Properties

Level 10
Symmetry odd
Weight 0
Character \( \chi_{10}(9,\cdot) \)
Multiplicity 2
Precision 1e-11
Fricke Eigenvalue 0
Atkin-Lehner Eigenvalues 1,0,0,0

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Spectral parameter

$R= 7.59598562264$

The first few Fourier Coefficients

n \( c_{ 1 } (n) \) \( c_{ 2 } (n) \)
0  0  0
1 1 1
2 6.893417509e-12 + 0.7071067811i 6.896369557e-12 - 0.7071067811i
3 3.688372198e-11 - 1.569460200i 3.688955137e-11 + 1.569460200i
4 -0.4999999999 - 5.328174474e-12i -0.4999999999 + 5.327941040e-12i
5 0.0927365902 - 0.995690677i 0.0927365902 + 0.995690677i
6 1.109775950 + 6.858943095e-12i 1.109775950 - 6.860044243e-12i
7 -2.224299423e-11 + 1.010699127i -2.224466681e-11 - 1.010699127i
8 -5.249654288e-13 - 0.3535533905i -5.217447590e-13 + 0.3535533905i
9 -1.463205320 - 1.241388980e-11i -1.463205320 + 1.241601731e-11i
10 0.704059629 + 0.0655746718i 0.704059629 - 0.0655746718i
11 -1.103280424 - 1.686066135e-11i -1.103280424 + 1.686011182e-11i
12 -5.413756040e-11 + 0.7847301002i -5.413734634e-11 - 0.7847301002i
13 8.949328090e-12 - 0.4222953600i 8.949447529e-12 + 0.4222953600i
14 -0.7146722065 - 5.328071902e-12i -0.7146722065 + 5.329363613e-12i
15 -1.562696889 - 0.1455463875i -1.562696889 + 0.1455463875i
16 0.2499999999 + 4.358852124e-12i 0.2499999999 - 4.360134093e-12i
17 -2.885544805e-11 + 1.059898078i -2.885787781e-11 - 1.059898078i
18 2.585697553e-11 - 1.034642404i 2.586230968e-11 + 1.034642404i
19 -1.043162719 - 1.070677131e-11i -1.043162719 + 1.070885853e-11i
20 -0.0463682951 + 0.497845338i -0.0463682951 - 0.497845338i
21 1.586252054 + 1.767106309e-11i 1.586252054 - 1.767404474e-11i
22 3.995634280e-11 - 0.7801370700i 3.996533348e-11 + 0.7801370700i
23 -1.212737115e-11 + 0.7199345218i -1.213100831e-11 - 0.7199345218i
24 -0.5548879752 - 3.138496258e-12i -0.5548879752 + 3.136236278e-12i
25 -0.982799849 - 0.184673916i -0.982799849 + 0.184673916i
26 0.2986079127 + 1.595290306e-11i 0.2986079127 - 1.595039703e-11i
27 1.606790574e-11 + 0.7269823154i 1.607578768e-11 - 0.7269823154i
28 -8.545628825e-12 - 0.5053495635i -8.551184632e-12 + 0.5053495635i
29 -0.9886893141 - 1.350131809e-11i -0.9886893141 + 1.350174686e-11i
30 0.1029168376 - 1.104993567i 0.1029168376 + 1.104993567i
31 1.157719180 + 3.799187172e-11i 1.157719180 - 3.799673155e-11i
32 -1.205486888e-10 + 0.1767766953i -1.205560150e-10 - 0.1767766953i
33 -1.829255876e-11 + 1.731554716i -1.828589420e-11 - 1.731554716i
34 -0.7494611186 - 6.469371535e-11i -0.7494611186 + 6.472309801e-11i
35 1.006343698 + 0.09372879077i 1.006343698 - 0.09372879077i
36 0.7316026601 + 7.033162926e-11i 0.7316026601 - 7.034928425e-11i
37 1.943812895e-10 - 1.281225125i 1.944253033e-10 + 1.281225125i
38 1.127857181e-10 - 0.7376274324i 1.128060902e-10 + 0.7376274324i
39 -0.6627757615 + 1.968714940e-10i -0.6627757615 - 1.970552661e-10i
40 -0.352029814 - 0.0327873359i -0.352029814 + 0.0327873359i
