Properties

Label 768.4.a.q.1.2
Level 768768
Weight 44
Character 768.1
Self dual yes
Analytic conductor 45.31345.313
Analytic rank 11
Dimension 33
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] = [768,4,Mod(1,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("768.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 768=283 768 = 2^{8} \cdot 3
Weight: k k == 4 4
Character orbit: [χ][\chi] == 768.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-9,0,-10,0,14,0,27,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 45.313466884445.3134668844
Analytic rank: 11
Dimension: 33
Coefficient field: 3.3.1436.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x311x12 x^{3} - 11x - 12 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 25 2^{5}
Twist minimal: no (minimal twist has level 24)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.2
Root 3.766443.76644 of defining polynomial
Character χ\chi == 768.1

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) == q3.00000q30.612661q522.7441q7+9.00000q9+60.2630q1152.9062q13+1.83798q15+47.1643q17+29.1643q19+68.2324q21+109.488q23124.625q2527.0000q27+10.4250q29+220.881q31180.789q33+13.9345q35408.348q37+158.718q39+360.742q41236.414q435.51395q45129.113q47+174.296q49141.493q51117.819q5336.9208q5587.4928q57262.854q59273.465q61204.697q63+32.4135q65+89.4077q67328.465q69350.521q71532.610q73+373.874q751370.63q77+166.561q79+81.0000q81361.934q8328.8957q8531.2750q8740.3285q89+1203.31q91662.642q9317.8678q95614.921q97+542.367q99+O(q100)q-3.00000 q^{3} -0.612661 q^{5} -22.7441 q^{7} +9.00000 q^{9} +60.2630 q^{11} -52.9062 q^{13} +1.83798 q^{15} +47.1643 q^{17} +29.1643 q^{19} +68.2324 q^{21} +109.488 q^{23} -124.625 q^{25} -27.0000 q^{27} +10.4250 q^{29} +220.881 q^{31} -180.789 q^{33} +13.9345 q^{35} -408.348 q^{37} +158.718 q^{39} +360.742 q^{41} -236.414 q^{43} -5.51395 q^{45} -129.113 q^{47} +174.296 q^{49} -141.493 q^{51} -117.819 q^{53} -36.9208 q^{55} -87.4928 q^{57} -262.854 q^{59} -273.465 q^{61} -204.697 q^{63} +32.4135 q^{65} +89.4077 q^{67} -328.465 q^{69} -350.521 q^{71} -532.610 q^{73} +373.874 q^{75} -1370.63 q^{77} +166.561 q^{79} +81.0000 q^{81} -361.934 q^{83} -28.8957 q^{85} -31.2750 q^{87} -40.3285 q^{89} +1203.31 q^{91} -662.642 q^{93} -17.8678 q^{95} -614.921 q^{97} +542.367 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 3q9q310q5+14q7+27q952q13+30q15+26q1728q1942q21+164q23+53q2581q27174q29+318q31+92q35296q37+156q39+1222q97+O(q100) 3 q - 9 q^{3} - 10 q^{5} + 14 q^{7} + 27 q^{9} - 52 q^{13} + 30 q^{15} + 26 q^{17} - 28 q^{19} - 42 q^{21} + 164 q^{23} + 53 q^{25} - 81 q^{27} - 174 q^{29} + 318 q^{31} + 92 q^{35} - 296 q^{37} + 156 q^{39}+ \cdots - 1222 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 −3.00000 −0.577350
44 0 0
55 −0.612661 −0.0547981 −0.0273990 0.999625i 0.508722π-0.508722\pi
−0.0273990 + 0.999625i 0.508722π0.508722\pi
66 0 0
77 −22.7441 −1.22807 −0.614034 0.789279i 0.710454π-0.710454\pi
−0.614034 + 0.789279i 0.710454π0.710454\pi
88 0 0
99 9.00000 0.333333
1010 0 0
1111 60.2630 1.65182 0.825908 0.563805i 0.190663π-0.190663\pi
0.825908 + 0.563805i 0.190663π0.190663\pi
1212 0 0
1313 −52.9062 −1.12873 −0.564367 0.825524i 0.690880π-0.690880\pi
−0.564367 + 0.825524i 0.690880π0.690880\pi
1414 0 0
1515 1.83798 0.0316377
1616 0 0
1717 47.1643 0.672883 0.336442 0.941704i 0.390777π-0.390777\pi
0.336442 + 0.941704i 0.390777π0.390777\pi
1818 0 0
1919 29.1643 0.352144 0.176072 0.984377i 0.443661π-0.443661\pi
0.176072 + 0.984377i 0.443661π0.443661\pi
2020 0 0
2121 68.2324 0.709026
2222 0 0
2323 109.488 0.992604 0.496302 0.868150i 0.334691π-0.334691\pi
0.496302 + 0.868150i 0.334691π0.334691\pi
2424 0 0
2525 −124.625 −0.996997
2626 0 0
2727 −27.0000 −0.192450
2828 0 0
2929 10.4250 0.0667542 0.0333771 0.999443i 0.489374π-0.489374\pi
0.0333771 + 0.999443i 0.489374π0.489374\pi
3030 0 0
3131 220.881 1.27972 0.639860 0.768492i 0.278992π-0.278992\pi
0.639860 + 0.768492i 0.278992π0.278992\pi
3232 0 0
3333 −180.789 −0.953676
3434 0 0
3535 13.9345 0.0672958
3636 0 0
3737 −408.348 −1.81438 −0.907188 0.420725i 0.861776π-0.861776\pi
−0.907188 + 0.420725i 0.861776π0.861776\pi
3838 0 0
3939 158.718 0.651674
4040 0 0
4141 360.742 1.37411 0.687054 0.726606i 0.258903π-0.258903\pi
0.687054 + 0.726606i 0.258903π0.258903\pi
4242 0 0
4343 −236.414 −0.838436 −0.419218 0.907886i 0.637696π-0.637696\pi
−0.419218 + 0.907886i 0.637696π0.637696\pi
4444 0 0
4545 −5.51395 −0.0182660
4646 0 0
