Properties

Label 1620.4.a.j.1.4
Level 16201620
Weight 44
Character 1620.1
Self dual yes
Analytic conductor 95.58395.583
Analytic rank 11
Dimension 66
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] = [1620,4,Mod(1,1620)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1620, 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("1620.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 1620=22345 1620 = 2^{2} \cdot 3^{4} \cdot 5
Weight: k k == 4 4
Character orbit: [χ][\chi] == 1620.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,30,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 95.583094209395.5830942093
Analytic rank: 11
Dimension: 66
Coefficient field: Q[x]/(x6)\mathbb{Q}[x]/(x^{6} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x62x5151x4212x3+4412x2+8656x19148 x^{6} - 2x^{5} - 151x^{4} - 212x^{3} + 4412x^{2} + 8656x - 19148 Copy content Toggle raw display
Coefficient ring: Z[a1,,a17]\Z[a_1, \ldots, a_{17}]
Coefficient ring index: 2235 2^{2}\cdot 3^{5}
Twist minimal: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.4
Root 4.35846-4.35846 of defining polynomial
Character χ\chi == 1620.1

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+5.00000q5+6.21414q728.3492q118.60548q1390.1352q17+114.170q1948.4106q23+25.0000q25+305.958q29+93.5957q31+31.0707q35282.892q3728.7685q41354.443q43522.998q47304.384q4966.9285q53141.746q557.63223q59+9.41124q6143.0274q65494.391q67560.709q71+1116.68q73176.166q77+1041.87q79+45.5107q83450.676q85357.159q8953.4757q91+570.849q95120.006q97+O(q100)q+5.00000 q^{5} +6.21414 q^{7} -28.3492 q^{11} -8.60548 q^{13} -90.1352 q^{17} +114.170 q^{19} -48.4106 q^{23} +25.0000 q^{25} +305.958 q^{29} +93.5957 q^{31} +31.0707 q^{35} -282.892 q^{37} -28.7685 q^{41} -354.443 q^{43} -522.998 q^{47} -304.384 q^{49} -66.9285 q^{53} -141.746 q^{55} -7.63223 q^{59} +9.41124 q^{61} -43.0274 q^{65} -494.391 q^{67} -560.709 q^{71} +1116.68 q^{73} -176.166 q^{77} +1041.87 q^{79} +45.5107 q^{83} -450.676 q^{85} -357.159 q^{89} -53.4757 q^{91} +570.849 q^{95} -120.006 q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q+30q5+12q784q1312q17114q1930q23+150q25168q29324q31+60q35492q37312q41156q43462q47588q491014q53+1188q97+O(q100) 6 q + 30 q^{5} + 12 q^{7} - 84 q^{13} - 12 q^{17} - 114 q^{19} - 30 q^{23} + 150 q^{25} - 168 q^{29} - 324 q^{31} + 60 q^{35} - 492 q^{37} - 312 q^{41} - 156 q^{43} - 462 q^{47} - 588 q^{49} - 1014 q^{53}+ \cdots - 1188 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 5.00000 0.447214
66 0 0
77 6.21414 0.335532 0.167766 0.985827i 0.446345π-0.446345\pi
0.167766 + 0.985827i 0.446345π0.446345\pi
88 0 0
99 0 0
1010 0 0
1111 −28.3492 −0.777054 −0.388527 0.921437i 0.627016π-0.627016\pi
−0.388527 + 0.921437i 0.627016π0.627016\pi
1212 0 0
1313 −8.60548 −0.183595 −0.0917973 0.995778i 0.529261π-0.529261\pi
−0.0917973 + 0.995778i 0.529261π0.529261\pi
1414 0 0
1515 0 0
1616 0 0
1717 −90.1352 −1.28594 −0.642970 0.765891i 0.722298π-0.722298\pi
−0.642970 + 0.765891i 0.722298π0.722298\pi
1818 0 0
1919 114.170 1.37854 0.689272 0.724503i 0.257931π-0.257931\pi
0.689272 + 0.724503i 0.257931π0.257931\pi
2020 0 0
2121 0 0
2222 0 0
2323 −48.4106 −0.438883 −0.219441 0.975626i 0.570423π-0.570423\pi
−0.219441 + 0.975626i 0.570423π0.570423\pi
2424 0 0
2525 25.0000 0.200000
2626 0 0
2727 0 0
2828 0 0
2929 305.958 1.95914 0.979568 0.201113i 0.0644557π-0.0644557\pi
0.979568 + 0.201113i 0.0644557π0.0644557\pi
3030 0 0
3131 93.5957 0.542267 0.271134 0.962542i 0.412601π-0.412601\pi
0.271134 + 0.962542i 0.412601π0.412601\pi
3232 0 0
3333 0 0
3434 0 0
3535 31.0707 0.150055
3636 0 0
3737 −282.892 −1.25695 −0.628475 0.777830i 0.716320π-0.716320\pi
−0.628475 + 0.777830i 0.716320π0.716320\pi
3838 0 0
3939 0 0
4040 0 0
4141 −28.7685 −0.109583 −0.0547913 0.998498i 0.517449π-0.517449\pi
−0.0547913 + 0.998498i 0.517449π0.517449\pi
4242 0 0
4343 −354.443 −1.25703 −0.628513 0.777799i 0.716336π-0.716336\pi
−0.628513 + 0.777799i 0.716336π0.716336\pi
4444 0 0
4545 0 0
4646 0 0
4747 −522.998 −1.62313 −0.811565 0.584262i 0.801384π-0.801384\pi
