Properties

Label 5225.2.a.j.1.2
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $5$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.68251\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.0881559 q^{2} +3.20362 q^{3} -1.99223 q^{4} -0.282418 q^{6} +1.95185 q^{7} +0.351939 q^{8} +7.26315 q^{9} +O(q^{10})\) \(q-0.0881559 q^{2} +3.20362 q^{3} -1.99223 q^{4} -0.282418 q^{6} +1.95185 q^{7} +0.351939 q^{8} +7.26315 q^{9} -1.00000 q^{11} -6.38233 q^{12} -3.20786 q^{13} -0.172067 q^{14} +3.95343 q^{16} -0.503983 q^{17} -0.640290 q^{18} +1.00000 q^{19} +6.25297 q^{21} +0.0881559 q^{22} -1.05529 q^{23} +1.12748 q^{24} +0.282792 q^{26} +13.6575 q^{27} -3.88853 q^{28} -7.91058 q^{29} +9.52800 q^{31} -1.05240 q^{32} -3.20362 q^{33} +0.0444291 q^{34} -14.4699 q^{36} +5.22650 q^{37} -0.0881559 q^{38} -10.2768 q^{39} +8.32622 q^{41} -0.551237 q^{42} +8.04835 q^{43} +1.99223 q^{44} +0.0930303 q^{46} +9.39522 q^{47} +12.6653 q^{48} -3.19028 q^{49} -1.61457 q^{51} +6.39079 q^{52} -2.37178 q^{53} -1.20399 q^{54} +0.686931 q^{56} +3.20362 q^{57} +0.697365 q^{58} -2.80063 q^{59} -1.53640 q^{61} -0.839949 q^{62} +14.1766 q^{63} -7.81409 q^{64} +0.282418 q^{66} +9.79354 q^{67} +1.00405 q^{68} -3.38075 q^{69} -4.45027 q^{71} +2.55618 q^{72} +1.55991 q^{73} -0.460747 q^{74} -1.99223 q^{76} -1.95185 q^{77} +0.905957 q^{78} +3.00849 q^{79} +21.9639 q^{81} -0.734005 q^{82} +2.22730 q^{83} -12.4574 q^{84} -0.709509 q^{86} -25.3425 q^{87} -0.351939 q^{88} +6.10788 q^{89} -6.26126 q^{91} +2.10238 q^{92} +30.5240 q^{93} -0.828244 q^{94} -3.37147 q^{96} +4.69521 q^{97} +0.281242 q^{98} -7.26315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9} - 5 q^{11} + 7 q^{12} - q^{13} - 3 q^{16} + 3 q^{17} + 7 q^{18} + 5 q^{19} + 11 q^{21} - 3 q^{22} + 8 q^{23} + 9 q^{24} - 16 q^{26} + 10 q^{27} + 22 q^{28} + 11 q^{29} - 5 q^{31} + 2 q^{32} - 7 q^{33} + 4 q^{34} - 3 q^{36} + 9 q^{37} + 3 q^{38} - 8 q^{39} + 15 q^{41} - 11 q^{42} + 13 q^{43} - 5 q^{44} + 18 q^{46} + 20 q^{47} + 20 q^{48} + 20 q^{49} + 24 q^{51} - q^{52} + 5 q^{53} + 17 q^{54} + 7 q^{57} + 33 q^{58} - 17 q^{59} + 3 q^{61} - 14 q^{62} + 22 q^{63} - 17 q^{64} - 2 q^{66} + 28 q^{67} + 25 q^{68} - 2 q^{69} - 6 q^{71} + 26 q^{72} + 16 q^{73} - 21 q^{74} + 5 q^{76} - 11 q^{77} - 29 q^{78} + 3 q^{79} + q^{81} - 2 q^{82} + 33 q^{83} + 33 q^{84} + 10 q^{86} + 3 q^{88} - 16 q^{89} - 22 q^{91} + 19 q^{92} + 26 q^{93} - 10 q^{94} + 5 q^{96} + 14 q^{97} - 10 q^{98} - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0881559 −0.0623356 −0.0311678 0.999514i \(-0.509923\pi\)
−0.0311678 + 0.999514i \(0.509923\pi\)
\(3\) 3.20362 1.84961 0.924804 0.380443i \(-0.124229\pi\)
0.924804 + 0.380443i \(0.124229\pi\)
\(4\) −1.99223 −0.996114
\(5\) 0 0
\(6\) −0.282418 −0.115297
\(7\) 1.95185 0.737730 0.368865 0.929483i \(-0.379747\pi\)
0.368865 + 0.929483i \(0.379747\pi\)
\(8\) 0.351939 0.124429
\(9\) 7.26315 2.42105
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −6.38233 −1.84242
\(13\) −3.20786 −0.889700 −0.444850 0.895605i \(-0.646743\pi\)
−0.444850 + 0.895605i \(0.646743\pi\)
\(14\) −0.172067 −0.0459869
\(15\) 0 0
\(16\) 3.95343 0.988358
\(17\) −0.503983 −0.122234 −0.0611169 0.998131i \(-0.519466\pi\)
−0.0611169 + 0.998131i \(0.519466\pi\)
\(18\) −0.640290 −0.150918
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 6.25297 1.36451
\(22\) 0.0881559 0.0187949
\(23\) −1.05529 −0.220044 −0.110022 0.993929i \(-0.535092\pi\)
−0.110022 + 0.993929i \(0.535092\pi\)
\(24\) 1.12748 0.230145
\(25\) 0 0
\(26\) 0.282792 0.0554601
\(27\) 13.6575 2.62839
\(28\) −3.88853 −0.734863
\(29\) −7.91058 −1.46896 −0.734479 0.678631i \(-0.762574\pi\)
−0.734479 + 0.678631i \(0.762574\pi\)
\(30\) 0 0
\(31\) 9.52800 1.71128 0.855639 0.517573i \(-0.173164\pi\)
0.855639 + 0.517573i \(0.173164\pi\)
\(32\) −1.05240 −0.186039
\(33\) −3.20362 −0.557678
\(34\) 0.0444291 0.00761953
\(35\) 0 0
\(36\) −14.4699 −2.41164
\(37\) 5.22650 0.859231 0.429615 0.903012i \(-0.358649\pi\)
0.429615 + 0.903012i \(0.358649\pi\)
\(38\) −0.0881559 −0.0143008
\(39\) −10.2768 −1.64560
\(40\) 0 0
\(41\) 8.32622 1.30034 0.650168 0.759790i \(-0.274698\pi\)
0.650168 + 0.759790i \(0.274698\pi\)
\(42\) −0.551237 −0.0850577
\(43\) 8.04835 1.22736 0.613681 0.789554i \(-0.289688\pi\)
0.613681 + 0.789554i \(0.289688\pi\)
\(44\) 1.99223 0.300340
\(45\) 0 0
\(46\) 0.0930303 0.0137166
\(47\) 9.39522 1.37043 0.685216 0.728339i \(-0.259708\pi\)
0.685216 + 0.728339i \(0.259708\pi\)
\(48\) 12.6653 1.82808
\(49\) −3.19028 −0.455755
\(50\) 0 0
\(51\) −1.61457 −0.226085
\(52\) 6.39079 0.886243
\(53\) −2.37178 −0.325789 −0.162895 0.986643i \(-0.552083\pi\)
−0.162895 + 0.986643i \(0.552083\pi\)
\(54\) −1.20399 −0.163842
\(55\) 0 0
\(56\) 0.686931 0.0917950
\(57\) 3.20362 0.424329
\(58\) 0.697365 0.0915685
\(59\) −2.80063 −0.364611 −0.182305 0.983242i \(-0.558356\pi\)
