Properties

Label 3549.2.a.z.1.2
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3549,2,Mod(1,3549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3549.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.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: \(28.3389076774\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.121819537.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 25x^{4} + 55x^{3} + 224x^{2} - 252x - 728 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.06987\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.24698 q^{2} -1.00000 q^{3} -0.445042 q^{4} +3.06987 q^{5} +1.24698 q^{6} +1.00000 q^{7} +3.04892 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.24698 q^{2} -1.00000 q^{3} -0.445042 q^{4} +3.06987 q^{5} +1.24698 q^{6} +1.00000 q^{7} +3.04892 q^{8} +1.00000 q^{9} -3.82806 q^{10} -3.50559 q^{11} +0.445042 q^{12} -1.24698 q^{14} -3.06987 q^{15} -2.91185 q^{16} +7.13559 q^{17} -1.24698 q^{18} -3.36622 q^{19} -1.36622 q^{20} -1.00000 q^{21} +4.37139 q^{22} -1.88076 q^{23} -3.04892 q^{24} +4.42409 q^{25} -1.00000 q^{27} -0.445042 q^{28} -4.40064 q^{29} +3.82806 q^{30} -2.60803 q^{31} -2.46681 q^{32} +3.50559 q^{33} -8.89793 q^{34} +3.06987 q^{35} -0.445042 q^{36} -5.25483 q^{37} +4.19761 q^{38} +9.35977 q^{40} +6.41744 q^{41} +1.24698 q^{42} -10.3388 q^{43} +1.56013 q^{44} +3.06987 q^{45} +2.34527 q^{46} -12.1076 q^{47} +2.91185 q^{48} +1.00000 q^{49} -5.51675 q^{50} -7.13559 q^{51} -10.3647 q^{53} +1.24698 q^{54} -10.7617 q^{55} +3.04892 q^{56} +3.36622 q^{57} +5.48751 q^{58} +12.6505 q^{59} +1.36622 q^{60} -12.5783 q^{61} +3.25215 q^{62} +1.00000 q^{63} +8.89977 q^{64} -4.37139 q^{66} +1.93301 q^{67} -3.17563 q^{68} +1.88076 q^{69} -3.82806 q^{70} -11.4790 q^{71} +3.04892 q^{72} +4.91575 q^{73} +6.55266 q^{74} -4.42409 q^{75} +1.49811 q^{76} -3.50559 q^{77} -3.41810 q^{79} -8.93901 q^{80} +1.00000 q^{81} -8.00242 q^{82} +7.10798 q^{83} +0.445042 q^{84} +21.9053 q^{85} +12.8923 q^{86} +4.40064 q^{87} -10.6882 q^{88} -0.730586 q^{89} -3.82806 q^{90} +0.837017 q^{92} +2.60803 q^{93} +15.0980 q^{94} -10.3339 q^{95} +2.46681 q^{96} -6.18414 q^{97} -1.24698 q^{98} -3.50559 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} - 6 q^{3} - 2 q^{4} - 3 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} - 6 q^{3} - 2 q^{4} - 3 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{9} - q^{10} + 2 q^{12} + 2 q^{14} + 3 q^{15} - 10 q^{16} - 9 q^{17} + 2 q^{18} - 11 q^{19} + q^{20} - 6 q^{21} + 7 q^{22} - 11 q^{23} + 29 q^{25} - 6 q^{27} - 2 q^{28} - 10 q^{29} + q^{30} - 7 q^{31} - 8 q^{32} - 10 q^{34} - 3 q^{35} - 2 q^{36} + 20 q^{37} - 6 q^{38} + 10 q^{41} - 2 q^{42} - 9 q^{43} - 3 q^{45} + 8 q^{46} - 36 q^{47} + 10 q^{48} + 6 q^{49} - 9 q^{50} + 9 q^{51} - 12 q^{53} - 2 q^{54} - 29 q^{55} + 11 q^{57} + 6 q^{58} + 41 q^{59} - q^{60} - 24 q^{61} + 6 q^{63} + 8 q^{64} - 7 q^{66} - 15 q^{67} + 10 q^{68} + 11 q^{69} - q^{70} - 13 q^{71} - 25 q^{73} + 2 q^{74} - 29 q^{75} - q^{76} - 16 q^{79} + 5 q^{80} + 6 q^{81} + q^{82} + 2 q^{83} + 2 q^{84} + 33 q^{85} + 4 q^{86} + 10 q^{87} - 14 q^{88} - 15 q^{89} - q^{90} + 13 q^{92} + 7 q^{93} - 33 q^{94} - 23 q^{95} + 8 q^{96} - 10 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.24698 −0.881748 −0.440874 0.897569i \(-0.645331\pi\)
−0.440874 + 0.897569i \(0.645331\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.445042 −0.222521
\(5\) 3.06987 1.37289 0.686443 0.727183i \(-0.259171\pi\)
0.686443 + 0.727183i \(0.259171\pi\)
\(6\) 1.24698 0.509077
\(7\) 1.00000 0.377964
\(8\) 3.04892 1.07796
\(9\) 1.00000 0.333333
\(10\) −3.82806 −1.21054
\(11\) −3.50559 −1.05697 −0.528487 0.848941i \(-0.677240\pi\)
−0.528487 + 0.848941i \(0.677240\pi\)
\(12\) 0.445042 0.128473
\(13\) 0 0
\(14\) −1.24698 −0.333269
\(15\) −3.06987 −0.792637
\(16\) −2.91185 −0.727963
\(17\) 7.13559 1.73063 0.865317 0.501225i \(-0.167117\pi\)
0.865317 + 0.501225i \(0.167117\pi\)
\(18\) −1.24698 −0.293916
\(19\) −3.36622 −0.772264 −0.386132 0.922444i \(-0.626189\pi\)
−0.386132 + 0.922444i \(0.626189\pi\)
\(20\) −1.36622 −0.305496
\(21\) −1.00000 −0.218218
\(22\) 4.37139 0.931984
\(23\) −1.88076 −0.392166 −0.196083 0.980587i \(-0.562822\pi\)
−0.196083 + 0.980587i \(0.562822\pi\)
\(24\) −3.04892 −0.622358
\(25\) 4.42409 0.884818
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −0.445042 −0.0841050
\(29\) −4.40064 −0.817178 −0.408589 0.912718i \(-0.633979\pi\)
−0.408589 + 0.912718i \(0.633979\pi\)
\(30\) 3.82806 0.698905
\(31\) −2.60803 −0.468415 −0.234208 0.972187i \(-0.575250\pi\)
−0.234208 + 0.972187i \(0.575250\pi\)
\(32\) −2.46681 −0.436075
\(33\) 3.50559 0.610244
\(34\) −8.89793 −1.52598
\(35\) 3.06987 0.518902
\(36\) −0.445042 −0.0741736
\(37\) −5.25483 −0.863888 −0.431944 0.901900i \(-0.642172\pi\)
−0.431944 + 0.901900i \(0.642172\pi\)
\(38\) 4.19761 0.680942
\(39\) 0 0
\(40\) 9.35977 1.47991
\(41\) 6.41744 1.00224 0.501118 0.865379i \(-0.332922\pi\)
0.501118 + 0.865379i \(0.332922\pi\)
\(42\) 1.24698 0.192413
\(43\) −10.3388 −1.57666 −0.788328 0.615255i \(-0.789053\pi\)
−0.788328 + 0.615255i \(0.789053\pi\)
