Properties

Level 10
Symmetry odd
Weight 0
Character \( \chi_{10}(9,\cdot) \)
Multiplicity 2
Precision 6e-14
Fricke Eigenvalue 0
Atkin-Lehner Eigenvalues 1,0,0,0

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Spectral parameter

$R= 8.24048538145$

The first few Fourier Coefficients

n \( c_{ 1 } (n) \) \( c_{ 2 } (n) \)
0  0  0
1 1 1
2 -3.992254285e-11 - 0.7071067813i -3.991471933e-11 + 0.7071067813i
3 -2.006055075e-11 - 0.09663020335i -2.005918300e-11 + 0.09663020335i
4 -0.5000000000 + 6.254479470e-12i -0.5000000000 - 6.253641535e-12i
5 0.916587554 - 0.399834033i 0.916587554 + 0.399834033i
6 -0.06832787207 + 6.076840402e-12i -0.06832787207 - 6.075771445e-12i
7 2.926099969e-11 + 1.278108214i 2.924810903e-11 - 1.278108214i
8 1.101638606e-11 + 0.3535533907i 1.101367389e-11 - 0.3535533907i
9 0.9906626037 + 5.150512645e-11i 0.9906626037 - 5.150009550e-11i
10 -0.282725356 - 0.648125275i -0.282725356 + 0.648125275i
11 0.2834165499 - 7.087936904e-12i 0.2834165499 + 7.086828732e-12i
12 -2.159013591e-11 + 0.04831510168i -2.158970677e-11 - 0.04831510168i
13 -1.483104167e-10 - 1.076865344i -1.482940441e-10 + 1.076865344i
14 0.9037589851 + 3.585319639e-11i 0.9037589851 - 3.584727229e-11i
15 -0.0386360439 - 0.0885700418i -0.0386360439 + 0.0885700418i
16 0.2500000000 + 3.949840836e-12i 0.2500000000 - 3.949465707e-12i
17 -4.857642180e-11 - 1.658044685i -4.855892099e-11 + 1.658044685i
18 -2.390016961e-11 - 0.7005042452i -2.389297044e-11 + 0.7005042452i
19 0.7375706009 - 9.406164356e-12i 0.7375706009 + 9.405327652e-12i
20 -0.458293777 + 0.199917016i -0.458293777 - 0.199917016i
21 0.1235038565 + 3.219143128e-11i 0.1235038565 - 3.218813073e-11i
22 -2.094359110e-11 - 0.2004057644i -2.094159693e-11 + 0.2004057644i
23 9.464458512e-12 + 0.9583384933i 9.455529252e-12 - 0.9583384933i
24 0.03416393605 - 1.443448935e-11i 0.03416393605 + 1.443247267e-11i
25 0.680265490 - 0.732965798i 0.680265490 + 0.732965798i
26 -0.7614587872 - 9.717940629e-12i -0.7614587872 + 9.718597905e-12i
27 3.224522981e-11 - 0.1923581322i 3.224415353e-11 + 0.1923581322i
28 -1.633965502e-11 - 0.6390541072i -1.633374444e-11 + 0.6390541072i
29 -0.2294576934 + 5.586022416e-11i -0.2294576934 - 5.585542806e-11i
30 -0.0626284770 + 0.0273198086i -0.0626284770 - 0.0273198086i
31 1.451861404 + 6.282009714e-12i 1.451861404 - 6.280173628e-12i
32 -4.759764401e-11 - 0.1767766952i -4.758848625e-11 + 0.1767766952i
33 7.418123250e-11 - 0.02738659892i 7.417348267e-11 + 0.02738659892i
34 -1.172414640 - 5.768201139e-11i -1.172414640 + 5.767945048e-11i
35 0.5110311627 + 1.171498083i 0.5110311627 - 1.171498083i
36 -0.4953313017 - 9.055047166e-12i -0.4953313017 + 9.053682279e-12i
37 3.853663967e-11 + 0.1773603090i 3.851846951e-11 - 0.1773603090i
38 4.058841820e-11 - 0.5215411736i 4.061619593e-11 + 0.5215411736i
39 -0.1040577169 + 1.469519384e-10i -0.1040577169 - 1.469402232e-10i
40 0.141362678 + 0.324062637i 0.141362678 - 0.324062637i
