Properties

Label 3840.2.d.bl.2689.4
Level 38403840
Weight 22
Character 3840.2689
Analytic conductor 30.66330.663
Analytic rank 00
Dimension 66
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3840,2,Mod(2689,3840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3840.2689");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 3840=2835 3840 = 2^{8} \cdot 3 \cdot 5
Weight: k k == 2 2
Character orbit: [χ][\chi] == 3840.d (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: 30.662554376230.6625543762
Analytic rank: 00
Dimension: 66
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x62x5+2x4+2x3+4x24x+2 x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 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 1920)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 2689.4
Root 0.8546380.854638i-0.854638 - 0.854638i of defining polynomial
Character χ\chi == 3840.2689
Dual form 3840.2.d.bl.2689.3

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+1.00000q3+(0.539189+2.17009i)q5+2.34017iq7+1.00000q93.07838iq11+0.921622q13+(0.539189+2.17009i)q157.75872iq174.00000iq19+2.34017iq212.15676iq23+(4.418552.34017i)q25+1.00000q276.49693iq292.00000q313.07838iq33+(5.078381.26180i)q35+3.07838q37+0.921622q3910.6803q41+2.15676q43+(0.539189+2.17009i)q458.68035iq47+1.52359q497.75872iq512.92162q53+(6.68035+1.65983i)q554.00000iq57+11.7587iq596.00000iq61+2.34017iq63+(0.496928+2.00000i)q656.83710q672.15676iq69+11.5174q716.52359iq73+(4.418552.34017i)q75+7.20394q7715.3607q79+1.00000q81+1.84324q83+(16.8371+4.18342i)q856.49693iq87+6.00000q89+2.15676iq912.00000q93+(8.68035+2.15676i)q953.07838iq99+O(q100)q+1.00000 q^{3} +(-0.539189 + 2.17009i) q^{5} +2.34017i q^{7} +1.00000 q^{9} -3.07838i q^{11} +0.921622 q^{13} +(-0.539189 + 2.17009i) q^{15} -7.75872i q^{17} -4.00000i q^{19} +2.34017i q^{21} -2.15676i q^{23} +(-4.41855 - 2.34017i) q^{25} +1.00000 q^{27} -6.49693i q^{29} -2.00000 q^{31} -3.07838i q^{33} +(-5.07838 - 1.26180i) q^{35} +3.07838 q^{37} +0.921622 q^{39} -10.6803 q^{41} +2.15676 q^{43} +(-0.539189 + 2.17009i) q^{45} -8.68035i q^{47} +1.52359 q^{49} -7.75872i q^{51} -2.92162 q^{53} +(6.68035 + 1.65983i) q^{55} -4.00000i q^{57} +11.7587i q^{59} -6.00000i q^{61} +2.34017i q^{63} +(-0.496928 + 2.00000i) q^{65} -6.83710 q^{67} -2.15676i q^{69} +11.5174 q^{71} -6.52359i q^{73} +(-4.41855 - 2.34017i) q^{75} +7.20394 q^{77} -15.3607 q^{79} +1.00000 q^{81} +1.84324 q^{83} +(16.8371 + 4.18342i) q^{85} -6.49693i q^{87} +6.00000 q^{89} +2.15676i q^{91} -2.00000 q^{93} +(8.68035 + 2.15676i) q^{95} -3.07838i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q+6q3+6q9+12q13+2q25+6q2712q3124q35+12q37+12q3920q4122q4924q534q55+32q65+16q6732q71+2q7532q77++8q95+O(q100) 6 q + 6 q^{3} + 6 q^{9} + 12 q^{13} + 2 q^{25} + 6 q^{27} - 12 q^{31} - 24 q^{35} + 12 q^{37} + 12 q^{39} - 20 q^{41} - 22 q^{49} - 24 q^{53} - 4 q^{55} + 32 q^{65} + 16 q^{67} - 32 q^{71} + 2 q^{75} - 32 q^{77}+ \cdots + 8 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/3840Z)×\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times.

nn 511511 15371537 25612561 28212821
χ(n)\chi(n) 11 1-1 11 1-1

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 1.00000 0.577350
44 0 0
55 −0.539189 + 2.17009i −0.241133 + 0.970492i
66 0 0
77 2.34017i 0.884502i 0.896891 + 0.442251i 0.145820π0.145820\pi
−0.896891 + 0.442251i 0.854180π0.854180\pi
88 0 0
99 1.00000 0.333333
1010 0 0
1111 3.07838i 0.928166i −0.885792 0.464083i 0.846384π-0.846384\pi
0.885792 0.464083i 0.153616π-0.153616\pi
1212 0 0
1313 0.921622 0.255612 0.127806 0.991799i 0.459207π-0.459207\pi
0.127806 + 0.991799i 0.459207π0.459207\pi
1414 0 0
1515 −0.539189 + 2.17009i −0.139218 + 0.560314i
1616 0 0
1717 7.75872i 1.88177i −0.338730 0.940883i 0.609997π-0.609997\pi
0.338730 0.940883i 0.390003π-0.390003\pi
1818 0 0
1919 4.00000i 0.917663i −0.888523 0.458831i 0.848268π-0.848268\pi
0.888523 0.458831i 0.151732π-0.151732\pi
2020 0 0
2121 2.34017i 0.510668i
2222 0 0
2323 2.15676i 0.449715i −0.974392 0.224857i 0.927808π-0.927808\pi
0.974392 0.224857i 0.0721916π-0.0721916\pi
2424 0 0
2525 −4.41855 2.34017i −0.883710 0.468035i
2626 0 0
2727 1.00000 0.192450
2828 0 0
2929 6.49693i 1.20645i −0.797571 0.603225i 0.793882π-0.793882\pi
0.797571 0.603225i 0.206118π-0.206118\pi
3030 0 0
3131 −2.00000 −0.359211 −0.179605 0.983739i 0.557482π-0.557482\pi
−0.179605 + 0.983739i 0.557482π0.557482\pi
3232 0 0
3333 3.07838i 0.535877i
3434 0 0
3535 −5.07838 1.26180i −0.858403 0.213282i
3636 0 0
3737 3.07838 0.506082 0.253041 0.967456i 0.418569π-0.418569\pi
0.253041 + 0.967456i 0.418569π0.418569\pi
3838 0 0
3939 0.921622 0.147578
4040 0 0
4141 −10.6803 −1.66799 −0.833995 0.551772i 0.813952π-0.813952\pi
−0.833995 + 0.551772i 0.813952π0.813952\pi
4242 0 0
4343 2.15676 0.328902 0.164451 0.986385i 0.447415π-0.447415\pi
0.164451 + 0.986385i 0.447415π0.447415\pi
4444 0 0
