Properties

Level 10
Symmetry odd
Weight 0
Character \( \chi_{10}(9,\cdot) \)
Multiplicity 2
Precision 7e-12
Fricke Eigenvalue 0
Atkin-Lehner Eigenvalues 1,0,0,0

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Spectral parameter

$R= 9.6936342693$

The first few Fourier Coefficients

n \( c_{ 1 } (n) \) \( c_{ 2 } (n) \)
0  0  0
1 1 1
2 7.701364518e-12 - 0.7071067811i 7.704964154e-12 + 0.7071067811i
3 3.413724370e-13 - 1.210190293i 3.462220558e-13 + 1.210190293i
4 -0.5000000000 - 4.100380026e-12i -0.5000000000 + 4.100482527e-12i
5 -0.620939467 - 0.783858518i -0.620939467 + 0.783858518i
6 -0.8557337631 - 3.243730753e-11i -0.8557337631 + 3.242415477e-11i
7 1.923728179e-11 + 1.334476372i 1.921222108e-11 - 1.334476372i
8 -7.706003436e-12 + 0.3535533906i -7.694454490e-12 - 0.3535533906i
9 -0.4645605467 + 1.694247120e-11i -0.4645605467 - 1.691780962e-11i
10 -0.554271674 + 0.439070508i -0.554271674 - 0.439070508i
11 -1.336725214 + 2.181139273e-10i -1.336725214 - 2.181603166e-10i
12 -3.162146543e-11 + 0.6050951467i -3.161045554e-11 - 0.6050951467i
13 1.148512917e-11 - 0.6505978924i 1.148309611e-11 + 0.6505978924i
14 0.9436172922 - 3.838888081e-11i 0.9436172922 + 3.839232132e-11i
15 -0.948617970 + 0.751454916i -0.948617970 - 0.751454916i
16 0.2500000000 + 1.964072757e-10i 0.2500000000 - 1.964032519e-10i
17 9.035575719e-12 + 0.3998107152i 9.069654502e-12 - 0.3998107152i
18 1.299803521e-11 + 0.3284939128i 1.299430569e-11 - 0.3284939128i
19 0.8179201764 + 8.351437262e-12i 0.8179201764 - 8.336083848e-12i
20 0.310469733 + 0.391929259i 0.310469733 - 0.391929259i
21 1.614970352 - 1.703358681e-12i 1.614970352 + 1.707443694e-12i
22 3.750312370e-11 + 0.9452074639i 3.750396323e-11 - 0.9452074639i
23 -4.229940653e-12 + 0.2469952050i -4.222136256e-12 - 0.2469952050i
24 0.4278668815 - 5.256307975e-11i 0.4278668815 + 5.256235172e-11i
25 -0.228868354 + 0.973457382i -0.228868354 - 0.973457382i
26 -0.4600421815 + 1.088244051e-10i -0.4600421815 - 1.088225536e-10i
27 2.535902821e-11 - 0.6479836291i 2.535911878e-11 + 0.6479836291i
28 -4.313549076e-11 - 0.6672381863i -4.313036791e-11 + 0.6672381863i
29 1.702878299 + 1.493264014e-10i 1.702878299 - 1.493481547e-10i
30 0.531358867 + 0.670774199i 0.531358867 - 0.670774199i
31 -1.553868124 - 1.853808582e-10i -1.553868124 + 1.853897949e-10i
32 -1.204103379e-11 - 0.1767766949i -1.195519894e-11 + 0.1767766949i
33 4.318670623e-11 + 1.617691879i 4.314388921e-11 - 1.617691879i
34 0.2827088678 - 3.394465284e-10i 0.2827088678 + 3.394863227e-10i
35 1.046040672 - 0.8286290484i 1.046040672 + 0.8286290484i
36 0.2322802733 - 2.173875255e-10i 0.2322802733 + 2.174067619e-10i
37 2.735939426e-10 + 0.7452159456i 2.734558556e-10 - 0.7452159456i
38 -1.372953280e-10 - 0.5783569024i -1.370742286e-10 + 0.5783569024i
39 -0.7873472546 - 1.184220158e-10i -0.7873472546 + 1.181796716e-10i
40 0.277135837 - 0.219535255i 0.277135837 + 0.219535255i
