Properties

Label 169.2.b.b.168.5
Level 169169
Weight 22
Character 169.168
Analytic conductor 1.3491.349
Analytic rank 00
Dimension 66
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,2,Mod(168,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.168");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 169=132 169 = 13^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 169.b (of order 22, degree 11, not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 1.349471794161.34947179416
Analytic rank: 00
Dimension: 66
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x6+5x4+6x2+1 x^{6} + 5x^{4} + 6x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 168.5
Root 1.24698i1.24698i of defining polynomial
Character χ\chi == 169.168
Dual form 169.2.b.b.168.2

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+0.801938iq22.24698q3+1.35690q4+0.246980iq51.80194iq6+2.35690iq7+2.69202iq8+2.04892q90.198062q10+4.24698iq113.04892q121.89008q140.554958iq15+0.554958q162.15883q17+1.64310iq180.0881460iq19+0.335126iq205.29590iq213.40581q221.49396q236.04892iq24+4.93900q25+2.13706q27+3.19806iq28+4.63102q29+0.445042q306.63102iq31+5.82908iq329.54288iq331.73125iq340.582105q35+2.78017q365.69202iq37+0.0706876q380.664874q4011.5918iq41+4.24698q42+0.295897q43+5.76271iq44+0.506041iq451.19806iq46+7.35690iq471.24698q48+1.44504q49+3.96077iq50+4.85086q5110.3937q53+1.71379iq541.04892q556.34481q56+0.198062iq57+3.71379iq58+6.78017iq590.753020iq60+3.47219q61+5.31767q62+4.82908iq633.56465q64+7.65279q66+7.67994iq672.92931q68+3.35690q690.466812iq708.66487iq71+5.51573iq726.73556iq73+4.56465q7411.0978q750.119605iq7610.0097q77+9.97046q79+0.137063iq8010.9487q81+9.29590q82+1.60925iq837.18598iq840.533188iq85+0.237291iq8610.4058q8711.4330q88+2.88471iq890.405813q902.02715q92+14.8998iq935.89977q94+0.0217703q9513.0978iq968.05861iq97+1.15883iq98+8.70171iq99+O(q100)q+0.801938i q^{2} -2.24698 q^{3} +1.35690 q^{4} +0.246980i q^{5} -1.80194i q^{6} +2.35690i q^{7} +2.69202i q^{8} +2.04892 q^{9} -0.198062 q^{10} +4.24698i q^{11} -3.04892 q^{12} -1.89008 q^{14} -0.554958i q^{15} +0.554958 q^{16} -2.15883 q^{17} +1.64310i q^{18} -0.0881460i q^{19} +0.335126i q^{20} -5.29590i q^{21} -3.40581 q^{22} -1.49396 q^{23} -6.04892i q^{24} +4.93900 q^{25} +2.13706 q^{27} +3.19806i q^{28} +4.63102 q^{29} +0.445042 q^{30} -6.63102i q^{31} +5.82908i q^{32} -9.54288i q^{33} -1.73125i q^{34} -0.582105 q^{35} +2.78017 q^{36} -5.69202i q^{37} +0.0706876 q^{38} -0.664874 q^{40} -11.5918i q^{41} +4.24698 q^{42} +0.295897 q^{43} +5.76271i q^{44} +0.506041i q^{45} -1.19806i q^{46} +7.35690i q^{47} -1.24698 q^{48} +1.44504 q^{49} +3.96077i q^{50} +4.85086 q^{51} -10.3937 q^{53} +1.71379i q^{54} -1.04892 q^{55} -6.34481 q^{56} +0.198062i q^{57} +3.71379i q^{58} +6.78017i q^{59} -0.753020i q^{60} +3.47219 q^{61} +5.31767 q^{62} +4.82908i q^{63} -3.56465 q^{64} +7.65279 q^{66} +7.67994i q^{67} -2.92931 q^{68} +3.35690 q^{69} -0.466812i q^{70} -8.66487i q^{71} +5.51573i q^{72} -6.73556i q^{73} +4.56465 q^{74} -11.0978 q^{75} -0.119605i q^{76} -10.0097 q^{77} +9.97046 q^{79} +0.137063i q^{80} -10.9487 q^{81} +9.29590 q^{82} +1.60925i q^{83} -7.18598i q^{84} -0.533188i q^{85} +0.237291i q^{86} -10.4058 q^{87} -11.4330 q^{88} +2.88471i q^{89} -0.405813 q^{90} -2.02715 q^{92} +14.8998i q^{93} -5.89977 q^{94} +0.0217703 q^{95} -13.0978i q^{96} -8.05861i q^{97} +1.15883i q^{98} +8.70171i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q4q36q910q1010q14+4q16+4q17+6q22+10q23+10q25+2q272q29+2q30+8q35+14q3624q386q40+16q4226q43+6q95+O(q100) 6 q - 4 q^{3} - 6 q^{9} - 10 q^{10} - 10 q^{14} + 4 q^{16} + 4 q^{17} + 6 q^{22} + 10 q^{23} + 10 q^{25} + 2 q^{27} - 2 q^{29} + 2 q^{30} + 8 q^{35} + 14 q^{36} - 24 q^{38} - 6 q^{40} + 16 q^{42} - 26 q^{43}+ \cdots - 6 q^{95}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/169Z)×\left(\mathbb{Z}/169\mathbb{Z}\right)^\times.

nn 22
χ(n)\chi(n) 1-1

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 0.801938i 0.567056i 0.958964 + 0.283528i 0.0915048π0.0915048\pi
−0.958964 + 0.283528i 0.908495π0.908495\pi
33 −2.24698 −1.29729 −0.648647 0.761089i 0.724665π-0.724665\pi
−0.648647 + 0.761089i 0.724665π0.724665\pi
44 1.35690 0.678448
55 0.246980i 0.110453i 0.998474 + 0.0552263i 0.0175880π0.0175880\pi
−0.998474 + 0.0552263i 0.982412π0.982412\pi
66 − 1.80194i − 0.735638i
77 2.35690i 0.890823i 0.895326 + 0.445411i 0.146943π0.146943\pi
−0.895326 + 0.445411i 0.853057π0.853057\pi
88 2.69202i 0.951773i
99 2.04892 0.682972
1010 −0.198062 −0.0626328
1111 4.24698i 1.28051i 0.768161 + 0.640256i 0.221172π0.221172\pi
−0.768161 + 0.640256i 0.778828π0.778828\pi
1212 −3.04892 −0.880147
1313 0 0
1414 −1.89008 −0.505146
1515 − 0.554958i − 0.143290i
1616 0.554958 0.138740
1717 −2.15883 −0.523594 −0.261797 0.965123i 0.584315π-0.584315\pi
−0.261797 + 0.965123i 0.584315π0.584315\pi
1818 1.64310i 0.387283i
1919 − 0.0881460i − 0.0202221i −0.999949 0.0101110i 0.996782π-0.996782\pi
0.999949 0.0101110i 0.00321850π-0.00321850\pi
2020 0.335126i 0.0749364i
2121 − 5.29590i − 1.15566i
2222 −3.40581 −0.726122
2323 −1.49396 −0.311512 −0.155756 0.987796i 0.549781π-0.549781\pi
−0.155756 + 0.987796i 0.549781π0.549781\pi
2424 − 6.04892i − 1.23473i
2525 4.93900 0.987800
2626 0 0
2727 2.13706 0.411278
2828 3.19806i 0.604377i
2929 4.63102 0.859959 0.429980 0.902839i 0.358521π-0.358521\pi
0.429980 + 0.902839i 0.358521π0.358521\pi
3030 0.445042 0.0812532
3131 − 6.63102i − 1.19097i −0.803368 0.595483i 0.796961π-0.796961\pi
0.803368 0.595483i 0.203039π-0.203039\pi
3232 5.82908i 1.03045i
3333 − 9.54288i − 1.66120i
3434 − 1.73125i − 0.296907i
3535 −0.582105 −0.0983937
3636 2.78017 0.463361
