Properties

Label 7500.2.a.g.1.1
Level $7500$
Weight $2$
Character 7500.1
Self dual yes
Analytic conductor $59.888$
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] = [7500,2,Mod(1,7500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7500, 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("7500.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7500.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: \(59.8878015160\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.95630\) of defining polynomial
Character \(\chi\) \(=\) 7500.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -4.78339 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -4.78339 q^{7} +1.00000 q^{9} +1.95630 q^{11} +1.07985 q^{13} -3.80573 q^{17} +4.25085 q^{19} -4.78339 q^{21} -5.86324 q^{23} +1.00000 q^{27} +10.5501 q^{29} -4.31592 q^{31} +1.95630 q^{33} +5.65983 q^{37} +1.07985 q^{39} +0.858568 q^{41} -10.8764 q^{43} +3.00158 q^{47} +15.8808 q^{49} -3.80573 q^{51} +4.22384 q^{53} +4.25085 q^{57} -4.77018 q^{59} +3.63184 q^{61} -4.78339 q^{63} +6.93046 q^{67} -5.86324 q^{69} -11.9971 q^{71} -16.9560 q^{73} -9.35772 q^{77} -6.62059 q^{79} +1.00000 q^{81} -2.58189 q^{83} +10.5501 q^{87} -0.832738 q^{89} -5.16535 q^{91} -4.31592 q^{93} -9.11698 q^{97} +1.95630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9} - q^{11} + 5 q^{13} - 4 q^{17} - 5 q^{19} - 4 q^{21} - 9 q^{23} + 4 q^{27} - 4 q^{29} - 9 q^{31} - q^{33} - 2 q^{37} + 5 q^{39} - 34 q^{43} - 6 q^{47} - 4 q^{49} - 4 q^{51} + 2 q^{53} - 5 q^{57} + q^{59} - 2 q^{61} - 4 q^{63} + 4 q^{67} - 9 q^{69} - 20 q^{71} - 12 q^{73} - 9 q^{77} - 3 q^{79} + 4 q^{81} - 14 q^{83} - 4 q^{87} + 15 q^{89} - 10 q^{91} - 9 q^{93} - 28 q^{97} - q^{99}+O(q^{100}) \) 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 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.78339 −1.80795 −0.903975 0.427585i \(-0.859364\pi\)
−0.903975 + 0.427585i \(0.859364\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.95630 0.589845 0.294923 0.955521i \(-0.404706\pi\)
0.294923 + 0.955521i \(0.404706\pi\)
\(12\) 0 0
\(13\) 1.07985 0.299497 0.149749 0.988724i \(-0.452154\pi\)
0.149749 + 0.988724i \(0.452154\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.80573 −0.923024 −0.461512 0.887134i \(-0.652693\pi\)
−0.461512 + 0.887134i \(0.652693\pi\)
\(18\) 0 0
\(19\) 4.25085 0.975212 0.487606 0.873064i \(-0.337870\pi\)
0.487606 + 0.873064i \(0.337870\pi\)
\(20\) 0 0
\(21\) −4.78339 −1.04382
\(22\) 0 0
\(23\) −5.86324 −1.22257 −0.611285 0.791411i \(-0.709347\pi\)
−0.611285 + 0.791411i \(0.709347\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 10.5501 1.95910 0.979550 0.201200i \(-0.0644841\pi\)
0.979550 + 0.201200i \(0.0644841\pi\)
\(30\) 0 0
\(31\) −4.31592 −0.775162 −0.387581 0.921836i \(-0.626689\pi\)
−0.387581 + 0.921836i \(0.626689\pi\)
\(32\) 0 0
\(33\) 1.95630 0.340547
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.65983 0.930470 0.465235 0.885187i \(-0.345970\pi\)
0.465235 + 0.885187i \(0.345970\pi\)
\(38\) 0 0
\(39\) 1.07985 0.172915
\(40\) 0 0
\(41\) 0.858568 0.134086 0.0670429 0.997750i \(-0.478644\pi\)
0.0670429 + 0.997750i \(0.478644\pi\)
\(42\) 0 0
\(43\) −10.8764 −1.65864 −0.829321 0.558773i \(-0.811272\pi\)
−0.829321 + 0.558773i \(0.811272\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00158 0.437825 0.218913 0.975744i \(-0.429749\pi\)
0.218913 + 0.975744i \(0.429749\pi\)
\(48\) 0 0
\(49\) 15.8808 2.26868
\(50\) 0 0
\(51\) −3.80573 −0.532908
\(52\) 0 0
\(53\) 4.22384 0.580189 0.290095 0.956998i \(-0.406313\pi\)
0.290095 + 0.956998i \(0.406313\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.25085 0.563039
\(58\) 0 0
\(59\) −4.77018 −0.621025 −0.310512 0.950569i \(-0.600501\pi\)
−0.310512 + 0.950569i \(0.600501\pi\)
\(60\) 0 0
\(61\) 3.63184 0.465010 0.232505 0.972595i \(-0.425308\pi\)
0.232505 + 0.972595i \(0.425308\pi\)
\(62\) 0 0
\(63\) −4.78339 −0.602650
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.93046 0.846691 0.423346 0.905968i \(-0.360856\pi\)
0.423346 + 0.905968i \(0.360856\pi\)
\(68\) 0 0
\(69\) −5.86324 −0.705851
\(70\) 0 0
\(71\) −11.9971 −1.42380 −0.711898 0.702283i \(-0.752164\pi\)