41 -1.241816999 - 6.083609310e-10i -1.241816999 + 6.084569907e-10i
42 -3.216702513e-10 - 1.141956485i -3.208431042e-10 + 1.141956485i
43 1.328754788e-10 + 0.6850341331i 1.325638683e-10 - 0.6850341331i
44 0.6683626077 - 2.647517342e-9i 0.6683626077 + 2.648530683e-9i
45 0.288463978 + 0.364149748i 0.288463978 - 0.364149748i
46 0.1746519845 + 7.563306562e-10i 0.1746519845 - 7.570465572e-10i
47 1.895046189e-9 + 0.4268680599i 1.891721833e-9 - 0.4268680599i
48 -1.357781732e-9 - 0.3025475658i -1.355891133e-9 + 0.3025475658i
49 -0.7808271941 - 8.100117978e-9i -0.7808271941 + 8.094643637e-9i
50 0.688338319 + 0.161834348i 0.688338319 - 0.161834348i
51 0.4838470595 + 5.150456708e-8i 0.4838470595 - 5.149732541e-8i
52 -1.055301567e-8 + 0.3252989642i -1.054149111e-8 - 0.3252989642i
53 -9.891667398e-9 - 0.1097470909i -9.891850427e-9 + 0.1097470909i
54 -0.4581936010 - 6.118804804e-8i -0.4581936010 + 6.120896062e-8i
55 0.8300254281 + 1.047803403i 0.8300254281 - 1.047803403i
56 -0.4718086601 + 1.067622061e-7i -0.4718086601 - 1.067906025e-7i
57 2.349408831e-10 - 0.9898390582i 2.348060765e-10 + 0.9898390582i
58 -3.894704916e-10 - 1.204116793i -3.893630051e-10 + 1.204116793i
59 -0.9046584156 - 1.666428014e-10i -0.9046584156 + 1.665346917e-10i
60 0.474308985 - 0.375727458i 0.474308985 + 0.375727458i
61 -0.6445612206 + 9.979104740e-10i -0.6445612206 - 9.973509528e-10i
62 -1.167420198e-9 + 1.098750686i -1.166922410e-9 - 1.098750686i
63 -1.072457244e-9 - 0.6199450747i -1.072010499e-9 + 0.6199450747i
64 -0.1249999975 + 3.022797571e-9i -0.1249999975 - 3.021123398e-9i
65 -0.509976701 + 0.403981909i -0.509976701 - 0.403981909i
66 1.143880897 + 1.698091500e-9i 1.143880897 - 1.699526031e-9i
67 1.431963851e-9 + 1.280238822i 1.430827781e-9 - 1.280238822i
68 6.120007514e-11 - 0.1999053666i 5.887507702e-11 + 0.1999053666i
69 0.2989111976 + 2.507575721e-9i 0.2989111976 - 2.510189456e-9i
70 -0.585929220 - 0.739662443i -0.585929220 + 0.739662443i
71 0.6119589690 + 7.949935400e-9i 0.6119589690 - 7.943340398e-9i
72 -5.657710453e-9 - 0.1642469671i -5.656782630e-9 + 0.1642469671i
73 -6.586872614e-9 + 0.6234721429i -6.583090725e-9 - 0.6234721429i
74 0.5269472642 + 1.805978160e-8i 0.5269472642 - 1.804962844e-8i
75 1.178068676 + 0.2769742741i 1.178068676 - 0.2769742741i
76 -0.4089600965 - 2.548744726e-8i -0.4089600965 + 2.548028814e-8i
77 -2.868647933e-8 - 1.783828240i -2.867311808e-8 + 1.783828240i
78 1.508430009e-8 + 0.5567386230i 1.507403886e-8 - 0.5567386230i
79 -1.051489425 + 6.391462243e-8i -1.051489425 - 6.388405184e-8i
80 -0.155234895 - 0.195964670i -0.155234895 + 0.195964670i
81 -1.248744073 - 7.738515783e-8i -1.248744073 + 7.739408180e-8i
82 -7.263903727e-10 + 0.8780972185i -7.266463302e-10 - 0.8780972185i
83 -1.322944984e-10 + 1.171012626i -1.328609409e-10 - 1.171012626i
84 -0.8074851774 + 8.167698860e-9i -0.8074851774 - 8.168577457e-9i
85 0.313395037 - 0.248258245i 0.313395037 + 0.248258245i
