Properties

Label 1305.2.a.r.1.4
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $0$
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] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.75660\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.75660 q^{2} +5.59883 q^{4} +1.00000 q^{5} +0.393832 q^{7} +9.92054 q^{8} +O(q^{10})\) \(q+2.75660 q^{2} +5.59883 q^{4} +1.00000 q^{5} +0.393832 q^{7} +9.92054 q^{8} +2.75660 q^{10} +0.393832 q^{11} -2.56511 q^{13} +1.08564 q^{14} +16.1493 q^{16} -2.07830 q^{17} -0.958939 q^{19} +5.59883 q^{20} +1.08564 q^{22} -6.15661 q^{23} +1.00000 q^{25} -7.07097 q^{26} +2.20500 q^{28} +1.00000 q^{29} -10.1566 q^{31} +24.6760 q^{32} -5.72905 q^{34} +0.393832 q^{35} -7.34192 q^{37} -2.64341 q^{38} +9.92054 q^{40} +1.65745 q^{41} +10.3279 q^{43} +2.20500 q^{44} -16.9713 q^{46} +11.5915 q^{47} -6.84490 q^{49} +2.75660 q^{50} -14.3616 q^{52} +12.3279 q^{53} +0.393832 q^{55} +3.90703 q^{56} +2.75660 q^{58} +9.54022 q^{59} -6.25340 q^{61} -27.9977 q^{62} +35.7232 q^{64} -2.56511 q^{65} -7.42023 q^{67} -11.6361 q^{68} +1.08564 q^{70} -5.98533 q^{71} +3.34192 q^{73} -20.2387 q^{74} -5.36894 q^{76} +0.155104 q^{77} -2.06745 q^{79} +16.1493 q^{80} +4.56892 q^{82} -6.41000 q^{83} -2.07830 q^{85} +28.4698 q^{86} +3.90703 q^{88} -15.8302 q^{89} -1.01022 q^{91} -34.4698 q^{92} +31.9531 q^{94} -0.958939 q^{95} -18.4575 q^{97} -18.8686 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 5 q^{4} + 4 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 5 q^{4} + 4 q^{5} + 2 q^{7} + 12 q^{8} + 3 q^{10} + 2 q^{11} - 8 q^{13} + 3 q^{14} + 11 q^{16} + 10 q^{17} - 2 q^{19} + 5 q^{20} + 3 q^{22} + 12 q^{23} + 4 q^{25} + 7 q^{26} - 9 q^{28} + 4 q^{29} - 4 q^{31} + 17 q^{32} - q^{34} + 2 q^{35} - 16 q^{37} + 10 q^{38} + 12 q^{40} + 12 q^{41} + 2 q^{43} - 9 q^{44} - 8 q^{46} + 12 q^{47} + 6 q^{49} + 3 q^{50} - 3 q^{52} + 10 q^{53} + 2 q^{55} + 3 q^{58} - 2 q^{59} - 26 q^{61} - 20 q^{62} + 34 q^{64} - 8 q^{65} + 2 q^{67} - 9 q^{68} + 3 q^{70} + 10 q^{71} - 48 q^{74} + 16 q^{76} + 34 q^{77} + 22 q^{79} + 11 q^{80} + 38 q^{82} + 10 q^{83} + 10 q^{85} + 4 q^{86} + 4 q^{89} - 8 q^{91} - 28 q^{92} + 39 q^{94} - 2 q^{95} - 22 q^{97} - 34 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.75660 1.94921 0.974605 0.223932i \(-0.0718895\pi\)
0.974605 + 0.223932i \(0.0718895\pi\)
\(3\) 0 0
\(4\) 5.59883 2.79942
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.393832 0.148855 0.0744273 0.997226i \(-0.476287\pi\)
0.0744273 + 0.997226i \(0.476287\pi\)
\(8\) 9.92054 3.50744
\(9\) 0 0
\(10\) 2.75660 0.871713
\(11\) 0.393832 0.118745 0.0593725 0.998236i \(-0.481090\pi\)
0.0593725 + 0.998236i \(0.481090\pi\)
\(12\) 0 0
\(13\) −2.56511 −0.711433 −0.355716 0.934594i \(-0.615763\pi\)
−0.355716 + 0.934594i \(0.615763\pi\)
\(14\) 1.08564 0.290149
\(15\) 0 0
\(16\) 16.1493 4.03732
\(17\) −2.07830 −0.504063 −0.252031 0.967719i \(-0.581099\pi\)
−0.252031 + 0.967719i \(0.581099\pi\)
\(18\) 0 0
\(19\) −0.958939 −0.219996 −0.109998 0.993932i \(-0.535084\pi\)
−0.109998 + 0.993932i \(0.535084\pi\)
\(20\) 5.59883 1.25194
\(21\) 0 0
\(22\) 1.08564 0.231459
\(23\) −6.15661 −1.28374 −0.641871 0.766813i \(-0.721841\pi\)
−0.641871 + 0.766813i \(0.721841\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −7.07097 −1.38673
\(27\) 0 0
\(28\) 2.20500 0.416706
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −10.1566 −1.82418 −0.912090 0.409989i \(-0.865532\pi\)
−0.912090 + 0.409989i \(0.865532\pi\)
\(32\) 24.6760 4.36214
\(33\) 0 0
\(34\) −5.72905 −0.982524
\(35\) 0.393832 0.0665698
\(36\) 0 0
\(37\) −7.34192 −1.20700 −0.603502 0.797361i \(-0.706229\pi\)
