Properties

Label 431.2.a.e.1.2
Level 431431
Weight 22
Character 431.1
Self dual yes
Analytic conductor 3.4423.442
Analytic rank 11
Dimension 44
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [431,2,Mod(1,431)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(431, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("431.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 431 431
Weight: k k == 2 2
Character orbit: [χ][\chi] == 431.a (trivial)

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: yes
Analytic conductor: 3.441552327123.44155232712
Analytic rank: 11
Dimension: 44
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x33x2+x+1 x^{4} - x^{3} - 3x^{2} + x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.2
Root 0.7376400.737640 of defining polynomial
Character χ\chi == 431.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q0.737640q22.35567q31.45589q4+0.355674q5+1.73764q6+2.19353q7+2.54920q8+2.54920q90.262360q10+1.31313q11+3.42960q120.219819q131.61803q140.837853q15+1.03138q164.72333q171.88039q183.76902q190.517822q205.16724q210.968620q220.150986q236.00509q244.87350q25+0.162147q26+1.06193q273.19353q281.01939q29+0.618034q304.64433q315.85919q323.09331q33+3.48412q34+0.780181q353.71135q361.24723q37+2.78018q38+0.517822q39+0.906685q40+0.961212q41+3.81156q422.70626q431.91177q44+0.906685q45+0.111374q460.791342q472.42960q482.18844q49+3.59489q50+11.1266q51+0.320031q524.41446q530.783326q54+0.467048q55+5.59174q56+8.87858q57+0.751946q584.46097q59+1.21982q603.10155q61+3.42584q62+5.59174q63+2.25922q640.0781839q65+2.28175q66+9.44584q67+6.87664q68+0.355674q690.575493q70+2.60980q71+6.49843q7210.8065q73+0.920006q74+11.4804q75+5.48727q76+2.88039q770.381966q784.27546q79+0.366835q8010.1492q810.709029q82+5.53916q83+7.52291q841.67997q85+1.99625q86+2.40136q87+3.34744q8814.6037q890.668808q900.482178q91+0.219819q92+10.9405q93+0.583726q941.34054q95+13.8023q969.16806q97+1.61428q98+3.34744q99+O(q100)q-0.737640 q^{2} -2.35567 q^{3} -1.45589 q^{4} +0.355674 q^{5} +1.73764 q^{6} +2.19353 q^{7} +2.54920 q^{8} +2.54920 q^{9} -0.262360 q^{10} +1.31313 q^{11} +3.42960 q^{12} -0.219819 q^{13} -1.61803 q^{14} -0.837853 q^{15} +1.03138 q^{16} -4.72333 q^{17} -1.88039 q^{18} -3.76902 q^{19} -0.517822 q^{20} -5.16724 q^{21} -0.968620 q^{22} -0.150986 q^{23} -6.00509 q^{24} -4.87350 q^{25} +0.162147 q^{26} +1.06193 q^{27} -3.19353 q^{28} -1.01939 q^{29} +0.618034 q^{30} -4.64433 q^{31} -5.85919 q^{32} -3.09331 q^{33} +3.48412 q^{34} +0.780181 q^{35} -3.71135 q^{36} -1.24723 q^{37} +2.78018 q^{38} +0.517822 q^{39} +0.906685 q^{40} +0.961212 q^{41} +3.81156 q^{42} -2.70626 q^{43} -1.91177 q^{44} +0.906685 q^{45} +0.111374 q^{46} -0.791342 q^{47} -2.42960 q^{48} -2.18844 q^{49} +3.59489 q^{50} +11.1266 q^{51} +0.320031 q^{52} -4.41446 q^{53} -0.783326 q^{54} +0.467048 q^{55} +5.59174 q^{56} +8.87858 q^{57} +0.751946 q^{58} -4.46097 q^{59} +1.21982 q^{60} -3.10155 q^{61} +3.42584 q^{62} +5.59174 q^{63} +2.25922 q^{64} -0.0781839 q^{65} +2.28175 q^{66} +9.44584 q^{67} +6.87664 q^{68} +0.355674 q^{69} -0.575493 q^{70} +2.60980 q^{71} +6.49843 q^{72} -10.8065 q^{73} +0.920006 q^{74} +11.4804 q^{75} +5.48727 q^{76} +2.88039 q^{77} -0.381966 q^{78} -4.27546 q^{79} +0.366835 q^{80} -10.1492 q^{81} -0.709029 q^{82} +5.53916 q^{83} +7.52291 q^{84} -1.67997 q^{85} +1.99625 q^{86} +2.40136 q^{87} +3.34744 q^{88} -14.6037 q^{89} -0.668808 q^{90} -0.482178 q^{91} +0.219819 q^{92} +10.9405 q^{93} +0.583726 q^{94} -1.34054 q^{95} +13.8023 q^{96} -9.16806 q^{97} +1.61428 q^{98} +3.34744 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4qq23q3q45q5+5q6+2q73q83q93q10+q112q125q132q143q153q16+2q175q186q19+4q20++11q99+O(q100) 4 q - q^{2} - 3 q^{3} - q^{4} - 5 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} - 3 q^{9} - 3 q^{10} + q^{11} - 2 q^{12} - 5 q^{13} - 2 q^{14} - 3 q^{15} - 3 q^{16} + 2 q^{17} - 5 q^{18} - 6 q^{19} + 4 q^{20}+ \cdots + 11 q^{99}+O(q^{100}) Copy content Toggle raw display

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 −0.737640 −0.521590 −0.260795 0.965394i 0.583985π-0.583985\pi
−0.260795 + 0.965394i 0.583985π0.583985\pi
33 −2.35567 −1.36005 −0.680025 0.733189i 0.738031π-0.738031\pi
−0.680025 + 0.733189i 0.738031π0.738031\pi
44 −1.45589 −0.727943
55 0.355674 0.159062 0.0795312 0.996832i 0.474658π-0.474658\pi
0.0795312 + 0.996832i 0.474658π0.474658\pi
66 1.73764 0.709389
77 2.19353 0.829075 0.414538 0.910032i 0.363943π-0.363943\pi
0.414538 + 0.910032i 0.363943π0.363943\pi
88 2.54920 0.901279
99 2.54920 0.849734
1010 −0.262360 −0.0829654
1111 1.31313 0.395925 0.197962 0.980210i 0.436568π-0.436568\pi
0.197962 + 0.980210i 0.436568π0.436568\pi
1212 3.42960 0.990039
1313 −0.219819 −0.0609668 −0.0304834 0.999535i 0.509705π-0.509705\pi
−0.0304834 + 0.999535i 0.509705π0.509705\pi
1414 −1.61803 −0.432438
1515 −0.837853 −0.216333
1616 1.03138 0.257845
1717 −4.72333 −1.14558 −0.572788 0.819703i 0.694138π-0.694138\pi
−0.572788 + 0.819703i 0.694138π0.694138\pi
1818 −1.88039 −0.443213
1919 −3.76902 −0.864673 −0.432336 0.901712i 0.642311π-0.642311\pi
−0.432336 + 0.901712i 0.642311π0.642311\pi
2020 −0.517822 −0.115788
2121 −5.16724 −1.12758
2222 −0.968620 −0.206511
2323 −0.150986 −0.0314828 −0.0157414 0.999876i 0.505011π-0.505011\pi
−0.0157414 + 0.999876i 0.505011π0.505011\pi
2424 −6.00509 −1.22578
2525 −4.87350 −0.974699
2626 0.162147 0.0317997
2727 1.06193 0.204369
2828 −3.19353 −0.603520
2929 −1.01939 −0.189297 −0.0946483 0.995511i 0.530173π-0.530173\pi
