Properties

Label 4235.2.a.bp.1.3
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $0$
Dimension $18$
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] = [4235,2,Mod(1,4235)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, 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("4235.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.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: \(33.8166452560\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} - 28 x^{16} + 54 x^{15} + 317 x^{14} - 580 x^{13} - 1874 x^{12} + 3158 x^{11} + 6290 x^{10} - 9228 x^{9} - 12411 x^{8} + 14224 x^{7} + 14618 x^{6} - 10744 x^{5} + \cdots - 176 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.35862\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.35862 q^{2} +3.12567 q^{3} +3.56309 q^{4} +1.00000 q^{5} -7.37227 q^{6} +1.00000 q^{7} -3.68672 q^{8} +6.76983 q^{9} +O(q^{10})\) \(q-2.35862 q^{2} +3.12567 q^{3} +3.56309 q^{4} +1.00000 q^{5} -7.37227 q^{6} +1.00000 q^{7} -3.68672 q^{8} +6.76983 q^{9} -2.35862 q^{10} +11.1370 q^{12} -0.303615 q^{13} -2.35862 q^{14} +3.12567 q^{15} +1.56941 q^{16} +1.03365 q^{17} -15.9675 q^{18} +5.12290 q^{19} +3.56309 q^{20} +3.12567 q^{21} +8.52137 q^{23} -11.5235 q^{24} +1.00000 q^{25} +0.716112 q^{26} +11.7833 q^{27} +3.56309 q^{28} -4.31946 q^{29} -7.37227 q^{30} -2.64939 q^{31} +3.67181 q^{32} -2.43800 q^{34} +1.00000 q^{35} +24.1215 q^{36} -2.27545 q^{37} -12.0830 q^{38} -0.949001 q^{39} -3.68672 q^{40} +9.65845 q^{41} -7.37227 q^{42} -1.37993 q^{43} +6.76983 q^{45} -20.0987 q^{46} +1.07602 q^{47} +4.90546 q^{48} +1.00000 q^{49} -2.35862 q^{50} +3.23087 q^{51} -1.08181 q^{52} -9.21775 q^{53} -27.7923 q^{54} -3.68672 q^{56} +16.0125 q^{57} +10.1880 q^{58} -8.81202 q^{59} +11.1370 q^{60} +10.5463 q^{61} +6.24889 q^{62} +6.76983 q^{63} -11.7992 q^{64} -0.303615 q^{65} +13.6999 q^{67} +3.68300 q^{68} +26.6350 q^{69} -2.35862 q^{70} -7.14373 q^{71} -24.9585 q^{72} -16.1243 q^{73} +5.36691 q^{74} +3.12567 q^{75} +18.2533 q^{76} +2.23833 q^{78} +0.573402 q^{79} +1.56941 q^{80} +16.5212 q^{81} -22.7806 q^{82} -11.2134 q^{83} +11.1370 q^{84} +1.03365 q^{85} +3.25473 q^{86} -13.5012 q^{87} -14.3947 q^{89} -15.9675 q^{90} -0.303615 q^{91} +30.3624 q^{92} -8.28111 q^{93} -2.53791 q^{94} +5.12290 q^{95} +11.4769 q^{96} -3.94258 q^{97} -2.35862 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{2} + 5 q^{3} + 24 q^{4} + 18 q^{5} - q^{6} + 18 q^{7} + 6 q^{8} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 2 q^{2} + 5 q^{3} + 24 q^{4} + 18 q^{5} - q^{6} + 18 q^{7} + 6 q^{8} + 37 q^{9} + 2 q^{10} + 15 q^{12} + 8 q^{13} + 2 q^{14} + 5 q^{15} + 44 q^{16} - 5 q^{17} + 2 q^{18} + 15 q^{19} + 24 q^{20} + 5 q^{21} + 4 q^{23} + 8 q^{24} + 18 q^{25} + 14 q^{26} + 20 q^{27} + 24 q^{28} - 6 q^{29} - q^{30} + 22 q^{31} - 6 q^{32} + 44 q^{34} + 18 q^{35} + 83 q^{36} + 26 q^{37} - 11 q^{38} - 38 q^{39} + 6 q^{40} + 7 q^{41} - q^{42} + 10 q^{43} + 37 q^{45} - 40 q^{46} - q^{47} - 15 q^{48} + 18 q^{49} + 2 q^{50} - 11 q^{51} + 18 q^{52} + 23 q^{53} + 13 q^{54} + 6 q^{56} - 16 q^{57} + 2 q^{58} + 30 q^{59} + 15 q^{60} + 17 q^{61} + 57 q^{62} + 37 q^{63} + 64 q^{64} + 8 q^{65} + 29 q^{67} - 66 q^{68} + 54 q^{69} + 2 q^{70} - 2 q^{71} - 77 q^{72} + 3 q^{73} - 48 q^{74} + 5 q^{75} + 47 q^{76} + 10 q^{78} - 18 q^{79} + 44 q^{80} + 110 q^{81} + 56 q^{82} + 9 q^{83} + 15 q^{84} - 5 q^{85} + 25 q^{86} + 23 q^{87} + 59 q^{89} + 2 q^{90} + 8 q^{91} - 74 q^{92} + 33 q^{93} - 19 q^{94} + 15 q^{95} + 14 q^{96} + 30 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.35862 −1.66780 −0.833898 0.551919i \(-0.813896\pi\)
−0.833898 + 0.551919i \(0.813896\pi\)
\(3\) 3.12567 1.80461 0.902304 0.431100i \(-0.141874\pi\)
0.902304 + 0.431100i \(0.141874\pi\)
\(4\) 3.56309 1.78154
\(5\) 1.00000 0.447214
\(6\) −7.37227 −3.00972
\(7\) 1.00000 0.377964
\(8\) −3.68672 −1.30345
\(9\) 6.76983 2.25661
\(10\) −2.35862 −0.745861
\(11\) 0 0
\(12\) 11.1370 3.21499
\(13\) −0.303615 −0.0842076 −0.0421038 0.999113i \(-0.513406\pi\)
−0.0421038 + 0.999113i \(0.513406\pi\)
\(14\) −2.35862 −0.630368
\(15\) 3.12567 0.807045
\(16\) 1.56941 0.392352
\(17\) 1.03365 0.250698 0.125349 0.992113i \(-0.459995\pi\)
0.125349 + 0.992113i \(0.459995\pi\)
\(18\) −15.9675 −3.76357
\(19\) 5.12290 1.17527 0.587636 0.809125i \(-0.300059\pi\)
0.587636 + 0.809125i \(0.300059\pi\)
\(20\) 3.56309 0.796730
\(21\) 3.12567 0.682078
\(22\) 0 0
\(23\) 8.52137 1.77683 0.888414 0.459043i \(-0.151808\pi\)
0.888414 + 0.459043i \(0.151808\pi\)
\(24\) −11.5235 −2.35222
\(25\) 1.00000 0.200000
\(26\) 0.716112 0.140441
\(27\) 11.7833 2.26769
\(28\) 3.56309 0.673360
\(29\) −4.31946 −0.802103 −0.401052 0.916055i \(-0.631355\pi\)
−0.401052 + 0.916055i \(0.631355\pi\)
\(30\) −7.37227 −1.34599
\(31\) −2.64939 −0.475844 −0.237922 0.971284i \(-0.576466\pi\)
−0.237922 + 0.971284i \(0.576466\pi\)
\(32\) 3.67181 0.649091
\(33\) 0 0
\(34\) −2.43800 −0.418113
\(35\) 1.00000 0.169031
\(36\) 24.1215 4.02025
\(37\) −2.27545 −0.374081 −0.187041 0.982352i \(-0.559890\pi\)
−0.187041 + 0.982352i \(0.559890\pi\)
\(38\) −12.0830 −1.96011
