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
2 1.342406731
3 -0.577350269
4 0.802055834
5 -0.0624853167
6 -0.775038885
7 0.753838133
8 -0.265721491
9 0.333332814
10 -0.0838776769
11 -0.408349071
12 -0.463067151
13 0.527645078
14 1.011957405
15 0.0360759141
16 -1.158762273
17 -0.967583197
18 0.447468910
19 1.560205624
20 -0.0501167124
21 -0.435228657
22 -0.548146758
23 0.285124949
24 0.153414426
25 -0.996095585
26 0.708314305
27 -0.192450089
28 0.604620284
29 -0.216424251
30 0.0484285500
31 1.154221603
32 -1.289808695
33 0.235750217
34 -1.298890198
35 -0.0471038150
36 0.267351944
37 0.254344444
38 2.094430534
39 -0.304636027
40 0.0166036970
41 -1.105075659
42 -0.584253881
43 -0.126149776
44 -0.327504541
45 -0.0208284407
46 0.382753645
47 1.008106189
48 0.669011723
49 -0.431728034
50 -1.337165440
51 0.558634401
52 0.423200841
53 0.247646995
54 -0.258346252
55 0.0255146117
56 -0.200311064
57 -0.900785137
58 -0.290529372
59 0.0689514842
60 0.0289348976
61 0.485289996
62 1.549434850
63 0.251279382
64 -0.572685602
65 -0.0329700686
66 0.316472679
67 -0.692441699
68 -0.776055750
69 -0.164616966
70 -0.0632324762
71 -1.106816140
72 -0.0885738618
73 1.507834831
74 0.341433698
75 0.575096056
76 1.251372019
77 -0.307815754
78 -0.408945455
79 -1.278016269
80 0.0724056223
81 0.111111099
82 -1.483461002
83 1.611629487
84 -0.349077669
85 0.0604596848
86 -0.169344399
87 0.124952628
88 0.108502588
89 -0.414761438
90 -0.0279604065
91 0.397758953
92 0.228686240
93 -0.666389909
94 1.353290009
95 -0.0974895890
96 0.744671392
97 -0.0802764464
98 -0.579554644
99 -0.136110442
100 -0.798924279
101 -1.225551630
102 0.749914614
103 -0.951498165
104 -0.140206665
105 0.0271953784
106 0.332442913
107 1.925346998
108 -0.154355727
109 1.398246600
110 0.0342510829
111 -0.146845784
112 -0.873519252
113 0.139699984
114 -1.209220031
115 -0.0178161729
116 -0.173584240
117 0.175881580
118 0.0925610361
119 -0.729401174
120 -0.00958607257
121 -0.833265497
122 0.651456386
123 0.638015997
124 0.925750117
125 0.124727211
126 0.337319072
127 -0.806325713
128 0.521031907
129 0.0728330343
130 -0.0442611053
131 -1.093897488
132 0.189085103
133 1.176142515
134 -0.929538412
135 0.0120253086
136 0.257107712
137 -0.0796738031
138 -0.220982932
139 0.516707932
140 -0.0377798934
141 -0.582030405
142 -1.485797428
143 -0.215454088
144 -0.386254047
145 0.0135233459
146 2.024127691
147 0.249258339
148 0.203998458
149 1.206022461
150 0.772012753
151 -0.297011115
152 -0.414580356
153 -0.322527720
154 -0.413213973
155 -0.0721216344
156 -0.244334957
157 -0.435163360
158 -1.715617583
159 -0.142978959
160 0.0805929616
161 0.214937794
162 0.149156281
163 0.772708312
164 -0.886331232
165 -0.0147309340
166 2.163462278
167 0.710903524
168 0.115649676
169 -0.721590667
170 0.0811614830
171 0.520068466
172 -0.101179212
173 -1.148051509
174 0.167737338
175 -0.750894758
176 0.473159038
177 -0.0398092422
178 -0.556778528
179 0.820122827
180 -0.0167056522
181 -0.796799267
182 0.533954595
183 -0.280182061
184 -0.0757638851
185 -0.0158929185
186 -0.894566999
187 0.395094298
188 0.808557562
189 -0.145075914
190 -0.130870375
191 -0.869847100
192 0.330639856
193 1.902885263
194 -0.107763794
195 0.0190352266
196 -0.346269940
197 0.177586448
198 -0.182715419
199 0.588553095
200 0.264683977
201 0.399781146
202 -1.645188966
203 -0.163148865
204 0.448056195
205 0.0690514060
206 -1.277297281
207 0.0950416441
208 -0.611415755
209 -0.637081321
210 0.0365070601
211 -1.128067857
212 0.198627360
213 0.639021056
214 2.584599070
215 0.00788227218
216 0.0511374893
217 0.870095585
218 1.877015575
219 -0.870548213
220 0.0204642247
221 -0.510540447
222 -0.197126837
223 -1.076530856
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226 0.187534148
227 0.784357915
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229 -1.027329122
230 -0.0239165777
231 0.177717326
232 0.0575085118
233 0.836205668
234 0.236105013
235 -0.0629915889
236 0.0553031008
237 0.737863386
238 -0.979152782
239 0.653409812
240 -0.0418031861
241 0.397367376
242 -1.118581185
243 -0.0641499920
244 0.389229673
245 0.0269766149
246 0.856476578
247 0.823234871
248 -0.306701614
249 -0.930474719
250 0.167433993
251 0.439771749
252 0.201539963
