Properties

Label 4788.2.a.q.1.4
Level 47884788
Weight 22
Character 4788.1
Self dual yes
Analytic conductor 38.23238.232
Analytic rank 11
Dimension 44
Inner twists 22

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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] = [4788,2,Mod(1,4788)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4788, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4788.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 4788=2232719 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 4788.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-4,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 38.232372487838.2323724878
Analytic rank: 11
Dimension: 44
Coefficient field: Q(2,5)\Q(\sqrt{2}, \sqrt{5})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x46x2+4 x^{4} - 6x^{2} + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 22 2^{2}
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.4
Root 2.288252.28825 of defining polynomial
Character χ\chi == 4788.1

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) == q+2.28825q51.00000q72.82843q113.23607q13+0.540182q17+1.00000q19+2.82843q23+0.236068q258.61280q29+7.70820q312.28825q35+4.47214q378.48528q415.23607q434.03631q47+1.00000q49+11.4412q536.47214q55+2.16073q593.52786q617.40492q6515.4164q67+12.5216q7110.9443q73+2.82843q7710.4721q797.94510q83+1.23607q858.48528q89+3.23607q91+2.28825q9512.4721q97+O(q100)q+2.28825 q^{5} -1.00000 q^{7} -2.82843 q^{11} -3.23607 q^{13} +0.540182 q^{17} +1.00000 q^{19} +2.82843 q^{23} +0.236068 q^{25} -8.61280 q^{29} +7.70820 q^{31} -2.28825 q^{35} +4.47214 q^{37} -8.48528 q^{41} -5.23607 q^{43} -4.03631 q^{47} +1.00000 q^{49} +11.4412 q^{53} -6.47214 q^{55} +2.16073 q^{59} -3.52786 q^{61} -7.40492 q^{65} -15.4164 q^{67} +12.5216 q^{71} -10.9443 q^{73} +2.82843 q^{77} -10.4721 q^{79} -7.94510 q^{83} +1.23607 q^{85} -8.48528 q^{89} +3.23607 q^{91} +2.28825 q^{95} -12.4721 q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q4q74q13+4q198q25+4q3112q43+4q498q5532q618q678q7324q794q85+4q9132q97+O(q100) 4 q - 4 q^{7} - 4 q^{13} + 4 q^{19} - 8 q^{25} + 4 q^{31} - 12 q^{43} + 4 q^{49} - 8 q^{55} - 32 q^{61} - 8 q^{67} - 8 q^{73} - 24 q^{79} - 4 q^{85} + 4 q^{91} - 32 q^{97}+O(q^{100}) Copy content Toggle raw display

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 2.28825 1.02333 0.511667 0.859184i 0.329028π-0.329028\pi
0.511667 + 0.859184i 0.329028π0.329028\pi
66 0 0
77 −1.00000 −0.377964
88 0 0
99 0 0
1010 0 0
1111 −2.82843 −0.852803 −0.426401 0.904534i 0.640219π-0.640219\pi
−0.426401 + 0.904534i 0.640219π0.640219\pi
1212 0 0
1313 −3.23607 −0.897524 −0.448762 0.893651i 0.648135π-0.648135\pi
−0.448762 + 0.893651i 0.648135π0.648135\pi
1414 0 0
1515 0 0
1616 0 0
1717 0.540182 0.131013 0.0655066 0.997852i 0.479134π-0.479134\pi
0.0655066 + 0.997852i 0.479134π0.479134\pi
1818 0 0
1919 1.00000 0.229416
2020 0 0
2121 0 0
2222 0 0
2323 2.82843 0.589768 0.294884 0.955533i 0.404719π-0.404719\pi
0.294884 + 0.955533i 0.404719π0.404719\pi
2424 0 0
2525 0.236068 0.0472136
2626 0 0
2727 0 0
2828 0 0
2929 −8.61280 −1.59936 −0.799678 0.600428i 0.794997π-0.794997\pi
−0.799678 + 0.600428i 0.794997π0.794997\pi
3030 0 0
3131 7.70820 1.38443 0.692217 0.721689i 0.256634π-0.256634\pi
0.692217 + 0.721689i 0.256634π0.256634\pi
3232 0 0
3333 0 0
3434 0 0
3535 −2.28825 −0.386784
3636 0 0
3737 4.47214 0.735215 0.367607 0.929981i 0.380177π-0.380177\pi
0.367607 + 0.929981i 0.380177π0.380177\pi
3838 0 0
3939 0 0
4040 0 0
4141 −8.48528 −1.32518 −0.662589 0.748983i 0.730542π-0.730542\pi
−0.662589 + 0.748983i 0.730542π0.730542\pi
4242 0 0
4343 −5.23607 −0.798493 −0.399246 0.916844i 0.630728π-0.630728\pi
−0.399246 + 0.916844i 0.630728π0.630728\pi
4444 0 0
4545 0 0
4646 0 0
4747 −4.03631 −0.588756 −0.294378 0.955689i 0.595113π-0.595113\pi
−0.294378 + 0.955689i 0.595113π0.595113\pi
4848 0 0
4949 1.00000 0.142857
5050 0 0
5151 0 0
5252 0 0
5353 11.4412 1.57157 0.785787 0.618497i 0.212258π-0.212258\pi