−0.711898 + 0.702283i \(0.752164\pi\)
\(72\) 0 0
\(73\) −16.9560 −1.98455 −0.992273 0.124075i \(-0.960404\pi\)
−0.992273 + 0.124075i \(0.960404\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −9.35772 −1.06641
\(78\) 0 0
\(79\) −6.62059 −0.744875 −0.372437 0.928057i \(-0.621478\pi\)
−0.372437 + 0.928057i \(0.621478\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.58189 −0.283399 −0.141699 0.989910i \(-0.545257\pi\)
−0.141699 + 0.989910i \(0.545257\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 10.5501 1.13109
\(88\) 0 0
\(89\) −0.832738 −0.0882700 −0.0441350 0.999026i \(-0.514053\pi\)
−0.0441350 + 0.999026i \(0.514053\pi\)
\(90\) 0 0
\(91\) −5.16535 −0.541476
\(92\) 0 0
\(93\) −4.31592 −0.447540
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.11698 −0.925689 −0.462844 0.886440i \(-0.653171\pi\)
−0.462844 + 0.886440i \(0.653171\pi\)
\(98\) 0 0
\(99\) 1.95630 0.196615
\(100\) 0 0
\(101\) 1.49541 0.148799 0.0743994 0.997229i \(-0.476296\pi\)
0.0743994 + 0.997229i \(0.476296\pi\)
\(102\) 0 0
\(103\) 10.8623 1.07029 0.535147 0.844759i \(-0.320256\pi\)
0.535147 + 0.844759i \(0.320256\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −19.8390 −1.91791 −0.958954 0.283563i \(-0.908484\pi\)
−0.958954 + 0.283563i \(0.908484\pi\)
\(108\) 0 0
\(109\) −2.34391 −0.224506 −0.112253 0.993680i \(-0.535807\pi\)
−0.112253 + 0.993680i \(0.535807\pi\)
\(110\) 0 0
\(111\) 5.65983 0.537207
\(112\) 0 0
\(113\) −3.22851 −0.303713 −0.151856 0.988403i \(-0.548525\pi\)
−0.151856 + 0.988403i \(0.548525\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.07985 0.0998324
\(118\) 0 0
\(119\) 18.2043 1.66878
\(120\) 0 0
\(121\) −7.17291 −0.652083
\(122\) 0 0
\(123\) 0.858568 0.0774145
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.38631 0.655429 0.327714 0.944777i \(-0.393722\pi\)
0.327714 + 0.944777i \(0.393722\pi\)
\(128\) 0 0
\(129\) −10.8764 −0.957617
\(130\) 0 0
\(131\) −16.8839 −1.47515 −0.737575 0.675265i \(-0.764029\pi\)
−0.737575 + 0.675265i \(0.764029\pi\)
\(132\) 0 0
\(133\) −20.3335 −1.76313
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.839318 0.0717078 0.0358539 0.999357i \(-0.488585\pi\)
0.0358539 + 0.999357i \(0.488585\pi\)
\(138\) 0 0
\(139\) −11.1580 −0.946409 −0.473205 0.880953i \(-0.656903\pi\)
−0.473205 + 0.880953i \(0.656903\pi\)
\(140\) 0 0
\(141\) 3.00158 0.252779
\(142\) 0 0
\(143\) 2.11251 0.176657
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 15.8808 1.30982
\(148\) 0 0
\(149\) −22.0277 −1.80458 −0.902288 0.431134i \(-0.858114\pi\)
−0.902288 + 0.431134i \(0.858114\pi\)
\(150\) 0 0
\(151\) 11.7890 0.959378 0.479689 0.877439i \(-0.340750\pi\)
0.479689 + 0.877439i \(0.340750\pi\)
\(152\) 0 0
\(153\) −3.80573 −0.307675
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.2769 0.899994 0.449997 0.893030i \(-0.351425\pi\)
0.449997 + 0.893030i \(0.351425\pi\)
\(158\) 0 0
\(159\) 4.22384 0.334972
\(160\) 0 0
\(161\) 28.0461 2.21035
\(162\) 0 0
\(163\) 9.89630 0.775138 0.387569 0.921841i \(-0.373315\pi\)
0.387569 + 0.921841i \(0.373315\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.51564 −0.581577 −0.290789 0.956787i \(-0.593918\pi\)
−0.290789 + 0.956787i \(0.593918\pi\)
\(168\) 0 0
\(169\) −11.8339 −0.910301
\(170\) 0 0
\(171\) 4.25085 0.325071
\(172\) 0 0
\(173\) 12.2783 0.933499 0.466749 0.884390i \(-0.345425\pi\)
0.466749 + 0.884390i \(0.345425\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.77018 −0.358549
\(178\) 0 0
\(179\) 15.2089 1.13677 0.568384 0.822763i \(-0.307569\pi\)
0.568384 + 0.822763i \(0.307569\pi\)
\(180\) 0 0
\(181\) −20.8755 −1.55166 −0.775831 0.630941i \(-0.782669\pi\)
−0.775831 + 0.630941i \(0.782669\pi\)
\(182\) 0 0
\(183\) 3.63184 0.268473
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −7.44512 −0.544441
\(188\) 0 0
\(189\) −4.78339 −0.347940
\(190\) 0 0
\(191\) −5.25085 −0.379938 −0.189969 0.981790i \(-0.560839\pi\)
−0.189969 + 0.981790i \(0.560839\pi\)
\(192\) 0 0
\(193\) 7.94565 0.571940 0.285970 0.958239i \(-0.407684\pi\)
0.285970 + 0.958239i \(0.407684\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.0059 −0.926633 −0.463317 0.886193i \(-0.653341\pi\)