−0.182305 + 0.983242i \(0.558356\pi\)
\(60\) 0 0
\(61\) −1.53640 −0.196715 −0.0983577 0.995151i \(-0.531359\pi\)
−0.0983577 + 0.995151i \(0.531359\pi\)
\(62\) −0.839949 −0.106674
\(63\) 14.1766 1.78608
\(64\) −7.81409 −0.976761
\(65\) 0 0
\(66\) 0.282418 0.0347632
\(67\) 9.79354 1.19647 0.598236 0.801320i \(-0.295869\pi\)
0.598236 + 0.801320i \(0.295869\pi\)
\(68\) 1.00405 0.121759
\(69\) −3.38075 −0.406995
\(70\) 0 0
\(71\) −4.45027 −0.528150 −0.264075 0.964502i \(-0.585067\pi\)
−0.264075 + 0.964502i \(0.585067\pi\)
\(72\) 2.55618 0.301249
\(73\) 1.55991 0.182573 0.0912866 0.995825i \(-0.470902\pi\)
0.0912866 + 0.995825i \(0.470902\pi\)
\(74\) −0.460747 −0.0535607
\(75\) 0 0
\(76\) −1.99223 −0.228524
\(77\) −1.95185 −0.222434
\(78\) 0.905957 0.102579
\(79\) 3.00849 0.338482 0.169241 0.985575i \(-0.445868\pi\)
0.169241 + 0.985575i \(0.445868\pi\)
\(80\) 0 0
\(81\) 21.9639 2.44044
\(82\) −0.734005 −0.0810573
\(83\) 2.22730 0.244478 0.122239 0.992501i \(-0.460992\pi\)
0.122239 + 0.992501i \(0.460992\pi\)
\(84\) −12.4574 −1.35921
\(85\) 0 0
\(86\) −0.709509 −0.0765084
\(87\) −25.3425 −2.71700
\(88\) −0.351939 −0.0375168
\(89\) 6.10788 0.647434 0.323717 0.946154i \(-0.395067\pi\)
0.323717 + 0.946154i \(0.395067\pi\)
\(90\) 0 0
\(91\) −6.26126 −0.656358
\(92\) 2.10238 0.219189
\(93\) 30.5240 3.16519
\(94\) −0.828244 −0.0854268
\(95\) 0 0
\(96\) −3.37147 −0.344099
\(97\) 4.69521 0.476726 0.238363 0.971176i \(-0.423389\pi\)
0.238363 + 0.971176i \(0.423389\pi\)
\(98\) 0.281242 0.0284098
\(99\) −7.26315 −0.729974
\(100\) 0 0
\(101\) 0.160574 0.0159777 0.00798885 0.999968i \(-0.497457\pi\)
0.00798885 + 0.999968i \(0.497457\pi\)
\(102\) 0.142334 0.0140931
\(103\) 12.1749 1.19962 0.599812 0.800141i \(-0.295242\pi\)
0.599812 + 0.800141i \(0.295242\pi\)
\(104\) −1.12897 −0.110705
\(105\) 0 0
\(106\) 0.209087 0.0203083
\(107\) −16.2082 −1.56690 −0.783452 0.621452i \(-0.786543\pi\)
−0.783452 + 0.621452i \(0.786543\pi\)
\(108\) −27.2089 −2.61817
\(109\) 17.2276 1.65010 0.825050 0.565059i \(-0.191147\pi\)
0.825050 + 0.565059i \(0.191147\pi\)
\(110\) 0 0
\(111\) 16.7437 1.58924
\(112\) 7.71650 0.729141
\(113\) 9.74136 0.916390 0.458195 0.888852i \(-0.348496\pi\)
0.458195 + 0.888852i \(0.348496\pi\)
\(114\) −0.282418 −0.0264508
\(115\) 0 0
\(116\) 15.7597 1.46325
\(117\) −23.2992 −2.15401
\(118\) 0.246892 0.0227283
\(119\) −0.983699 −0.0901756
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0.135442 0.0122624
\(123\) 26.6740 2.40511
\(124\) −18.9819 −1.70463
\(125\) 0 0
\(126\) −1.24975 −0.111337
\(127\) 6.14600 0.545370 0.272685 0.962103i \(-0.412088\pi\)
0.272685 + 0.962103i \(0.412088\pi\)
\(128\) 2.79365 0.246926
\(129\) 25.7838 2.27014
\(130\) 0 0
\(131\) −13.4596 −1.17597 −0.587987 0.808871i \(-0.700079\pi\)
−0.587987 + 0.808871i \(0.700079\pi\)
\(132\) 6.38233 0.555511
\(133\) 1.95185 0.169247
\(134\) −0.863359 −0.0745828
\(135\) 0 0
\(136\) −0.177371 −0.0152095
\(137\) −7.45968 −0.637324 −0.318662 0.947868i \(-0.603233\pi\)
−0.318662 + 0.947868i \(0.603233\pi\)
\(138\) 0.298033 0.0253703
\(139\) −18.3550 −1.55685 −0.778425 0.627738i \(-0.783981\pi\)
−0.778425 + 0.627738i \(0.783981\pi\)
\(140\) 0 0
\(141\) 30.0987 2.53476
\(142\) 0.392318 0.0329226
\(143\) 3.20786 0.268255
\(144\) 28.7144 2.39286
\(145\) 0 0
\(146\) −0.137515 −0.0113808
\(147\) −10.2204 −0.842968
\(148\) −10.4124 −0.855892
\(149\) −19.2387 −1.57610 −0.788048 0.615614i \(-0.788908\pi\)
−0.788048 + 0.615614i \(0.788908\pi\)
\(150\) 0 0
\(151\) 7.76410 0.631833 0.315917 0.948787i \(-0.397688\pi\)
0.315917 + 0.948787i \(0.397688\pi\)
\(152\) 0.351939 0.0285460
\(153\) −3.66051 −0.295934
\(154\) 0.172067 0.0138656
\(155\) 0 0
\(156\) 20.4736 1.63920
\(157\) 4.82520 0.385093 0.192546 0.981288i \(-0.438325\pi\)
0.192546 + 0.981288i \(0.438325\pi\)
\(158\) −0.265216 −0.0210995
\(159\) −7.59828 −0.602583
\(160\) 0 0
\(161\) −2.05977 −0.162333
\(162\) −1.93625 −0.152126
\(163\) −16.0681 −1.25855 −0.629276 0.777182i \(-0.716648\pi\)
−0.629276 + 0.777182i \(0.716648\pi\)
\(164\) −16.5877 −1.29528
\(165\) 0 0
\(166\) −0.196350 −0.0152397
\(167\) 19.8521 1.53620 0.768102 0.640327i \(-0.221201\pi\)
0.768102 + 0.640327i \(0.221201\pi\)
\(168\) 2.20066 0.169785
\(169\) −2.70963 −0.208433
\(170\) 0 0
\(171\) 7.26315 0.555427
\(172\) −16.0341 −1.22259
\(173\) 23.1362 1.75901 0.879507 0.475886i \(-0.157873\pi\)
0.879507 + 0.475886i \(0.157873\pi\)
\(174\) 2.23409 0.169366
\(175\) 0 0
\(176\) −3.95343 −0.298001
\(177\) −8.97214 −0.674387
\(178\) −0.538446 −0.0403582
\(179\) 4.66377 0.348586 0.174293 0.984694i \(-0.444236\pi\)
0.174293 + 0.984694i \(0.444236\pi\)
\(180\) 0 0
\(181\) 23.7883 1.76817 0.884084 0.467328i \(-0.154783\pi\)
0.884084 + 0.467328i \(0.154783\pi\)
\(182\) 0.551967 0.0409145
\(183\) −4.92202 −0.363846
