# Properties

 Label 460.2.m.b Level $460$ Weight $2$ Character orbit 460.m Analytic conductor $3.673$ Analytic rank $0$ Dimension $50$ 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: $$50$$ Relative dimension: $$5$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$50q + 5q^{5} - q^{7} - 25q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$50q + 5q^{5} - q^{7} - 25q^{9} - 6q^{13} + 12q^{17} + 19q^{19} + 39q^{21} - 16q^{23} - 5q^{25} + 21q^{27} - 6q^{29} + 34q^{31} + 50q^{33} - 10q^{35} + 7q^{37} - 70q^{39} - 51q^{41} - 18q^{43} - 74q^{45} + 30q^{47} - 16q^{49} - 80q^{51} - 23q^{53} - 33q^{55} + 27q^{57} - 18q^{59} + 76q^{61} + 138q^{63} + 6q^{65} + 25q^{67} - 30q^{69} - 37q^{71} + 20q^{73} + 92q^{77} + 18q^{79} + 25q^{81} - 22q^{83} - 12q^{85} - 109q^{87} + 8q^{89} + 110q^{91} + 64q^{93} + 3q^{95} - 38q^{97} - 126q^{99} + O(q^{100})$$

## 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 −2.08327 + 2.40422i 0 0.142315 0.989821i 0 −0.796185 1.74340i 0 −1.01332 7.04777i 0
41.2 0 −0.671975 + 0.775501i 0 0.142315 0.989821i 0 −1.80075 3.94308i 0 0.277094 + 1.92723i 0
41.3 0 −0.249442 + 0.287871i 0 0.142315 0.989821i 0 0.948129 + 2.07611i 0 0.406296 + 2.82585i 0
41.4 0 1.10811 1.27883i 0 0.142315 0.989821i 0 −1.08165 2.36848i 0 0.0194536 + 0.135303i 0
41.5 0 1.89657 2.18876i 0 0.142315 0.989821i 0 1.51786 + 3.32365i 0 −0.766744 5.33282i 0
81.1 0 −0.431764 + 3.00299i 0 0.959493 0.281733i 0 −2.67351 3.08540i 0 −5.95302 1.74797i 0
81.2 0 −0.282231 + 1.96296i 0 0.959493 0.281733i 0 2.71449 + 3.13269i 0 −0.895066 0.262815i 0
81.3 0 0.0891733 0.620214i 0 0.959493 0.281733i 0 0.926390 + 1.06911i 0 2.50177 + 0.734585i 0
81.4 0 0.147968 1.02914i 0 0.959493 0.281733i 0 −0.636768 0.734870i 0 1.84124 + 0.540637i 0
81.5 0 0.476853 3.31659i 0 0.959493 0.281733i 0 −0.777545 0.897335i 0 −7.89387 2.31785i 0
101.1 0 −2.08327 2.40422i 0 0.142315 + 0.989821i 0 −0.796185 + 1.74340i 0 −1.01332 + 7.04777i 0
101.2 0 −0.671975 0.775501i 0 0.142315 + 0.989821i 0 −1.80075 + 3.94308i 0 0.277094 1.92723i 0
101.3 0 −0.249442 0.287871i 0 0.142315 + 0.989821i 0 0.948129 2.07611i 0 0.406296 2.82585i 0
101.4 0 1.10811 + 1.27883i 0 0.142315 + 0.989821i 0 −1.08165 + 2.36848i 0 0.0194536 0.135303i 0
101.5 0 1.89657 + 2.18876i 0 0.142315 + 0.989821i 0 1.51786 3.32365i 0 −0.766744 + 5.33282i 0
121.1 0 −2.70621 0.794614i 0 −0.841254 + 0.540641i 0 −0.171423 + 1.19227i 0 4.16838 + 2.67886i 0
121.2 0 −1.08763 0.319356i 0 −0.841254 + 0.540641i 0 0.0860775 0.598682i 0 −1.44281 0.927240i 0
121.3 0 −0.169429 0.0497487i 0 −0.841254 + 0.540641i 0 0.0167209 0.116296i 0 −2.49753 1.60506i 0
121.4 0 1.36238 + 0.400031i 0 −0.841254 + 0.540641i 0 −0.649858 + 4.51987i 0 −0.827709 0.531936i 0
121.5 0 2.60088 + 0.763688i 0 −0.841254 + 0.540641i 0 0.742558 5.16460i 0 3.65762 + 2.35061i 0
See all 50 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 441.5 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.b 50
23.c even 11 1 inner 460.2.m.b 50

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.m.b 50 1.a even 1 1 trivial
460.2.m.b 50 23.c even 11 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$13\!\cdots\!39$$$$T_{3}^{17} + 332871021338 T_{3}^{16} - 89584124958 T_{3}^{15} +$$$$11\!\cdots\!47$$$$T_{3}^{14} - 332156586329 T_{3}^{13} +$$$$13\!\cdots\!47$$$$T_{3}^{12} + 111793858928 T_{3}^{11} +$$$$38\!\cdots\!75$$$$T_{3}^{10} -$$$$11\!\cdots\!66$$$$T_{3}^{9} +$$$$14\!\cdots\!30$$$$T_{3}^{8} - 211089748287 T_{3}^{7} + 75087422905 T_{3}^{6} - 4848564160 T_{3}^{5} + 36257950443 T_{3}^{4} + 3002513481 T_{3}^{3} + 1144917972 T_{3}^{2} + 887465304 T_{3} + 118352641$$">$$T_{3}^{50} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(460, [\chi])$$.