\(44\) 1.56013 0.235199
\(45\) 3.06987 0.457629
\(46\) 2.34527 0.345791
\(47\) −12.1076 −1.76608 −0.883039 0.469300i \(-0.844506\pi\)
−0.883039 + 0.469300i \(0.844506\pi\)
\(48\) 2.91185 0.420290
\(49\) 1.00000 0.142857
\(50\) −5.51675 −0.780186
\(51\) −7.13559 −0.999182
\(52\) 0 0
\(53\) −10.3647 −1.42371 −0.711854 0.702328i \(-0.752144\pi\)
−0.711854 + 0.702328i \(0.752144\pi\)
\(54\) 1.24698 0.169692
\(55\) −10.7617 −1.45111
\(56\) 3.04892 0.407429
\(57\) 3.36622 0.445867
\(58\) 5.48751 0.720545
\(59\) 12.6505 1.64695 0.823477 0.567350i \(-0.192031\pi\)
0.823477 + 0.567350i \(0.192031\pi\)
\(60\) 1.36622 0.176378
\(61\) −12.5783 −1.61049 −0.805245 0.592943i \(-0.797966\pi\)
−0.805245 + 0.592943i \(0.797966\pi\)
\(62\) 3.25215 0.413024
\(63\) 1.00000 0.125988
\(64\) 8.89977 1.11247
\(65\) 0 0
\(66\) −4.37139 −0.538081
\(67\) 1.93301 0.236155 0.118077 0.993004i \(-0.462327\pi\)
0.118077 + 0.993004i \(0.462327\pi\)
\(68\) −3.17563 −0.385102
\(69\) 1.88076 0.226417
\(70\) −3.82806 −0.457541
\(71\) −11.4790 −1.36231 −0.681154 0.732140i \(-0.738522\pi\)
−0.681154 + 0.732140i \(0.738522\pi\)
\(72\) 3.04892 0.359318
\(73\) 4.91575 0.575346 0.287673 0.957729i \(-0.407118\pi\)
0.287673 + 0.957729i \(0.407118\pi\)
\(74\) 6.55266 0.761732
\(75\) −4.42409 −0.510850
\(76\) 1.49811 0.171845
\(77\) −3.50559 −0.399499
\(78\) 0 0
\(79\) −3.41810 −0.384566 −0.192283 0.981339i \(-0.561589\pi\)
−0.192283 + 0.981339i \(0.561589\pi\)
\(80\) −8.93901 −0.999411
\(81\) 1.00000 0.111111
\(82\) −8.00242 −0.883719
\(83\) 7.10798 0.780202 0.390101 0.920772i \(-0.372440\pi\)
0.390101 + 0.920772i \(0.372440\pi\)
\(84\) 0.445042 0.0485580
\(85\) 21.9053 2.37596
\(86\) 12.8923 1.39021
\(87\) 4.40064 0.471798
\(88\) −10.6882 −1.13937
\(89\) −0.730586 −0.0774420 −0.0387210 0.999250i \(-0.512328\pi\)
−0.0387210 + 0.999250i \(0.512328\pi\)
\(90\) −3.82806 −0.403513
\(91\) 0 0
\(92\) 0.837017 0.0872650
\(93\) 2.60803 0.270440
\(94\) 15.0980 1.55724
\(95\) −10.3339 −1.06023
\(96\) 2.46681 0.251768
\(97\) −6.18414 −0.627904 −0.313952 0.949439i \(-0.601653\pi\)
−0.313952 + 0.949439i \(0.601653\pi\)
\(98\) −1.24698 −0.125964
\(99\) −3.50559 −0.352325
\(100\) −1.96891 −0.196891
\(101\) 11.6216 1.15639 0.578197 0.815897i \(-0.303757\pi\)
0.578197 + 0.815897i \(0.303757\pi\)
\(102\) 8.89793 0.881026
\(103\) 12.3453 1.21642 0.608208 0.793778i \(-0.291889\pi\)
0.608208 + 0.793778i \(0.291889\pi\)
\(104\) 0 0
\(105\) −3.06987 −0.299588
\(106\) 12.9246 1.25535
\(107\) −10.5138 −1.01641 −0.508203 0.861237i \(-0.669690\pi\)
−0.508203 + 0.861237i \(0.669690\pi\)
\(108\) 0.445042 0.0428242
\(109\) −0.707801 −0.0677950 −0.0338975 0.999425i \(-0.510792\pi\)
−0.0338975 + 0.999425i \(0.510792\pi\)
\(110\) 13.4196 1.27951
\(111\) 5.25483 0.498766
\(112\) −2.91185 −0.275144
\(113\) −6.54416 −0.615623 −0.307812 0.951447i \(-0.599597\pi\)
−0.307812 + 0.951447i \(0.599597\pi\)
\(114\) −4.19761 −0.393142
\(115\) −5.77368 −0.538399
\(116\) 1.95847 0.181839
\(117\) 0 0
\(118\) −15.7749 −1.45220
\(119\) 7.13559 0.654118
\(120\) −9.35977 −0.854427
\(121\) 1.28913 0.117194
\(122\) 15.6849 1.42005
\(123\) −6.41744 −0.578641
\(124\) 1.16068 0.104232
\(125\) −1.76796 −0.158132
\(126\) −1.24698 −0.111090
\(127\) −3.11279 −0.276216 −0.138108 0.990417i \(-0.544102\pi\)
−0.138108 + 0.990417i \(0.544102\pi\)
\(128\) −6.16421 −0.544844
\(129\) 10.3388 0.910283
\(130\) 0 0
\(131\) −5.35550 −0.467913 −0.233956 0.972247i \(-0.575167\pi\)
−0.233956 + 0.972247i \(0.575167\pi\)
\(132\) −1.56013 −0.135792
\(133\) −3.36622 −0.291888
\(134\) −2.41042 −0.208229
\(135\) −3.06987 −0.264212
\(136\) 21.7558 1.86555
\(137\) 15.6533 1.33735 0.668674 0.743556i \(-0.266862\pi\)
0.668674 + 0.743556i \(0.266862\pi\)
\(138\) −2.34527 −0.199643
\(139\) −7.69765 −0.652907 −0.326453 0.945213i \(-0.605854\pi\)
−0.326453 + 0.945213i \(0.605854\pi\)
\(140\) −1.36622 −0.115467
\(141\) 12.1076 1.01965
\(142\) 14.3141 1.20121
\(143\) 0 0
\(144\) −2.91185 −0.242654
\(145\) −13.5094 −1.12189
\(146\) −6.12985 −0.507310
\(147\) −1.00000 −0.0824786
\(148\) 2.33862 0.192233
\(149\) 14.3574 1.17620 0.588101 0.808787i \(-0.299876\pi\)
0.588101 + 0.808787i \(0.299876\pi\)
\(150\) 5.51675 0.450441
\(151\) 18.2801 1.48762 0.743808 0.668394i \(-0.233018\pi\)
0.743808 + 0.668394i \(0.233018\pi\)
\(152\) −10.2633 −0.832466
\(153\) 7.13559 0.576878
\(154\) 4.37139 0.352257
\(155\) −8.00629 −0.643081
\(156\) 0 0
\(157\) 9.14988 0.730240 0.365120 0.930960i \(-0.381028\pi\)
0.365120 + 0.930960i \(0.381028\pi\)
\(158\) 4.26230 0.339090
\(159\) 10.3647 0.821978
\(160\) −7.57279 −0.598681
\(161\) −1.88076 −0.148225
\(162\) −1.24698 −0.0979720
\(163\) −13.6215 −1.06691 −0.533457 0.845827i \(-0.679108\pi\)
−0.533457 + 0.845827i \(0.679108\pi\)
\(164\) −2.85603 −0.223018
\(165\) 10.7617 0.837796
\(166\) −8.86350 −0.687941
\(167\) −3.58810 −0.277655 −0.138828 0.990317i \(-0.544333\pi\)
−0.138828 + 0.990317i \(0.544333\pi\)
\(168\) −3.04892 −0.235229
\(169\) 0 0
\(170\) −27.3155 −2.09500
\(171\) −3.36622 −0.257421
\(172\) 4.60121 0.350839
\(173\) −5.00471 −0.380501 −0.190251 0.981736i \(-0.560930\pi\)