−0.811565 + 0.584262i 0.801384π0.801384\pi
4848 0 0
4949 −304.384 −0.887418
5050 0 0
5151 0 0
5252 0 0
5353 −66.9285 −0.173459 −0.0867296 0.996232i 0.527642π-0.527642\pi
−0.0867296 + 0.996232i 0.527642π0.527642\pi
5454 0 0
5555 −141.746 −0.347509
5656 0 0
5757 0 0
5858 0 0
5959 −7.63223 −0.0168412 −0.00842061 0.999965i 0.502680π-0.502680\pi
−0.00842061 + 0.999965i 0.502680π0.502680\pi
6060 0 0
6161 9.41124 0.0197539 0.00987693 0.999951i 0.496856π-0.496856\pi
0.00987693 + 0.999951i 0.496856π0.496856\pi
6262 0 0
6363 0 0
6464 0 0
6565 −43.0274 −0.0821060
6666 0 0
6767 −494.391 −0.901485 −0.450743 0.892654i 0.648841π-0.648841\pi
−0.450743 + 0.892654i 0.648841π0.648841\pi
6868 0 0
6969 0 0
7070 0 0
7171 −560.709 −0.937239 −0.468619 0.883400i 0.655248π-0.655248\pi
−0.468619 + 0.883400i 0.655248π0.655248\pi
7272 0 0
7373 1116.68 1.79038 0.895192 0.445680i 0.147038π-0.147038\pi
0.895192 + 0.445680i 0.147038π0.147038\pi
7474 0 0
7575 0 0
7676 0 0
7777 −176.166 −0.260726
7878 0 0
7979 1041.87 1.48379 0.741896 0.670515i 0.233927π-0.233927\pi
0.741896 + 0.670515i 0.233927π0.233927\pi
8080 0 0
8181 0 0
8282 0 0
8383 45.5107 0.0601862 0.0300931 0.999547i 0.490420π-0.490420\pi
0.0300931 + 0.999547i 0.490420π0.490420\pi
8484 0 0
8585 −450.676 −0.575090
8686 0 0
8787 0 0
8888 0 0
8989 −357.159 −0.425379 −0.212690 0.977120i 0.568222π-0.568222\pi
−0.212690 + 0.977120i 0.568222π0.568222\pi
9090 0 0
9191 −53.4757 −0.0616019
9292 0 0
9393 0 0
9494 0 0
9595 570.849 0.616503
9696 0 0
9797 −120.006 −0.125616 −0.0628079 0.998026i 0.520006π-0.520006\pi
−0.0628079 + 0.998026i 0.520006π0.520006\pi
9898 0 0
9999 0 0
100100 0 0
101101 −498.559 −0.491173 −0.245586 0.969375i 0.578980π-0.578980\pi
−0.245586 + 0.969375i 0.578980π0.578980\pi
102102 0 0
103103 −254.686 −0.243641 −0.121820 0.992552i 0.538873π-0.538873\pi
−0.121820 + 0.992552i 0.538873π0.538873\pi
104104 0 0
105105 0 0
106106 0 0
107107 311.077 0.281056 0.140528 0.990077i 0.455120π-0.455120\pi
0.140528 + 0.990077i 0.455120π0.455120\pi
108108 0 0
109109 −661.173 −0.580999 −0.290500 0.956875i 0.593822π-0.593822\pi
−0.290500 + 0.956875i 0.593822π0.593822\pi
110110 0 0
111111 0 0
112112 0 0
113113 −565.537 −0.470807 −0.235404 0.971898i 0.575641π-0.575641\pi
−0.235404 + 0.971898i 0.575641π0.575641\pi
114114 0 0
115115 −242.053 −0.196274
116116 0 0
117117 0 0
118118 0 0
119119 −560.113 −0.431474
120120 0 0
121121 −527.326 −0.396188
122122 0 0
123123 0 0
124124 0 0
125125 125.000 0.0894427
126126 0 0
127127 1809.60 1.26438 0.632190 0.774814i 0.282156π-0.282156\pi
0.632190 + 0.774814i 0.282156π0.282156\pi
128128 0 0
129129 0 0
130130 0 0
131131 −2029.83 −1.35379 −0.676897 0.736078i 0.736676π-0.736676\pi
−0.676897 + 0.736078i 0.736676π0.736676\pi
132132 0 0
133133 709.467 0.462546
134134 0 0
135135 0 0
136136 0 0
137137 545.350 0.340091 0.170045 0.985436i 0.445609π-0.445609\pi
0.170045 + 0.985436i 0.445609π0.445609\pi
138138 0 0
139139 −1762.64 −1.07558 −0.537788 0.843080i 0.680740π-0.680740\pi
−0.537788 + 0.843080i 0.680740π0.680740\pi
140140 0 0
141141 0 0
142142 0 0
143143 243.958 0.142663
144144 0 0
145145 1529.79 0.876152
146146 0 0
147147 0 0
148148 0 0
149149 −1223.53 −0.672722 −0.336361 0.941733i 0.609196π-0.609196\pi
−0.336361 + 0.941733i 0.609196π0.609196\pi
150150 0 0
151151 2849.03 1.53543 0.767716 0.640790i 0.221393π-0.221393\pi
0.767716 + 0.640790i 0.221393π0.221393\pi
152152 0 0
153153 0 0
154154 0 0
155155 467.978 0.242509
156156 0 0
157157 −942.831 −0.479275 −0.239637 0.970862i 0.577029π-0.577029\pi
−0.239637 + 0.970862i 0.577029π0.577029\pi
158158 0 0
159159 0 0
160160 0 0
161161 −300.830 −0.147259
162162 0 0
163163 417.774 0.200752 0.100376 0.994950i 0.467995π-0.467995\pi
0.100376 + 0.994950i 0.467995π0.467995\pi
164164 0 0
165165 0 0
166166 0 0
167167 −1392.76 −0.645360 −0.322680 0.946508i 0.604584π-0.604584\pi
−0.322680 + 0.946508i 0.604584π0.604584\pi
168168 0 0
169169 −2122.95 −0.966293
170170 0 0
171171 0 0
172172 0 0
173173 −3671.35 −1.61345 −0.806727 0.590924i 0.798763π-0.798763\pi
−0.806727 + 0.590924i 0.798763π0.798763\pi
