Properties

Label 252.8.k.b.109.1
Level 252252
Weight 88
Character 252.109
Analytic conductor 78.72178.721
Analytic rank 00
Dimension 88
Inner twists 22

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,8,Mod(37,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.37");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: N N == 252=22327 252 = 2^{2} \cdot 3^{2} \cdot 7
Weight: k k == 8 8
Character orbit: [χ][\chi] == 252.k (of order 33, degree 22, 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: 78.721026422078.7210264220
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x8)\mathbb{Q}[x]/(x^{8} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x82x7+659x6+12718x5+417701x4+3735784x3+32480596x2+479136x+7056 x^{8} - 2x^{7} + 659x^{6} + 12718x^{5} + 417701x^{4} + 3735784x^{3} + 32480596x^{2} + 479136x + 7056 Copy content Toggle raw display
Coefficient ring: Z[a1,,a19]\Z[a_1, \ldots, a_{19}]
Coefficient ring index: 263572 2^{6}\cdot 3^{5}\cdot 7^{2}
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

Embedding invariants

Embedding label 109.1
Root 0.00737575+0.0127752i-0.00737575 + 0.0127752i of defining polynomial
Character χ\chi == 252.109
Dual form 252.8.k.b.37.1

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+(80.0854138.712i)q5+(254.682871.022i)q7+(1581.932739.98i)q114771.53q13+(6504.73+11266.5i)q17+(22058.138205.8i)q19+(32533.056348.8i)q23+(26235.245440.6i)q25+246115.q29+(150328.+260376.i)q31+(100425.+105084.i)q35+(258479.447698.i)q37+377844.q4171420.3q43+(558378.+967140.i)q47+(693817.+443667.i)q49+(184365.+319330.i)q53506758.q55+(515977.893698.i)q59+(161341.+279450.i)q61+(382130.+661869.i)q65+(11763.820375.5i)q672.84199e6q71+(336523.+582875.i)q73+(2.78947e6680072.i)q77+(429448.+743826.i)q797.32276e6q83+2.08373e6q85+(2.31400e6+4.00797e6i)q89+(1.21522e6+4.15611e6i)q91+(3.53307e6+6.11946e6i)q95+951821.q97+O(q100)q+(-80.0854 - 138.712i) q^{5} +(-254.682 - 871.022i) q^{7} +(1581.93 - 2739.98i) q^{11} -4771.53 q^{13} +(-6504.73 + 11266.5i) q^{17} +(-22058.1 - 38205.8i) q^{19} +(-32533.0 - 56348.8i) q^{23} +(26235.2 - 45440.6i) q^{25} +246115. q^{29} +(-150328. + 260376. i) q^{31} +(-100425. + 105084. i) q^{35} +(-258479. - 447698. i) q^{37} +377844. q^{41} -71420.3 q^{43} +(558378. + 967140. i) q^{47} +(-693817. + 443667. i) q^{49} +(-184365. + 319330. i) q^{53} -506758. q^{55} +(515977. - 893698. i) q^{59} +(161341. + 279450. i) q^{61} +(382130. + 661869. i) q^{65} +(11763.8 - 20375.5i) q^{67} -2.84199e6 q^{71} +(-336523. + 582875. i) q^{73} +(-2.78947e6 - 680072. i) q^{77} +(429448. + 743826. i) q^{79} -7.32276e6 q^{83} +2.08373e6 q^{85} +(2.31400e6 + 4.00797e6i) q^{89} +(1.21522e6 + 4.15611e6i) q^{91} +(-3.53307e6 + 6.11946e6i) q^{95} +951821. q^{97} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+196q5434q7406q11+3948q13+7436q17+15874q19+6788q23+69898q25+189088q2955890q31+750596q35+93742q3713944q41487844q43++23722580q97+O(q100) 8 q + 196 q^{5} - 434 q^{7} - 406 q^{11} + 3948 q^{13} + 7436 q^{17} + 15874 q^{19} + 6788 q^{23} + 69898 q^{25} + 189088 q^{29} - 55890 q^{31} + 750596 q^{35} + 93742 q^{37} - 13944 q^{41} - 487844 q^{43}+ \cdots + 23722580 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/252Z)×\left(\mathbb{Z}/252\mathbb{Z}\right)^\times.

nn 2929 7373 127127
χ(n)\chi(n) 11 e(23)e\left(\frac{2}{3}\right) 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 −80.0854 138.712i −0.286522 0.496271i 0.686455 0.727172i 0.259166π-0.259166\pi
−0.972977 + 0.230901i 0.925833π0.925833\pi
66 0 0
77 −254.682 871.022i −0.280644 0.959812i
88 0 0
99 0 0
1010 0 0
1111 1581.93 2739.98i 0.358354 0.620688i −0.629332 0.777137i 0.716671π-0.716671\pi
0.987686 + 0.156449i 0.0500046π0.0500046\pi
1212 0 0
1313 −4771.53 −0.602360 −0.301180 0.953567i 0.597380π-0.597380\pi
−0.301180 + 0.953567i 0.597380π0.597380\pi
1414 0 0
1515 0 0
1616 0 0
1717 −6504.73 + 11266.5i −0.321113 + 0.556184i −0.980718 0.195429i 0.937390π-0.937390\pi
0.659605 + 0.751612i 0.270724π0.270724\pi
1818 0 0
1919 −22058.1 38205.8i −0.737787 1.27789i −0.953490 0.301426i 0.902537π-0.902537\pi
0.215702 0.976459i 0.430796π-0.430796\pi
2020 0 0
2121 0 0
2222 0 0
2323 −32533.0 56348.8i −0.557541 0.965690i −0.997701 0.0677701i 0.978412π-0.978412\pi
0.440160 0.897919i 0.354922π-0.354922\pi
2424 0 0
2525 26235.2 45440.6i 0.335810 0.581640i
2626 0 0
2727 0 0
2828 0 0
2929 246115. 1.87390 0.936949 0.349466i 0.113637π-0.113637\pi
0.936949 + 0.349466i 0.113637π0.113637\pi
3030 0 0
3131 −150328. + 260376.i −0.906306 + 1.56977i −0.0871514 + 0.996195i 0.527776π0.527776\pi
−0.819155 + 0.573573i 0.805557π0.805557\pi
3232 0 0
3333 0 0
3434 0 0
3535 −100425. + 105084.i −0.395916 + 0.414283i
3636 0 0
3737 −258479. 447698.i −0.838916 1.45305i −0.890801 0.454393i 0.849856π-0.849856\pi
0.0518851 0.998653i 0.483477π-0.483477\pi
3838 0 0
3939 0 0
4040 0 0
4141 377844. 0.856188 0.428094 0.903734i 0.359185π-0.359185\pi
0.428094 + 0.903734i 0.359185π0.359185\pi
4242 0 0