−0.463317 + 0.886193i \(0.653341\pi\)
\(198\) 0 0
\(199\) 1.06434 0.0754490 0.0377245 0.999288i \(-0.487989\pi\)
0.0377245 + 0.999288i \(0.487989\pi\)
\(200\) 0 0
\(201\) 6.93046 0.488837
\(202\) 0 0
\(203\) −50.4651 −3.54196
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −5.86324 −0.407523
\(208\) 0 0
\(209\) 8.31592 0.575224
\(210\) 0 0
\(211\) −20.4928 −1.41079 −0.705393 0.708817i \(-0.749229\pi\)
−0.705393 + 0.708817i \(0.749229\pi\)
\(212\) 0 0
\(213\) −11.9971 −0.822029
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 20.6447 1.40145
\(218\) 0 0
\(219\) −16.9560 −1.14578
\(220\) 0 0
\(221\) −4.10962 −0.276443
\(222\) 0 0
\(223\) 10.3121 0.690549 0.345275 0.938502i \(-0.387786\pi\)
0.345275 + 0.938502i \(0.387786\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −18.1409 −1.20405 −0.602027 0.798476i \(-0.705640\pi\)
−0.602027 + 0.798476i \(0.705640\pi\)
\(228\) 0 0
\(229\) −9.71827 −0.642202 −0.321101 0.947045i \(-0.604053\pi\)
−0.321101 + 0.947045i \(0.604053\pi\)
\(230\) 0 0
\(231\) −9.35772 −0.615692
\(232\) 0 0
\(233\) −22.2184 −1.45558 −0.727788 0.685802i \(-0.759451\pi\)
−0.727788 + 0.685802i \(0.759451\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −6.62059 −0.430054
\(238\) 0 0
\(239\) 15.9293 1.03038 0.515190 0.857076i \(-0.327721\pi\)
0.515190 + 0.857076i \(0.327721\pi\)
\(240\) 0 0
\(241\) −4.93147 −0.317664 −0.158832 0.987306i \(-0.550773\pi\)
−0.158832 + 0.987306i \(0.550773\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.59029 0.292073
\(248\) 0 0
\(249\) −2.58189 −0.163620
\(250\) 0 0
\(251\) 15.4351 0.974252 0.487126 0.873332i \(-0.338045\pi\)
0.487126 + 0.873332i \(0.338045\pi\)
\(252\) 0 0
\(253\) −11.4702 −0.721127
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.1828 1.32134 0.660672 0.750674i \(-0.270271\pi\)
0.660672 + 0.750674i \(0.270271\pi\)
\(258\) 0 0
\(259\) −27.0731 −1.68224
\(260\) 0 0
\(261\) 10.5501 0.653033
\(262\) 0 0
\(263\) −20.9264 −1.29038 −0.645188 0.764023i \(-0.723221\pi\)
−0.645188 + 0.764023i \(0.723221\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −0.832738 −0.0509627
\(268\) 0 0
\(269\) −13.7837 −0.840405 −0.420202 0.907430i \(-0.638041\pi\)
−0.420202 + 0.907430i \(0.638041\pi\)
\(270\) 0 0
\(271\) 10.5495 0.640837 0.320418 0.947276i \(-0.396177\pi\)
0.320418 + 0.947276i \(0.396177\pi\)
\(272\) 0 0
\(273\) −5.16535 −0.312621
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 17.9622 1.07924 0.539622 0.841907i \(-0.318567\pi\)
0.539622 + 0.841907i \(0.318567\pi\)
\(278\) 0 0
\(279\) −4.31592 −0.258387
\(280\) 0 0
\(281\) −0.926728 −0.0552840 −0.0276420 0.999618i \(-0.508800\pi\)
−0.0276420 + 0.999618i \(0.508800\pi\)
\(282\) 0 0
\(283\) −6.57749 −0.390991 −0.195496 0.980705i \(-0.562632\pi\)
−0.195496 + 0.980705i \(0.562632\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.10686 −0.242420
\(288\) 0 0
\(289\) −2.51644 −0.148026
\(290\) 0 0
\(291\) −9.11698 −0.534447
\(292\) 0 0
\(293\) 25.6208 1.49678 0.748391 0.663257i \(-0.230827\pi\)
0.748391 + 0.663257i \(0.230827\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.95630 0.113516
\(298\) 0 0
\(299\) −6.33143 −0.366156
\(300\) 0 0
\(301\) 52.0262 2.99874
\(302\) 0 0
\(303\) 1.49541 0.0859091
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −6.18420 −0.352951 −0.176476 0.984305i \(-0.556470\pi\)
−0.176476 + 0.984305i \(0.556470\pi\)
\(308\) 0 0
\(309\) 10.8623 0.617935
\(310\) 0 0
\(311\) −24.2382 −1.37442 −0.687210 0.726459i \(-0.741165\pi\)
−0.687210 + 0.726459i \(0.741165\pi\)
\(312\) 0 0
\(313\) −29.9612 −1.69351 −0.846755 0.531984i \(-0.821447\pi\)
−0.846755 + 0.531984i \(0.821447\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.46267 0.138317 0.0691586 0.997606i \(-0.477969\pi\)
0.0691586 + 0.997606i \(0.477969\pi\)
\(318\) 0 0
\(319\) 20.6391 1.15557
\(320\) 0 0
\(321\) −19.8390 −1.10730
\(322\) 0 0
\(323\) −16.1776 −0.900145
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.34391 −0.129618
\(328\) 0 0
\(329\) −14.3577 −0.791566
\(330\) 0 0
\(331\) −26.7867 −1.47233 −0.736166 0.676801i \(-0.763366\pi\)
−0.736166 + 0.676801i \(0.763366\pi\)
\(332\) 0 0
\(333\) 5.65983 0.310157