41 -0.8098781524 - 3.170539642e-10i -0.8098781524 + 3.174036420e-10i
42 -1.161134036e-9 + 1.121649584i -1.161429804e-9 - 1.121649584i
43 -9.206306116e-10 - 0.1248055200i -9.205731730e-10 + 0.1248055200i
44 0.5516402130 - 1.608362883e-9i 0.5516402130 + 1.608541572e-9i
45 -0.1356926757 + 1.456899897i -0.1356926757 - 1.456899897i
46 -0.5090705842 + 3.532906640e-9i -0.5090705842 - 3.533443391e-9i
47 9.052282690e-9 - 0.8014096362i 9.053579822e-9 + 0.8014096362i
48 1.036900759e-8 - 0.3923650551i 1.036840920e-8 + 0.3923650551i
49 -0.02151274138 - 5.261155640e-9i -0.02151274139 + 5.257755790e-9i
50 0.130584162 - 0.694944424i 0.130584162 + 0.694944424i
51 1.663467872 - 3.701925115e-9i 1.663467872 + 3.706160301e-9i
52 -1.712982569e-9 + 0.2111476885i -1.708897220e-9 - 0.2111476885i
53 -5.764970407e-8 - 1.319493622i -5.765949237e-8 + 1.319493622i
54 -0.5140540461 - 5.245230112e-8i -0.5140540461 + 5.246100490e-8i
55 -0.1023144647 + 1.098526033i -0.1023144647 - 1.098526033i
56 0.3573361033 + 1.752512843e-10i 0.3573361033 - 1.752356703e-10i
57 4.692248429e-11 + 1.637202370i 4.691027982e-11 - 1.637202370i
58 -1.387431852e-10 - 0.6991089187i -1.388674031e-10 + 0.6991089187i
59 -0.7878540945 - 7.642525442e-11i -0.7878540945 + 7.639682909e-11i
60 0.781348444 + 0.0727731936i 0.781348444 - 0.0727731936i
61 -0.7467215694 + 4.801399897e-10i -0.7467215694 - 4.800161210e-10i
62 -3.180645680e-10 + 0.8186310836i -3.179473830e-10 - 0.8186310836i
63 -1.478860341i 1.478860341i
64 -0.1250000004 - 8.321593850e-10i -0.1250000004 + 8.320599196e-10i
65 -0.420475554 - 0.0391622311i -0.420475554 + 0.0391622311i
66 -1.224394079 + 1.109579886e-9i -1.224394079 - 1.109121529e-9i
67 -1.985660417e-9 - 0.3951875094i -1.985839912e-9 + 0.3951875094i
68 2.103355843e-10 - 0.5299490403i 2.104410628e-10 + 0.5299490403i
69 1.129908583 - 5.727515788e-10i 1.129908583 + 5.733757714e-10i
70 -0.0662762704 + 0.711592455i -0.0662762704 - 0.711592455i
71 0.7822691205 + 1.306070613e-9i 0.7822691205 - 1.305936089e-9i
72 3.726532340e-9 + 0.5173211958i 3.725670851e-9 - 0.5173211958i
73 -3.473997601e-9 - 0.1452143960i -3.472150988e-9 + 0.1452143960i
74 0.9059629775 + 5.769868351e-9i 0.9059629775 - 5.769564618e-9i
75 -0.2898383552 + 1.542465244i -0.2898383552 - 1.542465244i
76 0.5215813506 + 3.230545294e-9i 0.5215813506 - 3.231232113e-9i
77 -2.037381353e-8 - 1.115084575i -2.038193442e-8 + 1.115084575i
78 2.324180865e-8 - 0.4686532288i 2.324936134e-8 + 0.4686532288i
79 -0.5538521613 + 2.078593074e-8i -0.5538521613 - 2.078344095e-8i
80 0.0231841463 - 0.248922677i 0.0231841463 + 0.248922677i
81 -0.3222355162 - 1.381497050e-8i -0.3222355162 + 1.381414055e-8i
82 -5.066273633e-8 - 0.5726703565i -5.069721988e-8 + 0.5726703565i
83 7.555038858e-8 - 0.1082037796i 7.556151814e-8 + 0.1082037796i
84 -0.7931260261 - 1.050768529e-10i -0.7931260261 + 1.053344832e-10i
85 1.055330637 + 0.09829133419i 1.055330637 - 0.09829133419i
86 0.08825083017 + 5.367398867e-10i 0.08825083017 - 5.367941042e-10i
