Properties

Label 132.4.i.b.37.2
Level 132132
Weight 44
Character 132.37
Analytic conductor 7.7887.788
Analytic rank 00
Dimension 88
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [132,4,Mod(25,132)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(132, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("132.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 132=22311 132 = 2^{2} \cdot 3 \cdot 11
Weight: k k == 4 4
Character orbit: [χ][\chi] == 132.i (of order 55, degree 44, 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: 7.788252120767.78825212076
Analytic rank: 00
Dimension: 88
Relative dimension: 22 over Q(ζ5)\Q(\zeta_{5})
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: x8x7+58x6115x5+2421x4+3435x3+83262x2+124773x+4791721 x^{8} - x^{7} + 58x^{6} - 115x^{5} + 2421x^{4} + 3435x^{3} + 83262x^{2} + 124773x + 4791721 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C5]\mathrm{SU}(2)[C_{5}]

Embedding invariants

Embedding label 37.2
Root 2.06788+6.36429i2.06788 + 6.36429i of defining polynomial
Character χ\chi == 132.37
Dual form 132.4.i.b.25.2

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+(0.927051+2.85317i)q3+(1.604771.16593i)q5+(3.78898+11.6613i)q7+(7.281155.29007i)q9+(8.21788+35.5453i)q11+(5.26513+3.82534i)q13+(1.83890+5.65957i)q15+(62.9784+45.7565i)q17+(28.2631+86.9850i)q1936.7842q2120.2371q23+(37.4112+115.140i)q25+(21.843515.8702i)q27+(23.583872.5834i)q29+(92.8764+67.4787i)q31+(93.798356.3993i)q33+(19.6767+14.2960i)q35+(27.6238+85.0174i)q37+(15.7954+11.4760i)q39+(30.465893.7641i)q41+488.233q4317.8525q45+(33.0191101.622i)q47+(155.864113.242i)q49+(72.1668222.106i)q51+(447.009324.771i)q53+(28.2556+66.6235i)q55+(221.982161.279i)q57+(52.3705161.180i)q59+(235.345170.988i)q61+(34.1008104.952i)q63+12.9094q65+107.351q67+(18.760857.7398i)q69+(589.515428.308i)q71+(110.836+341.119i)q73+(293.832213.481i)q75+(445.641+38.8493i)q77+(413.815+300.654i)q79+(25.0304+77.0356i)q81+(515.712374.687i)q83+(47.7168+146.857i)q85+228.956q87830.847q89+(24.6589+75.8924i)q91+(278.629+202.436i)q93+(56.0629+172.544i)q95+(642.830467.043i)q97+(247.873215.337i)q99+O(q100)q+(-0.927051 + 2.85317i) q^{3} +(1.60477 - 1.16593i) q^{5} +(3.78898 + 11.6613i) q^{7} +(-7.28115 - 5.29007i) q^{9} +(-8.21788 + 35.5453i) q^{11} +(5.26513 + 3.82534i) q^{13} +(1.83890 + 5.65957i) q^{15} +(-62.9784 + 45.7565i) q^{17} +(-28.2631 + 86.9850i) q^{19} -36.7842 q^{21} -20.2371 q^{23} +(-37.4112 + 115.140i) q^{25} +(21.8435 - 15.8702i) q^{27} +(-23.5838 - 72.5834i) q^{29} +(92.8764 + 67.4787i) q^{31} +(-93.7983 - 56.3993i) q^{33} +(19.6767 + 14.2960i) q^{35} +(27.6238 + 85.0174i) q^{37} +(-15.7954 + 11.4760i) q^{39} +(30.4658 - 93.7641i) q^{41} +488.233 q^{43} -17.8525 q^{45} +(33.0191 - 101.622i) q^{47} +(155.864 - 113.242i) q^{49} +(-72.1668 - 222.106i) q^{51} +(-447.009 - 324.771i) q^{53} +(28.2556 + 66.6235i) q^{55} +(-221.982 - 161.279i) q^{57} +(-52.3705 - 161.180i) q^{59} +(235.345 - 170.988i) q^{61} +(34.1008 - 104.952i) q^{63} +12.9094 q^{65} +107.351 q^{67} +(18.7608 - 57.7398i) q^{69} +(589.515 - 428.308i) q^{71} +(110.836 + 341.119i) q^{73} +(-293.832 - 213.481i) q^{75} +(-445.641 + 38.8493i) q^{77} +(413.815 + 300.654i) q^{79} +(25.0304 + 77.0356i) q^{81} +(515.712 - 374.687i) q^{83} +(-47.7168 + 146.857i) q^{85} +228.956 q^{87} -830.847 q^{89} +(-24.6589 + 75.8924i) q^{91} +(-278.629 + 202.436i) q^{93} +(56.0629 + 172.544i) q^{95} +(-642.830 - 467.043i) q^{97} +(247.873 - 215.337i) q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+6q327q5+13q718q945q1189q1399q1524q1728q19+66q21348q239q25+54q27249q29217q31540q33+630q35+1215q99+O(q100) 8 q + 6 q^{3} - 27 q^{5} + 13 q^{7} - 18 q^{9} - 45 q^{11} - 89 q^{13} - 99 q^{15} - 24 q^{17} - 28 q^{19} + 66 q^{21} - 348 q^{23} - 9 q^{25} + 54 q^{27} - 249 q^{29} - 217 q^{31} - 540 q^{33} + 630 q^{35}+ \cdots - 1215 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/132Z)×\left(\mathbb{Z}/132\mathbb{Z}\right)^\times.

nn 1313 6767 8989
χ(n)\chi(n) e(15)e\left(\frac{1}{5}\right) 11 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 −0.927051 + 2.85317i −0.178411 + 0.549093i
44 0 0
55 1.60477 1.16593i 0.143535 0.104284i −0.513700 0.857970i 0.671726π-0.671726\pi
0.657235 + 0.753685i 0.271726π0.271726\pi
66 0 0
77 3.78898 + 11.6613i 0.204586 + 0.629650i 0.999730 + 0.0232301i 0.00739502π0.00739502\pi
−0.795144 + 0.606420i 0.792605π0.792605\pi
88 0 0
99 −7.28115 5.29007i −0.269672 0.195928i
1010 0 0
1111 −8.21788 + 35.5453i −0.225253 + 0.974300i
1212 0 0
1313 5.26513 + 3.82534i 0.112330 + 0.0816123i 0.642532 0.766259i 0.277884π-0.277884\pi
−0.530202 + 0.847871i 0.677884π0.677884\pi
1414 0 0
1515 1.83890 + 5.65957i 0.0316535 + 0.0974196i
1616 0 0
1717 −62.9784 + 45.7565i −0.898500 + 0.652798i −0.938080 0.346418i 0.887398π-0.887398\pi
0.0395805 + 0.999216i 0.487398π0.487398\pi
1818 0 0
1919 −28.2631 + 86.9850i −0.341264 + 1.05030i 0.622290 + 0.782787i 0.286202π0.286202\pi
−0.963554 + 0.267515i 0.913798π0.913798\pi
2020 0 0
2121 −36.7842 −0.382237
2222 0 0
2323 −20.2371 −0.183466 −0.0917331 0.995784i 0.529241π-0.529241\pi
−0.0917331 + 0.995784i 0.529241π0.529241\pi
2424 0 0
2525 −37.4112 + 115.140i −0.299290 + 0.921120i
2626 0 0
2727 21.8435 15.8702i 0.155695 0.113119i
2828 0 0
2929 −23.5838 72.5834i −0.151014 0.464773i 0.846721 0.532036i 0.178573π-0.178573\pi
−0.997735 + 0.0672639i 0.978573π0.978573\pi
3030 0 0
3131 92.8764 + 67.4787i 0.538100 + 0.390953i 0.823379 0.567492i 0.192086π-0.192086\pi
−0.285279 + 0.958445i 0.592086π0.592086\pi
3232 0 0
3333 −93.7983 56.3993i −0.494794 0.297511i
3434 0 0
3535 19.6767 + 14.2960i 0.0950279 + 0.0690418i
3636 0 0
