Properties

Label 625.2.a.e.1.3
Level $625$
Weight $2$
Character 625.1
Self dual yes
Analytic conductor $4.991$
Analytic rank $1$
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] = [625,2,Mod(1,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.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: \(4.99065012633\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.6152203125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.32610\) of defining polynomial
Character \(\chi\) \(=\) 625.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.32610 q^{2} +2.30231 q^{3} +3.41075 q^{4} -5.35542 q^{6} -3.59425 q^{7} -3.28154 q^{8} +2.30065 q^{9} +O(q^{10})\) \(q-2.32610 q^{2} +2.30231 q^{3} +3.41075 q^{4} -5.35542 q^{6} -3.59425 q^{7} -3.28154 q^{8} +2.30065 q^{9} -0.497788 q^{11} +7.85261 q^{12} -2.64789 q^{13} +8.36058 q^{14} +0.811708 q^{16} -5.10719 q^{17} -5.35154 q^{18} -0.987277 q^{19} -8.27508 q^{21} +1.15790 q^{22} -6.41382 q^{23} -7.55514 q^{24} +6.15926 q^{26} -1.61013 q^{27} -12.2591 q^{28} -5.57001 q^{29} +6.05507 q^{31} +4.67497 q^{32} -1.14606 q^{33} +11.8798 q^{34} +7.84693 q^{36} +4.59612 q^{37} +2.29651 q^{38} -6.09627 q^{39} -2.87475 q^{41} +19.2487 q^{42} +9.48858 q^{43} -1.69783 q^{44} +14.9192 q^{46} -5.36834 q^{47} +1.86881 q^{48} +5.91861 q^{49} -11.7583 q^{51} -9.03129 q^{52} +0.307600 q^{53} +3.74532 q^{54} +11.7947 q^{56} -2.27302 q^{57} +12.9564 q^{58} -1.26645 q^{59} -6.22625 q^{61} -14.0847 q^{62} -8.26910 q^{63} -12.4979 q^{64} +2.66586 q^{66} +5.28626 q^{67} -17.4193 q^{68} -14.7666 q^{69} -0.151963 q^{71} -7.54968 q^{72} +14.8741 q^{73} -10.6910 q^{74} -3.36735 q^{76} +1.78917 q^{77} +14.1805 q^{78} -16.5886 q^{79} -10.6090 q^{81} +6.68696 q^{82} -14.5960 q^{83} -28.2242 q^{84} -22.0714 q^{86} -12.8239 q^{87} +1.63351 q^{88} +11.3822 q^{89} +9.51717 q^{91} -21.8759 q^{92} +13.9407 q^{93} +12.4873 q^{94} +10.7633 q^{96} -0.849192 q^{97} -13.7673 q^{98} -1.14523 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9} + q^{11} - 10 q^{12} - 10 q^{13} - 8 q^{14} + 13 q^{16} - 15 q^{17} + 5 q^{18} - 10 q^{19} - 14 q^{21} + 5 q^{22} - 30 q^{23} + 5 q^{24} + 11 q^{26} - 20 q^{27} + 5 q^{28} + 10 q^{29} - 9 q^{31} - 30 q^{32} - 5 q^{33} + 7 q^{34} + 3 q^{36} + 10 q^{37} - 20 q^{38} + 8 q^{39} - 4 q^{41} + 35 q^{42} - 18 q^{44} - 9 q^{46} - 30 q^{47} - 5 q^{48} - 4 q^{49} - 14 q^{51} - 5 q^{52} - 10 q^{53} - 20 q^{54} + 10 q^{57} + 30 q^{58} - 5 q^{59} + 6 q^{61} - 10 q^{62} - 9 q^{64} - 18 q^{66} - 10 q^{67} - 40 q^{68} + 3 q^{69} - 9 q^{71} + 15 q^{72} - 18 q^{74} - 10 q^{76} - 5 q^{77} + 30 q^{78} - 20 q^{79} + 8 q^{81} + 45 q^{82} - 40 q^{83} - 28 q^{84} - 24 q^{86} - 40 q^{87} + 40 q^{88} - 5 q^{89} + 6 q^{91} - 15 q^{92} + 40 q^{93} + 47 q^{94} + 71 q^{96} + 30 q^{98} - 22 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.32610 −1.64480 −0.822401 0.568908i \(-0.807366\pi\)
−0.822401 + 0.568908i \(0.807366\pi\)
\(3\) 2.30231 1.32924 0.664621 0.747181i \(-0.268593\pi\)
0.664621 + 0.747181i \(0.268593\pi\)
\(4\) 3.41075 1.70537
\(5\) 0 0
\(6\) −5.35542 −2.18634
\(7\) −3.59425 −1.35850 −0.679249 0.733908i \(-0.737694\pi\)
−0.679249 + 0.733908i \(0.737694\pi\)
\(8\) −3.28154 −1.16020
\(9\) 2.30065 0.766883
\(10\) 0 0
\(11\) −0.497788 −0.150089 −0.0750443 0.997180i \(-0.523910\pi\)
−0.0750443 + 0.997180i \(0.523910\pi\)
\(12\) 7.85261 2.26685
\(13\) −2.64789 −0.734392 −0.367196 0.930143i \(-0.619682\pi\)
−0.367196 + 0.930143i \(0.619682\pi\)
\(14\) 8.36058 2.23446
\(15\) 0 0
\(16\) 0.811708 0.202927
\(17\) −5.10719 −1.23867 −0.619337 0.785125i \(-0.712599\pi\)
−0.619337 + 0.785125i \(0.712599\pi\)
\(18\) −5.35154 −1.26137
\(19\) −0.987277 −0.226497 −0.113248 0.993567i \(-0.536126\pi\)
−0.113248 + 0.993567i \(0.536126\pi\)
\(20\) 0 0
\(21\) −8.27508 −1.80577
\(22\) 1.15790 0.246866
\(23\) −6.41382 −1.33737 −0.668687 0.743544i \(-0.733143\pi\)
−0.668687 + 0.743544i \(0.733143\pi\)
\(24\) −7.55514 −1.54219
\(25\) 0 0
\(26\) 6.15926 1.20793
\(27\) −1.61013 −0.309869
\(28\) −12.2591 −2.31675
\(29\) −5.57001 −1.03432 −0.517162 0.855887i \(-0.673012\pi\)
−0.517162 + 0.855887i \(0.673012\pi\)
\(30\) 0 0
\(31\) 6.05507 1.08752 0.543762 0.839240i \(-0.317000\pi\)
0.543762 + 0.839240i \(0.317000\pi\)
\(32\) 4.67497 0.826426
\(33\) −1.14606 −0.199504
\(34\) 11.8798 2.03738
\(35\) 0 0
\(36\) 7.84693 1.30782
\(37\) 4.59612 0.755598 0.377799 0.925888i \(-0.376681\pi\)
0.377799 + 0.925888i \(0.376681\pi\)
\(38\) 2.29651 0.372543
\(39\) −6.09627 −0.976185
\(40\) 0 0
\(41\) −2.87475 −0.448960 −0.224480 0.974479i \(-0.572068\pi\)
−0.224480 + 0.974479i \(0.572068\pi\)
\(42\) 19.2487 2.97014
\(43\) 9.48858 1.44700 0.723498 0.690327i \(-0.242533\pi\)
0.723498 + 0.690327i \(0.242533\pi\)
\(44\) −1.69783 −0.255957
\(45\) 0 0
\(46\) 14.9192 2.19972
\(47\) −5.36834 −0.783053 −0.391527 0.920167i \(-0.628053\pi\)
−0.391527 + 0.920167i \(0.628053\pi\)
\(48\) 1.86881 0.269739
\(49\) 5.91861 0.845515
\(50\) 0 0
\(51\) −11.7583 −1.64650
\(52\) −9.03129 −1.25241