4747 −129.113 −0.400703 −0.200352 0.979724i 0.564208π-0.564208\pi
−0.200352 + 0.979724i 0.564208π0.564208\pi
4848 0 0
4949 174.296 0.508152
5050 0 0
5151 −141.493 −0.388489
5252 0 0
5353 −117.819 −0.305353 −0.152677 0.988276i 0.548789π-0.548789\pi
−0.152677 + 0.988276i 0.548789π0.548789\pi
5454 0 0
5555 −36.9208 −0.0905163
5656 0 0
5757 −87.4928 −0.203311
5858 0 0
5959 −262.854 −0.580012 −0.290006 0.957025i 0.593657π-0.593657\pi
−0.290006 + 0.957025i 0.593657π0.593657\pi
6060 0 0
6161 −273.465 −0.573993 −0.286996 0.957932i 0.592657π-0.592657\pi
−0.286996 + 0.957932i 0.592657π0.592657\pi
6262 0 0
6363 −204.697 −0.409356
6464 0 0
6565 32.4135 0.0618524
6666 0 0
6767 89.4077 0.163028 0.0815141 0.996672i 0.474024π-0.474024\pi
0.0815141 + 0.996672i 0.474024π0.474024\pi
6868 0 0
6969 −328.465 −0.573080
7070 0 0
7171 −350.521 −0.585904 −0.292952 0.956127i 0.594638π-0.594638\pi
−0.292952 + 0.956127i 0.594638π0.594638\pi
7272 0 0
7373 −532.610 −0.853936 −0.426968 0.904267i 0.640418π-0.640418\pi
−0.426968 + 0.904267i 0.640418π0.640418\pi
7474 0 0
7575 373.874 0.575617
7676 0 0
7777 −1370.63 −2.02854
7878 0 0
7979 166.561 0.237210 0.118605 0.992942i 0.462158π-0.462158\pi
0.118605 + 0.992942i 0.462158π0.462158\pi
8080 0 0
8181 81.0000 0.111111
8282 0 0
8383 −361.934 −0.478644 −0.239322 0.970940i 0.576925π-0.576925\pi
−0.239322 + 0.970940i 0.576925π0.576925\pi
8484 0 0
8585 −28.8957 −0.0368727
8686 0 0
8787 −31.2750 −0.0385405
8888 0 0
8989 −40.3285 −0.0480316 −0.0240158 0.999712i 0.507645π-0.507645\pi
−0.0240158 + 0.999712i 0.507645π0.507645\pi
9090 0 0
9191 1203.31 1.38616
9292 0 0
9393 −662.642 −0.738846
9494 0 0
9595 −17.8678 −0.0192968
9696 0 0
9797 −614.921 −0.643667 −0.321834 0.946796i 0.604299π-0.604299\pi
−0.321834 + 0.946796i 0.604299π0.604299\pi
9898 0 0
9999 542.367 0.550605
100100 0 0
101101 −1664.99 −1.64033 −0.820163 0.572130i 0.806117π-0.806117\pi
−0.820163 + 0.572130i 0.806117π0.806117\pi
102102 0 0
103103 396.858 0.379647 0.189823 0.981818i 0.439208π-0.439208\pi
0.189823 + 0.981818i 0.439208π0.439208\pi
104104 0 0
105105 −41.8034 −0.0388532
106106 0 0
107107 350.630 0.316791 0.158396 0.987376i 0.449368π-0.449368\pi
0.158396 + 0.987376i 0.449368π0.449368\pi
108108 0 0
109109 −597.009 −0.524615 −0.262308 0.964984i 0.584484π-0.584484\pi
−0.262308 + 0.964984i 0.584484π0.584484\pi
110110 0 0
111111 1225.04 1.04753
112112 0 0
113113 496.422 0.413270 0.206635 0.978418i 0.433749π-0.433749\pi
0.206635 + 0.978418i 0.433749π0.433749\pi
114114 0 0
115115 −67.0792 −0.0543928
116116 0 0
117117 −476.155 −0.376244
118118 0 0
119119 −1072.71 −0.826346
120120 0 0
121121 2300.63 1.72849
122122 0 0
123123 −1082.23 −0.793342
124124 0 0
125125 152.935 0.109432
126126 0 0
127127 −1799.85 −1.25756 −0.628782 0.777581i 0.716446π-0.716446\pi
−0.628782 + 0.777581i 0.716446π0.716446\pi
128128 0 0
129129 709.241 0.484071
130130 0 0
131131 1121.45 0.747949 0.373974 0.927439i 0.377995π-0.377995\pi
0.373974 + 0.927439i 0.377995π0.377995\pi
132132 0 0
133133 −663.316 −0.432457
134134 0 0
135135 16.5419 0.0105459
136136 0 0
137137 −2449.55 −1.52759 −0.763793 0.645461i 0.776665π-0.776665\pi
−0.763793 + 0.645461i 0.776665π0.776665\pi
138138 0 0
139139 −2457.56 −1.49962 −0.749811 0.661652i 0.769856π-0.769856\pi
−0.749811 + 0.661652i 0.769856π0.769856\pi
140140 0 0
141141 387.339 0.231346
142142 0 0
143143 −3188.28 −1.86446
144144 0 0
145145 −6.38698 −0.00365800
146146 0 0
147147 −522.888 −0.293382
148148 0 0
149149 −2084.96 −1.14635 −0.573177 0.819432i 0.694289π-0.694289\pi
−0.573177 + 0.819432i 0.694289π0.694289\pi
150150 0 0
151151 1057.80 0.570084 0.285042 0.958515i 0.407992π-0.407992\pi
0.285042 + 0.958515i 0.407992π0.407992\pi
152152 0 0
153153 424.478 0.224294
154154 0 0
155155 −135.325 −0.0701262
156156 0 0
157157 −3193.01 −1.62312 −0.811559 0.584270i 0.801381π-0.801381\pi
−0.811559 + 0.584270i 0.801381π0.801381\pi
158158 0 0
159159 353.458 0.176296
160160 0 0
161161 −2490.22 −1.21899
162162 0 0
163163 −846.854 −0.406937 −0.203469 0.979081i 0.565221π-0.565221\pi
−0.203469 + 0.979081i 0.565221π0.565221\pi
164164 0 0
165165 110.762 0.0522596
166166 0 0
167167 −2630.15 −1.21873 −0.609363 0.792892i 0.708575π-0.708575\pi
−0.609363 + 0.792892i 0.708575π0.708575\pi
168168 0 0
169169 602.062 0.274038
170170 0 0
171171 262.478 0.117381
172172 0 0
173173 −429.843 −0.188904 −0.0944519 0.995529i 0.530110π-0.530110\pi
−0.0944519 + 0.995529i 0.530110π0.530110\pi
174174 0 0
175175 2834.48 1.22438
176176 0 0
