Properties

Label 30.1
Level $30$
Weight $0$
Character 30.1
Symmetry odd
\(R\) 0.946903
Fricke sign $-1$

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Maass form invariants

Level: \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( 0 \)
Character: 30.1
Symmetry: odd
Fricke sign: $-1$
Spectral parameter: \(0.946903799785436212673503914063 \pm 3 \cdot 10^{-10}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= +0.70710678 \pm 1.0 \cdot 10^{-8} \) \(a_{3}= -0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= +0.5 \) \(a_{5}= -0.44721360 \pm 1.0 \cdot 10^{-8} \) \(a_{6}= -0.40824829 \pm 1.0 \cdot 10^{-8} \)
\(a_{7}= -0.84779434 \pm 1.0 \cdot 10^{-8} \) \(a_{8}= +0.35355339 \pm 4.2 \cdot 10^{-8} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= -0.31622777 \pm 1.0 \cdot 10^{-8} \) \(a_{11}= +0.97917151 \pm 1 \cdot 10^{-8} \) \(a_{12}= -0.28867513 \pm 5.2 \cdot 10^{-8} \)
\(a_{13}= +0.14415432 \pm 1 \cdot 10^{-8} \) \(a_{14}= -0.59948113 \pm 2.0 \cdot 10^{-8} \) \(a_{15}= +0.25819889 \pm 1.0 \cdot 10^{-8} \)
\(a_{16}= +0.25 \) \(a_{17}= -0.38651276 \pm 1 \cdot 10^{-8} \) \(a_{18}= +0.23570226 \pm 7.3 \cdot 10^{-8} \)
\(a_{19}= +1.08485840 \pm 1 \cdot 10^{-8} \) \(a_{20}= -0.22360680 \pm 8.4 \cdot 10^{-8} \) \(a_{21}= +0.48947429 \pm 2.0 \cdot 10^{-8} \)
\(a_{22}= +0.69237882 \pm 1.9 \cdot 10^{-8} \) \(a_{23}= -0.80749794 \pm 1 \cdot 10^{-8} \) \(a_{24}= -0.20412415 \pm 9.4 \cdot 10^{-8} \)
\(a_{25}= +0.2 \) \(a_{26}= +0.10193250 \pm 1.7 \cdot 10^{-8} \) \(a_{27}= -0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= -0.42389717 \pm 2.0 \cdot 10^{-8} \) \(a_{29}= -1.28512431 \pm 1 \cdot 10^{-8} \) \(a_{30}= +0.18257419 \pm 1.0 \cdot 10^{-8} \)
\(a_{31}= -1.57899587 \pm 1.1 \cdot 10^{-8} \) \(a_{32}= +0.17677670 \pm 1.1 \cdot 10^{-7} \) \(a_{33}= -0.56532494 \pm 1.9 \cdot 10^{-8} \)
\(a_{34}= -0.27330579 \pm 1.9 \cdot 10^{-8} \) \(a_{35}= +0.37914516 \pm 2.0 \cdot 10^{-8} \) \(a_{36}= +0.16666667 \pm 1.0 \cdot 10^{-7} \)
\(a_{37}= +1.33246222 \pm 1 \cdot 10^{-8} \) \(a_{38}= +0.76711073 \pm 1.7 \cdot 10^{-8} \) \(a_{39}= -0.08322754 \pm 1.7 \cdot 10^{-8} \)
\(a_{40}= -0.15811388 \pm 1.2 \cdot 10^{-7} \) \(a_{41}= +0.39745108 \pm 1 \cdot 10^{-8} \) \(a_{42}= +0.34611059 \pm 2.0 \cdot 10^{-8} \)
\(a_{43}= +0.16592527 \pm 1 \cdot 10^{-8} \) \(a_{44}= +0.48958576 \pm 1.9 \cdot 10^{-8} \) \(a_{45}= -0.14907120 \pm 1.4 \cdot 10^{-7} \)
\(a_{46}= -0.57098727 \pm 1.9 \cdot 10^{-8} \) \(a_{47}= +1.15405969 \pm 1 \cdot 10^{-8} \) \(a_{48}= -0.14433757 \pm 1.5 \cdot 10^{-7} \)
\(a_{49}= -0.28124475 \pm 1 \cdot 10^{-8} \) \(a_{50}= +0.14142136 \pm 1.5 \cdot 10^{-7} \) \(a_{51}= +0.22315325 \pm 1.9 \cdot 10^{-8} \)
\(a_{52}= +0.07207716 \pm 1.7 \cdot 10^{-8} \) \(a_{53}= +0.75702868 \pm 1 \cdot 10^{-8} \) \(a_{54}= -0.13608276 \pm 1.6 \cdot 10^{-7} \)
\(a_{55}= -0.43789881 \pm 1.9 \cdot 10^{-8} \) \(a_{56}= -0.29974056 \pm 2.0 \cdot 10^{-8} \) \(a_{57}= -0.62634329 \pm 1.7 \cdot 10^{-8} \)
\(a_{58}= -0.90872011 \pm 2.0 \cdot 10^{-8} \) \(a_{59}= -0.45847930 \pm 1 \cdot 10^{-8} \) \(a_{60}= +0.12909944 \pm 1.7 \cdot 10^{-7} \)

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