\(53\) 0.307600 0.0422521 0.0211261 0.999777i \(-0.493275\pi\)
0.0211261 + 0.999777i \(0.493275\pi\)
\(54\) 3.74532 0.509673
\(55\) 0 0
\(56\) 11.7947 1.57613
\(57\) −2.27302 −0.301069
\(58\) 12.9564 1.70126
\(59\) −1.26645 −0.164878 −0.0824389 0.996596i \(-0.526271\pi\)
−0.0824389 + 0.996596i \(0.526271\pi\)
\(60\) 0 0
\(61\) −6.22625 −0.797190 −0.398595 0.917127i \(-0.630502\pi\)
−0.398595 + 0.917127i \(0.630502\pi\)
\(62\) −14.0847 −1.78876
\(63\) −8.26910 −1.04181
\(64\) −12.4979 −1.56223
\(65\) 0 0
\(66\) 2.66586 0.328145
\(67\) 5.28626 0.645819 0.322910 0.946430i \(-0.395339\pi\)
0.322910 + 0.946430i \(0.395339\pi\)
\(68\) −17.4193 −2.11240
\(69\) −14.7666 −1.77769
\(70\) 0 0
\(71\) −0.151963 −0.0180347 −0.00901734 0.999959i \(-0.502870\pi\)
−0.00901734 + 0.999959i \(0.502870\pi\)
\(72\) −7.54968 −0.889738
\(73\) 14.8741 1.74088 0.870439 0.492276i \(-0.163835\pi\)
0.870439 + 0.492276i \(0.163835\pi\)
\(74\) −10.6910 −1.24281
\(75\) 0 0
\(76\) −3.36735 −0.386262
\(77\) 1.78917 0.203895
\(78\) 14.1805 1.60563
\(79\) −16.5886 −1.86636 −0.933181 0.359406i \(-0.882979\pi\)
−0.933181 + 0.359406i \(0.882979\pi\)
\(80\) 0 0
\(81\) −10.6090 −1.17877
\(82\) 6.68696 0.738451
\(83\) −14.5960 −1.60212 −0.801058 0.598587i \(-0.795729\pi\)
−0.801058 + 0.598587i \(0.795729\pi\)
\(84\) −28.2242 −3.07952
\(85\) 0 0
\(86\) −22.0714 −2.38002
\(87\) −12.8239 −1.37487
\(88\) 1.63351 0.174133
\(89\) 11.3822 1.20652 0.603258 0.797546i \(-0.293869\pi\)
0.603258 + 0.797546i \(0.293869\pi\)
\(90\) 0 0
\(91\) 9.51717 0.997670
\(92\) −21.8759 −2.28072
\(93\) 13.9407 1.44558
\(94\) 12.4873 1.28797
\(95\) 0 0
\(96\) 10.7633 1.09852
\(97\) −0.849192 −0.0862223 −0.0431112 0.999070i \(-0.513727\pi\)
−0.0431112 + 0.999070i \(0.513727\pi\)
\(98\) −13.7673 −1.39070
\(99\) −1.14523 −0.115100
\(100\) 0 0
\(101\) −13.2498 −1.31841 −0.659203 0.751965i \(-0.729106\pi\)
−0.659203 + 0.751965i \(0.729106\pi\)
\(102\) 27.3511 2.70816
\(103\) −0.830909 −0.0818719 −0.0409360 0.999162i \(-0.513034\pi\)
−0.0409360 + 0.999162i \(0.513034\pi\)
\(104\) 8.68917 0.852043
\(105\) 0 0
\(106\) −0.715509 −0.0694964
\(107\) 0.0722844 0.00698800 0.00349400 0.999994i \(-0.498888\pi\)
0.00349400 + 0.999994i \(0.498888\pi\)
\(108\) −5.49174 −0.528443
\(109\) 5.59621 0.536020 0.268010 0.963416i \(-0.413634\pi\)
0.268010 + 0.963416i \(0.413634\pi\)
\(110\) 0 0
\(111\) 10.5817 1.00437
\(112\) −2.91748 −0.275676
\(113\) 2.60239 0.244812 0.122406 0.992480i \(-0.460939\pi\)
0.122406 + 0.992480i \(0.460939\pi\)
\(114\) 5.28728 0.495199
\(115\) 0 0
\(116\) −18.9979 −1.76391
\(117\) −6.09186 −0.563193
\(118\) 2.94589 0.271191
\(119\) 18.3565 1.68274
\(120\) 0 0
\(121\) −10.7522 −0.977473
\(122\) 14.4829 1.31122
\(123\) −6.61857 −0.596777
\(124\) 20.6523 1.85463
\(125\) 0 0
\(126\) 19.2348 1.71357
\(127\) −4.94483 −0.438783 −0.219391 0.975637i \(-0.570407\pi\)
−0.219391 + 0.975637i \(0.570407\pi\)
\(128\) 19.7214 1.74314
\(129\) 21.8457 1.92341
\(130\) 0 0
\(131\) 2.70342 0.236199 0.118099 0.993002i \(-0.462320\pi\)
0.118099 + 0.993002i \(0.462320\pi\)
\(132\) −3.90893 −0.340229
\(133\) 3.54852 0.307695
\(134\) −12.2964 −1.06224
\(135\) 0 0
\(136\) 16.7595 1.43711
\(137\) 2.24838 0.192092 0.0960459 0.995377i \(-0.469380\pi\)
0.0960459 + 0.995377i \(0.469380\pi\)
\(138\) 34.3487 2.92395
\(139\) −10.8032 −0.916317 −0.458159 0.888870i \(-0.651491\pi\)
−0.458159 + 0.888870i \(0.651491\pi\)
\(140\) 0 0
\(141\) −12.3596 −1.04087
\(142\) 0.353481 0.0296635
\(143\) 1.31809 0.110224
\(144\) 1.86745 0.155621
\(145\) 0 0
\(146\) −34.5986 −2.86340
\(147\) 13.6265 1.12389
\(148\) 15.6762 1.28858
\(149\) 12.1878 0.998460 0.499230 0.866469i \(-0.333616\pi\)
0.499230 + 0.866469i \(0.333616\pi\)
\(150\) 0 0
\(151\) 17.0860 1.39044 0.695220 0.718797i \(-0.255307\pi\)
0.695220 + 0.718797i \(0.255307\pi\)
\(152\) 3.23979 0.262782
\(153\) −11.7498 −0.949918
\(154\) −4.16179 −0.335367
\(155\) 0 0
\(156\) −20.7929 −1.66476
\(157\) 7.49835 0.598433 0.299217 0.954185i \(-0.403275\pi\)
0.299217 + 0.954185i \(0.403275\pi\)
\(158\) 38.5868 3.06980
\(159\) 0.708192 0.0561633
\(160\) 0 0
\(161\) 23.0529 1.81682
\(162\) 24.6775 1.93885
\(163\) −1.95259 −0.152939 −0.0764693 0.997072i \(-0.524365\pi\)
−0.0764693 + 0.997072i \(0.524365\pi\)
\(164\) −9.80504 −0.765645
\(165\) 0 0
\(166\) 33.9517 2.63516
\(167\) 0.356578 0.0275928 0.0137964 0.999905i \(-0.495608\pi\)
0.0137964 + 0.999905i \(0.495608\pi\)
\(168\) 27.1550 2.09506
\(169\) −5.98868 −0.460668
\(170\) 0 0
\(171\) −2.27138 −0.173697
\(172\) 32.3632 2.46767
\(173\) 9.95032 0.756509 0.378254 0.925702i \(-0.376524\pi\)
0.378254 + 0.925702i \(0.376524\pi\)
\(174\) 29.8297 2.26138
\(175\) 0 0
\(176\) −0.404058 −0.0304570
\(177\) −2.91577 −0.219162
\(178\) −26.4763 −1.98448
\(179\) 15.3824 1.14973 0.574867 0.818247i \(-0.305054\pi\)
0.574867 + 0.818247i \(0.305054\pi\)
\(180\) 0 0
\(181\) 8.91917 0.662957 0.331478 0.943463i \(-0.392453\pi\)
0.331478 + 0.943463i \(0.392453\pi\)
\(182\) −22.1379 −1.64097