−0.0946483 + 0.995511i 0.530173π0.530173\pi
3030 0.618034 0.112837
3131 −4.64433 −0.834146 −0.417073 0.908873i 0.636944π-0.636944\pi
−0.417073 + 0.908873i 0.636944π0.636944\pi
3232 −5.85919 −1.03577
3333 −3.09331 −0.538477
3434 3.48412 0.597522
3535 0.780181 0.131875
3636 −3.71135 −0.618558
3737 −1.24723 −0.205043 −0.102522 0.994731i 0.532691π-0.532691\pi
−0.102522 + 0.994731i 0.532691π0.532691\pi
3838 2.78018 0.451005
3939 0.517822 0.0829178
4040 0.906685 0.143360
4141 0.961212 0.150116 0.0750581 0.997179i 0.476086π-0.476086\pi
0.0750581 + 0.997179i 0.476086π0.476086\pi
4242 3.81156 0.588137
4343 −2.70626 −0.412701 −0.206350 0.978478i 0.566159π-0.566159\pi
−0.206350 + 0.978478i 0.566159π0.566159\pi
4444 −1.91177 −0.288211
4545 0.906685 0.135161
4646 0.111374 0.0164211
4747 −0.791342 −0.115429 −0.0577146 0.998333i 0.518381π-0.518381\pi
−0.0577146 + 0.998333i 0.518381π0.518381\pi
4848 −2.42960 −0.350682
4949 −2.18844 −0.312634
5050 3.59489 0.508394
5151 11.1266 1.55804
5252 0.320031 0.0443804
5353 −4.41446 −0.606373 −0.303187 0.952931i 0.598051π-0.598051\pi
−0.303187 + 0.952931i 0.598051π0.598051\pi
5454 −0.783326 −0.106597
5555 0.467048 0.0629767
5656 5.59174 0.747228
5757 8.87858 1.17600
5858 0.751946 0.0987353
5959 −4.46097 −0.580769 −0.290385 0.956910i 0.593783π-0.593783\pi
−0.290385 + 0.956910i 0.593783π0.593783\pi
6060 1.21982 0.157478
6161 −3.10155 −0.397112 −0.198556 0.980089i 0.563625π-0.563625\pi
−0.198556 + 0.980089i 0.563625π0.563625\pi
6262 3.42584 0.435082
6363 5.59174 0.704493
6464 2.25922 0.282402
6565 −0.0781839 −0.00969752
6666 2.28175 0.280864
6767 9.44584 1.15399 0.576997 0.816746i 0.304225π-0.304225\pi
0.576997 + 0.816746i 0.304225π0.304225\pi
6868 6.87664 0.833915
6969 0.355674 0.0428182
7070 −0.575493 −0.0687846
7171 2.60980 0.309726 0.154863 0.987936i 0.450506π-0.450506\pi
0.154863 + 0.987936i 0.450506π0.450506\pi
7272 6.49843 0.765847
7373 −10.8065 −1.26480 −0.632401 0.774641i 0.717930π-0.717930\pi
−0.632401 + 0.774641i 0.717930π0.717930\pi
7474 0.920006 0.106949
7575 11.4804 1.32564
7676 5.48727 0.629433
7777 2.88039 0.328251
7878 −0.381966 −0.0432491
7979 −4.27546 −0.481027 −0.240514 0.970646i 0.577316π-0.577316\pi
−0.240514 + 0.970646i 0.577316π0.577316\pi
8080 0.366835 0.0410134
8181 −10.1492 −1.12769
8282 −0.709029 −0.0782992
8383 5.53916 0.608002 0.304001 0.952672i 0.401677π-0.401677\pi
0.304001 + 0.952672i 0.401677π0.401677\pi
8484 7.52291 0.820817
8585 −1.67997 −0.182218
8686 1.99625 0.215261
8787 2.40136 0.257453
8888 3.34744 0.356838
8989 −14.6037 −1.54799 −0.773996 0.633190i 0.781745π-0.781745\pi
−0.773996 + 0.633190i 0.781745π0.781745\pi
9090 −0.668808 −0.0704985
9191 −0.482178 −0.0505460
9292 0.219819 0.0229177
9393 10.9405 1.13448
9494 0.583726 0.0602067
9595 −1.34054 −0.137537
9696 13.8023 1.40870
9797 −9.16806 −0.930875 −0.465438 0.885081i 0.654103π-0.654103\pi
−0.465438 + 0.885081i 0.654103π0.654103\pi
9898 1.61428 0.163067
9999 3.34744 0.336431
100100 7.09526 0.709526
101101 5.98194 0.595225 0.297613 0.954687i 0.403810π-0.403810\pi
0.297613 + 0.954687i 0.403810π0.403810\pi
102102 −8.20746 −0.812659
103103 −10.5400 −1.03854 −0.519268 0.854612i 0.673795π-0.673795\pi
−0.519268 + 0.854612i 0.673795π0.673795\pi
104104 −0.560362 −0.0549481
105105 −1.83785 −0.179356
106106 3.25629 0.316279
107107 10.8015 1.04422 0.522111 0.852877i 0.325145π-0.325145\pi
0.522111 + 0.852877i 0.325145π0.325145\pi
108108 −1.54606 −0.148769
109109 −19.3739 −1.85568 −0.927840 0.372979i 0.878336π-0.878336\pi
−0.927840 + 0.372979i 0.878336π0.878336\pi
110110 −0.344513 −0.0328481
111111 2.93807 0.278869
112112 2.26236 0.213773
113113 11.7803 1.10820 0.554099 0.832451i 0.313063π-0.313063\pi
0.554099 + 0.832451i 0.313063π0.313063\pi
114114 −6.54920 −0.613389
115115 −0.0537019 −0.00500773
116116 1.48412 0.137797
117117 −0.560362 −0.0518055
118118 3.29059 0.302924
119119 −10.3608 −0.949770
120120 −2.13586 −0.194976
121121 −9.27568 −0.843244
122122 2.28783 0.207130
123123 −2.26430 −0.204165
124124 6.76161 0.607211
125125 −3.51175 −0.314100
126126 −4.12469 −0.367457
127127 11.0823 0.983394 0.491697 0.870766i 0.336377π-0.336377\pi
0.491697 + 0.870766i 0.336377π0.336377\pi
128128 10.0519 0.888470
129129 6.37507 0.561293
130130 0.0576716 0.00505813
131131 9.41103 0.822245 0.411123 0.911580i 0.365137π-0.365137\pi
0.411123 + 0.911580i 0.365137π0.365137\pi
132132 4.50352 0.391981
133133 −8.26745 −0.716879
134134 −6.96764 −0.601912
135135 0.377703 0.0325075
136136 −12.0407 −1.03248
137137 14.4355 1.23331 0.616654 0.787234i 0.288488π-0.288488\pi
0.616654 + 0.787234i 0.288488π0.288488\pi
138138 −0.262360 −0.0223335
139139 −6.29486 −0.533923 −0.266961 0.963707i 0.586020π-0.586020\pi
−0.266961 + 0.963707i 0.586020π0.586020\pi
140140 −1.13586 −0.0959973
141141 1.86414 0.156989
142142 −1.92509 −0.161550
143143 −0.288651 −0.0241382
144144 2.62920 0.219100
145145 −0.362572 −0.0301100
146146 7.97129 0.659709
147147 5.15525 0.425198
148148 1.81582 0.149260
149149 1.27486 0.104440 0.0522201 0.998636i 0.483370π-0.483370\pi
0.0522201 + 0.998636i 0.483370π0.483370\pi
150150 −8.46838 −0.691441
151151 −13.7654 −1.12021 −0.560106 0.828421i 0.689240π-0.689240\pi
−0.560106 + 0.828421i 0.689240π0.689240\pi
152152 −9.60799 −0.779311
153153 −12.0407 −0.973435
154154 −2.12469 −0.171213
155155 −1.65187 −0.132681
156156 −0.753889 −0.0603595
157157 0.224691 0.0179323 0.00896613 0.999960i 0.497146π-0.497146\pi
0.00896613 + 0.999960i 0.497146π0.497146\pi