174174 0 0
175175 155.354 0.0671064
176176 0 0
177177 0 0
178178 0 0
179179 −2387.27 −0.996834 −0.498417 0.866938i 0.666085π-0.666085\pi
−0.498417 + 0.866938i 0.666085π0.666085\pi
180180 0 0
181181 2078.63 0.853608 0.426804 0.904344i 0.359639π-0.359639\pi
0.426804 + 0.904344i 0.359639π0.359639\pi
182182 0 0
183183 0 0
184184 0 0
185185 −1414.46 −0.562125
186186 0 0
187187 2555.26 0.999245
188188 0 0
189189 0 0
190190 0 0
191191 −2991.03 −1.13311 −0.566553 0.824026i 0.691723π-0.691723\pi
−0.566553 + 0.824026i 0.691723π0.691723\pi
192192 0 0
193193 −5055.79 −1.88561 −0.942807 0.333338i 0.891825π-0.891825\pi
−0.942807 + 0.333338i 0.891825π0.891825\pi
194194 0 0
195195 0 0
196196 0 0
197197 −3665.56 −1.32569 −0.662843 0.748758i 0.730651π-0.730651\pi
−0.662843 + 0.748758i 0.730651π0.730651\pi
198198 0 0
199199 3217.92 1.14629 0.573147 0.819452i 0.305722π-0.305722\pi
0.573147 + 0.819452i 0.305722π0.305722\pi
200200 0 0
201201 0 0
202202 0 0
203203 1901.27 0.657353
204204 0 0
205205 −143.843 −0.0490068
206206 0 0
207207 0 0
208208 0 0
209209 −3236.61 −1.07120
210210 0 0
211211 −3909.15 −1.27544 −0.637718 0.770270i 0.720121π-0.720121\pi
−0.637718 + 0.770270i 0.720121π0.720121\pi
212212 0 0
213213 0 0
214214 0 0
215215 −1772.22 −0.562159
216216 0 0
217217 581.617 0.181948
218218 0 0
219219 0 0
220220 0 0
221221 775.657 0.236092
222222 0 0
223223 808.723 0.242852 0.121426 0.992600i 0.461253π-0.461253\pi
0.121426 + 0.992600i 0.461253π0.461253\pi
224224 0 0
225225 0 0
226226 0 0
227227 18.7516 0.00548275 0.00274138 0.999996i 0.499127π-0.499127\pi
0.00274138 + 0.999996i 0.499127π0.499127\pi
228228 0 0
229229 −1022.99 −0.295202 −0.147601 0.989047i 0.547155π-0.547155\pi
−0.147601 + 0.989047i 0.547155π0.547155\pi
230230 0 0
231231 0 0
232232 0 0
233233 2881.43 0.810166 0.405083 0.914280i 0.367243π-0.367243\pi
0.405083 + 0.914280i 0.367243π0.367243\pi
234234 0 0
235235 −2614.99 −0.725886
236236 0 0
237237 0 0
238238 0 0
239239 114.828 0.0310779 0.0155390 0.999879i 0.495054π-0.495054\pi
0.0155390 + 0.999879i 0.495054π0.495054\pi
240240 0 0
241241 −1968.74 −0.526214 −0.263107 0.964767i 0.584747π-0.584747\pi
−0.263107 + 0.964767i 0.584747π0.584747\pi
242242 0 0
243243 0 0
244244 0 0
245245 −1521.92 −0.396865
246246 0 0
247247 −982.485 −0.253093
248248 0 0
249249 0 0
250250 0 0
251251 −1706.47 −0.429129 −0.214564 0.976710i 0.568833π-0.568833\pi
−0.214564 + 0.976710i 0.568833π0.568833\pi
252252 0 0
253253 1372.40 0.341035
254254 0 0
255255 0 0
256256 0 0
257257 −6191.41 −1.50276 −0.751380 0.659869i 0.770612π-0.770612\pi
−0.751380 + 0.659869i 0.770612π0.770612\pi
258258 0 0
259259 −1757.93 −0.421747
260260 0 0
261261 0 0
262262 0 0
263263 138.736 0.0325278 0.0162639 0.999868i 0.494823π-0.494823\pi
0.0162639 + 0.999868i 0.494823π0.494823\pi
264264 0 0
265265 −334.643 −0.0775733
266266 0 0
267267 0 0
268268 0 0
269269 −3362.16 −0.762061 −0.381030 0.924562i 0.624431π-0.624431\pi
−0.381030 + 0.924562i 0.624431π0.624431\pi
270270 0 0
271271 3502.68 0.785140 0.392570 0.919722i 0.371586π-0.371586\pi
0.392570 + 0.919722i 0.371586π0.371586\pi
272272 0 0
273273 0 0
274274 0 0
275275 −708.729 −0.155411
276276 0 0
277277 3853.36 0.835833 0.417917 0.908485i 0.362760π-0.362760\pi
0.417917 + 0.908485i 0.362760π0.362760\pi
278278 0 0
279279 0 0
280280 0 0
281281 8259.91 1.75354 0.876770 0.480910i 0.159694π-0.159694\pi
0.876770 + 0.480910i 0.159694π0.159694\pi
282282 0 0
283283 −5171.06 −1.08618 −0.543088 0.839676i 0.682745π-0.682745\pi
−0.543088 + 0.839676i 0.682745π0.682745\pi
284284 0 0
285285 0 0
286286 0 0
287287 −178.772 −0.0367685
288288 0 0
289289 3211.35 0.653644
290290 0 0
291291 0 0
292292 0 0
293293 −4190.68 −0.835571 −0.417786 0.908546i 0.637194π-0.637194\pi
−0.417786 + 0.908546i 0.637194π0.637194\pi
294294 0 0
295295 −38.1611 −0.00753162
296296 0 0
297297 0 0
298298 0 0
299299 416.596 0.0805765
300300 0 0
301301 −2202.56 −0.421772
302302 0 0
303303 0 0
304304 0 0
305305 47.0562 0.00883420
306306 0 0
307307 3745.74 0.696355 0.348177 0.937429i 0.386801π-0.386801\pi