−0.190251 + 0.981736i \(0.560930\pi\)
\(174\) −5.48751 −0.416007
\(175\) 4.42409 0.334430
\(176\) 10.2078 0.769438
\(177\) −12.6505 −0.950869
\(178\) 0.911026 0.0682843
\(179\) −6.82831 −0.510372 −0.255186 0.966892i \(-0.582137\pi\)
−0.255186 + 0.966892i \(0.582137\pi\)
\(180\) −1.36622 −0.101832
\(181\) 25.1122 1.86658 0.933289 0.359125i \(-0.116925\pi\)
0.933289 + 0.359125i \(0.116925\pi\)
\(182\) 0 0
\(183\) 12.5783 0.929817
\(184\) −5.73428 −0.422737
\(185\) −16.1316 −1.18602
\(186\) −3.25215 −0.238460
\(187\) −25.0144 −1.82924
\(188\) 5.38840 0.392989
\(189\) −1.00000 −0.0727393
\(190\) 12.8861 0.934856
\(191\) −17.4749 −1.26444 −0.632218 0.774790i \(-0.717855\pi\)
−0.632218 + 0.774790i \(0.717855\pi\)
\(192\) −8.89977 −0.642286
\(193\) 7.75149 0.557964 0.278982 0.960296i \(-0.410003\pi\)
0.278982 + 0.960296i \(0.410003\pi\)
\(194\) 7.71150 0.553653
\(195\) 0 0
\(196\) −0.445042 −0.0317887
\(197\) −10.0904 −0.718908 −0.359454 0.933163i \(-0.617037\pi\)
−0.359454 + 0.933163i \(0.617037\pi\)
\(198\) 4.37139 0.310661
\(199\) 21.9594 1.55666 0.778331 0.627854i \(-0.216067\pi\)
0.778331 + 0.627854i \(0.216067\pi\)
\(200\) 13.4887 0.953794
\(201\) −1.93301 −0.136344
\(202\) −14.4919 −1.01965
\(203\) −4.40064 −0.308864
\(204\) 3.17563 0.222339
\(205\) 19.7007 1.37596
\(206\) −15.3943 −1.07257
\(207\) −1.88076 −0.130722
\(208\) 0 0
\(209\) 11.8006 0.816263
\(210\) 3.82806 0.264161
\(211\) −19.3081 −1.32922 −0.664612 0.747189i \(-0.731403\pi\)
−0.664612 + 0.747189i \(0.731403\pi\)
\(212\) 4.61274 0.316805
\(213\) 11.4790 0.786529
\(214\) 13.1105 0.896214
\(215\) −31.7388 −2.16457
\(216\) −3.04892 −0.207453
\(217\) −2.60803 −0.177044
\(218\) 0.882613 0.0597781
\(219\) −4.91575 −0.332176
\(220\) 4.78940 0.322901
\(221\) 0 0
\(222\) −6.55266 −0.439786
\(223\) 18.9063 1.26606 0.633031 0.774126i \(-0.281810\pi\)
0.633031 + 0.774126i \(0.281810\pi\)
\(224\) −2.46681 −0.164821
\(225\) 4.42409 0.294939
\(226\) 8.16044 0.542824
\(227\) 9.41062 0.624605 0.312302 0.949983i \(-0.398900\pi\)
0.312302 + 0.949983i \(0.398900\pi\)
\(228\) −1.49811 −0.0992147
\(229\) −10.7573 −0.710861 −0.355431 0.934703i \(-0.615666\pi\)
−0.355431 + 0.934703i \(0.615666\pi\)
\(230\) 7.19967 0.474732
\(231\) 3.50559 0.230651
\(232\) −13.4172 −0.880881
\(233\) −22.6327 −1.48272 −0.741358 0.671110i \(-0.765818\pi\)
−0.741358 + 0.671110i \(0.765818\pi\)
\(234\) 0 0
\(235\) −37.1688 −2.42463
\(236\) −5.63000 −0.366482
\(237\) 3.41810 0.222029
\(238\) −8.89793 −0.576767
\(239\) −21.6067 −1.39762 −0.698810 0.715308i \(-0.746287\pi\)
−0.698810 + 0.715308i \(0.746287\pi\)
\(240\) 8.93901 0.577010
\(241\) 2.91822 0.187979 0.0939895 0.995573i \(-0.470038\pi\)
0.0939895 + 0.995573i \(0.470038\pi\)
\(242\) −1.60752 −0.103336
\(243\) −1.00000 −0.0641500
\(244\) 5.59788 0.358368
\(245\) 3.06987 0.196127
\(246\) 8.00242 0.510215
\(247\) 0 0
\(248\) −7.95165 −0.504930
\(249\) −7.10798 −0.450450
\(250\) 2.20462 0.139432
\(251\) −18.4508 −1.16460 −0.582302 0.812973i \(-0.697848\pi\)
−0.582302 + 0.812973i \(0.697848\pi\)
\(252\) −0.445042 −0.0280350
\(253\) 6.59317 0.414509
\(254\) 3.88159 0.243552
\(255\) −21.9053 −1.37176
\(256\) −10.1129 −0.632056
\(257\) −15.6082 −0.973615 −0.486807 0.873509i \(-0.661839\pi\)
−0.486807 + 0.873509i \(0.661839\pi\)
\(258\) −12.8923 −0.802640
\(259\) −5.25483 −0.326519
\(260\) 0 0
\(261\) −4.40064 −0.272393
\(262\) 6.67820 0.412581
\(263\) −16.8076 −1.03640 −0.518200 0.855259i \(-0.673398\pi\)
−0.518200 + 0.855259i \(0.673398\pi\)
\(264\) 10.6882 0.657816
\(265\) −31.8184 −1.95459
\(266\) 4.19761 0.257372
\(267\) 0.730586 0.0447111
\(268\) −0.860271 −0.0525494
\(269\) −20.3448 −1.24045 −0.620223 0.784426i \(-0.712958\pi\)
−0.620223 + 0.784426i \(0.712958\pi\)
\(270\) 3.82806 0.232968
\(271\) 5.26295 0.319702 0.159851 0.987141i \(-0.448899\pi\)
0.159851 + 0.987141i \(0.448899\pi\)
\(272\) −20.7778 −1.25984
\(273\) 0 0
\(274\) −19.5193 −1.17920
\(275\) −15.5090 −0.935230
\(276\) −0.837017 −0.0503825
\(277\) −27.0717 −1.62658 −0.813292 0.581856i \(-0.802327\pi\)
−0.813292 + 0.581856i \(0.802327\pi\)
\(278\) 9.59882 0.575699
\(279\) −2.60803 −0.156138
\(280\) 9.35977 0.559354
\(281\) −22.6562 −1.35156 −0.675779 0.737104i \(-0.736193\pi\)
−0.675779 + 0.737104i \(0.736193\pi\)
\(282\) −15.0980 −0.899070
\(283\) 2.68158 0.159403 0.0797016 0.996819i \(-0.474603\pi\)
0.0797016 + 0.996819i \(0.474603\pi\)
\(284\) 5.10864 0.303142
\(285\) 10.3339 0.612125
\(286\) 0 0
\(287\) 6.41744 0.378810
\(288\) −2.46681 −0.145358
\(289\) 33.9166 1.99509
\(290\) 16.8459 0.989227
\(291\) 6.18414 0.362521
\(292\) −2.18772 −0.128026
\(293\) −24.8294 −1.45055 −0.725276 0.688458i \(-0.758288\pi\)
−0.725276 + 0.688458i \(0.758288\pi\)
\(294\) 1.24698 0.0727253
\(295\) 38.8353 2.26108
\(296\) −16.0215 −0.931233
\(297\) 3.50559 0.203415
\(298\) −17.9034 −1.03711
\(299\) 0 0
\(300\) 1.96891 0.113675
\(301\) −10.3388 −0.595920
\(302\) −22.7949 −1.31170
\(303\) −11.6216 −0.667644
\(304\) 9.80194 0.562180
\(305\) −38.6138 −2.21102
\(306\) −8.89793 −0.508661