0.785787 + 0.618497i 0.212258π0.212258\pi
5454 0 0
5555 −6.47214 −0.872703
5656 0 0
5757 0 0
5858 0 0
5959 2.16073 0.281303 0.140651 0.990059i 0.455080π-0.455080\pi
0.140651 + 0.990059i 0.455080π0.455080\pi
6060 0 0
6161 −3.52786 −0.451697 −0.225848 0.974162i 0.572515π-0.572515\pi
−0.225848 + 0.974162i 0.572515π0.572515\pi
6262 0 0
6363 0 0
6464 0 0
6565 −7.40492 −0.918467
6666 0 0
6767 −15.4164 −1.88341 −0.941707 0.336434i 0.890779π-0.890779\pi
−0.941707 + 0.336434i 0.890779π0.890779\pi
6868 0 0
6969 0 0
7070 0 0
7171 12.5216 1.48604 0.743020 0.669270i 0.233393π-0.233393\pi
0.743020 + 0.669270i 0.233393π0.233393\pi
7272 0 0
7373 −10.9443 −1.28093 −0.640465 0.767987i 0.721258π-0.721258\pi
−0.640465 + 0.767987i 0.721258π0.721258\pi
7474 0 0
7575 0 0
7676 0 0
7777 2.82843 0.322329
7878 0 0
7979 −10.4721 −1.17821 −0.589104 0.808057i 0.700519π-0.700519\pi
−0.589104 + 0.808057i 0.700519π0.700519\pi
8080 0 0
8181 0 0
8282 0 0
8383 −7.94510 −0.872088 −0.436044 0.899925i 0.643621π-0.643621\pi
−0.436044 + 0.899925i 0.643621π0.643621\pi
8484 0 0
8585 1.23607 0.134070
8686 0 0
8787 0 0
8888 0 0
8989 −8.48528 −0.899438 −0.449719 0.893170i 0.648476π-0.648476\pi
−0.449719 + 0.893170i 0.648476π0.648476\pi
9090 0 0
9191 3.23607 0.339232
9292 0 0
9393 0 0
9494 0 0
9595 2.28825 0.234769
9696 0 0
9797 −12.4721 −1.26635 −0.633177 0.774007i 0.718249π-0.718249\pi
−0.633177 + 0.774007i 0.718249π0.718249\pi
9898 0 0
9999 0 0
100100 0 0
101101 −2.70091 −0.268750 −0.134375 0.990931i 0.542903π-0.542903\pi
−0.134375 + 0.990931i 0.542903π0.542903\pi
102102 0 0
103103 4.94427 0.487174 0.243587 0.969879i 0.421676π-0.421676\pi
0.243587 + 0.969879i 0.421676π0.421676\pi
104104 0 0
105105 0 0
106106 0 0
107107 7.53244 0.728188 0.364094 0.931362i 0.381379π-0.381379\pi
0.364094 + 0.931362i 0.381379π0.381379\pi
108108 0 0
109109 −1.05573 −0.101120 −0.0505602 0.998721i 0.516101π-0.516101\pi
−0.0505602 + 0.998721i 0.516101π0.516101\pi
110110 0 0
111111 0 0
112112 0 0
113113 −14.2697 −1.34238 −0.671188 0.741287i 0.734216π-0.734216\pi
−0.671188 + 0.741287i 0.734216π0.734216\pi
114114 0 0
115115 6.47214 0.603530
116116 0 0
117117 0 0
118118 0 0
119119 −0.540182 −0.0495184
120120 0 0
121121 −3.00000 −0.272727
122122 0 0
123123 0 0
124124 0 0
125125 −10.9010 −0.975019
126126 0 0
127127 −19.4164 −1.72293 −0.861464 0.507819i 0.830452π-0.830452\pi
−0.861464 + 0.507819i 0.830452π0.830452\pi
128128 0 0
129129 0 0
130130 0 0
131131 −12.5216 −1.09402 −0.547008 0.837127i 0.684233π-0.684233\pi
−0.547008 + 0.837127i 0.684233π0.684233\pi
132132 0 0
133133 −1.00000 −0.0867110
134134 0 0
135135 0 0
136136 0 0
137137 14.8098 1.26529 0.632645 0.774442i 0.281969π-0.281969\pi
0.632645 + 0.774442i 0.281969π0.281969\pi
138138 0 0
139139 12.9443 1.09792 0.548959 0.835849i 0.315024π-0.315024\pi
0.548959 + 0.835849i 0.315024π0.315024\pi
140140 0 0
141141 0 0
142142 0 0
143143 9.15298 0.765411
144144 0 0
145145 −19.7082 −1.63668
146146 0 0
147147 0 0
148148 0 0
149149 −13.7295 −1.12476 −0.562381 0.826878i 0.690115π-0.690115\pi
−0.562381 + 0.826878i 0.690115π0.690115\pi
150150 0 0
151151 −10.4721 −0.852210 −0.426105 0.904674i 0.640115π-0.640115\pi
−0.426105 + 0.904674i 0.640115π0.640115\pi
152152 0 0
153153 0 0
154154 0 0
155155 17.6383 1.41674
156156 0 0
157157 −6.00000 −0.478852 −0.239426 0.970915i 0.576959π-0.576959\pi
−0.239426 + 0.970915i 0.576959π0.576959\pi
158158 0 0
159159 0 0
160160 0 0
161161 −2.82843 −0.222911
162162 0 0
163163 15.1246 1.18465 0.592326 0.805699i 0.298210π-0.298210\pi
0.592326 + 0.805699i 0.298210π0.298210\pi
164164 0 0
165165 0 0
166166 0 0
167167 10.2333 0.791880 0.395940 0.918276i 0.370419π-0.370419\pi
0.395940 + 0.918276i 0.370419π0.370419\pi
168168 0 0
169169 −2.52786 −0.194451
170170 0 0
171171 0 0
172172 0 0
173173 −4.16383 −0.316570 −0.158285 0.987393i 0.550596π-0.550596\pi
−0.158285 + 0.987393i 0.550596π0.550596\pi
174174 0 0
175175 −0.236068 −0.0178451
176176 0 0
177177 0 0
178178 0 0