177177 788.563 0.334870
178178 0 0
179179 −1516.30 −0.633149 −0.316574 0.948568i 0.602533π-0.602533\pi
−0.316574 + 0.948568i 0.602533π0.602533\pi
180180 0 0
181181 −3380.20 −1.38811 −0.694056 0.719921i 0.744178π-0.744178\pi
−0.694056 + 0.719921i 0.744178π0.744178\pi
182182 0 0
183183 820.394 0.331395
184184 0 0
185185 250.179 0.0994244
186186 0 0
187187 2842.26 1.11148
188188 0 0
189189 614.092 0.236342
190190 0 0
191191 2799.71 1.06063 0.530314 0.847801i 0.322074π-0.322074\pi
0.530314 + 0.847801i 0.322074π0.322074\pi
192192 0 0
193193 624.106 0.232768 0.116384 0.993204i 0.462870π-0.462870\pi
0.116384 + 0.993204i 0.462870π0.462870\pi
194194 0 0
195195 −97.2406 −0.0357105
196196 0 0
197197 4779.25 1.72846 0.864232 0.503094i 0.167805π-0.167805\pi
0.864232 + 0.503094i 0.167805π0.167805\pi
198198 0 0
199199 2615.92 0.931846 0.465923 0.884825i 0.345722π-0.345722\pi
0.465923 + 0.884825i 0.345722π0.345722\pi
200200 0 0
201201 −268.223 −0.0941244
202202 0 0
203203 −237.107 −0.0819787
204204 0 0
205205 −221.013 −0.0752985
206206 0 0
207207 985.395 0.330868
208208 0 0
209209 1757.52 0.581677
210210 0 0
211211 −1745.78 −0.569595 −0.284798 0.958588i 0.591926π-0.591926\pi
−0.284798 + 0.958588i 0.591926π0.591926\pi
212212 0 0
213213 1051.56 0.338272
214214 0 0
215215 144.841 0.0459447
216216 0 0
217217 −5023.74 −1.57158
218218 0 0
219219 1597.83 0.493020
220220 0 0
221221 −2495.28 −0.759505
222222 0 0
223223 3385.60 1.01667 0.508333 0.861161i 0.330262π-0.330262\pi
0.508333 + 0.861161i 0.330262π0.330262\pi
224224 0 0
225225 −1121.62 −0.332332
226226 0 0
227227 3847.72 1.12503 0.562515 0.826787i 0.309834π-0.309834\pi
0.562515 + 0.826787i 0.309834π0.309834\pi
228228 0 0
229229 1335.15 0.385279 0.192640 0.981270i 0.438295π-0.438295\pi
0.192640 + 0.981270i 0.438295π0.438295\pi
230230 0 0
231231 4111.89 1.17118
232232 0 0
233233 5146.38 1.44700 0.723499 0.690325i 0.242532π-0.242532\pi
0.723499 + 0.690325i 0.242532π0.242532\pi
234234 0 0
235235 79.1025 0.0219578
236236 0 0
237237 −499.683 −0.136953
238238 0 0
239239 −7085.07 −1.91755 −0.958777 0.284160i 0.908285π-0.908285\pi
−0.958777 + 0.284160i 0.908285π0.908285\pi
240240 0 0
241241 2538.40 0.678476 0.339238 0.940701i 0.389831π-0.389831\pi
0.339238 + 0.940701i 0.389831π0.389831\pi
242242 0 0
243243 −243.000 −0.0641500
244244 0 0
245245 −106.784 −0.0278458
246246 0 0
247247 −1542.97 −0.397477
248248 0 0
249249 1085.80 0.276345
250250 0 0
251251 1696.99 0.426746 0.213373 0.976971i 0.431555π-0.431555\pi
0.213373 + 0.976971i 0.431555π0.431555\pi
252252 0 0
253253 6598.09 1.63960
254254 0 0
255255 86.6871 0.0212885
256256 0 0
257257 −382.902 −0.0929369 −0.0464685 0.998920i 0.514797π-0.514797\pi
−0.0464685 + 0.998920i 0.514797π0.514797\pi
258258 0 0
259259 9287.52 2.22818
260260 0 0
261261 93.8249 0.0222514
262262 0 0
263263 −5002.02 −1.17277 −0.586383 0.810034i 0.699449π-0.699449\pi
−0.586383 + 0.810034i 0.699449π0.699449\pi
264264 0 0
265265 72.1833 0.0167328
266266 0 0
267267 120.986 0.0277311
268268 0 0
269269 6117.47 1.38658 0.693288 0.720661i 0.256161π-0.256161\pi
0.693288 + 0.720661i 0.256161π0.256161\pi
270270 0 0
271271 3956.12 0.886780 0.443390 0.896329i 0.353776π-0.353776\pi
0.443390 + 0.896329i 0.353776π0.353776\pi
272272 0 0
273273 −3609.92 −0.800301
274274 0 0
275275 −7510.25 −1.64686
276276 0 0
277277 4842.17 1.05032 0.525158 0.851004i 0.324006π-0.324006\pi
0.525158 + 0.851004i 0.324006π0.324006\pi
278278 0 0
279279 1987.92 0.426573
280280 0 0
281281 −1878.68 −0.398835 −0.199417 0.979915i 0.563905π-0.563905\pi
−0.199417 + 0.979915i 0.563905π0.563905\pi
282282 0 0
283283 −5724.87 −1.20250 −0.601251 0.799060i 0.705331π-0.705331\pi
−0.601251 + 0.799060i 0.705331π0.705331\pi
284284 0 0
285285 53.6034 0.0111410
286286 0 0
287287 −8204.77 −1.68750
288288 0 0
289289 −2688.53 −0.547228
290290 0 0
291291 1844.76 0.371622
292292 0 0
293293 −5088.75 −1.01464 −0.507318 0.861759i 0.669363π-0.669363\pi
−0.507318 + 0.861759i 0.669363π0.669363\pi
294294 0 0
295295 161.041 0.0317836
296296 0 0
297297 −1627.10 −0.317892
298298 0 0
299299 −5792.61 −1.12038
300300 0 0
301301 5377.02 1.02966
302302 0 0
303303 4994.98 0.947043
304304 0 0
305305 167.541 0.0314537
306306 0 0
307307 −7219.21 −1.34209 −0.671046 0.741416i 0.734155π-0.734155\pi
−0.671046 + 0.741416i 0.734155π0.734155\pi
308308 0 0
309309 −1190.58 −0.219189
310310 0 0
311311 −1537.06 −0.280252 −0.140126 0.990134i 0.544751π-0.544751\pi
−0.140126 + 0.990134i 0.544751π0.544751\pi