\(184\) −0.371398 −0.0273798
\(185\) 0 0
\(186\) −2.69087 −0.197304
\(187\) 0.503983 0.0368549
\(188\) −18.7174 −1.36511
\(189\) 26.6574 1.93904
\(190\) 0 0
\(191\) −4.12335 −0.298355 −0.149178 0.988810i \(-0.547663\pi\)
−0.149178 + 0.988810i \(0.547663\pi\)
\(192\) −25.0333 −1.80663
\(193\) −16.6564 −1.19895 −0.599477 0.800392i \(-0.704625\pi\)
−0.599477 + 0.800392i \(0.704625\pi\)
\(194\) −0.413910 −0.0297170
\(195\) 0 0
\(196\) 6.35578 0.453984
\(197\) −12.7284 −0.906862 −0.453431 0.891291i \(-0.649800\pi\)
−0.453431 + 0.891291i \(0.649800\pi\)
\(198\) 0.640290 0.0455034
\(199\) −17.0542 −1.20894 −0.604471 0.796627i \(-0.706615\pi\)
−0.604471 + 0.796627i \(0.706615\pi\)
\(200\) 0 0
\(201\) 31.3747 2.21300
\(202\) −0.0141555 −0.000995980 0
\(203\) −15.4403 −1.08369
\(204\) 3.21659 0.225206
\(205\) 0 0
\(206\) −1.07329 −0.0747794
\(207\) −7.66475 −0.532737
\(208\) −12.6821 −0.879342
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −15.0057 −1.03304 −0.516518 0.856276i \(-0.672772\pi\)
−0.516518 + 0.856276i \(0.672772\pi\)
\(212\) 4.72513 0.324523
\(213\) −14.2570 −0.976871
\(214\) 1.42885 0.0976740
\(215\) 0 0
\(216\) 4.80660 0.327048
\(217\) 18.5972 1.26246
\(218\) −1.51871 −0.102860
\(219\) 4.99734 0.337689
\(220\) 0 0
\(221\) 1.61671 0.108752
\(222\) −1.47605 −0.0990663
\(223\) −4.21760 −0.282432 −0.141216 0.989979i \(-0.545101\pi\)
−0.141216 + 0.989979i \(0.545101\pi\)
\(224\) −2.05412 −0.137246
\(225\) 0 0
\(226\) −0.858759 −0.0571238
\(227\) −18.5091 −1.22849 −0.614245 0.789115i \(-0.710539\pi\)
−0.614245 + 0.789115i \(0.710539\pi\)
\(228\) −6.38233 −0.422680
\(229\) 10.0978 0.667282 0.333641 0.942700i \(-0.391723\pi\)
0.333641 + 0.942700i \(0.391723\pi\)
\(230\) 0 0
\(231\) −6.25297 −0.411416
\(232\) −2.78404 −0.182781
\(233\) 5.28276 0.346085 0.173043 0.984914i \(-0.444640\pi\)
0.173043 + 0.984914i \(0.444640\pi\)
\(234\) 2.05396 0.134272
\(235\) 0 0
\(236\) 5.57949 0.363194
\(237\) 9.63804 0.626058
\(238\) 0.0867189 0.00562115
\(239\) 6.01161 0.388859 0.194429 0.980917i \(-0.437714\pi\)
0.194429 + 0.980917i \(0.437714\pi\)
\(240\) 0 0
\(241\) −24.1747 −1.55723 −0.778613 0.627504i \(-0.784076\pi\)
−0.778613 + 0.627504i \(0.784076\pi\)
\(242\) −0.0881559 −0.00566688
\(243\) 29.3915 1.88546
\(244\) 3.06085 0.195951
\(245\) 0 0
\(246\) −2.35147 −0.149924
\(247\) −3.20786 −0.204111
\(248\) 3.35327 0.212933
\(249\) 7.13543 0.452189
\(250\) 0 0
\(251\) −21.5063 −1.35747 −0.678734 0.734385i \(-0.737471\pi\)
−0.678734 + 0.734385i \(0.737471\pi\)
\(252\) −28.2430 −1.77914
\(253\) 1.05529 0.0663457
\(254\) −0.541807 −0.0339960
\(255\) 0 0
\(256\) 15.3819 0.961369
\(257\) 8.04652 0.501928 0.250964 0.967996i \(-0.419252\pi\)
0.250964 + 0.967996i \(0.419252\pi\)
\(258\) −2.27300 −0.141511
\(259\) 10.2013 0.633880
\(260\) 0 0
\(261\) −57.4558 −3.55642
\(262\) 1.18655 0.0733050
\(263\) 19.5827 1.20752 0.603762 0.797165i \(-0.293668\pi\)
0.603762 + 0.797165i \(0.293668\pi\)
\(264\) −1.12748 −0.0693913
\(265\) 0 0
\(266\) −0.172067 −0.0105501
\(267\) 19.5673 1.19750
\(268\) −19.5110 −1.19182
\(269\) −27.2287 −1.66017 −0.830083 0.557640i \(-0.811707\pi\)
−0.830083 + 0.557640i \(0.811707\pi\)
\(270\) 0 0
\(271\) 16.4952 1.00201 0.501006 0.865444i \(-0.332963\pi\)
0.501006 + 0.865444i \(0.332963\pi\)
\(272\) −1.99246 −0.120811
\(273\) −20.0587 −1.21401
\(274\) 0.657615 0.0397280
\(275\) 0 0
\(276\) 6.73523 0.405413
\(277\) 14.0482 0.844077 0.422039 0.906578i \(-0.361315\pi\)
0.422039 + 0.906578i \(0.361315\pi\)
\(278\) 1.61810 0.0970472
\(279\) 69.2033 4.14309
\(280\) 0 0
\(281\) −4.14119 −0.247043 −0.123521 0.992342i \(-0.539419\pi\)
−0.123521 + 0.992342i \(0.539419\pi\)
\(282\) −2.65337 −0.158006
\(283\) −4.15687 −0.247100 −0.123550 0.992338i \(-0.539428\pi\)
−0.123550 + 0.992338i \(0.539428\pi\)
\(284\) 8.86596 0.526098
\(285\) 0 0
\(286\) −0.282792 −0.0167218
\(287\) 16.2515 0.959297
\(288\) −7.64371 −0.450410
\(289\) −16.7460 −0.985059
\(290\) 0 0
\(291\) 15.0416 0.881757
\(292\) −3.10769 −0.181864
\(293\) 5.34865 0.312471 0.156236 0.987720i \(-0.450064\pi\)
0.156236 + 0.987720i \(0.450064\pi\)
\(294\) 0.900993 0.0525470
\(295\) 0 0
\(296\) 1.83941 0.106913
\(297\) −13.6575 −0.792489
\(298\) 1.69601 0.0982470
\(299\) 3.38523 0.195773
\(300\) 0 0
\(301\) 15.7092 0.905461
\(302\) −0.684451 −0.0393858
\(303\) 0.514417 0.0295525
\(304\) 3.95343 0.226745
\(305\) 0 0
\(306\) 0.322695 0.0184473
\(307\) 24.7343 1.41166 0.705830 0.708381i \(-0.250574\pi\)
0.705830 + 0.708381i \(0.250574\pi\)
\(308\) 3.88853 0.221570
\(309\) 39.0036 2.21884
\(310\) 0 0
\(311\) 17.9991 1.02063 0.510317 0.859986i \(-0.329528\pi\)
0.510317 + 0.859986i \(0.329528\pi\)
\(312\) −3.61679 −0.204760
\(313\) 34.3706 1.94274 0.971372 0.237564i \(-0.0763490\pi\)
0.971372 + 0.237564i \(0.0763490\pi\)
\(314\) −0.425370 −0.0240050