41 -0.1276630891 + 5.374415382e-10i -0.1276630891 - 5.373897817e-10i
42 1.957199553e-10 - 0.08733041402i 1.957832601e-10 + 0.08733041402i
43 2.468425077e-9 - 0.7932832256i 2.468259974e-9 + 0.7932832256i
44 -0.1417082782 + 6.799212316e-10i -0.1417082782 - 6.798604470e-10i
45 0.908029014 - 0.396100624i 0.908029014 + 0.396100624i
46 0.6776476541 - 2.750430196e-9i 0.6776476541 + 2.750091962e-9i
47 -4.287576870e-9 + 0.3314267919i -4.287845977e-9 - 0.3314267919i
48 -1.215315551e-8 - 0.02415754192i -1.215224481e-8 + 0.02415754192i
49 -0.6335606016 + 1.211862280e-9i -0.6335606016 - 1.211811183e-9i
50 -0.518285076 - 0.481020355i -0.518285076 + 0.481020355i
51 -0.1602171955 - 2.002416203e-9i -0.1602171955 + 2.002851608e-9i
52 2.025210280e-8 + 0.5384326738i 2.025147003e-8 - 0.5384326738i
53 -2.323218054e-8 + 0.9414014178i -2.323460830e-8 - 0.9414014178i
54 -0.1360177719 - 6.445869956e-9i -0.1360177719 + 6.446292509e-9i
55 0.259776051 - 0.113319566i 0.259776051 + 0.113319566i
56 -0.4518794927 + 6.893788401e-11i -0.4518794927 - 6.892833475e-11i
57 1.316198805e-10 - 0.07127159723i 1.316115798e-10 + 0.07127159723i
58 -2.772229804e-10 + 0.1622510911i -2.772213287e-10 - 0.1622510911i
59 -0.04127566375 - 2.376115423e-10i -0.04127566375 + 2.376158357e-10i
60 0.0193180221 + 0.0442850208i 0.0193180221 - 0.0442850208i
61 1.239192526 - 4.839017560e-11i 1.239192526 + 4.835966283e-11i
62 -9.173251680e-10 - 1.026621044i -9.172141165e-10 + 1.026621044i
63 -2.160633866e-9 + 1.266174012i -2.160561474e-9 - 1.266174012i
64 -0.1249999979 - 8.943642770e-10i -0.1249999979 + 8.941975211e-10i
65 -0.430567413 - 0.987041373i -0.430567413 + 0.987041373i
66 -0.01936524988 - 1.146536459e-10i -0.01936524988 + 1.146832276e-10i
67 3.025283911e-10 - 1.842254857i 3.024257684e-10 + 1.842254857i
68 8.089875236e-10 + 0.8290223430i 8.090767157e-10 - 0.8290223430i
69 0.09260443954 + 2.530716420e-9i 0.09260443954 - 2.530514243e-9i
70 0.828374230 - 0.361353598i 0.828374230 + 0.361353598i
71 -1.106794788 - 2.956890216e-9i -1.106794788 + 2.956536052e-9i
72 2.163963496e-9 + 0.3502521221i 2.163930772e-9 - 0.3502521221i
73 7.539002510e-9 + 1.511190363i 7.538436601e-9 - 1.511190363i
74 0.1254126839 - 1.578432911e-9i 0.1254126839 + 1.578204190e-9i
75 -0.0708266670 - 0.0657341704i -0.0708266670 + 0.0657341704i
76 -0.3687852959 - 6.640159301e-9i -0.3687852959 + 6.639986184e-9i
77 -1.167034287e-8 + 0.3622370195i -1.167052798e-8 - 0.3622370195i
78 1.034789753e-8 + 0.07357992358i 1.034781758e-8 - 0.07357992358i
79 -0.1118319298 - 3.535714007e-8i -0.1118319298 + 3.535331372e-8i
80 0.229146846 - 0.0999585084i 0.229146846 + 0.0999585084i
81 0.9720749637 + 1.821279640e-8i 0.9720749637 - 1.821021457e-8i
82 -3.580975435e-8 + 0.09027146317i -3.580511007e-8 - 0.09027146317i
83 3.132324802e-10 + 0.7708554925i 3.132660806e-10 - 0.7708554925i
84 -0.06175192675 - 3.183466594e-10i -0.06175192675 + 3.183874134e-10i
85 -0.6629426926 - 1.519743124i -0.6629426926 + 1.519743124i