179179 10.7735 0.805251 0.402626 0.915365i 0.368098π-0.368098\pi
0.402626 + 0.915365i 0.368098π0.368098\pi
180180 0 0
181181 2.94427 0.218846 0.109423 0.993995i 0.465100π-0.465100\pi
0.109423 + 0.993995i 0.465100π0.465100\pi
182182 0 0
183183 0 0
184184 0 0
185185 10.2333 0.752371
186186 0 0
187187 −1.52786 −0.111728
188188 0 0
189189 0 0
190190 0 0
191191 13.3168 0.963571 0.481785 0.876289i 0.339988π-0.339988\pi
0.481785 + 0.876289i 0.339988π0.339988\pi
192192 0 0
193193 −18.0000 −1.29567 −0.647834 0.761781i 0.724325π-0.724325\pi
−0.647834 + 0.761781i 0.724325π0.724325\pi
194194 0 0
195195 0 0
196196 0 0
197197 20.4667 1.45819 0.729096 0.684412i 0.239941π-0.239941\pi
0.729096 + 0.684412i 0.239941π0.239941\pi
198198 0 0
199199 8.00000 0.567105 0.283552 0.958957i 0.408487π-0.408487\pi
0.283552 + 0.958957i 0.408487π0.408487\pi
200200 0 0
201201 0 0
202202 0 0
203203 8.61280 0.604500
204204 0 0
205205 −19.4164 −1.35610
206206 0 0
207207 0 0
208208 0 0
209209 −2.82843 −0.195646
210210 0 0
211211 10.4721 0.720932 0.360466 0.932772i 0.382618π-0.382618\pi
0.360466 + 0.932772i 0.382618π0.382618\pi
212212 0 0
213213 0 0
214214 0 0
215215 −11.9814 −0.817125
216216 0 0
217217 −7.70820 −0.523267
218218 0 0
219219 0 0
220220 0 0
221221 −1.74806 −0.117588
222222 0 0
223223 5.23607 0.350633 0.175317 0.984512i 0.443905π-0.443905\pi
0.175317 + 0.984512i 0.443905π0.443905\pi
224224 0 0
225225 0 0
226226 0 0
227227 11.5687 0.767845 0.383922 0.923365i 0.374573π-0.374573\pi
0.383922 + 0.923365i 0.374573π0.374573\pi
228228 0 0
229229 −26.3607 −1.74196 −0.870981 0.491316i 0.836516π-0.836516\pi
−0.870981 + 0.491316i 0.836516π0.836516\pi
230230 0 0
231231 0 0
232232 0 0
233233 12.3941 0.811963 0.405981 0.913881i 0.366930π-0.366930\pi
0.405981 + 0.913881i 0.366930π0.366930\pi
234234 0 0
235235 −9.23607 −0.602495
236236 0 0
237237 0 0
238238 0 0
239239 0.412662 0.0266929 0.0133464 0.999911i 0.495752π-0.495752\pi
0.0133464 + 0.999911i 0.495752π0.495752\pi
240240 0 0
241241 11.8885 0.765808 0.382904 0.923788i 0.374924π-0.374924\pi
0.382904 + 0.923788i 0.374924π0.374924\pi
242242 0 0
243243 0 0
244244 0 0
245245 2.28825 0.146191
246246 0 0
247247 −3.23607 −0.205906
248248 0 0
249249 0 0
250250 0 0
251251 29.0795 1.83548 0.917741 0.397180i 0.130011π-0.130011\pi
0.917741 + 0.397180i 0.130011π0.130011\pi
252252 0 0
253253 −8.00000 −0.502956
254254 0 0
255255 0 0
256256 0 0
257257 −18.7186 −1.16764 −0.583818 0.811885i 0.698442π-0.698442\pi
−0.583818 + 0.811885i 0.698442π0.698442\pi
258258 0 0
259259 −4.47214 −0.277885
260260 0 0
261261 0 0
262262 0 0
263263 −24.3755 −1.50306 −0.751528 0.659701i 0.770683π-0.770683\pi
−0.751528 + 0.659701i 0.770683π0.770683\pi
264264 0 0
265265 26.1803 1.60825
266266 0 0
267267 0 0
268268 0 0
269269 13.0618 0.796390 0.398195 0.917301i 0.369637π-0.369637\pi
0.398195 + 0.917301i 0.369637π0.369637\pi
270270 0 0
271271 −26.8328 −1.62998 −0.814989 0.579477i 0.803257π-0.803257\pi
−0.814989 + 0.579477i 0.803257π0.803257\pi
272272 0 0
273273 0 0
274274 0 0
275275 −0.667701 −0.0402639
276276 0 0
277277 −24.7639 −1.48792 −0.743960 0.668224i 0.767055π-0.767055\pi
−0.743960 + 0.668224i 0.767055π0.767055\pi
278278 0 0
279279 0 0
280280 0 0
281281 −10.1058 −0.602863 −0.301432 0.953488i 0.597464π-0.597464\pi
−0.301432 + 0.953488i 0.597464π0.597464\pi
282282 0 0
283283 6.47214 0.384729 0.192364 0.981324i 0.438384π-0.438384\pi
0.192364 + 0.981324i 0.438384π0.438384\pi
284284 0 0
285285 0 0
286286 0 0
287287 8.48528 0.500870
288288 0 0
289289 −16.7082 −0.982836
290290 0 0
291291 0 0
292292 0 0
293293 22.4698 1.31270 0.656350 0.754457i 0.272100π-0.272100\pi
0.656350 + 0.754457i 0.272100π0.272100\pi
294294 0 0
295295 4.94427 0.287867
296296 0 0
297297 0 0
298298 0 0
299299 −9.15298 −0.529331
300300 0 0
301301 5.23607 0.301802
302302 0 0
303303 0 0
304304 0 0
305305 −8.07262 −0.462237
306306 0 0
307307 −3.70820 −0.211638 −0.105819 0.994385i 0.533746π-0.533746\pi
−0.105819 + 0.994385i 0.533746π0.533746\pi
308308 0 0