\(39\) −0.949001 −0.151962
\(40\) −3.68672 −0.582922
\(41\) 9.65845 1.50840 0.754198 0.656647i \(-0.228026\pi\)
0.754198 + 0.656647i \(0.228026\pi\)
\(42\) −7.37227 −1.13757
\(43\) −1.37993 −0.210438 −0.105219 0.994449i \(-0.533554\pi\)
−0.105219 + 0.994449i \(0.533554\pi\)
\(44\) 0 0
\(45\) 6.76983 1.00919
\(46\) −20.0987 −2.96339
\(47\) 1.07602 0.156953 0.0784765 0.996916i \(-0.474994\pi\)
0.0784765 + 0.996916i \(0.474994\pi\)
\(48\) 4.90546 0.708042
\(49\) 1.00000 0.142857
\(50\) −2.35862 −0.333559
\(51\) 3.23087 0.452412
\(52\) −1.08181 −0.150019
\(53\) −9.21775 −1.26616 −0.633078 0.774088i \(-0.718209\pi\)
−0.633078 + 0.774088i \(0.718209\pi\)
\(54\) −27.7923 −3.78205
\(55\) 0 0
\(56\) −3.68672 −0.492659
\(57\) 16.0125 2.12091
\(58\) 10.1880 1.33774
\(59\) −8.81202 −1.14723 −0.573614 0.819126i \(-0.694459\pi\)
−0.573614 + 0.819126i \(0.694459\pi\)
\(60\) 11.1370 1.43779
\(61\) 10.5463 1.35031 0.675155 0.737675i \(-0.264077\pi\)
0.675155 + 0.737675i \(0.264077\pi\)
\(62\) 6.24889 0.793610
\(63\) 6.76983 0.852919
\(64\) −11.7992 −1.47490
\(65\) −0.303615 −0.0376588
\(66\) 0 0
\(67\) 13.6999 1.67371 0.836854 0.547427i \(-0.184392\pi\)
0.836854 + 0.547427i \(0.184392\pi\)
\(68\) 3.68300 0.446629
\(69\) 26.6350 3.20648
\(70\) −2.35862 −0.281909
\(71\) −7.14373 −0.847805 −0.423903 0.905708i \(-0.639340\pi\)
−0.423903 + 0.905708i \(0.639340\pi\)
\(72\) −24.9585 −2.94139
\(73\) −16.1243 −1.88721 −0.943605 0.331075i \(-0.892589\pi\)
−0.943605 + 0.331075i \(0.892589\pi\)
\(74\) 5.36691 0.623891
\(75\) 3.12567 0.360922
\(76\) 18.2533 2.09380
\(77\) 0 0
\(78\) 2.23833 0.253441
\(79\) 0.573402 0.0645127 0.0322564 0.999480i \(-0.489731\pi\)
0.0322564 + 0.999480i \(0.489731\pi\)
\(80\) 1.56941 0.175465
\(81\) 16.5212 1.83568
\(82\) −22.7806 −2.51570
\(83\) −11.2134 −1.23083 −0.615413 0.788205i \(-0.711011\pi\)
−0.615413 + 0.788205i \(0.711011\pi\)
\(84\) 11.1370 1.21515
\(85\) 1.03365 0.112116
\(86\) 3.25473 0.350967
\(87\) −13.5012 −1.44748
\(88\) 0 0
\(89\) −14.3947 −1.52583 −0.762917 0.646496i \(-0.776234\pi\)
−0.762917 + 0.646496i \(0.776234\pi\)
\(90\) −15.9675 −1.68312
\(91\) −0.303615 −0.0318275
\(92\) 30.3624 3.16550
\(93\) −8.28111 −0.858712
\(94\) −2.53791 −0.261766
\(95\) 5.12290 0.525598
\(96\) 11.4769 1.17135
\(97\) −3.94258 −0.400308 −0.200154 0.979764i \(-0.564144\pi\)
−0.200154 + 0.979764i \(0.564144\pi\)
\(98\) −2.35862 −0.238257
\(99\) 0 0
\(100\) 3.56309 0.356309
\(101\) 10.1902 1.01397 0.506983 0.861956i \(-0.330761\pi\)
0.506983 + 0.861956i \(0.330761\pi\)
\(102\) −7.62038 −0.754530
\(103\) 18.4463 1.81757 0.908783 0.417269i \(-0.137013\pi\)
0.908783 + 0.417269i \(0.137013\pi\)
\(104\) 1.11934 0.109761
\(105\) 3.12567 0.305034
\(106\) 21.7412 2.11169
\(107\) −17.9196 −1.73235 −0.866176 0.499738i \(-0.833430\pi\)
−0.866176 + 0.499738i \(0.833430\pi\)
\(108\) 41.9848 4.03999
\(109\) −1.01822 −0.0975276 −0.0487638 0.998810i \(-0.515528\pi\)
−0.0487638 + 0.998810i \(0.515528\pi\)
\(110\) 0 0
\(111\) −7.11230 −0.675070
\(112\) 1.56941 0.148295
\(113\) −1.79756 −0.169100 −0.0845501 0.996419i \(-0.526945\pi\)
−0.0845501 + 0.996419i \(0.526945\pi\)
\(114\) −37.7674 −3.53724
\(115\) 8.52137 0.794622
\(116\) −15.3906 −1.42898
\(117\) −2.05542 −0.190024
\(118\) 20.7842 1.91334
\(119\) 1.03365 0.0947549
\(120\) −11.5235 −1.05195
\(121\) 0 0
\(122\) −24.8746 −2.25204
\(123\) 30.1892 2.72206
\(124\) −9.43999 −0.847736
\(125\) 1.00000 0.0894427
\(126\) −15.9675 −1.42249
\(127\) 11.2516 0.998417 0.499209 0.866482i \(-0.333624\pi\)
0.499209 + 0.866482i \(0.333624\pi\)
\(128\) 20.4863 1.81075
\(129\) −4.31321 −0.379757
\(130\) 0.716112 0.0628072
\(131\) −3.75923 −0.328445 −0.164223 0.986423i \(-0.552512\pi\)
−0.164223 + 0.986423i \(0.552512\pi\)
\(132\) 0 0
\(133\) 5.12290 0.444211
\(134\) −32.3128 −2.79140
\(135\) 11.7833 1.01414
\(136\) −3.81080 −0.326773
\(137\) 15.0871 1.28898 0.644490 0.764613i \(-0.277070\pi\)
0.644490 + 0.764613i \(0.277070\pi\)
\(138\) −62.8219 −5.34775
\(139\) 13.3265 1.13034 0.565171 0.824974i \(-0.308810\pi\)
0.565171 + 0.824974i \(0.308810\pi\)
\(140\) 3.56309 0.301136
\(141\) 3.36327 0.283239
\(142\) 16.8493 1.41397
\(143\) 0 0
\(144\) 10.6246 0.885386
\(145\) −4.31946 −0.358712
\(146\) 38.0311 3.14748
\(147\) 3.12567 0.257801
\(148\) −8.10761 −0.666441
\(149\) −2.12769 −0.174307 −0.0871534 0.996195i \(-0.527777\pi\)
−0.0871534 + 0.996195i \(0.527777\pi\)
\(150\) −7.37227 −0.601944
\(151\) 15.7506 1.28177 0.640883 0.767639i \(-0.278568\pi\)
0.640883 + 0.767639i \(0.278568\pi\)
\(152\) −18.8867 −1.53191
\(153\) 6.99767 0.565728
\(154\) 0 0
\(155\) −2.64939 −0.212804
\(156\) −3.38137 −0.270726
\(157\) −8.52511 −0.680378 −0.340189 0.940357i \(-0.610491\pi\)
−0.340189 + 0.940357i \(0.610491\pi\)
\(158\) −1.35244 −0.107594
\(159\) −28.8117 −2.28491
\(160\) 3.67181 0.290282
\(161\) 8.52137 0.671578
\(162\) −38.9671 −3.06155
\(163\) −12.1811 −0.954097 −0.477049 0.878877i \(-0.658293\pi\)
−0.477049 + 0.878877i \(0.658293\pi\)
\(164\) 34.4139 2.68727
\(165\) 0 0
\(166\) 26.4480 2.05277
\(167\) −8.44915 −0.653815 −0.326908 0.945056i \(-0.606007\pi\)
