Properties

Label 2175.2.a.bd.1.5
Level $2175$
Weight $2$
Character 2175.1
Self dual yes
Analytic conductor $17.367$
Analytic rank $0$
Dimension $8$
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] = [2175,2,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, 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("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.510732\) of defining polynomial
Character \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.510732 q^{2} +1.00000 q^{3} -1.73915 q^{4} +0.510732 q^{6} -4.82343 q^{7} -1.90970 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.510732 q^{2} +1.00000 q^{3} -1.73915 q^{4} +0.510732 q^{6} -4.82343 q^{7} -1.90970 q^{8} +1.00000 q^{9} +4.88439 q^{11} -1.73915 q^{12} -4.59669 q^{13} -2.46348 q^{14} +2.50296 q^{16} -6.50987 q^{17} +0.510732 q^{18} +3.09205 q^{19} -4.82343 q^{21} +2.49461 q^{22} +5.62549 q^{23} -1.90970 q^{24} -2.34767 q^{26} +1.00000 q^{27} +8.38869 q^{28} +1.00000 q^{29} +9.24375 q^{31} +5.09775 q^{32} +4.88439 q^{33} -3.32480 q^{34} -1.73915 q^{36} +11.1261 q^{37} +1.57921 q^{38} -4.59669 q^{39} -2.84537 q^{41} -2.46348 q^{42} +4.58611 q^{43} -8.49470 q^{44} +2.87311 q^{46} -3.62713 q^{47} +2.50296 q^{48} +16.2655 q^{49} -6.50987 q^{51} +7.99434 q^{52} +0.967845 q^{53} +0.510732 q^{54} +9.21132 q^{56} +3.09205 q^{57} +0.510732 q^{58} -0.298882 q^{59} -0.786908 q^{61} +4.72108 q^{62} -4.82343 q^{63} -2.40234 q^{64} +2.49461 q^{66} -4.86742 q^{67} +11.3217 q^{68} +5.62549 q^{69} +0.741689 q^{71} -1.90970 q^{72} -5.52981 q^{73} +5.68246 q^{74} -5.37755 q^{76} -23.5595 q^{77} -2.34767 q^{78} -2.96278 q^{79} +1.00000 q^{81} -1.45322 q^{82} +13.6633 q^{83} +8.38869 q^{84} +2.34227 q^{86} +1.00000 q^{87} -9.32773 q^{88} +3.67835 q^{89} +22.1718 q^{91} -9.78358 q^{92} +9.24375 q^{93} -1.85249 q^{94} +5.09775 q^{96} +2.87658 q^{97} +8.30730 q^{98} +4.88439 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 8 q^{3} + 12 q^{4} + 2 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 8 q^{3} + 12 q^{4} + 2 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9} + 6 q^{11} + 12 q^{12} - 6 q^{13} + 9 q^{14} + 32 q^{16} + 12 q^{17} + 2 q^{18} - 2 q^{21} - 3 q^{22} + 14 q^{23} + 3 q^{24} + 18 q^{26} + 8 q^{27} - 14 q^{28} + 8 q^{29} + 8 q^{31} - 2 q^{32} + 6 q^{33} - 13 q^{34} + 12 q^{36} - 4 q^{37} + 26 q^{38} - 6 q^{39} + 2 q^{41} + 9 q^{42} - 2 q^{43} - 15 q^{44} + 24 q^{46} + 12 q^{47} + 32 q^{48} + 38 q^{49} + 12 q^{51} - 49 q^{52} + 4 q^{53} + 2 q^{54} + 58 q^{56} + 2 q^{58} + 18 q^{59} + 12 q^{61} - 4 q^{62} - 2 q^{63} + 21 q^{64} - 3 q^{66} - 26 q^{67} + 81 q^{68} + 14 q^{69} + 24 q^{71} + 3 q^{72} + 14 q^{73} - 22 q^{74} - 26 q^{77} + 18 q^{78} + 10 q^{79} + 8 q^{81} - 48 q^{82} + 40 q^{83} - 14 q^{84} + 8 q^{86} + 8 q^{87} + 10 q^{88} + 34 q^{89} + 26 q^{91} - 18 q^{92} + 8 q^{93} - 43 q^{94} - 2 q^{96} - 30 q^{97} + 29 q^{98} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.510732 0.361142 0.180571 0.983562i \(-0.442205\pi\)
0.180571 + 0.983562i \(0.442205\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.73915 −0.869577
\(5\) 0 0
\(6\) 0.510732 0.208505
\(7\) −4.82343 −1.82309 −0.911543 0.411205i \(-0.865108\pi\)
−0.911543 + 0.411205i \(0.865108\pi\)
\(8\) −1.90970 −0.675182
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.88439 1.47270 0.736349 0.676602i \(-0.236548\pi\)
0.736349 + 0.676602i \(0.236548\pi\)
\(12\) −1.73915 −0.502050
\(13\) −4.59669 −1.27489 −0.637446 0.770495i \(-0.720009\pi\)
−0.637446 + 0.770495i \(0.720009\pi\)
\(14\) −2.46348 −0.658392
\(15\) 0 0
\(16\) 2.50296 0.625740
\(17\) −6.50987 −1.57888 −0.789438 0.613830i \(-0.789628\pi\)
−0.789438 + 0.613830i \(0.789628\pi\)
\(18\) 0.510732 0.120381
\(19\) 3.09205 0.709364 0.354682 0.934987i \(-0.384589\pi\)
0.354682 + 0.934987i \(0.384589\pi\)
\(20\) 0 0
\(21\) −4.82343 −1.05256
\(22\) 2.49461 0.531853
\(23\) 5.62549 1.17299 0.586497 0.809951i \(-0.300506\pi\)
0.586497 + 0.809951i \(0.300506\pi\)
\(24\) −1.90970 −0.389817
\(25\) 0 0
\(26\) −2.34767 −0.460416
\(27\) 1.00000 0.192450
\(28\) 8.38869 1.58531
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 9.24375 1.66023 0.830113 0.557595i \(-0.188276\pi\)
0.830113 + 0.557595i \(0.188276\pi\)
\(32\) 5.09775 0.901163
\(33\) 4.88439 0.850262
\(34\) −3.32480 −0.570198
\(35\) 0 0
\(36\) −1.73915 −0.289859
\(37\) 11.1261 1.82912 0.914561 0.404448i \(-0.132536\pi\)
0.914561 + 0.404448i \(0.132536\pi\)
\(38\) 1.57921 0.256181
\(39\) −4.59669 −0.736059
\(40\) 0 0
\(41\) −2.84537 −0.444372 −0.222186 0.975004i \(-0.571319\pi\)
−0.222186 + 0.975004i \(0.571319\pi\)
\(42\) −2.46348 −0.380123
\(43\) 4.58611 0.699375 0.349688 0.936866i \(-0.386288\pi\)
0.349688 + 0.936866i \(0.386288\pi\)
\(44\) −8.49470 −1.28062
\(45\) 0 0
\(46\) 2.87311 0.423617
\(47\) −3.62713 −0.529071 −0.264536 0.964376i \(-0.585219\pi\)
−0.264536 + 0.964376i \(0.585219\pi\)