\(315\) 0 0
\(316\) −5.99360 −0.337166
\(317\) 13.1196 0.736870 0.368435 0.929654i \(-0.379894\pi\)
0.368435 + 0.929654i \(0.379894\pi\)
\(318\) 0.669833 0.0375624
\(319\) 7.91058 0.442908
\(320\) 0 0
\(321\) −51.9248 −2.89816
\(322\) 0.181581 0.0101191
\(323\) −0.503983 −0.0280424
\(324\) −43.7572 −2.43095
\(325\) 0 0
\(326\) 1.41650 0.0784527
\(327\) 55.1905 3.05204
\(328\) 2.93032 0.161800
\(329\) 18.3380 1.01101
\(330\) 0 0
\(331\) −0.911287 −0.0500889 −0.0250444 0.999686i \(-0.507973\pi\)
−0.0250444 + 0.999686i \(0.507973\pi\)
\(332\) −4.43730 −0.243528
\(333\) 37.9608 2.08024
\(334\) −1.75008 −0.0957603
\(335\) 0 0
\(336\) 24.7207 1.34863
\(337\) −12.9894 −0.707577 −0.353789 0.935325i \(-0.615107\pi\)
−0.353789 + 0.935325i \(0.615107\pi\)
\(338\) 0.238870 0.0129928
\(339\) 31.2076 1.69496
\(340\) 0 0
\(341\) −9.52800 −0.515970
\(342\) −0.640290 −0.0346229
\(343\) −19.8899 −1.07395
\(344\) 2.83252 0.152719
\(345\) 0 0
\(346\) −2.03959 −0.109649
\(347\) 8.50005 0.456307 0.228153 0.973625i \(-0.426731\pi\)
0.228153 + 0.973625i \(0.426731\pi\)
\(348\) 50.4880 2.70644
\(349\) −22.9281 −1.22731 −0.613657 0.789573i \(-0.710302\pi\)
−0.613657 + 0.789573i \(0.710302\pi\)
\(350\) 0 0
\(351\) −43.8114 −2.33848
\(352\) 1.05240 0.0560929
\(353\) 32.5037 1.73000 0.864998 0.501775i \(-0.167319\pi\)
0.864998 + 0.501775i \(0.167319\pi\)
\(354\) 0.790947 0.0420384
\(355\) 0 0
\(356\) −12.1683 −0.644919
\(357\) −3.15139 −0.166789
\(358\) −0.411139 −0.0217293
\(359\) −7.05497 −0.372347 −0.186174 0.982517i \(-0.559609\pi\)
−0.186174 + 0.982517i \(0.559609\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −2.09708 −0.110220
\(363\) 3.20362 0.168146
\(364\) 12.4739 0.653808
\(365\) 0 0
\(366\) 0.433905 0.0226806
\(367\) −14.8301 −0.774127 −0.387063 0.922053i \(-0.626511\pi\)
−0.387063 + 0.922053i \(0.626511\pi\)
\(368\) −4.17203 −0.217482
\(369\) 60.4746 3.14818
\(370\) 0 0
\(371\) −4.62936 −0.240344
\(372\) −60.8109 −3.15290
\(373\) −9.69631 −0.502056 −0.251028 0.967980i \(-0.580769\pi\)
−0.251028 + 0.967980i \(0.580769\pi\)
\(374\) −0.0444291 −0.00229737
\(375\) 0 0
\(376\) 3.30654 0.170522
\(377\) 25.3761 1.30693
\(378\) −2.35001 −0.120871
\(379\) 1.03558 0.0531941 0.0265970 0.999646i \(-0.491533\pi\)
0.0265970 + 0.999646i \(0.491533\pi\)
\(380\) 0 0
\(381\) 19.6894 1.00872
\(382\) 0.363498 0.0185982
\(383\) −1.35607 −0.0692922 −0.0346461 0.999400i \(-0.511030\pi\)
−0.0346461 + 0.999400i \(0.511030\pi\)
\(384\) 8.94978 0.456716
\(385\) 0 0
\(386\) 1.46836 0.0747376
\(387\) 58.4564 2.97150
\(388\) −9.35393 −0.474874
\(389\) −34.2838 −1.73826 −0.869129 0.494585i \(-0.835320\pi\)
−0.869129 + 0.494585i \(0.835320\pi\)
\(390\) 0 0
\(391\) 0.531850 0.0268968
\(392\) −1.12278 −0.0567092
\(393\) −43.1195 −2.17509
\(394\) 1.12208 0.0565298
\(395\) 0 0
\(396\) 14.4699 0.727138
\(397\) 35.4936 1.78137 0.890687 0.454617i \(-0.150224\pi\)
0.890687 + 0.454617i \(0.150224\pi\)
\(398\) 1.50343 0.0753601
\(399\) 6.25297 0.313040
\(400\) 0 0
\(401\) 28.4914 1.42279 0.711396 0.702791i \(-0.248063\pi\)
0.711396 + 0.702791i \(0.248063\pi\)
\(402\) −2.76587 −0.137949
\(403\) −30.5645 −1.52253
\(404\) −0.319900 −0.0159156
\(405\) 0 0
\(406\) 1.36115 0.0675528
\(407\) −5.22650 −0.259068
\(408\) −0.568229 −0.0281315
\(409\) 31.1180 1.53869 0.769343 0.638836i \(-0.220584\pi\)
0.769343 + 0.638836i \(0.220584\pi\)
\(410\) 0 0
\(411\) −23.8980 −1.17880
\(412\) −24.2551 −1.19496
\(413\) −5.46641 −0.268984
\(414\) 0.675693 0.0332085
\(415\) 0 0
\(416\) 3.37594 0.165519
\(417\) −58.8023 −2.87956
\(418\) 0.0881559 0.00431185
\(419\) −23.8806 −1.16664 −0.583322 0.812241i \(-0.698247\pi\)
−0.583322 + 0.812241i \(0.698247\pi\)
\(420\) 0 0
\(421\) −1.13683 −0.0554059 −0.0277029 0.999616i \(-0.508819\pi\)
−0.0277029 + 0.999616i \(0.508819\pi\)
\(422\) 1.32284 0.0643950
\(423\) 68.2389 3.31789
\(424\) −0.834722 −0.0405377
\(425\) 0 0
\(426\) 1.25684 0.0608939
\(427\) −2.99881 −0.145123
\(428\) 32.2904 1.56082
\(429\) 10.2768 0.496166
\(430\) 0 0
\(431\) −0.598886 −0.0288473 −0.0144237 0.999896i \(-0.504591\pi\)
−0.0144237 + 0.999896i \(0.504591\pi\)
\(432\) 53.9940 2.59779
\(433\) −32.3456 −1.55443 −0.777216 0.629234i \(-0.783369\pi\)
−0.777216 + 0.629234i \(0.783369\pi\)
\(434\) −1.63945 −0.0786963
\(435\) 0 0
\(436\) −34.3212 −1.64369
\(437\) −1.05529 −0.0504815
\(438\) −0.440545 −0.0210500
\(439\) 9.36506 0.446970 0.223485 0.974707i \(-0.428257\pi\)
0.223485 + 0.974707i \(0.428257\pi\)
\(440\) 0 0
\(441\) −23.1715 −1.10341
\(442\) −0.142522 −0.00677910
\(443\) −10.5557 −0.501517 −0.250759 0.968050i \(-0.580680\pi\)
−0.250759 + 0.968050i \(0.580680\pi\)
\(444\) −33.3572 −1.58306
\(445\) 0 0
\(446\) 0.371807 0.0176056
\(447\) −61.6334 −2.91516
\(448\) −15.2519 −0.720586
\(449\) −13.4399 −0.634269 −0.317135 0.948381i \(-0.602721\pi\)