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.43659 0.132730 0.0663648 0.997795i \(-0.478860\pi\)
0.0663648 + 0.997795i \(0.478860\pi\)
\(338\) 0 0
\(339\) −3.22851 −0.175349
\(340\) 0 0
\(341\) −8.44321 −0.457226
\(342\) 0 0
\(343\) −42.4802 −2.29372
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.63749 −0.410002 −0.205001 0.978762i \(-0.565720\pi\)
−0.205001 + 0.978762i \(0.565720\pi\)
\(348\) 0 0
\(349\) −20.2283 −1.08280 −0.541398 0.840766i \(-0.682105\pi\)
−0.541398 + 0.840766i \(0.682105\pi\)
\(350\) 0 0
\(351\) 1.07985 0.0576383
\(352\) 0 0
\(353\) −28.2922 −1.50584 −0.752922 0.658110i \(-0.771356\pi\)
−0.752922 + 0.658110i \(0.771356\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 18.2043 0.963472
\(358\) 0 0
\(359\) 22.8318 1.20502 0.602509 0.798112i \(-0.294168\pi\)
0.602509 + 0.798112i \(0.294168\pi\)
\(360\) 0 0
\(361\) −0.930261 −0.0489611
\(362\) 0 0
\(363\) −7.17291 −0.376480
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2.51795 −0.131436 −0.0657181 0.997838i \(-0.520934\pi\)
−0.0657181 + 0.997838i \(0.520934\pi\)
\(368\) 0 0
\(369\) 0.858568 0.0446953
\(370\) 0 0
\(371\) −20.2043 −1.04895
\(372\) 0 0
\(373\) 5.65398 0.292752 0.146376 0.989229i \(-0.453239\pi\)
0.146376 + 0.989229i \(0.453239\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11.3925 0.586745
\(378\) 0 0
\(379\) −14.3917 −0.739254 −0.369627 0.929180i \(-0.620515\pi\)
−0.369627 + 0.929180i \(0.620515\pi\)
\(380\) 0 0
\(381\) 7.38631 0.378412
\(382\) 0 0
\(383\) −8.19992 −0.418996 −0.209498 0.977809i \(-0.567183\pi\)
−0.209498 + 0.977809i \(0.567183\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −10.8764 −0.552881
\(388\) 0 0
\(389\) 4.73067 0.239855 0.119927 0.992783i \(-0.461734\pi\)
0.119927 + 0.992783i \(0.461734\pi\)
\(390\) 0 0
\(391\) 22.3139 1.12846
\(392\) 0 0
\(393\) −16.8839 −0.851679
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7.58326 −0.380593 −0.190297 0.981727i \(-0.560945\pi\)
−0.190297 + 0.981727i \(0.560945\pi\)
\(398\) 0 0
\(399\) −20.3335 −1.01795
\(400\) 0 0
\(401\) 10.1128 0.505011 0.252506 0.967595i \(-0.418745\pi\)
0.252506 + 0.967595i \(0.418745\pi\)
\(402\) 0 0
\(403\) −4.66056 −0.232159
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.0723 0.548833
\(408\) 0 0
\(409\) 3.97924 0.196761 0.0983804 0.995149i \(-0.468634\pi\)
0.0983804 + 0.995149i \(0.468634\pi\)
\(410\) 0 0
\(411\) 0.839318 0.0414005
\(412\) 0 0
\(413\) 22.8176 1.12278
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −11.1580 −0.546410
\(418\) 0 0
\(419\) 1.79655 0.0877671 0.0438835 0.999037i \(-0.486027\pi\)
0.0438835 + 0.999037i \(0.486027\pi\)
\(420\) 0 0
\(421\) 15.6701 0.763716 0.381858 0.924221i \(-0.375284\pi\)
0.381858 + 0.924221i \(0.375284\pi\)
\(422\) 0 0
\(423\) 3.00158 0.145942
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −17.3725 −0.840714
\(428\) 0 0
\(429\) 2.11251 0.101993
\(430\) 0 0
\(431\) −38.5760 −1.85814 −0.929071 0.369901i \(-0.879391\pi\)
−0.929071 + 0.369901i \(0.879391\pi\)
\(432\) 0 0
\(433\) 9.16949 0.440657 0.220329 0.975426i \(-0.429287\pi\)
0.220329 + 0.975426i \(0.429287\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −24.9238 −1.19227
\(438\) 0 0
\(439\) 17.6882 0.844212 0.422106 0.906546i \(-0.361291\pi\)
0.422106 + 0.906546i \(0.361291\pi\)
\(440\) 0 0
\(441\) 15.8808 0.756228
\(442\) 0 0
\(443\) −18.6673 −0.886912 −0.443456 0.896296i \(-0.646248\pi\)
−0.443456 + 0.896296i \(0.646248\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −22.0277 −1.04187
\(448\) 0 0
\(449\) 41.2111 1.94487 0.972436 0.233171i \(-0.0749102\pi\)
0.972436 + 0.233171i \(0.0749102\pi\)
\(450\) 0 0
\(451\) 1.67961 0.0790899
\(452\) 0 0
\(453\) 11.7890 0.553897
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5.56092 −0.260129 −0.130064 0.991506i \(-0.541518\pi\)
−0.130064 + 0.991506i \(0.541518\pi\)
\(458\) 0 0
\(459\) −3.80573 −0.177636
\(460\) 0 0
\(461\) 21.1627 0.985646 0.492823 0.870130i \(-0.335965\pi\)
0.492823 + 0.870130i \(0.335965\pi\)
\(462\) 0 0
\(463\) −7.74566 −0.359971 −0.179986 0.983669i \(-0.557605\pi\)
−0.179986 + 0.983669i \(0.557605\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.43205 −0.112542 −0.0562709 0.998416i \(-0.517921\pi\)