4545 −0.539189 + 2.17009i −0.0803775 + 0.323497i
4646 0 0
4747 8.68035i 1.26616i −0.774087 0.633079i 0.781791π-0.781791\pi
0.774087 0.633079i 0.218209π-0.218209\pi
4848 0 0
4949 1.52359 0.217656
5050 0 0
5151 7.75872i 1.08644i
5252 0 0
5353 −2.92162 −0.401316 −0.200658 0.979661i 0.564308π-0.564308\pi
−0.200658 + 0.979661i 0.564308π0.564308\pi
5454 0 0
5555 6.68035 + 1.65983i 0.900778 + 0.223811i
5656 0 0
5757 4.00000i 0.529813i
5858 0 0
5959 11.7587i 1.53086i 0.643522 + 0.765428i 0.277473π0.277473\pi
−0.643522 + 0.765428i 0.722527π0.722527\pi
6060 0 0
6161 6.00000i 0.768221i −0.923287 0.384111i 0.874508π-0.874508\pi
0.923287 0.384111i 0.125492π-0.125492\pi
6262 0 0
6363 2.34017i 0.294834i
6464 0 0
6565 −0.496928 + 2.00000i −0.0616364 + 0.248069i
6666 0 0
6767 −6.83710 −0.835285 −0.417642 0.908611i 0.637144π-0.637144\pi
−0.417642 + 0.908611i 0.637144π0.637144\pi
6868 0 0
6969 2.15676i 0.259643i
7070 0 0
7171 11.5174 1.36687 0.683435 0.730012i 0.260485π-0.260485\pi
0.683435 + 0.730012i 0.260485π0.260485\pi
7272 0 0
7373 6.52359i 0.763529i −0.924260 0.381764i 0.875317π-0.875317\pi
0.924260 0.381764i 0.124683π-0.124683\pi
7474 0 0
7575 −4.41855 2.34017i −0.510210 0.270220i
7676 0 0
7777 7.20394 0.820965
7878 0 0
7979 −15.3607 −1.72821 −0.864106 0.503309i 0.832116π-0.832116\pi
−0.864106 + 0.503309i 0.832116π0.832116\pi
8080 0 0
8181 1.00000 0.111111
8282 0 0
8383 1.84324 0.202322 0.101161 0.994870i 0.467744π-0.467744\pi
0.101161 + 0.994870i 0.467744π0.467744\pi
8484 0 0
8585 16.8371 + 4.18342i 1.82624 + 0.453755i
8686 0 0
8787 6.49693i 0.696544i
8888 0 0
8989 6.00000 0.635999 0.317999 0.948091i 0.396989π-0.396989\pi
0.317999 + 0.948091i 0.396989π0.396989\pi
9090 0 0
9191 2.15676i 0.226089i
9292 0 0
9393 −2.00000 −0.207390
9494 0 0
9595 8.68035 + 2.15676i 0.890585 + 0.221278i
9696 0 0
9797 0 0 1.00000 00
−1.00000 π\pi
9898 0 0
9999 3.07838i 0.309389i
100100 0 0
101101 8.34017i 0.829878i 0.909849 + 0.414939i 0.136197π0.136197\pi
−0.909849 + 0.414939i 0.863803π0.863803\pi
102102 0 0
103103 10.3402i 1.01885i −0.860516 0.509424i 0.829859π-0.829859\pi
0.860516 0.509424i 0.170141π-0.170141\pi
104104 0 0
105105 −5.07838 1.26180i −0.495599 0.123139i
106106 0 0
107107 13.3607 1.29163 0.645813 0.763495i 0.276518π-0.276518\pi
0.645813 + 0.763495i 0.276518π0.276518\pi
108108 0 0
109109 5.31965i 0.509530i 0.967003 + 0.254765i 0.0819982π0.0819982\pi
−0.967003 + 0.254765i 0.918002π0.918002\pi
110110 0 0
111111 3.07838 0.292187
112112 0 0
113113 19.7587i 1.85874i −0.369144 0.929372i 0.620349π-0.620349\pi
0.369144 0.929372i 0.379651π-0.379651\pi
114114 0 0
115115 4.68035 + 1.16290i 0.436445 + 0.108441i
116116 0 0
117117 0.921622 0.0852040
118118 0 0
119119 18.1568 1.66443
120120 0 0
121121 1.52359 0.138508
122122 0 0
123123 −10.6803 −0.963014
124124 0 0
125125 7.46081 8.32684i 0.667315 0.744775i
126126 0 0
127127 19.3340i 1.71562i 0.513969 + 0.857809i 0.328175π0.328175\pi
−0.513969 + 0.857809i 0.671825π0.671825\pi
128128 0 0
129129 2.15676 0.189892
130130 0 0
131131 7.44521i 0.650491i −0.945630 0.325246i 0.894553π-0.894553\pi
0.945630 0.325246i 0.105447π-0.105447\pi
132132 0 0
133133 9.36069 0.811675
134134 0 0
135135 −0.539189 + 2.17009i −0.0464060 + 0.186771i
136136 0 0
137137 3.44521i 0.294344i −0.989111 0.147172i 0.952983π-0.952983\pi
0.989111 0.147172i 0.0470171π-0.0470171\pi
138138 0 0
139139 19.2039i 1.62886i 0.580264 + 0.814428i 0.302949π0.302949\pi
−0.580264 + 0.814428i 0.697051π0.697051\pi
140140 0 0
141141 8.68035i 0.731017i
142142 0 0
143143 2.83710i 0.237250i
144144 0 0
145145 14.0989 + 3.50307i 1.17085 + 0.290914i
146146 0 0
147147 1.52359 0.125664
148148 0 0
149149 9.81658i 0.804206i −0.915594 0.402103i 0.868279π-0.868279\pi
0.915594 0.402103i 0.131721π-0.131721\pi
150150 0 0
151151 −23.6742 −1.92658 −0.963290 0.268464i 0.913484π-0.913484\pi
−0.963290 + 0.268464i 0.913484π0.913484\pi
152152 0 0
153153 7.75872i 0.627256i
154154 0 0
155155 1.07838 4.34017i 0.0866174 0.348611i
156156 0 0
157157 11.0784 0.884151 0.442075 0.896978i 0.354242π-0.354242\pi
0.442075 + 0.896978i 0.354242π0.354242\pi
158158 0 0
159159 −2.92162 −0.231700
160160 0 0
161161 5.04718 0.397774
162162 0 0
163163 12.9939 1.01776 0.508879 0.860838i 0.330060π-0.330060\pi
0.508879 + 0.860838i 0.330060π0.330060\pi
164164 0 0
165165 6.68035 + 1.65983i 0.520064 + 0.129217i
166166 0 0
167167 12.0000i 0.928588i 0.885681 + 0.464294i 0.153692π0.153692\pi
−0.885681 + 0.464294i 0.846308π0.846308\pi
168168 0 0
169169 −12.1506 −0.934662
170170 0 0
171171 4.00000i 0.305888i
172172 0 0
173173 18.4391 1.40190 0.700948 0.713212i 0.252760π-0.252760\pi
0.700948 + 0.713212i 0.252760π0.252760\pi
174174 0 0
175175 5.47641 10.3402i 0.413978 0.781644i
176176 0 0
177177 11.7587i 0.883840i
178178 0 0
179179 1.91548i 0.143170i 0.997435 + 0.0715848i 0.0228057π0.0228057\pi
−0.997435 + 0.0715848i 0.977194π0.977194\pi
