Properties

Label 1800.4.a.g.1.1
Level 18001800
Weight 44
Character 1800.1
Self dual yes
Analytic conductor 106.203106.203
Analytic rank 00
Dimension 11
CM no
Inner twists 11

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1800,4,Mod(1,1800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1800.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 1800=233252 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 1800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,-19,0,0,0,-22,0,1,0,0,0,58] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 106.203438010106.203438010
Analytic rank: 00
Dimension: 11
Coefficient field: Q\mathbb{Q}
Coefficient ring: Z\mathbb{Z}
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 600)
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.1
Character χ\chi == 1800.1

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
f(q)f(q) == q19.0000q722.0000q11+1.00000q13+58.0000q1753.0000q1958.0000q2322.0000q2935.0000q31270.000q37+468.000q41431.000q43+230.000q47+18.0000q49446.000q59+127.000q61811.000q6736.0000q71+522.000q73+418.000q77+1368.00q79+1138.00q83144.000q8919.0000q911079.00q97+O(q100)q-19.0000 q^{7} -22.0000 q^{11} +1.00000 q^{13} +58.0000 q^{17} -53.0000 q^{19} -58.0000 q^{23} -22.0000 q^{29} -35.0000 q^{31} -270.000 q^{37} +468.000 q^{41} -431.000 q^{43} +230.000 q^{47} +18.0000 q^{49} -446.000 q^{59} +127.000 q^{61} -811.000 q^{67} -36.0000 q^{71} +522.000 q^{73} +418.000 q^{77} +1368.00 q^{79} +1138.00 q^{83} -144.000 q^{89} -19.0000 q^{91} -1079.00 q^{97} +O(q^{100})

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 0 0
33 0 0
44 0 0
55 0 0
66 0 0
77 −19.0000 −1.02590 −0.512952 0.858417i 0.671448π-0.671448\pi
−0.512952 + 0.858417i 0.671448π0.671448\pi
88 0 0
99 0 0
1010 0 0
1111 −22.0000 −0.603023 −0.301511 0.953463i 0.597491π-0.597491\pi
−0.301511 + 0.953463i 0.597491π0.597491\pi
1212 0 0
1313 1.00000 0.0213346 0.0106673 0.999943i 0.496604π-0.496604\pi
0.0106673 + 0.999943i 0.496604π0.496604\pi
1414 0 0
1515 0 0
1616 0 0
1717 58.0000 0.827474 0.413737 0.910396i 0.364223π-0.364223\pi
0.413737 + 0.910396i 0.364223π0.364223\pi
1818 0 0
1919 −53.0000 −0.639949 −0.319975 0.947426i 0.603674π-0.603674\pi
−0.319975 + 0.947426i 0.603674π0.603674\pi
2020 0 0
2121 0 0
2222 0 0
2323 −58.0000 −0.525819 −0.262909 0.964821i 0.584682π-0.584682\pi
−0.262909 + 0.964821i 0.584682π0.584682\pi
2424 0 0
2525 0 0
2626 0 0
2727 0 0
2828 0 0
2929 −22.0000 −0.140872 −0.0704362 0.997516i 0.522439π-0.522439\pi
−0.0704362 + 0.997516i 0.522439π0.522439\pi
3030 0 0
3131 −35.0000 −0.202780 −0.101390 0.994847i 0.532329π-0.532329\pi
−0.101390 + 0.994847i 0.532329π0.532329\pi
3232 0 0
3333 0 0
3434 0 0
3535 0 0
3636 0 0
3737 −270.000 −1.19967 −0.599834 0.800124i 0.704767π-0.704767\pi
−0.599834 + 0.800124i 0.704767π0.704767\pi
3838 0 0
3939 0 0
4040 0 0
4141 468.000 1.78267 0.891333 0.453349i 0.149771π-0.149771\pi
0.891333 + 0.453349i 0.149771π0.149771\pi
4242 0 0
4343 −431.000 −1.52853 −0.764266 0.644901i 0.776899π-0.776899\pi
−0.764266 + 0.644901i 0.776899π0.776899\pi
4444 0 0
4545 0 0
4646 0 0
4747 230.000 0.713807 0.356904 0.934141i 0.383832π-0.383832\pi
0.356904 + 0.934141i 0.383832π0.383832\pi
4848 0 0
4949 18.0000 0.0524781
5050 0 0
5151 0 0
5252 0 0
5353 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
5454 0 0
5555 0 0
5656 0 0
5757 0 0
5858 0 0
5959 −446.000 −0.984140 −0.492070 0.870556i 0.663760π-0.663760\pi
−0.492070 + 0.870556i 0.663760π0.663760\pi
6060 0 0
6161 127.000 0.266569 0.133284 0.991078i 0.457448π-0.457448\pi
0.133284 + 0.991078i 0.457448π0.457448\pi
6262 0 0
6363 0 0
6464 0 0
6565 0 0
6666 0 0
6767 −811.000 −1.47880 −0.739399 0.673268i 0.764890π-0.764890\pi
−0.739399 + 0.673268i 0.764890π0.764890\pi
6868 0 0
6969 0 0
7070 0 0
7171 −36.0000 −0.0601748 −0.0300874 0.999547i 0.509579π-0.509579\pi
