Properties

Label 1440.2.k.e.721.3
Level 14401440
Weight 22
Character 1440.721
Analytic conductor 11.49811.498
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1440,2,Mod(721,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1440.721");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1440=25325 1440 = 2^{5} \cdot 3^{2} \cdot 5
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1440.k (of order 22, degree 11, not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 11.498457891111.4984578911
Analytic rank: 00
Dimension: 44
Coefficient field: Q(ζ12)\Q(\zeta_{12})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x2+1 x^{4} - x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 22 2^{2}
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 721.3
Root 0.866025+0.500000i0.866025 + 0.500000i of defining polynomial
Character χ\chi == 1440.721
Dual form 1440.2.k.e.721.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+1.00000iq50.732051q72.00000iq113.46410iq133.46410q170.535898iq196.19615q231.00000q256.92820iq29+5.46410q310.732051iq352.00000iq371.46410q41+5.26795iq43+3.26795q476.46410q4911.4641iq53+2.00000q557.46410iq598.92820iq61+3.46410q6510.7321iq67+5.46410q71+7.46410q73+1.46410iq77+1.07180q79+1.26795iq833.46410iq858.92820q89+2.53590iq91+0.535898q9514.3923q97+O(q100)q+1.00000i q^{5} -0.732051 q^{7} -2.00000i q^{11} -3.46410i q^{13} -3.46410 q^{17} -0.535898i q^{19} -6.19615 q^{23} -1.00000 q^{25} -6.92820i q^{29} +5.46410 q^{31} -0.732051i q^{35} -2.00000i q^{37} -1.46410 q^{41} +5.26795i q^{43} +3.26795 q^{47} -6.46410 q^{49} -11.4641i q^{53} +2.00000 q^{55} -7.46410i q^{59} -8.92820i q^{61} +3.46410 q^{65} -10.7321i q^{67} +5.46410 q^{71} +7.46410 q^{73} +1.46410i q^{77} +1.07180 q^{79} +1.26795i q^{83} -3.46410i q^{85} -8.92820 q^{89} +2.53590i q^{91} +0.535898 q^{95} -14.3923 q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+4q74q234q25+8q31+8q41+20q4712q49+8q55+8q71+16q73+32q798q89+16q9516q97+O(q100) 4 q + 4 q^{7} - 4 q^{23} - 4 q^{25} + 8 q^{31} + 8 q^{41} + 20 q^{47} - 12 q^{49} + 8 q^{55} + 8 q^{71} + 16 q^{73} + 32 q^{79} - 8 q^{89} + 16 q^{95} - 16 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1440Z)×\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times.

nn 577577 641641 901901 991991
χ(n)\chi(n) 11 11 1-1 11

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 1.00000i 0.447214i
66 0 0
77 −0.732051 −0.276689 −0.138345 0.990384i 0.544178π-0.544178\pi
−0.138345 + 0.990384i 0.544178π0.544178\pi
88 0 0
99 0 0
1010 0 0
1111 − 2.00000i − 0.603023i −0.953463 0.301511i 0.902509π-0.902509\pi
0.953463 0.301511i 0.0974911π-0.0974911\pi
1212 0 0
1313 − 3.46410i − 0.960769i −0.877058 0.480384i 0.840497π-0.840497\pi
0.877058 0.480384i 0.159503π-0.159503\pi
1414 0 0
1515 0 0
1616 0 0
1717 −3.46410 −0.840168 −0.420084 0.907485i 0.637999π-0.637999\pi
−0.420084 + 0.907485i 0.637999π0.637999\pi
1818 0 0
1919 − 0.535898i − 0.122944i −0.998109 0.0614718i 0.980421π-0.980421\pi
0.998109 0.0614718i 0.0195794π-0.0195794\pi
2020 0 0
2121 0 0
2222 0 0
2323 −6.19615 −1.29199 −0.645994 0.763343i 0.723557π-0.723557\pi
−0.645994 + 0.763343i 0.723557π0.723557\pi
2424 0 0
2525 −1.00000 −0.200000
2626 0 0
2727 0 0
2828 0 0
2929 − 6.92820i − 1.28654i −0.765641 0.643268i 0.777578π-0.777578\pi
0.765641 0.643268i 0.222422π-0.222422\pi
3030 0 0
3131 5.46410 0.981382 0.490691 0.871334i 0.336744π-0.336744\pi
0.490691 + 0.871334i 0.336744π0.336744\pi
3232 0 0
3333 0 0
3434 0 0
3535 − 0.732051i − 0.123739i
3636 0 0
3737 − 2.00000i − 0.328798i −0.986394 0.164399i 0.947432π-0.947432\pi
0.986394 0.164399i 0.0525685π-0.0525685\pi
3838 0 0
3939 0 0
4040 0 0
4141 −1.46410 −0.228654 −0.114327 0.993443i 0.536471π-0.536471\pi
−0.114327 + 0.993443i 0.536471π0.536471\pi
4242 0 0
4343 5.26795i 0.803355i 0.915781 + 0.401677i 0.131573π0.131573\pi
−0.915781 + 0.401677i 0.868427π0.868427\pi
4444 0 0
4545 0 0
4646 0 0
4747 3.26795 0.476679 0.238340 0.971182i 0.423397π-0.423397\pi
0.238340 + 0.971182i 0.423397π0.423397\pi
4848 0 0
4949 −6.46410 −0.923443
5050 0 0
5151 0 0
5252 0 0
5353 − 11.4641i − 1.57472i −0.616496 0.787358i 0.711449π-0.711449\pi
0.616496 0.787358i 0.288551π-0.288551\pi
5454 0 0
5555 2.00000 0.269680
5656 0 0
5757 0 0
5858 0 0
5959 − 7.46410i − 0.971743i −0.874030 0.485872i 0.838502π-0.838502\pi
0.874030 0.485872i 0.161498π-0.161498\pi
6060 0 0
