Properties

Label 3.33
Level $3$
Weight $0$
Character 3.1
Symmetry even
\(R\) 15.79949
Fricke sign $+1$

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Maass form invariants

Level: \( 3 \)
Weight: \( 0 \)
Character: 3.1
Symmetry: even
Fricke sign: $+1$
Spectral parameter: \(15.7994939260844159351480665632 \pm 2 \cdot 10^{-7}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= +1.88979233 \pm 8.0 \cdot 10^{-7} \) \(a_{3}= -0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= +2.57131506 \pm 4.9 \cdot 10^{-7} \) \(a_{5}= +0.58480210 \pm 6.9 \cdot 10^{-8} \) \(a_{6}= -1.09107211 \pm 8.1 \cdot 10^{-7} \)
\(a_{7}= -0.03104569 \pm 4.4 \cdot 10^{-7} \) \(a_{8}= +2.96945915 \pm 1.8 \cdot 10^{-7} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= +1.10515453 \pm 8.3 \cdot 10^{-8} \) \(a_{11}= +0.08493308 \pm 2.4 \cdot 10^{-7} \) \(a_{12}= -1.48454944 \pm 5.0 \cdot 10^{-7} \)
\(a_{13}= +1.52538435 \pm 3.2 \cdot 10^{-7} \) \(a_{14}= -0.05866991 \pm 6.0 \cdot 10^{-7} \) \(a_{15}= -0.33763565 \pm 8.0 \cdot 10^{-8} \)
\(a_{16}= +3.04034608 \pm 6.8 \cdot 10^{-7} \) \(a_{17}= -0.67682700 \pm 5.7 \cdot 10^{-7} \) \(a_{18}= +0.62993078 \pm 8.1 \cdot 10^{-7} \)
\(a_{19}= +0.04700919 \pm 9.1 \cdot 10^{-7} \) \(a_{20}= +1.50371045 \pm 8.7 \cdot 10^{-8} \) \(a_{21}= +0.01792424 \pm 4.6 \cdot 10^{-7} \)
\(a_{22}= +0.16050589 \pm 3.2 \cdot 10^{-7} \) \(a_{23}= +0.83713434 \pm 2.0 \cdot 10^{-7} \) \(a_{24}= -1.71441804 \pm 1.9 \cdot 10^{-7} \)
\(a_{25}= -0.65800650 \pm 5.8 \cdot 10^{-7} \) \(a_{26}= +2.88265965 \pm 4.5 \cdot 10^{-7} \) \(a_{27}= -0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= -0.07982825 \pm 3.7 \cdot 10^{-7} \) \(a_{29}= -1.08143461 \pm 1.5 \cdot 10^{-7} \) \(a_{30}= -0.63806127 \pm 8.8 \cdot 10^{-7} \)
\(a_{31}= -0.73514243 \pm 6.8 \cdot 10^{-7} \) \(a_{32}= +2.77616356 \pm 7.6 \cdot 10^{-7} \) \(a_{33}= -0.04903614 \pm 2.5 \cdot 10^{-7} \)
\(a_{34}= -1.27906248 \pm 7.6 \cdot 10^{-7} \) \(a_{35}= -0.01815559 \pm 6.1 \cdot 10^{-8} \) \(a_{36}= +0.85710502 \pm 5.0 \cdot 10^{-7} \)
\(a_{37}= -0.82980604 \pm 1.5 \cdot 10^{-7} \) \(a_{38}= +0.08883762 \pm 1.2 \cdot 10^{-6} \) \(a_{39}= -0.88068107 \pm 3.4 \cdot 10^{-7} \)
\(a_{40}= +1.73654596 \pm 5.3 \cdot 10^{-8} \) \(a_{41}= -0.61266330 \pm 6.5 \cdot 10^{-7} \) \(a_{42}= +0.03387309 \pm 1.2 \cdot 10^{-6} \)
\(a_{43}= -0.72099717 \pm 1.1 \cdot 10^{-7} \) \(a_{44}= +0.21838972 \pm 1.9 \cdot 10^{-7} \) \(a_{45}= +0.19493403 \pm 8.0 \cdot 10^{-8} \)
\(a_{46}= +1.58201006 \pm 2.9 \cdot 10^{-7} \) \(a_{47}= +0.23706617 \pm 5.9 \cdot 10^{-7} \) \(a_{48}= -1.75534463 \pm 6.9 \cdot 10^{-7} \)
\(a_{49}= -0.99903617 \pm 2.5 \cdot 10^{-7} \) \(a_{50}= -1.24349564 \pm 7.9 \cdot 10^{-7} \) \(a_{51}= +0.39076625 \pm 5.8 \cdot 10^{-7} \)
\(a_{52}= +3.92224376 \pm 3.1 \cdot 10^{-7} \) \(a_{53}= -0.94364515 \pm 1.7 \cdot 10^{-7} \) \(a_{54}= -0.36369070 \pm 8.1 \cdot 10^{-7} \)
\(a_{55}= +0.04966905 \pm 2.4 \cdot 10^{-8} \) \(a_{56}= -0.09218891 \pm 1.3 \cdot 10^{-7} \) \(a_{57}= -0.02714077 \pm 9.3 \cdot 10^{-7} \)
\(a_{58}= -2.04368684 \pm 2.2 \cdot 10^{-7} \) \(a_{59}= +1.03520149 \pm 5.5 \cdot 10^{-8} \) \(a_{60}= -0.86816764 \pm 5.7 \cdot 10^{-7} \)

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