Properties

Label 21.11
Level $21$
Weight $0$
Character 21.1
Symmetry odd
\(R\) 3.272721
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: \(3.27272100042075640598436233531 \pm 6 \cdot 10^{-11}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= -0.20541682 \pm 4.8 \cdot 10^{-7} \) \(a_{3}= +0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= -0.95780393 \pm 5.6 \cdot 10^{-7} \) \(a_{5}= -1.66412663 \pm 3.8 \cdot 10^{-7} \) \(a_{6}= -0.11859746 \pm 4.9 \cdot 10^{-7} \)
\(a_{7}= +0.37796447 \pm 1.0 \cdot 10^{-8} \) \(a_{8}= +0.40216586 \pm 5.6 \cdot 10^{-7} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= +0.34183960 \pm 4.7 \cdot 10^{-7} \) \(a_{11}= -0.88239856 \pm 3.7 \cdot 10^{-7} \) \(a_{12}= -0.55298836 \pm 5.7 \cdot 10^{-7} \)
\(a_{13}= +0.36743886 \pm 4.3 \cdot 10^{-7} \) \(a_{14}= -0.07764026 \pm 4.9 \cdot 10^{-7} \) \(a_{15}= -0.96078396 \pm 3.9 \cdot 10^{-7} \)
\(a_{16}= +0.87519230 \pm 4.6 \cdot 10^{-7} \) \(a_{17}= -0.70352362 \pm 4.2 \cdot 10^{-7} \) \(a_{18}= -0.06847227 \pm 4.9 \cdot 10^{-7} \)
\(a_{19}= -0.94609832 \pm 4.1 \cdot 10^{-7} \) \(a_{20}= +1.59390702 \pm 5.4 \cdot 10^{-7} \) \(a_{21}= +0.21821789 \pm 1.0 \cdot 10^{-8} \)
\(a_{22}= +0.18125951 \pm 4.9 \cdot 10^{-7} \) \(a_{23}= -1.57575420 \pm 4.0 \cdot 10^{-7} \) \(a_{24}= +0.23219057 \pm 5.7 \cdot 10^{-7} \)
\(a_{25}= +1.76931744 \pm 3.8 \cdot 10^{-7} \) \(a_{26}= -0.07547812 \pm 4.7 \cdot 10^{-7} \) \(a_{27}= +0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= -0.36201586 \pm 5.7 \cdot 10^{-7} \) \(a_{29}= -0.13149234 \pm 4.2 \cdot 10^{-7} \) \(a_{30}= +0.19736119 \pm 8.7 \cdot 10^{-7} \)
\(a_{31}= +1.01385587 \pm 4.4 \cdot 10^{-7} \) \(a_{32}= -0.58194508 \pm 4.3 \cdot 10^{-7} \) \(a_{33}= -0.50945305 \pm 3.8 \cdot 10^{-7} \)
\(a_{34}= +0.14451559 \pm 3.9 \cdot 10^{-7} \) \(a_{35}= -0.62898074 \pm 3.9 \cdot 10^{-7} \) \(a_{36}= -0.31926798 \pm 5.7 \cdot 10^{-7} \)
\(a_{37}= -1.48476363 \pm 3.8 \cdot 10^{-7} \) \(a_{38}= +0.19434451 \pm 4.7 \cdot 10^{-7} \) \(a_{39}= +0.21214092 \pm 4.4 \cdot 10^{-7} \)
\(a_{40}= -0.66925492 \pm 5.1 \cdot 10^{-7} \) \(a_{41}= +0.78213203 \pm 3.2 \cdot 10^{-7} \) \(a_{42}= -0.04482563 \pm 4.9 \cdot 10^{-7} \)
\(a_{43}= +0.43530151 \pm 4.3 \cdot 10^{-7} \) \(a_{44}= +0.84516481 \pm 5.6 \cdot 10^{-7} \) \(a_{45}= -0.55470888 \pm 3.9 \cdot 10^{-7} \)
\(a_{46}= +0.32368642 \pm 4.7 \cdot 10^{-7} \) \(a_{47}= +0.27021586 \pm 3.9 \cdot 10^{-7} \) \(a_{48}= +0.50529251 \pm 4.7 \cdot 10^{-7} \)
\(a_{49}= +0.14285714 \pm 1.5 \cdot 10^{-7} \) \(a_{50}= -0.36344757 \pm 3.7 \cdot 10^{-7} \) \(a_{51}= -0.40617955 \pm 4.3 \cdot 10^{-7} \)
\(a_{52}= -0.35193438 \pm 5.3 \cdot 10^{-7} \) \(a_{53}= +0.30476280 \pm 4.3 \cdot 10^{-7} \) \(a_{54}= -0.03953249 \pm 4.9 \cdot 10^{-7} \)
\(a_{55}= +1.46842294 \pm 3.3 \cdot 10^{-7} \) \(a_{56}= +0.15200441 \pm 5.7 \cdot 10^{-7} \) \(a_{57}= -0.54623012 \pm 4.2 \cdot 10^{-7} \)
\(a_{58}= +0.02701074 \pm 4.1 \cdot 10^{-7} \) \(a_{59}= +0.22030638 \pm 2.9 \cdot 10^{-7} \) \(a_{60}= +0.92024265 \pm 9.5 \cdot 10^{-7} \)

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