\(48\) 2.50296 0.361271
\(49\) 16.2655 2.32364
\(50\) 0 0
\(51\) −6.50987 −0.911564
\(52\) 7.99434 1.10862
\(53\) 0.967845 0.132944 0.0664719 0.997788i \(-0.478826\pi\)
0.0664719 + 0.997788i \(0.478826\pi\)
\(54\) 0.510732 0.0695018
\(55\) 0 0
\(56\) 9.21132 1.23091
\(57\) 3.09205 0.409552
\(58\) 0.510732 0.0670623
\(59\) −0.298882 −0.0389112 −0.0194556 0.999811i \(-0.506193\pi\)
−0.0194556 + 0.999811i \(0.506193\pi\)
\(60\) 0 0
\(61\) −0.786908 −0.100753 −0.0503766 0.998730i \(-0.516042\pi\)
−0.0503766 + 0.998730i \(0.516042\pi\)
\(62\) 4.72108 0.599577
\(63\) −4.82343 −0.607695
\(64\) −2.40234 −0.300293
\(65\) 0 0
\(66\) 2.49461 0.307065
\(67\) −4.86742 −0.594650 −0.297325 0.954776i \(-0.596094\pi\)
−0.297325 + 0.954776i \(0.596094\pi\)
\(68\) 11.3217 1.37295
\(69\) 5.62549 0.677229
\(70\) 0 0
\(71\) 0.741689 0.0880223 0.0440112 0.999031i \(-0.485986\pi\)
0.0440112 + 0.999031i \(0.485986\pi\)
\(72\) −1.90970 −0.225061
\(73\) −5.52981 −0.647215 −0.323608 0.946191i \(-0.604896\pi\)
−0.323608 + 0.946191i \(0.604896\pi\)
\(74\) 5.68246 0.660572
\(75\) 0 0
\(76\) −5.37755 −0.616847
\(77\) −23.5595 −2.68485
\(78\) −2.34767 −0.265822
\(79\) −2.96278 −0.333339 −0.166669 0.986013i \(-0.553301\pi\)
−0.166669 + 0.986013i \(0.553301\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −1.45322 −0.160481
\(83\) 13.6633 1.49975 0.749874 0.661581i \(-0.230114\pi\)
0.749874 + 0.661581i \(0.230114\pi\)
\(84\) 8.38869 0.915281
\(85\) 0 0
\(86\) 2.34227 0.252574
\(87\) 1.00000 0.107211
\(88\) −9.32773 −0.994339
\(89\) 3.67835 0.389905 0.194952 0.980813i \(-0.437545\pi\)
0.194952 + 0.980813i \(0.437545\pi\)
\(90\) 0 0
\(91\) 22.1718 2.32424
\(92\) −9.78358 −1.02001
\(93\) 9.24375 0.958532
\(94\) −1.85249 −0.191070
\(95\) 0 0
\(96\) 5.09775 0.520287
\(97\) 2.87658 0.292072 0.146036 0.989279i \(-0.453348\pi\)
0.146036 + 0.989279i \(0.453348\pi\)
\(98\) 8.30730 0.839164
\(99\) 4.88439 0.490899
\(100\) 0 0
\(101\) −14.6780 −1.46051 −0.730255 0.683174i \(-0.760599\pi\)
−0.730255 + 0.683174i \(0.760599\pi\)
\(102\) −3.32480 −0.329204
\(103\) 2.48154 0.244514 0.122257 0.992498i \(-0.460987\pi\)
0.122257 + 0.992498i \(0.460987\pi\)
\(104\) 8.77831 0.860784
\(105\) 0 0
\(106\) 0.494309 0.0480115
\(107\) 4.48423 0.433507 0.216753 0.976226i \(-0.430453\pi\)
0.216753 + 0.976226i \(0.430453\pi\)
\(108\) −1.73915 −0.167350
\(109\) 3.28140 0.314301 0.157151 0.987575i \(-0.449769\pi\)
0.157151 + 0.987575i \(0.449769\pi\)
\(110\) 0 0
\(111\) 11.1261 1.05604
\(112\) −12.0729 −1.14078
\(113\) 17.9820 1.69160 0.845800 0.533500i \(-0.179123\pi\)
0.845800 + 0.533500i \(0.179123\pi\)
\(114\) 1.57921 0.147906
\(115\) 0 0
\(116\) −1.73915 −0.161476
\(117\) −4.59669 −0.424964
\(118\) −0.152649 −0.0140524
\(119\) 31.3999 2.87843
\(120\) 0 0
\(121\) 12.8572 1.16884
\(122\) −0.401899 −0.0363862
\(123\) −2.84537 −0.256558
\(124\) −16.0763 −1.44369
\(125\) 0 0
\(126\) −2.46348 −0.219464
\(127\) 13.5099 1.19881 0.599405 0.800446i \(-0.295404\pi\)
0.599405 + 0.800446i \(0.295404\pi\)
\(128\) −11.4224 −1.00961
\(129\) 4.58611 0.403784
\(130\) 0 0
\(131\) 12.1450 1.06111 0.530556 0.847650i \(-0.321983\pi\)
0.530556 + 0.847650i \(0.321983\pi\)
\(132\) −8.49470 −0.739368
\(133\) −14.9143 −1.29323
\(134\) −2.48594 −0.214753
\(135\) 0 0
\(136\) 12.4319 1.06603
\(137\) 9.97466 0.852193 0.426096 0.904678i \(-0.359888\pi\)
0.426096 + 0.904678i \(0.359888\pi\)
\(138\) 2.87311 0.244576
\(139\) 2.82971 0.240013 0.120007 0.992773i \(-0.461708\pi\)
0.120007 + 0.992773i \(0.461708\pi\)
\(140\) 0 0
\(141\) −3.62713 −0.305459
\(142\) 0.378804 0.0317885
\(143\) −22.4520 −1.87753
\(144\) 2.50296 0.208580
\(145\) 0 0
\(146\) −2.82425 −0.233736
\(147\) 16.2655 1.34155
\(148\) −19.3500 −1.59056
\(149\) −22.5396 −1.84652 −0.923259 0.384178i \(-0.874485\pi\)
−0.923259 + 0.384178i \(0.874485\pi\)
\(150\) 0 0
\(151\) −14.2205 −1.15725 −0.578626 0.815593i \(-0.696411\pi\)
−0.578626 + 0.815593i \(0.696411\pi\)
\(152\) −5.90490 −0.478950
\(153\) −6.50987 −0.526292
\(154\) −12.0326 −0.969613
\(155\) 0 0
\(156\) 7.99434 0.640059
\(157\) −10.7712 −0.859632 −0.429816 0.902917i \(-0.641421\pi\)
−0.429816 + 0.902917i \(0.641421\pi\)
\(158\) −1.51319 −0.120383
\(159\) 0.967845 0.0767551
\(160\) 0 0
\(161\) −27.1341 −2.13847
\(162\) 0.510732 0.0401269
\(163\) 10.6908 0.837365 0.418682 0.908133i \(-0.362492\pi\)
0.418682 + 0.908133i \(0.362492\pi\)
\(164\) 4.94853 0.386415
\(165\) 0 0
\(166\) 6.97830 0.541621
\(167\) −7.90643 −0.611818 −0.305909 0.952061i \(-0.598960\pi\)
−0.305909 + 0.952061i \(0.598960\pi\)
\(168\) 9.21132 0.710669
\(169\) 8.12952 0.625347
\(170\) 0 0
\(171\) 3.09205 0.236455
\(172\) −7.97595 −0.608160
\(173\) −3.76760 −0.286445 −0.143223 0.989690i \(-0.545747\pi\)
−0.143223 + 0.989690i \(0.545747\pi\)
\(174\) 0.510732 0.0387185
\(175\) 0 0
\(176\) 12.2254 0.921526
\(177\) −0.298882 −0.0224654
