# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.z (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.

 $$\operatorname{Tr}(f)(q) =$$ $$648q - 8q^{2} - 13q^{3} + 44q^{4} + 3q^{5} + 2q^{6} - 7q^{7} - 16q^{8} - 39q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$648q - 8q^{2} - 13q^{3} + 44q^{4} + 3q^{5} + 2q^{6} - 7q^{7} - 16q^{8} - 39q^{9} - 22q^{10} - 5q^{11} + 2q^{12} - 4q^{13} + 52q^{14} - 4q^{15} + 38q^{16} - 37q^{17} - 16q^{18} - 14q^{19} - 11q^{20} - 6q^{21} - q^{22} - 20q^{23} - 26q^{24} - 97q^{25} - 44q^{26} + 14q^{27} - 22q^{28} - 53q^{29} - 9q^{30} + 10q^{31} - 10q^{32} + 2q^{33} - 7q^{34} - 32q^{35} - 22q^{36} - 39q^{37} + 59q^{38} - 26q^{39} - 19q^{40} - 17q^{41} - 83q^{42} - q^{43} - 31q^{44} + 81q^{45} + 56q^{46} + 50q^{47} - 23q^{48} - 19q^{49} - 21q^{50} - 30q^{51} - 9q^{52} + 50q^{53} + 48q^{54} + 20q^{55} - 239q^{56} + 33q^{57} + 17q^{58} - 37q^{59} - 152q^{60} - 105q^{61} - 30q^{62} + 6q^{63} - 88q^{64} - 5q^{65} - 99q^{66} + 13q^{67} - 290q^{68} + 64q^{69} + 17q^{70} - 19q^{71} + 11q^{72} - 40q^{73} - 57q^{74} + 69q^{75} - 41q^{76} + 19q^{77} + 81q^{78} + 13q^{79} + 163q^{80} - 179q^{81} - 28q^{82} + 73q^{83} - 341q^{84} - 10q^{85} + 15q^{86} + 13q^{87} - 13q^{88} - 104q^{89} + 171q^{90} - 49q^{91} - 91q^{92} - 275q^{93} - 4q^{94} - 29q^{95} + 231q^{96} - 11q^{97} - 33q^{98} + 62q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −0.209537 + 2.79608i −0.318381 1.70254i −5.79651 0.873684i −0.429093 + 1.87998i 4.82715 0.533474i 0.156954 2.64109i 2.40961 10.5572i −2.79727 + 1.08411i −5.16666 1.59370i
4.2 −0.193400 + 2.58074i 1.69682 + 0.347567i −4.64515 0.700143i 0.742208 3.25183i −1.22514 + 4.31183i 2.07445 1.64215i 1.55350 6.80632i 2.75839 + 1.17952i 8.24857 + 2.54435i
4.3 −0.192530 + 2.56913i 1.55341 0.766101i −4.58569 0.691182i −0.144697 + 0.633958i 1.66913 + 4.13841i −1.19412 + 2.36095i 1.51204 6.62468i 1.82618 2.38014i −1.60086 0.493800i
4.4 −0.187690 + 2.50455i −1.73007 + 0.0827751i −4.25986 0.642071i 0.223906 0.980997i 0.117403 4.34858i −2.61535 + 0.399936i 1.28987 5.65131i 2.98630 0.286414i 2.41493 + 0.744907i
4.5 −0.173812 + 2.31936i 0.977861 + 1.42961i −3.37158 0.508183i −0.186808 + 0.818458i −3.48575 + 2.01953i −2.56874 0.633687i 0.729575 3.19648i −1.08757 + 2.79592i −1.86583 0.575533i
4.6 −0.173112 + 2.31002i −0.732646 + 1.56947i −3.32857 0.501701i −0.445172 + 1.95043i −3.49867 1.96412i 0.693731 2.55318i 0.704215 3.08537i −1.92646 2.29973i −4.42846 1.36600i
4.7 −0.168873 + 2.25346i −1.65465 + 0.511977i −3.07188 0.463011i 0.626084 2.74306i −0.874290 3.81515i 2.34287 1.22921i 0.556439 2.43792i 2.47576 1.69429i 6.07562 + 1.87408i
4.8 −0.163627 + 2.18345i −1.55326 0.766401i −2.76302 0.416458i −0.523133 + 2.29200i 1.92755 3.26607i 1.85748 + 1.88409i 0.386968 1.69542i 1.82526 + 2.38085i −4.91886 1.51727i
