Properties

Label 3025.2.a.bd.1.4
Level $3025$
Weight $2$
Character 3025.1
Self dual yes
Analytic conductor $24.155$
Analytic rank $1$
Dimension $4$
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] = [3025,2,Mod(1,3025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.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: \(24.1547466114\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.09529\) of defining polynomial
Character \(\chi\) \(=\) 3025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.09529 q^{2} -1.91300 q^{3} +2.39026 q^{4} -4.00829 q^{6} +3.06719 q^{7} +0.817703 q^{8} +0.659557 q^{9} +O(q^{10})\) \(q+2.09529 q^{2} -1.91300 q^{3} +2.39026 q^{4} -4.00829 q^{6} +3.06719 q^{7} +0.817703 q^{8} +0.659557 q^{9} -4.57255 q^{12} -3.04981 q^{13} +6.42666 q^{14} -3.06719 q^{16} +0.463845 q^{17} +1.38197 q^{18} -7.89563 q^{19} -5.86752 q^{21} +1.39026 q^{23} -1.56426 q^{24} -6.39026 q^{26} +4.47726 q^{27} +7.33136 q^{28} +3.72162 q^{29} -10.4765 q^{31} -8.06206 q^{32} +0.971892 q^{34} +1.57651 q^{36} -1.84453 q^{37} -16.5437 q^{38} +5.83428 q^{39} -4.40763 q^{41} -12.2942 q^{42} +1.31478 q^{43} +2.91300 q^{46} +2.98018 q^{47} +5.86752 q^{48} +2.40763 q^{49} -0.887334 q^{51} -7.28984 q^{52} -4.18814 q^{53} +9.38118 q^{54} +2.50805 q^{56} +15.1043 q^{57} +7.79789 q^{58} +2.81502 q^{59} -2.01737 q^{61} -21.9513 q^{62} +2.02298 q^{63} -10.7580 q^{64} +6.75753 q^{67} +1.10871 q^{68} -2.65956 q^{69} -6.52195 q^{71} +0.539322 q^{72} +9.87581 q^{73} -3.86484 q^{74} -18.8726 q^{76} +12.2245 q^{78} -11.5579 q^{79} -10.5437 q^{81} -9.23528 q^{82} -8.91861 q^{83} -14.0249 q^{84} +2.75485 q^{86} -7.11945 q^{87} -6.76978 q^{89} -9.35435 q^{91} +3.32307 q^{92} +20.0415 q^{93} +6.24436 q^{94} +15.4227 q^{96} -15.3543 q^{97} +5.04469 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{4} - q^{6} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - q^{4} - q^{6} + 3 q^{7} + 3 q^{8} - 8 q^{12} + q^{13} + 2 q^{14} - 3 q^{16} - q^{17} + 10 q^{18} - 20 q^{19} - 10 q^{21} - 5 q^{23} - 11 q^{24} - 15 q^{26} + 15 q^{27} + 13 q^{28} - 12 q^{29} - 5 q^{31} - 8 q^{32} + 2 q^{34} - 7 q^{37} - 20 q^{38} - 7 q^{39} - 11 q^{41} - 12 q^{42} + 19 q^{43} + 4 q^{46} - 5 q^{47} + 10 q^{48} + 3 q^{49} - 7 q^{51} - 11 q^{52} + 11 q^{53} + 8 q^{54} + 11 q^{56} - 5 q^{57} + 14 q^{58} + 9 q^{59} - 12 q^{61} - 35 q^{62} - 5 q^{63} - 3 q^{64} + 19 q^{67} - 3 q^{68} - 8 q^{69} + 5 q^{71} - 25 q^{72} + 11 q^{73} + 8 q^{78} - 34 q^{79} + 4 q^{81} + 6 q^{82} - 11 q^{83} - 11 q^{84} + q^{86} - 19 q^{87} - 8 q^{89} - 8 q^{91} + 12 q^{92} + 5 q^{93} + q^{94} + 34 q^{96} - 32 q^{97} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.09529 1.48160 0.740798 0.671728i \(-0.234447\pi\)
0.740798 + 0.671728i \(0.234447\pi\)
\(3\) −1.91300 −1.10447 −0.552235 0.833689i \(-0.686225\pi\)
−0.552235 + 0.833689i \(0.686225\pi\)
\(4\) 2.39026 1.19513
\(5\) 0 0
\(6\) −4.00829 −1.63638
\(7\) 3.06719 1.15929 0.579644 0.814870i \(-0.303192\pi\)
0.579644 + 0.814870i \(0.303192\pi\)
\(8\) 0.817703 0.289102
\(9\) 0.659557 0.219852
\(10\) 0 0
\(11\) 0 0
\(12\) −4.57255 −1.31998
\(13\) −3.04981 −0.845866 −0.422933 0.906161i \(-0.638999\pi\)
−0.422933 + 0.906161i \(0.638999\pi\)
\(14\) 6.42666 1.71760
\(15\) 0 0
\(16\) −3.06719 −0.766796
\(17\) 0.463845 0.112499 0.0562495 0.998417i \(-0.482086\pi\)
0.0562495 + 0.998417i \(0.482086\pi\)
\(18\) 1.38197 0.325733
\(19\) −7.89563 −1.81138 −0.905690 0.423940i \(-0.860647\pi\)
−0.905690 + 0.423940i \(0.860647\pi\)
\(20\) 0 0
\(21\) −5.86752 −1.28040
\(22\) 0 0
\(23\) 1.39026 0.289889 0.144944 0.989440i \(-0.453700\pi\)
0.144944 + 0.989440i \(0.453700\pi\)
\(24\) −1.56426 −0.319304
\(25\) 0 0
\(26\) −6.39026 −1.25323
\(27\) 4.47726 0.861649
\(28\) 7.33136 1.38550
\(29\) 3.72162 0.691087 0.345544 0.938403i \(-0.387695\pi\)
0.345544 + 0.938403i \(0.387695\pi\)
\(30\) 0 0
\(31\) −10.4765 −1.88163 −0.940815 0.338921i \(-0.889938\pi\)
−0.940815 + 0.338921i \(0.889938\pi\)
\(32\) −8.06206 −1.42518
\(33\) 0 0
\(34\) 0.971892 0.166678
\(35\) 0 0
\(36\) 1.57651 0.262752
\(37\) −1.84453 −0.303239 −0.151620 0.988439i \(-0.548449\pi\)
−0.151620 + 0.988439i \(0.548449\pi\)
\(38\) −16.5437 −2.68374
\(39\) 5.83428 0.934233
\(40\) 0 0
\(41\) −4.40763 −0.688356 −0.344178 0.938904i \(-0.611842\pi\)
−0.344178 + 0.938904i \(0.611842\pi\)
\(42\) −12.2942 −1.89703
\(43\) 1.31478 0.200502 0.100251 0.994962i \(-0.468035\pi\)
0.100251 + 0.994962i \(0.468035\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 2.91300 0.429498
\(47\) 2.98018 0.434704 0.217352 0.976093i \(-0.430258\pi\)
0.217352 + 0.976093i \(0.430258\pi\)
\(48\) 5.86752 0.846903
\(49\) 2.40763 0.343947
\(50\) 0 0
\(51\) −0.887334 −0.124252
\(52\) −7.28984 −1.01092
\(53\) −4.18814 −0.575286 −0.287643 0.957738i \(-0.592872\pi\)
−0.287643 + 0.957738i \(0.592872\pi\)
\(54\) 9.38118 1.27662
\(55\) 0 0
\(56\) 2.50805 0.335152