309309 0 0
310310 0 0
311311 −12.5216 −0.710034 −0.355017 0.934860i 0.615525π-0.615525\pi
−0.355017 + 0.934860i 0.615525π0.615525\pi
312312 0 0
313313 15.8885 0.898074 0.449037 0.893513i 0.351767π-0.351767\pi
0.449037 + 0.893513i 0.351767π0.351767\pi
314314 0 0
315315 0 0
316316 0 0
317317 28.8245 1.61894 0.809471 0.587159i 0.199754π-0.199754\pi
0.809471 + 0.587159i 0.199754π0.199754\pi
318318 0 0
319319 24.3607 1.36394
320320 0 0
321321 0 0
322322 0 0
323323 0.540182 0.0300565
324324 0 0
325325 −0.763932 −0.0423753
326326 0 0
327327 0 0
328328 0 0
329329 4.03631 0.222529
330330 0 0
331331 −21.8885 −1.20310 −0.601552 0.798834i 0.705451π-0.705451\pi
−0.601552 + 0.798834i 0.705451π0.705451\pi
332332 0 0
333333 0 0
334334 0 0
335335 −35.2765 −1.92736
336336 0 0
337337 −6.94427 −0.378279 −0.189139 0.981950i 0.560570π-0.560570\pi
−0.189139 + 0.981950i 0.560570π0.560570\pi
338338 0 0
339339 0 0
340340 0 0
341341 −21.8021 −1.18065
342342 0 0
343343 −1.00000 −0.0539949
344344 0 0
345345 0 0
346346 0 0
347347 −17.8933 −0.960563 −0.480281 0.877114i 0.659465π-0.659465\pi
−0.480281 + 0.877114i 0.659465π0.659465\pi
348348 0 0
349349 −17.4164 −0.932279 −0.466139 0.884711i 0.654356π-0.654356\pi
−0.466139 + 0.884711i 0.654356π0.654356\pi
350350 0 0
351351 0 0
352352 0 0
353353 −5.78437 −0.307871 −0.153936 0.988081i 0.549195π-0.549195\pi
−0.153936 + 0.988081i 0.549195π0.549195\pi
354354 0 0
355355 28.6525 1.52072
356356 0 0
357357 0 0
358358 0 0
359359 2.00310 0.105720 0.0528599 0.998602i 0.483166π-0.483166\pi
0.0528599 + 0.998602i 0.483166π0.483166\pi
360360 0 0
361361 1.00000 0.0526316
362362 0 0
363363 0 0
364364 0 0
365365 −25.0432 −1.31082
366366 0 0
367367 −23.4164 −1.22233 −0.611163 0.791505i 0.709298π-0.709298\pi
−0.611163 + 0.791505i 0.709298π0.709298\pi
368368 0 0
369369 0 0
370370 0 0
371371 −11.4412 −0.593999
372372 0 0
373373 −5.05573 −0.261776 −0.130888 0.991397i 0.541783π-0.541783\pi
−0.130888 + 0.991397i 0.541783π0.541783\pi
374374 0 0
375375 0 0
376376 0 0
377377 27.8716 1.43546
378378 0 0
379379 27.4164 1.40829 0.704143 0.710058i 0.251331π-0.251331\pi
0.704143 + 0.710058i 0.251331π0.251331\pi
380380 0 0
381381 0 0
382382 0 0
383383 28.5393 1.45829 0.729145 0.684359i 0.239918π-0.239918\pi
0.729145 + 0.684359i 0.239918π0.239918\pi
384384 0 0
385385 6.47214 0.329851
386386 0 0
387387 0 0
388388 0 0
389389 26.3786 1.33745 0.668724 0.743511i 0.266841π-0.266841\pi
0.668724 + 0.743511i 0.266841π0.266841\pi
390390 0 0
391391 1.52786 0.0772674
392392 0 0
393393 0 0
394394 0 0
395395 −23.9628 −1.20570
396396 0 0
397397 −18.0000 −0.903394 −0.451697 0.892171i 0.649181π-0.649181\pi
−0.451697 + 0.892171i 0.649181π0.649181\pi
398398 0 0
399399 0 0
400400 0 0
401401 3.62365 0.180956 0.0904782 0.995898i 0.471160π-0.471160\pi
0.0904782 + 0.995898i 0.471160π0.471160\pi
402402 0 0
403403 −24.9443 −1.24256
404404 0 0
405405 0 0
406406 0 0
407407 −12.6491 −0.626993
408408 0 0
409409 −25.1246 −1.24233 −0.621166 0.783679i 0.713341π-0.713341\pi
−0.621166 + 0.783679i 0.713341π0.713341\pi
410410 0 0
411411 0 0
412412 0 0
413413 −2.16073 −0.106322
414414 0 0
415415 −18.1803 −0.892438
416416 0 0
417417 0 0
418418 0 0
419419 −36.0717 −1.76222 −0.881110 0.472911i 0.843203π-0.843203\pi
−0.881110 + 0.472911i 0.843203π0.843203\pi
420420 0 0
421421 −2.00000 −0.0974740 −0.0487370 0.998812i 0.515520π-0.515520\pi
−0.0487370 + 0.998812i 0.515520π0.515520\pi
422422 0 0
423423 0 0
424424 0 0
425425 0.127520 0.00618561
426426 0 0
427427 3.52786 0.170725
428428 0 0
429429 0 0
430430 0 0
431431 11.0286 0.531227 0.265614 0.964080i 0.414425π-0.414425\pi
0.265614 + 0.964080i 0.414425π0.414425\pi
432432 0 0
433433 −11.8885 −0.571327 −0.285663 0.958330i 0.592214π-0.592214\pi
−0.285663 + 0.958330i 0.592214π0.592214\pi
434434 0 0
435435 0 0
436436 0 0
437437 2.82843 0.135302
438438 0 0
439439 15.1246 0.721858 0.360929 0.932593i 0.382460π-0.382460\pi
0.360929 + 0.932593i 0.382460π0.382460\pi
440440 0 0
441441 0 0
442442 0 0