3737 − 5.69202i − 0.935763i −0.883791 0.467881i 0.845017π-0.845017\pi
0.883791 0.467881i 0.154983π-0.154983\pi
3838 0.0706876 0.0114670
3939 0 0
4040 −0.664874 −0.105126
4141 − 11.5918i − 1.81033i −0.425056 0.905167i 0.639746π-0.639746\pi
0.425056 0.905167i 0.360254π-0.360254\pi
4242 4.24698 0.655323
4343 0.295897 0.0451239 0.0225619 0.999745i 0.492818π-0.492818\pi
0.0225619 + 0.999745i 0.492818π0.492818\pi
4444 5.76271i 0.868761i
4545 0.506041i 0.0754361i
4646 − 1.19806i − 0.176645i
4747 7.35690i 1.07311i 0.843864 + 0.536557i 0.180275π0.180275\pi
−0.843864 + 0.536557i 0.819725π0.819725\pi
4848 −1.24698 −0.179986
4949 1.44504 0.206435
5050 3.96077i 0.560138i
5151 4.85086 0.679256
5252 0 0
5353 −10.3937 −1.42769 −0.713844 0.700304i 0.753048π-0.753048\pi
−0.713844 + 0.700304i 0.753048π0.753048\pi
5454 1.71379i 0.233218i
5555 −1.04892 −0.141436
5656 −6.34481 −0.847861
5757 0.198062i 0.0262340i
5858 3.71379i 0.487645i
5959 6.78017i 0.882703i 0.897334 + 0.441351i 0.145501π0.145501\pi
−0.897334 + 0.441351i 0.854499π0.854499\pi
6060 − 0.753020i − 0.0972145i
6161 3.47219 0.444568 0.222284 0.974982i 0.428649π-0.428649\pi
0.222284 + 0.974982i 0.428649π0.428649\pi
6262 5.31767 0.675344
6363 4.82908i 0.608407i
6464 −3.56465 −0.445581
6565 0 0
6666 7.65279 0.941994
6767 7.67994i 0.938254i 0.883131 + 0.469127i 0.155431π0.155431\pi
−0.883131 + 0.469127i 0.844569π0.844569\pi
6868 −2.92931 −0.355231
6969 3.35690 0.404123
7070 − 0.466812i − 0.0557947i
7171 − 8.66487i − 1.02833i −0.857691 0.514166i 0.828102π-0.828102\pi
0.857691 0.514166i 0.171898π-0.171898\pi
7272 5.51573i 0.650035i
7373 − 6.73556i − 0.788338i −0.919038 0.394169i 0.871032π-0.871032\pi
0.919038 0.394169i 0.128968π-0.128968\pi
7474 4.56465 0.530629
7575 −11.0978 −1.28147
7676 − 0.119605i − 0.0137196i
7777 −10.0097 −1.14071
7878 0 0
7979 9.97046 1.12176 0.560882 0.827896i 0.310462π-0.310462\pi
0.560882 + 0.827896i 0.310462π0.310462\pi
8080 0.137063i 0.0153241i
8181 −10.9487 −1.21652
8282 9.29590 1.02656
8383 1.60925i 0.176638i 0.996092 + 0.0883192i 0.0281495π0.0281495\pi
−0.996092 + 0.0883192i 0.971850π0.971850\pi
8484 − 7.18598i − 0.784055i
8585 − 0.533188i − 0.0578323i
8686 0.237291i 0.0255877i
8787 −10.4058 −1.11562
8888 −11.4330 −1.21876
8989 2.88471i 0.305778i 0.988243 + 0.152889i 0.0488577π0.0488577\pi
−0.988243 + 0.152889i 0.951142π0.951142\pi
9090 −0.405813 −0.0427765
9191 0 0
9292 −2.02715 −0.211345
9393 14.8998i 1.54503i
9494 −5.89977 −0.608515
9595 0.0217703 0.00223358
9696 − 13.0978i − 1.33679i
9797 − 8.05861i − 0.818227i −0.912483 0.409114i 0.865838π-0.865838\pi
0.912483 0.409114i 0.134162π-0.134162\pi
9898 1.15883i 0.117060i
9999 8.70171i 0.874555i
100100 6.70171 0.670171
101101 13.3545 1.32882 0.664411 0.747367i 0.268682π-0.268682\pi
0.664411 + 0.747367i 0.268682π0.268682\pi
102102 3.89008i 0.385176i
103103 −1.36227 −0.134229 −0.0671144 0.997745i 0.521379π-0.521379\pi
−0.0671144 + 0.997745i 0.521379π0.521379\pi
104104 0 0
105105 1.30798 0.127646
106106 − 8.33513i − 0.809579i
107107 3.26875 0.316002 0.158001 0.987439i 0.449495π-0.449495\pi
0.158001 + 0.987439i 0.449495π0.449495\pi
108108 2.89977 0.279031
109109 15.7017i 1.50395i 0.659191 + 0.751976i 0.270899π0.270899\pi
−0.659191 + 0.751976i 0.729101π0.729101\pi
110110 − 0.841166i − 0.0802021i
111111 12.7899i 1.21396i
112112 1.30798i 0.123592i
113113 12.0489 1.13347 0.566733 0.823901i 0.308207π-0.308207\pi
0.566733 + 0.823901i 0.308207π0.308207\pi
114114 −0.158834 −0.0148761
115115 − 0.368977i − 0.0344073i
116116 6.28382 0.583438
117117 0 0
118118 −5.43727 −0.500541
119119 − 5.08815i − 0.466430i
120120 1.49396 0.136379
121121 −7.03684 −0.639712
122122 2.78448i 0.252095i
123123 26.0465i 2.34854i
124124 − 8.99761i − 0.808009i
125125 2.45473i 0.219558i
126126 −3.87263 −0.345001
127127 9.80731 0.870258 0.435129 0.900368i 0.356703π-0.356703\pi
0.435129 + 0.900368i 0.356703π0.356703\pi
128128 8.79954i 0.777777i
129129 −0.664874 −0.0585389
130130 0 0
131131 −6.57673 −0.574611 −0.287306 0.957839i 0.592760π-0.592760\pi
−0.287306 + 0.957839i 0.592760π0.592760\pi
132132 − 12.9487i − 1.12704i
133133 0.207751 0.0180143
134134 −6.15883 −0.532042
135135 0.527811i 0.0454267i
136136 − 5.81163i − 0.498343i
137137 − 6.21983i − 0.531396i −0.964056 0.265698i 0.914398π-0.914398\pi
0.964056 0.265698i 0.0856024π-0.0856024\pi
138138 2.69202i 0.229160i
139139 −14.7071 −1.24744 −0.623719 0.781648i 0.714379π-0.714379\pi
−0.623719 + 0.781648i 0.714379π0.714379\pi
140140 −0.789856 −0.0667550
141141 − 16.5308i − 1.39214i
142142 6.94869 0.583121
143143 0 0
144144 1.13706 0.0947553
145145 1.14377i 0.0949848i
146146 5.40150 0.447031
147147 −3.24698 −0.267806
148148 − 7.72348i − 0.634866i
149149 4.33513i 0.355147i 0.984108 + 0.177574i 0.0568248π0.0568248\pi
−0.984108 + 0.177574i 0.943175π0.943175\pi
150150 − 8.89977i − 0.726663i
151151 − 3.94438i − 0.320989i −0.987037 0.160494i 0.948691π-0.948691\pi
0.987037 0.160494i 0.0513089π-0.0513089\pi
152152 0.237291 0.0192468
153153 −4.42327 −0.357600
154154 − 8.02715i − 0.646846i
155155 1.63773 0.131545
156156 0 0
157157 4.45473 0.355526 0.177763 0.984073i 0.443114π-0.443114\pi
0.177763 + 0.984073i 0.443114π0.443114\pi
158158 7.99569i 0.636103i
159159 23.3545 1.85213
160160 −1.43967 −0.113816
161161 − 3.52111i − 0.277502i
162162 − 8.78017i − 0.689835i
163163 − 16.1588i − 1.26566i −0.774292 0.632829i 0.781894π-0.781894\pi
0.774292 0.632829i 0.218106π-0.218106\pi
164164 − 15.7289i − 1.22822i
165165 2.35690 0.183484
166166 −1.29052 −0.100164
167167 − 16.1172i − 1.24719i −0.781749 0.623594i 0.785672π-0.785672\pi
0.781749 0.623594i 0.214328π-0.214328\pi
168168 14.2567 1.09993