3737 27.6238 + 85.0174i 0.122739 + 0.377751i 0.993482 0.113987i 0.0363621π-0.0363621\pi
−0.870744 + 0.491737i 0.836362π0.836362\pi
3838 0 0
3939 −15.7954 + 11.4760i −0.0648535 + 0.0471189i
4040 0 0
4141 30.4658 93.7641i 0.116048 0.357159i −0.876116 0.482100i 0.839874π-0.839874\pi
0.992164 + 0.124941i 0.0398743π0.0398743\pi
4242 0 0
4343 488.233 1.73151 0.865753 0.500471i 0.166840π-0.166840\pi
0.865753 + 0.500471i 0.166840π0.166840\pi
4444 0 0
4545 −17.8525 −0.0591397
4646 0 0
4747 33.0191 101.622i 0.102475 0.315386i −0.886654 0.462433i 0.846977π-0.846977\pi
0.989130 + 0.147046i 0.0469766π0.0469766\pi
4848 0 0
4949 155.864 113.242i 0.454413 0.330150i
5050 0 0
5151 −72.1668 222.106i −0.198144 0.609826i
5252 0 0
5353 −447.009 324.771i −1.15852 0.841712i −0.168928 0.985628i 0.554030π-0.554030\pi
−0.989590 + 0.143916i 0.954030π0.954030\pi
5454 0 0
5555 28.2556 + 66.6235i 0.0692725 + 0.163337i
5656 0 0
5757 −221.982 161.279i −0.515828 0.374771i
5858 0 0
5959 −52.3705 161.180i −0.115560 0.355658i 0.876503 0.481396i 0.159870π-0.159870\pi
−0.992063 + 0.125738i 0.959870π0.959870\pi
6060 0 0
6161 235.345 170.988i 0.493982 0.358899i −0.312732 0.949841i 0.601244π-0.601244\pi
0.806713 + 0.590943i 0.201244π0.201244\pi
6262 0 0
6363 34.1008 104.952i 0.0681953 0.209883i
6464 0 0
6565 12.9094 0.0246341
6666 0 0
6767 107.351 0.195747 0.0978736 0.995199i 0.468796π-0.468796\pi
0.0978736 + 0.995199i 0.468796π0.468796\pi
6868 0 0
6969 18.7608 57.7398i 0.0327324 0.100740i
7070 0 0
7171 589.515 428.308i 0.985389 0.715927i 0.0264824 0.999649i 0.491569π-0.491569\pi
0.958906 + 0.283722i 0.0915694π0.0915694\pi
7272 0 0
7373 110.836 + 341.119i 0.177704 + 0.546917i 0.999747 0.0225091i 0.00716546π-0.00716546\pi
−0.822043 + 0.569426i 0.807165π0.807165\pi
7474 0 0
7575 −293.832 213.481i −0.452383 0.328676i
7676 0 0
7777 −445.641 + 38.8493i −0.659552 + 0.0574973i
7878 0 0
7979 413.815 + 300.654i 0.589339 + 0.428180i 0.842079 0.539354i 0.181332π-0.181332\pi
−0.252740 + 0.967534i 0.581332π0.581332\pi
8080 0 0
8181 25.0304 + 77.0356i 0.0343352 + 0.105673i
8282 0 0
8383 515.712 374.687i 0.682010 0.495509i −0.192014 0.981392i 0.561502π-0.561502\pi
0.874024 + 0.485883i 0.161502π0.161502\pi
8484 0 0
8585 −47.7168 + 146.857i −0.0608896 + 0.187399i
8686 0 0
8787 228.956 0.282146
8888 0 0
8989 −830.847 −0.989547 −0.494773 0.869022i 0.664749π-0.664749\pi
−0.494773 + 0.869022i 0.664749π0.664749\pi
9090 0 0
9191 −24.6589 + 75.8924i −0.0284061 + 0.0874251i
9292 0 0
9393 −278.629 + 202.436i −0.310672 + 0.225717i
9494 0 0
9595 56.0629 + 172.544i 0.0605467 + 0.186344i
9696 0 0
9797 −642.830 467.043i −0.672882 0.488877i 0.198107 0.980180i 0.436521π-0.436521\pi
−0.870989 + 0.491303i 0.836521π0.836521\pi
9898 0 0
9999 247.873 215.337i 0.251638 0.218608i
100100 0 0
101101 381.819 + 277.407i 0.376162 + 0.273298i 0.759762 0.650202i 0.225316π-0.225316\pi
−0.383599 + 0.923500i 0.625316π0.625316\pi
102102 0 0
103103 571.034 + 1757.46i 0.546268 + 1.68124i 0.717955 + 0.696090i 0.245078π0.245078\pi
−0.171686 + 0.985152i 0.554922π0.554922\pi
104104 0 0
105105 −59.0302 + 42.8880i −0.0548644 + 0.0398613i
106106 0 0
107107 −287.507 + 884.855i −0.259760 + 0.799460i 0.733094 + 0.680127i 0.238075π0.238075\pi
−0.992854 + 0.119332i 0.961925π0.961925\pi
108108 0 0
109109 433.332 0.380786 0.190393 0.981708i 0.439024π-0.439024\pi
0.190393 + 0.981708i 0.439024π0.439024\pi
110110 0 0
111111 −268.178 −0.229318
112112 0 0
113113 −520.776 + 1602.78i −0.433544 + 1.33431i 0.461026 + 0.887386i 0.347481π0.347481\pi
−0.894571 + 0.446926i 0.852519π0.852519\pi
114114 0 0
115115 −32.4759 + 23.5951i −0.0263338 + 0.0191327i
116116 0 0
117117 −18.0999 55.7058i −0.0143020 0.0440171i
118118 0 0
119119 −772.203 561.038i −0.594855 0.432187i
120120 0 0
121121 −1195.93 584.214i −0.898522 0.438928i
122122 0 0
123123 239.282 + 173.848i 0.175409 + 0.127442i
124124 0 0
125125 150.830 + 464.208i 0.107925 + 0.332160i
126126 0 0
127127 −100.382 + 72.9320i −0.0701377 + 0.0509580i −0.622302 0.782778i 0.713802π-0.713802\pi
0.552164 + 0.833736i 0.313802π0.313802\pi
128128 0 0
129129 −452.617 + 1393.01i −0.308920 + 0.950758i
130130 0 0
131131 −140.240 −0.0935328 −0.0467664 0.998906i 0.514892π-0.514892\pi
−0.0467664 + 0.998906i 0.514892π0.514892\pi
132132 0 0
133133 −1121.45 −0.731140
134134 0 0
135135 16.5501 50.9361i 0.0105512 0.0324732i
136136 0 0
137137 610.321 443.424i 0.380608 0.276528i −0.380988 0.924580i 0.624416π-0.624416\pi
0.761596 + 0.648052i 0.224416π0.224416\pi
138138 0 0
139139 −367.333 1130.53i −0.224149 0.689861i −0.998377 0.0569534i 0.981861π-0.981861\pi
0.774227 0.632907i 0.218139π-0.218139\pi
140140 0 0
141141 259.336 + 188.418i 0.154894 + 0.112537i
142142 0 0
143143 −179.241 + 155.714i −0.104817 + 0.0910594i
144144 0 0
145145 −122.474 88.9827i −0.0701443 0.0509628i
146146 0 0
147147 178.604 + 549.686i 0.100211 + 0.308417i
148148 0 0
149149 1173.73 852.763i 0.645339 0.468866i −0.216341 0.976318i 0.569412π-0.569412\pi
0.861680 + 0.507452i 0.169412π0.169412\pi
150150 0 0
151151 −870.060 + 2677.77i −0.468904 + 1.44314i 0.385102 + 0.922874i 0.374166π0.374166\pi
−0.854006 + 0.520263i 0.825834π0.825834\pi
152152 0 0
153153 700.610 0.370202
154154 0 0
155155 227.721 0.118006
156156 0 0
157157 742.481 2285.12i 0.377429 1.16161i −0.564396 0.825504i 0.690891π-0.690891\pi
0.941825 0.336104i 0.109109π-0.109109\pi
158158 0 0
159159 1341.03 974.314i 0.668871 0.485963i
160160 0 0
161161 −76.6779 235.990i −0.0375346 0.115520i
162162 0 0
163163 −1694.23 1230.93i −0.814123 0.591495i 0.100900 0.994897i 0.467828π-0.467828\pi
−0.915023 + 0.403402i 0.867828π0.867828\pi
164164 0 0
165165 −216.283 + 18.8547i −0.102046 + 0.00889598i
166166 0 0