443443 −3.08347 −0.146500 −0.0732500 0.997314i 0.523337π-0.523337\pi
−0.0732500 + 0.997314i 0.523337π0.523337\pi
444444 0 0
445445 −19.4164 −0.920426
446446 0 0
447447 0 0
448448 0 0
449449 −40.8059 −1.92575 −0.962874 0.269952i 0.912992π-0.912992\pi
−0.962874 + 0.269952i 0.912992π0.912992\pi
450450 0 0
451451 24.0000 1.13012
452452 0 0
453453 0 0
454454 0 0
455455 7.40492 0.347148
456456 0 0
457457 31.5967 1.47803 0.739017 0.673687i 0.235290π-0.235290\pi
0.739017 + 0.673687i 0.235290π0.235290\pi
458458 0 0
459459 0 0
460460 0 0
461461 −32.8307 −1.52908 −0.764538 0.644579i 0.777033π-0.777033\pi
−0.764538 + 0.644579i 0.777033π0.777033\pi
462462 0 0
463463 −21.8885 −1.01725 −0.508623 0.860989i 0.669845π-0.669845\pi
−0.508623 + 0.860989i 0.669845π0.669845\pi
464464 0 0
465465 0 0
466466 0 0
467467 −20.3392 −0.941185 −0.470592 0.882351i 0.655960π-0.655960\pi
−0.470592 + 0.882351i 0.655960π0.655960\pi
468468 0 0
469469 15.4164 0.711864
470470 0 0
471471 0 0
472472 0 0
473473 14.8098 0.680957
474474 0 0
475475 0.236068 0.0108315
476476 0 0
477477 0 0
478478 0 0
479479 31.4953 1.43906 0.719528 0.694464i 0.244358π-0.244358\pi
0.719528 + 0.694464i 0.244358π0.244358\pi
480480 0 0
481481 −14.4721 −0.659873
482482 0 0
483483 0 0
484484 0 0
485485 −28.5393 −1.29590
486486 0 0
487487 −2.11146 −0.0956792 −0.0478396 0.998855i 0.515234π-0.515234\pi
−0.0478396 + 0.998855i 0.515234π0.515234\pi
488488 0 0
489489 0 0
490490 0 0
491491 −26.5362 −1.19756 −0.598781 0.800913i 0.704348π-0.704348\pi
−0.598781 + 0.800913i 0.704348π0.704348\pi
492492 0 0
493493 −4.65248 −0.209537
494494 0 0
495495 0 0
496496 0 0
497497 −12.5216 −0.561670
498498 0 0
499499 17.8885 0.800801 0.400401 0.916340i 0.368871π-0.368871\pi
0.400401 + 0.916340i 0.368871π0.368871\pi
500500 0 0
501501 0 0
502502 0 0
503503 6.60970 0.294712 0.147356 0.989084i 0.452924π-0.452924\pi
0.147356 + 0.989084i 0.452924π0.452924\pi
504504 0 0
505505 −6.18034 −0.275022
506506 0 0
507507 0 0
508508 0 0
509509 39.1853 1.73686 0.868429 0.495813i 0.165130π-0.165130\pi
0.868429 + 0.495813i 0.165130π0.165130\pi
510510 0 0
511511 10.9443 0.484146
512512 0 0
513513 0 0
514514 0 0
515515 11.3137 0.498542
516516 0 0
517517 11.4164 0.502093
518518 0 0
519519 0 0
520520 0 0
521521 3.90879 0.171247 0.0856236 0.996328i 0.472712π-0.472712\pi
0.0856236 + 0.996328i 0.472712π0.472712\pi
522522 0 0
523523 6.18034 0.270247 0.135124 0.990829i 0.456857π-0.456857\pi
0.135124 + 0.990829i 0.456857π0.456857\pi
524524 0 0
525525 0 0
526526 0 0
527527 4.16383 0.181379
528528 0 0
529529 −15.0000 −0.652174
530530 0 0
531531 0 0
532532 0 0
533533 27.4589 1.18938
534534 0 0
535535 17.2361 0.745180
536536 0 0
537537 0 0
538538 0 0
539539 −2.82843 −0.121829
540540 0 0
541541 24.4721 1.05214 0.526070 0.850441i 0.323665π-0.323665\pi
0.526070 + 0.850441i 0.323665π0.323665\pi
542542 0 0
543543 0 0
544544 0 0
545545 −2.41577 −0.103480
546546 0 0
547547 43.4164 1.85635 0.928176 0.372142i 0.121377π-0.121377\pi
0.928176 + 0.372142i 0.121377π0.121377\pi
548548 0 0
549549 0 0
550550 0 0
551551 −8.61280 −0.366918
552552 0 0
553553 10.4721 0.445321
554554 0 0
555555 0 0
556556 0 0
557557 2.41577 0.102359 0.0511796 0.998689i 0.483702π-0.483702\pi
0.0511796 + 0.998689i 0.483702π0.483702\pi
558558 0 0
559559 16.9443 0.716666
560560 0 0
561561 0 0
562562 0 0
563563 −9.40802 −0.396501 −0.198250 0.980151i 0.563526π-0.563526\pi
−0.198250 + 0.980151i 0.563526π0.563526\pi
564564 0 0
565565 −32.6525 −1.37370
566566 0 0
567567 0 0
568568 0 0
569569 12.5216 0.524932 0.262466 0.964941i 0.415464π-0.415464\pi
0.262466 + 0.964941i 0.415464π0.415464\pi
570570 0 0
571571 2.11146 0.0883617 0.0441808 0.999024i 0.485932π-0.485932\pi
0.0441808 + 0.999024i 0.485932π0.485932\pi
572572 0 0
573573 0 0
574574 0 0
575575 0.667701 0.0278451
576576 0 0
577577 28.8328 1.20033 0.600163 0.799878i 0.295102π-0.295102\pi
0.600163 + 0.799878i 0.295102π0.295102\pi
578578 0 0
579579 0 0
580580 0 0
581581 7.94510 0.329618
582582 0 0
