# Properties

 Label 441.2.bd.a Level $441$ Weight $2$ Character orbit 441.bd 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.bd (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.

 $$\operatorname{Tr}(f)(q) =$$ $$648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9} - 22 q^{10} - 21 q^{11} + 24 q^{12} - 10 q^{13} - 24 q^{14} - 10 q^{15} + 42 q^{16} + 9 q^{17} + 12 q^{18} - 36 q^{19} - 15 q^{20} - 38 q^{21} - 9 q^{22} - 84 q^{23} - 38 q^{24} - 97 q^{25} - 6 q^{26} - 20 q^{27} - 26 q^{28} - 123 q^{29} - q^{30} - 6 q^{31} - 30 q^{32} - 8 q^{33} - q^{34} + 70 q^{36} - 13 q^{37} - 75 q^{38} - 24 q^{39} - 7 q^{40} - 15 q^{41} - 133 q^{42} - 9 q^{43} - 51 q^{44} - 41 q^{45} - 108 q^{46} - 36 q^{47} + 15 q^{48} + 5 q^{49} - 45 q^{50} + 10 q^{51} - 35 q^{52} + 12 q^{53} - 50 q^{54} + 14 q^{55} + 33 q^{56} - 7 q^{57} + 33 q^{58} - 3 q^{59} + 180 q^{60} + 55 q^{61} - 66 q^{62} - 20 q^{63} + 64 q^{64} + 3 q^{65} + 285 q^{66} + 13 q^{67} + 6 q^{68} + 82 q^{69} + 11 q^{70} + 63 q^{71} - 21 q^{72} - 22 q^{73} - 21 q^{74} - 107 q^{75} + 21 q^{76} - 69 q^{77} + 63 q^{78} + q^{79} + 45 q^{80} - 211 q^{81} - 28 q^{82} + 9 q^{83} + 239 q^{84} - 10 q^{85} - 21 q^{86} - 109 q^{87} - 29 q^{88} - 132 q^{89} - 275 q^{90} - 13 q^{91} + 195 q^{92} + 19 q^{93} - 4 q^{94} - 129 q^{95} - 5 q^{96} - 3 q^{97} - 21 q^{98} - 148 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 −1.83532 1.97800i 0.292304 1.70721i −0.394641 + 5.26612i −0.817866 1.02557i −3.91333 + 2.55509i 0.0523706 + 2.64523i 6.92143 5.51966i −2.82912 0.998047i −0.527539 + 3.49999i
47.2 −1.83191 1.97433i −1.17858 1.26924i −0.392623 + 5.23919i 1.47204 + 1.84588i −0.346843 + 4.65202i −1.31601 2.29524i 6.85171 5.46406i −0.221916 + 2.99178i 0.947728 6.28776i
47.3 −1.79697 1.93667i −1.59909 + 0.665506i −0.372140 + 4.96586i −2.12889 2.66954i 4.16239 + 1.90103i 2.41582 1.07881i 6.15488 4.90835i 2.11420 2.12841i −1.34448 + 8.92003i
47.4 −1.75148 1.88764i 0.0116260 + 1.73201i −0.346064 + 4.61791i −0.695350 0.871941i 3.24905 3.05552i −2.60184 + 0.480046i 5.29657 4.22388i −2.99973 + 0.0402729i −0.428024 + 2.83975i
47.5 −1.67227 1.80228i −0.998913 + 1.41498i −0.302261 + 4.03339i 2.61954 + 3.28480i 4.22064 0.565913i 2.36845 + 1.17916i 3.93034 3.13434i −1.00435 2.82689i 1.53954 10.2142i
47.6 −1.59981 1.72418i 1.65781 + 0.501676i −0.263960 + 3.52230i 2.05109 + 2.57199i −1.78719 3.66094i −2.34210 + 1.23069i 2.81755 2.24692i 2.49664 + 1.66336i 1.15322 7.65114i
47.7 −1.59975 1.72412i 1.51926 0.831777i −0.263930 + 3.52190i 0.503410 + 0.631256i −3.86451 1.28875i 0.617740 2.57262i 2.81669 2.24623i 1.61629 2.52737i 0.283031 1.87779i
47.8 −1.58104 1.70396i 1.73188 0.0242085i −0.254319 + 3.39365i −1.54809 1.94124i −2.77943 2.91278i 1.62028 + 2.09157i 2.55004 2.03359i 2.99883 0.0838526i −0.860202 + 5.70707i