167167 3089.83 + 2244.90i 1.43173 + 1.04021i 0.989692 + 0.143214i 0.0457437π0.0457437\pi
0.442036 + 0.896997i 0.354256π0.354256\pi
168168 0 0
169169 −665.822 2049.19i −0.303060 0.932722i
170170 0 0
171171 665.945 483.837i 0.297813 0.216374i
172172 0 0
173173 −611.373 + 1881.61i −0.268681 + 0.826916i 0.722141 + 0.691746i 0.243158π0.243158\pi
−0.990822 + 0.135170i 0.956842π0.956842\pi
174174 0 0
175175 −1484.43 −0.641214
176176 0 0
177177 508.424 0.215907
178178 0 0
179179 818.393 2518.76i 0.341729 1.05174i −0.621582 0.783349i 0.713510π-0.713510\pi
0.963311 0.268386i 0.0864903π-0.0864903\pi
180180 0 0
181181 3446.85 2504.29i 1.41548 1.02841i 0.422988 0.906135i 0.360981π-0.360981\pi
0.992496 0.122274i 0.0390188π-0.0390188\pi
182182 0 0
183183 269.682 + 829.995i 0.108937 + 0.335273i
184184 0 0
185185 143.455 + 104.226i 0.0570108 + 0.0414208i
186186 0 0
187187 −1108.88 2614.60i −0.433632 1.02245i
188188 0 0
189189 267.831 + 194.591i 0.103079 + 0.0748910i
190190 0 0
191191 −407.397 1253.84i −0.154336 0.474998i 0.843757 0.536726i 0.180339π-0.180339\pi
−0.998093 + 0.0617276i 0.980339π0.980339\pi
192192 0 0
193193 2230.05 1620.23i 0.831724 0.604283i −0.0883223 0.996092i 0.528151π-0.528151\pi
0.920047 + 0.391809i 0.128151π0.128151\pi
194194 0 0
195195 −11.9677 + 36.8328i −0.00439500 + 0.0135264i
196196 0 0
197197 1901.61 0.687738 0.343869 0.939018i 0.388262π-0.388262\pi
0.343869 + 0.939018i 0.388262π0.388262\pi
198198 0 0
199199 −3994.66 −1.42299 −0.711493 0.702693i 0.751981π-0.751981\pi
−0.711493 + 0.702693i 0.751981π0.751981\pi
200200 0 0
201201 −99.5203 + 306.292i −0.0349235 + 0.107483i
202202 0 0
203203 757.058 550.034i 0.261749 0.190172i
204204 0 0
205205 −60.4322 185.991i −0.0205891 0.0633668i
206206 0 0
207207 147.349 + 107.056i 0.0494758 + 0.0359462i
208208 0 0
209209 −2859.64 1719.45i −0.946438 0.569077i
210210 0 0
211211 −417.464 303.305i −0.136206 0.0989592i 0.517596 0.855625i 0.326827π-0.326827\pi
−0.653802 + 0.756666i 0.726827π0.726827\pi
212212 0 0
213213 675.525 + 2079.05i 0.217306 + 0.668799i
214214 0 0
215215 783.502 569.247i 0.248532 0.180569i
216216 0 0
217217 −434.981 + 1338.73i −0.136076 + 0.418798i
218218 0 0
219219 −1076.02 −0.332013
220220 0 0
221221 −506.624 −0.154204
222222 0 0
223223 −271.131 + 834.455i −0.0814182 + 0.250579i −0.983477 0.181034i 0.942056π-0.942056\pi
0.902059 + 0.431614i 0.142056π0.142056\pi
224224 0 0
225225 881.495 640.444i 0.261184 0.189761i
226226 0 0
227227 −1238.72 3812.39i −0.362189 1.11470i −0.951723 0.306959i 0.900689π-0.900689\pi
0.589534 0.807743i 0.299311π-0.299311\pi
228228 0 0
229229 −1386.93 1007.66i −0.400222 0.290779i 0.369409 0.929267i 0.379560π-0.379560\pi
−0.769632 + 0.638488i 0.779560π0.779560\pi
230230 0 0
231231 302.288 1307.50i 0.0861000 0.372413i
232232 0 0
233233 5121.82 + 3721.22i 1.44009 + 1.04629i 0.988023 + 0.154305i 0.0493137π0.0493137\pi
0.452068 + 0.891983i 0.350686π0.350686\pi
234234 0 0
235235 −65.4970 201.579i −0.0181811 0.0559556i
236236 0 0
237237 −1241.44 + 901.962i −0.340255 + 0.247210i
238238 0 0
239239 −1863.05 + 5733.87i −0.504228 + 1.55185i 0.297837 + 0.954617i 0.403735π0.403735\pi
−0.802065 + 0.597237i 0.796265π0.796265\pi
240240 0 0
241241 2528.58 0.675852 0.337926 0.941173i 0.390275π-0.390275\pi
0.337926 + 0.941173i 0.390275π0.390275\pi
242242 0 0
243243 −243.000 −0.0641500
244244 0 0
245245 118.093 363.454i 0.0307947 0.0947763i
246246 0 0
247247 −481.557 + 349.871i −0.124051 + 0.0901287i
248248 0 0
249249 590.954 + 1818.77i 0.150402 + 0.462891i
250250 0 0
251251 −2313.85 1681.11i −0.581867 0.422751i 0.257530 0.966270i 0.417092π-0.417092\pi
−0.839397 + 0.543519i 0.817092π0.817092\pi
252252 0 0
253253 166.306 719.333i 0.0413263 0.178751i
254254 0 0
255255 −374.773 272.288i −0.0920360 0.0668681i
256256 0 0
257257 1454.89 + 4477.69i 0.353127 + 1.08681i 0.957088 + 0.289798i 0.0935882π0.0935882\pi
−0.603961 + 0.797014i 0.706412π0.706412\pi
258258 0 0
259259 −886.746 + 644.259i −0.212740 + 0.154565i
260260 0 0
261261 −212.254 + 653.251i −0.0503379 + 0.154924i
262262 0 0
263263 −1322.27 −0.310018 −0.155009 0.987913i 0.549541π-0.549541\pi
−0.155009 + 0.987913i 0.549541π0.549541\pi
264264 0 0
265265 −1096.01 −0.254065
266266 0 0
267267 770.238 2370.55i 0.176546 0.543353i
268268 0 0
269269 −4208.37 + 3057.56i −0.953863 + 0.693022i −0.951717 0.306976i 0.900683π-0.900683\pi
−0.00214547 + 0.999998i 0.500683π0.500683\pi
270270 0 0
271271 1664.83 + 5123.81i 0.373177 + 1.14852i 0.944700 + 0.327935i 0.106353π0.106353\pi
−0.571523 + 0.820586i 0.693647π0.693647\pi
272272 0 0
273273 −193.674 140.712i −0.0429365 0.0311952i
274274 0 0
275275 −3785.24 2276.00i −0.830031 0.499083i
276276 0 0
277277 2523.21 + 1833.22i 0.547310 + 0.397644i 0.826793 0.562507i 0.190163π-0.190163\pi
−0.279483 + 0.960151i 0.590163π0.590163\pi
278278 0 0
279279 −319.281 982.645i −0.0685120 0.210858i
280280 0 0
281281 5524.63 4013.88i 1.17285 0.852127i 0.181505 0.983390i 0.441903π-0.441903\pi
0.991348 + 0.131263i 0.0419031π0.0419031\pi
282282 0 0
283283 −2827.14 + 8701.05i −0.593838 + 1.82765i −0.0334114 + 0.999442i 0.510637π0.510637\pi
−0.560427 + 0.828204i 0.689363π0.689363\pi
284284 0 0
285285 −544.271 −0.113122
286286 0 0
287287 1208.84 0.248627
288288 0 0
289289 354.420 1090.79i 0.0721391 0.222021i
290290 0 0
291291 1928.49 1401.13i 0.388488 0.282253i
292292 0 0
293293 1000.77 + 3080.04i 0.199541 + 0.614123i 0.999894 + 0.0145935i 0.00464542π0.00464542\pi
−0.800353 + 0.599529i 0.795355π0.795355\pi
294294 0 0
295295 −271.968 197.596i −0.0536766 0.0389983i
296296 0 0
297297 384.604 + 906.851i 0.0751413 + 0.177175i
298298 0 0
299299 −106.551 77.4138i −0.0206087 0.0149731i
300300 0 0