\(178\) 1.87865 0.140811
\(179\) 10.3282 0.771964 0.385982 0.922506i \(-0.373863\pi\)
0.385982 + 0.922506i \(0.373863\pi\)
\(180\) 0 0
\(181\) 14.4533 1.07431 0.537154 0.843484i \(-0.319500\pi\)
0.537154 + 0.843484i \(0.319500\pi\)
\(182\) 11.3238 0.839378
\(183\) −0.786908 −0.0581699
\(184\) −10.7430 −0.791985
\(185\) 0 0
\(186\) 4.72108 0.346166
\(187\) −31.7967 −2.32521
\(188\) 6.30813 0.460068
\(189\) −4.82343 −0.350853
\(190\) 0 0
\(191\) 8.81550 0.637867 0.318934 0.947777i \(-0.396675\pi\)
0.318934 + 0.947777i \(0.396675\pi\)
\(192\) −2.40234 −0.173374
\(193\) −17.2543 −1.24199 −0.620996 0.783814i \(-0.713272\pi\)
−0.620996 + 0.783814i \(0.713272\pi\)
\(194\) 1.46916 0.105479
\(195\) 0 0
\(196\) −28.2882 −2.02058
\(197\) 22.5719 1.60818 0.804090 0.594508i \(-0.202653\pi\)
0.804090 + 0.594508i \(0.202653\pi\)
\(198\) 2.49461 0.177284
\(199\) −13.7975 −0.978078 −0.489039 0.872262i \(-0.662652\pi\)
−0.489039 + 0.872262i \(0.662652\pi\)
\(200\) 0 0
\(201\) −4.86742 −0.343321
\(202\) −7.49649 −0.527451
\(203\) −4.82343 −0.338538
\(204\) 11.3217 0.792675
\(205\) 0 0
\(206\) 1.26740 0.0883041
\(207\) 5.62549 0.390998
\(208\) −11.5053 −0.797751
\(209\) 15.1028 1.04468
\(210\) 0 0
\(211\) 16.4705 1.13387 0.566937 0.823761i \(-0.308128\pi\)
0.566937 + 0.823761i \(0.308128\pi\)
\(212\) −1.68323 −0.115605
\(213\) 0.741689 0.0508197
\(214\) 2.29024 0.156557
\(215\) 0 0
\(216\) −1.90970 −0.129939
\(217\) −44.5866 −3.02674
\(218\) 1.67592 0.113507
\(219\) −5.52981 −0.373670
\(220\) 0 0
\(221\) 29.9238 2.01289
\(222\) 5.68246 0.381382
\(223\) 4.58903 0.307304 0.153652 0.988125i \(-0.450896\pi\)
0.153652 + 0.988125i \(0.450896\pi\)
\(224\) −24.5886 −1.64290
\(225\) 0 0
\(226\) 9.18396 0.610908
\(227\) 25.5254 1.69418 0.847089 0.531452i \(-0.178353\pi\)
0.847089 + 0.531452i \(0.178353\pi\)
\(228\) −5.37755 −0.356137
\(229\) −8.04982 −0.531947 −0.265974 0.963980i \(-0.585693\pi\)
−0.265974 + 0.963980i \(0.585693\pi\)
\(230\) 0 0
\(231\) −23.5595 −1.55010
\(232\) −1.90970 −0.125378
\(233\) −3.75903 −0.246262 −0.123131 0.992390i \(-0.539294\pi\)
−0.123131 + 0.992390i \(0.539294\pi\)
\(234\) −2.34767 −0.153472
\(235\) 0 0
\(236\) 0.519802 0.0338362
\(237\) −2.96278 −0.192453
\(238\) 16.0369 1.03952
\(239\) −3.53192 −0.228461 −0.114231 0.993454i \(-0.536440\pi\)
−0.114231 + 0.993454i \(0.536440\pi\)
\(240\) 0 0
\(241\) 3.05553 0.196824 0.0984118 0.995146i \(-0.468624\pi\)
0.0984118 + 0.995146i \(0.468624\pi\)
\(242\) 6.56659 0.422117
\(243\) 1.00000 0.0641500
\(244\) 1.36855 0.0876126
\(245\) 0 0
\(246\) −1.45322 −0.0926538
\(247\) −14.2132 −0.904363
\(248\) −17.6528 −1.12096
\(249\) 13.6633 0.865880
\(250\) 0 0
\(251\) −23.3283 −1.47247 −0.736233 0.676728i \(-0.763397\pi\)
−0.736233 + 0.676728i \(0.763397\pi\)
\(252\) 8.38869 0.528437
\(253\) 27.4770 1.72747
\(254\) 6.89994 0.432941
\(255\) 0 0
\(256\) −1.02912 −0.0643202
\(257\) −28.2268 −1.76074 −0.880369 0.474289i \(-0.842705\pi\)
−0.880369 + 0.474289i \(0.842705\pi\)
\(258\) 2.34227 0.145823
\(259\) −53.6660 −3.33465
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 6.20283 0.383212
\(263\) 27.6688 1.70613 0.853067 0.521802i \(-0.174740\pi\)
0.853067 + 0.521802i \(0.174740\pi\)
\(264\) −9.32773 −0.574082
\(265\) 0 0
\(266\) −7.61719 −0.467040
\(267\) 3.67835 0.225112
\(268\) 8.46519 0.517094
\(269\) 0.839052 0.0511579 0.0255790 0.999673i \(-0.491857\pi\)
0.0255790 + 0.999673i \(0.491857\pi\)
\(270\) 0 0
\(271\) 1.27423 0.0774037 0.0387018 0.999251i \(-0.487678\pi\)
0.0387018 + 0.999251i \(0.487678\pi\)
\(272\) −16.2940 −0.987966
\(273\) 22.1718 1.34190
\(274\) 5.09437 0.307762
\(275\) 0 0
\(276\) −9.78358 −0.588902
\(277\) 26.7387 1.60657 0.803287 0.595592i \(-0.203083\pi\)
0.803287 + 0.595592i \(0.203083\pi\)
\(278\) 1.44522 0.0866788
\(279\) 9.24375 0.553409
\(280\) 0 0
\(281\) −16.3446 −0.975038 −0.487519 0.873112i \(-0.662098\pi\)
−0.487519 + 0.873112i \(0.662098\pi\)
\(282\) −1.85249 −0.110314
\(283\) −23.7207 −1.41005 −0.705025 0.709183i \(-0.749064\pi\)
−0.705025 + 0.709183i \(0.749064\pi\)
\(284\) −1.28991 −0.0765422
\(285\) 0 0
\(286\) −11.4669 −0.678054
\(287\) 13.7244 0.810127
\(288\) 5.09775 0.300388
\(289\) 25.3784 1.49285
\(290\) 0 0
\(291\) 2.87658 0.168628
\(292\) 9.61718 0.562803
\(293\) 8.28535 0.484035 0.242018 0.970272i \(-0.422191\pi\)
0.242018 + 0.970272i \(0.422191\pi\)
\(294\) 8.30730 0.484491
\(295\) 0 0
\(296\) −21.2476 −1.23499
\(297\) 4.88439 0.283421
\(298\) −11.5117 −0.666855
\(299\) −25.8586 −1.49544
\(300\) 0 0
\(301\) −22.1208 −1.27502
\(302\) −7.26288 −0.417932
\(303\) −14.6780 −0.843226
\(304\) 7.73927 0.443878
\(305\) 0 0
\(306\) −3.32480 −0.190066
\(307\) 16.3024 0.930425 0.465213 0.885199i \(-0.345978\pi\)
0.465213 + 0.885199i \(0.345978\pi\)
\(308\) 40.9736 2.33469
\(309\) 2.48154 0.141170
\(310\) 0 0
\(311\) −7.72085 −0.437809 −0.218905 0.975746i \(-0.570248\pi\)