169169 0 0
170170 0.427583 0.0327942
171171 − 0.180604i − 0.0138111i
172172 0.401501 0.0306142
173173 21.5362 1.63736 0.818682 0.574247i 0.194705π-0.194705\pi
0.818682 + 0.574247i 0.194705π0.194705\pi
174174 − 8.34481i − 0.632619i
175175 11.6407i 0.879955i
176176 2.35690i 0.177658i
177177 − 15.2349i − 1.14513i
178178 −2.31336 −0.173393
179179 −11.4330 −0.854540 −0.427270 0.904124i 0.640525π-0.640525\pi
−0.427270 + 0.904124i 0.640525π0.640525\pi
180180 0.686645i 0.0511795i
181181 −20.9705 −1.55872 −0.779361 0.626575i 0.784456π-0.784456\pi
−0.779361 + 0.626575i 0.784456π0.784456\pi
182182 0 0
183183 −7.80194 −0.576736
184184 − 4.02177i − 0.296489i
185185 1.40581 0.103357
186186 −11.9487 −0.876120
187187 − 9.16852i − 0.670469i
188188 9.98254i 0.728052i
189189 5.03684i 0.366376i
190190 0.0174584i 0.00126657i
191191 −14.4373 −1.04464 −0.522322 0.852748i 0.674934π-0.674934\pi
−0.522322 + 0.852748i 0.674934π0.674934\pi
192192 8.00969 0.578049
193193 13.5797i 0.977489i 0.872427 + 0.488745i 0.162545π0.162545\pi
−0.872427 + 0.488745i 0.837455π0.837455\pi
194194 6.46250 0.463980
195195 0 0
196196 1.96077 0.140055
197197 − 0.560335i − 0.0399222i −0.999801 0.0199611i 0.993646π-0.993646\pi
0.999801 0.0199611i 0.00635424π-0.00635424\pi
198198 −6.97823 −0.495921
199199 −11.4916 −0.814616 −0.407308 0.913291i 0.633532π-0.633532\pi
−0.407308 + 0.913291i 0.633532π0.633532\pi
200200 13.2959i 0.940162i
201201 − 17.2567i − 1.21719i
202202 10.7095i 0.753516i
203203 10.9148i 0.766071i
204204 6.58211 0.460840
205205 2.86294 0.199956
206206 − 1.09246i − 0.0761151i
207207 −3.06100 −0.212754
208208 0 0
209209 0.374354 0.0258946
210210 1.04892i 0.0723822i
211211 8.78448 0.604748 0.302374 0.953189i 0.402221π-0.402221\pi
0.302374 + 0.953189i 0.402221π0.402221\pi
212212 −14.1032 −0.968613
213213 19.4698i 1.33405i
214214 2.62133i 0.179191i
215215 0.0730805i 0.00498405i
216216 5.75302i 0.391443i
217217 15.6286 1.06094
218218 −12.5918 −0.852824
219219 15.1347i 1.02271i
220220 −1.42327 −0.0959570
221221 0 0
222222 −10.2567 −0.688383
223223 − 2.25906i − 0.151278i −0.997135 0.0756390i 0.975900π-0.975900\pi
0.997135 0.0756390i 0.0240996π-0.0240996\pi
224224 −13.7385 −0.917945
225225 10.1196 0.674640
226226 9.66248i 0.642739i
227227 − 6.96615i − 0.462359i −0.972911 0.231180i 0.925741π-0.925741\pi
0.972911 0.231180i 0.0742585π-0.0742585\pi
228228 0.268750i 0.0177984i
229229 24.1739i 1.59746i 0.601692 + 0.798728i 0.294493π0.294493\pi
−0.601692 + 0.798728i 0.705507π0.705507\pi
230230 0.295897 0.0195109
231231 22.4916 1.47984
232232 12.4668i 0.818486i
233233 3.06100 0.200533 0.100266 0.994961i 0.468031π-0.468031\pi
0.100266 + 0.994961i 0.468031π0.468031\pi
234234 0 0
235235 −1.81700 −0.118528
236236 9.19998i 0.598868i
237237 −22.4034 −1.45526
238238 4.08038 0.264492
239239 − 25.1468i − 1.62661i −0.581839 0.813304i 0.697667π-0.697667\pi
0.581839 0.813304i 0.302333π-0.302333\pi
240240 − 0.307979i − 0.0198799i
241241 20.2664i 1.30547i 0.757586 + 0.652735i 0.226379π0.226379\pi
−0.757586 + 0.652735i 0.773621π0.773621\pi
242242 − 5.64310i − 0.362752i
243243 18.1903 1.16691
244244 4.71140 0.301616
245245 0.356896i 0.0228012i
246246 −20.8877 −1.33175
247247 0 0
248248 17.8509 1.13353
249249 − 3.61596i − 0.229152i
250250 −1.96854 −0.124501
251251 23.7211 1.49726 0.748631 0.662987i 0.230712π-0.230712\pi
0.748631 + 0.662987i 0.230712π0.230712\pi
252252 6.55257i 0.412773i
253253 − 6.34481i − 0.398895i
254254 7.86486i 0.493485i
255255 1.19806i 0.0750256i
256256 −14.1860 −0.886624
257257 −14.2241 −0.887278 −0.443639 0.896206i 0.646313π-0.646313\pi
−0.443639 + 0.896206i 0.646313π0.646313\pi
258258 − 0.533188i − 0.0331948i
259259 13.4155 0.833599
260260 0 0
261261 9.48858 0.587329
262262 − 5.27413i − 0.325837i
263263 −17.0954 −1.05415 −0.527075 0.849819i 0.676711π-0.676711\pi
−0.527075 + 0.849819i 0.676711π0.676711\pi
264264 25.6896 1.58109
265265 − 2.56704i − 0.157692i
266266 0.166603i 0.0102151i
267267 − 6.48188i − 0.396684i
268268 10.4209i 0.636556i
269269 −6.46681 −0.394288 −0.197144 0.980374i 0.563167π-0.563167\pi
−0.197144 + 0.980374i 0.563167π0.563167\pi
270270 −0.423272 −0.0257595
271271 − 6.44803i − 0.391690i −0.980635 0.195845i 0.937255π-0.937255\pi
0.980635 0.195845i 0.0627449π-0.0627449\pi
272272 −1.19806 −0.0726432
273273 0 0
274274 4.98792 0.301331
275275 20.9758i 1.26489i
276276 4.55496 0.274176
277277 −13.4601 −0.808739 −0.404370 0.914596i 0.632509π-0.632509\pi
−0.404370 + 0.914596i 0.632509π0.632509\pi
278278 − 11.7942i − 0.707367i
279279 − 13.5864i − 0.813398i
280280 − 1.56704i − 0.0936485i
281281 − 5.03684i − 0.300472i −0.988650 0.150236i 0.951997π-0.951997\pi
0.988650 0.150236i 0.0480034π-0.0480034\pi
282282 13.2567 0.789423
283283 −22.1280 −1.31537 −0.657686 0.753293i 0.728464π-0.728464\pi
−0.657686 + 0.753293i 0.728464π0.728464\pi
284284 − 11.7573i − 0.697669i
285285 −0.0489173 −0.00289761
286286 0 0
287287 27.3207 1.61269
288288 11.9433i 0.703766i
289289 −12.3394 −0.725849
290290 −0.917231 −0.0538616
291291 18.1075i 1.06148i
292292 − 9.13946i − 0.534846i
293293 − 14.9463i − 0.873172i −0.899663 0.436586i 0.856187π-0.856187\pi
0.899663 0.436586i 0.143813π-0.143813\pi
294294 − 2.60388i − 0.151861i
295295 −1.67456 −0.0974968
296296 15.3230 0.890634
297297 9.07606i 0.526647i
298298 −3.47650 −0.201388
299299 0 0
300300 −15.0586 −0.869409
301301 0.697398i 0.0401974i
302302 3.16315 0.182019
303303 −30.0073 −1.72387
304304 − 0.0489173i − 0.00280560i
305305 0.857560i 0.0491037i
306306 − 3.54719i − 0.202779i
307307 − 19.1293i − 1.09177i −0.837861 0.545883i 0.816194π-0.816194\pi
0.837861 0.545883i 0.183806π-0.183806\pi
308308 −13.5821 −0.773912