−0.0300874 + 0.999547i 0.509579π0.509579\pi
7272 0 0
7373 522.000 0.836924 0.418462 0.908234i 0.362569π-0.362569\pi
0.418462 + 0.908234i 0.362569π0.362569\pi
7474 0 0
7575 0 0
7676 0 0
7777 418.000 0.618643
7878 0 0
7979 1368.00 1.94825 0.974127 0.226002i 0.0725657π-0.0725657\pi
0.974127 + 0.226002i 0.0725657π0.0725657\pi
8080 0 0
8181 0 0
8282 0 0
8383 1138.00 1.50496 0.752480 0.658615i 0.228857π-0.228857\pi
0.752480 + 0.658615i 0.228857π0.228857\pi
8484 0 0
8585 0 0
8686 0 0
8787 0 0
8888 0 0
8989 −144.000 −0.171505 −0.0857526 0.996316i 0.527329π-0.527329\pi
−0.0857526 + 0.996316i 0.527329π0.527329\pi
9090 0 0
9191 −19.0000 −0.0218873
9292 0 0
9393 0 0
9494 0 0
9595 0 0
9696 0 0
9797 −1079.00 −1.12944 −0.564721 0.825282i 0.691016π-0.691016\pi
−0.564721 + 0.825282i 0.691016π0.691016\pi
9898 0 0
9999 0 0
100100 0 0
101101 1440.00 1.41867 0.709333 0.704873i 0.248996π-0.248996\pi
0.709333 + 0.704873i 0.248996π0.248996\pi
102102 0 0
103103 124.000 0.118622 0.0593111 0.998240i 0.481110π-0.481110\pi
0.0593111 + 0.998240i 0.481110π0.481110\pi
104104 0 0
105105 0 0
106106 0 0
107107 432.000 0.390309 0.195154 0.980773i 0.437479π-0.437479\pi
0.195154 + 0.980773i 0.437479π0.437479\pi
108108 0 0
109109 701.000 0.615997 0.307998 0.951387i 0.400341π-0.400341\pi
0.307998 + 0.951387i 0.400341π0.400341\pi
110110 0 0
111111 0 0
112112 0 0
113113 1044.00 0.869126 0.434563 0.900641i 0.356903π-0.356903\pi
0.434563 + 0.900641i 0.356903π0.356903\pi
114114 0 0
115115 0 0
116116 0 0
117117 0 0
118118 0 0
119119 −1102.00 −0.848909
120120 0 0
121121 −847.000 −0.636364
122122 0 0
123123 0 0
124124 0 0
125125 0 0
126126 0 0
127127 504.000 0.352148 0.176074 0.984377i 0.443660π-0.443660\pi
0.176074 + 0.984377i 0.443660π0.443660\pi
128128 0 0
129129 0 0
130130 0 0
131131 −72.0000 −0.0480204 −0.0240102 0.999712i 0.507643π-0.507643\pi
−0.0240102 + 0.999712i 0.507643π0.507643\pi
132132 0 0
133133 1007.00 0.656526
134134 0 0
135135 0 0
136136 0 0
137137 1850.00 1.15369 0.576847 0.816852i 0.304283π-0.304283\pi
0.576847 + 0.816852i 0.304283π0.304283\pi
138138 0 0
139139 1836.00 1.12034 0.560171 0.828377i 0.310735π-0.310735\pi
0.560171 + 0.828377i 0.310735π0.310735\pi
140140 0 0
141141 0 0
142142 0 0
143143 −22.0000 −0.0128653
144144 0 0
145145 0 0
146146 0 0
147147 0 0
148148 0 0
149149 2398.00 1.31847 0.659234 0.751938i 0.270881π-0.270881\pi
0.659234 + 0.751938i 0.270881π0.270881\pi
150150 0 0
151151 1871.00 1.00834 0.504172 0.863604i 0.331798π-0.331798\pi
0.504172 + 0.863604i 0.331798π0.331798\pi
152152 0 0
153153 0 0
154154 0 0
155155 0 0
156156 0 0
157157 −3293.00 −1.67395 −0.836975 0.547242i 0.815678π-0.815678\pi
−0.836975 + 0.547242i 0.815678π0.815678\pi
158158 0 0
159159 0 0
160160 0 0
161161 1102.00 0.539440
162162 0 0
163163 883.000 0.424306 0.212153 0.977236i 0.431952π-0.431952\pi
0.212153 + 0.977236i 0.431952π0.431952\pi
164164 0 0
165165 0 0
166166 0 0
167167 −4104.00 −1.90166 −0.950830 0.309715i 0.899766π-0.899766\pi
−0.950830 + 0.309715i 0.899766π0.899766\pi
168168 0 0
169169 −2196.00 −0.999545
170170 0 0
171171 0 0
172172 0 0
173173 −1706.00 −0.749739 −0.374869 0.927078i 0.622312π-0.622312\pi
−0.374869 + 0.927078i 0.622312π0.622312\pi
174174 0 0
175175 0 0
176176 0 0
177177 0 0
178178 0 0
179179 −662.000 −0.276426 −0.138213 0.990403i 0.544136π-0.544136\pi
−0.138213 + 0.990403i 0.544136π0.544136\pi
180180 0 0
181181 4121.00 1.69233 0.846164 0.532922i 0.178906π-0.178906\pi
0.846164 + 0.532922i 0.178906π0.178906\pi
182182 0 0
183183 0 0
184184 0 0
185185 0 0
186186 0 0
187187 −1276.00 −0.498986
188188 0 0
189189 0 0
190190 0 0
191191 958.000 0.362924 0.181462 0.983398i 0.441917π-0.441917\pi
0.181462 + 0.983398i 0.441917π0.441917\pi
192192 0 0
193193 3187.00 1.18863 0.594314 0.804233i 0.297424π-0.297424\pi
0.594314 + 0.804233i 0.297424π0.297424\pi
194194 0 0
195195 0 0
196196 0 0