583583 −32.3607 −1.34024
584584 0 0
585585 0 0
586586 0 0
587587 1.62054 0.0668870 0.0334435 0.999441i 0.489353π-0.489353\pi
0.0334435 + 0.999441i 0.489353π0.489353\pi
588588 0 0
589589 7.70820 0.317611
590590 0 0
591591 0 0
592592 0 0
593593 6.60970 0.271428 0.135714 0.990748i 0.456667π-0.456667\pi
0.135714 + 0.990748i 0.456667π0.456667\pi
594594 0 0
595595 −1.23607 −0.0506738
596596 0 0
597597 0 0
598598 0 0
599599 26.9188 1.09987 0.549935 0.835207i 0.314652π-0.314652\pi
0.549935 + 0.835207i 0.314652π0.314652\pi
600600 0 0
601601 2.58359 0.105387 0.0526935 0.998611i 0.483219π-0.483219\pi
0.0526935 + 0.998611i 0.483219π0.483219\pi
602602 0 0
603603 0 0
604604 0 0
605605 −6.86474 −0.279091
606606 0 0
607607 12.0000 0.487065 0.243532 0.969893i 0.421694π-0.421694\pi
0.243532 + 0.969893i 0.421694π0.421694\pi
608608 0 0
609609 0 0
610610 0 0
611611 13.0618 0.528423
612612 0 0
613613 −7.81966 −0.315833 −0.157917 0.987452i 0.550478π-0.550478\pi
−0.157917 + 0.987452i 0.550478π0.550478\pi
614614 0 0
615615 0 0
616616 0 0
617617 −23.1375 −0.931480 −0.465740 0.884921i 0.654212π-0.654212\pi
−0.465740 + 0.884921i 0.654212π0.654212\pi
618618 0 0
619619 −35.7771 −1.43800 −0.719001 0.695009i 0.755400π-0.755400\pi
−0.719001 + 0.695009i 0.755400π0.755400\pi
620620 0 0
621621 0 0
622622 0 0
623623 8.48528 0.339956
624624 0 0
625625 −26.1246 −1.04498
626626 0 0
627627 0 0
628628 0 0
629629 2.41577 0.0963229
630630 0 0
631631 37.5967 1.49670 0.748351 0.663302i 0.230846π-0.230846\pi
0.748351 + 0.663302i 0.230846π0.230846\pi
632632 0 0
633633 0 0
634634 0 0
635635 −44.4295 −1.76313
636636 0 0
637637 −3.23607 −0.128218
638638 0 0
639639 0 0
640640 0 0
641641 −14.5247 −0.573691 −0.286845 0.957977i 0.592607π-0.592607\pi
−0.286845 + 0.957977i 0.592607π0.592607\pi
642642 0 0
643643 46.4721 1.83268 0.916341 0.400399i 0.131128π-0.131128\pi
0.916341 + 0.400399i 0.131128π0.131128\pi
644644 0 0
645645 0 0
646646 0 0
647647 −15.3500 −0.603471 −0.301736 0.953392i 0.597566π-0.597566\pi
−0.301736 + 0.953392i 0.597566π0.597566\pi
648648 0 0
649649 −6.11146 −0.239896
650650 0 0
651651 0 0
652652 0 0
653653 −46.5902 −1.82322 −0.911608 0.411060i 0.865159π-0.865159\pi
−0.911608 + 0.411060i 0.865159π0.865159\pi
654654 0 0
655655 −28.6525 −1.11954
656656 0 0
657657 0 0
658658 0 0
659659 19.2588 0.750217 0.375108 0.926981i 0.377606π-0.377606\pi
0.375108 + 0.926981i 0.377606π0.377606\pi
660660 0 0
661661 19.2361 0.748196 0.374098 0.927389i 0.377952π-0.377952\pi
0.374098 + 0.927389i 0.377952π0.377952\pi
662662 0 0
663663 0 0
664664 0 0
665665 −2.28825 −0.0887344
666666 0 0
667667 −24.3607 −0.943249
668668 0 0
669669 0 0
670670 0 0
671671 9.97831 0.385208
672672 0 0
673673 28.8328 1.11142 0.555712 0.831375i 0.312446π-0.312446\pi
0.555712 + 0.831375i 0.312446π0.312446\pi
674674 0 0
675675 0 0
676676 0 0
677677 −38.3600 −1.47429 −0.737147 0.675732i 0.763828π-0.763828\pi
−0.737147 + 0.675732i 0.763828π0.763828\pi
678678 0 0
679679 12.4721 0.478637
680680 0 0
681681 0 0
682682 0 0
683683 −23.1676 −0.886484 −0.443242 0.896402i 0.646172π-0.646172\pi
−0.443242 + 0.896402i 0.646172π0.646172\pi
684684 0 0
685685 33.8885 1.29481
686686 0 0
687687 0 0
688688 0 0
689689 −37.0246 −1.41052
690690 0 0
691691 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
692692 0 0
693693 0 0
694694 0 0
695695 29.6197 1.12354
696696 0 0
697697 −4.58359 −0.173616
698698 0 0
699699 0 0
700700 0 0
701701 30.9551 1.16916 0.584579 0.811337i 0.301260π-0.301260\pi
0.584579 + 0.811337i 0.301260π0.301260\pi
702702 0 0
703703 4.47214 0.168670
704704 0 0
705705 0 0
706706 0 0
707707 2.70091 0.101578
708708 0 0
709709 −14.2918 −0.536740 −0.268370 0.963316i 0.586485π-0.586485\pi
−0.268370 + 0.963316i 0.586485π0.586485\pi
710710 0 0
711711 0 0
712712 0 0
713713 21.8021 0.816495
714714 0 0
715715 20.9443 0.783271
716716 0 0
717717 0 0
718718 0 0
719719 49.9588 1.86315 0.931575 0.363549i 0.118435π-0.118435\pi
0.931575 + 0.363549i 0.118435π0.118435\pi
720720 0 0