−0.326908 + 0.945056i \(0.606007\pi\)
\(168\) −11.5235 −0.889057
\(169\) −12.9078 −0.992909
\(170\) −2.43800 −0.186986
\(171\) 34.6812 2.65213
\(172\) −4.91681 −0.374904
\(173\) −8.85874 −0.673517 −0.336759 0.941591i \(-0.609331\pi\)
−0.336759 + 0.941591i \(0.609331\pi\)
\(174\) 31.8442 2.41411
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −27.5435 −2.07030
\(178\) 33.9516 2.54478
\(179\) 8.08279 0.604136 0.302068 0.953286i \(-0.402323\pi\)
0.302068 + 0.953286i \(0.402323\pi\)
\(180\) 24.1215 1.79791
\(181\) −1.35065 −0.100393 −0.0501966 0.998739i \(-0.515985\pi\)
−0.0501966 + 0.998739i \(0.515985\pi\)
\(182\) 0.716112 0.0530818
\(183\) 32.9642 2.43678
\(184\) −31.4159 −2.31601
\(185\) −2.27545 −0.167294
\(186\) 19.5320 1.43216
\(187\) 0 0
\(188\) 3.83394 0.279618
\(189\) 11.7833 0.857107
\(190\) −12.0830 −0.876590
\(191\) 12.2917 0.889396 0.444698 0.895680i \(-0.353311\pi\)
0.444698 + 0.895680i \(0.353311\pi\)
\(192\) −36.8805 −2.66162
\(193\) 18.5658 1.33640 0.668198 0.743983i \(-0.267066\pi\)
0.668198 + 0.743983i \(0.267066\pi\)
\(194\) 9.29904 0.667633
\(195\) −0.949001 −0.0679594
\(196\) 3.56309 0.254506
\(197\) 12.8716 0.917062 0.458531 0.888678i \(-0.348376\pi\)
0.458531 + 0.888678i \(0.348376\pi\)
\(198\) 0 0
\(199\) 9.93245 0.704093 0.352046 0.935983i \(-0.385486\pi\)
0.352046 + 0.935983i \(0.385486\pi\)
\(200\) −3.68672 −0.260691
\(201\) 42.8214 3.02039
\(202\) −24.0349 −1.69109
\(203\) −4.31946 −0.303167
\(204\) 11.5119 0.805991
\(205\) 9.65845 0.674575
\(206\) −43.5078 −3.03133
\(207\) 57.6882 4.00961
\(208\) −0.476496 −0.0330390
\(209\) 0 0
\(210\) −7.37227 −0.508735
\(211\) 17.1478 1.18050 0.590252 0.807219i \(-0.299028\pi\)
0.590252 + 0.807219i \(0.299028\pi\)
\(212\) −32.8436 −2.25571
\(213\) −22.3290 −1.52996
\(214\) 42.2655 2.88921
\(215\) −1.37993 −0.0941105
\(216\) −43.4417 −2.95583
\(217\) −2.64939 −0.179852
\(218\) 2.40159 0.162656
\(219\) −50.3994 −3.40567
\(220\) 0 0
\(221\) −0.313833 −0.0211107
\(222\) 16.7752 1.12588
\(223\) 15.3328 1.02676 0.513380 0.858161i \(-0.328393\pi\)
0.513380 + 0.858161i \(0.328393\pi\)
\(224\) 3.67181 0.245333
\(225\) 6.76983 0.451322
\(226\) 4.23976 0.282025
\(227\) 2.01861 0.133980 0.0669899 0.997754i \(-0.478660\pi\)
0.0669899 + 0.997754i \(0.478660\pi\)
\(228\) 57.0539 3.77849
\(229\) −0.312306 −0.0206377 −0.0103189 0.999947i \(-0.503285\pi\)
−0.0103189 + 0.999947i \(0.503285\pi\)
\(230\) −20.0987 −1.32527
\(231\) 0 0
\(232\) 15.9247 1.04550
\(233\) −17.3822 −1.13874 −0.569371 0.822080i \(-0.692813\pi\)
−0.569371 + 0.822080i \(0.692813\pi\)
\(234\) 4.84796 0.316921
\(235\) 1.07602 0.0701915
\(236\) −31.3980 −2.04383
\(237\) 1.79227 0.116420
\(238\) −2.43800 −0.158032
\(239\) −22.6580 −1.46562 −0.732812 0.680431i \(-0.761793\pi\)
−0.732812 + 0.680431i \(0.761793\pi\)
\(240\) 4.90546 0.316646
\(241\) −6.28549 −0.404884 −0.202442 0.979294i \(-0.564888\pi\)
−0.202442 + 0.979294i \(0.564888\pi\)
\(242\) 0 0
\(243\) 16.2899 1.04500
\(244\) 37.5772 2.40564
\(245\) 1.00000 0.0638877
\(246\) −71.2047 −4.53985
\(247\) −1.55539 −0.0989669
\(248\) 9.76755 0.620240
\(249\) −35.0493 −2.22116
\(250\) −2.35862 −0.149172
\(251\) −10.6760 −0.673861 −0.336931 0.941529i \(-0.609389\pi\)
−0.336931 + 0.941529i \(0.609389\pi\)
\(252\) 24.1215 1.51951
\(253\) 0 0
\(254\) −26.5382 −1.66516
\(255\) 3.23087 0.202325
\(256\) −24.7208 −1.54505
\(257\) −17.6734 −1.10244 −0.551220 0.834360i \(-0.685837\pi\)
−0.551220 + 0.834360i \(0.685837\pi\)
\(258\) 10.1732 0.633358
\(259\) −2.27545 −0.141389
\(260\) −1.08181 −0.0670908
\(261\) −29.2420 −1.81004
\(262\) 8.86659 0.547780
\(263\) −12.2684 −0.756499 −0.378250 0.925704i \(-0.623474\pi\)
−0.378250 + 0.925704i \(0.623474\pi\)
\(264\) 0 0
\(265\) −9.21775 −0.566242
\(266\) −12.0830 −0.740854
\(267\) −44.9931 −2.75353
\(268\) 48.8139 2.98178
\(269\) 21.5065 1.31128 0.655639 0.755075i \(-0.272399\pi\)
0.655639 + 0.755075i \(0.272399\pi\)
\(270\) −27.7923 −1.69138
\(271\) 4.38256 0.266221 0.133111 0.991101i \(-0.457503\pi\)
0.133111 + 0.991101i \(0.457503\pi\)
\(272\) 1.62223 0.0983619
\(273\) −0.949001 −0.0574362
\(274\) −35.5848 −2.14976
\(275\) 0 0
\(276\) 94.9028 5.71248
\(277\) −11.4465 −0.687756 −0.343878 0.939014i \(-0.611741\pi\)
−0.343878 + 0.939014i \(0.611741\pi\)
\(278\) −31.4322 −1.88518
\(279\) −17.9359 −1.07379
\(280\) −3.68672 −0.220324
\(281\) −11.4639 −0.683879 −0.341940 0.939722i \(-0.611084\pi\)
−0.341940 + 0.939722i \(0.611084\pi\)
\(282\) −7.93268 −0.472384
\(283\) 12.6804 0.753772 0.376886 0.926260i \(-0.376995\pi\)
0.376886 + 0.926260i \(0.376995\pi\)
\(284\) −25.4537 −1.51040
\(285\) 16.0125 0.948498
\(286\) 0 0
\(287\) 9.65845 0.570120
\(288\) 24.8576 1.46475
\(289\) −15.9316 −0.937151
\(290\) 10.1880 0.598258
\(291\) −12.3232 −0.722400
\(292\) −57.4523 −3.36214
\(293\) −8.25289 −0.482139 −0.241070 0.970508i \(-0.577498\pi\)
−0.241070 + 0.970508i \(0.577498\pi\)
\(294\) −7.37227 −0.429960
\(295\) −8.81202 −0.513056
\(296\) 8.38894 0.487597
\(297\) 0 0
\(298\) 5.01840 0.290708
\(299\) −2.58721 −0.149622
\(300\) 11.1370 0.642997
\(301\) −1.37993 −0.0795379
