# Properties

 Label 1045.2.f.a Level $1045$ Weight $2$ Character orbit 1045.f Analytic conductor $8.344$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.34436701122$$ Analytic rank: $$0$$ Dimension: $$40$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 40 q^{4} - 40 q^{5} - 28 q^{9}+O(q^{10})$$ 40 * q + 40 * q^4 - 40 * q^5 - 28 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 40 q^{4} - 40 q^{5} - 28 q^{9} - 4 q^{11} + 32 q^{16} - 40 q^{20} - 16 q^{23} + 40 q^{25} + 8 q^{26} + 8 q^{36} + 28 q^{38} - 84 q^{42} - 48 q^{44} + 28 q^{45} + 32 q^{47} - 20 q^{49} + 4 q^{55} - 20 q^{58} + 72 q^{64} + 36 q^{66} + 16 q^{77} - 32 q^{80} + 16 q^{81} + 16 q^{82} - 20 q^{92} - 8 q^{93}+O(q^{100})$$ 40 * q + 40 * q^4 - 40 * q^5 - 28 * q^9 - 4 * q^11 + 32 * q^16 - 40 * q^20 - 16 * q^23 + 40 * q^25 + 8 * q^26 + 8 * q^36 + 28 * q^38 - 84 * q^42 - 48 * q^44 + 28 * q^45 + 32 * q^47 - 20 * q^49 + 4 * q^55 - 20 * q^58 + 72 * q^64 + 36 * q^66 + 16 * q^77 - 32 * q^80 + 16 * q^81 + 16 * q^82 - 20 * q^92 - 8 * q^93

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
626.1 −2.71965 2.28311i 5.39648 −1.00000 6.20925i 2.59337i −9.23723 −2.21259 2.71965
626.2 −2.71965 2.28311i 5.39648 −1.00000 6.20925i 2.59337i −9.23723 −2.21259 2.71965
626.3 −2.52972 0.384887i 4.39948 −1.00000 0.973657i 1.35297i −6.07000 2.85186 2.52972
626.4 −2.52972 0.384887i 4.39948 −1.00000 0.973657i 1.35297i −6.07000 2.85186 2.52972
626.5 −2.08652 2.80799i 2.35358 −1.00000 5.85893i 0.950248i −0.737755 −4.88478 2.08652
626.6 −2.08652 2.80799i 2.35358 −1.00000 5.85893i 0.950248i −0.737755 −4.88478 2.08652
626.7 −2.04337 0.838315i 2.17535 −1.00000 1.71299i 4.78057i −0.358306 2.29723 2.04337
626.8 −2.04337 0.838315i 2.17535 −1.00000 1.71299i 4.78057i −0.358306 2.29723 2.04337
626.9 −1.55769 2.37450i 0.426394 −1.00000 3.69874i 4.79166i 2.45119 −2.63826 1.55769
626.10 −1.55769 2.37450i 0.426394 −1.00000 3.69874i 4.79166i 2.45119 −2.63826 1.55769
626.11 −1.53521 1.36588i 0.356884 −1.00000 2.09692i 1.36858i 2.52254 1.13438 1.53521
626.12 −1.53521 1.36588i 0.356884 −1.00000 2.09692i 1.36858i 2.52254 1.13438 1.53521
626.13 −1.42582 1.05301i 0.0329672 −1.00000 1.50140i 0.901840i 2.80464 1.89118 1.42582
626.14 −1.42582 1.05301i 0.0329672 −1.00000 1.50140i 0.901840i 2.80464 1.89118 1.42582
626.15 −0.743263 1.99663i −1.44756 −1.00000 1.48402i 2.79593i 2.56244 −0.986525 0.743263
626.16 −0.743263 1.99663i −1.44756 −1.00000 1.48402i 2.79593i 2.56244 −0.986525 0.743263
626.17 −0.512926 3.19882i −1.73691 −1.00000 1.64076i 1.77223i 1.91676 −7.23242 0.512926
626.18 −0.512926 3.19882i −1.73691 −1.00000 1.64076i 1.77223i 1.91676 −7.23242 0.512926
626.19 −0.208163 0.469109i −1.95667 −1.00000 0.0976512i 2.46634i 0.823632 2.77994 0.208163
626.20 −0.208163 0.469109i −1.95667 −1.00000 0.0976512i 2.46634i 0.823632 2.77994 0.208163
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 626.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
19.b odd 2 1 inner
209.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.f.a 40
11.b odd 2 1 inner 1045.2.f.a 40
19.b odd 2 1 inner 1045.2.f.a 40
209.d even 2 1 inner 1045.2.f.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.f.a 40 1.a even 1 1 trivial
1045.2.f.a 40 11.b odd 2 1 inner
1045.2.f.a 40 19.b odd 2 1 inner
1045.2.f.a 40 209.d even 2 1 inner