# Properties

 Label 1110.2.u.e 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 24q^{7} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 24q^{7} - 8q^{9} + 40q^{10} - 8q^{12} - 16q^{13} - 40q^{16} + 8q^{18} + 16q^{19} - 24q^{31} + 24q^{33} - 8q^{34} + 16q^{39} - 20q^{42} - 32q^{43} - 8q^{45} + 72q^{46} + 96q^{49} + 20q^{51} + 16q^{52} + 24q^{54} - 16q^{57} + 40q^{61} - 24q^{63} - 44q^{66} - 24q^{70} + 8q^{72} + 8q^{75} - 16q^{76} + 48q^{79} + 24q^{81} - 56q^{82} - 8q^{84} + 12q^{87} - 8q^{90} - 64q^{91} + 12q^{93} + 24q^{94} + 8q^{97} + O(q^{100})$$

## 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 −1.73082 + 0.0652635i 1.00000i −0.707107 + 0.707107i 1.27002 + 1.17773i −5.10391 0.707107 0.707107i 2.99148 0.225919i 1.00000
191.2 −0.707107 0.707107i 0.347239 + 1.69689i 1.00000i −0.707107 + 0.707107i 0.954345 1.44542i −3.94859 0.707107 0.707107i −2.75885 + 1.17845i 1.00000
191.3 −0.707107 0.707107i 1.64725 + 0.535321i 1.00000i −0.707107 + 0.707107i −0.786252 1.54331i −2.53955 0.707107 0.707107i 2.42686 + 1.76361i 1.00000
191.4 −0.707107 0.707107i 0.878679 + 1.49262i 1.00000i −0.707107 + 0.707107i 0.434124 1.67676i 2.38487 0.707107 0.707107i −1.45585 + 2.62307i 1.00000
191.5 −0.707107 0.707107i −1.08352 1.35129i 1.00000i −0.707107 + 0.707107i −0.189341 + 1.72167i −3.77650 0.707107 0.707107i −0.651967 + 2.92830i 1.00000
191.6 −0.707107 0.707107i −0.912132 + 1.47242i 1.00000i −0.707107 + 0.707107i 1.68613 0.396182i 0.363832 0.707107 0.707107i −1.33603 2.68608i 1.00000
191.7 −0.707107 0.707107i 0.544013 1.64440i 1.00000i −0.707107 + 0.707107i −1.54744 + 0.778090i −1.56844 0.707107 0.707107i −2.40810 1.78915i 1.00000
191.8 −0.707107 0.707107i −1.59695 + 0.670623i 1.00000i −0.707107 + 0.707107i 1.60342 + 0.655016i 3.28633 0.707107 0.707107i 2.10053 2.14191i 1.00000
191.9 −0.707107 0.707107i 0.349356 1.69645i 1.00000i −0.707107 + 0.707107i −1.44660 + 0.952541i 2.97676 0.707107 0.707107i −2.75590 1.18533i 1.00000
191.10 −0.707107 0.707107i 1.55689 + 0.759006i 1.00000i −0.707107 + 0.707107i −0.564190 1.63759i 1.92521 0.707107 0.707107i 1.84782 + 2.36338i 1.00000
191.11 0.707107 + 0.707107i 1.08352 1.35129i 1.00000i 0.707107 0.707107i 1.72167 0.189341i −3.77650 −0.707107 + 0.707107i −0.651967 2.92830i 1.00000
191.12 0.707107 + 0.707107i −0.347239 + 1.69689i 1.00000i 0.707107 0.707107i −1.44542 + 0.954345i −3.94859 −0.707107 + 0.707107i −2.75885 1.17845i 1.00000
191.13 0.707107 + 0.707107i −0.544013 1.64440i 1.00000i 0.707107 0.707107i 0.778090 1.54744i −1.56844 −0.707107 + 0.707107i −2.40810 + 1.78915i 1.00000
191.14 0.707107 + 0.707107i −1.55689 + 0.759006i 1.00000i 0.707107 0.707107i −1.63759 0.564190i 1.92521 −0.707107 + 0.707107i 1.84782 2.36338i 1.00000
191.15 0.707107 + 0.707107i 1.73082 + 0.0652635i 1.00000i 0.707107 0.707107i 1.17773 + 1.27002i −5.10391 −0.707107 + 0.707107i 2.99148 + 0.225919i 1.00000
191.16 0.707107 + 0.707107i −1.64725 + 0.535321i 1.00000i 0.707107 0.707107i −1.54331 0.786252i −2.53955 −0.707107 + 0.707107i 2.42686 1.76361i 1.00000
191.17 0.707107 + 0.707107i 0.912132 + 1.47242i 1.00000i 0.707107 0.707107i −0.396182 + 1.68613i 0.363832 −0.707107 + 0.707107i −1.33603 + 2.68608i 1.00000
191.18 0.707107 + 0.707107i −0.349356 1.69645i 1.00000i 0.707107 0.707107i 0.952541 1.44660i 2.97676 −0.707107 + 0.707107i −2.75590 + 1.18533i 1.00000
191.19 0.707107 + 0.707107i 1.59695 + 0.670623i 1.00000i 0.707107 0.707107i 0.655016 + 1.60342i 3.28633 −0.707107 + 0.707107i 2.10053 + 2.14191i 1.00000
191.20 0.707107 + 0.707107i −0.878679 + 1.49262i 1.00000i 0.707107 0.707107i −1.67676 + 0.434124i 2.38487 −0.707107 + 0.707107i −1.45585 2.62307i 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.e 40
3.b odd 2 1 inner 1110.2.u.e 40
37.d odd 4 1 inner 1110.2.u.e 40
111.g even 4 1 inner 1110.2.u.e 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.u.e 40 1.a even 1 1 trivial
1110.2.u.e 40 3.b odd 2 1 inner
1110.2.u.e 40 37.d odd 4 1 inner
1110.2.u.e 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 $$S_{2}^{\mathrm{new}}(1110, [\chi])$$.