# Properties

 Label 462.2.y.b Level $462$ Weight $2$ Character orbit 462.y Analytic conductor $3.689$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.y (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$3$$ over $$\Q(\zeta_{15})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 3q^{2} + 3q^{3} + 3q^{4} + 5q^{5} - 6q^{6} - 5q^{7} - 6q^{8} + 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 3q^{2} + 3q^{3} + 3q^{4} + 5q^{5} - 6q^{6} - 5q^{7} - 6q^{8} + 3q^{9} + 10q^{10} - 2q^{11} - 12q^{12} + 10q^{13} + 7q^{14} + 20q^{15} + 3q^{16} + 8q^{17} + 3q^{18} + 7q^{19} - 10q^{20} + 8q^{21} - 6q^{22} + 24q^{23} + 3q^{24} + 8q^{25} - 6q^{27} - 2q^{28} + 2q^{29} - 10q^{30} - 19q^{31} - 12q^{32} - 7q^{33} + 24q^{34} + 14q^{35} - 6q^{36} - q^{37} - 8q^{38} - 5q^{39} - 10q^{40} + 16q^{41} + 10q^{42} - 68q^{43} - 2q^{44} + 10q^{45} - 6q^{46} + 25q^{47} - 6q^{48} - 21q^{49} + 14q^{50} - 2q^{51} - 5q^{53} - 12q^{54} - 10q^{55} - 10q^{56} + 16q^{57} - q^{58} - 36q^{59} + 5q^{60} - q^{61} - 52q^{62} + 7q^{63} - 6q^{64} + 6q^{65} + 8q^{66} + 56q^{67} + 8q^{68} + 12q^{69} - 17q^{70} - 6q^{71} + 3q^{72} + 35q^{73} - q^{74} - 7q^{75} - 4q^{76} + 38q^{77} - 20q^{78} - 3q^{79} - 10q^{80} + 3q^{81} + 2q^{82} - 28q^{83} + 7q^{84} - 80q^{85} - 11q^{86} + 4q^{87} - 7q^{88} - 6q^{89} - 10q^{90} - 24q^{91} + 12q^{92} - 19q^{93} - 30q^{94} - 44q^{95} + 3q^{96} - 8q^{97} - 6q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 0.913545 + 0.406737i −0.978148 0.207912i 0.669131 + 0.743145i −0.225801 + 2.14835i −0.809017 0.587785i −2.17096 + 1.51227i 0.309017 + 0.951057i 0.913545 + 0.406737i −1.08009 + 1.87078i
25.2 0.913545 + 0.406737i −0.978148 0.207912i 0.669131 + 0.743145i −0.122006 + 1.16081i −0.809017 0.587785i 2.38913 1.13668i 0.309017 + 0.951057i 0.913545 + 0.406737i −0.583600 + 1.01083i
25.3 0.913545 + 0.406737i −0.978148 0.207912i 0.669131 + 0.743145i 0.258529 2.45974i −0.809017 0.587785i −0.613647 + 2.57360i 0.309017 + 0.951057i 0.913545 + 0.406737i 1.23664 2.14193i
37.1 0.913545 0.406737i −0.978148 + 0.207912i 0.669131 0.743145i −0.225801 2.14835i −0.809017 + 0.587785i −2.17096 1.51227i 0.309017 0.951057i 0.913545 0.406737i −1.08009 1.87078i
37.2 0.913545 0.406737i −0.978148 + 0.207912i 0.669131 0.743145i −0.122006 1.16081i −0.809017 + 0.587785i 2.38913 + 1.13668i 0.309017 0.951057i 0.913545 0.406737i −0.583600 1.01083i
37.3 0.913545 0.406737i −0.978148 + 0.207912i 0.669131 0.743145i 0.258529 + 2.45974i −0.809017 + 0.587785i −0.613647 2.57360i 0.309017 0.951057i 0.913545 0.406737i 1.23664 + 2.14193i
163.1 0.669131 + 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i 0.282912 + 0.0601348i 0.309017 + 0.951057i 1.34567 2.27797i −0.809017 + 0.587785i 0.669131 + 0.743145i 0.144616 + 0.250483i
163.2 0.669131 + 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i 1.46774 + 0.311978i 0.309017 + 0.951057i 1.41096 + 2.23812i −0.809017 + 0.587785i 0.669131 + 0.743145i 0.750265 + 1.29950i