4343 −71420.3 −0.136988 −0.0684940 0.997652i 0.521819π-0.521819\pi
−0.0684940 + 0.997652i 0.521819π0.521819\pi
4444 0 0
4545 0 0
4646 0 0
4747 558378. + 967140.i 0.784488 + 1.35877i 0.929305 + 0.369314i 0.120407π0.120407\pi
−0.144817 + 0.989458i 0.546259π0.546259\pi
4848 0 0
4949 −693817. + 443667.i −0.842478 + 0.538730i
5050 0 0
5151 0 0
5252 0 0
5353 −184365. + 319330.i −0.170104 + 0.294628i −0.938456 0.345399i 0.887744π-0.887744\pi
0.768352 + 0.640027i 0.221077π0.221077\pi
5454 0 0
5555 −506758. −0.410706
5656 0 0
5757 0 0
5858 0 0
5959 515977. 893698.i 0.327076 0.566512i −0.654855 0.755755i 0.727270π-0.727270\pi
0.981930 + 0.189243i 0.0606035π0.0606035\pi
6060 0 0
6161 161341. + 279450.i 0.0910101 + 0.157634i 0.907936 0.419108i 0.137657π-0.137657\pi
−0.816926 + 0.576742i 0.804324π0.804324\pi
6262 0 0
6363 0 0
6464 0 0
6565 382130. + 661869.i 0.172590 + 0.298934i
6666 0 0
6767 11763.8 20375.5i 0.00477844 0.00827650i −0.863626 0.504133i 0.831812π-0.831812\pi
0.868405 + 0.495856i 0.165146π0.165146\pi
6868 0 0
6969 0 0
7070 0 0
7171 −2.84199e6 −0.942363 −0.471181 0.882036i 0.656172π-0.656172\pi
−0.471181 + 0.882036i 0.656172π0.656172\pi
7272 0 0
7373 −336523. + 582875.i −0.101248 + 0.175366i −0.912199 0.409748i 0.865617π-0.865617\pi
0.810951 + 0.585114i 0.198950π0.198950\pi
7474 0 0
7575 0 0
7676 0 0
7777 −2.78947e6 680072.i −0.696314 0.169761i
7878 0 0
7979 429448. + 743826.i 0.0979977 + 0.169737i 0.910856 0.412725i 0.135423π-0.135423\pi
−0.812858 + 0.582462i 0.802090π0.802090\pi
8080 0 0
8181 0 0
8282 0 0
8383 −7.32276e6 −1.40573 −0.702864 0.711324i 0.748096π-0.748096\pi
−0.702864 + 0.711324i 0.748096π0.748096\pi
8484 0 0
8585 2.08373e6 0.368024
8686 0 0
8787 0 0
8888 0 0
8989 2.31400e6 + 4.00797e6i 0.347935 + 0.602641i 0.985883 0.167439i 0.0535496π-0.0535496\pi
−0.637947 + 0.770080i 0.720216π0.720216\pi
9090 0 0
9191 1.21522e6 + 4.15611e6i 0.169048 + 0.578152i
9292 0 0
9393 0 0
9494 0 0
9595 −3.53307e6 + 6.11946e6i −0.422785 + 0.732285i
9696 0 0
9797 951821. 0.105890 0.0529449 0.998597i 0.483139π-0.483139\pi
0.0529449 + 0.998597i 0.483139π0.483139\pi
9898 0 0
9999 0 0
100100 0 0
101101 −8.54896e6 + 1.48072e7i −0.825635 + 1.43004i 0.0757975 + 0.997123i 0.475850π0.475850\pi
−0.901433 + 0.432919i 0.857484π0.857484\pi
102102 0 0
103103 2.02111e6 + 3.50067e6i 0.182247 + 0.315661i 0.942645 0.333796i 0.108330π-0.108330\pi
−0.760398 + 0.649457i 0.774996π0.774996\pi
104104 0 0
105105 0 0
106106 0 0
107107 −3.96974e6 6.87579e6i −0.313270 0.542600i 0.665798 0.746132i 0.268091π-0.268091\pi
−0.979068 + 0.203532i 0.934758π0.934758\pi
108108 0 0
109109 3.49719e6 6.05732e6i 0.258659 0.448010i −0.707224 0.706989i 0.750053π-0.750053\pi
0.965883 + 0.258979i 0.0833862π0.0833862\pi
110110 0 0
111111 0 0
112112 0 0
113113 −2.09031e7 −1.36281 −0.681407 0.731904i 0.738632π-0.738632\pi
−0.681407 + 0.731904i 0.738632π0.738632\pi
114114 0 0
115115 −5.21084e6 + 9.02544e6i −0.319496 + 0.553383i
116116 0 0
117117 0 0
118118 0 0
119119 1.14700e7 + 2.79638e6i 0.623950 + 0.152119i
120120 0 0
121121 4.73858e6 + 8.20747e6i 0.243164 + 0.421173i
122122 0 0
123123 0 0
124124 0 0
125125 −2.09176e7 −0.957913
126126 0 0
127127 −1.32156e7 −0.572496 −0.286248 0.958155i 0.592408π-0.592408\pi
−0.286248 + 0.958155i 0.592408π0.592408\pi
128128 0 0
129129 0 0
130130 0 0
131131 7.97459e6 + 1.38124e7i 0.309926 + 0.536808i 0.978346 0.206976i 0.0663622π-0.0663622\pi
−0.668420 + 0.743784i 0.733029π0.733029\pi
132132 0 0
133133 −2.76603e7 + 2.89435e7i −1.01947 + 1.06677i
134134 0 0
135135 0 0
136136 0 0
137137 −1.78389e7 + 3.08978e7i −0.592714 + 1.02661i 0.401151 + 0.916012i 0.368610π0.368610\pi
−0.993865 + 0.110599i 0.964723π0.964723\pi
138138 0 0
139139 −1.12444e7 −0.355127 −0.177563 0.984109i 0.556821π-0.556821\pi
−0.177563 + 0.984109i 0.556821π0.556821\pi
140140 0 0
141141 0 0
142142 0 0
143143 −7.54823e6 + 1.30739e7i −0.215858 + 0.373878i
144144 0 0
145145 −1.97103e7 3.41392e7i −0.536913 0.929961i
146146 0 0
147147 0 0
148148 0 0
149149 9.72724e6 + 1.68481e7i 0.240900 + 0.417252i 0.960971 0.276649i 0.0892240π-0.0892240\pi
−0.720071 + 0.693901i 0.755891π0.755891\pi
150150 0 0
151151 −4.01078e7 + 6.94687e7i −0.948002 + 1.64199i −0.198375 + 0.980126i 0.563566π0.563566\pi
−0.749627 + 0.661861i 0.769767π0.769767\pi
152152 0 0
153153 0 0
154154 0 0
155155 4.81564e7 1.03871
156156 0 0
157157 2.44031e7 4.22674e7i 0.503264 0.871679i −0.496729 0.867906i 0.665466π-0.665466\pi
0.999993 0.00377328i 0.00120108π-0.00120108\pi
158158 0 0
159159 0 0
160160 0 0
161161 −4.07955e7 + 4.26880e7i −0.770410 + 0.806149i
162162 0 0
163163 −6.59019e6 1.14145e7i −0.119190 0.206444i 0.800257 0.599658i 0.204697π-0.204697\pi
−0.919447 + 0.393214i 0.871363π0.871363\pi
164164 0 0
165165 0 0
166166 0 0
167167 6.54304e7 1.08711 0.543553 0.839375i 0.317079π-0.317079\pi
0.543553 + 0.839375i 0.317079π0.317079\pi
168168 0 0
169169 −3.99810e7 −0.637163
170170 0 0
171171 0 0
172172 0 0
