Properties

Level 1
Symmetry odd
Weight 0
Character \( \chi_{1}(1,\cdot) \)
Multiplicity 1
Precision 5e-14
Fricke Eigenvalue undefined
Atkin-Lehner Eigenvalues 1

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Spectral parameter

$R= 12.1730083247$

The first few Fourier Coefficients

n c(n)
0  0
1 1
2 0.289251871
3 -1.201858761
4 -0.916333354
5 0.0395527072
6 -0.347639895
7 0.448133104
8 -0.554303009
9 0.444464481
10 0.0114406946
11 -0.691450783
12 1.101303270
13 -0.802780003
14 0.129623339
15 -0.0475367677
16 0.756000172
17 -1.037675373
18 0.128562183
19 0.637178807
20 -0.0362434649
21 -0.538592697
22 -0.200003433
23 0.508798692
24 0.666193927
25 -0.998435583
26 -0.232205618
27 0.667675230
28 -0.410639311
29 0.779996881
30 -0.0137500990
31 -1.598013583
32 0.772977473
33 0.831026181
34 -0.300149543
35 0.0177248775
36 -0.407277629
37 -0.429215160
38 0.184305162
39 0.964828180
40 -0.0219241846
41 -0.548524187
42 -0.155788945
43 -0.489355022
44 0.633599416
45 0.0175797734
46 0.147170974
47 0.816868005
48 -0.908605430
49 -0.799176720
50 -0.288799360
51 1.247139238
52 0.735614092
53 -0.978756047
54 0.193126313
55 -0.0273487635
56 -0.248401523
57 -0.765798931
58 0.225615557
59 1.342931384
60 0.0435595258
61 -0.451739345
62 -0.462228419
63 0.199179247
64 -0.532414991
65 -0.0317521224
66 0.240375878
67 0.407059646
68 0.950856556
69 -0.611504166
70 0.00512695399
71 -1.218362389
72 -0.246367999
73 0.847623229
74 -0.124151288
75 1.199978553
76 -0.583868194
77 -0.309861986
78 0.279078356
79 -0.152197859
80 0.0299018535
81 -1.246915806
82 -0.158661647
83 1.512681485
84 0.493530453
85 -0.0410428702
86 -0.141546855
87 -0.937446085
88 0.383273250
89 -0.351411890
90 0.00508498046
91 -0.359752294
92 -0.466229211
93 1.920586624
94 0.236280600
95 0.0252021467
96 -0.929009751
97 -0.434877624
98 -0.231163364
99 -0.307325314
100 0.914899827
101 -0.428540668
102 0.360737358
103 0.136140070
104 0.444983371
105 -0.0213027993
106 -0.283107018
107 0.318195539
108 -0.611813083
109 -1.051614670
110 -0.00791067726
111 0.515856001
112 0.338788704
113 1.230036743
114 -0.221508774
115 0.0201243657
116 -0.714737158
117 -0.356807197
118 0.388445416
119 -0.465016686
120 0.0263497735
121 -0.521895814
122 -0.130666451
123 0.659248600
124 1.464313148
125 -0.0790435381
126 0.0576129708
127 -1.036459334
128 -0.926979506
129 0.588135621
130 -0.00918436151
131 0.314430908
132 -0.761497011
133 0.285540917
134 0.117742765
135 0.0264083628
136 0.575186579
137 1.154299357
138 -0.176878723
139 -1.144207004
140 -0.0162419027
141 -0.981759969
142 -0.352413601
143 0.555082862
144 0.336015224
145 0.0308509881
146 0.245176605
147 0.960497543
148 0.393304168
149 -1.414908027
150 0.347096042
151 0.208857471
152 -0.353190130
153 -0.461209847
154 -0.0896281587
155 -0.0632057634
156 -0.884104243
157 1.143195921
158 -0.0440235158
159 1.176326530
160 0.0305733517
161 0.228009537
162 -0.360672730
163 -1.433929039
164 0.502631011
165 0.0328693343
166 0.437545951
167 -1.044990120
168 0.298543554
169 -0.355544269
170 -0.0118717318
171 0.283203350
172 0.448412325
173 1.904636093
174 -0.271158041
175 -0.447432045
176 -0.522736891
177 -1.614013848
178 -0.101646547
179 0.537915463
180 -0.0161089329
181 0.336326778
182 -0.104059024
183 0.542926890
184 -0.282028646
185 -0.0169766218
186 0.555533275
187 0.717501450
188 -0.748523400
189 0.299207373
190 0.00728976686
191 -1.254886027
192 0.639887621
193 -1.044430269
194 -0.125789168
195 0.0381615664
196 0.732312286
197 0.354186207
198 -0.0888944213
199 0.618998534
200 0.553435848
201 -0.489228206
202 -0.123956192
203 0.349542424
204 -1.142795285
205 -0.0216956107
206 0.0393787745
207 0.226142940
208 -0.606901822
209 -0.440577785
210 -0.00616187657
211 0.0232932731
212 0.896866804
213 1.464299511
214 0.0920386548
215 -0.0193553153
216 -0.370094389
217 -0.716122788
218 -0.304181511
219 -1.018723404
220 0.0250605738
221 0.833025039
222 0.149212314
223 -0.354326068
224 0.346396793
225 -0.443769151
226 0.355790429
227 -0.998837987
228 0.701727103
229 1.637393927
230 0.00582100961
231 0.372410344
232 -0.432354620
233 -1.210009705
234 -0.103207147
235 0.0323093414
236 -1.230572825
237 0.182920325
238 -0.134506933
239 1.433407802
240 -0.0359378046
241 -0.358208763
242 -0.150959333
243 0.830941447
244 0.413943829