253 -0.116425537
254 -1.082417335
255 -0.0349064390
256 1.272122046
257 -1.197697596
258 0.0977710748
259 0.191734432
260 -0.0264442492
261 -0.0721414725
262 -1.468452592
263 -1.228125242
264 -0.0626439713
265 -0.0154745934
266 1.578861747
267 0.239462228
268 -0.555376031
269 1.697438032
270 0.0161428844
271 0.408358648
272 1.121198857
273 -0.229646237
274 -0.106954643
275 0.406737000
276 -0.132032022
277 -0.989424487
278 0.693632188
279 0.384740488
280 0.0125165057
281 -1.342783165
282 -0.781321578
283 0.477636027
284 -0.887728410
285 0.0562859485
286 -0.289227002
287 -0.833048133
288 -0.429936081
289 -0.0637826867
290 0.0181535752
291 0.0463475694
292 1.209367820
293 0.442779999
294 0.334605953
295 -0.00430805953
296 -0.0675843505
297 0.0785831676
298 1.618972857
299 0.150444910
300 0.461259179
301 -0.0950966759
302 -0.398709968
303 0.707572687
304 -1.807907539
305 -0.0303236409
306 -0.432963237
307 -0.581760437
308 -0.246885557
309 0.549347718
310 -0.0968166824
311 -0.189702832
312 0.0809480215
313 1.237173273
314 -0.584165416
315 -0.0157014424
316 -1.025040611
317 1.623571511
318 -0.191935726
319 0.0883723971
320 0.0357844068
321 -1.111599227
322 0.288534120
323 -1.509629287
324 0.0891174814
325 -0.525584648
326 1.037293058
327 -0.807278664
328 0.293643065
329 0.759949681
330 -0.0197753749
331 0.394459783
332 1.292616850
333 0.0847813790
334 0.954321501
335 0.0432674921
336 0.504326579
337 -0.128547995
338 -0.968668167
339 -0.0806556374
340 0.0484921089
341 -0.471305002
342 0.698143536
343 -1.079291165
344 0.0335205531
345 0.0102860306
346 -1.541152045
347 0.226857659
348 0.100218809
349 -0.225472255
350 -1.008005908
351 -0.101545365
352 0.526669082
353 -0.471510224
354 -0.0534400007
355 0.0691594967
356 -0.332661943
357 0.421120055
358 1.100939115
359 1.363390675
360 0.00553469345
361 1.434242177
362 -1.069628726
363 0.481086032
364 0.319024956
365 -0.0942172962
366 -0.376119096
367 0.784712794
368 -0.330391987
369 -0.368358465
370 -0.0213344110
371 0.186685802
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373 -1.748413726
374 0.530377450
375 -0.0720111435
376 -0.267875732
377 -0.114195260
378 -0.194751218
379 -0.367644992
380 -0.0781921460
381 0.465532744
382 -1.167688882
383 0.970787001
384 -0.300817941
385 0.0192336972
386 2.554446501
387 -0.0420501374
388 -0.0643863792
389 -1.803787843
390 0.0255532089
391 -0.275881756
392 0.114719485
393 0.631560936
394 0.238392956
395 0.0798569185
396 -0.109168568
397 0.000333383989
398 0.790077275
399 -0.679045935
400 1.154238042
401 -1.173383133
402 0.536669896
403 0.609019918
404 -0.982960355
405 -0.00694313917
406 -0.219012100
407 -0.103856771
408 -0.148441335
409 1.793048405
410 0.0926944571
411 0.0459996874
412 -0.763154555
413 0.0519781609
414 0.127584667
415 -0.100703223
416 -0.680561261
417 -0.298321350
418 -0.855221790
419 -0.933503818
420 0.0218123295
421 -0.583139464
422 -1.514325732
423 0.336035247
424 -0.0658050983
425 0.963805513
426 0.857825324
427 0.365830275
428 1.544235652
429 0.124392324
430 0.0105817090
431 0.573317536
432 0.223003975
433 -0.0523646822
434 1.168022842
435 -0.00780741321
436 1.121471650
437 0.444853539
438 -1.168630401
439 -1.318987800
440 -0.00677970343
441 -0.143909505
442 -0.685353194
443 0.393875622
444 -0.117778641
445 0.0259166085
446 -1.445142096
447 -0.696297665
448 -0.431711604
449 -0.0453311717
450 -0.445721875
451 0.451237181
452 0.112047066
453 0.171479666
454 1.052927299
455 -0.0248541196
456 0.239358050
457 0.202827835
458 -1.379093656
459 0.186211501
460 -0.0142895746
461 1.123425832
462 0.238569277
463 -0.572140943
464 0.250784496
465 0.0416394111
466 1.122528378
467 0.369110994
468 0.141066848
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471 0.251241368
472 -0.0183218296
473 0.0515109053
474 0.990512322
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482 0.533428900
483 -0.124094793
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485 0.00501622146
486 -0.0861153994
487 1.351068767
488 -0.128951776
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490 0.0362140299
491 -1.436767801
492 0.511724460
493 0.209408590
494 1.105115862
495 0.00850554454
496 -1.337468582
497 -0.834360102
498 -1.249075031
499 0.843031458