721721 −4.94427 −0.184134
722722 0 0
723723 0 0
724724 0 0
725725 −2.03321 −0.0755114
726726 0 0
727727 −14.4721 −0.536742 −0.268371 0.963316i 0.586485π-0.586485\pi
−0.268371 + 0.963316i 0.586485π0.586485\pi
728728 0 0
729729 0 0
730730 0 0
731731 −2.82843 −0.104613
732732 0 0
733733 −40.8328 −1.50819 −0.754097 0.656763i 0.771925π-0.771925\pi
−0.754097 + 0.656763i 0.771925π0.771925\pi
734734 0 0
735735 0 0
736736 0 0
737737 43.6042 1.60618
738738 0 0
739739 15.1246 0.556368 0.278184 0.960528i 0.410268π-0.410268\pi
0.278184 + 0.960528i 0.410268π0.410268\pi
740740 0 0
741741 0 0
742742 0 0
743743 −16.4304 −0.602772 −0.301386 0.953502i 0.597449π-0.597449\pi
−0.301386 + 0.953502i 0.597449π0.597449\pi
744744 0 0
745745 −31.4164 −1.15101
746746 0 0
747747 0 0
748748 0 0
749749 −7.53244 −0.275229
750750 0 0
751751 28.3607 1.03490 0.517448 0.855715i 0.326882π-0.326882\pi
0.517448 + 0.855715i 0.326882π0.326882\pi
752752 0 0
753753 0 0
754754 0 0
755755 −23.9628 −0.872096
756756 0 0
757757 −19.3050 −0.701650 −0.350825 0.936441i 0.614099π-0.614099\pi
−0.350825 + 0.936441i 0.614099π0.614099\pi
758758 0 0
759759 0 0
760760 0 0
761761 46.7178 1.69352 0.846759 0.531977i 0.178551π-0.178551\pi
0.846759 + 0.531977i 0.178551π0.178551\pi
762762 0 0
763763 1.05573 0.0382199
764764 0 0
765765 0 0
766766 0 0
767767 −6.99226 −0.252476
768768 0 0
769769 4.11146 0.148263 0.0741315 0.997248i 0.476382π-0.476382\pi
0.0741315 + 0.997248i 0.476382π0.476382\pi
770770 0 0
771771 0 0
772772 0 0
773773 44.2719 1.59235 0.796175 0.605067i 0.206854π-0.206854\pi
0.796175 + 0.605067i 0.206854π0.206854\pi
774774 0 0
775775 1.81966 0.0653641
776776 0 0
777777 0 0
778778 0 0
779779 −8.48528 −0.304017
780780 0 0
781781 −35.4164 −1.26730
782782 0 0
783783 0 0
784784 0 0
785785 −13.7295 −0.490026
786786 0 0
787787 12.8754 0.458958 0.229479 0.973314i 0.426298π-0.426298\pi
0.229479 + 0.973314i 0.426298π0.426298\pi
788788 0 0
789789 0 0
790790 0 0
791791 14.2697 0.507371
792792 0 0
793793 11.4164 0.405409
794794 0 0
795795 0 0
796796 0 0
797797 14.6522 0.519008 0.259504 0.965742i 0.416441π-0.416441\pi
0.259504 + 0.965742i 0.416441π0.416441\pi
798798 0 0
799799 −2.18034 −0.0771349
800800 0 0
801801 0 0
802802 0 0
803803 30.9551 1.09238
804804 0 0
805805 −6.47214 −0.228113
806806 0 0
807807 0 0
808808 0 0
809809 33.1158 1.16429 0.582145 0.813085i 0.302214π-0.302214\pi
0.582145 + 0.813085i 0.302214π0.302214\pi
810810 0 0
811811 20.0000 0.702295 0.351147 0.936320i 0.385792π-0.385792\pi
0.351147 + 0.936320i 0.385792π0.385792\pi
812812 0 0
813813 0 0
814814 0 0
815815 34.6088 1.21229
816816 0 0
817817 −5.23607 −0.183187
818818 0 0
819819 0 0
820820 0 0
821821 6.73722 0.235131 0.117565 0.993065i 0.462491π-0.462491\pi
0.117565 + 0.993065i 0.462491π0.462491\pi
822822 0 0
823823 −22.1803 −0.773158 −0.386579 0.922256i 0.626343π-0.626343\pi
−0.386579 + 0.922256i 0.626343π0.626343\pi
824824 0 0
825825 0 0
826826 0 0
827827 −18.4335 −0.640995 −0.320498 0.947249i 0.603850π-0.603850\pi
−0.320498 + 0.947249i 0.603850π0.603850\pi
828828 0 0
829829 28.4721 0.988878 0.494439 0.869212i 0.335374π-0.335374\pi
0.494439 + 0.869212i 0.335374π0.335374\pi
830830 0 0
831831 0 0
832832 0 0
833833 0.540182 0.0187162
834834 0 0
835835 23.4164 0.810358
836836 0 0
837837 0 0
838838 0 0
839839 21.8021 0.752692 0.376346 0.926479i 0.377180π-0.377180\pi
0.376346 + 0.926479i 0.377180π0.377180\pi
840840 0 0
841841 45.1803 1.55794
842842 0 0
843843 0 0
844844 0 0
845845 −5.78437 −0.198989
846846 0 0
847847 3.00000 0.103081
848848 0 0
849849 0 0
850850 0 0
851851 12.6491 0.433606
852852 0 0
853853 26.3607 0.902572 0.451286 0.892379i 0.350965π-0.350965\pi
0.451286 + 0.892379i 0.350965π0.350965\pi
854854 0 0
855855 0 0
856856 0 0
857857 −18.4636 −0.630704 −0.315352 0.948975i 0.602123π-0.602123\pi
−0.315352 + 0.948975i 0.602123π0.602123\pi
858858 0 0
859859 −21.3050 −0.726916 −0.363458 0.931611i 0.618404π-0.618404\pi
−0.363458 + 0.931611i 0.618404π0.618404\pi
860860 0 0
861861 0 0