\(307\) −22.7954 −1.30100 −0.650501 0.759505i \(-0.725441\pi\)
−0.650501 + 0.759505i \(0.725441\pi\)
\(308\) 1.56013 0.0888968
\(309\) −12.3453 −0.702298
\(310\) 9.98368 0.567035
\(311\) 23.1401 1.31216 0.656078 0.754693i \(-0.272214\pi\)
0.656078 + 0.754693i \(0.272214\pi\)
\(312\) 0 0
\(313\) −17.5357 −0.991175 −0.495588 0.868558i \(-0.665047\pi\)
−0.495588 + 0.868558i \(0.665047\pi\)
\(314\) −11.4097 −0.643887
\(315\) 3.06987 0.172967
\(316\) 1.52120 0.0855741
\(317\) 21.1340 1.18700 0.593500 0.804834i \(-0.297746\pi\)
0.593500 + 0.804834i \(0.297746\pi\)
\(318\) −12.9246 −0.724777
\(319\) 15.4268 0.863736
\(320\) 27.3211 1.52730
\(321\) 10.5138 0.586823
\(322\) 2.34527 0.130697
\(323\) −24.0200 −1.33651
\(324\) −0.445042 −0.0247245
\(325\) 0 0
\(326\) 16.9857 0.940750
\(327\) 0.707801 0.0391415
\(328\) 19.5662 1.08037
\(329\) −12.1076 −0.667515
\(330\) −13.4196 −0.738725
\(331\) −19.6453 −1.07980 −0.539901 0.841728i \(-0.681538\pi\)
−0.539901 + 0.841728i \(0.681538\pi\)
\(332\) −3.16335 −0.173611
\(333\) −5.25483 −0.287963
\(334\) 4.47428 0.244822
\(335\) 5.93409 0.324214
\(336\) 2.91185 0.158855
\(337\) 0.587432 0.0319994 0.0159997 0.999872i \(-0.494907\pi\)
0.0159997 + 0.999872i \(0.494907\pi\)
\(338\) 0 0
\(339\) 6.54416 0.355430
\(340\) −9.74878 −0.528702
\(341\) 9.14266 0.495103
\(342\) 4.19761 0.226981
\(343\) 1.00000 0.0539949
\(344\) −31.5222 −1.69956
\(345\) 5.77368 0.310845
\(346\) 6.24077 0.335506
\(347\) −16.4236 −0.881663 −0.440831 0.897590i \(-0.645316\pi\)
−0.440831 + 0.897590i \(0.645316\pi\)
\(348\) −1.95847 −0.104985
\(349\) 34.5166 1.84763 0.923815 0.382838i \(-0.125053\pi\)
0.923815 + 0.382838i \(0.125053\pi\)
\(350\) −5.51675 −0.294883
\(351\) 0 0
\(352\) 8.64762 0.460920
\(353\) 29.2191 1.55518 0.777589 0.628773i \(-0.216443\pi\)
0.777589 + 0.628773i \(0.216443\pi\)
\(354\) 15.7749 0.838427
\(355\) −35.2391 −1.87030
\(356\) 0.325141 0.0172325
\(357\) −7.13559 −0.377655
\(358\) 8.51477 0.450019
\(359\) −8.37106 −0.441808 −0.220904 0.975296i \(-0.570901\pi\)
−0.220904 + 0.975296i \(0.570901\pi\)
\(360\) 9.35977 0.493303
\(361\) −7.66856 −0.403609
\(362\) −31.3145 −1.64585
\(363\) −1.28913 −0.0676620
\(364\) 0 0
\(365\) 15.0907 0.789884
\(366\) −15.6849 −0.819864
\(367\) −3.23730 −0.168986 −0.0844928 0.996424i \(-0.526927\pi\)
−0.0844928 + 0.996424i \(0.526927\pi\)
\(368\) 5.47650 0.285482
\(369\) 6.41744 0.334079
\(370\) 20.1158 1.04577
\(371\) −10.3647 −0.538111
\(372\) −1.16068 −0.0601785
\(373\) 20.8028 1.07713 0.538564 0.842585i \(-0.318967\pi\)
0.538564 + 0.842585i \(0.318967\pi\)
\(374\) 31.1925 1.61292
\(375\) 1.76796 0.0912973
\(376\) −36.9151 −1.90375
\(377\) 0 0
\(378\) 1.24698 0.0641377
\(379\) 34.5507 1.77475 0.887375 0.461049i \(-0.152527\pi\)
0.887375 + 0.461049i \(0.152527\pi\)
\(380\) 4.59900 0.235924
\(381\) 3.11279 0.159473
\(382\) 21.7908 1.11491
\(383\) 19.7920 1.01132 0.505662 0.862732i \(-0.331248\pi\)
0.505662 + 0.862732i \(0.331248\pi\)
\(384\) 6.16421 0.314566
\(385\) −10.7617 −0.548466
\(386\) −9.66595 −0.491984
\(387\) −10.3388 −0.525552
\(388\) 2.75220 0.139722
\(389\) −31.7604 −1.61032 −0.805159 0.593059i \(-0.797920\pi\)
−0.805159 + 0.593059i \(0.797920\pi\)
\(390\) 0 0
\(391\) −13.4203 −0.678695
\(392\) 3.04892 0.153994
\(393\) 5.35550 0.270149
\(394\) 12.5825 0.633896
\(395\) −10.4931 −0.527966
\(396\) 1.56013 0.0783996
\(397\) 2.45734 0.123330 0.0616651 0.998097i \(-0.480359\pi\)
0.0616651 + 0.998097i \(0.480359\pi\)
\(398\) −27.3829 −1.37258
\(399\) 3.36622 0.168522
\(400\) −12.8823 −0.644115
\(401\) 7.91185 0.395099 0.197549 0.980293i \(-0.436702\pi\)
0.197549 + 0.980293i \(0.436702\pi\)
\(402\) 2.41042 0.120221
\(403\) 0 0
\(404\) −5.17210 −0.257322
\(405\) 3.06987 0.152543
\(406\) 5.48751 0.272340
\(407\) 18.4212 0.913107
\(408\) −21.7558 −1.07707
\(409\) 0.291281 0.0144029 0.00720147 0.999974i \(-0.497708\pi\)
0.00720147 + 0.999974i \(0.497708\pi\)
\(410\) −24.5664 −1.21325
\(411\) −15.6533 −0.772118
\(412\) −5.49416 −0.270678
\(413\) 12.6505 0.622490
\(414\) 2.34527 0.115264
\(415\) 21.8206 1.07113
\(416\) 0 0
\(417\) 7.69765 0.376956
\(418\) −14.7151 −0.719738
\(419\) −25.9101 −1.26579 −0.632895 0.774238i \(-0.718134\pi\)
−0.632895 + 0.774238i \(0.718134\pi\)
\(420\) 1.36622 0.0666647
\(421\) −29.4450 −1.43506 −0.717530 0.696527i \(-0.754727\pi\)
−0.717530 + 0.696527i \(0.754727\pi\)
\(422\) 24.0768 1.17204
\(423\) −12.1076 −0.588693
\(424\) −31.6012 −1.53469
\(425\) 31.5685 1.53130
\(426\) −14.3141 −0.693520
\(427\) −12.5783 −0.608708
\(428\) 4.67908 0.226172
\(429\) 0 0
\(430\) 39.5777 1.90860
\(431\) −13.6974 −0.659778 −0.329889 0.944020i \(-0.607011\pi\)
−0.329889 + 0.944020i \(0.607011\pi\)
\(432\) 2.91185 0.140097
\(433\) 11.0908 0.532989 0.266494 0.963836i \(-0.414135\pi\)
0.266494 + 0.963836i \(0.414135\pi\)
\(434\) 3.25215 0.156108
\(435\) 13.5094 0.647725
\(436\) 0.315001 0.0150858
\(437\) 6.33105 0.302855
\(438\) 6.12985 0.292895
\(439\) −5.41203 −0.258302 −0.129151 0.991625i \(-0.541225\pi\)
−0.129151 + 0.991625i \(0.541225\pi\)
\(440\) −32.8115 −1.56423
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 30.5902 1.45339 0.726693 0.686962i \(-0.241056\pi\)
