Properties

Label 21.72
Level $21$
Weight $0$
Character 21.1
Symmetry odd
\(R\) 8.450640
Fricke sign $-1$

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Maass form invariants

Level: \( 21 = 3 \cdot 7 \)
Weight: \( 0 \)
Character: 21.1
Symmetry: odd
Fricke sign: $-1$
Spectral parameter: \(8.45064079787709261875461823034 \pm 2 \cdot 10^{-9}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= +0.92799433 \pm 2.8 \cdot 10^{-6} \) \(a_{3}= +0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= -0.13882652 \pm 3.2 \cdot 10^{-6} \) \(a_{5}= +0.23410446 \pm 2.2 \cdot 10^{-6} \) \(a_{6}= +0.53577778 \pm 2.8 \cdot 10^{-6} \)
\(a_{7}= -0.37796447 \pm 1.0 \cdot 10^{-8} \) \(a_{8}= -1.05682456 \pm 3.2 \cdot 10^{-6} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= +0.21724761 \pm 2.7 \cdot 10^{-6} \) \(a_{11}= +1.74846782 \pm 2.1 \cdot 10^{-6} \) \(a_{12}= -0.08015153 \pm 3.2 \cdot 10^{-6} \)
\(a_{13}= +0.90643397 \pm 2.5 \cdot 10^{-6} \) \(a_{14}= -0.35074889 \pm 2.8 \cdot 10^{-6} \) \(a_{15}= +0.13516027 \pm 2.2 \cdot 10^{-6} \)
\(a_{16}= -0.84190068 \pm 2.7 \cdot 10^{-6} \) \(a_{17}= +0.05612656 \pm 2.4 \cdot 10^{-6} \) \(a_{18}= +0.30933144 \pm 2.8 \cdot 10^{-6} \)
\(a_{19}= +0.97500210 \pm 2.3 \cdot 10^{-6} \) \(a_{20}= -0.03249991 \pm 3.1 \cdot 10^{-6} \) \(a_{21}= -0.21821789 \pm 1.0 \cdot 10^{-8} \)
\(a_{22}= +1.62256823 \pm 2.8 \cdot 10^{-6} \) \(a_{23}= +0.57238150 \pm 2.3 \cdot 10^{-6} \) \(a_{24}= -0.61015794 \pm 3.2 \cdot 10^{-6} \)
\(a_{25}= -0.94519510 \pm 2.2 \cdot 10^{-6} \) \(a_{26}= +0.84116558 \pm 2.7 \cdot 10^{-6} \) \(a_{27}= +0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= +0.05247149 \pm 3.2 \cdot 10^{-6} \) \(a_{29}= +0.91313285 \pm 2.5 \cdot 10^{-6} \) \(a_{30}= +0.12542797 \pm 5.0 \cdot 10^{-6} \)
\(a_{31}= +0.86897075 \pm 2.6 \cdot 10^{-6} \) \(a_{32}= +0.27554550 \pm 2.5 \cdot 10^{-6} \) \(a_{33}= +1.00947837 \pm 2.1 \cdot 10^{-6} \)
\(a_{34}= +0.05208513 \pm 2.3 \cdot 10^{-6} \) \(a_{35}= -0.08848317 \pm 2.2 \cdot 10^{-6} \) \(a_{36}= -0.04627551 \pm 3.2 \cdot 10^{-6} \)
\(a_{37}= +0.79629886 \pm 2.2 \cdot 10^{-6} \) \(a_{38}= +0.90479643 \pm 2.7 \cdot 10^{-6} \) \(a_{39}= +0.52332989 \pm 2.5 \cdot 10^{-6} \)
\(a_{40}= -0.24740734 \pm 3.0 \cdot 10^{-6} \) \(a_{41}= -1.15208149 \pm 1.8 \cdot 10^{-6} \) \(a_{42}= -0.20250497 \pm 2.8 \cdot 10^{-6} \)
\(a_{43}= +1.90883842 \pm 2.5 \cdot 10^{-6} \) \(a_{44}= -0.24273370 \pm 3.2 \cdot 10^{-6} \) \(a_{45}= +0.07803482 \pm 2.2 \cdot 10^{-6} \)
\(a_{46}= +0.53116679 \pm 2.7 \cdot 10^{-6} \) \(a_{47}= +0.60839088 \pm 2.3 \cdot 10^{-6} \) \(a_{48}= -0.48607158 \pm 2.7 \cdot 10^{-6} \)
\(a_{49}= +0.14285714 \pm 1.5 \cdot 10^{-7} \) \(a_{50}= -0.87713570 \pm 2.1 \cdot 10^{-6} \) \(a_{51}= +0.03240468 \pm 2.4 \cdot 10^{-6} \)
\(a_{52}= -0.12583707 \pm 3.1 \cdot 10^{-6} \) \(a_{53}= -0.68748089 \pm 2.5 \cdot 10^{-6} \) \(a_{54}= +0.17859259 \pm 2.8 \cdot 10^{-6} \)
\(a_{55}= +0.40932412 \pm 1.9 \cdot 10^{-6} \) \(a_{56}= +0.39944214 \pm 3.2 \cdot 10^{-6} \) \(a_{57}= +0.56291773 \pm 2.4 \cdot 10^{-6} \)
\(a_{58}= +0.84738211 \pm 2.4 \cdot 10^{-6} \) \(a_{59}= -1.59867646 \pm 1.7 \cdot 10^{-6} \) \(a_{60}= -0.01876383 \pm 5.5 \cdot 10^{-6} \)

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