Properties

Label 196.8.e.b.165.1
Level 196196
Weight 88
Character 196.165
Analytic conductor 61.22761.227
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] = [196,8,Mod(165,196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.165");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: N N == 196=2272 196 = 2^{2} \cdot 7^{2}
Weight: k k == 8 8
Character orbit: [χ][\chi] == 196.e (of order 33, degree 22, 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: 61.227464994961.2274649949
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q(3,1009)\Q(\sqrt{-3}, \sqrt{1009})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x3+253x2+252x+63504 x^{4} - x^{3} + 253x^{2} + 252x + 63504 Copy content Toggle raw display
Coefficient ring: Z[a1,,a9]\Z[a_1, \ldots, a_{9}]
Coefficient ring index: 22 2^{2}
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

Embedding invariants

Embedding label 165.1
Root 8.1911914.1876i8.19119 - 14.1876i of defining polynomial
Character χ\chi == 196.165
Dual form 196.8.e.b.177.1

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+(19.3824+33.5713i)q3+(248.206+429.906i)q5+(342.147+592.615i)q9+(2017.59+3494.57i)q1114469.7q1319243.3q15+(13375.123166.3i)q17+(18305.031705.2i)q19+(17460.0+30241.7i)q23+(84150.1+145752.i)q25111305.q27+35874.3q29+(99299.2+171991.i)q31+(78211.5135466.i)q33+(57206.6+99084.7i)q37+(280458.485767.i)q39+245990.q41+30224.0q43+(169846.+294182.i)q45+(152616.+264339.i)q47+(518481.+898036.i)q51+(605264.1.04835e6i)q532.00312e6q55+1.41918e6q57+(767785.1.32984e6i)q59+(22543.2+39045.9i)q61+(3.59147e66.22061e6i)q65+(1.54250e6+2.67169e6i)q671.35367e6q691.69051e6q71+(1.93466e63.35093e6i)q73+(3.26206e65.65005e6i)q75+(1.00455e61.73993e6i)q79+(1.40908e62.44060e6i)q818.28720e6q83+1.32791e7q85+(695328.+1.20434e6i)q87+(3.25312e6+5.63457e6i)q89+(3.84931e66.66720e6i)q93+(9.08683e61.57389e7i)q951.14882e7q972.76125e6q99+O(q100)q+(-19.3824 + 33.5713i) q^{3} +(248.206 + 429.906i) q^{5} +(342.147 + 592.615i) q^{9} +(-2017.59 + 3494.57i) q^{11} -14469.7 q^{13} -19243.3 q^{15} +(13375.1 - 23166.3i) q^{17} +(-18305.0 - 31705.2i) q^{19} +(17460.0 + 30241.7i) q^{23} +(-84150.1 + 145752. i) q^{25} -111305. q^{27} +35874.3 q^{29} +(-99299.2 + 171991. i) q^{31} +(-78211.5 - 135466. i) q^{33} +(57206.6 + 99084.7i) q^{37} +(280458. - 485767. i) q^{39} +245990. q^{41} +30224.0 q^{43} +(-169846. + 294182. i) q^{45} +(152616. + 264339. i) q^{47} +(518481. + 898036. i) q^{51} +(605264. - 1.04835e6i) q^{53} -2.00312e6 q^{55} +1.41918e6 q^{57} +(767785. - 1.32984e6i) q^{59} +(22543.2 + 39045.9i) q^{61} +(-3.59147e6 - 6.22061e6i) q^{65} +(-1.54250e6 + 2.67169e6i) q^{67} -1.35367e6 q^{69} -1.69051e6 q^{71} +(1.93466e6 - 3.35093e6i) q^{73} +(-3.26206e6 - 5.65005e6i) q^{75} +(-1.00455e6 - 1.73993e6i) q^{79} +(1.40908e6 - 2.44060e6i) q^{81} -8.28720e6 q^{83} +1.32791e7 q^{85} +(-695328. + 1.20434e6i) q^{87} +(3.25312e6 + 5.63457e6i) q^{89} +(-3.84931e6 - 6.66720e6i) q^{93} +(9.08683e6 - 1.57389e7i) q^{95} -1.14882e7 q^{97} -2.76125e6 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q14q3+294q5+2258q9+3492q1132340q1348512q15+29232q17+3206q19+9360q23131146q2536344q27+369408q29165060q31342832q33++18168664q99+O(q100) 4 q - 14 q^{3} + 294 q^{5} + 2258 q^{9} + 3492 q^{11} - 32340 q^{13} - 48512 q^{15} + 29232 q^{17} + 3206 q^{19} + 9360 q^{23} - 131146 q^{25} - 36344 q^{27} + 369408 q^{29} - 165060 q^{31} - 342832 q^{33}+ \cdots + 18168664 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/196Z)×\left(\mathbb{Z}/196\mathbb{Z}\right)^\times.

nn 9999 101101
χ(n)\chi(n) 11 e(23)e\left(\frac{2}{3}\right)

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 −19.3824 + 33.5713i −0.414460 + 0.717866i −0.995372 0.0961008i 0.969363π-0.969363\pi
0.580912 + 0.813967i 0.302696π0.302696\pi
44 0 0
55 248.206 + 429.906i 0.888009 + 1.53808i 0.842225 + 0.539126i 0.181245π0.181245\pi
0.0457844 + 0.998951i 0.485421π0.485421\pi
66 0 0
77 0 0
88 0 0
99 342.147 + 592.615i 0.156446 + 0.270972i
1010 0 0
1111 −2017.59 + 3494.57i −0.457045 + 0.791626i −0.998803 0.0489089i 0.984426π-0.984426\pi
0.541758 + 0.840535i 0.317759π0.317759\pi
1212 0 0
1313 −14469.7 −1.82666 −0.913331 0.407217i 0.866499π-0.866499\pi
−0.913331 + 0.407217i 0.866499π0.866499\pi
1414 0 0
1515 −19243.3 −1.47218
1616 0 0
1717 13375.1 23166.3i 0.660275 1.14363i −0.320268 0.947327i 0.603773π-0.603773\pi
0.980543 0.196303i 0.0628936π-0.0628936\pi
1818 0 0
1919 −18305.0 31705.2i −0.612255 1.06046i −0.990859 0.134898i 0.956929π-0.956929\pi
0.378605 0.925558i 0.376404π-0.376404\pi
2020 0 0
2121 0 0
2222 0 0
2323 17460.0 + 30241.7i 0.299225 + 0.518272i 0.975959 0.217955i 0.0699386π-0.0699386\pi
−0.676734 + 0.736228i 0.736605π0.736605\pi
2424 0 0
2525 −84150.1 + 145752.i −1.07712 + 1.86563i
2626 0 0
2727 −111305. −1.08828
2828 0 0
2929 35874.3 0.273143 0.136571 0.990630i 0.456392π-0.456392\pi
0.136571 + 0.990630i 0.456392π0.456392\pi
3030 0 0
3131 −99299.2 + 171991.i −0.598660 + 1.03691i 0.394360 + 0.918956i 0.370966π0.370966\pi
−0.993019 + 0.117953i 0.962367π0.962367\pi
3232 0 0
3333 −78211.5 135466.i −0.378854 0.656194i
3434 0 0
3535 0 0
3636 0 0
3737 57206.6 + 99084.7i 0.185669 + 0.321589i 0.943802 0.330512i 0.107221π-0.107221\pi
−0.758133 + 0.652100i 0.773888π0.773888\pi
3838 0 0
3939 280458. 485767.i 0.757079 1.31130i
4040 0 0
4141 245990. 0.557409 0.278704 0.960377i 0.410095π-0.410095\pi