−0.0562709 + 0.998416i \(0.517921\pi\)
\(468\) 0 0
\(469\) −33.1511 −1.53078
\(470\) 0 0
\(471\) 11.2769 0.519611
\(472\) 0 0
\(473\) −21.2775 −0.978342
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.22384 0.193396
\(478\) 0 0
\(479\) 34.1291 1.55940 0.779699 0.626155i \(-0.215372\pi\)
0.779699 + 0.626155i \(0.215372\pi\)
\(480\) 0 0
\(481\) 6.11178 0.278673
\(482\) 0 0
\(483\) 28.0461 1.27614
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −32.0358 −1.45168 −0.725841 0.687863i \(-0.758549\pi\)
−0.725841 + 0.687863i \(0.758549\pi\)
\(488\) 0 0
\(489\) 9.89630 0.447526
\(490\) 0 0
\(491\) 29.4011 1.32685 0.663426 0.748242i \(-0.269102\pi\)
0.663426 + 0.748242i \(0.269102\pi\)
\(492\) 0 0
\(493\) −40.1507 −1.80830
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 57.3868 2.57415
\(498\) 0 0
\(499\) −9.04298 −0.404819 −0.202410 0.979301i \(-0.564877\pi\)
−0.202410 + 0.979301i \(0.564877\pi\)
\(500\) 0 0
\(501\) −7.51564 −0.335774
\(502\) 0 0
\(503\) 3.07701 0.137197 0.0685985 0.997644i \(-0.478147\pi\)
0.0685985 + 0.997644i \(0.478147\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −11.8339 −0.525563
\(508\) 0 0
\(509\) 26.2553 1.16375 0.581873 0.813279i \(-0.302320\pi\)
0.581873 + 0.813279i \(0.302320\pi\)
\(510\) 0 0
\(511\) 81.1069 3.58796
\(512\) 0 0
\(513\) 4.25085 0.187680
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.87198 0.258249
\(518\) 0 0
\(519\) 12.2783 0.538956
\(520\) 0 0
\(521\) 20.7473 0.908956 0.454478 0.890758i \(-0.349826\pi\)
0.454478 + 0.890758i \(0.349826\pi\)
\(522\) 0 0
\(523\) −22.4498 −0.981663 −0.490831 0.871255i \(-0.663307\pi\)
−0.490831 + 0.871255i \(0.663307\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.4252 0.715494
\(528\) 0 0
\(529\) 11.3776 0.494677
\(530\) 0 0
\(531\) −4.77018 −0.207008
\(532\) 0 0
\(533\) 0.927127 0.0401583
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.2089 0.656314
\(538\) 0 0
\(539\) 31.0675 1.33817
\(540\) 0 0
\(541\) −25.3337 −1.08918 −0.544591 0.838702i \(-0.683315\pi\)
−0.544591 + 0.838702i \(0.683315\pi\)
\(542\) 0 0
\(543\) −20.8755 −0.895852
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −11.8073 −0.504844 −0.252422 0.967617i \(-0.581227\pi\)
−0.252422 + 0.967617i \(0.581227\pi\)
\(548\) 0 0
\(549\) 3.63184 0.155003
\(550\) 0 0
\(551\) 44.8468 1.91054
\(552\) 0 0
\(553\) 31.6688 1.34670
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.93356 0.251413 0.125706 0.992067i \(-0.459880\pi\)
0.125706 + 0.992067i \(0.459880\pi\)
\(558\) 0 0
\(559\) −11.7450 −0.496759
\(560\) 0 0
\(561\) −7.44512 −0.314333
\(562\) 0 0
\(563\) 13.7426 0.579183 0.289592 0.957150i \(-0.406480\pi\)
0.289592 + 0.957150i \(0.406480\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −4.78339 −0.200883
\(568\) 0 0
\(569\) 33.1078 1.38795 0.693975 0.719999i \(-0.255858\pi\)
0.693975 + 0.719999i \(0.255858\pi\)
\(570\) 0 0
\(571\) −25.6092 −1.07171 −0.535855 0.844310i \(-0.680011\pi\)
−0.535855 + 0.844310i \(0.680011\pi\)
\(572\) 0 0
\(573\) −5.25085 −0.219357
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 3.48758 0.145190 0.0725949 0.997362i \(-0.476872\pi\)
0.0725949 + 0.997362i \(0.476872\pi\)
\(578\) 0 0
\(579\) 7.94565 0.330210
\(580\) 0 0
\(581\) 12.3502 0.512371
\(582\) 0 0
\(583\) 8.26308 0.342222
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 34.4479 1.42182 0.710909 0.703284i \(-0.248284\pi\)
0.710909 + 0.703284i \(0.248284\pi\)
\(588\) 0 0
\(589\) −18.3463 −0.755948
\(590\) 0 0
\(591\) −13.0059 −0.534992
\(592\) 0 0
\(593\) 35.7248 1.46704 0.733522 0.679666i \(-0.237875\pi\)
0.733522 + 0.679666i \(0.237875\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.06434 0.0435605
\(598\) 0 0
\(599\) 39.7697 1.62495 0.812473 0.582998i \(-0.198121\pi\)
0.812473 + 0.582998i \(0.198121\pi\)
\(600\) 0 0
\(601\) −15.5134 −0.632805 −0.316402 0.948625i \(-0.602475\pi\)
−0.316402 + 0.948625i \(0.602475\pi\)
\(602\) 0 0
\(603\) 6.93046 0.282230
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 15.1855 0.616358 0.308179 0.951328i \(-0.400280\pi\)
0.308179 + 0.951328i \(0.400280\pi\)
\(608\) 0 0