0.726693 + 0.686962i \(0.241056\pi\)
\(444\) −2.33862 −0.110986
\(445\) −2.24280 −0.106319
\(446\) −23.5758 −1.11635
\(447\) −14.3574 −0.679081
\(448\) 8.89977 0.420475
\(449\) 9.50119 0.448389 0.224195 0.974544i \(-0.428025\pi\)
0.224195 + 0.974544i \(0.428025\pi\)
\(450\) −5.51675 −0.260062
\(451\) −22.4969 −1.05934
\(452\) 2.91243 0.136989
\(453\) −18.2801 −0.858875
\(454\) −11.7349 −0.550744
\(455\) 0 0
\(456\) 10.2633 0.480624
\(457\) −26.5823 −1.24347 −0.621734 0.783229i \(-0.713571\pi\)
−0.621734 + 0.783229i \(0.713571\pi\)
\(458\) 13.4141 0.626800
\(459\) −7.13559 −0.333061
\(460\) 2.56953 0.119805
\(461\) 11.1580 0.519679 0.259839 0.965652i \(-0.416330\pi\)
0.259839 + 0.965652i \(0.416330\pi\)
\(462\) −4.37139 −0.203376
\(463\) 2.23413 0.103829 0.0519144 0.998652i \(-0.483468\pi\)
0.0519144 + 0.998652i \(0.483468\pi\)
\(464\) 12.8140 0.594876
\(465\) 8.00629 0.371283
\(466\) 28.2225 1.30738
\(467\) −19.9539 −0.923355 −0.461677 0.887048i \(-0.652752\pi\)
−0.461677 + 0.887048i \(0.652752\pi\)
\(468\) 0 0
\(469\) 1.93301 0.0892581
\(470\) 46.3487 2.13791
\(471\) −9.14988 −0.421604
\(472\) 38.5703 1.77534
\(473\) 36.2436 1.66648
\(474\) −4.26230 −0.195774
\(475\) −14.8925 −0.683313
\(476\) −3.17563 −0.145555
\(477\) −10.3647 −0.474569
\(478\) 26.9431 1.23235
\(479\) −8.85542 −0.404614 −0.202307 0.979322i \(-0.564844\pi\)
−0.202307 + 0.979322i \(0.564844\pi\)
\(480\) 7.57279 0.345649
\(481\) 0 0
\(482\) −3.63896 −0.165750
\(483\) 1.88076 0.0855775
\(484\) −0.573718 −0.0260781
\(485\) −18.9845 −0.862041
\(486\) 1.24698 0.0565641
\(487\) 16.7789 0.760323 0.380161 0.924920i \(-0.375868\pi\)
0.380161 + 0.924920i \(0.375868\pi\)
\(488\) −38.3503 −1.73604
\(489\) 13.6215 0.615983
\(490\) −3.82806 −0.172934
\(491\) −6.15382 −0.277718 −0.138859 0.990312i \(-0.544343\pi\)
−0.138859 + 0.990312i \(0.544343\pi\)
\(492\) 2.85603 0.128760
\(493\) −31.4011 −1.41424
\(494\) 0 0
\(495\) −10.7617 −0.483702
\(496\) 7.59419 0.340989
\(497\) −11.4790 −0.514904
\(498\) 8.86350 0.397183
\(499\) 32.9818 1.47647 0.738234 0.674544i \(-0.235660\pi\)
0.738234 + 0.674544i \(0.235660\pi\)
\(500\) 0.786818 0.0351876
\(501\) 3.58810 0.160304
\(502\) 23.0078 1.02689
\(503\) 30.6076 1.36473 0.682363 0.731013i \(-0.260952\pi\)
0.682363 + 0.731013i \(0.260952\pi\)
\(504\) 3.04892 0.135810
\(505\) 35.6768 1.58760
\(506\) −8.22154 −0.365492
\(507\) 0 0
\(508\) 1.38532 0.0614638
\(509\) −0.286603 −0.0127035 −0.00635174 0.999980i \(-0.502022\pi\)
−0.00635174 + 0.999980i \(0.502022\pi\)
\(510\) 27.3155 1.20955
\(511\) 4.91575 0.217460
\(512\) 24.9390 1.10216
\(513\) 3.36622 0.148622
\(514\) 19.4631 0.858483
\(515\) 37.8984 1.67000
\(516\) −4.60121 −0.202557
\(517\) 42.4443 1.86670
\(518\) 6.55266 0.287907
\(519\) 5.00471 0.219683
\(520\) 0 0
\(521\) −15.3464 −0.672339 −0.336169 0.941802i \(-0.609131\pi\)
−0.336169 + 0.941802i \(0.609131\pi\)
\(522\) 5.48751 0.240182
\(523\) 24.4508 1.06916 0.534580 0.845118i \(-0.320470\pi\)
0.534580 + 0.845118i \(0.320470\pi\)
\(524\) 2.38342 0.104120
\(525\) −4.42409 −0.193083
\(526\) 20.9587 0.913844
\(527\) −18.6098 −0.810655
\(528\) −10.2078 −0.444236
\(529\) −19.4627 −0.846206
\(530\) 39.6769 1.72345
\(531\) 12.6505 0.548985
\(532\) 1.49811 0.0649513
\(533\) 0 0
\(534\) −0.911026 −0.0394239
\(535\) −32.2760 −1.39541
\(536\) 5.89359 0.254564
\(537\) 6.82831 0.294663
\(538\) 25.3696 1.09376
\(539\) −3.50559 −0.150996
\(540\) 1.36622 0.0587927
\(541\) 8.48224 0.364680 0.182340 0.983236i \(-0.441633\pi\)
0.182340 + 0.983236i \(0.441633\pi\)
\(542\) −6.56280 −0.281896
\(543\) −25.1122 −1.07767
\(544\) −17.6022 −0.754686
\(545\) −2.17286 −0.0930749
\(546\) 0 0
\(547\) 23.2091 0.992352 0.496176 0.868222i \(-0.334737\pi\)
0.496176 + 0.868222i \(0.334737\pi\)
\(548\) −6.96635 −0.297588
\(549\) −12.5783 −0.536830
\(550\) 19.3394 0.824637
\(551\) 14.8135 0.631077
\(552\) 5.73428 0.244067
\(553\) −3.41810 −0.145352
\(554\) 33.7579 1.43424
\(555\) 16.1316 0.684749
\(556\) 3.42578 0.145285
\(557\) 23.1593 0.981292 0.490646 0.871359i \(-0.336761\pi\)
0.490646 + 0.871359i \(0.336761\pi\)
\(558\) 3.25215 0.137675
\(559\) 0 0
\(560\) −8.93901 −0.377742
\(561\) 25.0144 1.05611
\(562\) 28.2519 1.19173
\(563\) −45.3459 −1.91110 −0.955552 0.294824i \(-0.904739\pi\)
−0.955552 + 0.294824i \(0.904739\pi\)
\(564\) −5.38840 −0.226892
\(565\) −20.0897 −0.845181
\(566\) −3.34387 −0.140553
\(567\) 1.00000 0.0419961
\(568\) −34.9986 −1.46851
\(569\) 1.97526 0.0828073 0.0414037 0.999143i \(-0.486817\pi\)
0.0414037 + 0.999143i \(0.486817\pi\)
\(570\) −12.8861 −0.539739
\(571\) 25.0965 1.05025 0.525127 0.851024i \(-0.324018\pi\)
0.525127 + 0.851024i \(0.324018\pi\)
\(572\) 0 0
\(573\) 17.4749 0.730023
\(574\) −8.00242 −0.334014
\(575\) −8.32065 −0.346995
\(576\) 8.89977 0.370824
\(577\) −21.2775 −0.885793 −0.442897 0.896573i \(-0.646049\pi\)
−0.442897 + 0.896573i \(0.646049\pi\)
\(578\) −42.2933 −1.75917
\(579\) −7.75149 −0.322141
\(580\) 6.01224 0.249645
\(581\) 7.10798 0.294889
\(582\) −7.71150 −0.319652
\(583\) 36.3345 1.50482
\(584\) 14.9877 0.620197