−0.317135 + 0.948381i \(0.602721\pi\)
\(450\) 0 0
\(451\) −8.32622 −0.392066
\(452\) −19.4070 −0.912829
\(453\) 24.8732 1.16864
\(454\) 1.63168 0.0765787
\(455\) 0 0
\(456\) 1.12748 0.0527989
\(457\) −3.88536 −0.181750 −0.0908748 0.995862i \(-0.528966\pi\)
−0.0908748 + 0.995862i \(0.528966\pi\)
\(458\) −0.890182 −0.0415955
\(459\) −6.88315 −0.321278
\(460\) 0 0
\(461\) 15.7252 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(462\) 0.551237 0.0256459
\(463\) −35.4757 −1.64869 −0.824347 0.566084i \(-0.808458\pi\)
−0.824347 + 0.566084i \(0.808458\pi\)
\(464\) −31.2740 −1.45186
\(465\) 0 0
\(466\) −0.465707 −0.0215734
\(467\) −13.5528 −0.627148 −0.313574 0.949564i \(-0.601526\pi\)
−0.313574 + 0.949564i \(0.601526\pi\)
\(468\) 46.4173 2.14564
\(469\) 19.1155 0.882673
\(470\) 0 0
\(471\) 15.4581 0.712271
\(472\) −0.985649 −0.0453682
\(473\) −8.04835 −0.370063
\(474\) −0.849651 −0.0390257
\(475\) 0 0
\(476\) 1.95975 0.0898252
\(477\) −17.2266 −0.788752
\(478\) −0.529959 −0.0242398
\(479\) 39.8445 1.82054 0.910271 0.414012i \(-0.135873\pi\)
0.910271 + 0.414012i \(0.135873\pi\)
\(480\) 0 0
\(481\) −16.7659 −0.764458
\(482\) 2.13114 0.0970707
\(483\) −6.59872 −0.300252
\(484\) −1.99223 −0.0905558
\(485\) 0 0
\(486\) −2.59103 −0.117532
\(487\) −35.9781 −1.63032 −0.815161 0.579234i \(-0.803352\pi\)
−0.815161 + 0.579234i \(0.803352\pi\)
\(488\) −0.540717 −0.0244771
\(489\) −51.4761 −2.32783
\(490\) 0 0
\(491\) −37.4470 −1.68996 −0.844980 0.534798i \(-0.820388\pi\)
−0.844980 + 0.534798i \(0.820388\pi\)
\(492\) −53.1407 −2.39577
\(493\) 3.98680 0.179557
\(494\) 0.282792 0.0127234
\(495\) 0 0
\(496\) 37.6683 1.69136
\(497\) −8.68626 −0.389632
\(498\) −0.629030 −0.0281875
\(499\) 28.2460 1.26446 0.632232 0.774779i \(-0.282139\pi\)
0.632232 + 0.774779i \(0.282139\pi\)
\(500\) 0 0
\(501\) 63.5986 2.84138
\(502\) 1.89591 0.0846186
\(503\) 15.8937 0.708667 0.354333 0.935119i \(-0.384708\pi\)
0.354333 + 0.935119i \(0.384708\pi\)
\(504\) 4.98929 0.222240
\(505\) 0 0
\(506\) −0.0930303 −0.00413570
\(507\) −8.68062 −0.385520
\(508\) −12.2442 −0.543251
\(509\) 1.13300 0.0502195 0.0251097 0.999685i \(-0.492006\pi\)
0.0251097 + 0.999685i \(0.492006\pi\)
\(510\) 0 0
\(511\) 3.04470 0.134690
\(512\) −6.94330 −0.306854
\(513\) 13.6575 0.602994
\(514\) −0.709348 −0.0312880
\(515\) 0 0
\(516\) −51.3672 −2.26132
\(517\) −9.39522 −0.413201
\(518\) −0.899308 −0.0395133
\(519\) 74.1195 3.25349
\(520\) 0 0
\(521\) −35.7749 −1.56732 −0.783662 0.621187i \(-0.786651\pi\)
−0.783662 + 0.621187i \(0.786651\pi\)
\(522\) 5.06507 0.221692
\(523\) 11.7973 0.515862 0.257931 0.966163i \(-0.416959\pi\)
0.257931 + 0.966163i \(0.416959\pi\)
\(524\) 26.8147 1.17140
\(525\) 0 0
\(526\) −1.72633 −0.0752718
\(527\) −4.80195 −0.209176
\(528\) −12.6653 −0.551185
\(529\) −21.8864 −0.951581
\(530\) 0 0
\(531\) −20.3414 −0.882741
\(532\) −3.88853 −0.168589
\(533\) −26.7093 −1.15691
\(534\) −1.72497 −0.0746469
\(535\) 0 0
\(536\) 3.44673 0.148876
\(537\) 14.9409 0.644748
\(538\) 2.40037 0.103488
\(539\) 3.19028 0.137415
\(540\) 0 0
\(541\) −29.1292 −1.25236 −0.626181 0.779678i \(-0.715383\pi\)
−0.626181 + 0.779678i \(0.715383\pi\)
\(542\) −1.45415 −0.0624611
\(543\) 76.2085 3.27042
\(544\) 0.530390 0.0227403
\(545\) 0 0
\(546\) 1.76829 0.0756758
\(547\) −25.3904 −1.08561 −0.542807 0.839857i \(-0.682639\pi\)
−0.542807 + 0.839857i \(0.682639\pi\)
\(548\) 14.8614 0.634847
\(549\) −11.1591 −0.476258
\(550\) 0 0
\(551\) −7.91058 −0.337002
\(552\) −1.18982 −0.0506420
\(553\) 5.87212 0.249708
\(554\) −1.23844 −0.0526161
\(555\) 0 0
\(556\) 36.5673 1.55080
\(557\) 2.80423 0.118819 0.0594096 0.998234i \(-0.481078\pi\)
0.0594096 + 0.998234i \(0.481078\pi\)
\(558\) −6.10068 −0.258262
\(559\) −25.8180 −1.09198
\(560\) 0 0
\(561\) 1.61457 0.0681671
\(562\) 0.365070 0.0153996
\(563\) −14.6693 −0.618239 −0.309119 0.951023i \(-0.600034\pi\)
−0.309119 + 0.951023i \(0.600034\pi\)
\(564\) −59.9634 −2.52491
\(565\) 0 0
\(566\) 0.366453 0.0154031
\(567\) 42.8703 1.80038
\(568\) −1.56622 −0.0657172
\(569\) −38.3728 −1.60867 −0.804336 0.594175i \(-0.797479\pi\)
−0.804336 + 0.594175i \(0.797479\pi\)
\(570\) 0 0
\(571\) 25.0213 1.04711 0.523554 0.851993i \(-0.324606\pi\)
0.523554 + 0.851993i \(0.324606\pi\)
\(572\) −6.39079 −0.267212
\(573\) −13.2096 −0.551840
\(574\) −1.43267 −0.0597984
\(575\) 0 0
\(576\) −56.7549 −2.36479
\(577\) 28.9574 1.20551 0.602756 0.797925i \(-0.294069\pi\)
0.602756 + 0.797925i \(0.294069\pi\)
\(578\) 1.47626 0.0614043
\(579\) −53.3607 −2.21760
\(580\) 0 0
\(581\) 4.34736 0.180359
\(582\) −1.32601 −0.0549649
\(583\) 2.37178 0.0982292
\(584\) 0.548991 0.0227174
\(585\) 0 0
\(586\) −0.471515 −0.0194781
\(587\) 45.1372 1.86301 0.931506 0.363726i \(-0.118496\pi\)
0.931506 + 0.363726i \(0.118496\pi\)
\(588\) 20.3615 0.839693
\(589\) 9.52800 0.392594