180180 0 0
181181 6.99386i 0.519849i −0.965629 0.259925i 0.916302π-0.916302\pi
0.965629 0.259925i 0.0836977π-0.0836977\pi
182182 0 0
183183 6.00000i 0.443533i
184184 0 0
185185 −1.65983 + 6.68035i −0.122033 + 0.491149i
186186 0 0
187187 −23.8843 −1.74659
188188 0 0
189189 2.34017i 0.170223i
190190 0 0
191191 −6.15676 −0.445487 −0.222744 0.974877i 0.571501π-0.571501\pi
−0.222744 + 0.974877i 0.571501π0.571501\pi
192192 0 0
193193 14.5236i 1.04543i −0.852507 0.522715i 0.824919π-0.824919\pi
0.852507 0.522715i 0.175081π-0.175081\pi
194194 0 0
195195 −0.496928 + 2.00000i −0.0355858 + 0.143223i
196196 0 0
197197 24.5958 1.75238 0.876190 0.481966i 0.160077π-0.160077\pi
0.876190 + 0.481966i 0.160077π0.160077\pi
198198 0 0
199199 −3.36069 −0.238233 −0.119117 0.992880i 0.538006π-0.538006\pi
−0.119117 + 0.992880i 0.538006π0.538006\pi
200200 0 0
201201 −6.83710 −0.482252
202202 0 0
203203 15.2039 1.06711
204204 0 0
205205 5.75872 23.1773i 0.402207 1.61877i
206206 0 0
207207 2.15676i 0.149905i
208208 0 0
209209 −12.3135 −0.851743
210210 0 0
211211 11.2039i 0.771311i 0.922643 + 0.385655i 0.126025π0.126025\pi
−0.922643 + 0.385655i 0.873975π0.873975\pi
212212 0 0
213213 11.5174 0.789162
214214 0 0
215215 −1.16290 + 4.68035i −0.0793090 + 0.319197i
216216 0 0
217217 4.68035i 0.317723i
218218 0 0
219219 6.52359i 0.440823i
220220 0 0
221221 7.15061i 0.481002i
222222 0 0
223223 6.65368i 0.445564i 0.974868 + 0.222782i 0.0715138π0.0715138\pi
−0.974868 + 0.222782i 0.928486π0.928486\pi
224224 0 0
225225 −4.41855 2.34017i −0.294570 0.156012i
226226 0 0
227227 −9.84324 −0.653319 −0.326660 0.945142i 0.605923π-0.605923\pi
−0.326660 + 0.945142i 0.605923π0.605923\pi
228228 0 0
229229 1.31965i 0.0872052i −0.999049 0.0436026i 0.986116π-0.986116\pi
0.999049 0.0436026i 0.0138835π-0.0138835\pi
230230 0 0
231231 7.20394 0.473984
232232 0 0
233233 2.39803i 0.157100i −0.996910 0.0785501i 0.974971π-0.974971\pi
0.996910 0.0785501i 0.0250291π-0.0250291\pi
234234 0 0
235235 18.8371 + 4.68035i 1.22880 + 0.305312i
236236 0 0
237237 −15.3607 −0.997784
238238 0 0
239239 −6.15676 −0.398247 −0.199124 0.979974i 0.563810π-0.563810\pi
−0.199124 + 0.979974i 0.563810π0.563810\pi
240240 0 0
241241 0.639308 0.0411815 0.0205907 0.999788i 0.493445π-0.493445\pi
0.0205907 + 0.999788i 0.493445π0.493445\pi
242242 0 0
243243 1.00000 0.0641500
244244 0 0
245245 −0.821503 + 3.30632i −0.0524839 + 0.211233i
246246 0 0
247247 3.68649i 0.234566i
248248 0 0
249249 1.84324 0.116811
250250 0 0
251251 12.9216i 0.815606i −0.913070 0.407803i 0.866295π-0.866295\pi
0.913070 0.407803i 0.133705π-0.133705\pi
252252 0 0
253253 −6.63931 −0.417410
254254 0 0
255255 16.8371 + 4.18342i 1.05438 + 0.261976i
256256 0 0
257257 14.9627i 0.933345i −0.884430 0.466673i 0.845453π-0.845453\pi
0.884430 0.466673i 0.154547π-0.154547\pi
258258 0 0
259259 7.20394i 0.447631i
260260 0 0
261261 6.49693i 0.402150i
262262 0 0
263263 13.3607i 0.823856i 0.911216 + 0.411928i 0.135144π0.135144\pi
−0.911216 + 0.411928i 0.864856π0.864856\pi
264264 0 0
265265 1.57531 6.34017i 0.0967703 0.389474i
266266 0 0
267267 6.00000 0.367194
268268 0 0
269269 27.5441i 1.67939i 0.543055 + 0.839697i 0.317267π0.317267\pi
−0.543055 + 0.839697i 0.682733π0.682733\pi
270270 0 0
271271 −10.0000 −0.607457 −0.303728 0.952759i 0.598232π-0.598232\pi
−0.303728 + 0.952759i 0.598232π0.598232\pi
272272 0 0
273273 2.15676i 0.130533i
274274 0 0
275275 −7.20394 + 13.6020i −0.434414 + 0.820230i
276276 0 0
277277 4.12556 0.247881 0.123940 0.992290i 0.460447π-0.460447\pi
0.123940 + 0.992290i 0.460447π0.460447\pi
278278 0 0
279279 −2.00000 −0.119737
280280 0 0
281281 11.3607 0.677722 0.338861 0.940836i 0.389958π-0.389958\pi
0.338861 + 0.940836i 0.389958π0.389958\pi
282282 0 0
283283 −10.5236 −0.625563 −0.312781 0.949825i 0.601261π-0.601261\pi
−0.312781 + 0.949825i 0.601261π0.601261\pi
284284 0 0
285285 8.68035 + 2.15676i 0.514179 + 0.127755i
286286 0 0
287287 24.9939i 1.47534i
288288 0 0
289289 −43.1978 −2.54105
290290 0 0
291291 0 0
292292 0 0
293293 −32.5958 −1.90427 −0.952134 0.305680i 0.901116π-0.901116\pi
−0.952134 + 0.305680i 0.901116π0.901116\pi
294294 0 0
295295 −25.5174 6.34017i −1.48568 0.369139i
296296 0 0
297297 3.07838i 0.178626i
298298 0 0
299299 1.98771i 0.114952i
300300 0 0
301301 5.04718i 0.290915i
302302 0 0
303303 8.34017i 0.479130i
304304 0 0
305305 13.0205 + 3.23513i 0.745553 + 0.185243i
306306 0 0
307307 29.9877 1.71149 0.855745 0.517398i 0.173099π-0.173099\pi
0.855745 + 0.517398i 0.173099π0.173099\pi
308308 0 0
309309 10.3402i 0.588232i
310310 0 0
311311 −27.2039 −1.54259 −0.771297 0.636476i 0.780392π-0.780392\pi
−0.771297 + 0.636476i 0.780392π0.780392\pi
312312 0 0
313313 26.8371i 1.51692i −0.651718 0.758461i 0.725951π-0.725951\pi
0.651718 0.758461i 0.274049π-0.274049\pi
314314 0 0
315315 −5.07838 1.26180i −0.286134 0.0710941i
316316 0 0