\(183\) −14.3348 −1.05966
\(184\) 21.0472 1.55162
\(185\) 0 0
\(186\) −32.4274 −2.37769
\(187\) 2.54230 0.185911
\(188\) −18.3101 −1.33540
\(189\) 5.78719 0.420956
\(190\) 0 0
\(191\) −11.3887 −0.824056 −0.412028 0.911171i \(-0.635179\pi\)
−0.412028 + 0.911171i \(0.635179\pi\)
\(192\) −28.7740 −2.07659
\(193\) −17.3321 −1.24759 −0.623795 0.781588i \(-0.714410\pi\)
−0.623795 + 0.781588i \(0.714410\pi\)
\(194\) 1.97531 0.141819
\(195\) 0 0
\(196\) 20.1869 1.44192
\(197\) −25.9609 −1.84964 −0.924820 0.380405i \(-0.875785\pi\)
−0.924820 + 0.380405i \(0.875785\pi\)
\(198\) 2.66393 0.189317
\(199\) −8.38571 −0.594447 −0.297223 0.954808i \(-0.596061\pi\)
−0.297223 + 0.954808i \(0.596061\pi\)
\(200\) 0 0
\(201\) 12.1706 0.858450
\(202\) 30.8204 2.16852
\(203\) 20.0200 1.40513
\(204\) −40.1048 −2.80790
\(205\) 0 0
\(206\) 1.93278 0.134663
\(207\) −14.7559 −1.02561
\(208\) −2.14931 −0.149028
\(209\) 0.491454 0.0339946
\(210\) 0 0
\(211\) 15.9135 1.09553 0.547765 0.836632i \(-0.315479\pi\)
0.547765 + 0.836632i \(0.315479\pi\)
\(212\) 1.04915 0.0720557
\(213\) −0.349866 −0.0239724
\(214\) −0.168141 −0.0114939
\(215\) 0 0
\(216\) 5.28370 0.359510
\(217\) −21.7634 −1.47740
\(218\) −13.0174 −0.881647
\(219\) 34.2448 2.31405
\(220\) 0 0
\(221\) 13.5233 0.909674
\(222\) −24.6142 −1.65199
\(223\) 0.379706 0.0254270 0.0127135 0.999919i \(-0.495953\pi\)
0.0127135 + 0.999919i \(0.495953\pi\)
\(224\) −16.8030 −1.12270
\(225\) 0 0
\(226\) −6.05343 −0.402668
\(227\) −20.3009 −1.34742 −0.673710 0.738996i \(-0.735300\pi\)
−0.673710 + 0.738996i \(0.735300\pi\)
\(228\) −7.75270 −0.513435
\(229\) −11.9107 −0.787078 −0.393539 0.919308i \(-0.628749\pi\)
−0.393539 + 0.919308i \(0.628749\pi\)
\(230\) 0 0
\(231\) 4.11923 0.271026
\(232\) 18.2782 1.20002
\(233\) 19.0610 1.24873 0.624364 0.781133i \(-0.285358\pi\)
0.624364 + 0.781133i \(0.285358\pi\)
\(234\) 14.1703 0.926341
\(235\) 0 0
\(236\) −4.31954 −0.281178
\(237\) −38.1922 −2.48085
\(238\) −42.6991 −2.76777
\(239\) 4.12493 0.266819 0.133410 0.991061i \(-0.457407\pi\)
0.133410 + 0.991061i \(0.457407\pi\)
\(240\) 0 0
\(241\) 16.1858 1.04262 0.521310 0.853367i \(-0.325444\pi\)
0.521310 + 0.853367i \(0.325444\pi\)
\(242\) 25.0107 1.60775
\(243\) −19.5948 −1.25701
\(244\) −21.2362 −1.35951
\(245\) 0 0
\(246\) 15.3955 0.981580
\(247\) 2.61420 0.166338
\(248\) −19.8700 −1.26175
\(249\) −33.6045 −2.12960
\(250\) 0 0
\(251\) 11.8718 0.749344 0.374672 0.927157i \(-0.377755\pi\)
0.374672 + 0.927157i \(0.377755\pi\)
\(252\) −28.2038 −1.77667
\(253\) 3.19272 0.200725
\(254\) 11.5022 0.721711
\(255\) 0 0
\(256\) −20.8782 −1.30489
\(257\) 18.9164 1.17997 0.589986 0.807414i \(-0.299133\pi\)
0.589986 + 0.807414i \(0.299133\pi\)
\(258\) −50.8153 −3.16362
\(259\) −16.5196 −1.02648
\(260\) 0 0
\(261\) −12.8146 −0.793206
\(262\) −6.28843 −0.388501
\(263\) 6.74703 0.416040 0.208020 0.978125i \(-0.433298\pi\)
0.208020 + 0.978125i \(0.433298\pi\)
\(264\) 3.76086 0.231465
\(265\) 0 0
\(266\) −8.25421 −0.506098
\(267\) 26.2055 1.60375
\(268\) 18.0301 1.10136
\(269\) −25.3329 −1.54457 −0.772286 0.635275i \(-0.780887\pi\)
−0.772286 + 0.635275i \(0.780887\pi\)
\(270\) 0 0
\(271\) 9.40735 0.571456 0.285728 0.958311i \(-0.407765\pi\)
0.285728 + 0.958311i \(0.407765\pi\)
\(272\) −4.14555 −0.251361
\(273\) 21.9115 1.32614
\(274\) −5.22995 −0.315953
\(275\) 0 0
\(276\) −50.3653 −3.03163
\(277\) 5.96736 0.358544 0.179272 0.983800i \(-0.442626\pi\)
0.179272 + 0.983800i \(0.442626\pi\)
\(278\) 25.1294 1.50716
\(279\) 13.9306 0.834003
\(280\) 0 0
\(281\) 11.0723 0.660516 0.330258 0.943891i \(-0.392864\pi\)
0.330258 + 0.943891i \(0.392864\pi\)
\(282\) 28.7497 1.71202
\(283\) 3.02406 0.179762 0.0898810 0.995953i \(-0.471351\pi\)
0.0898810 + 0.995953i \(0.471351\pi\)
\(284\) −0.518307 −0.0307559
\(285\) 0 0
\(286\) −3.06600 −0.181297
\(287\) 10.3326 0.609911
\(288\) 10.7555 0.633772
\(289\) 9.08337 0.534316
\(290\) 0 0
\(291\) −1.95511 −0.114610
\(292\) 50.7317 2.96885
\(293\) −6.26426 −0.365962 −0.182981 0.983116i \(-0.558575\pi\)
−0.182981 + 0.983116i \(0.558575\pi\)
\(294\) −31.6966 −1.84858
\(295\) 0 0
\(296\) −15.0824 −0.876646
\(297\) 0.801501 0.0465078
\(298\) −28.3500 −1.64227
\(299\) 16.9831 0.982158
\(300\) 0 0
\(301\) −34.1043 −1.96574
\(302\) −39.7438 −2.28700
\(303\) −30.5052 −1.75248
\(304\) −0.801381 −0.0459623
\(305\) 0 0
\(306\) 27.3313 1.56243
\(307\) −25.8734 −1.47667 −0.738337 0.674432i \(-0.764388\pi\)
−0.738337 + 0.674432i \(0.764388\pi\)
\(308\) 6.10241 0.347717
\(309\) −1.91301 −0.108828
\(310\) 0 0
\(311\) −9.00277 −0.510500 −0.255250 0.966875i \(-0.582158\pi\)
−0.255250 + 0.966875i \(0.582158\pi\)
\(312\) 20.0052 1.13257
\(313\) 33.2274 1.87812 0.939062 0.343748i \(-0.111696\pi\)
0.939062 + 0.343748i \(0.111696\pi\)
\(314\) −17.4419 −0.984304
\(315\) 0 0
\(316\) −56.5795 −3.18285
\(317\) 25.9693 1.45858 0.729290 0.684204i \(-0.239850\pi\)
0.729290 + 0.684204i \(0.239850\pi\)
\(318\) −1.64733 −0.0923774
\(319\) 2.77268 0.155240
\(320\) 0 0
\(321\) 0.166421 0.00928874