6161 − 8.92820i − 1.14314i −0.820554 0.571570i 0.806335π-0.806335\pi
0.820554 0.571570i 0.193665π-0.193665\pi
6262 0 0
6363 0 0
6464 0 0
6565 3.46410 0.429669
6666 0 0
6767 − 10.7321i − 1.31113i −0.755139 0.655564i 0.772431π-0.772431\pi
0.755139 0.655564i 0.227569π-0.227569\pi
6868 0 0
6969 0 0
7070 0 0
7171 5.46410 0.648470 0.324235 0.945977i 0.394893π-0.394893\pi
0.324235 + 0.945977i 0.394893π0.394893\pi
7272 0 0
7373 7.46410 0.873607 0.436804 0.899557i 0.356111π-0.356111\pi
0.436804 + 0.899557i 0.356111π0.356111\pi
7474 0 0
7575 0 0
7676 0 0
7777 1.46410i 0.166850i
7878 0 0
7979 1.07180 0.120587 0.0602933 0.998181i 0.480796π-0.480796\pi
0.0602933 + 0.998181i 0.480796π0.480796\pi
8080 0 0
8181 0 0
8282 0 0
8383 1.26795i 0.139176i 0.997576 + 0.0695878i 0.0221684π0.0221684\pi
−0.997576 + 0.0695878i 0.977832π0.977832\pi
8484 0 0
8585 − 3.46410i − 0.375735i
8686 0 0
8787 0 0
8888 0 0
8989 −8.92820 −0.946388 −0.473194 0.880958i 0.656899π-0.656899\pi
−0.473194 + 0.880958i 0.656899π0.656899\pi
9090 0 0
9191 2.53590i 0.265834i
9292 0 0
9393 0 0
9494 0 0
9595 0.535898 0.0549820
9696 0 0
9797 −14.3923 −1.46132 −0.730659 0.682743i 0.760787π-0.760787\pi
−0.730659 + 0.682743i 0.760787π0.760787\pi
9898 0 0
9999 0 0
100100 0 0
101101 − 2.92820i − 0.291367i −0.989331 0.145684i 0.953462π-0.953462\pi
0.989331 0.145684i 0.0465381π-0.0465381\pi
102102 0 0
103103 −15.6603 −1.54305 −0.771525 0.636199i 0.780506π-0.780506\pi
−0.771525 + 0.636199i 0.780506π0.780506\pi
104104 0 0
105105 0 0
106106 0 0
107107 2.73205i 0.264117i 0.991242 + 0.132059i 0.0421587π0.0421587\pi
−0.991242 + 0.132059i 0.957841π0.957841\pi
108108 0 0
109109 16.9282i 1.62143i 0.585443 + 0.810714i 0.300921π0.300921\pi
−0.585443 + 0.810714i 0.699079π0.699079\pi
110110 0 0
111111 0 0
112112 0 0
113113 12.9282 1.21618 0.608092 0.793867i 0.291935π-0.291935\pi
0.608092 + 0.793867i 0.291935π0.291935\pi
114114 0 0
115115 − 6.19615i − 0.577794i
116116 0 0
117117 0 0
118118 0 0
119119 2.53590 0.232465
120120 0 0
121121 7.00000 0.636364
122122 0 0
123123 0 0
124124 0 0
125125 − 1.00000i − 0.0894427i
126126 0 0
127127 −16.7321 −1.48473 −0.742365 0.669996i 0.766296π-0.766296\pi
−0.742365 + 0.669996i 0.766296π0.766296\pi
128128 0 0
129129 0 0
130130 0 0
131131 − 19.8564i − 1.73486i −0.497557 0.867431i 0.665770π-0.665770\pi
0.497557 0.867431i 0.334230π-0.334230\pi
132132 0 0
133133 0.392305i 0.0340171i
134134 0 0
135135 0 0
136136 0 0
137137 −4.92820 −0.421045 −0.210522 0.977589i 0.567516π-0.567516\pi
−0.210522 + 0.977589i 0.567516π0.567516\pi
138138 0 0
139139 0.535898i 0.0454543i 0.999742 + 0.0227272i 0.00723490π0.00723490\pi
−0.999742 + 0.0227272i 0.992765π0.992765\pi
140140 0 0
141141 0 0
142142 0 0
143143 −6.92820 −0.579365
144144 0 0
145145 6.92820 0.575356
146146 0 0
147147 0 0
148148 0 0
149149 7.85641i 0.643622i 0.946804 + 0.321811i 0.104292π0.104292\pi
−0.946804 + 0.321811i 0.895708π0.895708\pi
150150 0 0
151151 12.3923 1.00847 0.504236 0.863566i 0.331774π-0.331774\pi
0.504236 + 0.863566i 0.331774π0.331774\pi
152152 0 0
153153 0 0
154154 0 0
155155 5.46410i 0.438887i
156156 0 0
157157 − 3.07180i − 0.245156i −0.992459 0.122578i 0.960884π-0.960884\pi
0.992459 0.122578i 0.0391162π-0.0391162\pi
158158 0 0
159159 0 0
160160 0 0
161161 4.53590 0.357479
162162 0 0
163163 0.196152i 0.0153638i 0.999970 + 0.00768192i 0.00244526π0.00244526\pi
−0.999970 + 0.00768192i 0.997555π0.997555\pi
164164 0 0
165165 0 0
166166 0 0
167167 −9.80385 −0.758645 −0.379322 0.925265i 0.623843π-0.623843\pi
−0.379322 + 0.925265i 0.623843π0.623843\pi
168168 0 0
169169 1.00000 0.0769231
170170 0 0
171171 0 0
172172 0 0
173173 − 2.00000i − 0.152057i −0.997106 0.0760286i 0.975776π-0.975776\pi
0.997106 0.0760286i 0.0242240π-0.0242240\pi
174174 0 0
175175 0.732051 0.0553378
176176 0 0
177177 0 0
178178 0 0
179179 − 8.53590i − 0.638003i −0.947754 0.319002i 0.896652π-0.896652\pi
0.947754 0.319002i 0.103348π-0.103348\pi
180180 0 0
181181 16.0000i 1.18927i 0.803996 + 0.594635i 0.202704π0.202704\pi
−0.803996 + 0.594635i 0.797296π0.797296\pi
182182 0 0
183183 0 0
184184 0 0
185185 2.00000 0.147043
186186 0 0
187187 6.92820i 0.506640i
188188 0 0
189189 0 0
190190 0 0
191191 15.3205 1.10855 0.554277 0.832333i 0.312995π-0.312995\pi
0.554277 + 0.832333i 0.312995π0.312995\pi
192192 0 0
193193 0.535898 0.0385748 0.0192874 0.999814i 0.493860π-0.493860\pi
0.0192874 + 0.999814i 0.493860π0.493860\pi