−0.603502 + 0.797361i \(0.706229\pi\)
\(38\) −2.64341 −0.428818
\(39\) 0 0
\(40\) 9.92054 1.56857
\(41\) 1.65745 0.258850 0.129425 0.991589i \(-0.458687\pi\)
0.129425 + 0.991589i \(0.458687\pi\)
\(42\) 0 0
\(43\) 10.3279 1.57499 0.787494 0.616323i \(-0.211378\pi\)
0.787494 + 0.616323i \(0.211378\pi\)
\(44\) 2.20500 0.332417
\(45\) 0 0
\(46\) −16.9713 −2.50228
\(47\) 11.5915 1.69079 0.845397 0.534138i \(-0.179364\pi\)
0.845397 + 0.534138i \(0.179364\pi\)
\(48\) 0 0
\(49\) −6.84490 −0.977842
\(50\) 2.75660 0.389842
\(51\) 0 0
\(52\) −14.3616 −1.99160
\(53\) 12.3279 1.69336 0.846682 0.532099i \(-0.178596\pi\)
0.846682 + 0.532099i \(0.178596\pi\)
\(54\) 0 0
\(55\) 0.393832 0.0531043
\(56\) 3.90703 0.522099
\(57\) 0 0
\(58\) 2.75660 0.361959
\(59\) 9.54022 1.24203 0.621015 0.783798i \(-0.286720\pi\)
0.621015 + 0.783798i \(0.286720\pi\)
\(60\) 0 0
\(61\) −6.25340 −0.800665 −0.400333 0.916370i \(-0.631105\pi\)
−0.400333 + 0.916370i \(0.631105\pi\)
\(62\) −27.9977 −3.55571
\(63\) 0 0
\(64\) 35.7232 4.46540
\(65\) −2.56511 −0.318162
\(66\) 0 0
\(67\) −7.42023 −0.906525 −0.453262 0.891377i \(-0.649740\pi\)
−0.453262 + 0.891377i \(0.649740\pi\)
\(68\) −11.6361 −1.41108
\(69\) 0 0
\(70\) 1.08564 0.129758
\(71\) −5.98533 −0.710328 −0.355164 0.934804i \(-0.615575\pi\)
−0.355164 + 0.934804i \(0.615575\pi\)
\(72\) 0 0
\(73\) 3.34192 0.391142 0.195571 0.980690i \(-0.437344\pi\)
0.195571 + 0.980690i \(0.437344\pi\)
\(74\) −20.2387 −2.35270
\(75\) 0 0
\(76\) −5.36894 −0.615860
\(77\) 0.155104 0.0176757
\(78\) 0 0
\(79\) −2.06745 −0.232607 −0.116303 0.993214i \(-0.537104\pi\)
−0.116303 + 0.993214i \(0.537104\pi\)
\(80\) 16.1493 1.80554
\(81\) 0 0
\(82\) 4.56892 0.504553
\(83\) −6.41000 −0.703589 −0.351795 0.936077i \(-0.614428\pi\)
−0.351795 + 0.936077i \(0.614428\pi\)
\(84\) 0 0
\(85\) −2.07830 −0.225424
\(86\) 28.4698 3.06998
\(87\) 0 0
\(88\) 3.90703 0.416491
\(89\) −15.8302 −1.67800 −0.839000 0.544131i \(-0.816860\pi\)
−0.839000 + 0.544131i \(0.816860\pi\)
\(90\) 0 0
\(91\) −1.01022 −0.105900
\(92\) −34.4698 −3.59373
\(93\) 0 0
\(94\) 31.9531 3.29571
\(95\) −0.958939 −0.0983851
\(96\) 0 0
\(97\) −18.4575 −1.87407 −0.937036 0.349233i \(-0.886442\pi\)
−0.937036 + 0.349233i \(0.886442\pi\)
\(98\) −18.8686 −1.90602
\(99\) 0 0
\(100\) 5.59883 0.559883
\(101\) 12.8038 1.27403 0.637015 0.770852i \(-0.280169\pi\)
0.637015 + 0.770852i \(0.280169\pi\)
\(102\) 0 0
\(103\) 4.86979 0.479834 0.239917 0.970793i \(-0.422880\pi\)
0.239917 + 0.970793i \(0.422880\pi\)
\(104\) −25.4472 −2.49531
\(105\) 0 0
\(106\) 33.9830 3.30072
\(107\) 2.34255 0.226463 0.113231 0.993569i \(-0.463880\pi\)
0.113231 + 0.993569i \(0.463880\pi\)
\(108\) 0 0
\(109\) 8.55044 0.818984 0.409492 0.912314i \(-0.365706\pi\)
0.409492 + 0.912314i \(0.365706\pi\)
\(110\) 1.08564 0.103511
\(111\) 0 0
\(112\) 6.36011 0.600973
\(113\) 11.2085 1.05441 0.527204 0.849739i \(-0.323240\pi\)
0.527204 + 0.849739i \(0.323240\pi\)
\(114\) 0 0
\(115\) −6.15661 −0.574107
\(116\) 5.59883 0.519839
\(117\) 0 0
\(118\) 26.2985 2.42098
\(119\) −0.818503 −0.0750321
\(120\) 0 0
\(121\) −10.8449 −0.985900
\(122\) −17.2381 −1.56066
\(123\) 0 0
\(124\) −56.8652 −5.10664
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 20.1566 1.78861 0.894305 0.447458i \(-0.147671\pi\)
0.894305 + 0.447458i \(0.147671\pi\)
\(128\) 49.1226 4.34187
\(129\) 0 0
\(130\) −7.07097 −0.620165
\(131\) 5.50235 0.480742 0.240371 0.970681i \(-0.422731\pi\)
0.240371 + 0.970681i \(0.422731\pi\)
\(132\) 0 0
\(133\) −0.377661 −0.0327474
\(134\) −20.4546 −1.76701
\(135\) 0 0
\(136\) −20.6179 −1.76797
\(137\) −7.49853 −0.640643 −0.320321 0.947309i \(-0.603791\pi\)