\(57\) 15.1043 2.00061
\(58\) 7.79789 1.02391
\(59\) 2.81502 0.366485 0.183242 0.983068i \(-0.441341\pi\)
0.183242 + 0.983068i \(0.441341\pi\)
\(60\) 0 0
\(61\) −2.01737 −0.258298 −0.129149 0.991625i \(-0.541225\pi\)
−0.129149 + 0.991625i \(0.541225\pi\)
\(62\) −21.9513 −2.78782
\(63\) 2.02298 0.254872
\(64\) −10.7580 −1.34475
\(65\) 0 0
\(66\) 0 0
\(67\) 6.75753 0.825564 0.412782 0.910830i \(-0.364557\pi\)
0.412782 + 0.910830i \(0.364557\pi\)
\(68\) 1.10871 0.134451
\(69\) −2.65956 −0.320173
\(70\) 0 0
\(71\) −6.52195 −0.774013 −0.387007 0.922077i \(-0.626491\pi\)
−0.387007 + 0.922077i \(0.626491\pi\)
\(72\) 0.539322 0.0635597
\(73\) 9.87581 1.15588 0.577938 0.816081i \(-0.303858\pi\)
0.577938 + 0.816081i \(0.303858\pi\)
\(74\) −3.86484 −0.449278
\(75\) 0 0
\(76\) −18.8726 −2.16483
\(77\) 0 0
\(78\) 12.2245 1.38416
\(79\) −11.5579 −1.30036 −0.650180 0.759780i \(-0.725307\pi\)
−0.650180 + 0.759780i \(0.725307\pi\)
\(80\) 0 0
\(81\) −10.5437 −1.17152
\(82\) −9.23528 −1.01987
\(83\) −8.91861 −0.978945 −0.489472 0.872019i \(-0.662811\pi\)
−0.489472 + 0.872019i \(0.662811\pi\)
\(84\) −14.0249 −1.53024
\(85\) 0 0
\(86\) 2.75485 0.297063
\(87\) −7.11945 −0.763285
\(88\) 0 0
\(89\) −6.76978 −0.717595 −0.358797 0.933415i \(-0.616813\pi\)
−0.358797 + 0.933415i \(0.616813\pi\)
\(90\) 0 0
\(91\) −9.35435 −0.980602
\(92\) 3.32307 0.346454
\(93\) 20.0415 2.07820
\(94\) 6.24436 0.644056
\(95\) 0 0
\(96\) 15.4227 1.57407
\(97\) −15.3543 −1.55900 −0.779499 0.626404i \(-0.784526\pi\)
−0.779499 + 0.626404i \(0.784526\pi\)
\(98\) 5.04469 0.509591
\(99\) 0 0
\(100\) 0 0
\(101\) 11.7326 1.16744 0.583718 0.811956i \(-0.301597\pi\)
0.583718 + 0.811956i \(0.301597\pi\)
\(102\) −1.85923 −0.184091
\(103\) −13.8881 −1.36843 −0.684215 0.729280i \(-0.739855\pi\)
−0.684215 + 0.729280i \(0.739855\pi\)
\(104\) −2.49384 −0.244541
\(105\) 0 0
\(106\) −8.77539 −0.852341
\(107\) 7.32100 0.707748 0.353874 0.935293i \(-0.384864\pi\)
0.353874 + 0.935293i \(0.384864\pi\)
\(108\) 10.7018 1.02978
\(109\) −7.43306 −0.711958 −0.355979 0.934494i \(-0.615853\pi\)
−0.355979 + 0.934494i \(0.615853\pi\)
\(110\) 0 0
\(111\) 3.52859 0.334919
\(112\) −9.40763 −0.888937
\(113\) 3.03640 0.285640 0.142820 0.989749i \(-0.454383\pi\)
0.142820 + 0.989749i \(0.454383\pi\)
\(114\) 31.6480 2.96410
\(115\) 0 0
\(116\) 8.89563 0.825938
\(117\) −2.01153 −0.185966
\(118\) 5.89830 0.542983
\(119\) 1.42270 0.130419
\(120\) 0 0
\(121\) 0 0
\(122\) −4.22699 −0.382693
\(123\) 8.43178 0.760268
\(124\) −25.0415 −2.24879
\(125\) 0 0
\(126\) 4.23875 0.377618
\(127\) 0.451597 0.0400728 0.0200364 0.999799i \(-0.493622\pi\)
0.0200364 + 0.999799i \(0.493622\pi\)
\(128\) −6.41709 −0.567196
\(129\) −2.51517 −0.221448
\(130\) 0 0
\(131\) 0.629003 0.0549563 0.0274781 0.999622i \(-0.491252\pi\)
0.0274781 + 0.999622i \(0.491252\pi\)
\(132\) 0 0
\(133\) −24.2173 −2.09991
\(134\) 14.1590 1.22315
\(135\) 0 0
\(136\) 0.379287 0.0325236
\(137\) −11.1834 −0.955462 −0.477731 0.878506i \(-0.658541\pi\)
−0.477731 + 0.878506i \(0.658541\pi\)
\(138\) −5.57255 −0.474367
\(139\) −2.37495 −0.201441 −0.100720 0.994915i \(-0.532115\pi\)
−0.100720 + 0.994915i \(0.532115\pi\)
\(140\) 0 0
\(141\) −5.70108 −0.480118
\(142\) −13.6654 −1.14678
\(143\) 0 0
\(144\) −2.02298 −0.168582
\(145\) 0 0
\(146\) 20.6927 1.71254
\(147\) −4.60579 −0.379879
\(148\) −4.40891 −0.362410
\(149\) −8.72034 −0.714398 −0.357199 0.934028i \(-0.616268\pi\)
−0.357199 + 0.934028i \(0.616268\pi\)
\(150\) 0 0
\(151\) 11.9354 0.971291 0.485646 0.874156i \(-0.338585\pi\)
0.485646 + 0.874156i \(0.338585\pi\)
\(152\) −6.45628 −0.523673
\(153\) 0.305932 0.0247332
\(154\) 0 0
\(155\) 0 0
\(156\) 13.9454 1.11653
\(157\) −3.87532 −0.309284 −0.154642 0.987971i \(-0.549422\pi\)
−0.154642 + 0.987971i \(0.549422\pi\)
\(158\) −24.2171 −1.92661
\(159\) 8.01190 0.635385
\(160\) 0 0
\(161\) 4.26418 0.336064
\(162\) −22.0921 −1.73572
\(163\) 9.93621 0.778264 0.389132 0.921182i \(-0.372775\pi\)
0.389132 + 0.921182i \(0.372775\pi\)
\(164\) −10.5354 −0.822674
\(165\) 0 0
\(166\) −18.6871 −1.45040
\(167\) −13.9200 −1.07716 −0.538581 0.842574i \(-0.681040\pi\)
−0.538581 + 0.842574i \(0.681040\pi\)
\(168\) −4.79789 −0.370165
\(169\) −3.69863 −0.284510
\(170\) 0 0
\(171\) −5.20762 −0.398236
\(172\) 3.14266 0.239626
\(173\) −10.8311 −0.823475 −0.411737 0.911303i \(-0.635078\pi\)
−0.411737 + 0.911303i \(0.635078\pi\)
\(174\) −14.9173 −1.13088
\(175\) 0 0
\(176\) 0 0
\(177\) −5.38513 −0.404771
\(178\) −14.1847 −1.06319
\(179\) 22.7335 1.69918 0.849589 0.527445i \(-0.176850\pi\)
0.849589 + 0.527445i \(0.176850\pi\)
\(180\) 0 0
\(181\) 2.39831 0.178265 0.0891327 0.996020i \(-0.471590\pi\)
0.0891327 + 0.996020i \(0.471590\pi\)
\(182\) −19.6001 −1.45286
\(183\) 3.85923 0.285282
\(184\) 1.13682 0.0838073
\(185\) 0 0
\(186\) 41.9927 3.07906
\(187\) 0 0
\(188\) 7.12340 0.519527
\(189\) 13.7326 0.998899
\(190\) 0 0
