Properties

Label 15.99
Level $15$
Weight $0$
Character 15.1
Symmetry odd
\(R\) 12.22540
Fricke sign $+1$

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Maass form invariants

Level: \( 15 = 3 \cdot 5 \)
Weight: \( 0 \)
Character: 15.1
Symmetry: odd
Fricke sign: $+1$
Spectral parameter: \(12.2254089157236274658501505337 \pm 2 \cdot 10^{-8}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= +0.08727140 \pm 6.3 \cdot 10^{-5} \) \(a_{3}= +0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= -0.99238370 \pm 6.3 \cdot 10^{-5} \) \(a_{5}= +0.44721360 \pm 1.0 \cdot 10^{-8} \) \(a_{6}= +0.05038616 \pm 6.3 \cdot 10^{-5} \)
\(a_{7}= -0.44505338 \pm 5.4 \cdot 10^{-5} \) \(a_{8}= -0.17387811 \pm 4.5 \cdot 10^{-5} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= +0.03902895 \pm 6.3 \cdot 10^{-5} \) \(a_{11}= +0.71673436 \pm 5.5 \cdot 10^{-5} \) \(a_{12}= -0.57295300 \pm 6.3 \cdot 10^{-5} \)
\(a_{13}= +0.24291178 \pm 5.4 \cdot 10^{-5} \) \(a_{14}= -0.03884043 \pm 6.4 \cdot 10^{-5} \) \(a_{15}= +0.25819889 \pm 1.0 \cdot 10^{-8} \)
\(a_{16}= +0.97720912 \pm 5.3 \cdot 10^{-5} \) \(a_{17}= -0.03834011 \pm 4.7 \cdot 10^{-5} \) \(a_{18}= +0.02909047 \pm 6.3 \cdot 10^{-5} \)
\(a_{19}= -1.36361919 \pm 4.7 \cdot 10^{-5} \) \(a_{20}= -0.44380748 \pm 6.3 \cdot 10^{-5} \) \(a_{21}= -0.25695169 \pm 5.4 \cdot 10^{-5} \)
\(a_{22}= +0.06255041 \pm 6.5 \cdot 10^{-5} \) \(a_{23}= -0.11093054 \pm 4.2 \cdot 10^{-5} \) \(a_{24}= -0.10038857 \pm 4.5 \cdot 10^{-5} \)
\(a_{25}= +0.2 \) \(a_{26}= +0.02119925 \pm 7.0 \cdot 10^{-5} \) \(a_{27}= +0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= +0.44166372 \pm 7.0 \cdot 10^{-5} \) \(a_{29}= -0.55720135 \pm 3.5 \cdot 10^{-5} \) \(a_{30}= +0.02253338 \pm 6.3 \cdot 10^{-5} \)
\(a_{31}= +0.51708233 \pm 2.8 \cdot 10^{-5} \) \(a_{32}= +0.25916051 \pm 4.8 \cdot 10^{-5} \) \(a_{33}= +0.41380678 \pm 5.5 \cdot 10^{-5} \)
\(a_{34}= -0.00334599 \pm 6.0 \cdot 10^{-5} \) \(a_{35}= -0.19903392 \pm 5.4 \cdot 10^{-5} \) \(a_{36}= -0.33079457 \pm 6.3 \cdot 10^{-5} \)
\(a_{37}= +1.36437633 \pm 4.1 \cdot 10^{-5} \) \(a_{38}= -0.11900495 \pm 5.3 \cdot 10^{-5} \) \(a_{39}= +0.14024518 \pm 5.4 \cdot 10^{-5} \)
\(a_{40}= -0.07776065 \pm 4.5 \cdot 10^{-5} \) \(a_{41}= -1.42026118 \pm 4.8 \cdot 10^{-5} \) \(a_{42}= -0.02242453 \pm 1.1 \cdot 10^{-4} \)
\(a_{43}= -1.28002518 \pm 5.2 \cdot 10^{-5} \) \(a_{44}= -0.71127550 \pm 5.1 \cdot 10^{-5} \) \(a_{45}= +0.14907120 \pm 1.4 \cdot 10^{-7} \)
\(a_{46}= -0.00968106 \pm 3.1 \cdot 10^{-5} \) \(a_{47}= -1.62606054 \pm 6.3 \cdot 10^{-5} \) \(a_{48}= +0.56419195 \pm 5.3 \cdot 10^{-5} \)
\(a_{49}= -0.80192749 \pm 4.0 \cdot 10^{-5} \) \(a_{50}= +0.01745428 \pm 6.3 \cdot 10^{-5} \) \(a_{51}= -0.02213567 \pm 4.8 \cdot 10^{-5} \)
\(a_{52}= -0.24106169 \pm 5.6 \cdot 10^{-5} \) \(a_{53}= -0.51990822 \pm 6.7 \cdot 10^{-5} \) \(a_{54}= +0.01679539 \pm 6.3 \cdot 10^{-5} \)
\(a_{55}= +0.32053335 \pm 5.5 \cdot 10^{-5} \) \(a_{56}= +0.07738504 \pm 4.6 \cdot 10^{-5} \) \(a_{57}= -0.78728591 \pm 4.7 \cdot 10^{-5} \)
\(a_{58}= -0.04862774 \pm 4.6 \cdot 10^{-5} \) \(a_{59}= -1.10187802 \pm 4.4 \cdot 10^{-5} \) \(a_{60}= -0.25623237 \pm 6.3 \cdot 10^{-5} \)

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