\(322\) −53.6233 −2.98831
\(323\) 5.04221 0.280556
\(324\) −36.1845 −2.01025
\(325\) 0 0
\(326\) 4.54192 0.251554
\(327\) 12.8842 0.712500
\(328\) 9.43361 0.520884
\(329\) 19.2951 1.06378
\(330\) 0 0
\(331\) 12.1284 0.666636 0.333318 0.942814i \(-0.391832\pi\)
0.333318 + 0.942814i \(0.391832\pi\)
\(332\) −49.7832 −2.73221
\(333\) 10.5741 0.579455
\(334\) −0.829436 −0.0453847
\(335\) 0 0
\(336\) −6.71695 −0.366440
\(337\) 30.2519 1.64793 0.823963 0.566644i \(-0.191759\pi\)
0.823963 + 0.566644i \(0.191759\pi\)
\(338\) 13.9303 0.757707
\(339\) 5.99152 0.325415
\(340\) 0 0
\(341\) −3.01414 −0.163225
\(342\) 5.28345 0.285696
\(343\) 3.88680 0.209867
\(344\) −31.1372 −1.67881
\(345\) 0 0
\(346\) −23.1455 −1.24431
\(347\) 13.3283 0.715502 0.357751 0.933817i \(-0.383544\pi\)
0.357751 + 0.933817i \(0.383544\pi\)
\(348\) −43.7391 −2.34466
\(349\) −27.4444 −1.46906 −0.734532 0.678574i \(-0.762598\pi\)
−0.734532 + 0.678574i \(0.762598\pi\)
\(350\) 0 0
\(351\) 4.26344 0.227566
\(352\) −2.32714 −0.124037
\(353\) −15.5564 −0.827983 −0.413992 0.910281i \(-0.635866\pi\)
−0.413992 + 0.910281i \(0.635866\pi\)
\(354\) 6.78237 0.360479
\(355\) 0 0
\(356\) 38.8220 2.05756
\(357\) 42.2624 2.23676
\(358\) −35.7810 −1.89109
\(359\) −21.9829 −1.16021 −0.580107 0.814540i \(-0.696989\pi\)
−0.580107 + 0.814540i \(0.696989\pi\)
\(360\) 0 0
\(361\) −18.0253 −0.948699
\(362\) −20.7469 −1.09043
\(363\) −24.7550 −1.29930
\(364\) 32.4607 1.70140
\(365\) 0 0
\(366\) 33.3442 1.74293
\(367\) −12.1103 −0.632155 −0.316078 0.948733i \(-0.602366\pi\)
−0.316078 + 0.948733i \(0.602366\pi\)
\(368\) −5.20615 −0.271389
\(369\) −6.61379 −0.344300
\(370\) 0 0
\(371\) −1.10559 −0.0573994
\(372\) 47.5481 2.46526
\(373\) −7.08604 −0.366901 −0.183451 0.983029i \(-0.558727\pi\)
−0.183451 + 0.983029i \(0.558727\pi\)
\(374\) −5.91364 −0.305787
\(375\) 0 0
\(376\) 17.6164 0.908499
\(377\) 14.7488 0.759600
\(378\) −13.4616 −0.692390
\(379\) −21.5030 −1.10453 −0.552266 0.833668i \(-0.686237\pi\)
−0.552266 + 0.833668i \(0.686237\pi\)
\(380\) 0 0
\(381\) −11.3845 −0.583248
\(382\) 26.4912 1.35541
\(383\) −24.8816 −1.27139 −0.635696 0.771940i \(-0.719287\pi\)
−0.635696 + 0.771940i \(0.719287\pi\)
\(384\) 45.4048 2.31706
\(385\) 0 0
\(386\) 40.3161 2.05204
\(387\) 21.8299 1.10968
\(388\) −2.89638 −0.147041
\(389\) −35.4142 −1.79557 −0.897785 0.440434i \(-0.854825\pi\)
−0.897785 + 0.440434i \(0.854825\pi\)
\(390\) 0 0
\(391\) 32.7566 1.65657
\(392\) −19.4222 −0.980967
\(393\) 6.22412 0.313965
\(394\) 60.3878 3.04229
\(395\) 0 0
\(396\) −3.90611 −0.196289
\(397\) −5.49705 −0.275889 −0.137944 0.990440i \(-0.544050\pi\)
−0.137944 + 0.990440i \(0.544050\pi\)
\(398\) 19.5060 0.977748
\(399\) 8.16980 0.409001
\(400\) 0 0
\(401\) 24.8463 1.24077 0.620383 0.784299i \(-0.286977\pi\)
0.620383 + 0.784299i \(0.286977\pi\)
\(402\) −28.3101 −1.41198
\(403\) −16.0332 −0.798669
\(404\) −45.1918 −2.24838
\(405\) 0 0
\(406\) −46.5685 −2.31116
\(407\) −2.28789 −0.113407
\(408\) 38.5855 1.91027
\(409\) 3.93157 0.194404 0.0972019 0.995265i \(-0.469011\pi\)
0.0972019 + 0.995265i \(0.469011\pi\)
\(410\) 0 0
\(411\) 5.17647 0.255336
\(412\) −2.83402 −0.139622
\(413\) 4.55194 0.223986
\(414\) 34.3238 1.68692
\(415\) 0 0
\(416\) −12.3788 −0.606921
\(417\) −24.8724 −1.21801
\(418\) −1.14317 −0.0559144
\(419\) −4.76604 −0.232836 −0.116418 0.993200i \(-0.537141\pi\)
−0.116418 + 0.993200i \(0.537141\pi\)
\(420\) 0 0
\(421\) 16.0581 0.782625 0.391312 0.920258i \(-0.372021\pi\)
0.391312 + 0.920258i \(0.372021\pi\)
\(422\) −37.0164 −1.80193
\(423\) −12.3507 −0.600510
\(424\) −1.00940 −0.0490209
\(425\) 0 0
\(426\) 0.813824 0.0394299
\(427\) 22.3787 1.08298
\(428\) 0.246544 0.0119172
\(429\) 3.03465 0.146514
\(430\) 0 0
\(431\) −41.3693 −1.99269 −0.996344 0.0854360i \(-0.972772\pi\)
−0.996344 + 0.0854360i \(0.972772\pi\)
\(432\) −1.30695 −0.0628808
\(433\) −15.1854 −0.729763 −0.364881 0.931054i \(-0.618890\pi\)
−0.364881 + 0.931054i \(0.618890\pi\)
\(434\) 50.6239 2.43003
\(435\) 0 0
\(436\) 19.0873 0.914115
\(437\) 6.33222 0.302911
\(438\) −79.6568 −3.80615
\(439\) 17.1194 0.817063 0.408532 0.912744i \(-0.366041\pi\)
0.408532 + 0.912744i \(0.366041\pi\)
\(440\) 0 0
\(441\) 13.6166 0.648411
\(442\) −31.4565 −1.49623
\(443\) −35.3909 −1.68147 −0.840736 0.541445i \(-0.817877\pi\)
−0.840736 + 0.541445i \(0.817877\pi\)
\(444\) 36.0916 1.71283
\(445\) 0 0
\(446\) −0.883234 −0.0418223
\(447\) 28.0601 1.32719
\(448\) 44.9204 2.12229
\(449\) 20.6830 0.976090 0.488045 0.872818i \(-0.337710\pi\)
0.488045 + 0.872818i \(0.337710\pi\)
\(450\) 0 0
\(451\) 1.43101 0.0673838
\(452\) 8.87610 0.417497
\(453\) 39.3374 1.84823
\(454\) 47.2220 2.21624
\(455\) 0 0
\(456\) 7.45902 0.349301
\(457\) −41.7664 −1.95375 −0.976876 0.213807i \(-0.931414\pi\)
−0.976876 + 0.213807i \(0.931414\pi\)
\(458\) 27.7054 1.29459
\(459\) 8.22322 0.383827
\(460\) 0 0
\(461\) −10.6783 −0.497340 −0.248670 0.968588i \(-0.579993\pi\)
−0.248670 + 0.968588i \(0.579993\pi\)
\(462\) −9.58176 −0.445784