\(191\) 17.2462 1.24789 0.623947 0.781466i \(-0.285528\pi\)
0.623947 + 0.781466i \(0.285528\pi\)
\(192\) 20.5801 1.48524
\(193\) 2.58574 0.186125 0.0930627 0.995660i \(-0.470334\pi\)
0.0930627 + 0.995660i \(0.470334\pi\)
\(194\) −32.1719 −2.30981
\(195\) 0 0
\(196\) 5.75485 0.411061
\(197\) −0.144731 −0.0103116 −0.00515582 0.999987i \(-0.501641\pi\)
−0.00515582 + 0.999987i \(0.501641\pi\)
\(198\) 0 0
\(199\) −7.54177 −0.534622 −0.267311 0.963610i \(-0.586135\pi\)
−0.267311 + 0.963610i \(0.586135\pi\)
\(200\) 0 0
\(201\) −12.9271 −0.911810
\(202\) 24.5832 1.72967
\(203\) 11.4149 0.801169
\(204\) −2.12096 −0.148497
\(205\) 0 0
\(206\) −29.0996 −2.02746
\(207\) 0.916954 0.0637327
\(208\) 9.35435 0.648607
\(209\) 0 0
\(210\) 0 0
\(211\) −2.26881 −0.156191 −0.0780957 0.996946i \(-0.524884\pi\)
−0.0780957 + 0.996946i \(0.524884\pi\)
\(212\) −10.0107 −0.687540
\(213\) 12.4765 0.854874
\(214\) 15.3397 1.04860
\(215\) 0 0
\(216\) 3.66107 0.249104
\(217\) −32.1333 −2.18135
\(218\) −15.5744 −1.05483
\(219\) −18.8924 −1.27663
\(220\) 0 0
\(221\) −1.41464 −0.0951591
\(222\) 7.39342 0.496214
\(223\) −8.57968 −0.574538 −0.287269 0.957850i \(-0.592747\pi\)
−0.287269 + 0.957850i \(0.592747\pi\)
\(224\) −24.7278 −1.65220
\(225\) 0 0
\(226\) 6.36215 0.423204
\(227\) −6.20039 −0.411534 −0.205767 0.978601i \(-0.565969\pi\)
−0.205767 + 0.978601i \(0.565969\pi\)
\(228\) 36.1032 2.39099
\(229\) 23.1659 1.53084 0.765422 0.643528i \(-0.222530\pi\)
0.765422 + 0.643528i \(0.222530\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.04318 0.199794
\(233\) 27.8627 1.82535 0.912673 0.408690i \(-0.134014\pi\)
0.912673 + 0.408690i \(0.134014\pi\)
\(234\) −4.21474 −0.275526
\(235\) 0 0
\(236\) 6.72863 0.437997
\(237\) 22.1102 1.43621
\(238\) 2.98097 0.193228
\(239\) −16.2862 −1.05347 −0.526734 0.850030i \(-0.676584\pi\)
−0.526734 + 0.850030i \(0.676584\pi\)
\(240\) 0 0
\(241\) 4.39063 0.282826 0.141413 0.989951i \(-0.454836\pi\)
0.141413 + 0.989951i \(0.454836\pi\)
\(242\) 0 0
\(243\) 6.73820 0.432256
\(244\) −4.82204 −0.308699
\(245\) 0 0
\(246\) 17.6671 1.12641
\(247\) 24.0802 1.53219
\(248\) −8.56664 −0.543982
\(249\) 17.0613 1.08121
\(250\) 0 0
\(251\) −2.66668 −0.168319 −0.0841597 0.996452i \(-0.526821\pi\)
−0.0841597 + 0.996452i \(0.526821\pi\)
\(252\) 4.83545 0.304605
\(253\) 0 0
\(254\) 0.946229 0.0593717
\(255\) 0 0
\(256\) 8.07035 0.504397
\(257\) 21.6327 1.34941 0.674704 0.738088i \(-0.264271\pi\)
0.674704 + 0.738088i \(0.264271\pi\)
\(258\) −5.27002 −0.328097
\(259\) −5.65752 −0.351541
\(260\) 0 0
\(261\) 2.45462 0.151937
\(262\) 1.31795 0.0814230
\(263\) −22.1392 −1.36516 −0.682581 0.730810i \(-0.739142\pi\)
−0.682581 + 0.730810i \(0.739142\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −50.7425 −3.11122
\(267\) 12.9506 0.792562
\(268\) 16.1522 0.986655
\(269\) 20.7184 1.26322 0.631611 0.775285i \(-0.282394\pi\)
0.631611 + 0.775285i \(0.282394\pi\)
\(270\) 0 0
\(271\) −0.423112 −0.0257022 −0.0128511 0.999917i \(-0.504091\pi\)
−0.0128511 + 0.999917i \(0.504091\pi\)
\(272\) −1.42270 −0.0862638
\(273\) 17.8948 1.08304
\(274\) −23.4325 −1.41561
\(275\) 0 0
\(276\) −6.35702 −0.382648
\(277\) 8.57255 0.515075 0.257537 0.966268i \(-0.417089\pi\)
0.257537 + 0.966268i \(0.417089\pi\)
\(278\) −4.97623 −0.298454
\(279\) −6.90983 −0.413681
\(280\) 0 0
\(281\) −7.01108 −0.418246 −0.209123 0.977889i \(-0.567061\pi\)
−0.209123 + 0.977889i \(0.567061\pi\)
\(282\) −11.9454 −0.711341
\(283\) −9.95317 −0.591655 −0.295827 0.955241i \(-0.595595\pi\)
−0.295827 + 0.955241i \(0.595595\pi\)
\(284\) −15.5891 −0.925045
\(285\) 0 0
\(286\) 0 0
\(287\) −13.5190 −0.798002
\(288\) −5.31739 −0.313330
\(289\) −16.7848 −0.987344
\(290\) 0 0
\(291\) 29.3728 1.72186
\(292\) 23.6057 1.38142
\(293\) −21.8209 −1.27479 −0.637394 0.770538i \(-0.719988\pi\)
−0.637394 + 0.770538i \(0.719988\pi\)
\(294\) −9.65048 −0.562827
\(295\) 0 0
\(296\) −1.50828 −0.0876670
\(297\) 0 0
\(298\) −18.2717 −1.05845
\(299\) −4.24002 −0.245207
\(300\) 0 0
\(301\) 4.03268 0.232440
\(302\) 25.0082 1.43906
\(303\) −22.4444 −1.28940
\(304\) 24.2173 1.38896
\(305\) 0 0
\(306\) 0.641018 0.0366446
\(307\) 30.8674 1.76170 0.880849 0.473397i \(-0.156972\pi\)
0.880849 + 0.473397i \(0.156972\pi\)
\(308\) 0 0
\(309\) 26.5678 1.51139
\(310\) 0 0
\(311\) −19.4150 −1.10092 −0.550462 0.834860i \(-0.685549\pi\)
−0.550462 + 0.834860i \(0.685549\pi\)
\(312\) 4.77071 0.270088
\(313\) −1.05147 −0.0594326 −0.0297163 0.999558i \(-0.509460\pi\)
−0.0297163 + 0.999558i \(0.509460\pi\)
\(314\) −8.11993 −0.458234
\(315\) 0 0
\(316\) −27.6263 −1.55410
\(317\) 23.8314 1.33851 0.669253 0.743034i \(-0.266614\pi\)
0.669253 + 0.743034i \(0.266614\pi\)
\(318\) 16.7873 0.941385
\(319\) 0 0
\(320\) 0 0
\(321\) −14.0051 −0.781686
\(322\) 8.93470 0.497912
\(323\) −3.66235 −0.203778
\(324\) −25.2020 −1.40011
\(325\) 0 0
\(326\) 20.8193 1.15307
\(327\) 14.2194 0.786336