−0.320321 + 0.947309i \(0.603791\pi\)
\(138\) 0 0
\(139\) −9.35277 −0.793292 −0.396646 0.917972i \(-0.629826\pi\)
−0.396646 + 0.917972i \(0.629826\pi\)
\(140\) 2.20500 0.186357
\(141\) 0 0
\(142\) −16.4992 −1.38458
\(143\) −1.01022 −0.0844790
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 9.21234 0.762418
\(147\) 0 0
\(148\) −41.1062 −3.37891
\(149\) 11.5402 0.945411 0.472706 0.881220i \(-0.343277\pi\)
0.472706 + 0.881220i \(0.343277\pi\)
\(150\) 0 0
\(151\) 6.08212 0.494956 0.247478 0.968894i \(-0.420398\pi\)
0.247478 + 0.968894i \(0.420398\pi\)
\(152\) −9.51320 −0.771622
\(153\) 0 0
\(154\) 0.427559 0.0344537
\(155\) −10.1566 −0.815798
\(156\) 0 0
\(157\) −11.5953 −0.925407 −0.462704 0.886513i \(-0.653121\pi\)
−0.462704 + 0.886513i \(0.653121\pi\)
\(158\) −5.69914 −0.453399
\(159\) 0 0
\(160\) 24.6760 1.95081
\(161\) −2.42467 −0.191091
\(162\) 0 0
\(163\) −0.855118 −0.0669780 −0.0334890 0.999439i \(-0.510662\pi\)
−0.0334890 + 0.999439i \(0.510662\pi\)
\(164\) 9.27979 0.724630
\(165\) 0 0
\(166\) −17.6698 −1.37144
\(167\) −7.73194 −0.598315 −0.299158 0.954204i \(-0.596706\pi\)
−0.299158 + 0.954204i \(0.596706\pi\)
\(168\) 0 0
\(169\) −6.42023 −0.493863
\(170\) −5.72905 −0.439398
\(171\) 0 0
\(172\) 57.8241 4.40905
\(173\) −8.07449 −0.613892 −0.306946 0.951727i \(-0.599307\pi\)
−0.306946 + 0.951727i \(0.599307\pi\)
\(174\) 0 0
\(175\) 0.393832 0.0297709
\(176\) 6.36011 0.479411
\(177\) 0 0
\(178\) −43.6376 −3.27077
\(179\) −12.4100 −0.927567 −0.463784 0.885949i \(-0.653508\pi\)
−0.463784 + 0.885949i \(0.653508\pi\)
\(180\) 0 0
\(181\) −2.49765 −0.185649 −0.0928246 0.995682i \(-0.529590\pi\)
−0.0928246 + 0.995682i \(0.529590\pi\)
\(182\) −2.78478 −0.206421
\(183\) 0 0
\(184\) −61.0769 −4.50265
\(185\) −7.34192 −0.539789
\(186\) 0 0
\(187\) −0.818503 −0.0598549
\(188\) 64.8989 4.73324
\(189\) 0 0
\(190\) −2.64341 −0.191773
\(191\) 6.32085 0.457361 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(192\) 0 0
\(193\) 25.2921 1.82057 0.910284 0.413984i \(-0.135863\pi\)
0.910284 + 0.413984i \(0.135863\pi\)
\(194\) −50.8798 −3.65296
\(195\) 0 0
\(196\) −38.3234 −2.73739
\(197\) 21.2047 1.51077 0.755386 0.655280i \(-0.227449\pi\)
0.755386 + 0.655280i \(0.227449\pi\)
\(198\) 0 0
\(199\) −2.64723 −0.187657 −0.0938285 0.995588i \(-0.529911\pi\)
−0.0938285 + 0.995588i \(0.529911\pi\)
\(200\) 9.92054 0.701488
\(201\) 0 0
\(202\) 35.2950 2.48335
\(203\) 0.393832 0.0276416
\(204\) 0 0
\(205\) 1.65745 0.115761
\(206\) 13.4240 0.935297
\(207\) 0 0
\(208\) −41.4246 −2.87228
\(209\) −0.377661 −0.0261234
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 69.0218 4.74043
\(213\) 0 0
\(214\) 6.45747 0.441423
\(215\) 10.3279 0.704356
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 23.5701 1.59637
\(219\) 0 0
\(220\) 2.20500 0.148661
\(221\) 5.33107 0.358607
\(222\) 0 0
\(223\) 12.5504 0.840440 0.420220 0.907422i \(-0.361953\pi\)
0.420220 + 0.907422i \(0.361953\pi\)
\(224\) 9.71820 0.649324
\(225\) 0 0
\(226\) 30.8974 2.05526
\(227\) −1.30149 −0.0863829 −0.0431914 0.999067i \(-0.513753\pi\)
−0.0431914 + 0.999067i \(0.513753\pi\)
\(228\) 0 0
\(229\) 3.28682 0.217199 0.108600 0.994086i \(-0.465363\pi\)
0.108600 + 0.994086i \(0.465363\pi\)
\(230\) −16.9713 −1.11905
\(231\) 0 0
\(232\) 9.92054 0.651315
\(233\) −17.7115 −1.16032 −0.580159 0.814503i \(-0.697010\pi\)
−0.580159 + 0.814503i \(0.697010\pi\)
\(234\) 0 0
\(235\) 11.5915 0.756146
\(236\) 53.4141 3.47696
\(237\) 0 0
\(238\) −2.25628 −0.146253
\(239\) −21.2651 −1.37553 −0.687763 0.725935i \(-0.741407\pi\)
−0.687763 + 0.725935i \(0.741407\pi\)
\(240\) 0 0
\(241\) 15.3177 0.986697 0.493349 0.869832i \(-0.335773\pi\)