309309 3.06100 0.174134
310310 1.31336i 0.0745936i
311311 0.269815 0.0152998 0.00764990 0.999971i 0.497565π-0.497565\pi
0.00764990 + 0.999971i 0.497565π0.497565\pi
312312 0 0
313313 −23.3937 −1.32229 −0.661146 0.750257i 0.729930π-0.729930\pi
−0.661146 + 0.750257i 0.729930π0.729930\pi
314314 3.57242i 0.201603i
315315 −1.19269 −0.0672002
316316 13.5289 0.761059
317317 − 13.9952i − 0.786050i −0.919528 0.393025i 0.871429π-0.871429\pi
0.919528 0.393025i 0.128571π-0.128571\pi
318318 18.7289i 1.05026i
319319 19.6679i 1.10119i
320320 − 0.880395i − 0.0492156i
321321 −7.34481 −0.409948
322322 2.82371 0.157359
323323 0.190293i 0.0105882i
324324 −14.8562 −0.825346
325325 0 0
326326 12.9584 0.717698
327327 − 35.2814i − 1.95107i
328328 31.2054 1.72303
329329 −17.3394 −0.955954
330330 1.89008i 0.104046i
331331 17.8213i 0.979548i 0.871849 + 0.489774i 0.162921π0.162921\pi
−0.871849 + 0.489774i 0.837079π0.837079\pi
332332 2.18359i 0.119840i
333333 − 11.6625i − 0.639100i
334334 12.9250 0.707225
335335 −1.89679 −0.103633
336336 − 2.93900i − 0.160336i
337337 27.8485 1.51700 0.758501 0.651672i 0.225932π-0.225932\pi
0.758501 + 0.651672i 0.225932π0.225932\pi
338338 0 0
339339 −27.0737 −1.47044
340340 − 0.723480i − 0.0392362i
341341 28.1618 1.52505
342342 0.144833 0.00783167
343343 19.9041i 1.07472i
344344 0.796561i 0.0429477i
345345 0.829085i 0.0446364i
346346 17.2707i 0.928477i
347347 1.50365 0.0807200 0.0403600 0.999185i 0.487150π-0.487150\pi
0.0403600 + 0.999185i 0.487150π0.487150\pi
348348 −14.1196 −0.756890
349349 14.1860i 0.759358i 0.925118 + 0.379679i 0.123966π0.123966\pi
−0.925118 + 0.379679i 0.876034π0.876034\pi
350350 −9.33513 −0.498983
351351 0 0
352352 −24.7560 −1.31950
353353 − 7.16852i − 0.381542i −0.981635 0.190771i 0.938901π-0.938901\pi
0.981635 0.190771i 0.0610988π-0.0610988\pi
354354 12.2174 0.649350
355355 2.14005 0.113582
356356 3.91425i 0.207455i
357357 11.4330i 0.605096i
358358 − 9.16852i − 0.484571i
359359 − 19.8853i − 1.04951i −0.851255 0.524753i 0.824158π-0.824158\pi
0.851255 0.524753i 0.175842π-0.175842\pi
360360 −1.36227 −0.0717981
361361 18.9922 0.999591
362362 − 16.8170i − 0.883882i
363363 15.8116 0.829895
364364 0 0
365365 1.66355 0.0870740
366366 − 6.25667i − 0.327041i
367367 1.08383 0.0565757 0.0282878 0.999600i 0.490994π-0.490994\pi
0.0282878 + 0.999600i 0.490994π0.490994\pi
368368 −0.829085 −0.0432190
369369 − 23.7506i − 1.23641i
370370 1.12737i 0.0586094i
371371 − 24.4969i − 1.27182i
372372 20.2174i 1.04823i
373373 −6.13036 −0.317418 −0.158709 0.987325i 0.550733π-0.550733\pi
−0.158709 + 0.987325i 0.550733π0.550733\pi
374374 7.35258 0.380193
375375 − 5.51573i − 0.284831i
376376 −19.8049 −1.02136
377377 0 0
378378 −4.03923 −0.207756
379379 − 2.40880i − 0.123732i −0.998084 0.0618658i 0.980295π-0.980295\pi
0.998084 0.0618658i 0.0197051π-0.0197051\pi
380380 0.0295400 0.00151537
381381 −22.0368 −1.12898
382382 − 11.5778i − 0.592371i
383383 − 30.3913i − 1.55292i −0.630164 0.776462i 0.717012π-0.717012\pi
0.630164 0.776462i 0.282988π-0.282988\pi
384384 − 19.7724i − 1.00901i
385385 − 2.47219i − 0.125994i
386386 −10.8901 −0.554291
387387 0.606268 0.0308184
388388 − 10.9347i − 0.555125i
389389 15.9409 0.808237 0.404118 0.914707i 0.367578π-0.367578\pi
0.404118 + 0.914707i 0.367578π0.367578\pi
390390 0 0
391391 3.22521 0.163106
392392 3.89008i 0.196479i
393393 14.7778 0.745440
394394 0.449354 0.0226381
395395 2.46250i 0.123902i
396396 11.8073i 0.593340i
397397 − 16.9148i − 0.848931i −0.905444 0.424466i 0.860462π-0.860462\pi
0.905444 0.424466i 0.139538π-0.139538\pi
398398 − 9.21552i − 0.461932i
399399 −0.466812 −0.0233698
400400 2.74094 0.137047
401401 − 26.6625i − 1.33146i −0.746192 0.665730i 0.768120π-0.768120\pi
0.746192 0.665730i 0.231880π-0.231880\pi
402402 13.8388 0.690215
403403 0 0
404404 18.1207 0.901537
405405 − 2.70410i − 0.134368i
406406 −8.75302 −0.434405
407407 24.1739 1.19826
408408 13.0586i 0.646497i
409409 28.5163i 1.41004i 0.709187 + 0.705021i 0.249062π0.249062\pi
−0.709187 + 0.705021i 0.750938π0.750938\pi
410410 2.29590i 0.113386i
411411 13.9758i 0.689377i
412412 −1.84846 −0.0910672
413413 −15.9801 −0.786332
414414 − 2.45473i − 0.120643i
415415 −0.397452 −0.0195102
416416 0 0
417417 33.0465 1.61830
418418 0.300209i 0.0146837i
419419 −29.6093 −1.44651 −0.723253 0.690583i 0.757354π-0.757354\pi
−0.723253 + 0.690583i 0.757354π0.757354\pi
420420 1.77479 0.0866009
421421 − 11.6606i − 0.568301i −0.958780 0.284151i 0.908288π-0.908288\pi
0.958780 0.284151i 0.0917115π-0.0917115\pi
422422 7.04461i 0.342926i
423423 15.0737i 0.732907i
424424 − 27.9801i − 1.35884i
425425 −10.6625 −0.517206
426426 −15.6136 −0.756480
427427 8.18359i 0.396032i
428428 4.43535 0.214391
429429 0 0
430430 −0.0586060 −0.00282623
431431 4.34913i 0.209490i 0.994499 + 0.104745i 0.0334026π0.0334026\pi
−0.994499 + 0.104745i 0.966597π0.966597\pi
432432 1.18598 0.0570605
433433 14.3884 0.691460 0.345730 0.938334i 0.387631π-0.387631\pi
0.345730 + 0.938334i 0.387631π0.387631\pi
434434 12.5332i 0.601612i
435435 − 2.57002i − 0.123223i
436436 21.3056i 1.02035i
437437 0.131687i 0.00629942i
438438 −12.1371 −0.579931
439439 20.2325 0.965645 0.482822 0.875718i 0.339612π-0.339612\pi
0.482822 + 0.875718i 0.339612π0.339612\pi
440440 − 2.82371i − 0.134615i
441441 2.96077 0.140989
442442 0 0
443443 8.12200 0.385888 0.192944 0.981210i 0.438196π-0.438196\pi
0.192944 + 0.981210i 0.438196π0.438196\pi
444444 17.3545i 0.823608i
445445 −0.712464 −0.0337740
446446 1.81163 0.0857830
447447 − 9.74094i − 0.460731i
448448 − 8.40150i − 0.396934i
449449 − 12.4916i − 0.589513i −0.955572 0.294757i 0.904761π-0.904761\pi
0.955572 0.294757i 0.0952386π-0.0952386\pi
