# Properties

 Label 462.2.bf.a Level $462$ Weight $2$ Character orbit 462.bf Analytic conductor $3.689$ Analytic rank $0$ Dimension $256$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.bf (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$256$$ Relative dimension: $$32$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$256q - 32q^{4} + 4q^{7} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$256q - 32q^{4} + 4q^{7} - 12q^{9} + 12q^{10} + 24q^{15} + 32q^{16} - 8q^{18} - 16q^{21} - 12q^{22} + 48q^{25} + 6q^{28} + 18q^{31} - 132q^{33} + 16q^{36} + 4q^{37} - 18q^{40} - 4q^{42} + 64q^{43} - 48q^{45} + 8q^{46} + 76q^{49} - 8q^{51} - 88q^{57} + 46q^{58} - 8q^{60} - 12q^{63} + 64q^{64} - 120q^{66} - 32q^{67} - 58q^{70} - 12q^{72} - 96q^{73} - 204q^{75} - 32q^{78} - 4q^{79} - 64q^{81} + 24q^{82} - 36q^{84} + 232q^{85} - 228q^{87} - 6q^{88} + 40q^{91} - 2q^{93} - 144q^{94} + 160q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −0.207912 + 0.978148i −1.63536 0.570609i −0.913545 0.406737i −2.60800 2.89648i 0.898150 1.48099i −2.64502 0.0622836i 0.587785 0.809017i 2.34881 + 1.86630i 3.37542 1.94880i
5.2 −0.207912 + 0.978148i −1.51046 + 0.847653i −0.913545 0.406737i −1.24203 1.37941i −0.515088 1.65369i 2.15154 + 1.53976i 0.587785 0.809017i 1.56297 2.56069i 1.60750 0.928093i
5.3 −0.207912 + 0.978148i −1.46408 0.925462i −0.913545 0.406737i 0.383830 + 0.426287i 1.20964 1.23967i −2.12466 + 1.57664i 0.587785 0.809017i 1.28704 + 2.70989i −0.496774 + 0.286813i
5.4 −0.207912 + 0.978148i −1.37785 + 1.04954i −0.913545 0.406737i 2.40395 + 2.66986i −0.740133 1.56595i −0.383771 + 2.61777i 0.587785 0.809017i 0.796937 2.89221i −3.11133 + 1.79633i
5.5 −0.207912 + 0.978148i −1.26644 1.18158i −0.913545 0.406737i 2.46318 + 2.73564i 1.41907 0.993102i 2.33823 1.23802i 0.587785 0.809017i 0.207743 + 2.99280i −3.18798 + 1.84058i
5.6 −0.207912 + 0.978148i −0.610492 + 1.62089i −0.913545 0.406737i −1.62461 1.80431i −1.45855 0.934154i 1.18467 2.36571i 0.587785 0.809017i −2.25460 1.97909i 2.10266 1.21397i
5.7 −0.207912 + 0.978148i −0.596483 + 1.62610i −0.913545 0.406737i 0.0808680 + 0.0898130i −1.46655 0.921534i −2.64561 0.0274346i 0.587785 0.809017i −2.28842 1.93988i −0.104664 + 0.0604276i
5.8 −0.207912 + 0.978148i −0.442840 1.67448i −0.913545 0.406737i −0.833125 0.925279i 1.72996 0.0850182i 0.162919 2.64073i 0.587785 0.809017i −2.60779 + 1.48306i 1.07828 0.622543i
5.9 −0.207912 + 0.978148i 0.263570 1.71188i −0.913545 0.406737i −1.94189 2.15669i 1.61967 + 0.613730i 2.26273 + 1.37115i 0.587785 0.809017i −2.86106 0.902401i 2.51330 1.45105i
5.10 −0.207912 + 0.978148i 0.665114 1.59926i −0.913545 0.406737i 1.17526 + 1.30525i 1.42602 + 0.983084i −1.84250 1.89873i 0.587785 0.809017i −2.11525 2.12738i −1.52108 + 0.878197i
