# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.bh (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$648$$ Relative dimension: $$54$$ over $$\Q(\zeta_{42})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$648 q - 15 q^{2} - 14 q^{3} - 57 q^{4} - 21 q^{5} + 14 q^{6} - 5 q^{7} - 20 q^{9} - 28 q^{10} - 15 q^{11} + 21 q^{12} - 7 q^{13} - 114 q^{14} - 10 q^{15} + 39 q^{16} - 18 q^{18} - 21 q^{20} + 10 q^{21} + 3 q^{22} + 30 q^{23} - 14 q^{24} + 41 q^{25} + 7 q^{27} - 20 q^{28} + 75 q^{29} - 70 q^{30} - 39 q^{32} - 14 q^{33} - 7 q^{34} - 128 q^{36} - 10 q^{37} + 21 q^{38} - 36 q^{39} - 7 q^{40} - 21 q^{41} + 104 q^{42} + 3 q^{43} - 35 q^{45} - 72 q^{46} - 147 q^{47} - 13 q^{49} - 18 q^{50} + 22 q^{51} - 35 q^{52} - 14 q^{54} - 112 q^{55} - 63 q^{56} - 16 q^{57} + 33 q^{58} - 21 q^{59} - 90 q^{60} - 56 q^{61} - 38 q^{63} + 52 q^{64} + 27 q^{65} - 42 q^{66} - 26 q^{67} - 182 q^{69} - 25 q^{70} + 24 q^{72} - 28 q^{73} + 33 q^{74} - 14 q^{75} + 21 q^{76} + 3 q^{77} + 90 q^{78} - 2 q^{79} + 56 q^{81} - 28 q^{82} - 21 q^{83} + 116 q^{84} + 5 q^{85} - 123 q^{86} - 70 q^{87} - 41 q^{88} - 224 q^{90} - 4 q^{91} - 225 q^{92} + 112 q^{93} - 7 q^{94} - 12 q^{95} - 371 q^{96} + 224 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
20.1 −1.54230 2.26214i −1.06685 1.36449i −2.00790 + 5.11604i 2.32352 2.15591i −1.44125 + 4.51782i 2.26676 1.36447i 9.33151 2.12986i −0.723646 + 2.91141i −8.46053 1.93106i
20.2 −1.53439 2.25054i 1.47003 + 0.915972i −1.97989 + 5.04466i −0.202216 + 0.187629i −0.194174 4.71382i −1.68008 2.04385i 9.08004 2.07246i 1.32199 + 2.69302i 0.732545 + 0.167199i
20.3 −1.48443 2.17726i 1.22179 1.22769i −1.80625 + 4.60225i −2.33048 + 2.16237i −4.48667 0.837728i 2.63962 + 0.180026i 7.56341 1.72630i −0.0144623 2.99997i 8.16748 + 1.86417i
20.4 −1.41329 2.07291i −0.175117 + 1.72318i −1.56890 + 3.99749i −1.66577 + 1.54561i 3.81948 2.07234i 0.387642 + 2.61720i 5.61184 1.28087i −2.93867 0.603513i 5.55812 + 1.26861i
20.5 −1.35898 1.99325i −1.18305 + 1.26507i −1.39555 + 3.55581i −0.596575 + 0.553541i 4.12933 + 0.638906i 0.195045 2.63855i 4.28023 0.976936i −0.200808 2.99327i 1.91408 + 0.436875i
20.6 −1.33942 1.96457i −1.60889 + 0.641453i −1.33480 + 3.40102i 2.36654 2.19583i 3.41516 + 2.30161i −2.59454 + 0.518026i 3.83317 0.874895i 2.17708 2.06406i −7.48364 1.70809i
20.7 −1.31233 1.92483i 1.51789 + 0.834281i −1.25210 + 3.19029i 2.00895 1.86404i −0.386114 4.01653i −0.201552 + 2.63806i 3.24148 0.739847i 1.60795 + 2.53268i −6.22437 1.42067i
20.8 −1.29533 1.89990i −0.181065 1.72256i −1.20106 + 3.06026i −0.929807 + 0.862735i −3.03816 + 2.57529i −2.62877 + 0.299281i 2.88636 0.658794i −2.93443 + 0.623789i 2.84352 + 0.649015i
