# Properties

 Label 462.2.u.a Level $462$ Weight $2$ Character orbit 462.u Analytic conductor $3.689$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.u (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 8q^{4} - 10q^{5} + 8q^{6} - 10q^{7} + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 8q^{4} - 10q^{5} + 8q^{6} - 10q^{7} + 8q^{9} - 4q^{10} + 8q^{11} - 2q^{14} + 6q^{15} - 8q^{16} + 12q^{17} + 16q^{19} - 10q^{20} + 8q^{21} - 4q^{22} + 8q^{23} - 8q^{24} + 6q^{25} + 20q^{29} + 50q^{31} + 16q^{33} + 32q^{35} - 8q^{36} - 16q^{37} - 6q^{40} - 40q^{41} - 10q^{42} + 12q^{44} - 28q^{49} + 40q^{51} + 32q^{54} + 40q^{55} - 8q^{56} + 10q^{58} - 60q^{59} + 4q^{60} + 4q^{61} - 20q^{62} - 10q^{63} + 8q^{64} - 8q^{66} - 16q^{67} - 12q^{68} - 30q^{69} - 18q^{70} - 48q^{71} + 74q^{73} - 40q^{74} + 24q^{76} - 70q^{77} - 60q^{79} - 8q^{81} - 20q^{82} - 4q^{83} + 2q^{84} - 10q^{85} - 36q^{86} - 20q^{87} - 16q^{88} + 4q^{90} - 60q^{91} - 8q^{92} - 10q^{93} - 20q^{95} + 8q^{96} - 60q^{97} + 40q^{98} - 8q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −0.951057 0.309017i 0.587785 + 0.809017i 0.809017 + 0.587785i −3.76443 + 1.22314i −0.309017 0.951057i −1.44620 2.21552i −0.587785 0.809017i −0.309017 + 0.951057i 3.95815
13.2 −0.951057 0.309017i 0.587785 + 0.809017i 0.809017 + 0.587785i −0.0789999 + 0.0256686i −0.309017 0.951057i −1.18860 + 2.36373i −0.587785 0.809017i −0.309017 + 0.951057i 0.0830654
13.3 −0.951057 0.309017i 0.587785 + 0.809017i 0.809017 + 0.587785i 1.43844 0.467378i −0.309017 0.951057i 2.01686 + 1.71239i −0.587785 0.809017i −0.309017 + 0.951057i −1.51247
13.4 −0.951057 0.309017i 0.587785 + 0.809017i 0.809017 + 0.587785i 2.13481 0.693642i −0.309017 0.951057i 1.63277 2.08184i −0.587785 0.809017i −0.309017 + 0.951057i −2.24467
13.5 0.951057 + 0.309017i −0.587785 0.809017i 0.809017 + 0.587785i −3.18685 + 1.03547i −0.309017 0.951057i 2.59478 + 0.516820i 0.587785 + 0.809017i −0.309017 + 0.951057i −3.35085
13.6 0.951057 + 0.309017i −0.587785 0.809017i 0.809017 + 0.587785i −2.46351 + 0.800443i −0.309017 0.951057i −2.50308 + 0.857092i 0.587785 + 0.809017i −0.309017 + 0.951057i −2.59029
13.7 0.951057 + 0.309017i −0.587785 0.809017i 0.809017 + 0.587785i 0.169394 0.0550394i −0.309017 0.951057i −0.364877 2.62047i 0.587785 + 0.809017i −0.309017 + 0.951057i 0.178111
13.8 0.951057 + 0.309017i −0.587785 0.809017i 0.809017 + 0.587785i 2.13310 0.693087i −0.309017 0.951057i 0.112437 + 2.64336i 0.587785 + 0.809017i −0.309017 + 0.951057i 2.24288
139.1 −0.587785 + 0.809017i −0.951057 0.309017i −0.309017 0.951057i −2.23816 3.08056i 0.809017 0.587785i −1.26203 2.32536i 0.951057 + 0.309017i 0.809017 + 0.587785i 3.80779
139.2 −0.587785 + 0.809017i −0.951057 0.309017i −0.309017 0.951057i −0.884493 1.21740i 0.809017 0.587785i −1.60058 + 2.10669i 0.951057 + 0.309017i 0.809017 + 0.587785i 1.50479
139.3 −0.587785 + 0.809017i −0.951057 0.309017i −0.309017 0.951057i 0.946241 + 1.30239i 0.809017 0.587785i −1.50361 + 2.17696i 0.951057 + 0.309017i 0.809017 + 0.587785i −1.60984
139.4 −0.587785 + 0.809017i −0.951057 0.309017i −0.309017 0.951057i 1.12216 + 1.54452i 0.809017 0.587785i 0.488107 2.60034i 0.951057 + 0.309017i 0.809017 + 0.587785i −1.90913
139.5 0.587785 0.809017i 0.951057 + 0.309017i −0.309017 0.951057i −2.18694 3.01006i 0.809017 0.587785i −2.63889 + 0.190406i −0.951057 0.309017i 0.809017 + 0.587785i −3.72064
139.6 0.587785 0.809017i 0.951057 + 0.309017i −0.309017 0.951057i −0.531600 0.731684i 0.809017 0.587785i 0.261116 2.63283i −0.951057 0.309017i 0.809017 + 0.587785i −0.904411
139.7 0.587785 0.809017i 0.951057 + 0.309017i −0.309017 0.951057i 1.08816 + 1.49773i 0.809017 0.587785i −2.23342 + 1.41839i −0.951057 0.309017i 0.809017 + 0.587785i 1.85129
139.8 0.587785 0.809017i 0.951057 + 0.309017i −0.309017 0.951057i 1.30266 + 1.79296i 0.809017 0.587785i 2.63520 0.236035i −0.951057 0.309017i 0.809017 + 0.587785i 2.21622
349.1 −0.587785 0.809017i −0.951057 + 0.309017i −0.309017 + 0.951057i −2.23816 + 3.08056i 0.809017 + 0.587785i −1.26203 + 2.32536i 0.951057 0.309017i 0.809017 0.587785i 3.80779
349.2 −0.587785 0.809017i −0.951057 + 0.309017i −0.309017 + 0.951057i −0.884493 + 1.21740i 0.809017 + 0.587785i −1.60058 2.10669i 0.951057 0.309017i 0.809017 0.587785i 1.50479
349.3 −0.587785 0.809017i −0.951057 + 0.309017i −0.309017 + 0.951057i 0.946241 1.30239i 0.809017 + 0.587785i −1.50361 2.17696i 0.951057 0.309017i 0.809017 0.587785i −1.60984
349.4 −0.587785 0.809017i −0.951057 + 0.309017i −0.309017 + 0.951057i 1.12216 1.54452i 0.809017 + 0.587785i 0.488107 + 2.60034i 0.951057 0.309017i 0.809017 0.587785i −1.90913
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.l even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.u.a 32
7.b odd 2 1 462.2.u.b yes 32
11.d odd 10 1 462.2.u.b yes 32
77.l even 10 1 inner 462.2.u.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.u.a 32 1.a even 1 1 trivial
462.2.u.a 32 77.l even 10 1 inner
462.2.u.b yes 32 7.b odd 2 1
462.2.u.b yes 32 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{32} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.