# Properties

 Label 552.2.x.a Level $552$ Weight $2$ Character orbit 552.x Analytic conductor $4.408$ Analytic rank $0$ Dimension $920$ CM no Inner twists $8$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$552 = 2^{3} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 552.x (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$920$$ Relative dimension: $$92$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$920q - 18q^{3} - 14q^{4} - 16q^{6} - 18q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$920q - 18q^{3} - 14q^{4} - 16q^{6} - 18q^{9} - 14q^{10} - 6q^{12} - 30q^{16} - 16q^{18} - 52q^{19} - 32q^{22} - 26q^{24} - 112q^{25} - 30q^{27} - 34q^{28} + 11q^{30} - 30q^{33} - 88q^{34} - 18q^{36} + 124q^{40} - 3q^{42} - 36q^{43} - 110q^{46} + 32q^{49} - 30q^{51} + 90q^{52} - 39q^{54} - 6q^{57} - 68q^{58} + 13q^{60} + 28q^{64} - 46q^{66} - 100q^{67} - 92q^{70} + 29q^{72} - 36q^{73} + 14q^{75} - 50q^{76} - 86q^{78} - 2q^{81} - 12q^{82} - 151q^{84} - 42q^{88} - 196q^{90} - 136q^{91} - 68q^{94} - 175q^{96} - 36q^{97} - 54q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −1.41407 + 0.0201587i −0.595463 + 1.62648i 1.99919 0.0570116i 0.0333805 0.00980139i 0.809239 2.31195i 0.316707 0.274428i −2.82584 + 0.120919i −2.29085 1.93701i −0.0470047 + 0.0145328i
35.2 −1.41396 0.0268897i 0.618438 1.61788i 1.99855 + 0.0760418i 1.48175 0.435081i −0.917950 + 2.27098i 3.65626 3.16816i −2.82383 0.161260i −2.23507 2.00112i −2.10683 + 0.575343i
35.3 −1.41384 0.0325375i −0.335489 1.69925i 1.99788 + 0.0920055i −3.41280 + 1.00209i 0.419038 + 2.41338i −1.52640 + 1.32263i −2.82169 0.195087i −2.77489 + 1.14016i 4.85775 1.30575i
35.4 −1.41269 0.0656090i 1.50835 0.851390i 1.99139 + 0.185370i 2.73552 0.803222i −2.18670 + 1.10379i −3.00502 + 2.60386i −2.80106 0.392524i 1.55027 2.56840i −3.91714 + 0.955229i
35.5 −1.40292 + 0.178405i 1.61394 + 0.628649i 1.93634 0.500576i −2.04785 + 0.601303i −2.37637 0.594005i −1.23389 + 1.06917i −2.62722 + 1.04772i 2.20960 + 2.02920i 2.76568 1.20892i
35.6 −1.39813 + 0.212705i −1.49325 + 0.877613i 1.90951 0.594777i 0.229886 0.0675006i 1.90108 1.54464i 1.84453 1.59829i −2.54323 + 1.23774i 1.45959 2.62099i −0.307052 + 0.143272i
35.7 −1.37018 0.350170i −0.141521 + 1.72626i 1.75476 + 0.959590i 2.19665 0.644996i 0.798394 2.31572i −2.54738 + 2.20732i −2.06831 1.92927i −2.95994 0.488605i −3.23566 + 0.114554i
35.8 −1.33616 0.463327i 0.833138 + 1.51851i 1.57066 + 1.23816i −1.70423 + 0.500408i −0.409640 2.41499i 2.57338 2.22984i −1.52498 2.38211i −1.61176 + 2.53026i 2.50898 + 0.120991i
35.9 −1.33559 + 0.464960i 1.06576 + 1.36534i 1.56763 1.24199i 3.82579 1.12335i −2.05825 1.32801i 2.32451 2.01420i −1.51623 + 2.38768i −0.728327 + 2.91025i −4.58739 + 3.27918i
35.10 −1.32803 + 0.486139i −1.68129 0.416231i 1.52734 1.29122i −0.607897 + 0.178495i 2.43516 0.264575i −1.35812 + 1.17682i −1.40064 + 2.45728i 2.65350 + 1.39962i 0.720533 0.532569i
