# Properties

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

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## Newspace parameters

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

 $$\operatorname{Tr}(f)(q) =$$ $$648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9} - 22 q^{10} - 18 q^{11} - 72 q^{12} - 4 q^{13} + 66 q^{14} - 10 q^{15} - 105 q^{16} - 9 q^{17} - 27 q^{18} - 36 q^{19} - 27 q^{20} - 11 q^{21} - 9 q^{22} - 27 q^{23} - 8 q^{24} + 38 q^{25} + 6 q^{26} - 29 q^{27} - 26 q^{28} + 3 q^{29} - 16 q^{30} - 21 q^{32} - 11 q^{33} - 13 q^{34} + 28 q^{36} - 13 q^{37} - 90 q^{38} - 15 q^{39} - 31 q^{40} - 27 q^{41} - 4 q^{42} - 9 q^{43} + 51 q^{44} - 11 q^{45} - 108 q^{46} + 75 q^{47} - 15 q^{48} - 13 q^{49} - 45 q^{50} - 38 q^{51} + 64 q^{52} - 12 q^{53} - 41 q^{54} + 14 q^{55} + 3 q^{56} - 7 q^{57} - 90 q^{58} + 15 q^{59} - 69 q^{60} - 56 q^{61} + 66 q^{62} + 13 q^{63} + 64 q^{64} - 21 q^{65} - 204 q^{66} - 26 q^{67} + 3 q^{68} + 58 q^{69} - 22 q^{70} - 63 q^{71} - 18 q^{72} - 22 q^{73} - 12 q^{74} + 118 q^{75} - 63 q^{76} - 69 q^{77} - 147 q^{78} - 2 q^{79} - 45 q^{80} + 29 q^{81} - 28 q^{82} - 51 q^{83} - 31 q^{84} - 10 q^{85} - 72 q^{86} - 67 q^{87} + 4 q^{88} + 132 q^{89} + 58 q^{90} - 13 q^{91} - 15 q^{92} + 217 q^{93} - 7 q^{94} - 21 q^{95} - 44 q^{96} + 3 q^{97} + 21 q^{98} - 148 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −2.16353 + 1.72536i 0.581018 1.63169i 1.25896 5.51585i 2.91211 + 1.98544i 1.55820 + 4.53267i −2.07232 + 1.64484i 4.39168 + 9.11943i −2.32484 1.89608i −9.72603 + 0.728865i
5.2 −2.15589 + 1.71927i −1.44111 0.960829i 1.24695 5.46324i −1.16954 0.797380i 4.75880 0.406215i 1.71430 2.01524i 4.31162 + 8.95316i 1.15362 + 2.76933i 3.89231 0.291689i
5.3 −2.01930 + 1.61034i 0.161619 + 1.72449i 1.03935 4.55368i 0.846045 + 0.576824i −3.10338 3.22201i −1.90950 1.83134i 2.99296 + 6.21494i −2.94776 + 0.557422i −2.63730 + 0.197639i
5.4 −1.98600 + 1.58378i 1.73017 + 0.0806248i 0.990793 4.34095i −3.40470 2.32129i −3.56382 + 2.58010i −2.11613 1.58809i 2.70311 + 5.61307i 2.98700 + 0.278990i 10.4382 0.782232i
5.5 −1.96009 + 1.56312i 1.60753 + 0.644869i 0.953564 4.17784i 0.688287 + 0.469266i −4.15890 + 1.24876i 1.84507 + 1.89623i 2.48585 + 5.16192i 2.16829 + 2.07329i −2.08262 + 0.156071i
5.6 −1.86842 + 1.49001i −1.41768 + 0.995078i 0.825803 3.61808i −1.88505 1.28520i 1.16614 3.97158i −0.922050 + 2.47988i 1.77425 + 3.68427i 1.01964 2.82141i 5.43702 0.407448i
5.7 −1.70698 + 1.36127i 0.921293 1.46670i 0.615683 2.69748i −1.79313 1.22254i 0.423952 + 3.75777i 2.16539 + 1.52022i 0.726444 + 1.50848i −1.30244 2.70253i 4.72505 0.354094i
5.8 −1.65706 + 1.32146i −1.64577 0.539842i 0.554552 2.42965i 0.208441 + 0.142112i 3.44053 1.28028i −2.12758 + 1.57270i 0.452565 + 0.939762i 2.41714 + 1.77691i −0.533196 + 0.0399575i
