Properties

 Label 441.2.ba.a Level $441$ Weight $2$ Character orbit 441.ba Analytic conductor $3.521$ Analytic rank $0$ Dimension $648$ CM no Inner twists $4$

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Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.ba (of order $$21$$, degree $$12$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$648$$ Relative dimension: $$54$$ over $$\Q(\zeta_{21})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$648q - 5q^{2} - 10q^{3} + 47q^{4} - 9q^{5} - 34q^{6} - 7q^{7} - 28q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$648q - 5q^{2} - 10q^{3} + 47q^{4} - 9q^{5} - 34q^{6} - 7q^{7} - 28q^{8} - 28q^{10} - 5q^{11} + 5q^{12} - 7q^{13} - 38q^{14} - 22q^{15} + 47q^{16} - 16q^{17} - 46q^{18} - 44q^{19} - 29q^{20} - 12q^{21} - 13q^{22} - 20q^{23} - 2q^{24} + 41q^{25} - 20q^{26} - 13q^{27} - 28q^{28} - 35q^{29} - 6q^{30} - 20q^{31} - 25q^{32} - 22q^{33} - q^{34} - 44q^{35} - 28q^{36} - 30q^{37} - 7q^{38} - 8q^{39} + 5q^{40} - 29q^{41} - 104q^{42} - 13q^{43} - 88q^{44} + 63q^{45} + 32q^{46} - 55q^{47} - 14q^{48} - q^{49} + 6q^{50} - 18q^{51} + 3q^{52} - 136q^{53} + 30q^{54} - 100q^{55} + 145q^{56} - 96q^{57} + 17q^{58} - 19q^{59} + 118q^{60} + 42q^{61} + 96q^{62} - 6q^{63} - 124q^{64} - 11q^{65} + 120q^{66} - 26q^{67} + 166q^{68} - 134q^{69} - 7q^{70} - 22q^{71} + 32q^{72} + 8q^{73} - 45q^{74} - 36q^{75} - 41q^{76} + q^{77} - 18q^{78} - 26q^{79} - 440q^{80} + 64q^{81} - 28q^{82} + 61q^{83} - 86q^{84} + 5q^{85} + 15q^{86} + 34q^{87} - q^{88} + 22q^{89} + 96q^{90} - 16q^{91} - 43q^{92} - 110q^{93} - q^{94} - 38q^{95} - 165q^{96} - 14q^{97} - 144q^{98} - 196q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1 −2.28558 1.55828i −0.781588 1.54568i 2.06496 + 5.26143i −0.806726 0.748533i −0.622222 + 4.75071i 1.94410 + 1.79457i 2.24807 9.84945i −1.77824 + 2.41617i 0.677414 + 2.96794i
22.2 −2.20796 1.50536i −1.57686 + 0.716598i 1.87830 + 4.78582i 2.79913 + 2.59721i 4.56039 + 0.791525i −2.53111 0.770385i 1.86790 8.18382i 1.97297 2.25995i −2.27063 9.94826i
22.3 −2.13496 1.45559i 1.68474 + 0.402062i 1.70863 + 4.35352i 2.61229 + 2.42385i −3.01161 3.31068i 2.64469 + 0.0747901i 1.53912 6.74332i 2.67669 + 1.35474i −2.04900 8.97724i
22.4 −2.12732 1.45038i 1.48498 + 0.891527i 1.69120 + 4.30910i −2.50301 2.32245i −1.86598 4.05036i −2.17833 + 1.50162i 1.50627 6.59941i 1.41036 + 2.64781i 1.95625 + 8.57090i
22.5 −2.11562 1.44241i −0.314751 + 1.70321i 1.66464 + 4.24143i −1.01272 0.939663i 3.12262 3.14936i 1.62237 2.08995i 1.45657 6.38167i −2.80186 1.07218i 0.787149 + 3.44872i
22.6 −2.08565 1.42197i 1.06450 1.36632i 1.59724 + 4.06970i 0.525543 + 0.487633i −4.16304 + 1.33598i −2.64570 0.0162007i 1.33231 5.83723i −0.733680 2.90890i −0.402698 1.76434i
22.7 −1.80236 1.22883i −1.63261 + 0.578434i 1.00781 + 2.56785i −1.67523 1.55438i 3.65335 + 0.963652i −0.0954394 + 2.64403i 0.368203 1.61320i 2.33083 1.88871i 1.10930 + 4.86014i