158158 3.15375 0.250899
159159 10.3990 0.824698
160160 −2.08396 −0.164752
161161 −0.331192 −0.0261016
162162 7.48644 0.588190
163163 10.7653 0.843201 0.421600 0.906782i 0.361469π-0.361469\pi
0.421600 + 0.906782i 0.361469π0.361469\pi
164164 −1.39942 −0.109276
165165 −1.10021 −0.0856514
166166 −4.08591 −0.317128
167167 5.99953 0.464257 0.232129 0.972685i 0.425431π-0.425431\pi
0.232129 + 0.972685i 0.425431π0.425431\pi
168168 −13.1723 −1.01627
169169 −12.9517 −0.996283
170170 1.23921 0.0950433
171171 −9.60799 −0.734741
172172 3.94001 0.300423
173173 −3.24856 −0.246984 −0.123492 0.992346i 0.539409π-0.539409\pi
−0.123492 + 0.992346i 0.539409π0.539409\pi
174174 −1.77134 −0.134285
175175 −10.6901 −0.808099
176176 1.35434 0.102087
177177 10.5086 0.789875
178178 10.7723 0.807418
179179 −4.62709 −0.345845 −0.172923 0.984935i 0.555321π-0.555321\pi
−0.172923 + 0.984935i 0.555321π0.555321\pi
180180 −1.32003 −0.0983893
181181 −9.89238 −0.735295 −0.367647 0.929965i 0.619837π-0.619837\pi
−0.367647 + 0.929965i 0.619837π0.619837\pi
182182 0.355674 0.0263643
183183 7.30624 0.540092
184184 −0.384894 −0.0283748
185185 −0.443607 −0.0326147
186186 −8.07017 −0.591733
187187 −6.20237 −0.453562
188188 1.15210 0.0840259
189189 2.32938 0.169438
190190 0.988839 0.0717379
191191 8.84323 0.639874 0.319937 0.947439i 0.396338π-0.396338\pi
0.319937 + 0.947439i 0.396338π0.396338\pi
192192 −5.32197 −0.384080
193193 9.17608 0.660508 0.330254 0.943892i 0.392866π-0.392866\pi
0.330254 + 0.943892i 0.392866π0.392866\pi
194194 6.76273 0.485536
195195 0.184176 0.0131891
196196 3.18612 0.227580
197197 −0.248564 −0.0177094 −0.00885472 0.999961i 0.502819π-0.502819\pi
−0.00885472 + 0.999961i 0.502819π0.502819\pi
198198 −2.46921 −0.175479
199199 5.15802 0.365642 0.182821 0.983146i 0.441477π-0.441477\pi
0.182821 + 0.983146i 0.441477π0.441477\pi
200200 −12.4235 −0.878476
201201 −22.2513 −1.56949
202202 −4.41252 −0.310464
203203 −2.23607 −0.156941
204204 −16.1991 −1.13417
205205 0.341879 0.0238778
206206 7.77472 0.541690
207207 −0.384894 −0.0267520
208208 −0.226717 −0.0157200
209209 −4.94923 −0.342345
210210 1.35567 0.0935504
211211 17.5140 1.20571 0.602856 0.797850i 0.294029π-0.294029\pi
0.602856 + 0.797850i 0.294029π0.294029\pi
212212 6.42696 0.441405
213213 −6.14784 −0.421243
214214 −7.96764 −0.544656
215215 −0.962547 −0.0656452
216216 2.70709 0.184194
217217 −10.1875 −0.691569
218218 14.2909 0.967905
219219 25.4565 1.72019
220220 −0.679969 −0.0458435
221221 1.03828 0.0698421
222222 −2.16724 −0.145455
223223 −18.8858 −1.26469 −0.632344 0.774687i 0.717907π-0.717907\pi
−0.632344 + 0.774687i 0.717907π0.717907\pi
224224 −12.8523 −0.858730
225225 −12.4235 −0.828235
226226 −8.68964 −0.578026
227227 4.29688 0.285194 0.142597 0.989781i 0.454455π-0.454455\pi
0.142597 + 0.989781i 0.454455π0.454455\pi
228228 −12.9262 −0.856059
229229 5.94217 0.392670 0.196335 0.980537i 0.437096π-0.437096\pi
0.196335 + 0.980537i 0.437096π0.437096\pi
230230 0.0396127 0.00261198
231231 −6.78527 −0.446438
232232 −2.59864 −0.170609
233233 2.48037 0.162494 0.0812472 0.996694i 0.474110π-0.474110\pi
0.0812472 + 0.996694i 0.474110π0.474110\pi
234234 0.413346 0.0270213
235235 −0.281460 −0.0183604
236236 6.49467 0.422767
237237 10.0716 0.654221
238238 7.64252 0.495391
239239 −20.0791 −1.29881 −0.649406 0.760442i 0.724982π-0.724982\pi
−0.649406 + 0.760442i 0.724982π0.724982\pi
240240 −0.864145 −0.0557803
241241 17.7876 1.14580 0.572899 0.819626i 0.305819π-0.305819\pi
0.572899 + 0.819626i 0.305819π0.305819\pi
242242 6.84212 0.439828
243243 20.7223 1.32934
244244 4.51550 0.289075
245245 −0.778371 −0.0497283
246246 1.67024 0.106491
247247 0.828502 0.0527163
248248 −11.8393 −0.751798
249249 −13.0485 −0.826912
250250 2.59041 0.163832
251251 −7.46524 −0.471202 −0.235601 0.971850i 0.575706π-0.575706\pi
−0.235601 + 0.971850i 0.575706π0.575706\pi
252252 −8.14094 −0.512831
253253 −0.198265 −0.0124648
254254 −8.17474 −0.512929
255255 3.95746 0.247826
256256 −11.9331 −0.745819
257257 −3.16460 −0.197402 −0.0987012 0.995117i 0.531469π-0.531469\pi
−0.0987012 + 0.995117i 0.531469π0.531469\pi
258258 −4.70251 −0.292765
259259 −2.73583 −0.169996
260260 0.113827 0.00705924
261261 −2.59864 −0.160852
262262 −6.94195 −0.428875
263263 −7.20962 −0.444564 −0.222282 0.974982i 0.571351π-0.571351\pi
−0.222282 + 0.974982i 0.571351π0.571351\pi
264264 −7.88548 −0.485318
265265 −1.57011 −0.0964512
266266 6.09840 0.373917
267267 34.4016 2.10535
268268 −13.7521 −0.840042
269269 1.81673 0.110768 0.0553840 0.998465i 0.482362π-0.482362\pi
0.0553840 + 0.998465i 0.482362π0.482362\pi
270270 −0.278609 −0.0169556
271271 15.4315 0.937399 0.468700 0.883358i 0.344723π-0.344723\pi
0.468700 + 0.883358i 0.344723π0.344723\pi
272272 −4.87155 −0.295381
273273 1.13586 0.0687451
274274 −10.6482 −0.643282
275275 −6.39955 −0.385907
276276 −0.517822 −0.0311692
277277 −3.68852 −0.221621 −0.110811 0.993842i 0.535345π-0.535345\pi
−0.110811 + 0.993842i 0.535345π0.535345\pi
278278 4.64334 0.278489
279279 −11.8393 −0.708802
280280 1.98884 0.118856
281281 −28.8380 −1.72033 −0.860165 0.510016i 0.829640π-0.829640\pi
−0.860165 + 0.510016i 0.829640π0.829640\pi
282282 −1.37507 −0.0818841
283283 −19.5214 −1.16043 −0.580214 0.814464i 0.697031π-0.697031\pi
−0.580214 + 0.814464i 0.697031π0.697031\pi
284284 −3.79958 −0.225463
285285 3.15788 0.187057
286286 0.212921 0.0125903
287287 2.10845 0.124458
288288 −14.9363 −0.880127
289289 5.30989 0.312346
290290 0.267448 0.0157051
