Properties

Level 3
Symmetry even
Weight 0
Character \( \chi_{3}(1,\cdot) \)
Multiplicity 1
Precision 5e-12
Fricke Eigenvalue 1
Atkin-Lehner Eigenvalues 1,1

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Spectral parameter

$R= 5.09874190873$

The first few Fourier Coefficients

n c(n)
0  0
1  1.0
2  1.3424067
3 -0.5773502
4  0.8020558
5 -0.0624853
6 -0.7750388
7  0.7538381
8 -0.2657214
9  0.3333328
10 -0.0838776
11 -0.408349
12 -0.4630671
13  0.527645
14  1.0119574
15  0.0360759
16 -1.1587622
17 -0.9675831
18  0.4474689
19  1.5602056
20 -0.0501167
21 -0.4352286
22 -0.5481467
23  0.2851249
24  0.1534144
25 -0.9960955
26  0.7083143
27 -0.19245
28  0.6046202
29 -0.2164242
30  0.0484285
31  1.1542216
32 -1.2898086
33  0.2357502
34 -1.2988901
35 -0.0471038
36  0.2673519
37  0.2543444
38  2.0944305
39 -0.304636
40  0.0166036
41 -1.1050756
42 -0.5842538
43 -0.1261497
44 -0.3275045
45 -0.0208284
46  0.3827536
47  1.0081061
48  0.6690117
49 -0.431728
50 -1.3371654
51  0.5586344
52  0.4232008
53  0.2476469
54 -0.2583462
55  0.0255146
56 -0.200311
57 -0.9007851
58 -0.2905293
59  0.0689514
60  0.0289348
61  0.4852899
62  1.5494348
63  0.2512793
64 -0.5726856
65 -0.03297
66  0.3164726
67 -0.6924416
68 -0.7760557
69 -0.1646169
70 -0.0632324
71 -1.1068161
72 -0.0885738
73  1.5078348
74  0.3414336
75  0.575096
76  1.251372
77 -0.3078157
78 -0.4089454
79 -1.2780162
80  0.0724056
81  0.111111
82 -1.483461
83  1.6116294
84 -0.3490776
85  0.0604596
86 -0.1693443
87  0.1249526
88  0.1085025
89 -0.4147614
90 -0.0279604
91  0.3977589
92  0.2286862
93 -0.6663899
94  1.35329
95 -0.0974895
96  0.7446713
97 -0.0802764
98 -0.5795546
99 -0.1361104
100 -0.7989242
101 -1.2255516
102  0.7499146
103 -0.9514981
104 -0.1402066
105  0.0271953
106  0.3324429
107  1.9253469
108 -0.1543557
109  1.3982466
110  0.034251
111 -0.1468457
112 -0.8735192
113  0.1396999
114 -1.20922
115 -0.0178161
116 -0.1735842
117  0.1758815
118  0.092561
119 -0.7294011
120 -0.009586
121 -0.8332654
122  0.6514563
123  0.6380159
124  0.9257501
125  0.1247272
126  0.337319
127 -0.8063257
128  0.5210319
129  0.072833
130 -0.0442611
131 -1.0938974
132  0.1890851
133  1.1761425
134 -0.9295384
135  0.0120253
136  0.2571077
137 -0.0796738
138 -0.2209829
139  0.5167079
140 -0.0377798
141 -0.5820304
142 -1.4857974
143 -0.215454
144 -0.386254
145  0.0135233
146  2.0241276
147  0.2492583
148  0.2039984
149  1.2060224
150  0.7720127
151 -0.2970111
152 -0.4145803
153 -0.3225277
154 -0.4132139
155 -0.0721216
156 -0.2443349
157 -0.4351633
158 -1.7156175
159 -0.1429789
160  0.0805929
161  0.2149377
162  0.1491562
163  0.7727083
164 -0.8863312
165 -0.0147309
166  2.1634622
167  0.7109035
168  0.1156496
169 -0.7215906
170  0.0811614
171  0.5200684
172 -0.1011792
173 -1.1480515
174  0.1677373
175 -0.7508947
176  0.473159
177 -0.0398092
178 -0.5567785
179  0.8201228
180 -0.0167056
181 -0.7967992
182  0.5339545
183 -0.280182
184 -0.0757638
185 -0.0158929
186 -0.8945669
187  0.3950942
188  0.8085575
189 -0.1450759
190 -0.1308703
191 -0.8698471
192  0.3306398
193  1.9028852
194 -0.1077637
195  0.0190352
196 -0.3462699
197  0.1775864
198 -0.1827154
199  0.588553
200  0.2646839
201  0.3997811
202 -1.6451889
203 -0.1631488
204  0.4480561
205  0.0690514
206 -1.2772972
207  0.0950416
208 -0.6114157
209 -0.6370813
210  0.036507
211 -1.1280678
212  0.1986273
213  0.639021
214  2.584599
215  0.0078822
216  0.0511374
217  0.8700955
218  1.8770155
219 -0.8705482
220  0.0204642
221 -0.5105404
222 -0.1971268
223 -1.0765308
224 -0.972307
225 -0.3320319
226  0.1875341
227  0.7843579
228 -0.7224799
229 -1.0273291
230 -0.0239165
231  0.1777173
232  0.0575085
233  0.8362056
234  0.236105
235 -0.0629915
236  0.0553031
237  0.7378633
238 -0.9791527
239  0.6534098
240 -0.0418031
241  0.3973673
242 -1.1185811
243 -0.0641499