\(328\) −3.60413 −0.199005
\(329\) 9.14077 0.503947
\(330\) 0 0
\(331\) 25.6693 1.41091 0.705457 0.708753i \(-0.250742\pi\)
0.705457 + 0.708753i \(0.250742\pi\)
\(332\) −21.3178 −1.16996
\(333\) −1.21657 −0.0666679
\(334\) −29.1665 −1.59592
\(335\) 0 0
\(336\) 17.9968 0.981804
\(337\) 23.8922 1.30149 0.650744 0.759297i \(-0.274457\pi\)
0.650744 + 0.759297i \(0.274457\pi\)
\(338\) −7.74973 −0.421530
\(339\) −5.80862 −0.315481
\(340\) 0 0
\(341\) 0 0
\(342\) −10.9115 −0.590026
\(343\) −14.0857 −0.760554
\(344\) 1.07510 0.0579655
\(345\) 0 0
\(346\) −22.6944 −1.22006
\(347\) 0.332152 0.0178309 0.00891543 0.999960i \(-0.497162\pi\)
0.00891543 + 0.999960i \(0.497162\pi\)
\(348\) −17.0173 −0.912223
\(349\) 1.22149 0.0653846 0.0326923 0.999465i \(-0.489592\pi\)
0.0326923 + 0.999465i \(0.489592\pi\)
\(350\) 0 0
\(351\) −13.6548 −0.728840
\(352\) 0 0
\(353\) 25.7038 1.36808 0.684039 0.729446i \(-0.260222\pi\)
0.684039 + 0.729446i \(0.260222\pi\)
\(354\) −11.2834 −0.599708
\(355\) 0 0
\(356\) −16.1815 −0.857618
\(357\) −2.72162 −0.144043
\(358\) 47.6333 2.51750
\(359\) −17.9315 −0.946387 −0.473193 0.880959i \(-0.656899\pi\)
−0.473193 + 0.880959i \(0.656899\pi\)
\(360\) 0 0
\(361\) 43.3409 2.28110
\(362\) 5.02517 0.264117
\(363\) 0 0
\(364\) −22.3593 −1.17195
\(365\) 0 0
\(366\) 8.08621 0.422673
\(367\) 8.49091 0.443222 0.221611 0.975135i \(-0.428869\pi\)
0.221611 + 0.975135i \(0.428869\pi\)
\(368\) −4.26418 −0.222286
\(369\) −2.90708 −0.151337
\(370\) 0 0
\(371\) −12.8458 −0.666921
\(372\) 47.9042 2.48372
\(373\) −35.8450 −1.85598 −0.927991 0.372604i \(-0.878465\pi\)
−0.927991 + 0.372604i \(0.878465\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 2.43690 0.125674
\(377\) −11.3502 −0.584567
\(378\) 28.7738 1.47997
\(379\) 17.5516 0.901564 0.450782 0.892634i \(-0.351145\pi\)
0.450782 + 0.892634i \(0.351145\pi\)
\(380\) 0 0
\(381\) −0.863904 −0.0442592
\(382\) 36.1360 1.84888
\(383\) 21.6250 1.10499 0.552494 0.833517i \(-0.313676\pi\)
0.552494 + 0.833517i \(0.313676\pi\)
\(384\) 12.2759 0.626450
\(385\) 0 0
\(386\) 5.41788 0.275763
\(387\) 0.867173 0.0440809
\(388\) −36.7008 −1.86320
\(389\) 35.7416 1.81217 0.906086 0.423093i \(-0.139056\pi\)
0.906086 + 0.423093i \(0.139056\pi\)
\(390\) 0 0
\(391\) 0.644864 0.0326122
\(392\) 1.96872 0.0994356
\(393\) −1.20328 −0.0606975
\(394\) −0.303254 −0.0152777
\(395\) 0 0
\(396\) 0 0
\(397\) 20.0447 1.00601 0.503007 0.864282i \(-0.332227\pi\)
0.503007 + 0.864282i \(0.332227\pi\)
\(398\) −15.8022 −0.792094
\(399\) 46.3277 2.31929
\(400\) 0 0
\(401\) −20.8987 −1.04363 −0.521815 0.853059i \(-0.674745\pi\)
−0.521815 + 0.853059i \(0.674745\pi\)
\(402\) −27.0861 −1.35093
\(403\) 31.9513 1.59161
\(404\) 28.0439 1.39524
\(405\) 0 0
\(406\) 23.9176 1.18701
\(407\) 0 0
\(408\) −0.725576 −0.0359214
\(409\) 23.4982 1.16191 0.580955 0.813936i \(-0.302679\pi\)
0.580955 + 0.813936i \(0.302679\pi\)
\(410\) 0 0
\(411\) 21.3938 1.05528
\(412\) −33.1960 −1.63545
\(413\) 8.63420 0.424861
\(414\) 1.92129 0.0944261
\(415\) 0 0
\(416\) 24.5878 1.20552
\(417\) 4.54328 0.222485
\(418\) 0 0
\(419\) −10.1128 −0.494043 −0.247022 0.969010i \(-0.579452\pi\)
−0.247022 + 0.969010i \(0.579452\pi\)
\(420\) 0 0
\(421\) 8.92283 0.434872 0.217436 0.976075i \(-0.430231\pi\)
0.217436 + 0.976075i \(0.430231\pi\)
\(422\) −4.75383 −0.231413
\(423\) 1.96560 0.0955708
\(424\) −3.42466 −0.166316
\(425\) 0 0
\(426\) 26.1419 1.26658
\(427\) −6.18765 −0.299442
\(428\) 17.4991 0.845850
\(429\) 0 0
\(430\) 0 0
\(431\) −17.6122 −0.848352 −0.424176 0.905580i \(-0.639436\pi\)
−0.424176 + 0.905580i \(0.639436\pi\)
\(432\) −13.7326 −0.660709
\(433\) −9.14397 −0.439431 −0.219716 0.975564i \(-0.570513\pi\)
−0.219716 + 0.975564i \(0.570513\pi\)
\(434\) −67.3287 −3.23188
\(435\) 0 0
\(436\) −17.7669 −0.850881
\(437\) −10.9769 −0.525099
\(438\) −39.5851 −1.89145
\(439\) 6.46946 0.308770 0.154385 0.988011i \(-0.450660\pi\)
0.154385 + 0.988011i \(0.450660\pi\)
\(440\) 0 0
\(441\) 1.58797 0.0756176
\(442\) −2.96409 −0.140987
\(443\) 40.8842 1.94247 0.971233 0.238132i \(-0.0765350\pi\)
0.971233 + 0.238132i \(0.0765350\pi\)
\(444\) 8.43423 0.400271
\(445\) 0 0
\(446\) −17.9769 −0.851233
\(447\) 16.6820 0.789031
\(448\) −32.9968 −1.55895
\(449\) −12.1608 −0.573902 −0.286951 0.957945i \(-0.592642\pi\)
−0.286951 + 0.957945i \(0.592642\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 7.25777 0.341377
\(453\) −22.8324 −1.07276
\(454\) −12.9916 −0.609728
\(455\) 0 0
\(456\) 12.3508 0.578381
\(457\) 9.90240 0.463215 0.231607 0.972809i \(-0.425602\pi\)
0.231607 + 0.972809i \(0.425602\pi\)
\(458\) 48.5393 2.26809
\(459\) 2.07676 0.0969346
\(460\) 0 0
\(461\) −3.12529 −0.145559 −0.0727796 0.997348i \(-0.523187\pi\)
−0.0727796 + 0.997348i \(0.523187\pi\)
\(462\) 0 0
\(463\) 24.3518 1.13173 0.565863 0.824499i \(-0.308543\pi\)
0.565863 + 0.824499i \(0.308543\pi\)
\(464\) −11.4149 −0.529923