450450 8.11529i 0.382559i
451451 49.2301 2.31816
452452 16.3491 0.768998
453453 8.86294i 0.416417i
454454 5.58642 0.262184
455455 0 0
456456 −0.533188 −0.0249688
457457 5.98121i 0.279789i 0.990166 + 0.139895i 0.0446764π0.0446764\pi
−0.990166 + 0.139895i 0.955324π0.955324\pi
458458 −19.3860 −0.905847
459459 −4.61356 −0.215343
460460 − 0.500664i − 0.0233436i
461461 − 2.05669i − 0.0957895i −0.998852 0.0478947i 0.984749π-0.984749\pi
0.998852 0.0478947i 0.0152512π-0.0152512\pi
462462 18.0368i 0.839150i
463463 8.44935i 0.392675i 0.980536 + 0.196337i 0.0629048π0.0629048\pi
−0.980536 + 0.196337i 0.937095π0.937095\pi
464464 2.57002 0.119310
465465 −3.67994 −0.170653
466466 2.45473i 0.113713i
467467 −33.5139 −1.55084 −0.775420 0.631446i 0.782462π-0.782462\pi
−0.775420 + 0.631446i 0.782462π0.782462\pi
468468 0 0
469469 −18.1008 −0.835818
470470 − 1.45712i − 0.0672121i
471471 −10.0097 −0.461222
472472 −18.2524 −0.840133
473473 1.25667i 0.0577817i
474474 − 17.9661i − 0.825213i
475475 − 0.435353i − 0.0199754i
476476 − 6.90408i − 0.316448i
477477 −21.2959 −0.975072
478478 20.1661 0.922377
479479 24.7313i 1.13000i 0.825091 + 0.565000i 0.191124π0.191124\pi
−0.825091 + 0.565000i 0.808876π0.808876\pi
480480 3.23490 0.147652
481481 0 0
482482 −16.2524 −0.740275
483483 7.91185i 0.360002i
484484 −9.54825 −0.434012
485485 1.99031 0.0903754
486486 14.5875i 0.661702i
487487 − 37.7555i − 1.71087i −0.517913 0.855433i 0.673291π-0.673291\pi
0.517913 0.855433i 0.326709π-0.326709\pi
488488 9.34721i 0.423128i
489489 36.3086i 1.64193i
490490 −0.286208 −0.0129296
491491 −31.3110 −1.41304 −0.706522 0.707691i 0.749737π-0.749737\pi
−0.706522 + 0.707691i 0.749737π0.749737\pi
492492 35.3424i 1.59336i
493493 −9.99761 −0.450270
494494 0 0
495495 −2.14914 −0.0965969
496496 − 3.67994i − 0.165234i
497497 20.4222 0.916061
498498 2.89977 0.129942
499499 21.4873i 0.961902i 0.876748 + 0.480951i 0.159708π0.159708\pi
−0.876748 + 0.480951i 0.840292π0.840292\pi
500500 3.33081i 0.148959i
501501 36.2150i 1.61797i
502502 19.0228i 0.849031i
503503 37.5924 1.67616 0.838081 0.545546i 0.183678π-0.183678\pi
0.838081 + 0.545546i 0.183678π0.183678\pi
504504 −13.0000 −0.579066
505505 3.29829i 0.146772i
506506 5.08815 0.226196
507507 0 0
508508 13.3075 0.590425
509509 − 17.1075i − 0.758278i −0.925340 0.379139i 0.876220π-0.876220\pi
0.925340 0.379139i 0.123780π-0.123780\pi
510510 −0.960771 −0.0425437
511511 15.8750 0.702269
512512 6.22282i 0.275012i
513513 − 0.188374i − 0.00831690i
514514 − 11.4069i − 0.503136i
515515 − 0.336454i − 0.0148259i
516516 −0.902165 −0.0397156
517517 −31.2446 −1.37414
518518 10.7584i 0.472697i
519519 −48.3913 −2.12414
520520 0 0
521521 −19.8465 −0.869493 −0.434746 0.900553i 0.643162π-0.643162\pi
−0.434746 + 0.900553i 0.643162π0.643162\pi
522522 7.60925i 0.333048i
523523 −11.4300 −0.499798 −0.249899 0.968272i 0.580397π-0.580397\pi
−0.249899 + 0.968272i 0.580397π0.580397\pi
524524 −8.92394 −0.389844
525525 − 26.1564i − 1.14156i
526526 − 13.7095i − 0.597762i
527527 14.3153i 0.623583i
528528 − 5.29590i − 0.230474i
529529 −20.7681 −0.902960
530530 2.05861 0.0894201
531531 13.8920i 0.602862i
532532 0.281896 0.0122218
533533 0 0
534534 5.19806 0.224942
535535 0.807315i 0.0349033i
536536 −20.6746 −0.893005
537537 25.6896 1.10859
538538 − 5.18598i − 0.223584i
539539 6.13706i 0.264342i
540540 0.716185i 0.0308197i
541541 − 16.1884i − 0.695993i −0.937496 0.347996i 0.886862π-0.886862\pi
0.937496 0.347996i 0.113138π-0.113138\pi
542542 5.17092 0.222110
543543 47.1202 2.02212
544544 − 12.5840i − 0.539536i
545545 −3.87800 −0.166115
546546 0 0
547547 5.33081 0.227929 0.113965 0.993485i 0.463645π-0.463645\pi
0.113965 + 0.993485i 0.463645π0.463645\pi
548548 − 8.43967i − 0.360525i
549549 7.11423 0.303628
550550 −16.8213 −0.717263
551551 − 0.408206i − 0.0173902i
552552 9.03684i 0.384633i
553553 23.4993i 0.999293i
554554 − 10.7942i − 0.458600i
555555 −3.15883 −0.134085
556556 −19.9560 −0.846322
557557 − 7.39075i − 0.313156i −0.987666 0.156578i 0.949954π-0.949954\pi
0.987666 0.156578i 0.0500463π-0.0500463\pi
558558 10.8955 0.461242
559559 0 0
560560 −0.323044 −0.0136511
561561 20.6015i 0.869795i
562562 4.03923 0.170385
563563 9.47889 0.399488 0.199744 0.979848i 0.435989π-0.435989\pi
0.199744 + 0.979848i 0.435989π0.435989\pi
564564 − 22.4306i − 0.944497i
565565 2.97584i 0.125194i
566566 − 17.7453i − 0.745889i
567567 − 25.8049i − 1.08370i
568568 23.3260 0.978738
569569 10.1438 0.425249 0.212624 0.977134i 0.431799π-0.431799\pi
0.212624 + 0.977134i 0.431799π0.431799\pi
570570 − 0.0392287i − 0.00164311i
571571 14.0925 0.589751 0.294876 0.955536i 0.404722π-0.404722\pi
0.294876 + 0.955536i 0.404722π0.404722\pi
572572 0 0
573573 32.4403 1.35521
574574 21.9095i 0.914483i
575575 −7.37867 −0.307712
576576 −7.30367 −0.304319
577577 25.1545i 1.04720i 0.851965 + 0.523598i 0.175411π0.175411\pi
−0.851965 + 0.523598i 0.824589π0.824589\pi
578578 − 9.89546i − 0.411597i
579579 − 30.5133i − 1.26809i
580580 1.55197i 0.0644422i
581581 −3.79284 −0.157354
582582 −14.5211 −0.601919
583583 − 44.1420i − 1.82817i
584584 18.1323 0.750319
585585 0 0
586586 11.9860 0.495137
587587 43.8353i 1.80928i 0.426180 + 0.904639i 0.359859π0.359859\pi
−0.426180 + 0.904639i 0.640141π0.640141\pi
588588 −4.40581 −0.181693
589589 −0.584498 −0.0240838
590590 − 1.34290i − 0.0552861i
591591 1.25906i 0.0517909i
592592 − 3.15883i − 0.129827i
593593 24.9965i 1.02648i 0.858244 + 0.513242i 0.171556π0.171556\pi
−0.858244 + 0.513242i 0.828444π0.828444\pi
594594 −7.27844 −0.298638
595595 1.25667 0.0515184
596596 5.88231i 0.240949i
597597 25.8213 1.05680
598598 0 0
599599 −6.24027 −0.254971 −0.127485 0.991840i 0.540691π-0.540691\pi