173173 8.07134e6 + 1.39800e7i 0.118518 + 0.205279i 0.919181 0.393836i 0.128852π-0.128852\pi
−0.800663 + 0.599116i 0.795519π0.795519\pi
174174 0 0
175175 −4.62614e7 1.12785e7i −0.652508 0.159081i
176176 0 0
177177 0 0
178178 0 0
179179 −4.82833e7 + 8.36291e7i −0.629233 + 1.08986i 0.358473 + 0.933540i 0.383297π0.383297\pi
−0.987706 + 0.156323i 0.950036π0.950036\pi
180180 0 0
181181 7.18706e7 0.900899 0.450450 0.892802i 0.351264π-0.351264\pi
0.450450 + 0.892802i 0.351264π0.351264\pi
182182 0 0
183183 0 0
184184 0 0
185185 −4.14007e7 + 7.17082e7i −0.480736 + 0.832660i
186186 0 0
187187 2.05800e7 + 3.56457e7i 0.230144 + 0.398622i
188188 0 0
189189 0 0
190190 0 0
191191 −8.37396e7 1.45041e8i −0.869589 1.50617i −0.862417 0.506198i 0.831050π-0.831050\pi
−0.00717183 0.999974i 0.502283π-0.502283\pi
192192 0 0
193193 −8.45041e6 + 1.46365e7i −0.0846111 + 0.146551i −0.905225 0.424932i 0.860298π-0.860298\pi
0.820614 + 0.571482i 0.193631π0.193631\pi
194194 0 0
195195 0 0
196196 0 0
197197 5.41597e7 0.504713 0.252357 0.967634i 0.418794π-0.418794\pi
0.252357 + 0.967634i 0.418794π0.418794\pi
198198 0 0
199199 −7.69956e6 + 1.33360e7i −0.0692596 + 0.119961i −0.898576 0.438819i 0.855397π-0.855397\pi
0.829316 + 0.558780i 0.188730π0.188730\pi
200200 0 0
201201 0 0
202202 0 0
203203 −6.26812e7 2.14372e8i −0.525897 1.79859i
204204 0 0
205205 −3.02598e7 5.24115e7i −0.245317 0.424901i
206206 0 0
207207 0 0
208208 0 0
209209 −1.39578e8 −1.05756
210210 0 0
211211 −1.35532e8 −0.993238 −0.496619 0.867969i 0.665425π-0.665425\pi
−0.496619 + 0.867969i 0.665425π0.665425\pi
212212 0 0
213213 0 0
214214 0 0
215215 5.71973e6 + 9.90686e6i 0.0392501 + 0.0679832i
216216 0 0
217217 2.65079e8 + 6.46262e7i 1.76103 + 0.429338i
218218 0 0
219219 0 0
220220 0 0
221221 3.10375e7 5.37585e7i 0.193426 0.335023i
222222 0 0
223223 2.14853e8 1.29740 0.648701 0.761044i 0.275313π-0.275313\pi
0.648701 + 0.761044i 0.275313π0.275313\pi
224224 0 0
225225 0 0
226226 0 0
227227 −5.08630e7 + 8.80973e7i −0.288610 + 0.499887i −0.973478 0.228780i 0.926526π-0.926526\pi
0.684868 + 0.728667i 0.259860π0.259860\pi
228228 0 0
229229 6.57822e7 + 1.13938e8i 0.361980 + 0.626967i 0.988287 0.152610i 0.0487677π-0.0487677\pi
−0.626307 + 0.779576i 0.715434π0.715434\pi
230230 0 0
231231 0 0
232232 0 0
233233 2.75854e7 + 4.77793e7i 0.142867 + 0.247454i 0.928575 0.371144i 0.121034π-0.121034\pi
−0.785708 + 0.618598i 0.787701π0.787701\pi
234234 0 0
235235 8.94359e7 1.54908e8i 0.449546 0.778637i
236236 0 0
237237 0 0
238238 0 0
239239 1.89001e8 0.895513 0.447757 0.894155i 0.352223π-0.352223\pi
0.447757 + 0.894155i 0.352223π0.352223\pi
240240 0 0
241241 1.49348e8 2.58678e8i 0.687290 1.19042i −0.285422 0.958402i 0.592134π-0.592134\pi
0.972711 0.232019i 0.0745330π-0.0745330\pi
242242 0 0
243243 0 0
244244 0 0
245245 1.17107e8 + 6.07095e7i 0.508745 + 0.263739i
246246 0 0
247247 1.05251e8 + 1.82300e8i 0.444414 + 0.769747i
248248 0 0
249249 0 0
250250 0 0
251251 −2.63594e7 −0.105215 −0.0526074 0.998615i 0.516753π-0.516753\pi
−0.0526074 + 0.998615i 0.516753π0.516753\pi
252252 0 0
253253 −2.05860e8 −0.799189
254254 0 0
255255 0 0
256256 0 0
257257 −4.05761e7 7.02799e7i −0.149109 0.258265i 0.781789 0.623543i 0.214307π-0.214307\pi
−0.930899 + 0.365278i 0.880974π0.880974\pi
258258 0 0
259259 −3.24125e8 + 3.39161e8i −1.15921 + 1.21299i
260260 0 0
261261 0 0
262262 0 0
263263 2.71901e8 4.70947e8i 0.921651 1.59635i 0.124790 0.992183i 0.460174π-0.460174\pi
0.796861 0.604163i 0.206492π-0.206492\pi
264264 0 0
265265 5.90599e7 0.194954
266266 0 0
267267 0 0
268268 0 0
269269 −1.06642e7 + 1.84709e7i −0.0334037 + 0.0578569i −0.882244 0.470793i 0.843968π-0.843968\pi
0.848840 + 0.528649i 0.177301π0.177301\pi
270270 0 0
271271 −2.35124e8 4.07246e8i −0.717636 1.24298i −0.961934 0.273281i 0.911891π-0.911891\pi
0.244298 0.969700i 0.421442π-0.421442\pi
272272 0 0
273273 0 0
274274 0 0
275275 −8.30043e7 1.43768e8i −0.240678 0.416866i
276276 0 0
277277 4.46999e7 7.74225e7i 0.126365 0.218871i −0.795901 0.605427i 0.793002π-0.793002\pi
0.922266 + 0.386557i 0.126336π0.126336\pi
278278 0 0
279279 0 0
280280 0 0
281281 6.84556e8 1.84051 0.920253 0.391324i 0.127983π-0.127983\pi
0.920253 + 0.391324i 0.127983π0.127983\pi
282282 0 0
283283 3.43984e8 5.95798e8i 0.902165 1.56260i 0.0774878 0.996993i 0.475310π-0.475310\pi
0.824678 0.565603i 0.191357π-0.191357\pi
284284 0 0
285285 0 0
286286 0 0
287287 −9.62300e7 3.29111e8i −0.240284 0.821779i
288288 0 0
289289 1.20546e8 + 2.08793e8i 0.293773 + 0.508830i
290290 0 0
291291 0 0
292292 0 0
293293 5.46969e8 1.27036 0.635179 0.772365i 0.280926π-0.280926\pi
0.635179 + 0.772365i 0.280926π0.280926\pi
294294 0 0
295295 −1.65289e8 −0.374858
296296 0 0
297297 0 0
298298 0 0
299299 1.55232e8 + 2.68870e8i 0.335840 + 0.581693i
300300 0 0
301301 1.81895e7 + 6.22087e7i 0.0384448 + 0.131483i
302302 0 0
303303 0 0
304304 0 0
305305 2.58421e7 4.47598e7i 0.0521529 0.0903314i
306306 0 0
307307 −5.33032e8 −1.05140 −0.525702 0.850669i 0.676197π-0.676197\pi