317317 24.5958 1.38144 0.690720 0.723123i 0.257294π-0.257294\pi
0.690720 + 0.723123i 0.257294π0.257294\pi
318318 0 0
319319 −20.0000 −1.11979
320320 0 0
321321 13.3607 0.745721
322322 0 0
323323 −31.0349 −1.72683
324324 0 0
325325 −4.07223 2.15676i −0.225887 0.119635i
326326 0 0
327327 5.31965i 0.294178i
328328 0 0
329329 20.3135 1.11992
330330 0 0
331331 12.3135i 0.676812i −0.941000 0.338406i 0.890112π-0.890112\pi
0.941000 0.338406i 0.109888π-0.109888\pi
332332 0 0
333333 3.07838 0.168694
334334 0 0
335335 3.68649 14.8371i 0.201414 0.810637i
336336 0 0
337337 9.47641i 0.516213i −0.966116 0.258106i 0.916901π-0.916901\pi
0.966116 0.258106i 0.0830985π-0.0830985\pi
338338 0 0
339339 19.7587i 1.07315i
340340 0 0
341341 6.15676i 0.333407i
342342 0 0
343343 19.9467i 1.07702i
344344 0 0
345345 4.68035 + 1.16290i 0.251981 + 0.0626084i
346346 0 0
347347 −7.51745 −0.403558 −0.201779 0.979431i 0.564672π-0.564672\pi
−0.201779 + 0.979431i 0.564672π0.564672\pi
348348 0 0
349349 27.6742i 1.48137i 0.671855 + 0.740683i 0.265498π0.265498\pi
−0.671855 + 0.740683i 0.734502π0.734502\pi
350350 0 0
351351 0.921622 0.0491926
352352 0 0
353353 21.6020i 1.14976i −0.818239 0.574878i 0.805049π-0.805049\pi
0.818239 0.574878i 0.194951π-0.194951\pi
354354 0 0
355355 −6.21008 + 24.9939i −0.329597 + 1.32654i
356356 0 0
357357 18.1568 0.960957
358358 0 0
359359 12.0000 0.633336 0.316668 0.948536i 0.397436π-0.397436\pi
0.316668 + 0.948536i 0.397436π0.397436\pi
360360 0 0
361361 3.00000 0.157895
362362 0 0
363363 1.52359 0.0799678
364364 0 0
365365 14.1568 + 3.51745i 0.740998 + 0.184112i
366366 0 0
367367 23.0205i 1.20166i 0.799376 + 0.600831i 0.205163π0.205163\pi
−0.799376 + 0.600831i 0.794837π0.794837\pi
368368 0 0
369369 −10.6803 −0.555997
370370 0 0
371371 6.83710i 0.354965i
372372 0 0
373373 −26.5958 −1.37708 −0.688540 0.725199i 0.741748π-0.741748\pi
−0.688540 + 0.725199i 0.741748π0.741748\pi
374374 0 0
375375 7.46081 8.32684i 0.385275 0.429996i
376376 0 0
377377 5.98771i 0.308383i
378378 0 0
379379 36.1445i 1.85662i 0.371811 + 0.928308i 0.378737π0.378737\pi
−0.371811 + 0.928308i 0.621263π0.621263\pi
380380 0 0
381381 19.3340i 0.990512i
382382 0 0
383383 6.83710i 0.349360i −0.984625 0.174680i 0.944111π-0.944111\pi
0.984625 0.174680i 0.0558890π-0.0558890\pi
384384 0 0
385385 −3.88428 + 15.6332i −0.197961 + 0.796740i
386386 0 0
387387 2.15676 0.109634
388388 0 0
389389 16.7070i 0.847079i −0.905878 0.423539i 0.860787π-0.860787\pi
0.905878 0.423539i 0.139213π-0.139213\pi
390390 0 0
391391 −16.7337 −0.846258
392392 0 0
393393 7.44521i 0.375561i
394394 0 0
395395 8.28231 33.3340i 0.416728 1.67722i
396396 0 0
397397 3.56093 0.178718 0.0893590 0.995999i 0.471518π-0.471518\pi
0.0893590 + 0.995999i 0.471518π0.471518\pi
398398 0 0
399399 9.36069 0.468621
400400 0 0
401401 −25.7152 −1.28416 −0.642079 0.766639i 0.721928π-0.721928\pi
−0.642079 + 0.766639i 0.721928π0.721928\pi
402402 0 0
403403 −1.84324 −0.0918185
404404 0 0
405405 −0.539189 + 2.17009i −0.0267925 + 0.107832i
406406 0 0
407407 9.47641i 0.469728i
408408 0 0
409409 28.8371 1.42590 0.712951 0.701213i 0.247358π-0.247358\pi
0.712951 + 0.701213i 0.247358π0.247358\pi
410410 0 0
411411 3.44521i 0.169940i
412412 0 0
413413 −27.5174 −1.35405
414414 0 0
415415 −0.993857 + 4.00000i −0.0487865 + 0.196352i
416416 0 0
417417 19.2039i 0.940421i
418418 0 0
419419 6.28231i 0.306911i −0.988156 0.153456i 0.950960π-0.950960\pi
0.988156 0.153456i 0.0490402π-0.0490402\pi
420420 0 0
421421 10.9939i 0.535808i −0.963446 0.267904i 0.913669π-0.913669\pi
0.963446 0.267904i 0.0863310π-0.0863310\pi
422422 0 0
423423 8.68035i 0.422053i
424424 0 0
425425 −18.1568 + 34.2823i −0.880732 + 1.66294i
426426 0 0
427427 14.0410 0.679493
428428 0 0
429429 2.83710i 0.136977i
430430 0 0
431431 21.3607 1.02891 0.514454 0.857518i 0.327995π-0.327995\pi
0.514454 + 0.857518i 0.327995π0.327995\pi
432432 0 0
433433 18.4703i 0.887624i −0.896120 0.443812i 0.853626π-0.853626\pi
0.896120 0.443812i 0.146374π-0.146374\pi
434434 0 0
435435 14.0989 + 3.50307i 0.675990 + 0.167959i
436436 0 0
437437 −8.62702 −0.412686
438438 0 0
439439 −6.31351 −0.301327 −0.150664 0.988585i 0.548141π-0.548141\pi
−0.150664 + 0.988585i 0.548141π0.548141\pi
440440 0 0
441441 1.52359 0.0725519
442442 0 0
443443 −27.8310 −1.32229 −0.661144 0.750259i 0.729929π-0.729929\pi
−0.661144 + 0.750259i 0.729929π0.729929\pi
444444 0 0
445445 −3.23513 + 13.0205i −0.153360 + 0.617232i
446446 0 0
447447 9.81658i 0.464308i
448448 0 0
449449 −14.6803 −0.692808 −0.346404 0.938085i 0.612597π-0.612597\pi
−0.346404 + 0.938085i 0.612597π0.612597\pi
450450 0 0
451451 32.8781i 1.54817i
452452 0 0
453453 −23.6742 −1.11231
454454 0 0
455455 −4.68035 1.16290i −0.219418 0.0545175i
456456 0 0
457457 19.6865i 0.920895i 0.887687 + 0.460448i 0.152311π0.152311\pi
−0.887687 + 0.460448i 0.847689π0.847689\pi
458458 0 0
459459 7.75872i 0.362146i
460460 0 0
