# Properties

 Label 552.2.j.c Level $552$ Weight $2$ Character orbit 552.j Analytic conductor $4.408$ Analytic rank $0$ Dimension $42$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$552 = 2^{3} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 552.j (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$42$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$42q + 2q^{3} + 4q^{4} - 8q^{5} + q^{6} + 9q^{8} + 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$42q + 2q^{3} + 4q^{4} - 8q^{5} + q^{6} + 9q^{8} + 2q^{9} - 4q^{10} + 4q^{14} + 8q^{15} - 12q^{16} + 16q^{18} + 4q^{19} + 2q^{20} - 8q^{21} + 18q^{22} - 42q^{23} - 24q^{24} + 22q^{25} - 11q^{26} - 16q^{27} + 6q^{28} - 24q^{30} + 20q^{32} + 12q^{33} + 14q^{34} + 15q^{36} + 22q^{38} - 8q^{39} + 4q^{40} + 36q^{42} + 28q^{43} - 56q^{44} + 8q^{45} - 9q^{48} - 50q^{49} - 20q^{50} + 28q^{51} - q^{52} - 24q^{53} - 24q^{54} + 34q^{56} - 8q^{57} - 21q^{58} + 18q^{60} + 79q^{62} + 16q^{63} + 7q^{64} + 16q^{66} - 4q^{67} - 20q^{68} - 2q^{69} - 8q^{70} - 62q^{72} + 4q^{73} - 36q^{74} - 6q^{75} + 14q^{76} - 32q^{77} - 62q^{78} + 52q^{80} + 18q^{81} + 11q^{82} + 66q^{84} + 28q^{86} + 48q^{87} - 38q^{88} - 8q^{91} - 4q^{92} + 22q^{93} + q^{94} + 16q^{95} - 54q^{96} + 20q^{97} - 64q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
323.1 −1.39813 0.212673i 1.67830 + 0.428159i 1.90954 + 0.594688i 1.58050 −2.25542 0.955550i 2.65755i −2.54331 1.23756i 2.63336 + 1.43716i −2.20975 0.336130i
323.2 −1.39813 + 0.212673i 1.67830 0.428159i 1.90954 0.594688i 1.58050 −2.25542 + 0.955550i 2.65755i −2.54331 + 1.23756i 2.63336 1.43716i −2.20975 + 0.336130i
323.3 −1.29483 0.568707i −1.25835 + 1.19019i 1.35315 + 1.47275i −0.156681 2.30622 0.825452i 3.17007i −0.914523 2.67650i 0.166904 2.99535i 0.202875 + 0.0891057i
323.4 −1.29483 + 0.568707i −1.25835 1.19019i 1.35315 1.47275i −0.156681 2.30622 + 0.825452i 3.17007i −0.914523 + 2.67650i 0.166904 + 2.99535i 0.202875 0.0891057i
323.5 −1.29433 0.569844i 0.611727 + 1.62043i 1.35056 + 1.47513i −2.61830 0.131618 2.44595i 0.280990i −0.907465 2.67890i −2.25158 + 1.98252i 3.38893 + 1.49202i
323.6 −1.29433 + 0.569844i 0.611727 1.62043i 1.35056 1.47513i −2.61830 0.131618 + 2.44595i 0.280990i −0.907465 + 2.67890i −2.25158 1.98252i 3.38893 1.49202i
323.7 −1.28262 0.595732i −1.03555 1.38840i 1.29021 + 1.52819i 1.31940 0.501098 + 2.39769i 1.36534i −0.744447 2.72870i −0.855283 + 2.87550i −1.69228 0.786008i
323.8 −1.28262 + 0.595732i −1.03555 + 1.38840i 1.29021 1.52819i 1.31940 0.501098 2.39769i 1.36534i −0.744447 + 2.72870i −0.855283 2.87550i −1.69228 + 0.786008i
