Properties

Label 13.178
Level $13$
Weight $0$
Character 13.1
Symmetry even
\(R\) 14.21380
Fricke sign $+1$

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Maass form invariants

Level: \( 13 \)
Weight: \( 0 \)
Character: 13.1
Symmetry: even
Fricke sign: $+1$
Spectral parameter: \(14.2138017412218413679300317845 \pm 9 \cdot 10^{-8}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= +1.24750239 \pm 1.2 \cdot 10^{-5} \) \(a_{3}= -0.95454844 \pm 9.1 \cdot 10^{-6} \)
\(a_{4}= +0.55626221 \pm 1.3 \cdot 10^{-5} \) \(a_{5}= -1.41742099 \pm 1.0 \cdot 10^{-5} \) \(a_{6}= -1.19080146 \pm 1.1 \cdot 10^{-5} \)
\(a_{7}= +0.57765953 \pm 9.2 \cdot 10^{-6} \) \(a_{8}= -0.55356396 \pm 1.1 \cdot 10^{-5} \) \(a_{9}= -0.08883727 \pm 8.2 \cdot 10^{-6} \)
\(a_{10}= -1.76823607 \pm 9.2 \cdot 10^{-6} \) \(a_{11}= -0.71121320 \pm 6.8 \cdot 10^{-6} \) \(a_{12}= -0.53097922 \pm 1.1 \cdot 10^{-5} \)
\(a_{13}= -0.27735010 \pm 1.0 \cdot 10^{-8} \) \(a_{14}= +0.72063165 \pm 1.3 \cdot 10^{-5} \) \(a_{15}= +1.35299700 \pm 7.5 \cdot 10^{-6} \)
\(a_{16}= -1.24683456 \pm 8.3 \cdot 10^{-6} \) \(a_{17}= -1.18399962 \pm 9.0 \cdot 10^{-6} \) \(a_{18}= -0.11082471 \pm 9.6 \cdot 10^{-6} \)
\(a_{19}= -0.05551808 \pm 9.8 \cdot 10^{-6} \) \(a_{20}= -0.78845773 \pm 1.1 \cdot 10^{-5} \) \(a_{21}= -0.55140401 \pm 7.5 \cdot 10^{-6} \)
\(a_{22}= -0.88724017 \pm 6.5 \cdot 10^{-6} \) \(a_{23}= -0.92207407 \pm 6.6 \cdot 10^{-6} \) \(a_{24}= +0.52840361 \pm 1.0 \cdot 10^{-5} \)
\(a_{25}= +1.00908226 \pm 1.1 \cdot 10^{-5} \) \(a_{26}= -0.34599491 \pm 1.2 \cdot 10^{-5} \) \(a_{27}= +1.03934792 \pm 9.9 \cdot 10^{-6} \)
\(a_{28}= +0.32133017 \pm 1.4 \cdot 10^{-5} \) \(a_{29}= +1.54562156 \pm 8.7 \cdot 10^{-6} \) \(a_{30}= +1.68786699 \pm 6.2 \cdot 10^{-6} \)
\(a_{31}= +0.70170323 \pm 6.8 \cdot 10^{-6} \) \(a_{32}= -1.00186514 \pm 9.0 \cdot 10^{-6} \) \(a_{33}= +0.67888746 \pm 8.1 \cdot 10^{-6} \)
\(a_{34}= -1.47704236 \pm 1.1 \cdot 10^{-5} \) \(a_{35}= -0.81878675 \pm 8.3 \cdot 10^{-6} \) \(a_{36}= -0.04941682 \pm 8.0 \cdot 10^{-6} \)
\(a_{37}= +0.78429404 \pm 8.0 \cdot 10^{-6} \) \(a_{38}= -0.06925893 \pm 1.3 \cdot 10^{-5} \) \(a_{39}= +0.26474410 \pm 9.2 \cdot 10^{-6} \)
\(a_{40}= +0.78463317 \pm 1.0 \cdot 10^{-5} \) \(a_{41}= +1.41787188 \pm 1.3 \cdot 10^{-5} \) \(a_{42}= -0.68787782 \pm 1.0 \cdot 10^{-5} \)
\(a_{43}= -1.26754883 \pm 7.6 \cdot 10^{-6} \) \(a_{44}= -0.39562103 \pm 5.3 \cdot 10^{-6} \) \(a_{45}= +0.12591981 \pm 7.0 \cdot 10^{-6} \)
\(a_{46}= -1.15028961 \pm 7.8 \cdot 10^{-6} \) \(a_{47}= +0.37300103 \pm 7.0 \cdot 10^{-6} \) \(a_{48}= +1.19016399 \pm 7.4 \cdot 10^{-6} \)
\(a_{49}= -0.66630946 \pm 7.0 \cdot 10^{-6} \) \(a_{50}= +1.25883253 \pm 9.0 \cdot 10^{-6} \) \(a_{51}= +1.13018499 \pm 9.8 \cdot 10^{-6} \)
\(a_{52}= -0.15427938 \pm 1.3 \cdot 10^{-5} \) \(a_{53}= -0.34469640 \pm 9.6 \cdot 10^{-6} \) \(a_{54}= +1.29658901 \pm 1.0 \cdot 10^{-5} \)
\(a_{55}= +1.00808852 \pm 6.5 \cdot 10^{-6} \) \(a_{56}= -0.31977150 \pm 1.1 \cdot 10^{-5} \) \(a_{57}= +0.05299469 \pm 5.5 \cdot 10^{-6} \)
\(a_{58}= +1.92816658 \pm 1.4 \cdot 10^{-5} \) \(a_{59}= -0.23659806 \pm 8.0 \cdot 10^{-6} \) \(a_{60}= +0.75262110 \pm 7.2 \cdot 10^{-6} \)

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