461461 13.5031i 0.628901i −0.949274 0.314450i 0.898180π-0.898180\pi
0.949274 0.314450i 0.101820π-0.101820\pi
462462 0 0
463463 6.02666i 0.280083i −0.990146 0.140041i 0.955276π-0.955276\pi
0.990146 0.140041i 0.0447236π-0.0447236\pi
464464 0 0
465465 1.07838 4.34017i 0.0500086 0.201271i
466466 0 0
467467 −19.8310 −0.917667 −0.458834 0.888522i 0.651733π-0.651733\pi
−0.458834 + 0.888522i 0.651733π0.651733\pi
468468 0 0
469469 16.0000i 0.738811i
470470 0 0
471471 11.0784 0.510465
472472 0 0
473473 6.63931i 0.305276i
474474 0 0
475475 −9.36069 + 17.6742i −0.429498 + 0.810948i
476476 0 0
477477 −2.92162 −0.133772
478478 0 0
479479 23.2039 1.06021 0.530107 0.847930i 0.322152π-0.322152\pi
0.530107 + 0.847930i 0.322152π0.322152\pi
480480 0 0
481481 2.83710 0.129361
482482 0 0
483483 5.04718 0.229655
484484 0 0
485485 0 0
486486 0 0
487487 23.3874i 1.05978i −0.848066 0.529891i 0.822233π-0.822233\pi
0.848066 0.529891i 0.177767π-0.177767\pi
488488 0 0
489489 12.9939 0.587603
490490 0 0
491491 30.2290i 1.36422i −0.731252 0.682108i 0.761064π-0.761064\pi
0.731252 0.682108i 0.238936π-0.238936\pi
492492 0 0
493493 −50.4079 −2.27026
494494 0 0
495495 6.68035 + 1.65983i 0.300259 + 0.0746037i
496496 0 0
497497 26.9528i 1.20900i
498498 0 0
499499 1.36069i 0.0609129i 0.999536 + 0.0304565i 0.00969609π0.00969609\pi
−0.999536 + 0.0304565i 0.990304π0.990304\pi
500500 0 0
501501 12.0000i 0.536120i
502502 0 0
503503 16.6803i 0.743740i 0.928285 + 0.371870i 0.121283π0.121283\pi
−0.928285 + 0.371870i 0.878717π0.878717\pi
504504 0 0
505505 −18.0989 4.49693i −0.805390 0.200111i
506506 0 0
507507 −12.1506 −0.539628
508508 0 0
509509 14.0144i 0.621176i −0.950545 0.310588i 0.899474π-0.899474\pi
0.950545 0.310588i 0.100526π-0.100526\pi
510510 0 0
511511 15.2663 0.675343
512512 0 0
513513 4.00000i 0.176604i
514514 0 0
515515 22.4391 + 5.57531i 0.988784 + 0.245677i
516516 0 0
517517 −26.7214 −1.17521
518518 0 0
519519 18.4391 0.809385
520520 0 0
521521 −6.68035 −0.292671 −0.146336 0.989235i 0.546748π-0.546748\pi
−0.146336 + 0.989235i 0.546748π0.546748\pi
522522 0 0
523523 26.0410 1.13870 0.569348 0.822097i 0.307196π-0.307196\pi
0.569348 + 0.822097i 0.307196π0.307196\pi
524524 0 0
525525 5.47641 10.3402i 0.239010 0.451282i
526526 0 0
527527 15.5174i 0.675951i
528528 0 0
529529 18.3484 0.797757
530530 0 0
531531 11.7587i 0.510285i
532532 0 0
533533 −9.84324 −0.426358
534534 0 0
535535 −7.20394 + 28.9939i −0.311453 + 1.25351i
536536 0 0
537537 1.91548i 0.0826590i
538538 0 0
539539 4.69019i 0.202021i
540540 0 0
541541 14.9939i 0.644636i 0.946631 + 0.322318i 0.104462π0.104462\pi
−0.946631 + 0.322318i 0.895538π0.895538\pi
542542 0 0
543543 6.99386i 0.300135i
544544 0 0
545545 −11.5441 2.86830i −0.494495 0.122864i
546546 0 0
547547 26.1568 1.11838 0.559191 0.829039i 0.311112π-0.311112\pi
0.559191 + 0.829039i 0.311112π0.311112\pi
548548 0 0
549549 6.00000i 0.256074i
550550 0 0
551551 −25.9877 −1.10711
552552 0 0
553553 35.9467i 1.52861i
554554 0 0
555555 −1.65983 + 6.68035i −0.0704557 + 0.283565i
556556 0 0
557557 14.7526 0.625087 0.312543 0.949903i 0.398819π-0.398819\pi
0.312543 + 0.949903i 0.398819π0.398819\pi
558558 0 0
559559 1.98771 0.0840713
560560 0 0
561561 −23.8843 −1.00840
562562 0 0
563563 38.5523 1.62479 0.812394 0.583109i 0.198164π-0.198164\pi
0.812394 + 0.583109i 0.198164π0.198164\pi
564564 0 0
565565 42.8781 + 10.6537i 1.80390 + 0.448204i
566566 0 0
567567 2.34017i 0.0982780i
568568 0 0
569569 14.6803 0.615432 0.307716 0.951478i 0.400435π-0.400435\pi
0.307716 + 0.951478i 0.400435π0.400435\pi
570570 0 0
571571 13.6742i 0.572248i 0.958193 + 0.286124i 0.0923669π0.0923669\pi
−0.958193 + 0.286124i 0.907633π0.907633\pi
572572 0 0
573573 −6.15676 −0.257202
574574 0 0
575575 −5.04718 + 9.52973i −0.210482 + 0.397417i
576576 0 0
577577 22.6681i 0.943684i 0.881683 + 0.471842i 0.156411π0.156411\pi
−0.881683 + 0.471842i 0.843589π0.843589\pi
578578 0 0
579579 14.5236i 0.603580i
580580 0 0
581581 4.31351i 0.178955i
582582 0 0
583583 8.99386i 0.372487i
584584 0 0
585585 −0.496928 + 2.00000i −0.0205455 + 0.0826898i
586586 0 0
587587 37.3607 1.54204 0.771020 0.636810i 0.219747π-0.219747\pi
0.771020 + 0.636810i 0.219747π0.219747\pi
588588 0 0
589589 8.00000i 0.329634i
590590 0 0
591591 24.5958 1.01174
592592 0 0
593593 15.5897i 0.640192i 0.947385 + 0.320096i 0.103715π0.103715\pi
−0.947385 + 0.320096i 0.896285π0.896285\pi
594594 0 0
595595 −9.78992 + 39.4017i −0.401348 + 1.61531i
596596 0 0
597597 −3.36069 −0.137544
598598 0 0
599599 45.3607 1.85339 0.926694 0.375817i 0.122638π-0.122638\pi
0.926694 + 0.375817i 0.122638π0.122638\pi
600600 0 0
601601 −16.5236 −0.674011 −0.337006 0.941503i 0.609414π-0.609414\pi
−0.337006 + 0.941503i 0.609414π0.609414\pi
602602 0 0
603603 −6.83710 −0.278428
604604 0 0
605605 −0.821503 + 3.30632i −0.0333988 + 0.134421i
606606 0 0
607607 30.0554i 1.21991i −0.792435 0.609956i 0.791187π-0.791187\pi