\(302\) −37.1497 −2.13772
\(303\) 31.8513 1.82981
\(304\) 8.03991 0.461121
\(305\) 10.5463 0.603877
\(306\) −16.5048 −0.943519
\(307\) −1.76640 −0.100814 −0.0504070 0.998729i \(-0.516052\pi\)
−0.0504070 + 0.998729i \(0.516052\pi\)
\(308\) 0 0
\(309\) 57.6570 3.27999
\(310\) 6.24889 0.354913
\(311\) −4.51683 −0.256126 −0.128063 0.991766i \(-0.540876\pi\)
−0.128063 + 0.991766i \(0.540876\pi\)
\(312\) 3.49871 0.198075
\(313\) −6.29171 −0.355628 −0.177814 0.984064i \(-0.556903\pi\)
−0.177814 + 0.984064i \(0.556903\pi\)
\(314\) 20.1075 1.13473
\(315\) 6.76983 0.381437
\(316\) 2.04308 0.114932
\(317\) 3.69981 0.207802 0.103901 0.994588i \(-0.466867\pi\)
0.103901 + 0.994588i \(0.466867\pi\)
\(318\) 67.9558 3.81077
\(319\) 0 0
\(320\) −11.7992 −0.659597
\(321\) −56.0108 −3.12622
\(322\) −20.0987 −1.12005
\(323\) 5.29530 0.294639
\(324\) 58.8663 3.27035
\(325\) −0.303615 −0.0168415
\(326\) 28.7306 1.59124
\(327\) −3.18262 −0.175999
\(328\) −35.6080 −1.96612
\(329\) 1.07602 0.0593227
\(330\) 0 0
\(331\) −20.6957 −1.13754 −0.568769 0.822497i \(-0.692580\pi\)
−0.568769 + 0.822497i \(0.692580\pi\)
\(332\) −39.9541 −2.19277
\(333\) −15.4044 −0.844156
\(334\) 19.9283 1.09043
\(335\) 13.6999 0.748505
\(336\) 4.90546 0.267615
\(337\) −13.8060 −0.752061 −0.376030 0.926607i \(-0.622711\pi\)
−0.376030 + 0.926607i \(0.622711\pi\)
\(338\) 30.4446 1.65597
\(339\) −5.61858 −0.305160
\(340\) 3.68300 0.199739
\(341\) 0 0
\(342\) −81.7996 −4.42322
\(343\) 1.00000 0.0539949
\(344\) 5.08743 0.274296
\(345\) 26.6350 1.43398
\(346\) 20.8944 1.12329
\(347\) −2.92092 −0.156803 −0.0784016 0.996922i \(-0.524982\pi\)
−0.0784016 + 0.996922i \(0.524982\pi\)
\(348\) −48.1060 −2.57875
\(349\) −14.2115 −0.760727 −0.380363 0.924837i \(-0.624201\pi\)
−0.380363 + 0.924837i \(0.624201\pi\)
\(350\) −2.35862 −0.126074
\(351\) −3.57758 −0.190957
\(352\) 0 0
\(353\) −24.1977 −1.28792 −0.643958 0.765061i \(-0.722709\pi\)
−0.643958 + 0.765061i \(0.722709\pi\)
\(354\) 64.9646 3.45283
\(355\) −7.14373 −0.379150
\(356\) −51.2895 −2.71834
\(357\) 3.23087 0.170996
\(358\) −19.0642 −1.00758
\(359\) 17.3847 0.917532 0.458766 0.888557i \(-0.348292\pi\)
0.458766 + 0.888557i \(0.348292\pi\)
\(360\) −24.9585 −1.31543
\(361\) 7.24405 0.381266
\(362\) 3.18568 0.167435
\(363\) 0 0
\(364\) −1.08181 −0.0567020
\(365\) −16.1243 −0.843986
\(366\) −77.7500 −4.06406
\(367\) −8.65437 −0.451755 −0.225877 0.974156i \(-0.572525\pi\)
−0.225877 + 0.974156i \(0.572525\pi\)
\(368\) 13.3735 0.697142
\(369\) 65.3861 3.40386
\(370\) 5.36691 0.279012
\(371\) −9.21775 −0.478562
\(372\) −29.5063 −1.52983
\(373\) −11.7827 −0.610087 −0.305044 0.952338i \(-0.598671\pi\)
−0.305044 + 0.952338i \(0.598671\pi\)
\(374\) 0 0
\(375\) 3.12567 0.161409
\(376\) −3.96697 −0.204581
\(377\) 1.31145 0.0675432
\(378\) −27.7923 −1.42948
\(379\) 3.97512 0.204188 0.102094 0.994775i \(-0.467446\pi\)
0.102094 + 0.994775i \(0.467446\pi\)
\(380\) 18.2533 0.936375
\(381\) 35.1688 1.80175
\(382\) −28.9914 −1.48333
\(383\) −18.7450 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(384\) 64.0333 3.26769
\(385\) 0 0
\(386\) −43.7897 −2.22884
\(387\) −9.34191 −0.474876
\(388\) −14.0477 −0.713166
\(389\) 2.31336 0.117292 0.0586459 0.998279i \(-0.481322\pi\)
0.0586459 + 0.998279i \(0.481322\pi\)
\(390\) 2.23833 0.113342
\(391\) 8.80815 0.445447
\(392\) −3.68672 −0.186208
\(393\) −11.7501 −0.592715
\(394\) −30.3592 −1.52947
\(395\) 0.573402 0.0288510
\(396\) 0 0
\(397\) 22.3968 1.12406 0.562032 0.827116i \(-0.310020\pi\)
0.562032 + 0.827116i \(0.310020\pi\)
\(398\) −23.4269 −1.17428
\(399\) 16.0125 0.801628
\(400\) 1.56941 0.0784704
\(401\) 24.2091 1.20894 0.604472 0.796626i \(-0.293384\pi\)
0.604472 + 0.796626i \(0.293384\pi\)
\(402\) −100.999 −5.03739
\(403\) 0.804393 0.0400697
\(404\) 36.3086 1.80642
\(405\) 16.5212 0.820943
\(406\) 10.1880 0.505620
\(407\) 0 0
\(408\) −11.9113 −0.589698
\(409\) 17.4252 0.861622 0.430811 0.902442i \(-0.358228\pi\)
0.430811 + 0.902442i \(0.358228\pi\)
\(410\) −22.7806 −1.12505
\(411\) 47.1574 2.32610
\(412\) 65.7257 3.23807
\(413\) −8.81202 −0.433611
\(414\) −136.065 −6.68721
\(415\) −11.2134 −0.550442
\(416\) −1.11482 −0.0546584
\(417\) 41.6544 2.03982
\(418\) 0 0
\(419\) 18.0980 0.884145 0.442073 0.896979i \(-0.354243\pi\)
0.442073 + 0.896979i \(0.354243\pi\)
\(420\) 11.1370 0.543432
\(421\) 19.4325 0.947082 0.473541 0.880772i \(-0.342976\pi\)
0.473541 + 0.880772i \(0.342976\pi\)
\(422\) −40.4452 −1.96884
\(423\) 7.28445 0.354182
\(424\) 33.9833 1.65037
\(425\) 1.03365 0.0501396
\(426\) 52.6656 2.55165
\(427\) 10.5463 0.510370
\(428\) −63.8491 −3.08626
\(429\) 0 0
\(430\) 3.25473 0.156957
\(431\) −7.90822 −0.380926 −0.190463 0.981694i \(-0.560999\pi\)
−0.190463 + 0.981694i \(0.560999\pi\)
\(432\) 18.4928 0.889734
\(433\) 18.2620 0.877618 0.438809 0.898580i \(-0.355400\pi\)
0.438809 + 0.898580i \(0.355400\pi\)
\(434\) 6.24889 0.299956
\(435\) −13.5012 −0.647334
\(436\) −3.62800 −0.173750
\(437\) 43.6541 2.08826
\(438\) 118.873 5.67997
\(439\) 24.6331 1.17567 0.587837 0.808979i \(-0.299980\pi\)
0.587837 + 0.808979i \(0.299980\pi\)
\(440\) 0 0