86 0.4843922844 + 3.505365636e-9i 0.4843922844 - 3.505094692e-9i
87 1.375127253e-9 - 2.060806789i 1.374228276e-9 + 2.060806789i
88 1.941668929e-9 - 0.4726037498i 1.938501263e-9 + 0.4726037498i
89 0.4577681988 - 6.648495415e-9i 0.4577681988 + 6.647499474e-9i
90 0.257492742 - 0.203974831i 0.257492742 + 0.203974831i
91 0.8682074984 - 2.167569739e-8i 0.8682074984 + 2.166761459e-8i
92 3.771798383e-9 - 0.1234975702i 3.773752608e-9 + 0.1234975702i
93 1.920975968e-8 + 1.880476156i 1.920723939e-8 - 1.880476156i
94 0.3018413220 - 3.423730258e-8i 0.3018413220 + 3.424531820e-8i
95 -0.507878894 - 0.641133681i -0.507878894 + 0.641133681i
96 -0.2139334442 - 1.286795310e-8i -0.2139334442 + 1.286203728e-8i
97 -1.543423353e-8 + 1.288245764i -1.542905321e-8 - 1.288245764i
98 -4.477121519e-8 + 0.5521282683i -4.474910312e-8 - 0.5521282683i
99 0.6209897310 - 8.626062317e-8i 0.6209897310 + 8.624505812e-8i
100 0.114434169 - 0.486728744i 0.114434169 + 0.486728744i
101 -0.3640080630 + 1.015913101e-7i -0.3640080630 - 1.015602776e-7i
102 1.125177555e-7 - 0.3421316856i 1.124797801e-7 + 0.3421316856i
103 1.211680209e-9 + 1.258461940i 1.210262864e-9 - 1.258461940i
104 0.2300210954 - 9.436084538e-11i 0.2300210954 + 9.712734985e-11i
105 -1.002798844 - 1.265908282i -1.002798844 + 1.265908282i
106 -0.07760288140 - 7.549545352e-9i -0.07760288141 + 7.550606176e-9i
107 -3.289580416e-9 - 1.130601715i -3.280090542e-9 + 1.130601715i
108 -3.834257234e-9 + 0.3239918262i -3.829996155e-9 - 0.3239918262i
109 -0.8129912443 + 2.993370446e-9i -0.8129912443 - 2.972883731e-9i
110 0.740908893 - 0.586916661i 0.740908893 + 0.586916661i
111 0.9018530928 - 8.811213108e-8i 0.9018530928 + 8.810913539e-8i
112 2.381479855e-8 + 0.3336191006i 2.380263103e-8 - 0.3336191006i
113 3.297911651e-9 - 0.5396526583i 3.301409064e-9 + 0.5396526583i
114 -0.6999219153 - 2.096955049e-8i -0.6999219153 + 2.096701714e-8i
115 0.193609288 - 0.153369105i 0.193609288 + 0.153369105i
116 -0.8514391396 + 6.316115183e-9i -0.8514391396 - 6.305646929e-9i
117 9.796998210e-9 + 0.3022420968i 9.787122108e-9 - 0.3022420968i
118 -9.866027091e-10 + 0.6396901300i -9.818355887e-10 - 0.6396901300i
119 -0.5335379628 + 4.160404311e-8i -0.5335379628 - 4.161956838e-8i
120 -0.265679451 - 0.335387127i -0.265679451 + 0.335387127i
121 0.7868342972 + 1.406013563e-8i 0.7868342972 - 1.405936125e-8i
122 -3.558355453e-9 + 0.4557735631i -3.563455750e-9 - 0.4557735631i
123 1.973371769e-8 + 1.502834832i 1.971000898e-8 - 1.502834832i
124 0.7769340932 + 2.881338077e-8i 0.7769340932 - 2.878387632e-8i
125 0.905166262 - 0.425057707i 0.905166262 + 0.425057707i
126 -0.4383673629 + 1.736693195e-8i -0.4383673629 - 1.736721121e-8i
127 -6.465242920e-9 - 0.8253696526i -6.462636189e-9 + 0.8253696526i
128 5.029306599e-9 + 0.08838833199i 5.024688047e-9 - 0.08838833199i
129 0.8290216594 - 2.716873071e-8i 0.8290216594 + 2.717197014e-8i
130 0.285658353 + 0.360607971i 0.285658353 - 0.360607971i