301301 1849.90 + 5693.42i 0.354242 + 1.09024i
302302 0 0
303303 −1145.46 + 832.222i −0.217177 + 0.157789i
304304 0 0
305305 178.314 548.794i 0.0334762 0.103029i
306306 0 0
307307 −3909.16 −0.726735 −0.363367 0.931646i 0.618373π-0.618373\pi
−0.363367 + 0.931646i 0.618373π0.618373\pi
308308 0 0
309309 −5543.71 −1.02062
310310 0 0
311311 −2705.87 + 8327.83i −0.493364 + 1.51842i 0.326128 + 0.945326i 0.394256π0.394256\pi
−0.819491 + 0.573091i 0.805744π0.805744\pi
312312 0 0
313313 4483.78 3257.66i 0.809707 0.588286i −0.104039 0.994573i 0.533177π-0.533177\pi
0.913746 + 0.406287i 0.133177π0.133177\pi
314314 0 0
315315 −67.6426 208.183i −0.0120991 0.0372373i
316316 0 0
317317 −6558.45 4765.00i −1.16202 0.844255i −0.171985 0.985099i 0.555018π-0.555018\pi
−0.990032 + 0.140845i 0.955018π0.955018\pi
318318 0 0
319319 2773.81 241.810i 0.486844 0.0424413i
320320 0 0
321321 −2258.11 1640.61i −0.392633 0.285265i
322322 0 0
323323 −2200.16 6771.39i −0.379010 1.16647i
324324 0 0
325325 −637.425 + 463.116i −0.108794 + 0.0790433i
326326 0 0
327327 −401.721 + 1236.37i −0.0679365 + 0.209087i
328328 0 0
329329 1310.16 0.219548
330330 0 0
331331 2603.54 0.432336 0.216168 0.976356i 0.430644π-0.430644\pi
0.216168 + 0.976356i 0.430644π0.430644\pi
332332 0 0
333333 248.614 765.156i 0.0409129 0.125917i
334334 0 0
335335 172.274 125.165i 0.0280966 0.0204134i
336336 0 0
337337 −238.684 734.594i −0.0385814 0.118741i 0.929911 0.367785i 0.119884π-0.119884\pi
−0.968492 + 0.249044i 0.919884π0.919884\pi
338338 0 0
339339 −4090.23 2971.73i −0.655312 0.476112i
340340 0 0
341341 −3161.80 + 2746.79i −0.502114 + 0.436208i
342342 0 0
343343 5313.56 + 3860.53i 0.836458 + 0.607722i
344344 0 0
345345 −37.2141 114.533i −0.00580735 0.0178732i
346346 0 0
347347 3408.56 2476.46i 0.527323 0.383123i −0.292032 0.956408i 0.594332π-0.594332\pi
0.819355 + 0.573286i 0.194332π0.194332\pi
348348 0 0
349349 1483.19 4564.79i 0.227488 0.700137i −0.770541 0.637390i 0.780014π-0.780014\pi
0.998029 0.0627467i 0.0199860π-0.0199860\pi
350350 0 0
351351 175.718 0.0267211
352352 0 0
353353 −448.615 −0.0676412 −0.0338206 0.999428i 0.510767π-0.510767\pi
−0.0338206 + 0.999428i 0.510767π0.510767\pi
354354 0 0
355355 446.658 1374.67i 0.0667779 0.205521i
356356 0 0
357357 2316.61 1683.11i 0.343440 0.249523i
358358 0 0
359359 −3435.35 10572.9i −0.505045 1.55437i −0.800695 0.599072i 0.795536π-0.795536\pi
0.295651 0.955296i 0.404464π-0.404464\pi
360360 0 0
361361 −1218.54 885.320i −0.177655 0.129074i
362362 0 0
363363 2775.55 2870.60i 0.401319 0.415062i
364364 0 0
365365 575.589 + 418.190i 0.0825417 + 0.0599700i
366366 0 0
367367 3149.15 + 9692.08i 0.447913 + 1.37854i 0.879257 + 0.476348i 0.158040π0.158040\pi
−0.431344 + 0.902188i 0.641960π0.641960\pi
368368 0 0
369369 −717.845 + 521.545i −0.101272 + 0.0735787i
370370 0 0
371371 2093.54 6443.25i 0.292968 0.901663i
372372 0 0
373373 11174.0 1.55112 0.775560 0.631274i 0.217468π-0.217468\pi
0.775560 + 0.631274i 0.217468π0.217468\pi
374374 0 0
375375 −1464.29 −0.201642
376376 0 0
377377 153.485 472.377i 0.0209678 0.0645323i
378378 0 0
379379 7717.65 5607.20i 1.04599 0.759954i 0.0745413 0.997218i 0.476251π-0.476251\pi
0.971445 + 0.237264i 0.0762507π0.0762507\pi
380380 0 0
381381 −115.028 354.019i −0.0154673 0.0476036i
382382 0 0
383383 9441.77 + 6859.85i 1.25967 + 0.915201i 0.998741 0.0501584i 0.0159726π-0.0159726\pi
0.260925 + 0.965359i 0.415973π0.415973\pi
384384 0 0
385385 −669.856 + 581.932i −0.0886728 + 0.0770338i
386386 0 0
387387 −3554.90 2582.78i −0.466939 0.339251i
388388 0 0
389389 −3268.87 10060.6i −0.426063 1.31129i −0.901973 0.431793i 0.857881π-0.857881\pi
0.475910 0.879494i 0.342119π-0.342119\pi
390390 0 0
391391 1274.50 925.977i 0.164844 0.119766i
392392 0 0
393393 130.009 400.128i 0.0166873 0.0513582i
394394 0 0
395395 1014.62 0.129243
396396 0 0
397397 −337.577 −0.0426763 −0.0213381 0.999772i 0.506793π-0.506793\pi
−0.0213381 + 0.999772i 0.506793π0.506793\pi
398398 0 0
399399 1039.64 3199.67i 0.130443 0.401464i
400400 0 0
401401 −4762.00 + 3459.79i −0.593024 + 0.430857i −0.843396 0.537292i 0.819447π-0.819447\pi
0.250372 + 0.968150i 0.419447π0.419447\pi
402402 0 0
403403 230.878 + 710.569i 0.0285381 + 0.0878311i
404404 0 0
405405 129.986 + 94.4407i 0.0159483 + 0.0115872i
406406 0 0
407407 −3248.98 + 283.233i −0.395690 + 0.0344947i
408408 0 0
409409 −10318.7 7496.99i −1.24750 0.906363i −0.249427 0.968394i 0.580242π-0.580242\pi
−0.998074 + 0.0620309i 0.980242π0.980242\pi
410410 0 0
411411 699.366 + 2152.43i 0.0839347 + 0.258325i
412412 0 0
413413 1681.13 1221.41i 0.200298 0.145525i
414414 0 0
415415 390.740 1202.57i 0.0462185 0.142246i
416416 0 0
417417 3566.14 0.418788
418418 0 0
419419 4686.15 0.546380 0.273190 0.961960i 0.411921π-0.411921\pi
0.273190 + 0.961960i 0.411921π0.411921\pi
420420 0 0
421421 −479.525 + 1475.83i −0.0555121 + 0.170849i −0.974968 0.222344i 0.928629π-0.928629\pi
0.919456 + 0.393192i 0.128629π0.128629\pi
422422 0 0
423423 −778.007 + 565.255i −0.0894279 + 0.0649732i
424424 0 0
425425 −2912.30 8963.13i −0.332393 1.02300i
426426 0 0
427427 2885.66 + 2096.56i 0.327042 + 0.237610i
428428 0 0
429429 −278.114 655.760i −0.0312994 0.0738005i
430430 0 0
431431 9433.01 + 6853.48i 1.05423 + 0.765941i 0.973012 0.230756i 0.0741198π-0.0741198\pi
0.0812156 + 0.996697i 0.474120π0.474120\pi
432432 0 0
433433 785.662 + 2418.02i 0.0871975 + 0.268366i 0.985142 0.171743i 0.0549398π-0.0549398\pi
−0.897944 + 0.440109i 0.854940π0.854940\pi
434434 0 0
435435 367.422 266.948i 0.0404978 0.0294234i
436436 0 0
437437 571.963 1760.32i 0.0626103 0.192695i
438438 0 0
439439 −5101.99 −0.554681 −0.277340 0.960772i 0.589453π-0.589453\pi
−0.277340 + 0.960772i 0.589453π0.589453\pi