0.278704 + 0.960377i 0.410095π0.410095\pi
4242 0 0
4343 30224.0 0.0579711 0.0289856 0.999580i 0.490772π-0.490772\pi
0.0289856 + 0.999580i 0.490772π0.490772\pi
4444 0 0
4545 −169846. + 294182.i −0.277850 + 0.481251i
4646 0 0
4747 152616. + 264339.i 0.214417 + 0.371381i 0.953092 0.302681i 0.0978816π-0.0978816\pi
−0.738675 + 0.674061i 0.764548π0.764548\pi
4848 0 0
4949 0 0
5050 0 0
5151 518481. + 898036.i 0.547315 + 0.947978i
5252 0 0
5353 605264. 1.04835e6i 0.558444 0.967253i −0.439183 0.898398i 0.644732π-0.644732\pi
0.997627 0.0688551i 0.0219346π-0.0219346\pi
5454 0 0
5555 −2.00312e6 −1.62344
5656 0 0
5757 1.41918e6 1.01502
5858 0 0
5959 767785. 1.32984e6i 0.486696 0.842981i −0.513188 0.858277i 0.671535π-0.671535\pi
0.999883 + 0.0152952i 0.00486879π0.00486879\pi
6060 0 0
6161 22543.2 + 39045.9i 0.0127163 + 0.0220253i 0.872314 0.488947i 0.162619π-0.162619\pi
−0.859597 + 0.510972i 0.829285π0.829285\pi
6262 0 0
6363 0 0
6464 0 0
6565 −3.59147e6 6.22061e6i −1.62209 2.80955i
6666 0 0
6767 −1.54250e6 + 2.67169e6i −0.626562 + 1.08524i 0.361674 + 0.932305i 0.382205π0.382205\pi
−0.988236 + 0.152933i 0.951128π0.951128\pi
6868 0 0
6969 −1.35367e6 −0.496067
7070 0 0
7171 −1.69051e6 −0.560550 −0.280275 0.959920i 0.590426π-0.590426\pi
−0.280275 + 0.959920i 0.590426π0.590426\pi
7272 0 0
7373 1.93466e6 3.35093e6i 0.582069 1.00817i −0.413165 0.910656i 0.635577π-0.635577\pi
0.995234 0.0975172i 0.0310901π-0.0310901\pi
7474 0 0
7575 −3.26206e6 5.65005e6i −0.892848 1.54646i
7676 0 0
7777 0 0
7878 0 0
7979 −1.00455e6 1.73993e6i −0.229232 0.397042i 0.728349 0.685207i 0.240288π-0.240288\pi
−0.957581 + 0.288165i 0.906955π0.906955\pi
8080 0 0
8181 1.40908e6 2.44060e6i 0.294604 0.510269i
8282 0 0
8383 −8.28720e6 −1.59087 −0.795435 0.606039i 0.792758π-0.792758\pi
−0.795435 + 0.606039i 0.792758π0.792758\pi
8484 0 0
8585 1.32791e7 2.34532
8686 0 0
8787 −695328. + 1.20434e6i −0.113207 + 0.196080i
8888 0 0
8989 3.25312e6 + 5.63457e6i 0.489142 + 0.847219i 0.999922 0.0124923i 0.00397654π-0.00397654\pi
−0.510780 + 0.859712i 0.670643π0.670643\pi
9090 0 0
9191 0 0
9292 0 0
9393 −3.84931e6 6.66720e6i −0.496241 0.859515i
9494 0 0
9595 9.08683e6 1.57389e7i 1.08738 1.88339i
9696 0 0
9797 −1.14882e7 −1.27806 −0.639028 0.769183i 0.720663π-0.720663\pi
−0.639028 + 0.769183i 0.720663π0.720663\pi
9898 0 0
9999 −2.76125e6 −0.286011
100100 0 0
101101 3.45226e6 5.97950e6i 0.333410 0.577484i −0.649768 0.760133i 0.725134π-0.725134\pi
0.983178 + 0.182649i 0.0584672π0.0584672\pi
102102 0 0
103103 −2.08698e6 3.61475e6i −0.188186 0.325948i 0.756459 0.654041i 0.226928π-0.226928\pi
−0.944646 + 0.328093i 0.893594π0.893594\pi
104104 0 0
105105 0 0
106106 0 0
107107 3.32843e6 + 5.76501e6i 0.262661 + 0.454942i 0.966948 0.254973i 0.0820665π-0.0820665\pi
−0.704287 + 0.709915i 0.748733π0.748733\pi
108108 0 0
109109 5.56972e6 9.64704e6i 0.411946 0.713512i −0.583156 0.812360i 0.698182π-0.698182\pi
0.995103 + 0.0988480i 0.0315158π0.0315158\pi
110110 0 0
111111 −4.43520e6 −0.307810
112112 0 0
113113 4.18354e6 0.272753 0.136377 0.990657i 0.456454π-0.456454\pi
0.136377 + 0.990657i 0.456454π0.456454\pi
114114 0 0
115115 −8.66737e6 + 1.50123e7i −0.531429 + 0.920462i
116116 0 0
117117 −4.95077e6 8.57498e6i −0.285773 0.494974i
118118 0 0
119119 0 0
120120 0 0
121121 1.60222e6 + 2.77513e6i 0.0822193 + 0.142408i
122122 0 0
123123 −4.76787e6 + 8.25819e6i −0.231024 + 0.400145i
124124 0 0
125125 −4.47641e7 −2.04996
126126 0 0
127127 3.65300e7 1.58247 0.791237 0.611510i 0.209438π-0.209438\pi
0.791237 + 0.611510i 0.209438π0.209438\pi
128128 0 0
129129 −585812. + 1.01466e6i −0.0240267 + 0.0416155i
130130 0 0
131131 −2.54084e7 4.40087e7i −0.987479 1.71036i −0.630353 0.776308i 0.717090π-0.717090\pi
−0.357126 0.934056i 0.616243π-0.616243\pi
132132 0 0
133133 0 0
134134 0 0
135135 −2.76266e7 4.78507e7i −0.966405 1.67386i
136136 0 0
137137 −1.51976e7 + 2.63230e7i −0.504955 + 0.874608i 0.495028 + 0.868877i 0.335158π0.335158\pi
−0.999984 + 0.00573133i 0.998176π0.998176\pi
138138 0 0
139139 −2.92052e7 −0.922377 −0.461189 0.887302i 0.652577π-0.652577\pi
−0.461189 + 0.887302i 0.652577π0.652577\pi
140140 0 0
141141 −1.18323e7 −0.355469
142142 0 0
143143 2.91940e7 5.05655e7i 0.834867 1.44603i
144144 0 0
145145 8.90421e6 + 1.54225e7i 0.242553 + 0.420115i
146146 0 0
147147 0 0
148148 0 0
149149 1.85475e7 + 3.21252e7i 0.459339 + 0.795598i 0.998926 0.0463317i 0.0147531π-0.0147531\pi
−0.539588 + 0.841930i 0.681420π0.681420\pi
150150 0 0
151151 −9.11971e6 + 1.57958e7i −0.215557 + 0.373356i −0.953445 0.301568i 0.902490π-0.902490\pi
0.737888 + 0.674923i 0.235823π0.235823\pi
152152 0 0
153153 1.83049e7 0.413189
154154 0 0
155155 −9.85867e7 −2.12646
156156 0 0
157157 −1.06032e7 + 1.83653e7i −0.218670 + 0.378747i −0.954402 0.298526i 0.903505π-0.903505\pi
0.735732 + 0.677273i 0.236838π0.236838\pi
158158 0 0
159159 2.34629e7 + 4.06390e7i 0.462905 + 0.801775i
160160 0 0
161161 0 0
162162 0 0
163163 −2.25476e7 3.90537e7i −0.407797 0.706326i 0.586845 0.809699i 0.300370π-0.300370\pi
−0.994643 + 0.103373i 0.967036π0.967036\pi
164164 0 0
165165 3.88252e7 6.72472e7i 0.672852 1.16541i
166166 0 0
167167 6.32823e7 1.05142 0.525708 0.850665i 0.323801π-0.323801\pi
0.525708 + 0.850665i 0.323801π0.323801\pi
168168 0 0
169169 1.46624e8 2.33670
170170 0 0
171171 1.25260e7 2.16957e7i 0.191569 0.331808i
172172 0 0