291291 21.5970 1.26604
292292 15.7330 0.920704
293293 19.2030 1.12185 0.560926 0.827866i 0.310445π-0.310445\pi
0.560926 + 0.827866i 0.310445π0.310445\pi
294294 −3.80272 −0.221779
295295 −1.58665 −0.0923786
296296 −3.17944 −0.184801
297297 1.39446 0.0809149
298298 −0.940385 −0.0544750
299299 0.0331896 0.00191940
300300 −16.7141 −0.964990
301301 −5.93626 −0.342160
302302 10.1539 0.584292
303303 −14.0915 −0.809536
304304 −3.88729 −0.222951
305305 −1.10314 −0.0631657
306306 8.88173 0.507735
307307 8.44718 0.482106 0.241053 0.970512i 0.422507π-0.422507\pi
0.241053 + 0.970512i 0.422507π0.422507\pi
308308 −4.19353 −0.238948
309309 24.8288 1.41246
310310 1.21848 0.0692052
311311 24.4490 1.38637 0.693187 0.720758i 0.256206π-0.256206\pi
0.693187 + 0.720758i 0.256206π0.256206\pi
312312 1.32003 0.0747321
313313 −4.86660 −0.275076 −0.137538 0.990496i 0.543919π-0.543919\pi
−0.137538 + 0.990496i 0.543919π0.543919\pi
314314 −0.165741 −0.00935329
315315 1.98884 0.112058
316316 6.22459 0.350161
317317 34.5036 1.93792 0.968959 0.247221i 0.0795175π-0.0795175\pi
0.968959 + 0.247221i 0.0795175π0.0795175\pi
318318 −7.67075 −0.430154
319319 −1.33860 −0.0749472
320320 0.803545 0.0449195
321321 −25.4449 −1.42019
322322 0.244301 0.0136144
323323 17.8023 0.990549
324324 14.7761 0.820892
325325 1.07129 0.0594243
326326 −7.94089 −0.439805
327327 45.6385 2.52382
328328 2.45032 0.135296
329329 −1.73583 −0.0956994
330330 0.811561 0.0446750
331331 −5.74204 −0.315611 −0.157805 0.987470i 0.550442π-0.550442\pi
−0.157805 + 0.987470i 0.550442π0.550442\pi
332332 −8.06439 −0.442591
333333 −3.17944 −0.174232
334334 −4.42549 −0.242152
335335 3.35964 0.183557
336336 −5.32938 −0.290742
337337 −12.3242 −0.671343 −0.335671 0.941979i 0.608963π-0.608963\pi
−0.335671 + 0.941979i 0.608963π0.608963\pi
338338 9.55368 0.519652
339339 −27.7506 −1.50720
340340 2.44584 0.132645
341341 −6.09862 −0.330259
342342 7.08724 0.383234
343343 −20.1551 −1.08827
344344 −6.89880 −0.371959
345345 0.126504 0.00681076
346346 2.39627 0.128824
347347 11.8379 0.635489 0.317745 0.948176i 0.397075π-0.397075\pi
0.317745 + 0.948176i 0.397075π0.397075\pi
348348 −3.49611 −0.187411
349349 0.350720 0.0187736 0.00938680 0.999956i 0.497012π-0.497012\pi
0.00938680 + 0.999956i 0.497012π0.497012\pi
350350 7.88548 0.421497
351351 −0.233433 −0.0124597
352352 −7.69390 −0.410086
353353 7.12371 0.379157 0.189578 0.981866i 0.439288π-0.439288\pi
0.189578 + 0.981866i 0.439288π0.439288\pi
354354 −7.75157 −0.411991
355355 0.928239 0.0492658
356356 21.2614 1.12685
357357 24.4066 1.29173
358358 3.41313 0.180389
359359 −19.6384 −1.03647 −0.518237 0.855237i 0.673412π-0.673412\pi
−0.518237 + 0.855237i 0.673412π0.673412\pi
360360 2.31132 0.121817
361361 −4.79449 −0.252341
362362 7.29702 0.383523
363363 21.8505 1.14685
364364 0.701997 0.0367947
365365 −3.84358 −0.201182
366366 −5.38937 −0.281707
367367 −1.92083 −0.100267 −0.0501333 0.998743i 0.515965π-0.515965\pi
−0.0501333 + 0.998743i 0.515965π0.515965\pi
368368 −0.155724 −0.00811768
369369 2.45032 0.127559
370370 0.327223 0.0170115
371371 −9.68325 −0.502729
372372 −15.9282 −0.825836
373373 38.1400 1.97481 0.987406 0.158206i 0.0505710π-0.0505710\pi
0.987406 + 0.158206i 0.0505710π0.0505710\pi
374374 4.57512 0.236574
375375 8.27254 0.427192
376376 −2.01729 −0.104034
377377 0.224082 0.0115408
378378 −1.71825 −0.0883771
379379 37.7074 1.93690 0.968448 0.249215i 0.0801724π-0.0801724\pi
0.968448 + 0.249215i 0.0801724π0.0801724\pi
380380 1.95168 0.100119
381381 −26.1063 −1.33746
382382 −6.52313 −0.333752
383383 7.78393 0.397740 0.198870 0.980026i 0.436273π-0.436273\pi
0.198870 + 0.980026i 0.436273π0.436273\pi
384384 −23.6790 −1.20836
385385 1.02448 0.0522124
386386 −6.76864 −0.344515
387387 −6.89880 −0.350686
388388 13.3477 0.677625
389389 −15.3669 −0.779132 −0.389566 0.920998i 0.627375π-0.627375\pi
−0.389566 + 0.920998i 0.627375π0.627375\pi
390390 −0.135855 −0.00687931
391391 0.713158 0.0360660
392392 −5.57877 −0.281771
393393 −22.1693 −1.11829
394394 0.183351 0.00923708
395395 −1.52067 −0.0765133
396396 −4.87350 −0.244902
397397 8.41124 0.422148 0.211074 0.977470i 0.432304π-0.432304\pi
0.211074 + 0.977470i 0.432304π0.432304\pi
398398 −3.80476 −0.190715
399399 19.4754 0.974990
400400 −5.02643 −0.251321
401401 35.1299 1.75430 0.877152 0.480213i 0.159440π-0.159440\pi
0.877152 + 0.480213i 0.159440π0.159440\pi
402402 16.4135 0.818630
403403 1.02091 0.0508552
404404 −8.70903 −0.433290
405405 −3.60980 −0.179372
406406 1.64941 0.0818590
407407 −1.63778 −0.0811816
408408 28.3640 1.40423
409409 30.9050 1.52816 0.764078 0.645124i 0.223194π-0.223194\pi
0.764078 + 0.645124i 0.223194π0.223194\pi
410410 −0.252183 −0.0124544
411411 −34.0054 −1.67736
412412 15.3450 0.755995
413413 −9.78527 −0.481502
414414 0.283913 0.0139536
415415 1.97014 0.0967102
416416 1.28796 0.0631474
417417 14.8286 0.726161
418418 3.65075 0.178564
419419 34.1501 1.66834 0.834172 0.551505i 0.185946π-0.185946\pi
0.834172 + 0.551505i 0.185946π0.185946\pi
420420 2.67571 0.130561
421421 −10.1575 −0.495048 −0.247524 0.968882i 0.579617π-0.579617\pi
−0.247524 + 0.968882i 0.579617π0.579617\pi
422422 −12.9190 −0.628888
423423 −2.01729 −0.0980840
424424 −11.2534 −0.546511
425425 23.0192 1.11659
426426 4.53490 0.219716
427427 −6.80333 −0.329236
428428 −15.7258 −0.760135
429429 0.679969 0.0328292
430430 0.710014 0.0342399
431431 −1.00000 −0.0481683
432432 1.09526 0.0526956
433433 5.06771 0.243539 0.121769 0.992558i 0.461143π-0.461143\pi