−0.218905 + 0.975746i \(0.570248\pi\)
\(312\) 8.77831 0.496974
\(313\) 6.22393 0.351797 0.175899 0.984408i \(-0.443717\pi\)
0.175899 + 0.984408i \(0.443717\pi\)
\(314\) −5.50117 −0.310449
\(315\) 0 0
\(316\) 5.15273 0.289864
\(317\) 31.0672 1.74491 0.872454 0.488696i \(-0.162527\pi\)
0.872454 + 0.488696i \(0.162527\pi\)
\(318\) 0.494309 0.0277195
\(319\) 4.88439 0.273473
\(320\) 0 0
\(321\) 4.48423 0.250285
\(322\) −13.8583 −0.772291
\(323\) −20.1288 −1.12000
\(324\) −1.73915 −0.0966196
\(325\) 0 0
\(326\) 5.46011 0.302407
\(327\) 3.28140 0.181462
\(328\) 5.43381 0.300032
\(329\) 17.4952 0.964542
\(330\) 0 0
\(331\) 34.8731 1.91680 0.958399 0.285431i \(-0.0921366\pi\)
0.958399 + 0.285431i \(0.0921366\pi\)
\(332\) −23.7627 −1.30415
\(333\) 11.1261 0.609707
\(334\) −4.03806 −0.220953
\(335\) 0 0
\(336\) −12.0729 −0.658628
\(337\) 16.5875 0.903579 0.451790 0.892124i \(-0.350786\pi\)
0.451790 + 0.892124i \(0.350786\pi\)
\(338\) 4.15200 0.225839
\(339\) 17.9820 0.976646
\(340\) 0 0
\(341\) 45.1500 2.44501
\(342\) 1.57921 0.0853937
\(343\) −44.6914 −2.41311
\(344\) −8.75811 −0.472206
\(345\) 0 0
\(346\) −1.92423 −0.103447
\(347\) −0.413605 −0.0222035 −0.0111017 0.999938i \(-0.503534\pi\)
−0.0111017 + 0.999938i \(0.503534\pi\)
\(348\) −1.73915 −0.0932284
\(349\) 35.5590 1.90343 0.951715 0.306982i \(-0.0993190\pi\)
0.951715 + 0.306982i \(0.0993190\pi\)
\(350\) 0 0
\(351\) −4.59669 −0.245353
\(352\) 24.8994 1.32714
\(353\) −3.08030 −0.163948 −0.0819740 0.996634i \(-0.526122\pi\)
−0.0819740 + 0.996634i \(0.526122\pi\)
\(354\) −0.152649 −0.00811318
\(355\) 0 0
\(356\) −6.39722 −0.339052
\(357\) 31.3999 1.66186
\(358\) 5.27492 0.278788
\(359\) −14.3219 −0.755879 −0.377940 0.925830i \(-0.623367\pi\)
−0.377940 + 0.925830i \(0.623367\pi\)
\(360\) 0 0
\(361\) −9.43924 −0.496802
\(362\) 7.38177 0.387977
\(363\) 12.8572 0.674829
\(364\) −38.5601 −2.02110
\(365\) 0 0
\(366\) −0.401899 −0.0210076
\(367\) −17.3009 −0.903102 −0.451551 0.892245i \(-0.649129\pi\)
−0.451551 + 0.892245i \(0.649129\pi\)
\(368\) 14.0804 0.733990
\(369\) −2.84537 −0.148124
\(370\) 0 0
\(371\) −4.66833 −0.242368
\(372\) −16.0763 −0.833517
\(373\) −31.8314 −1.64817 −0.824083 0.566469i \(-0.808309\pi\)
−0.824083 + 0.566469i \(0.808309\pi\)
\(374\) −16.2396 −0.839729
\(375\) 0 0
\(376\) 6.92674 0.357219
\(377\) −4.59669 −0.236741
\(378\) −2.46348 −0.126708
\(379\) −18.1855 −0.934127 −0.467064 0.884224i \(-0.654688\pi\)
−0.467064 + 0.884224i \(0.654688\pi\)
\(380\) 0 0
\(381\) 13.5099 0.692134
\(382\) 4.50236 0.230361
\(383\) −32.3026 −1.65059 −0.825293 0.564704i \(-0.808990\pi\)
−0.825293 + 0.564704i \(0.808990\pi\)
\(384\) −11.4224 −0.582899
\(385\) 0 0
\(386\) −8.81232 −0.448535
\(387\) 4.58611 0.233125
\(388\) −5.00281 −0.253979
\(389\) −26.5696 −1.34713 −0.673567 0.739126i \(-0.735239\pi\)
−0.673567 + 0.739126i \(0.735239\pi\)
\(390\) 0 0
\(391\) −36.6212 −1.85201
\(392\) −31.0622 −1.56888
\(393\) 12.1450 0.612634
\(394\) 11.5282 0.580781
\(395\) 0 0
\(396\) −8.49470 −0.426875
\(397\) −8.87832 −0.445590 −0.222795 0.974865i \(-0.571518\pi\)
−0.222795 + 0.974865i \(0.571518\pi\)
\(398\) −7.04681 −0.353225
\(399\) −14.9143 −0.746648
\(400\) 0 0
\(401\) 25.6502 1.28091 0.640455 0.767995i \(-0.278746\pi\)
0.640455 + 0.767995i \(0.278746\pi\)
\(402\) −2.48594 −0.123988
\(403\) −42.4906 −2.11661
\(404\) 25.5272 1.27003
\(405\) 0 0
\(406\) −2.46348 −0.122260
\(407\) 54.3442 2.69374
\(408\) 12.4319 0.615472
\(409\) 9.92575 0.490797 0.245398 0.969422i \(-0.421081\pi\)
0.245398 + 0.969422i \(0.421081\pi\)
\(410\) 0 0
\(411\) 9.97466 0.492014
\(412\) −4.31579 −0.212624
\(413\) 1.44164 0.0709384
\(414\) 2.87311 0.141206
\(415\) 0 0
\(416\) −23.4327 −1.14888
\(417\) 2.82971 0.138572
\(418\) 7.71345 0.377277
\(419\) 8.12620 0.396991 0.198495 0.980102i \(-0.436394\pi\)
0.198495 + 0.980102i \(0.436394\pi\)
\(420\) 0 0
\(421\) −8.37826 −0.408332 −0.204166 0.978936i \(-0.565448\pi\)
−0.204166 + 0.978936i \(0.565448\pi\)
\(422\) 8.41198 0.409489
\(423\) −3.62713 −0.176357
\(424\) −1.84830 −0.0897612
\(425\) 0 0
\(426\) 0.378804 0.0183531
\(427\) 3.79559 0.183682
\(428\) −7.79876 −0.376967
\(429\) −22.4520 −1.08399
\(430\) 0 0
\(431\) −32.3877 −1.56006 −0.780030 0.625742i \(-0.784796\pi\)
−0.780030 + 0.625742i \(0.784796\pi\)
\(432\) 2.50296 0.120424
\(433\) −9.89328 −0.475441 −0.237720 0.971334i \(-0.576400\pi\)
−0.237720 + 0.971334i \(0.576400\pi\)
\(434\) −22.7718 −1.09308
\(435\) 0 0
\(436\) −5.70686 −0.273309
\(437\) 17.3943 0.832081
\(438\) −2.82425 −0.134948
\(439\) 5.84726 0.279075 0.139537 0.990217i \(-0.455438\pi\)
0.139537 + 0.990217i \(0.455438\pi\)
\(440\) 0 0
\(441\) 16.2655 0.774547
\(442\) 15.2830 0.726940
\(443\) −3.14176 −0.149270 −0.0746348 0.997211i \(-0.523779\pi\)
−0.0746348 + 0.997211i \(0.523779\pi\)
\(444\) −19.3500 −0.918311
\(445\) 0 0
\(446\) 2.34376 0.110980
\(447\) −22.5396 −1.06609