173173 1.67096e7 + 2.89420e7i 0.245361 + 0.424978i 0.962233 0.272227i 0.0877601π-0.0877601\pi
−0.716872 + 0.697205i 0.754427π0.754427\pi
174174 0 0
175175 0 0
176176 0 0
177177 2.97630e7 + 5.15510e7i 0.403432 + 0.698764i
178178 0 0
179179 6.49827e7 1.12553e8i 0.846861 1.46681i −0.0371339 0.999310i 0.511823π-0.511823\pi
0.883995 0.467496i 0.154844π-0.154844\pi
180180 0 0
181181 1.91607e7 0.240180 0.120090 0.992763i 0.461682π-0.461682\pi
0.120090 + 0.992763i 0.461682π0.461682\pi
182182 0 0
183183 −1.74776e6 −0.0210816
184184 0 0
185185 −2.83981e7 + 4.91869e7i −0.329752 + 0.571147i
186186 0 0
187187 5.39709e7 + 9.34803e7i 0.603551 + 1.04538i
188188 0 0
189189 0 0
190190 0 0
191191 −1.77182e7 3.06888e7i −0.183994 0.318686i 0.759243 0.650807i 0.225569π-0.225569\pi
−0.943237 + 0.332121i 0.892236π0.892236\pi
192192 0 0
193193 −9.01695e7 + 1.56178e8i −0.902837 + 1.56376i −0.0790419 + 0.996871i 0.525186π0.525186\pi
−0.823795 + 0.566888i 0.808147π0.808147\pi
194194 0 0
195195 2.78445e8 2.68917
196196 0 0
197197 −1.39463e8 −1.29965 −0.649826 0.760083i 0.725158π-0.725158\pi
−0.649826 + 0.760083i 0.725158π0.725158\pi
198198 0 0
199199 −7.31562e7 + 1.26710e8i −0.658060 + 1.13979i 0.323057 + 0.946379i 0.395289π0.395289\pi
−0.981117 + 0.193414i 0.938044π0.938044\pi
200200 0 0
201201 −5.97948e7 1.03568e8i −0.519370 0.899576i
202202 0 0
203203 0 0
204204 0 0
205205 6.10562e7 + 1.05752e8i 0.494984 + 0.857338i
206206 0 0
207207 −1.19478e7 + 2.06942e7i −0.0936248 + 0.162163i
208208 0 0
209209 1.47728e8 1.11931
210210 0 0
211211 −1.02935e8 −0.754351 −0.377176 0.926142i 0.623105π-0.623105\pi
−0.377176 + 0.926142i 0.623105π0.623105\pi
212212 0 0
213213 3.27662e7 5.67527e7i 0.232326 0.402400i
214214 0 0
215215 7.50177e6 + 1.29934e7i 0.0514789 + 0.0891641i
216216 0 0
217217 0 0
218218 0 0
219219 7.49966e7 + 1.29898e8i 0.482489 + 0.835695i
220220 0 0
221221 −1.93533e8 + 3.35210e8i −1.20610 + 2.08903i
222222 0 0
223223 −3.26446e8 −1.97126 −0.985632 0.168906i 0.945977π-0.945977\pi
−0.985632 + 0.168906i 0.945977π0.945977\pi
224224 0 0
225225 −1.15167e8 −0.674044
226226 0 0
227227 4.49769e7 7.79023e7i 0.255211 0.442038i −0.709742 0.704462i 0.751188π-0.751188\pi
0.964953 + 0.262424i 0.0845218π0.0845218\pi
228228 0 0
229229 5.09646e7 + 8.82734e7i 0.280443 + 0.485742i 0.971494 0.237064i 0.0761853π-0.0761853\pi
−0.691051 + 0.722806i 0.742852π0.742852\pi
230230 0 0
231231 0 0
232232 0 0
233233 2.13457e7 + 3.69719e7i 0.110552 + 0.191481i 0.915993 0.401195i 0.131405π-0.131405\pi
−0.805441 + 0.592676i 0.798072π0.798072\pi
234234 0 0
235235 −7.57606e7 + 1.31221e8i −0.380808 + 0.659579i
236236 0 0
237237 7.78821e7 0.380030
238238 0 0
239239 −6.02163e7 −0.285313 −0.142656 0.989772i 0.545564π-0.545564\pi
−0.142656 + 0.989772i 0.545564π0.545564\pi
240240 0 0
241241 2.35768e7 4.08362e7i 0.108499 0.187926i −0.806663 0.591011i 0.798729π-0.798729\pi
0.915162 + 0.403086i 0.132062π0.132062\pi
242242 0 0
243243 −6.70893e7 1.16202e8i −0.299938 0.519508i
244244 0 0
245245 0 0
246246 0 0
247247 2.64868e8 + 4.58765e8i 1.11838 + 1.93710i
248248 0 0
249249 1.60626e8 2.78212e8i 0.659352 1.14203i
250250 0 0
251251 1.94014e8 0.774416 0.387208 0.921992i 0.373440π-0.373440\pi
0.387208 + 0.921992i 0.373440π0.373440\pi
252252 0 0
253253 −1.40909e8 −0.547037
254254 0 0
255255 −2.57381e8 + 4.45796e8i −0.972042 + 1.68363i
256256 0 0
257257 3.39371e7 + 5.87809e7i 0.124712 + 0.216008i 0.921620 0.388092i 0.126866π-0.126866\pi
−0.796908 + 0.604101i 0.793533π0.793533\pi
258258 0 0
259259 0 0
260260 0 0
261261 1.22743e7 + 2.12596e7i 0.0427320 + 0.0740140i
262262 0 0
263263 −8.05998e7 + 1.39603e8i −0.273205 + 0.473205i −0.969681 0.244375i 0.921417π-0.921417\pi
0.696476 + 0.717580i 0.254750π0.254750\pi
264264 0 0
265265 6.00921e8 1.98361
266266 0 0
267267 −2.52213e8 −0.810920
268268 0 0
269269 1.32052e8 2.28721e8i 0.413631 0.716429i −0.581653 0.813437i 0.697594π-0.697594\pi
0.995284 + 0.0970078i 0.0309272π0.0309272\pi
270270 0 0
271271 2.46217e7 + 4.26460e7i 0.0751493 + 0.130162i 0.901151 0.433505i 0.142723π-0.142723\pi
−0.826002 + 0.563667i 0.809390π0.809390\pi
272272 0 0
273273 0 0
274274 0 0
275275 −3.39561e8 5.88138e8i −0.984587 1.70535i
276276 0 0
277277 1.42902e8 2.47513e8i 0.403978 0.699711i −0.590224 0.807240i 0.700960π-0.700960\pi
0.994202 + 0.107529i 0.0342938π0.0342938\pi
278278 0 0
279279 −1.35900e8 −0.374631
280280 0 0
281281 5.50944e8 1.48127 0.740637 0.671905i 0.234524π-0.234524\pi
0.740637 + 0.671905i 0.234524π0.234524\pi
282282 0 0
283283 −1.42073e8 + 2.46077e8i −0.372613 + 0.645385i −0.989967 0.141301i 0.954871π-0.954871\pi
0.617354 + 0.786686i 0.288205π0.288205\pi
284284 0 0
285285 3.52249e8 + 6.10113e8i 0.901348 + 1.56118i
286286 0 0
287287 0 0
288288 0 0
289289 −1.52616e8 2.64338e8i −0.371926 0.644195i
290290 0 0
291291 2.22668e8 3.85673e8i 0.529704 0.917474i
292292 0 0
293293 4.33632e7 0.100713 0.0503564 0.998731i 0.483964π-0.483964\pi
0.0503564 + 0.998731i 0.483964π0.483964\pi
294294 0 0
295295 7.62276e8 1.72876
296296 0 0
297297 2.24568e8 3.88964e8i 0.497394 0.861512i
298298 0 0
299299 −2.52642e8 4.37588e8i −0.546583 0.946709i
300300 0 0
301301 0 0
302302 0 0
303303 1.33826e8 + 2.31794e8i 0.276371 + 0.478688i
304304 0 0
305305 −1.11907e7 + 1.93829e7i −0.0225844 + 0.0391173i
306306 0 0
307307 −4.07582e8 −0.803954 −0.401977 0.915650i 0.631677π-0.631677\pi