131 -0.7583050648 + 1.092749857e-8i -0.7583050648 - 1.092914703e-8i
132 -1.029802667e-8 - 0.8088458961i -1.028413092e-8 + 0.8088458961i
133 4.490676384e-9 + 1.091495190i 4.493651458e-9 - 1.091495190i
134 0.9052655409 - 1.016442609e-8i 0.9052655409 + 1.015509360e-8i
135 -0.507927472 + 0.402358591i -0.507927472 - 0.402358591i
136 -0.1413544399 - 1.096986438e-7i -0.1413544399 + 1.097058964e-7i
137 1.098809810e-9 - 1.516579229i 1.094116301e-9 + 1.516579229i
138 1.028551695e-9 - 0.2113620903i 1.037476502e-9 + 0.2113620903i
139 -1.430826490 + 7.854273133e-8i -1.430826490 - 7.856439123e-8i
140 -0.523020277 + 0.414314723i -0.523020277 - 0.414314723i
141 0.5165915944 + 7.624218481e-9i 0.5165915944 - 7.628306224e-9i
142 -5.093495666e-9 - 0.4327204794i -5.107520366e-9 + 0.4327204794i
143 -7.929396282e-9 + 0.8696705840i -7.926771226e-9 - 0.8696705840i
144 -0.1161401364 + 1.624587810e-8i -0.1161401364 - 1.624750837e-8i
145 -1.057384344 - 1.334815644i -1.057384344 + 1.334815644i
146 0.4408613929 - 1.895351792e-8i 0.4408613928 + 1.895507402e-8i
147 9.567746677e-10 + 0.9449494940i 9.521425989e-10 - 0.9449494940i
148 6.217583843e-10 - 0.3726079977i 6.137851576e-10 + 0.3726079977i
149 0.8637369319 - 4.175167939e-8i 0.8637369319 + 4.174173053e-8i
150 0.195850373 - 0.833020304i 0.195850373 + 0.833020304i
151 -0.1157421751 + 5.797880876e-8i -0.1157421750 - 5.797316531e-8i
152 8.012400206e-9 + 0.2891784640i 8.008199315e-9 - 0.2891784640i
153 3.892182191e-9 - 0.1857362107i 3.900186555e-9 + 0.1857362107i
154 -1.261357048 - 1.300029305e-7i -1.261357048 + 1.300090593e-7i
155 0.9648580556 + 1.218012719i 0.9648580556 - 1.218012719i
156 0.3936736221 + 6.873581988e-9i 0.3936736221 - 6.885159289e-9i
157 -1.083766813e-8 - 0.1941958603i -1.083273205e-8 + 0.1941958603i
158 4.523007509e-9 + 0.7435153689i 4.524977860e-9 - 0.7435153689i
159 -0.1328148082 - 1.987817555e-8i -0.1328148082 + 1.988276829e-8i
160 -0.138567925 + 0.109767600i -0.138567925 - 0.109767600i
161 -0.3296092665 + 1.142281881e-8i -0.3296092665 - 1.142011690e-8i
162 6.658863615e-9 + 0.8829953689i 6.652152337e-9 - 0.8829953689i
163 1.337817658e-8 - 0.7400106444i 1.337578346e-8 + 0.7400106444i
164 0.6209085001 + 4.490881885e-8i 0.6209085001 - 4.492113713e-8i
165 1.268041552 - 1.004488752i 1.268041552 + 1.004488752i
166 0.8280309599 - 1.642056174e-8i 0.8280309599 + 1.641629546e-8i
167 3.116664072e-9 + 0.2165282747i 3.112088223e-9 - 0.2165282747i
168 6.755533891e-9 + 0.5709781909i 6.736212016e-9 - 0.5709781909i
169 0.5767223947 + 5.838174092e-8i 0.5767223947 - 5.836635257e-8i
170 -0.175545083 - 0.221603745i -0.175545083 + 0.221603745i
171 -0.3799734679 - 1.913696319e-8i -0.3799734678 + 1.911053679e-8i
172 -2.066360674e-8 - 0.3425169905i -2.062872525e-8 + 0.3425169905i
173 -9.015763275e-11 - 0.2294679340i -9.189776573e-11 + 0.2294679340i
174 -1.457210458 - 1.268124324e-8i -1.457210458 + 1.268055021e-8i
