# Properties

 Label 460.2.s.a Level $460$ Weight $2$ Character orbit 460.s Analytic conductor $3.673$ Analytic rank $0$ Dimension $120$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$120$$ Relative dimension: $$12$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$120q + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$120q + 20q^{9} - 4q^{11} + 9q^{15} + 8q^{19} + 14q^{25} + 10q^{29} - 18q^{31} + 10q^{35} - 60q^{39} + 2q^{41} - 2q^{45} - 28q^{49} + 24q^{51} + 6q^{55} - 36q^{61} + 39q^{65} + 118q^{69} - 76q^{71} + 83q^{75} + 64q^{79} - 160q^{81} + 38q^{85} - 48q^{89} - 80q^{91} + 21q^{95} - 142q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 0 −2.47974 + 2.14871i 0 1.82066 1.29815i 0 −0.253328 0.115691i 0 1.10522 7.68699i 0
9.2 0 −1.90027 + 1.64660i 0 −2.22785 + 0.191526i 0 −1.95670 0.893593i 0 0.472816 3.28851i 0
9.3 0 −1.79458 + 1.55501i 0 0.246165 + 2.22248i 0 1.58902 + 0.725681i 0 0.375508 2.61172i 0
9.4 0 −0.786010 + 0.681082i 0 2.20308 0.382657i 0 −3.36943 1.53877i 0 −0.273005 + 1.89879i 0
9.5 0 −0.410666 + 0.355844i 0 1.87331 + 1.22094i 0 2.52593 + 1.15355i 0 −0.384923 + 2.67720i 0
9.6 0 −0.391893 + 0.339577i 0 −0.773120 2.09816i 0 −1.88813 0.862281i 0 −0.388677 + 2.70331i 0
9.7 0 0.391893 0.339577i 0 0.150682 2.23099i 0 1.88813 + 0.862281i 0 −0.388677 + 2.70331i 0
9.8 0 0.410666 0.355844i 0 −1.45345 + 1.69926i 0 −2.52593 1.15355i 0 −0.384923 + 2.67720i 0
9.9 0 0.786010 0.681082i 0 −2.22165 + 0.253523i 0 3.36943 + 1.53877i 0 −0.273005 + 1.89879i 0
9.10 0 1.79458 1.55501i 0 0.389951 + 2.20180i 0 −1.58902 0.725681i 0 0.375508 2.61172i 0
9.11 0 1.90027 1.64660i 0 2.19157 0.443890i 0 1.95670 + 0.893593i 0 0.472816 3.28851i 0
9.12 0 2.47974 2.14871i 0 −2.11264 0.732624i 0 0.253328 + 0.115691i 0 1.10522 7.68699i 0
29.1 0 −0.822669 + 2.80175i 0 −0.608360 2.15172i 0 −0.516845 0.0743111i 0 −4.64928 2.98791i 0
29.2 0 −0.625007 + 2.12858i 0 0.203721 + 2.22677i 0 3.14354 + 0.451973i 0 −1.61646 1.03883i 0
29.3 0 −0.545020 + 1.85617i 0 2.03988 0.915906i 0 3.48084 + 0.500469i 0 −0.624547 0.401372i 0
29.4 0 −0.384428 + 1.30924i 0 2.22428 0.229290i 0 −2.93912 0.422582i 0 0.957431 + 0.615303i 0
29.5 0 −0.277288 + 0.944356i 0 −1.52418 1.63612i 0 −1.03968 0.149483i 0 1.70884 + 1.09821i 0
29.6 0 −0.244021 + 0.831059i 0 −1.24767 + 1.85562i 0 −3.43041 0.493219i 0 1.89265 + 1.21633i 0
29.7 0 0.244021 0.831059i 0 −2.20623 + 0.364069i 0 3.43041 + 0.493219i 0 1.89265 + 1.21633i 0
29.8 0 0.277288 0.944356i 0 0.855101 + 2.06611i 0 1.03968 + 0.149483i 0 1.70884 + 1.09821i 0
See next 80 embeddings (of 120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.12 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
23.c even 11 1 inner
115.j 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.s.a 120
5.b even 2 1 inner 460.2.s.a 120
23.c even 11 1 inner 460.2.s.a 120
115.j even 22 1 inner 460.2.s.a 120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.s.a 120 1.a even 1 1 trivial
460.2.s.a 120 5.b even 2 1 inner
460.2.s.a 120 23.c even 11 1 inner
460.2.s.a 120 115.j even 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(460, [\chi])$$.