5.9 −1.53265 + 1.22225i −0.923591 + 1.46526i 0.410090 1.79672i −2.34879 1.60137i −0.375366 3.37459i 1.90530 1.83571i −0.133600 0.277423i −1.29396 2.70660i 5.55716 0.416451i
5.10 −1.52262 + 1.21425i 0.820482 + 1.52539i 0.398931 1.74783i 0.733166 + 0.499864i −3.10149 1.32632i 2.50828 0.841742i −0.175099 0.363598i −1.65362 + 2.50311i −1.72330 + 0.129143i
5.11 −1.50680 + 1.20163i −0.277954 1.70960i 0.381481 1.67138i 0.918605 + 0.626294i 2.47313 + 2.24203i 0.418961 2.61237i −0.238856 0.495990i −2.84548 + 0.950381i −2.13673 + 0.160126i
5.12 −1.50264 + 1.19832i −1.36206 1.06995i 0.376930 1.65144i 2.30957 + 1.57464i 3.32883 0.0244174i 1.77434 + 1.96258i −0.255253 0.530039i 0.710396 + 2.91468i −5.35737 + 0.401480i
5.13 −1.47430 + 1.17571i 1.71213 0.261949i 0.346209 1.51684i 3.14585 + 2.14480i −2.21621 + 2.39916i 0.361407 2.62095i −0.363391 0.754589i 2.86277 0.896980i −7.15958 + 0.536537i
5.14 −1.33541 + 1.06496i 1.50168 0.863111i 0.204153 0.894452i 0.337281 + 0.229955i −1.08619 + 2.75183i −2.64519 + 0.0547315i −0.802272 1.66593i 1.51008 2.59223i −0.695301 + 0.0521056i
5.15 −1.24435 + 0.992336i −0.323588 + 1.70156i 0.118634 0.519770i 3.37511 + 2.30111i −1.28586 2.43844i −0.700800 + 2.55125i −1.01296 2.10343i −2.79058 1.10121i −6.48329 + 0.485855i
5.16 −1.14829 + 0.915734i −0.441504 1.67484i 0.0349690 0.153209i −1.40572 0.958401i 2.04068 + 1.51890i −2.52036 0.804862i −1.17437 2.43860i −2.61015 + 1.47889i 2.49182 0.186736i
5.17 −1.08970 + 0.869006i 0.851533 + 1.50827i −0.0127693 + 0.0559462i −2.47373 1.68656i −2.23861 0.903578i −0.888042 + 2.49226i −1.24418 2.58356i −1.54978 + 2.56869i 4.16125 0.311843i
5.18 −0.920093 + 0.733749i −1.68844 + 0.386212i −0.136860 + 0.599621i 1.62477 + 1.10775i 1.27014 1.59424i 1.62874 2.08500i −1.33527 2.77272i 2.70168 1.30419i −2.30775 + 0.172942i
5.19 −0.781758 + 0.623431i −1.03674 1.38750i −0.222563 + 0.975111i −3.13042 2.13429i 1.67549 + 0.438356i 2.20194 + 1.46679i −1.30161 2.70282i −0.850336 + 2.87697i 3.77782 0.283108i
5.20 −0.744514 + 0.593730i −1.04233 + 1.38331i −0.243256 + 1.06577i 0.397639 + 0.271106i −0.0452820 1.64876i −2.38053 1.15459i −1.27802 2.65384i −0.827090 2.88373i −0.457012 + 0.0342483i
See next 80 embeddings (of 648 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 416.54 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.bn 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.bn.a yes 648
9.d odd 6 1 441.2.bd.a 648
49.h odd 42 1 441.2.bd.a 648
441.bn even 42 1 inner 441.2.bn.a yes 648

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.bd.a 648 9.d odd 6 1
441.2.bd.a 648 49.h odd 42 1
441.2.bn.a yes 648 1.a even 1 1 trivial
441.2.bn.a yes 648 441.bn even 42 1 inner

## Hecke kernels

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