\(609\) −50.4651 −2.04495
\(610\) 0 0
\(611\) 3.24126 0.131128
\(612\) 0 0
\(613\) −32.1345 −1.29790 −0.648951 0.760831i \(-0.724792\pi\)
−0.648951 + 0.760831i \(0.724792\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.32238 0.174012 0.0870061 0.996208i \(-0.472270\pi\)
0.0870061 + 0.996208i \(0.472270\pi\)
\(618\) 0 0
\(619\) 33.3967 1.34233 0.671163 0.741310i \(-0.265795\pi\)
0.671163 + 0.741310i \(0.265795\pi\)
\(620\) 0 0
\(621\) −5.86324 −0.235284
\(622\) 0 0
\(623\) 3.98331 0.159588
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 8.31592 0.332106
\(628\) 0 0
\(629\) −21.5398 −0.858847
\(630\) 0 0
\(631\) 10.9937 0.437654 0.218827 0.975764i \(-0.429777\pi\)
0.218827 + 0.975764i \(0.429777\pi\)
\(632\) 0 0
\(633\) −20.4928 −0.814517
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 17.1489 0.679464
\(638\) 0 0
\(639\) −11.9971 −0.474598
\(640\) 0 0
\(641\) −6.86056 −0.270976 −0.135488 0.990779i \(-0.543260\pi\)
−0.135488 + 0.990779i \(0.543260\pi\)
\(642\) 0 0
\(643\) 23.1923 0.914615 0.457308 0.889309i \(-0.348814\pi\)
0.457308 + 0.889309i \(0.348814\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −30.7846 −1.21027 −0.605135 0.796123i \(-0.706881\pi\)
−0.605135 + 0.796123i \(0.706881\pi\)
\(648\) 0 0
\(649\) −9.33188 −0.366309
\(650\) 0 0
\(651\) 20.6447 0.809130
\(652\) 0 0
\(653\) −7.80686 −0.305506 −0.152753 0.988264i \(-0.548814\pi\)
−0.152753 + 0.988264i \(0.548814\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −16.9560 −0.661515
\(658\) 0 0
\(659\) 5.27044 0.205307 0.102654 0.994717i \(-0.467267\pi\)
0.102654 + 0.994717i \(0.467267\pi\)
\(660\) 0 0
\(661\) −36.8954 −1.43506 −0.717532 0.696526i \(-0.754728\pi\)
−0.717532 + 0.696526i \(0.754728\pi\)
\(662\) 0 0
\(663\) −4.10962 −0.159605
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −61.8576 −2.39514
\(668\) 0 0
\(669\) 10.3121 0.398689
\(670\) 0 0
\(671\) 7.10495 0.274284
\(672\) 0 0
\(673\) −45.0209 −1.73543 −0.867714 0.497063i \(-0.834412\pi\)
−0.867714 + 0.497063i \(0.834412\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 31.5396 1.21217 0.606083 0.795402i \(-0.292740\pi\)
0.606083 + 0.795402i \(0.292740\pi\)
\(678\) 0 0
\(679\) 43.6100 1.67360
\(680\) 0 0
\(681\) −18.1409 −0.695161
\(682\) 0 0
\(683\) −20.6888 −0.791636 −0.395818 0.918329i \(-0.629539\pi\)
−0.395818 + 0.918329i \(0.629539\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −9.71827 −0.370775
\(688\) 0 0
\(689\) 4.56113 0.173765
\(690\) 0 0
\(691\) −34.1801 −1.30027 −0.650137 0.759817i \(-0.725288\pi\)
−0.650137 + 0.759817i \(0.725288\pi\)
\(692\) 0 0
\(693\) −9.35772 −0.355470
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −3.26748 −0.123764
\(698\) 0 0
\(699\) −22.2184 −0.840377
\(700\) 0 0
\(701\) −16.6859 −0.630216 −0.315108 0.949056i \(-0.602041\pi\)
−0.315108 + 0.949056i \(0.602041\pi\)
\(702\) 0 0
\(703\) 24.0591 0.907406
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.15312 −0.269021
\(708\) 0 0
\(709\) 47.3162 1.77700 0.888499 0.458878i \(-0.151749\pi\)
0.888499 + 0.458878i \(0.151749\pi\)
\(710\) 0 0
\(711\) −6.62059 −0.248292
\(712\) 0 0
\(713\) 25.3053 0.947690
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 15.9293 0.594890
\(718\) 0 0
\(719\) 5.61236 0.209306 0.104653 0.994509i \(-0.466627\pi\)
0.104653 + 0.994509i \(0.466627\pi\)
\(720\) 0 0
\(721\) −51.9586 −1.93504
\(722\) 0 0
\(723\) −4.93147 −0.183403
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −14.6366 −0.542841 −0.271420 0.962461i \(-0.587493\pi\)
−0.271420 + 0.962461i \(0.587493\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 41.3928 1.53097
\(732\) 0 0
\(733\) 32.2577 1.19147 0.595733 0.803183i \(-0.296862\pi\)
0.595733 + 0.803183i \(0.296862\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13.5580 0.499417
\(738\) 0 0
\(739\) −23.8507 −0.877361 −0.438680 0.898643i \(-0.644554\pi\)
−0.438680 + 0.898643i \(0.644554\pi\)
\(740\) 0 0
\(741\) 4.59029 0.168629
\(742\) 0 0
\(743\) 7.48188 0.274483 0.137242 0.990538i \(-0.456176\pi\)
0.137242 + 0.990538i \(0.456176\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −2.58189 −0.0944663
\(748\) 0 0
\(749\) 94.8975 3.46748
