# Properties

 Label 460.2.m.a Level $460$ Weight $2$ Character orbit 460.m Analytic conductor $3.673$ Analytic rank $0$ Dimension $30$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$460 = 2^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 460.m (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$3$$ over $$\Q(\zeta_{11})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{5} + q^{7} + 21q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{5} + q^{7} + 21q^{9} + 2q^{13} + 10q^{17} + 3q^{19} + 39q^{21} + 10q^{23} - 3q^{25} + 21q^{27} + 14q^{29} - 2q^{31} - 50q^{33} - 10q^{35} + 9q^{37} + 38q^{39} - 3q^{41} - 50q^{43} + 10q^{45} - 6q^{47} - 36q^{49} - 36q^{51} - 5q^{53} - 11q^{55} + 23q^{57} + 14q^{59} - 16q^{61} - 52q^{63} + 2q^{65} + 27q^{67} + 42q^{69} + 19q^{71} + 24q^{73} - 10q^{77} - 22q^{79} + 35q^{81} + 36q^{83} + 10q^{85} - 3q^{87} - 28q^{89} - 98q^{91} - 60q^{93} - 19q^{95} - 2q^{97} - 14q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 0 −1.26886 + 1.46434i 0 −0.142315 + 0.989821i 0 0.0191296 + 0.0418879i 0 −0.107349 0.746632i 0
41.2 0 0.329283 0.380013i 0 −0.142315 + 0.989821i 0 −0.847137 1.85497i 0 0.390962 + 2.71920i 0
41.3 0 0.939578 1.08433i 0 −0.142315 + 0.989821i 0 2.04060 + 4.46829i 0 0.133978 + 0.931838i 0
81.1 0 −0.161849 + 1.12569i 0 −0.959493 + 0.281733i 0 0.847514 + 0.978083i 0 1.63751 + 0.480815i 0
81.2 0 −0.0800965 + 0.557084i 0 −0.959493 + 0.281733i 0 −2.98213 3.44157i 0 2.57455 + 0.755957i 0
81.3 0 0.241946 1.68277i 0 −0.959493 + 0.281733i 0 2.58157 + 2.97929i 0 0.105306 + 0.0309206i 0
101.1 0 −1.26886 1.46434i 0 −0.142315 0.989821i 0 0.0191296 0.0418879i 0 −0.107349 + 0.746632i 0
101.2 0 0.329283 + 0.380013i 0 −0.142315 0.989821i 0 −0.847137 + 1.85497i 0 0.390962 2.71920i 0
101.3 0 0.939578 + 1.08433i 0 −0.142315 0.989821i 0 2.04060 4.46829i 0 0.133978 0.931838i 0
121.1 0 −2.73464 0.802964i 0 0.841254 0.540641i 0 −0.359196 + 2.49826i 0 4.30976 + 2.76972i 0
121.2 0 0.0447663 + 0.0131446i 0 0.841254 0.540641i 0 0.423834 2.94783i 0 −2.52193 1.62075i 0
121.3 0 2.68988 + 0.789819i 0 0.841254 0.540641i 0 −0.0887134 + 0.617015i 0 4.08786 + 2.62711i 0
141.1 0 −1.08673 + 2.37960i 0 −0.654861 + 0.755750i 0 0.0544585 0.0349983i 0 −2.51694 2.90471i 0
141.2 0 0.256560 0.561788i 0 −0.654861 + 0.755750i 0 −1.01950 + 0.655191i 0 1.71480 + 1.97898i 0
141.3 0 0.830167 1.81781i 0 −0.654861 + 0.755750i 0 2.04574 1.31472i 0 −0.650684 0.750930i 0
261.1 0 −1.08673 2.37960i 0 −0.654861 0.755750i 0 0.0544585 + 0.0349983i 0 −2.51694 + 2.90471i 0
261.2 0 0.256560 + 0.561788i 0 −0.654861 0.755750i 0 −1.01950 0.655191i 0 1.71480 1.97898i 0
261.3 0 0.830167 + 1.81781i 0 −0.654861 0.755750i 0 2.04574 + 1.31472i 0 −0.650684 + 0.750930i 0
301.1 0 −0.161849 1.12569i 0 −0.959493 0.281733i 0 0.847514 0.978083i 0 1.63751 0.480815i 0
301.2 0 −0.0800965 0.557084i 0 −0.959493 0.281733i 0 −2.98213 + 3.44157i 0 2.57455 0.755957i 0
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 441.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.m.a 30
23.c even 11 1 inner 460.2.m.a 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.m.a 30 1.a even 1 1 trivial
460.2.m.a 30 23.c even 11 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{30} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(460, [\chi])$$.