\(465\) 0 0
\(466\) 58.3806 2.70443
\(467\) −33.1737 −1.53510 −0.767548 0.640991i \(-0.778523\pi\)
−0.767548 + 0.640991i \(0.778523\pi\)
\(468\) −4.80806 −0.222253
\(469\) 20.7266 0.957065
\(470\) 0 0
\(471\) 7.41347 0.341595
\(472\) 2.30185 0.105951
\(473\) 0 0
\(474\) 46.3273 2.12788
\(475\) 0 0
\(476\) 3.40062 0.155867
\(477\) −2.76232 −0.126478
\(478\) −34.1244 −1.56082
\(479\) −17.2977 −0.790353 −0.395176 0.918605i \(-0.629317\pi\)
−0.395176 + 0.918605i \(0.629317\pi\)
\(480\) 0 0
\(481\) 5.62548 0.256500
\(482\) 9.19967 0.419033
\(483\) −8.15736 −0.371173
\(484\) 0 0
\(485\) 0 0
\(486\) 14.1185 0.640429
\(487\) −7.64061 −0.346229 −0.173114 0.984902i \(-0.555383\pi\)
−0.173114 + 0.984902i \(0.555383\pi\)
\(488\) −1.64961 −0.0746744
\(489\) −19.0079 −0.859569
\(490\) 0 0
\(491\) −30.6563 −1.38350 −0.691750 0.722137i \(-0.743160\pi\)
−0.691750 + 0.722137i \(0.743160\pi\)
\(492\) 20.1541 0.908618
\(493\) 1.72625 0.0777466
\(494\) 50.4551 2.27008
\(495\) 0 0
\(496\) 32.1333 1.44283
\(497\) −20.0040 −0.897303
\(498\) 35.7484 1.60192
\(499\) −37.4783 −1.67776 −0.838879 0.544317i \(-0.816789\pi\)
−0.838879 + 0.544317i \(0.816789\pi\)
\(500\) 0 0
\(501\) 26.6289 1.18969
\(502\) −5.58748 −0.249381
\(503\) −8.47695 −0.377969 −0.188984 0.981980i \(-0.560520\pi\)
−0.188984 + 0.981980i \(0.560520\pi\)
\(504\) 1.65420 0.0736839
\(505\) 0 0
\(506\) 0 0
\(507\) 7.07548 0.314233
\(508\) 1.07943 0.0478921
\(509\) −1.69723 −0.0752286 −0.0376143 0.999292i \(-0.511976\pi\)
−0.0376143 + 0.999292i \(0.511976\pi\)
\(510\) 0 0
\(511\) 30.2909 1.33999
\(512\) 29.7439 1.31451
\(513\) −35.3508 −1.56077
\(514\) 45.3268 1.99928
\(515\) 0 0
\(516\) −6.01190 −0.264659
\(517\) 0 0
\(518\) −11.8542 −0.520843
\(519\) 20.7199 0.909502
\(520\) 0 0
\(521\) −37.0929 −1.62507 −0.812535 0.582912i \(-0.801913\pi\)
−0.812535 + 0.582912i \(0.801913\pi\)
\(522\) 5.14315 0.225110
\(523\) 18.0818 0.790662 0.395331 0.918539i \(-0.370630\pi\)
0.395331 + 0.918539i \(0.370630\pi\)
\(524\) 1.50348 0.0656798
\(525\) 0 0
\(526\) −46.3881 −2.02262
\(527\) −4.85946 −0.211681
\(528\) 0 0
\(529\) −21.0672 −0.915965
\(530\) 0 0
\(531\) 1.85667 0.0805726
\(532\) −57.8857 −2.50966
\(533\) 13.4424 0.582257
\(534\) 27.1352 1.17426
\(535\) 0 0
\(536\) 5.52565 0.238672
\(537\) −43.4890 −1.87669
\(538\) 43.4111 1.87159
\(539\) 0 0
\(540\) 0 0
\(541\) 11.7524 0.505275 0.252638 0.967561i \(-0.418702\pi\)
0.252638 + 0.967561i \(0.418702\pi\)
\(542\) −0.886544 −0.0380803
\(543\) −4.58797 −0.196889
\(544\) −3.73955 −0.160332
\(545\) 0 0
\(546\) 37.4949 1.60464
\(547\) 21.7569 0.930256 0.465128 0.885243i \(-0.346008\pi\)
0.465128 + 0.885243i \(0.346008\pi\)
\(548\) −26.7312 −1.14190
\(549\) −1.33057 −0.0567874
\(550\) 0 0
\(551\) −29.3845 −1.25182
\(552\) −2.17473 −0.0925626
\(553\) −35.4501 −1.50749
\(554\) 17.9620 0.763133
\(555\) 0 0
\(556\) −5.67675 −0.240748
\(557\) −4.83432 −0.204837 −0.102418 0.994741i \(-0.532658\pi\)
−0.102418 + 0.994741i \(0.532658\pi\)
\(558\) −14.4781 −0.612908
\(559\) −4.00984 −0.169598
\(560\) 0 0
\(561\) 0 0
\(562\) −14.6903 −0.619672
\(563\) −4.77199 −0.201115 −0.100558 0.994931i \(-0.532063\pi\)
−0.100558 + 0.994931i \(0.532063\pi\)
\(564\) −13.6270 −0.573802
\(565\) 0 0
\(566\) −20.8548 −0.876594
\(567\) −32.3394 −1.35813
\(568\) −5.33302 −0.223768
\(569\) 35.7187 1.49741 0.748703 0.662905i \(-0.230677\pi\)
0.748703 + 0.662905i \(0.230677\pi\)
\(570\) 0 0
\(571\) −33.9838 −1.42218 −0.711090 0.703101i \(-0.751798\pi\)
−0.711090 + 0.703101i \(0.751798\pi\)
\(572\) 0 0
\(573\) −32.9920 −1.37826
\(574\) −28.3263 −1.18232
\(575\) 0 0
\(576\) −7.09553 −0.295647
\(577\) 20.6579 0.860000 0.430000 0.902829i \(-0.358514\pi\)
0.430000 + 0.902829i \(0.358514\pi\)
\(578\) −35.1692 −1.46285
\(579\) −4.94650 −0.205570
\(580\) 0 0
\(581\) −27.3550 −1.13488
\(582\) 61.5447 2.55111
\(583\) 0 0
\(584\) 8.07548 0.334166
\(585\) 0 0
\(586\) −45.7211 −1.88872
\(587\) −13.3600 −0.551428 −0.275714 0.961240i \(-0.588914\pi\)
−0.275714 + 0.961240i \(0.588914\pi\)
\(588\) −11.0090 −0.454004
\(589\) 82.7183 3.40835
\(590\) 0 0
\(591\) 0.276870 0.0113889
\(592\) 5.65752 0.232523
\(593\) −20.8062 −0.854410 −0.427205 0.904155i \(-0.640502\pi\)
−0.427205 + 0.904155i \(0.640502\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −20.8439 −0.853798
\(597\) 14.4274 0.590473
\(598\) −8.88410 −0.363298
\(599\) −14.2456 −0.582061 −0.291030 0.956714i \(-0.593998\pi\)
−0.291030 + 0.956714i \(0.593998\pi\)
\(600\) 0 0
\(601\) −20.7462 −0.846255 −0.423127 0.906070i \(-0.639068\pi\)
−0.423127 + 0.906070i \(0.639068\pi\)
\(602\) 8.44964 0.344382
\(603\) 4.45698 0.181502
\(604\) 28.5287 1.16082
\(605\) 0 0
\(606\) −47.0276 −1.91037
\(607\) −17.9219 −0.727428 −0.363714 0.931511i \(-0.618492\pi\)
−0.363714 + 0.931511i \(0.618492\pi\)
\(608\) 63.6550 2.58155
\(609\) −21.8367 −0.884866
\(610\) 0 0
\(611\) −9.08900 −0.367702
