Properties

Label 21.79
Level $21$
Weight $0$
Character 21.1
Symmetry even
\(R\) 8.842904
Fricke sign $+1$

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Maass form invariants

Level: \( 21 = 3 \cdot 7 \)
Weight: \( 0 \)
Character: 21.1
Symmetry: even
Fricke sign: $+1$
Spectral parameter: \(8.84290412194447989199750287474 \pm 2 \cdot 10^{-9}\)

Maass form coefficients

The coefficients here are shown to at most $8$ digits of precision. Full precision coefficients are available in the downloads.

\(a_{1}= +1 \) \(a_{2}= +0.33735720 \pm 5.1 \cdot 10^{-8} \) \(a_{3}= +0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= -0.88619012 \pm 4.9 \cdot 10^{-8} \) \(a_{5}= +1.83774686 \pm 4.7 \cdot 10^{-8} \) \(a_{6}= +0.19477327 \pm 6.2 \cdot 10^{-8} \)
\(a_{7}= +0.37796447 \pm 1.0 \cdot 10^{-8} \) \(a_{8}= -0.63631982 \pm 3.9 \cdot 10^{-8} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= +0.61997714 \pm 5.5 \cdot 10^{-8} \) \(a_{11}= +1.17992931 \pm 3.8 \cdot 10^{-8} \) \(a_{12}= -0.51164210 \pm 6.0 \cdot 10^{-8} \)
\(a_{13}= -0.82800991 \pm 3.6 \cdot 10^{-8} \) \(a_{14}= +0.12750904 \pm 6.2 \cdot 10^{-8} \) \(a_{15}= +1.06102364 \pm 5.7 \cdot 10^{-8} \)
\(a_{16}= +0.67152304 \pm 4.0 \cdot 10^{-8} \) \(a_{17}= +0.38091271 \pm 4.1 \cdot 10^{-8} \) \(a_{18}= +0.11245240 \pm 6.2 \cdot 10^{-8} \)
\(a_{19}= -0.59621833 \pm 3.1 \cdot 10^{-8} \) \(a_{20}= -1.62859310 \pm 5.1 \cdot 10^{-8} \) \(a_{21}= +0.21821789 \pm 1.0 \cdot 10^{-8} \)
\(a_{22}= +0.39805765 \pm 4.6 \cdot 10^{-8} \) \(a_{23}= -0.65673033 \pm 4.5 \cdot 10^{-8} \) \(a_{24}= -0.36737942 \pm 5.0 \cdot 10^{-8} \)
\(a_{25}= +2.37731351 \pm 4.0 \cdot 10^{-8} \) \(a_{26}= -0.27933511 \pm 3.9 \cdot 10^{-8} \) \(a_{27}= +0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= -0.33494838 \pm 6.0 \cdot 10^{-8} \) \(a_{29}= -0.39657525 \pm 3.9 \cdot 10^{-8} \) \(a_{30}= +0.35794397 \pm 1.0 \cdot 10^{-7} \)
\(a_{31}= +1.36187498 \pm 3.4 \cdot 10^{-8} \) \(a_{32}= +0.86286296 \pm 4.6 \cdot 10^{-8} \) \(a_{33}= +0.68123251 \pm 4.9 \cdot 10^{-8} \)
\(a_{34}= +0.12850365 \pm 5.2 \cdot 10^{-8} \) \(a_{35}= +0.69460302 \pm 5.7 \cdot 10^{-8} \) \(a_{36}= -0.29539671 \pm 6.0 \cdot 10^{-8} \)
\(a_{37}= +0.92274722 \pm 3.6 \cdot 10^{-8} \) \(a_{38}= -0.20113855 \pm 4.4 \cdot 10^{-8} \) \(a_{39}= -0.47805175 \pm 4.6 \cdot 10^{-8} \)
\(a_{40}= -1.16939475 \pm 4.0 \cdot 10^{-8} \) \(a_{41}= +1.40806085 \pm 3.8 \cdot 10^{-8} \) \(a_{42}= +0.07361738 \pm 6.2 \cdot 10^{-8} \)
\(a_{43}= +1.57064151 \pm 3.9 \cdot 10^{-8} \) \(a_{44}= -1.04564170 \pm 4.2 \cdot 10^{-8} \) \(a_{45}= +0.61258229 \pm 5.7 \cdot 10^{-8} \)
\(a_{46}= -0.22155271 \pm 6.5 \cdot 10^{-8} \) \(a_{47}= +0.15755537 \pm 3.9 \cdot 10^{-8} \) \(a_{48}= +0.38770401 \pm 5.0 \cdot 10^{-8} \)
\(a_{49}= +0.14285714 \pm 1.5 \cdot 10^{-7} \) \(a_{50}= +0.80200383 \pm 4.5 \cdot 10^{-8} \) \(a_{51}= +0.21992006 \pm 5.2 \cdot 10^{-8} \)
\(a_{52}= +0.73377420 \pm 3.9 \cdot 10^{-8} \) \(a_{53}= -1.81235988 \pm 2.4 \cdot 10^{-8} \) \(a_{54}= +0.06492442 \pm 6.2 \cdot 10^{-8} \)
\(a_{55}= +2.16841139 \pm 4.5 \cdot 10^{-8} \) \(a_{56}= -0.24050629 \pm 5.0 \cdot 10^{-8} \) \(a_{57}= -0.34422681 \pm 4.1 \cdot 10^{-8} \)
\(a_{58}= -0.13378752 \pm 5.0 \cdot 10^{-8} \) \(a_{59}= +0.61690625 \pm 4.2 \cdot 10^{-8} \) \(a_{60}= -0.94026867 \pm 1.0 \cdot 10^{-7} \)

Displaying $a_n$ with $n$ up to: 60 180 1000