# Properties

 Label 460.2.o.a Level $460$ Weight $2$ Character orbit 460.o Analytic conductor $3.673$ Analytic rank $0$ Dimension $40$ CM discriminant -20 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$460 = 2^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 460.o (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$4$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 8q^{4} - 8q^{6} + 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 8q^{4} - 8q^{6} + 4q^{9} - 16q^{16} - 16q^{24} + 20q^{25} + 24q^{29} + 8q^{36} + 48q^{41} - 4q^{46} + 100q^{49} - 276q^{54} - 264q^{56} - 32q^{64} - 4q^{69} - 40q^{70} + 20q^{81} + 352q^{84} + 396q^{86} - 56q^{94} - 32q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −0.587486 1.28641i −2.19630 + 0.644892i −1.30972 + 1.51150i −1.20891 + 1.88110i 2.11989 + 2.44649i −4.60217 + 0.661692i 2.71386 + 0.796860i 1.88410 1.21083i 3.13009 + 0.450039i
19.2 −0.587486 1.28641i 3.32368 0.975920i −1.30972 + 1.51150i 1.20891 1.88110i −3.20805 3.70229i −3.71125 + 0.533598i 2.71386 + 0.796860i 7.57066 4.86537i −3.13009 0.450039i
19.3 0.587486 + 1.28641i −3.32368 + 0.975920i −1.30972 + 1.51150i 1.20891 1.88110i −3.20805 3.70229i 3.71125 0.533598i −2.71386 0.796860i 7.57066 4.86537i 3.13009 + 0.450039i
19.4 0.587486 + 1.28641i 2.19630 0.644892i −1.30972 + 1.51150i −1.20891 + 1.88110i 2.11989 + 2.44649i 4.60217 0.661692i −2.71386 0.796860i 1.88410 1.21083i −3.13009 0.450039i
79.1 −0.926113 1.06879i −2.78960 1.79277i −0.284630 + 1.97964i −2.03400 0.928896i 0.667391 + 4.64180i −0.518076 + 1.76441i 2.37942 1.52916i 3.32161 + 7.27330i 0.890917 + 3.03418i
79.2 −0.926113 1.06879i 1.23141 + 0.791378i −0.284630 + 1.97964i 2.03400 + 0.928896i −0.294605 2.04902i 1.19158 4.05815i 2.37942 1.52916i −0.356159 0.779879i −0.890917 3.03418i
79.3 0.926113 + 1.06879i −1.23141 0.791378i −0.284630 + 1.97964i 2.03400 + 0.928896i −0.294605 2.04902i −1.19158 + 4.05815i −2.37942 + 1.52916i −0.356159 0.779879i 0.890917 + 3.03418i
79.4 0.926113 + 1.06879i 2.78960 + 1.79277i −0.284630 + 1.97964i −2.03400 0.928896i 0.667391 + 4.64180i 0.518076 1.76441i −2.37942 + 1.52916i 3.32161 + 7.27330i −0.890917 3.03418i
99.1 −0.926113 + 1.06879i −2.78960 + 1.79277i −0.284630 1.97964i −2.03400 + 0.928896i 0.667391 4.64180i −0.518076 1.76441i 2.37942 + 1.52916i 3.32161 7.27330i 0.890917 3.03418i
99.2 −0.926113 + 1.06879i 1.23141 0.791378i −0.284630 1.97964i 2.03400 0.928896i −0.294605 + 2.04902i 1.19158 + 4.05815i 2.37942 + 1.52916i −0.356159 + 0.779879i −0.890917 + 3.03418i
99.3 0.926113 1.06879i −1.23141 + 0.791378i −0.284630 1.97964i 2.03400 0.928896i −0.294605 + 2.04902i −1.19158 4.05815i −2.37942 1.52916i −0.356159 + 0.779879i 0.890917 3.03418i
