Properties

Label 33.27
Level $33$
Weight $0$
Character 33.1
Symmetry even
\(R\) 4.040894
Fricke sign $+1$

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Maass form invariants

Level: \( 33 = 3 \cdot 11 \)
Weight: \( 0 \)
Character: 33.1
Symmetry: even
Fricke sign: $+1$
Spectral parameter: \(4.0408941784612013888821346379 \pm 6 \cdot 10^{-10}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= +1.88906787 \pm 5.6 \cdot 10^{-8} \) \(a_{3}= -0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= +2.56857740 \pm 6.5 \cdot 10^{-8} \) \(a_{5}= +1.61074356 \pm 4.8 \cdot 10^{-8} \) \(a_{6}= -1.09065384 \pm 6.7 \cdot 10^{-8} \)
\(a_{7}= -0.75114038 \pm 4.6 \cdot 10^{-8} \) \(a_{8}= +2.96314916 \pm 6.9 \cdot 10^{-8} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= +3.04280390 \pm 4.9 \cdot 10^{-8} \) \(a_{11}= -0.30151134 \pm 1.0 \cdot 10^{-8} \) \(a_{12}= -1.48296885 \pm 7.5 \cdot 10^{-8} \)
\(a_{13}= +0.36877752 \pm 4.6 \cdot 10^{-8} \) \(a_{14}= -1.41895515 \pm 6.0 \cdot 10^{-8} \) \(a_{15}= -0.92996323 \pm 5.8 \cdot 10^{-8} \)
\(a_{16}= +3.02901246 \pm 7.4 \cdot 10^{-8} \) \(a_{17}= -0.91056548 \pm 5.0 \cdot 10^{-8} \) \(a_{18}= +0.62968929 \pm 6.7 \cdot 10^{-8} \)
\(a_{19}= -0.21047495 \pm 4.8 \cdot 10^{-8} \) \(a_{20}= +4.13731951 \pm 5.5 \cdot 10^{-8} \) \(a_{21}= +0.43367110 \pm 5.7 \cdot 10^{-8} \)
\(a_{22}= -0.56957539 \pm 6.7 \cdot 10^{-8} \) \(a_{23}= -1.18596833 \pm 4.3 \cdot 10^{-8} \) \(a_{24}= -1.71077497 \pm 7.9 \cdot 10^{-8} \)
\(a_{25}= +1.59449483 \pm 4.6 \cdot 10^{-8} \) \(a_{26}= +0.69664576 \pm 5.3 \cdot 10^{-8} \) \(a_{27}= -0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= -1.92936219 \pm 6.2 \cdot 10^{-8} \) \(a_{29}= -0.70753034 \pm 4.1 \cdot 10^{-8} \) \(a_{30}= -1.75676365 \pm 1.1 \cdot 10^{-7} \)
\(a_{31}= -0.69685594 \pm 3.5 \cdot 10^{-8} \) \(a_{32}= +2.75886094 \pm 7.6 \cdot 10^{-8} \) \(a_{33}= +0.17407766 \pm 1.0 \cdot 10^{-8} \)
\(a_{34}= -1.72011999 \pm 5.8 \cdot 10^{-8} \) \(a_{35}= -1.20989452 \pm 4.1 \cdot 10^{-8} \) \(a_{36}= +0.85619247 \pm 7.5 \cdot 10^{-8} \)
\(a_{37}= +1.30365060 \pm 4.3 \cdot 10^{-8} \) \(a_{38}= -0.39760147 \pm 5.1 \cdot 10^{-8} \) \(a_{39}= -0.21291380 \pm 5.7 \cdot 10^{-8} \)
\(a_{40}= +4.77287344 \pm 5.3 \cdot 10^{-8} \) \(a_{41}= +0.44106481 \pm 4.2 \cdot 10^{-8} \) \(a_{42}= +0.81923414 \pm 1.1 \cdot 10^{-7} \)
\(a_{43}= -1.05440167 \pm 5.4 \cdot 10^{-8} \) \(a_{44}= -0.77445523 \pm 7.5 \cdot 10^{-8} \) \(a_{45}= +0.53691452 \pm 5.8 \cdot 10^{-8} \)
\(a_{46}= -2.24037467 \pm 5.0 \cdot 10^{-8} \) \(a_{47}= -0.15337268 \pm 4.1 \cdot 10^{-8} \) \(a_{48}= -1.74880116 \pm 8.4 \cdot 10^{-8} \)
\(a_{49}= -0.43578814 \pm 4.6 \cdot 10^{-8} \) \(a_{50}= +3.01210894 \pm 5.4 \cdot 10^{-8} \) \(a_{51}= +0.52571522 \pm 6.1 \cdot 10^{-8} \)
\(a_{52}= +0.94723360 \pm 6.0 \cdot 10^{-8} \) \(a_{53}= +0.14119538 \pm 3.6 \cdot 10^{-8} \) \(a_{54}= -0.36355128 \pm 6.7 \cdot 10^{-8} \)
\(a_{55}= -0.48565746 \pm 5.8 \cdot 10^{-8} \) \(a_{56}= -2.22574097 \pm 6.3 \cdot 10^{-8} \) \(a_{57}= +0.12151777 \pm 5.8 \cdot 10^{-8} \)
\(a_{58}= -1.33657282 \pm 5.7 \cdot 10^{-8} \) \(a_{59}= +1.74391282 \pm 4.2 \cdot 10^{-8} \) \(a_{60}= -2.38868253 \pm 1.2 \cdot 10^{-7} \)

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