Properties

Label 13.154
Level $13$
Weight $0$
Character 13.1
Symmetry even
\(R\) 13.14458
Fricke sign $+1$

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Maass form invariants

Level: \( 13 \)
Weight: \( 0 \)
Character: 13.1
Symmetry: even
Fricke sign: $+1$
Spectral parameter: \(13.1445892937610080453709428861 \pm 3 \cdot 10^{-8}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= -0.30285279 \pm 1.1 \cdot 10^{-6} \) \(a_{3}= +0.21575518 \pm 8.8 \cdot 10^{-7} \)
\(a_{4}= -0.90828019 \pm 1.2 \cdot 10^{-6} \) \(a_{5}= -1.02721771 \pm 9.7 \cdot 10^{-7} \) \(a_{6}= -0.06534206 \pm 1.1 \cdot 10^{-6} \)
\(a_{7}= +1.43198668 \pm 8.8 \cdot 10^{-7} \) \(a_{8}= +0.57792798 \pm 1.1 \cdot 10^{-6} \) \(a_{9}= -0.95344970 \pm 7.8 \cdot 10^{-7} \)
\(a_{10}= +0.31109575 \pm 8.8 \cdot 10^{-7} \) \(a_{11}= -1.39422847 \pm 6.6 \cdot 10^{-7} \) \(a_{12}= -0.19596616 \pm 1.1 \cdot 10^{-6} \)
\(a_{13}= -0.27735010 \pm 1.0 \cdot 10^{-8} \) \(a_{14}= -0.43368116 \pm 1.2 \cdot 10^{-6} \) \(a_{15}= -0.22162754 \pm 7.2 \cdot 10^{-7} \)
\(a_{16}= +0.73325309 \pm 7.9 \cdot 10^{-7} \) \(a_{17}= -0.54669496 \pm 8.6 \cdot 10^{-7} \) \(a_{18}= +0.28875490 \pm 9.2 \cdot 10^{-7} \)
\(a_{19}= -1.85071656 \pm 9.4 \cdot 10^{-7} \) \(a_{20}= +0.93300149 \pm 1.0 \cdot 10^{-6} \) \(a_{21}= +0.30895854 \pm 7.2 \cdot 10^{-7} \)
\(a_{22}= +0.42224598 \pm 6.3 \cdot 10^{-7} \) \(a_{23}= +0.22745624 \pm 6.3 \cdot 10^{-7} \) \(a_{24}= +0.12469096 \pm 1.0 \cdot 10^{-6} \)
\(a_{25}= +0.05517621 \pm 1.0 \cdot 10^{-6} \) \(a_{26}= +0.08399625 \pm 1.1 \cdot 10^{-6} \) \(a_{27}= -0.42146689 \pm 9.5 \cdot 10^{-7} \)
\(a_{28}= -1.30064513 \pm 1.3 \cdot 10^{-6} \) \(a_{29}= +0.32871316 \pm 8.3 \cdot 10^{-7} \) \(a_{30}= +0.06712052 \pm 6.0 \cdot 10^{-7} \)
\(a_{31}= -0.70094149 \pm 6.5 \cdot 10^{-7} \) \(a_{32}= -0.79999572 \pm 8.6 \cdot 10^{-7} \) \(a_{33}= -0.30081202 \pm 7.7 \cdot 10^{-7} \)
\(a_{34}= +0.16556809 \pm 1.1 \cdot 10^{-6} \) \(a_{35}= -1.47096207 \pm 8.0 \cdot 10^{-7} \) \(a_{36}= +0.86599947 \pm 7.7 \cdot 10^{-7} \)
\(a_{37}= +1.58937437 \pm 7.7 \cdot 10^{-7} \) \(a_{38}= +0.56049467 \pm 1.2 \cdot 10^{-6} \) \(a_{39}= -0.05983972 \pm 8.9 \cdot 10^{-7} \)
\(a_{40}= -0.59365785 \pm 1.0 \cdot 10^{-6} \) \(a_{41}= -1.40541165 \pm 1.2 \cdot 10^{-6} \) \(a_{42}= -0.09356896 \pm 1.0 \cdot 10^{-6} \)
\(a_{43}= +0.69052302 \pm 7.3 \cdot 10^{-7} \) \(a_{44}= +1.26635010 \pm 5.1 \cdot 10^{-7} \) \(a_{45}= +0.97940042 \pm 6.7 \cdot 10^{-7} \)
\(a_{46}= -0.06888576 \pm 7.5 \cdot 10^{-7} \) \(a_{47}= +0.38806137 \pm 6.7 \cdot 10^{-7} \) \(a_{48}= +0.15820315 \pm 7.1 \cdot 10^{-7} \)
\(a_{49}= +1.05058585 \pm 6.7 \cdot 10^{-7} \) \(a_{50}= -0.01671027 \pm 8.6 \cdot 10^{-7} \) \(a_{51}= -0.11795227 \pm 9.4 \cdot 10^{-7} \)
\(a_{52}= +0.25191160 \pm 1.2 \cdot 10^{-6} \) \(a_{53}= +1.10079537 \pm 9.3 \cdot 10^{-7} \) \(a_{54}= +0.12764242 \pm 1.0 \cdot 10^{-6} \)
\(a_{55}= +1.43217617 \pm 6.2 \cdot 10^{-7} \) \(a_{56}= +0.82758517 \pm 1.0 \cdot 10^{-6} \) \(a_{57}= -0.39930169 \pm 5.3 \cdot 10^{-7} \)
\(a_{58}= -0.09955170 \pm 1.3 \cdot 10^{-6} \) \(a_{59}= -0.65845074 \pm 7.7 \cdot 10^{-7} \) \(a_{60}= +0.20129991 \pm 7.0 \cdot 10^{-7} \)

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