862862 0 0
863863 −45.4798 −1.54815 −0.774075 0.633094i 0.781785π-0.781785\pi
−0.774075 + 0.633094i 0.781785π0.781785\pi
864864 0 0
865865 −9.52786 −0.323957
866866 0 0
867867 0 0
868868 0 0
869869 29.6197 1.00478
870870 0 0
871871 49.8885 1.69041
872872 0 0
873873 0 0
874874 0 0
875875 10.9010 0.368523
876876 0 0
877877 6.00000 0.202606 0.101303 0.994856i 0.467699π-0.467699\pi
0.101303 + 0.994856i 0.467699π0.467699\pi
878878 0 0
879879 0 0
880880 0 0
881881 53.9651 1.81813 0.909064 0.416656i 0.136798π-0.136798\pi
0.909064 + 0.416656i 0.136798π0.136798\pi
882882 0 0
883883 35.0557 1.17972 0.589860 0.807506i 0.299183π-0.299183\pi
0.589860 + 0.807506i 0.299183π0.299183\pi
884884 0 0
885885 0 0
886886 0 0
887887 −22.3724 −0.751191 −0.375595 0.926784i 0.622562π-0.622562\pi
−0.375595 + 0.926784i 0.622562π0.622562\pi
888888 0 0
889889 19.4164 0.651205
890890 0 0
891891 0 0
892892 0 0
893893 −4.03631 −0.135070
894894 0 0
895895 24.6525 0.824041
896896 0 0
897897 0 0
898898 0 0
899899 −66.3892 −2.21420
900900 0 0
901901 6.18034 0.205897
902902 0 0
903903 0 0
904904 0 0
905905 6.73722 0.223953
906906 0 0
907907 −3.05573 −0.101464 −0.0507319 0.998712i 0.516155π-0.516155\pi
−0.0507319 + 0.998712i 0.516155π0.516155\pi
908908 0 0
909909 0 0
910910 0 0
911911 −9.02546 −0.299027 −0.149513 0.988760i 0.547771π-0.547771\pi
−0.149513 + 0.988760i 0.547771π0.547771\pi
912912 0 0
913913 22.4721 0.743719
914914 0 0
915915 0 0
916916 0 0
917917 12.5216 0.413499
918918 0 0
919919 −28.9443 −0.954783 −0.477392 0.878691i 0.658418π-0.658418\pi
−0.477392 + 0.878691i 0.658418π0.658418\pi
920920 0 0
921921 0 0
922922 0 0
923923 −40.5207 −1.33376
924924 0 0
925925 1.05573 0.0347121
926926 0 0
927927 0 0
928928 0 0
929929 22.3423 0.733026 0.366513 0.930413i 0.380551π-0.380551\pi
0.366513 + 0.930413i 0.380551π0.380551\pi
930930 0 0
931931 1.00000 0.0327737
932932 0 0
933933 0 0
934934 0 0
935935 −3.49613 −0.114336
936936 0 0
937937 −10.9443 −0.357534 −0.178767 0.983891i 0.557211π-0.557211\pi
−0.178767 + 0.983891i 0.557211π0.557211\pi
938938 0 0
939939 0 0
940940 0 0
941941 −46.1776 −1.50535 −0.752673 0.658395i 0.771236π-0.771236\pi
−0.752673 + 0.658395i 0.771236π0.771236\pi
942942 0 0
943943 −24.0000 −0.781548
944944 0 0
945945 0 0
946946 0 0
947947 3.65375 0.118731 0.0593655 0.998236i 0.481092π-0.481092\pi
0.0593655 + 0.998236i 0.481092π0.481092\pi
948948 0 0
949949 35.4164 1.14967
950950 0 0
951951 0 0
952952 0 0
953953 −13.8570 −0.448872 −0.224436 0.974489i 0.572054π-0.572054\pi
−0.224436 + 0.974489i 0.572054π0.572054\pi
954954 0 0
955955 30.4721 0.986055
956956 0 0
957957 0 0
958958 0 0
959959 −14.8098 −0.478235
960960 0 0
961961 28.4164 0.916658
962962 0 0
963963 0 0
964964 0 0
965965 −41.1884 −1.32590
966966 0 0
967967 −50.5410 −1.62529 −0.812645 0.582759i 0.801973π-0.801973\pi
−0.812645 + 0.582759i 0.801973π0.801973\pi
968968 0 0
969969 0 0
970970 0 0
971971 11.3137 0.363074 0.181537 0.983384i 0.441893π-0.441893\pi
0.181537 + 0.983384i 0.441893π0.441893\pi
972972 0 0
973973 −12.9443 −0.414974
974974 0 0
975975 0 0
976976 0 0
977977 0.952843 0.0304842 0.0152421 0.999884i 0.495148π-0.495148\pi
0.0152421 + 0.999884i 0.495148π0.495148\pi
978978 0 0
979979 24.0000 0.767043
980980 0 0
981981 0 0
982982 0 0
983983 29.6197 0.944721 0.472360 0.881405i 0.343402π-0.343402\pi
0.472360 + 0.881405i 0.343402π0.343402\pi
984984 0 0
985985 46.8328 1.49222
986986 0 0
987987 0 0
988988 0 0
989989 −14.8098 −0.470925
990990 0 0
991991 −29.8885 −0.949441 −0.474720 0.880137i 0.657451π-0.657451\pi
−0.474720 + 0.880137i 0.657451π0.657451\pi
992992 0 0
993993 0 0
994994 0 0
995995 18.3060 0.580338
996996 0 0
997997 53.7771 1.70314 0.851569 0.524243i 0.175652π-0.175652\pi
0.851569 + 0.524243i 0.175652π0.175652\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 4788.2.a.q.1.4 yes 4
3.2 odd 2 inner 4788.2.a.q.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4788.2.a.q.1.1 4 3.2 odd 2 inner
4788.2.a.q.1.4 yes 4 1.1 even 1 trivial