\(441\) 6.76983 0.322373
\(442\) 0.740212 0.0352083
\(443\) −16.6461 −0.790881 −0.395441 0.918492i \(-0.629408\pi\)
−0.395441 + 0.918492i \(0.629408\pi\)
\(444\) −25.3417 −1.20267
\(445\) −14.3947 −0.682374
\(446\) −36.1642 −1.71243
\(447\) −6.65045 −0.314555
\(448\) −11.7992 −0.557461
\(449\) −8.05832 −0.380296 −0.190148 0.981755i \(-0.560897\pi\)
−0.190148 + 0.981755i \(0.560897\pi\)
\(450\) −15.9675 −0.752713
\(451\) 0 0
\(452\) −6.40486 −0.301259
\(453\) 49.2312 2.31309
\(454\) −4.76113 −0.223451
\(455\) −0.303615 −0.0142337
\(456\) −59.0337 −2.76450
\(457\) −22.9712 −1.07455 −0.537273 0.843408i \(-0.680546\pi\)
−0.537273 + 0.843408i \(0.680546\pi\)
\(458\) 0.736610 0.0344195
\(459\) 12.1798 0.568506
\(460\) 30.3624 1.41565
\(461\) 15.0286 0.699951 0.349975 0.936759i \(-0.386190\pi\)
0.349975 + 0.936759i \(0.386190\pi\)
\(462\) 0 0
\(463\) −22.9066 −1.06456 −0.532279 0.846569i \(-0.678664\pi\)
−0.532279 + 0.846569i \(0.678664\pi\)
\(464\) −6.77900 −0.314707
\(465\) −8.28111 −0.384027
\(466\) 40.9979 1.89919
\(467\) 1.64223 0.0759932 0.0379966 0.999278i \(-0.487902\pi\)
0.0379966 + 0.999278i \(0.487902\pi\)
\(468\) −7.32365 −0.338536
\(469\) 13.6999 0.632602
\(470\) −2.53791 −0.117065
\(471\) −26.6467 −1.22781
\(472\) 32.4875 1.49536
\(473\) 0 0
\(474\) −4.22728 −0.194165
\(475\) 5.12290 0.235055
\(476\) 3.68300 0.168810
\(477\) −62.4026 −2.85722
\(478\) 53.4416 2.44436
\(479\) −6.50665 −0.297296 −0.148648 0.988890i \(-0.547492\pi\)
−0.148648 + 0.988890i \(0.547492\pi\)
\(480\) 11.4769 0.523846
\(481\) 0.690859 0.0315005
\(482\) 14.8251 0.675264
\(483\) 26.6350 1.21194
\(484\) 0 0
\(485\) −3.94258 −0.179023
\(486\) −38.4217 −1.74284
\(487\) 31.1060 1.40955 0.704774 0.709432i \(-0.251048\pi\)
0.704774 + 0.709432i \(0.251048\pi\)
\(488\) −38.8812 −1.76007
\(489\) −38.0741 −1.72177
\(490\) −2.35862 −0.106552
\(491\) 19.2332 0.867984 0.433992 0.900917i \(-0.357105\pi\)
0.433992 + 0.900917i \(0.357105\pi\)
\(492\) 107.567 4.84947
\(493\) −4.46483 −0.201086
\(494\) 3.66857 0.165057
\(495\) 0 0
\(496\) −4.15797 −0.186698
\(497\) −7.14373 −0.320440
\(498\) 82.6679 3.70444
\(499\) 12.4567 0.557636 0.278818 0.960344i \(-0.410057\pi\)
0.278818 + 0.960344i \(0.410057\pi\)
\(500\) 3.56309 0.159346
\(501\) −26.4093 −1.17988
\(502\) 25.1806 1.12386
\(503\) −0.952980 −0.0424913 −0.0212456 0.999774i \(-0.506763\pi\)
−0.0212456 + 0.999774i \(0.506763\pi\)
\(504\) −24.9585 −1.11174
\(505\) 10.1902 0.453459
\(506\) 0 0
\(507\) −40.3456 −1.79181
\(508\) 40.0904 1.77872
\(509\) 30.0463 1.33178 0.665890 0.746050i \(-0.268052\pi\)
0.665890 + 0.746050i \(0.268052\pi\)
\(510\) −7.62038 −0.337436
\(511\) −16.1243 −0.713298
\(512\) 17.3345 0.766085
\(513\) 60.3645 2.66516
\(514\) 41.6849 1.83864
\(515\) 18.4463 0.812840
\(516\) −15.3684 −0.676554
\(517\) 0 0
\(518\) 5.36691 0.235809
\(519\) −27.6895 −1.21544
\(520\) 1.11934 0.0490865
\(521\) 35.3266 1.54769 0.773844 0.633376i \(-0.218331\pi\)
0.773844 + 0.633376i \(0.218331\pi\)
\(522\) 68.9708 3.01877
\(523\) 3.82511 0.167261 0.0836303 0.996497i \(-0.473349\pi\)
0.0836303 + 0.996497i \(0.473349\pi\)
\(524\) −13.3945 −0.585140
\(525\) 3.12567 0.136416
\(526\) 28.9364 1.26169
\(527\) −2.73855 −0.119293
\(528\) 0 0
\(529\) 49.6137 2.15712
\(530\) 21.7412 0.944376
\(531\) −59.6559 −2.58885
\(532\) 18.2533 0.791382
\(533\) −2.93245 −0.127018
\(534\) 106.122 4.59233
\(535\) −17.9196 −0.774732
\(536\) −50.5077 −2.18160
\(537\) 25.2642 1.09023
\(538\) −50.7258 −2.18694
\(539\) 0 0
\(540\) 41.9848 1.80674
\(541\) −1.78673 −0.0768175 −0.0384087 0.999262i \(-0.512229\pi\)
−0.0384087 + 0.999262i \(0.512229\pi\)
\(542\) −10.3368 −0.444003
\(543\) −4.22170 −0.181170
\(544\) 3.79538 0.162726
\(545\) −1.01822 −0.0436157
\(546\) 2.23833 0.0957918
\(547\) −33.1680 −1.41816 −0.709081 0.705127i \(-0.750890\pi\)
−0.709081 + 0.705127i \(0.750890\pi\)
\(548\) 53.7567 2.29637
\(549\) 71.3965 3.04713
\(550\) 0 0
\(551\) −22.1281 −0.942690
\(552\) −98.1960 −4.17950
\(553\) 0.573402 0.0243835
\(554\) 26.9980 1.14704
\(555\) −7.11230 −0.301900
\(556\) 47.4836 2.01375
\(557\) 2.03928 0.0864070 0.0432035 0.999066i \(-0.486244\pi\)
0.0432035 + 0.999066i \(0.486244\pi\)
\(558\) 42.3040 1.79087
\(559\) 0.418968 0.0177204
\(560\) 1.56941 0.0663196
\(561\) 0 0
\(562\) 27.0390 1.14057
\(563\) 18.5776 0.782953 0.391476 0.920188i \(-0.371964\pi\)
0.391476 + 0.920188i \(0.371964\pi\)
\(564\) 11.9836 0.504602
\(565\) −1.79756 −0.0756239
\(566\) −29.9083 −1.25714
\(567\) 16.5212 0.693823
\(568\) 26.3370 1.10507
\(569\) −16.4095 −0.687920 −0.343960 0.938984i \(-0.611768\pi\)
−0.343960 + 0.938984i \(0.611768\pi\)
\(570\) −37.7674 −1.58190
\(571\) −18.5329 −0.775578 −0.387789 0.921748i \(-0.626761\pi\)
−0.387789 + 0.921748i \(0.626761\pi\)
\(572\) 0 0
\(573\) 38.4198 1.60501
\(574\) −22.7806 −0.950844
\(575\) 8.52137 0.355366
\(576\) −79.8788 −3.32828
\(577\) −23.7459 −0.988554 −0.494277 0.869304i \(-0.664567\pi\)
−0.494277 + 0.869304i \(0.664567\pi\)
\(578\) 37.5765 1.56298
\(579\) 58.0307 2.41167
\(580\) −15.3906 −0.639060
\(581\) −11.2134 −0.465208
\(582\) 29.0658 1.20482
\(583\) 0 0