\(612\) 0.731257 0.0295593
\(613\) 28.2019 1.13906 0.569531 0.821970i \(-0.307125\pi\)
0.569531 + 0.821970i \(0.307125\pi\)
\(614\) 64.6764 2.61013
\(615\) 0 0
\(616\) 0 0
\(617\) 4.72930 0.190394 0.0951972 0.995458i \(-0.469652\pi\)
0.0951972 + 0.995458i \(0.469652\pi\)
\(618\) 55.6674 2.23927
\(619\) −30.0575 −1.20811 −0.604056 0.796942i \(-0.706450\pi\)
−0.604056 + 0.796942i \(0.706450\pi\)
\(620\) 0 0
\(621\) 6.22454 0.249782
\(622\) −40.6802 −1.63113
\(623\) −20.7642 −0.831899
\(624\) −17.8948 −0.716367
\(625\) 0 0
\(626\) −2.20314 −0.0880551
\(627\) 0 0
\(628\) −9.26301 −0.369634
\(629\) −0.855577 −0.0341141
\(630\) 0 0
\(631\) −31.7036 −1.26210 −0.631050 0.775742i \(-0.717376\pi\)
−0.631050 + 0.775742i \(0.717376\pi\)
\(632\) −9.45090 −0.375936
\(633\) 4.34023 0.172509
\(634\) 49.9338 1.98313
\(635\) 0 0
\(636\) 19.1505 0.759367
\(637\) −7.34282 −0.290933
\(638\) 0 0
\(639\) −4.30160 −0.170169
\(640\) 0 0
\(641\) −18.9573 −0.748770 −0.374385 0.927273i \(-0.622146\pi\)
−0.374385 + 0.927273i \(0.622146\pi\)
\(642\) −29.3447 −1.15814
\(643\) 36.2489 1.42952 0.714758 0.699372i \(-0.246537\pi\)
0.714758 + 0.699372i \(0.246537\pi\)
\(644\) 10.1925 0.401640
\(645\) 0 0
\(646\) −7.67369 −0.301917
\(647\) 34.9519 1.37410 0.687050 0.726610i \(-0.258905\pi\)
0.687050 + 0.726610i \(0.258905\pi\)
\(648\) −8.62158 −0.338688
\(649\) 0 0
\(650\) 0 0
\(651\) 61.4709 2.40923
\(652\) 23.7501 0.930126
\(653\) 29.7893 1.16574 0.582872 0.812564i \(-0.301929\pi\)
0.582872 + 0.812564i \(0.301929\pi\)
\(654\) 29.7939 1.16503
\(655\) 0 0
\(656\) 13.5190 0.527829
\(657\) 6.51366 0.254122
\(658\) 19.1526 0.746646
\(659\) −7.30532 −0.284575 −0.142287 0.989825i \(-0.545446\pi\)
−0.142287 + 0.989825i \(0.545446\pi\)
\(660\) 0 0
\(661\) −22.7352 −0.884296 −0.442148 0.896942i \(-0.645783\pi\)
−0.442148 + 0.896942i \(0.645783\pi\)
\(662\) 53.7848 2.09040
\(663\) 2.70620 0.105100
\(664\) −7.29277 −0.283014
\(665\) 0 0
\(666\) −2.54908 −0.0987749
\(667\) 5.17401 0.200338
\(668\) −33.2724 −1.28735
\(669\) 16.4129 0.634559
\(670\) 0 0
\(671\) 0 0
\(672\) 47.3043 1.82480
\(673\) 17.7451 0.684024 0.342012 0.939696i \(-0.388892\pi\)
0.342012 + 0.939696i \(0.388892\pi\)
\(674\) 50.0611 1.92828
\(675\) 0 0
\(676\) −8.84069 −0.340026
\(677\) −8.16216 −0.313697 −0.156849 0.987623i \(-0.550133\pi\)
−0.156849 + 0.987623i \(0.550133\pi\)
\(678\) −12.1708 −0.467416
\(679\) −47.0946 −1.80733
\(680\) 0 0
\(681\) 11.8613 0.454527
\(682\) 0 0
\(683\) 6.19100 0.236892 0.118446 0.992960i \(-0.462209\pi\)
0.118446 + 0.992960i \(0.462209\pi\)
\(684\) −12.4475 −0.475944
\(685\) 0 0
\(686\) −29.5136 −1.12683
\(687\) −44.3163 −1.69077
\(688\) −4.03268 −0.153744
\(689\) 12.7731 0.486615
\(690\) 0 0
\(691\) 23.6051 0.897981 0.448990 0.893537i \(-0.351784\pi\)
0.448990 + 0.893537i \(0.351784\pi\)
\(692\) −25.8892 −0.984158
\(693\) 0 0
\(694\) 0.695956 0.0264181
\(695\) 0 0
\(696\) −5.82159 −0.220667
\(697\) −2.04446 −0.0774393
\(698\) 2.55937 0.0968737
\(699\) −53.3013 −2.01604
\(700\) 0 0
\(701\) 37.2284 1.40610 0.703049 0.711142i \(-0.251822\pi\)
0.703049 + 0.711142i \(0.251822\pi\)
\(702\) −28.6108 −1.07985
\(703\) 14.5637 0.549282
\(704\) 0 0
\(705\) 0 0
\(706\) 53.8571 2.02694
\(707\) 35.9860 1.35339
\(708\) −12.8719 −0.483754
\(709\) 18.2537 0.685534 0.342767 0.939420i \(-0.388636\pi\)
0.342767 + 0.939420i \(0.388636\pi\)
\(710\) 0 0
\(711\) −7.62307 −0.285887
\(712\) −5.53567 −0.207458
\(713\) −14.5650 −0.545463
\(714\) −5.70259 −0.213414
\(715\) 0 0
\(716\) 54.3388 2.03074
\(717\) 31.1555 1.16352
\(718\) −37.5717 −1.40216
\(719\) 9.57389 0.357046 0.178523 0.983936i \(-0.442868\pi\)
0.178523 + 0.983936i \(0.442868\pi\)
\(720\) 0 0
\(721\) −42.5972 −1.58640
\(722\) 90.8119 3.37967
\(723\) −8.39927 −0.312372
\(724\) 5.73259 0.213050
\(725\) 0 0
\(726\) 0 0
\(727\) 14.0175 0.519882 0.259941 0.965625i \(-0.416297\pi\)
0.259941 + 0.965625i \(0.416297\pi\)
\(728\) −7.64908 −0.283494
\(729\) 18.7408 0.694104
\(730\) 0 0
\(731\) 0.609854 0.0225563
\(732\) 9.22454 0.340949
\(733\) −9.28772 −0.343050 −0.171525 0.985180i \(-0.554869\pi\)
−0.171525 + 0.985180i \(0.554869\pi\)
\(734\) 17.7909 0.656676
\(735\) 0 0
\(736\) −11.2083 −0.413145
\(737\) 0 0
\(738\) −6.09119 −0.224220
\(739\) −38.9586 −1.43312 −0.716558 0.697527i \(-0.754284\pi\)
−0.716558 + 0.697527i \(0.754284\pi\)
\(740\) 0 0
\(741\) −46.0653 −1.69225
\(742\) −26.9158 −0.988108
\(743\) −45.9296 −1.68499 −0.842497 0.538701i \(-0.818915\pi\)
−0.842497 + 0.538701i \(0.818915\pi\)
\(744\) 16.3880 0.600812
\(745\) 0 0
\(746\) −75.1057 −2.74982
\(747\) −5.88233 −0.215223
\(748\) 0 0
\(749\) 22.4549 0.820483
\(750\) 0 0
\(751\) 23.1928 0.846318 0.423159 0.906055i \(-0.360921\pi\)
0.423159 + 0.906055i \(0.360921\pi\)
\(752\) −9.14077 −0.333330
\(753\) 5.10135 0.185904
\(754\) −23.7821 −0.866093
\(755\) 0 0
\(756\) 32.8244 1.19381