86 -0.5609359491 + 7.313280816e-10i -0.5609359491 - 7.312517782e-10i
87 -1.011711426e-9 + 0.02217254533i -1.011770739e-9 - 0.02217254533i
88 7.205880396e-10 + 0.1002028827i 7.204148945e-10 - 0.1002028827i
89 -0.3565684618 + 3.128793070e-10i -0.3565684618 - 3.126859277e-10i
90 -0.280085433 - 0.642073477i -0.280085433 + 0.642073477i
91 1.376350446 - 1.451399278e-9i 1.376350446 + 1.451484378e-9i
92 -2.700297907e-9 - 0.4791692455i -2.700069805e-9 + 0.4791692455i
93 -1.096825926e-8 - 0.1402936581i -1.096754770e-8 + 0.1402936581i
94 0.2343541340 + 2.682339267e-9i 0.2343541340 - 2.682529452e-9i
95 0.676048065 - 0.294905835i 0.676048065 + 0.294905835i
96 -0.01708199136 + 5.717097704e-9i -0.01708199136 - 5.716024071e-9i
97 -5.364861112e-9 + 0.3408829297i -5.365601184e-9 - 0.3408829297i
98 2.265549489e-8 + 0.4479949887i 2.265443668e-8 - 0.4479949887i
99 0.2807701924 - 2.975197121e-9i 0.2807701924 + 2.974323847e-9i
100 -0.340132796 + 0.366482915i -0.340132796 - 0.366482915i
101 -1.625139887 + 3.178779635e-8i -1.625139887 - 3.178736652e-8i
102 8.495219816e-8 + 0.1132906434i 8.495020554e-8 - 0.1132906434i
103 -3.794759589e-8 + 1.676337808i -3.794693767e-8 - 1.676337808i
104 0.3807293952 - 2.166331051e-9i 0.3807293952 + 2.166176069e-9i
105 0.113202091 - 0.0493810414i 0.113202091 + 0.0493810414i
106 0.6656713170 + 2.944786127e-9i 0.6656713170 - 2.944692161e-9i
107 -6.417746701e-9 - 1.162878859i -6.417383916e-9 + 1.162878859i
108 4.511765814e-9 + 0.09617906162i 4.511506032e-9 - 0.09617906162i
109 -1.413578122 - 3.013431878e-9i -1.413578122 + 3.012807227e-9i
110 -0.0801290413 - 0.183689435i -0.0801290413 + 0.183689435i
111 0.01713837964 - 1.069653567e-8i 0.01713837964 + 1.069564072e-8i
112 -1.771717540e-8 + 0.3195270636i -1.771629715e-8 - 0.3195270636i
113 2.731447960e-8 - 0.3194958885i 2.731495916e-8 + 0.3194958885i
114 -0.05039666454 + 2.697340747e-8i -0.05039666454 - 2.697059296e-8i
115 0.383176288 + 0.878401155i 0.383176288 - 0.878401155i
116 0.1147288746 - 1.736763049e-8i 0.1147288746 + 1.736647048e-8i
117 6.050661654e-8 - 1.066810243i 6.050273718e-8 + 1.066810243i
118 8.414207971e-8 + 0.02918625710i 8.413778397e-8 - 0.02918625711i
119 2.119160463 - 1.265368527e-9i 2.119160463 + 1.268764983e-9i
120 0.0313142367 - 0.0136599045i 0.0313142367 + 0.0136599045i
121 -0.9196750515 - 7.061249012e-10i -0.9196750515 + 7.057425776e-10i
122 -3.501953116e-9 - 0.8762414363i -3.502532500e-9 + 0.8762414363i
123 3.116259218e-10 + 0.01233611373i 3.109888588e-10 - 0.01233611373i
124 -0.7259307026 + 7.440024736e-10i -0.7259307026 - 7.444702977e-10i
125 0.330458212 - 0.943820630i 0.330458212 + 0.943820630i
126 0.8953202367 - 7.043669291e-9i 0.8953202367 + 7.044101510e-9i
127 -1.618006880e-8 + 0.6708410709i -1.617852502e-8 - 0.6708410709i
128 6.152014726e-9 + 0.08838835049i 6.150880685e-9 - 0.08838835049i
129 -0.07665513436 + 1.346272407e-8i -0.07665513436 - 1.346308663e-8i
130 -0.697943647 + 0.304457146i -0.697943647 - 0.304457146i