22.8 −1.76857 1.20579i −1.66628 0.472785i 0.943225 + 2.40330i −0.0377034 0.0349836i 2.37684 + 2.84533i 1.13677 2.38909i 0.277100 1.21405i 2.55295 + 1.57558i 0.0244981 + 0.107333i
22.9 −1.71062 1.16628i −0.227483 1.71705i 0.835328 + 2.12838i 1.88227 + 1.74649i −1.61342 + 3.20252i 1.73079 2.00109i 0.131960 0.578153i −2.89650 + 0.781197i −1.18295 5.18284i
22.10 −1.58020 1.07736i −0.0140747 1.73199i 0.605641 + 1.54315i −3.06138 2.84054i −1.84374 + 2.75206i −1.13241 2.39116i −0.145657 + 0.638167i −2.99960 + 0.0487547i 1.77730 + 7.78684i
22.11 −1.53495 1.04651i 1.72901 0.102560i 0.530198 + 1.35092i −0.256118 0.237642i −2.76127 1.65200i −0.648127 2.56514i −0.226850 + 0.993895i 2.97896 0.354656i 0.144432 + 0.632798i
22.12 −1.50607 1.02682i −0.181074 + 1.72256i 0.483202 + 1.23118i 0.989195 + 0.917838i 2.04147 2.40836i −2.54327 + 0.729243i −0.274759 + 1.20380i −2.93442 0.623821i −0.547340 2.39805i
22.13 −1.48693 1.01377i 1.13536 1.30804i 0.452539 + 1.15305i 1.39969 + 1.29873i −3.01424 + 0.793970i −0.229184 + 2.63581i −0.304877 + 1.33575i −0.421935 2.97018i −0.764633 3.35008i
22.14 −1.31403 0.895892i 1.71871 + 0.214588i 0.193375 + 0.492712i −1.33434 1.23808i −2.06619 1.82175i 2.50568 + 0.849447i −0.520469 + 2.28032i 2.90790 + 0.737628i 0.644172 + 2.82230i
22.15 −1.21486 0.828276i −1.14380 + 1.30067i 0.0591562 + 0.150728i 2.23394 + 2.07279i 2.46686 0.632745i 2.42421 + 1.05980i −0.601388 + 2.63485i −0.383462 2.97539i −0.997073 4.36846i
22.16 −1.20704 0.822949i −1.50963 0.849130i 0.0490286 + 0.124923i −0.415450 0.385481i 1.12340 + 2.26728i −2.62355 + 0.341999i −0.606532 + 2.65739i 1.55796 + 2.56374i 0.184235 + 0.807187i
22.17 −1.11972 0.763412i 0.735246 + 1.56825i −0.0597084 0.152135i −1.91052 1.77271i 0.373953 2.31730i 2.60358 + 0.470493i −0.652406 + 2.85838i −1.91883 + 2.30610i 0.785945 + 3.44345i
22.18 −0.946246 0.645140i 1.51934 + 0.831632i −0.251505 0.640825i 1.33307 + 1.23691i −0.901149 1.76711i −1.78159 + 1.95600i −0.685118 + 3.00170i 1.61678 + 2.52706i −0.463433 2.03043i
22.19 −0.912794 0.622332i −0.609005 + 1.62145i −0.284787 0.725626i −1.69424 1.57203i 1.56498 1.10105i −1.84008 1.90108i −0.683292 + 2.99370i −2.25823 1.97495i 0.568171 + 2.48932i
22.20 −0.819051 0.558419i −0.438134 1.67572i −0.371670 0.947000i −1.06857 0.991491i −0.576900 + 1.61716i 0.639861 + 2.56721i −0.665577 + 2.91608i −2.61608 + 1.46838i 0.321548 + 1.40879i
See next 80 embeddings (of 648 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 421.54 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
49.e even 7 1 inner
441.ba even 21 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.ba.a 648
9.c even 3 1 inner 441.2.ba.a 648
49.e even 7 1 inner 441.2.ba.a 648
441.ba even 21 1 inner 441.2.ba.a 648

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.ba.a 648 1.a even 1 1 trivial
441.2.ba.a 648 9.c even 3 1 inner
441.2.ba.a 648 49.e even 7 1 inner
441.2.ba.a 648 441.ba even 21 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(441, [\chi])$$.