\(590\) 0 0
\(591\) −40.7769 −1.67734
\(592\) 20.6626 0.849227
\(593\) −8.57775 −0.352246 −0.176123 0.984368i \(-0.556356\pi\)
−0.176123 + 0.984368i \(0.556356\pi\)
\(594\) 1.20399 0.0494003
\(595\) 0 0
\(596\) 38.3279 1.56997
\(597\) −54.6351 −2.23607
\(598\) −0.298428 −0.0122036
\(599\) 6.04473 0.246981 0.123490 0.992346i \(-0.460591\pi\)
0.123490 + 0.992346i \(0.460591\pi\)
\(600\) 0 0
\(601\) 22.5599 0.920237 0.460118 0.887858i \(-0.347807\pi\)
0.460118 + 0.887858i \(0.347807\pi\)
\(602\) −1.38486 −0.0564425
\(603\) 71.1320 2.89672
\(604\) −15.4679 −0.629378
\(605\) 0 0
\(606\) −0.0453489 −0.00184217
\(607\) −38.8854 −1.57831 −0.789155 0.614194i \(-0.789481\pi\)
−0.789155 + 0.614194i \(0.789481\pi\)
\(608\) −1.05240 −0.0426803
\(609\) −49.4647 −2.00441
\(610\) 0 0
\(611\) −30.1385 −1.21927
\(612\) 7.29257 0.294785
\(613\) 26.0557 1.05238 0.526191 0.850367i \(-0.323620\pi\)
0.526191 + 0.850367i \(0.323620\pi\)
\(614\) −2.18047 −0.0879967
\(615\) 0 0
\(616\) −0.686931 −0.0276772
\(617\) −1.39432 −0.0561331 −0.0280665 0.999606i \(-0.508935\pi\)
−0.0280665 + 0.999606i \(0.508935\pi\)
\(618\) −3.43840 −0.138313
\(619\) 9.82493 0.394897 0.197449 0.980313i \(-0.436734\pi\)
0.197449 + 0.980313i \(0.436734\pi\)
\(620\) 0 0
\(621\) −14.4127 −0.578360
\(622\) −1.58673 −0.0636219
\(623\) 11.9217 0.477632
\(624\) −40.6284 −1.62644
\(625\) 0 0
\(626\) −3.02998 −0.121102
\(627\) −3.20362 −0.127940
\(628\) −9.61290 −0.383596
\(629\) −2.63407 −0.105027
\(630\) 0 0
\(631\) −39.0163 −1.55321 −0.776607 0.629986i \(-0.783061\pi\)
−0.776607 + 0.629986i \(0.783061\pi\)
\(632\) 1.05880 0.0421169
\(633\) −48.0726 −1.91071
\(634\) −1.15657 −0.0459333
\(635\) 0 0
\(636\) 15.1375 0.600241
\(637\) 10.2340 0.405485
\(638\) −0.697365 −0.0276089
\(639\) −32.3230 −1.27868
\(640\) 0 0
\(641\) 24.6377 0.973130 0.486565 0.873644i \(-0.338250\pi\)
0.486565 + 0.873644i \(0.338250\pi\)
\(642\) 4.57748 0.180659
\(643\) −31.8076 −1.25437 −0.627185 0.778871i \(-0.715793\pi\)
−0.627185 + 0.778871i \(0.715793\pi\)
\(644\) 4.10354 0.161702
\(645\) 0 0
\(646\) 0.0444291 0.00174804
\(647\) −18.5631 −0.729790 −0.364895 0.931049i \(-0.618895\pi\)
−0.364895 + 0.931049i \(0.618895\pi\)
\(648\) 7.72996 0.303661
\(649\) 2.80063 0.109934
\(650\) 0 0
\(651\) 59.5783 2.33506
\(652\) 32.0114 1.25366
\(653\) −11.2747 −0.441212 −0.220606 0.975363i \(-0.570804\pi\)
−0.220606 + 0.975363i \(0.570804\pi\)
\(654\) −4.86537 −0.190251
\(655\) 0 0
\(656\) 32.9171 1.28520
\(657\) 11.3298 0.442019
\(658\) −1.61661 −0.0630219
\(659\) −39.1971 −1.52690 −0.763451 0.645866i \(-0.776497\pi\)
−0.763451 + 0.645866i \(0.776497\pi\)
\(660\) 0 0
\(661\) −42.1179 −1.63820 −0.819099 0.573652i \(-0.805526\pi\)
−0.819099 + 0.573652i \(0.805526\pi\)
\(662\) 0.0803354 0.00312232
\(663\) 5.17931 0.201148
\(664\) 0.783874 0.0304202
\(665\) 0 0
\(666\) −3.34647 −0.129673
\(667\) 8.34798 0.323235
\(668\) −39.5500 −1.53024
\(669\) −13.5116 −0.522388
\(670\) 0 0
\(671\) 1.53640 0.0593119
\(672\) −6.58060 −0.253852
\(673\) −25.5042 −0.983116 −0.491558 0.870845i \(-0.663572\pi\)
−0.491558 + 0.870845i \(0.663572\pi\)
\(674\) 1.14509 0.0441073
\(675\) 0 0
\(676\) 5.39821 0.207623
\(677\) −48.5446 −1.86572 −0.932861 0.360237i \(-0.882696\pi\)
−0.932861 + 0.360237i \(0.882696\pi\)
\(678\) −2.75113 −0.105657
\(679\) 9.16434 0.351695
\(680\) 0 0
\(681\) −59.2960 −2.27223
\(682\) 0.839949 0.0321633
\(683\) −22.0305 −0.842975 −0.421488 0.906834i \(-0.638492\pi\)
−0.421488 + 0.906834i \(0.638492\pi\)
\(684\) −14.4699 −0.553269
\(685\) 0 0
\(686\) 1.75341 0.0669456
\(687\) 32.3495 1.23421
\(688\) 31.8186 1.21307
\(689\) 7.60834 0.289855
\(690\) 0 0
\(691\) −23.7336 −0.902870 −0.451435 0.892304i \(-0.649088\pi\)
−0.451435 + 0.892304i \(0.649088\pi\)
\(692\) −46.0926 −1.75218
\(693\) −14.1766 −0.538524
\(694\) −0.749330 −0.0284442
\(695\) 0 0
\(696\) −8.91899 −0.338074
\(697\) −4.19627 −0.158945
\(698\) 2.02125 0.0765054
\(699\) 16.9239 0.640122
\(700\) 0 0
\(701\) 3.46600 0.130909 0.0654546 0.997856i \(-0.479150\pi\)
0.0654546 + 0.997856i \(0.479150\pi\)
\(702\) 3.86223 0.145771
\(703\) 5.22650 0.197121
\(704\) 7.81409 0.294505
\(705\) 0 0
\(706\) −2.86539 −0.107840
\(707\) 0.313416 0.0117872
\(708\) 17.8746 0.671767
\(709\) 11.4552 0.430208 0.215104 0.976591i \(-0.430991\pi\)
0.215104 + 0.976591i \(0.430991\pi\)
\(710\) 0 0
\(711\) 21.8511 0.819481
\(712\) 2.14960 0.0805597
\(713\) −10.0548 −0.376556
\(714\) 0.277814 0.0103969
\(715\) 0 0
\(716\) −9.29129 −0.347232
\(717\) 19.2589 0.719237
\(718\) 0.621938 0.0232105
\(719\) 12.8364 0.478718 0.239359 0.970931i \(-0.423063\pi\)
0.239359 + 0.970931i \(0.423063\pi\)
\(720\) 0 0
\(721\) 23.7635 0.884998
\(722\) −0.0881559 −0.00328082
\(723\) −77.4463 −2.88026
\(724\) −47.3917 −1.76130
\(725\) 0 0
\(726\) −0.282418 −0.0104815