440440 0 0
441441 −1733.92 −0.187228
442442 0 0
443443 846.872 2606.40i 0.0908264 0.279535i −0.895317 0.445429i 0.853051π-0.853051\pi
0.986144 + 0.165894i 0.0530511π0.0530511\pi
444444 0 0
445445 −1333.32 + 968.714i −0.142035 + 0.103194i
446446 0 0
447447 1344.97 + 4139.40i 0.142315 + 0.438002i
448448 0 0
449449 −8818.11 6406.73i −0.926843 0.673391i 0.0183750 0.999831i 0.494151π-0.494151\pi
−0.945218 + 0.326440i 0.894151π0.894151\pi
450450 0 0
451451 3082.51 + 1853.46i 0.321840 + 0.193517i
452452 0 0
453453 −6833.54 4964.86i −0.708758 0.514943i
454454 0 0
455455 48.9136 + 150.541i 0.00503979 + 0.0155109i
456456 0 0
457457 −11106.2 + 8069.11i −1.13682 + 0.825946i −0.986673 0.162718i 0.947974π-0.947974\pi
−0.150144 + 0.988664i 0.547974π0.547974\pi
458458 0 0
459459 −649.501 + 1998.96i −0.0660482 + 0.203275i
460460 0 0
461461 3104.40 0.313636 0.156818 0.987628i 0.449876π-0.449876\pi
0.156818 + 0.987628i 0.449876π0.449876\pi
462462 0 0
463463 3771.28 0.378545 0.189272 0.981925i 0.439387π-0.439387\pi
0.189272 + 0.981925i 0.439387π0.439387\pi
464464 0 0
465465 −211.109 + 649.727i −0.0210537 + 0.0647965i
466466 0 0
467467 −9914.73 + 7203.47i −0.982439 + 0.713784i −0.958252 0.285924i 0.907700π-0.907700\pi
−0.0241865 + 0.999707i 0.507700π0.507700\pi
468468 0 0
469469 406.753 + 1251.86i 0.0400471 + 0.123252i
470470 0 0
471471 5831.52 + 4236.85i 0.570493 + 0.414487i
472472 0 0
473473 −4012.24 + 17354.4i −0.390027 + 1.68701i
474474 0 0
475475 −8958.09 6508.43i −0.865316 0.628689i
476476 0 0
477477 1536.68 + 4729.42i 0.147505 + 0.453973i
478478 0 0
479479 843.488 612.830i 0.0804592 0.0584570i −0.546828 0.837245i 0.684165π-0.684165\pi
0.627288 + 0.778788i 0.284165π0.284165\pi
480480 0 0
481481 −179.778 + 553.298i −0.0170419 + 0.0524496i
482482 0 0
483483 744.405 0.0701275
484484 0 0
485485 −1576.14 −0.147564
486486 0 0
487487 536.464 1651.07i 0.0499169 0.153628i −0.922991 0.384822i 0.874263π-0.874263\pi
0.972908 + 0.231193i 0.0742630π0.0742630\pi
488488 0 0
489489 5082.68 3692.78i 0.470034 0.341500i
490490 0 0
491491 3489.83 + 10740.6i 0.320761 + 0.987202i 0.973318 + 0.229462i 0.0736966π0.0736966\pi
−0.652556 + 0.757740i 0.726303π0.726303\pi
492492 0 0
493493 4806.43 + 3492.07i 0.439089 + 0.319016i
494494 0 0
495495 146.709 634.570i 0.0133214 0.0576198i
496496 0 0
497497 7228.28 + 5251.66i 0.652380 + 0.473982i
498498 0 0
499499 −4140.83 12744.2i −0.371481 1.14330i −0.945822 0.324686i 0.894741π-0.894741\pi
0.574341 0.818616i 0.305259π-0.305259\pi
500500 0 0
501501 −9269.50 + 6734.69i −0.826608 + 0.600566i
502502 0 0
503503 2410.37 7418.36i 0.213664 0.657591i −0.785582 0.618758i 0.787636π-0.787636\pi
0.999246 0.0388326i 0.0123639π-0.0123639\pi
504504 0 0
505505 936.171 0.0824932
506506 0 0
507507 6463.94 0.566220
508508 0 0
509509 −6733.56 + 20723.8i −0.586365 + 1.80465i 0.00735129 + 0.999973i 0.497660π0.497660\pi
−0.593717 + 0.804674i 0.702340π0.702340\pi
510510 0 0
511511 −3557.93 + 2584.99i −0.308011 + 0.223783i
512512 0 0
513513 763.105 + 2348.60i 0.0656762 + 0.202131i
514514 0 0
515515 2965.46 + 2154.54i 0.253736 + 0.184350i
516516 0 0
517517 3340.85 + 2008.80i 0.284198 + 0.170883i
518518 0 0
519519 −4801.79 3488.70i −0.406118 0.295062i
520520 0 0
521521 −258.126 794.431i −0.0217058 0.0668036i 0.939617 0.342228i 0.111182π-0.111182\pi
−0.961323 + 0.275425i 0.911182π0.911182\pi
522522 0 0
523523 −664.886 + 483.068i −0.0555898 + 0.0403883i −0.615233 0.788345i 0.710938π-0.710938\pi
0.559643 + 0.828734i 0.310938π0.310938\pi
524524 0 0
525525 1376.14 4235.33i 0.114400 0.352086i
526526 0 0
527527 −8936.79 −0.738696
528528 0 0
529529 −11757.5 −0.966340
530530 0 0
531531 −471.335 + 1450.62i −0.0385201 + 0.118553i
532532 0 0
533533 519.087 377.138i 0.0421841 0.0306486i
534534 0 0
535535 570.300 + 1755.20i 0.0460864 + 0.141839i
536536 0 0
537537 6427.74 + 4670.03i 0.516532 + 0.375282i
538538 0 0
539539 2744.33 + 6470.82i 0.219308 + 0.517102i
540540 0 0
541541 −16604.9 12064.2i −1.31960 0.958744i −0.999937 0.0112114i 0.996431π-0.996431\pi
−0.319660 0.947532i 0.603569π-0.603569\pi
542542 0 0
543543 3949.74 + 12156.1i 0.312154 + 0.960712i
544544 0 0
545545 695.399 505.237i 0.0546562 0.0397101i
546546 0 0
547547 1843.40 5673.41i 0.144092 0.443469i −0.852801 0.522236i 0.825098π-0.825098\pi
0.996893 + 0.0787666i 0.0250982π0.0250982\pi
548548 0 0
549549 −2618.12 −0.203532
550550 0 0
551551 6980.22 0.539687
552552 0 0
553553 −1938.08 + 5964.79i −0.149033 + 0.458677i
554554 0 0
555555 −430.364 + 312.678i −0.0329152 + 0.0239143i
556556 0 0
557557 −451.401 1389.27i −0.0343383 0.105683i 0.932418 0.361381i 0.117695π-0.117695\pi
−0.966757 + 0.255698i 0.917695π0.917695\pi
558558 0 0
559559 2570.61 + 1867.66i 0.194499 + 0.141312i
560560 0 0
561561 8487.89 739.943i 0.638786 0.0556870i
562562 0 0
563563 20000.9 + 14531.5i 1.49723 + 1.08780i 0.971469 + 0.237167i 0.0762187π0.0762187\pi
0.525759 + 0.850633i 0.323781π0.323781\pi
564564 0 0
565565 1033.02 + 3179.29i 0.0769191 + 0.236733i
566566 0 0
567567 −803.494 + 583.773i −0.0595125 + 0.0432384i
568568 0 0
569569 7905.61 24331.0i 0.582461 1.79263i −0.0267756 0.999641i 0.508524π-0.508524\pi
0.609236 0.792989i 0.291476π-0.291476\pi
570570 0 0
571571 −8840.84 −0.647947 −0.323973 0.946066i 0.605019π-0.605019\pi
−0.323973 + 0.946066i 0.605019π0.605019\pi
572572 0 0
573573 3955.09 0.288353
574574 0 0
575575 757.094 2330.10i 0.0549096 0.168994i
576576 0 0
577577 −14479.3 + 10519.8i −1.04468 + 0.759003i −0.971193 0.238293i 0.923412π-0.923412\pi
−0.0734847 + 0.997296i 0.523412π0.523412\pi
578578 0 0
579579 2555.41 + 7864.76i 0.183419 + 0.564505i
580580 0 0
581581 6323.36 + 4594.19i 0.451527 + 0.328053i
582582 0 0