47.9 −1.34426 1.44876i −1.69107 + 0.374554i −0.142431 + 1.90061i 0.213240 + 0.267394i 2.81587 + 1.94646i −2.30163 1.30480i −0.145343 + 0.115907i 2.71942 1.26679i 0.100742 0.668380i
47.10 −1.29221 1.39267i −1.37046 1.05916i −0.120265 + 1.60482i 0.149465 + 0.187424i 0.295858 + 3.27726i 2.64547 0.0386573i −0.580288 + 0.462764i 0.756341 + 2.90309i 0.0678787 0.450346i
47.11 −1.29218 1.39264i 1.30169 + 1.14263i −0.120254 + 1.60468i −2.30579 2.89137i −0.0907452 3.28927i −0.473280 2.60308i −0.580487 + 0.462923i 0.388790 + 2.97470i −1.04714 + 6.94730i
47.12 −1.22870 1.32422i 0.168894 + 1.72380i −0.0944028 + 1.25972i 0.684410 + 0.858223i 2.07517 2.34167i 1.06533 2.42179i −1.04054 + 0.829802i −2.94295 + 0.582277i 0.295544 1.96080i
47.13 −1.22764 1.32308i −1.33805 1.09983i −0.0939836 + 1.25412i −2.40505 3.01584i 0.187488 + 3.12054i −1.92992 + 1.80981i −1.04756 + 0.835402i 0.580768 + 2.94325i −1.03766 + 6.88444i
47.14 −1.18529 1.27744i 0.137429 1.72659i −0.0774791 + 1.03389i 2.21900 + 2.78254i −2.36851 + 1.87096i −1.21027 + 2.35271i −1.31233 + 1.04655i −2.96223 0.474566i 0.924366 6.13277i
47.15 −1.08388 1.16814i 0.336533 1.69904i −0.0403058 + 0.537843i 0.551923 + 0.692089i −2.34949 + 1.44844i 2.64276 + 0.125839i −1.81979 + 1.45123i −2.77349 1.14357i 0.210242 1.39487i
47.16 −0.950574 1.02447i −0.660575 + 1.60114i 0.00350198 0.0467306i −1.62704 2.04025i 2.26825 0.845256i 1.86948 + 1.87217i −2.23650 + 1.78355i −2.12728 2.11534i −0.543558 + 3.60627i
47.17 −0.932976 1.00551i −1.41782 + 0.994877i 0.00885622 0.118178i 0.700700 + 0.878650i 2.32315 + 0.497436i −1.28096 + 2.31498i −2.27193 + 1.81180i 1.02044 2.82112i 0.229754 1.52432i
47.18 −0.742532 0.800259i 1.56641 + 0.739164i 0.0603993 0.805973i 1.83554 + 2.30169i −0.571585 1.80238i 2.47412 0.937401i −2.39686 + 1.91143i 1.90727 + 2.31567i 0.479004 3.17799i
47.19 −0.740834 0.798429i 1.60616 0.648268i 0.0608061 0.811401i −1.45215 1.82094i −1.70749 0.802146i −1.33635 + 2.28346i −2.39601 + 1.91076i 2.15950 2.08244i −0.378088 + 2.50845i
47.20 −0.578520 0.623496i −0.623215 1.61605i 0.0953980 1.27300i 0.0546144 + 0.0684843i −0.647056 + 1.32349i −1.63124 2.08304i −2.17887 + 1.73759i −2.22320 + 2.01429i 0.0111042 0.0736714i
See next 80 embeddings (of 648 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 437.54 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.bd 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.bd.a 648
9.d odd 6 1 441.2.bn.a yes 648
49.h odd 42 1 441.2.bn.a yes 648
441.bd even 42 1 inner 441.2.bd.a 648

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

## Hecke kernels

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