\(585\) 0 0
\(586\) 30.9618 1.27902
\(587\) −22.4224 −0.925470 −0.462735 0.886497i \(-0.653132\pi\)
−0.462735 + 0.886497i \(0.653132\pi\)
\(588\) 0.445042 0.0183532
\(589\) 8.77919 0.361740
\(590\) −48.4269 −1.99370
\(591\) 10.0904 0.415062
\(592\) 15.3013 0.628879
\(593\) 11.0541 0.453937 0.226969 0.973902i \(-0.427119\pi\)
0.226969 + 0.973902i \(0.427119\pi\)
\(594\) −4.37139 −0.179360
\(595\) 21.9053 0.898030
\(596\) −6.38964 −0.261730
\(597\) −21.9594 −0.898739
\(598\) 0 0
\(599\) −26.6837 −1.09027 −0.545134 0.838349i \(-0.683521\pi\)
−0.545134 + 0.838349i \(0.683521\pi\)
\(600\) −13.4887 −0.550673
\(601\) 21.0761 0.859714 0.429857 0.902897i \(-0.358564\pi\)
0.429857 + 0.902897i \(0.358564\pi\)
\(602\) 12.8923 0.525451
\(603\) 1.93301 0.0787183
\(604\) −8.13542 −0.331026
\(605\) 3.95747 0.160894
\(606\) 14.4919 0.588694
\(607\) 16.7427 0.679564 0.339782 0.940504i \(-0.389647\pi\)
0.339782 + 0.940504i \(0.389647\pi\)
\(608\) 8.30383 0.336765
\(609\) 4.40064 0.178323
\(610\) 48.1506 1.94956
\(611\) 0 0
\(612\) −3.17563 −0.128367
\(613\) −40.9378 −1.65346 −0.826730 0.562598i \(-0.809802\pi\)
−0.826730 + 0.562598i \(0.809802\pi\)
\(614\) 28.4254 1.14716
\(615\) −19.7007 −0.794409
\(616\) −10.6882 −0.430642
\(617\) −7.98627 −0.321515 −0.160758 0.986994i \(-0.551394\pi\)
−0.160758 + 0.986994i \(0.551394\pi\)
\(618\) 15.3943 0.619250
\(619\) 11.7272 0.471357 0.235679 0.971831i \(-0.424269\pi\)
0.235679 + 0.971831i \(0.424269\pi\)
\(620\) 3.56314 0.143099
\(621\) 1.88076 0.0754723
\(622\) −28.8552 −1.15699
\(623\) −0.730586 −0.0292703
\(624\) 0 0
\(625\) −27.5479 −1.10191
\(626\) 21.8666 0.873967
\(627\) −11.8006 −0.471270
\(628\) −4.07208 −0.162494
\(629\) −37.4963 −1.49507
\(630\) −3.82806 −0.152514
\(631\) 23.2438 0.925320 0.462660 0.886536i \(-0.346895\pi\)
0.462660 + 0.886536i \(0.346895\pi\)
\(632\) −10.4215 −0.414545
\(633\) 19.3081 0.767428
\(634\) −26.3536 −1.04664
\(635\) −9.55586 −0.379213
\(636\) −4.61274 −0.182907
\(637\) 0 0
\(638\) −19.2369 −0.761597
\(639\) −11.4790 −0.454103
\(640\) −18.9233 −0.748010
\(641\) −2.88394 −0.113909 −0.0569544 0.998377i \(-0.518139\pi\)
−0.0569544 + 0.998377i \(0.518139\pi\)
\(642\) −13.1105 −0.517430
\(643\) 2.36239 0.0931636 0.0465818 0.998914i \(-0.485167\pi\)
0.0465818 + 0.998914i \(0.485167\pi\)
\(644\) 0.837017 0.0329831
\(645\) 31.7388 1.24972
\(646\) 29.9524 1.17846
\(647\) 7.21370 0.283600 0.141800 0.989895i \(-0.454711\pi\)
0.141800 + 0.989895i \(0.454711\pi\)
\(648\) 3.04892 0.119773
\(649\) −44.3474 −1.74079
\(650\) 0 0
\(651\) 2.60803 0.102217
\(652\) 6.06212 0.237411
\(653\) −9.44736 −0.369704 −0.184852 0.982766i \(-0.559181\pi\)
−0.184852 + 0.982766i \(0.559181\pi\)
\(654\) −0.882613 −0.0345129
\(655\) −16.4407 −0.642391
\(656\) −18.6866 −0.729591
\(657\) 4.91575 0.191782
\(658\) 15.0980 0.588580
\(659\) −25.1033 −0.977884 −0.488942 0.872316i \(-0.662617\pi\)
−0.488942 + 0.872316i \(0.662617\pi\)
\(660\) −4.78940 −0.186427
\(661\) 30.8959 1.20171 0.600855 0.799358i \(-0.294827\pi\)
0.600855 + 0.799358i \(0.294827\pi\)
\(662\) 24.4973 0.952113
\(663\) 0 0
\(664\) 21.6716 0.841023
\(665\) −10.3339 −0.400730
\(666\) 6.55266 0.253911
\(667\) 8.27654 0.320469
\(668\) 1.59685 0.0617841
\(669\) −18.9063 −0.730961
\(670\) −7.39969 −0.285875
\(671\) 44.0944 1.70225
\(672\) 2.46681 0.0951593
\(673\) 5.84117 0.225161 0.112580 0.993643i \(-0.464088\pi\)
0.112580 + 0.993643i \(0.464088\pi\)
\(674\) −0.732515 −0.0282154
\(675\) −4.42409 −0.170283
\(676\) 0 0
\(677\) 17.7932 0.683848 0.341924 0.939728i \(-0.388921\pi\)
0.341924 + 0.939728i \(0.388921\pi\)
\(678\) −8.16044 −0.313400
\(679\) −6.18414 −0.237325
\(680\) 66.7875 2.56118
\(681\) −9.41062 −0.360616
\(682\) −11.4007 −0.436556
\(683\) −26.0580 −0.997081 −0.498541 0.866866i \(-0.666131\pi\)
−0.498541 + 0.866866i \(0.666131\pi\)
\(684\) 1.49811 0.0572816
\(685\) 48.0534 1.83603
\(686\) −1.24698 −0.0476099
\(687\) 10.7573 0.410416
\(688\) 30.1051 1.14775
\(689\) 0 0
\(690\) −7.19967 −0.274087
\(691\) 30.3760 1.15556 0.577779 0.816193i \(-0.303920\pi\)
0.577779 + 0.816193i \(0.303920\pi\)
\(692\) 2.22731 0.0846695
\(693\) −3.50559 −0.133166
\(694\) 20.4798 0.777404
\(695\) −23.6308 −0.896367
\(696\) 13.4172 0.508577
\(697\) 45.7922 1.73450
\(698\) −43.0415 −1.62914
\(699\) 22.6327 0.856046
\(700\) −1.96891 −0.0744176
\(701\) −11.1900 −0.422641 −0.211321 0.977417i \(-0.567776\pi\)
−0.211321 + 0.977417i \(0.567776\pi\)
\(702\) 0 0
\(703\) 17.6889 0.667150
\(704\) −31.1989 −1.17585
\(705\) 37.1688 1.39986
\(706\) −36.4357 −1.37127
\(707\) 11.6216 0.437076
\(708\) 5.63000 0.211588
\(709\) −48.9630 −1.83884 −0.919422 0.393273i \(-0.871343\pi\)
−0.919422 + 0.393273i \(0.871343\pi\)
\(710\) 43.9424 1.64913
\(711\) −3.41810 −0.128189
\(712\) −2.22750 −0.0834790
\(713\) 4.90507 0.183696
\(714\) 8.89793 0.332997
\(715\) 0 0
\(716\) 3.03888 0.113568
\(717\) 21.6067 0.806916
\(718\) 10.4385 0.389563
\(719\) −13.9878 −0.521655 −0.260828 0.965385i \(-0.583995\pi\)
−0.260828 + 0.965385i \(0.583995\pi\)
\(720\) −8.93901 −0.333137
\(721\) 12.3453 0.459762