583583 15217.6 13220.1i 1.08104 0.939146i
584584 0 0
585585 −93.9956 68.2918i −0.00664314 0.00482653i
586586 0 0
587587 −8445.52 25992.6i −0.593840 1.82765i −0.560419 0.828209i 0.689360π-0.689360\pi
−0.0334206 0.999441i 0.510640π-0.510640\pi
588588 0 0
589589 −8494.61 + 6171.70i −0.594252 + 0.431749i
590590 0 0
591591 −1762.89 + 5425.63i −0.122700 + 0.377632i
592592 0 0
593593 −389.353 −0.0269626 −0.0134813 0.999909i 0.504291π-0.504291\pi
−0.0134813 + 0.999909i 0.504291π0.504291\pi
594594 0 0
595595 −1893.34 −0.130453
596596 0 0
597597 3703.26 11397.5i 0.253876 0.781351i
598598 0 0
599599 −20429.9 + 14843.2i −1.39356 + 1.01248i −0.398096 + 0.917344i 0.630329π0.630329\pi
−0.995464 + 0.0951368i 0.969671π0.969671\pi
600600 0 0
601601 −2181.63 6714.37i −0.148071 0.455715i 0.849322 0.527875i 0.177011π-0.177011\pi
−0.997393 + 0.0721596i 0.977011π0.977011\pi
602602 0 0
603603 −781.642 567.896i −0.0527876 0.0383524i
604604 0 0
605605 −2600.35 + 456.850i −0.174743 + 0.0307002i
606606 0 0
607607 −12283.3 8924.34i −0.821357 0.596751i 0.0957435 0.995406i 0.469477π-0.469477\pi
−0.917101 + 0.398655i 0.869477π0.869477\pi
608608 0 0
609609 867.511 + 2669.92i 0.0577230 + 0.177653i
610610 0 0
611611 562.591 408.746i 0.0372504 0.0270640i
612612 0 0
613613 4796.66 14762.6i 0.316044 0.972685i −0.659278 0.751899i 0.729138π-0.729138\pi
0.975322 0.220785i 0.0708620π-0.0708620\pi
614614 0 0
615615 586.688 0.0384676
616616 0 0
617617 27430.4 1.78980 0.894899 0.446269i 0.147248π-0.147248\pi
0.894899 + 0.446269i 0.147248π0.147248\pi
618618 0 0
619619 6668.83 20524.5i 0.433025 1.33272i −0.462071 0.886843i 0.652894π-0.652894\pi
0.895097 0.445872i 0.147106π-0.147106\pi
620620 0 0
621621 −442.048 + 321.167i −0.0285648 + 0.0207536i
622622 0 0
623623 −3148.07 9688.75i −0.202447 0.623068i
624624 0 0
625625 −11459.7 8325.96i −0.733421 0.532862i
626626 0 0
627627 7556.93 6565.02i 0.481331 0.418153i
628628 0 0
629629 −5629.80 4090.29i −0.356875 0.259285i
630630 0 0
631631 −2320.52 7141.83i −0.146400 0.450573i 0.850788 0.525509i 0.176125π-0.176125\pi
−0.997188 + 0.0749353i 0.976125π0.976125\pi
632632 0 0
633633 1252.39 909.916i 0.0786384 0.0571341i
634634 0 0
635635 −76.0566 + 234.078i −0.00475310 + 0.0146285i
636636 0 0
637637 1253.83 0.0779883
638638 0 0
639639 −6558.13 −0.406003
640640 0 0
641641 −2424.82 + 7462.83i −0.149414 + 0.459850i −0.997552 0.0699253i 0.977724π-0.977724\pi
0.848138 + 0.529776i 0.177724π0.177724\pi
642642 0 0
643643 706.659 513.418i 0.0433404 0.0314887i −0.565904 0.824471i 0.691473π-0.691473\pi
0.609245 + 0.792982i 0.291473π0.291473\pi
644644 0 0
645645 897.813 + 2763.18i 0.0548083 + 0.168683i
646646 0 0
647647 20663.0 + 15012.6i 1.25556 + 0.912217i 0.998531 0.0541866i 0.0172566π-0.0172566\pi
0.257028 + 0.966404i 0.417257π0.417257\pi
648648 0 0
649649 6159.56 536.967i 0.372548 0.0324774i
650650 0 0
651651 −3416.39 2482.15i −0.205682 0.149436i
652652 0 0
653653 −3190.62 9819.73i −0.191208 0.588477i −1.00000 0.000382475i 0.999878π-0.999878\pi
0.808792 0.588095i 0.200122π-0.200122\pi
654654 0 0
655655 −225.053 + 163.510i −0.0134252 + 0.00975401i
656656 0 0
657657 997.526 3070.07i 0.0592347 0.182306i
658658 0 0
659659 −28905.7 −1.70866 −0.854329 0.519733i 0.826031π-0.826031\pi
−0.854329 + 0.519733i 0.826031π0.826031\pi
660660 0 0
661661 28041.5 1.65006 0.825028 0.565092i 0.191159π-0.191159\pi
0.825028 + 0.565092i 0.191159π0.191159\pi
662662 0 0
663663 469.666 1445.48i 0.0275118 0.0846726i
664664 0 0
665665 −1799.66 + 1307.53i −0.104944 + 0.0762465i
666666 0 0
667667 477.267 + 1468.88i 0.0277059 + 0.0852701i
668668 0 0
669669 −2129.49 1547.16i −0.123065 0.0894123i
670670 0 0
671671 4143.79 + 9770.57i 0.238404 + 0.562129i
672672 0 0
673673 −7569.87 5499.83i −0.433576 0.315012i 0.349501 0.936936i 0.386351π-0.386351\pi
−0.783077 + 0.621924i 0.786351π0.786351\pi
674674 0 0
675675 1010.10 + 3108.78i 0.0575984 + 0.177270i
676676 0 0
677677 18651.9 13551.4i 1.05886 0.769309i 0.0849856 0.996382i 0.472916π-0.472916\pi
0.973878 + 0.227073i 0.0729156π0.0729156\pi
678678 0 0
679679 3010.66 9265.84i 0.170160 0.523697i
680680 0 0
681681 12025.8 0.676693
682682 0 0
683683 18443.9 1.03329 0.516645 0.856200i 0.327181π-0.327181\pi
0.516645 + 0.856200i 0.327181π0.327181\pi
684684 0 0
685685 462.422 1423.19i 0.0257931 0.0793829i
686686 0 0
687687 4160.79 3022.99i 0.231068 0.167881i
688688 0 0
689689 −1111.20 3419.93i −0.0614418 0.189098i
690690 0 0
691691 20208.4 + 14682.3i 1.11254 + 0.808306i 0.983061 0.183276i 0.0586703π-0.0586703\pi
0.129476 + 0.991582i 0.458670π0.458670\pi
692692 0 0
693693 3450.30 + 2074.60i 0.189128 + 0.113720i
694694 0 0
695695 −1907.61 1385.96i −0.104115 0.0756440i
696696 0 0
697697 2371.63 + 7299.12i 0.128884 + 0.396663i
698698 0 0
699699 −15365.5 + 11163.7i −0.831437 + 0.604075i
700700 0 0
701701 7753.13 23861.7i 0.417734 1.28565i −0.492048 0.870568i 0.663751π-0.663751\pi
0.909782 0.415086i 0.136249π-0.136249\pi
702702 0 0
703703 −8175.97 −0.438638
704704 0 0
705705 635.858 0.0339685
706706 0 0
707707 −1788.22 + 5503.59i −0.0951246 + 0.292763i
708708 0 0
709709 12414.6 9019.76i 0.657604 0.477777i −0.208249 0.978076i 0.566776π-0.566776\pi
0.865853 + 0.500298i 0.166776π0.166776\pi
710710 0 0
711711 −1422.57 4378.22i −0.0750359 0.230937i
712712 0 0
713713 −1879.55 1365.57i −0.0987232 0.0717266i
714714 0 0
715715 −106.088 + 458.869i −0.00554891 + 0.0240010i
716716 0 0
717717 −14632.6 10631.2i −0.762152 0.553736i
718718 0 0
719719 −9216.75 28366.3i −0.478063 1.47133i −0.841783 0.539817i 0.818494π-0.818494\pi
0.363720 0.931508i 0.381506π-0.381506\pi
720720 0 0
721721 −18330.6 + 13318.0i −0.946835 + 0.687916i
722722 0 0