323.9 −1.03749 0.961053i 1.07871 1.35513i 0.152756 + 1.99416i 4.03392 −2.42150 + 0.369231i 0.282887i 1.75801 2.21572i −0.672762 2.92359i −4.18514 3.87681i
323.10 −1.03749 + 0.961053i 1.07871 + 1.35513i 0.152756 1.99416i 4.03392 −2.42150 0.369231i 0.282887i 1.75801 + 2.21572i −0.672762 + 2.92359i −4.18514 + 3.87681i
323.11 −1.03694 0.961640i −0.303431 1.70527i 0.150496 + 1.99433i −3.63397 −1.32521 + 2.06005i 2.53044i 1.76177 2.21273i −2.81586 + 1.03486i 3.76822 + 3.49457i
323.12 −1.03694 + 0.961640i −0.303431 + 1.70527i 0.150496 1.99433i −3.63397 −1.32521 2.06005i 2.53044i 1.76177 + 2.21273i −2.81586 1.03486i 3.76822 3.49457i
323.13 −0.866210 1.11789i −1.39323 + 1.02904i −0.499360 + 1.93666i −3.31842 2.35718 + 0.666110i 4.93974i 2.59752 1.11932i 0.882155 2.86737i 2.87445 + 3.70963i
323.14 −0.866210 + 1.11789i −1.39323 1.02904i −0.499360 1.93666i −3.31842 2.35718 0.666110i 4.93974i 2.59752 + 1.11932i 0.882155 + 2.86737i 2.87445 3.70963i
323.15 −0.610595 1.27561i 0.0769352 + 1.73034i −1.25435 + 1.55776i 2.14134 2.16026 1.15468i 2.49021i 2.75299 + 0.648892i −2.98816 + 0.266248i −1.30749 2.73151i
323.16 −0.610595 + 1.27561i 0.0769352 1.73034i −1.25435 1.55776i 2.14134 2.16026 + 1.15468i 2.49021i 2.75299 0.648892i −2.98816 0.266248i −1.30749 + 2.73151i
323.17 −0.581428 1.28916i 1.42468 0.985030i −1.32388 + 1.49911i −1.37188 −2.09821 1.26392i 0.424283i 2.70234 + 0.835077i 1.05943 2.80671i 0.797651 + 1.76858i
323.18 −0.581428 + 1.28916i 1.42468 + 0.985030i −1.32388 1.49911i −1.37188 −2.09821 + 1.26392i 0.424283i 2.70234 0.835077i 1.05943 + 2.80671i 0.797651 1.76858i
323.19 −0.484105 1.32877i −1.59465 0.676097i −1.53128 + 1.28653i 1.67234 −0.126404 + 2.44623i 2.28745i 2.45082 + 1.41191i 2.08579 + 2.15627i −0.809588 2.22216i
323.20 −0.484105 + 1.32877i −1.59465 + 0.676097i −1.53128 1.28653i 1.67234 −0.126404 2.44623i 2.28745i 2.45082 1.41191i 2.08579 2.15627i −0.809588 + 2.22216i
See all 42 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.42 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.j.c 42
3.b odd 2 1 552.2.j.d yes 42
4.b odd 2 1 2208.2.j.c 42
8.b even 2 1 2208.2.j.d 42
8.d odd 2 1 552.2.j.d yes 42
12.b even 2 1 2208.2.j.d 42
24.f even 2 1 inner 552.2.j.c 42
24.h odd 2 1 2208.2.j.c 42

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.j.c 42 1.a even 1 1 trivial
552.2.j.c 42 24.f even 2 1 inner
552.2.j.d yes 42 3.b odd 2 1
552.2.j.d yes 42 8.d odd 2 1
2208.2.j.c 42 4.b odd 2 1
2208.2.j.c 42 24.h odd 2 1
2208.2.j.d 42 8.b even 2 1
2208.2.j.d 42 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{21} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(552, [\chi])$$.