Properties

Label 21.22
Level $21$
Weight $0$
Character 21.1
Symmetry odd
\(R\) 4.604671
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: \(4.6046719045231397441513613654 \pm 9 \cdot 10^{-11}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= -1.64850787 \pm 2.3 \cdot 10^{-7} \) \(a_{3}= +0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= +1.71757821 \pm 2.7 \cdot 10^{-7} \) \(a_{5}= -0.16160013 \pm 1.8 \cdot 10^{-7} \) \(a_{6}= -0.95176646 \pm 2.4 \cdot 10^{-7} \)
\(a_{7}= +0.37796447 \pm 1.0 \cdot 10^{-8} \) \(a_{8}= -1.18293333 \pm 2.6 \cdot 10^{-7} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= +0.26639909 \pm 2.2 \cdot 10^{-7} \) \(a_{11}= -1.17046880 \pm 1.7 \cdot 10^{-7} \) \(a_{12}= +0.99164424 \pm 2.8 \cdot 10^{-7} \)
\(a_{13}= +0.02700853 \pm 2.0 \cdot 10^{-7} \) \(a_{14}= -0.62307741 \pm 2.4 \cdot 10^{-7} \) \(a_{15}= -0.09329988 \pm 1.9 \cdot 10^{-7} \)
\(a_{16}= +0.23249669 \pm 2.2 \cdot 10^{-7} \) \(a_{17}= -1.77300275 \pm 2.0 \cdot 10^{-7} \) \(a_{18}= -0.54950262 \pm 2.4 \cdot 10^{-7} \)
\(a_{19}= +1.07147196 \pm 1.9 \cdot 10^{-7} \) \(a_{20}= -0.27756087 \pm 2.6 \cdot 10^{-7} \) \(a_{21}= +0.21821789 \pm 1.0 \cdot 10^{-8} \)
\(a_{22}= +1.92952704 \pm 2.3 \cdot 10^{-7} \) \(a_{23}= -1.01202874 \pm 1.9 \cdot 10^{-7} \) \(a_{24}= -0.68296687 \pm 2.8 \cdot 10^{-7} \)
\(a_{25}= -0.97388540 \pm 1.8 \cdot 10^{-7} \) \(a_{26}= -0.04452377 \pm 2.2 \cdot 10^{-7} \) \(a_{27}= +0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= +0.64918354 \pm 2.8 \cdot 10^{-7} \) \(a_{29}= +0.36953769 \pm 2.0 \cdot 10^{-7} \) \(a_{30}= +0.15380559 \pm 4.2 \cdot 10^{-7} \)
\(a_{31}= -1.90571387 \pm 2.1 \cdot 10^{-7} \) \(a_{32}= +0.79966070 \pm 2.0 \cdot 10^{-7} \) \(a_{33}= -0.67577048 \pm 1.8 \cdot 10^{-7} \)
\(a_{34}= +2.92280900 \pm 1.8 \cdot 10^{-7} \) \(a_{35}= -0.06107911 \pm 1.9 \cdot 10^{-7} \) \(a_{36}= +0.57252607 \pm 2.8 \cdot 10^{-7} \)
\(a_{37}= +0.74736299 \pm 1.8 \cdot 10^{-7} \) \(a_{38}= -1.76632996 \pm 2.2 \cdot 10^{-7} \) \(a_{39}= +0.01559338 \pm 2.1 \cdot 10^{-7} \)
\(a_{40}= +0.19116218 \pm 2.4 \cdot 10^{-7} \) \(a_{41}= -0.96082226 \pm 1.5 \cdot 10^{-7} \) \(a_{42}= -0.35973391 \pm 2.4 \cdot 10^{-7} \)
\(a_{43}= -1.12942737 \pm 2.0 \cdot 10^{-7} \) \(a_{44}= -2.01037171 \pm 2.6 \cdot 10^{-7} \) \(a_{45}= -0.05386671 \pm 1.9 \cdot 10^{-7} \)
\(a_{46}= +1.66833734 \pm 2.2 \cdot 10^{-7} \) \(a_{47}= -0.07092855 \pm 1.8 \cdot 10^{-7} \) \(a_{48}= +0.13423203 \pm 2.3 \cdot 10^{-7} \)
\(a_{49}= +0.14285714 \pm 1.5 \cdot 10^{-7} \) \(a_{50}= +1.60545774 \pm 1.7 \cdot 10^{-7} \) \(a_{51}= -1.02364362 \pm 2.1 \cdot 10^{-7} \)
\(a_{52}= +0.04638926 \pm 2.5 \cdot 10^{-7} \) \(a_{53}= +1.45980343 \pm 2.1 \cdot 10^{-7} \) \(a_{54}= -0.31725549 \pm 2.4 \cdot 10^{-7} \)
\(a_{55}= +0.18914791 \pm 1.6 \cdot 10^{-7} \) \(a_{56}= -0.44710677 \pm 2.8 \cdot 10^{-7} \) \(a_{57}= +0.61861462 \pm 2.0 \cdot 10^{-7} \)
\(a_{58}= -0.60918579 \pm 1.9 \cdot 10^{-7} \) \(a_{59}= +0.03024355 \pm 1.4 \cdot 10^{-7} \) \(a_{60}= -0.16024984 \pm 4.6 \cdot 10^{-7} \)

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