197197 2282.00 0.825308 0.412654 0.910888i 0.364602π-0.364602\pi
0.412654 + 0.910888i 0.364602π0.364602\pi
198198 0 0
199199 1043.00 0.371539 0.185770 0.982593i 0.440522π-0.440522\pi
0.185770 + 0.982593i 0.440522π0.440522\pi
200200 0 0
201201 0 0
202202 0 0
203203 418.000 0.144521
204204 0 0
205205 0 0
206206 0 0
207207 0 0
208208 0 0
209209 1166.00 0.385904
210210 0 0
211211 4139.00 1.35043 0.675214 0.737621i 0.264051π-0.264051\pi
0.675214 + 0.737621i 0.264051π0.264051\pi
212212 0 0
213213 0 0
214214 0 0
215215 0 0
216216 0 0
217217 665.000 0.208033
218218 0 0
219219 0 0
220220 0 0
221221 58.0000 0.0176539
222222 0 0
223223 413.000 0.124020 0.0620101 0.998076i 0.480249π-0.480249\pi
0.0620101 + 0.998076i 0.480249π0.480249\pi
224224 0 0
225225 0 0
226226 0 0
227227 −5652.00 −1.65258 −0.826292 0.563242i 0.809554π-0.809554\pi
−0.826292 + 0.563242i 0.809554π0.809554\pi
228228 0 0
229229 −4391.00 −1.26710 −0.633549 0.773703i 0.718403π-0.718403\pi
−0.633549 + 0.773703i 0.718403π0.718403\pi
230230 0 0
231231 0 0
232232 0 0
233233 2052.00 0.576957 0.288479 0.957486i 0.406851π-0.406851\pi
0.288479 + 0.957486i 0.406851π0.406851\pi
234234 0 0
235235 0 0
236236 0 0
237237 0 0
238238 0 0
239239 4320.00 1.16919 0.584597 0.811324i 0.301252π-0.301252\pi
0.584597 + 0.811324i 0.301252π0.301252\pi
240240 0 0
241241 4265.00 1.13997 0.569985 0.821655i 0.306949π-0.306949\pi
0.569985 + 0.821655i 0.306949π0.306949\pi
242242 0 0
243243 0 0
244244 0 0
245245 0 0
246246 0 0
247247 −53.0000 −0.0136531
248248 0 0
249249 0 0
250250 0 0
251251 2412.00 0.606550 0.303275 0.952903i 0.401920π-0.401920\pi
0.303275 + 0.952903i 0.401920π0.401920\pi
252252 0 0
253253 1276.00 0.317081
254254 0 0
255255 0 0
256256 0 0
257257 −6948.00 −1.68640 −0.843199 0.537601i 0.819331π-0.819331\pi
−0.843199 + 0.537601i 0.819331π0.819331\pi
258258 0 0
259259 5130.00 1.23074
260260 0 0
261261 0 0
262262 0 0
263263 3564.00 0.835611 0.417805 0.908537i 0.362799π-0.362799\pi
0.417805 + 0.908537i 0.362799π0.362799\pi
264264 0 0
265265 0 0
266266 0 0
267267 0 0
268268 0 0
269269 1534.00 0.347694 0.173847 0.984773i 0.444380π-0.444380\pi
0.173847 + 0.984773i 0.444380π0.444380\pi
270270 0 0
271271 7704.00 1.72688 0.863440 0.504451i 0.168305π-0.168305\pi
0.863440 + 0.504451i 0.168305π0.168305\pi
272272 0 0
273273 0 0
274274 0 0
275275 0 0
276276 0 0
277277 4877.00 1.05787 0.528936 0.848662i 0.322591π-0.322591\pi
0.528936 + 0.848662i 0.322591π0.322591\pi
278278 0 0
279279 0 0
280280 0 0
281281 3758.00 0.797806 0.398903 0.916993i 0.369391π-0.369391\pi
0.398903 + 0.916993i 0.369391π0.369391\pi
282282 0 0
283283 935.000 0.196396 0.0981978 0.995167i 0.468692π-0.468692\pi
0.0981978 + 0.995167i 0.468692π0.468692\pi
284284 0 0
285285 0 0
286286 0 0
287287 −8892.00 −1.82884
288288 0 0
289289 −1549.00 −0.315286
290290 0 0
291291 0 0
292292 0 0
293293 −6214.00 −1.23900 −0.619498 0.784998i 0.712664π-0.712664\pi
−0.619498 + 0.784998i 0.712664π0.712664\pi
294294 0 0
295295 0 0
296296 0 0
297297 0 0
298298 0 0
299299 −58.0000 −0.0112181
300300 0 0
301301 8189.00 1.56813
302302 0 0
303303 0 0
304304 0 0
305305 0 0
306306 0 0
307307 −3905.00 −0.725961 −0.362981 0.931797i 0.618241π-0.618241\pi
−0.362981 + 0.931797i 0.618241π0.618241\pi
308308 0 0
309309 0 0
310310 0 0
311311 −9662.00 −1.76168 −0.880839 0.473416i 0.843021π-0.843021\pi
−0.880839 + 0.473416i 0.843021π0.843021\pi
312312 0 0
313313 5147.00 0.929475 0.464737 0.885449i 0.346149π-0.346149\pi
0.464737 + 0.885449i 0.346149π0.346149\pi
314314 0 0
315315 0 0
316316 0 0
317317 9216.00 1.63288 0.816439 0.577432i 0.195945π-0.195945\pi
0.816439 + 0.577432i 0.195945π0.195945\pi
318318 0 0
319319 484.000 0.0849492
320320 0 0
321321 0 0
322322 0 0
323323 −3074.00 −0.529542
324324 0 0
325325 0 0
326326 0 0
327327 0 0
328328 0 0
329329 −4370.00 −0.732298
330330 0 0
331331 2196.00 0.364662 0.182331 0.983237i 0.441636π-0.441636\pi