4.9 −0.162086 + 2.16289i 0.451822 1.67208i −2.67415 0.403063i 0.397052 1.73960i 3.54329 + 1.24826i 1.01797 + 2.44208i 0.339946 1.48940i −2.59171 1.51097i 3.69820 + 1.14074i
4.10 −0.152487 + 2.03479i −0.743876 1.56418i −2.13947 0.322473i 0.723985 3.17199i 3.29621 1.27512i −2.58955 0.542443i 0.0742997 0.325528i −1.89330 + 2.32711i 6.34394 + 1.95685i
4.11 −0.132809 + 1.77221i 1.41906 0.993117i −1.14543 0.172646i −0.788881 + 3.45631i 1.57155 + 2.64676i 2.57823 0.593925i −0.332830 + 1.45822i 1.02744 2.81858i −6.02055 1.85709i
4.12 −0.129339 + 1.72591i −1.02279 + 1.39782i −0.984361 0.148369i −0.845939 + 3.70630i −2.28022 1.94603i −1.47863 + 2.19400i −0.386868 + 1.69498i −0.907804 2.85935i −6.28731 1.93938i
4.13 −0.122748 + 1.63795i 0.324451 + 1.70139i −0.690165 0.104026i 0.354390 1.55268i −2.82663 + 0.322594i 2.24116 + 1.40613i −0.475897 + 2.08504i −2.78946 + 1.10404i 2.49973 + 0.771063i
4.14 −0.109094 + 1.45576i 1.61060 + 0.637154i −0.129671 0.0195447i −0.0506126 + 0.221748i −1.10325 + 2.27514i 1.03301 2.43575i −0.607092 + 2.65984i 2.18807 + 2.05240i −0.317290 0.0978711i
4.15 −0.107721 + 1.43743i 0.0199394 1.73194i −0.0769418 0.0115971i −0.762536 + 3.34089i 2.48739 + 0.215227i −2.60405 0.467908i −0.616552 + 2.70129i −2.99920 0.0690675i −4.72015 1.45597i
4.16 −0.105772 + 1.41143i 1.42810 0.980073i −0.00328183 0.000494657i 0.365770 1.60254i 1.23225 + 2.11932i −2.11011 1.59607i −0.628861 + 2.75522i 1.07891 2.79928i 2.22319 + 0.685763i
4.17 −0.101230 + 1.35082i 1.66963 + 0.460807i 0.163204 + 0.0245990i −0.157203 + 0.688750i −0.791482 + 2.20871i −1.05230 + 2.42748i −0.652605 + 2.85925i 2.57531 + 1.53875i −0.914461 0.282074i
4.18 −0.0832951 + 1.11150i −0.959106 + 1.44226i 0.749175 + 0.112920i 0.757564 3.31911i −1.52318 1.18618i −1.18524 2.36542i −0.683962 + 2.99663i −1.16023 2.76656i 3.62607 + 1.11850i
4.19 −0.0806131 + 1.07571i −1.11699 1.32376i 0.827015 + 0.124652i 0.478314 2.09563i 1.51402 1.09484i 2.41117 1.08917i −0.680835 + 2.98293i −0.504687 + 2.95724i 2.21572 + 0.683461i
4.20 −0.0535758 + 0.714919i 1.52049 0.829523i 1.46942 + 0.221480i 0.990551 4.33989i 0.511580 + 1.13147i 1.86250 + 1.87912i −0.556127 + 2.43655i 1.62378 2.52256i 3.04960 + 0.940676i
See next 80 embeddings (of 648 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 394.54 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
441.z 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.z.a yes 648
9.c even 3 1 441.2.y.a 648
49.g even 21 1 441.2.y.a 648
441.z even 21 1 inner 441.2.z.a yes 648

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.y.a 648 9.c even 3 1
441.2.y.a 648 49.g even 21 1
441.2.z.a yes 648 1.a even 1 1 trivial
441.2.z.a yes 648 441.z even 21 1 inner

## Hecke kernels

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