131 -0.3883019540 + 4.690234500e-9i -0.3883019540 - 4.690723781e-9i
132 2.368489768e-9 + 0.01369327891i 2.371411793e-9 - 0.01369327891i
133 2.141755574e-8 + 0.9426950293i 2.142058370e-8 - 0.9426950293i
134 -1.302670873 - 4.976842242e-9i -1.302670873 + 4.977942651e-9i
135 -0.0769112129 - 0.176313100i -0.0769112129 + 0.176313100i
136 0.5862072300 + 6.260258562e-8i 0.5862072300 - 6.260086915e-8i
137 -3.253600046e-8 + 0.9664497399i -3.254148647e-8 - 0.9664497399i
138 -5.965604980e-8 - 0.06548123156i -5.966010407e-8 + 0.06548123156i
139 0.4935002287 - 6.902403861e-9i 0.4935002287 + 6.908704719e-9i
140 -0.255515476 - 0.585749092i -0.255515476 + 0.585749092i
141 0.03202585918 - 1.275876371e-8i 0.03202585919 + 1.276637140e-8i
142 6.572390225e-8 + 0.7826221193i 6.572506705e-8 - 0.7826221193i
143 -5.686460234e-9 - 0.3052014654i -5.687120023e-9 + 0.3052014654i
144 0.2476656845 - 1.194920060e-8i 0.2476656845 + 1.194813828e-8i
145 -0.210318063 + 0.0917449913i -0.210318063 - 0.0917449913i
146 1.068572904 + 1.865025483e-8i 1.068572904 - 1.864816740e-8i
147 -1.865859010e-9 + 0.06122110174i -1.864406244e-9 - 0.06122110174i
148 7.713554731e-9 - 0.08868014625i 7.713752308e-9 + 0.08868014625i
149 -0.02381296824 + 1.001471867e-9i -0.02381296824 - 1.001106370e-9i
150 -0.0464810496 + 0.0500819635i -0.0464810496 - 0.0500819635i
151 -1.279998171 - 3.450236009e-8i -1.279998171 + 3.450001308e-8i
152 -4.466155059e-8 + 0.2607705915i -4.466129014e-8 - 0.2607705915i
153 5.514981442e-9 - 1.642562866i 5.514688376e-9 + 1.642562866i
154 0.2561402606 - 3.991533346e-9i 0.2561402606 + 3.990907109e-9i
155 1.330758092 - 0.5805035975i 1.330758092 + 0.5805035975i
156 0.05202883992 + 5.698322425e-9i 0.05202883992 - 5.697177761e-9i
157 -1.974699253e-8 - 0.3327987452i -1.974567802e-8 + 0.3327987452i
158 2.217734488e-8 + 0.07907716903i 2.217798244e-8 - 0.07907716903i
159 0.09096784896 - 1.113883554e-8i 0.09096784896 + 1.113718417e-8i
160 -0.0706813597 - 0.162031305i -0.0706813597 + 0.162031305i
161 -1.224860331 + 7.065141597e-9i -1.224860331 - 7.064897312e-9i
162 1.561200782e-8 - 0.6873608306i 1.560999581e-8 + 0.6873608306i
163 2.691769577e-8 + 1.323355033i 2.691682222e-8 - 1.323355033i
164 0.06383152174 - 3.950691104e-9i 0.06383152174 + 3.953078203e-9i
165 -0.0109501380 - 0.0251022065i -0.0109501380 + 0.0251022065i
166 0.5450771672 + 6.258463647e-9i 0.5450771672 - 6.259723462e-9i
167 3.919725192e-8 - 0.1891231551i 3.919673563e-8 + 0.1891231551i
168 -1.115795893e-8 + 0.04366520255i -1.115857670e-8 - 0.04366520256i
169 -0.1596389496 + 1.322314030e-8i -0.1596389496 - 1.322319835e-8i
170 -1.074620611 + 0.4687712585i -1.074620611 - 0.4687712585i
171 0.7306834841 + 2.899536554e-8i 0.7306834841 - 2.898798422e-8i
172 -5.975212691e-8 + 0.3966416112i -5.975140220e-8 - 0.3966416112i
173 8.535387127e-8 + 0.6075505436i 8.535096160e-8 - 0.6075505437i
174 0.01567835182 + 4.648413833e-8i 0.01567835182 - 4.648870165e-8i