0.182331 + 0.983237i 0.441636π0.441636\pi
332332 0 0
333333 0 0
334334 0 0
335335 0 0
336336 0 0
337337 2521.00 0.407500 0.203750 0.979023i 0.434687π-0.434687\pi
0.203750 + 0.979023i 0.434687π0.434687\pi
338338 0 0
339339 0 0
340340 0 0
341341 770.000 0.122281
342342 0 0
343343 6175.00 0.972066
344344 0 0
345345 0 0
346346 0 0
347347 −7034.00 −1.08820 −0.544099 0.839021i 0.683129π-0.683129\pi
−0.544099 + 0.839021i 0.683129π0.683129\pi
348348 0 0
349349 7362.00 1.12917 0.564583 0.825376i 0.309037π-0.309037\pi
0.564583 + 0.825376i 0.309037π0.309037\pi
350350 0 0
351351 0 0
352352 0 0
353353 9382.00 1.41460 0.707300 0.706914i 0.249913π-0.249913\pi
0.707300 + 0.706914i 0.249913π0.249913\pi
354354 0 0
355355 0 0
356356 0 0
357357 0 0
358358 0 0
359359 10116.0 1.48719 0.743596 0.668629i 0.233119π-0.233119\pi
0.743596 + 0.668629i 0.233119π0.233119\pi
360360 0 0
361361 −4050.00 −0.590465
362362 0 0
363363 0 0
364364 0 0
365365 0 0
366366 0 0
367367 −3277.00 −0.466098 −0.233049 0.972465i 0.574870π-0.574870\pi
−0.233049 + 0.972465i 0.574870π0.574870\pi
368368 0 0
369369 0 0
370370 0 0
371371 0 0
372372 0 0
373373 10891.0 1.51184 0.755918 0.654667i 0.227191π-0.227191\pi
0.755918 + 0.654667i 0.227191π0.227191\pi
374374 0 0
375375 0 0
376376 0 0
377377 −22.0000 −0.00300546
378378 0 0
379379 −2591.00 −0.351163 −0.175581 0.984465i 0.556181π-0.556181\pi
−0.175581 + 0.984465i 0.556181π0.556181\pi
380380 0 0
381381 0 0
382382 0 0
383383 612.000 0.0816494 0.0408247 0.999166i 0.487001π-0.487001\pi
0.0408247 + 0.999166i 0.487001π0.487001\pi
384384 0 0
385385 0 0
386386 0 0
387387 0 0
388388 0 0
389389 −3708.00 −0.483298 −0.241649 0.970364i 0.577688π-0.577688\pi
−0.241649 + 0.970364i 0.577688π0.577688\pi
390390 0 0
391391 −3364.00 −0.435102
392392 0 0
393393 0 0
394394 0 0
395395 0 0
396396 0 0
397397 −3833.00 −0.484566 −0.242283 0.970206i 0.577896π-0.577896\pi
−0.242283 + 0.970206i 0.577896π0.577896\pi
398398 0 0
399399 0 0
400400 0 0
401401 9288.00 1.15666 0.578330 0.815803i 0.303705π-0.303705\pi
0.578330 + 0.815803i 0.303705π0.303705\pi
402402 0 0
403403 −35.0000 −0.00432624
404404 0 0
405405 0 0
406406 0 0
407407 5940.00 0.723427
408408 0 0
409409 −755.000 −0.0912771 −0.0456386 0.998958i 0.514532π-0.514532\pi
−0.0456386 + 0.998958i 0.514532π0.514532\pi
410410 0 0
411411 0 0
412412 0 0
413413 8474.00 1.00963
414414 0 0
415415 0 0
416416 0 0
417417 0 0
418418 0 0
419419 9576.00 1.11651 0.558256 0.829669i 0.311471π-0.311471\pi
0.558256 + 0.829669i 0.311471π0.311471\pi
420420 0 0
421421 9414.00 1.08981 0.544905 0.838498i 0.316566π-0.316566\pi
0.544905 + 0.838498i 0.316566π0.316566\pi
422422 0 0
423423 0 0
424424 0 0
425425 0 0
426426 0 0
427427 −2413.00 −0.273474
428428 0 0
429429 0 0
430430 0 0
431431 2254.00 0.251906 0.125953 0.992036i 0.459801π-0.459801\pi
0.125953 + 0.992036i 0.459801π0.459801\pi
432432 0 0
433433 −6301.00 −0.699323 −0.349661 0.936876i 0.613703π-0.613703\pi
−0.349661 + 0.936876i 0.613703π0.613703\pi
434434 0 0
435435 0 0
436436 0 0
437437 3074.00 0.336497
438438 0 0
439439 −3779.00 −0.410847 −0.205423 0.978673i 0.565857π-0.565857\pi
−0.205423 + 0.978673i 0.565857π0.565857\pi
440440 0 0
441441 0 0
442442 0 0
443443 −7632.00 −0.818527 −0.409263 0.912416i 0.634214π-0.634214\pi
−0.409263 + 0.912416i 0.634214π0.634214\pi
444444 0 0
445445 0 0
446446 0 0
447447 0 0
448448 0 0
449449 −2988.00 −0.314059 −0.157029 0.987594i 0.550192π-0.550192\pi
−0.157029 + 0.987594i 0.550192π0.550192\pi
450450 0 0
451451 −10296.0 −1.07499
452452 0 0
453453 0 0
454454 0 0
455455 0 0
456456 0 0
457457 −1370.00 −0.140232 −0.0701159 0.997539i 0.522337π-0.522337\pi
−0.0701159 + 0.997539i 0.522337π0.522337\pi
458458 0 0
459459 0 0
460460 0 0
461461 7992.00 0.807429 0.403714 0.914885i 0.367719π-0.367719\pi
0.403714 + 0.914885i 0.367719π0.367719\pi
462462 0 0