0.348177 + 0.937429i 0.386801π0.386801\pi
308308 0 0
309309 0 0
310310 0 0
311311 −4854.78 −0.885175 −0.442588 0.896725i 0.645939π-0.645939\pi
−0.442588 + 0.896725i 0.645939π0.645939\pi
312312 0 0
313313 3171.06 0.572649 0.286324 0.958133i 0.407567π-0.407567\pi
0.286324 + 0.958133i 0.407567π0.407567\pi
314314 0 0
315315 0 0
316316 0 0
317317 −5837.07 −1.03420 −0.517101 0.855924i 0.672989π-0.672989\pi
−0.517101 + 0.855924i 0.672989π0.672989\pi
318318 0 0
319319 −8673.65 −1.52235
320320 0 0
321321 0 0
322322 0 0
323323 −10290.7 −1.77273
324324 0 0
325325 −215.137 −0.0367189
326326 0 0
327327 0 0
328328 0 0
329329 −3249.99 −0.544612
330330 0 0
331331 5595.31 0.929142 0.464571 0.885536i 0.346209π-0.346209\pi
0.464571 + 0.885536i 0.346209π0.346209\pi
332332 0 0
333333 0 0
334334 0 0
335335 −2471.96 −0.403157
336336 0 0
337337 −8661.15 −1.40001 −0.700004 0.714139i 0.746819π-0.746819\pi
−0.700004 + 0.714139i 0.746819π0.746819\pi
338338 0 0
339339 0 0
340340 0 0
341341 −2653.36 −0.421371
342342 0 0
343343 −4022.94 −0.633289
344344 0 0
345345 0 0
346346 0 0
347347 10555.3 1.63297 0.816483 0.577369i 0.195921π-0.195921\pi
0.816483 + 0.577369i 0.195921π0.195921\pi
348348 0 0
349349 6399.71 0.981572 0.490786 0.871280i 0.336710π-0.336710\pi
0.490786 + 0.871280i 0.336710π0.336710\pi
350350 0 0
351351 0 0
352352 0 0
353353 −9778.11 −1.47432 −0.737162 0.675716i 0.763835π-0.763835\pi
−0.737162 + 0.675716i 0.763835π0.763835\pi
354354 0 0
355355 −2803.55 −0.419146
356356 0 0
357357 0 0
358358 0 0
359359 2893.60 0.425399 0.212700 0.977118i 0.431774π-0.431774\pi
0.212700 + 0.977118i 0.431774π0.431774\pi
360360 0 0
361361 6175.72 0.900382
362362 0 0
363363 0 0
364364 0 0
365365 5583.42 0.800684
366366 0 0
367367 −7307.82 −1.03941 −0.519707 0.854345i 0.673959π-0.673959\pi
−0.519707 + 0.854345i 0.673959π0.673959\pi
368368 0 0
369369 0 0
370370 0 0
371371 −415.903 −0.0582012
372372 0 0
373373 8272.24 1.14831 0.574156 0.818746i 0.305330π-0.305330\pi
0.574156 + 0.818746i 0.305330π0.305330\pi
374374 0 0
375375 0 0
376376 0 0
377377 −2632.91 −0.359687
378378 0 0
379379 −740.953 −0.100423 −0.0502113 0.998739i 0.515989π-0.515989\pi
−0.0502113 + 0.998739i 0.515989π0.515989\pi
380380 0 0
381381 0 0
382382 0 0
383383 −1233.49 −0.164566 −0.0822828 0.996609i 0.526221π-0.526221\pi
−0.0822828 + 0.996609i 0.526221π0.526221\pi
384384 0 0
385385 −880.828 −0.116600
386386 0 0
387387 0 0
388388 0 0
389389 6958.20 0.906927 0.453464 0.891275i 0.350188π-0.350188\pi
0.453464 + 0.891275i 0.350188π0.350188\pi
390390 0 0
391391 4363.50 0.564377
392392 0 0
393393 0 0
394394 0 0
395395 5209.35 0.663572
396396 0 0
397397 5973.68 0.755190 0.377595 0.925971i 0.376751π-0.376751\pi
0.377595 + 0.925971i 0.376751π0.376751\pi
398398 0 0
399399 0 0
400400 0 0
401401 −4716.88 −0.587406 −0.293703 0.955897i 0.594888π-0.594888\pi
−0.293703 + 0.955897i 0.594888π0.594888\pi
402402 0 0
403403 −805.436 −0.0995574
404404 0 0
405405 0 0
406406 0 0
407407 8019.74 0.976717
408408 0 0
409409 7815.88 0.944915 0.472458 0.881353i 0.343367π-0.343367\pi
0.472458 + 0.881353i 0.343367π0.343367\pi
410410 0 0
411411 0 0
412412 0 0
413413 −47.4277 −0.00565077
414414 0 0
415415 227.554 0.0269161
416416 0 0
417417 0 0
418418 0 0
419419 −2955.17 −0.344557 −0.172278 0.985048i 0.555113π-0.555113\pi
−0.172278 + 0.985048i 0.555113π0.555113\pi
420420 0 0
421421 −6770.08 −0.783737 −0.391869 0.920021i 0.628171π-0.628171\pi
−0.391869 + 0.920021i 0.628171π0.628171\pi
422422 0 0
423423 0 0
424424 0 0
425425 −2253.38 −0.257188
426426 0 0
427427 58.4828 0.00662805
428428 0 0
429429 0 0
430430 0 0
431431 6082.40 0.679765 0.339883 0.940468i 0.389613π-0.389613\pi
0.339883 + 0.940468i 0.389613π0.389613\pi
432432 0 0
433433 −8993.04 −0.998102 −0.499051 0.866573i 0.666318π-0.666318\pi
−0.499051 + 0.866573i 0.666318π0.666318\pi
434434 0 0
435435 0 0
436436 0 0
437437 −5527.02 −0.605019
438438 0 0
439439 17180.5 1.86784 0.933919 0.357485i 0.116366π-0.116366\pi
0.933919 + 0.357485i 0.116366π0.116366\pi
440440 0 0
441441 0 0
442442 0 0