312312 0 0
313313 2200.93 0.397456 0.198728 0.980055i 0.436319π-0.436319\pi
0.198728 + 0.980055i 0.436319π0.436319\pi
314314 0 0
315315 125.410 0.0224319
316316 0 0
317317 2840.41 0.503260 0.251630 0.967824i 0.419033π-0.419033\pi
0.251630 + 0.967824i 0.419033π0.419033\pi
318318 0 0
319319 628.240 0.110266
320320 0 0
321321 −1051.89 −0.182899
322322 0 0
323323 1375.51 0.236952
324324 0 0
325325 6593.41 1.12534
326326 0 0
327327 1791.03 0.302887
328328 0 0
329329 2936.56 0.492091
330330 0 0
331331 2118.52 0.351795 0.175898 0.984408i 0.443717π-0.443717\pi
0.175898 + 0.984408i 0.443717π0.443717\pi
332332 0 0
333333 −3675.13 −0.604792
334334 0 0
335335 −54.7766 −0.00893364
336336 0 0
337337 −659.599 −0.106619 −0.0533096 0.998578i 0.516977π-0.516977\pi
−0.0533096 + 0.998578i 0.516977π0.516977\pi
338338 0 0
339339 −1489.27 −0.238601
340340 0 0
341341 13310.9 2.11386
342342 0 0
343343 3837.03 0.604023
344344 0 0
345345 201.238 0.0314037
346346 0 0
347347 8377.42 1.29603 0.648017 0.761626i 0.275599π-0.275599\pi
0.648017 + 0.761626i 0.275599π0.275599\pi
348348 0 0
349349 3254.18 0.499119 0.249559 0.968360i 0.419714π-0.419714\pi
0.249559 + 0.968360i 0.419714π0.419714\pi
350350 0 0
351351 1428.47 0.217225
352352 0 0
353353 11117.5 1.67627 0.838137 0.545459i 0.183645π-0.183645\pi
0.838137 + 0.545459i 0.183645π0.183645\pi
354354 0 0
355355 214.750 0.0321064
356356 0 0
357357 3218.13 0.477091
358358 0 0
359359 −4756.56 −0.699281 −0.349640 0.936884i 0.613696π-0.613696\pi
−0.349640 + 0.936884i 0.613696π0.613696\pi
360360 0 0
361361 −6008.45 −0.875994
362362 0 0
363363 −6901.88 −0.997946
364364 0 0
365365 326.310 0.0467940
366366 0 0
367367 −1837.40 −0.261339 −0.130670 0.991426i 0.541713π-0.541713\pi
−0.130670 + 0.991426i 0.541713π0.541713\pi
368368 0 0
369369 3246.68 0.458036
370370 0 0
371371 2679.70 0.374995
372372 0 0
373373 5598.07 0.777097 0.388549 0.921428i 0.372977π-0.372977\pi
0.388549 + 0.921428i 0.372977π0.372977\pi
374374 0 0
375375 −458.806 −0.0631804
376376 0 0
377377 −551.546 −0.0753476
378378 0 0
379379 3460.18 0.468965 0.234482 0.972120i 0.424661π-0.424661\pi
0.234482 + 0.972120i 0.424661π0.424661\pi
380380 0 0
381381 5399.55 0.726055
382382 0 0
383383 −5059.63 −0.675027 −0.337513 0.941321i 0.609586π-0.609586\pi
−0.337513 + 0.941321i 0.609586π0.609586\pi
384384 0 0
385385 839.732 0.111160
386386 0 0
387387 −2127.72 −0.279479
388388 0 0
389389 2192.22 0.285732 0.142866 0.989742i 0.454368π-0.454368\pi
0.142866 + 0.989742i 0.454368π0.454368\pi
390390 0 0
391391 5163.93 0.667906
392392 0 0
393393 −3364.34 −0.431828
394394 0 0
395395 −102.045 −0.0129986
396396 0 0
397397 −5519.94 −0.697828 −0.348914 0.937155i 0.613449π-0.613449\pi
−0.348914 + 0.937155i 0.613449π0.613449\pi
398398 0 0
399399 1989.95 0.249679
400400 0 0
401401 −7352.64 −0.915645 −0.457822 0.889044i 0.651370π-0.651370\pi
−0.457822 + 0.889044i 0.651370π0.651370\pi
402402 0 0
403403 −11685.9 −1.44446
404404 0 0
405405 −49.6256 −0.00608868
406406 0 0
407407 −24608.2 −2.99702
408408 0 0
409409 11311.2 1.36749 0.683745 0.729721i 0.260350π-0.260350\pi
0.683745 + 0.729721i 0.260350π0.260350\pi
410410 0 0
411411 7348.66 0.881952
412412 0 0
413413 5978.40 0.712295
414414 0 0
415415 221.743 0.0262288
416416 0 0
417417 7372.68 0.865808
418418 0 0
419419 13042.2 1.52065 0.760325 0.649543i 0.225040π-0.225040\pi
0.760325 + 0.649543i 0.225040π0.225040\pi
420420 0 0
421421 −4544.38 −0.526080 −0.263040 0.964785i 0.584725π-0.584725\pi
−0.263040 + 0.964785i 0.584725π0.584725\pi
422422 0 0
423423 −1162.02 −0.133568
424424 0 0
425425 −5877.83 −0.670863
426426 0 0
427427 6219.72 0.704903
428428 0 0
429429 9564.85 1.07645
430430 0 0
431431 −7713.83 −0.862093 −0.431047 0.902330i 0.641856π-0.641856\pi
−0.431047 + 0.902330i 0.641856π0.641856\pi
432432 0 0
433433 −15068.3 −1.67237 −0.836183 0.548451i 0.815218π-0.815218\pi
−0.836183 + 0.548451i 0.815218π0.815218\pi
434434 0 0
435435 19.1609 0.00211195
436436 0 0
437437 3193.14 0.349540
438438 0 0
439439 11004.7 1.19642 0.598208 0.801341i 0.295880π-0.295880\pi
0.598208 + 0.801341i 0.295880π0.295880\pi
440440 0 0
441441 1568.67 0.169384
442442 0 0
443443 −2513.04 −0.269522 −0.134761 0.990878i 0.543027π-0.543027\pi
−0.134761 + 0.990878i 0.543027π0.543027\pi
444444 0 0
445445 24.7077 0.00263204
446446 0 0
447447 6254.89 0.661848
448448 0 0
449449 −15752.7 −1.65571 −0.827855 0.560942i 0.810439π-0.810439\pi
−0.827855 + 0.560942i 0.810439π0.810439\pi
450450 0 0
451451 21739.4 2.26977