\(757\) 6.52202 0.237047 0.118523 0.992951i \(-0.462184\pi\)
0.118523 + 0.992951i \(0.462184\pi\)
\(758\) 36.7757 1.33575
\(759\) 0 0
\(760\) 0 0
\(761\) −6.56682 −0.238047 −0.119024 0.992891i \(-0.537976\pi\)
−0.119024 + 0.992891i \(0.537976\pi\)
\(762\) −1.81013 −0.0655742
\(763\) −22.7986 −0.825364
\(764\) 41.2230 1.49139
\(765\) 0 0
\(766\) 45.3108 1.63715
\(767\) −8.58530 −0.309997
\(768\) −15.4386 −0.557091
\(769\) 12.5950 0.454188 0.227094 0.973873i \(-0.427078\pi\)
0.227094 + 0.973873i \(0.427078\pi\)
\(770\) 0 0
\(771\) −41.3832 −1.49038
\(772\) 6.18057 0.222444
\(773\) −21.9448 −0.789299 −0.394650 0.918832i \(-0.629134\pi\)
−0.394650 + 0.918832i \(0.629134\pi\)
\(774\) 1.81698 0.0653101
\(775\) 0 0
\(776\) −12.5553 −0.450709
\(777\) 10.8228 0.388267
\(778\) 74.8892 2.68491
\(779\) 34.8010 1.24687
\(780\) 0 0
\(781\) 0 0
\(782\) 1.35118 0.0483181
\(783\) 16.6627 0.595475
\(784\) −7.38464 −0.263737
\(785\) 0 0
\(786\) −2.52123 −0.0899292
\(787\) −16.7298 −0.596354 −0.298177 0.954511i \(-0.596379\pi\)
−0.298177 + 0.954511i \(0.596379\pi\)
\(788\) −0.345944 −0.0123237
\(789\) 42.3522 1.50778
\(790\) 0 0
\(791\) 9.31320 0.331139
\(792\) 0 0
\(793\) 6.15261 0.218486
\(794\) 41.9995 1.49051
\(795\) 0 0
\(796\) −18.0268 −0.638942
\(797\) 11.7707 0.416940 0.208470 0.978029i \(-0.433152\pi\)
0.208470 + 0.978029i \(0.433152\pi\)
\(798\) 97.0702 3.43625
\(799\) 1.38234 0.0489038
\(800\) 0 0
\(801\) −4.46505 −0.157765
\(802\) −43.7889 −1.54624
\(803\) 0 0
\(804\) −30.8992 −1.08973
\(805\) 0 0
\(806\) 66.9473 2.35812
\(807\) −39.6342 −1.39519
\(808\) 9.59377 0.337508
\(809\) 1.19595 0.0420472 0.0210236 0.999779i \(-0.493307\pi\)
0.0210236 + 0.999779i \(0.493307\pi\)
\(810\) 0 0
\(811\) 35.8253 1.25800 0.628998 0.777407i \(-0.283465\pi\)
0.628998 + 0.777407i \(0.283465\pi\)
\(812\) 27.2845 0.957499
\(813\) 0.809412 0.0283873
\(814\) 0 0
\(815\) 0 0
\(816\) 2.72162 0.0952757
\(817\) −10.3810 −0.363186
\(818\) 49.2356 1.72148
\(819\) −6.16973 −0.215588
\(820\) 0 0
\(821\) 5.49200 0.191672 0.0958360 0.995397i \(-0.469448\pi\)
0.0958360 + 0.995397i \(0.469448\pi\)
\(822\) 44.8263 1.56350
\(823\) −42.7252 −1.48931 −0.744654 0.667451i \(-0.767386\pi\)
−0.744654 + 0.667451i \(0.767386\pi\)
\(824\) −11.3563 −0.395616
\(825\) 0 0
\(826\) 18.0912 0.629473
\(827\) −32.4208 −1.12738 −0.563691 0.825986i \(-0.690619\pi\)
−0.563691 + 0.825986i \(0.690619\pi\)
\(828\) 2.19176 0.0761688
\(829\) −2.91149 −0.101120 −0.0505600 0.998721i \(-0.516101\pi\)
−0.0505600 + 0.998721i \(0.516101\pi\)
\(830\) 0 0
\(831\) −16.3993 −0.568884
\(832\) 32.8100 1.13748
\(833\) 1.11677 0.0386937
\(834\) 9.51950 0.329633
\(835\) 0 0
\(836\) 0 0
\(837\) −46.9059 −1.62130
\(838\) −21.1893 −0.731973
\(839\) −20.8465 −0.719701 −0.359850 0.933010i \(-0.617172\pi\)
−0.359850 + 0.933010i \(0.617172\pi\)
\(840\) 0 0
\(841\) −15.1496 −0.522398
\(842\) 18.6960 0.644305
\(843\) 13.4122 0.461940
\(844\) −5.42304 −0.186669
\(845\) 0 0
\(846\) 4.11851 0.141597
\(847\) 0 0
\(848\) 12.8458 0.441127
\(849\) 19.0404 0.653464
\(850\) 0 0
\(851\) −2.56437 −0.0879056
\(852\) 29.8220 1.02168
\(853\) −34.5509 −1.18300 −0.591500 0.806305i \(-0.701464\pi\)
−0.591500 + 0.806305i \(0.701464\pi\)
\(854\) −12.9650 −0.443652
\(855\) 0 0
\(856\) 5.98641 0.204611
\(857\) −33.2969 −1.13740 −0.568699 0.822545i \(-0.692553\pi\)
−0.568699 + 0.822545i \(0.692553\pi\)
\(858\) 0 0
\(859\) −16.7665 −0.572067 −0.286034 0.958220i \(-0.592337\pi\)
−0.286034 + 0.958220i \(0.592337\pi\)
\(860\) 0 0
\(861\) 25.8618 0.881369
\(862\) −36.9028 −1.25692
\(863\) 1.48415 0.0505211 0.0252605 0.999681i \(-0.491958\pi\)
0.0252605 + 0.999681i \(0.491958\pi\)
\(864\) −36.0959 −1.22801
\(865\) 0 0
\(866\) −19.1593 −0.651060
\(867\) 32.1094 1.09049
\(868\) −76.8068 −2.60699
\(869\) 0 0
\(870\) 0 0
\(871\) −20.6092 −0.698316
\(872\) −6.07803 −0.205828
\(873\) −10.1271 −0.342749
\(874\) −22.9999 −0.777984
\(875\) 0 0
\(876\) −45.1577 −1.52574
\(877\) 24.8615 0.839513 0.419756 0.907637i \(-0.362115\pi\)
0.419756 + 0.907637i \(0.362115\pi\)
\(878\) 13.5554 0.457473
\(879\) 41.7433 1.40797
\(880\) 0 0
\(881\) 32.6968 1.10158 0.550792 0.834643i \(-0.314326\pi\)
0.550792 + 0.834643i \(0.314326\pi\)
\(882\) 3.32726 0.112035
\(883\) −47.6218 −1.60260 −0.801300 0.598263i \(-0.795858\pi\)
−0.801300 + 0.598263i \(0.795858\pi\)
\(884\) −3.38136 −0.113727
\(885\) 0 0
\(886\) 85.6644 2.87795
\(887\) −59.0960 −1.98425 −0.992125 0.125251i \(-0.960026\pi\)
−0.992125 + 0.125251i \(0.960026\pi\)
\(888\) 2.88533 0.0968255
\(889\) 1.38513 0.0464559
\(890\) 0 0
\(891\) 0 0
\(892\) −20.5076 −0.686646
\(893\) −23.5304 −0.787415
\(894\) 34.9537 1.16903
\(895\) 0 0
\(896\) −19.6824 −0.657543
\(897\) 8.11115 0.270824
\(898\) −25.4804 −0.850291
\(899\) −38.9894 −1.30037
\(900\) 0 0
\(901\) −1.94265 −0.0647190
\(902\) 0 0
\(903\) −7.71450 −0.256722
\(904\) 2.48287 0.0825791
\(905\) 0 0