87 -5.385500526e-10 + 1.551708527i -5.395216927e-10 - 1.551708527i
88 -2.298250998e-9 + 0.3900685353i -2.298969147e-9 - 0.3900685353i
89 -1.619344988 + 4.522445967e-10i -1.619344988 - 4.525390154e-10i
90 -1.030183790 - 0.09594920703i -1.030183790 + 0.09594920703i
91 0.4268135509 + 1.528114373e-9i 0.4268135509 - 1.528008817e-9i
92 -4.876425387e-9 - 0.3599672623i -4.877602707e-9 + 0.3599672623i
93 3.814634128e-9 - 1.816994176i 3.814512739e-9 + 1.816994176i
94 0.5666821892 - 5.040081320e-9i 0.5666821892 + 5.039920292e-9i
95 -0.09673935516 + 1.038667395i -0.09673935516 - 1.038667395i
96 0.2774439836 - 5.156199734e-9i 0.2774439836 + 5.155379966e-9i
97 1.620977080e-8 - 0.2026129774i 1.621426040e-8 + 0.2026129774i
98 -9.001316408e-9 - 0.01521178737i -9.001673141e-9 + 0.01521178737i
99 1.614325783 + 2.524621674e-9i 1.614325783 - 2.524921520e-9i
100 0.491399944 + 0.0923369647i 0.491399944 - 0.0923369647i
101 -0.9777796913 - 1.832127888e-8i -0.9777796913 + 1.831245667e-8i
102 -1.754344933e-8 + 1.176249408i -1.754582932e-8 - 1.176249408i
103 4.051596870e-8 + 0.5257829114i 4.052250669e-8 - 0.5257829114i
104 -0.1493039040 + 1.548617065e-8i -0.1493039039 - 1.547765345e-8i
105 0.1471035727 - 1.579416369i 0.1471035727 + 1.579416369i
106 0.9330229006 - 4.225409090e-8i 0.9330229006 + 4.225401644e-8i
107 4.462186244e-8 + 0.2018527707i 4.463868623e-8 - 0.2018527707i
108 -1.039570104e-7 - 0.3634912034i -1.039822822e-7 + 0.3634912033i
109 1.528465765 + 3.954164299e-8i 1.528465766 - 3.953161943e-8i
110 -0.776775207 - 0.0723472548i -0.776775207 + 0.0723472548i
111 -2.010831840 - 3.176693445e-10i -2.010831840 + 3.178653879e-10i
112 -8.553833534e-10 + 0.2526747852i -8.552859100e-10 - 0.2526747852i
113 4.095306527e-9 - 0.4971103106i 4.097482587e-9 + 0.4971103106i
114 -1.157676899 + 1.668214098e-9i -1.157676899 - 1.668525218e-9i
115 0.716832078 + 0.0667642725i 0.716832078 - 0.0667642725i
116 0.4943446622 + 3.182071060e-9i 0.4943446622 - 3.181694209e-9i
117 5.995303837e-9 + 0.6179048122i 5.996094836e-9 - 0.6179048122i
118 -1.810920025e-8 - 0.5570969689i -1.811259677e-8 + 0.5570969689i
119 -1.071238049 - 8.409372761e-9i -1.071238049 + 8.412502025e-9i
120 -0.0514583977 + 0.552496768i -0.0514583977 - 0.552496768i
121 0.2172276678 + 2.192644702e-8i 0.2172276678 - 2.192931756e-8i
122 3.454712699e-8 - 0.5280118905i 3.455800275e-8 + 0.5280118905i
123 1.249632766e-8 + 1.271071537i 1.249144896e-8 - 1.271071537i
124 -0.5788596580 + 5.829579870e-9i -0.5788596580 - 5.838184619e-9i
125 -0.275019564 + 0.961438596i -0.275019564 - 0.961438596i
126 1.045712188 - 1.427475908e-8i 1.045712188 + 1.427803657e-8i
127 -6.077963516e-8 - 0.8170054993i -6.077734913e-8 + 0.8170054992i
128 -4.379098025e-9 - 0.08838831864i -4.385360231e-9 + 0.08838831864i
129 -0.1958772691 + 1.351035793e-8i -0.1958772691 - 1.350600767e-8i
130 0.0276918562 - 0.297321092i 0.0276918561 + 0.297321092i
