# Properties

 Label 460.2.g.c Level $460$ Weight $2$ Character orbit 460.g Analytic conductor $3.673$ Analytic rank $0$ Dimension $56$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$460 = 2^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 460.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$56$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q - 8q^{4} - 8q^{6} + 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q - 8q^{4} - 8q^{6} + 16q^{9} - 8q^{16} - 100q^{24} - 24q^{25} - 24q^{26} - 16q^{29} + 104q^{41} - 8q^{46} + 32q^{49} - 32q^{50} + 52q^{54} - 92q^{64} + 32q^{69} - 44q^{70} + 24q^{81} + 56q^{85} + 28q^{94} + 88q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
459.1 −1.35976 0.388659i 1.11016 1.69789 + 1.05697i −1.71804 + 1.43120i −1.50955 0.431474i 4.32159i −1.89792 2.09712i −1.76755 2.89237 1.27835i
459.2 −1.35976 0.388659i 1.11016 1.69789 + 1.05697i 1.71804 1.43120i −1.50955 0.431474i 4.32159i −1.89792 2.09712i −1.76755 −2.89237 + 1.27835i
459.3 −1.35976 + 0.388659i 1.11016 1.69789 1.05697i −1.71804 1.43120i −1.50955 + 0.431474i 4.32159i −1.89792 + 2.09712i −1.76755 2.89237 + 1.27835i
459.4 −1.35976 + 0.388659i 1.11016 1.69789 1.05697i 1.71804 + 1.43120i −1.50955 + 0.431474i 4.32159i −1.89792 + 2.09712i −1.76755 −2.89237 1.27835i
459.5 −1.28797 0.584066i 2.56426 1.31773 + 1.50452i −0.662839 2.13557i −3.30269 1.49770i 1.62105i −0.818466 2.70742i 3.57544 −0.393594 + 3.13769i
459.6 −1.28797 0.584066i 2.56426 1.31773 + 1.50452i 0.662839 + 2.13557i −3.30269 1.49770i 1.62105i −0.818466 2.70742i 3.57544 0.393594 3.13769i
459.7 −1.28797 + 0.584066i 2.56426 1.31773 1.50452i −0.662839 + 2.13557i −3.30269 + 1.49770i 1.62105i −0.818466 + 2.70742i 3.57544 −0.393594 3.13769i
459.8 −1.28797 + 0.584066i 2.56426 1.31773 1.50452i 0.662839 2.13557i −3.30269 + 1.49770i 1.62105i −0.818466 + 2.70742i 3.57544 0.393594 + 3.13769i
459.9 −1.20833 0.734800i −1.77158 0.920137 + 1.77577i −1.90499 1.17090i 2.14066 + 1.30176i 1.27817i 0.193001 2.82183i 0.138505 1.44148 + 2.81463i
459.10 −1.20833 0.734800i −1.77158 0.920137 + 1.77577i 1.90499 + 1.17090i 2.14066 + 1.30176i 1.27817i 0.193001 2.82183i 0.138505 −1.44148 2.81463i
459.11 −1.20833 + 0.734800i −1.77158 0.920137 1.77577i −1.90499 + 1.17090i 2.14066 1.30176i 1.27817i 0.193001 + 2.82183i 0.138505 1.44148 2.81463i
459.12 −1.20833 + 0.734800i −1.77158 0.920137 1.77577i 1.90499 1.17090i 2.14066 1.30176i 1.27817i 0.193001 + 2.82183i 0.138505 −1.44148 + 2.81463i
459.13 −0.968186 1.03083i 0.281229 −0.125231 + 1.99608i −1.92344 + 1.14033i −0.272282 0.289900i 0.654652i 2.17887 1.80348i −2.92091 3.03775 + 0.878693i
459.14 −0.968186 1.03083i 0.281229 −0.125231 + 1.99608i 1.92344 1.14033i −0.272282 0.289900i 0.654652i 2.17887 1.80348i −2.92091 −3.03775 0.878693i
459.15 −0.968186 + 1.03083i 0.281229 −0.125231 1.99608i −1.92344 1.14033i −0.272282 + 0.289900i 0.654652i 2.17887 + 1.80348i −2.92091 3.03775 0.878693i
459.16 −0.968186 + 1.03083i 0.281229 −0.125231 1.99608i 1.92344 + 1.14033i −0.272282 + 0.289900i 0.654652i 2.17887 + 1.80348i −2.92091 −3.03775 + 0.878693i
459.17 −0.627129 1.26756i −3.03928 −1.21342 + 1.58985i −0.951754 + 2.02340i 1.90602 + 3.85247i 1.96560i 2.77620 + 0.541043i 6.23720 3.16166 0.0625285i
459.18 −0.627129 1.26756i −3.03928 −1.21342 + 1.58985i 0.951754 2.02340i 1.90602 + 3.85247i 1.96560i 2.77620 + 0.541043i 6.23720 −3.16166 + 0.0625285i
459.19 −0.627129 + 1.26756i −3.03928 −1.21342 1.58985i −0.951754 2.02340i 1.90602 3.85247i 1.96560i 2.77620 0.541043i 6.23720 3.16166 + 0.0625285i
459.20 −0.627129 + 1.26756i −3.03928 −1.21342 1.58985i 0.951754 + 2.02340i 1.90602 3.85247i 1.96560i 2.77620 0.541043i 6.23720 −3.16166 0.0625285i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 459.56 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner
115.c odd 2 1 inner
460.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.g.c 56
4.b odd 2 1 inner 460.2.g.c 56
5.b even 2 1 inner 460.2.g.c 56
20.d odd 2 1 inner 460.2.g.c 56
23.b odd 2 1 inner 460.2.g.c 56
92.b even 2 1 inner 460.2.g.c 56
115.c odd 2 1 inner 460.2.g.c 56
460.g even 2 1 inner 460.2.g.c 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.g.c 56 1.a even 1 1 trivial
460.2.g.c 56 4.b odd 2 1 inner
460.2.g.c 56 5.b even 2 1 inner
460.2.g.c 56 20.d odd 2 1 inner
460.2.g.c 56 23.b odd 2 1 inner
460.2.g.c 56 92.b even 2 1 inner
460.2.g.c 56 115.c odd 2 1 inner
460.2.g.c 56 460.g even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{14} - 23 T_{3}^{12} + 191 T_{3}^{10} - 712 T_{3}^{8} + 1213 T_{3}^{6} - 805 T_{3}^{4} + 107 T_{3}^{2} - 4$$ acting on $$S_{2}^{\mathrm{new}}(460, [\chi])$$.