194194 0 0
195195 0 0
196196 0 0
197197 19.4641i 1.38676i 0.720572 + 0.693380i 0.243879π0.243879\pi
−0.720572 + 0.693380i 0.756121π0.756121\pi
198198 0 0
199199 1.85641 0.131597 0.0657986 0.997833i 0.479041π-0.479041\pi
0.0657986 + 0.997833i 0.479041π0.479041\pi
200200 0 0
201201 0 0
202202 0 0
203203 5.07180i 0.355970i
204204 0 0
205205 − 1.46410i − 0.102257i
206206 0 0
207207 0 0
208208 0 0
209209 −1.07180 −0.0741377
210210 0 0
211211 − 26.7846i − 1.84393i −0.387275 0.921964i 0.626584π-0.626584\pi
0.387275 0.921964i 0.373416π-0.373416\pi
212212 0 0
213213 0 0
214214 0 0
215215 −5.26795 −0.359271
216216 0 0
217217 −4.00000 −0.271538
218218 0 0
219219 0 0
220220 0 0
221221 12.0000i 0.807207i
222222 0 0
223223 5.80385 0.388654 0.194327 0.980937i 0.437748π-0.437748\pi
0.194327 + 0.980937i 0.437748π0.437748\pi
224224 0 0
225225 0 0
226226 0 0
227227 10.0526i 0.667212i 0.942713 + 0.333606i 0.108265π0.108265\pi
−0.942713 + 0.333606i 0.891735π0.891735\pi
228228 0 0
229229 − 4.00000i − 0.264327i −0.991228 0.132164i 0.957808π-0.957808\pi
0.991228 0.132164i 0.0421925π-0.0421925\pi
230230 0 0
231231 0 0
232232 0 0
233233 5.32051 0.348558 0.174279 0.984696i 0.444241π-0.444241\pi
0.174279 + 0.984696i 0.444241π0.444241\pi
234234 0 0
235235 3.26795i 0.213177i
236236 0 0
237237 0 0
238238 0 0
239239 −20.0000 −1.29369 −0.646846 0.762620i 0.723912π-0.723912\pi
−0.646846 + 0.762620i 0.723912π0.723912\pi
240240 0 0
241241 16.3923 1.05592 0.527961 0.849269i 0.322957π-0.322957\pi
0.527961 + 0.849269i 0.322957π0.322957\pi
242242 0 0
243243 0 0
244244 0 0
245245 − 6.46410i − 0.412976i
246246 0 0
247247 −1.85641 −0.118120
248248 0 0
249249 0 0
250250 0 0
251251 24.9282i 1.57345i 0.617301 + 0.786727i 0.288226π0.288226\pi
−0.617301 + 0.786727i 0.711774π0.711774\pi
252252 0 0
253253 12.3923i 0.779098i
254254 0 0
255255 0 0
256256 0 0
257257 2.00000 0.124757 0.0623783 0.998053i 0.480131π-0.480131\pi
0.0623783 + 0.998053i 0.480131π0.480131\pi
258258 0 0
259259 1.46410i 0.0909748i
260260 0 0
261261 0 0
262262 0 0
263263 −11.6603 −0.719002 −0.359501 0.933145i 0.617053π-0.617053\pi
−0.359501 + 0.933145i 0.617053π0.617053\pi
264264 0 0
265265 11.4641 0.704234
266266 0 0
267267 0 0
268268 0 0
269269 8.92820i 0.544362i 0.962246 + 0.272181i 0.0877450π0.0877450\pi
−0.962246 + 0.272181i 0.912255π0.912255\pi
270270 0 0
271271 19.3205 1.17364 0.586819 0.809718i 0.300380π-0.300380\pi
0.586819 + 0.809718i 0.300380π0.300380\pi
272272 0 0
273273 0 0
274274 0 0
275275 2.00000i 0.120605i
276276 0 0
277277 2.00000i 0.120168i 0.998193 + 0.0600842i 0.0191369π0.0191369\pi
−0.998193 + 0.0600842i 0.980863π0.980863\pi
278278 0 0
279279 0 0
280280 0 0
281281 −10.5359 −0.628519 −0.314260 0.949337i 0.601756π-0.601756\pi
−0.314260 + 0.949337i 0.601756π0.601756\pi
282282 0 0
283283 9.66025i 0.574242i 0.957894 + 0.287121i 0.0926983π0.0926983\pi
−0.957894 + 0.287121i 0.907302π0.907302\pi
284284 0 0
285285 0 0
286286 0 0
287287 1.07180 0.0632662
288288 0 0
289289 −5.00000 −0.294118
290290 0 0
291291 0 0
292292 0 0
293293 15.8564i 0.926341i 0.886269 + 0.463171i 0.153288π0.153288\pi
−0.886269 + 0.463171i 0.846712π0.846712\pi
294294 0 0
295295 7.46410 0.434577
296296 0 0
297297 0 0
298298 0 0
299299 21.4641i 1.24130i
300300 0 0
301301 − 3.85641i − 0.222280i
302302 0 0
303303 0 0
304304 0 0
305305 8.92820 0.511227
306306 0 0
307307 24.9808i 1.42573i 0.701303 + 0.712864i 0.252602π0.252602\pi
−0.701303 + 0.712864i 0.747398π0.747398\pi
308308 0 0
309309 0 0
310310 0 0
311311 31.3205 1.77602 0.888012 0.459821i 0.152086π-0.152086\pi
0.888012 + 0.459821i 0.152086π0.152086\pi
312312 0 0
313313 −4.14359 −0.234210 −0.117105 0.993120i 0.537361π-0.537361\pi
−0.117105 + 0.993120i 0.537361π0.537361\pi
314314 0 0
315315 0 0
316316 0 0
317317 − 8.53590i − 0.479424i −0.970844 0.239712i 0.922947π-0.922947\pi
0.970844 0.239712i 0.0770530π-0.0770530\pi
318318 0 0
319319 −13.8564 −0.775810
320320 0 0
321321 0 0
322322 0 0
323323 1.85641i 0.103293i
324324 0 0
325325 3.46410i 0.192154i
326326 0 0
327327 0 0
328328 0 0
329329 −2.39230 −0.131892
330330 0 0
331331 14.0000i 0.769510i 0.923019 + 0.384755i 0.125714π0.125714\pi
−0.923019 + 0.384755i 0.874286π0.874286\pi
332332 0 0
333333 0 0
334334 0 0
335335 10.7321 0.586355
336336 0 0
337337 −19.8564 −1.08165 −0.540824 0.841136i 0.681887π-0.681887\pi
−0.540824 + 0.841136i 0.681887π0.681887\pi
