Properties

Label 47.2
Level $47$
Weight $0$
Character 47.1
Symmetry even
\(R\) 1.111574
Fricke sign $+1$

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Maass form invariants

Level: \( 47 \)
Weight: \( 0 \)
Character: 47.1
Symmetry: even
Fricke sign: $+1$
Spectral parameter: \(1.11157408467367535651204324816 \pm 4 \cdot 10^{-9}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= +0.05629776 \pm 9.3 \cdot 10^{-6} \) \(a_{3}= -1.06670731 \pm 8.6 \cdot 10^{-6} \)
\(a_{4}= -0.99683056 \pm 1.0 \cdot 10^{-5} \) \(a_{5}= +0.48104477 \pm 7.5 \cdot 10^{-6} \) \(a_{6}= -0.06005323 \pm 1.1 \cdot 10^{-5} \)
\(a_{7}= +1.74740970 \pm 8.1 \cdot 10^{-6} \) \(a_{8}= -0.11241709 \pm 1.0 \cdot 10^{-5} \) \(a_{9}= +0.13786449 \pm 7.9 \cdot 10^{-6} \)
\(a_{10}= +0.02708174 \pm 9.4 \cdot 10^{-6} \) \(a_{11}= -0.02720978 \pm 7.2 \cdot 10^{-6} \) \(a_{12}= +1.06332645 \pm 1.3 \cdot 10^{-5} \)
\(a_{13}= -1.20297660 \pm 7.4 \cdot 10^{-6} \) \(a_{14}= +0.09837525 \pm 8.8 \cdot 10^{-6} \) \(a_{15}= -0.51313397 \pm 8.5 \cdot 10^{-6} \)
\(a_{16}= +0.99050173 \pm 9.9 \cdot 10^{-6} \) \(a_{17}= +0.68144778 \pm 7.6 \cdot 10^{-6} \) \(a_{18}= +0.00776146 \pm 1.0 \cdot 10^{-5} \)
\(a_{19}= -0.20551900 \pm 7.5 \cdot 10^{-6} \) \(a_{20}= -0.47952013 \pm 1.0 \cdot 10^{-5} \) \(a_{21}= -1.86397470 \pm 9.1 \cdot 10^{-6} \)
\(a_{22}= -0.00153185 \pm 8.8 \cdot 10^{-6} \) \(a_{23}= +1.81118463 \pm 8.2 \cdot 10^{-6} \) \(a_{24}= +0.11991613 \pm 1.4 \cdot 10^{-5} \)
\(a_{25}= -0.76859593 \pm 7.2 \cdot 10^{-6} \) \(a_{26}= -0.06772489 \pm 9.1 \cdot 10^{-6} \) \(a_{27}= +0.91964625 \pm 8.3 \cdot 10^{-6} \)
\(a_{28}= -1.74187139 \pm 9.4 \cdot 10^{-6} \) \(a_{29}= -0.37389657 \pm 7.1 \cdot 10^{-6} \) \(a_{30}= -0.02888829 \pm 1.1 \cdot 10^{-5} \)
\(a_{31}= -0.90378829 \pm 7.5 \cdot 10^{-6} \) \(a_{32}= +0.16818012 \pm 9.4 \cdot 10^{-6} \) \(a_{33}= +0.02902487 \pm 8.3 \cdot 10^{-6} \)
\(a_{34}= +0.03836398 \pm 9.2 \cdot 10^{-6} \) \(a_{35}= +0.84058229 \pm 8.3 \cdot 10^{-6} \) \(a_{36}= -0.13742754 \pm 1.2 \cdot 10^{-5} \)
\(a_{37}= -0.68588149 \pm 7.0 \cdot 10^{-6} \) \(a_{38}= -0.01157026 \pm 8.7 \cdot 10^{-6} \) \(a_{39}= +1.28322393 \pm 8.4 \cdot 10^{-6} \)
\(a_{40}= -0.05407765 \pm 1.0 \cdot 10^{-5} \) \(a_{41}= +1.27690819 \pm 8.0 \cdot 10^{-6} \) \(a_{42}= -0.10493760 \pm 1.0 \cdot 10^{-5} \)
\(a_{43}= +0.25729750 \pm 7.3 \cdot 10^{-6} \) \(a_{44}= +0.02712354 \pm 9.2 \cdot 10^{-6} \) \(a_{45}= +0.06631899 \pm 8.0 \cdot 10^{-6} \)
\(a_{46}= +0.10196564 \pm 7.7 \cdot 10^{-6} \) \(a_{47}= -0.14586499 \pm 1.0 \cdot 10^{-8} \) \(a_{48}= -1.05657544 \pm 1.3 \cdot 10^{-5} \)
\(a_{49}= +2.05344065 \pm 7.8 \cdot 10^{-6} \) \(a_{50}= -0.04327023 \pm 8.7 \cdot 10^{-6} \) \(a_{51}= -0.72690533 \pm 7.5 \cdot 10^{-6} \)
\(a_{52}= +1.19916384 \pm 9.6 \cdot 10^{-6} \) \(a_{53}= -1.11129749 \pm 7.1 \cdot 10^{-6} \) \(a_{54}= +0.05177403 \pm 9.2 \cdot 10^{-6} \)
\(a_{55}= -0.01308912 \pm 6.8 \cdot 10^{-6} \) \(a_{56}= -0.19643871 \pm 8.7 \cdot 10^{-6} \) \(a_{57}= +0.21922862 \pm 7.8 \cdot 10^{-6} \)
\(a_{58}= -0.02104954 \pm 8.9 \cdot 10^{-6} \) \(a_{59}= +0.37138481 \pm 8.0 \cdot 10^{-6} \) \(a_{60}= +0.51150763 \pm 1.3 \cdot 10^{-5} \)

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