0.493349 + 0.869832i \(0.335773\pi\)
\(242\) −29.8950 −1.92172
\(243\) 0 0
\(244\) −35.0117 −2.24140
\(245\) −6.84490 −0.437304
\(246\) 0 0
\(247\) 2.45978 0.156512
\(248\) −100.759 −6.39820
\(249\) 0 0
\(250\) 2.75660 0.174343
\(251\) −26.8917 −1.69739 −0.848696 0.528882i \(-0.822612\pi\)
−0.848696 + 0.528882i \(0.822612\pi\)
\(252\) 0 0
\(253\) −2.42467 −0.152438
\(254\) 55.5637 3.48637
\(255\) 0 0
\(256\) 63.9648 3.99780
\(257\) 6.85512 0.427611 0.213805 0.976876i \(-0.431414\pi\)
0.213805 + 0.976876i \(0.431414\pi\)
\(258\) 0 0
\(259\) −2.89149 −0.179668
\(260\) −14.3616 −0.890669
\(261\) 0 0
\(262\) 15.1678 0.937067
\(263\) 13.6294 0.840423 0.420212 0.907426i \(-0.361956\pi\)
0.420212 + 0.907426i \(0.361956\pi\)
\(264\) 0 0
\(265\) 12.3279 0.757296
\(266\) −1.04106 −0.0638315
\(267\) 0 0
\(268\) −41.5446 −2.53774
\(269\) 8.46129 0.515894 0.257947 0.966159i \(-0.416954\pi\)
0.257947 + 0.966159i \(0.416954\pi\)
\(270\) 0 0
\(271\) 6.34255 0.385282 0.192641 0.981269i \(-0.438295\pi\)
0.192641 + 0.981269i \(0.438295\pi\)
\(272\) −33.5631 −2.03506
\(273\) 0 0
\(274\) −20.6704 −1.24875
\(275\) 0.393832 0.0237490
\(276\) 0 0
\(277\) −17.1464 −1.03023 −0.515113 0.857122i \(-0.672250\pi\)
−0.515113 + 0.857122i \(0.672250\pi\)
\(278\) −25.7818 −1.54629
\(279\) 0 0
\(280\) 3.90703 0.233490
\(281\) 12.2985 0.733670 0.366835 0.930286i \(-0.380441\pi\)
0.366835 + 0.930286i \(0.380441\pi\)
\(282\) 0 0
\(283\) 25.1830 1.49697 0.748487 0.663149i \(-0.230781\pi\)
0.748487 + 0.663149i \(0.230781\pi\)
\(284\) −33.5109 −1.98851
\(285\) 0 0
\(286\) −2.78478 −0.164667
\(287\) 0.652757 0.0385311
\(288\) 0 0
\(289\) −12.6807 −0.745921
\(290\) 2.75660 0.161873
\(291\) 0 0
\(292\) 18.7109 1.09497
\(293\) −12.3170 −0.719569 −0.359784 0.933035i \(-0.617150\pi\)
−0.359784 + 0.933035i \(0.617150\pi\)
\(294\) 0 0
\(295\) 9.54022 0.555453
\(296\) −72.8358 −4.23350
\(297\) 0 0
\(298\) 31.8117 1.84280
\(299\) 15.7924 0.913296
\(300\) 0 0
\(301\) 4.06745 0.234444
\(302\) 16.7660 0.964773
\(303\) 0 0
\(304\) −15.4862 −0.888193
\(305\) −6.25340 −0.358068
\(306\) 0 0
\(307\) −22.7379 −1.29772 −0.648860 0.760908i \(-0.724754\pi\)
−0.648860 + 0.760908i \(0.724754\pi\)
\(308\) 0.868401 0.0494817
\(309\) 0 0
\(310\) −27.9977 −1.59016
\(311\) 5.33810 0.302696 0.151348 0.988481i \(-0.451639\pi\)
0.151348 + 0.988481i \(0.451639\pi\)
\(312\) 0 0
\(313\) −1.96338 −0.110977 −0.0554885 0.998459i \(-0.517672\pi\)
−0.0554885 + 0.998459i \(0.517672\pi\)
\(314\) −31.9636 −1.80381
\(315\) 0 0
\(316\) −11.5753 −0.651163
\(317\) −1.29064 −0.0724895 −0.0362448 0.999343i \(-0.511540\pi\)
−0.0362448 + 0.999343i \(0.511540\pi\)
\(318\) 0 0
\(319\) 0.393832 0.0220504
\(320\) 35.7232 1.99699
\(321\) 0 0
\(322\) −6.68384 −0.372476
\(323\) 1.99297 0.110892
\(324\) 0 0
\(325\) −2.56511 −0.142287
\(326\) −2.35722 −0.130554
\(327\) 0 0
\(328\) 16.4428 0.907902
\(329\) 4.56511 0.251683
\(330\) 0 0
\(331\) 28.9971 1.59382 0.796911 0.604096i \(-0.206466\pi\)
0.796911 + 0.604096i \(0.206466\pi\)
\(332\) −35.8885 −1.96964
\(333\) 0 0
\(334\) −21.3138 −1.16624
\(335\) −7.42023 −0.405410
\(336\) 0 0
\(337\) −16.7854 −0.914356 −0.457178 0.889375i \(-0.651140\pi\)
−0.457178 + 0.889375i \(0.651140\pi\)
\(338\) −17.6980 −0.962643
\(339\) 0 0
\(340\) −11.6361 −0.631055
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −5.45257 −0.294411
\(344\) 102.458 5.52417
\(345\) 0 0
\(346\) −22.2581 −1.19660
\(347\) 17.8681 0.959210 0.479605 0.877485i \(-0.340780\pi\)
0.479605 + 0.877485i \(0.340780\pi\)
\(348\) 0 0
\(349\) 8.73789 0.467728 0.233864 0.972269i \(-0.424863\pi\)
0.233864 + 0.972269i \(0.424863\pi\)