\(727\) 22.3762 0.829887 0.414943 0.909847i \(-0.363801\pi\)
0.414943 + 0.909847i \(0.363801\pi\)
\(728\) −2.20358 −0.0816701
\(729\) 28.2672 1.04693
\(730\) 0 0
\(731\) −4.05623 −0.150025
\(732\) 9.80579 0.362433
\(733\) −35.6014 −1.31497 −0.657483 0.753469i \(-0.728379\pi\)
−0.657483 + 0.753469i \(0.728379\pi\)
\(734\) 1.30736 0.0482557
\(735\) 0 0
\(736\) 1.11059 0.0409367
\(737\) −9.79354 −0.360750
\(738\) −5.33119 −0.196244
\(739\) 24.3156 0.894465 0.447233 0.894418i \(-0.352410\pi\)
0.447233 + 0.894418i \(0.352410\pi\)
\(740\) 0 0
\(741\) −10.2768 −0.377526
\(742\) 0.408106 0.0149820
\(743\) −34.2814 −1.25766 −0.628831 0.777542i \(-0.716466\pi\)
−0.628831 + 0.777542i \(0.716466\pi\)
\(744\) 10.7426 0.393842
\(745\) 0 0
\(746\) 0.854787 0.0312960
\(747\) 16.1773 0.591895
\(748\) −1.00405 −0.0367117
\(749\) −31.6359 −1.15595
\(750\) 0 0
\(751\) 32.2464 1.17669 0.588345 0.808610i \(-0.299780\pi\)
0.588345 + 0.808610i \(0.299780\pi\)
\(752\) 37.1433 1.35448
\(753\) −68.8980 −2.51078
\(754\) −2.23705 −0.0814685
\(755\) 0 0
\(756\) −53.1076 −1.93151
\(757\) −2.94119 −0.106899 −0.0534497 0.998571i \(-0.517022\pi\)
−0.0534497 + 0.998571i \(0.517022\pi\)
\(758\) −0.0912923 −0.00331589
\(759\) 3.38075 0.122714
\(760\) 0 0
\(761\) 26.6626 0.966519 0.483260 0.875477i \(-0.339453\pi\)
0.483260 + 0.875477i \(0.339453\pi\)
\(762\) −1.73574 −0.0628792
\(763\) 33.6256 1.21733
\(764\) 8.21466 0.297196
\(765\) 0 0
\(766\) 0.119546 0.00431937
\(767\) 8.98403 0.324394
\(768\) 49.2777 1.77816
\(769\) −7.23123 −0.260765 −0.130382 0.991464i \(-0.541621\pi\)
−0.130382 + 0.991464i \(0.541621\pi\)
\(770\) 0 0
\(771\) 25.7779 0.928370
\(772\) 33.1834 1.19430
\(773\) 8.64003 0.310760 0.155380 0.987855i \(-0.450340\pi\)
0.155380 + 0.987855i \(0.450340\pi\)
\(774\) −5.15327 −0.185231
\(775\) 0 0
\(776\) 1.65243 0.0593186
\(777\) 32.6811 1.17243
\(778\) 3.02232 0.108355
\(779\) 8.32622 0.298318
\(780\) 0 0
\(781\) 4.45027 0.159243
\(782\) −0.0468857 −0.00167663
\(783\) −108.039 −3.86099
\(784\) −12.6126 −0.450449
\(785\) 0 0
\(786\) 3.80124 0.135586
\(787\) −16.4120 −0.585025 −0.292512 0.956262i \(-0.594491\pi\)
−0.292512 + 0.956262i \(0.594491\pi\)
\(788\) 25.3579 0.903338
\(789\) 62.7356 2.23345
\(790\) 0 0
\(791\) 19.0137 0.676048
\(792\) −2.55618 −0.0908300
\(793\) 4.92854 0.175018
\(794\) −3.12897 −0.111043
\(795\) 0 0
\(796\) 33.9759 1.20424
\(797\) 54.6736 1.93664 0.968319 0.249716i \(-0.0803372\pi\)
0.968319 + 0.249716i \(0.0803372\pi\)
\(798\) −0.551237 −0.0195136
\(799\) −4.73503 −0.167513
\(800\) 0 0
\(801\) 44.3625 1.56747
\(802\) −2.51168 −0.0886907
\(803\) −1.55991 −0.0550479
\(804\) −62.5057 −2.20440
\(805\) 0 0
\(806\) 2.69444 0.0949076
\(807\) −87.2304 −3.07066
\(808\) 0.0565122 0.00198809
\(809\) −20.6416 −0.725719 −0.362859 0.931844i \(-0.618199\pi\)
−0.362859 + 0.931844i \(0.618199\pi\)
\(810\) 0 0
\(811\) 0.728429 0.0255786 0.0127893 0.999918i \(-0.495929\pi\)
0.0127893 + 0.999918i \(0.495929\pi\)
\(812\) 30.7605 1.07948
\(813\) 52.8443 1.85333
\(814\) 0.460747 0.0161492
\(815\) 0 0
\(816\) −6.38309 −0.223453
\(817\) 8.04835 0.281576
\(818\) −2.74324 −0.0959150
\(819\) −45.4765 −1.58908
\(820\) 0 0
\(821\) 7.88685 0.275253 0.137626 0.990484i \(-0.456053\pi\)
0.137626 + 0.990484i \(0.456053\pi\)
\(822\) 2.10675 0.0734812
\(823\) −21.3819 −0.745327 −0.372663 0.927967i \(-0.621555\pi\)
−0.372663 + 0.927967i \(0.621555\pi\)
\(824\) 4.28480 0.149268
\(825\) 0 0
\(826\) 0.481896 0.0167673
\(827\) 18.0869 0.628943 0.314471 0.949267i \(-0.398173\pi\)
0.314471 + 0.949267i \(0.398173\pi\)
\(828\) 15.2699 0.530667
\(829\) 38.7216 1.34486 0.672429 0.740162i \(-0.265251\pi\)
0.672429 + 0.740162i \(0.265251\pi\)
\(830\) 0 0
\(831\) 45.0052 1.56121
\(832\) 25.0665 0.869025
\(833\) 1.60785 0.0557087
\(834\) 5.18377 0.179499
\(835\) 0 0
\(836\) 1.99223 0.0689027
\(837\) 130.129 4.49790
\(838\) 2.10522 0.0727235
\(839\) 11.1336 0.384375 0.192187 0.981358i \(-0.438442\pi\)
0.192187 + 0.981358i \(0.438442\pi\)
\(840\) 0 0
\(841\) 33.5774 1.15784
\(842\) 0.100219 0.00345376
\(843\) −13.2668 −0.456932
\(844\) 29.8948 1.02902
\(845\) 0 0
\(846\) −6.01566 −0.206823
\(847\) 1.95185 0.0670663
\(848\) −9.37668 −0.321996
\(849\) −13.3170 −0.457039
\(850\) 0 0
\(851\) −5.51548 −0.189068
\(852\) 28.4031 0.973075
\(853\) 37.4831 1.28340 0.641698 0.766957i \(-0.278230\pi\)
0.641698 + 0.766957i \(0.278230\pi\)
\(854\) 0.264363 0.00904632
\(855\) 0 0
\(856\) −5.70429 −0.194968
\(857\) −38.7649 −1.32418 −0.662091 0.749423i \(-0.730331\pi\)
−0.662091 + 0.749423i \(0.730331\pi\)
\(858\) −0.905957 −0.0309288
\(859\) 14.9492 0.510060 0.255030 0.966933i \(-0.417915\pi\)
0.255030 + 0.966933i \(0.417915\pi\)
\(860\) 0 0
\(861\) 52.0636 1.77432
\(862\) 0.0527954 0.00179822
\(863\) 45.1615 1.53732 0.768658 0.639660i \(-0.220925\pi\)
0.768658 + 0.639660i \(0.220925\pi\)