723723 −2344.13 + 7214.48i −0.120580 + 0.371106i
724724 0 0
725725 9239.55 0.473308
726726 0 0
727727 −19398.7 −0.989626 −0.494813 0.869000i 0.664763π-0.664763\pi
−0.494813 + 0.869000i 0.664763π0.664763\pi
728728 0 0
729729 225.273 693.320i 0.0114451 0.0352243i
730730 0 0
731731 −30748.1 + 22339.8i −1.55576 + 1.13032i
732732 0 0
733733 10180.5 + 31332.4i 0.512996 + 1.57884i 0.786900 + 0.617081i 0.211685π0.211685\pi
−0.273904 + 0.961757i 0.588315π0.588315\pi
734734 0 0
735735 927.516 + 673.880i 0.0465469 + 0.0338183i
736736 0 0
737737 −882.201 + 3815.84i −0.0440927 + 0.190717i
738738 0 0
739739 5707.17 + 4146.50i 0.284089 + 0.206403i 0.720699 0.693248i 0.243821π-0.243821\pi
−0.436610 + 0.899651i 0.643821π0.643821\pi
740740 0 0
741741 −551.815 1698.31i −0.0273568 0.0841957i
742742 0 0
743743 −4025.05 + 2924.37i −0.198741 + 0.144394i −0.682705 0.730694i 0.739197π-0.739197\pi
0.483964 + 0.875088i 0.339197π0.339197\pi
744744 0 0
745745 889.298 2736.98i 0.0437334 0.134598i
746746 0 0
747747 −5737.10 −0.281003
748748 0 0
749749 −11407.9 −0.556523
750750 0 0
751751 10127.4 31168.8i 0.492081 1.51447i −0.329376 0.944199i 0.606838π-0.606838\pi
0.821457 0.570270i 0.193162π-0.193162\pi
752752 0 0
753753 6941.54 5043.32i 0.335941 0.244076i
754754 0 0
755755 1725.86 + 5311.64i 0.0831925 + 0.256040i
756756 0 0
757757 25632.9 + 18623.4i 1.23070 + 0.894158i 0.996943 0.0781357i 0.0248968π-0.0248968\pi
0.233761 + 0.972294i 0.424897π0.424897\pi
758758 0 0
759759 1898.20 + 1141.36i 0.0907779 + 0.0545832i
760760 0 0
761761 16845.6 + 12239.0i 0.802432 + 0.583001i 0.911627 0.411019i 0.134827π-0.134827\pi
−0.109195 + 0.994020i 0.534827π0.534827\pi
762762 0 0
763763 1641.89 + 5053.21i 0.0779035 + 0.239762i
764764 0 0
765765 1124.32 816.865i 0.0531370 0.0386063i
766766 0 0
767767 340.831 1048.97i 0.0160452 0.0493821i
768768 0 0
769769 −15356.6 −0.720123 −0.360062 0.932929i 0.617244π-0.617244\pi
−0.360062 + 0.932929i 0.617244π0.617244\pi
770770 0 0
771771 −14124.4 −0.659762
772772 0 0
773773 −7407.09 + 22796.7i −0.344650 + 1.06072i 0.617121 + 0.786868i 0.288299π0.288299\pi
−0.961771 + 0.273855i 0.911701π0.911701\pi
774774 0 0
775775 −11244.1 + 8169.33i −0.521162 + 0.378646i
776776 0 0
777777 −1016.12 3127.30i −0.0469152 0.144390i
778778 0 0
779779 7295.01 + 5300.14i 0.335521 + 0.243770i
780780 0 0
781781 10379.8 + 24474.3i 0.475566 + 1.12133i
782782 0 0
783783 −1667.07 1211.19i −0.0760869 0.0552804i
784784 0 0
785785 −1472.79 4532.78i −0.0669632 0.206092i
786786 0 0
787787 −20832.6 + 15135.7i −0.943584 + 0.685554i −0.949281 0.314430i 0.898187π-0.898187\pi
0.00569705 + 0.999984i 0.498187π0.498187\pi
788788 0 0
789789 1225.81 3772.67i 0.0553107 0.170229i
790790 0 0
791791 −20663.7 −0.928847
792792 0 0
793793 1893.21 0.0847793
794794 0 0
795795 1016.06 3127.10i 0.0453281 0.139505i
796796 0 0
797797 −18104.8 + 13153.9i −0.804651 + 0.584613i −0.912275 0.409579i 0.865676π-0.865676\pi
0.107624 + 0.994192i 0.465676π0.465676\pi
798798 0 0
799799 2570.39 + 7910.86i 0.113810 + 0.350270i
800800 0 0
801801 6049.53 + 4395.24i 0.266853 + 0.193880i
802802 0 0
803803 −13036.0 + 1136.43i −0.572890 + 0.0499424i
804804 0 0
805805 −398.200 289.309i −0.0174344 0.0126668i
806806 0 0
807807 −4822.37 14841.7i −0.210354 0.647402i
808808 0 0
809809 327.625 238.034i 0.0142382 0.0103446i −0.580643 0.814158i 0.697199π-0.697199\pi
0.594882 + 0.803813i 0.297199π0.297199\pi
810810 0 0
811811 −1964.56 + 6046.29i −0.0850616 + 0.261793i −0.984536 0.175180i 0.943949π-0.943949\pi
0.899475 + 0.436973i 0.143949π0.143949\pi
812812 0 0
813813 −16162.5 −0.697224
814814 0 0
815815 −4154.03 −0.178539
816816 0 0
817817 −13799.0 + 42468.9i −0.590900 + 1.81860i
818818 0 0
819819 581.021 422.137i 0.0247894 0.0180106i
820820 0 0
821821 −13029.9 40102.0i −0.553895 1.70471i −0.698844 0.715274i 0.746302π-0.746302\pi
0.144949 0.989439i 0.453698π-0.453698\pi
822822 0 0
823823 12019.8 + 8732.87i 0.509092 + 0.369877i 0.812479 0.582991i 0.198118π-0.198118\pi
−0.303387 + 0.952867i 0.598118π0.598118\pi
824824 0 0
825825 10002.9 8689.96i 0.422130 0.366722i
826826 0 0
827827 296.623 + 215.509i 0.0124723 + 0.00906167i 0.594004 0.804462i 0.297546π-0.297546\pi
−0.581532 + 0.813524i 0.697546π0.697546\pi
828828 0 0
829829 9464.11 + 29127.5i 0.396504 + 1.22032i 0.927784 + 0.373118i 0.121711π0.121711\pi
−0.531279 + 0.847197i 0.678289π0.678289\pi
830830 0 0
831831 −7569.62 + 5499.65i −0.315989 + 0.229580i
832832 0 0
833833 −4634.50 + 14263.5i −0.192768 + 0.593280i
834834 0 0
835835 7575.88 0.313981
836836 0 0
837837 3099.64 0.128004
838838 0 0
839839 6047.10 18611.1i 0.248831 0.765822i −0.746152 0.665776i 0.768101π-0.768101\pi
0.994983 0.100047i 0.0318992π-0.0318992\pi
840840 0 0
841841 15019.0 10911.9i 0.615809 0.447411i
842842 0 0
843843 6330.66 + 19483.8i 0.258647 + 0.796034i
844844 0 0
845845 −3457.71 2512.18i −0.140768 0.102274i
846846 0 0
847847 2281.32 16159.7i 0.0925466 0.655553i
848848 0 0
849849 −22204.7 16132.6i −0.897600 0.652144i
850850 0 0
851851 −559.025 1720.50i −0.0225184 0.0693045i
852852 0 0
853853 6707.36 4873.18i 0.269233 0.195609i −0.444975 0.895543i 0.646787π-0.646787\pi
0.714207 + 0.699934i 0.246787π0.246787\pi
854854 0 0
855855 504.567 1552.90i 0.0201822 0.0621145i
856856 0 0
857857 36841.7 1.46848 0.734240 0.678890i 0.237539π-0.237539\pi
0.734240 + 0.678890i 0.237539π0.237539\pi
858858 0 0
859859 24003.0 0.953400 0.476700 0.879066i 0.341833π-0.341833\pi
0.476700 + 0.879066i 0.341833π0.341833\pi
860860 0 0
861861 −1120.66 + 3449.04i −0.0443577 + 0.136519i
862862 0 0
863863 −15761.4 + 11451.3i −0.621697 + 0.451689i −0.853514 0.521070i 0.825533π-0.825533\pi
0.231817 + 0.972759i 0.425533π0.425533\pi
864864 0 0