0.121769 + 0.992558i 0.461143π0.461143\pi
434434 7.51468 0.360716
435435 0.854102 0.0409511
436436 28.2062 1.35083
437437 0.569070 0.0272223
438438 −18.7778 −0.897236
439439 −21.0411 −1.00424 −0.502118 0.864799i 0.667446π-0.667446\pi
−0.502118 + 0.864799i 0.667446π0.667446\pi
440440 1.19060 0.0567596
441441 −5.57877 −0.265656
442442 −0.765876 −0.0364290
443443 16.4981 0.783846 0.391923 0.919998i 0.371810π-0.371810\pi
0.391923 + 0.919998i 0.371810π0.371810\pi
444444 −4.27749 −0.203001
445445 −5.19417 −0.246227
446446 13.9310 0.659650
447447 −3.00314 −0.142044
448448 4.95565 0.234132
449449 30.4997 1.43937 0.719684 0.694301i 0.244286π-0.244286\pi
0.719684 + 0.694301i 0.244286π0.244286\pi
450450 9.16409 0.431999
451451 1.26220 0.0594347
452452 −17.1508 −0.806706
453453 32.4268 1.52354
454454 −3.16955 −0.148755
455455 −0.171498 −0.00803997
456456 22.6333 1.05990
457457 11.8416 0.553929 0.276964 0.960880i 0.410672π-0.410672\pi
0.276964 + 0.960880i 0.410672π0.410672\pi
458458 −4.38318 −0.204813
459459 −5.01587 −0.234121
460460 0.0781839 0.00364534
461461 −11.9837 −0.558138 −0.279069 0.960271i 0.590026π-0.590026\pi
−0.279069 + 0.960271i 0.590026π0.590026\pi
462462 5.00509 0.232858
463463 4.46877 0.207682 0.103841 0.994594i 0.466887π-0.466887\pi
0.103841 + 0.994594i 0.466887π0.466887\pi
464464 −1.05138 −0.0488092
465465 3.89126 0.180453
466466 −1.82962 −0.0847555
467467 −29.9185 −1.38446 −0.692231 0.721676i 0.743372π-0.743372\pi
−0.692231 + 0.721676i 0.743372π0.743372\pi
468468 0.815824 0.0377115
469469 20.7197 0.956748
470470 0.207616 0.00957663
471471 −0.529298 −0.0243887
472472 −11.3719 −0.523435
473473 −3.55368 −0.163398
474474 −7.42922 −0.341235
475475 18.3683 0.842796
476476 15.0841 0.691378
477477 −11.2534 −0.515256
478478 14.8112 0.677448
479479 −33.6205 −1.53616 −0.768080 0.640354i 0.778788π-0.778788\pi
−0.768080 + 0.640354i 0.778788π0.778788\pi
480480 4.90914 0.224071
481481 0.274164 0.0125008
482482 −13.1208 −0.597637
483483 0.780181 0.0354995
484484 13.5043 0.613834
485485 −3.26084 −0.148067
486486 −15.2856 −0.693371
487487 −7.66808 −0.347474 −0.173737 0.984792i 0.555584π-0.555584\pi
−0.173737 + 0.984792i 0.555584π0.555584\pi
488488 −7.90647 −0.357909
489489 −25.3595 −1.14679
490490 0.574158 0.0259378
491491 −23.7671 −1.07259 −0.536296 0.844030i 0.680177π-0.680177\pi
−0.536296 + 0.844030i 0.680177π0.680177\pi
492492 3.29657 0.148621
493493 4.81494 0.216854
494494 −0.611136 −0.0274963
495495 1.19060 0.0535134
496496 −4.79006 −0.215080
497497 5.72467 0.256787
498498 9.62507 0.431310
499499 29.6770 1.32853 0.664263 0.747499i 0.268746π-0.268746\pi
0.664263 + 0.747499i 0.268746π0.268746\pi
500500 5.11271 0.228647
501501 −14.1329 −0.631413
502502 5.50666 0.245774
503503 −36.5805 −1.63105 −0.815523 0.578725i 0.803550π-0.803550\pi
−0.815523 + 0.578725i 0.803550π0.803550\pi
504504 14.2545 0.634945
505505 2.12762 0.0946780
506506 0.146248 0.00650153
507507 30.5099 1.35499
508508 −16.1346 −0.715855
509509 −38.2366 −1.69481 −0.847403 0.530951i 0.821835π-0.821835\pi
−0.847403 + 0.530951i 0.821835π0.821835\pi
510510 −2.91918 −0.129264
511511 −23.7043 −1.04862
512512 −11.3014 −0.499458
513513 −4.00245 −0.176713
514514 2.33434 0.102963
515515 −3.74880 −0.165192
516516 −9.28138 −0.408590
517517 −1.03914 −0.0457012
518518 2.01806 0.0886684
519519 7.65256 0.335910
520520 −0.199306 −0.00874017
521521 −31.7558 −1.39125 −0.695624 0.718406i 0.744872π-0.744872\pi
−0.695624 + 0.718406i 0.744872π0.744872\pi
522522 1.91686 0.0838988
523523 4.30013 0.188031 0.0940157 0.995571i 0.470030π-0.470030\pi
0.0940157 + 0.995571i 0.470030π0.470030\pi
524524 −13.7014 −0.598548
525525 25.1825 1.09905
526526 5.31810 0.231880
527527 21.9367 0.955578
528528 −3.19038 −0.138844
529529 −22.9772 −0.999009
530530 1.15818 0.0503080
531531 −11.3719 −0.493499
532532 12.0365 0.521847
533533 −0.211293 −0.00915210
534534 −25.3760 −1.09813
535535 3.84182 0.166096
536536 24.0794 1.04007
537537 10.8999 0.470366
538538 −1.34009 −0.0577756
539539 −2.87371 −0.123780
540540 −0.549893 −0.0236636
541541 −25.1363 −1.08069 −0.540347 0.841443i 0.681707π-0.681707\pi
−0.540347 + 0.841443i 0.681707π0.681707\pi
542542 −11.3829 −0.488938
543543 23.3032 1.00004
544544 27.6749 1.18655
545545 −6.89079 −0.295169
546546 −0.837853 −0.0358568
547547 −12.8712 −0.550333 −0.275167 0.961397i 0.588733π-0.588733\pi
−0.275167 + 0.961397i 0.588733π0.588733\pi
548548 −21.0165 −0.897779
549549 −7.90647 −0.337440
550550 4.72057 0.201286
551551 3.84212 0.163680
552552 0.906685 0.0385911
553553 −9.37835 −0.398808
554554 2.72080 0.115596
555555 1.04499 0.0443575
556556 9.16460 0.388666
557557 −3.93164 −0.166589 −0.0832945 0.996525i 0.526544π-0.526544\pi
−0.0832945 + 0.996525i 0.526544π0.526544\pi
558558 8.73316 0.369704
559559 0.594887 0.0251610
560560 0.804663 0.0340032
561561 14.6108 0.616867
562562 21.2721 0.897308
563563 −17.0326 −0.717838 −0.358919 0.933369i 0.616855π-0.616855\pi
−0.358919 + 0.933369i 0.616855π0.616855\pi
564564 −2.71398 −0.114279
565565 4.18996 0.176273
566566 14.3998 0.605268
567567 −22.2625 −0.934937
568568 6.65291 0.279150
569569 18.0132 0.755154 0.377577 0.925978i 0.376757π-0.376757\pi
0.377577 + 0.925978i 0.376757π0.376757\pi
570570 −2.32938 −0.0975671
571571 −30.4440 −1.27404 −0.637020 0.770847i 0.719833π-0.719833\pi
−0.637020 + 0.770847i 0.719833π0.719833\pi
572572 0.420244 0.0175713
573573 −20.8318 −0.870260
574574 −1.55527 −0.0649159
575575 0.735831 0.0306863
576576 5.75919 0.239966