452452 0 0
453453 −3173.40 −0.329138
454454 0 0
455455 −737.218 −0.0759590
456456 0 0
457457 5257.06 0.538107 0.269053 0.963125i 0.413289π-0.413289\pi
0.269053 + 0.963125i 0.413289π0.413289\pi
458458 0 0
459459 −1273.43 −0.129496
460460 0 0
461461 8066.31 0.814936 0.407468 0.913220i 0.366412π-0.366412\pi
0.407468 + 0.913220i 0.366412π0.366412\pi
462462 0 0
463463 −5683.43 −0.570478 −0.285239 0.958456i 0.592073π-0.592073\pi
−0.285239 + 0.958456i 0.592073π0.592073\pi
464464 0 0
465465 405.975 0.0404874
466466 0 0
467467 −11139.3 −1.10378 −0.551891 0.833916i 0.686094π-0.686094\pi
−0.551891 + 0.833916i 0.686094π0.686094\pi
468468 0 0
469469 −2033.50 −0.200210
470470 0 0
471471 9579.02 0.937108
472472 0 0
473473 −14247.0 −1.38494
474474 0 0
475475 −3634.59 −0.351087
476476 0 0
477477 −1060.37 −0.101784
478478 0 0
479479 −3477.35 −0.331699 −0.165850 0.986151i 0.553037π-0.553037\pi
−0.165850 + 0.986151i 0.553037π0.553037\pi
480480 0 0
481481 21604.1 2.04795
482482 0 0
483483 7470.65 0.703782
484484 0 0
485485 376.738 0.0352717
486486 0 0
487487 −478.797 −0.0445510 −0.0222755 0.999752i 0.507091π-0.507091\pi
−0.0222755 + 0.999752i 0.507091π0.507091\pi
488488 0 0
489489 2540.56 0.234945
490490 0 0
491491 −16601.8 −1.52592 −0.762961 0.646444i 0.776255π-0.776255\pi
−0.762961 + 0.646444i 0.776255π0.776255\pi
492492 0 0
493493 491.687 0.0449178
494494 0 0
495495 −332.287 −0.0301721
496496 0 0
497497 7972.29 0.719530
498498 0 0
499499 −9482.20 −0.850664 −0.425332 0.905037i 0.639843π-0.639843\pi
−0.425332 + 0.905037i 0.639843π0.639843\pi
500500 0 0
501501 7890.45 0.703631
502502 0 0
503503 16561.2 1.46805 0.734023 0.679124i 0.237640π-0.237640\pi
0.734023 + 0.679124i 0.237640π0.237640\pi
504504 0 0
505505 1020.08 0.0898867
506506 0 0
507507 −1806.19 −0.158216
508508 0 0
509509 −4197.35 −0.365509 −0.182755 0.983159i 0.558501π-0.558501\pi
−0.182755 + 0.983159i 0.558501π0.558501\pi
510510 0 0
511511 12113.8 1.04869
512512 0 0
513513 −787.435 −0.0677702
514514 0 0
515515 −243.140 −0.0208039
516516 0 0
517517 −7780.73 −0.661888
518518 0 0
519519 1289.53 0.109064
520520 0 0
521521 15755.5 1.32488 0.662440 0.749115i 0.269521π-0.269521\pi
0.662440 + 0.749115i 0.269521π0.269521\pi
522522 0 0
523523 −11555.1 −0.966098 −0.483049 0.875593i 0.660471π-0.660471\pi
−0.483049 + 0.875593i 0.660471π0.660471\pi
524524 0 0
525525 −8503.44 −0.706897
526526 0 0
527527 10417.7 0.861102
528528 0 0
529529 −179.314 −0.0147378
530530 0 0
531531 −2365.69 −0.193337
532532 0 0
533533 −19085.5 −1.55100
534534 0 0
535535 −214.817 −0.0173595
536536 0 0
537537 4548.90 0.365549
538538 0 0
539539 10503.6 0.839373
540540 0 0
541541 −7475.65 −0.594091 −0.297045 0.954863i 0.596001π-0.596001\pi
−0.297045 + 0.954863i 0.596001π0.596001\pi
542542 0 0
543543 10140.6 0.801427
544544 0 0
545545 365.764 0.0287479
546546 0 0
547547 −6028.08 −0.471192 −0.235596 0.971851i 0.575704π-0.575704\pi
−0.235596 + 0.971851i 0.575704π0.575704\pi
548548 0 0
549549 −2461.18 −0.191331
550550 0 0
551551 304.037 0.0235071
552552 0 0
553553 −3788.29 −0.291310
554554 0 0
555555 −750.536 −0.0574027
556556 0 0
557557 −19381.5 −1.47436 −0.737181 0.675696i 0.763843π-0.763843\pi
−0.737181 + 0.675696i 0.763843π0.763843\pi
558558 0 0
559559 12507.7 0.946370
560560 0 0
561561 −8526.77 −0.641712
562562 0 0
563563 −20565.0 −1.53946 −0.769728 0.638372i 0.779608π-0.779608\pi
−0.769728 + 0.638372i 0.779608π0.779608\pi
564564 0 0
565565 −304.139 −0.0226464
566566 0 0
567567 −1842.28 −0.136452
568568 0 0
569569 −15252.9 −1.12379 −0.561895 0.827209i 0.689927π-0.689927\pi
−0.561895 + 0.827209i 0.689927π0.689927\pi
570570 0 0
571571 16492.8 1.20876 0.604379 0.796697i 0.293421π-0.293421\pi
0.604379 + 0.796697i 0.293421π0.293421\pi
572572 0 0
573573 −8399.13 −0.612354
574574 0 0
575575 −13644.9 −0.989623
576576 0 0
577577 −10298.2 −0.743016 −0.371508 0.928430i 0.621159π-0.621159\pi
−0.371508 + 0.928430i 0.621159π0.621159\pi
578578 0 0
579579 −1872.32 −0.134388
580580 0 0
581581 8231.89 0.587808
582582 0 0
583583 −7100.14 −0.504387
584584 0 0
585585 291.722 0.0206175
586586 0 0
587587 13104.8 0.921453 0.460727 0.887542i 0.347589π-0.347589\pi
0.460727 + 0.887542i 0.347589π0.347589\pi
588588 0 0
589589 6441.82 0.450646
590590 0 0
591591 −14337.7 −0.997929
592592 0 0
593593 4163.34 0.288310 0.144155 0.989555i 0.453954π-0.453954\pi
0.144155 + 0.989555i 0.453954π0.453954\pi
594594 0 0
595595 657.208 0.0452822
596596 0 0
597597 −7847.75 −0.538001
598598 0 0
