# Properties

 Label 552.2.j.d 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.

 $$\operatorname{Tr}(f)(q) =$$ $$42q + 2q^{3} + 4q^{4} + 8q^{5} + 4q^{6} - 9q^{8} + 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$42q + 2q^{3} + 4q^{4} + 8q^{5} + 4q^{6} - 9q^{8} + 2q^{9} - 4q^{10} - 21q^{12} - 4q^{14} - 8q^{15} - 12q^{16} - 11q^{18} + 4q^{19} - 2q^{20} + 8q^{21} + 18q^{22} + 42q^{23} + 28q^{24} + 22q^{25} + 11q^{26} - 16q^{27} + 6q^{28} + 2q^{30} - 20q^{32} + 12q^{33} + 14q^{34} - 24q^{36} - 22q^{38} + 8q^{39} + 4q^{40} - 44q^{42} + 28q^{43} + 56q^{44} - 8q^{45} + 30q^{48} - 50q^{49} + 20q^{50} + 28q^{51} - q^{52} + 24q^{53} + 52q^{54} - 34q^{56} - 8q^{57} - 21q^{58} - 42q^{60} - 79q^{62} - 16q^{63} + 7q^{64} - 62q^{66} - 4q^{67} + 20q^{68} + 2q^{69} - 8q^{70} + 22q^{72} + 4q^{73} + 36q^{74} - 6q^{75} + 14q^{76} + 32q^{77} + 27q^{78} - 52q^{80} + 18q^{81} + 11q^{82} - 80q^{84} - 28q^{86} - 48q^{87} - 38q^{88} - 46q^{90} - 8q^{91} + 4q^{92} - 22q^{93} + q^{94} - 16q^{95} + 9q^{96} + 20q^{97} + 64q^{98} + O(q^{100})$$

## 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 −1.41381 0.0336448i 0.124259 1.72759i 1.99774 + 0.0951350i 1.31688 −0.233803 + 2.43831i 2.00880i −2.82122 0.201717i −2.96912 0.429336i −1.86182 0.0443062i
323.2 −1.41381 + 0.0336448i 0.124259 + 1.72759i 1.99774 0.0951350i 1.31688 −0.233803 2.43831i 2.00880i −2.82122 + 0.201717i −2.96912 + 0.429336i −1.86182 + 0.0443062i
323.3 −1.40766 0.136034i −1.50629 0.855045i 1.96299 + 0.382978i −2.98020 2.00402 + 1.40851i 1.36209i −2.71112 0.806135i 1.53780 + 2.57588i 4.19509 + 0.405409i
323.4 −1.40766 + 0.136034i −1.50629 + 0.855045i 1.96299 0.382978i −2.98020 2.00402 1.40851i 1.36209i −2.71112 + 0.806135i 1.53780 2.57588i 4.19509 0.405409i
323.5 −1.33226 0.474424i 0.873799 1.49549i 1.54984 + 1.26411i −1.38336 −1.87362 + 1.57783i 5.18286i −1.46507 2.41942i −1.47295 2.61351i 1.84300 + 0.656301i
323.6 −1.33226 + 0.474424i 0.873799 + 1.49549i 1.54984 1.26411i −1.38336 −1.87362 1.57783i 5.18286i −1.46507 + 2.41942i −1.47295 + 2.61351i 1.84300 0.656301i
323.7 −1.30785 0.538067i −1.71722 + 0.226158i 1.42097 + 1.40743i 3.49837 2.36757 + 0.628199i 1.83575i −1.10113 2.60529i 2.89771 0.776727i −4.57536 1.88236i
323.8 −1.30785 + 0.538067i −1.71722 0.226158i 1.42097 1.40743i 3.49837 2.36757 0.628199i 1.83575i −1.10113 + 2.60529i 2.89771 + 0.776727i −4.57536 + 1.88236i
323.9 −1.07807 0.915298i 1.49079 + 0.881788i 0.324458 + 1.97351i 1.83050 −0.800072 2.31514i 4.30356i 1.45656 2.42455i 1.44490 + 2.62912i −1.97340 1.67545i
323.10 −1.07807 + 0.915298i 1.49079 0.881788i 0.324458 1.97351i 1.83050 −0.800072 + 2.31514i 4.30356i 1.45656 + 2.42455i 1.44490 2.62912i −1.97340 + 1.67545i
323.11 −0.975618 1.02380i 1.69183 + 0.371101i −0.0963383 + 1.99768i −1.56730 −1.27065 2.09415i 0.668312i 2.13922 1.85034i 2.72457 + 1.25568i 1.52908 + 1.60460i
323.12 −0.975618 + 1.02380i 1.69183 0.371101i −0.0963383 1.99768i −1.56730 −1.27065 + 2.09415i 0.668312i 2.13922 + 1.85034i 2.72457 1.25568i 1.52908 1.60460i
323.13 −0.964429 1.03435i −0.513686 + 1.65412i −0.139754 + 1.99511i 2.51134 2.20635 1.06396i 2.77384i 2.19842 1.77959i −2.47225 1.69940i −2.42201 2.59760i
323.14 −0.964429 + 1.03435i −0.513686 1.65412i −0.139754 1.99511i 2.51134 2.20635 + 1.06396i 2.77384i 2.19842 + 1.77959i −2.47225 + 1.69940i −2.42201 + 2.59760i
323.15 −0.734855 1.20830i −1.73179 0.0302019i −0.919975 + 1.77585i −0.871841 1.23612 + 2.11471i 3.09231i 2.82181 0.193388i 2.99818 + 0.104607i 0.640677 + 1.05344i
323.16 −0.734855 + 1.20830i −1.73179 + 0.0302019i −0.919975 1.77585i −0.871841 1.23612 2.11471i 3.09231i 2.82181 + 0.193388i 2.99818 0.104607i 0.640677 1.05344i
323.17 −0.242573 1.39325i 1.04780 + 1.37917i −1.88232 + 0.675933i −2.15601 1.66736 1.79441i 2.22228i 1.39835 + 2.45858i −0.804216 + 2.89020i 0.522991 + 3.00387i
323.18 −0.242573 + 1.39325i 1.04780 1.37917i −1.88232 0.675933i −2.15601 1.66736 + 1.79441i 2.22228i 1.39835 2.45858i −0.804216 2.89020i 0.522991 3.00387i
323.19 −0.218758 1.39719i 0.262136 1.71210i −1.90429 + 0.611292i −0.130526 −2.44948 + 0.00827965i 3.77863i 1.27067 + 2.52693i −2.86257 0.897607i 0.0285535 + 0.182369i
323.20 −0.218758 + 1.39719i 0.262136 + 1.71210i −1.90429 0.611292i −0.130526 −2.44948 0.00827965i 3.77863i 1.27067 2.52693i −2.86257 + 0.897607i 0.0285535 0.182369i
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.d yes 42
3.b odd 2 1 552.2.j.c 42
4.b odd 2 1 2208.2.j.d 42
8.b even 2 1 2208.2.j.c 42
8.d odd 2 1 552.2.j.c 42
12.b even 2 1 2208.2.j.c 42
24.f even 2 1 inner 552.2.j.d yes 42
24.h odd 2 1 2208.2.j.d 42

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

## Hecke kernels

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