−0.401977 + 0.915650i 0.631677π0.631677\pi
308308 0 0
309309 1.61802e8 0.311983
310310 0 0
311311 −3.84132e8 + 6.65336e8i −0.724134 + 1.25424i 0.235195 + 0.971948i 0.424427π0.424427\pi
−0.959329 + 0.282289i 0.908906π0.908906\pi
312312 0 0
313313 −2.15655e8 3.73525e8i −0.397515 0.688517i 0.595903 0.803056i 0.296794π-0.296794\pi
−0.993419 + 0.114539i 0.963461π0.963461\pi
314314 0 0
315315 0 0
316316 0 0
317317 −7.89918e7 1.36818e8i −0.139276 0.241232i 0.787947 0.615743i 0.211144π-0.211144\pi
−0.927223 + 0.374511i 0.877811π0.877811\pi
318318 0 0
319319 −7.23797e7 + 1.25365e8i −0.124839 + 0.216227i
320320 0 0
321321 −2.58051e8 −0.435450
322322 0 0
323323 −9.79323e8 −1.61703
324324 0 0
325325 1.21763e9 2.10899e9i 1.96754 3.40787i
326326 0 0
327327 2.15909e8 + 3.73965e8i 0.341471 + 0.591445i
328328 0 0
329329 0 0
330330 0 0
331331 −4.87683e8 8.44692e8i −0.739162 1.28027i −0.952873 0.303370i 0.901888π-0.901888\pi
0.213711 0.976897i 0.431445π-0.431445\pi
332332 0 0
333333 −3.91461e7 + 6.78030e7i −0.0580943 + 0.100622i
334334 0 0
335335 −1.53144e9 −2.22557
336336 0 0
337337 −9.07880e8 −1.29218 −0.646091 0.763260i 0.723597π-0.723597\pi
−0.646091 + 0.763260i 0.723597π0.723597\pi
338338 0 0
339339 −8.10870e7 + 1.40447e8i −0.113045 + 0.195800i
340340 0 0
341341 −4.00691e8 6.94017e8i −0.547229 0.947828i
342342 0 0
343343 0 0
344344 0 0
345345 −3.35989e8 5.81949e8i −0.440512 0.762989i
346346 0 0
347347 2.89860e7 5.02052e7i 0.0372422 0.0645053i −0.846804 0.531906i 0.821476π-0.821476\pi
0.884046 + 0.467400i 0.154809π0.154809\pi
348348 0 0
349349 −1.33879e9 −1.68587 −0.842935 0.538016i 0.819174π-0.819174\pi
−0.842935 + 0.538016i 0.819174π0.819174\pi
350350 0 0
351351 1.61055e9 1.98792
352352 0 0
353353 −2.91117e8 + 5.04230e8i −0.352254 + 0.610122i −0.986644 0.162891i 0.947918π-0.947918\pi
0.634390 + 0.773013i 0.281251π0.281251\pi
354354 0 0
355355 −4.19596e8 7.26762e8i −0.497774 0.862170i
356356 0 0
357357 0 0
358358 0 0
359359 1.60217e8 + 2.77505e8i 0.182759 + 0.316548i 0.942819 0.333305i 0.108164π-0.108164\pi
−0.760060 + 0.649853i 0.774830π0.774830\pi
360360 0 0
361361 −2.23210e8 + 3.86612e8i −0.249712 + 0.432514i
362362 0 0
363363 −1.24219e8 −0.136306
364364 0 0
365365 1.92078e9 2.06753
366366 0 0
367367 −3.27639e8 + 5.67487e8i −0.345990 + 0.599273i −0.985533 0.169483i 0.945790π-0.945790\pi
0.639543 + 0.768755i 0.279124π0.279124\pi
368368 0 0
369369 8.41646e7 + 1.45777e8i 0.0872042 + 0.151042i
370370 0 0
371371 0 0
372372 0 0
373373 2.71464e8 + 4.70189e8i 0.270851 + 0.469128i 0.969080 0.246746i 0.0793615π-0.0793615\pi
−0.698229 + 0.715875i 0.746028π0.746028\pi
374374 0 0
375375 8.67635e8 1.50279e9i 0.849626 1.47159i
376376 0 0
377377 −5.19090e8 −0.498940
378378 0 0
379379 −1.57480e9 −1.48590 −0.742949 0.669348i 0.766573π-0.766573\pi
−0.742949 + 0.669348i 0.766573π0.766573\pi
380380 0 0
381381 −7.08038e8 + 1.22636e9i −0.655872 + 1.13600i
382382 0 0
383383 4.44411e8 + 7.69742e8i 0.404193 + 0.700083i 0.994227 0.107295i 0.0342190π-0.0342190\pi
−0.590034 + 0.807378i 0.700886π0.700886\pi
384384 0 0
385385 0 0
386386 0 0
387387 1.03410e7 + 1.79112e7i 0.00906933 + 0.0157085i
388388 0 0
389389 −6.73363e8 + 1.16630e9i −0.579997 + 1.00458i 0.415482 + 0.909601i 0.363613π0.363613\pi
−0.995479 + 0.0949829i 0.969720π0.969720\pi
390390 0 0
391391 9.34116e8 0.790282
392392 0 0
393393 1.96990e9 1.63708
394394 0 0
395395 4.98670e8 8.63722e8i 0.407121 0.705154i
396396 0 0
397397 −1.07964e9 1.86998e9i −0.865985 1.49993i −0.866066 0.499930i 0.833359π-0.833359\pi
8.13447e−5 1.00000i 0.499974π-0.499974\pi
398398 0 0
399399 0 0
400400 0 0
401401 9.38083e8 + 1.62481e9i 0.726500 + 1.25834i 0.958354 + 0.285585i 0.0921877π0.0921877\pi
−0.231853 + 0.972751i 0.574479π0.574479\pi
402402 0 0
403403 1.43683e9 2.48867e9i 1.09355 1.89408i
404404 0 0
405405 1.39897e9 1.04644
406406 0 0
407407 −4.61678e8 −0.339437
408408 0 0
409409 −1.13724e6 + 1.96976e6i −0.000821906 + 0.00142358i −0.866436 0.499288i 0.833595π-0.833595\pi
0.865614 + 0.500712i 0.166928π0.166928\pi
410410 0 0
411411 −5.89131e8 1.02041e9i −0.418568 0.724980i
412412 0 0
413413 0 0
414414 0 0
415415 −2.05694e9 3.56272e9i −1.41271 2.44688i
416416 0 0
417417 5.66067e8 9.80456e8i 0.382289 0.662143i
418418 0 0
419419 −2.40944e9 −1.60017 −0.800087 0.599884i 0.795213π-0.795213\pi
−0.800087 + 0.599884i 0.795213π0.795213\pi
420420 0 0
421421 −1.08307e9 −0.707404 −0.353702 0.935358i 0.615077π-0.615077\pi
−0.353702 + 0.935358i 0.615077π0.615077\pi
422422 0 0
423423 −1.04434e8 + 1.80886e8i −0.0670891 + 0.116202i
424424 0 0
425425 2.25103e9 + 3.89889e9i 1.42239 + 2.46366i
426426 0 0
427427 0 0
428428 0 0
429429 1.13170e9 + 1.96016e9i 0.692038 + 1.19865i
430430 0 0
431431 −2.52993e8 + 4.38197e8i −0.152208 + 0.263633i −0.932039 0.362358i 0.881972π-0.881972\pi
0.779831 + 0.625991i 0.215305π0.215305\pi
432432 0 0
433433 3.73323e8 0.220992 0.110496 0.993877i 0.464756π-0.464756\pi
0.110496 + 0.993877i 0.464756π0.464756\pi
434434 0 0
435435 −6.90339e8 −0.402115
436436 0 0
437437 6.39212e8 1.10715e9i 0.366404 0.634630i
438438 0 0
439439 4.44465e6 + 7.69836e6i 0.00250733 + 0.00434283i 0.867276 0.497827i 0.165869π-0.165869\pi
−0.864769 + 0.502170i 0.832535π0.832535\pi
440440 0 0
441441 0 0
442442 0 0
443443 −5.12024e8 8.86851e8i −0.279819 0.484661i 0.691521 0.722357i 0.256941π-0.256941\pi
−0.971340 + 0.237696i 0.923608π0.923608\pi
444444 0 0