\(463\) 7.83033 0.363906 0.181953 0.983307i \(-0.441758\pi\)
0.181953 + 0.983307i \(0.441758\pi\)
\(464\) −4.52122 −0.209892
\(465\) 0 0
\(466\) −44.3379 −2.05391
\(467\) −4.98100 −0.230493 −0.115246 0.993337i \(-0.536766\pi\)
−0.115246 + 0.993337i \(0.536766\pi\)
\(468\) −20.7778 −0.960455
\(469\) −19.0001 −0.877344
\(470\) 0 0
\(471\) 17.2635 0.795462
\(472\) 4.15591 0.191291
\(473\) −4.72330 −0.217178
\(474\) 88.8388 4.08050
\(475\) 0 0
\(476\) 62.6094 2.86970
\(477\) 0.707679 0.0324024
\(478\) −9.59500 −0.438865
\(479\) −16.3981 −0.749247 −0.374623 0.927177i \(-0.622228\pi\)
−0.374623 + 0.927177i \(0.622228\pi\)
\(480\) 0 0
\(481\) −12.1700 −0.554906
\(482\) −37.6498 −1.71490
\(483\) 53.0749 2.41499
\(484\) −36.6731 −1.66696
\(485\) 0 0
\(486\) 45.5794 2.06753
\(487\) −0.876508 −0.0397184 −0.0198592 0.999803i \(-0.506322\pi\)
−0.0198592 + 0.999803i \(0.506322\pi\)
\(488\) 20.4317 0.924901
\(489\) −4.49547 −0.203292
\(490\) 0 0
\(491\) 12.2118 0.551112 0.275556 0.961285i \(-0.411138\pi\)
0.275556 + 0.961285i \(0.411138\pi\)
\(492\) −22.5743 −1.01773
\(493\) 28.4471 1.28119
\(494\) −6.08090 −0.273592
\(495\) 0 0
\(496\) 4.91495 0.220688
\(497\) 0.546192 0.0245001
\(498\) 78.1675 3.50277
\(499\) −3.34603 −0.149789 −0.0748945 0.997191i \(-0.523862\pi\)
−0.0748945 + 0.997191i \(0.523862\pi\)
\(500\) 0 0
\(501\) 0.820954 0.0366775
\(502\) −27.6151 −1.23252
\(503\) −11.8820 −0.529792 −0.264896 0.964277i \(-0.585338\pi\)
−0.264896 + 0.964277i \(0.585338\pi\)
\(504\) 27.1354 1.20871
\(505\) 0 0
\(506\) −7.42660 −0.330152
\(507\) −13.7878 −0.612339
\(508\) −16.8656 −0.748289
\(509\) 36.3183 1.60978 0.804890 0.593424i \(-0.202224\pi\)
0.804890 + 0.593424i \(0.202224\pi\)
\(510\) 0 0
\(511\) −53.4611 −2.36498
\(512\) 9.12203 0.403140
\(513\) 1.58964 0.0701844
\(514\) −44.0014 −1.94082
\(515\) 0 0
\(516\) 74.5102 3.28013
\(517\) 2.67229 0.117527
\(518\) 38.4263 1.68835
\(519\) 22.9088 1.00558
\(520\) 0 0
\(521\) 0.204224 0.00894722 0.00447361 0.999990i \(-0.498576\pi\)
0.00447361 + 0.999990i \(0.498576\pi\)
\(522\) 29.8081 1.30467
\(523\) −28.0312 −1.22572 −0.612859 0.790192i \(-0.709981\pi\)
−0.612859 + 0.790192i \(0.709981\pi\)
\(524\) 9.22069 0.402808
\(525\) 0 0
\(526\) −15.6943 −0.684303
\(527\) −30.9244 −1.34709
\(528\) −0.930269 −0.0404848
\(529\) 18.1371 0.788570
\(530\) 0 0
\(531\) −2.91366 −0.126442
\(532\) 12.1031 0.524736
\(533\) 7.61202 0.329713
\(534\) −60.9567 −2.63785
\(535\) 0 0
\(536\) −17.3471 −0.749280
\(537\) 35.4151 1.52827
\(538\) 58.9268 2.54052
\(539\) −2.94621 −0.126902
\(540\) 0 0
\(541\) −9.32216 −0.400791 −0.200396 0.979715i \(-0.564223\pi\)
−0.200396 + 0.979715i \(0.564223\pi\)
\(542\) −21.8824 −0.939931
\(543\) 20.5347 0.881230
\(544\) −23.8760 −1.02367
\(545\) 0 0
\(546\) −50.9684 −2.18125
\(547\) 20.0378 0.856755 0.428378 0.903600i \(-0.359085\pi\)
0.428378 + 0.903600i \(0.359085\pi\)
\(548\) 7.66865 0.327588
\(549\) −14.3244 −0.611351
\(550\) 0 0
\(551\) 5.49914 0.234271
\(552\) 48.4573 2.06248
\(553\) 59.6235 2.53545
\(554\) −13.8807 −0.589734
\(555\) 0 0
\(556\) −36.8471 −1.56266
\(557\) 28.9839 1.22809 0.614043 0.789272i \(-0.289542\pi\)
0.614043 + 0.789272i \(0.289542\pi\)
\(558\) −32.4040 −1.37177
\(559\) −25.1247 −1.06266
\(560\) 0 0
\(561\) 5.85316 0.247121
\(562\) −25.7552 −1.08642
\(563\) −13.7025 −0.577492 −0.288746 0.957406i \(-0.593238\pi\)
−0.288746 + 0.957406i \(0.593238\pi\)
\(564\) −42.1555 −1.77507
\(565\) 0 0
\(566\) −7.03428 −0.295673
\(567\) 38.1312 1.60136
\(568\) 0.498673 0.0209239
\(569\) 34.6152 1.45115 0.725573 0.688146i \(-0.241575\pi\)
0.725573 + 0.688146i \(0.241575\pi\)
\(570\) 0 0
\(571\) −42.4197 −1.77521 −0.887604 0.460607i \(-0.847632\pi\)
−0.887604 + 0.460607i \(0.847632\pi\)
\(572\) 4.49566 0.187973
\(573\) −26.2203 −1.09537
\(574\) −24.0346 −1.00318
\(575\) 0 0
\(576\) −28.7532 −1.19805
\(577\) 34.3049 1.42813 0.714067 0.700078i \(-0.246851\pi\)
0.714067 + 0.700078i \(0.246851\pi\)
\(578\) −21.1288 −0.878844
\(579\) −39.9039 −1.65835
\(580\) 0 0
\(581\) 52.4615 2.17647
\(582\) 4.54777 0.188511
\(583\) −0.153120 −0.00634156
\(584\) −48.8099 −2.01977
\(585\) 0 0
\(586\) 14.5713 0.601935
\(587\) −29.7521 −1.22800 −0.614001 0.789305i \(-0.710441\pi\)
−0.614001 + 0.789305i \(0.710441\pi\)
\(588\) 46.4765 1.91666
\(589\) −5.97803 −0.246321
\(590\) 0 0
\(591\) −59.7702 −2.45862
\(592\) 3.73071 0.153331
\(593\) 14.8105 0.608195 0.304098 0.952641i \(-0.401645\pi\)
0.304098 + 0.952641i \(0.401645\pi\)
\(594\) −1.86437 −0.0764962
\(595\) 0 0
\(596\) 41.5694 1.70275
\(597\) −19.3065 −0.790163
\(598\) −39.5044 −1.61545
\(599\) 27.2394 1.11297 0.556486 0.830857i \(-0.312149\pi\)
0.556486 + 0.830857i \(0.312149\pi\)
\(600\) 0 0
\(601\) 33.1682 1.35296 0.676480 0.736461i \(-0.263505\pi\)
0.676480 + 0.736461i \(0.263505\pi\)
\(602\) 79.3301 3.23325
\(603\) 12.1618 0.495268
\(604\) 58.2761 2.37122
\(605\) 0 0
\(606\) 70.9583 2.88248
\(607\) −5.79849 −0.235353 −0.117677 0.993052i \(-0.537545\pi\)
−0.117677 + 0.993052i \(0.537545\pi\)