244  0.3892296
245  0.0269766
246  0.8564765
247  0.8232348
248 -0.3067016
249 -0.9304747
250  0.1674339
251  0.4397717
252  0.2015399
253 -0.1164255
254 -1.0824173
255 -0.0349064
256  1.272122
257 -1.1976975
258  0.097771
259  0.1917344
260 -0.0264442
261 -0.0721414
262 -1.4684525
263 -1.2281252
264 -0.0626439
265 -0.0154745
266  1.5788617
267  0.2394622
268 -0.555376
269  1.697438
270  0.0161428
271  0.4083586
272  1.1211988
273 -0.2296462
274 -0.1069546
275  0.406737
276 -0.132032
277 -0.9894244
278  0.6936321
279  0.3847404
280  0.0125165
281 -1.3427831
282 -0.7813215
283  0.477636
284 -0.8877284
285  0.0562859
286 -0.289227
287 -0.8330481
288 -0.429936
289 -0.0637826
290  0.0181535
291  0.0463475
292  1.2093678
293  0.4427799
294  0.3346059
295 -0.004308
296 -0.0675843
297  0.0785831
298  1.6189728
299  0.1504449
300  0.4612591
301 -0.0950966
302 -0.3987099
303  0.7075726
304 -1.8079075
305 -0.0303236
306 -0.4329632
307 -0.5817604
308 -0.2468855
309  0.5493477
310 -0.0968166
311 -0.1897028
312  0.080948
313  1.2371732
314 -0.5841654
315 -0.0157014
316 -1.0250406
317  1.6235715
318 -0.1919357
319  0.0883723
320  0.0357844
321 -1.1115992
322  0.2885341
323 -1.5096292
324  0.0891174
325 -0.5255846
326  1.037293
327 -0.8072786
328  0.293643
329  0.7599496
330 -0.0197753
331  0.3944597
332  1.2926168
333  0.0847813
334  0.9543215
335  0.0432674
336  0.5043265
337 -0.1285479
338 -0.9686681
339 -0.0806556
340  0.0484921
341 -0.471305
342  0.6981435
343 -1.0792911
344  0.0335205
345  0.010286
346 -1.541152
347  0.2268576
348  0.1002188
349 -0.2254722
350 -1.0080059
351 -0.1015453
352  0.526669
353 -0.4715102
354 -0.05344
355  0.0691594
356 -0.3326619
357  0.42112
358  1.1009391
359  1.3633906
360  0.0055346
361  1.4342421
362 -1.0696287
363  0.481086
364  0.3190249
365 -0.0942172
366 -0.376119
367  0.7847127
368 -0.3303919
369 -0.3683584
370 -0.0213344
371  0.1866858
372 -0.534482
373 -1.7484137
374  0.5303774
375 -0.0720111
376 -0.2678757
377 -0.1141952
378 -0.1947512
379 -0.3676449
380 -0.0781921
381  0.4655327
382 -1.1676888
383  0.970787
384 -0.3008179
385  0.0192336
386  2.5544465
387 -0.0420501
388 -0.0643863
389 -1.8037878
390  0.0255532
391 -0.2758817
392  0.1147194
393  0.6315609
394  0.2383929
395  0.0798569
396 -0.1091685
397  0.0003333
398  0.7900772
399 -0.6790459
400  1.154238
401 -1.1733831
402  0.5366698
403  0.6090199
404 -0.9829603
405 -0.0069431
406 -0.2190121
407 -0.1038567
408 -0.1484413
409  1.7930484
410  0.0926944
411  0.0459996
412 -0.7631545
413  0.0519781
414  0.1275846
415 -0.1007032
416 -0.6805612
417 -0.2983213
418 -0.8552217
419 -0.9335038
420  0.0218123
421 -0.5831394
422 -1.5143257
423  0.3360352
424 -0.065805
425  0.9638055
426  0.8578253
427  0.3658302
428  1.5442356
429  0.1243923
430  0.0105817
431  0.5733175
432  0.2230039
433 -0.0523646
434  1.1680228
435 -0.0078074
436  1.1214716
437  0.4448535
438 -1.1686304
439 -1.3189878
440 -0.0067797
441 -0.1439095
442 -0.6853531
443  0.3938756
444 -0.1177786
445  0.0259166
446 -1.445142
447 -0.6962976
448 -0.4317116
449 -0.0453311
450 -0.4457218
451  0.4512371
452  0.112047
453  0.1714796
454  1.0529272
455 -0.0248541
456  0.239358
457  0.2028278
458 -1.3790936
459  0.1862115
460 -0.0142895
461  1.1234258
462  0.2385692
463 -0.5721409
464  0.2507844
465  0.0416394
466  1.1225283
467  0.3691109
468  0.1410668
469 -0.5219889
470 -0.0845607
471  0.2512413
472 -0.0183218
473  0.0515109
474  0.9905123
475 -1.5541136
476 -0.5850206
477  0.0825493
478  0.8771415
479 -1.179756
480 -0.0465311
481  0.1342034
482  0.5334289
483 -0.1240947
484 -0.6683253
485  0.0050162
486 -0.0861153
487  1.3510687
488 -0.1289517
489 -0.4461259
490  0.036214
491 -1.4367678
492  0.5117244
493  0.2094085
494  1.1051158
495  0.0085055
496 -1.3374685
497 -0.8343601
498 -1.249075
499  0.8430314