0.792435 0.609956i 0.208813π-0.208813\pi
608608 0 0
609609 15.2039 0.616095
610610 0 0
611611 8.00000i 0.323645i
612612 0 0
613613 24.2700 0.980257 0.490129 0.871650i 0.336950π-0.336950\pi
0.490129 + 0.871650i 0.336950π0.336950\pi
614614 0 0
615615 5.75872 23.1773i 0.232214 0.934598i
616616 0 0
617617 0.0722347i 0.00290806i 0.999999 + 0.00145403i 0.000462832π0.000462832\pi
−0.999999 + 0.00145403i 0.999537π0.999537\pi
618618 0 0
619619 29.1917i 1.17331i −0.809836 0.586656i 0.800444π-0.800444\pi
0.809836 0.586656i 0.199556π-0.199556\pi
620620 0 0
621621 2.15676i 0.0865476i
622622 0 0
623623 14.0410i 0.562542i
624624 0 0
625625 14.0472 + 20.6803i 0.561887 + 0.827214i
626626 0 0
627627 −12.3135 −0.491754
628628 0 0
629629 23.8843i 0.952329i
630630 0 0
631631 3.36069 0.133787 0.0668935 0.997760i 0.478691π-0.478691\pi
0.0668935 + 0.997760i 0.478691π0.478691\pi
632632 0 0
633633 11.2039i 0.445316i
634634 0 0
635635 −41.9565 10.4247i −1.66499 0.413691i
636636 0 0
637637 1.40417 0.0556354
638638 0 0
639639 11.5174 0.455623
640640 0 0
641641 36.6681 1.44830 0.724151 0.689642i 0.242232π-0.242232\pi
0.724151 + 0.689642i 0.242232π0.242232\pi
642642 0 0
643643 −40.6803 −1.60428 −0.802138 0.597139i 0.796304π-0.796304\pi
−0.802138 + 0.597139i 0.796304π0.796304\pi
644644 0 0
645645 −1.16290 + 4.68035i −0.0457891 + 0.184288i
646646 0 0
647647 37.7275i 1.48322i 0.670830 + 0.741611i 0.265938π0.265938\pi
−0.670830 + 0.741611i 0.734062π0.734062\pi
648648 0 0
649649 36.1978 1.42089
650650 0 0
651651 4.68035i 0.183437i
652652 0 0
653653 −38.9216 −1.52312 −0.761560 0.648094i 0.775566π-0.775566\pi
−0.761560 + 0.648094i 0.775566π0.775566\pi
654654 0 0
655655 16.1568 + 4.01438i 0.631297 + 0.156855i
656656 0 0
657657 6.52359i 0.254510i
658658 0 0
659659 32.9504i 1.28356i −0.766887 0.641782i 0.778195π-0.778195\pi
0.766887 0.641782i 0.221805π-0.221805\pi
660660 0 0
661661 36.3012i 1.41195i 0.708235 + 0.705977i 0.249492π0.249492\pi
−0.708235 + 0.705977i 0.750508π0.750508\pi
662662 0 0
663663 7.15061i 0.277707i
664664 0 0
665665 −5.04718 + 20.3135i −0.195721 + 0.787724i
666666 0 0
667667 −14.0123 −0.542558
668668 0 0
669669 6.65368i 0.257246i
670670 0 0
671671 −18.4703 −0.713037
672672 0 0
673673 37.1917i 1.43363i 0.697262 + 0.716816i 0.254401π0.254401\pi
−0.697262 + 0.716816i 0.745599π0.745599\pi
674674 0 0
675675 −4.41855 2.34017i −0.170070 0.0900733i
676676 0 0
677677 9.07838 0.348910 0.174455 0.984665i 0.444184π-0.444184\pi
0.174455 + 0.984665i 0.444184π0.444184\pi
678678 0 0
679679 0 0
680680 0 0
681681 −9.84324 −0.377194
682682 0 0
683683 9.84324 0.376641 0.188321 0.982108i 0.439696π-0.439696\pi
0.188321 + 0.982108i 0.439696π0.439696\pi
684684 0 0
685685 7.47641 + 1.85762i 0.285659 + 0.0709760i
686686 0 0
687687 1.31965i 0.0503479i
688688 0 0
689689 −2.69263 −0.102581
690690 0 0
691691 1.67420i 0.0636897i −0.999493 0.0318448i 0.989862π-0.989862\pi
0.999493 0.0318448i 0.0101382π-0.0101382\pi
692692 0 0
693693 7.20394 0.273655
694694 0 0
695695 −41.6742 10.3545i −1.58079 0.392770i
696696 0 0
697697 82.8659i 3.13877i
698698 0 0
699699 2.39803i 0.0907019i
700700 0 0
701701 8.96719i 0.338686i 0.985557 + 0.169343i 0.0541646π0.0541646\pi
−0.985557 + 0.169343i 0.945835π0.945835\pi
702702 0 0
703703 12.3135i 0.464413i
704704 0 0
705705 18.8371 + 4.68035i 0.709446 + 0.176272i
706706 0 0
707707 −19.5174 −0.734029
708708 0 0
709709 25.7152i 0.965756i −0.875688 0.482878i 0.839591π-0.839591\pi
0.875688 0.482878i 0.160409π-0.160409\pi
710710 0 0
711711 −15.3607 −0.576071
712712 0 0
713713 4.31351i 0.161542i
714714 0 0
715715 6.15676 + 1.52973i 0.230250 + 0.0572088i
716716 0 0
717717 −6.15676 −0.229928
718718 0 0
719719 18.4079 0.686498 0.343249 0.939244i 0.388473π-0.388473\pi
0.343249 + 0.939244i 0.388473π0.388473\pi
720720 0 0
721721 24.1978 0.901173
722722 0 0
723723 0.639308 0.0237761
724724 0 0
725725 −15.2039 + 28.7070i −0.564660 + 1.06615i
726726 0 0
727727 42.3402i 1.57031i −0.619299 0.785155i 0.712583π-0.712583\pi
0.619299 0.785155i 0.287417π-0.287417\pi
728728 0 0
729729 1.00000 0.0370370
730730 0 0
731731 16.7337i 0.618917i
732732 0 0
733733 4.75258 0.175541 0.0877703 0.996141i 0.472026π-0.472026\pi
0.0877703 + 0.996141i 0.472026π0.472026\pi
734734 0 0
735735 −0.821503 + 3.30632i −0.0303016 + 0.121956i
736736 0 0
737737 21.0472i 0.775283i
738738 0 0
739739 21.0472i 0.774233i −0.922031 0.387117i 0.873471π-0.873471\pi
0.922031 0.387117i 0.126529π-0.126529\pi
740740 0 0
741741 3.68649i 0.135427i
742742 0 0
743743 49.5585i 1.81812i −0.416660 0.909062i 0.636800π-0.636800\pi
0.416660 0.909062i 0.363200π-0.363200\pi
744744 0 0
745745 21.3028 + 5.29299i 0.780475 + 0.193920i
746746 0 0
747747 1.84324 0.0674408
748748 0 0
749749 31.2663i 1.14245i
750750 0 0
751751 −41.6619 −1.52026 −0.760132 0.649768i 0.774866π-0.774866\pi
−0.760132 + 0.649768i 0.774866π0.774866\pi