131 0.6009035854 - 1.026320835e-7i 0.6009035854 + 1.026344103e-7i
132 2.724995959e-9 - 0.8657773550i 2.727863418e-9 + 0.8657773550i
133 -1.308587697e-8 - 1.054323647i -1.308924751e-8 + 1.054323647i
134 0.2794397659 + 2.222442918e-9i 0.2794397659 - 2.222468163e-9i
135 0.723849520 + 0.0674178603i 0.723849520 - 0.0674178603i
136 0.3747305686 - 1.528484898e-9i 0.3747305686 + 1.530073597e-9i
137 1.062147675e-8 - 0.3996758883i 1.062803154e-8 + 0.3996758883i
138 1.811280127e-8 + 0.7989660021i 1.811449027e-8 - 0.7989660021i
139 1.458497492 - 6.799844118e-9i 1.458497492 + 6.799142922e-9i
140 -0.503171863 - 0.0468643895i -0.503171863 + 0.0468643895i
141 -1.257780502 + 4.505709155e-10i -1.257780502 - 4.446152012e-10i
142 9.326427773e-9 + 0.5531478148i 9.335455434e-9 - 0.5531478148i
143 -4.704168179e-8 + 0.4659102247i -4.704658065e-8 - 0.4659102247i
144 -0.3658013671 - 6.067309394e-9i -0.3658013671 + 6.059676404e-9i
145 -0.0916877170 + 0.984428726i -0.0916877170 - 0.984428726i
146 0.1026819824 + 8.835363965e-9i 0.1026819823 - 8.853986422e-9i
147 1.885847043e-8 + 0.03376334852i 1.885459389e-8 - 0.03376334852i
148 1.395850152e-7 + 0.6406125508i 1.396243705e-7 - 0.6406125507i
149 0.4012538517 + 2.759569415e-8i 0.4012538518 - 2.757837074e-8i
150 -1.090687606 - 0.2049466833i -1.090687606 + 0.2049466833i
151 -0.9407972902 + 1.823139087e-9i -0.9407972902 - 1.821161409e-9i
152 -7.018046471e-9 + 0.3688137324i -7.016999083e-9 - 0.3688137324i
153 1.269682079e-8 - 1.550848495i 1.270519284e-8 + 1.550848495i
154 0.7884838626 + 4.502802353e-9i 0.7884838626 - 4.503243299e-9i
155 0.1073629012 - 1.152730209i 0.1073629012 + 1.152730209i
156 0.3313878838 - 2.751418357e-8i 0.3313878837 + 2.751660564e-8i
157 1.526770098e-8 - 0.6637863228i 1.526121725e-8 + 0.6637863228i
158 -2.263215285e-8 - 0.3916326005i -2.262584258e-8 + 0.3916326005i
159 -2.070892751 + 4.006177550e-8i -2.070892751 - 4.006389221e-8i
160 0.176014848 + 0.0163936997i 0.176014848 - 0.0163936997i
161 -0.7276372983 - 2.061763016e-10i -0.7276372983 + 1.879003957e-10i
162 -2.378143046e-8 - 0.2278549421i -2.379941711e-8 + 0.2278549421i
163 1.527371058e-7 + 1.933993434i 1.527655178e-7 - 1.933993434i
164 0.4049390818 + 4.433282502e-9i 0.4049390818 - 4.433048712e-9i
165 1.724092893 + 0.1605784824i 1.724092893 - 0.1605784824i
166 0.07651164751 - 2.425630903e-10i 0.07651164751 + 2.460002529e-10i
167 2.319304552e-8 + 1.645072020i 2.319732188e-8 - 1.645072020i
168 2.651762416e-8 - 0.5608247926i 2.652464794e-8 + 0.5608247926i
169 0.8216666087 - 3.649915066e-9i 0.8216666087 + 3.647227307e-9i
170 -0.0695025114 + 0.746231446i -0.0695025114 - 0.746231446i
171 1.526361197 + 4.016931976e-9i 1.526361197 - 4.025491385e-9i
172 -2.502977315e-8 + 0.06240276965i -2.503168389e-8 - 0.06240276966i
173 1.557856976e-8 - 1.520937234i 1.558977700e-8 + 1.520937234i
174 -1.097223584 - 5.616846572e-9i -1.097223584 + 5.624025590e-9i
175 0.186649811 - 0.993314958i 0.186649811 + 0.993314958i