338338 0 0
339339 0 0
340340 0 0
341341 − 10.9282i − 0.591795i
342342 0 0
343343 9.85641 0.532196
344344 0 0
345345 0 0
346346 0 0
347347 − 1.66025i − 0.0891271i −0.999007 0.0445636i 0.985810π-0.985810\pi
0.999007 0.0445636i 0.0141897π-0.0141897\pi
348348 0 0
349349 − 28.0000i − 1.49881i −0.662114 0.749403i 0.730341π-0.730341\pi
0.662114 0.749403i 0.269659π-0.269659\pi
350350 0 0
351351 0 0
352352 0 0
353353 −12.9282 −0.688099 −0.344049 0.938952i 0.611799π-0.611799\pi
−0.344049 + 0.938952i 0.611799π0.611799\pi
354354 0 0
355355 5.46410i 0.290004i
356356 0 0
357357 0 0
358358 0 0
359359 18.9282 0.998992 0.499496 0.866316i 0.333518π-0.333518\pi
0.499496 + 0.866316i 0.333518π0.333518\pi
360360 0 0
361361 18.7128 0.984885
362362 0 0
363363 0 0
364364 0 0
365365 7.46410i 0.390689i
366366 0 0
367367 −2.87564 −0.150107 −0.0750537 0.997179i 0.523913π-0.523913\pi
−0.0750537 + 0.997179i 0.523913π0.523913\pi
368368 0 0
369369 0 0
370370 0 0
371371 8.39230i 0.435707i
372372 0 0
373373 − 25.7128i − 1.33136i −0.746238 0.665679i 0.768142π-0.768142\pi
0.746238 0.665679i 0.231858π-0.231858\pi
374374 0 0
375375 0 0
376376 0 0
377377 −24.0000 −1.23606
378378 0 0
379379 36.2487i 1.86197i 0.365056 + 0.930986i 0.381050π0.381050\pi
−0.365056 + 0.930986i 0.618950π0.618950\pi
380380 0 0
381381 0 0
382382 0 0
383383 21.1244 1.07940 0.539702 0.841856i 0.318537π-0.318537\pi
0.539702 + 0.841856i 0.318537π0.318537\pi
384384 0 0
385385 −1.46410 −0.0746175
386386 0 0
387387 0 0
388388 0 0
389389 − 6.78461i − 0.343993i −0.985098 0.171997i 0.944978π-0.944978\pi
0.985098 0.171997i 0.0550218π-0.0550218\pi
390390 0 0
391391 21.4641 1.08549
392392 0 0
393393 0 0
394394 0 0
395395 1.07180i 0.0539279i
396396 0 0
397397 32.2487i 1.61852i 0.587453 + 0.809258i 0.300131π0.300131\pi
−0.587453 + 0.809258i 0.699869π0.699869\pi
398398 0 0
399399 0 0
400400 0 0
401401 7.85641 0.392330 0.196165 0.980571i 0.437151π-0.437151\pi
0.196165 + 0.980571i 0.437151π0.437151\pi
402402 0 0
403403 − 18.9282i − 0.942881i
404404 0 0
405405 0 0
406406 0 0
407407 −4.00000 −0.198273
408408 0 0
409409 −11.3205 −0.559763 −0.279882 0.960035i 0.590295π-0.590295\pi
−0.279882 + 0.960035i 0.590295π0.590295\pi
410410 0 0
411411 0 0
412412 0 0
413413 5.46410i 0.268871i
414414 0 0
415415 −1.26795 −0.0622412
416416 0 0
417417 0 0
418418 0 0
419419 18.3923i 0.898523i 0.893400 + 0.449261i 0.148313π0.148313\pi
−0.893400 + 0.449261i 0.851687π0.851687\pi
420420 0 0
421421 0.143594i 0.00699832i 0.999994 + 0.00349916i 0.00111382π0.00111382\pi
−0.999994 + 0.00349916i 0.998886π0.998886\pi
422422 0 0
423423 0 0
424424 0 0
425425 3.46410 0.168034
426426 0 0
427427 6.53590i 0.316294i
428428 0 0
429429 0 0
430430 0 0
431431 −21.4641 −1.03389 −0.516945 0.856019i 0.672931π-0.672931\pi
−0.516945 + 0.856019i 0.672931π0.672931\pi
432432 0 0
433433 −19.4641 −0.935385 −0.467693 0.883891i 0.654915π-0.654915\pi
−0.467693 + 0.883891i 0.654915π0.654915\pi
434434 0 0
435435 0 0
436436 0 0
437437 3.32051i 0.158841i
438438 0 0
439439 −40.7846 −1.94654 −0.973272 0.229657i 0.926240π-0.926240\pi
−0.973272 + 0.229657i 0.926240π0.926240\pi
440440 0 0
441441 0 0
442442 0 0
443443 20.9808i 0.996826i 0.866940 + 0.498413i 0.166084π0.166084\pi
−0.866940 + 0.498413i 0.833916π0.833916\pi
444444 0 0
445445 − 8.92820i − 0.423237i
446446 0 0
447447 0 0
448448 0 0
449449 23.3205 1.10056 0.550281 0.834979i 0.314520π-0.314520\pi
0.550281 + 0.834979i 0.314520π0.314520\pi
450450 0 0
451451 2.92820i 0.137884i
452452 0 0
453453 0 0
454454 0 0
455455 −2.53590 −0.118885
456456 0 0
457457 −26.7846 −1.25293 −0.626466 0.779449i 0.715499π-0.715499\pi
−0.626466 + 0.779449i 0.715499π0.715499\pi
458458 0 0
459459 0 0
460460 0 0
461461 10.9282i 0.508977i 0.967076 + 0.254489i 0.0819071π0.0819071\pi
−0.967076 + 0.254489i 0.918093π0.918093\pi
462462 0 0
463463 11.2679 0.523666 0.261833 0.965113i 0.415673π-0.415673\pi
0.261833 + 0.965113i 0.415673π0.415673\pi
464464 0 0
465465 0 0
466466 0 0
467467 − 25.6603i − 1.18741i −0.804681 0.593707i 0.797664π-0.797664\pi
0.804681 0.593707i 0.202336π-0.202336\pi
468468 0 0
469469 7.85641i 0.362775i
470470 0 0
471471 0 0
472472 0 0
473473 10.5359 0.484441
474474 0 0
475475 0.535898i 0.0245887i
476476 0 0
477477 0 0
478478 0 0
479479 5.85641 0.267586 0.133793 0.991009i 0.457284π-0.457284\pi
0.133793 + 0.991009i 0.457284π0.457284\pi
480480 0 0
481481 −6.92820 −0.315899