752752 0 0
753753 12.9216i 0.470890i
754754 0 0
755755 12.7649 51.3751i 0.464561 1.86973i
756756 0 0
757757 −21.9688 −0.798470 −0.399235 0.916849i 0.630724π-0.630724\pi
−0.399235 + 0.916849i 0.630724π0.630724\pi
758758 0 0
759759 −6.63931 −0.240992
760760 0 0
761761 11.3074 0.409892 0.204946 0.978773i 0.434298π-0.434298\pi
0.204946 + 0.978773i 0.434298π0.434298\pi
762762 0 0
763763 −12.4489 −0.450681
764764 0 0
765765 16.8371 + 4.18342i 0.608747 + 0.151252i
766766 0 0
767767 10.8371i 0.391305i
768768 0 0
769769 −6.19779 −0.223498 −0.111749 0.993736i 0.535645π-0.535645\pi
−0.111749 + 0.993736i 0.535645π0.535645\pi
770770 0 0
771771 14.9627i 0.538867i
772772 0 0
773773 −30.4391 −1.09482 −0.547409 0.836865i 0.684386π-0.684386\pi
−0.547409 + 0.836865i 0.684386π0.684386\pi
774774 0 0
775775 8.83710 + 4.68035i 0.317438 + 0.168123i
776776 0 0
777777 7.20394i 0.258440i
778778 0 0
779779 42.7214i 1.53065i
780780 0 0
781781 35.4551i 1.26868i
782782 0 0
783783 6.49693i 0.232181i
784784 0 0
785785 −5.97334 + 24.0410i −0.213198 + 0.858061i
786786 0 0
787787 36.9939 1.31869 0.659344 0.751841i 0.270834π-0.270834\pi
0.659344 + 0.751841i 0.270834π0.270834\pi
788788 0 0
789789 13.3607i 0.475653i
790790 0 0
791791 46.2388 1.64406
792792 0 0
793793 5.52973i 0.196367i
794794 0 0
795795 1.57531 6.34017i 0.0558704 0.224863i
796796 0 0
797797 −30.7526 −1.08931 −0.544656 0.838659i 0.683340π-0.683340\pi
−0.544656 + 0.838659i 0.683340π0.683340\pi
798798 0 0
799799 −67.3484 −2.38262
800800 0 0
801801 6.00000 0.212000
802802 0 0
803803 −20.0821 −0.708681
804804 0 0
805805 −2.72138 + 10.9528i −0.0959162 + 0.386036i
806806 0 0
807807 27.5441i 0.969599i
808808 0 0
809809 −5.31965 −0.187029 −0.0935145 0.995618i 0.529810π-0.529810\pi
−0.0935145 + 0.995618i 0.529810π0.529810\pi
810810 0 0
811811 49.9253i 1.75312i 0.481297 + 0.876558i 0.340166π0.340166\pi
−0.481297 + 0.876558i 0.659834π0.659834\pi
812812 0 0
813813 −10.0000 −0.350715
814814 0 0
815815 −7.00614 + 28.1978i −0.245414 + 0.987726i
816816 0 0
817817 8.62702i 0.301821i
818818 0 0
819819 2.15676i 0.0753631i
820820 0 0
821821 44.1711i 1.54158i −0.637087 0.770792i 0.719861π-0.719861\pi
0.637087 0.770792i 0.280139π-0.280139\pi
822822 0 0
823823 19.0738i 0.664872i 0.943126 + 0.332436i 0.107871π0.107871\pi
−0.943126 + 0.332436i 0.892129π0.892129\pi
824824 0 0
825825 −7.20394 + 13.6020i −0.250809 + 0.473560i
826826 0 0
827827 −43.0349 −1.49647 −0.748235 0.663434i 0.769098π-0.769098\pi
−0.748235 + 0.663434i 0.769098π0.769098\pi
828828 0 0
829829 36.0410i 1.25176i −0.779921 0.625878i 0.784741π-0.784741\pi
0.779921 0.625878i 0.215259π-0.215259\pi
830830 0 0
831831 4.12556 0.143114
832832 0 0
833833 11.8211i 0.409577i
834834 0 0
835835 −26.0410 6.47027i −0.901187 0.223913i
836836 0 0
837837 −2.00000 −0.0691301
838838 0 0
839839 43.0349 1.48573 0.742865 0.669441i 0.233467π-0.233467\pi
0.742865 + 0.669441i 0.233467π0.233467\pi
840840 0 0
841841 −13.2101 −0.455520
842842 0 0
843843 11.3607 0.391283
844844 0 0
845845 6.55148 26.3679i 0.225378 0.907083i
846846 0 0
847847 3.56547i 0.122511i
848848 0 0
849849 −10.5236 −0.361169
850850 0 0
851851 6.63931i 0.227593i
852852 0 0
853853 −47.1605 −1.61474 −0.807372 0.590043i 0.799111π-0.799111\pi
−0.807372 + 0.590043i 0.799111π0.799111\pi
854854 0 0
855855 8.68035 + 2.15676i 0.296862 + 0.0737595i
856856 0 0
857857 12.8059i 0.437441i 0.975788 + 0.218721i 0.0701884π0.0701884\pi
−0.975788 + 0.218721i 0.929812π0.929812\pi
858858 0 0
859859 14.4703i 0.493719i −0.969051 0.246860i 0.920601π-0.920601\pi
0.969051 0.246860i 0.0793986π-0.0793986\pi
860860 0 0
861861 24.9939i 0.851788i
862862 0 0
863863 24.3135i 0.827642i −0.910358 0.413821i 0.864194π-0.864194\pi
0.910358 0.413821i 0.135806π-0.135806\pi
864864 0 0
865865 −9.94214 + 40.0144i −0.338043 + 1.36053i
866866 0 0
867867 −43.1978 −1.46707
868868 0 0
869869 47.2860i 1.60407i
870870 0 0
871871 −6.30122 −0.213509
872872 0 0
873873 0 0
874874 0 0
875875 19.4863 + 17.4596i 0.658756 + 0.590242i
876876 0 0
877877 44.4391 1.50060 0.750300 0.661097i 0.229909π-0.229909\pi
0.750300 + 0.661097i 0.229909π0.229909\pi
878878 0 0
879879 −32.5958 −1.09943
880880 0 0
881881 6.62702 0.223270 0.111635 0.993749i 0.464391π-0.464391\pi
0.111635 + 0.993749i 0.464391π0.464391\pi
882882 0 0
883883 7.31965 0.246326 0.123163 0.992386i 0.460696π-0.460696\pi
0.123163 + 0.992386i 0.460696π0.460696\pi
884884 0 0
885885 −25.5174 6.34017i −0.857760 0.213123i
886886 0 0
887887 12.0000i 0.402921i −0.979497 0.201460i 0.935431π-0.935431\pi
0.979497 0.201460i 0.0645687π-0.0645687\pi
888888 0 0
889889 −45.2450 −1.51747
890890 0 0
891891 3.07838i 0.103130i
892892 0 0
893893 −34.7214 −1.16191
894894 0 0
895895 −4.15676 1.03281i −0.138945 0.0345229i
896896 0 0
897897 1.98771i 0.0663678i
898898 0 0
899899 12.9939i 0.433369i
900900 0 0
901901 22.6681i 0.755183i
902902 0 0