176 -0.2758201006 - 1.403230066e-8i -0.2758201006 + 1.403298323e-8i
177 -3.822055533e-8 + 1.236505652i -3.822728314e-8 - 1.236505652i
178 -3.999567872e-8 - 1.145049821i -4.001032545e-8 + 1.145049821i
179 0.1467112244 + 2.723577773e-9i 0.1467112244 - 2.724059284e-9i
180 0.0678463727 - 0.728449937i 0.0678463727 + 0.728449937i
181 0.03073235864 - 2.515340533e-9i 0.03073235864 + 2.515241535e-9i
182 6.941661664e-9 + 0.3018027525i 6.941814253e-9 - 0.3018027525i
183 -3.634719351e-9 + 1.171949783i -3.638790886e-9 - 1.171949783i
184 0.2545352987 + 8.209337741e-10i 0.2545352987 - 8.192981570e-10i
185 -1.275703907 - 0.1188164591i -1.275703907 + 0.1188164591i
186 1.284808902 + 5.937882075e-9i 1.284808902 - 5.937499374e-9i
187 -1.880795917e-8 - 1.169364793i -1.880808703e-8 + 1.169364793i
188 -2.721962325e-8 + 0.4007048334i -2.722375474e-8 - 0.4007048334i
189 -0.7347603954 - 3.239730219e-9i -0.7347603954 + 3.239564650e-9i
190 -0.734448751 - 0.0684050755i -0.734448751 + 0.0684050755i
191 -1.358790491 - 8.145682977e-9i -1.358790491 + 8.148767948e-9i
192 2.390808656e-8 + 0.1961825224i 2.390797502e-8 - 0.1961825224i
193 3.399230722e-8 + 0.4104666818i 3.400140912e-8 - 0.4104666818i
194 0.1432689855 + 2.156759371e-8i 0.1432689855 - 2.157148961e-8i
195 -0.0614635902 + 0.659919648i -0.0614635902 - 0.659919648i
196 0.01075637153 + 8.418080636e-9i 0.01075637153 - 8.416917227e-9i
197 1.402966409e-8 - 0.5604760728i 1.401961692e-8 + 0.5604760728i
198 3.074536539e-8 + 1.141500684i 3.074881312e-8 - 1.141500684i
199 1.496666689 - 2.644308007e-8i 1.496666689 + 2.643925461e-8i
200 -0.0652920899 + 0.347472213i -0.0652920899 - 0.347472213i
201 -0.6202311548 + 7.603777419e-8i -0.6202311548 - 7.605108911e-8i
202 -1.041081770e-7 - 0.6913945665i -1.041108605e-7 + 0.6913945665i
203 4.926602363e-10 - 0.9992674350i 5.083665125e-10 + 0.9992674350i
204 -0.8317338261 - 7.743423992e-8i -0.8317338261 + 7.744976350e-8i
205 -0.0751052176 + 0.806388123i -0.0751052176 - 0.806388123i
206 -0.3717846099 - 2.572534383e-8i -0.3717846099 + 2.573053045e-8i
207 5.346605868e-8 - 1.053412016i 5.346045014e-8 + 1.053412016i
208 6.093964616e-9 - 0.1055738425i 6.095459821e-9 + 0.1055738425i
209 1.150901023 - 1.959067159e-9i 1.150901023 + 1.962691796e-9i
210 1.116816040 + 0.1040179648i 1.116816040 - 0.1040179648i
211 0.6130678300 - 4.256195823e-9i 0.6130678300 + 4.258703600e-9i
212 -3.172622674e-8 + 0.6597468111i -3.173679886e-8 - 0.6597468112i
213 -3.142393414e-8 - 1.227740247i -3.142777773e-8 + 1.227740247i
214 -0.1427314648 + 1.411319348e-8i -0.1427314648 - 1.411611570e-8i
215 -0.124267697 - 0.0115740400i -0.124267697 + 0.0115740400i
216 0.2570270995 - 1.384266380e-9i 0.2570270995 + 1.392095795e-9i
217 2.155689829e-9 + 1.170105763i 2.156642620e-9 - 1.170105763i
218 3.825982727e-8 + 1.080788429i 3.825528505e-8 - 1.080788429i
219 -0.2279082151 - 5.548761859e-9i -0.2279082151 + 5.548243388e-9i
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