−0.127485 + 0.991840i 0.540691π0.540691\pi
600600 − 29.8756i − 1.21967i
601601 6.32975 0.258196 0.129098 0.991632i 0.458792π-0.458792\pi
0.129098 + 0.991632i 0.458792π0.458792\pi
602602 −0.559270 −0.0227941
603603 15.7356i 0.640802i
604604 − 5.35211i − 0.217774i
605605 − 1.73795i − 0.0706579i
606606 − 24.0640i − 0.977532i
607607 −43.6480 −1.77162 −0.885809 0.464050i 0.846396π-0.846396\pi
−0.885809 + 0.464050i 0.846396π0.846396\pi
608608 0.513811 0.0208378
609609 − 24.5254i − 0.993820i
610610 −0.687710 −0.0278445
611611 0 0
612612 −6.00192 −0.242613
613613 − 25.9541i − 1.04827i −0.851634 0.524137i 0.824388π-0.824388\pi
0.851634 0.524137i 0.175612π-0.175612\pi
614614 15.3405 0.619092
615615 −6.43296 −0.259402
616616 − 26.9463i − 1.08570i
617617 45.9396i 1.84946i 0.380626 + 0.924729i 0.375709π0.375709\pi
−0.380626 + 0.924729i 0.624291π0.624291\pi
618618 2.45473i 0.0987437i
619619 − 6.73556i − 0.270725i −0.990796 0.135363i 0.956780π-0.956780\pi
0.990796 0.135363i 0.0432199π-0.0432199\pi
620620 2.22223 0.0892467
621621 −3.19269 −0.128118
622622 0.216375i 0.00867583i
623623 −6.79895 −0.272394
624624 0 0
625625 24.0887 0.963549
626626 − 18.7603i − 0.749813i
627627 −0.841166 −0.0335930
628628 6.04461 0.241206
629629 12.2881i 0.489960i
630630 − 0.956459i − 0.0381063i
631631 − 45.0998i − 1.79539i −0.440614 0.897696i 0.645239π-0.645239\pi
0.440614 0.897696i 0.354761π-0.354761\pi
632632 26.8407i 1.06767i
633633 −19.7385 −0.784537
634634 11.2233 0.445734
635635 2.42221i 0.0961223i
636636 31.6896 1.25658
637637 0 0
638638 −15.7724 −0.624435
639639 − 17.7536i − 0.702322i
640640 −2.17331 −0.0859075
641641 −32.5821 −1.28692 −0.643458 0.765482i 0.722501π-0.722501\pi
−0.643458 + 0.765482i 0.722501π0.722501\pi
642642 − 5.89008i − 0.232463i
643643 25.5754i 1.00860i 0.863530 + 0.504298i 0.168249π0.168249\pi
−0.863530 + 0.504298i 0.831751π0.831751\pi
644644 − 4.77777i − 0.188271i
645645 − 0.164210i − 0.00646578i
646646 −0.152603 −0.00600408
647647 30.1715 1.18616 0.593082 0.805142i 0.297911π-0.297911\pi
0.593082 + 0.805142i 0.297911π0.297911\pi
648648 − 29.4741i − 1.15785i
649649 −28.7952 −1.13031
650650 0 0
651651 −35.1172 −1.37635
652652 − 21.9259i − 0.858683i
653653 36.9028 1.44412 0.722058 0.691832i 0.243196π-0.243196\pi
0.722058 + 0.691832i 0.243196π0.243196\pi
654654 28.2935 1.10636
655655 − 1.62432i − 0.0634673i
656656 − 6.43296i − 0.251165i
657657 − 13.8006i − 0.538413i
658658 − 13.9051i − 0.542079i
659659 23.6866 0.922701 0.461350 0.887218i 0.347365π-0.347365\pi
0.461350 + 0.887218i 0.347365π0.347365\pi
660660 3.19806 0.124484
661661 31.7590i 1.23528i 0.786460 + 0.617641i 0.211911π0.211911\pi
−0.786460 + 0.617641i 0.788089π0.788089\pi
662662 −14.2916 −0.555458
663663 0 0
664664 −4.33214 −0.168120
665665 0.0513102i 0.00198973i
666666 9.35258 0.362405
667667 −6.91856 −0.267888
668668 − 21.8694i − 0.846152i
669669 5.07606i 0.196252i
670670 − 1.52111i − 0.0587655i
671671 14.7463i 0.569275i
672672 30.8702 1.19085
673673 7.50232 0.289193 0.144597 0.989491i 0.453812π-0.453812\pi
0.144597 + 0.989491i 0.453812π0.453812\pi
674674 22.3327i 0.860225i
675675 10.5550 0.406261
676676 0 0
677677 −35.0315 −1.34637 −0.673184 0.739475i 0.735074π-0.735074\pi
−0.673184 + 0.739475i 0.735074π0.735074\pi
678678 − 21.7114i − 0.833821i
679679 18.9933 0.728896
680680 1.43535 0.0550433
681681 15.6528i 0.599816i
682682 22.5840i 0.864787i
683683 24.0834i 0.921524i 0.887524 + 0.460762i 0.152424π0.152424\pi
−0.887524 + 0.460762i 0.847576π0.847576\pi
684684 − 0.245061i − 0.00937013i
685685 1.53617 0.0586941
686686 −15.9618 −0.609426
687687 − 54.3183i − 2.07237i
688688 0.164210 0.00626046
689689 0 0
690690 −0.664874 −0.0253113
691691 2.01447i 0.0766342i 0.999266 + 0.0383171i 0.0121997π0.0121997\pi
−0.999266 + 0.0383171i 0.987800π0.987800\pi
692692 29.2223 1.11087
693693 −20.5090 −0.779073
694694 1.20583i 0.0457728i
695695 − 3.63235i − 0.137783i
696696 − 28.0127i − 1.06182i
697697 25.0248i 0.947880i
698698 −11.3763 −0.430598
699699 −6.87800 −0.260150
700700 15.7952i 0.597004i
701701 48.8189 1.84387 0.921933 0.387350i 0.126610π-0.126610\pi
0.921933 + 0.387350i 0.126610π0.126610\pi
702702 0 0
703703 −0.501729 −0.0189231
704704 − 15.1390i − 0.570572i
705705 4.08277 0.153766
706706 5.74871 0.216355
707707 31.4752i 1.18375i
708708 − 20.6722i − 0.776908i
709709 20.8060i 0.781385i 0.920521 + 0.390693i 0.127764π0.127764\pi
−0.920521 + 0.390693i 0.872236π0.872236\pi
710710 1.71618i 0.0644073i
711711 20.4286 0.766134
712712 −7.76569 −0.291032
713713 9.90648i 0.371000i
714714 −9.16852 −0.343123
715715 0 0
716716 −15.5133 −0.579761
717717 56.5042i 2.11019i
718718 15.9468 0.595128
719719 −21.4306 −0.799225 −0.399613 0.916684i 0.630855π-0.630855\pi
−0.399613 + 0.916684i 0.630855π0.630855\pi
720720 0.280831i 0.0104660i
721721 − 3.21073i − 0.119574i
722722 15.2306i 0.566824i
723723 − 45.5381i − 1.69358i
724724 −28.4547 −1.05751
725725 22.8726 0.849468
726726 12.6799i 0.470597i
727727 −13.4862 −0.500175 −0.250088 0.968223i 0.580459π-0.580459\pi
−0.250088 + 0.968223i 0.580459π0.580459\pi
728728 0 0
729729 −8.02715 −0.297302
730730 1.33406i 0.0493758i
731731 −0.638792 −0.0236266
732732 −10.5864 −0.391285
733733 − 43.5424i − 1.60828i −0.594443 0.804138i 0.702627π-0.702627\pi
0.594443 0.804138i 0.297373π-0.297373\pi
734734 0.869167i 0.0320816i
735735 − 0.801938i − 0.0295799i
736736 − 8.70841i − 0.320996i
737737 −32.6165 −1.20145
738738 19.0465 0.701112
739739 − 20.0543i − 0.737709i −0.929487 0.368855i 0.879750π-0.879750\pi
0.929487 0.368855i 0.120250π-0.120250\pi
740740 1.90754 0.0701226
741741 0 0
742742 19.6450 0.721191
