# Properties

 Label 252.4.w.a Level $252$ Weight $4$ Character orbit 252.w Analytic conductor $14.868$ Analytic rank $0$ Dimension $48$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.w (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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 6q^{7} + 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 6q^{7} + 12q^{9} + 12q^{11} - 36q^{13} + 66q^{15} - 72q^{17} + 24q^{21} + 30q^{23} - 600q^{25} - 396q^{27} + 42q^{29} + 390q^{35} + 84q^{37} - 840q^{39} - 618q^{41} - 42q^{43} + 366q^{45} + 396q^{47} + 318q^{49} - 738q^{51} - 1620q^{53} + 492q^{57} + 1500q^{59} + 672q^{63} - 588q^{67} - 924q^{69} + 564q^{75} - 2472q^{77} + 1608q^{79} + 2592q^{81} - 360q^{85} + 2640q^{87} + 1722q^{89} + 540q^{91} + 660q^{93} - 792q^{97} - 54q^{99} + O(q^{100})$$

## 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 −5.18021 + 0.406745i 0 −8.68596 15.0445i 0 6.03770 17.5085i 0 26.6691 4.21405i 0
5.2 0 −5.15302 + 0.668133i 0 9.18977 + 15.9171i 0 18.3240 2.68897i 0 26.1072 6.88581i 0
5.3 0 −4.97073 1.51388i 0 −3.24590 5.62207i 0 −7.45627 + 16.9530i 0 22.4163 + 15.0502i 0
5.4 0 −4.76836 2.06463i 0 9.23798 + 16.0007i 0 −13.5371 12.6391i 0 18.4746 + 19.6898i 0
5.5 0 −4.59055 + 2.43451i 0 1.72890 + 2.99454i 0 −17.7636 + 5.23980i 0 15.1463 22.3515i 0
5.6 0 −4.38438 2.78877i 0 −1.97944 3.42849i 0 15.7046 + 9.81667i 0 11.4456 + 24.4540i 0
5.7 0 −3.44710 + 3.88813i 0 −0.950243 1.64587i 0 14.2951 + 11.7749i 0 −3.23505 26.8055i 0
5.8 0 −2.54719 4.52900i 0 −3.81512 6.60797i 0 −15.6850 9.84781i 0 −14.0236 + 23.0725i 0
5.9 0 −1.96476 + 4.81038i 0 2.75454 + 4.77100i 0 −0.994545 18.4935i 0 −19.2795 18.9024i 0
5.10 0 −1.35688 5.01586i 0 3.33893 + 5.78320i 0 14.9872 10.8804i 0 −23.3178 + 13.6118i 0
5.11 0 −0.808952 + 5.13280i 0 −10.5226 18.2257i 0 −18.5183 + 0.266265i 0 −25.6912 8.30437i 0
5.12 0 −0.452892 5.17638i 0 6.54401 + 11.3346i 0 1.26985 + 18.4767i 0 −26.5898 + 4.68868i 0
5.13 0 0.569877 + 5.16481i 0 8.28296 + 14.3465i 0 −3.56863 + 18.1732i 0 −26.3505 + 5.88661i 0
5.14 0 0.973978 + 5.10405i 0 −6.29153 10.8972i 0 18.2954 + 2.87731i 0 −25.1027 + 9.94247i 0
5.15 0 1.05398 5.08813i 0 −9.99264 17.3078i 0 −0.138706 + 18.5197i 0 −24.7782 10.7256i 0
5.16 0 3.03995 4.21411i 0 5.20294 + 9.01176i 0 −16.5030 + 8.40550i 0 −8.51746 25.6213i 0
5.17 0 3.04499 4.21047i 0 −4.90701 8.49919i 0 13.1134 13.0781i 0 −8.45605 25.6417i 0
5.18 0 3.58371 + 3.76258i 0 −1.13691 1.96919i 0 −14.5181 + 11.4989i 0 −1.31403 + 26.9680i 0
5.19 0 3.79518 + 3.54917i 0 5.15592 + 8.93032i 0 −5.91622 17.5499i 0 1.80675 + 26.9395i 0
5.20 0 3.80953 + 3.53377i 0 −3.77250 6.53416i 0 14.8160 11.1124i 0 2.02498 + 26.9240i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i 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.w.a 48
3.b odd 2 1 756.4.w.a 48
7.d odd 6 1 252.4.bm.a yes 48
9.c even 3 1 756.4.bm.a 48
9.d odd 6 1 252.4.bm.a yes 48
21.g even 6 1 756.4.bm.a 48
63.i even 6 1 inner 252.4.w.a 48
63.t odd 6 1 756.4.w.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.w.a 48 1.a even 1 1 trivial
252.4.w.a 48 63.i even 6 1 inner
252.4.bm.a yes 48 7.d odd 6 1
252.4.bm.a yes 48 9.d odd 6 1
756.4.w.a 48 3.b odd 2 1
756.4.w.a 48 63.t odd 6 1
756.4.bm.a 48 9.c even 3 1
756.4.bm.a 48 21.g even 6 1

## Hecke kernels

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