\(350\) 1.08564 0.0580298
\(351\) 0 0
\(352\) 9.71820 0.517982
\(353\) 22.6017 1.20297 0.601484 0.798885i \(-0.294576\pi\)
0.601484 + 0.798885i \(0.294576\pi\)
\(354\) 0 0
\(355\) −5.98533 −0.317668
\(356\) −88.6308 −4.69742
\(357\) 0 0
\(358\) −34.2094 −1.80802
\(359\) 13.1830 0.695772 0.347886 0.937537i \(-0.386900\pi\)
0.347886 + 0.937537i \(0.386900\pi\)
\(360\) 0 0
\(361\) −18.0804 −0.951602
\(362\) −6.88503 −0.361869
\(363\) 0 0
\(364\) −5.65607 −0.296458
\(365\) 3.34192 0.174924
\(366\) 0 0
\(367\) 17.4511 0.910938 0.455469 0.890252i \(-0.349472\pi\)
0.455469 + 0.890252i \(0.349472\pi\)
\(368\) −99.4247 −5.18287
\(369\) 0 0
\(370\) −20.2387 −1.05216
\(371\) 4.85512 0.252065
\(372\) 0 0
\(373\) −32.2018 −1.66734 −0.833672 0.552260i \(-0.813766\pi\)
−0.833672 + 0.552260i \(0.813766\pi\)
\(374\) −2.25628 −0.116670
\(375\) 0 0
\(376\) 114.994 5.93036
\(377\) −2.56511 −0.132110
\(378\) 0 0
\(379\) 32.3660 1.66253 0.831265 0.555876i \(-0.187617\pi\)
0.831265 + 0.555876i \(0.187617\pi\)
\(380\) −5.36894 −0.275421
\(381\) 0 0
\(382\) 17.4240 0.891492
\(383\) 25.8739 1.32209 0.661047 0.750345i \(-0.270113\pi\)
0.661047 + 0.750345i \(0.270113\pi\)
\(384\) 0 0
\(385\) 0.155104 0.00790483
\(386\) 69.7203 3.54867
\(387\) 0 0
\(388\) −103.340 −5.24631
\(389\) −32.6103 −1.65341 −0.826703 0.562639i \(-0.809786\pi\)
−0.826703 + 0.562639i \(0.809786\pi\)
\(390\) 0 0
\(391\) 12.7953 0.647086
\(392\) −67.9051 −3.42972
\(393\) 0 0
\(394\) 58.4528 2.94481
\(395\) −2.06745 −0.104025
\(396\) 0 0
\(397\) 4.39534 0.220596 0.110298 0.993899i \(-0.464820\pi\)
0.110298 + 0.993899i \(0.464820\pi\)
\(398\) −7.29734 −0.365783
\(399\) 0 0
\(400\) 16.1493 0.807464
\(401\) 19.0658 0.952099 0.476049 0.879418i \(-0.342068\pi\)
0.476049 + 0.879418i \(0.342068\pi\)
\(402\) 0 0
\(403\) 26.0528 1.29778
\(404\) 71.6865 3.56654
\(405\) 0 0
\(406\) 1.08564 0.0538793
\(407\) −2.89149 −0.143326
\(408\) 0 0
\(409\) −1.38235 −0.0683530 −0.0341765 0.999416i \(-0.510881\pi\)
−0.0341765 + 0.999416i \(0.510881\pi\)
\(410\) 4.56892 0.225643
\(411\) 0 0
\(412\) 27.2651 1.34326
\(413\) 3.75725 0.184882
\(414\) 0 0
\(415\) −6.41000 −0.314655
\(416\) −63.2965 −3.10337
\(417\) 0 0
\(418\) −1.04106 −0.0509199
\(419\) −2.86979 −0.140198 −0.0700991 0.997540i \(-0.522332\pi\)
−0.0700991 + 0.997540i \(0.522332\pi\)
\(420\) 0 0
\(421\) 13.6440 0.664970 0.332485 0.943109i \(-0.392113\pi\)
0.332485 + 0.943109i \(0.392113\pi\)
\(422\) 5.51320 0.268378
\(423\) 0 0
\(424\) 122.299 5.93938
\(425\) −2.07830 −0.100813
\(426\) 0 0
\(427\) −2.46279 −0.119183
\(428\) 13.1155 0.633964
\(429\) 0 0
\(430\) 28.4698 1.37294
\(431\) 11.9707 0.576607 0.288303 0.957539i \(-0.406909\pi\)
0.288303 + 0.957539i \(0.406909\pi\)
\(432\) 0 0
\(433\) 17.9907 0.864576 0.432288 0.901736i \(-0.357706\pi\)
0.432288 + 0.901736i \(0.357706\pi\)
\(434\) −11.0264 −0.529284
\(435\) 0 0
\(436\) 47.8725 2.29268
\(437\) 5.90381 0.282418
\(438\) 0 0
\(439\) −16.2226 −0.774260 −0.387130 0.922025i \(-0.626534\pi\)
−0.387130 + 0.922025i \(0.626534\pi\)
\(440\) 3.90703 0.186260
\(441\) 0 0
\(442\) 14.6956 0.698999
\(443\) 35.3762 1.68078 0.840388 0.541986i \(-0.182327\pi\)
0.840388 + 0.541986i \(0.182327\pi\)
\(444\) 0 0
\(445\) −15.8302 −0.750425
\(446\) 34.5965 1.63819
\(447\) 0 0
\(448\) 14.0690 0.664696
\(449\) −41.6971 −1.96781 −0.983903 0.178702i \(-0.942810\pi\)
−0.983903 + 0.178702i \(0.942810\pi\)
\(450\) 0 0
\(451\) 0.652757 0.0307371
\(452\) 62.7546 2.95173
\(453\) 0 0
\(454\) −3.58768 −0.168378
\(455\) −1.01022 −0.0473599
\(456\) 0 0
\(457\) −16.1111 −0.753646 −0.376823 0.926285i \(-0.622983\pi\)
−0.376823 + 0.926285i \(0.622983\pi\)