\(750\) 0 0
\(751\) −32.3687 −1.18115 −0.590576 0.806982i \(-0.701099\pi\)
−0.590576 + 0.806982i \(0.701099\pi\)
\(752\) 0 0
\(753\) 15.4351 0.562485
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −10.2847 −0.373804 −0.186902 0.982379i \(-0.559845\pi\)
−0.186902 + 0.982379i \(0.559845\pi\)
\(758\) 0 0
\(759\) −11.4702 −0.416343
\(760\) 0 0
\(761\) 28.8044 1.04416 0.522080 0.852897i \(-0.325156\pi\)
0.522080 + 0.852897i \(0.325156\pi\)
\(762\) 0 0
\(763\) 11.2118 0.405895
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.15109 −0.185995
\(768\) 0 0
\(769\) −39.3743 −1.41987 −0.709936 0.704266i \(-0.751276\pi\)
−0.709936 + 0.704266i \(0.751276\pi\)
\(770\) 0 0
\(771\) 21.1828 0.762879
\(772\) 0 0
\(773\) 5.06407 0.182142 0.0910709 0.995844i \(-0.470971\pi\)
0.0910709 + 0.995844i \(0.470971\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −27.0731 −0.971244
\(778\) 0 0
\(779\) 3.64965 0.130762
\(780\) 0 0
\(781\) −23.4699 −0.839819
\(782\) 0 0
\(783\) 10.5501 0.377029
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5.33151 0.190048 0.0950239 0.995475i \(-0.469707\pi\)
0.0950239 + 0.995475i \(0.469707\pi\)
\(788\) 0 0
\(789\) −20.9264 −0.744999
\(790\) 0 0
\(791\) 15.4432 0.549098
\(792\) 0 0
\(793\) 3.92185 0.139269
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.4253 0.794345 0.397172 0.917744i \(-0.369991\pi\)
0.397172 + 0.917744i \(0.369991\pi\)
\(798\) 0 0
\(799\) −11.4232 −0.404124
\(800\) 0 0
\(801\) −0.832738 −0.0294233
\(802\) 0 0
\(803\) −33.1709 −1.17057
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −13.7837 −0.485208
\(808\) 0 0
\(809\) 12.0715 0.424411 0.212206 0.977225i \(-0.431935\pi\)
0.212206 + 0.977225i \(0.431935\pi\)
\(810\) 0 0
\(811\) 26.5422 0.932024 0.466012 0.884778i \(-0.345690\pi\)
0.466012 + 0.884778i \(0.345690\pi\)
\(812\) 0 0
\(813\) 10.5495 0.369987
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −46.2341 −1.61753
\(818\) 0 0
\(819\) −5.16535 −0.180492
\(820\) 0 0
\(821\) −1.79377 −0.0626031 −0.0313015 0.999510i \(-0.509965\pi\)
−0.0313015 + 0.999510i \(0.509965\pi\)
\(822\) 0 0
\(823\) 35.5881 1.24052 0.620261 0.784396i \(-0.287027\pi\)
0.620261 + 0.784396i \(0.287027\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.4424 0.954264 0.477132 0.878832i \(-0.341676\pi\)
0.477132 + 0.878832i \(0.341676\pi\)
\(828\) 0 0
\(829\) −38.0241 −1.32063 −0.660316 0.750988i \(-0.729578\pi\)
−0.660316 + 0.750988i \(0.729578\pi\)
\(830\) 0 0
\(831\) 17.9622 0.623102
\(832\) 0 0
\(833\) −60.4379 −2.09405
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.31592 −0.149180
\(838\) 0 0
\(839\) −36.1012 −1.24635 −0.623175 0.782082i \(-0.714158\pi\)
−0.623175 + 0.782082i \(0.714158\pi\)
\(840\) 0 0
\(841\) 82.3042 2.83807
\(842\) 0 0
\(843\) −0.926728 −0.0319182
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 34.3108 1.17893
\(848\) 0 0
\(849\) −6.57749 −0.225739
\(850\) 0 0
\(851\) −33.1849 −1.13756
\(852\) 0 0
\(853\) −32.9489 −1.12815 −0.564075 0.825724i \(-0.690767\pi\)
−0.564075 + 0.825724i \(0.690767\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −13.9514 −0.476572 −0.238286 0.971195i \(-0.576586\pi\)
−0.238286 + 0.971195i \(0.576586\pi\)
\(858\) 0 0
\(859\) −38.6514 −1.31877 −0.659384 0.751806i \(-0.729183\pi\)
−0.659384 + 0.751806i \(0.729183\pi\)
\(860\) 0 0
\(861\) −4.10686 −0.139962
\(862\) 0 0
\(863\) −3.57575 −0.121720 −0.0608599 0.998146i \(-0.519384\pi\)
−0.0608599 + 0.998146i \(0.519384\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −2.51644 −0.0854629
\(868\) 0 0
\(869\) −12.9518 −0.439361
\(870\) 0 0
\(871\) 7.48388 0.253582
\(872\) 0 0
\(873\) −9.11698 −0.308563
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −7.52136 −0.253978 −0.126989 0.991904i \(-0.540531\pi\)
−0.126989 + 0.991904i \(0.540531\pi\)
\(878\) 0 0
\(879\) 25.6208 0.864168
\(880\) 0 0
\(881\) −26.3654 −0.888272 −0.444136 0.895959i \(-0.646489\pi\)
−0.444136 + 0.895959i \(0.646489\pi\)
\(882\) 0 0
\(883\) −0.411659 −0.0138534 −0.00692672 0.999976i \(-0.502205\pi\)
−0.00692672 + 0.999976i \(0.502205\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 28.8013 0.967054 0.483527 0.875329i \(-0.339355\pi\)