599599 −5718.60 −0.390076 −0.195038 0.980796i 0.562483π-0.562483\pi
−0.195038 + 0.980796i 0.562483π0.562483\pi
600600 0 0
601601 −17473.0 −1.18592 −0.592959 0.805233i 0.702040π-0.702040\pi
−0.592959 + 0.805233i 0.702040π0.702040\pi
602602 0 0
603603 804.670 0.0543428
604604 0 0
605605 −1409.50 −0.0947181
606606 0 0
607607 5647.60 0.377643 0.188821 0.982011i 0.439533π-0.439533\pi
0.188821 + 0.982011i 0.439533π0.439533\pi
608608 0 0
609609 711.322 0.0473304
610610 0 0
611611 6830.87 0.452287
612612 0 0
613613 16023.6 1.05577 0.527884 0.849317i 0.322986π-0.322986\pi
0.527884 + 0.849317i 0.322986π0.322986\pi
614614 0 0
615615 663.038 0.0434736
616616 0 0
617617 21022.1 1.37167 0.685834 0.727758i 0.259438π-0.259438\pi
0.685834 + 0.727758i 0.259438π0.259438\pi
618618 0 0
619619 −17824.2 −1.15737 −0.578686 0.815550i 0.696434π-0.696434\pi
−0.578686 + 0.815550i 0.696434π0.696434\pi
620620 0 0
621621 −2956.18 −0.191027
622622 0 0
623623 917.238 0.0589861
624624 0 0
625625 15484.4 0.991001
626626 0 0
627627 −5272.57 −0.335831
628628 0 0
629629 −19259.4 −1.22086
630630 0 0
631631 −22339.0 −1.40935 −0.704677 0.709528i 0.748908π-0.748908\pi
−0.704677 + 0.709528i 0.748908π0.748908\pi
632632 0 0
633633 5237.35 0.328856
634634 0 0
635635 1102.70 0.0689121
636636 0 0
637637 −9221.34 −0.573568
638638 0 0
639639 −3154.69 −0.195301
640640 0 0
641641 −5268.43 −0.324634 −0.162317 0.986739i 0.551897π-0.551897\pi
−0.162317 + 0.986739i 0.551897π0.551897\pi
642642 0 0
643643 21965.8 1.34719 0.673597 0.739099i 0.264748π-0.264748\pi
0.673597 + 0.739099i 0.264748π0.264748\pi
644644 0 0
645645 −434.524 −0.0265262
646646 0 0
647647 3165.40 0.192341 0.0961706 0.995365i 0.469341π-0.469341\pi
0.0961706 + 0.995365i 0.469341π0.469341\pi
648648 0 0
649649 −15840.4 −0.958073
650650 0 0
651651 15071.2 0.907354
652652 0 0
653653 −12094.7 −0.724814 −0.362407 0.932020i 0.618045π-0.618045\pi
−0.362407 + 0.932020i 0.618045π0.618045\pi
654654 0 0
655655 −687.067 −0.0409861
656656 0 0
657657 −4793.49 −0.284645
658658 0 0
659659 7523.17 0.444706 0.222353 0.974966i 0.428626π-0.428626\pi
0.222353 + 0.974966i 0.428626π0.428626\pi
660660 0 0
661661 −24141.1 −1.42054 −0.710271 0.703928i 0.751428π-0.751428\pi
−0.710271 + 0.703928i 0.751428π0.751428\pi
662662 0 0
663663 7485.84 0.438501
664664 0 0
665665 406.388 0.0236978
666666 0 0
667667 1141.41 0.0662604
668668 0 0
669669 −10156.8 −0.586972
670670 0 0
671671 −16479.8 −0.948130
672672 0 0
673673 944.143 0.0540773 0.0270387 0.999634i 0.491392π-0.491392\pi
0.0270387 + 0.999634i 0.491392π0.491392\pi
674674 0 0
675675 3364.87 0.191872
676676 0 0
677677 −4450.51 −0.252654 −0.126327 0.991989i 0.540319π-0.540319\pi
−0.126327 + 0.991989i 0.540319π0.540319\pi
678678 0 0
679679 13985.8 0.790468
680680 0 0
681681 −11543.1 −0.649536
682682 0 0
683683 −1726.40 −0.0967189 −0.0483594 0.998830i 0.515399π-0.515399\pi
−0.0483594 + 0.998830i 0.515399π0.515399\pi
684684 0 0
685685 1500.75 0.0837088
686686 0 0
687687 −4005.44 −0.222441
688688 0 0
689689 6233.37 0.344662
690690 0 0
691691 683.143 0.0376092 0.0188046 0.999823i 0.494014π-0.494014\pi
0.0188046 + 0.999823i 0.494014π0.494014\pi
692692 0 0
693693 −12335.7 −0.676181
694694 0 0
695695 1505.65 0.0821764
696696 0 0
697697 17014.1 0.924614
698698 0 0
699699 −15439.1 −0.835425
700700 0 0
701701 −19533.2 −1.05244 −0.526220 0.850349i 0.676391π-0.676391\pi
−0.526220 + 0.850349i 0.676391π0.676391\pi
702702 0 0
703703 −11909.2 −0.638922
704704 0 0
705705 −237.307 −0.0126773
706706 0 0
707707 37868.8 2.01443
708708 0 0
709709 −26081.9 −1.38156 −0.690779 0.723066i 0.742732π-0.742732\pi
−0.690779 + 0.723066i 0.742732π0.742732\pi
710710 0 0
711711 1499.05 0.0790699
712712 0 0
713713 24183.8 1.27025
714714 0 0
715715 1953.34 0.102169
716716 0 0
717717 21255.2 1.10710
718718 0 0
719719 30077.3 1.56007 0.780036 0.625734i 0.215200π-0.215200\pi
0.780036 + 0.625734i 0.215200π0.215200\pi
720720 0 0
721721 −9026.21 −0.466232
722722 0 0
723723 −7615.20 −0.391718
724724 0 0
725725 −1299.21 −0.0665537
726726 0 0
727727 23049.9 1.17589 0.587946 0.808900i 0.299937π-0.299937\pi
0.587946 + 0.808900i 0.299937π0.299937\pi
728728 0 0
729729 729.000 0.0370370
730730 0 0
731731 −11150.3 −0.564169
732732 0 0
733733 −4444.57 −0.223962 −0.111981 0.993710i 0.535720π-0.535720\pi
−0.111981 + 0.993710i 0.535720π0.535720\pi
734734 0 0
735735 320.353 0.0160768
736736 0 0
737737 5387.98 0.269293
738738 0 0
739739 −28465.1 −1.41692 −0.708462 0.705749i 0.750610π-0.750610\pi