\(458\) 9.06045 0.423367
\(459\) 0 0
\(460\) −34.4698 −1.60716
\(461\) −17.3396 −0.807586 −0.403793 0.914850i \(-0.632308\pi\)
−0.403793 + 0.914850i \(0.632308\pi\)
\(462\) 0 0
\(463\) 19.3177 0.897768 0.448884 0.893590i \(-0.351822\pi\)
0.448884 + 0.893590i \(0.351822\pi\)
\(464\) 16.1493 0.749711
\(465\) 0 0
\(466\) −48.8235 −2.26170
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −2.92232 −0.134940
\(470\) 31.9531 1.47389
\(471\) 0 0
\(472\) 94.6441 4.35635
\(473\) 4.06745 0.187022
\(474\) 0 0
\(475\) −0.958939 −0.0439992
\(476\) −4.58266 −0.210046
\(477\) 0 0
\(478\) −58.6194 −2.68119
\(479\) 40.3877 1.84536 0.922681 0.385565i \(-0.125994\pi\)
0.922681 + 0.385565i \(0.125994\pi\)
\(480\) 0 0
\(481\) 18.8328 0.858702
\(482\) 42.2246 1.92328
\(483\) 0 0
\(484\) −60.7188 −2.75994
\(485\) −18.4575 −0.838110
\(486\) 0 0
\(487\) 2.84171 0.128770 0.0643850 0.997925i \(-0.479491\pi\)
0.0643850 + 0.997925i \(0.479491\pi\)
\(488\) −62.0371 −2.80829
\(489\) 0 0
\(490\) −18.8686 −0.852398
\(491\) −0.157863 −0.00712428 −0.00356214 0.999994i \(-0.501134\pi\)
−0.00356214 + 0.999994i \(0.501134\pi\)
\(492\) 0 0
\(493\) −2.07830 −0.0936021
\(494\) 6.78063 0.305075
\(495\) 0 0
\(496\) −164.022 −7.36480
\(497\) −2.35722 −0.105736
\(498\) 0 0
\(499\) 2.64723 0.118506 0.0592531 0.998243i \(-0.481128\pi\)
0.0592531 + 0.998243i \(0.481128\pi\)
\(500\) 5.59883 0.250387
\(501\) 0 0
\(502\) −74.1297 −3.30857
\(503\) −13.9545 −0.622200 −0.311100 0.950377i \(-0.600697\pi\)
−0.311100 + 0.950377i \(0.600697\pi\)
\(504\) 0 0
\(505\) 12.8038 0.569763
\(506\) −6.68384 −0.297133
\(507\) 0 0
\(508\) 112.853 5.00706
\(509\) 16.4921 0.731001 0.365500 0.930811i \(-0.380898\pi\)
0.365500 + 0.930811i \(0.380898\pi\)
\(510\) 0 0
\(511\) 1.31616 0.0582233
\(512\) 78.0802 3.45069
\(513\) 0 0
\(514\) 18.8968 0.833502
\(515\) 4.86979 0.214588
\(516\) 0 0
\(517\) 4.56511 0.200773
\(518\) −7.97066 −0.350211
\(519\) 0 0
\(520\) −25.4472 −1.11594
\(521\) 12.2253 0.535601 0.267800 0.963474i \(-0.413703\pi\)
0.267800 + 0.963474i \(0.413703\pi\)
\(522\) 0 0
\(523\) 1.66659 0.0728748 0.0364374 0.999336i \(-0.488399\pi\)
0.0364374 + 0.999336i \(0.488399\pi\)
\(524\) 30.8067 1.34580
\(525\) 0 0
\(526\) 37.5707 1.63816
\(527\) 21.1085 0.919501
\(528\) 0 0
\(529\) 14.9038 0.647992
\(530\) 33.9830 1.47613
\(531\) 0 0
\(532\) −2.11446 −0.0916736
\(533\) −4.25154 −0.184155
\(534\) 0 0
\(535\) 2.34255 0.101277
\(536\) −73.6126 −3.17958
\(537\) 0 0
\(538\) 23.3244 1.00558
\(539\) −2.69574 −0.116114
\(540\) 0 0
\(541\) −34.6558 −1.48997 −0.744984 0.667083i \(-0.767543\pi\)
−0.744984 + 0.667083i \(0.767543\pi\)
\(542\) 17.4839 0.750996
\(543\) 0 0
\(544\) −51.2842 −2.19879
\(545\) 8.55044 0.366261
\(546\) 0 0
\(547\) 38.2530 1.63558 0.817791 0.575515i \(-0.195199\pi\)
0.817791 + 0.575515i \(0.195199\pi\)
\(548\) −41.9830 −1.79343
\(549\) 0 0
\(550\) 1.08564 0.0462917
\(551\) −0.958939 −0.0408522
\(552\) 0 0
\(553\) −0.814230 −0.0346246
\(554\) −47.2657 −2.00813
\(555\) 0 0
\(556\) −52.3646 −2.22075
\(557\) 28.7760 1.21928 0.609639 0.792679i \(-0.291314\pi\)
0.609639 + 0.792679i \(0.291314\pi\)
\(558\) 0 0
\(559\) −26.4921 −1.12050
\(560\) 6.36011 0.268764
\(561\) 0 0
\(562\) 33.9022 1.43008
\(563\) 32.9809 1.38998 0.694989 0.719020i \(-0.255409\pi\)
0.694989 + 0.719020i \(0.255409\pi\)
\(564\) 0 0
\(565\) 11.2085 0.471546
\(566\) 69.4194 2.91792
\(567\) 0 0
\(568\) −59.3777 −2.49143
\(569\) −16.8038 −0.704453 −0.352227 0.935915i \(-0.614575\pi\)
−0.352227 + 0.935915i \(0.614575\pi\)
\(570\) 0 0
\(571\) −6.21703 −0.260175 −0.130087 0.991503i \(-0.541526\pi\)
−0.130087 + 0.991503i \(0.541526\pi\)