\(584\) 59.4459 2.45989
\(585\) −2.05542 −0.0849813
\(586\) 19.4654 0.804110
\(587\) 21.0434 0.868554 0.434277 0.900779i \(-0.357004\pi\)
0.434277 + 0.900779i \(0.357004\pi\)
\(588\) 11.1370 0.459284
\(589\) −13.5725 −0.559246
\(590\) 20.7842 0.855672
\(591\) 40.2324 1.65494
\(592\) −3.57110 −0.146771
\(593\) −46.2299 −1.89843 −0.949217 0.314621i \(-0.898122\pi\)
−0.949217 + 0.314621i \(0.898122\pi\)
\(594\) 0 0
\(595\) 1.03365 0.0423757
\(596\) −7.58112 −0.310535
\(597\) 31.0456 1.27061
\(598\) 6.10225 0.249540
\(599\) 16.7230 0.683282 0.341641 0.939831i \(-0.389017\pi\)
0.341641 + 0.939831i \(0.389017\pi\)
\(600\) −11.5235 −0.470445
\(601\) 10.4475 0.426161 0.213080 0.977035i \(-0.431650\pi\)
0.213080 + 0.977035i \(0.431650\pi\)
\(602\) 3.25473 0.132653
\(603\) 92.7460 3.77691
\(604\) 56.1207 2.28352
\(605\) 0 0
\(606\) −75.1251 −3.05175
\(607\) 19.9139 0.808279 0.404139 0.914697i \(-0.367571\pi\)
0.404139 + 0.914697i \(0.367571\pi\)
\(608\) 18.8103 0.762859
\(609\) −13.5012 −0.547097
\(610\) −24.8746 −1.00714
\(611\) −0.326694 −0.0132166
\(612\) 24.9333 1.00787
\(613\) −17.4111 −0.703229 −0.351615 0.936145i \(-0.614367\pi\)
−0.351615 + 0.936145i \(0.614367\pi\)
\(614\) 4.16627 0.168137
\(615\) 30.1892 1.21734
\(616\) 0 0
\(617\) −13.6972 −0.551427 −0.275713 0.961240i \(-0.588914\pi\)
−0.275713 + 0.961240i \(0.588914\pi\)
\(618\) −135.991 −5.47036
\(619\) −31.9100 −1.28257 −0.641286 0.767302i \(-0.721599\pi\)
−0.641286 + 0.767302i \(0.721599\pi\)
\(620\) −9.43999 −0.379119
\(621\) 100.410 4.02930
\(622\) 10.6535 0.427165
\(623\) −14.3947 −0.576711
\(624\) −1.48937 −0.0596225
\(625\) 1.00000 0.0400000
\(626\) 14.8397 0.593115
\(627\) 0 0
\(628\) −30.3757 −1.21212
\(629\) −2.35203 −0.0937814
\(630\) −15.9675 −0.636159
\(631\) 22.1491 0.881742 0.440871 0.897570i \(-0.354670\pi\)
0.440871 + 0.897570i \(0.354670\pi\)
\(632\) −2.11397 −0.0840894
\(633\) 53.5985 2.13035
\(634\) −8.72644 −0.346571
\(635\) 11.2516 0.446506
\(636\) −102.658 −4.07067
\(637\) −0.303615 −0.0120297
\(638\) 0 0
\(639\) −48.3619 −1.91317
\(640\) 20.4863 0.809790
\(641\) 47.0120 1.85686 0.928431 0.371506i \(-0.121158\pi\)
0.928431 + 0.371506i \(0.121158\pi\)
\(642\) 132.108 5.21389
\(643\) −16.2825 −0.642117 −0.321059 0.947059i \(-0.604039\pi\)
−0.321059 + 0.947059i \(0.604039\pi\)
\(644\) 30.3624 1.19644
\(645\) −4.31321 −0.169833
\(646\) −12.4896 −0.491397
\(647\) −40.8269 −1.60507 −0.802536 0.596604i \(-0.796516\pi\)
−0.802536 + 0.596604i \(0.796516\pi\)
\(648\) −60.9089 −2.39273
\(649\) 0 0
\(650\) 0.716112 0.0280882
\(651\) −8.28111 −0.324562
\(652\) −43.4023 −1.69976
\(653\) 23.2768 0.910893 0.455447 0.890263i \(-0.349480\pi\)
0.455447 + 0.890263i \(0.349480\pi\)
\(654\) 7.50658 0.293531
\(655\) −3.75923 −0.146885
\(656\) 15.1580 0.591822
\(657\) −109.159 −4.25870
\(658\) −2.53791 −0.0989381
\(659\) −8.17158 −0.318320 −0.159160 0.987253i \(-0.550879\pi\)
−0.159160 + 0.987253i \(0.550879\pi\)
\(660\) 0 0
\(661\) 7.64860 0.297496 0.148748 0.988875i \(-0.452476\pi\)
0.148748 + 0.988875i \(0.452476\pi\)
\(662\) 48.8133 1.89718
\(663\) −0.980939 −0.0380965
\(664\) 41.3405 1.60432
\(665\) 5.12290 0.198657
\(666\) 36.3331 1.40788
\(667\) −36.8077 −1.42520
\(668\) −30.1051 −1.16480
\(669\) 47.9253 1.85290
\(670\) −32.3128 −1.24835
\(671\) 0 0
\(672\) 11.4769 0.442730
\(673\) 14.2835 0.550590 0.275295 0.961360i \(-0.411224\pi\)
0.275295 + 0.961360i \(0.411224\pi\)
\(674\) 32.5631 1.25428
\(675\) 11.7833 0.453538
\(676\) −45.9917 −1.76891
\(677\) −28.7655 −1.10555 −0.552774 0.833331i \(-0.686431\pi\)
−0.552774 + 0.833331i \(0.686431\pi\)
\(678\) 13.2521 0.508944
\(679\) −3.94258 −0.151302
\(680\) −3.81080 −0.146137
\(681\) 6.30952 0.241781
\(682\) 0 0
\(683\) −22.0519 −0.843794 −0.421897 0.906644i \(-0.638636\pi\)
−0.421897 + 0.906644i \(0.638636\pi\)
\(684\) 123.572 4.72489
\(685\) 15.0871 0.576449
\(686\) −2.35862 −0.0900525
\(687\) −0.976166 −0.0372430
\(688\) −2.16568 −0.0825656
\(689\) 2.79865 0.106620
\(690\) −62.8219 −2.39159
\(691\) 8.92737 0.339613 0.169807 0.985477i \(-0.445686\pi\)
0.169807 + 0.985477i \(0.445686\pi\)
\(692\) −31.5644 −1.19990
\(693\) 0 0
\(694\) 6.88934 0.261516
\(695\) 13.3265 0.505504
\(696\) 49.7753 1.88673
\(697\) 9.98350 0.378152
\(698\) 33.5196 1.26874
\(699\) −54.3309 −2.05499
\(700\) 3.56309 0.134672
\(701\) −0.0337866 −0.00127610 −0.000638050 1.00000i \(-0.500203\pi\)
−0.000638050 1.00000i \(0.500203\pi\)
\(702\) 8.43814 0.318477
\(703\) −11.6569 −0.439647
\(704\) 0 0
\(705\) 3.36327 0.126668
\(706\) 57.0733 2.14798
\(707\) 10.1902 0.383243
\(708\) −98.1398 −3.68832
\(709\) −2.86944 −0.107764 −0.0538821 0.998547i \(-0.517160\pi\)
−0.0538821 + 0.998547i \(0.517160\pi\)
\(710\) 16.8493 0.632345
\(711\) 3.88184 0.145580
\(712\) 53.0693 1.98886
\(713\) −22.5764 −0.845492
\(714\) −7.62038 −0.285186
\(715\) 0 0
\(716\) 28.7997 1.07629
\(717\) −70.8215 −2.64488
\(718\) −41.0040 −1.53026
\(719\) 1.67521 0.0624749 0.0312375 0.999512i \(-0.490055\pi\)
0.0312375 + 0.999512i \(0.490055\pi\)
\(720\) 10.6246 0.395957
\(721\) 18.4463 0.686975
\(722\) −17.0860 −0.635874