175 -1.299055869 - 0.3054193865i -1.299055869 + 0.3054193865i
176 -0.3341812998 + 9.018001482e-9i -0.3341812998 - 9.015899678e-9i
177 -3.088353135e-9 + 1.094808847i -3.085309267e-9 - 1.094808847i
178 -1.578900746e-10 - 0.3236909909i -1.556634362e-10 + 0.3236909909i
179 0.8931457485 - 2.935587059e-8i 0.8931457485 + 2.935060228e-8i
180 -0.144231996 - 0.182074895i -0.144231996 + 0.182074895i
181 -1.744845662 - 6.772663569e-9i -1.744845662 + 6.781531950e-9i
182 -2.664702455e-10 - 0.6139154243i -2.713743037e-10 + 0.6139154243i
183 5.349096067e-9 + 0.7800417820i 5.348634657e-9 - 0.7800417820i
184 -0.08732598508 + 3.893184428e-8i -0.08732598508 - 3.893249064e-8i
185 0.584143845 - 0.462734019i 0.584143845 + 0.462734019i
186 1.329697430 + 1.239961405e-8i 1.329697430 - 1.239325656e-8i
187 -1.552784489e-8 - 0.5344371605i -1.552892797e-8 + 0.5344371605i
188 -1.555488185e-8 - 0.2134340874i -1.555036521e-8 + 0.2134340874i
189 0.8647188720 + 9.052152689e-8i 0.8647188720 - 9.050157284e-8i
190 -0.453349987 + 0.359124622i -0.453349987 - 0.359124622i
191 -1.404597748 + 7.916377078e-9i -1.404597748 - 7.908825628e-9i
192 8.514744407e-9 + 0.1512737938i 8.506639335e-9 - 0.1512737938i
193 2.255403247e-9 - 0.5063668046i 2.250908784e-9 + 0.5063668046i
194 0.9109273196 - 7.890282102e-9i 0.9109273196 + 7.882633098e-9i
195 0.488894962 + 0.617168830i 0.488894962 - 0.617168830i
196 0.3904135917 + 1.177320453e-8i 0.3904135917 - 1.177968730e-8i
197 8.327972095e-9 - 0.7605020437i 8.324305397e-9 + 0.7605020437i
198 3.417572027e-9 - 0.4391060963i 3.413639960e-9 + 0.4391060963i
199 -0.9729784100 - 2.794694622e-9i -0.9729784100 + 2.810942293e-9i
200 -0.344169146 - 0.0809171821i -0.344169146 + 0.0809171821i
201 1.549332595 + 8.556619142e-9i 1.549332595 - 8.560692100e-9i
202 -1.108600984e-8 + 0.2573926200i -1.107066835e-8 - 0.2573926200i
203 -1.692216313e-8 + 2.272450865i -1.690184205e-8 - 2.272450865i
204 -0.2419235226 - 6.216985117e-9i -0.2419235226 + 6.215852048e-9i
205 0.771093196 + 0.973408824i 0.771093196 - 0.973408824i
206 0.8898669861 - 1.203250742e-8i 0.8898669861 + 1.204666714e-8i
207 1.707130819e-8 - 0.1147442366i 1.705094640e-8 + 0.1147442366i
208 -1.289042129e-8 - 0.1626494702i -1.288266790e-8 + 0.1626494702i
209 -1.093334553 - 1.015947640e-8i -1.093334553 + 1.013165116e-8i
210 -0.895132315 + 0.709085871i -0.895132315 - 0.709085871i
211 -0.5061289139 + 4.294671462e-9i -0.5061289139 - 4.293458269e-9i
212 -7.651253690e-9 + 0.05487349624i -7.651761182e-9 - 0.05487349624i
213 -5.377082700e-9 - 0.7405868345i -5.378949378e-9 + 0.7405868345i
214 -0.7994561692 + 6.928299465e-9i -0.7994561692 - 6.927304574e-9i
215 0.536969851 - 0.425364724i 0.536969851 + 0.425364724i
216 0.2290968150 + 3.960741082e-8i 0.2290968150 - 3.960776449e-8i
217 -2.496409904e-9 - 2.073600286i -2.492372557e-9 + 2.073600286i
218 1.954120916e-9 + 0.5748716537i 1.953257666e-9 - 0.5748716537i
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