\(572\) −5.65607 −0.236492
\(573\) 0 0
\(574\) 1.79939 0.0751051
\(575\) −6.15661 −0.256748
\(576\) 0 0
\(577\) 11.9977 0.499470 0.249735 0.968314i \(-0.419656\pi\)
0.249735 + 0.968314i \(0.419656\pi\)
\(578\) −34.9555 −1.45396
\(579\) 0 0
\(580\) 5.59883 0.232479
\(581\) −2.52447 −0.104733
\(582\) 0 0
\(583\) 4.85512 0.201078
\(584\) 33.1537 1.37191
\(585\) 0 0
\(586\) −33.9531 −1.40259
\(587\) 11.2194 0.463073 0.231536 0.972826i \(-0.425625\pi\)
0.231536 + 0.972826i \(0.425625\pi\)
\(588\) 0 0
\(589\) 9.73957 0.401312
\(590\) 26.2985 1.08269
\(591\) 0 0
\(592\) −118.567 −4.87306
\(593\) 7.69682 0.316071 0.158035 0.987433i \(-0.449484\pi\)
0.158035 + 0.987433i \(0.449484\pi\)
\(594\) 0 0
\(595\) −0.818503 −0.0335554
\(596\) 64.6118 2.64660
\(597\) 0 0
\(598\) 43.5332 1.78020
\(599\) −20.0543 −0.819396 −0.409698 0.912221i \(-0.634366\pi\)
−0.409698 + 0.912221i \(0.634366\pi\)
\(600\) 0 0
\(601\) 3.10851 0.126799 0.0633995 0.997988i \(-0.479806\pi\)
0.0633995 + 0.997988i \(0.479806\pi\)
\(602\) 11.2123 0.456981
\(603\) 0 0
\(604\) 34.0528 1.38559
\(605\) −10.8449 −0.440908
\(606\) 0 0
\(607\) 37.0441 1.50357 0.751786 0.659407i \(-0.229193\pi\)
0.751786 + 0.659407i \(0.229193\pi\)
\(608\) −23.6628 −0.959652
\(609\) 0 0
\(610\) −17.2381 −0.697950
\(611\) −29.7334 −1.20289
\(612\) 0 0
\(613\) −13.3839 −0.540569 −0.270284 0.962781i \(-0.587118\pi\)
−0.270284 + 0.962781i \(0.587118\pi\)
\(614\) −62.6792 −2.52953
\(615\) 0 0
\(616\) 1.53871 0.0619966
\(617\) 12.6364 0.508721 0.254361 0.967109i \(-0.418135\pi\)
0.254361 + 0.967109i \(0.418135\pi\)
\(618\) 0 0
\(619\) −41.2447 −1.65776 −0.828882 0.559424i \(-0.811022\pi\)
−0.828882 + 0.559424i \(0.811022\pi\)
\(620\) −56.8652 −2.28376
\(621\) 0 0
\(622\) 14.7150 0.590018
\(623\) −6.23446 −0.249778
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −5.41226 −0.216318
\(627\) 0 0
\(628\) −64.9203 −2.59060
\(629\) 15.2587 0.608406
\(630\) 0 0
\(631\) 29.8009 1.18635 0.593177 0.805072i \(-0.297873\pi\)
0.593177 + 0.805072i \(0.297873\pi\)
\(632\) −20.5103 −0.815854
\(633\) 0 0
\(634\) −3.55777 −0.141297
\(635\) 20.1566 0.799891
\(636\) 0 0
\(637\) 17.5579 0.695669
\(638\) 1.08564 0.0429808
\(639\) 0 0
\(640\) 49.1226 1.94174
\(641\) −17.7774 −0.702167 −0.351083 0.936344i \(-0.614187\pi\)
−0.351083 + 0.936344i \(0.614187\pi\)
\(642\) 0 0
\(643\) −44.5534 −1.75702 −0.878508 0.477727i \(-0.841461\pi\)
−0.878508 + 0.477727i \(0.841461\pi\)
\(644\) −13.5753 −0.534943
\(645\) 0 0
\(646\) 5.49381 0.216151
\(647\) −42.9339 −1.68790 −0.843952 0.536418i \(-0.819777\pi\)
−0.843952 + 0.536418i \(0.819777\pi\)
\(648\) 0 0
\(649\) 3.75725 0.147485
\(650\) −7.07097 −0.277346
\(651\) 0 0
\(652\) −4.78766 −0.187499
\(653\) 19.8426 0.776500 0.388250 0.921554i \(-0.373080\pi\)
0.388250 + 0.921554i \(0.373080\pi\)
\(654\) 0 0
\(655\) 5.50235 0.214994
\(656\) 26.7666 1.04506
\(657\) 0 0
\(658\) 12.5842 0.490582
\(659\) −23.0483 −0.897836 −0.448918 0.893573i \(-0.648190\pi\)
−0.448918 + 0.893573i \(0.648190\pi\)
\(660\) 0 0
\(661\) −36.2079 −1.40832 −0.704162 0.710040i \(-0.748677\pi\)
−0.704162 + 0.710040i \(0.748677\pi\)
\(662\) 79.9332 3.10669
\(663\) 0 0
\(664\) −63.5907 −2.46780
\(665\) −0.377661 −0.0146451
\(666\) 0 0
\(667\) −6.15661 −0.238385
\(668\) −43.2898 −1.67493
\(669\) 0 0
\(670\) −20.4546 −0.790229
\(671\) −2.46279 −0.0950749
\(672\) 0 0
\(673\) 22.9000 0.882731 0.441365 0.897327i \(-0.354494\pi\)
0.441365 + 0.897327i \(0.354494\pi\)
\(674\) −46.2705 −1.78227
\(675\) 0 0
\(676\) −35.9458 −1.38253
\(677\) −36.0068 −1.38385 −0.691927 0.721967i \(-0.743238\pi\)
−0.691927 + 0.721967i \(0.743238\pi\)
\(678\) 0 0
\(679\) −7.26915 −0.278964