0.483527 + 0.875329i \(0.339355\pi\)
\(888\) 0 0
\(889\) −35.3316 −1.18498
\(890\) 0 0
\(891\) 1.95630 0.0655384
\(892\) 0 0
\(893\) 12.7593 0.426973
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −6.33143 −0.211400
\(898\) 0 0
\(899\) −45.5333 −1.51862
\(900\) 0 0
\(901\) −16.0748 −0.535529
\(902\) 0 0
\(903\) 52.0262 1.73132
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.80122 −0.0598085 −0.0299042 0.999553i \(-0.509520\pi\)
−0.0299042 + 0.999553i \(0.509520\pi\)
\(908\) 0 0
\(909\) 1.49541 0.0495996
\(910\) 0 0
\(911\) 4.33268 0.143548 0.0717741 0.997421i \(-0.477134\pi\)
0.0717741 + 0.997421i \(0.477134\pi\)
\(912\) 0 0
\(913\) −5.05093 −0.167161
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 80.7621 2.66700
\(918\) 0 0
\(919\) −7.79390 −0.257097 −0.128548 0.991703i \(-0.541032\pi\)
−0.128548 + 0.991703i \(0.541032\pi\)
\(920\) 0 0
\(921\) −6.18420 −0.203776
\(922\) 0 0
\(923\) −12.9551 −0.426423
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 10.8623 0.356765
\(928\) 0 0
\(929\) −33.3560 −1.09437 −0.547187 0.837010i \(-0.684301\pi\)
−0.547187 + 0.837010i \(0.684301\pi\)
\(930\) 0 0
\(931\) 67.5069 2.21245
\(932\) 0 0
\(933\) −24.2382 −0.793522
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 33.0108 1.07842 0.539209 0.842172i \(-0.318723\pi\)
0.539209 + 0.842172i \(0.318723\pi\)
\(938\) 0 0
\(939\) −29.9612 −0.977748
\(940\) 0 0
\(941\) −51.1774 −1.66834 −0.834168 0.551510i \(-0.814052\pi\)
−0.834168 + 0.551510i \(0.814052\pi\)
\(942\) 0 0
\(943\) −5.03399 −0.163929
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −7.16556 −0.232849 −0.116425 0.993200i \(-0.537143\pi\)
−0.116425 + 0.993200i \(0.537143\pi\)
\(948\) 0 0
\(949\) −18.3099 −0.594366
\(950\) 0 0
\(951\) 2.46267 0.0798575
\(952\) 0 0
\(953\) −13.4032 −0.434171 −0.217085 0.976153i \(-0.569655\pi\)
−0.217085 + 0.976153i \(0.569655\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 20.6391 0.667166
\(958\) 0 0
\(959\) −4.01478 −0.129644
\(960\) 0 0
\(961\) −12.3728 −0.399124
\(962\) 0 0
\(963\) −19.8390 −0.639302
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −11.5961 −0.372907 −0.186453 0.982464i \(-0.559699\pi\)
−0.186453 + 0.982464i \(0.559699\pi\)
\(968\) 0 0
\(969\) −16.1776 −0.519699
\(970\) 0 0
\(971\) −7.62019 −0.244544 −0.122272 0.992497i \(-0.539018\pi\)
−0.122272 + 0.992497i \(0.539018\pi\)
\(972\) 0 0
\(973\) 53.3730 1.71106
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 43.4103 1.38882 0.694409 0.719580i \(-0.255666\pi\)
0.694409 + 0.719580i \(0.255666\pi\)
\(978\) 0 0
\(979\) −1.62908 −0.0520657
\(980\) 0 0
\(981\) −2.34391 −0.0748352
\(982\) 0 0
\(983\) −54.7992 −1.74782 −0.873912 0.486085i \(-0.838425\pi\)
−0.873912 + 0.486085i \(0.838425\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −14.3577 −0.457011
\(988\) 0 0
\(989\) 63.7712 2.02781
\(990\) 0 0
\(991\) 21.9440 0.697074 0.348537 0.937295i \(-0.386678\pi\)
0.348537 + 0.937295i \(0.386678\pi\)
\(992\) 0 0
\(993\) −26.7867 −0.850051
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 39.0854 1.23785 0.618924 0.785451i \(-0.287569\pi\)
0.618924 + 0.785451i \(0.287569\pi\)
\(998\) 0 0
\(999\) 5.65983 0.179069
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 7500.2.a.g.1.1 4
5.2 odd 4 7500.2.d.d.1249.1 8
5.3 odd 4 7500.2.d.d.1249.8 8
5.4 even 2 7500.2.a.d.1.4 4
25.3 odd 20 1500.2.o.a.49.2 16
25.4 even 10 1500.2.m.b.1201.2 8
25.6 even 5 300.2.m.a.61.1 8
25.8 odd 20 1500.2.o.a.949.4 16
25.17 odd 20 1500.2.o.a.949.1 16
25.19 even 10 1500.2.m.b.301.2 8
25.21 even 5 300.2.m.a.241.1 yes 8
25.22 odd 20 1500.2.o.a.49.3 16
75.56 odd 10 900.2.n.a.361.2 8
75.71 odd 10 900.2.n.a.541.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.m.a.61.1 8 25.6 even 5
300.2.m.a.241.1 yes 8 25.21 even 5
900.2.n.a.361.2 8 75.56 odd 10
900.2.n.a.541.2 8 75.71 odd 10
1500.2.m.b.301.2 8 25.19 even 10
1500.2.m.b.1201.2 8 25.4 even 10
1500.2.o.a.49.2 16 25.3 odd 20
1500.2.o.a.49.3 16 25.22 odd 20
1500.2.o.a.949.1 16 25.17 odd 20
1500.2.o.a.949.4 16 25.8 odd 20
7500.2.a.d.1.4 4 5.4 even 2
7500.2.a.g.1.1 4 1.1 even 1 trivial
7500.2.d.d.1249.1 8 5.2 odd 4
7500.2.d.d.1249.8 8 5.3 odd 4