−0.708462 + 0.705749i 0.750610π0.750610\pi
740740 0 0
741741 4628.91 0.229483
742742 0 0
743743 4389.76 0.216749 0.108374 0.994110i 0.465435π-0.465435\pi
0.108374 + 0.994110i 0.465435π0.465435\pi
744744 0 0
745745 1277.38 0.0628180
746746 0 0
747747 −3257.41 −0.159548
748748 0 0
749749 −7974.77 −0.389041
750750 0 0
751751 −14121.9 −0.686171 −0.343085 0.939304i 0.611472π-0.611472\pi
−0.343085 + 0.939304i 0.611472π0.611472\pi
752752 0 0
753753 −5090.98 −0.246382
754754 0 0
755755 −648.074 −0.0312395
756756 0 0
757757 −17006.3 −0.816516 −0.408258 0.912866i 0.633864π-0.633864\pi
−0.408258 + 0.912866i 0.633864π0.633864\pi
758758 0 0
759759 −19794.3 −0.946622
760760 0 0
761761 4603.38 0.219280 0.109640 0.993971i 0.465030π-0.465030\pi
0.109640 + 0.993971i 0.465030π0.465030\pi
762762 0 0
763763 13578.5 0.644264
764764 0 0
765765 −260.061 −0.0122909
766766 0 0
767767 13906.6 0.654679
768768 0 0
769769 −12459.1 −0.584248 −0.292124 0.956380i 0.594362π-0.594362\pi
−0.292124 + 0.956380i 0.594362π0.594362\pi
770770 0 0
771771 1148.71 0.0536571
772772 0 0
773773 −27068.1 −1.25947 −0.629737 0.776808i 0.716837π-0.716837\pi
−0.629737 + 0.776808i 0.716837π0.716837\pi
774774 0 0
775775 −27527.2 −1.27588
776776 0 0
777777 −27862.6 −1.28644
778778 0 0
779779 10520.8 0.483884
780780 0 0
781781 −21123.4 −0.967804
782782 0 0
783783 −281.475 −0.0128468
784784 0 0
785785 1956.23 0.0889438
786786 0 0
787787 −6986.86 −0.316461 −0.158230 0.987402i 0.550579π-0.550579\pi
−0.158230 + 0.987402i 0.550579π0.550579\pi
788788 0 0
789789 15006.1 0.677097
790790 0 0
791791 −11290.7 −0.507523
792792 0 0
793793 14468.0 0.647885
794794 0 0
795795 −216.550 −0.00966067
796796 0 0
797797 27271.5 1.21205 0.606027 0.795444i 0.292762π-0.292762\pi
0.606027 + 0.795444i 0.292762π0.292762\pi
798798 0 0
799799 −6089.52 −0.269627
800800 0 0
801801 −362.957 −0.0160105
802802 0 0
803803 −32096.7 −1.41054
804804 0 0
805805 1525.66 0.0667981
806806 0 0
807807 −18352.4 −0.800540
808808 0 0
809809 −23785.9 −1.03371 −0.516853 0.856074i 0.672897π-0.672897\pi
−0.516853 + 0.856074i 0.672897π0.672897\pi
810810 0 0
811811 −21703.5 −0.939718 −0.469859 0.882741i 0.655695π-0.655695\pi
−0.469859 + 0.882741i 0.655695π0.655695\pi
812812 0 0
813813 −11868.4 −0.511982
814814 0 0
815815 518.835 0.0222994
816816 0 0
817817 −6894.83 −0.295250
818818 0 0
819819 10829.7 0.462054
820820 0 0
821821 33240.4 1.41303 0.706515 0.707698i 0.250266π-0.250266\pi
0.706515 + 0.707698i 0.250266π0.250266\pi
822822 0 0
823823 −17227.5 −0.729665 −0.364832 0.931073i 0.618874π-0.618874\pi
−0.364832 + 0.931073i 0.618874π0.618874\pi
824824 0 0
825825 22530.8 0.950812
826826 0 0
827827 −21678.4 −0.911528 −0.455764 0.890101i 0.650634π-0.650634\pi
−0.455764 + 0.890101i 0.650634π0.650634\pi
828828 0 0
829829 −34269.8 −1.43575 −0.717877 0.696170i 0.754886π-0.754886\pi
−0.717877 + 0.696170i 0.754886π0.754886\pi
830830 0 0
831831 −14526.5 −0.606401
832832 0 0
833833 8220.55 0.341927
834834 0 0
835835 1611.39 0.0667838
836836 0 0
837837 −5963.77 −0.246282
838838 0 0
839839 33444.1 1.37618 0.688092 0.725623i 0.258448π-0.258448\pi
0.688092 + 0.725623i 0.258448π0.258448\pi
840840 0 0
841841 −24280.3 −0.995544
842842 0 0
843843 5636.04 0.230267
844844 0 0
845845 −368.860 −0.0150168
846846 0 0
847847 −52325.8 −2.12271
848848 0 0
849849 17174.6 0.694265
850850 0 0
851851 −44709.3 −1.80096
852852 0 0
853853 37037.3 1.48667 0.743337 0.668917i 0.233242π-0.233242\pi
0.743337 + 0.668917i 0.233242π0.233242\pi
854854 0 0
855855 −160.810 −0.00643227
856856 0 0
857857 −35794.2 −1.42673 −0.713365 0.700793i 0.752830π-0.752830\pi
−0.713365 + 0.700793i 0.752830π0.752830\pi
858858 0 0
859859 20582.1 0.817522 0.408761 0.912641i 0.365961π-0.365961\pi
0.408761 + 0.912641i 0.365961π0.365961\pi
860860 0 0
861861 24614.3 0.974278
862862 0 0
863863 25677.7 1.01284 0.506420 0.862287i 0.330969π-0.330969\pi
0.506420 + 0.862287i 0.330969π0.330969\pi
864864 0 0
865865 263.348 0.0103516
866866 0 0
867867 8065.60 0.315942
868868 0 0
869869 10037.5 0.391827
870870 0 0
871871 −4730.22 −0.184015
872872 0 0
873873 −5534.29 −0.214556
874874 0 0
875875 −3478.38 −0.134389
876876 0 0
877877 −32783.0 −1.26226 −0.631131 0.775676i 0.717409π-0.717409\pi
−0.631131 + 0.775676i 0.717409π0.717409\pi
878878 0 0
879879 15266.3 0.585800
880880 0 0
881881 26615.7 1.01783 0.508914 0.860818i 0.330047π-0.330047\pi
0.508914 + 0.860818i 0.330047π0.330047\pi
882882 0 0