−0.525702 + 0.850669i 0.676197π0.676197\pi
308308 0 0
309309 0 0
310310 0 0
311311 7.53305e7 1.30476e8i 0.142007 0.245963i −0.786245 0.617914i 0.787978π-0.787978\pi
0.928252 + 0.371951i 0.121311π0.121311\pi
312312 0 0
313313 −2.13226e8 3.69318e8i −0.393038 0.680762i 0.599810 0.800142i 0.295243π-0.295243\pi
−0.992849 + 0.119380i 0.961909π0.961909\pi
314314 0 0
315315 0 0
316316 0 0
317317 −5.17845e8 8.96934e8i −0.913045 1.58144i −0.809738 0.586791i 0.800391π-0.800391\pi
−0.103307 0.994650i 0.532942π-0.532942\pi
318318 0 0
319319 3.89337e8 6.74352e8i 0.671519 1.16311i
320320 0 0
321321 0 0
322322 0 0
323323 5.73929e8 0.947652
324324 0 0
325325 −1.25182e8 + 2.16821e8i −0.202278 + 0.350357i
326326 0 0
327327 0 0
328328 0 0
329329 7.00192e8 7.32673e8i 1.08400 1.13429i
330330 0 0
331331 −7.15548e7 1.23937e8i −0.108453 0.187846i 0.806691 0.590974i 0.201256π-0.201256\pi
−0.915144 + 0.403128i 0.867923π0.867923\pi
332332 0 0
333333 0 0
334334 0 0
335335 −3.76844e6 −0.00547652
336336 0 0
337337 −8.02807e8 −1.14263 −0.571316 0.820730i 0.693567π-0.693567\pi
−0.571316 + 0.820730i 0.693567π0.693567\pi
338338 0 0
339339 0 0
340340 0 0
341341 4.75618e8 + 8.23794e8i 0.649557 + 1.12507i
342342 0 0
343343 5.63147e8 + 4.91336e8i 0.753516 + 0.657430i
344344 0 0
345345 0 0
346346 0 0
347347 −5.45919e8 + 9.45559e8i −0.701415 + 1.21489i 0.266555 + 0.963820i 0.414115π0.414115\pi
−0.967970 + 0.251066i 0.919219π0.919219\pi
348348 0 0
349349 1.03523e9 1.30361 0.651805 0.758386i 0.274012π-0.274012\pi
0.651805 + 0.758386i 0.274012π0.274012\pi
350350 0 0
351351 0 0
352352 0 0
353353 −7.35935e8 + 1.27468e9i −0.890487 + 1.54237i −0.0511955 + 0.998689i 0.516303π0.516303\pi
−0.839292 + 0.543681i 0.817030π0.817030\pi
354354 0 0
355355 2.27602e8 + 3.94218e8i 0.270008 + 0.467667i
356356 0 0
357357 0 0
358358 0 0
359359 −5.08408e8 8.80589e8i −0.579939 1.00448i −0.995486 0.0949118i 0.969743π-0.969743\pi
0.415547 0.909572i 0.363590π-0.363590\pi
360360 0 0
361361 −5.26187e8 + 9.11383e8i −0.588661 + 1.01959i
362362 0 0
363363 0 0
364364 0 0
365365 1.07802e8 0.116039
366366 0 0
367367 3.81967e8 6.61586e8i 0.403362 0.698643i −0.590768 0.806842i 0.701175π-0.701175\pi
0.994129 + 0.108199i 0.0345084π0.0345084\pi
368368 0 0
369369 0 0
370370 0 0
371371 3.25099e8 + 7.92588e7i 0.330527 + 0.0805821i
372372 0 0
373373 5.15720e8 + 8.93254e8i 0.514557 + 0.891239i 0.999857 + 0.0168911i 0.00537687π0.00537687\pi
−0.485301 + 0.874347i 0.661290π0.661290\pi
374374 0 0
375375 0 0
376376 0 0
377377 −1.17435e9 −1.12876
378378 0 0
379379 5.88248e8 0.555038 0.277519 0.960720i 0.410488π-0.410488\pi
0.277519 + 0.960720i 0.410488π0.410488\pi
380380 0 0
381381 0 0
382382 0 0
383383 −6.19545e8 1.07308e9i −0.563479 0.975974i −0.997189 0.0749216i 0.976129π-0.976129\pi
0.433711 0.901052i 0.357204π-0.357204\pi
384384 0 0
385385 1.29062e8 + 4.41398e8i 0.115262 + 0.394201i
386386 0 0
387387 0 0
388388 0 0
389389 2.55841e8 4.43130e8i 0.220367 0.381687i −0.734552 0.678552i 0.762608π-0.762608\pi
0.954920 + 0.296865i 0.0959411π0.0959411\pi
390390 0 0
391391 8.46473e8 0.716135
392392 0 0
393393 0 0
394394 0 0
395395 6.87850e7 1.19139e8i 0.0561570 0.0972668i
396396 0 0
397397 −6.28008e8 1.08774e9i −0.503730 0.872487i −0.999991 0.00431285i 0.998627π-0.998627\pi
0.496260 0.868174i 0.334706π-0.334706\pi
398398 0 0
399399 0 0
400400 0 0
401401 5.87326e8 + 1.01728e9i 0.454856 + 0.787833i 0.998680 0.0513658i 0.0163575π-0.0163575\pi
−0.543824 + 0.839199i 0.683024π0.683024\pi
402402 0 0
403403 7.17296e8 1.24239e9i 0.545922 0.945565i
404404 0 0
405405 0 0
406406 0 0
407407 −1.63558e9 −1.20252
408408 0 0
409409 −1.08611e9 + 1.88120e9i −0.784953 + 1.35958i 0.144075 + 0.989567i 0.453979π0.453979\pi
−0.929027 + 0.370011i 0.879354π0.879354\pi
410410 0 0
411411 0 0
412412 0 0
413413 −9.09841e8 2.21819e8i −0.635536 0.154943i
414414 0 0
415415 5.86446e8 + 1.01575e9i 0.402772 + 0.697622i
416416 0 0
417417 0 0
418418 0 0
419419 −1.97088e9 −1.30892 −0.654458 0.756098i 0.727103π-0.727103\pi
−0.654458 + 0.756098i 0.727103π0.727103\pi
420420 0 0
421421 −1.34491e9 −0.878430 −0.439215 0.898382i 0.644743π-0.644743\pi
−0.439215 + 0.898382i 0.644743π0.644743\pi
422422 0 0
423423 0 0
424424 0 0
425425 3.41305e8 + 5.91157e8i 0.215666 + 0.373544i
426426 0 0
427427 2.02317e8 2.11702e8i 0.125758 0.131592i
428428 0 0
429429 0 0
430430 0 0
431431 −3.40654e8 + 5.90031e8i −0.204948 + 0.354980i −0.950116 0.311896i 0.899036π-0.899036\pi
0.745168 + 0.666877i 0.232369π0.232369\pi
432432 0 0
433433 −1.29953e9 −0.769272 −0.384636 0.923068i 0.625673π-0.625673\pi
−0.384636 + 0.923068i 0.625673π0.625673\pi
434434 0 0
435435 0 0
436436 0 0
437437 −1.43524e9 + 2.48590e9i −0.822694 + 1.42495i
438438 0 0
439439 −8.64679e8 1.49767e9i −0.487786 0.844870i 0.512115 0.858917i 0.328862π-0.328862\pi
−0.999901 + 0.0140465i 0.995529π0.995529\pi
440440 0 0
441441 0 0
442442 0 0
443443 4.60591e8 + 7.97768e8i 0.251711 + 0.435977i 0.963997 0.265913i 0.0856733π-0.0856733\pi