163.3 0.669131 + 0.743145i 0.913545 + 0.406737i −0.104528 + 0.994522i 3.97552 + 0.845024i 0.309017 + 0.951057i −2.27849 1.34480i −0.809017 + 0.587785i 0.669131 + 0.743145i 2.03217 + 3.51982i
235.1 −0.104528 + 0.994522i 0.669131 + 0.743145i −0.978148 0.207912i −2.25946 1.00598i −0.809017 + 0.587785i −1.01628 2.44278i 0.309017 0.951057i −0.104528 + 0.994522i 1.23664 2.14193i
235.2 −0.104528 + 0.994522i 0.669131 + 0.743145i −0.978148 0.207912i 1.06629 + 0.474743i −0.809017 + 0.587785i −1.26472 + 2.32389i 0.309017 0.951057i −0.104528 + 0.994522i −0.583600 + 1.01083i
235.3 −0.104528 + 0.994522i 0.669131 + 0.743145i −0.978148 0.207912i 1.97343 + 0.878627i −0.809017 + 0.587785i 0.867452 2.49951i 0.309017 0.951057i −0.104528 + 0.994522i −1.08009 + 1.87078i
247.1 −0.978148 0.207912i −0.104528 + 0.994522i 0.913545 + 0.406737i −2.71957 3.02039i 0.309017 0.951057i −1.98308 + 1.75140i −0.809017 0.587785i −0.978148 0.207912i 2.03217 + 3.51982i
247.2 −0.978148 0.207912i −0.104528 + 0.994522i 0.913545 + 0.406737i −1.00405 1.11511i 0.309017 0.951057i 2.56459 0.650286i −0.809017 0.587785i −0.978148 0.207912i 0.750265 + 1.29950i
247.3 −0.978148 0.207912i −0.104528 + 0.994522i 0.913545 + 0.406737i −0.193534 0.214942i 0.309017 0.951057i −1.75065 1.98374i −0.809017 0.587785i −0.978148 0.207912i 0.144616 + 0.250483i
289.1 −0.104528 0.994522i 0.669131 0.743145i −0.978148 + 0.207912i −2.25946 + 1.00598i −0.809017 0.587785i −1.01628 + 2.44278i 0.309017 + 0.951057i −0.104528 0.994522i 1.23664 + 2.14193i
289.2 −0.104528 0.994522i 0.669131 0.743145i −0.978148 + 0.207912i 1.06629 0.474743i −0.809017 0.587785i −1.26472 2.32389i 0.309017 + 0.951057i −0.104528 0.994522i −0.583600 1.01083i
289.3 −0.104528 0.994522i 0.669131 0.743145i −0.978148 + 0.207912i 1.97343 0.878627i −0.809017 0.587785i 0.867452 + 2.49951i 0.309017 + 0.951057i −0.104528 0.994522i −1.08009 1.87078i
361.1 −0.978148 + 0.207912i −0.104528 0.994522i 0.913545 0.406737i −2.71957 + 3.02039i 0.309017 + 0.951057i −1.98308 1.75140i −0.809017 + 0.587785i −0.978148 + 0.207912i 2.03217 3.51982i
361.2 −0.978148 + 0.207912i −0.104528 0.994522i 0.913545 0.406737i −1.00405 + 1.11511i 0.309017 + 0.951057i 2.56459 + 0.650286i −0.809017 + 0.587785i −0.978148 + 0.207912i 0.750265 1.29950i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 445.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.y.b 24
7.c even 3 1 inner 462.2.y.b 24
11.c even 5 1 inner 462.2.y.b 24
77.m even 15 1 inner 462.2.y.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.y.b 24 1.a even 1 1 trivial
462.2.y.b 24 7.c even 3 1 inner
462.2.y.b 24 11.c even 5 1 inner
462.2.y.b 24 77.m even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{24} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.