443443 −12231.3 −1.31180 −0.655901 0.754847i 0.727711π-0.727711\pi
−0.655901 + 0.754847i 0.727711π0.727711\pi
444444 0 0
445445 −1785.79 −0.190235
446446 0 0
447447 0 0
448448 0 0
449449 −1182.08 −0.124245 −0.0621224 0.998069i 0.519787π-0.519787\pi
−0.0621224 + 0.998069i 0.519787π0.519787\pi
450450 0 0
451451 815.563 0.0851515
452452 0 0
453453 0 0
454454 0 0
455455 −267.378 −0.0275492
456456 0 0
457457 −7353.03 −0.752648 −0.376324 0.926488i 0.622812π-0.622812\pi
−0.376324 + 0.926488i 0.622812π0.622812\pi
458458 0 0
459459 0 0
460460 0 0
461461 −3706.94 −0.374510 −0.187255 0.982311i 0.559959π-0.559959\pi
−0.187255 + 0.982311i 0.559959π0.559959\pi
462462 0 0
463463 −14015.4 −1.40680 −0.703401 0.710794i 0.748336π-0.748336\pi
−0.703401 + 0.710794i 0.748336π0.748336\pi
464464 0 0
465465 0 0
466466 0 0
467467 12187.1 1.20760 0.603801 0.797135i 0.293652π-0.293652\pi
0.603801 + 0.797135i 0.293652π0.293652\pi
468468 0 0
469469 −3072.22 −0.302477
470470 0 0
471471 0 0
472472 0 0
473473 10048.2 0.976776
474474 0 0
475475 2854.24 0.275709
476476 0 0
477477 0 0
478478 0 0
479479 −64.6375 −0.00616568 −0.00308284 0.999995i 0.500981π-0.500981\pi
−0.00308284 + 0.999995i 0.500981π0.500981\pi
480480 0 0
481481 2434.42 0.230769
482482 0 0
483483 0 0
484484 0 0
485485 −600.029 −0.0561771
486486 0 0
487487 −3763.31 −0.350168 −0.175084 0.984553i 0.556020π-0.556020\pi
−0.175084 + 0.984553i 0.556020π0.556020\pi
488488 0 0
489489 0 0
490490 0 0
491491 −2106.94 −0.193655 −0.0968277 0.995301i 0.530870π-0.530870\pi
−0.0968277 + 0.995301i 0.530870π0.530870\pi
492492 0 0
493493 −27577.6 −2.51933
494494 0 0
495495 0 0
496496 0 0
497497 −3484.33 −0.314474
498498 0 0
499499 2130.97 0.191173 0.0955866 0.995421i 0.469527π-0.469527\pi
0.0955866 + 0.995421i 0.469527π0.469527\pi
500500 0 0
501501 0 0
502502 0 0
503503 12920.0 1.14528 0.572639 0.819808i 0.305920π-0.305920\pi
0.572639 + 0.819808i 0.305920π0.305920\pi
504504 0 0
505505 −2492.79 −0.219659
506506 0 0
507507 0 0
508508 0 0
509509 685.808 0.0597209 0.0298604 0.999554i 0.490494π-0.490494\pi
0.0298604 + 0.999554i 0.490494π0.490494\pi
510510 0 0
511511 6939.24 0.600731
512512 0 0
513513 0 0
514514 0 0
515515 −1273.43 −0.108959
516516 0 0
517517 14826.6 1.26126
518518 0 0
519519 0 0
520520 0 0
521521 −21259.3 −1.78769 −0.893845 0.448377i 0.852002π-0.852002\pi
−0.893845 + 0.448377i 0.852002π0.852002\pi
522522 0 0
523523 6099.14 0.509937 0.254968 0.966949i 0.417935π-0.417935\pi
0.254968 + 0.966949i 0.417935π0.417935\pi
524524 0 0
525525 0 0
526526 0 0
527527 −8436.26 −0.697323
528528 0 0
529529 −9823.42 −0.807382
530530 0 0
531531 0 0
532532 0 0
533533 247.567 0.0201188
534534 0 0
535535 1555.39 0.125692
536536 0 0
537537 0 0
538538 0 0
539539 8629.04 0.689572
540540 0 0
541541 9552.07 0.759104 0.379552 0.925170i 0.376078π-0.376078\pi
0.379552 + 0.925170i 0.376078π0.376078\pi
542542 0 0
543543 0 0
544544 0 0
545545 −3305.87 −0.259831
546546 0 0
547547 24878.0 1.94462 0.972309 0.233698i 0.0750828π-0.0750828\pi
0.972309 + 0.233698i 0.0750828π0.0750828\pi
548548 0 0
549549 0 0
550550 0 0
551551 34931.1 2.70075
552552 0 0
553553 6474.33 0.497860
554554 0 0
555555 0 0
556556 0 0
557557 −10625.7 −0.808306 −0.404153 0.914692i 0.632434π-0.632434\pi
−0.404153 + 0.914692i 0.632434π0.632434\pi
558558 0 0
559559 3050.16 0.230783
560560 0 0
561561 0 0
562562 0 0
563563 16188.0 1.21180 0.605900 0.795541i 0.292813π-0.292813\pi
0.605900 + 0.795541i 0.292813π0.292813\pi
564564 0 0
565565 −2827.68 −0.210551
566566 0 0
567567 0 0
568568 0 0
569569 −7919.46 −0.583482 −0.291741 0.956497i 0.594234π-0.594234\pi
−0.291741 + 0.956497i 0.594234π0.594234\pi
570570 0 0
571571 13505.3 0.989808 0.494904 0.868948i 0.335203π-0.335203\pi
0.494904 + 0.868948i 0.335203π0.335203\pi
572572 0 0
573573 0 0
574574 0 0
575575 −1210.26 −0.0877765
576576 0 0
577577 −7873.07 −0.568042 −0.284021 0.958818i 0.591669π-0.591669\pi
−0.284021 + 0.958818i 0.591669π0.591669\pi
578578 0 0
579579 0 0
580580 0 0
581581 282.810 0.0201944
582582 0 0
583583 1897.37 0.134787
584584 0 0
585585 0 0
586586 0 0