\(448\) 11.5875 0.547459
\(449\) 27.8643 1.31500 0.657500 0.753455i \(-0.271614\pi\)
0.657500 + 0.753455i \(0.271614\pi\)
\(450\) 0 0
\(451\) −13.8979 −0.654425
\(452\) −31.2734 −1.47098
\(453\) −14.2205 −0.668139
\(454\) 13.0366 0.611838
\(455\) 0 0
\(456\) −5.90490 −0.276522
\(457\) 31.8928 1.49188 0.745940 0.666013i \(-0.232000\pi\)
0.745940 + 0.666013i \(0.232000\pi\)
\(458\) −4.11130 −0.192108
\(459\) −6.50987 −0.303855
\(460\) 0 0
\(461\) 5.64660 0.262989 0.131494 0.991317i \(-0.458022\pi\)
0.131494 + 0.991317i \(0.458022\pi\)
\(462\) −12.0326 −0.559806
\(463\) −32.7602 −1.52250 −0.761248 0.648460i \(-0.775413\pi\)
−0.761248 + 0.648460i \(0.775413\pi\)
\(464\) 2.50296 0.116197
\(465\) 0 0
\(466\) −1.91985 −0.0889355
\(467\) 37.4129 1.73126 0.865632 0.500680i \(-0.166917\pi\)
0.865632 + 0.500680i \(0.166917\pi\)
\(468\) 7.99434 0.369539
\(469\) 23.4777 1.08410
\(470\) 0 0
\(471\) −10.7712 −0.496308
\(472\) 0.570777 0.0262721
\(473\) 22.4003 1.02997
\(474\) −1.51319 −0.0695029
\(475\) 0 0
\(476\) −54.6093 −2.50301
\(477\) 0.967845 0.0443146
\(478\) −1.80387 −0.0825069
\(479\) −13.0770 −0.597503 −0.298751 0.954331i \(-0.596570\pi\)
−0.298751 + 0.954331i \(0.596570\pi\)
\(480\) 0 0
\(481\) −51.1432 −2.33193
\(482\) 1.56055 0.0710813
\(483\) −27.1341 −1.23465
\(484\) −22.3607 −1.01639
\(485\) 0 0
\(486\) 0.510732 0.0231673
\(487\) −19.4492 −0.881328 −0.440664 0.897672i \(-0.645257\pi\)
−0.440664 + 0.897672i \(0.645257\pi\)
\(488\) 1.50276 0.0680268
\(489\) 10.6908 0.483453
\(490\) 0 0
\(491\) 15.8221 0.714041 0.357020 0.934097i \(-0.383793\pi\)
0.357020 + 0.934097i \(0.383793\pi\)
\(492\) 4.94853 0.223097
\(493\) −6.50987 −0.293190
\(494\) −7.25912 −0.326603
\(495\) 0 0
\(496\) 23.1367 1.03887
\(497\) −3.57749 −0.160472
\(498\) 6.97830 0.312705
\(499\) 4.88592 0.218724 0.109362 0.994002i \(-0.465119\pi\)
0.109362 + 0.994002i \(0.465119\pi\)
\(500\) 0 0
\(501\) −7.90643 −0.353233
\(502\) −11.9145 −0.531769
\(503\) 16.9719 0.756742 0.378371 0.925654i \(-0.376484\pi\)
0.378371 + 0.925654i \(0.376484\pi\)
\(504\) 9.21132 0.410305
\(505\) 0 0
\(506\) 14.0334 0.623860
\(507\) 8.12952 0.361045
\(508\) −23.4958 −1.04246
\(509\) −7.53223 −0.333860 −0.166930 0.985969i \(-0.553385\pi\)
−0.166930 + 0.985969i \(0.553385\pi\)
\(510\) 0 0
\(511\) 26.6726 1.17993
\(512\) 22.3193 0.986383
\(513\) 3.09205 0.136517
\(514\) −14.4163 −0.635876
\(515\) 0 0
\(516\) −7.97595 −0.351122
\(517\) −17.7163 −0.779162
\(518\) −27.4089 −1.20428
\(519\) −3.76760 −0.165379
\(520\) 0 0
\(521\) −0.748655 −0.0327992 −0.0163996 0.999866i \(-0.505220\pi\)
−0.0163996 + 0.999866i \(0.505220\pi\)
\(522\) 0.510732 0.0223541
\(523\) −18.3909 −0.804178 −0.402089 0.915601i \(-0.631716\pi\)
−0.402089 + 0.915601i \(0.631716\pi\)
\(524\) −21.1220 −0.922719
\(525\) 0 0
\(526\) 14.1313 0.616156
\(527\) −60.1756 −2.62129
\(528\) 12.2254 0.532043
\(529\) 8.64609 0.375917
\(530\) 0 0
\(531\) −0.298882 −0.0129704
\(532\) 25.9382 1.12456
\(533\) 13.0793 0.566525
\(534\) 1.87865 0.0812972
\(535\) 0 0
\(536\) 9.29533 0.401497
\(537\) 10.3282 0.445693
\(538\) 0.428531 0.0184753
\(539\) 79.4469 3.42202
\(540\) 0 0
\(541\) 33.4160 1.43667 0.718334 0.695698i \(-0.244905\pi\)
0.718334 + 0.695698i \(0.244905\pi\)
\(542\) 0.650787 0.0279537
\(543\) 14.4533 0.620251
\(544\) −33.1857 −1.42282
\(545\) 0 0
\(546\) 11.3238 0.484615
\(547\) −3.00020 −0.128279 −0.0641396 0.997941i \(-0.520430\pi\)
−0.0641396 + 0.997941i \(0.520430\pi\)
\(548\) −17.3475 −0.741047
\(549\) −0.786908 −0.0335844
\(550\) 0 0
\(551\) 3.09205 0.131726
\(552\) −10.7430 −0.457253
\(553\) 14.2908 0.607705
\(554\) 13.6563 0.580201
\(555\) 0 0
\(556\) −4.92131 −0.208710
\(557\) −4.11112 −0.174194 −0.0870969 0.996200i \(-0.527759\pi\)
−0.0870969 + 0.996200i \(0.527759\pi\)
\(558\) 4.72108 0.199859
\(559\) −21.0809 −0.891627
\(560\) 0 0
\(561\) −31.7967 −1.34246
\(562\) −8.34771 −0.352127
\(563\) 11.5127 0.485204 0.242602 0.970126i \(-0.421999\pi\)
0.242602 + 0.970126i \(0.421999\pi\)
\(564\) 6.30813 0.265620
\(565\) 0 0
\(566\) −12.1149 −0.509228
\(567\) −4.82343 −0.202565
\(568\) −1.41641 −0.0594311
\(569\) 32.9470 1.38121 0.690605 0.723232i \(-0.257344\pi\)
0.690605 + 0.723232i \(0.257344\pi\)
\(570\) 0 0
\(571\) −22.3213 −0.934117 −0.467059 0.884226i \(-0.654686\pi\)
−0.467059 + 0.884226i \(0.654686\pi\)
\(572\) 39.0474 1.63266
\(573\) 8.81550 0.368273
\(574\) 7.00950 0.292571
\(575\) 0 0
\(576\) −2.40234 −0.100098
\(577\) 39.8197 1.65772 0.828859 0.559458i \(-0.188991\pi\)
0.828859 + 0.559458i \(0.188991\pi\)
\(578\) 12.9616 0.539130
\(579\) −17.2543 −0.717064
\(580\) 0 0
\(581\) −65.9042 −2.73417
\(582\) 1.46916 0.0608986
\(583\) 4.72733 0.195786
\(584\) 10.5603 0.436988
\(585\) 0 0
\(586\) 4.23159 0.174805
\(587\) 28.7487 1.18659 0.593293 0.804987i \(-0.297828\pi\)
0.593293 + 0.804987i \(0.297828\pi\)
\(588\) −28.2882 −1.16658
\(589\) 28.5821 1.17771