\(723\) −19.6464 −0.730657
\(724\) −4.81249 −0.178855
\(725\) −4.31946 −0.160421
\(726\) 0 0
\(727\) 0.612732 0.0227250 0.0113625 0.999935i \(-0.496383\pi\)
0.0113625 + 0.999935i \(0.496383\pi\)
\(728\) 1.11934 0.0414857
\(729\) 1.35351 0.0501298
\(730\) 38.0311 1.40760
\(731\) −1.42637 −0.0527563
\(732\) 117.454 4.34123
\(733\) −36.4732 −1.34717 −0.673584 0.739111i \(-0.735246\pi\)
−0.673584 + 0.739111i \(0.735246\pi\)
\(734\) 20.4124 0.753434
\(735\) 3.12567 0.115292
\(736\) 31.2889 1.15332
\(737\) 0 0
\(738\) −154.221 −5.67695
\(739\) 44.7363 1.64565 0.822826 0.568293i \(-0.192396\pi\)
0.822826 + 0.568293i \(0.192396\pi\)
\(740\) −8.10761 −0.298042
\(741\) −4.86163 −0.178597
\(742\) 21.7412 0.798143
\(743\) 13.6791 0.501838 0.250919 0.968008i \(-0.419267\pi\)
0.250919 + 0.968008i \(0.419267\pi\)
\(744\) 30.5302 1.11929
\(745\) −2.12769 −0.0779523
\(746\) 27.7910 1.01750
\(747\) −75.9125 −2.77749
\(748\) 0 0
\(749\) −17.9196 −0.654768
\(750\) −7.37227 −0.269197
\(751\) −46.1908 −1.68553 −0.842764 0.538284i \(-0.819073\pi\)
−0.842764 + 0.538284i \(0.819073\pi\)
\(752\) 1.68871 0.0615808
\(753\) −33.3696 −1.21606
\(754\) −3.09322 −0.112648
\(755\) 15.7506 0.573223
\(756\) 41.9848 1.52697
\(757\) 51.4344 1.86942 0.934708 0.355417i \(-0.115661\pi\)
0.934708 + 0.355417i \(0.115661\pi\)
\(758\) −9.37579 −0.340544
\(759\) 0 0
\(760\) −18.8867 −0.685093
\(761\) −32.2686 −1.16974 −0.584869 0.811128i \(-0.698854\pi\)
−0.584869 + 0.811128i \(0.698854\pi\)
\(762\) −82.9498 −3.00495
\(763\) −1.01822 −0.0368620
\(764\) 43.7964 1.58450
\(765\) 6.99767 0.253001
\(766\) 44.2124 1.59746
\(767\) 2.67546 0.0966053
\(768\) −77.2692 −2.78821
\(769\) −20.8369 −0.751399 −0.375699 0.926742i \(-0.622597\pi\)
−0.375699 + 0.926742i \(0.622597\pi\)
\(770\) 0 0
\(771\) −55.2414 −1.98947
\(772\) 66.1516 2.38085
\(773\) −39.5002 −1.42072 −0.710361 0.703838i \(-0.751468\pi\)
−0.710361 + 0.703838i \(0.751468\pi\)
\(774\) 22.0340 0.791996
\(775\) −2.64939 −0.0951687
\(776\) 14.5352 0.521783
\(777\) −7.11230 −0.255152
\(778\) −5.45633 −0.195619
\(779\) 49.4792 1.77278
\(780\) −3.38137 −0.121073
\(781\) 0 0
\(782\) −20.7751 −0.742915
\(783\) −50.8974 −1.81892
\(784\) 1.56941 0.0560503
\(785\) −8.52511 −0.304274
\(786\) 27.7141 0.988528
\(787\) 23.4515 0.835954 0.417977 0.908458i \(-0.362739\pi\)
0.417977 + 0.908458i \(0.362739\pi\)
\(788\) 45.8626 1.63379
\(789\) −38.3469 −1.36519
\(790\) −1.35244 −0.0481175
\(791\) −1.79756 −0.0639139
\(792\) 0 0
\(793\) −3.20200 −0.113706
\(794\) −52.8255 −1.87471
\(795\) −28.8117 −1.02184
\(796\) 35.3902 1.25437
\(797\) 19.6169 0.694868 0.347434 0.937705i \(-0.387053\pi\)
0.347434 + 0.937705i \(0.387053\pi\)
\(798\) −37.7674 −1.33695
\(799\) 1.11223 0.0393478
\(800\) 3.67181 0.129818
\(801\) −97.4497 −3.44322
\(802\) −57.1001 −2.01627
\(803\) 0 0
\(804\) 152.576 5.38095
\(805\) 8.52137 0.300339
\(806\) −1.89726 −0.0668280
\(807\) 67.2224 2.36634
\(808\) −37.5686 −1.32166
\(809\) 1.20245 0.0422757 0.0211379 0.999777i \(-0.493271\pi\)
0.0211379 + 0.999777i \(0.493271\pi\)
\(810\) −38.9671 −1.36916
\(811\) 15.3789 0.540025 0.270012 0.962857i \(-0.412972\pi\)
0.270012 + 0.962857i \(0.412972\pi\)
\(812\) −15.3906 −0.540104
\(813\) 13.6984 0.480425
\(814\) 0 0
\(815\) −12.1811 −0.426685
\(816\) 5.07055 0.177505
\(817\) −7.06924 −0.247322
\(818\) −41.0995 −1.43701
\(819\) −2.05542 −0.0718223
\(820\) 34.4139 1.20178
\(821\) 36.6757 1.27999 0.639996 0.768378i \(-0.278936\pi\)
0.639996 + 0.768378i \(0.278936\pi\)
\(822\) −111.226 −3.87947
\(823\) −41.8158 −1.45761 −0.728804 0.684722i \(-0.759924\pi\)
−0.728804 + 0.684722i \(0.759924\pi\)
\(824\) −68.0063 −2.36911
\(825\) 0 0
\(826\) 20.7842 0.723175
\(827\) −54.0547 −1.87966 −0.939832 0.341636i \(-0.889019\pi\)
−0.939832 + 0.341636i \(0.889019\pi\)
\(828\) 205.548 7.14329
\(829\) 26.8067 0.931034 0.465517 0.885039i \(-0.345868\pi\)
0.465517 + 0.885039i \(0.345868\pi\)
\(830\) 26.4480 0.918025
\(831\) −35.7782 −1.24113
\(832\) 3.58242 0.124198
\(833\) 1.03365 0.0358140
\(834\) −98.2469 −3.40201
\(835\) −8.44915 −0.292395
\(836\) 0 0
\(837\) −31.2184 −1.07907
\(838\) −42.6863 −1.47457
\(839\) 16.2884 0.562338 0.281169 0.959658i \(-0.409278\pi\)
0.281169 + 0.959658i \(0.409278\pi\)
\(840\) −11.5235 −0.397598
\(841\) −10.3423 −0.356630
\(842\) −45.8339 −1.57954
\(843\) −35.8324 −1.23413
\(844\) 61.0991 2.10312
\(845\) −12.9078 −0.444042
\(846\) −17.1812 −0.590703
\(847\) 0 0
\(848\) −14.4664 −0.496779
\(849\) 39.6348 1.36026
\(850\) −2.43800 −0.0836226
\(851\) −19.3899 −0.664678
\(852\) −79.5601 −2.72568
\(853\) 30.6496 1.04942 0.524710 0.851281i \(-0.324174\pi\)
0.524710 + 0.851281i \(0.324174\pi\)
\(854\) −24.8746 −0.851192
\(855\) 34.6812 1.18607
\(856\) 66.0646 2.25804
\(857\) −42.2735 −1.44404 −0.722018 0.691875i \(-0.756785\pi\)
−0.722018 + 0.691875i \(0.756785\pi\)
\(858\) 0 0
\(859\) −13.3365 −0.455036 −0.227518 0.973774i \(-0.573061\pi\)
−0.227518 + 0.973774i \(0.573061\pi\)
\(860\) −4.91681 −0.167662
\(861\) 30.1892 1.02884
\(862\) 18.6525 0.635306
\(863\) −37.7743 −1.28585 −0.642926 0.765929i \(-0.722280\pi\)