445445 −1.61489e9 + 2.79707e9i −0.868726 + 1.50468i
446446 0 0
447447 −1.43798e9 −0.761510
448448 0 0
449449 8.32711e8 0.434142 0.217071 0.976156i 0.430350π-0.430350\pi
0.217071 + 0.976156i 0.430350π0.430350\pi
450450 0 0
451451 −4.96308e8 + 8.59630e8i −0.254761 + 0.441259i
452452 0 0
453453 −3.53524e8 6.12321e8i −0.178679 0.309482i
454454 0 0
455455 0 0
456456 0 0
457457 1.14391e9 + 1.98130e9i 0.560639 + 0.971055i 0.997441 + 0.0714976i 0.0227778π0.0227778\pi
−0.436802 + 0.899558i 0.643889π0.643889\pi
458458 0 0
459459 −1.48871e9 + 2.57852e9i −0.718565 + 1.24459i
460460 0 0
461461 7.84459e8 0.372921 0.186461 0.982462i 0.440298π-0.440298\pi
0.186461 + 0.982462i 0.440298π0.440298\pi
462462 0 0
463463 −2.31590e9 −1.08439 −0.542196 0.840252i 0.682407π-0.682407\pi
−0.542196 + 0.840252i 0.682407π0.682407\pi
464464 0 0
465465 1.91085e9 3.30968e9i 0.881333 1.52651i
466466 0 0
467467 3.06330e8 + 5.30578e8i 0.139181 + 0.241068i 0.927187 0.374599i 0.122220π-0.122220\pi
−0.788006 + 0.615668i 0.788886π0.788886\pi
468468 0 0
469469 0 0
470470 0 0
471471 −4.11031e8 7.11927e8i −0.181260 0.313951i
472472 0 0
473473 −6.09796e7 + 1.05620e8i −0.0264954 + 0.0458914i
474474 0 0
475475 6.16147e9 2.63789
476476 0 0
477477 8.28356e8 0.349464
478478 0 0
479479 −3.64475e8 + 6.31289e8i −0.151528 + 0.262454i −0.931789 0.362999i 0.881753π-0.881753\pi
0.780261 + 0.625454i 0.215086π0.215086\pi
480480 0 0
481481 −8.27763e8 1.43373e9i −0.339155 0.587434i
482482 0 0
483483 0 0
484484 0 0
485485 −2.85144e9 4.93883e9i −1.13493 1.96575i
486486 0 0
487487 4.34192e8 7.52043e8i 0.170346 0.295047i −0.768195 0.640216i 0.778845π-0.778845\pi
0.938541 + 0.345169i 0.112178π0.112178\pi
488488 0 0
489489 1.74811e9 0.676063
490490 0 0
491491 −3.47503e9 −1.32487 −0.662435 0.749119i 0.730477π-0.730477\pi
−0.662435 + 0.749119i 0.730477π0.730477\pi
492492 0 0
493493 4.79821e8 8.31074e8i 0.180349 0.312374i
494494 0 0
495495 −6.85360e8 1.18708e9i −0.253980 0.439907i
496496 0 0
497497 0 0
498498 0 0
499499 1.84086e9 + 3.18846e9i 0.663236 + 1.14876i 0.979760 + 0.200174i 0.0641507π0.0641507\pi
−0.316524 + 0.948584i 0.602516π0.602516\pi
500500 0 0
501501 −1.22656e9 + 2.12447e9i −0.435770 + 0.754775i
502502 0 0
503503 3.79099e9 1.32820 0.664102 0.747642i 0.268814π-0.268814\pi
0.664102 + 0.747642i 0.268814π0.268814\pi
504504 0 0
505505 3.42749e9 1.18429
506506 0 0
507507 −2.84193e9 + 4.92236e9i −0.968467 + 1.67743i
508508 0 0
509509 2.04656e9 + 3.54475e9i 0.687880 + 1.19144i 0.972522 + 0.232810i 0.0747920π0.0747920\pi
−0.284642 + 0.958634i 0.591875π0.591875\pi
510510 0 0
511511 0 0
512512 0 0
513513 2.03744e9 + 3.52895e9i 0.666306 + 1.15408i
514514 0 0
515515 1.03600e9 1.79441e9i 0.334222 0.578890i
516516 0 0
517517 −1.23167e9 −0.391993
518518 0 0
519519 −1.29549e9 −0.406770
520520 0 0
521521 −8.47729e8 + 1.46831e9i −0.262618 + 0.454868i −0.966937 0.255016i 0.917919π-0.917919\pi
0.704319 + 0.709884i 0.251253π0.251253\pi
522522 0 0
523523 7.68571e8 + 1.33120e9i 0.234924 + 0.406901i 0.959251 0.282556i 0.0911824π-0.0911824\pi
−0.724326 + 0.689457i 0.757849π0.757849\pi
524524 0 0
525525 0 0
526526 0 0
527527 2.65627e9 + 4.60079e9i 0.790560 + 1.36929i
528528 0 0
529529 1.09271e9 1.89263e9i 0.320929 0.555866i
530530 0 0
531531 1.05078e9 0.304566
532532 0 0
533533 −3.55940e9 −1.01820
534534 0 0
535535 −1.65227e9 + 2.86182e9i −0.466491 + 0.807986i
536536 0 0
537537 2.51904e9 + 4.36310e9i 0.701980 + 1.21587i
538538 0 0
539539 0 0
540540 0 0
541541 2.17255e9 + 3.76296e9i 0.589901 + 1.02174i 0.994245 + 0.107131i 0.0341664π0.0341664\pi
−0.404344 + 0.914607i 0.632500π0.632500\pi
542542 0 0
543543 −3.71380e8 + 6.43249e8i −0.0995449 + 0.172417i
544544 0 0
545545 5.52976e9 1.46325
546546 0 0
547547 2.30195e9 0.601367 0.300683 0.953724i 0.402785π-0.402785\pi
0.300683 + 0.953724i 0.402785π0.402785\pi
548548 0 0
549549 −1.54262e7 + 2.67189e7i −0.00397882 + 0.00689152i
550550 0 0
551551 −6.56678e8 1.13740e9i −0.167233 0.289656i
552552 0 0
553553 0 0
554554 0 0
555555 −1.10084e9 1.90672e9i −0.273338 0.473435i
556556 0 0
557557 −8.55534e8 + 1.48183e9i −0.209770 + 0.363333i −0.951642 0.307209i 0.900605π-0.900605\pi
0.741872 + 0.670542i 0.233938π0.233938\pi
558558 0 0
559559 −4.37332e8 −0.105894
560560 0 0
561561 −4.18434e9 −1.00059
562562 0 0
563563 −1.08317e9 + 1.87610e9i −0.255809 + 0.443074i −0.965115 0.261827i 0.915675π-0.915675\pi
0.709306 + 0.704901i 0.249008π0.249008\pi
564564 0 0
565565 1.03838e9 + 1.79853e9i 0.242207 + 0.419515i
566566 0 0
567567 0 0
568568 0 0
569569 1.38031e9 + 2.39077e9i 0.314112 + 0.544059i 0.979248 0.202664i 0.0649598π-0.0649598\pi
−0.665136 + 0.746722i 0.731626π0.731626\pi
570570 0 0
571571 −3.62954e8 + 6.28654e8i −0.0815877 + 0.141314i −0.903932 0.427676i 0.859332π-0.859332\pi
0.822344 + 0.568990i 0.192666π0.192666\pi
572572 0 0
573573 1.37368e9 0.305032
574574 0 0
575575 −5.87705e9 −1.28921
576576 0 0
577577 2.68517e9 4.65086e9i 0.581911 1.00790i −0.413341 0.910576i 0.635638π-0.635638\pi
0.995253 0.0973240i 0.0310283π-0.0310283\pi
578578 0 0
579579 −3.49540e9 6.05421e9i −0.748380 1.29623i
580580 0 0
581581 0 0
582582 0 0
583583 2.44235e9 + 4.23028e9i 0.510468 + 0.884157i
584584 0 0
585585 2.45762e9 4.25672e9i 0.507539 0.879083i
586586 0 0
587587 −7.86145e9 −1.60424 −0.802120 0.597163i 0.796295π-0.796295\pi
−0.802120 + 0.597163i 0.796295π0.796295\pi
588588 0 0