20.9 −1.16564 1.70968i −1.66545 0.475684i −0.833613 + 2.12401i −2.61384 + 2.42529i 1.12805 + 3.40187i 0.957095 2.46657i 0.568369 0.129727i 2.54745 + 1.58446i 7.19327 + 1.64182i
20.10 −1.09567 1.60705i 1.70741 0.291147i −0.651445 + 1.65985i 1.64275 1.52425i −2.33864 2.42489i 2.30011 1.30747i −0.411271 + 0.0938698i 2.83047 0.994211i −4.24947 0.969913i
20.11 −1.07603 1.57825i −1.16503 1.28168i −0.602346 + 1.53475i 0.0991137 0.0919641i −0.769190 + 3.21784i 1.56054 + 2.13652i −0.654172 + 0.149311i −0.285388 + 2.98639i −0.251792 0.0574699i
20.12 −1.01131 1.48331i 1.68226 0.412308i −0.446798 + 1.13842i −1.62492 + 1.50771i −2.31286 2.07835i −2.44442 1.01232i −1.36001 + 0.310414i 2.66000 1.38722i 3.87969 + 0.885515i
20.13 −0.979822 1.43714i −1.66301 + 0.484159i −0.374624 + 0.954527i 0.820873 0.761659i 2.32525 + 1.91558i 1.31110 + 2.29805i −1.65267 + 0.377211i 2.53118 1.61032i −1.89892 0.433415i
20.14 −0.961845 1.41077i 0.474232 1.66586i −0.334437 + 0.852132i 1.96011 1.81872i −2.80629 + 0.933272i −0.724718 2.54456i −1.80546 + 0.412084i −2.55021 1.58001i −4.45112 1.01594i
20.15 −0.866176 1.27045i 0.194311 + 1.72112i −0.133092 + 0.339112i 2.18164 2.02427i 2.01828 1.73765i −0.899701 2.48808i −2.45204 + 0.559663i −2.92449 + 0.668863i −4.46141 1.01829i
20.16 −0.813844 1.19369i 0.511765 + 1.65472i −0.0318715 + 0.0812073i 0.0488140 0.0452928i 1.55873 1.95757i −2.34525 + 1.22466i −2.69414 + 0.614919i −2.47619 + 1.69365i −0.0937925 0.0214075i
20.17 −0.808964 1.18653i 1.30536 + 1.13844i −0.0227537 + 0.0579756i −2.55074 + 2.36674i 0.294803 2.46980i 2.61161 + 0.423656i −2.71292 + 0.619207i 0.407924 + 2.97214i 4.87167 + 1.11193i
20.18 −0.620172 0.909626i −0.824177 + 1.52340i 0.287877 0.733498i −0.377097 + 0.349895i 1.89685 0.195075i 2.54802 0.712459i −2.99238 + 0.682992i −1.64146 2.51109i 0.552139 + 0.126022i
20.19 −0.598227 0.877437i −1.65613 + 0.507172i 0.318661 0.811934i −2.90237 + 2.69301i 1.43575 + 1.14975i −1.67397 + 2.04886i −2.97373 + 0.678735i 2.48555 1.67989i 4.09922 + 0.935620i
20.20 −0.558540 0.819228i 0.799969 1.53625i 0.371515 0.946605i 0.352468 0.327042i −1.70535 + 0.202698i 2.11315 + 1.59204i −2.91630 + 0.665627i −1.72010 2.45790i −0.464790 0.106085i
See next 80 embeddings (of 648 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.54 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.d odd 6 1 inner
49.f odd 14 1 inner
441.bh even 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.bh.a 648
9.d odd 6 1 inner 441.2.bh.a 648
49.f odd 14 1 inner 441.2.bh.a 648
441.bh even 42 1 inner 441.2.bh.a 648

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.bh.a 648 1.a even 1 1 trivial
441.2.bh.a 648 9.d odd 6 1 inner
441.2.bh.a 648 49.f odd 14 1 inner
441.2.bh.a 648 441.bh even 42 1 inner

## Hecke kernels

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