175 0.936809644 + 0.869452866i 0.936809644 - 0.869452866i
176 0.07085415307 - 5.395008900e-9i 0.07085415307 + 5.395182792e-9i
177 7.015355699e-9 + 0.003988480593i 7.014949467e-9 - 0.003988480592i
178 4.102949410e-9 + 0.2521319818i 4.102401467e-9 - 0.2521319818i
179 -0.03807905191 + 1.184683269e-8i -0.03807905191 - 1.184664584e-8i
180 -0.454014499 + 0.198050306i -0.454014499 - 0.198050306i
181 -0.9598962897 - 1.323595462e-8i -0.9598962897 + 1.323545207e-8i
182 -5.223104251e-8 - 0.9732267293i -5.222828811e-8 + 0.9732267293i
183 -4.971760023e-8 - 0.1197434143i -4.971431084e-8 + 0.1197434143i
184 -0.3388237519 - 1.684680713e-8i -0.3388237519 + 1.684598117e-8i
185 0.0709147356 + 0.162566264i 0.0709147356 - 0.162566264i
186 -0.09920253992 - 6.894133493e-9i -0.09920253993 + 6.890255178e-9i
187 3.263212990e-8 - 0.4699173083i 3.263147650e-8 + 0.4699173083i
188 -9.063156978e-8 - 0.1657133660i -9.062960956e-8 + 0.1657133660i
189 0.2458544747 - 4.905065712e-8i 0.2458544747 + 4.905256464e-8i
190 -0.208530035 - 0.478038149i -0.208530035 + 0.478038149i
191 1.009404700 + 2.293367327e-9i 1.009404700 - 2.290937240e-9i
192 9.427095532e-8 + 0.01207879883i 9.426586118e-8 - 0.01207879884i
193 -1.607146774e-8 - 0.1235924164i -1.607105920e-8 + 0.1235924164i
194 0.2410406338 + 3.750139977e-9i 0.2410406338 - 3.749096946e-9i
195 -0.0953780014 + 0.0416058083i -0.0953780014 - 0.0416058083i
196 0.3167803275 - 6.495744689e-9i 0.3167803275 + 6.495577816e-9i
197 1.161847246e-8 - 1.466310822i 1.161848674e-8 + 1.466310822i
198 -1.788764969e-8 - 0.1985344853i -1.788838840e-8 + 0.1985344853i
199 -0.3752488489 + 1.174929289e-8i -0.3752488489 - 1.175095668e-8i
200 0.259142549 + 0.240510168i 0.259142549 - 0.240510168i
201 -0.1780174566 + 3.677308419e-9i -0.1780174566 - 3.678492901e-9i
202 2.991925435e-8 + 1.149147365i 2.991957869e-8 - 1.149147365i
203 -5.599679530e-9 - 0.2932717802i -5.598492561e-9 + 0.2932717802i
204 0.08010860341 - 1.944433698e-8i 0.08010860341 + 1.944293727e-8i
205 -0.117014378 + 0.0510440420i -0.117014378 - 0.0510440420i
206 1.185349785 + 4.913587428e-8i 1.185349785 - 4.913475143e-8i
207 -3.415030003e-8 + 0.9493901574i -3.414914919e-8 - 0.9493901574i
208 -3.693166198e-9 - 0.2692163207i -3.690847648e-9 + 0.2692163207i
209 0.2090397103 - 1.432548757e-8i 0.2090397103 + 1.433078888e-8i
210 -0.0349176483 - 0.0800459983i -0.0349176483 + 0.0800459983i
211 -0.7864181103 - 5.251497120e-8i -0.7864181103 + 5.251644831e-8i
212 2.114568561e-8 - 0.4707007038i 2.114499913e-8 + 0.4707007038i
213 1.887674502e-8 + 0.1069497938i 1.887495605e-8 - 0.1069497938i
214 -0.8222795493 + 3.740351795e-9i -0.8222795493 - 3.740058834e-9i
215 -0.317181604 - 0.727113540i -0.317181604 + 0.727113540i
216 0.06800886642 - 1.647720593e-11i 0.06800886642 + 1.658560794e-11i
217 -3.776599310e-8 + 1.855635993i -3.776416635e-8 - 1.855635993i
218 5.259991981e-12 + 0.9995506738i 6.494825691e-12 - 0.9995506738i
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