\(608\) −4.61549 −0.187183
\(609\) 46.0923 1.86775
\(610\) 0 0
\(611\) 14.2148 0.575068
\(612\) −40.0758 −1.61997
\(613\) −4.04653 −0.163438 −0.0817189 0.996655i \(-0.526041\pi\)
−0.0817189 + 0.996655i \(0.526041\pi\)
\(614\) 60.1842 2.42884
\(615\) 0 0
\(616\) −5.87125 −0.236559
\(617\) −37.1749 −1.49661 −0.748303 0.663357i \(-0.769131\pi\)
−0.748303 + 0.663357i \(0.769131\pi\)
\(618\) 4.44986 0.179000
\(619\) −37.7118 −1.51576 −0.757882 0.652391i \(-0.773766\pi\)
−0.757882 + 0.652391i \(0.773766\pi\)
\(620\) 0 0
\(621\) 10.3271 0.414411
\(622\) 20.9413 0.839672
\(623\) −40.9106 −1.63905
\(624\) −4.94839 −0.198094
\(625\) 0 0
\(626\) −77.2903 −3.08914
\(627\) 1.13148 0.0451870
\(628\) 25.5750 1.02055
\(629\) −23.4733 −0.935940
\(630\) 0 0
\(631\) 18.5225 0.737368 0.368684 0.929555i \(-0.379808\pi\)
0.368684 + 0.929555i \(0.379808\pi\)
\(632\) 54.4362 2.16536
\(633\) 36.6379 1.45622
\(634\) −60.4072 −2.39908
\(635\) 0 0
\(636\) 2.41546 0.0957794
\(637\) −15.6718 −0.620940
\(638\) −6.44954 −0.255340
\(639\) −0.349613 −0.0138305
\(640\) 0 0
\(641\) −19.7116 −0.778560 −0.389280 0.921120i \(-0.627276\pi\)
−0.389280 + 0.921120i \(0.627276\pi\)
\(642\) −0.387113 −0.0152781
\(643\) −42.5897 −1.67957 −0.839787 0.542916i \(-0.817320\pi\)
−0.839787 + 0.542916i \(0.817320\pi\)
\(644\) 78.6275 3.09836
\(645\) 0 0
\(646\) −11.7287 −0.461459
\(647\) −27.4018 −1.07728 −0.538638 0.842537i \(-0.681061\pi\)
−0.538638 + 0.842537i \(0.681061\pi\)
\(648\) 34.8138 1.36761
\(649\) 0.630424 0.0247463
\(650\) 0 0
\(651\) −50.1062 −1.96382
\(652\) −6.65979 −0.260818
\(653\) −21.4414 −0.839068 −0.419534 0.907740i \(-0.637807\pi\)
−0.419534 + 0.907740i \(0.637807\pi\)
\(654\) −29.9701 −1.17192
\(655\) 0 0
\(656\) −2.33346 −0.0911062
\(657\) 34.2200 1.33505
\(658\) −44.8825 −1.74970
\(659\) −36.7342 −1.43096 −0.715480 0.698633i \(-0.753792\pi\)
−0.715480 + 0.698633i \(0.753792\pi\)
\(660\) 0 0
\(661\) −15.7515 −0.612664 −0.306332 0.951925i \(-0.599102\pi\)
−0.306332 + 0.951925i \(0.599102\pi\)
\(662\) −28.2118 −1.09648
\(663\) 31.1348 1.20918
\(664\) 47.8973 1.85878
\(665\) 0 0
\(666\) −24.5963 −0.953089
\(667\) 35.7250 1.38328
\(668\) 1.21620 0.0470561
\(669\) 0.874201 0.0337986
\(670\) 0 0
\(671\) 3.09935 0.119649
\(672\) −38.6858 −1.49234
\(673\) −31.0107 −1.19537 −0.597687 0.801730i \(-0.703913\pi\)
−0.597687 + 0.801730i \(0.703913\pi\)
\(674\) −70.3690 −2.71051
\(675\) 0 0
\(676\) −20.4259 −0.785611
\(677\) 9.59882 0.368912 0.184456 0.982841i \(-0.440948\pi\)
0.184456 + 0.982841i \(0.440948\pi\)
\(678\) −13.9369 −0.535243
\(679\) 3.05220 0.117133
\(680\) 0 0
\(681\) −46.7391 −1.79105
\(682\) 7.01120 0.268473
\(683\) 0.868350 0.0332265 0.0166132 0.999862i \(-0.494712\pi\)
0.0166132 + 0.999862i \(0.494712\pi\)
\(684\) −7.74710 −0.296218
\(685\) 0 0
\(686\) −9.04109 −0.345190
\(687\) −27.4221 −1.04622
\(688\) 7.70196 0.293635
\(689\) −0.814491 −0.0310296
\(690\) 0 0
\(691\) −13.1375 −0.499772 −0.249886 0.968275i \(-0.580393\pi\)
−0.249886 + 0.968275i \(0.580393\pi\)
\(692\) 33.9380 1.29013
\(693\) 4.11625 0.156364
\(694\) −31.0030 −1.17686
\(695\) 0 0
\(696\) 42.0822 1.59512
\(697\) 14.6819 0.556116
\(698\) 63.8384 2.41632
\(699\) 43.8844 1.65986
\(700\) 0 0
\(701\) 7.13602 0.269524 0.134762 0.990878i \(-0.456973\pi\)
0.134762 + 0.990878i \(0.456973\pi\)
\(702\) −9.91719 −0.374300
\(703\) −4.53765 −0.171141
\(704\) 6.22129 0.234474
\(705\) 0 0
\(706\) 36.1858 1.36187
\(707\) 47.6231 1.79105
\(708\) −9.94495 −0.373754
\(709\) −8.00513 −0.300639 −0.150319 0.988637i \(-0.548030\pi\)
−0.150319 + 0.988637i \(0.548030\pi\)
\(710\) 0 0
\(711\) −38.1645 −1.43128
\(712\) −37.3513 −1.39980
\(713\) −38.8362 −1.45443
\(714\) −98.3066 −3.67903
\(715\) 0 0
\(716\) 52.4655 1.96073
\(717\) 9.49688 0.354667
\(718\) 51.1345 1.90832
\(719\) −41.2567 −1.53861 −0.769307 0.638879i \(-0.779398\pi\)
−0.769307 + 0.638879i \(0.779398\pi\)
\(720\) 0 0
\(721\) 2.98649 0.111223
\(722\) 41.9286 1.56042
\(723\) 37.2648 1.38589
\(724\) 30.4210 1.13059
\(725\) 0 0
\(726\) 57.5825 2.13709
\(727\) −39.2602 −1.45608 −0.728040 0.685534i \(-0.759569\pi\)
−0.728040 + 0.685534i \(0.759569\pi\)
\(728\) −31.2310 −1.15750
\(729\) −13.2864 −0.492090
\(730\) 0 0
\(731\) −48.4600 −1.79236
\(732\) −48.8924 −1.80711
\(733\) −0.912055 −0.0336875 −0.0168438 0.999858i \(-0.505362\pi\)
−0.0168438 + 0.999858i \(0.505362\pi\)
\(734\) 28.1699 1.03977
\(735\) 0 0
\(736\) −29.9844 −1.10524
\(737\) −2.63143 −0.0969301
\(738\) 15.3843 0.566305
\(739\) −11.6941 −0.430174 −0.215087 0.976595i \(-0.569003\pi\)
−0.215087 + 0.976595i \(0.569003\pi\)
\(740\) 0 0
\(741\) 6.01871 0.221103
\(742\) 2.57171 0.0944106
\(743\) −11.5631 −0.424211 −0.212105 0.977247i \(-0.568032\pi\)
−0.212105 + 0.977247i \(0.568032\pi\)
\(744\) −45.7470 −1.67716
\(745\) 0 0
\(746\) 16.4828 0.603480
\(747\) −33.5802 −1.22863
\(748\) 8.67113 0.317048
\(749\) −0.259808 −0.00949318
\(750\) 0 0
\(751\) −27.2430 −0.994112 −0.497056 0.867718i \(-0.665586\pi\)