587587 20117.0 1.41451 0.707254 0.706959i 0.249934π-0.249934\pi
0.707254 + 0.706959i 0.249934π0.249934\pi
588588 0 0
589589 10685.8 0.747539
590590 0 0
591591 0 0
592592 0 0
593593 4394.15 0.304294 0.152147 0.988358i 0.451381π-0.451381\pi
0.152147 + 0.988358i 0.451381π0.451381\pi
594594 0 0
595595 −2800.56 −0.192961
596596 0 0
597597 0 0
598598 0 0
599599 14244.7 0.971660 0.485830 0.874053i 0.338517π-0.338517\pi
0.485830 + 0.874053i 0.338517π0.338517\pi
600600 0 0
601601 16435.6 1.11551 0.557755 0.830006i 0.311663π-0.311663\pi
0.557755 + 0.830006i 0.311663π0.311663\pi
602602 0 0
603603 0 0
604604 0 0
605605 −2636.63 −0.177180
606606 0 0
607607 27431.2 1.83426 0.917131 0.398586i 0.130499π-0.130499\pi
0.917131 + 0.398586i 0.130499π0.130499\pi
608608 0 0
609609 0 0
610610 0 0
611611 4500.65 0.297998
612612 0 0
613613 −15161.9 −0.998995 −0.499497 0.866315i 0.666482π-0.666482\pi
−0.499497 + 0.866315i 0.666482π0.666482\pi
614614 0 0
615615 0 0
616616 0 0
617617 −6160.44 −0.401961 −0.200981 0.979595i 0.564413π-0.564413\pi
−0.200981 + 0.979595i 0.564413π0.564413\pi
618618 0 0
619619 −26799.2 −1.74015 −0.870074 0.492922i 0.835929π-0.835929\pi
−0.870074 + 0.492922i 0.835929π0.835929\pi
620620 0 0
621621 0 0
622622 0 0
623623 −2219.44 −0.142728
624624 0 0
625625 625.000 0.0400000
626626 0 0
627627 0 0
628628 0 0
629629 25498.5 1.61636
630630 0 0
631631 −26100.0 −1.64663 −0.823315 0.567585i 0.807878π-0.807878\pi
−0.823315 + 0.567585i 0.807878π0.807878\pi
632632 0 0
633633 0 0
634634 0 0
635635 9048.01 0.565448
636636 0 0
637637 2619.37 0.162925
638638 0 0
639639 0 0
640640 0 0
641641 −3235.84 −0.199389 −0.0996944 0.995018i 0.531787π-0.531787\pi
−0.0996944 + 0.995018i 0.531787π0.531787\pi
642642 0 0
643643 −18448.4 −1.13147 −0.565733 0.824589i 0.691407π-0.691407\pi
−0.565733 + 0.824589i 0.691407π0.691407\pi
644644 0 0
645645 0 0
646646 0 0
647647 −17173.6 −1.04353 −0.521765 0.853089i 0.674726π-0.674726\pi
−0.521765 + 0.853089i 0.674726π0.674726\pi
648648 0 0
649649 216.367 0.0130865
650650 0 0
651651 0 0
652652 0 0
653653 26039.4 1.56049 0.780245 0.625474i 0.215094π-0.215094\pi
0.780245 + 0.625474i 0.215094π0.215094\pi
654654 0 0
655655 −10149.1 −0.605435
656656 0 0
657657 0 0
658658 0 0
659659 2706.12 0.159963 0.0799815 0.996796i 0.474514π-0.474514\pi
0.0799815 + 0.996796i 0.474514π0.474514\pi
660660 0 0
661661 27271.4 1.60474 0.802371 0.596825i 0.203571π-0.203571\pi
0.802371 + 0.596825i 0.203571π0.203571\pi
662662 0 0
663663 0 0
664664 0 0
665665 3547.33 0.206857
666666 0 0
667667 −14811.6 −0.859831
668668 0 0
669669 0 0
670670 0 0
671671 −266.801 −0.0153498
672672 0 0
673673 −12793.0 −0.732738 −0.366369 0.930470i 0.619399π-0.619399\pi
−0.366369 + 0.930470i 0.619399π0.619399\pi
674674 0 0
675675 0 0
676676 0 0
677677 27114.1 1.53926 0.769629 0.638491i 0.220441π-0.220441\pi
0.769629 + 0.638491i 0.220441π0.220441\pi
678678 0 0
679679 −745.733 −0.0421481
680680 0 0
681681 0 0
682682 0 0
683683 33488.7 1.87615 0.938075 0.346433i 0.112607π-0.112607\pi
0.938075 + 0.346433i 0.112607π0.112607\pi
684684 0 0
685685 2726.75 0.152093
686686 0 0
687687 0 0
688688 0 0
689689 575.952 0.0318462
690690 0 0
691691 −1728.99 −0.0951864 −0.0475932 0.998867i 0.515155π-0.515155\pi
−0.0475932 + 0.998867i 0.515155π0.515155\pi
692692 0 0
693693 0 0
694694 0 0
695695 −8813.19 −0.481012
696696 0 0
697697 2593.05 0.140917
698698 0 0
699699 0 0
700700 0 0
701701 9451.65 0.509250 0.254625 0.967040i 0.418048π-0.418048\pi
0.254625 + 0.967040i 0.418048π0.418048\pi
702702 0 0
703703 −32297.7 −1.73276
704704 0 0
705705 0 0
706706 0 0
707707 −3098.11 −0.164804
708708 0 0
709709 −23271.7 −1.23271 −0.616353 0.787470i 0.711391π-0.711391\pi
−0.616353 + 0.787470i 0.711391π0.711391\pi
710710 0 0
711711 0 0
712712 0 0
713713 −4531.02 −0.237992
714714 0 0
715715 1219.79 0.0638008
716716 0 0
717717 0 0
718718 0 0
719719 −24974.3 −1.29539 −0.647695 0.761900i 0.724267π-0.724267\pi
−0.647695 + 0.761900i 0.724267π0.724267\pi
720720 0 0
721721 −1582.66 −0.0817493
722722 0 0
723723 0 0
724724 0 0
725725 7648.95 0.391827