\(590\) 0 0
\(591\) 22.5719 0.928483
\(592\) 27.8482 1.14455
\(593\) −27.5400 −1.13093 −0.565467 0.824771i \(-0.691304\pi\)
−0.565467 + 0.824771i \(0.691304\pi\)
\(594\) 2.49461 0.102355
\(595\) 0 0
\(596\) 39.1999 1.60569
\(597\) −13.7975 −0.564693
\(598\) −13.2068 −0.540066
\(599\) −4.11339 −0.168069 −0.0840343 0.996463i \(-0.526781\pi\)
−0.0840343 + 0.996463i \(0.526781\pi\)
\(600\) 0 0
\(601\) −0.656799 −0.0267914 −0.0133957 0.999910i \(-0.504264\pi\)
−0.0133957 + 0.999910i \(0.504264\pi\)
\(602\) −11.2978 −0.460463
\(603\) −4.86742 −0.198217
\(604\) 24.7317 1.00632
\(605\) 0 0
\(606\) −7.49649 −0.304524
\(607\) 18.4210 0.747684 0.373842 0.927492i \(-0.378040\pi\)
0.373842 + 0.927492i \(0.378040\pi\)
\(608\) 15.7625 0.639253
\(609\) −4.82343 −0.195455
\(610\) 0 0
\(611\) 16.6728 0.674508
\(612\) 11.3217 0.457651
\(613\) −21.3356 −0.861737 −0.430869 0.902415i \(-0.641793\pi\)
−0.430869 + 0.902415i \(0.641793\pi\)
\(614\) 8.32613 0.336015
\(615\) 0 0
\(616\) 44.9917 1.81277
\(617\) 23.2463 0.935861 0.467931 0.883765i \(-0.345000\pi\)
0.467931 + 0.883765i \(0.345000\pi\)
\(618\) 1.26740 0.0509824
\(619\) −4.11413 −0.165361 −0.0826804 0.996576i \(-0.526348\pi\)
−0.0826804 + 0.996576i \(0.526348\pi\)
\(620\) 0 0
\(621\) 5.62549 0.225743
\(622\) −3.94328 −0.158111
\(623\) −17.7423 −0.710830
\(624\) −11.5053 −0.460582
\(625\) 0 0
\(626\) 3.17876 0.127049
\(627\) 15.1028 0.603146
\(628\) 18.7327 0.747515
\(629\) −72.4296 −2.88796
\(630\) 0 0
\(631\) −45.7221 −1.82017 −0.910083 0.414426i \(-0.863982\pi\)
−0.910083 + 0.414426i \(0.863982\pi\)
\(632\) 5.65803 0.225065
\(633\) 16.4705 0.654642
\(634\) 15.8670 0.630159
\(635\) 0 0
\(636\) −1.68323 −0.0667444
\(637\) −74.7673 −2.96239
\(638\) 2.49461 0.0987626
\(639\) 0.741689 0.0293408
\(640\) 0 0
\(641\) 28.3443 1.11953 0.559767 0.828650i \(-0.310891\pi\)
0.559767 + 0.828650i \(0.310891\pi\)
\(642\) 2.29024 0.0903884
\(643\) 17.4545 0.688337 0.344168 0.938908i \(-0.388161\pi\)
0.344168 + 0.938908i \(0.388161\pi\)
\(644\) 47.1904 1.85956
\(645\) 0 0
\(646\) −10.2804 −0.404478
\(647\) −17.3874 −0.683569 −0.341784 0.939778i \(-0.611031\pi\)
−0.341784 + 0.939778i \(0.611031\pi\)
\(648\) −1.90970 −0.0750202
\(649\) −1.45986 −0.0573044
\(650\) 0 0
\(651\) −44.5866 −1.74749
\(652\) −18.5929 −0.728153
\(653\) 30.2052 1.18202 0.591010 0.806664i \(-0.298729\pi\)
0.591010 + 0.806664i \(0.298729\pi\)
\(654\) 1.67592 0.0655335
\(655\) 0 0
\(656\) −7.12184 −0.278061
\(657\) −5.52981 −0.215738
\(658\) 8.93535 0.348336
\(659\) 26.3659 1.02707 0.513535 0.858069i \(-0.328336\pi\)
0.513535 + 0.858069i \(0.328336\pi\)
\(660\) 0 0
\(661\) 15.8187 0.615274 0.307637 0.951504i \(-0.400462\pi\)
0.307637 + 0.951504i \(0.400462\pi\)
\(662\) 17.8108 0.692236
\(663\) 29.9238 1.16215
\(664\) −26.0929 −1.01260
\(665\) 0 0
\(666\) 5.68246 0.220191
\(667\) 5.62549 0.217820
\(668\) 13.7505 0.532023
\(669\) 4.58903 0.177422
\(670\) 0 0
\(671\) −3.84356 −0.148379
\(672\) −24.5886 −0.948527
\(673\) −43.1070 −1.66165 −0.830827 0.556530i \(-0.812132\pi\)
−0.830827 + 0.556530i \(0.812132\pi\)
\(674\) 8.47177 0.326320
\(675\) 0 0
\(676\) −14.1385 −0.543788
\(677\) −40.6978 −1.56415 −0.782073 0.623187i \(-0.785837\pi\)
−0.782073 + 0.623187i \(0.785837\pi\)
\(678\) 9.18396 0.352708
\(679\) −13.8750 −0.532472
\(680\) 0 0
\(681\) 25.5254 0.978134
\(682\) 23.0596 0.882996
\(683\) 21.0217 0.804372 0.402186 0.915558i \(-0.368250\pi\)
0.402186 + 0.915558i \(0.368250\pi\)
\(684\) −5.37755 −0.205616
\(685\) 0 0
\(686\) −22.8253 −0.871475
\(687\) −8.04982 −0.307120
\(688\) 11.4789 0.437627
\(689\) −4.44888 −0.169489
\(690\) 0 0
\(691\) 6.98613 0.265765 0.132882 0.991132i \(-0.457577\pi\)
0.132882 + 0.991132i \(0.457577\pi\)
\(692\) 6.55244 0.249086
\(693\) −23.5595 −0.894951
\(694\) −0.211241 −0.00801860
\(695\) 0 0
\(696\) −1.90970 −0.0723871
\(697\) 18.5230 0.701608
\(698\) 18.1611 0.687408
\(699\) −3.75903 −0.142179
\(700\) 0 0
\(701\) 1.40572 0.0530934 0.0265467 0.999648i \(-0.491549\pi\)
0.0265467 + 0.999648i \(0.491549\pi\)
\(702\) −2.34767 −0.0886072
\(703\) 34.4025 1.29751
\(704\) −11.7340 −0.442240
\(705\) 0 0
\(706\) −1.57321 −0.0592085
\(707\) 70.7981 2.66264
\(708\) 0.519802 0.0195354
\(709\) 1.27110 0.0477372 0.0238686 0.999715i \(-0.492402\pi\)
0.0238686 + 0.999715i \(0.492402\pi\)
\(710\) 0 0
\(711\) −2.96278 −0.111113
\(712\) −7.02457 −0.263257
\(713\) 52.0006 1.94744
\(714\) 16.0369 0.600167
\(715\) 0 0
\(716\) −17.9623 −0.671282
\(717\) −3.53192 −0.131902
\(718\) −7.31463 −0.272979
\(719\) 27.8439 1.03840 0.519201 0.854652i \(-0.326230\pi\)
0.519201 + 0.854652i \(0.326230\pi\)
\(720\) 0 0
\(721\) −11.9696 −0.445770
\(722\) −4.82092 −0.179416
\(723\) 3.05553 0.113636
\(724\) −25.1365 −0.934192
\(725\) 0 0
\(726\) 6.56659 0.243709
\(727\) 8.97787 0.332971 0.166485 0.986044i \(-0.446758\pi\)
0.166485 + 0.986044i \(0.446758\pi\)