589589 7.27069e9 1.46613
590590 0 0
591591 2.70313e9 4.68195e9i 0.538654 0.932976i
592592 0 0
593593 −2.72206e9 4.71475e9i −0.536051 0.928468i −0.999112 0.0421412i 0.986582π-0.986582\pi
0.463060 0.886327i 0.346751π-0.346751\pi
594594 0 0
595595 0 0
596596 0 0
597597 −2.83588e9 4.91189e9i −0.545479 0.944798i
598598 0 0
599599 −3.98704e9 + 6.90575e9i −0.757978 + 1.31286i 0.185902 + 0.982568i 0.440479π0.440479\pi
−0.943880 + 0.330288i 0.892854π0.892854\pi
600600 0 0
601601 −6.27378e8 −0.117888 −0.0589439 0.998261i 0.518773π-0.518773\pi
−0.0589439 + 0.998261i 0.518773π0.518773\pi
602602 0 0
603603 −2.11105e9 −0.392092
604604 0 0
605605 −7.95362e8 + 1.37761e9i −0.146023 + 0.252919i
606606 0 0
607607 −1.89879e9 3.28880e9i −0.344601 0.596866i 0.640680 0.767808i 0.278652π-0.278652\pi
−0.985281 + 0.170941i 0.945319π0.945319\pi
608608 0 0
609609 0 0
610610 0 0
611611 −2.20832e9 3.82491e9i −0.391667 0.678387i
612612 0 0
613613 −5.30599e9 + 9.19025e9i −0.930369 + 1.61145i −0.147677 + 0.989036i 0.547180π0.547180\pi
−0.782691 + 0.622410i 0.786154π0.786154\pi
614614 0 0
615615 −4.73366e9 −0.820605
616616 0 0
617617 2.26828e8 0.0388776 0.0194388 0.999811i 0.493812π-0.493812\pi
0.0194388 + 0.999811i 0.493812π0.493812\pi
618618 0 0
619619 −2.45900e9 + 4.25912e9i −0.416718 + 0.721776i −0.995607 0.0936299i 0.970153π-0.970153\pi
0.578889 + 0.815406i 0.303486π0.303486\pi
620620 0 0
621621 −1.94339e9 3.36605e9i −0.325641 0.564027i
622622 0 0
623623 0 0
624624 0 0
625625 −4.53650e9 7.85745e9i −0.743260 1.28736i
626626 0 0
627627 −2.86332e9 + 4.95942e9i −0.463910 + 0.803516i
628628 0 0
629629 3.06057e9 0.490371
630630 0 0
631631 6.22228e9 0.985931 0.492966 0.870049i 0.335913π-0.335913\pi
0.492966 + 0.870049i 0.335913π0.335913\pi
632632 0 0
633633 1.99512e9 3.45565e9i 0.312648 0.541523i
634634 0 0
635635 9.06697e9 + 1.57044e10i 1.40525 + 2.43397i
636636 0 0
637637 0 0
638638 0 0
639639 −5.78404e8 1.00182e9i −0.0876957 0.151893i
640640 0 0
641641 1.35677e9 2.35000e9i 0.203472 0.352424i −0.746173 0.665752i 0.768111π-0.768111\pi
0.949645 + 0.313328i 0.101444π0.101444\pi
642642 0 0
643643 −5.96086e9 −0.884240 −0.442120 0.896956i 0.645774π-0.645774\pi
−0.442120 + 0.896956i 0.645774π0.645774\pi
644644 0 0
645645 −5.81609e8 −0.0853438
646646 0 0
647647 −9.35684e8 + 1.62065e9i −0.135820 + 0.235247i −0.925910 0.377743i 0.876700π-0.876700\pi
0.790090 + 0.612990i 0.210034π0.210034\pi
648648 0 0
649649 3.09815e9 + 5.36616e9i 0.444884 + 0.770561i
650650 0 0
651651 0 0
652652 0 0
653653 −4.34121e9 7.51920e9i −0.610120 1.05676i −0.991220 0.132225i 0.957788π-0.957788\pi
0.381100 0.924534i 0.375545π-0.375545\pi
654654 0 0
655655 1.26131e10 2.18464e10i 1.75378 3.03764i
656656 0 0
657657 2.64775e9 0.364249
658658 0 0
659659 1.17028e10 1.59291 0.796457 0.604696i 0.206705π-0.206705\pi
0.796457 + 0.604696i 0.206705π0.206705\pi
660660 0 0
661661 2.11762e9 3.66782e9i 0.285195 0.493973i −0.687461 0.726221i 0.741275π-0.741275\pi
0.972657 + 0.232248i 0.0746082π0.0746082\pi
662662 0 0
663663 −7.50228e9 1.29943e10i −0.999760 1.73164i
664664 0 0
665665 0 0
666666 0 0
667667 6.26365e8 + 1.08490e9i 0.0817311 + 0.141562i
668668 0 0
669669 6.32731e9 1.09592e10i 0.817010 1.41510i
670670 0 0
671671 −1.81932e8 −0.0232477
672672 0 0
673673 −1.64366e9 −0.207854 −0.103927 0.994585i 0.533141π-0.533141\pi
−0.103927 + 0.994585i 0.533141π0.533141\pi
674674 0 0
675675 9.36633e9 1.62230e10i 1.17221 2.03033i
676676 0 0
677677 −5.44190e9 9.42565e9i −0.674048 1.16748i −0.976746 0.214398i 0.931221π-0.931221\pi
0.302699 0.953086i 0.402112π-0.402112\pi
678678 0 0
679679 0 0
680680 0 0
681681 1.74352e9 + 3.01986e9i 0.211549 + 0.366414i
682682 0 0
683683 1.92639e9 3.33660e9i 0.231351 0.400711i −0.726855 0.686791i 0.759019π-0.759019\pi
0.958206 + 0.286080i 0.0923522π0.0923522\pi
684684 0 0
685685 −1.50885e10 −1.79362
686686 0 0
687687 −3.95126e9 −0.464930
688688 0 0
689689 −8.75800e9 + 1.51693e10i −1.02009 + 1.76684i
690690 0 0
691691 7.77195e9 + 1.34614e10i 0.896100 + 1.55209i 0.832437 + 0.554120i 0.186945π0.186945\pi
0.0636633 + 0.997971i 0.479722π0.479722\pi
692692 0 0
693693 0 0
694694 0 0
695695 −7.24892e9 1.25555e10i −0.819080 1.41869i
696696 0 0
697697 3.29013e9 5.69868e9i 0.368043 0.637469i
698698 0 0
699699 −1.65493e9 −0.183277
700700 0 0
701701 −7.75099e9 −0.849854 −0.424927 0.905228i 0.639700π-0.639700\pi
−0.424927 + 0.905228i 0.639700π0.639700\pi
702702 0 0
703703 2.09433e9 3.62749e9i 0.227354 0.393788i
704704 0 0
705705 −2.93684e9 5.08676e9i −0.315660 0.546738i
706706 0 0
707707 0 0
708708 0 0
709709 5.73320e9 + 9.93020e9i 0.604137 + 1.04640i 0.992187 + 0.124758i 0.0398154π0.0398154\pi
−0.388050 + 0.921638i 0.626851π0.626851\pi
710710 0 0
711711 6.87405e8 1.19062e9i 0.0717248 0.124231i
712712 0 0
713713 −6.93507e9 −0.716535
714714 0 0
715715 2.89845e10 2.96548
716716 0 0
717717 1.16713e9 2.02154e9i 0.118251 0.204816i
718718 0 0
719719 8.19259e9 + 1.41900e10i 0.821997 + 1.42374i 0.904193 + 0.427125i 0.140473π0.140473\pi
−0.0821957 + 0.996616i 0.526193π0.526193\pi
720720 0 0
721721 0 0
722722 0 0
723723 9.13949e8 + 1.58301e9i 0.0899369 + 0.155775i
724724 0 0
725725 −3.01882e9 + 5.22875e9i −0.294208 + 0.509583i
726726 0 0
727727 1.66277e10 1.60495 0.802474 0.596687i 0.203517π-0.203517\pi
0.802474 + 0.596687i 0.203517π0.203517\pi
728728 0 0
729729 1.13647e10 1.08646
730730 0 0