726726 0 0
727727 1666.52 0.0850177 0.0425089 0.999096i 0.486465π-0.486465\pi
0.0425089 + 0.999096i 0.486465π0.486465\pi
728728 0 0
729729 0 0
730730 0 0
731731 31947.8 1.61646
732732 0 0
733733 −8694.92 −0.438137 −0.219068 0.975710i 0.570302π-0.570302\pi
−0.219068 + 0.975710i 0.570302π0.570302\pi
734734 0 0
735735 0 0
736736 0 0
737737 14015.6 0.700503
738738 0 0
739739 −19373.0 −0.964342 −0.482171 0.876077i 0.660152π-0.660152\pi
−0.482171 + 0.876077i 0.660152π0.660152\pi
740740 0 0
741741 0 0
742742 0 0
743743 27444.0 1.35508 0.677538 0.735488i 0.263047π-0.263047\pi
0.677538 + 0.735488i 0.263047π0.263047\pi
744744 0 0
745745 −6117.65 −0.300850
746746 0 0
747747 0 0
748748 0 0
749749 1933.08 0.0943032
750750 0 0
751751 −3861.56 −0.187630 −0.0938152 0.995590i 0.529906π-0.529906\pi
−0.0938152 + 0.995590i 0.529906π0.529906\pi
752752 0 0
753753 0 0
754754 0 0
755755 14245.1 0.686666
756756 0 0
757757 19846.9 0.952903 0.476452 0.879201i 0.341923π-0.341923\pi
0.476452 + 0.879201i 0.341923π0.341923\pi
758758 0 0
759759 0 0
760760 0 0
761761 −30259.4 −1.44140 −0.720698 0.693250i 0.756178π-0.756178\pi
−0.720698 + 0.693250i 0.756178π0.756178\pi
762762 0 0
763763 −4108.63 −0.194944
764764 0 0
765765 0 0
766766 0 0
767767 65.6790 0.00309196
768768 0 0
769769 29877.1 1.40104 0.700518 0.713634i 0.252952π-0.252952\pi
0.700518 + 0.713634i 0.252952π0.252952\pi
770770 0 0
771771 0 0
772772 0 0
773773 −9509.77 −0.442487 −0.221244 0.975219i 0.571012π-0.571012\pi
−0.221244 + 0.975219i 0.571012π0.571012\pi
774774 0 0
775775 2339.89 0.108453
776776 0 0
777777 0 0
778778 0 0
779779 −3284.49 −0.151064
780780 0 0
781781 15895.6 0.728285
782782 0 0
783783 0 0
784784 0 0
785785 −4714.16 −0.214338
786786 0 0
787787 −13809.0 −0.625461 −0.312730 0.949842i 0.601244π-0.601244\pi
−0.312730 + 0.949842i 0.601244π0.601244\pi
788788 0 0
789789 0 0
790790 0 0
791791 −3514.33 −0.157971
792792 0 0
793793 −80.9882 −0.00362670
794794 0 0
795795 0 0
796796 0 0
797797 16821.9 0.747631 0.373815 0.927503i 0.378049π-0.378049\pi
0.373815 + 0.927503i 0.378049π0.378049\pi
798798 0 0
799799 47140.5 2.08725
800800 0 0
801801 0 0
802802 0 0
803803 −31657.1 −1.39122
804804 0 0
805805 −1504.15 −0.0658563
806806 0 0
807807 0 0
808808 0 0
809809 −28306.2 −1.23015 −0.615075 0.788468i 0.710874π-0.710874\pi
−0.615075 + 0.788468i 0.710874π0.710874\pi
810810 0 0
811811 13617.1 0.589594 0.294797 0.955560i 0.404748π-0.404748\pi
0.294797 + 0.955560i 0.404748π0.404748\pi
812812 0 0
813813 0 0
814814 0 0
815815 2088.87 0.0897790
816816 0 0
817817 −40466.7 −1.73286
818818 0 0
819819 0 0
820820 0 0
821821 30903.3 1.31368 0.656841 0.754029i 0.271892π-0.271892\pi
0.656841 + 0.754029i 0.271892π0.271892\pi
822822 0 0
823823 20273.3 0.858665 0.429333 0.903146i 0.358749π-0.358749\pi
0.429333 + 0.903146i 0.358749π0.358749\pi
824824 0 0
825825 0 0
826826 0 0
827827 32437.6 1.36393 0.681963 0.731387i 0.261127π-0.261127\pi
0.681963 + 0.731387i 0.261127π0.261127\pi
828828 0 0
829829 −7539.68 −0.315879 −0.157940 0.987449i 0.550485π-0.550485\pi
−0.157940 + 0.987449i 0.550485π0.550485\pi
830830 0 0
831831 0 0
832832 0 0
833833 27435.7 1.14117
834834 0 0
835835 −6963.81 −0.288614
836836 0 0
837837 0 0
838838 0 0
839839 20839.4 0.857518 0.428759 0.903419i 0.358951π-0.358951\pi
0.428759 + 0.903419i 0.358951π0.358951\pi
840840 0 0
841841 69221.2 2.83822
842842 0 0
843843 0 0
844844 0 0
845845 −10614.7 −0.432139
846846 0 0
847847 −3276.88 −0.132934
848848 0 0
849849 0 0
850850 0 0
851851 13695.0 0.551653
852852 0 0
853853 −36319.7 −1.45787 −0.728934 0.684584i 0.759984π-0.759984\pi
−0.728934 + 0.684584i 0.759984π0.759984\pi
854854 0 0
855855 0 0
856856 0 0
857857 10265.3 0.409168 0.204584 0.978849i 0.434416π-0.434416\pi
0.204584 + 0.978849i 0.434416π0.434416\pi
858858 0 0
859859 38218.5 1.51804 0.759022 0.651065i 0.225677π-0.225677\pi
0.759022 + 0.651065i 0.225677π0.225677\pi
860860 0 0
861861 0 0
862862 0 0
863863 27385.0 1.08018 0.540090 0.841608i 0.318390π-0.318390\pi
0.540090 + 0.841608i 0.318390π0.318390\pi
864864 0 0
865865 −18356.7 −0.721559