−0.712286 + 0.701890i 0.752340π0.752340\pi
444444 0 0
445445 3.70635e8 6.41959e8i 0.199382 0.345340i
446446 0 0
447447 0 0
448448 0 0
449449 1.26736e9 0.660751 0.330376 0.943850i 0.392825π-0.392825\pi
0.330376 + 0.943850i 0.392825π0.392825\pi
450450 0 0
451451 5.97722e8 1.03529e9i 0.306819 0.531425i
452452 0 0
453453 0 0
454454 0 0
455455 4.79181e8 5.01410e8i 0.238484 0.249547i
456456 0 0
457457 6.62958e8 + 1.14828e9i 0.324922 + 0.562782i 0.981497 0.191479i 0.0613285π-0.0613285\pi
−0.656574 + 0.754261i 0.727995π0.727995\pi
458458 0 0
459459 0 0
460460 0 0
461461 1.44980e9 0.689215 0.344608 0.938747i 0.388012π-0.388012\pi
0.344608 + 0.938747i 0.388012π0.388012\pi
462462 0 0
463463 −3.49096e9 −1.63460 −0.817299 0.576214i 0.804530π-0.804530\pi
−0.817299 + 0.576214i 0.804530π0.804530\pi
464464 0 0
465465 0 0
466466 0 0
467467 −4.23615e8 7.33723e8i −0.192470 0.333368i 0.753598 0.657335i 0.228316π-0.228316\pi
−0.946068 + 0.323968i 0.894983π0.894983\pi
468468 0 0
469469 −2.07436e7 5.05726e6i −0.00928492 0.00226366i
470470 0 0
471471 0 0
472472 0 0
473473 −1.12982e8 + 1.95690e8i −0.0490902 + 0.0850268i
474474 0 0
475475 −2.31479e9 −0.991026
476476 0 0
477477 0 0
478478 0 0
479479 −1.41330e9 + 2.44791e9i −0.587572 + 1.01770i 0.406977 + 0.913438i 0.366583π0.366583\pi
−0.994549 + 0.104266i 0.966751π0.966751\pi
480480 0 0
481481 1.23334e9 + 2.13621e9i 0.505330 + 0.875257i
482482 0 0
483483 0 0
484484 0 0
485485 −7.62269e7 1.32029e8i −0.0303398 0.0525500i
486486 0 0
487487 −1.92842e9 + 3.34012e9i −0.756572 + 1.31042i 0.188016 + 0.982166i 0.439794π0.439794\pi
−0.944589 + 0.328256i 0.893539π0.893539\pi
488488 0 0
489489 0 0
490490 0 0
491491 −1.98226e9 −0.755746 −0.377873 0.925857i 0.623344π-0.623344\pi
−0.377873 + 0.925857i 0.623344π0.623344\pi
492492 0 0
493493 −1.60091e9 + 2.77286e9i −0.601733 + 1.04223i
494494 0 0
495495 0 0
496496 0 0
497497 7.23803e8 + 2.47544e9i 0.264468 + 0.904491i
498498 0 0
499499 −1.75830e9 3.04547e9i −0.633492 1.09724i −0.986832 0.161746i 0.948287π-0.948287\pi
0.353340 0.935495i 0.385046π-0.385046\pi
500500 0 0
501501 0 0
502502 0 0
503503 5.01692e9 1.75772 0.878859 0.477081i 0.158305π-0.158305\pi
0.878859 + 0.477081i 0.158305π0.158305\pi
504504 0 0
505505 2.73859e9 0.946252
506506 0 0
507507 0 0
508508 0 0
509509 2.77610e9 + 4.80834e9i 0.933088 + 1.61616i 0.778008 + 0.628254i 0.216230π0.216230\pi
0.155080 + 0.987902i 0.450436π0.450436\pi
510510 0 0
511511 5.93404e8 + 1.44671e8i 0.196733 + 0.0479634i
512512 0 0
513513 0 0
514514 0 0
515515 3.23723e8 5.60705e8i 0.104436 0.180888i
516516 0 0
517517 3.53326e9 1.12450
518518 0 0
519519 0 0
520520 0 0
521521 −8.99849e8 + 1.55858e9i −0.278765 + 0.482835i −0.971078 0.238762i 0.923258π-0.923258\pi
0.692313 + 0.721597i 0.256592π0.256592\pi
522522 0 0
523523 6.38363e8 + 1.10568e9i 0.195124 + 0.337965i 0.946941 0.321407i 0.104156π-0.104156\pi
−0.751817 + 0.659372i 0.770822π0.770822\pi
524524 0 0
525525 0 0
526526 0 0
527527 −1.95569e9 3.38735e9i −0.582053 1.00815i
528528 0 0
529529 −4.14382e8 + 7.17730e8i −0.121704 + 0.210798i
530530 0 0
531531 0 0
532532 0 0
533533 −1.80289e9 −0.515733
534534 0 0
535535 −6.35837e8 + 1.10130e9i −0.179518 + 0.310934i
536536 0 0
537537 0 0
538538 0 0
539539 1.18071e8 + 2.60290e9i 0.0324776 + 0.715973i
540540 0 0
541541 −3.19521e9 5.53426e9i −0.867579 1.50269i −0.864464 0.502695i 0.832342π-0.832342\pi
−0.00311482 0.999995i 0.500991π-0.500991\pi
542542 0 0
543543 0 0
544544 0 0
545545 −1.12030e9 −0.296446
546546 0 0
547547 −4.83104e9 −1.26207 −0.631037 0.775752i 0.717371π-0.717371\pi
−0.631037 + 0.775752i 0.717371π0.717371\pi
548548 0 0
549549 0 0
550550 0 0
551551 −5.42885e9 9.40304e9i −1.38254 2.39463i
552552 0 0
553553 5.38516e8 5.63498e8i 0.135413 0.141695i
554554 0 0
555555 0 0
556556 0 0
557557 −5.26607e6 + 9.12110e6i −0.00129120 + 0.00223642i −0.866670 0.498881i 0.833744π-0.833744\pi
0.865379 + 0.501118i 0.167078π0.167078\pi
558558 0 0
559559 3.40784e8 0.0825160
560560 0 0
561561 0 0
562562 0 0
563563 1.21000e9 2.09579e9i 0.285764 0.494957i −0.687030 0.726629i 0.741086π-0.741086\pi
0.972794 + 0.231671i 0.0744194π0.0744194\pi
564564 0 0
565565 1.67404e9 + 2.89952e9i 0.390477 + 0.676326i
566566 0 0
567567 0 0
568568 0 0
569569 −1.21669e8 2.10737e8i −0.0276877 0.0479565i 0.851850 0.523787i 0.175481π-0.175481\pi
−0.879537 + 0.475830i 0.842148π0.842148\pi
570570 0 0
571571 2.81425e9 4.87442e9i 0.632610 1.09571i −0.354406 0.935092i 0.615317π-0.615317\pi
0.987016 0.160621i 0.0513497π-0.0513497\pi
572572 0 0
573573 0 0
574574 0 0
575575 −3.41403e9 −0.748911
576576 0 0
577577 1.89860e9 3.28847e9i 0.411451 0.712653i −0.583598 0.812043i 0.698356π-0.698356\pi
0.995049 + 0.0993895i 0.0316890π0.0316890\pi
578578 0 0
579579 0 0
580580 0 0
581581 1.86497e9 + 6.37829e9i 0.394509 + 1.34923i
582582 0 0
583583 5.83306e8 + 1.01032e9i 0.121915 + 0.211163i
584584 0 0
585585 0 0
586586 0 0
587587 2.00190e9 0.408516 0.204258 0.978917i 0.434522π-0.434522\pi