\(906\) −47.8407 −1.58940
\(907\) −34.6576 −1.15079 −0.575393 0.817877i \(-0.695151\pi\)
−0.575393 + 0.817877i \(0.695151\pi\)
\(908\) −14.8205 −0.491836
\(909\) 7.73831 0.256664
\(910\) 0 0
\(911\) −10.1883 −0.337552 −0.168776 0.985654i \(-0.553981\pi\)
−0.168776 + 0.985654i \(0.553981\pi\)
\(912\) −46.3277 −1.53406
\(913\) 0 0
\(914\) 20.7484 0.686298
\(915\) 0 0
\(916\) 55.3724 1.82956
\(917\) 1.92927 0.0637101
\(918\) 4.35141 0.143618
\(919\) −36.1289 −1.19178 −0.595891 0.803065i \(-0.703201\pi\)
−0.595891 + 0.803065i \(0.703201\pi\)
\(920\) 0 0
\(921\) −59.0493 −1.94574
\(922\) −6.54840 −0.215660
\(923\) 19.8907 0.654711
\(924\) 0 0
\(925\) 0 0
\(926\) 51.0242 1.67676
\(927\) −9.15997 −0.300853
\(928\) −30.0039 −0.984927
\(929\) 28.1240 0.922717 0.461359 0.887214i \(-0.347362\pi\)
0.461359 + 0.887214i \(0.347362\pi\)
\(930\) 0 0
\(931\) −19.0097 −0.623019
\(932\) 66.5990 2.18152
\(933\) 37.1409 1.21594
\(934\) −69.5087 −2.27439
\(935\) 0 0
\(936\) −1.64483 −0.0537630
\(937\) −0.0851677 −0.00278231 −0.00139115 0.999999i \(-0.500443\pi\)
−0.00139115 + 0.999999i \(0.500443\pi\)
\(938\) 43.4283 1.41798
\(939\) 2.01146 0.0656414
\(940\) 0 0
\(941\) −41.8154 −1.36314 −0.681572 0.731751i \(-0.738703\pi\)
−0.681572 + 0.731751i \(0.738703\pi\)
\(942\) 15.5334 0.506106
\(943\) −6.12774 −0.199547
\(944\) −8.63420 −0.281019
\(945\) 0 0
\(946\) 0 0
\(947\) −8.92463 −0.290012 −0.145006 0.989431i \(-0.546320\pi\)
−0.145006 + 0.989431i \(0.546320\pi\)
\(948\) 52.8489 1.71645
\(949\) −30.1194 −0.977716
\(950\) 0 0
\(951\) −45.5894 −1.47834
\(952\) 1.16335 0.0377042
\(953\) −5.26383 −0.170512 −0.0852561 0.996359i \(-0.527171\pi\)
−0.0852561 + 0.996359i \(0.527171\pi\)
\(954\) −5.78787 −0.187389
\(955\) 0 0
\(956\) −38.9283 −1.25903
\(957\) 0 0
\(958\) −36.2438 −1.17098
\(959\) −34.3016 −1.10765
\(960\) 0 0
\(961\) 78.7564 2.54053
\(962\) 11.7870 0.380029
\(963\) 4.82862 0.155600
\(964\) 10.4947 0.338013
\(965\) 0 0
\(966\) −17.0921 −0.549928
\(967\) 18.5421 0.596275 0.298138 0.954523i \(-0.403635\pi\)
0.298138 + 0.954523i \(0.403635\pi\)
\(968\) 0 0
\(969\) 7.00606 0.225067
\(970\) 0 0
\(971\) −24.2230 −0.777354 −0.388677 0.921374i \(-0.627068\pi\)
−0.388677 + 0.921374i \(0.627068\pi\)
\(972\) 16.1060 0.516601
\(973\) −7.28442 −0.233528
\(974\) −16.0093 −0.512972
\(975\) 0 0
\(976\) 6.18765 0.198062
\(977\) −49.0618 −1.56963 −0.784813 0.619732i \(-0.787241\pi\)
−0.784813 + 0.619732i \(0.787241\pi\)
\(978\) −39.8272 −1.27353
\(979\) 0 0
\(980\) 0 0
\(981\) −4.90253 −0.156526
\(982\) −64.2340 −2.04979
\(983\) −49.1394 −1.56731 −0.783653 0.621199i \(-0.786646\pi\)
−0.783653 + 0.621199i \(0.786646\pi\)
\(984\) 6.89469 0.219795
\(985\) 0 0
\(986\) 3.61701 0.115189
\(987\) −17.4863 −0.556594
\(988\) 57.5578 1.83116
\(989\) 1.82788 0.0581233
\(990\) 0 0
\(991\) 30.9620 0.983541 0.491771 0.870725i \(-0.336350\pi\)
0.491771 + 0.870725i \(0.336350\pi\)
\(992\) 84.4619 2.68167
\(993\) −49.1053 −1.55831
\(994\) −41.9143 −1.32944
\(995\) 0 0
\(996\) 40.7808 1.29219
\(997\) −22.8939 −0.725058 −0.362529 0.931972i \(-0.618087\pi\)
−0.362529 + 0.931972i \(0.618087\pi\)
\(998\) −78.5280 −2.48576
\(999\) −8.25845 −0.261286
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 3025.2.a.bd.1.4 4
5.4 even 2 605.2.a.j.1.1 4
11.3 even 5 275.2.h.a.251.2 8
11.4 even 5 275.2.h.a.126.2 8
11.10 odd 2 3025.2.a.w.1.1 4
15.14 odd 2 5445.2.a.bp.1.4 4
20.19 odd 2 9680.2.a.cn.1.1 4
55.3 odd 20 275.2.z.a.174.1 16
55.4 even 10 55.2.g.b.16.1 8
55.9 even 10 605.2.g.m.81.2 8
55.14 even 10 55.2.g.b.31.1 yes 8
55.19 odd 10 605.2.g.k.251.2 8
55.24 odd 10 605.2.g.e.81.1 8
55.29 odd 10 605.2.g.k.511.2 8
55.37 odd 20 275.2.z.a.49.1 16
55.39 odd 10 605.2.g.e.366.1 8
55.47 odd 20 275.2.z.a.174.4 16
55.48 odd 20 275.2.z.a.49.4 16
55.49 even 10 605.2.g.m.366.2 8
55.54 odd 2 605.2.a.k.1.4 4
165.14 odd 10 495.2.n.e.361.2 8
165.59 odd 10 495.2.n.e.181.2 8
165.164 even 2 5445.2.a.bi.1.1 4
220.59 odd 10 880.2.bo.h.401.2 8
220.179 odd 10 880.2.bo.h.801.2 8
220.219 even 2 9680.2.a.cm.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.g.b.16.1 8 55.4 even 10
55.2.g.b.31.1 yes 8 55.14 even 10
275.2.h.a.126.2 8 11.4 even 5
275.2.h.a.251.2 8 11.3 even 5
275.2.z.a.49.1 16 55.37 odd 20
275.2.z.a.49.4 16 55.48 odd 20
275.2.z.a.174.1 16 55.3 odd 20
275.2.z.a.174.4 16 55.47 odd 20
495.2.n.e.181.2 8 165.59 odd 10
495.2.n.e.361.2 8 165.14 odd 10
605.2.a.j.1.1 4 5.4 even 2
605.2.a.k.1.4 4 55.54 odd 2
605.2.g.e.81.1 8 55.24 odd 10
605.2.g.e.366.1 8 55.39 odd 10
605.2.g.k.251.2 8 55.19 odd 10
605.2.g.k.511.2 8 55.29 odd 10
605.2.g.m.81.2 8 55.9 even 10
605.2.g.m.366.2 8 55.49 even 10
880.2.bo.h.401.2 8 220.59 odd 10
880.2.bo.h.801.2 8 220.179 odd 10
3025.2.a.w.1.1 4 11.10 odd 2
3025.2.a.bd.1.4 4 1.1 even 1 trivial
5445.2.a.bi.1.1 4 165.164 even 2
5445.2.a.bp.1.4 4 15.14 odd 2
9680.2.a.cm.1.1 4 220.219 even 2
9680.2.a.cn.1.1 4 20.19 odd 2