731731 4.04247e8 7.00177e8i 0.0382769 0.0662975i
732732 0 0
733733 −7.28863e9 1.26243e10i −0.683568 1.18398i −0.973884 0.227044i 0.927094π-0.927094\pi
0.290316 0.956931i 0.406240π-0.406240\pi
734734 0 0
735735 0 0
736736 0 0
737737 −6.22429e9 1.07808e10i −0.572735 0.992006i
738738 0 0
739739 −2.15001e9 + 3.72393e9i −0.195968 + 0.339427i −0.947218 0.320592i 0.896118π-0.896118\pi
0.751249 + 0.660019i 0.229452π0.229452\pi
740740 0 0
741741 −2.05351e10 −1.85410
742742 0 0
743743 −9.14731e9 −0.818150 −0.409075 0.912501i 0.634148π-0.634148\pi
−0.409075 + 0.912501i 0.634148π0.634148\pi
744744 0 0
745745 −9.20720e9 + 1.59473e10i −0.815794 + 1.41300i
746746 0 0
747747 −2.83544e9 4.91113e9i −0.248885 0.431081i
748748 0 0
749749 0 0
750750 0 0
751751 −8.59368e8 1.48847e9i −0.0740354 0.128233i 0.826631 0.562744i 0.190254π-0.190254\pi
−0.900666 + 0.434511i 0.856921π0.856921\pi
752752 0 0
753753 −3.76045e9 + 6.51328e9i −0.320964 + 0.555927i
754754 0 0
755755 −9.05428e9 −0.765666
756756 0 0
757757 −1.58386e10 −1.32703 −0.663517 0.748161i 0.730937π-0.730937\pi
−0.663517 + 0.748161i 0.730937π0.730937\pi
758758 0 0
759759 2.73115e9 4.73049e9i 0.226725 0.392699i
760760 0 0
761761 −1.64664e9 2.85207e9i −0.135442 0.234592i 0.790324 0.612689i 0.209912π-0.209912\pi
−0.925766 + 0.378096i 0.876579π0.876579\pi
762762 0 0
763763 0 0
764764 0 0
765765 4.54340e9 + 7.86940e9i 0.366915 + 0.635516i
766766 0 0
767767 −1.11096e10 + 1.92424e10i −0.889028 + 1.53984i
768768 0 0
769769 −1.80581e10 −1.43196 −0.715978 0.698123i 0.754019π-0.754019\pi
−0.715978 + 0.698123i 0.754019π0.754019\pi
770770 0 0
771771 −2.63113e9 −0.206753
772772 0 0
773773 7.87646e9 1.36424e10i 0.613342 1.06234i −0.377331 0.926079i 0.623158π-0.623158\pi
0.990673 0.136261i 0.0435087π-0.0435087\pi
774774 0 0
775775 −1.67121e10 2.89462e10i −1.28966 2.23375i
776776 0 0
777777 0 0
778778 0 0
779779 −4.50285e9 7.79916e9i −0.341276 0.591108i
780780 0 0
781781 3.41077e9 5.90763e9i 0.256197 0.443746i
782782 0 0
783783 −3.99298e9 −0.297257
784784 0 0
785785 −1.05271e10 −0.776724
786786 0 0
787787 −5.11120e9 + 8.85285e9i −0.373776 + 0.647398i −0.990143 0.140060i 0.955270π-0.955270\pi
0.616367 + 0.787459i 0.288604π0.288604\pi
788788 0 0
789789 −3.12443e9 5.41167e9i −0.226465 0.392249i
790790 0 0
791791 0 0
792792 0 0
793793 −3.26193e8 5.64984e8i −0.0232284 0.0402328i
794794 0 0
795795 −1.16473e10 + 2.01737e10i −0.822128 + 1.42397i
796796 0 0
797797 1.44391e10 1.01027 0.505134 0.863041i 0.331443π-0.331443\pi
0.505134 + 0.863041i 0.331443π0.331443\pi
798798 0 0
799799 8.16502e9 0.566296
800800 0 0
801801 −2.22609e9 + 3.85570e9i −0.153048 + 0.265088i
802802 0 0
803803 7.80672e9 + 1.35216e10i 0.532064 + 0.921562i
804804 0 0
805805 0 0
806806 0 0
807807 5.11897e9 + 8.86632e9i 0.342867 + 0.593863i
808808 0 0
809809 5.04427e9 8.73692e9i 0.334949 0.580148i −0.648526 0.761192i 0.724614π-0.724614\pi
0.983475 + 0.181044i 0.0579477π0.0579477\pi
810810 0 0
811811 1.35981e9 0.0895172 0.0447586 0.998998i 0.485748π-0.485748\pi
0.0447586 + 0.998998i 0.485748π0.485748\pi
812812 0 0
813813 −1.90890e9 −0.124585
814814 0 0
815815 1.11929e10 1.93867e10i 0.724256 1.25445i
816816 0 0
817817 −5.53250e8 9.58256e8i −0.0354931 0.0614758i
818818 0 0
819819 0 0
820820 0 0
821821 7.49011e9 + 1.29733e10i 0.472375 + 0.818178i 0.999500 0.0316097i 0.0100633π-0.0100633\pi
−0.527125 + 0.849788i 0.676730π0.676730\pi
822822 0 0
823823 9.80080e9 1.69755e10i 0.612861 1.06151i −0.377895 0.925849i 0.623352π-0.623352\pi
0.990756 0.135658i 0.0433148π-0.0433148\pi
824824 0 0
825825 2.63260e10 1.63229
826826 0 0
827827 1.68768e9 0.103758 0.0518789 0.998653i 0.483479π-0.483479\pi
0.0518789 + 0.998653i 0.483479π0.483479\pi
828828 0 0
829829 8.22703e8 1.42496e9i 0.0501536 0.0868686i −0.839859 0.542805i 0.817362π-0.817362\pi
0.890012 + 0.455936i 0.150696π0.150696\pi
830830 0 0
831831 5.53955e9 + 9.59478e9i 0.334866 + 0.580005i
832832 0 0
833833 0 0
834834 0 0
835835 1.57071e10 + 2.72054e10i 0.933667 + 1.61716i
836836 0 0
837837 1.10525e10 1.91435e10i 0.651510 1.12845i
838838 0 0
839839 −3.49725e9 −0.204437 −0.102219 0.994762i 0.532594π-0.532594\pi
−0.102219 + 0.994762i 0.532594π0.532594\pi
840840 0 0
841841 −1.59629e10 −0.925393
842842 0 0
843843 −1.06786e10 + 1.84959e10i −0.613929 + 1.06336i
844844 0 0
845845 3.63930e10 + 6.30346e10i 2.07501 + 3.59402i
846846 0 0
847847 0 0
848848 0 0
849849 −5.50741e9 9.53912e9i −0.308866 0.534972i
850850 0 0
851851 −1.99766e9 + 3.46004e9i −0.111114 + 0.192454i
852852 0 0
853853 2.86798e10 1.58218 0.791088 0.611702i 0.209515π-0.209515\pi
0.791088 + 0.611702i 0.209515π0.209515\pi
854854 0 0
855855 1.24361e10 0.680461
856856 0 0
857857 −1.83667e10 + 3.18120e10i −0.996776 + 1.72647i −0.428901 + 0.903351i 0.641099π0.641099\pi
−0.567875 + 0.823115i 0.692234π0.692234\pi
858858 0 0
859859 −2.77367e9 4.80413e9i −0.149306 0.258606i 0.781665 0.623699i 0.214371π-0.214371\pi
−0.930971 + 0.365092i 0.881037π0.881037\pi
860860 0 0
861861 0 0
862862 0 0
863863 −2.63029e9 4.55579e9i −0.139305 0.241283i 0.787929 0.615766i 0.211153π-0.211153\pi
−0.927234 + 0.374484i 0.877820π0.877820\pi
864864 0 0
865865 −8.29487e9 + 1.43671e10i −0.435766 + 0.754769i
866866 0 0
867867 1.18322e10 0.616594
868868 0 0
869869 8.10707e9 0.419078
870870 0 0
871871 2.23196e10 3.86587e10i 1.14452 1.98236i
872872 0 0
873873 −3.93064e9 6.80807e9i −0.199946 0.346317i