0.204258 + 0.978917i 0.434522π0.434522\pi
588588 0 0
589589 1.32639e10 2.67464
590590 0 0
591591 0 0
592592 0 0
593593 −8.06563e8 1.39701e9i −0.158835 0.275111i 0.775614 0.631208i 0.217441π-0.217441\pi
−0.934449 + 0.356097i 0.884107π0.884107\pi
594594 0 0
595595 −5.30690e8 1.81498e9i −0.103284 0.353234i
596596 0 0
597597 0 0
598598 0 0
599599 4.69338e9 8.12917e9i 0.892261 1.54544i 0.0551024 0.998481i 0.482451π-0.482451\pi
0.837158 0.546960i 0.184215π-0.184215\pi
600600 0 0
601601 −4.79642e9 −0.901274 −0.450637 0.892707i 0.648803π-0.648803\pi
−0.450637 + 0.892707i 0.648803π0.648803\pi
602602 0 0
603603 0 0
604604 0 0
605605 7.58983e8 1.31460e9i 0.139344 0.241351i
606606 0 0
607607 2.00844e9 + 3.47872e9i 0.364500 + 0.631333i 0.988696 0.149935i 0.0479064π-0.0479064\pi
−0.624195 + 0.781268i 0.714573π0.714573\pi
608608 0 0
609609 0 0
610610 0 0
611611 −2.66432e9 4.61474e9i −0.472544 0.818470i
612612 0 0
613613 −1.16046e9 + 2.00997e9i −0.203478 + 0.352434i −0.949647 0.313323i 0.898558π-0.898558\pi
0.746169 + 0.665757i 0.231891π0.231891\pi
614614 0 0
615615 0 0
616616 0 0
617617 −7.65061e9 −1.31129 −0.655644 0.755070i 0.727603π-0.727603\pi
−0.655644 + 0.755070i 0.727603π0.727603\pi
618618 0 0
619619 −2.52375e9 + 4.37126e9i −0.427689 + 0.740780i −0.996667 0.0815729i 0.974006π-0.974006\pi
0.568978 + 0.822353i 0.307339π0.307339\pi
620620 0 0
621621 0 0
622622 0 0
623623 2.90169e9 3.03630e9i 0.480777 0.503080i
624624 0 0
625625 −3.74430e8 6.48532e8i −0.0613466 0.106255i
626626 0 0
627627 0 0
628628 0 0
629629 6.72533e9 1.07755
630630 0 0
631631 9.61793e9 1.52398 0.761989 0.647590i 0.224223π-0.224223\pi
0.761989 + 0.647590i 0.224223π0.224223\pi
632632 0 0
633633 0 0
634634 0 0
635635 1.05837e9 + 1.83316e9i 0.164033 + 0.284113i
636636 0 0
637637 3.31057e9 2.11697e9i 0.507475 0.324509i
638638 0 0
639639 0 0
640640 0 0
641641 −2.57668e9 + 4.46294e9i −0.386418 + 0.669296i −0.991965 0.126514i 0.959621π-0.959621\pi
0.605547 + 0.795810i 0.292955π0.292955\pi
642642 0 0
643643 6.36581e9 0.944311 0.472155 0.881515i 0.343476π-0.343476\pi
0.472155 + 0.881515i 0.343476π0.343476\pi
644644 0 0
645645 0 0
646646 0 0
647647 2.22790e9 3.85884e9i 0.323394 0.560134i −0.657792 0.753199i 0.728510π-0.728510\pi
0.981186 + 0.193065i 0.0618428π0.0618428\pi
648648 0 0
649649 −1.63248e9 2.82753e9i −0.234418 0.406024i
650650 0 0
651651 0 0
652652 0 0
653653 2.47326e8 + 4.28382e8i 0.0347596 + 0.0602053i 0.882882 0.469595i 0.155600π-0.155600\pi
−0.848122 + 0.529801i 0.822267π0.822267\pi
654654 0 0
655655 1.27730e9 2.21234e9i 0.177602 0.307615i
656656 0 0
657657 0 0
658658 0 0
659659 7.33078e9 0.997818 0.498909 0.866654i 0.333734π-0.333734\pi
0.498909 + 0.866654i 0.333734π0.333734\pi
660660 0 0
661661 −3.49714e9 + 6.05723e9i −0.470986 + 0.815772i −0.999449 0.0331843i 0.989435π-0.989435\pi
0.528463 + 0.848956i 0.322769π0.322769\pi
662662 0 0
663663 0 0
664664 0 0
665665 6.22999e9 + 1.51887e9i 0.821508 + 0.200283i
666666 0 0
667667 −8.00688e9 1.38683e10i −1.04478 1.80960i
668668 0 0
669669 0 0
670670 0 0
671671 1.02092e9 0.130455
672672 0 0
673673 −2.05913e9 −0.260394 −0.130197 0.991488i 0.541561π-0.541561\pi
−0.130197 + 0.991488i 0.541561π0.541561\pi
674674 0 0
675675 0 0
676676 0 0
677677 −1.01355e9 1.75552e9i −0.125541 0.217443i 0.796403 0.604766i 0.206733π-0.206733\pi
−0.921944 + 0.387323i 0.873400π0.873400\pi
678678 0 0
679679 −2.42412e8 8.29057e8i −0.0297173 0.101634i
680680 0 0
681681 0 0
682682 0 0
683683 4.72964e9 8.19198e9i 0.568009 0.983821i −0.428753 0.903422i 0.641047π-0.641047\pi
0.996763 0.0803996i 0.0256196π-0.0256196\pi
684684 0 0
685685 5.71454e9 0.679304
686686 0 0
687687 0 0
688688 0 0
689689 8.79706e8 1.52370e9i 0.102464 0.177472i
690690 0 0
691691 −5.57572e9 9.65742e9i −0.642876 1.11349i −0.984788 0.173762i 0.944408π-0.944408\pi
0.341911 0.939732i 0.388926π-0.388926\pi
692692 0 0
693693 0 0
694694 0 0
695695 9.00510e8 + 1.55973e9i 0.101752 + 0.176239i
696696 0 0
697697 −2.45777e9 + 4.25698e9i −0.274933 + 0.476198i
698698 0 0
699699 0 0
700700 0 0
701701 −1.01555e10 −1.11349 −0.556746 0.830683i 0.687950π-0.687950\pi
−0.556746 + 0.830683i 0.687950π0.687950\pi
702702 0 0
703703 −1.14031e10 + 1.97508e10i −1.23788 + 2.14408i
704704 0 0
705705 0 0
706706 0 0
707707 1.50747e10 + 3.67520e9i 1.60428 + 0.391122i
708708 0 0
709709 −5.65609e9 9.79664e9i −0.596012 1.03232i −0.993403 0.114674i 0.963418π-0.963418\pi
0.397391 0.917649i 0.369916π-0.369916\pi
710710 0 0
711711 0 0
712712 0 0
713713 1.95625e10 2.02121
714714 0 0
715715 2.41801e9 0.247393
716716 0 0
717717 0 0
718718 0 0
719719 −4.36466e9 7.55981e9i −0.437924 0.758507i 0.559605 0.828760i 0.310953π-0.310953\pi
−0.997529 + 0.0702522i 0.977620π0.977620\pi
720720 0 0
721721 2.53442e9 2.65199e9i 0.251829 0.263511i
722722 0 0
723723 0 0
724724 0 0
725725 6.45688e9 1.11836e10i 0.629274 1.08993i
726726 0 0
727727 5.43142e9 0.524255 0.262128 0.965033i 0.415576π-0.415576\pi
0.262128 + 0.965033i 0.415576π0.415576\pi
728728 0 0