482482 0 0
483483 0 0
484484 0 0
485485 − 14.3923i − 0.653521i
486486 0 0
487487 −6.58846 −0.298551 −0.149276 0.988796i 0.547694π-0.547694\pi
−0.149276 + 0.988796i 0.547694π0.547694\pi
488488 0 0
489489 0 0
490490 0 0
491491 − 16.9282i − 0.763959i −0.924171 0.381980i 0.875242π-0.875242\pi
0.924171 0.381980i 0.124758π-0.124758\pi
492492 0 0
493493 24.0000i 1.08091i
494494 0 0
495495 0 0
496496 0 0
497497 −4.00000 −0.179425
498498 0 0
499499 − 31.4641i − 1.40853i −0.709939 0.704263i 0.751277π-0.751277\pi
0.709939 0.704263i 0.248723π-0.248723\pi
500500 0 0
501501 0 0
502502 0 0
503503 −0.339746 −0.0151485 −0.00757426 0.999971i 0.502411π-0.502411\pi
−0.00757426 + 0.999971i 0.502411π0.502411\pi
504504 0 0
505505 2.92820 0.130303
506506 0 0
507507 0 0
508508 0 0
509509 1.85641i 0.0822838i 0.999153 + 0.0411419i 0.0130996π0.0130996\pi
−0.999153 + 0.0411419i 0.986900π0.986900\pi
510510 0 0
511511 −5.46410 −0.241718
512512 0 0
513513 0 0
514514 0 0
515515 − 15.6603i − 0.690073i
516516 0 0
517517 − 6.53590i − 0.287448i
518518 0 0
519519 0 0
520520 0 0
521521 43.8564 1.92138 0.960692 0.277616i 0.0895444π-0.0895444\pi
0.960692 + 0.277616i 0.0895444π0.0895444\pi
522522 0 0
523523 11.8038i 0.516146i 0.966125 + 0.258073i 0.0830875π0.0830875\pi
−0.966125 + 0.258073i 0.916912π0.916912\pi
524524 0 0
525525 0 0
526526 0 0
527527 −18.9282 −0.824525
528528 0 0
529529 15.3923 0.669231
530530 0 0
531531 0 0
532532 0 0
533533 5.07180i 0.219684i
534534 0 0
535535 −2.73205 −0.118117
536536 0 0
537537 0 0
538538 0 0
539539 12.9282i 0.556857i
540540 0 0
541541 − 26.9282i − 1.15773i −0.815422 0.578867i 0.803495π-0.803495\pi
0.815422 0.578867i 0.196505π-0.196505\pi
542542 0 0
543543 0 0
544544 0 0
545545 −16.9282 −0.725125
546546 0 0
547547 − 33.2679i − 1.42243i −0.702972 0.711217i 0.748144π-0.748144\pi
0.702972 0.711217i 0.251856π-0.251856\pi
548548 0 0
549549 0 0
550550 0 0
551551 −3.71281 −0.158171
552552 0 0
553553 −0.784610 −0.0333650
554554 0 0
555555 0 0
556556 0 0
557557 − 14.7846i − 0.626444i −0.949680 0.313222i 0.898592π-0.898592\pi
0.949680 0.313222i 0.101408π-0.101408\pi
558558 0 0
559559 18.2487 0.771838
560560 0 0
561561 0 0
562562 0 0
563563 − 22.0526i − 0.929405i −0.885467 0.464702i 0.846161π-0.846161\pi
0.885467 0.464702i 0.153839π-0.153839\pi
564564 0 0
565565 12.9282i 0.543894i
566566 0 0
567567 0 0
568568 0 0
569569 13.4641 0.564445 0.282222 0.959349i 0.408928π-0.408928\pi
0.282222 + 0.959349i 0.408928π0.408928\pi
570570 0 0
571571 − 6.78461i − 0.283927i −0.989872 0.141964i 0.954658π-0.954658\pi
0.989872 0.141964i 0.0453416π-0.0453416\pi
572572 0 0
573573 0 0
574574 0 0
575575 6.19615 0.258397
576576 0 0
577577 39.5692 1.64729 0.823644 0.567107i 0.191937π-0.191937\pi
0.823644 + 0.567107i 0.191937π0.191937\pi
578578 0 0
579579 0 0
580580 0 0
581581 − 0.928203i − 0.0385084i
582582 0 0
583583 −22.9282 −0.949589
584584 0 0
585585 0 0
586586 0 0
587587 − 3.80385i − 0.157002i −0.996914 0.0785008i 0.974987π-0.974987\pi
0.996914 0.0785008i 0.0250133π-0.0250133\pi
588588 0 0
589589 − 2.92820i − 0.120655i
590590 0 0
591591 0 0
592592 0 0
593593 −32.6410 −1.34041 −0.670203 0.742178i 0.733793π-0.733793\pi
−0.670203 + 0.742178i 0.733793π0.733793\pi
594594 0 0
595595 2.53590i 0.103962i
596596 0 0
597597 0 0
598598 0 0
599599 −34.6410 −1.41539 −0.707697 0.706516i 0.750266π-0.750266\pi
−0.707697 + 0.706516i 0.750266π0.750266\pi
600600 0 0
601601 18.5359 0.756095 0.378048 0.925786i 0.376596π-0.376596\pi
0.378048 + 0.925786i 0.376596π0.376596\pi
602602 0 0
603603 0 0
604604 0 0
605605 7.00000i 0.284590i
606606 0 0
607607 −30.9808 −1.25747 −0.628735 0.777619i 0.716427π-0.716427\pi
−0.628735 + 0.777619i 0.716427π0.716427\pi
608608 0 0
609609 0 0
610610 0 0
611611 − 11.3205i − 0.457979i
612612 0 0
613613 26.3923i 1.06598i 0.846123 + 0.532988i 0.178931π0.178931\pi
−0.846123 + 0.532988i 0.821069π0.821069\pi
614614 0 0
615615 0 0
616616 0 0
617617 20.5359 0.826744 0.413372 0.910562i 0.364351π-0.364351\pi
0.413372 + 0.910562i 0.364351π0.364351\pi
618618 0 0
619619 − 1.32051i − 0.0530757i −0.999648 0.0265379i 0.991552π-0.991552\pi
0.999648 0.0265379i 0.00844825π-0.00844825\pi
620620 0 0
621621 0 0
622622 0 0
623623 6.53590 0.261855
624624 0 0
625625 1.00000 0.0400000
626626 0 0
627627 0 0
628628 0 0
629629 6.92820i 0.276246i
630630 0 0
631631 23.3205 0.928375 0.464187 0.885737i 0.346346π-0.346346\pi