874874 0 0
875875 0 0
876876 0 0
877877 −1.41580e10 2.45224e10i −0.708767 1.22762i −0.965315 0.261089i 0.915918π-0.915918\pi
0.256548 0.966532i 0.417415π-0.417415\pi
878878 0 0
879879 −8.40482e8 + 1.45576e9i −0.0417414 + 0.0722983i
880880 0 0
881881 −5.58205e9 −0.275029 −0.137514 0.990500i 0.543911π-0.543911\pi
−0.137514 + 0.990500i 0.543911π0.543911\pi
882882 0 0
883883 −2.81316e10 −1.37509 −0.687547 0.726140i 0.741312π-0.741312\pi
−0.687547 + 0.726140i 0.741312π0.741312\pi
884884 0 0
885885 −1.47747e10 + 2.55906e10i −0.716502 + 1.24102i
886886 0 0
887887 4.96052e6 + 8.59188e6i 0.000238668 + 0.000413386i 0.866145 0.499793i 0.166591π-0.166591\pi
−0.865906 + 0.500207i 0.833257π0.833257\pi
888888 0 0
889889 0 0
890890 0 0
891891 5.68590e9 + 9.84828e9i 0.269295 + 0.466432i
892892 0 0
893893 5.58729e9 9.67746e9i 0.262555 0.454759i
894894 0 0
895895 6.45164e10 3.00808
896896 0 0
897897 1.95872e10 0.906147
898898 0 0
899899 −3.56229e9 + 6.17006e9i −0.163520 + 0.283224i
900900 0 0
901901 −1.61909e10 2.80435e10i −0.737453 1.27731i
902902 0 0
903903 0 0
904904 0 0
905905 4.75580e9 + 8.23730e9i 0.213282 + 0.369415i
906906 0 0
907907 8.05863e8 1.39579e9i 0.0358621 0.0621149i −0.847537 0.530736i 0.821916π-0.821916\pi
0.883399 + 0.468621i 0.155249π0.155249\pi
908908 0 0
909909 4.72472e9 0.208642
910910 0 0
911911 −2.31768e10 −1.01564 −0.507818 0.861464i 0.669548π-0.669548\pi
−0.507818 + 0.861464i 0.669548π0.669548\pi
912912 0 0
913913 1.67202e10 2.89602e10i 0.727100 1.25937i
914914 0 0
915915 −4.33805e8 7.51373e8i −0.0187207 0.0324251i
916916 0 0
917917 0 0
918918 0 0
919919 6.83205e9 + 1.18335e10i 0.290367 + 0.502930i 0.973896 0.226993i 0.0728894π-0.0728894\pi
−0.683530 + 0.729923i 0.739556π0.739556\pi
920920 0 0
921921 7.89992e9 1.36831e10i 0.333207 0.577131i
922922 0 0
923923 2.44613e10 1.02394
924924 0 0
925925 −1.92558e10 −0.799953
926926 0 0
927927 1.42811e9 2.47355e9i 0.0588818 0.101986i
928928 0 0
929929 1.67141e10 + 2.89497e10i 0.683956 + 1.18465i 0.973764 + 0.227562i 0.0730753π0.0730753\pi
−0.289808 + 0.957085i 0.593591π0.593591\pi
930930 0 0
931931 0 0
932932 0 0
933933 −1.48908e10 2.57916e10i −0.600250 1.03966i
934934 0 0
935935 −2.67918e10 + 4.64048e10i −1.07192 + 1.85662i
936936 0 0
937937 1.04114e10 0.413450 0.206725 0.978399i 0.433720π-0.433720\pi
0.206725 + 0.978399i 0.433720π0.433720\pi
938938 0 0
939939 1.67196e10 0.659017
940940 0 0
941941 −2.50008e10 + 4.33027e10i −0.978117 + 1.69415i −0.308877 + 0.951102i 0.599953π0.599953\pi
−0.669240 + 0.743047i 0.733380π0.733380\pi
942942 0 0
943943 4.29499e9 + 7.43914e9i 0.166790 + 0.288890i
944944 0 0
945945 0 0
946946 0 0
947947 9.13649e9 + 1.58249e10i 0.349586 + 0.605501i 0.986176 0.165702i 0.0529888π-0.0529888\pi
−0.636590 + 0.771203i 0.719656π0.719656\pi
948948 0 0
949949 −2.79940e10 + 4.84870e10i −1.06324 + 1.84159i
950950 0 0
951951 6.12420e9 0.230897
952952 0 0
953953 −1.24663e10 −0.466565 −0.233282 0.972409i 0.574947π-0.574947\pi
−0.233282 + 0.972409i 0.574947π0.574947\pi
954954 0 0
955955 8.79553e9 1.52343e10i 0.326776 0.565993i
956956 0 0
957957 −2.80578e9 4.85975e9i −0.103481 0.179235i
958958 0 0
959959 0 0
960960 0 0
961961 −5.96436e9 1.03306e10i −0.216786 0.375485i
962962 0 0
963963 −2.27762e9 + 3.94495e9i −0.0821844 + 0.142348i
964964 0 0
965965 −8.95225e10 −3.20691
966966 0 0
967967 4.12045e9 0.146539 0.0732693 0.997312i 0.476657π-0.476657\pi
0.0732693 + 0.997312i 0.476657π0.476657\pi
968968 0 0
969969 1.89816e10 3.28771e10i 0.670193 1.16081i
970970 0 0
971971 6.21190e9 + 1.07593e10i 0.217749 + 0.377153i 0.954120 0.299426i 0.0967951π-0.0967951\pi
−0.736370 + 0.676579i 0.763462π0.763462\pi
972972 0 0
973973 0 0
974974 0 0
975975 4.72011e10 + 8.17547e10i 1.63093 + 2.82486i
976976 0 0
977977 −2.25652e10 + 3.90840e10i −0.774119 + 1.34081i 0.161170 + 0.986927i 0.448473π0.448473\pi
−0.935289 + 0.353886i 0.884860π0.884860\pi
978978 0 0
979979 −2.62539e10 −0.894241
980980 0 0
981981 7.62265e9 0.257789
982982 0 0
983983 −2.87779e9 + 4.98448e9i −0.0966321 + 0.167372i −0.910289 0.413974i 0.864140π-0.864140\pi
0.813656 + 0.581346i 0.197474π0.197474\pi
984984 0 0
985985 −3.46156e10 5.99560e10i −1.15410 1.99897i
986986 0 0
987987 0 0
988988 0 0
989989 5.27711e8 + 9.14022e8i 0.0173464 + 0.0300448i
990990 0 0
991991 1.82054e10 3.15327e10i 0.594214 1.02921i −0.399443 0.916758i 0.630797π-0.630797\pi
0.993657 0.112451i 0.0358701π-0.0358701\pi
992992 0 0
993993 3.78098e10 1.22541
994994 0 0
995995 −7.26313e10 −2.33745
996996 0 0
997997 −5.65165e9 + 9.78895e9i −0.180610 + 0.312826i −0.942089 0.335364i 0.891141π-0.891141\pi
0.761478 + 0.648190i 0.224474π0.224474\pi
998998 0 0
999999 −6.36738e9 1.10286e10i −0.202060 0.349979i
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 196.8.e.b.165.1 4
7.2 even 3 inner 196.8.e.b.177.1 4
7.3 odd 6 196.8.a.a.1.1 2
7.4 even 3 28.8.a.b.1.2 2
7.5 odd 6 196.8.e.c.177.2 4
7.6 odd 2 196.8.e.c.165.2 4
21.11 odd 6 252.8.a.f.1.2 2
28.11 odd 6 112.8.a.h.1.1 2
56.11 odd 6 448.8.a.q.1.2 2
56.53 even 6 448.8.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.8.a.b.1.2 2 7.4 even 3
112.8.a.h.1.1 2 28.11 odd 6
196.8.a.a.1.1 2 7.3 odd 6
196.8.e.b.165.1 4 1.1 even 1 trivial
196.8.e.b.177.1 4 7.2 even 3 inner
196.8.e.c.165.2 4 7.6 odd 2
196.8.e.c.177.2 4 7.5 odd 6
252.8.a.f.1.2 2 21.11 odd 6
448.8.a.o.1.1 2 56.53 even 6
448.8.a.q.1.2 2 56.11 odd 6