729729 0 0
730730 0 0
731731 4.64570e8 8.04658e8i 0.0439886 0.0761905i
732732 0 0
733733 −1.84738e9 3.19976e9i −0.173258 0.300092i 0.766299 0.642484i 0.222096π-0.222096\pi
−0.939557 + 0.342392i 0.888763π0.888763\pi
734734 0 0
735735 0 0
736736 0 0
737737 −3.72190e7 6.44652e7i −0.00342475 0.00593184i
738738 0 0
739739 5.50425e9 9.53363e9i 0.501698 0.868966i −0.498300 0.867005i 0.666042π-0.666042\pi
0.999998 0.00196163i 0.000624407π-0.000624407\pi
740740 0 0
741741 0 0
742742 0 0
743743 −1.40281e10 −1.25469 −0.627346 0.778741i 0.715859π-0.715859\pi
−0.627346 + 0.778741i 0.715859π0.715859\pi
744744 0 0
745745 1.55802e9 2.69857e9i 0.138047 0.239104i
746746 0 0
747747 0 0
748748 0 0
749749 −4.97795e9 + 5.20887e9i −0.432876 + 0.452957i
750750 0 0
751751 2.72345e9 + 4.71716e9i 0.234628 + 0.406388i 0.959165 0.282849i 0.0912794π-0.0912794\pi
−0.724536 + 0.689237i 0.757946π0.757946\pi
752752 0 0
753753 0 0
754754 0 0
755755 1.28482e10 1.08649
756756 0 0
757757 −1.04051e10 −0.871785 −0.435892 0.899999i 0.643567π-0.643567\pi
−0.435892 + 0.899999i 0.643567π0.643567\pi
758758 0 0
759759 0 0
760760 0 0
761761 4.35065e8 + 7.53554e8i 0.0357855 + 0.0619823i 0.883363 0.468689i 0.155273π-0.155273\pi
−0.847578 + 0.530671i 0.821940π0.821940\pi
762762 0 0
763763 −6.16673e9 1.50344e9i −0.502596 0.122533i
764764 0 0
765765 0 0
766766 0 0
767767 −2.46200e9 + 4.26431e9i −0.197017 + 0.341244i
768768 0 0
769769 1.17740e9 0.0933643 0.0466822 0.998910i 0.485135π-0.485135\pi
0.0466822 + 0.998910i 0.485135π0.485135\pi
770770 0 0
771771 0 0
772772 0 0
773773 −8.40843e9 + 1.45638e10i −0.654767 + 1.13409i 0.327185 + 0.944960i 0.393900π0.393900\pi
−0.981952 + 0.189129i 0.939433π0.939433\pi
774774 0 0
775775 7.88777e9 + 1.36620e10i 0.608693 + 1.05429i
776776 0 0
777777 0 0
778778 0 0
779779 −8.33453e9 1.44358e10i −0.631684 1.09411i
780780 0 0
781781 −4.49583e9 + 7.78700e9i −0.337700 + 0.584913i
782782 0 0
783783 0 0
784784 0 0
785785 −7.81733e9 −0.576786
786786 0 0
787787 7.43513e9 1.28780e10i 0.543722 0.941755i −0.454964 0.890510i 0.650348π-0.650348\pi
0.998686 0.0512448i 0.0163189π-0.0163189\pi
788788 0 0
789789 0 0
790790 0 0
791791 5.32365e9 + 1.82071e10i 0.382465 + 1.30805i
792792 0 0
793793 −7.69843e8 1.33341e9i −0.0548208 0.0949525i
794794 0 0
795795 0 0
796796 0 0
797797 1.62043e10 1.13377 0.566886 0.823796i 0.308148π-0.308148\pi
0.566886 + 0.823796i 0.308148π0.308148\pi
798798 0 0
799799 −1.45284e10 −1.00764
800800 0 0
801801 0 0
802802 0 0
803803 1.06471e9 + 1.84414e9i 0.0725651 + 0.125686i
804804 0 0
805805 9.18847e9 + 2.24014e9i 0.620808 + 0.151353i
806806 0 0
807807 0 0
808808 0 0
809809 1.10784e9 1.91883e9i 0.0735624 0.127414i −0.826898 0.562352i 0.809897π-0.809897\pi
0.900460 + 0.434938i 0.143230π0.143230\pi
810810 0 0
811811 4.87572e9 0.320971 0.160485 0.987038i 0.448694π-0.448694\pi
0.160485 + 0.987038i 0.448694π0.448694\pi
812812 0 0
813813 0 0
814814 0 0
815815 −1.05556e9 + 1.82828e9i −0.0683014 + 0.118301i
816816 0 0
817817 1.57540e9 + 2.72867e9i 0.101068 + 0.175055i
818818 0 0
819819 0 0
820820 0 0
821821 −5.48439e9 9.49925e9i −0.345882 0.599084i 0.639632 0.768681i 0.279087π-0.279087\pi
−0.985514 + 0.169597i 0.945753π0.945753\pi
822822 0 0
823823 −1.08079e8 + 1.87198e8i −0.00675834 + 0.0117058i −0.869385 0.494136i 0.835485π-0.835485\pi
0.862626 + 0.505841i 0.168818π0.168818\pi
824824 0 0
825825 0 0
826826 0 0
827827 −2.31243e10 −1.42167 −0.710836 0.703357i 0.751683π-0.751683\pi
−0.710836 + 0.703357i 0.751683π0.751683\pi
828828 0 0
829829 −1.17820e10 + 2.04071e10i −0.718256 + 1.24406i 0.243434 + 0.969917i 0.421726π0.421726\pi
−0.961690 + 0.274139i 0.911607π0.911607\pi
830830 0 0
831831 0 0
832832 0 0
833833 −4.85496e8 1.07028e10i −0.0291024 0.641566i
834834 0 0
835835 −5.24002e9 9.07598e9i −0.311480 0.539499i
836836 0 0
837837 0 0
838838 0 0
839839 −2.50625e10 −1.46507 −0.732534 0.680730i 0.761663π-0.761663\pi
−0.732534 + 0.680730i 0.761663π0.761663\pi
840840 0 0
841841 4.33229e10 2.51149
842842 0 0
843843 0 0
844844 0 0
845845 3.20190e9 + 5.54585e9i 0.182561 + 0.316205i
846846 0 0
847847 5.94206e9 6.21771e9i 0.336004 0.351591i
848848 0 0
849849 0 0
850850 0 0
851851 −1.68182e10 + 2.91299e10i −0.935461 + 1.62027i
852852 0 0
853853 −1.90748e10 −1.05230 −0.526149 0.850392i 0.676365π-0.676365\pi
−0.526149 + 0.850392i 0.676365π0.676365\pi
854854 0 0
855855 0 0
856856 0 0
857857 −1.67098e10 + 2.89422e10i −0.906856 + 1.57072i −0.0884496 + 0.996081i 0.528191π0.528191\pi
−0.818406 + 0.574640i 0.805142π0.805142\pi
858858 0 0
859859 3.98428e9 + 6.90097e9i 0.214473 + 0.371479i 0.953110 0.302625i 0.0978631π-0.0978631\pi
−0.738636 + 0.674104i 0.764530π0.764530\pi
860860 0 0
861861 0 0
862862 0 0
863863 −1.08562e10 1.88034e10i −0.574961 0.995861i −0.996046 0.0888399i 0.971684π-0.971684\pi
0.421085 0.907021i 0.361649π-0.361649\pi
864864 0 0
865865 1.29279e9 2.23918e9i 0.0679161 0.117634i
866866 0 0
867867 0 0
868868 0 0
869869 2.71743e9 0.140472
870870 0 0