−0.642926 + 0.765929i \(0.722280\pi\)
\(864\) 43.2660 1.47194
\(865\) −8.85874 −0.301206
\(866\) −43.0732 −1.46369
\(867\) −49.7968 −1.69119
\(868\) −9.43999 −0.320414
\(869\) 0 0
\(870\) 31.8442 1.07962
\(871\) −4.15949 −0.140939
\(872\) 3.75389 0.127123
\(873\) −26.6906 −0.903340
\(874\) −102.963 −3.48279
\(875\) 1.00000 0.0338062
\(876\) −179.577 −6.06735
\(877\) −8.69917 −0.293750 −0.146875 0.989155i \(-0.546922\pi\)
−0.146875 + 0.989155i \(0.546922\pi\)
\(878\) −58.1001 −1.96078
\(879\) −25.7958 −0.870072
\(880\) 0 0
\(881\) −3.89312 −0.131163 −0.0655814 0.997847i \(-0.520890\pi\)
−0.0655814 + 0.997847i \(0.520890\pi\)
\(882\) −15.9675 −0.537652
\(883\) 45.1520 1.51948 0.759742 0.650225i \(-0.225325\pi\)
0.759742 + 0.650225i \(0.225325\pi\)
\(884\) −1.11821 −0.0376096
\(885\) −27.5435 −0.925864
\(886\) 39.2619 1.31903
\(887\) −15.6274 −0.524718 −0.262359 0.964970i \(-0.584500\pi\)
−0.262359 + 0.964970i \(0.584500\pi\)
\(888\) 26.2211 0.879922
\(889\) 11.2516 0.377366
\(890\) 33.9516 1.13806
\(891\) 0 0
\(892\) 54.6321 1.82922
\(893\) 5.51231 0.184463
\(894\) 15.6859 0.524614
\(895\) 8.08279 0.270178
\(896\) 20.4863 0.684398
\(897\) −8.08679 −0.270010
\(898\) 19.0065 0.634256
\(899\) 11.4439 0.381676
\(900\) 24.1215 0.804050
\(901\) −9.52797 −0.317423
\(902\) 0 0
\(903\) −4.31321 −0.143535
\(904\) 6.62711 0.220414
\(905\) −1.35065 −0.0448972
\(906\) −116.118 −3.85775
\(907\) 22.9194 0.761027 0.380513 0.924775i \(-0.375747\pi\)
0.380513 + 0.924775i \(0.375747\pi\)
\(908\) 7.19248 0.238691
\(909\) 68.9861 2.28813
\(910\) 0.716112 0.0237389
\(911\) 39.8282 1.31957 0.659784 0.751455i \(-0.270648\pi\)
0.659784 + 0.751455i \(0.270648\pi\)
\(912\) 25.1301 0.832142
\(913\) 0 0
\(914\) 54.1803 1.79212
\(915\) 32.9642 1.08976
\(916\) −1.11277 −0.0367670
\(917\) −3.75923 −0.124141
\(918\) −28.7276 −0.948152
\(919\) −9.94569 −0.328078 −0.164039 0.986454i \(-0.552452\pi\)
−0.164039 + 0.986454i \(0.552452\pi\)
\(920\) −31.4159 −1.03575
\(921\) −5.52120 −0.181930
\(922\) −35.4467 −1.16737
\(923\) 2.16894 0.0713917
\(924\) 0 0
\(925\) −2.27545 −0.0748162
\(926\) 54.0278 1.77547
\(927\) 124.878 4.10154
\(928\) −15.8602 −0.520638
\(929\) −52.2385 −1.71389 −0.856944 0.515409i \(-0.827640\pi\)
−0.856944 + 0.515409i \(0.827640\pi\)
\(930\) 19.5320 0.640479
\(931\) 5.12290 0.167896
\(932\) −61.9341 −2.02872
\(933\) −14.1181 −0.462207
\(934\) −3.87339 −0.126741
\(935\) 0 0
\(936\) 7.57778 0.247687
\(937\) 5.94143 0.194098 0.0970490 0.995280i \(-0.469060\pi\)
0.0970490 + 0.995280i \(0.469060\pi\)
\(938\) −32.3128 −1.05505
\(939\) −19.6658 −0.641770
\(940\) 3.83394 0.125049
\(941\) 15.7759 0.514281 0.257140 0.966374i \(-0.417220\pi\)
0.257140 + 0.966374i \(0.417220\pi\)
\(942\) 62.8494 2.04774
\(943\) 82.3032 2.68016
\(944\) −13.8297 −0.450117
\(945\) 11.7833 0.383310
\(946\) 0 0
\(947\) 45.1583 1.46745 0.733723 0.679448i \(-0.237781\pi\)
0.733723 + 0.679448i \(0.237781\pi\)
\(948\) 6.38600 0.207408
\(949\) 4.89558 0.158917
\(950\) −12.0830 −0.392023
\(951\) 11.5644 0.375001
\(952\) −3.81080 −0.123509
\(953\) 38.0837 1.23365 0.616825 0.787100i \(-0.288418\pi\)
0.616825 + 0.787100i \(0.288418\pi\)
\(954\) 147.184 4.76526
\(955\) 12.2917 0.397750
\(956\) −80.7324 −2.61107
\(957\) 0 0
\(958\) 15.3467 0.495830
\(959\) 15.0871 0.487189
\(960\) −36.8805 −1.19031
\(961\) −23.9808 −0.773573
\(962\) −1.62947 −0.0525364
\(963\) −121.313 −3.90925
\(964\) −22.3957 −0.721318
\(965\) 18.5658 0.597655
\(966\) −62.8219 −2.02126
\(967\) −15.4735 −0.497593 −0.248796 0.968556i \(-0.580035\pi\)
−0.248796 + 0.968556i \(0.580035\pi\)
\(968\) 0 0
\(969\) 16.5514 0.531707
\(970\) 9.29904 0.298574
\(971\) 0.473814 0.0152054 0.00760270 0.999971i \(-0.497580\pi\)
0.00760270 + 0.999971i \(0.497580\pi\)
\(972\) 58.0424 1.86171
\(973\) 13.3265 0.427229
\(974\) −73.3673 −2.35084
\(975\) −0.949001 −0.0303924
\(976\) 16.5514 0.529797
\(977\) −46.1591 −1.47676 −0.738380 0.674384i \(-0.764409\pi\)
−0.738380 + 0.674384i \(0.764409\pi\)
\(978\) 89.8024 2.87156
\(979\) 0 0
\(980\) 3.56309 0.113819
\(981\) −6.89317 −0.220082
\(982\) −45.3639 −1.44762
\(983\) −51.4375 −1.64060 −0.820300 0.571934i \(-0.806194\pi\)
−0.820300 + 0.571934i \(0.806194\pi\)
\(984\) −111.299 −3.54809
\(985\) 12.8716 0.410123
\(986\) 10.5308 0.335370
\(987\) 3.36327 0.107054
\(988\) −5.54198 −0.176314
\(989\) −11.7589 −0.373911
\(990\) 0 0
\(991\) 28.7757 0.914091 0.457046 0.889443i \(-0.348908\pi\)
0.457046 + 0.889443i \(0.348908\pi\)
\(992\) −9.72805 −0.308866
\(993\) −64.6880 −2.05281
\(994\) 16.8493 0.534429
\(995\) 9.93245 0.314880
\(996\) −124.884 −3.95709
\(997\) −45.9499 −1.45525 −0.727625 0.685975i \(-0.759376\pi\)
−0.727625 + 0.685975i \(0.759376\pi\)
\(998\) −29.3805 −0.930024
\(999\) −26.8122 −0.848300
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 4235.2.a.bp.1.3 18
11.7 odd 10 385.2.n.f.71.8 36
11.8 odd 10 385.2.n.f.141.8 yes 36
11.10 odd 2 4235.2.a.bo.1.16 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.n.f.71.8 36 11.7 odd 10
385.2.n.f.141.8 yes 36 11.8 odd 10
4235.2.a.bo.1.16 18 11.10 odd 2
4235.2.a.bp.1.3 18 1.1 even 1 trivial