−0.497056 + 0.867718i \(0.665586\pi\)
\(752\) −4.35753 −0.158903
\(753\) 27.3327 0.996059
\(754\) −34.3071 −1.24939
\(755\) 0 0
\(756\) 19.7387 0.717888
\(757\) −17.1080 −0.621800 −0.310900 0.950443i \(-0.600630\pi\)
−0.310900 + 0.950443i \(0.600630\pi\)
\(758\) 50.0181 1.81674
\(759\) 7.35065 0.266812
\(760\) 0 0
\(761\) 37.7173 1.36725 0.683625 0.729833i \(-0.260402\pi\)
0.683625 + 0.729833i \(0.260402\pi\)
\(762\) 26.4816 0.959328
\(763\) −20.1142 −0.728182
\(764\) −38.8439 −1.40532
\(765\) 0 0
\(766\) 57.8772 2.09119
\(767\) 3.35342 0.121085
\(768\) −48.0682 −1.73451
\(769\) −5.93347 −0.213966 −0.106983 0.994261i \(-0.534119\pi\)
−0.106983 + 0.994261i \(0.534119\pi\)
\(770\) 0 0
\(771\) 43.5514 1.56847
\(772\) −59.1153 −2.12761
\(773\) −23.5908 −0.848503 −0.424252 0.905544i \(-0.639463\pi\)
−0.424252 + 0.905544i \(0.639463\pi\)
\(774\) −50.7786 −1.82520
\(775\) 0 0
\(776\) 2.78666 0.100035
\(777\) −38.0333 −1.36444
\(778\) 82.3770 2.95336
\(779\) 2.83817 0.101688
\(780\) 0 0
\(781\) 0.0756453 0.00270680
\(782\) −76.1952 −2.72473
\(783\) 8.96842 0.320505
\(784\) 4.80418 0.171578
\(785\) 0 0
\(786\) −14.4779 −0.516411
\(787\) 18.2307 0.649854 0.324927 0.945739i \(-0.394660\pi\)
0.324927 + 0.945739i \(0.394660\pi\)
\(788\) −88.5462 −3.15433
\(789\) 15.5338 0.553017
\(790\) 0 0
\(791\) −9.35364 −0.332577
\(792\) 3.75814 0.133540
\(793\) 16.4864 0.585451
\(794\) 12.7867 0.453783
\(795\) 0 0
\(796\) −28.6015 −1.01375
\(797\) 10.8296 0.383603 0.191801 0.981434i \(-0.438567\pi\)
0.191801 + 0.981434i \(0.438567\pi\)
\(798\) −19.0038 −0.672727
\(799\) 27.4171 0.969948
\(800\) 0 0
\(801\) 26.1865 0.925256
\(802\) −57.7950 −2.04081
\(803\) −7.40413 −0.261286
\(804\) 41.5109 1.46398
\(805\) 0 0
\(806\) 37.2948 1.31365
\(807\) −58.3242 −2.05311
\(808\) 43.4799 1.52962
\(809\) 44.8177 1.57571 0.787854 0.615863i \(-0.211192\pi\)
0.787854 + 0.615863i \(0.211192\pi\)
\(810\) 0 0
\(811\) −3.06296 −0.107555 −0.0537775 0.998553i \(-0.517126\pi\)
−0.0537775 + 0.998553i \(0.517126\pi\)
\(812\) 68.2831 2.39627
\(813\) 21.6587 0.759602
\(814\) 5.32187 0.186532
\(815\) 0 0
\(816\) −9.54435 −0.334119
\(817\) −9.36786 −0.327740
\(818\) −9.14524 −0.319756
\(819\) 21.8957 0.765096
\(820\) 0 0
\(821\) 20.3210 0.709208 0.354604 0.935017i \(-0.384616\pi\)
0.354604 + 0.935017i \(0.384616\pi\)
\(822\) −12.0410 −0.419978
\(823\) 50.0369 1.74418 0.872089 0.489348i \(-0.162765\pi\)
0.872089 + 0.489348i \(0.162765\pi\)
\(824\) 2.72666 0.0949879
\(825\) 0 0
\(826\) −10.5883 −0.368413
\(827\) 31.6293 1.09986 0.549929 0.835211i \(-0.314655\pi\)
0.549929 + 0.835211i \(0.314655\pi\)
\(828\) −50.3288 −1.74905
\(829\) 33.1369 1.15089 0.575446 0.817840i \(-0.304829\pi\)
0.575446 + 0.817840i \(0.304829\pi\)
\(830\) 0 0
\(831\) 13.7387 0.476591
\(832\) 33.0930 1.14729
\(833\) −30.2274 −1.04732
\(834\) 57.8557 2.00338
\(835\) 0 0
\(836\) 1.67623 0.0579735
\(837\) −9.74944 −0.336990
\(838\) 11.0863 0.382970
\(839\) 10.3854 0.358543 0.179271 0.983800i \(-0.442626\pi\)
0.179271 + 0.983800i \(0.442626\pi\)
\(840\) 0 0
\(841\) 2.02500 0.0698275
\(842\) −37.3528 −1.28726
\(843\) 25.4918 0.877986
\(844\) 54.2769 1.86829
\(845\) 0 0
\(846\) 28.7289 0.987720
\(847\) 38.6461 1.32790
\(848\) 0.249681 0.00857410
\(849\) 6.96235 0.238947
\(850\) 0 0
\(851\) −29.4787 −1.01052
\(852\) −1.19331 −0.0408820
\(853\) 9.31456 0.318924 0.159462 0.987204i \(-0.449024\pi\)
0.159462 + 0.987204i \(0.449024\pi\)
\(854\) −52.0551 −1.78129
\(855\) 0 0
\(856\) −0.237204 −0.00810748
\(857\) 34.0314 1.16249 0.581245 0.813729i \(-0.302566\pi\)
0.581245 + 0.813729i \(0.302566\pi\)
\(858\) −7.05890 −0.240987
\(859\) 33.4243 1.14042 0.570211 0.821498i \(-0.306861\pi\)
0.570211 + 0.821498i \(0.306861\pi\)
\(860\) 0 0
\(861\) 23.7888 0.810719
\(862\) 96.2291 3.27758
\(863\) −16.5900 −0.564730 −0.282365 0.959307i \(-0.591119\pi\)
−0.282365 + 0.959307i \(0.591119\pi\)
\(864\) −7.52730 −0.256084
\(865\) 0 0
\(866\) 35.3227 1.20032
\(867\) 20.9128 0.710235
\(868\) −74.2296 −2.51952
\(869\) 8.25760 0.280120
\(870\) 0 0
\(871\) −13.9974 −0.474285
\(872\) −18.3642 −0.621891
\(873\) −1.95369 −0.0661224
\(874\) −14.7294 −0.498229
\(875\) 0 0
\(876\) 116.800 3.94632
\(877\) 32.9538 1.11277 0.556385 0.830925i \(-0.312188\pi\)
0.556385 + 0.830925i \(0.312188\pi\)
\(878\) −39.8214 −1.34391
\(879\) −14.4223 −0.486452
\(880\) 0 0
\(881\) −50.0755 −1.68709 −0.843544 0.537061i \(-0.819535\pi\)
−0.843544 + 0.537061i \(0.819535\pi\)
\(882\) −31.6737 −1.06651
\(883\) −12.5335 −0.421786 −0.210893 0.977509i \(-0.567637\pi\)
−0.210893 + 0.977509i \(0.567637\pi\)
\(884\) 46.1245 1.55133
\(885\) 0 0
\(886\) 82.3228 2.76569
\(887\) −26.9358 −0.904415 −0.452207 0.891913i \(-0.649363\pi\)
−0.452207 + 0.891913i \(0.649363\pi\)
\(888\) −34.7244 −1.16527
\(889\) 17.7729 0.596085
\(890\) 0 0
\(891\) 5.28101 0.176921
\(892\) 1.29508 0.0433625
\(893\) 5.30004 0.177359
\(894\) −65.2705 −2.18297
\(895\) 0 0
\(896\) −70.8835 −2.36805
\(897\) 39.1004 1.30552
\(898\) −48.1107 −1.60548