0.464187 + 0.885737i 0.346346π0.346346\pi
632632 0 0
633633 0 0
634634 0 0
635635 − 16.7321i − 0.663991i
636636 0 0
637637 22.3923i 0.887215i
638638 0 0
639639 0 0
640640 0 0
641641 −0.392305 −0.0154951 −0.00774755 0.999970i 0.502466π-0.502466\pi
−0.00774755 + 0.999970i 0.502466π0.502466\pi
642642 0 0
643643 − 39.1244i − 1.54291i −0.636281 0.771457i 0.719528π-0.719528\pi
0.636281 0.771457i 0.280472π-0.280472\pi
644644 0 0
645645 0 0
646646 0 0
647647 16.7321 0.657805 0.328902 0.944364i 0.393321π-0.393321\pi
0.328902 + 0.944364i 0.393321π0.393321\pi
648648 0 0
649649 −14.9282 −0.585983
650650 0 0
651651 0 0
652652 0 0
653653 − 12.2487i − 0.479329i −0.970856 0.239665i 0.922963π-0.922963\pi
0.970856 0.239665i 0.0770375π-0.0770375\pi
654654 0 0
655655 19.8564 0.775854
656656 0 0
657657 0 0
658658 0 0
659659 − 17.3205i − 0.674711i −0.941377 0.337356i 0.890468π-0.890468\pi
0.941377 0.337356i 0.109532π-0.109532\pi
660660 0 0
661661 − 8.14359i − 0.316749i −0.987379 0.158375i 0.949375π-0.949375\pi
0.987379 0.158375i 0.0506253π-0.0506253\pi
662662 0 0
663663 0 0
664664 0 0
665665 −0.392305 −0.0152129
666666 0 0
667667 42.9282i 1.66219i
668668 0 0
669669 0 0
670670 0 0
671671 −17.8564 −0.689339
672672 0 0
673673 12.5359 0.483223 0.241612 0.970373i 0.422324π-0.422324\pi
0.241612 + 0.970373i 0.422324π0.422324\pi
674674 0 0
675675 0 0
676676 0 0
677677 − 17.6077i − 0.676719i −0.941017 0.338359i 0.890128π-0.890128\pi
0.941017 0.338359i 0.109872π-0.109872\pi
678678 0 0
679679 10.5359 0.404331
680680 0 0
681681 0 0
682682 0 0
683683 − 16.9808i − 0.649751i −0.945757 0.324875i 0.894678π-0.894678\pi
0.945757 0.324875i 0.105322π-0.105322\pi
684684 0 0
685685 − 4.92820i − 0.188297i
686686 0 0
687687 0 0
688688 0 0
689689 −39.7128 −1.51294
690690 0 0
691691 − 18.0000i − 0.684752i −0.939563 0.342376i 0.888768π-0.888768\pi
0.939563 0.342376i 0.111232π-0.111232\pi
692692 0 0
693693 0 0
694694 0 0
695695 −0.535898 −0.0203278
696696 0 0
697697 5.07180 0.192108
698698 0 0
699699 0 0
700700 0 0
701701 − 19.0718i − 0.720332i −0.932888 0.360166i 0.882720π-0.882720\pi
0.932888 0.360166i 0.117280π-0.117280\pi
702702 0 0
703703 −1.07180 −0.0404236
704704 0 0
705705 0 0
706706 0 0
707707 2.14359i 0.0806181i
708708 0 0
709709 − 12.7846i − 0.480136i −0.970756 0.240068i 0.922830π-0.922830\pi
0.970756 0.240068i 0.0771698π-0.0771698\pi
710710 0 0
711711 0 0
712712 0 0
713713 −33.8564 −1.26793
714714 0 0
715715 − 6.92820i − 0.259100i
716716 0 0
717717 0 0
718718 0 0
719719 −1.85641 −0.0692323 −0.0346161 0.999401i 0.511021π-0.511021\pi
−0.0346161 + 0.999401i 0.511021π0.511021\pi
720720 0 0
721721 11.4641 0.426945
722722 0 0
723723 0 0
724724 0 0
725725 6.92820i 0.257307i
726726 0 0
727727 24.0526 0.892060 0.446030 0.895018i 0.352837π-0.352837\pi
0.446030 + 0.895018i 0.352837π0.352837\pi
728728 0 0
729729 0 0
730730 0 0
731731 − 18.2487i − 0.674953i
732732 0 0
733733 − 35.0718i − 1.29541i −0.761893 0.647703i 0.775730π-0.775730\pi
0.761893 0.647703i 0.224270π-0.224270\pi
734734 0 0
735735 0 0
736736 0 0
737737 −21.4641 −0.790640
738738 0 0
739739 − 29.3205i − 1.07857i −0.842123 0.539286i 0.818694π-0.818694\pi
0.842123 0.539286i 0.181306π-0.181306\pi
740740 0 0
741741 0 0
742742 0 0
743743 10.9808 0.402845 0.201423 0.979504i 0.435444π-0.435444\pi
0.201423 + 0.979504i 0.435444π0.435444\pi
744744 0 0
745745 −7.85641 −0.287836
746746 0 0
747747 0 0
748748 0 0
749749 − 2.00000i − 0.0730784i
750750 0 0
751751 −26.2487 −0.957829 −0.478915 0.877862i 0.658970π-0.658970\pi
−0.478915 + 0.877862i 0.658970π0.658970\pi
752752 0 0
753753 0 0
754754 0 0
755755 12.3923i 0.451002i
756756 0 0
757757 − 19.0718i − 0.693176i −0.938017 0.346588i 0.887340π-0.887340\pi
0.938017 0.346588i 0.112660π-0.112660\pi
758758 0 0
759759 0 0
760760 0 0
761761 5.71281 0.207089 0.103545 0.994625i 0.466982π-0.466982\pi
0.103545 + 0.994625i 0.466982π0.466982\pi
762762 0 0
763763 − 12.3923i − 0.448632i
764764 0 0
765765 0 0
766766 0 0
767767 −25.8564 −0.933621
768768 0 0
769769 12.9282 0.466203 0.233101 0.972452i 0.425113π-0.425113\pi
0.233101 + 0.972452i 0.425113π0.425113\pi
770770 0 0
771771 0 0
772772 0 0
773773 22.3923i 0.805395i 0.915333 + 0.402698i 0.131927π0.131927\pi
−0.915333 + 0.402698i 0.868073π0.868073\pi
774774 0 0
775775 −5.46410 −0.196276
776776 0 0
777777 0 0
778778 0 0
779779 0.784610i 0.0281116i
780780 0 0