463463 −3096.00 −0.310763 −0.155382 0.987855i 0.549661π-0.549661\pi
−0.155382 + 0.987855i 0.549661π0.549661\pi
464464 0 0
465465 0 0
466466 0 0
467467 8338.00 0.826203 0.413101 0.910685i 0.364446π-0.364446\pi
0.413101 + 0.910685i 0.364446π0.364446\pi
468468 0 0
469469 15409.0 1.51710
470470 0 0
471471 0 0
472472 0 0
473473 9482.00 0.921740
474474 0 0
475475 0 0
476476 0 0
477477 0 0
478478 0 0
479479 −4918.00 −0.469121 −0.234561 0.972101i 0.575365π-0.575365\pi
−0.234561 + 0.972101i 0.575365π0.575365\pi
480480 0 0
481481 −270.000 −0.0255945
482482 0 0
483483 0 0
484484 0 0
485485 0 0
486486 0 0
487487 −19637.0 −1.82718 −0.913591 0.406635i 0.866702π-0.866702\pi
−0.913591 + 0.406635i 0.866702π0.866702\pi
488488 0 0
489489 0 0
490490 0 0
491491 −3096.00 −0.284563 −0.142282 0.989826i 0.545444π-0.545444\pi
−0.142282 + 0.989826i 0.545444π0.545444\pi
492492 0 0
493493 −1276.00 −0.116568
494494 0 0
495495 0 0
496496 0 0
497497 684.000 0.0617336
498498 0 0
499499 6875.00 0.616768 0.308384 0.951262i 0.400212π-0.400212\pi
0.308384 + 0.951262i 0.400212π0.400212\pi
500500 0 0
501501 0 0
502502 0 0
503503 −11268.0 −0.998838 −0.499419 0.866361i 0.666453π-0.666453\pi
−0.499419 + 0.866361i 0.666453π0.666453\pi
504504 0 0
505505 0 0
506506 0 0
507507 0 0
508508 0 0
509509 −16078.0 −1.40009 −0.700044 0.714100i 0.746836π-0.746836\pi
−0.700044 + 0.714100i 0.746836π0.746836\pi
510510 0 0
511511 −9918.00 −0.858604
512512 0 0
513513 0 0
514514 0 0
515515 0 0
516516 0 0
517517 −5060.00 −0.430442
518518 0 0
519519 0 0
520520 0 0
521521 17366.0 1.46030 0.730152 0.683285i 0.239449π-0.239449\pi
0.730152 + 0.683285i 0.239449π0.239449\pi
522522 0 0
523523 −4913.00 −0.410766 −0.205383 0.978682i 0.565844π-0.565844\pi
−0.205383 + 0.978682i 0.565844π0.565844\pi
524524 0 0
525525 0 0
526526 0 0
527527 −2030.00 −0.167795
528528 0 0
529529 −8803.00 −0.723514
530530 0 0
531531 0 0
532532 0 0
533533 468.000 0.0380325
534534 0 0
535535 0 0
536536 0 0
537537 0 0
538538 0 0
539539 −396.000 −0.0316455
540540 0 0
541541 17605.0 1.39907 0.699536 0.714597i 0.253390π-0.253390\pi
0.699536 + 0.714597i 0.253390π0.253390\pi
542542 0 0
543543 0 0
544544 0 0
545545 0 0
546546 0 0
547547 −14560.0 −1.13810 −0.569050 0.822303i 0.692689π-0.692689\pi
−0.569050 + 0.822303i 0.692689π0.692689\pi
548548 0 0
549549 0 0
550550 0 0
551551 1166.00 0.0901511
552552 0 0
553553 −25992.0 −1.99872
554554 0 0
555555 0 0
556556 0 0
557557 20614.0 1.56812 0.784060 0.620685i 0.213145π-0.213145\pi
0.784060 + 0.620685i 0.213145π0.213145\pi
558558 0 0
559559 −431.000 −0.0326107
560560 0 0
561561 0 0
562562 0 0
563563 −3442.00 −0.257661 −0.128830 0.991667i 0.541122π-0.541122\pi
−0.128830 + 0.991667i 0.541122π0.541122\pi
564564 0 0
565565 0 0
566566 0 0
567567 0 0
568568 0 0
569569 −22082.0 −1.62693 −0.813467 0.581611i 0.802423π-0.802423\pi
−0.813467 + 0.581611i 0.802423π0.802423\pi
570570 0 0
571571 451.000 0.0330539 0.0165269 0.999863i 0.494739π-0.494739\pi
0.0165269 + 0.999863i 0.494739π0.494739\pi
572572 0 0
573573 0 0
574574 0 0
575575 0 0
576576 0 0
577577 20987.0 1.51421 0.757106 0.653292i 0.226613π-0.226613\pi
0.757106 + 0.653292i 0.226613π0.226613\pi
578578 0 0
579579 0 0
580580 0 0
581581 −21622.0 −1.54394
582582 0 0
583583 0 0
584584 0 0
585585 0 0
586586 0 0
587587 −3888.00 −0.273381 −0.136691 0.990614i 0.543647π-0.543647\pi
−0.136691 + 0.990614i 0.543647π0.543647\pi
588588 0 0
589589 1855.00 0.129769
590590 0 0
591591 0 0
592592 0 0
593593 11268.0 0.780306 0.390153 0.920750i 0.372422π-0.372422\pi
0.390153 + 0.920750i 0.372422π0.372422\pi
594594 0 0
595595 0 0
596596 0 0
597597 0 0
598598 0 0
599599 3996.00 0.272575 0.136287 0.990669i 0.456483π-0.456483\pi
0.136287 + 0.990669i 0.456483π0.456483\pi
600600 0 0
601601 −24965.0 −1.69442 −0.847208 0.531262i 0.821718π-0.821718\pi
−0.847208 + 0.531262i 0.821718π0.821718\pi
602602 0 0