\(899\) −33.7268 −1.12485
\(900\) 0 0
\(901\) −1.57097 −0.0523366
\(902\) −3.32869 −0.110833
\(903\) −78.5188 −2.61294
\(904\) −8.53986 −0.284032
\(905\) 0 0
\(906\) −91.5027 −3.03997
\(907\) 28.6510 0.951342 0.475671 0.879623i \(-0.342205\pi\)
0.475671 + 0.879623i \(0.342205\pi\)
\(908\) −69.2414 −2.29786
\(909\) −30.4832 −1.01106
\(910\) 0 0
\(911\) −8.90071 −0.294894 −0.147447 0.989070i \(-0.547106\pi\)
−0.147447 + 0.989070i \(0.547106\pi\)
\(912\) −1.84503 −0.0610950
\(913\) 7.26569 0.240459
\(914\) 97.1530 3.21354
\(915\) 0 0
\(916\) −40.6242 −1.34226
\(917\) −9.71676 −0.320876
\(918\) −19.1280 −0.631320
\(919\) 17.1901 0.567049 0.283525 0.958965i \(-0.408496\pi\)
0.283525 + 0.958965i \(0.408496\pi\)
\(920\) 0 0
\(921\) −59.5687 −1.96286
\(922\) 24.8389 0.818025
\(923\) 0.402381 0.0132445
\(924\) 14.0497 0.462200
\(925\) 0 0
\(926\) −18.2141 −0.598554
\(927\) −1.91163 −0.0627861
\(928\) −26.0396 −0.854793
\(929\) 34.7789 1.14106 0.570530 0.821277i \(-0.306738\pi\)
0.570530 + 0.821277i \(0.306738\pi\)
\(930\) 0 0
\(931\) −5.84330 −0.191507
\(932\) 65.0123 2.12955
\(933\) −20.7272 −0.678578
\(934\) 11.5863 0.379115
\(935\) 0 0
\(936\) 19.9907 0.653417
\(937\) 16.5386 0.540292 0.270146 0.962819i \(-0.412928\pi\)
0.270146 + 0.962819i \(0.412928\pi\)
\(938\) 44.1962 1.44306
\(939\) 76.4999 2.49648
\(940\) 0 0
\(941\) 41.1982 1.34302 0.671512 0.740994i \(-0.265645\pi\)
0.671512 + 0.740994i \(0.265645\pi\)
\(942\) −40.1568 −1.30838
\(943\) 18.4381 0.600428
\(944\) −1.02799 −0.0334582
\(945\) 0 0
\(946\) 10.9869 0.357214
\(947\) 37.1110 1.20594 0.602972 0.797762i \(-0.293983\pi\)
0.602972 + 0.797762i \(0.293983\pi\)
\(948\) −130.264 −4.23077
\(949\) −39.3849 −1.27849
\(950\) 0 0
\(951\) 59.7895 1.93881
\(952\) −60.2376 −1.95231
\(953\) −20.8962 −0.676895 −0.338447 0.940985i \(-0.609902\pi\)
−0.338447 + 0.940985i \(0.609902\pi\)
\(954\) −1.64613 −0.0532956
\(955\) 0 0
\(956\) 14.0691 0.455027
\(957\) 6.38358 0.206352
\(958\) 38.1436 1.23236
\(959\) −8.08122 −0.260956
\(960\) 0 0
\(961\) 5.66391 0.182707
\(962\) 28.3087 0.912710
\(963\) 0.166301 0.00535897
\(964\) 55.2057 1.77806
\(965\) 0 0
\(966\) −123.458 −3.97218
\(967\) −8.28057 −0.266285 −0.133143 0.991097i \(-0.542507\pi\)
−0.133143 + 0.991097i \(0.542507\pi\)
\(968\) 35.2838 1.13407
\(969\) 11.6087 0.372927
\(970\) 0 0
\(971\) −26.4077 −0.847463 −0.423731 0.905788i \(-0.639280\pi\)
−0.423731 + 0.905788i \(0.639280\pi\)
\(972\) −66.8329 −2.14366
\(973\) 38.8294 1.24481
\(974\) 2.03885 0.0653289
\(975\) 0 0
\(976\) −5.05390 −0.161771
\(977\) −25.3280 −0.810314 −0.405157 0.914247i \(-0.632783\pi\)
−0.405157 + 0.914247i \(0.632783\pi\)
\(978\) 10.4569 0.334376
\(979\) −5.66594 −0.181084
\(980\) 0 0
\(981\) 12.8749 0.411065
\(982\) −28.4059 −0.906470
\(983\) 5.12811 0.163561 0.0817807 0.996650i \(-0.473939\pi\)
0.0817807 + 0.996650i \(0.473939\pi\)
\(984\) 21.7191 0.692381
\(985\) 0 0
\(986\) −66.1708 −2.10731
\(987\) 44.4235 1.41401
\(988\) 8.91638 0.283668
\(989\) −60.8581 −1.93517
\(990\) 0 0
\(991\) −26.5396 −0.843059 −0.421530 0.906815i \(-0.638507\pi\)
−0.421530 + 0.906815i \(0.638507\pi\)
\(992\) 28.3073 0.898758
\(993\) 27.9233 0.886120
\(994\) −1.27050 −0.0402978
\(995\) 0 0
\(996\) −114.616 −3.63176
\(997\) −44.4973 −1.40924 −0.704622 0.709583i \(-0.748883\pi\)
−0.704622 + 0.709583i \(0.748883\pi\)
\(998\) 7.78321 0.246373
\(999\) −7.40034 −0.234136
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 625.2.a.e.1.3 8
3.2 odd 2 5625.2.a.be.1.6 8
4.3 odd 2 10000.2.a.bn.1.1 8
5.2 odd 4 625.2.b.d.624.3 16
5.3 odd 4 625.2.b.d.624.14 16
5.4 even 2 625.2.a.g.1.6 yes 8
15.14 odd 2 5625.2.a.s.1.3 8
20.19 odd 2 10000.2.a.be.1.8 8
25.2 odd 20 625.2.e.k.124.2 32
25.3 odd 20 625.2.e.j.249.7 32
25.4 even 10 625.2.d.m.376.2 16
25.6 even 5 625.2.d.q.251.3 16
25.8 odd 20 625.2.e.j.374.2 32
25.9 even 10 625.2.d.n.126.3 16
25.11 even 5 625.2.d.p.501.2 16
25.12 odd 20 625.2.e.k.499.7 32
25.13 odd 20 625.2.e.k.499.2 32
25.14 even 10 625.2.d.n.501.3 16
25.16 even 5 625.2.d.p.126.2 16
25.17 odd 20 625.2.e.j.374.7 32
25.19 even 10 625.2.d.m.251.2 16
25.21 even 5 625.2.d.q.376.3 16
25.22 odd 20 625.2.e.j.249.2 32
25.23 odd 20 625.2.e.k.124.7 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
625.2.a.e.1.3 8 1.1 even 1 trivial
625.2.a.g.1.6 yes 8 5.4 even 2
625.2.b.d.624.3 16 5.2 odd 4
625.2.b.d.624.14 16 5.3 odd 4
625.2.d.m.251.2 16 25.19 even 10
625.2.d.m.376.2 16 25.4 even 10
625.2.d.n.126.3 16 25.9 even 10
625.2.d.n.501.3 16 25.14 even 10
625.2.d.p.126.2 16 25.16 even 5
625.2.d.p.501.2 16 25.11 even 5
625.2.d.q.251.3 16 25.6 even 5
625.2.d.q.376.3 16 25.21 even 5
625.2.e.j.249.2 32 25.22 odd 20
625.2.e.j.249.7 32 25.3 odd 20
625.2.e.j.374.2 32 25.8 odd 20
625.2.e.j.374.7 32 25.17 odd 20
625.2.e.k.124.2 32 25.2 odd 20
625.2.e.k.124.7 32 25.23 odd 20
625.2.e.k.499.2 32 25.13 odd 20
625.2.e.k.499.7 32 25.12 odd 20
5625.2.a.s.1.3 8 15.14 odd 2
5625.2.a.be.1.6 8 3.2 odd 2
10000.2.a.be.1.8 8 20.19 odd 2
10000.2.a.bn.1.1 8 4.3 odd 2