Properties

Label 53.1
Level $53$
Weight $0$
Character 53.1
Symmetry odd
\(R\) 0.803989
Fricke sign $-1$

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Maass form invariants

Level: \( 53 \)
Weight: \( 0 \)
Character: 53.1
Symmetry: odd
Fricke sign: $-1$
Spectral parameter: \(0.8039894595747808092764914832 \pm 2 \cdot 10^{-8}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= +0.19839667 \pm 1.5 \cdot 10^{-5} \) \(a_{3}= +0.56992146 \pm 1.3 \cdot 10^{-5} \)
\(a_{4}= -0.96063876 \pm 1.6 \cdot 10^{-5} \) \(a_{5}= +1.87231256 \pm 1.4 \cdot 10^{-5} \) \(a_{6}= +0.11307052 \pm 1.6 \cdot 10^{-5} \)
\(a_{7}= -0.78114997 \pm 1.3 \cdot 10^{-5} \) \(a_{8}= -0.38898420 \pm 1.7 \cdot 10^{-5} \) \(a_{9}= -0.67518953 \pm 1.3 \cdot 10^{-5} \)
\(a_{10}= +0.37146057 \pm 1.5 \cdot 10^{-5} \) \(a_{11}= -0.24837676 \pm 1.2 \cdot 10^{-5} \) \(a_{12}= -0.54748864 \pm 1.7 \cdot 10^{-5} \)
\(a_{13}= -0.55291421 \pm 1.2 \cdot 10^{-5} \) \(a_{14}= -0.15497755 \pm 1.5 \cdot 10^{-5} \) \(a_{15}= +1.06707110 \pm 1.4 \cdot 10^{-5} \)
\(a_{16}= +0.88346559 \pm 1.7 \cdot 10^{-5} \) \(a_{17}= -0.07290918 \pm 1.1 \cdot 10^{-5} \) \(a_{18}= -0.13395535 \pm 1.4 \cdot 10^{-5} \)
\(a_{19}= -0.28025419 \pm 1.1 \cdot 10^{-5} \) \(a_{20}= -1.79861602 \pm 1.6 \cdot 10^{-5} \) \(a_{21}= -0.44519413 \pm 1.3 \cdot 10^{-5} \)
\(a_{22}= -0.04927712 \pm 1.4 \cdot 10^{-5} \) \(a_{23}= +0.49755067 \pm 1.2 \cdot 10^{-5} \) \(a_{24}= -0.22169044 \pm 1.9 \cdot 10^{-5} \)
\(a_{25}= +2.50555433 \pm 1.4 \cdot 10^{-5} \) \(a_{26}= -0.10969634 \pm 1.4 \cdot 10^{-5} \) \(a_{27}= -0.95472646 \pm 1.3 \cdot 10^{-5} \)
\(a_{28}= +0.75040294 \pm 1.6 \cdot 10^{-5} \) \(a_{29}= +1.17440971 \pm 1.2 \cdot 10^{-5} \) \(a_{30}= +0.21170335 \pm 1.7 \cdot 10^{-5} \)
\(a_{31}= +0.89077647 \pm 1.1 \cdot 10^{-5} \) \(a_{32}= +0.56426083 \pm 1.6 \cdot 10^{-5} \) \(a_{33}= -0.14155525 \pm 1.4 \cdot 10^{-5} \)
\(a_{34}= -0.01446494 \pm 1.5 \cdot 10^{-5} \) \(a_{35}= -1.46255690 \pm 1.3 \cdot 10^{-5} \) \(a_{36}= +0.64861324 \pm 1.4 \cdot 10^{-5} \)
\(a_{37}= -0.09000535 \pm 1.2 \cdot 10^{-5} \) \(a_{38}= -0.05560150 \pm 1.5 \cdot 10^{-5} \) \(a_{39}= -0.31511767 \pm 1.4 \cdot 10^{-5} \)
\(a_{40}= -0.72830000 \pm 1.4 \cdot 10^{-5} \) \(a_{41}= -0.89343632 \pm 1.3 \cdot 10^{-5} \) \(a_{42}= -0.08832503 \pm 1.6 \cdot 10^{-5} \)
\(a_{43}= -1.01192436 \pm 1.2 \cdot 10^{-5} \) \(a_{44}= +0.23860034 \pm 1.6 \cdot 10^{-5} \) \(a_{45}= -1.26416585 \pm 1.4 \cdot 10^{-5} \)
\(a_{46}= +0.09871240 \pm 1.5 \cdot 10^{-5} \) \(a_{47}= +1.40263937 \pm 1.2 \cdot 10^{-5} \) \(a_{48}= +0.50350600 \pm 1.9 \cdot 10^{-5} \)
\(a_{49}= -0.38980473 \pm 1.2 \cdot 10^{-5} \) \(a_{50}= +0.49709363 \pm 1.6 \cdot 10^{-5} \) \(a_{51}= -0.04155251 \pm 1.3 \cdot 10^{-5} \)
\(a_{52}= +0.53115082 \pm 1.3 \cdot 10^{-5} \) \(a_{53}= +0.13736056 \pm 1.0 \cdot 10^{-8} \) \(a_{54}= -0.18941455 \pm 1.3 \cdot 10^{-5} \)
\(a_{55}= -0.46503893 \pm 1.2 \cdot 10^{-5} \) \(a_{56}= +0.30385499 \pm 1.6 \cdot 10^{-5} \) \(a_{57}= -0.15972288 \pm 1.3 \cdot 10^{-5} \)
\(a_{58}= +0.23299897 \pm 1.5 \cdot 10^{-5} \) \(a_{59}= -0.08142825 \pm 1.2 \cdot 10^{-5} \) \(a_{60}= -1.02506986 \pm 1.8 \cdot 10^{-5} \)

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