Properties

Label 13.75
Level $13$
Weight $0$
Character 13.1
Symmetry odd
\(R\) 9.481478
Fricke sign $-1$

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Maass form invariants

Level: \( 13 \)
Weight: \( 0 \)
Character: 13.1
Symmetry: odd
Fricke sign: $-1$
Spectral parameter: \(9.48147888733239185975363636513 \pm 3 \cdot 10^{-10}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= -0.38189823 \pm 9.8 \cdot 10^{-7} \) \(a_{3}= +1.74853633 \pm 9.5 \cdot 10^{-7} \)
\(a_{4}= -0.85415374 \pm 8.9 \cdot 10^{-7} \) \(a_{5}= -0.41535158 \pm 8.4 \cdot 10^{-7} \) \(a_{6}= -0.66776293 \pm 1.0 \cdot 10^{-6} \)
\(a_{7}= -0.31001498 \pm 8.7 \cdot 10^{-7} \) \(a_{8}= +0.70809803 \pm 7.4 \cdot 10^{-7} \) \(a_{9}= +2.05737931 \pm 9.2 \cdot 10^{-7} \)
\(a_{10}= +0.15862203 \pm 9.6 \cdot 10^{-7} \) \(a_{11}= +0.09183794 \pm 8.0 \cdot 10^{-7} \) \(a_{12}= -1.49351885 \pm 9.6 \cdot 10^{-7} \)
\(a_{13}= +0.27735010 \pm 1.0 \cdot 10^{-8} \) \(a_{14}= +0.11839417 \pm 9.4 \cdot 10^{-7} \) \(a_{15}= -0.72625732 \pm 1.0 \cdot 10^{-6} \)
\(a_{16}= +0.58373236 \pm 7.7 \cdot 10^{-7} \) \(a_{17}= -0.51752466 \pm 8.7 \cdot 10^{-7} \) \(a_{18}= -0.78570952 \pm 1.0 \cdot 10^{-6} \)
\(a_{19}= +1.51633675 \pm 7.1 \cdot 10^{-7} \) \(a_{20}= +0.35477410 \pm 8.7 \cdot 10^{-7} \) \(a_{21}= -0.54207245 \pm 7.4 \cdot 10^{-7} \)
\(a_{22}= -0.03507275 \pm 8.8 \cdot 10^{-7} \) \(a_{23}= +0.91220643 \pm 7.2 \cdot 10^{-7} \) \(a_{24}= +1.23813514 \pm 7.2 \cdot 10^{-7} \)
\(a_{25}= -0.82748307 \pm 6.5 \cdot 10^{-7} \) \(a_{26}= -0.10591951 \pm 9.9 \cdot 10^{-7} \) \(a_{27}= +1.84886615 \pm 7.7 \cdot 10^{-7} \)
\(a_{28}= +0.26480045 \pm 8.1 \cdot 10^{-7} \) \(a_{29}= +1.54381337 \pm 6.4 \cdot 10^{-7} \) \(a_{30}= +0.27735639 \pm 1.2 \cdot 10^{-6} \)
\(a_{31}= -0.20517106 \pm 9.2 \cdot 10^{-7} \) \(a_{32}= -0.93102439 \pm 1.0 \cdot 10^{-6} \) \(a_{33}= +0.16058198 \pm 6.9 \cdot 10^{-7} \)
\(a_{34}= +0.19764175 \pm 9.5 \cdot 10^{-7} \) \(a_{35}= +0.12876521 \pm 7.0 \cdot 10^{-7} \) \(a_{36}= -1.75731824 \pm 8.7 \cdot 10^{-7} \)
\(a_{37}= +1.91501927 \pm 1.0 \cdot 10^{-6} \) \(a_{38}= -0.57908632 \pm 6.8 \cdot 10^{-7} \) \(a_{39}= +0.48495672 \pm 9.6 \cdot 10^{-7} \)
\(a_{40}= -0.29410963 \pm 7.4 \cdot 10^{-7} \) \(a_{41}= +1.83331789 \pm 7.8 \cdot 10^{-7} \) \(a_{42}= +0.20701651 \pm 6.2 \cdot 10^{-7} \)
\(a_{43}= +0.41048428 \pm 8.7 \cdot 10^{-7} \) \(a_{44}= -0.07844372 \pm 8.3 \cdot 10^{-7} \) \(a_{45}= -0.85453574 \pm 9.3 \cdot 10^{-7} \)
\(a_{46}= -0.34837002 \pm 9.3 \cdot 10^{-7} \) \(a_{47}= -0.09763336 \pm 9.0 \cdot 10^{-7} \) \(a_{48}= +1.02067724 \pm 8.8 \cdot 10^{-7} \)
\(a_{49}= -0.90389071 \pm 7.9 \cdot 10^{-7} \) \(a_{50}= +0.31601432 \pm 7.8 \cdot 10^{-7} \) \(a_{51}= -0.90491067 \pm 1.0 \cdot 10^{-6} \)
\(a_{52}= -0.23689962 \pm 9.0 \cdot 10^{-7} \) \(a_{53}= +0.54767265 \pm 6.6 \cdot 10^{-7} \) \(a_{54}= -0.70607871 \pm 8.9 \cdot 10^{-7} \)
\(a_{55}= -0.03814503 \pm 5.7 \cdot 10^{-7} \) \(a_{56}= -0.21952100 \pm 6.5 \cdot 10^{-7} \) \(a_{57}= +2.65136990 \pm 6.9 \cdot 10^{-7} \)
\(a_{58}= -0.58957959 \pm 6.3 \cdot 10^{-7} \) \(a_{59}= -0.63026236 \pm 8.0 \cdot 10^{-7} \) \(a_{60}= +0.62033541 \pm 1.0 \cdot 10^{-6} \)

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