Properties

Label 31.1
Level $31$
Weight $0$
Character 31.1
Symmetry even
\(R\) 0.789356
Fricke sign $+1$

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The Maass form of level 31 with the smallest eigenvalue. This Maass form is even, and for all smaller prime levels the Maass form with the smallest eigenvalue is odd.

Maass form invariants

Level: \( 31 \)
Weight: \( 0 \)
Character: 31.1
Symmetry: even
Fricke sign: $+1$
Spectral parameter: \(0.789356177738072661896459134574 \pm 3 \cdot 10^{-10}\)

Maass form coefficients

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

\(a_{1}= +1 \) \(a_{2}= -1.37431044 \pm 3.4 \cdot 10^{-8} \) \(a_{3}= -0.90270409 \pm 3.1 \cdot 10^{-8} \)
\(a_{4}= +0.88872919 \pm 3.5 \cdot 10^{-8} \) \(a_{5}= -0.73004223 \pm 3.0 \cdot 10^{-8} \) \(a_{6}= +1.24059565 \pm 3.5 \cdot 10^{-8} \)
\(a_{7}= +1.12629679 \pm 3.0 \cdot 10^{-8} \) \(a_{8}= +0.15292063 \pm 3.4 \cdot 10^{-8} \) \(a_{9}= -0.18512533 \pm 2.9 \cdot 10^{-8} \)
\(a_{10}= +1.00330466 \pm 2.9 \cdot 10^{-8} \) \(a_{11}= -0.21446791 \pm 3.0 \cdot 10^{-8} \) \(a_{12}= -0.80225948 \pm 3.3 \cdot 10^{-8} \)
\(a_{13}= -0.75720569 \pm 2.4 \cdot 10^{-8} \) \(a_{14}= -1.54788144 \pm 4.1 \cdot 10^{-8} \) \(a_{15}= +0.65901211 \pm 3.2 \cdot 10^{-8} \)
\(a_{16}= -1.09888961 \pm 3.6 \cdot 10^{-8} \) \(a_{17}= +0.29447442 \pm 2.8 \cdot 10^{-8} \) \(a_{18}= +0.25441968 \pm 2.6 \cdot 10^{-8} \)
\(a_{19}= +1.66968382 \pm 2.7 \cdot 10^{-8} \) \(a_{20}= -0.64880984 \pm 3.0 \cdot 10^{-8} \) \(a_{21}= -1.01671271 \pm 3.0 \cdot 10^{-8} \)
\(a_{22}= +0.29474549 \pm 3.5 \cdot 10^{-8} \) \(a_{23}= -0.18660272 \pm 2.4 \cdot 10^{-8} \) \(a_{24}= -0.13804208 \pm 3.6 \cdot 10^{-8} \)
\(a_{25}= -0.46703834 \pm 3.2 \cdot 10^{-8} \) \(a_{26}= +1.04063569 \pm 2.7 \cdot 10^{-8} \) \(a_{27}= +1.06981748 \pm 3.1 \cdot 10^{-8} \)
\(a_{28}= +1.00097284 \pm 4.3 \cdot 10^{-8} \) \(a_{29}= +0.62388537 \pm 2.8 \cdot 10^{-8} \) \(a_{30}= -0.90568722 \pm 3.2 \cdot 10^{-8} \)
\(a_{31}= -0.17960530 \pm 1.0 \cdot 10^{-8} \) \(a_{32}= +1.35729484 \pm 3.5 \cdot 10^{-8} \) \(a_{33}= +0.19360106 \pm 3.4 \cdot 10^{-8} \)
\(a_{34}= -0.40469928 \pm 3.0 \cdot 10^{-8} \) \(a_{35}= -0.82224422 \pm 3.1 \cdot 10^{-8} \) \(a_{36}= -0.16452629 \pm 2.6 \cdot 10^{-8} \)
\(a_{37}= -0.55042516 \pm 3.0 \cdot 10^{-8} \) \(a_{38}= -2.29466391 \pm 3.5 \cdot 10^{-8} \) \(a_{39}= +0.68353267 \pm 3.0 \cdot 10^{-8} \)
\(a_{40}= -0.11163852 \pm 2.8 \cdot 10^{-8} \) \(a_{41}= +0.28216170 \pm 2.3 \cdot 10^{-8} \) \(a_{42}= +1.39727890 \pm 4.0 \cdot 10^{-8} \)
\(a_{43}= -1.75438082 \pm 3.0 \cdot 10^{-8} \) \(a_{44}= -0.19060389 \pm 3.7 \cdot 10^{-8} \) \(a_{45}= +0.13514931 \pm 3.0 \cdot 10^{-8} \)
\(a_{46}= +0.25645006 \pm 3.0 \cdot 10^{-8} \) \(a_{47}= +0.91350446 \pm 2.6 \cdot 10^{-8} \) \(a_{48}= +0.99197215 \pm 3.3 \cdot 10^{-8} \)
\(a_{49}= +0.26854446 \pm 2.6 \cdot 10^{-8} \) \(a_{50}= +0.64185567 \pm 3.0 \cdot 10^{-8} \) \(a_{51}= -0.26582327 \pm 2.7 \cdot 10^{-8} \)
\(a_{52}= -0.67295080 \pm 2.6 \cdot 10^{-8} \) \(a_{53}= -0.05436985 \pm 3.0 \cdot 10^{-8} \) \(a_{54}= -1.47026134 \pm 3.3 \cdot 10^{-8} \)
\(a_{55}= +0.15657063 \pm 3.1 \cdot 10^{-8} \) \(a_{56}= +0.17223401 \pm 4.4 \cdot 10^{-8} \) \(a_{57}= -1.50723041 \pm 2.5 \cdot 10^{-8} \)
\(a_{58}= -0.85741218 \pm 3.5 \cdot 10^{-8} \) \(a_{59}= +0.11224316 \pm 2.4 \cdot 10^{-8} \) \(a_{60}= +0.58568330 \pm 3.0 \cdot 10^{-8} \)

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