342i 1.00000i 0.707107 0.707107i 1.47855 0.902163i 2.94059 0.707107 0.707107i −2.66778 1.37220i −1.00000
191.7 −0.707107 0.707107i 1.70946 + 0.278802i 1.00000i 0.707107 0.707107i −1.01163 1.40592i −0.348034 0.707107 0.707107i 2.84454 + 0.953204i −1.00000
191.8 −0.707107 0.707107i −1.72452 0.161390i 1.00000i 0.707107 0.707107i 1.10530 + 1.33354i 0.212791 0.707107 0.707107i 2.94791 + 0.556641i −1.00000
191.9 −0.707107 0.707107i 1.65121 0.522966i 1.00000i 0.707107 0.707107i −1.53738 0.797792i 3.44243 0.707107 0.707107i 2.45301 1.72706i −1.00000
191.10 −0.707107 0.707107i −1.55108 + 0.770820i 1.00000i 0.707107 0.707107i 1.64183 + 0.551724i 4.38167 0.707107 0.707107i 1.81167 2.39120i −1.00000
191.11 0.707107 + 0.707107i −1.33218 + 1.10694i 1.00000i −0.707107 + 0.707107i −1.72471 0.159268i −3.17089 −0.707107 + 0.707107i 0.549384 2.94927i −1.00000
191.12 0.707107 + 0.707107i 0.912090 1.47244i 1.00000i −0.707107 + 0.707107i 1.68612 0.396230i −2.79858 −0.707107 + 0.707107i −1.33618 2.68600i −1.00000
191.13 0.707107 + 0.707107i −1.70946 + 0.278802i 1.00000i −0.707107 + 0.707107i −1.40592 1.01163i −0.348034 −0.707107 + 0.707107i 2.84454 0.953204i −1.00000
191.14 0.707107 + 0.707107i 0.392697 1.68695i 1.00000i −0.707107 + 0.707107i 1.47053 0.915172i −0.873718 −0.707107 + 0.707107i −2.69158 1.32492i −1.00000
191.15 0.707107 + 0.707107i −1.65121 0.522966i 1.00000i −0.707107 + 0.707107i −0.797792 1.53738i 3.44243 −0.707107 + 0.707107i 2.45301 + 1.72706i −1.00000
191.16 0.707107 + 0.707107i −0.125486 + 1.72750i 1.00000i −0.707107 + 0.707107i −1.31026 + 1.13279i 0.386861 −0.707107 + 0.707107i −2.96851 0.433555i −1.00000
191.17 0.707107 + 0.707107i 1.55108 + 0.770820i 1.00000i −0.707107 + 0.707107i 0.551724 + 1.64183i 4.38167 −0.707107 + 0.707107i 1.81167 + 2.39120i −1.00000
191.18 0.707107 + 0.707107i −0.169604 1.72373i 1.00000i −0.707107 + 0.707107i 1.09893 1.33879i 1.82689 −0.707107 + 0.707107i −2.94247 + 0.584702i −1.00000
191.19 0.707107 + 0.707107i 1.72452 0.161390i 1.00000i −0.707107 + 0.707107i 1.33354 + 1.10530i 0.212791 −0.707107 + 0.707107i 2.94791 0.556641i −1.00000
191.20 0.707107 + 0.707107i 0.407565 + 1.68342i 1.00000i −0.707107 + 0.707107i −0.902163 + 1.47855i 2.94059 −0.707107 + 0.707107i −2.66778 + 1.37220i −1.00000
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 401.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.d odd 4 1 inner
111.g 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.u.f 40
3.b odd 2 1 inner 1110.2.u.f 40
37.d odd 4 1 inner 1110.2.u.f 40
111.g even 4 1 inner 1110.2.u.f 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.u.f 40 1.a even 1 1 trivial
1110.2.u.f 40 3.b odd 2 1 inner
1110.2.u.f 40 37.d odd 4 1 inner
1110.2.u.f 40 111.g even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{10} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1110, [\chi])$$.