603603 0 0
604604 0 0
605605 0 0
606606 0 0
607607 4176.00 0.279240 0.139620 0.990205i 0.455412π-0.455412\pi
0.139620 + 0.990205i 0.455412π0.455412\pi
608608 0 0
609609 0 0
610610 0 0
611611 230.000 0.0152288
612612 0 0
613613 −9558.00 −0.629762 −0.314881 0.949131i 0.601965π-0.601965\pi
−0.314881 + 0.949131i 0.601965π0.601965\pi
614614 0 0
615615 0 0
616616 0 0
617617 9000.00 0.587239 0.293619 0.955922i 0.405140π-0.405140\pi
0.293619 + 0.955922i 0.405140π0.405140\pi
618618 0 0
619619 −15625.0 −1.01457 −0.507287 0.861777i 0.669352π-0.669352\pi
−0.507287 + 0.861777i 0.669352π0.669352\pi
620620 0 0
621621 0 0
622622 0 0
623623 2736.00 0.175948
624624 0 0
625625 0 0
626626 0 0
627627 0 0
628628 0 0
629629 −15660.0 −0.992695
630630 0 0
631631 −31175.0 −1.96681 −0.983405 0.181424i 0.941929π-0.941929\pi
−0.983405 + 0.181424i 0.941929π0.941929\pi
632632 0 0
633633 0 0
634634 0 0
635635 0 0
636636 0 0
637637 18.0000 0.00111960
638638 0 0
639639 0 0
640640 0 0
641641 6732.00 0.414817 0.207409 0.978254i 0.433497π-0.433497\pi
0.207409 + 0.978254i 0.433497π0.433497\pi
642642 0 0
643643 −6228.00 −0.381973 −0.190986 0.981593i 0.561169π-0.561169\pi
−0.190986 + 0.981593i 0.561169π0.561169\pi
644644 0 0
645645 0 0
646646 0 0
647647 396.000 0.0240624 0.0120312 0.999928i 0.496170π-0.496170\pi
0.0120312 + 0.999928i 0.496170π0.496170\pi
648648 0 0
649649 9812.00 0.593459
650650 0 0
651651 0 0
652652 0 0
653653 4702.00 0.281782 0.140891 0.990025i 0.455003π-0.455003\pi
0.140891 + 0.990025i 0.455003π0.455003\pi
654654 0 0
655655 0 0
656656 0 0
657657 0 0
658658 0 0
659659 22140.0 1.30873 0.654364 0.756180i 0.272936π-0.272936\pi
0.654364 + 0.756180i 0.272936π0.272936\pi
660660 0 0
661661 −13518.0 −0.795445 −0.397723 0.917506i 0.630199π-0.630199\pi
−0.397723 + 0.917506i 0.630199π0.630199\pi
662662 0 0
663663 0 0
664664 0 0
665665 0 0
666666 0 0
667667 1276.00 0.0740733
668668 0 0
669669 0 0
670670 0 0
671671 −2794.00 −0.160747
672672 0 0
673673 −11250.0 −0.644362 −0.322181 0.946678i 0.604416π-0.604416\pi
−0.322181 + 0.946678i 0.604416π0.604416\pi
674674 0 0
675675 0 0
676676 0 0
677677 15170.0 0.861197 0.430599 0.902544i 0.358302π-0.358302\pi
0.430599 + 0.902544i 0.358302π0.358302\pi
678678 0 0
679679 20501.0 1.15870
680680 0 0
681681 0 0
682682 0 0
683683 −22680.0 −1.27061 −0.635305 0.772262i 0.719125π-0.719125\pi
−0.635305 + 0.772262i 0.719125π0.719125\pi
684684 0 0
685685 0 0
686686 0 0
687687 0 0
688688 0 0
689689 0 0
690690 0 0
691691 128.000 0.00704682 0.00352341 0.999994i 0.498878π-0.498878\pi
0.00352341 + 0.999994i 0.498878π0.498878\pi
692692 0 0
693693 0 0
694694 0 0
695695 0 0
696696 0 0
697697 27144.0 1.47511
698698 0 0
699699 0 0
700700 0 0
701701 −25682.0 −1.38373 −0.691866 0.722026i 0.743211π-0.743211\pi
−0.691866 + 0.722026i 0.743211π0.743211\pi
702702 0 0
703703 14310.0 0.767727
704704 0 0
705705 0 0
706706 0 0
707707 −27360.0 −1.45542
708708 0 0
709709 −4951.00 −0.262255 −0.131127 0.991366i 0.541860π-0.541860\pi
−0.131127 + 0.991366i 0.541860π0.541860\pi
710710 0 0
711711 0 0
712712 0 0
713713 2030.00 0.106626
714714 0 0
715715 0 0
716716 0 0
717717 0 0
718718 0 0
719719 22190.0 1.15097 0.575485 0.817812i 0.304813π-0.304813\pi
0.575485 + 0.817812i 0.304813π0.304813\pi
720720 0 0
721721 −2356.00 −0.121695
722722 0 0
723723 0 0
724724 0 0
725725 0 0
726726 0 0
727727 7685.00 0.392051 0.196025 0.980599i 0.437197π-0.437197\pi
0.196025 + 0.980599i 0.437197π0.437197\pi
728728 0 0
729729 0 0
730730 0 0
731731 −24998.0 −1.26482
732732 0 0
733733 29574.0 1.49023 0.745116 0.666934i 0.232394π-0.232394\pi
0.745116 + 0.666934i 0.232394π0.232394\pi
734734 0 0
735735 0 0
736736 0 0
737737 17842.0 0.891748
738738 0 0
739739 −32580.0 −1.62175 −0.810876 0.585218i 0.801009π-0.801009\pi
−0.810876 + 0.585218i 0.801009π0.801009\pi
740740 0 0
741741 0 0
742742 0 0