781781 − 10.9282i − 0.391042i
782782 0 0
783783 0 0
784784 0 0
785785 3.07180 0.109637
786786 0 0
787787 − 16.5885i − 0.591315i −0.955294 0.295657i 0.904461π-0.904461\pi
0.955294 0.295657i 0.0955387π-0.0955387\pi
788788 0 0
789789 0 0
790790 0 0
791791 −9.46410 −0.336505
792792 0 0
793793 −30.9282 −1.09829
794794 0 0
795795 0 0
796796 0 0
797797 − 50.1051i − 1.77481i −0.460986 0.887407i 0.652504π-0.652504\pi
0.460986 0.887407i 0.347496π-0.347496\pi
798798 0 0
799799 −11.3205 −0.400491
800800 0 0
801801 0 0
802802 0 0
803803 − 14.9282i − 0.526805i
804804 0 0
805805 4.53590i 0.159869i
806806 0 0
807807 0 0
808808 0 0
809809 −23.8564 −0.838747 −0.419373 0.907814i 0.637750π-0.637750\pi
−0.419373 + 0.907814i 0.637750π0.637750\pi
810810 0 0
811811 28.9282i 1.01581i 0.861414 + 0.507903i 0.169579π0.169579\pi
−0.861414 + 0.507903i 0.830421π0.830421\pi
812812 0 0
813813 0 0
814814 0 0
815815 −0.196152 −0.00687092
816816 0 0
817817 2.82309 0.0987673
818818 0 0
819819 0 0
820820 0 0
821821 − 34.7846i − 1.21399i −0.794705 0.606996i 0.792375π-0.792375\pi
0.794705 0.606996i 0.207625π-0.207625\pi
822822 0 0
823823 9.12436 0.318055 0.159028 0.987274i 0.449164π-0.449164\pi
0.159028 + 0.987274i 0.449164π0.449164\pi
824824 0 0
825825 0 0
826826 0 0
827827 23.1244i 0.804113i 0.915615 + 0.402056i 0.131704π0.131704\pi
−0.915615 + 0.402056i 0.868296π0.868296\pi
828828 0 0
829829 28.9282i 1.00472i 0.864659 + 0.502359i 0.167534π0.167534\pi
−0.864659 + 0.502359i 0.832466π0.832466\pi
830830 0 0
831831 0 0
832832 0 0
833833 22.3923 0.775847
834834 0 0
835835 − 9.80385i − 0.339276i
836836 0 0
837837 0 0
838838 0 0
839839 24.7846 0.855660 0.427830 0.903859i 0.359278π-0.359278\pi
0.427830 + 0.903859i 0.359278π0.359278\pi
840840 0 0
841841 −19.0000 −0.655172
842842 0 0
843843 0 0
844844 0 0
845845 1.00000i 0.0344010i
846846 0 0
847847 −5.12436 −0.176075
848848 0 0
849849 0 0
850850 0 0
851851 12.3923i 0.424803i
852852 0 0
853853 − 21.6077i − 0.739833i −0.929065 0.369917i 0.879386π-0.879386\pi
0.929065 0.369917i 0.120614π-0.120614\pi
854854 0 0
855855 0 0
856856 0 0
857857 19.8564 0.678282 0.339141 0.940736i 0.389864π-0.389864\pi
0.339141 + 0.940736i 0.389864π0.389864\pi
858858 0 0
859859 − 28.2487i − 0.963834i −0.876217 0.481917i 0.839941π-0.839941\pi
0.876217 0.481917i 0.160059π-0.160059\pi
860860 0 0
861861 0 0
862862 0 0
863863 47.6603 1.62237 0.811187 0.584787i 0.198822π-0.198822\pi
0.811187 + 0.584787i 0.198822π0.198822\pi
864864 0 0
865865 2.00000 0.0680020
866866 0 0
867867 0 0
868868 0 0
869869 − 2.14359i − 0.0727164i
870870 0 0
871871 −37.1769 −1.25969
872872 0 0
873873 0 0
874874 0 0
875875 0.732051i 0.0247478i
876876 0 0
877877 − 1.71281i − 0.0578376i −0.999582 0.0289188i 0.990794π-0.990794\pi
0.999582 0.0289188i 0.00920642π-0.00920642\pi
878878 0 0
879879 0 0
880880 0 0
881881 −9.46410 −0.318854 −0.159427 0.987210i 0.550965π-0.550965\pi
−0.159427 + 0.987210i 0.550965π0.550965\pi
882882 0 0
883883 27.9090i 0.939211i 0.882876 + 0.469606i 0.155604π0.155604\pi
−0.882876 + 0.469606i 0.844396π0.844396\pi
884884 0 0
885885 0 0
886886 0 0
887887 −13.9090 −0.467017 −0.233509 0.972355i 0.575021π-0.575021\pi
−0.233509 + 0.972355i 0.575021π0.575021\pi
888888 0 0
889889 12.2487 0.410809
890890 0 0
891891 0 0
892892 0 0
893893 − 1.75129i − 0.0586046i
894894 0 0
895895 8.53590 0.285324
896896 0 0
897897 0 0
898898 0 0
899899 − 37.8564i − 1.26258i
900900 0 0
901901 39.7128i 1.32303i
902902 0 0
903903 0 0
904904 0 0
905905 −16.0000 −0.531858
906906 0 0
907907 − 4.87564i − 0.161893i −0.996718 0.0809466i 0.974206π-0.974206\pi
0.996718 0.0809466i 0.0257943π-0.0257943\pi
908908 0 0
909909 0 0
910910 0 0
911911 −49.1769 −1.62930 −0.814652 0.579950i 0.803072π-0.803072\pi
−0.814652 + 0.579950i 0.803072π0.803072\pi
912912 0 0
913913 2.53590 0.0839260
914914 0 0
915915 0 0
916916 0 0
917917 14.5359i 0.480018i
918918 0 0
919919 38.9282 1.28412 0.642061 0.766653i 0.278079π-0.278079\pi
0.642061 + 0.766653i 0.278079π0.278079\pi
920920 0 0
921921 0 0
922922 0 0
923923 − 18.9282i − 0.623029i
924924 0 0
925925 2.00000i 0.0657596i
926926 0 0
927927 0 0
928928 0 0
929929 17.4641 0.572979 0.286489 0.958083i 0.407512π-0.407512\pi
0.286489 + 0.958083i 0.407512π0.407512\pi
930930 0 0
931931 3.46410i 0.113531i
932932 0 0
933933 0 0
934934 0 0
935935 −6.92820 −0.226576
936936 0 0
937937 4.24871 0.138799 0.0693997 0.997589i 0.477892π-0.477892\pi