245 -0.0316095880
246 0.190688891
247 -0.511514404
248 0.885783736
249 -1.818029495
250 -0.0228634909
251 0.140754032
252 -0.182514589
253 -0.351809253
254 -0.299797801
255 0.0493277325
256 0.264284433
257 1.521660531
258 0.170119329
259 -0.192345526
260 0.0290955281
261 0.346680909
262 0.0909497321
263 -0.957417253
264 -0.460640307
265 -0.0387124565
266 0.0825932501
267 0.422347462
268 -0.373002330
269 -0.209315948
270 0.00763865039
271 -0.119245916
272 -0.784482753
273 0.432371453
274 0.333883238
275 0.690369065
276 0.560341665
277 1.490381238
278 -0.330964015
279 -0.710260278
280 -0.00982495248
281 -1.779211530
282 -0.283975910
283 -0.683180752
284 1.116426094
285 -0.0302894207
286 0.160558759
287 -0.245811850
288 0.343561026
289 0.0767701823
290 0.00892370200
291 0.522661490
292 -0.776705434
293 1.153046776
294 0.277825710
295 0.0531165709
296 0.237915251
297 -0.461664554
298 -0.409264788
299 -0.408453415
300 -1.099580365
301 -0.219296204
302 0.0604124401
303 0.515045342
304 0.481707294
305 -0.0178675145
306 -0.133405811
307 0.490159654
308 0.283936873
309 -0.163621134
310 -0.0182823852
311 1.090177349
312 -0.534807164
313 -0.464620530
314 0.330671559
315 0.00787807701
316 0.139463974
317 -1.251819433
318 0.340254653
319 -0.539329451
320 -0.0210584588
321 -0.382426089
322 0.0659521803
323 -0.661184767
324 1.142590549
325 0.801524113
326 -0.414766651
327 1.263892317
328 0.304048600
329 0.366065596
330 0.00950750563
331 -0.716097552
332 -1.386120479
333 -0.190770915
334 -0.302265353
335 0.0161003095
336 -0.407176179
337 1.672811235
338 -0.102841844
339 -1.478330436
340 0.0376089511
341 1.104947744
342 0.0819170967
343 -0.806270649
344 0.271250961
345 -0.0241866454
346 0.550919550
347 1.292517720
348 0.859013113
349 -0.992972179
350 -0.129420556
351 -0.535996327
352 -0.534475878
353 -0.841434800
354 -0.466856527
355 -0.0481895306
356 0.322010437
357 0.558884378
358 0.155593058
359 0.199941682
360 -0.00974451440
361 -0.594003178
362 0.0972831618
363 0.627245061
364 0.329653007
365 0.0335258019
366 0.157042624
367 0.495808551
368 0.384651895
369 -0.243799497
370 -0.00491053342
371 -0.438612992
372 -1.759897567
373 -0.0346719904
374 0.207538635
375 0.0949991707
376 -0.452792395
377 -0.626165899
378 0.0865462926
379 1.731918080
380 -0.0230935692
381 1.245677730
382 -0.362978131
383 -0.960394592
384 1.114098438
385 -0.0122558788
386 -0.302103411
387 -0.217500927
388 0.398492875
389 -0.902617736
390 0.0110383027
391 -0.527967869
392 0.442986060
393 -0.377901541
394 0.102449025
395 -0.00601983777
396 0.281612445
397 -0.795400395
398 0.179046491
399 -0.343179858
400 -0.754817466
401 1.659325614
402 -0.141510172
403 1.282853347
404 0.392686111
405 -0.0493188879
406 0.101105797
407 0.296781165
408 -0.691293050
409 -0.824874134
410 -0.00627551312
411 -1.387304761
412 -0.124749696
413 0.601812022
414 0.0654122793
415 0.0598306376
416 -0.620530857
417 1.375175214
418 -0.127437948
419 -0.792174988
420 0.0195204678
421 0.215639198
422 0.00673762055
423 0.363068816
424 0.542527420
425 1.036052018
426 0.423551375
427 -0.202439350
428 -0.291573189
429 -0.667131197
430 -0.00559856135
431 -1.260010329
432 0.504762587
433 -1.481134828
434 -0.207139851
435 -0.0370785326
436 0.963629591
437 0.324195744
438 -0.294667650
439 1.201302548
440 0.0151594943
441 -0.355205670
442 0.240954064
443 0.00622425791
444 -0.472696068
445 -0.0138992897
446 -0.102489474
447 1.700519608
448 -0.238592778
449 -0.301747786
450 -0.128361061
451 0.379277476
452 -1.127123692
453 -0.251017158
454 -0.288915787
455 -0.0142291609
456 0.424484656
457 -1.350488659
458 0.473619268
459 -0.692830141
460 -0.0184406288
461 0.923284290
462 0.107720388
463 1.085943294
464 0.589677776
465 0.0759644021
466 -0.349997572
467 0.269404923
468 0.326954337
469 0.182416903
470 0.00934553731
471 -1.373960033
472 -0.744390909
473 0.338364915
474 0.0529100463
475 -0.636181990
476 0.426110301
477 -0.435022298
478 0.414615892
479 -0.331148013
480 -0.0367448532
481 0.344565345
482 -0.103612560
483 -0.274035259
484 0.478230546
485 -0.0172005852
486 0.240351368
487 0.261384215
488 0.250400482
489 1.723380175
490 -0.00914313557
491 -0.316255200
492 -0.604091481
493 -0.809383557
494 -0.147956502
495 -0.0121555398
496 -1.208098539
497 -0.545988523
498 -0.525868425
499 1.298984105