743743 3060.00 0.151091 0.0755454 0.997142i 0.475930π-0.475930\pi
0.0755454 + 0.997142i 0.475930π0.475930\pi
744744 0 0
745745 0 0
746746 0 0
747747 0 0
748748 0 0
749749 −8208.00 −0.400419
750750 0 0
751751 −7992.00 −0.388325 −0.194163 0.980969i 0.562199π-0.562199\pi
−0.194163 + 0.980969i 0.562199π0.562199\pi
752752 0 0
753753 0 0
754754 0 0
755755 0 0
756756 0 0
757757 22841.0 1.09666 0.548329 0.836263i 0.315264π-0.315264\pi
0.548329 + 0.836263i 0.315264π0.315264\pi
758758 0 0
759759 0 0
760760 0 0
761761 17172.0 0.817982 0.408991 0.912538i 0.365881π-0.365881\pi
0.408991 + 0.912538i 0.365881π0.365881\pi
762762 0 0
763763 −13319.0 −0.631953
764764 0 0
765765 0 0
766766 0 0
767767 −446.000 −0.0209963
768768 0 0
769769 −30869.0 −1.44755 −0.723774 0.690037i 0.757594π-0.757594\pi
−0.723774 + 0.690037i 0.757594π0.757594\pi
770770 0 0
771771 0 0
772772 0 0
773773 −34884.0 −1.62314 −0.811572 0.584252i 0.801388π-0.801388\pi
−0.811572 + 0.584252i 0.801388π0.801388\pi
774774 0 0
775775 0 0
776776 0 0
777777 0 0
778778 0 0
779779 −24804.0 −1.14082
780780 0 0
781781 792.000 0.0362868
782782 0 0
783783 0 0
784784 0 0
785785 0 0
786786 0 0
787787 5039.00 0.228235 0.114118 0.993467i 0.463596π-0.463596\pi
0.114118 + 0.993467i 0.463596π0.463596\pi
788788 0 0
789789 0 0
790790 0 0
791791 −19836.0 −0.891640
792792 0 0
793793 127.000 0.00568714
794794 0 0
795795 0 0
796796 0 0
797797 −540.000 −0.0239997 −0.0119999 0.999928i 0.503820π-0.503820\pi
−0.0119999 + 0.999928i 0.503820π0.503820\pi
798798 0 0
799799 13340.0 0.590657
800800 0 0
801801 0 0
802802 0 0
803803 −11484.0 −0.504684
804804 0 0
805805 0 0
806806 0 0
807807 0 0
808808 0 0
809809 41328.0 1.79606 0.898032 0.439931i 0.144997π-0.144997\pi
0.898032 + 0.439931i 0.144997π0.144997\pi
810810 0 0
811811 −12853.0 −0.556510 −0.278255 0.960507i 0.589756π-0.589756\pi
−0.278255 + 0.960507i 0.589756π0.589756\pi
812812 0 0
813813 0 0
814814 0 0
815815 0 0
816816 0 0
817817 22843.0 0.978183
818818 0 0
819819 0 0
820820 0 0
821821 −29470.0 −1.25275 −0.626376 0.779521i 0.715463π-0.715463\pi
−0.626376 + 0.779521i 0.715463π0.715463\pi
822822 0 0
823823 −24407.0 −1.03375 −0.516874 0.856062i 0.672904π-0.672904\pi
−0.516874 + 0.856062i 0.672904π0.672904\pi
824824 0 0
825825 0 0
826826 0 0
827827 −15048.0 −0.632733 −0.316367 0.948637i 0.602463π-0.602463\pi
−0.316367 + 0.948637i 0.602463π0.602463\pi
828828 0 0
829829 −28406.0 −1.19009 −0.595043 0.803694i 0.702865π-0.702865\pi
−0.595043 + 0.803694i 0.702865π0.702865\pi
830830 0 0
831831 0 0
832832 0 0
833833 1044.00 0.0434243
834834 0 0
835835 0 0
836836 0 0
837837 0 0
838838 0 0
839839 26914.0 1.10748 0.553739 0.832690i 0.313200π-0.313200\pi
0.553739 + 0.832690i 0.313200π0.313200\pi
840840 0 0
841841 −23905.0 −0.980155
842842 0 0
843843 0 0
844844 0 0
845845 0 0
846846 0 0
847847 16093.0 0.652848
848848 0 0
849849 0 0
850850 0 0
851851 15660.0 0.630808
852852 0 0
853853 21275.0 0.853977 0.426988 0.904257i 0.359575π-0.359575\pi
0.426988 + 0.904257i 0.359575π0.359575\pi
854854 0 0
855855 0 0
856856 0 0
857857 −39132.0 −1.55977 −0.779885 0.625922i 0.784723π-0.784723\pi
−0.779885 + 0.625922i 0.784723π0.784723\pi
858858 0 0
859859 −448.000 −0.0177946 −0.00889730 0.999960i 0.502832π-0.502832\pi
−0.00889730 + 0.999960i 0.502832π0.502832\pi
860860 0 0
861861 0 0
862862 0 0
863863 26856.0 1.05932 0.529658 0.848212i 0.322320π-0.322320\pi
0.529658 + 0.848212i 0.322320π0.322320\pi
864864 0 0
865865 0 0
866866 0 0
867867 0 0
868868 0 0
869869 −30096.0 −1.17484
870870 0 0
871871 −811.000 −0.0315496
872872 0 0
873873 0 0
874874 0 0
875875 0 0
876876 0 0
877877 −3653.00 −0.140653 −0.0703267 0.997524i 0.522404π-0.522404\pi
−0.0703267 + 0.997524i 0.522404π0.522404\pi
878878 0 0
879879 0 0
880880 0 0
881881 6552.00 0.250559 0.125280 0.992121i 0.460017π-0.460017\pi
0.125280 + 0.992121i 0.460017π0.460017\pi
882882 0 0