743743 − 33.1685i − 1.21684i −0.793617 0.608418i 0.791805π-0.791805\pi
0.793617 0.608418i 0.208195π-0.208195\pi
744744 −40.1105 −1.47052
745745 −1.07069 −0.0392270
746746 − 4.91617i − 0.179994i
747747 3.29722i 0.120639i
748748 − 12.4407i − 0.454878i
749749 7.70410i 0.281502i
750750 4.42327 0.161515
751751 −39.2814 −1.43340 −0.716700 0.697382i 0.754348π-0.754348\pi
−0.716700 + 0.697382i 0.754348π0.754348\pi
752752 4.08277i 0.148883i
753753 −53.3008 −1.94239
754754 0 0
755755 0.974181 0.0354541
756756 6.83446i 0.248567i
757757 −46.6426 −1.69526 −0.847628 0.530592i 0.821970π-0.821970\pi
−0.847628 + 0.530592i 0.821970π0.821970\pi
758758 1.93171 0.0701627
759759 14.2567i 0.517484i
760760 0.0586060i 0.00212586i
761761 21.8984i 0.793818i 0.917858 + 0.396909i 0.129917π0.129917\pi
−0.917858 + 0.396909i 0.870083π0.870083\pi
762762 − 17.6722i − 0.640195i
763763 −37.0073 −1.33975
764764 −19.5899 −0.708737
765765 − 1.09246i − 0.0394979i
766766 24.3720 0.880595
767767 0 0
768768 31.8756 1.15021
769769 46.7096i 1.68439i 0.539172 + 0.842196i 0.318737π0.318737\pi
−0.539172 + 0.842196i 0.681263π0.681263\pi
770770 1.98254 0.0714458
771771 31.9614 1.15106
772772 18.4263i 0.663175i
773773 − 30.2416i − 1.08771i −0.839178 0.543857i 0.816963π-0.816963\pi
0.839178 0.543857i 0.183037π-0.183037\pi
774774 0.486189i 0.0174757i
775775 − 32.7506i − 1.17644i
776776 21.6939 0.778767
777777 −30.1444 −1.08142
778778 12.7836i 0.458315i
779779 −1.02177 −0.0366087
780780 0 0
781781 36.7995 1.31679
782782 2.58642i 0.0924901i
783783 9.89679 0.353682
784784 0.801938 0.0286406
785785 1.10023i 0.0392688i
786786 11.8509i 0.422706i
787787 − 28.7023i − 1.02313i −0.859246 0.511563i 0.829067π-0.829067\pi
0.859246 0.511563i 0.170933π-0.170933\pi
788788 − 0.760316i − 0.0270851i
789789 38.4131 1.36754
790790 −1.97477 −0.0702592
791791 28.3980i 1.00972i
792792 −23.4252 −0.832378
793793 0 0
794794 13.5646 0.481391
795795 5.76809i 0.204573i
796796 −15.5929 −0.552674
797797 18.5418 0.656785 0.328392 0.944541i 0.393493π-0.393493\pi
0.328392 + 0.944541i 0.393493π0.393493\pi
798798 − 0.374354i − 0.0132520i
799799 − 15.8823i − 0.561876i
800800 28.7899i 1.01788i
801801 5.91053i 0.208838i
802802 21.3817 0.755012
803803 28.6058 1.00948
804804 − 23.4155i − 0.825801i
805805 0.869641 0.0306508
806806 0 0
807807 14.5308 0.511508
808808 35.9506i 1.26474i
809809 −10.0677 −0.353962 −0.176981 0.984214i 0.556633π-0.556633\pi
−0.176981 + 0.984214i 0.556633π0.556633\pi
810810 2.16852 0.0761941
811811 − 10.0285i − 0.352147i −0.984377 0.176074i 0.943660π-0.943660\pi
0.984377 0.176074i 0.0563397π-0.0563397\pi
812812 14.8103i 0.519740i
813813 14.4886i 0.508137i
814814 19.3860i 0.679478i
815815 3.99090 0.139795
816816 2.69202 0.0942396
817817 − 0.0260821i 0 0.000912498i
818818 −22.8683 −0.799572
819819 0 0
820820 3.88471 0.135660
821821 − 26.1704i − 0.913355i −0.889632 0.456677i 0.849039π-0.849039\pi
0.889632 0.456677i 0.150961π-0.150961\pi
822822 −11.2078 −0.390915
823823 −1.82238 −0.0635242 −0.0317621 0.999495i 0.510112π-0.510112\pi
−0.0317621 + 0.999495i 0.510112π0.510112\pi
824824 − 3.66727i − 0.127755i
825825 − 47.1323i − 1.64094i
826826 − 12.8151i − 0.445894i
827827 32.2941i 1.12298i 0.827485 + 0.561488i 0.189771π0.189771\pi
−0.827485 + 0.561488i 0.810229π0.810229\pi
828828 −4.15346 −0.144343
829829 −15.1002 −0.524453 −0.262226 0.965006i 0.584457π-0.584457\pi
−0.262226 + 0.965006i 0.584457π0.584457\pi
830830 − 0.318732i − 0.0110634i
831831 30.2446 1.04917
832832 0 0
833833 −3.11960 −0.108088
834834 26.5013i 0.917663i
835835 3.98062 0.137755
836836 0.507960 0.0175682
837837 − 14.1709i − 0.489818i
838838 − 23.7448i − 0.820250i
839839 32.9965i 1.13917i 0.821933 + 0.569584i 0.192896π0.192896\pi
−0.821933 + 0.569584i 0.807104π0.807104\pi
840840 3.52111i 0.121490i
841841 −7.55363 −0.260470
842842 9.35105 0.322258
843843 11.3177i 0.389801i
844844 11.9196 0.410290
845845 0 0
846846 −12.0881 −0.415599
847847 − 16.5851i − 0.569870i
848848 −5.76809 −0.198077
849849 49.7211 1.70642
850850 − 8.55065i − 0.293285i
851851 8.50365i 0.291501i
852852 26.4185i 0.905082i
853853 37.7802i 1.29357i 0.762673 + 0.646784i 0.223887π0.223887\pi
−0.762673 + 0.646784i 0.776113π0.776113\pi
854854 −6.56273 −0.224572
855855 0.0446055 0.00152547
856856 8.79954i 0.300762i
857857 −27.3623 −0.934677 −0.467339 0.884078i 0.654787π-0.654787\pi
−0.467339 + 0.884078i 0.654787π0.654787\pi
858858 0 0
859859 −20.0629 −0.684538 −0.342269 0.939602i 0.611195π-0.611195\pi
−0.342269 + 0.939602i 0.611195π0.611195\pi
860860 0.0991626i 0.00338142i
861861 −61.3889 −2.09213
862862 −3.48773 −0.118792
863863 − 6.14483i − 0.209173i −0.994516 0.104586i 0.966648π-0.966648\pi
0.994516 0.104586i 0.0333518π-0.0333518\pi
864864 12.4571i 0.423800i
865865 5.31900i 0.180851i
866866 11.5386i 0.392096i
867867 27.7265 0.941640
868868 21.2064 0.719793
869869 42.3443i 1.43643i
870870 2.06100 0.0698744
871871 0 0
872872 −42.2693 −1.43142
873873 − 16.5114i − 0.558827i
874874 −0.105604 −0.00357212
875875 −5.78554 −0.195587
876876 20.5362i 0.693853i
877877 − 13.5077i − 0.456123i −0.973647 0.228061i 0.926761π-0.926761\pi
0.973647 0.228061i 0.0732386π-0.0732386\pi
878878 16.2252i 0.547574i
879879 33.5840i 1.13276i
880880 −0.582105 −0.0196228
881881 5.23431 0.176348 0.0881741 0.996105i 0.471897π-0.471897\pi
0.0881741 + 0.996105i 0.471897π0.471897\pi
882882 2.37435i 0.0799487i
883883 4.57301 0.153894 0.0769470 0.997035i 0.475483π-0.475483\pi
0.0769470 + 0.997035i 0.475483π0.475483\pi
884884 0 0
885885 3.76271 0.126482
886886 6.51334i 0.218820i
887887 −1.64071 −0.0550897 −0.0275448 0.999621i 0.508769π-0.508769\pi
−0.0275448 + 0.999621i 0.508769π0.508769\pi
888888 −34.4306 −1.15541
