Properties

Label 21.18
Level $21$
Weight $0$
Character 21.1
Symmetry even
\(R\) 4.230378
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: \(4.2303786906247529873255258492 \pm 8 \cdot 10^{-11}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= -1.24395112 \pm 1 \cdot 10^{-8} \) \(a_{3}= -0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= +0.54741438 \pm 1 \cdot 10^{-8} \) \(a_{5}= -1.50378394 \pm 1 \cdot 10^{-8} \) \(a_{6}= +0.71819551 \pm 1.2 \cdot 10^{-8} \)
\(a_{7}= -0.37796447 \pm 1.0 \cdot 10^{-8} \) \(a_{8}= +0.56299439 \pm 1 \cdot 10^{-8} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= +1.87063371 \pm 1 \cdot 10^{-8} \) \(a_{11}= -1.10235492 \pm 1 \cdot 10^{-8} \) \(a_{12}= -0.31604984 \pm 1.2 \cdot 10^{-8} \)
\(a_{13}= +1.15027378 \pm 1 \cdot 10^{-8} \) \(a_{14}= +0.47016933 \pm 1.2 \cdot 10^{-8} \) \(a_{15}= +0.86821006 \pm 1.2 \cdot 10^{-8} \)
\(a_{16}= -1.24775188 \pm 1 \cdot 10^{-8} \) \(a_{17}= -1.22372126 \pm 1 \cdot 10^{-8} \) \(a_{18}= -0.41465037 \pm 1.2 \cdot 10^{-8} \)
\(a_{19}= +1.05730101 \pm 1 \cdot 10^{-8} \) \(a_{20}= -0.82319295 \pm 1 \cdot 10^{-8} \) \(a_{21}= +0.21821789 \pm 1.0 \cdot 10^{-8} \)
\(a_{22}= +1.37127563 \pm 1 \cdot 10^{-8} \) \(a_{23}= +0.21320830 \pm 1 \cdot 10^{-8} \) \(a_{24}= -0.32504496 \pm 1.2 \cdot 10^{-8} \)
\(a_{25}= +1.26136614 \pm 1 \cdot 10^{-8} \) \(a_{26}= -1.43088436 \pm 1 \cdot 10^{-8} \) \(a_{27}= -0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= -0.20690319 \pm 1.2 \cdot 10^{-8} \) \(a_{29}= +0.01309880 \pm 1 \cdot 10^{-8} \) \(a_{30}= -1.08001088 \pm 1.4 \cdot 10^{-8} \)
\(a_{31}= +0.36466877 \pm 1 \cdot 10^{-8} \) \(a_{32}= +0.98914795 \pm 1 \cdot 10^{-8} \) \(a_{33}= +0.63644491 \pm 1.1 \cdot 10^{-8} \)
\(a_{34}= +1.52224943 \pm 1 \cdot 10^{-8} \) \(a_{35}= +0.56837690 \pm 1.2 \cdot 10^{-8} \) \(a_{36}= +0.18247146 \pm 1.2 \cdot 10^{-8} \)
\(a_{37}= -1.57106614 \pm 1 \cdot 10^{-8} \) \(a_{38}= -1.31523076 \pm 1 \cdot 10^{-8} \) \(a_{39}= -0.66411088 \pm 1.1 \cdot 10^{-8} \)
\(a_{40}= -0.84662192 \pm 1 \cdot 10^{-8} \) \(a_{41}= +1.37456808 \pm 1 \cdot 10^{-8} \) \(a_{42}= -0.27145239 \pm 1.2 \cdot 10^{-8} \)
\(a_{43}= +0.36072142 \pm 1 \cdot 10^{-8} \) \(a_{44}= -0.60344493 \pm 1 \cdot 10^{-8} \) \(a_{45}= -0.50126131 \pm 1.2 \cdot 10^{-8} \)
\(a_{46}= -0.26522071 \pm 1 \cdot 10^{-8} \) \(a_{47}= -0.25075701 \pm 1 \cdot 10^{-8} \) \(a_{48}= +0.72038988 \pm 1.2 \cdot 10^{-8} \)
\(a_{49}= +0.14285714 \pm 1.5 \cdot 10^{-7} \) \(a_{50}= -1.56907782 \pm 1 \cdot 10^{-8} \) \(a_{51}= +0.70651580 \pm 1.2 \cdot 10^{-8} \)
\(a_{52}= +0.62967641 \pm 1 \cdot 10^{-8} \) \(a_{53}= +1.22594021 \pm 1 \cdot 10^{-8} \) \(a_{54}= +0.23939850 \pm 1.2 \cdot 10^{-8} \)
\(a_{55}= +1.65770362 \pm 1 \cdot 10^{-8} \) \(a_{56}= -0.21279188 \pm 1.2 \cdot 10^{-8} \) \(a_{57}= -0.61043302 \pm 1.1 \cdot 10^{-8} \)
\(a_{58}= -0.01629427 \pm 1 \cdot 10^{-8} \) \(a_{59}= +0.75882412 \pm 1 \cdot 10^{-8} \) \(a_{60}= +0.47527067 \pm 1.4 \cdot 10^{-8} \)

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