865865 1212.72 + 3732.38i 0.0476692 + 0.146711i
866866 0 0
867867 2783.65 + 2022.44i 0.109040 + 0.0792222i
868868 0 0
869869 −14087.5 + 12238.4i −0.549927 + 0.477745i
870870 0 0
871871 565.220 + 410.656i 0.0219882 + 0.0159754i
872872 0 0
873873 2209.85 + 6801.23i 0.0856726 + 0.263673i
874874 0 0
875875 −4841.76 + 3517.75i −0.187065 + 0.135910i
876876 0 0
877877 761.474 2343.58i 0.0293194 0.0902360i −0.935326 0.353787i 0.884894π-0.884894\pi
0.964645 + 0.263551i 0.0848938π0.0848938\pi
878878 0 0
879879 −9715.64 −0.372810
880880 0 0
881881 12953.1 0.495349 0.247675 0.968843i 0.420334π-0.420334\pi
0.247675 + 0.968843i 0.420334π0.420334\pi
882882 0 0
883883 3138.31 9658.74i 0.119607 0.368111i −0.873273 0.487231i 0.838007π-0.838007\pi
0.992880 + 0.119119i 0.0380071π0.0380071\pi
884884 0 0
885885 815.904 592.789i 0.0309902 0.0225157i
886886 0 0
887887 −3710.82 11420.7i −0.140470 0.432323i 0.855930 0.517091i 0.172985π-0.172985\pi
−0.996401 + 0.0847679i 0.972985π0.972985\pi
888888 0 0
889889 −1230.83 894.248i −0.0464349 0.0337369i
890890 0 0
891891 −2943.95 + 256.642i −0.110691 + 0.00964965i
892892 0 0
893893 7906.41 + 5744.34i 0.296280 + 0.215260i
894894 0 0
895895 −1623.37 4996.22i −0.0606294 0.186598i
896896 0 0
897897 319.653 232.241i 0.0118984 0.00864472i
898898 0 0
899899 2707.46 8332.69i 0.100444 0.309133i
900900 0 0
901901 43012.3 1.59040
902902 0 0
903903 −17959.2 −0.661845
904904 0 0
905905 2611.58 8037.61i 0.0959247 0.295226i
906906 0 0
907907 −5268.47 + 3827.77i −0.192874 + 0.140131i −0.680031 0.733183i 0.738034π-0.738034\pi
0.487157 + 0.873314i 0.338034π0.338034\pi
908908 0 0
909909 −1312.58 4039.69i −0.0478937 0.147402i
910910 0 0
911911 −13675.2 9935.64i −0.497345 0.361342i 0.310657 0.950522i 0.399451π-0.399451\pi
−0.808002 + 0.589180i 0.799451π0.799451\pi
912912 0 0
913913 9080.29 + 21410.3i 0.329150 + 0.776097i
914914 0 0
915915 1400.50 + 1017.52i 0.0506000 + 0.0367631i
916916 0 0
917917 −531.366 1635.38i −0.0191355 0.0588930i
918918 0 0
919919 27195.5 19758.7i 0.976166 0.709226i 0.0193174 0.999813i 0.493851π-0.493851\pi
0.956848 + 0.290587i 0.0938507π0.0938507\pi
920920 0 0
921921 3623.99 11153.5i 0.129658 0.399045i
922922 0 0
923923 4742.30 0.169117
924924 0 0
925925 −10822.3 −0.384688
926926 0 0
927927 5139.31 15817.2i 0.182089 0.560414i
928928 0 0
929929 14105.6 10248.3i 0.498160 0.361934i −0.310154 0.950686i 0.600381π-0.600381\pi
0.808314 + 0.588752i 0.200381π0.200381\pi
930930 0 0
931931 5445.12 + 16758.4i 0.191683 + 0.589939i
932932 0 0
933933 −21252.2 15440.6i −0.745730 0.541805i
934934 0 0
935935 −4827.95 2902.96i −0.168867 0.101537i
936936 0 0
937937 −32056.6 23290.5i −1.11766 0.812024i −0.133804 0.991008i 0.542719π-0.542719\pi
−0.983852 + 0.178983i 0.942719π0.942719\pi
938938 0 0
939939 5137.95 + 15813.0i 0.178563 + 0.549561i
940940 0 0
941941 1854.88 1347.65i 0.0642586 0.0466866i −0.555192 0.831722i 0.687355π-0.687355\pi
0.619451 + 0.785035i 0.287355π0.287355\pi
942942 0 0
943943 −616.539 + 1897.51i −0.0212909 + 0.0655265i
944944 0 0
945945 656.688 0.0226054
946946 0 0
947947 22750.9 0.780680 0.390340 0.920671i 0.372357π-0.372357\pi
0.390340 + 0.920671i 0.372357π0.372357\pi
948948 0 0
949949 −721.329 + 2220.02i −0.0246737 + 0.0759378i
950950 0 0
951951 19675.4 14295.0i 0.670891 0.487431i
952952 0 0
953953 −12914.9 39747.8i −0.438985 1.35106i −0.888947 0.458010i 0.848562π-0.848562\pi
0.449961 0.893048i 0.351438π-0.351438\pi
954954 0 0
955955 −2115.67 1537.13i −0.0716875 0.0520840i
956956 0 0
957957 −1881.54 + 8138.31i −0.0635542 + 0.274895i
958958 0 0
959959 7483.39 + 5437.00i 0.251983 + 0.183076i
960960 0 0
961961 −5133.26 15798.6i −0.172309 0.530313i
962962 0 0
963963 6774.32 4921.84i 0.226687 0.164698i
964964 0 0
965965 1689.65 5200.19i 0.0563644 0.173472i
966966 0 0
967967 −17075.1 −0.567836 −0.283918 0.958849i 0.591634π-0.591634\pi
−0.283918 + 0.958849i 0.591634π0.591634\pi
968968 0 0
969969 21359.6 0.708121
970970 0 0
971971 −5934.01 + 18263.0i −0.196119 + 0.603592i 0.803843 + 0.594842i 0.202785π0.202785\pi
−0.999962 + 0.00875008i 0.997215π0.997215\pi
972972 0 0
973973 11791.7 8567.15i 0.388513 0.282271i
974974 0 0
975975 −730.424 2248.01i −0.0239921 0.0738401i
976976 0 0
977977 −27764.0 20171.8i −0.909161 0.660544i 0.0316416 0.999499i 0.489926π-0.489926\pi
−0.940803 + 0.338955i 0.889926π0.889926\pi
978978 0 0
979979 6827.81 29532.7i 0.222899 0.964116i
980980 0 0
981981 −3155.16 2292.36i −0.102688 0.0746069i
982982 0 0
983983 1435.17 + 4416.99i 0.0465664 + 0.143317i 0.971636 0.236480i 0.0759939π-0.0759939\pi
−0.925070 + 0.379797i 0.875994π0.875994\pi
984984 0 0
985985 3051.66 2217.16i 0.0987146 0.0717203i
986986 0 0
987987 −1214.58 + 3738.10i −0.0391698 + 0.120552i
988988 0 0
989989 −9880.40 −0.317673
990990 0 0
991991 −14255.1 −0.456941 −0.228471 0.973551i 0.573372π-0.573372\pi
−0.228471 + 0.973551i 0.573372π0.573372\pi
992992 0 0
993993 −2413.61 + 7428.33i −0.0771335 + 0.237393i
994994 0 0
995995 −6410.52 + 4657.52i −0.204248 + 0.148395i
996996 0 0
997997 −4467.91 13750.8i −0.141926 0.436803i 0.854677 0.519160i 0.173755π-0.173755\pi
−0.996603 + 0.0823571i 0.973755π0.973755\pi
998998 0 0
999999 1952.64 + 1418.68i 0.0618407 + 0.0449299i
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 132.4.i.b.37.2 yes 8
3.2 odd 2 396.4.j.b.37.1 8
11.3 even 5 inner 132.4.i.b.25.2 8
11.5 even 5 1452.4.a.o.1.2 4
11.6 odd 10 1452.4.a.p.1.2 4
33.14 odd 10 396.4.j.b.289.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
132.4.i.b.25.2 8 11.3 even 5 inner
132.4.i.b.37.2 yes 8 1.1 even 1 trivial
396.4.j.b.37.1 8 3.2 odd 2
396.4.j.b.289.1 8 33.14 odd 10
1452.4.a.o.1.2 4 11.5 even 5
1452.4.a.p.1.2 4 11.6 odd 10