99.4 0.926113 1.06879i 2.78960 1.79277i −0.284630 1.97964i −2.03400 + 0.928896i 0.667391 4.64180i 0.518076 + 1.76441i −2.37942 1.52916i 3.32161 7.27330i −0.890917 + 3.03418i
159.1 −1.18971 0.764582i −0.0739958 0.514652i 0.830830 + 1.81926i 0.629973 2.14549i −0.305460 + 0.668863i −0.858171 0.743609i 0.402527 2.79964i 2.61909 0.769034i −2.38989 + 2.07085i
159.2 −1.18971 0.764582i 0.412623 + 2.86986i 0.830830 + 1.81926i −0.629973 + 2.14549i 1.70334 3.72979i −3.98826 3.45585i 0.402527 2.79964i −5.18734 + 1.52314i 2.38989 2.07085i
159.3 1.18971 + 0.764582i −0.412623 2.86986i 0.830830 + 1.81926i −0.629973 + 2.14549i 1.70334 3.72979i 3.98826 + 3.45585i −0.402527 + 2.79964i −5.18734 + 1.52314i −2.38989 + 2.07085i
159.4 1.18971 + 0.764582i 0.0739958 + 0.514652i 0.830830 + 1.81926i 0.629973 2.14549i −0.305460 + 0.668863i 0.858171 + 0.743609i −0.402527 + 2.79964i 2.61909 0.769034i 2.38989 2.07085i
199.1 −0.201264 + 1.39982i −1.38389 3.03031i −1.91899 0.563465i 1.68991 1.46431i 4.52041 1.32731i 2.67835 4.16759i 1.17497 2.57283i −5.30301 + 6.12000i 1.70966 + 2.66028i
199.2 −0.201264 + 1.39982i 1.21668 + 2.66415i −1.91899 0.563465i −1.68991 + 1.46431i −3.97421 + 1.16693i −0.198162 + 0.308347i 1.17497 2.57283i −3.65283 + 4.21559i −1.70966 2.66028i
199.3 0.201264 1.39982i −1.21668 2.66415i −1.91899 0.563465i −1.68991 + 1.46431i −3.97421 + 1.16693i 0.198162 0.308347i −1.17497 + 2.57283i −3.65283 + 4.21559i 1.70966 + 2.66028i
199.4 0.201264 1.39982i 1.38389 + 3.03031i −1.91899 0.563465i 1.68991 1.46431i 4.52041 1.32731i −2.67835 + 4.16759i −1.17497 + 2.57283i −5.30301 + 6.12000i −1.70966 2.66028i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
23.d odd 22 1 inner
92.h even 22 1 inner
115.i odd 22 1 inner
460.o even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.o.a 40
4.b odd 2 1 inner 460.2.o.a 40
5.b even 2 1 inner 460.2.o.a 40
20.d odd 2 1 CM 460.2.o.a 40
23.d odd 22 1 inner 460.2.o.a 40
92.h even 22 1 inner 460.2.o.a 40
115.i odd 22 1 inner 460.2.o.a 40
460.o even 22 1 inner 460.2.o.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.o.a 40 1.a even 1 1 trivial
460.2.o.a 40 4.b odd 2 1 inner
460.2.o.a 40 5.b even 2 1 inner
460.2.o.a 40 20.d odd 2 1 CM
460.2.o.a 40 23.d odd 22 1 inner
460.2.o.a 40 92.h even 22 1 inner
460.2.o.a 40 115.i odd 22 1 inner
460.2.o.a 40 460.o even 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$80\!\cdots\!35$$$$T_{3}^{12} -$$$$85\!\cdots\!40$$$$T_{3}^{10} +$$$$29\!\cdots\!31$$$$T_{3}^{8} +$$$$19\!\cdots\!08$$$$T_{3}^{6} +$$$$51\!\cdots\!86$$$$T_{3}^{4} + 994162360456 T_{3}^{2} + 123657019201$$">$$T_{3}^{40} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(460, [\chi])$$.