903903 5.04718i 0.167960i
904904 0 0
905905 15.1773 + 3.77101i 0.504510 + 0.125353i
906906 0 0
907907 −4.48255 −0.148841 −0.0744204 0.997227i 0.523711π-0.523711\pi
−0.0744204 + 0.997227i 0.523711π0.523711\pi
908908 0 0
909909 8.34017i 0.276626i
910910 0 0
911911 −8.73367 −0.289359 −0.144680 0.989479i 0.546215π-0.546215\pi
−0.144680 + 0.989479i 0.546215π0.546215\pi
912912 0 0
913913 5.67420i 0.187789i
914914 0 0
915915 13.0205 + 3.23513i 0.430445 + 0.106950i
916916 0 0
917917 17.4231 0.575361
918918 0 0
919919 −18.7337 −0.617967 −0.308983 0.951067i 0.599989π-0.599989\pi
−0.308983 + 0.951067i 0.599989π0.599989\pi
920920 0 0
921921 29.9877 0.988129
922922 0 0
923923 10.6147 0.349388
924924 0 0
925925 −13.6020 7.20394i −0.447230 0.236864i
926926 0 0
927927 10.3402i 0.339616i
928928 0 0
929929 20.0410 0.657525 0.328763 0.944413i 0.393368π-0.393368\pi
0.328763 + 0.944413i 0.393368π0.393368\pi
930930 0 0
931931 6.09436i 0.199735i
932932 0 0
933933 −27.2039 −0.890617
934934 0 0
935935 12.8781 51.8310i 0.421160 1.69505i
936936 0 0
937937 45.0472i 1.47163i 0.677184 + 0.735814i 0.263200π0.263200\pi
−0.677184 + 0.735814i 0.736800π0.736800\pi
938938 0 0
939939 26.8371i 0.875796i
940940 0 0
941941 19.3751i 0.631609i 0.948824 + 0.315805i 0.102274π0.102274\pi
−0.948824 + 0.315805i 0.897726π0.897726\pi
942942 0 0
943943 23.0349i 0.750119i
944944 0 0
945945 −5.07838 1.26180i −0.165200 0.0410462i
946946 0 0
947947 −6.95282 −0.225936 −0.112968 0.993599i 0.536036π-0.536036\pi
−0.112968 + 0.993599i 0.536036π0.536036\pi
948948 0 0
949949 6.01229i 0.195167i
950950 0 0
951951 24.5958 0.797574
952952 0 0
953953 29.2885i 0.948746i −0.880324 0.474373i 0.842675π-0.842675\pi
0.880324 0.474373i 0.157325π-0.157325\pi
954954 0 0
955955 3.31965 13.3607i 0.107421 0.432342i
956956 0 0
957957 −20.0000 −0.646508
958958 0 0
959959 8.06239 0.260348
960960 0 0
961961 −27.0000 −0.870968
962962 0 0
963963 13.3607 0.430542
964964 0 0
965965 31.5174 + 7.83096i 1.01458 + 0.252087i
966966 0 0
967967 47.3874i 1.52387i 0.647651 + 0.761937i 0.275752π0.275752\pi
−0.647651 + 0.761937i 0.724248π0.724248\pi
968968 0 0
969969 −31.0349 −0.996984
970970 0 0
971971 2.59583i 0.0833040i −0.999132 0.0416520i 0.986738π-0.986738\pi
0.999132 0.0416520i 0.0132621π-0.0132621\pi
972972 0 0
973973 −44.9405 −1.44073
974974 0 0
975975 −4.07223 2.15676i −0.130416 0.0690715i
976976 0 0
977977 28.3234i 0.906144i 0.891474 + 0.453072i 0.149672π0.149672\pi
−0.891474 + 0.453072i 0.850328π0.850328\pi
978978 0 0
979979 18.4703i 0.590312i
980980 0 0
981981 5.31965i 0.169843i
982982 0 0
983983 1.16290i 0.0370907i −0.999828 0.0185454i 0.994096π-0.994096\pi
0.999828 0.0185454i 0.00590351π-0.00590351\pi
984984 0 0
985985 −13.2618 + 53.3751i −0.422556 + 1.70067i
986986 0 0
987987 20.3135 0.646586
988988 0 0
989989 4.65159i 0.147912i
990990 0 0
991991 12.9528 0.411460 0.205730 0.978609i 0.434043π-0.434043\pi
0.205730 + 0.978609i 0.434043π0.434043\pi
992992 0 0
993993 12.3135i 0.390757i
994994 0 0
995995 1.81205 7.29299i 0.0574458 0.231203i
996996 0 0
997997 −19.0784 −0.604218 −0.302109 0.953273i 0.597691π-0.597691\pi
−0.302109 + 0.953273i 0.597691π0.597691\pi
998998 0 0
999999 3.07838 0.0973956
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 3840.2.d.bl.2689.4 6
4.3 odd 2 3840.2.d.bj.2689.4 6
5.4 even 2 3840.2.d.bi.2689.4 6
8.3 odd 2 3840.2.d.bk.2689.3 6
8.5 even 2 3840.2.d.bi.2689.3 6
16.3 odd 4 1920.2.f.p.769.3 yes 6
16.5 even 4 1920.2.f.n.769.1 yes 6
16.11 odd 4 1920.2.f.m.769.4 yes 6
16.13 even 4 1920.2.f.o.769.6 yes 6
20.19 odd 2 3840.2.d.bk.2689.4 6
40.19 odd 2 3840.2.d.bj.2689.3 6
40.29 even 2 inner 3840.2.d.bl.2689.3 6
80.3 even 4 9600.2.a.dt.1.3 3
80.13 odd 4 9600.2.a.dq.1.1 3
80.19 odd 4 1920.2.f.p.769.6 yes 6
80.27 even 4 9600.2.a.du.1.1 3
80.29 even 4 1920.2.f.o.769.3 yes 6
80.37 odd 4 9600.2.a.dp.1.3 3
80.43 even 4 9600.2.a.do.1.3 3
80.53 odd 4 9600.2.a.dv.1.1 3
80.59 odd 4 1920.2.f.m.769.1 6
80.67 even 4 9600.2.a.dr.1.1 3
80.69 even 4 1920.2.f.n.769.4 yes 6
80.77 odd 4 9600.2.a.ds.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1920.2.f.m.769.1 6 80.59 odd 4
1920.2.f.m.769.4 yes 6 16.11 odd 4
1920.2.f.n.769.1 yes 6 16.5 even 4
1920.2.f.n.769.4 yes 6 80.69 even 4
1920.2.f.o.769.3 yes 6 80.29 even 4
1920.2.f.o.769.6 yes 6 16.13 even 4
1920.2.f.p.769.3 yes 6 16.3 odd 4
1920.2.f.p.769.6 yes 6 80.19 odd 4
3840.2.d.bi.2689.3 6 8.5 even 2
3840.2.d.bi.2689.4 6 5.4 even 2
3840.2.d.bj.2689.3 6 40.19 odd 2
3840.2.d.bj.2689.4 6 4.3 odd 2
3840.2.d.bk.2689.3 6 8.3 odd 2
3840.2.d.bk.2689.4 6 20.19 odd 2
3840.2.d.bl.2689.3 6 40.29 even 2 inner
3840.2.d.bl.2689.4 6 1.1 even 1 trivial
9600.2.a.do.1.3 3 80.43 even 4
9600.2.a.dp.1.3 3 80.37 odd 4
9600.2.a.dq.1.1 3 80.13 odd 4
9600.2.a.dr.1.1 3 80.67 even 4
9600.2.a.ds.1.3 3 80.77 odd 4
9600.2.a.dt.1.3 3 80.3 even 4
9600.2.a.du.1.1 3 80.27 even 4
9600.2.a.dv.1.1 3 80.53 odd 4