Properties

Label 33.5
Level $33$
Weight $0$
Character 33.1
Symmetry odd
\(R\) 1.595991
Fricke sign $-1$

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Maass form invariants

Level: \( 33 = 3 \cdot 11 \)
Weight: \( 0 \)
Character: 33.1
Symmetry: odd
Fricke sign: $-1$
Spectral parameter: \(1.59599133120226005054961145514 \pm 4 \cdot 10^{-10}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= +1.39842087 \pm 1.9 \cdot 10^{-6} \) \(a_{3}= -0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= +0.95558092 \pm 1.8 \cdot 10^{-6} \) \(a_{5}= +0.65181333 \pm 1.7 \cdot 10^{-6} \) \(a_{6}= -0.80737866 \pm 1.9 \cdot 10^{-6} \)
\(a_{7}= -1.28674561 \pm 1.6 \cdot 10^{-6} \) \(a_{8}= -0.06211657 \pm 1.8 \cdot 10^{-6} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= +0.91150936 \pm 2.2 \cdot 10^{-6} \) \(a_{11}= +0.30151134 \pm 1.0 \cdot 10^{-8} \) \(a_{12}= -0.55170490 \pm 1.8 \cdot 10^{-6} \)
\(a_{13}= +1.83605543 \pm 1.6 \cdot 10^{-6} \) \(a_{14}= -1.79941191 \pm 1.8 \cdot 10^{-6} \) \(a_{15}= -0.37632460 \pm 1.7 \cdot 10^{-6} \)
\(a_{16}= -1.04244602 \pm 1.8 \cdot 10^{-6} \) \(a_{17}= +0.46014857 \pm 1.5 \cdot 10^{-6} \) \(a_{18}= +0.46614029 \pm 1.9 \cdot 10^{-6} \)
\(a_{19}= -0.29463055 \pm 1.7 \cdot 10^{-6} \) \(a_{20}= +0.62286038 \pm 1.8 \cdot 10^{-6} \) \(a_{21}= +0.74290292 \pm 1.6 \cdot 10^{-6} \)
\(a_{22}= +0.42163976 \pm 1.9 \cdot 10^{-6} \) \(a_{23}= +0.10737909 \pm 1.4 \cdot 10^{-6} \) \(a_{24}= +0.03586302 \pm 1.8 \cdot 10^{-6} \)
\(a_{25}= -0.57513938 \pm 1.7 \cdot 10^{-6} \) \(a_{26}= +2.56757822 \pm 2.2 \cdot 10^{-6} \) \(a_{27}= -0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= -1.22958956 \pm 1.6 \cdot 10^{-6} \) \(a_{29}= +0.59768488 \pm 1.6 \cdot 10^{-6} \) \(a_{30}= -0.52626018 \pm 3.7 \cdot 10^{-6} \)
\(a_{31}= -1.15141786 \pm 1.4 \cdot 10^{-6} \) \(a_{32}= -1.39566171 \pm 1.7 \cdot 10^{-6} \) \(a_{33}= -0.17407766 \pm 1.0 \cdot 10^{-8} \)
\(a_{34}= +0.64348137 \pm 1.7 \cdot 10^{-6} \) \(a_{35}= -0.83871794 \pm 1.7 \cdot 10^{-6} \) \(a_{36}= +0.31852697 \pm 1.8 \cdot 10^{-6} \)
\(a_{37}= -0.82176911 \pm 1.6 \cdot 10^{-6} \) \(a_{38}= -0.41201751 \pm 2.0 \cdot 10^{-6} \) \(a_{39}= -1.06004710 \pm 1.6 \cdot 10^{-6} \)
\(a_{40}= -0.04048841 \pm 1.8 \cdot 10^{-6} \) \(a_{41}= -0.10404782 \pm 1.4 \cdot 10^{-6} \) \(a_{42}= +1.03889095 \pm 3.5 \cdot 10^{-6} \)
\(a_{43}= +1.37823805 \pm 1.3 \cdot 10^{-6} \) \(a_{44}= +0.28811849 \pm 1.8 \cdot 10^{-6} \) \(a_{45}= +0.21727111 \pm 1.7 \cdot 10^{-6} \)
\(a_{46}= +0.15016116 \pm 1.6 \cdot 10^{-6} \) \(a_{47}= +0.93397174 \pm 1.6 \cdot 10^{-6} \) \(a_{48}= +0.60185649 \pm 1.8 \cdot 10^{-6} \)
\(a_{49}= +0.65571426 \pm 1.5 \cdot 10^{-6} \) \(a_{50}= -0.80428691 \pm 2.1 \cdot 10^{-6} \) \(a_{51}= -0.26566690 \pm 1.5 \cdot 10^{-6} \)
\(a_{52}= +1.75449954 \pm 2.1 \cdot 10^{-6} \) \(a_{53}= -1.12037691 \pm 1.7 \cdot 10^{-6} \) \(a_{54}= -0.26912622 \pm 1.9 \cdot 10^{-6} \)
\(a_{55}= +0.19652911 \pm 1.7 \cdot 10^{-6} \) \(a_{56}= +0.07992822 \pm 1.6 \cdot 10^{-6} \) \(a_{57}= +0.17010503 \pm 1.7 \cdot 10^{-6} \)
\(a_{58}= +0.83581501 \pm 1.7 \cdot 10^{-6} \) \(a_{59}= +0.86443557 \pm 1.6 \cdot 10^{-6} \) \(a_{60}= -0.35960861 \pm 3.6 \cdot 10^{-6} \)

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