889889 23.1148i 0.775246i
890890 − 0.571352i − 0.0191517i
891891 − 46.4989i − 1.55777i
892892 − 3.06531i − 0.102634i
893893 0.648481 0.0217006
894894 7.81163 0.261260
895895 − 2.82371i − 0.0943861i
896896 −20.7396 −0.692862
897897 0 0
898898 10.0175 0.334287
899899 − 30.7084i − 1.02418i
900900 13.7313 0.457708
901901 22.4383 0.747529
902902 39.4795i 1.31452i
903903 − 1.56704i − 0.0521478i
904904 32.4359i 1.07880i
905905 − 5.17928i − 0.172165i
906906 −7.10752 −0.236132
907907 −8.10215 −0.269027 −0.134514 0.990912i 0.542947π-0.542947\pi
−0.134514 + 0.990912i 0.542947π0.542947\pi
908908 − 9.45234i − 0.313687i
909909 27.3623 0.907549
910910 0 0
911911 −9.18119 −0.304187 −0.152093 0.988366i 0.548601π-0.548601\pi
−0.152093 + 0.988366i 0.548601π0.548601\pi
912912 0.109916i 0.00363969i
913913 −6.83446 −0.226188
914914 −4.79656 −0.158656
915915 − 1.92692i − 0.0637020i
916916 32.8015i 1.08379i
917917 − 15.5007i − 0.511877i
918918 − 3.69979i − 0.122111i
919919 27.5036 0.907262 0.453631 0.891190i 0.350128π-0.350128\pi
0.453631 + 0.891190i 0.350128π0.350128\pi
920920 0.993295 0.0327480
921921 42.9831i 1.41634i
922922 1.64933 0.0543180
923923 0 0
924924 30.5187 1.00399
925925 − 28.1129i − 0.924346i
926926 −6.77586 −0.222668
927927 −2.79118 −0.0916745
928928 26.9946i 0.886142i
929929 24.2131i 0.794407i 0.917731 + 0.397203i 0.130019π0.130019\pi
−0.917731 + 0.397203i 0.869981π0.869981\pi
930930 − 2.95108i − 0.0967698i
931931 − 0.127375i − 0.00417454i
932932 4.15346 0.136051
933933 −0.606268 −0.0198483
934934 − 26.8761i − 0.879412i
935935 2.26444 0.0740550
936936 0 0
937937 11.1830 0.365333 0.182666 0.983175i 0.441527π-0.441527\pi
0.182666 + 0.983175i 0.441527π0.441527\pi
938938 − 14.5157i − 0.473955i
939939 52.5652 1.71540
940940 −2.46548 −0.0804152
941941 − 15.9638i − 0.520404i −0.965554 0.260202i 0.916211π-0.916211\pi
0.965554 0.260202i 0.0837891π-0.0837891\pi
942942 − 8.02715i − 0.261539i
943943 17.3177i 0.563941i
944944 3.76271i 0.122466i
945945 −1.24400 −0.0404672
946946 −1.00777 −0.0327654
947947 − 6.51466i − 0.211698i −0.994382 0.105849i 0.966244π-0.966244\pi
0.994382 0.105849i 0.0337560π-0.0337560\pi
948948 −30.3991 −0.987317
949949 0 0
950950 0.349126 0.0113271
951951 31.4470i 1.01974i
952952 13.6974 0.443935
953953 −47.6469 −1.54344 −0.771718 0.635965i 0.780602π-0.780602\pi
−0.771718 + 0.635965i 0.780602π0.780602\pi
954954 − 17.0780i − 0.552920i
955955 − 3.56571i − 0.115384i
956956 − 34.1215i − 1.10357i
957957 − 44.1933i − 1.42857i
958958 −19.8329 −0.640773
959959 14.6595 0.473380
960960 1.97823i 0.0638471i
961961 −12.9705 −0.418402
962962 0 0
963963 6.69740 0.215821
964964 27.4993i 0.885694i
965965 −3.35391 −0.107966
966966 −6.34481 −0.204141
967967 − 43.8122i − 1.40891i −0.709751 0.704453i 0.751192π-0.751192\pi
0.709751 0.704453i 0.248808π-0.248808\pi
968968 − 18.9433i − 0.608861i
969969 − 0.427583i − 0.0137360i
970970 1.59611i 0.0512479i
971971 4.29483 0.137828 0.0689139 0.997623i 0.478047π-0.478047\pi
0.0689139 + 0.997623i 0.478047π0.478047\pi
972972 24.6823 0.791686
973973 − 34.6631i − 1.11125i
974974 30.2776 0.970156
975975 0 0
976976 1.92692 0.0616792
977977 26.8019i 0.857470i 0.903430 + 0.428735i 0.141041π0.141041\pi
−0.903430 + 0.428735i 0.858959π0.858959\pi
978978 −29.1172 −0.931066
979979 −12.2513 −0.391553
980980 0.484271i 0.0154695i
981981 32.1715i 1.02716i
982982 − 25.1094i − 0.801275i
983983 − 27.2495i − 0.869124i −0.900642 0.434562i 0.856903π-0.856903\pi
0.900642 0.434562i 0.143097π-0.143097\pi
984984 −70.1178 −2.23527
985985 0.138391 0.00440951
986986 − 8.01746i − 0.255328i
987987 38.9614 1.24015
988988 0 0
989989 −0.442058 −0.0140566
990990 − 1.72348i − 0.0547758i
991991 24.3889 0.774740 0.387370 0.921924i 0.373384π-0.373384\pi
0.387370 + 0.921924i 0.373384π0.373384\pi
992992 38.6528 1.22723
993993 − 40.0441i − 1.27076i
994994 16.3773i 0.519458i
995995 − 2.83818i − 0.0899764i
996996 − 4.90648i − 0.155468i
997997 31.3207 0.991935 0.495967 0.868341i 0.334814π-0.334814\pi
0.495967 + 0.868341i 0.334814π0.334814\pi
998998 −17.2314 −0.545452
999999 − 12.1642i − 0.384859i
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 169.2.b.b.168.5 6
3.2 odd 2 1521.2.b.l.1351.2 6
4.3 odd 2 2704.2.f.o.337.6 6
13.2 odd 12 169.2.c.c.22.1 6
13.3 even 3 169.2.e.b.147.2 12
13.4 even 6 169.2.e.b.23.2 12
13.5 odd 4 169.2.a.b.1.3 3
13.6 odd 12 169.2.c.c.146.1 6
13.7 odd 12 169.2.c.b.146.3 6
13.8 odd 4 169.2.a.c.1.1 yes 3
13.9 even 3 169.2.e.b.23.5 12
13.10 even 6 169.2.e.b.147.5 12
13.11 odd 12 169.2.c.b.22.3 6
13.12 even 2 inner 169.2.b.b.168.2 6
39.5 even 4 1521.2.a.r.1.1 3
39.8 even 4 1521.2.a.o.1.3 3
39.38 odd 2 1521.2.b.l.1351.5 6
52.31 even 4 2704.2.a.z.1.3 3
52.47 even 4 2704.2.a.ba.1.3 3
52.51 odd 2 2704.2.f.o.337.5 6
65.34 odd 4 4225.2.a.bb.1.3 3
65.44 odd 4 4225.2.a.bg.1.1 3
91.34 even 4 8281.2.a.bj.1.1 3
91.83 even 4 8281.2.a.bf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 13.5 odd 4
169.2.a.c.1.1 yes 3 13.8 odd 4
169.2.b.b.168.2 6 13.12 even 2 inner
169.2.b.b.168.5 6 1.1 even 1 trivial
169.2.c.b.22.3 6 13.11 odd 12
169.2.c.b.146.3 6 13.7 odd 12
169.2.c.c.22.1 6 13.2 odd 12
169.2.c.c.146.1 6 13.6 odd 12
169.2.e.b.23.2 12 13.4 even 6
169.2.e.b.23.5 12 13.9 even 3
169.2.e.b.147.2 12 13.3 even 3
169.2.e.b.147.5 12 13.10 even 6
1521.2.a.o.1.3 3 39.8 even 4
1521.2.a.r.1.1 3 39.5 even 4
1521.2.b.l.1351.2 6 3.2 odd 2
1521.2.b.l.1351.5 6 39.38 odd 2
2704.2.a.z.1.3 3 52.31 even 4
2704.2.a.ba.1.3 3 52.47 even 4
2704.2.f.o.337.5 6 52.51 odd 2
2704.2.f.o.337.6 6 4.3 odd 2
4225.2.a.bb.1.3 3 65.34 odd 4
4225.2.a.bg.1.1 3 65.44 odd 4
8281.2.a.bf.1.3 3 91.83 even 4
8281.2.a.bj.1.1 3 91.34 even 4