0.0693997 + 0.997589i 0.477892π0.477892\pi
938938 0 0
939939 0 0
940940 0 0
941941 32.0000i 1.04317i 0.853199 + 0.521585i 0.174659π0.174659\pi
−0.853199 + 0.521585i 0.825341π0.825341\pi
942942 0 0
943943 9.07180 0.295418
944944 0 0
945945 0 0
946946 0 0
947947 − 3.12436i − 0.101528i −0.998711 0.0507640i 0.983834π-0.983834\pi
0.998711 0.0507640i 0.0161656π-0.0161656\pi
948948 0 0
949949 − 25.8564i − 0.839334i
950950 0 0
951951 0 0
952952 0 0
953953 17.2154 0.557661 0.278831 0.960340i 0.410053π-0.410053\pi
0.278831 + 0.960340i 0.410053π0.410053\pi
954954 0 0
955955 15.3205i 0.495760i
956956 0 0
957957 0 0
958958 0 0
959959 3.60770 0.116499
960960 0 0
961961 −1.14359 −0.0368901
962962 0 0
963963 0 0
964964 0 0
965965 0.535898i 0.0172512i
966966 0 0
967967 −16.3397 −0.525451 −0.262725 0.964871i 0.584621π-0.584621\pi
−0.262725 + 0.964871i 0.584621π0.584621\pi
968968 0 0
969969 0 0
970970 0 0
971971 36.9282i 1.18508i 0.805540 + 0.592541i 0.201875π0.201875\pi
−0.805540 + 0.592541i 0.798125π0.798125\pi
972972 0 0
973973 − 0.392305i − 0.0125767i
974974 0 0
975975 0 0
976976 0 0
977977 24.5359 0.784973 0.392486 0.919758i 0.371615π-0.371615\pi
0.392486 + 0.919758i 0.371615π0.371615\pi
978978 0 0
979979 17.8564i 0.570693i
980980 0 0
981981 0 0
982982 0 0
983983 −48.7321 −1.55431 −0.777156 0.629309i 0.783338π-0.783338\pi
−0.777156 + 0.629309i 0.783338π0.783338\pi
984984 0 0
985985 −19.4641 −0.620178
986986 0 0
987987 0 0
988988 0 0
989989 − 32.6410i − 1.03792i
990990 0 0
991991 −41.4641 −1.31715 −0.658575 0.752515i 0.728841π-0.728841\pi
−0.658575 + 0.752515i 0.728841π0.728841\pi
992992 0 0
993993 0 0
994994 0 0
995995 1.85641i 0.0588520i
996996 0 0
997997 − 11.1769i − 0.353976i −0.984213 0.176988i 0.943365π-0.943365\pi
0.984213 0.176988i 0.0566354π-0.0566354\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 1440.2.k.e.721.3 4
3.2 odd 2 160.2.d.a.81.1 4
4.3 odd 2 360.2.k.e.181.4 4
5.2 odd 4 7200.2.d.o.2449.2 4
5.3 odd 4 7200.2.d.n.2449.3 4
5.4 even 2 7200.2.k.j.3601.3 4
8.3 odd 2 360.2.k.e.181.3 4
8.5 even 2 inner 1440.2.k.e.721.1 4
12.11 even 2 40.2.d.a.21.1 4
15.2 even 4 800.2.f.c.49.1 4
15.8 even 4 800.2.f.e.49.4 4
15.14 odd 2 800.2.d.e.401.4 4
20.3 even 4 1800.2.d.l.1549.3 4
20.7 even 4 1800.2.d.p.1549.2 4
20.19 odd 2 1800.2.k.j.901.1 4
24.5 odd 2 160.2.d.a.81.4 4
24.11 even 2 40.2.d.a.21.2 yes 4
40.3 even 4 1800.2.d.p.1549.1 4
40.13 odd 4 7200.2.d.o.2449.3 4
40.19 odd 2 1800.2.k.j.901.2 4
40.27 even 4 1800.2.d.l.1549.4 4
40.29 even 2 7200.2.k.j.3601.4 4
40.37 odd 4 7200.2.d.n.2449.2 4
48.5 odd 4 1280.2.a.d.1.1 2
48.11 even 4 1280.2.a.o.1.2 2
48.29 odd 4 1280.2.a.n.1.2 2
48.35 even 4 1280.2.a.a.1.1 2
60.23 odd 4 200.2.f.e.149.2 4
60.47 odd 4 200.2.f.c.149.3 4
60.59 even 2 200.2.d.f.101.4 4
120.29 odd 2 800.2.d.e.401.1 4
120.53 even 4 800.2.f.c.49.2 4
120.59 even 2 200.2.d.f.101.3 4
120.77 even 4 800.2.f.e.49.3 4
120.83 odd 4 200.2.f.c.149.4 4
120.107 odd 4 200.2.f.e.149.1 4
240.29 odd 4 6400.2.a.be.1.1 2
240.59 even 4 6400.2.a.z.1.1 2
240.149 odd 4 6400.2.a.cj.1.2 2
240.179 even 4 6400.2.a.ce.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.d.a.21.1 4 12.11 even 2
40.2.d.a.21.2 yes 4 24.11 even 2
160.2.d.a.81.1 4 3.2 odd 2
160.2.d.a.81.4 4 24.5 odd 2
200.2.d.f.101.3 4 120.59 even 2
200.2.d.f.101.4 4 60.59 even 2
200.2.f.c.149.3 4 60.47 odd 4
200.2.f.c.149.4 4 120.83 odd 4
200.2.f.e.149.1 4 120.107 odd 4
200.2.f.e.149.2 4 60.23 odd 4
360.2.k.e.181.3 4 8.3 odd 2
360.2.k.e.181.4 4 4.3 odd 2
800.2.d.e.401.1 4 120.29 odd 2
800.2.d.e.401.4 4 15.14 odd 2
800.2.f.c.49.1 4 15.2 even 4
800.2.f.c.49.2 4 120.53 even 4
800.2.f.e.49.3 4 120.77 even 4
800.2.f.e.49.4 4 15.8 even 4
1280.2.a.a.1.1 2 48.35 even 4
1280.2.a.d.1.1 2 48.5 odd 4
1280.2.a.n.1.2 2 48.29 odd 4
1280.2.a.o.1.2 2 48.11 even 4
1440.2.k.e.721.1 4 8.5 even 2 inner
1440.2.k.e.721.3 4 1.1 even 1 trivial
1800.2.d.l.1549.3 4 20.3 even 4
1800.2.d.l.1549.4 4 40.27 even 4
1800.2.d.p.1549.1 4 40.3 even 4
1800.2.d.p.1549.2 4 20.7 even 4
1800.2.k.j.901.1 4 20.19 odd 2
1800.2.k.j.901.2 4 40.19 odd 2
6400.2.a.z.1.1 2 240.59 even 4
6400.2.a.be.1.1 2 240.29 odd 4
6400.2.a.ce.1.2 2 240.179 even 4
6400.2.a.cj.1.2 2 240.149 odd 4
7200.2.d.n.2449.2 4 40.37 odd 4
7200.2.d.n.2449.3 4 5.3 odd 4
7200.2.d.o.2449.2 4 5.2 odd 4
7200.2.d.o.2449.3 4 40.13 odd 4
7200.2.k.j.3601.3 4 5.4 even 2
7200.2.k.j.3601.4 4 40.29 even 2