# Properties

 Label 252.4.bm.a Level $252$ Weight $4$ Character orbit 252.bm 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.bm (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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 6q^{7} - 30q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 6q^{7} - 30q^{9} + 36q^{13} + 66q^{15} + 72q^{17} + 126q^{21} + 1200q^{25} + 396q^{27} + 42q^{29} - 90q^{31} + 108q^{33} - 390q^{35} + 84q^{37} + 1014q^{39} + 618q^{41} - 42q^{43} - 1014q^{45} + 198q^{47} - 276q^{49} + 408q^{51} + 1620q^{53} + 492q^{57} + 750q^{59} - 1314q^{61} + 1542q^{63} + 564q^{65} + 294q^{67} + 924q^{69} - 1410q^{75} - 2448q^{77} - 804q^{79} - 666q^{81} - 360q^{85} + 1788q^{87} - 1722q^{89} + 540q^{91} + 1128q^{93} - 2946q^{95} + 792q^{97} - 54q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
173.1 0 −5.14829 0.703660i 0 5.50907 0 −15.5186 + 10.1081i 0 26.0097 + 7.24529i 0
173.2 0 −5.09076 + 1.04121i 0 −1.90049 0 3.04981 18.2674i 0 24.8318 10.6011i 0
173.3 0 −4.84961 1.86582i 0 −21.0452 0 9.48976 + 15.9042i 0 20.0374 + 18.0970i 0
173.4 0 −4.40362 + 2.75828i 0 3.45780 0 13.4196 + 12.7638i 0 11.7838 24.2928i 0
173.5 0 −4.18792 3.07593i 0 16.5659 0 17.5228 5.99607i 0 8.07729 + 25.7635i 0
173.6 0 −3.93325 3.39552i 0 −12.5831 0 −6.65587 17.2829i 0 3.94093 + 26.7108i 0
173.7 0 −3.15513 + 4.12858i 0 18.3795 0 −11.4907 14.5246i 0 −7.09031 26.0524i 0
173.8 0 −2.94236 + 4.28282i 0 −17.3719 0 −18.1816 + 3.52543i 0 −9.68509 25.2032i 0
173.9 0 −1.46664 4.98488i 0 −2.27383 0 17.2174 + 6.82357i 0 −22.6980 + 14.6220i 0
173.10 0 −1.17608 5.06131i 0 10.3118 0 −12.2405 + 13.8985i 0 −24.2336 + 11.9051i 0
173.11 0 −1.17430 + 5.06172i 0 −6.49181 0 18.4099 2.01918i 0 −24.2420 11.8880i 0
173.12 0 −1.15557 5.06603i 0 −7.54500 0 −17.0316 7.27489i 0 −24.3293 + 11.7083i 0
173.13 0 −0.596161 + 5.16184i 0 18.4760 0 −4.17729 + 18.0430i 0 −26.2892 6.15457i 0
173.14 0 0.222952 + 5.19137i 0 −3.95888 0 0.649207 18.5089i 0 −26.9006 + 2.31485i 0
173.15 0 2.56423 4.51937i 0 17.7911 0 −5.94198 17.5412i 0 −13.8494 23.1774i 0
173.16 0 2.64863 + 4.47043i 0 −7.63023 0 −0.685930 + 18.5076i 0 −12.9695 + 23.6810i 0
173.17 0 2.92280 4.29619i 0 −6.03810 0 10.8562 15.0047i 0 −9.91446 25.1138i 0
173.18 0 2.97726 4.25863i 0 −12.7833 0 2.83671 + 18.3017i 0 −9.27184 25.3581i 0
173.19 0 3.66543 + 3.68302i 0 6.67787 0 −16.9163 7.53913i 0 −0.129304 + 26.9997i 0
173.20 0 4.25643 + 2.98040i 0 13.0880 0 15.3663 10.3381i 0 9.23438 + 25.3718i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 185.24 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.s 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.bm.a yes 48
3.b odd 2 1 756.4.bm.a 48
7.d odd 6 1 252.4.w.a 48
9.c even 3 1 756.4.w.a 48
9.d odd 6 1 252.4.w.a 48
21.g even 6 1 756.4.w.a 48
63.k odd 6 1 756.4.bm.a 48
63.s even 6 1 inner 252.4.bm.a yes 48

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

## Hecke kernels

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