Properties

Label 21.76
Level $21$
Weight $0$
Character 21.1
Symmetry even
\(R\) 8.752370
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: \(8.75237080868404451593214088427 \pm 2 \cdot 10^{-9}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= -1.16278581 \pm 5.2 \cdot 10^{-8} \) \(a_{3}= -0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= +0.35207085 \pm 5.0 \cdot 10^{-8} \) \(a_{5}= +0.28199500 \pm 4.8 \cdot 10^{-8} \) \(a_{6}= +0.67133470 \pm 6.3 \cdot 10^{-8} \)
\(a_{7}= -0.37796447 \pm 1.0 \cdot 10^{-8} \) \(a_{8}= +0.75340283 \pm 4.0 \cdot 10^{-8} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= -0.32789978 \pm 5.6 \cdot 10^{-8} \) \(a_{11}= +1.34787761 \pm 3.9 \cdot 10^{-8} \) \(a_{12}= -0.20326820 \pm 6.1 \cdot 10^{-8} \)
\(a_{13}= -1.44366560 \pm 3.7 \cdot 10^{-8} \) \(a_{14}= +0.43949173 \pm 6.3 \cdot 10^{-8} \) \(a_{15}= -0.16280989 \pm 5.8 \cdot 10^{-8} \)
\(a_{16}= -1.22811697 \pm 4.0 \cdot 10^{-8} \) \(a_{17}= -1.78757535 \pm 4.2 \cdot 10^{-8} \) \(a_{18}= -0.38759527 \pm 6.3 \cdot 10^{-8} \)
\(a_{19}= +0.58243002 \pm 3.2 \cdot 10^{-8} \) \(a_{20}= +0.09928222 \pm 5.2 \cdot 10^{-8} \) \(a_{21}= +0.21821789 \pm 1.0 \cdot 10^{-8} \)
\(a_{22}= -1.56729297 \pm 4.6 \cdot 10^{-8} \) \(a_{23}= -0.95160994 \pm 4.6 \cdot 10^{-8} \) \(a_{24}= -0.43497733 \pm 5.1 \cdot 10^{-8} \)
\(a_{25}= -0.92047882 \pm 4.0 \cdot 10^{-8} \) \(a_{26}= +1.67867388 \pm 4.0 \cdot 10^{-8} \) \(a_{27}= -0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= -0.13307027 \pm 6.1 \cdot 10^{-8} \) \(a_{29}= +0.76321343 \pm 4.0 \cdot 10^{-8} \) \(a_{30}= +0.18931303 \pm 1.1 \cdot 10^{-7} \)
\(a_{31}= +1.86735199 \pm 3.4 \cdot 10^{-8} \) \(a_{32}= +0.67463415 \pm 4.7 \cdot 10^{-8} \) \(a_{33}= -0.77819750 \pm 4.9 \cdot 10^{-8} \)
\(a_{34}= +2.07856726 \pm 5.3 \cdot 10^{-8} \) \(a_{35}= -0.10658409 \pm 5.8 \cdot 10^{-8} \) \(a_{36}= +0.11735695 \pm 6.1 \cdot 10^{-8} \)
\(a_{37}= -0.80425651 \pm 3.7 \cdot 10^{-8} \) \(a_{38}= -0.67724137 \pm 4.5 \cdot 10^{-8} \) \(a_{39}= +0.83350072 \pm 4.7 \cdot 10^{-8} \)
\(a_{40}= +0.21245583 \pm 4.1 \cdot 10^{-8} \) \(a_{41}= -0.96472738 \pm 3.9 \cdot 10^{-8} \) \(a_{42}= -0.25374067 \pm 6.3 \cdot 10^{-8} \)
\(a_{43}= +0.35947710 \pm 4.0 \cdot 10^{-8} \) \(a_{44}= +0.47454841 \pm 4.3 \cdot 10^{-8} \) \(a_{45}= +0.09399833 \pm 5.8 \cdot 10^{-8} \)
\(a_{46}= +1.10651854 \pm 6.6 \cdot 10^{-8} \) \(a_{47}= +0.50393654 \pm 4.0 \cdot 10^{-8} \) \(a_{48}= +0.70905366 \pm 5.1 \cdot 10^{-8} \)
\(a_{49}= +0.14285714 \pm 1.5 \cdot 10^{-7} \) \(a_{50}= +1.07031971 \pm 4.6 \cdot 10^{-8} \) \(a_{51}= +1.03205711 \pm 5.2 \cdot 10^{-8} \)
\(a_{52}= -0.50827257 \pm 3.9 \cdot 10^{-8} \) \(a_{53}= -1.19716449 \pm 2.5 \cdot 10^{-8} \) \(a_{54}= +0.22377823 \pm 6.3 \cdot 10^{-8} \)
\(a_{55}= +0.38009475 \pm 4.6 \cdot 10^{-8} \) \(a_{56}= -0.28475950 \pm 5.1 \cdot 10^{-8} \) \(a_{57}= -0.33626613 \pm 4.2 \cdot 10^{-8} \)
\(a_{58}= -0.88745375 \pm 5.1 \cdot 10^{-8} \) \(a_{59}= +1.14402341 \pm 4.3 \cdot 10^{-8} \) \(a_{60}= -0.05732062 \pm 1.0 \cdot 10^{-7} \)

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