866866 0 0
867867 0 0
868868 0 0
869869 −29536.2 −1.15299
870870 0 0
871871 4254.48 0.165508
872872 0 0
873873 0 0
874874 0 0
875875 776.768 0.0300109
876876 0 0
877877 31319.6 1.20592 0.602958 0.797773i 0.293989π-0.293989\pi
0.602958 + 0.797773i 0.293989π0.293989\pi
878878 0 0
879879 0 0
880880 0 0
881881 6309.71 0.241293 0.120647 0.992696i 0.461503π-0.461503\pi
0.120647 + 0.992696i 0.461503π0.461503\pi
882882 0 0
883883 −24665.0 −0.940026 −0.470013 0.882660i 0.655751π-0.655751\pi
−0.470013 + 0.882660i 0.655751π0.655751\pi
884884 0 0
885885 0 0
886886 0 0
887887 27988.7 1.05949 0.529746 0.848156i 0.322287π-0.322287\pi
0.529746 + 0.848156i 0.322287π0.322287\pi
888888 0 0
889889 11245.1 0.424240
890890 0 0
891891 0 0
892892 0 0
893893 −59710.6 −2.23756
894894 0 0
895895 −11936.4 −0.445798
896896 0 0
897897 0 0
898898 0 0
899899 28636.3 1.06238
900900 0 0
901901 6032.62 0.223058
902902 0 0
903903 0 0
904904 0 0
905905 10393.1 0.381745
906906 0 0
907907 23263.2 0.851645 0.425822 0.904807i 0.359985π-0.359985\pi
0.425822 + 0.904807i 0.359985π0.359985\pi
908908 0 0
909909 0 0
910910 0 0
911911 38345.8 1.39457 0.697286 0.716793i 0.254391π-0.254391\pi
0.697286 + 0.716793i 0.254391π0.254391\pi
912912 0 0
913913 −1290.19 −0.0467679
914914 0 0
915915 0 0
916916 0 0
917917 −12613.6 −0.454241
918918 0 0
919919 −11675.3 −0.419077 −0.209539 0.977800i 0.567196π-0.567196\pi
−0.209539 + 0.977800i 0.567196π0.567196\pi
920920 0 0
921921 0 0
922922 0 0
923923 4825.17 0.172072
924924 0 0
925925 −7072.29 −0.251390
926926 0 0
927927 0 0
928928 0 0
929929 34050.4 1.20254 0.601269 0.799047i 0.294662π-0.294662\pi
0.601269 + 0.799047i 0.294662π0.294662\pi
930930 0 0
931931 −34751.5 −1.22334
932932 0 0
933933 0 0
934934 0 0
935935 12776.3 0.446876
936936 0 0
937937 15395.5 0.536766 0.268383 0.963312i 0.413511π-0.413511\pi
0.268383 + 0.963312i 0.413511π0.413511\pi
938938 0 0
939939 0 0
940940 0 0
941941 −25708.7 −0.890626 −0.445313 0.895375i 0.646908π-0.646908\pi
−0.445313 + 0.895375i 0.646908π0.646908\pi
942942 0 0
943943 1392.70 0.0480939
944944 0 0
945945 0 0
946946 0 0
947947 −7035.07 −0.241403 −0.120702 0.992689i 0.538514π-0.538514\pi
−0.120702 + 0.992689i 0.538514π0.538514\pi
948948 0 0
949949 −9609.61 −0.328705
950950 0 0
951951 0 0
952952 0 0
953953 15233.0 0.517782 0.258891 0.965907i 0.416643π-0.416643\pi
0.258891 + 0.965907i 0.416643π0.416643\pi
954954 0 0
955955 −14955.1 −0.506740
956956 0 0
957957 0 0
958958 0 0
959959 3388.88 0.114111
960960 0 0
961961 −21030.8 −0.705946
962962 0 0
963963 0 0
964964 0 0
965965 −25278.9 −0.843273
966966 0 0
967967 18311.7 0.608959 0.304479 0.952519i 0.401518π-0.401518\pi
0.304479 + 0.952519i 0.401518π0.401518\pi
968968 0 0
969969 0 0
970970 0 0
971971 −37567.0 −1.24159 −0.620795 0.783973i 0.713190π-0.713190\pi
−0.620795 + 0.783973i 0.713190π0.713190\pi
972972 0 0
973973 −10953.3 −0.360890
974974 0 0
975975 0 0
976976 0 0
977977 −50040.0 −1.63861 −0.819305 0.573357i 0.805641π-0.805641\pi
−0.819305 + 0.573357i 0.805641π0.805641\pi
978978 0 0
979979 10125.2 0.330543
980980 0 0
981981 0 0
982982 0 0
983983 3328.05 0.107984 0.0539921 0.998541i 0.482805π-0.482805\pi
0.0539921 + 0.998541i 0.482805π0.482805\pi
984984 0 0
985985 −18327.8 −0.592865
986986 0 0
987987 0 0
988988 0 0
989989 17158.8 0.551687
990990 0 0
991991 47485.9 1.52214 0.761069 0.648671i 0.224675π-0.224675\pi
0.761069 + 0.648671i 0.224675π0.224675\pi
992992 0 0
993993 0 0
994994 0 0
995995 16089.6 0.512639
996996 0 0
997997 101.392 0.00322076 0.00161038 0.999999i 0.499487π-0.499487\pi
0.00161038 + 0.999999i 0.499487π0.499487\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 1620.4.a.j.1.4 yes 6
3.2 odd 2 1620.4.a.i.1.4 6
9.2 odd 6 1620.4.i.x.1081.3 12
9.4 even 3 1620.4.i.w.541.3 12
9.5 odd 6 1620.4.i.x.541.3 12
9.7 even 3 1620.4.i.w.1081.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.4.a.i.1.4 6 3.2 odd 2
1620.4.a.j.1.4 yes 6 1.1 even 1 trivial
1620.4.i.w.541.3 12 9.4 even 3
1620.4.i.w.1081.3 12 9.7 even 3
1620.4.i.x.541.3 12 9.5 odd 6
1620.4.i.x.1081.3 12 9.2 odd 6