\(864\) −14.3731 −0.488983
\(865\) 0 0
\(866\) 2.85146 0.0968965
\(867\) −53.6478 −1.82197
\(868\) −37.0499 −1.25756
\(869\) −3.00849 −0.102056
\(870\) 0 0
\(871\) −31.4163 −1.06450
\(872\) 6.06304 0.205320
\(873\) 34.1020 1.15418
\(874\) 0.0930303 0.00314680
\(875\) 0 0
\(876\) −9.95584 −0.336377
\(877\) −13.1538 −0.444171 −0.222086 0.975027i \(-0.571286\pi\)
−0.222086 + 0.975027i \(0.571286\pi\)
\(878\) −0.825585 −0.0278622
\(879\) 17.1350 0.577949
\(880\) 0 0
\(881\) −19.8386 −0.668380 −0.334190 0.942506i \(-0.608463\pi\)
−0.334190 + 0.942506i \(0.608463\pi\)
\(882\) 2.04271 0.0687815
\(883\) −18.2493 −0.614138 −0.307069 0.951687i \(-0.599348\pi\)
−0.307069 + 0.951687i \(0.599348\pi\)
\(884\) −3.22085 −0.108329
\(885\) 0 0
\(886\) 0.930549 0.0312624
\(887\) 19.6448 0.659606 0.329803 0.944050i \(-0.393018\pi\)
0.329803 + 0.944050i \(0.393018\pi\)
\(888\) 5.89275 0.197748
\(889\) 11.9961 0.402335
\(890\) 0 0
\(891\) −21.9639 −0.735819
\(892\) 8.40243 0.281334
\(893\) 9.39522 0.314399
\(894\) 5.43335 0.181718
\(895\) 0 0
\(896\) 5.45278 0.182165
\(897\) 10.8450 0.362103
\(898\) 1.18481 0.0395376
\(899\) −75.3720 −2.51380
\(900\) 0 0
\(901\) 1.19534 0.0398225
\(902\) 0.734005 0.0244397
\(903\) 50.3261 1.67475
\(904\) 3.42836 0.114026
\(905\) 0 0
\(906\) −2.19272 −0.0728482
\(907\) 36.7829 1.22136 0.610678 0.791879i \(-0.290897\pi\)
0.610678 + 0.791879i \(0.290897\pi\)
\(908\) 36.8743 1.22372
\(909\) 1.16627 0.0386828
\(910\) 0 0
\(911\) −35.8447 −1.18759 −0.593794 0.804617i \(-0.702370\pi\)
−0.593794 + 0.804617i \(0.702370\pi\)
\(912\) 12.6653 0.419389
\(913\) −2.22730 −0.0737130
\(914\) 0.342518 0.0113295
\(915\) 0 0
\(916\) −20.1171 −0.664689
\(917\) −26.2712 −0.867550
\(918\) 0.606791 0.0200271
\(919\) −2.43716 −0.0803943 −0.0401972 0.999192i \(-0.512799\pi\)
−0.0401972 + 0.999192i \(0.512799\pi\)
\(920\) 0 0
\(921\) 79.2391 2.61102
\(922\) −1.38627 −0.0456544
\(923\) 14.2759 0.469895
\(924\) 12.4574 0.409817
\(925\) 0 0
\(926\) 3.12739 0.102772
\(927\) 88.4279 2.90435
\(928\) 8.32506 0.273284
\(929\) 50.7417 1.66478 0.832391 0.554189i \(-0.186971\pi\)
0.832391 + 0.554189i \(0.186971\pi\)
\(930\) 0 0
\(931\) −3.19028 −0.104557
\(932\) −10.5245 −0.344740
\(933\) 57.6621 1.88777
\(934\) 1.19476 0.0390937
\(935\) 0 0
\(936\) −8.19988 −0.268021
\(937\) 34.9861 1.14295 0.571473 0.820621i \(-0.306372\pi\)
0.571473 + 0.820621i \(0.306372\pi\)
\(938\) −1.68515 −0.0550220
\(939\) 110.110 3.59331
\(940\) 0 0
\(941\) −38.7579 −1.26347 −0.631736 0.775183i \(-0.717658\pi\)
−0.631736 + 0.775183i \(0.717658\pi\)
\(942\) −1.36272 −0.0443999
\(943\) −8.78660 −0.286131
\(944\) −11.0721 −0.360366
\(945\) 0 0
\(946\) 0.709509 0.0230681
\(947\) −33.8957 −1.10146 −0.550732 0.834682i \(-0.685651\pi\)
−0.550732 + 0.834682i \(0.685651\pi\)
\(948\) −19.2012 −0.623626
\(949\) −5.00396 −0.162435
\(950\) 0 0
\(951\) 42.0301 1.36292
\(952\) −0.346202 −0.0112205
\(953\) −56.6907 −1.83639 −0.918195 0.396128i \(-0.870354\pi\)
−0.918195 + 0.396128i \(0.870354\pi\)
\(954\) 1.51863 0.0491674
\(955\) 0 0
\(956\) −11.9765 −0.387348
\(957\) 25.3425 0.819206
\(958\) −3.51253 −0.113485
\(959\) −14.5602 −0.470173
\(960\) 0 0
\(961\) 59.7827 1.92847
\(962\) 1.47801 0.0476530
\(963\) −117.723 −3.79356
\(964\) 48.1614 1.55118
\(965\) 0 0
\(966\) 0.581716 0.0187164
\(967\) −45.0600 −1.44903 −0.724517 0.689257i \(-0.757937\pi\)
−0.724517 + 0.689257i \(0.757937\pi\)
\(968\) 0.351939 0.0113117
\(969\) −1.61457 −0.0518674
\(970\) 0 0
\(971\) −24.1262 −0.774247 −0.387124 0.922028i \(-0.626531\pi\)
−0.387124 + 0.922028i \(0.626531\pi\)
\(972\) −58.5546 −1.87814
\(973\) −35.8262 −1.14853
\(974\) 3.17168 0.101627
\(975\) 0 0
\(976\) −6.07404 −0.194425
\(977\) 12.1551 0.388875 0.194437 0.980915i \(-0.437712\pi\)
0.194437 + 0.980915i \(0.437712\pi\)
\(978\) 4.53792 0.145107
\(979\) −6.10788 −0.195209
\(980\) 0 0
\(981\) 125.126 3.99498
\(982\) 3.30118 0.105345
\(983\) 26.5655 0.847308 0.423654 0.905824i \(-0.360747\pi\)
0.423654 + 0.905824i \(0.360747\pi\)
\(984\) 9.38761 0.299266
\(985\) 0 0
\(986\) −0.351460 −0.0111928
\(987\) 58.7480 1.86997
\(988\) 6.39079 0.203318
\(989\) −8.49336 −0.270073
\(990\) 0 0
\(991\) 6.91424 0.219638 0.109819 0.993952i \(-0.464973\pi\)
0.109819 + 0.993952i \(0.464973\pi\)
\(992\) −10.0272 −0.318365
\(993\) −2.91941 −0.0926448
\(994\) 0.765745 0.0242880
\(995\) 0 0
\(996\) −14.2154 −0.450432
\(997\) −42.2221 −1.33719 −0.668593 0.743628i \(-0.733103\pi\)
−0.668593 + 0.743628i \(0.733103\pi\)
\(998\) −2.49005 −0.0788212
\(999\) 71.3809 2.25839
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 5225.2.a.j.1.2 5
5.4 even 2 1045.2.a.d.1.4 5
15.14 odd 2 9405.2.a.v.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.d.1.4 5 5.4 even 2
5225.2.a.j.1.2 5 1.1 even 1 trivial
9405.2.a.v.1.2 5 15.14 odd 2