883883 19203.4 0.731875 0.365937 0.930639i 0.380748π-0.380748\pi
0.365937 + 0.930639i 0.380748π0.380748\pi
884884 0 0
885885 −483.122 −0.0183503
886886 0 0
887887 23198.0 0.878144 0.439072 0.898452i 0.355307π-0.355307\pi
0.439072 + 0.898452i 0.355307π0.355307\pi
888888 0 0
889889 40936.0 1.54438
890890 0 0
891891 4881.30 0.183535
892892 0 0
893893 −3765.48 −0.141105
894894 0 0
895895 928.979 0.0346953
896896 0 0
897897 17377.8 0.646854
898898 0 0
899899 2302.68 0.0854266
900900 0 0
901901 −5556.86 −0.205467
902902 0 0
903903 −16131.1 −0.594472
904904 0 0
905905 2070.92 0.0760659
906906 0 0
907907 16151.3 0.591285 0.295643 0.955299i 0.404466π-0.404466\pi
0.295643 + 0.955299i 0.404466π0.404466\pi
908908 0 0
909909 −14984.9 −0.546776
910910 0 0
911911 3487.34 0.126829 0.0634143 0.997987i 0.479801π-0.479801\pi
0.0634143 + 0.997987i 0.479801π0.479801\pi
912912 0 0
913913 −21811.2 −0.790632
914914 0 0
915915 −502.624 −0.0181598
916916 0 0
917917 −25506.3 −0.918532
918918 0 0
919919 17055.5 0.612198 0.306099 0.952000i 0.400976π-0.400976\pi
0.306099 + 0.952000i 0.400976π0.400976\pi
920920 0 0
921921 21657.6 0.774857
922922 0 0
923923 18544.7 0.661329
924924 0 0
925925 50890.2 1.80893
926926 0 0
927927 3571.73 0.126549
928928 0 0
929929 −55339.3 −1.95438 −0.977192 0.212359i 0.931885π-0.931885\pi
−0.977192 + 0.212359i 0.931885π0.931885\pi
930930 0 0
931931 5083.22 0.178943
932932 0 0
933933 4611.17 0.161804
934934 0 0
935935 −1741.34 −0.0609069
936936 0 0
937937 20457.6 0.713255 0.356627 0.934247i 0.383927π-0.383927\pi
0.356627 + 0.934247i 0.383927π0.383927\pi
938938 0 0
939939 −6602.78 −0.229471
940940 0 0
941941 55891.0 1.93623 0.968116 0.250502i 0.0805958π-0.0805958\pi
0.968116 + 0.250502i 0.0805958π0.0805958\pi
942942 0 0
943943 39497.0 1.36395
944944 0 0
945945 −376.230 −0.0129511
946946 0 0
947947 54727.0 1.87792 0.938960 0.344027i 0.111791π-0.111791\pi
0.938960 + 0.344027i 0.111791π0.111791\pi
948948 0 0
949949 28178.4 0.963865
950950 0 0
951951 −8521.23 −0.290557
952952 0 0
953953 −29958.8 −1.01832 −0.509160 0.860672i 0.670044π-0.670044\pi
−0.509160 + 0.860672i 0.670044π0.670044\pi
954954 0 0
955955 −1715.27 −0.0581204
956956 0 0
957957 −1884.72 −0.0636619
958958 0 0
959959 55713.0 1.87598
960960 0 0
961961 18997.2 0.637682
962962 0 0
963963 3155.67 0.105597
964964 0 0
965965 −382.365 −0.0127552
966966 0 0
967967 13498.5 0.448896 0.224448 0.974486i 0.427942π-0.427942\pi
0.224448 + 0.974486i 0.427942π0.427942\pi
968968 0 0
969969 −4126.53 −0.136804
970970 0 0
971971 9652.36 0.319010 0.159505 0.987197i 0.449010π-0.449010\pi
0.159505 + 0.987197i 0.449010π0.449010\pi
972972 0 0
973973 55895.1 1.84164
974974 0 0
975975 −19780.2 −0.649717
976976 0 0
977977 12169.8 0.398511 0.199256 0.979948i 0.436148π-0.436148\pi
0.199256 + 0.979948i 0.436148π0.436148\pi
978978 0 0
979979 −2430.32 −0.0793394
980980 0 0
981981 −5373.08 −0.174872
982982 0 0
983983 47545.2 1.54268 0.771341 0.636423i 0.219587π-0.219587\pi
0.771341 + 0.636423i 0.219587π0.219587\pi
984984 0 0
985985 −2928.06 −0.0947165
986986 0 0
987987 −8809.69 −0.284109
988988 0 0
989989 −25884.5 −0.832234
990990 0 0
991991 892.350 0.0286039 0.0143019 0.999898i 0.495447π-0.495447\pi
0.0143019 + 0.999898i 0.495447π0.495447\pi
992992 0 0
993993 −6355.55 −0.203109
994994 0 0
995995 −1602.67 −0.0510634
996996 0 0
997997 −4458.57 −0.141629 −0.0708146 0.997489i 0.522560π-0.522560\pi
−0.0708146 + 0.997489i 0.522560π0.522560\pi
998998 0 0
999999 11025.4 0.349177
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 768.4.a.q.1.2 3
3.2 odd 2 2304.4.a.bw.1.2 3
4.3 odd 2 768.4.a.s.1.2 3
8.3 odd 2 768.4.a.r.1.2 3
8.5 even 2 768.4.a.t.1.2 3
12.11 even 2 2304.4.a.bv.1.2 3
16.3 odd 4 24.4.d.a.13.5 6
16.5 even 4 96.4.d.a.49.5 6
16.11 odd 4 24.4.d.a.13.6 yes 6
16.13 even 4 96.4.d.a.49.2 6
24.5 odd 2 2304.4.a.bu.1.2 3
24.11 even 2 2304.4.a.bt.1.2 3
48.5 odd 4 288.4.d.d.145.4 6
48.11 even 4 72.4.d.d.37.1 6
48.29 odd 4 288.4.d.d.145.3 6
48.35 even 4 72.4.d.d.37.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.d.a.13.5 6 16.3 odd 4
24.4.d.a.13.6 yes 6 16.11 odd 4
72.4.d.d.37.1 6 48.11 even 4
72.4.d.d.37.2 6 48.35 even 4
96.4.d.a.49.2 6 16.13 even 4
96.4.d.a.49.5 6 16.5 even 4
288.4.d.d.145.3 6 48.29 odd 4
288.4.d.d.145.4 6 48.5 odd 4
768.4.a.q.1.2 3 1.1 even 1 trivial
768.4.a.r.1.2 3 8.3 odd 2
768.4.a.s.1.2 3 4.3 odd 2
768.4.a.t.1.2 3 8.5 even 2
2304.4.a.bt.1.2 3 24.11 even 2
2304.4.a.bu.1.2 3 24.5 odd 2
2304.4.a.bv.1.2 3 12.11 even 2
2304.4.a.bw.1.2 3 3.2 odd 2