883883 4481.00 0.170779 0.0853894 0.996348i 0.472787π-0.472787\pi
0.0853894 + 0.996348i 0.472787π0.472787\pi
884884 0 0
885885 0 0
886886 0 0
887887 14666.0 0.555170 0.277585 0.960701i 0.410466π-0.410466\pi
0.277585 + 0.960701i 0.410466π0.410466\pi
888888 0 0
889889 −9576.00 −0.361270
890890 0 0
891891 0 0
892892 0 0
893893 −12190.0 −0.456800
894894 0 0
895895 0 0
896896 0 0
897897 0 0
898898 0 0
899899 770.000 0.0285661
900900 0 0
901901 0 0
902902 0 0
903903 0 0
904904 0 0
905905 0 0
906906 0 0
907907 51804.0 1.89650 0.948249 0.317528i 0.102853π-0.102853\pi
0.948249 + 0.317528i 0.102853π0.102853\pi
908908 0 0
909909 0 0
910910 0 0
911911 −31198.0 −1.13462 −0.567308 0.823505i 0.692015π-0.692015\pi
−0.567308 + 0.823505i 0.692015π0.692015\pi
912912 0 0
913913 −25036.0 −0.907525
914914 0 0
915915 0 0
916916 0 0
917917 1368.00 0.0492643
918918 0 0
919919 −27001.0 −0.969185 −0.484592 0.874740i 0.661032π-0.661032\pi
−0.484592 + 0.874740i 0.661032π0.661032\pi
920920 0 0
921921 0 0
922922 0 0
923923 −36.0000 −0.00128381
924924 0 0
925925 0 0
926926 0 0
927927 0 0
928928 0 0
929929 22694.0 0.801470 0.400735 0.916194i 0.368755π-0.368755\pi
0.400735 + 0.916194i 0.368755π0.368755\pi
930930 0 0
931931 −954.000 −0.0335833
932932 0 0
933933 0 0
934934 0 0
935935 0 0
936936 0 0
937937 29503.0 1.02862 0.514312 0.857603i 0.328047π-0.328047\pi
0.514312 + 0.857603i 0.328047π0.328047\pi
938938 0 0
939939 0 0
940940 0 0
941941 −14566.0 −0.504610 −0.252305 0.967648i 0.581189π-0.581189\pi
−0.252305 + 0.967648i 0.581189π0.581189\pi
942942 0 0
943943 −27144.0 −0.937360
944944 0 0
945945 0 0
946946 0 0
947947 −42322.0 −1.45225 −0.726125 0.687563i 0.758681π-0.758681\pi
−0.726125 + 0.687563i 0.758681π0.758681\pi
948948 0 0
949949 522.000 0.0178555
950950 0 0
951951 0 0
952952 0 0
953953 −43416.0 −1.47574 −0.737871 0.674942i 0.764169π-0.764169\pi
−0.737871 + 0.674942i 0.764169π0.764169\pi
954954 0 0
955955 0 0
956956 0 0
957957 0 0
958958 0 0
959959 −35150.0 −1.18358
960960 0 0
961961 −28566.0 −0.958880
962962 0 0
963963 0 0
964964 0 0
965965 0 0
966966 0 0
967967 21528.0 0.715919 0.357960 0.933737i 0.383473π-0.383473\pi
0.357960 + 0.933737i 0.383473π0.383473\pi
968968 0 0
969969 0 0
970970 0 0
971971 −27050.0 −0.894002 −0.447001 0.894533i 0.647508π-0.647508\pi
−0.447001 + 0.894533i 0.647508π0.647508\pi
972972 0 0
973973 −34884.0 −1.14936
974974 0 0
975975 0 0
976976 0 0
977977 24934.0 0.816489 0.408244 0.912873i 0.366141π-0.366141\pi
0.408244 + 0.912873i 0.366141π0.366141\pi
978978 0 0
979979 3168.00 0.103422
980980 0 0
981981 0 0
982982 0 0
983983 −8388.00 −0.272162 −0.136081 0.990698i 0.543451π-0.543451\pi
−0.136081 + 0.990698i 0.543451π0.543451\pi
984984 0 0
985985 0 0
986986 0 0
987987 0 0
988988 0 0
989989 24998.0 0.803731
990990 0 0
991991 29033.0 0.930639 0.465320 0.885143i 0.345939π-0.345939\pi
0.465320 + 0.885143i 0.345939π0.345939\pi
992992 0 0
993993 0 0
994994 0 0
995995 0 0
996996 0 0
997997 −25326.0 −0.804496 −0.402248 0.915531i 0.631771π-0.631771\pi
−0.402248 + 0.915531i 0.631771π0.631771\pi
998998 0 0
999999 0 0
Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1800.4.a.g.1.1 1
3.2 odd 2 600.4.a.j.1.1 yes 1
5.2 odd 4 1800.4.f.g.649.1 2
5.3 odd 4 1800.4.f.g.649.2 2
5.4 even 2 1800.4.a.bf.1.1 1
12.11 even 2 1200.4.a.n.1.1 1
15.2 even 4 600.4.f.h.49.1 2
15.8 even 4 600.4.f.h.49.2 2
15.14 odd 2 600.4.a.g.1.1 1
60.23 odd 4 1200.4.f.f.49.1 2
60.47 odd 4 1200.4.f.f.49.2 2
60.59 even 2 1200.4.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.4.a.g.1.1 1 15.14 odd 2
600.4.a.j.1.1 yes 1 3.2 odd 2
600.4.f.h.49.1 2 15.2 even 4
600.4.f.h.49.2 2 15.8 even 4
1200.4.a.n.1.1 1 12.11 even 2
1200.4.a.x.1.1 1 60.59 even 2
1200.4.f.f.49.1 2 60.23 odd 4
1200.4.f.f.49.2 2 60.47 odd 4
1800.4.a.g.1.1 1 1.1 even 1 trivial
1800.4.a.bf.1.1 1 5.4 even 2
1800.4.f.g.649.1 2 5.2 odd 4
1800.4.f.g.649.2 2 5.3 odd 4