# Properties

 Label 252.4.x.a Level $252$ Weight $4$ Character orbit 252.x Analytic conductor $14.868$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.x (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8684813214$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 6q^{7} + 60q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 6q^{7} + 60q^{9} - 12q^{11} + 192q^{15} - 72q^{21} - 408q^{23} - 600q^{25} - 84q^{29} + 336q^{37} + 36q^{39} + 84q^{43} + 318q^{49} - 1812q^{51} - 852q^{57} - 564q^{63} + 2964q^{65} - 588q^{67} + 2400q^{77} + 204q^{79} + 1980q^{81} - 360q^{85} - 1080q^{91} + 2496q^{93} + 300q^{95} - 4968q^{99} + O(q^{100})$$

## 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 −5.19208 0.205661i 0 2.34269 4.05766i 0 −10.9369 + 14.9461i 0 26.9154 + 2.13562i 0
41.2 0 −5.09172 1.03651i 0 −10.5571 + 18.2854i 0 −4.11592 + 18.0571i 0 24.8513 + 10.5552i 0
41.3 0 −4.81916 + 1.94311i 0 9.12012 15.7965i 0 −9.48165 15.9091i 0 19.4487 18.7283i 0
41.4 0 −4.66323 + 2.29222i 0 −5.16485 + 8.94579i 0 14.4986 11.5234i 0 16.4915 21.3783i 0
41.5 0 −4.37628 2.80147i 0 −2.20656 + 3.82187i 0 13.5618 12.6126i 0 11.3036 + 24.5200i 0
41.6 0 −4.09692 + 3.19613i 0 3.53447 6.12188i 0 10.8305 + 15.0234i 0 6.56950 26.1886i 0
41.7 0 −4.07854 3.21955i 0 0.330097 0.571745i 0 −15.6440 9.91296i 0 6.26899 + 26.2621i 0
41.8 0 −3.66558 3.68287i 0 8.29874 14.3738i 0 18.2297 + 3.26749i 0 −0.127054 + 26.9997i 0
41.9 0 −2.29905 + 4.65987i 0 −6.03570 + 10.4541i 0 −16.9871 7.37834i 0 −16.4287 21.4265i 0
41.10 0 −0.494604 5.17256i 0 2.99997 5.19610i 0 −0.375477 + 18.5165i 0 −26.5107 + 5.11673i 0
41.11 0 −0.316211 5.18652i 0 −5.49690 + 9.52092i 0 −18.3277 2.66385i 0 −26.8000 + 3.28007i 0
41.12 0 −0.0939923 5.19530i 0 −7.82452 + 13.5525i 0 17.6225 5.69627i 0 −26.9823 + 0.976636i 0
41.13 0 0.0939923 + 5.19530i 0 7.82452 13.5525i 0 −13.7444 + 12.4134i 0 −26.9823 + 0.976636i 0
41.14 0 0.316211 + 5.18652i 0 5.49690 9.52092i 0 6.85688 17.2042i 0 −26.8000 + 3.28007i 0
41.15 0 0.494604 + 5.17256i 0 −2.99997 + 5.19610i 0 16.2235 + 8.93305i 0 −26.5107 + 5.11673i 0
41.16 0 2.29905 4.65987i 0 6.03570 10.4541i 0 2.10370 18.4004i 0 −16.4287 21.4265i 0
41.17 0 3.66558 + 3.68287i 0 −8.29874 + 14.3738i 0 −6.28515 + 17.4212i 0 −0.127054 + 26.9997i 0
41.18 0 4.07854 + 3.21955i 0 −0.330097 + 0.571745i 0 −0.762896 18.5045i 0 6.26899 + 26.2621i 0
41.19 0 4.09692 3.19613i 0 −3.53447 + 6.12188i 0 7.59538 + 16.8911i 0 6.56950 26.1886i 0
41.20 0 4.37628 + 2.80147i 0 2.20656 3.82187i 0 −17.7037 + 5.43851i 0 11.3036 + 24.5200i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.24 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.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.x.a 48
3.b odd 2 1 756.4.x.a 48
7.b odd 2 1 inner 252.4.x.a 48
9.c even 3 1 756.4.x.a 48
9.c even 3 1 2268.4.f.a 48
9.d odd 6 1 inner 252.4.x.a 48
9.d odd 6 1 2268.4.f.a 48
21.c even 2 1 756.4.x.a 48
63.l odd 6 1 756.4.x.a 48
63.l odd 6 1 2268.4.f.a 48
63.o even 6 1 inner 252.4.x.a 48
63.o even 6 1 2268.4.f.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.x.a 48 1.a even 1 1 trivial
252.4.x.a 48 7.b odd 2 1 inner
252.4.x.a 48 9.d odd 6 1 inner
252.4.x.a 48 63.o even 6 1 inner
756.4.x.a 48 3.b odd 2 1
756.4.x.a 48 9.c even 3 1
756.4.x.a 48 21.c even 2 1
756.4.x.a 48 63.l odd 6 1
2268.4.f.a 48 9.c even 3 1
2268.4.f.a 48 9.d odd 6 1
2268.4.f.a 48 63.l odd 6 1
2268.4.f.a 48 63.o even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(252, [\chi])$$.