\(728\) −42.3416 −1.56928
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −29.8550 −1.10423
\(732\) 1.36855 0.0505832
\(733\) 34.7466 1.28339 0.641697 0.766958i \(-0.278231\pi\)
0.641697 + 0.766958i \(0.278231\pi\)
\(734\) −8.83614 −0.326148
\(735\) 0 0
\(736\) 28.6773 1.05706
\(737\) −23.7744 −0.875740
\(738\) −1.45322 −0.0534937
\(739\) −3.32518 −0.122319 −0.0611594 0.998128i \(-0.519480\pi\)
−0.0611594 + 0.998128i \(0.519480\pi\)
\(740\) 0 0
\(741\) −14.2132 −0.522134
\(742\) −2.38426 −0.0875291
\(743\) 1.06314 0.0390029 0.0195015 0.999810i \(-0.493792\pi\)
0.0195015 + 0.999810i \(0.493792\pi\)
\(744\) −17.6528 −0.647184
\(745\) 0 0
\(746\) −16.2573 −0.595222
\(747\) 13.6633 0.499916
\(748\) 55.2994 2.02195
\(749\) −21.6294 −0.790320
\(750\) 0 0
\(751\) −38.7756 −1.41494 −0.707472 0.706742i \(-0.750164\pi\)
−0.707472 + 0.706742i \(0.750164\pi\)
\(752\) −9.07856 −0.331061
\(753\) −23.3283 −0.850129
\(754\) −2.34767 −0.0854972
\(755\) 0 0
\(756\) 8.38869 0.305094
\(757\) −40.9695 −1.48906 −0.744531 0.667588i \(-0.767327\pi\)
−0.744531 + 0.667588i \(0.767327\pi\)
\(758\) −9.28792 −0.337352
\(759\) 27.4770 0.997354
\(760\) 0 0
\(761\) 13.8561 0.502283 0.251142 0.967950i \(-0.419194\pi\)
0.251142 + 0.967950i \(0.419194\pi\)
\(762\) 6.89994 0.249958
\(763\) −15.8276 −0.572998
\(764\) −15.3315 −0.554675
\(765\) 0 0
\(766\) −16.4980 −0.596096
\(767\) 1.37387 0.0496075
\(768\) −1.02912 −0.0371353
\(769\) −6.73919 −0.243021 −0.121511 0.992590i \(-0.538774\pi\)
−0.121511 + 0.992590i \(0.538774\pi\)
\(770\) 0 0
\(771\) −28.2268 −1.01656
\(772\) 30.0079 1.08001
\(773\) −7.45730 −0.268220 −0.134110 0.990966i \(-0.542818\pi\)
−0.134110 + 0.990966i \(0.542818\pi\)
\(774\) 2.34227 0.0841912
\(775\) 0 0
\(776\) −5.49341 −0.197202
\(777\) −53.6660 −1.92526
\(778\) −13.5700 −0.486506
\(779\) −8.79801 −0.315221
\(780\) 0 0
\(781\) 3.62270 0.129630
\(782\) −18.7036 −0.668839
\(783\) 1.00000 0.0357371
\(784\) 40.7119 1.45400
\(785\) 0 0
\(786\) 6.20283 0.221248
\(787\) 7.81608 0.278613 0.139307 0.990249i \(-0.455513\pi\)
0.139307 + 0.990249i \(0.455513\pi\)
\(788\) −39.2559 −1.39844
\(789\) 27.6688 0.985037
\(790\) 0 0
\(791\) −86.7347 −3.08393
\(792\) −9.32773 −0.331446
\(793\) 3.61717 0.128449
\(794\) −4.53444 −0.160921
\(795\) 0 0
\(796\) 23.9959 0.850514
\(797\) −21.8313 −0.773305 −0.386653 0.922225i \(-0.626369\pi\)
−0.386653 + 0.922225i \(0.626369\pi\)
\(798\) −7.61719 −0.269646
\(799\) 23.6121 0.835338
\(800\) 0 0
\(801\) 3.67835 0.129968
\(802\) 13.1004 0.462590
\(803\) −27.0097 −0.953152
\(804\) 8.46519 0.298544
\(805\) 0 0
\(806\) −21.7013 −0.764396
\(807\) 0.839052 0.0295360
\(808\) 28.0305 0.986111
\(809\) 5.71715 0.201004 0.100502 0.994937i \(-0.467955\pi\)
0.100502 + 0.994937i \(0.467955\pi\)
\(810\) 0 0
\(811\) −44.8661 −1.57546 −0.787732 0.616019i \(-0.788745\pi\)
−0.787732 + 0.616019i \(0.788745\pi\)
\(812\) 8.38869 0.294385
\(813\) 1.27423 0.0446890
\(814\) 27.7553 0.972823
\(815\) 0 0
\(816\) −16.2940 −0.570402
\(817\) 14.1805 0.496112
\(818\) 5.06939 0.177247
\(819\) 22.1718 0.774745
\(820\) 0 0
\(821\) 37.3699 1.30422 0.652110 0.758125i \(-0.273884\pi\)
0.652110 + 0.758125i \(0.273884\pi\)
\(822\) 5.09437 0.177687
\(823\) 31.6982 1.10493 0.552464 0.833537i \(-0.313688\pi\)
0.552464 + 0.833537i \(0.313688\pi\)
\(824\) −4.73901 −0.165091
\(825\) 0 0
\(826\) 0.736290 0.0256188
\(827\) 14.4089 0.501046 0.250523 0.968111i \(-0.419397\pi\)
0.250523 + 0.968111i \(0.419397\pi\)
\(828\) −9.78358 −0.340003
\(829\) −20.6622 −0.717630 −0.358815 0.933409i \(-0.616819\pi\)
−0.358815 + 0.933409i \(0.616819\pi\)
\(830\) 0 0
\(831\) 26.7387 0.927556
\(832\) 11.0428 0.382840
\(833\) −105.886 −3.66874
\(834\) 1.44522 0.0500440
\(835\) 0 0
\(836\) −26.2660 −0.908429
\(837\) 9.24375 0.319511
\(838\) 4.15031 0.143370
\(839\) 1.64970 0.0569541 0.0284771 0.999594i \(-0.490934\pi\)
0.0284771 + 0.999594i \(0.490934\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −4.27904 −0.147466
\(843\) −16.3446 −0.562938
\(844\) −28.6447 −0.985990
\(845\) 0 0
\(846\) −1.85249 −0.0636899
\(847\) −62.0159 −2.13089
\(848\) 2.42248 0.0831882
\(849\) −23.7207 −0.814092
\(850\) 0 0
\(851\) 62.5898 2.14555
\(852\) −1.28991 −0.0441916
\(853\) 23.4066 0.801428 0.400714 0.916203i \(-0.368762\pi\)
0.400714 + 0.916203i \(0.368762\pi\)
\(854\) 1.93853 0.0663351
\(855\) 0 0
\(856\) −8.56355 −0.292696
\(857\) −18.7479 −0.640415 −0.320207 0.947347i \(-0.603753\pi\)
−0.320207 + 0.947347i \(0.603753\pi\)
\(858\) −11.4669 −0.391475
\(859\) −29.1326 −0.993990 −0.496995 0.867753i \(-0.665563\pi\)
−0.496995 + 0.867753i \(0.665563\pi\)
\(860\) 0 0
\(861\) 13.7244 0.467727
\(862\) −16.5414 −0.563403
\(863\) −25.5033 −0.868142 −0.434071 0.900879i \(-0.642923\pi\)
−0.434071 + 0.900879i \(0.642923\pi\)
\(864\) 5.09775 0.173429
\(865\) 0 0
\(866\) −5.05281 −0.171701
\(867\) 25.3784 0.861897
\(868\) 77.5429 2.63198
