Properties

Label 13.195
Level $13$
Weight $0$
Character 13.1
Symmetry odd
\(R\) 14.68678
Fricke sign $-1$

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Maass form invariants

Level: \( 13 \)
Weight: \( 0 \)
Character: 13.1
Symmetry: odd
Fricke sign: $-1$
Spectral parameter: \(14.6867865799050458251438388791 \pm 2 \cdot 10^{-7}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= +1.94787247 \pm 1.1 \cdot 10^{-4} \) \(a_{3}= +1.55128076 \pm 1.0 \cdot 10^{-4} \)
\(a_{4}= +2.79420716 \pm 1.0 \cdot 10^{-4} \) \(a_{5}= +0.77122104 \pm 9.4 \cdot 10^{-5} \) \(a_{6}= +3.02169709 \pm 1.2 \cdot 10^{-4} \)
\(a_{7}= -0.43375318 \pm 9.7 \cdot 10^{-5} \) \(a_{8}= +3.49488674 \pm 8.3 \cdot 10^{-5} \) \(a_{9}= +1.40647200 \pm 1.0 \cdot 10^{-4} \)
\(a_{10}= +1.50224024 \pm 1.0 \cdot 10^{-4} \) \(a_{11}= -1.52707194 \pm 8.9 \cdot 10^{-5} \) \(a_{12}= +4.33459981 \pm 1.0 \cdot 10^{-4} \)
\(a_{13}= +0.27735010 \pm 1.0 \cdot 10^{-8} \) \(a_{14}= -0.84489589 \pm 1.0 \cdot 10^{-4} \) \(a_{15}= +1.19638036 \pm 1.1 \cdot 10^{-4} \)
\(a_{16}= +4.01338651 \pm 8.6 \cdot 10^{-5} \) \(a_{17}= +0.24554588 \pm 9.7 \cdot 10^{-5} \) \(a_{18}= +2.73962808 \pm 1.1 \cdot 10^{-4} \)
\(a_{19}= -0.91265373 \pm 7.9 \cdot 10^{-5} \) \(a_{20}= +2.15495136 \pm 9.8 \cdot 10^{-5} \) \(a_{21}= -0.67287297 \pm 8.3 \cdot 10^{-5} \)
\(a_{22}= -2.97454139 \pm 9.8 \cdot 10^{-5} \) \(a_{23}= -0.04233970 \pm 8.1 \cdot 10^{-5} \) \(a_{24}= +5.42155056 \pm 8.1 \cdot 10^{-5} \)
\(a_{25}= -0.40521811 \pm 7.3 \cdot 10^{-5} \) \(a_{26}= +0.54024262 \pm 1.1 \cdot 10^{-4} \) \(a_{27}= +0.63055219 \pm 8.6 \cdot 10^{-5} \)
\(a_{28}= -1.21199625 \pm 9.1 \cdot 10^{-5} \) \(a_{29}= +0.98052051 \pm 7.2 \cdot 10^{-5} \) \(a_{30}= +2.33039637 \pm 1.3 \cdot 10^{-4} \)
\(a_{31}= +0.70762566 \pm 1.0 \cdot 10^{-4} \) \(a_{32}= +4.32267836 \pm 1.1 \cdot 10^{-4} \) \(a_{33}= -2.36891732 \pm 7.7 \cdot 10^{-5} \)
\(a_{34}= +0.47829207 \pm 1.0 \cdot 10^{-4} \) \(a_{35}= -0.33451958 \pm 7.8 \cdot 10^{-5} \) \(a_{36}= +3.92997412 \pm 9.7 \cdot 10^{-5} \)
\(a_{37}= -1.45487692 \pm 1.2 \cdot 10^{-4} \) \(a_{38}= -1.77773308 \pm 7.6 \cdot 10^{-5} \) \(a_{39}= +0.43024787 \pm 1.0 \cdot 10^{-4} \)
\(a_{40}= +2.69533019 \pm 8.3 \cdot 10^{-5} \) \(a_{41}= +0.15920503 \pm 8.7 \cdot 10^{-5} \) \(a_{42}= -1.31067073 \pm 6.9 \cdot 10^{-5} \)
\(a_{43}= +0.15553059 \pm 9.7 \cdot 10^{-5} \) \(a_{44}= -4.26695535 \pm 9.3 \cdot 10^{-5} \) \(a_{45}= +1.08470080 \pm 1.0 \cdot 10^{-4} \)
\(a_{46}= -0.08247233 \pm 1.0 \cdot 10^{-4} \) \(a_{47}= -1.18094594 \pm 1.0 \cdot 10^{-4} \) \(a_{48}= +6.22588927 \pm 9.8 \cdot 10^{-5} \)
\(a_{49}= -0.81185818 \pm 8.8 \cdot 10^{-5} \) \(a_{50}= -0.78931319 \pm 8.7 \cdot 10^{-5} \) \(a_{51}= +0.38091060 \pm 1.1 \cdot 10^{-4} \)
\(a_{52}= +0.77497363 \pm 1.0 \cdot 10^{-4} \) \(a_{53}= -0.33552997 \pm 7.4 \cdot 10^{-5} \) \(a_{54}= +1.22823524 \pm 1.0 \cdot 10^{-4} \)
\(a_{55}= -1.17771001 \pm 6.4 \cdot 10^{-5} \) \(a_{56}= -1.51591825 \pm 7.3 \cdot 10^{-5} \) \(a_{57}= -1.41578218 \pm 7.7 \cdot 10^{-5} \)
\(a_{58}= +1.90992891 \pm 7.0 \cdot 10^{-5} \) \(a_{59}= +0.59839164 \pm 8.9 \cdot 10^{-5} \) \(a_{60}= +3.34293458 \pm 1.1 \cdot 10^{-4} \)

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