577577 34.3629 1.43055 0.715273 0.698846i 0.246303π-0.246303\pi
0.715273 + 0.698846i 0.246303π0.246303\pi
578578 −3.91679 −0.162917
579579 −21.6158 −0.898324
580580 0.527864 0.0219184
581581 12.1503 0.504079
582582 −15.9308 −0.660353
583583 −5.79678 −0.240078
584584 −27.5479 −1.13994
585585 −0.199306 −0.00824031
586586 −14.1649 −0.585147
587587 −35.3347 −1.45842 −0.729209 0.684291i 0.760112π-0.760112\pi
−0.729209 + 0.684291i 0.760112π0.760112\pi
588588 −7.50546 −0.309520
589589 17.5046 0.721263
590590 1.17038 0.0481838
591591 0.585536 0.0240857
592592 −1.28637 −0.0528694
593593 −9.31148 −0.382377 −0.191188 0.981553i 0.561234π-0.561234\pi
−0.191188 + 0.981553i 0.561234π0.561234\pi
594594 −1.02861 −0.0422044
595595 −3.68506 −0.151073
596596 −1.85605 −0.0760266
597597 −12.1506 −0.497291
598598 −0.0244820 −0.00100114
599599 15.9577 0.652013 0.326006 0.945368i 0.394297π-0.394297\pi
0.326006 + 0.945368i 0.394297π0.394297\pi
600600 29.2658 1.19477
601601 −40.6737 −1.65912 −0.829558 0.558421i 0.811407π-0.811407\pi
−0.829558 + 0.558421i 0.811407π0.811407\pi
602602 4.37882 0.178467
603603 24.0794 0.980587
604604 20.0409 0.815451
605605 −3.29912 −0.134128
606606 10.3945 0.422246
607607 −30.2347 −1.22719 −0.613594 0.789621i 0.710277π-0.710277\pi
−0.613594 + 0.789621i 0.710277π0.710277\pi
608608 22.0834 0.895600
609609 5.26745 0.213448
610610 0.813721 0.0329466
611611 0.173952 0.00703734
612612 17.5299 0.708606
613613 13.2408 0.534789 0.267395 0.963587i 0.413837π-0.413837\pi
0.267395 + 0.963587i 0.413837π0.413837\pi
614614 −6.23098 −0.251462
615615 −0.805354 −0.0324750
616616 7.34270 0.295846
617617 4.31606 0.173758 0.0868790 0.996219i 0.472311π-0.472311\pi
0.0868790 + 0.996219i 0.472311π0.472311\pi
618618 −18.3147 −0.736725
619619 −8.65208 −0.347757 −0.173878 0.984767i 0.555630π-0.555630\pi
−0.173878 + 0.984767i 0.555630π0.555630\pi
620620 2.40493 0.0965844
621621 −0.160337 −0.00643412
622622 −18.0345 −0.723119
623623 −32.0337 −1.28340
624624 0.534071 0.0213799
625625 23.1184 0.924738
626626 3.58980 0.143477
627627 11.6588 0.465606
628628 −0.327124 −0.0130537
629629 5.89108 0.234893
630630 −1.46705 −0.0584486
631631 4.03074 0.160461 0.0802305 0.996776i 0.474434π-0.474434\pi
0.0802305 + 0.996776i 0.474434π0.474434\pi
632632 −10.8990 −0.433540
633633 −41.2572 −1.63983
634634 −25.4513 −1.01080
635635 3.94168 0.156421
636636 −15.1398 −0.600333
637637 0.481060 0.0190603
638638 0.987405 0.0390918
639639 6.65291 0.263185
640640 3.57520 0.141322
641641 −42.7002 −1.68656 −0.843279 0.537476i 0.819378π-0.819378\pi
−0.843279 + 0.537476i 0.819378π0.819378\pi
642642 18.7692 0.740759
643643 −2.51260 −0.0990873 −0.0495436 0.998772i 0.515777π-0.515777\pi
−0.0495436 + 0.998772i 0.515777π0.515777\pi
644644 0.482178 0.0190005
645645 2.26745 0.0892807
646646 −13.1317 −0.516661
647647 −5.08862 −0.200054 −0.100027 0.994985i 0.531893π-0.531893\pi
−0.100027 + 0.994985i 0.531893π0.531893\pi
648648 −25.8723 −1.01636
649649 −5.85786 −0.229941
650650 −0.790224 −0.0309951
651651 23.9983 0.940568
652652 −15.6730 −0.613802
653653 −43.4653 −1.70093 −0.850465 0.526031i 0.823679π-0.823679\pi
−0.850465 + 0.526031i 0.823679π0.823679\pi
654654 −33.6648 −1.31640
655655 3.34726 0.130788
656656 0.991375 0.0387067
657657 −27.5479 −1.07474
658658 1.28042 0.0499159
659659 0.206667 0.00805059 0.00402530 0.999992i 0.498719π-0.498719\pi
0.00402530 + 0.999992i 0.498719π0.498719\pi
660660 1.60178 0.0623494
661661 35.9230 1.39724 0.698621 0.715492i 0.253797π-0.253797\pi
0.698621 + 0.715492i 0.253797π0.253797\pi
662662 4.23556 0.164620
663663 −2.44584 −0.0949887
664664 14.1204 0.547979
665665 −2.94052 −0.114028
666666 2.34528 0.0908778
667667 0.153914 0.00595959
668668 −8.73463 −0.337953
669669 44.4889 1.72004
670670 −2.47821 −0.0957416
671671 −4.07275 −0.157227
672672 30.2758 1.16791
673673 35.1183 1.35371 0.676855 0.736116i 0.263342π-0.263342\pi
0.676855 + 0.736116i 0.263342π0.263342\pi
674674 9.09084 0.350166
675675 −5.17533 −0.199199
676676 18.8562 0.725238
677677 −5.23428 −0.201170 −0.100585 0.994928i 0.532071π-0.532071\pi
−0.100585 + 0.994928i 0.532071π0.532071\pi
678678 20.4700 0.786144
679679 −20.1104 −0.771766
680680 −4.28258 −0.164229
681681 −10.1221 −0.387878
682682 4.49859 0.172260
683683 20.7013 0.792113 0.396057 0.918226i 0.370378π-0.370378\pi
0.396057 + 0.918226i 0.370378π0.370378\pi
684684 13.9881 0.534850
685685 5.13434 0.196173
686686 14.8672 0.567633
687687 −13.9978 −0.534050
688688 −2.79118 −0.106413
689689 0.970382 0.0369686
690690 −0.0933146 −0.00355243
691691 14.7800 0.562258 0.281129 0.959670i 0.409291π-0.409291\pi
0.281129 + 0.959670i 0.409291π0.409291\pi
692692 4.72954 0.179790
693693 7.34270 0.278926
694694 −8.73208 −0.331465
695695 −2.23892 −0.0849270
696696 6.12155 0.232037
697697 −4.54013 −0.171970
698698 −0.258705 −0.00979213
699699 −5.84294 −0.221000
700700 15.5636 0.588250
701701 34.1735 1.29072 0.645358 0.763881i 0.276708π-0.276708\pi
0.645358 + 0.763881i 0.276708π0.276708\pi
702702 0.172190 0.00649888
703703 4.70083 0.177295
704704 2.96665 0.111810
705705 0.663028 0.0249711
706706 −5.25474 −0.197765
707707 13.1215 0.493487
708708 −15.2993 −0.574984
709709 8.21388 0.308479 0.154239 0.988034i 0.450707π-0.450707\pi
0.154239 + 0.988034i 0.450707π0.450707\pi
710710 −0.684707 −0.0256966
711711 −10.8990 −0.408745
712712 −37.2278 −1.39517
713713 0.701229 0.0262612
714714 −18.0033 −0.673756
715715 −0.102666 −0.00383949
716716 6.73652 0.251756