871871 −5.61314e7 + 9.72224e7i −0.00287834 + 0.00498543i
872872 0 0
873873 0 0
874874 0 0
875875 5.32733e9 + 1.82197e10i 0.268832 + 0.919416i
876876 0 0
877877 9.33106e9 + 1.61619e10i 0.467124 + 0.809083i 0.999295 0.0375541i 0.0119567π-0.0119567\pi
−0.532170 + 0.846637i 0.678623π0.678623\pi
878878 0 0
879879 0 0
880880 0 0
881881 −1.50045e10 −0.739273 −0.369636 0.929176i 0.620518π-0.620518\pi
−0.369636 + 0.929176i 0.620518π0.620518\pi
882882 0 0
883883 −2.65989e10 −1.30017 −0.650086 0.759861i 0.725267π-0.725267\pi
−0.650086 + 0.759861i 0.725267π0.725267\pi
884884 0 0
885885 0 0
886886 0 0
887887 −6.16407e9 1.06765e10i −0.296575 0.513683i 0.678775 0.734346i 0.262511π-0.262511\pi
−0.975350 + 0.220663i 0.929178π0.929178\pi
888888 0 0
889889 3.36577e9 + 1.15111e10i 0.160667 + 0.549489i
890890 0 0
891891 0 0
892892 0 0
893893 2.46336e10 4.26666e10i 1.15757 2.00497i
894894 0 0
895895 1.54672e10 0.721157
896896 0 0
897897 0 0
898898 0 0
899899 −3.69981e10 + 6.40826e10i −1.69832 + 2.94158i
900900 0 0
901901 −2.39849e9 4.15431e9i −0.109245 0.189218i
902902 0 0
903903 0 0
904904 0 0
905905 −5.75579e9 9.96932e9i −0.258128 0.447090i
906906 0 0
907907 2.27456e9 3.93965e9i 0.101221 0.175320i −0.810967 0.585092i 0.801058π-0.801058\pi
0.912188 + 0.409772i 0.134392π0.134392\pi
908908 0 0
909909 0 0
910910 0 0
911911 8.78309e9 0.384887 0.192443 0.981308i 0.438359π-0.438359\pi
0.192443 + 0.981308i 0.438359π0.438359\pi
912912 0 0
913913 −1.15841e10 + 2.00642e10i −0.503749 + 0.872519i
914914 0 0
915915 0 0
916916 0 0
917917 9.99992e9 1.04638e10i 0.428256 0.448123i
918918 0 0
919919 1.15521e10 + 2.00088e10i 0.490971 + 0.850388i 0.999946 0.0103940i 0.00330859π-0.00330859\pi
−0.508974 + 0.860782i 0.669975π0.669975\pi
920920 0 0
921921 0 0
922922 0 0
923923 1.35606e10 0.567642
924924 0 0
925925 −2.71249e10 −1.12687
926926 0 0
927927 0 0
928928 0 0
929929 −1.42081e10 2.46091e10i −0.581407 1.00703i −0.995313 0.0967070i 0.969169π-0.969169\pi
0.413906 0.910320i 0.364164π-0.364164\pi
930930 0 0
931931 3.22550e10 + 1.67214e10i 1.31001 + 0.679122i
932932 0 0
933933 0 0
934934 0 0
935935 3.29632e9 5.70940e9i 0.131883 0.228428i
936936 0 0
937937 −2.66810e10 −1.05953 −0.529766 0.848144i 0.677720π-0.677720\pi
−0.529766 + 0.848144i 0.677720π0.677720\pi
938938 0 0
939939 0 0
940940 0 0
941941 −2.47193e10 + 4.28151e10i −0.967103 + 1.67507i −0.263249 + 0.964728i 0.584794π0.584794\pi
−0.703854 + 0.710344i 0.748539π0.748539\pi
942942 0 0
943943 −1.22924e10 2.12911e10i −0.477360 0.826811i
944944 0 0
945945 0 0
946946 0 0
947947 1.04792e10 + 1.81505e10i 0.400961 + 0.694486i 0.993842 0.110805i 0.0353428π-0.0353428\pi
−0.592881 + 0.805290i 0.702009π0.702009\pi
948948 0 0
949949 1.60573e9 2.78121e9i 0.0609875 0.105634i
950950 0 0
951951 0 0
952952 0 0
953953 1.44763e10 0.541791 0.270895 0.962609i 0.412680π-0.412680\pi
0.270895 + 0.962609i 0.412680π0.412680\pi
954954 0 0
955955 −1.34126e10 + 2.32314e10i −0.498313 + 0.863104i
956956 0 0
957957 0 0
958958 0 0
959959 3.14560e10 + 7.66894e9i 1.15170 + 0.280782i
960960 0 0
961961 −3.14409e10 5.44572e10i −1.14278 1.97935i
962962 0 0
963963 0 0
964964 0 0
965965 2.70702e9 0.0969718
966966 0 0
967967 −3.47169e10 −1.23466 −0.617332 0.786703i 0.711786π-0.711786\pi
−0.617332 + 0.786703i 0.711786π0.711786\pi
968968 0 0
969969 0 0
970970 0 0
971971 6.62040e9 + 1.14669e10i 0.232069 + 0.401955i 0.958417 0.285372i 0.0921172π-0.0921172\pi
−0.726348 + 0.687327i 0.758784π0.758784\pi
972972 0 0
973973 2.86374e9 + 9.79409e9i 0.0996640 + 0.340855i
974974 0 0
975975 0 0
976976 0 0
977977 −1.03510e10 + 1.79284e10i −0.355099 + 0.615050i −0.987135 0.159890i 0.948886π-0.948886\pi
0.632036 + 0.774939i 0.282220π0.282220\pi
978978 0 0
979979 1.46423e10 0.498736
980980 0 0
981981 0 0
982982 0 0
983983 2.02622e9 3.50951e9i 0.0680376 0.117845i −0.830000 0.557764i 0.811660π-0.811660\pi
0.898037 + 0.439919i 0.144993π0.144993\pi
984984 0 0
985985 −4.33741e9 7.51261e9i −0.144612 0.250475i
986986 0 0
987987 0 0
988988 0 0
989989 2.32352e9 + 4.02445e9i 0.0763764 + 0.132288i
990990 0 0
991991 2.45811e10 4.25757e10i 0.802313 1.38965i −0.115778 0.993275i 0.536936π-0.536936\pi
0.918090 0.396371i 0.129731π-0.129731\pi
992992 0 0
993993 0 0
994994 0 0
995995 2.46649e9 0.0793777
996996 0 0
997997 5.36989e9 9.30093e9i 0.171606 0.297230i −0.767375 0.641198i 0.778438π-0.778438\pi
0.938981 + 0.343968i 0.111771π0.111771\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 252.8.k.b.109.1 8
3.2 odd 2 84.8.i.a.25.4 8
7.2 even 3 inner 252.8.k.b.37.1 8
21.2 odd 6 84.8.i.a.37.4 yes 8
21.5 even 6 588.8.i.o.373.1 8
21.11 odd 6 588.8.a.i.1.1 4
21.17 even 6 588.8.a.j.1.4 4
21.20 even 2 588.8.i.o.361.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.i.a.25.4 8 3.2 odd 2
84.8.i.a.37.4 yes 8 21.2 odd 6
252.8.k.b.37.1 8 7.2 even 3 inner
252.8.k.b.109.1 8 1.1 even 1 trivial
588.8.a.i.1.1 4 21.11 odd 6
588.8.a.j.1.4 4 21.17 even 6
588.8.i.o.361.1 8 21.20 even 2
588.8.i.o.373.1 8 21.5 even 6