35.11 −1.29176 + 0.575639i −1.36388 1.06763i 1.33728 1.48717i 2.50603 0.735837i 2.37637 + 0.594013i −0.0911016 + 0.0789400i −0.871368 + 2.69086i 0.720344 + 2.91223i −2.81361 + 2.39309i
35.12 −1.28905 0.581685i −0.724881 1.57307i 1.32328 + 1.49964i 0.385158 0.113093i 0.0193752 + 2.44941i −0.150483 + 0.130394i −0.833459 2.70284i −1.94909 + 2.28058i −0.562271 0.0782590i
35.13 −1.28897 + 0.581846i 0.880176 1.49174i 1.32291 1.49997i −0.611012 + 0.179409i −0.266561 + 2.43494i −1.41902 + 1.22959i −0.832444 + 2.70315i −1.45058 2.62599i 0.683190 0.586769i
35.14 −1.28558 0.589316i −1.72832 0.113594i 1.30541 + 1.51522i −0.988646 + 0.290293i 2.15495 + 1.16456i −3.80564 + 3.29760i −0.785267 2.71723i 2.97419 + 0.392655i 1.44205 + 0.209431i
35.15 −1.27844 0.604647i 1.53786 0.796851i 1.26880 + 1.54601i −2.80420 + 0.823387i −2.44788 + 0.0888594i 0.808303 0.700399i −0.687299 2.74365i 1.73006 2.45090i 4.08285 + 0.642901i
35.16 −1.27084 0.620447i −1.72827 + 0.114318i 1.23009 + 1.57698i 4.02671 1.18235i 2.26730 + 0.927022i 1.49077 1.29176i −0.584820 2.76731i 2.97386 0.395146i −5.85090 0.995778i
35.17 −1.26653 0.629211i 1.61370 + 0.629273i 1.20819 + 1.59383i 1.81679 0.533458i −1.64785 1.81235i 0.219203 0.189940i −0.527349 2.77883i 2.20803 + 2.03091i −2.63667 0.467505i
35.18 −1.20136 + 0.746150i −0.910347 + 1.47352i 0.886521 1.79279i −3.47534 + 1.02045i −0.00581645 2.44948i −3.08702 + 2.67492i 0.272658 + 2.81525i −1.34254 2.68283i 3.41372 3.81905i
35.19 −1.17778 + 0.782832i 0.458332 + 1.67031i 0.774349 1.84401i −3.47534 + 1.02045i −1.84739 1.60847i 3.08702 2.67492i 0.531536 + 2.77803i −2.57986 + 1.53111i 3.29436 3.92248i
35.20 −1.06473 + 0.930785i −0.424251 1.67929i 0.267280 1.98206i −0.611012 + 0.179409i 2.01477 + 1.39309i 1.41902 1.22959i 1.56029 + 2.35913i −2.64002 + 1.42488i 0.483568 0.759742i
See next 80 embeddings (of 920 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 515.92 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
23.c even 11 1 inner
24.f even 2 1 inner
69.h odd 22 1 inner
184.k odd 22 1 inner
552.x even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.x.a 920
3.b odd 2 1 inner 552.2.x.a 920
8.d odd 2 1 inner 552.2.x.a 920
23.c even 11 1 inner 552.2.x.a 920
24.f even 2 1 inner 552.2.x.a 920
69.h odd 22 1 inner 552.2.x.a 920
184.k odd 22 1 inner 552.2.x.a 920
552.x even 22 1 inner 552.2.x.a 920

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.x.a 920 1.a even 1 1 trivial
552.2.x.a 920 3.b odd 2 1 inner
552.2.x.a 920 8.d odd 2 1 inner
552.2.x.a 920 23.c even 11 1 inner
552.2.x.a 920 24.f even 2 1 inner
552.2.x.a 920 69.h odd 22 1 inner
552.2.x.a 920 184.k odd 22 1 inner
552.2.x.a 920 552.x even 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(552, [\chi])$$.