717717 47.2999 1.76645
718718 14.4861 0.540615
719719 −33.7298 −1.25791 −0.628954 0.777443i 0.716517π-0.716517\pi
−0.628954 + 0.777443i 0.716517π0.716517\pi
720720 0.935137 0.0348505
721721 −23.1197 −0.861024
722722 3.53661 0.131619
723723 −41.9017 −1.55834
724724 14.4022 0.535253
725725 4.96801 0.184507
726726 −16.1178 −0.598188
727727 −21.4570 −0.795798 −0.397899 0.917429i 0.630260π-0.630260\pi
−0.397899 + 0.917429i 0.630260π0.630260\pi
728728 −1.22917 −0.0455561
729729 −18.3676 −0.680281
730730 2.83518 0.104935
731731 12.7826 0.472781
732732 −10.6371 −0.393157
733733 −25.9785 −0.959539 −0.479770 0.877395i 0.659280π-0.659280\pi
−0.479770 + 0.877395i 0.659280π0.659280\pi
734734 1.41688 0.0522981
735735 1.83359 0.0676330
736736 0.884657 0.0326089
737737 12.4037 0.456894
738738 −1.80746 −0.0665334
739739 34.0810 1.25369 0.626845 0.779144i 0.284346π-0.284346\pi
0.626845 + 0.779144i 0.284346π0.284346\pi
740740 0.645842 0.0237416
741741 −1.95168 −0.0716967
742742 7.14275 0.262219
743743 8.04036 0.294972 0.147486 0.989064i 0.452882π-0.452882\pi
0.147486 + 0.989064i 0.452882π0.452882\pi
744744 27.8896 1.02248
745745 0.453433 0.0166125
746746 −28.1336 −1.03004
747747 14.1204 0.516640
748748 9.02995 0.330168
749749 23.6934 0.865739
750750 −6.10216 −0.222819
751751 36.3502 1.32644 0.663220 0.748425i 0.269190π-0.269190\pi
0.663220 + 0.748425i 0.269190π0.269190\pi
752752 −0.816174 −0.0297628
753753 17.5857 0.640857
754754 −0.165292 −0.00601957
755755 −4.89600 −0.178184
756756 −3.39132 −0.123341
757757 −6.55362 −0.238195 −0.119098 0.992883i 0.538000π-0.538000\pi
−0.119098 + 0.992883i 0.538000π0.538000\pi
758758 −27.8145 −1.01027
759759 0.467048 0.0169528
760760 −3.41732 −0.123959
761761 −28.9735 −1.05029 −0.525144 0.851014i 0.675988π-0.675988\pi
−0.525144 + 0.851014i 0.675988π0.675988\pi
762762 19.2570 0.697609
763763 −42.4971 −1.53850
764764 −12.8747 −0.465792
765765 −4.28258 −0.154837
766766 −5.74174 −0.207458
767767 0.980606 0.0354076
768768 28.1105 1.01435
769769 −20.7996 −0.750052 −0.375026 0.927014i 0.622366π-0.622366\pi
−0.375026 + 0.927014i 0.622366π0.622366\pi
770770 −0.755699 −0.0272335
771771 7.45477 0.268477
772772 −13.3593 −0.480813
773773 34.7699 1.25059 0.625293 0.780390i 0.284980π-0.284980\pi
0.625293 + 0.780390i 0.284980π0.284980\pi
774774 5.08884 0.182914
775775 22.6341 0.813041
776776 −23.3712 −0.838978
777777 6.44473 0.231203
778778 11.3352 0.406388
779779 −3.62283 −0.129801
780780 −0.268139 −0.00960092
781781 3.42702 0.122628
782782 −0.526054 −0.0188117
783783 −1.08253 −0.0386865
784784 −2.25711 −0.0806111
785785 0.0799166 0.00285235
786786 16.3530 0.583291
787787 18.6268 0.663973 0.331986 0.943284i 0.392281π-0.392281\pi
0.331986 + 0.943284i 0.392281π0.392281\pi
788788 0.361881 0.0128915
789789 16.9835 0.604629
790790 1.12171 0.0399086
791791 25.8404 0.918780
792792 8.53330 0.303218
793793 0.681778 0.0242107
794794 −6.20447 −0.220188
795795 3.69867 0.131178
796796 −7.50949 −0.266167
797797 −8.55696 −0.303103 −0.151552 0.988449i 0.548427π-0.548427\pi
−0.151552 + 0.988449i 0.548427π0.548427\pi
798798 −14.3659 −0.508546
799799 3.73777 0.132233
800800 28.5547 1.00956
801801 −37.2278 −1.31538
802802 −25.9132 −0.915028
803803 −14.1903 −0.500766
804804 32.3954 1.14250
805805 −0.117797 −0.00415178
806806 −0.753064 −0.0265256
807807 −4.27963 −0.150650
808808 15.2492 0.536464
809809 36.5255 1.28417 0.642084 0.766634i 0.278070π-0.278070\pi
0.642084 + 0.766634i 0.278070π0.278070\pi
810810 2.66273 0.0935590
811811 0.291636 0.0102407 0.00512037 0.999987i 0.498370π-0.498370\pi
0.00512037 + 0.999987i 0.498370π0.498370\pi
812812 3.25546 0.114244
813813 −36.3517 −1.27491
814814 1.20809 0.0423436
815815 3.82893 0.134122
816816 11.4758 0.401733
817817 10.2000 0.356851
818818 −22.7968 −0.797072
819819 −1.22917 −0.0429507
820820 −0.497736 −0.0173817
821821 −24.0196 −0.838291 −0.419145 0.907919i 0.637670π-0.637670\pi
−0.419145 + 0.907919i 0.637670π0.637670\pi
822822 25.0837 0.874895
823823 −37.3060 −1.30040 −0.650202 0.759761i 0.725316π-0.725316\pi
−0.650202 + 0.759761i 0.725316π0.725316\pi
824824 −26.8685 −0.936010
825825 15.0753 0.524853
826826 7.21801 0.251147
827827 2.15858 0.0750610 0.0375305 0.999295i 0.488051π-0.488051\pi
0.0375305 + 0.999295i 0.488051π0.488051\pi
828828 0.560362 0.0194739
829829 39.2679 1.36383 0.681915 0.731431i 0.261147π-0.261147\pi
0.681915 + 0.731431i 0.261147π0.261147\pi
830830 −1.45325 −0.0504431
831831 8.68894 0.301416
832832 −0.496618 −0.0172171
833833 10.3367 0.358146
834834 −10.9382 −0.378759
835835 2.13388 0.0738459
836836 7.20551 0.249208
837837 −4.93197 −0.170474
838838 −25.1905 −0.870192
839839 19.4462 0.671358 0.335679 0.941976i 0.391034π-0.391034\pi
0.335679 + 0.941976i 0.391034π0.391034\pi
840840 −4.68506 −0.161650
841841 −27.9608 −0.964167
842842 7.49261 0.258213
843843 67.9329 2.33973
844844 −25.4984 −0.877690
845845 −4.60658 −0.158471
846846 1.48803 0.0511597
847847 −20.3465 −0.699113
848848 −4.55299 −0.156350
849849 45.9861 1.57824
850850 −16.9799 −0.582404
851851 0.188314 0.00645533
852852 8.95056 0.306641
853853 −31.5937 −1.08175 −0.540874 0.841104i 0.681906π-0.681906\pi
−0.540874 + 0.841104i 0.681906π0.681906\pi
854854 5.01841 0.171726
855855 −3.41732 −0.116870
856856 27.5352 0.941135
857857 −38.8076 −1.32564 −0.662822 0.748777i 0.730641π-0.730641\pi
−0.662822 + 0.748777i 0.730641π0.730641\pi
858858 −0.501572 −0.0171234
859859 −30.1896 −1.03005 −0.515027 0.857174i 0.672218π-0.672218\pi
