Properties

Label 3.37
Level $3$
Weight $0$
Character 3.1
Symmetry odd
\(R\) 16.99056
Fricke sign $-1$

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Maass form invariants

Level: \( 3 \)
Weight: \( 0 \)
Character: 3.1
Symmetry: odd
Fricke sign: $-1$
Spectral parameter: \(16.9905650918938697774436857509 \pm 3 \cdot 10^{-6}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= +1.59187815 \pm 2.6 \cdot 10^{-3} \) \(a_{3}= +0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= +1.53407605 \pm 4.8 \cdot 10^{-3} \) \(a_{5}= +1.84128560 \pm 3.6 \cdot 10^{-3} \) \(a_{6}= +0.91907128 \pm 2.6 \cdot 10^{-3} \)
\(a_{7}= -0.58632653 \pm 3.0 \cdot 10^{-3} \) \(a_{8}= +0.85018399 \pm 3.2 \cdot 10^{-3} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= +2.93110232 \pm 2.1 \cdot 10^{-3} \) \(a_{11}= -0.06533740 \pm 3.3 \cdot 10^{-3} \) \(a_{12}= +0.88569922 \pm 4.8 \cdot 10^{-3} \)
\(a_{13}= -0.62373348 \pm 3.2 \cdot 10^{-3} \) \(a_{14}= -0.93336040 \pm 2.0 \cdot 10^{-3} \) \(a_{15}= +1.06306674 \pm 3.6 \cdot 10^{-3} \)
\(a_{16}= -0.18068673 \pm 3.9 \cdot 10^{-3} \) \(a_{17}= +0.59673171 \pm 2.4 \cdot 10^{-3} \) \(a_{18}= +0.53062605 \pm 2.6 \cdot 10^{-3} \)
\(a_{19}= +0.12792492 \pm 2.0 \cdot 10^{-3} \) \(a_{20}= +2.82467213 \pm 3.4 \cdot 10^{-3} \) \(a_{21}= -0.33851578 \pm 3.0 \cdot 10^{-3} \)
\(a_{22}= -0.10400917 \pm 1.7 \cdot 10^{-3} \) \(a_{23}= -0.25387485 \pm 7.4 \cdot 10^{-3} \) \(a_{24}= +0.49085395 \pm 3.2 \cdot 10^{-3} \)
\(a_{25}= +2.39033266 \pm 3.0 \cdot 10^{-3} \) \(a_{26}= -0.99290770 \pm 1.1 \cdot 10^{-3} \) \(a_{27}= +0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= -0.89946949 \pm 2.0 \cdot 10^{-3} \) \(a_{29}= -0.67260630 \pm 6.9 \cdot 10^{-3} \) \(a_{30}= +1.69227271 \pm 6.3 \cdot 10^{-3} \)
\(a_{31}= +0.67027640 \pm 3.1 \cdot 10^{-3} \) \(a_{32}= -1.13781525 \pm 4.3 \cdot 10^{-3} \) \(a_{33}= -0.03772256 \pm 3.3 \cdot 10^{-3} \)
\(a_{34}= +0.94992417 \pm 2.9 \cdot 10^{-3} \) \(a_{35}= -1.07959460 \pm 1.3 \cdot 10^{-3} \) \(a_{36}= +0.51135868 \pm 4.8 \cdot 10^{-3} \)
\(a_{37}= -0.53701226 \pm 3.7 \cdot 10^{-3} \) \(a_{38}= +0.20364089 \pm 9.1 \cdot 10^{-4} \) \(a_{39}= -0.36011269 \pm 3.2 \cdot 10^{-3} \)
\(a_{40}= +1.56543153 \pm 2.0 \cdot 10^{-3} \) \(a_{41}= -0.93660130 \pm 7.1 \cdot 10^{-3} \) \(a_{42}= -0.53887588 \pm 5.7 \cdot 10^{-3} \)
\(a_{43}= +0.80999460 \pm 4.7 \cdot 10^{-3} \) \(a_{44}= -0.10023253 \pm 2.9 \cdot 10^{-3} \) \(a_{45}= +0.61376187 \pm 3.6 \cdot 10^{-3} \)
\(a_{46}= -0.40413783 \pm 2.1 \cdot 10^{-3} \) \(a_{47}= -1.72643484 \pm 5.8 \cdot 10^{-3} \) \(a_{48}= -0.10431953 \pm 3.9 \cdot 10^{-3} \)
\(a_{49}= -0.65622120 \pm 5.1 \cdot 10^{-3} \) \(a_{50}= +3.80511834 \pm 1.5 \cdot 10^{-3} \) \(a_{51}= +0.34452321 \pm 2.4 \cdot 10^{-3} \)
\(a_{52}= -0.95685459 \pm 3.0 \cdot 10^{-3} \) \(a_{53}= -1.56262872 \pm 2.9 \cdot 10^{-3} \) \(a_{54}= +0.30635709 \pm 2.6 \cdot 10^{-3} \)
\(a_{55}= -0.12030481 \pm 2.3 \cdot 10^{-3} \) \(a_{56}= -0.49848543 \pm 2.9 \cdot 10^{-3} \) \(a_{57}= +0.07385749 \pm 2.0 \cdot 10^{-3} \)
\(a_{58}= -1.07070727 \pm 2.8 \cdot 10^{-3} \) \(a_{59}= +1.46394498 \pm 4.7 \cdot 10^{-3} \) \(a_{60}= +1.63082522 \pm 8.5 \cdot 10^{-3} \)

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