\(722\) 9.56254 0.355881
\(723\) −2.91822 −0.108530
\(724\) −11.1760 −0.415353
\(725\) −19.4688 −0.723054
\(726\) 1.60752 0.0596608
\(727\) −10.7215 −0.397640 −0.198820 0.980036i \(-0.563711\pi\)
−0.198820 + 0.980036i \(0.563711\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −18.8178 −0.696479
\(731\) −73.7736 −2.72861
\(732\) −5.59788 −0.206904
\(733\) −16.6709 −0.615753 −0.307877 0.951426i \(-0.599618\pi\)
−0.307877 + 0.951426i \(0.599618\pi\)
\(734\) 4.03684 0.149003
\(735\) −3.06987 −0.113234
\(736\) 4.63948 0.171014
\(737\) −6.77633 −0.249609
\(738\) −8.00242 −0.294573
\(739\) −40.7050 −1.49736 −0.748678 0.662934i \(-0.769311\pi\)
−0.748678 + 0.662934i \(0.769311\pi\)
\(740\) 7.17925 0.263914
\(741\) 0 0
\(742\) 12.9246 0.474478
\(743\) 47.9898 1.76057 0.880287 0.474442i \(-0.157350\pi\)
0.880287 + 0.474442i \(0.157350\pi\)
\(744\) 7.95165 0.291522
\(745\) 44.0753 1.61479
\(746\) −25.9407 −0.949755
\(747\) 7.10798 0.260067
\(748\) 11.1325 0.407043
\(749\) −10.5138 −0.384166
\(750\) −2.20462 −0.0805012
\(751\) −8.38503 −0.305974 −0.152987 0.988228i \(-0.548889\pi\)
−0.152987 + 0.988228i \(0.548889\pi\)
\(752\) 35.2556 1.28564
\(753\) 18.4508 0.672384
\(754\) 0 0
\(755\) 56.1176 2.04233
\(756\) 0.445042 0.0161860
\(757\) −25.8828 −0.940727 −0.470363 0.882473i \(-0.655877\pi\)
−0.470363 + 0.882473i \(0.655877\pi\)
\(758\) −43.0840 −1.56488
\(759\) −6.59317 −0.239317
\(760\) −31.5071 −1.14288
\(761\) 31.1680 1.12984 0.564920 0.825146i \(-0.308907\pi\)
0.564920 + 0.825146i \(0.308907\pi\)
\(762\) −3.88159 −0.140615
\(763\) −0.707801 −0.0256241
\(764\) 7.77705 0.281364
\(765\) 21.9053 0.791988
\(766\) −24.6802 −0.891732
\(767\) 0 0
\(768\) 10.1129 0.364918
\(769\) 6.07920 0.219222 0.109611 0.993975i \(-0.465040\pi\)
0.109611 + 0.993975i \(0.465040\pi\)
\(770\) 13.4196 0.483609
\(771\) 15.6082 0.562117
\(772\) −3.44974 −0.124159
\(773\) −18.9735 −0.682428 −0.341214 0.939986i \(-0.610838\pi\)
−0.341214 + 0.939986i \(0.610838\pi\)
\(774\) 12.8923 0.463404
\(775\) −11.5381 −0.414462
\(776\) −18.8549 −0.676853
\(777\) 5.25483 0.188516
\(778\) 39.6046 1.41989
\(779\) −21.6025 −0.773990
\(780\) 0 0
\(781\) 40.2407 1.43992
\(782\) 16.7349 0.598438
\(783\) 4.40064 0.157266
\(784\) −2.91185 −0.103995
\(785\) 28.0889 1.00254
\(786\) −6.67820 −0.238204
\(787\) 11.5758 0.412634 0.206317 0.978485i \(-0.433852\pi\)
0.206317 + 0.978485i \(0.433852\pi\)
\(788\) 4.49063 0.159972
\(789\) 16.8076 0.598366
\(790\) 13.0847 0.465533
\(791\) −6.54416 −0.232684
\(792\) −10.6882 −0.379790
\(793\) 0 0
\(794\) −3.06425 −0.108746
\(795\) 31.8184 1.12848
\(796\) −9.77286 −0.346390
\(797\) 24.5622 0.870039 0.435020 0.900421i \(-0.356741\pi\)
0.435020 + 0.900421i \(0.356741\pi\)
\(798\) −4.19761 −0.148594
\(799\) −86.3950 −3.05643
\(800\) −10.9134 −0.385847
\(801\) −0.730586 −0.0258140
\(802\) −9.86591 −0.348378
\(803\) −17.2326 −0.608125
\(804\) 0.860271 0.0303394
\(805\) −5.77368 −0.203496
\(806\) 0 0
\(807\) 20.3448 0.716172
\(808\) 35.4333 1.24654
\(809\) −1.70053 −0.0597876 −0.0298938 0.999553i \(-0.509517\pi\)
−0.0298938 + 0.999553i \(0.509517\pi\)
\(810\) −3.82806 −0.134504
\(811\) −8.85787 −0.311042 −0.155521 0.987833i \(-0.549706\pi\)
−0.155521 + 0.987833i \(0.549706\pi\)
\(812\) 1.95847 0.0687288
\(813\) −5.26295 −0.184580
\(814\) −22.9709 −0.805130
\(815\) −41.8161 −1.46475
\(816\) 20.7778 0.727368
\(817\) 34.8028 1.21759
\(818\) −0.363222 −0.0126998
\(819\) 0 0
\(820\) −8.76763 −0.306179
\(821\) 34.4523 1.20239 0.601197 0.799101i \(-0.294691\pi\)
0.601197 + 0.799101i \(0.294691\pi\)
\(822\) 19.5193 0.680813
\(823\) 38.6660 1.34781 0.673906 0.738817i \(-0.264615\pi\)
0.673906 + 0.738817i \(0.264615\pi\)
\(824\) 37.6397 1.31124
\(825\) 15.5090 0.539955
\(826\) −15.7749 −0.548879
\(827\) 18.0478 0.627582 0.313791 0.949492i \(-0.398401\pi\)
0.313791 + 0.949492i \(0.398401\pi\)
\(828\) 0.837017 0.0290883
\(829\) −42.3679 −1.47150 −0.735749 0.677254i \(-0.763170\pi\)
−0.735749 + 0.677254i \(0.763170\pi\)
\(830\) −27.2098 −0.944466
\(831\) 27.0717 0.939108
\(832\) 0 0
\(833\) 7.13559 0.247233
\(834\) −9.59882 −0.332380
\(835\) −11.0150 −0.381189
\(836\) −5.25175 −0.181636
\(837\) 2.60803 0.0901465
\(838\) 32.3093 1.11611
\(839\) 16.0041 0.552523 0.276262 0.961082i \(-0.410904\pi\)
0.276262 + 0.961082i \(0.410904\pi\)
\(840\) −9.35977 −0.322943
\(841\) −9.63438 −0.332220
\(842\) 36.7173 1.26536
\(843\) 22.6562 0.780323
\(844\) 8.59291 0.295780
\(845\) 0 0
\(846\) 15.0980 0.519078
\(847\) 1.28913 0.0442952
\(848\) 30.1806 1.03641
\(849\) −2.68158 −0.0920315
\(850\) −39.3653 −1.35022
\(851\) 9.88307 0.338787
\(852\) −5.10864 −0.175019
\(853\) −45.4092 −1.55478 −0.777391 0.629017i \(-0.783457\pi\)
−0.777391 + 0.629017i \(0.783457\pi\)
\(854\) 15.6849 0.536727
\(855\) −10.3339 −0.353410
\(856\) −32.0557 −1.09564
\(857\) 4.77756 0.163198 0.0815991 0.996665i \(-0.473997\pi\)
0.0815991 + 0.996665i \(0.473997\pi\)
\(858\) 0 0
\(859\) 0.175468 0.00598688 0.00299344 0.999996i \(-0.499047\pi\)
0.00299344 + 0.999996i \(0.499047\pi\)
\(860\) 14.1251 0.481662
\(861\) −6.41744 −0.218706
\(862\) 17.0803 0.581758