−0.515027 + 0.857174i 0.672218π0.672218\pi
860860 1.40136 0.0477860
861861 −4.96681 −0.169268
862862 0.737640 0.0251241
863863 11.4522 0.389837 0.194918 0.980819i 0.437556π-0.437556\pi
0.194918 + 0.980819i 0.437556π0.437556\pi
864864 −6.22208 −0.211679
865865 −1.15543 −0.0392858
866866 −3.73815 −0.127028
867867 −12.5084 −0.424807
868868 14.8318 0.503423
869869 −5.61425 −0.190451
870870 −0.630020 −0.0213597
871871 −2.07637 −0.0703553
872872 −49.3879 −1.67248
873873 −23.3712 −0.790996
874874 −0.419769 −0.0141989
875875 −7.70312 −0.260413
876876 −37.0618 −1.25220
877877 −37.2682 −1.25846 −0.629229 0.777220i 0.716629π-0.716629\pi
−0.629229 + 0.777220i 0.716629π0.716629\pi
878878 15.5207 0.523800
879879 −45.2360 −1.52577
880880 0.481704 0.0162382
881881 −7.29814 −0.245880 −0.122940 0.992414i 0.539232π-0.539232\pi
−0.122940 + 0.992414i 0.539232π0.539232\pi
882882 4.11513 0.138564
883883 −19.7549 −0.664805 −0.332402 0.943138i 0.607859π-0.607859\pi
−0.332402 + 0.943138i 0.607859π0.607859\pi
884884 −1.51161 −0.0508411
885885 3.73764 0.125639
886886 −12.1696 −0.408847
887887 48.7214 1.63591 0.817953 0.575285i 0.195109π-0.195109\pi
0.817953 + 0.575285i 0.195109π0.195109\pi
888888 7.48972 0.251339
889889 24.3093 0.815308
890890 3.83143 0.128430
891891 −13.3272 −0.446479
892892 27.4956 0.920622
893893 2.98258 0.0998084
894894 2.21524 0.0740887
895895 −1.64574 −0.0550109
896896 22.0491 0.736609
897897 −0.0781839 −0.00261048
898898 −22.4978 −0.750761
899899 4.73440 0.157901
900900 18.0872 0.602908
901901 20.8510 0.694647
902902 −0.931050 −0.0310006
903903 13.9839 0.465355
904904 30.0304 0.998796
905905 −3.51847 −0.116958
906906 −23.9193 −0.794666
907907 7.89607 0.262185 0.131092 0.991370i 0.458152π-0.458152\pi
0.131092 + 0.991370i 0.458152π0.458152\pi
908908 −6.25578 −0.207605
909909 15.2492 0.505783
910910 0.126504 0.00419357
911911 41.9892 1.39117 0.695583 0.718446i 0.255146π-0.255146\pi
0.695583 + 0.718446i 0.255146π0.255146\pi
912912 9.15719 0.303225
913913 7.27365 0.240723
914914 −8.73487 −0.288924
915915 2.59864 0.0859084
916916 −8.65112 −0.285841
917917 20.6433 0.681703
918918 3.69991 0.122115
919919 47.6809 1.57285 0.786424 0.617687i 0.211930π-0.211930\pi
0.786424 + 0.617687i 0.211930π0.211930\pi
920920 −0.136897 −0.00451336
921921 −19.8988 −0.655688
922922 8.83968 0.291119
923923 −0.573683 −0.0188830
924924 9.87858 0.324982
925925 6.07837 0.199855
926926 −3.29635 −0.108325
927927 −26.8685 −0.882479
928928 5.97282 0.196067
929929 9.33964 0.306424 0.153212 0.988193i 0.451038π-0.451038\pi
0.153212 + 0.988193i 0.451038π0.451038\pi
930930 −2.87035 −0.0941225
931931 8.24827 0.270326
932932 −3.61114 −0.118287
933933 −57.5938 −1.88554
934934 22.0691 0.722122
935935 −2.20602 −0.0721447
936936 −1.42848 −0.0466912
937937 34.6393 1.13162 0.565809 0.824536i 0.308564π-0.308564\pi
0.565809 + 0.824536i 0.308564π0.308564\pi
938938 −15.2837 −0.499030
939939 11.4641 0.374117
940940 0.409774 0.0133654
941941 48.6872 1.58716 0.793579 0.608467i 0.208215π-0.208215\pi
0.793579 + 0.608467i 0.208215π0.208215\pi
942942 0.390431 0.0127209
943943 −0.145130 −0.00472608
944944 −4.60096 −0.149748
945945 0.828502 0.0269512
946946 2.62134 0.0852271
947947 −35.0995 −1.14058 −0.570290 0.821444i 0.693169π-0.693169\pi
−0.570290 + 0.821444i 0.693169π0.693169\pi
948948 −14.6631 −0.476236
949949 2.37547 0.0771109
950950 −13.5492 −0.439594
951951 −81.2794 −2.63566
952952 −26.4117 −0.856007
953953 41.5286 1.34524 0.672622 0.739986i 0.265168π-0.265168\pi
0.672622 + 0.739986i 0.265168π0.265168\pi
954954 8.30093 0.268753
955955 3.14531 0.101780
956956 29.2330 0.945461
957957 3.15331 0.101932
958958 24.7998 0.801246
959959 31.6647 1.02251
960960 −1.89289 −0.0610928
961961 −9.43024 −0.304201
962962 −0.202235 −0.00652031
963963 27.5352 0.887311
964964 −25.8967 −0.834076
965965 3.26369 0.105062
966966 −0.575493 −0.0185162
967967 25.9307 0.833875 0.416937 0.908935i 0.363103π-0.363103\pi
0.416937 + 0.908935i 0.363103π0.363103\pi
968968 −23.6456 −0.759998
969969 −41.9365 −1.34720
970970 2.40533 0.0772305
971971 −34.0711 −1.09339 −0.546697 0.837331i 0.684115π-0.684115\pi
−0.546697 + 0.837331i 0.684115π0.684115\pi
972972 −30.1694 −0.967684
973973 −13.8079 −0.442662
974974 5.65629 0.181239
975975 −2.52360 −0.0808199
976976 −3.19887 −0.102393
977977 57.9989 1.85555 0.927775 0.373139i 0.121719π-0.121719\pi
0.927775 + 0.373139i 0.121719π0.121719\pi
978978 18.7062 0.598157
979979 −19.1766 −0.612888
980980 1.13322 0.0361994
981981 −49.3879 −1.57683
982982 17.5315 0.559454
983983 −34.3211 −1.09467 −0.547337 0.836912i 0.684359π-0.684359\pi
−0.547337 + 0.836912i 0.684359π0.684359\pi
984984 −5.77216 −0.184010
985985 −0.0884078 −0.00281691
986986 −3.55169 −0.113109
987987 4.08905 0.130156
988988 −1.20620 −0.0383745
989989 0.408608 0.0129930
990990 −0.878234 −0.0279121
991991 49.4157 1.56974 0.784870 0.619660i 0.212730π-0.212730\pi
0.784870 + 0.619660i 0.212730π0.212730\pi
992992 27.2120 0.863981
993993 13.5264 0.429246
994994 −4.22275 −0.133937
995995 1.83457 0.0581599
996996 18.9971 0.601945
997997 −18.9248 −0.599354 −0.299677 0.954041i 0.596879π-0.596879\pi
−0.299677 + 0.954041i 0.596879π0.596879\pi
998998 −21.8910 −0.692946
999999 −1.32448 −0.0419046
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 431.2.a.e.1.2 4
3.2 odd 2 3879.2.a.m.1.3 4
4.3 odd 2 6896.2.a.o.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
431.2.a.e.1.2 4 1.1 even 1 trivial
3879.2.a.m.1.3 4 3.2 odd 2
6896.2.a.o.1.4 4 4.3 odd 2