\(869\) −14.4714 −0.490908
\(870\) 0 0
\(871\) 22.3740 0.758114
\(872\) −6.26651 −0.212211
\(873\) 2.87658 0.0973573
\(874\) 8.88380 0.300499
\(875\) 0 0
\(876\) 9.61718 0.324935
\(877\) −24.3134 −0.821005 −0.410502 0.911860i \(-0.634647\pi\)
−0.410502 + 0.911860i \(0.634647\pi\)
\(878\) 2.98638 0.100786
\(879\) 8.28535 0.279458
\(880\) 0 0
\(881\) 51.3387 1.72965 0.864823 0.502077i \(-0.167431\pi\)
0.864823 + 0.502077i \(0.167431\pi\)
\(882\) 8.30730 0.279721
\(883\) −17.1764 −0.578033 −0.289016 0.957324i \(-0.593328\pi\)
−0.289016 + 0.957324i \(0.593328\pi\)
\(884\) −52.0421 −1.75037
\(885\) 0 0
\(886\) −1.60460 −0.0539075
\(887\) −38.2392 −1.28395 −0.641974 0.766727i \(-0.721884\pi\)
−0.641974 + 0.766727i \(0.721884\pi\)
\(888\) −21.2476 −0.713022
\(889\) −65.1641 −2.18553
\(890\) 0 0
\(891\) 4.88439 0.163633
\(892\) −7.98103 −0.267225
\(893\) −11.2153 −0.375304
\(894\) −11.5117 −0.385009
\(895\) 0 0
\(896\) 55.0954 1.84061
\(897\) −25.8586 −0.863393
\(898\) 14.2312 0.474901
\(899\) 9.24375 0.308296
\(900\) 0 0
\(901\) −6.30055 −0.209902
\(902\) −7.09808 −0.236340
\(903\) −22.1208 −0.736134
\(904\) −34.3402 −1.14214
\(905\) 0 0
\(906\) −7.26288 −0.241293
\(907\) −9.37007 −0.311128 −0.155564 0.987826i \(-0.549719\pi\)
−0.155564 + 0.987826i \(0.549719\pi\)
\(908\) −44.3925 −1.47322
\(909\) −14.6780 −0.486837
\(910\) 0 0
\(911\) −10.5220 −0.348611 −0.174305 0.984692i \(-0.555768\pi\)
−0.174305 + 0.984692i \(0.555768\pi\)
\(912\) 7.73927 0.256273
\(913\) 66.7371 2.20867
\(914\) 16.2886 0.538780
\(915\) 0 0
\(916\) 13.9999 0.462569
\(917\) −58.5805 −1.93450
\(918\) −3.32480 −0.109735
\(919\) −47.9043 −1.58022 −0.790109 0.612967i \(-0.789976\pi\)
−0.790109 + 0.612967i \(0.789976\pi\)
\(920\) 0 0
\(921\) 16.3024 0.537181
\(922\) 2.88390 0.0949762
\(923\) −3.40931 −0.112219
\(924\) 40.9736 1.34793
\(925\) 0 0
\(926\) −16.7317 −0.549837
\(927\) 2.48154 0.0815046
\(928\) 5.09775 0.167342
\(929\) 32.0568 1.05175 0.525875 0.850562i \(-0.323738\pi\)
0.525875 + 0.850562i \(0.323738\pi\)
\(930\) 0 0
\(931\) 50.2937 1.64831
\(932\) 6.53752 0.214144
\(933\) −7.72085 −0.252769
\(934\) 19.1080 0.625232
\(935\) 0 0
\(936\) 8.77831 0.286928
\(937\) −23.8183 −0.778109 −0.389054 0.921215i \(-0.627198\pi\)
−0.389054 + 0.921215i \(0.627198\pi\)
\(938\) 11.9908 0.391513
\(939\) 6.22393 0.203110
\(940\) 0 0
\(941\) 45.7836 1.49250 0.746251 0.665664i \(-0.231852\pi\)
0.746251 + 0.665664i \(0.231852\pi\)
\(942\) −5.50117 −0.179238
\(943\) −16.0066 −0.521246
\(944\) −0.748091 −0.0243483
\(945\) 0 0
\(946\) 11.4406 0.371965
\(947\) 4.69130 0.152447 0.0762233 0.997091i \(-0.475714\pi\)
0.0762233 + 0.997091i \(0.475714\pi\)
\(948\) 5.15273 0.167353
\(949\) 25.4188 0.825129
\(950\) 0 0
\(951\) 31.0672 1.00742
\(952\) −59.9645 −1.94346
\(953\) −56.4381 −1.82821 −0.914104 0.405479i \(-0.867105\pi\)
−0.914104 + 0.405479i \(0.867105\pi\)
\(954\) 0.494309 0.0160038
\(955\) 0 0
\(956\) 6.14256 0.198665
\(957\) 4.88439 0.157890
\(958\) −6.67883 −0.215783
\(959\) −48.1121 −1.55362
\(960\) 0 0
\(961\) 54.4469 1.75635
\(962\) −26.1205 −0.842158
\(963\) 4.48423 0.144502
\(964\) −5.31403 −0.171153
\(965\) 0 0
\(966\) −13.8583 −0.445882
\(967\) 42.2611 1.35902 0.679512 0.733664i \(-0.262192\pi\)
0.679512 + 0.733664i \(0.262192\pi\)
\(968\) −24.5535 −0.789179
\(969\) −20.1288 −0.646631
\(970\) 0 0
\(971\) 25.6613 0.823510 0.411755 0.911295i \(-0.364916\pi\)
0.411755 + 0.911295i \(0.364916\pi\)
\(972\) −1.73915 −0.0557834
\(973\) −13.6489 −0.437565
\(974\) −9.93333 −0.318284
\(975\) 0 0
\(976\) −1.96960 −0.0630453
\(977\) −7.35242 −0.235225 −0.117612 0.993060i \(-0.537524\pi\)
−0.117612 + 0.993060i \(0.537524\pi\)
\(978\) 5.46011 0.174595
\(979\) 17.9665 0.574212
\(980\) 0 0
\(981\) 3.28140 0.104767
\(982\) 8.08084 0.257870
\(983\) −3.23911 −0.103312 −0.0516558 0.998665i \(-0.516450\pi\)
−0.0516558 + 0.998665i \(0.516450\pi\)
\(984\) 5.43381 0.173223
\(985\) 0 0
\(986\) −3.32480 −0.105883
\(987\) 17.4952 0.556879
\(988\) 24.7189 0.786413
\(989\) 25.7991 0.820364
\(990\) 0 0
\(991\) 3.19230 0.101407 0.0507034 0.998714i \(-0.483854\pi\)
0.0507034 + 0.998714i \(0.483854\pi\)
\(992\) 47.1223 1.49614
\(993\) 34.8731 1.10666
\(994\) −1.82714 −0.0579532
\(995\) 0 0
\(996\) −23.7627 −0.752949
\(997\) 49.0253 1.55265 0.776323 0.630335i \(-0.217083\pi\)
0.776323 + 0.630335i \(0.217083\pi\)
\(998\) 2.49539 0.0789903
\(999\) 11.1261 0.352015
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 2175.2.a.bd.1.5 yes 8
3.2 odd 2 6525.2.a.by.1.4 8
5.2 odd 4 2175.2.c.p.349.11 16
5.3 odd 4 2175.2.c.p.349.6 16
5.4 even 2 2175.2.a.bc.1.4 8
15.14 odd 2 6525.2.a.bz.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2175.2.a.bc.1.4 8 5.4 even 2
2175.2.a.bd.1.5 yes 8 1.1 even 1 trivial
2175.2.c.p.349.6 16 5.3 odd 4
2175.2.c.p.349.11 16 5.2 odd 4
6525.2.a.by.1.4 8 3.2 odd 2
6525.2.a.bz.1.5 8 15.14 odd 2