5.11 −0.207912 + 0.978148i 0.915662 + 1.47023i −0.913545 0.406737i −2.41026 2.67687i −1.62847 + 0.589975i −0.0504795 + 2.64527i 0.587785 0.809017i −1.32313 + 2.69246i 3.11949 1.80104i
5.12 −0.207912 + 0.978148i 0.940283 + 1.45460i −0.913545 0.406737i 1.15142 + 1.27878i −1.61831 + 0.617307i 2.35020 1.21514i 0.587785 0.809017i −1.23174 + 2.73548i −1.49023 + 0.860384i
5.13 −0.207912 + 0.978148i 1.30084 1.14360i −0.913545 0.406737i 1.92888 + 2.14224i 0.848146 + 1.51018i 2.60946 + 0.436733i 0.587785 0.809017i 0.384375 2.97527i −2.49647 + 1.44134i
5.14 −0.207912 + 0.978148i 1.38379 + 1.04170i −0.913545 0.406737i 1.51673 + 1.68449i −1.30664 + 1.13697i −2.61122 0.426069i 0.587785 0.809017i 0.829732 + 2.88298i −1.96303 + 1.13336i
5.15 −0.207912 + 0.978148i 1.70274 0.317274i −0.913545 0.406737i −1.03302 1.14729i −0.0436800 + 1.73150i −0.907001 + 2.48543i 0.587785 0.809017i 2.79867 1.08047i 1.33700 0.771915i
5.16 −0.207912 + 0.978148i 1.73200 + 0.0134239i −0.913545 0.406737i −1.28642 1.42871i −0.373233 + 1.69136i 0.715130 2.54727i 0.587785 0.809017i 2.99964 + 0.0465004i 1.66495 0.961262i
5.17 0.207912 0.978148i −1.71826 + 0.218129i −0.913545 0.406737i 2.60800 + 2.89648i −0.143884 + 1.72606i −2.64502 0.0622836i −0.587785 + 0.809017i 2.90484 0.749604i 3.37542 1.94880i
5.18 0.207912 0.978148i −1.62450 + 0.600840i −0.913545 0.406737i −0.383830 0.426287i 0.249958 + 1.71392i −2.12466 + 1.57664i −0.587785 + 0.809017i 2.27798 1.95213i −0.496774 + 0.286813i
5.19 0.207912 0.978148i −1.48443 + 0.892451i −0.913545 0.406737i −2.46318 2.73564i 0.564318 + 1.63754i 2.33823 1.23802i −0.587785 + 0.809017i 1.40706 2.64956i −3.18798 + 1.84058i
5.20 0.207912 0.978148i −1.30121 1.14317i −0.913545 0.406737i 1.24203 + 1.37941i −1.38873 + 1.03510i 2.15154 + 1.53976i −0.587785 + 0.809017i 0.386315 + 2.97502i 1.60750 0.928093i
See next 80 embeddings (of 256 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 383.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
11.c even 5 1 inner
21.g even 6 1 inner
33.h odd 10 1 inner
77.p odd 30 1 inner
231.bc even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.bf.a 256
3.b odd 2 1 inner 462.2.bf.a 256
7.d odd 6 1 inner 462.2.bf.a 256
11.c even 5 1 inner 462.2.bf.a 256
21.g even 6 1 inner 462.2.bf.a 256
33.h odd 10 1 inner 462.2.bf.a 256
77.p odd 30 1 inner 462.2.bf.a 256
231.bc even 30 1 inner 462.2.bf.a 256

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.bf.a 256 1.a even 1 1 trivial
462.2.bf.a 256 3.b odd 2 1 inner
462.2.bf.a 256 7.d odd 6 1 inner
462.2.bf.a 256 11.c even 5 1 inner
462.2.bf.a 256 21.g even 6 1 inner
462.2.bf.a 256 33.h odd 10 1 inner
462.2.bf.a 256 77.p odd 30 1 inner
462.2.bf.a 256 231.bc even 30 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(462, [\chi])$$.