Properties

Label 13.56
Level $13$
Weight $0$
Character 13.1
Symmetry even
\(R\) 8.322577
Fricke sign $+1$

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Maass form invariants

Level: \( 13 \)
Weight: \( 0 \)
Character: 13.1
Symmetry: even
Fricke sign: $+1$
Spectral parameter: \(8.32257722261542143711892684188 \pm 5 \cdot 10^{-11}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= -1.70965539 \pm 2.3 \cdot 10^{-8} \) \(a_{3}= -0.51125795 \pm 1.7 \cdot 10^{-8} \)
\(a_{4}= +1.92292157 \pm 2.5 \cdot 10^{-8} \) \(a_{5}= +1.36477620 \pm 1.9 \cdot 10^{-8} \) \(a_{6}= +0.87407492 \pm 2.2 \cdot 10^{-8} \)
\(a_{7}= +0.48375874 \pm 1.7 \cdot 10^{-8} \) \(a_{8}= -1.57787783 \pm 2.2 \cdot 10^{-8} \) \(a_{9}= -0.73861530 \pm 1.5 \cdot 10^{-8} \)
\(a_{10}= -2.33329699 \pm 1.7 \cdot 10^{-8} \) \(a_{11}= +1.17444023 \pm 1.3 \cdot 10^{-8} \) \(a_{12}= -0.98310895 \pm 2.2 \cdot 10^{-8} \)
\(a_{13}= -0.27735010 \pm 1.0 \cdot 10^{-8} \) \(a_{14}= -0.82706074 \pm 2.5 \cdot 10^{-8} \) \(a_{15}= -0.69775269 \pm 1.4 \cdot 10^{-8} \)
\(a_{16}= +0.77470578 \pm 1.6 \cdot 10^{-8} \) \(a_{17}= +0.96804829 \pm 1.7 \cdot 10^{-8} \) \(a_{18}= +1.26277764 \pm 1.8 \cdot 10^{-8} \)
\(a_{19}= +0.00002203 \pm 1.8 \cdot 10^{-8} \) \(a_{20}= +2.62435758 \pm 2.1 \cdot 10^{-8} \) \(a_{21}= -0.24732551 \pm 1.4 \cdot 10^{-8} \)
\(a_{22}= -2.00788807 \pm 1.2 \cdot 10^{-8} \) \(a_{23}= -1.27765062 \pm 1.2 \cdot 10^{-8} \) \(a_{24}= +0.80670259 \pm 2.0 \cdot 10^{-8} \)
\(a_{25}= +0.86261407 \pm 2.1 \cdot 10^{-8} \) \(a_{26}= +0.47417309 \pm 3.3 \cdot 10^{-8} \) \(a_{27}= +0.88888090 \pm 1.9 \cdot 10^{-8} \)
\(a_{28}= +0.93023012 \pm 2.7 \cdot 10^{-8} \) \(a_{29}= -1.69047324 \pm 1.6 \cdot 10^{-8} \) \(a_{30}= +1.19291665 \pm 1.2 \cdot 10^{-8} \)
\(a_{31}= +1.61060473 \pm 1.3 \cdot 10^{-8} \) \(a_{32}= +0.25339792 \pm 1.7 \cdot 10^{-8} \) \(a_{33}= -0.60044191 \pm 1.5 \cdot 10^{-8} \)
\(a_{34}= -1.65502898 \pm 2.2 \cdot 10^{-8} \) \(a_{35}= +0.66022242 \pm 1.6 \cdot 10^{-8} \) \(a_{36}= -1.42029930 \pm 1.5 \cdot 10^{-8} \)
\(a_{37}= +1.07027123 \pm 1.5 \cdot 10^{-8} \) \(a_{38}= -0.00003766 \pm 2.5 \cdot 10^{-8} \) \(a_{39}= +0.14179744 \pm 2.8 \cdot 10^{-8} \)
\(a_{40}= -2.15345011 \pm 2.0 \cdot 10^{-8} \) \(a_{41}= +0.52270619 \pm 2.5 \cdot 10^{-8} \) \(a_{42}= +0.42284138 \pm 2.0 \cdot 10^{-8} \)
\(a_{43}= +0.43744289 \pm 1.4 \cdot 10^{-8} \) \(a_{44}= +2.25835644 \pm 1.0 \cdot 10^{-8} \) \(a_{45}= -1.00804458 \pm 1.3 \cdot 10^{-8} \)
\(a_{46}= +2.18434227 \pm 1.5 \cdot 10^{-8} \) \(a_{47}= -0.44286120 \pm 1.3 \cdot 10^{-8} \) \(a_{48}= -0.39607449 \pm 1.4 \cdot 10^{-8} \)
\(a_{49}= -0.76597748 \pm 1.3 \cdot 10^{-8} \) \(a_{50}= -1.47477279 \pm 1.7 \cdot 10^{-8} \) \(a_{51}= -0.49492239 \pm 1.8 \cdot 10^{-8} \)
\(a_{52}= -0.53332248 \pm 3.5 \cdot 10^{-8} \) \(a_{53}= +1.62729118 \pm 1.8 \cdot 10^{-8} \) \(a_{54}= -1.51968003 \pm 2.0 \cdot 10^{-8} \)
\(a_{55}= +1.60284807 \pm 1.2 \cdot 10^{-8} \) \(a_{56}= -0.76331220 \pm 2.1 \cdot 10^{-8} \) \(a_{57}= -0.00001126 \pm 1.0 \cdot 10^{-8} \)
\(a_{58}= +2.89012670 \pm 2.7 \cdot 10^{-8} \) \(a_{59}= -0.00812788 \pm 1.5 \cdot 10^{-8} \) \(a_{60}= -1.34172369 \pm 1.4 \cdot 10^{-8} \)

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