\(863\) −23.4266 −0.797450 −0.398725 0.917071i \(-0.630547\pi\)
−0.398725 + 0.917071i \(0.630547\pi\)
\(864\) 2.46681 0.0839227
\(865\) −15.3638 −0.522385
\(866\) −13.8300 −0.469962
\(867\) −33.9166 −1.15187
\(868\) 1.16068 0.0393961
\(869\) 11.9824 0.406477
\(870\) −16.8459 −0.571130
\(871\) 0 0
\(872\) −2.15803 −0.0730800
\(873\) −6.18414 −0.209301
\(874\) −7.89469 −0.267042
\(875\) −1.76796 −0.0597681
\(876\) 2.18772 0.0739161
\(877\) −42.1109 −1.42199 −0.710993 0.703200i \(-0.751754\pi\)
−0.710993 + 0.703200i \(0.751754\pi\)
\(878\) 6.74868 0.227757
\(879\) 24.8294 0.837477
\(880\) 31.3365 1.05635
\(881\) 38.1193 1.28427 0.642137 0.766590i \(-0.278048\pi\)
0.642137 + 0.766590i \(0.278048\pi\)
\(882\) −1.24698 −0.0419880
\(883\) 53.4646 1.79923 0.899614 0.436686i \(-0.143848\pi\)
0.899614 + 0.436686i \(0.143848\pi\)
\(884\) 0 0
\(885\) −38.8353 −1.30544
\(886\) −38.1454 −1.28152
\(887\) 16.2289 0.544913 0.272457 0.962168i \(-0.412164\pi\)
0.272457 + 0.962168i \(0.412164\pi\)
\(888\) 16.0215 0.537647
\(889\) −3.11279 −0.104400
\(890\) 2.79673 0.0937466
\(891\) −3.50559 −0.117442
\(892\) −8.41411 −0.281725
\(893\) 40.7569 1.36388
\(894\) 17.9034 0.598778
\(895\) −20.9620 −0.700683
\(896\) −6.16421 −0.205932
\(897\) 0 0
\(898\) −11.8478 −0.395366
\(899\) 11.4770 0.382779
\(900\) −1.96891 −0.0656302
\(901\) −73.9585 −2.46392
\(902\) 28.0532 0.934068
\(903\) 10.3388 0.344055
\(904\) −19.9526 −0.663614
\(905\) 77.0913 2.56260
\(906\) 22.7949 0.757311
\(907\) −38.3992 −1.27503 −0.637513 0.770440i \(-0.720037\pi\)
−0.637513 + 0.770440i \(0.720037\pi\)
\(908\) −4.18812 −0.138988
\(909\) 11.6216 0.385465
\(910\) 0 0
\(911\) 27.7816 0.920447 0.460223 0.887803i \(-0.347769\pi\)
0.460223 + 0.887803i \(0.347769\pi\)
\(912\) −9.80194 −0.324575
\(913\) −24.9176 −0.824653
\(914\) 33.1476 1.09642
\(915\) 38.6138 1.27653
\(916\) 4.78744 0.158182
\(917\) −5.35550 −0.176854
\(918\) 8.89793 0.293675
\(919\) −2.60876 −0.0860550 −0.0430275 0.999074i \(-0.513700\pi\)
−0.0430275 + 0.999074i \(0.513700\pi\)
\(920\) −17.6035 −0.580370
\(921\) 22.7954 0.751134
\(922\) −13.9138 −0.458226
\(923\) 0 0
\(924\) −1.56013 −0.0513246
\(925\) −23.2478 −0.764384
\(926\) −2.78591 −0.0915508
\(927\) 12.3453 0.405472
\(928\) 10.8555 0.356351
\(929\) 1.72935 0.0567381 0.0283690 0.999598i \(-0.490969\pi\)
0.0283690 + 0.999598i \(0.490969\pi\)
\(930\) −9.98368 −0.327378
\(931\) −3.36622 −0.110323
\(932\) 10.0725 0.329935
\(933\) −23.1401 −0.757573
\(934\) 24.8821 0.814166
\(935\) −76.7910 −2.51133
\(936\) 0 0
\(937\) −10.0417 −0.328047 −0.164023 0.986456i \(-0.552447\pi\)
−0.164023 + 0.986456i \(0.552447\pi\)
\(938\) −2.41042 −0.0787032
\(939\) 17.5357 0.572255
\(940\) 16.5417 0.539530
\(941\) −41.5696 −1.35513 −0.677565 0.735463i \(-0.736965\pi\)
−0.677565 + 0.735463i \(0.736965\pi\)
\(942\) 11.4097 0.371749
\(943\) −12.0697 −0.393042
\(944\) −36.8364 −1.19892
\(945\) −3.06987 −0.0998628
\(946\) −45.1951 −1.46942
\(947\) 3.83868 0.124740 0.0623701 0.998053i \(-0.480134\pi\)
0.0623701 + 0.998053i \(0.480134\pi\)
\(948\) −1.52120 −0.0494062
\(949\) 0 0
\(950\) 18.5706 0.602510
\(951\) −21.1340 −0.685315
\(952\) 21.7558 0.705110
\(953\) 53.5364 1.73421 0.867107 0.498121i \(-0.165977\pi\)
0.867107 + 0.498121i \(0.165977\pi\)
\(954\) 12.9246 0.418450
\(955\) −53.6455 −1.73593
\(956\) 9.61587 0.311000
\(957\) −15.4268 −0.498678
\(958\) 11.0425 0.356768
\(959\) 15.6533 0.505470
\(960\) −27.3211 −0.881786
\(961\) −24.1982 −0.780587
\(962\) 0 0
\(963\) −10.5138 −0.338802
\(964\) −1.29873 −0.0418293
\(965\) 23.7961 0.766022
\(966\) −2.34527 −0.0754578
\(967\) 34.8048 1.11925 0.559624 0.828747i \(-0.310946\pi\)
0.559624 + 0.828747i \(0.310946\pi\)
\(968\) 3.93046 0.126330
\(969\) 24.0200 0.771632
\(970\) 23.6733 0.760103
\(971\) 0.107800 0.00345946 0.00172973 0.999999i \(-0.499449\pi\)
0.00172973 + 0.999999i \(0.499449\pi\)
\(972\) 0.445042 0.0142747
\(973\) −7.69765 −0.246775
\(974\) −20.9229 −0.670413
\(975\) 0 0
\(976\) 36.6262 1.17238
\(977\) 18.3161 0.585985 0.292992 0.956115i \(-0.405349\pi\)
0.292992 + 0.956115i \(0.405349\pi\)
\(978\) −16.9857 −0.543142
\(979\) 2.56113 0.0818541
\(980\) −1.36622 −0.0436423
\(981\) −0.707801 −0.0225983
\(982\) 7.67369 0.244877
\(983\) 25.6401 0.817791 0.408896 0.912581i \(-0.365914\pi\)
0.408896 + 0.912581i \(0.365914\pi\)
\(984\) −19.5662 −0.623749
\(985\) −30.9761 −0.986980
\(986\) 39.1566 1.24700
\(987\) 12.1076 0.385390
\(988\) 0 0
\(989\) 19.4448 0.618310
\(990\) 13.4196 0.426503
\(991\) 38.2347 1.21457 0.607283 0.794486i \(-0.292260\pi\)
0.607283 + 0.794486i \(0.292260\pi\)
\(992\) 6.43351 0.204264
\(993\) 19.6453 0.623424
\(994\) 14.3141 0.454016
\(995\) 67.4125 2.13712
\(996\) 3.16335 0.100235
\(997\) 45.5428 1.44235 0.721177 0.692751i \(-0.243602\pi\)
0.721177 + 0.692751i \(0.243602\pi\)
\(998\) −41.1276 −1.30187
\(999\) 5.25483 0.166255
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3549.2.a.z.1.2 yes 6
13.12 even 2 3549.2.a.y.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3549.2.a.y.1.5 6 13.12 even 2
3549.2.a.z.1.2 yes 6 1.1 even 1 trivial