Properties

Label 21.48
Level $21$
Weight $0$
Character 21.1
Symmetry odd
\(R\) 6.859098
Fricke sign $+1$

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Maass form invariants

Level: \( 21 = 3 \cdot 7 \)
Weight: \( 0 \)
Character: 21.1
Symmetry: odd
Fricke sign: $+1$
Spectral parameter: \(6.85909810893757400230158763128 \pm 4 \cdot 10^{-10}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= +1.20755961 \pm 1.7 \cdot 10^{-6} \) \(a_{3}= -0.57735027 \pm 1.0 \cdot 10^{-8} \)
\(a_{4}= +0.45820021 \pm 1.9 \cdot 10^{-6} \) \(a_{5}= +0.72418480 \pm 1.3 \cdot 10^{-6} \) \(a_{6}= -0.69718487 \pm 1.7 \cdot 10^{-6} \)
\(a_{7}= -0.37796447 \pm 1.0 \cdot 10^{-8} \) \(a_{8}= -0.65425554 \pm 1.9 \cdot 10^{-6} \) \(a_{9}= +0.33333333 \pm 4.2 \cdot 10^{-8} \)
\(a_{10}= +0.87449632 \pm 1.6 \cdot 10^{-6} \) \(a_{11}= +0.09359931 \pm 1.3 \cdot 10^{-6} \) \(a_{12}= -0.26454201 \pm 1.9 \cdot 10^{-6} \)
\(a_{13}= -1.27477243 \pm 1.5 \cdot 10^{-6} \) \(a_{14}= -0.45641463 \pm 1.7 \cdot 10^{-6} \) \(a_{15}= -0.41810829 \pm 1.3 \cdot 10^{-6} \)
\(a_{16}= -1.24825278 \pm 1.6 \cdot 10^{-6} \) \(a_{17}= -0.66253888 \pm 1.4 \cdot 10^{-6} \) \(a_{18}= +0.40251987 \pm 1.7 \cdot 10^{-6} \)
\(a_{19}= -0.49939365 \pm 1.4 \cdot 10^{-6} \) \(a_{20}= +0.33182163 \pm 1.9 \cdot 10^{-6} \) \(a_{21}= +0.21821789 \pm 1.0 \cdot 10^{-8} \)
\(a_{22}= +0.11302675 \pm 1.7 \cdot 10^{-6} \) \(a_{23}= -1.57396439 \pm 1.4 \cdot 10^{-6} \) \(a_{24}= +0.37773461 \pm 1.9 \cdot 10^{-6} \)
\(a_{25}= -0.47555637 \pm 1.3 \cdot 10^{-6} \) \(a_{26}= -1.53936370 \pm 1.6 \cdot 10^{-6} \) \(a_{27}= -0.19245009 \pm 9.4 \cdot 10^{-8} \)
\(a_{28}= -0.17318340 \pm 1.9 \cdot 10^{-6} \) \(a_{29}= -0.22794120 \pm 1.5 \cdot 10^{-6} \) \(a_{30}= -0.50489068 \pm 3.0 \cdot 10^{-6} \)
\(a_{31}= +1.40290068 \pm 1.5 \cdot 10^{-6} \) \(a_{32}= -0.85308409 \pm 1.5 \cdot 10^{-6} \) \(a_{33}= -0.05403959 \pm 1.3 \cdot 10^{-6} \)
\(a_{34}= -0.80005519 \pm 1.3 \cdot 10^{-6} \) \(a_{35}= -0.27371613 \pm 1.3 \cdot 10^{-6} \) \(a_{36}= +0.15273340 \pm 1.9 \cdot 10^{-6} \)
\(a_{37}= +1.61380135 \pm 1.3 \cdot 10^{-6} \) \(a_{38}= -0.60304760 \pm 1.6 \cdot 10^{-6} \) \(a_{39}= +0.73599021 \pm 1.5 \cdot 10^{-6} \)
\(a_{40}= -0.47380192 \pm 1.8 \cdot 10^{-6} \) \(a_{41}= -0.28987441 \pm 1.1 \cdot 10^{-6} \) \(a_{42}= +0.26351111 \pm 1.7 \cdot 10^{-6} \)
\(a_{43}= +0.48770373 \pm 1.5 \cdot 10^{-6} \) \(a_{44}= +0.04288722 \pm 1.9 \cdot 10^{-6} \) \(a_{45}= +0.24139493 \pm 1.3 \cdot 10^{-6} \)
\(a_{46}= -1.90065582 \pm 1.6 \cdot 10^{-6} \) \(a_{47}= -0.50665410 \pm 1.3 \cdot 10^{-6} \) \(a_{48}= +0.72067908 \pm 1.6 \cdot 10^{-6} \)
\(a_{49}= +0.14285714 \pm 1.5 \cdot 10^{-7} \) \(a_{50}= -0.57426267 \pm 1.3 \cdot 10^{-6} \) \(a_{51}= +0.38251700 \pm 1.4 \cdot 10^{-6} \)
\(a_{52}= -0.58410099 \pm 1.8 \cdot 10^{-6} \) \(a_{53}= +0.99961283 \pm 1.5 \cdot 10^{-6} \) \(a_{54}= -0.23239496 \pm 1.7 \cdot 10^{-6} \)
\(a_{55}= +0.06778320 \pm 1.1 \cdot 10^{-6} \) \(a_{56}= +0.24728535 \pm 1.9 \cdot 10^{-6} \) \(a_{57}= +0.28832506 \pm 1.4 \cdot 10^{-6} \)
\(a_{58}= -0.27525258 \pm 1.4 \cdot 10^{-6} \) \(a_{59}= -0.76470330 \pm 1.0 \cdot 10^{-6} \) \(a_{60}= -0.19157731 \pm 3.3 \cdot 10^{-6} \)

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