\(680\) −20.6179 −0.790660
\(681\) 0 0
\(682\) −11.0264 −0.422222
\(683\) −9.12896 −0.349310 −0.174655 0.984630i \(-0.555881\pi\)
−0.174655 + 0.984630i \(0.555881\pi\)
\(684\) 0 0
\(685\) −7.49853 −0.286504
\(686\) −15.0305 −0.573869
\(687\) 0 0
\(688\) 166.788 6.35873
\(689\) −31.6223 −1.20472
\(690\) 0 0
\(691\) 5.85956 0.222908 0.111454 0.993770i \(-0.464449\pi\)
0.111454 + 0.993770i \(0.464449\pi\)
\(692\) −45.2077 −1.71854
\(693\) 0 0
\(694\) 49.2552 1.86970
\(695\) −9.35277 −0.354771
\(696\) 0 0
\(697\) −3.44469 −0.130477
\(698\) 24.0868 0.911700
\(699\) 0 0
\(700\) 2.20500 0.0833412
\(701\) 7.56955 0.285898 0.142949 0.989730i \(-0.454342\pi\)
0.142949 + 0.989730i \(0.454342\pi\)
\(702\) 0 0
\(703\) 7.04046 0.265536
\(704\) 14.0690 0.530244
\(705\) 0 0
\(706\) 62.3039 2.34484
\(707\) 5.04256 0.189645
\(708\) 0 0
\(709\) 14.5209 0.545342 0.272671 0.962107i \(-0.412093\pi\)
0.272671 + 0.962107i \(0.412093\pi\)
\(710\) −16.4992 −0.619202
\(711\) 0 0
\(712\) −157.044 −5.88549
\(713\) 62.5302 2.34178
\(714\) 0 0
\(715\) −1.01022 −0.0377802
\(716\) −69.4815 −2.59665
\(717\) 0 0
\(718\) 36.3402 1.35621
\(719\) −12.2164 −0.455596 −0.227798 0.973708i \(-0.573153\pi\)
−0.227798 + 0.973708i \(0.573153\pi\)
\(720\) 0 0
\(721\) 1.91788 0.0714255
\(722\) −49.8405 −1.85487
\(723\) 0 0
\(724\) −13.9839 −0.519709
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) −40.8856 −1.51636 −0.758182 0.652044i \(-0.773912\pi\)
−0.758182 + 0.652044i \(0.773912\pi\)
\(728\) −10.0219 −0.371438
\(729\) 0 0
\(730\) 9.21234 0.340964
\(731\) −21.4645 −0.793892
\(732\) 0 0
\(733\) 25.4915 0.941550 0.470775 0.882253i \(-0.343974\pi\)
0.470775 + 0.882253i \(0.343974\pi\)
\(734\) 48.1056 1.77561
\(735\) 0 0
\(736\) −151.920 −5.59986
\(737\) −2.92232 −0.107645
\(738\) 0 0
\(739\) 9.56830 0.351975 0.175988 0.984392i \(-0.443688\pi\)
0.175988 + 0.984392i \(0.443688\pi\)
\(740\) −41.1062 −1.51109
\(741\) 0 0
\(742\) 13.3836 0.491328
\(743\) 18.0455 0.662025 0.331013 0.943626i \(-0.392610\pi\)
0.331013 + 0.943626i \(0.392610\pi\)
\(744\) 0 0
\(745\) 11.5402 0.422801
\(746\) −88.7673 −3.25000
\(747\) 0 0
\(748\) −4.58266 −0.167559
\(749\) 0.922572 0.0337100
\(750\) 0 0
\(751\) 11.9255 0.435168 0.217584 0.976042i \(-0.430182\pi\)
0.217584 + 0.976042i \(0.430182\pi\)
\(752\) 187.194 6.82627
\(753\) 0 0
\(754\) −7.07097 −0.257510
\(755\) 6.08212 0.221351
\(756\) 0 0
\(757\) 5.86152 0.213041 0.106520 0.994311i \(-0.466029\pi\)
0.106520 + 0.994311i \(0.466029\pi\)
\(758\) 89.2201 3.24062
\(759\) 0 0
\(760\) −9.51320 −0.345080
\(761\) −49.8739 −1.80793 −0.903963 0.427610i \(-0.859356\pi\)
−0.903963 + 0.427610i \(0.859356\pi\)
\(762\) 0 0
\(763\) 3.36744 0.121909
\(764\) 35.3894 1.28034
\(765\) 0 0
\(766\) 71.3239 2.57704
\(767\) −24.4717 −0.883621
\(768\) 0 0
\(769\) 29.5121 1.06423 0.532117 0.846671i \(-0.321396\pi\)
0.532117 + 0.846671i \(0.321396\pi\)
\(770\) 0.427559 0.0154082
\(771\) 0 0
\(772\) 141.607 5.09653
\(773\) 2.41935 0.0870180 0.0435090 0.999053i \(-0.486146\pi\)
0.0435090 + 0.999053i \(0.486146\pi\)
\(774\) 0 0
\(775\) −10.1566 −0.364836
\(776\) −183.108 −6.57320
\(777\) 0 0
\(778\) −89.8934 −3.22283
\(779\) −1.58939 −0.0569460
\(780\) 0 0
\(781\) −2.35722 −0.0843479
\(782\) 35.2715 1.26131
\(783\) 0 0
\(784\) −110.540 −3.94786
\(785\) −11.5953 −0.413855
\(786\) 0 0
\(787\) −51.1549 −1.82348 −0.911738 0.410772i \(-0.865259\pi\)
−0.911738 + 0.410772i \(0.865259\pi\)
\(788\) 118.722 4.22928
\(789\) 0 0
\(790\) −5.69914 −0.202766
\(791\) 4.41428 0.156954
\(792\) 0 0
\(793\) 16.0406 0.569619
\(794\) 12.1162 0.429987
\(795\) 0 0
\(796\) −14.8214 −0.525330
\(797\) −28.8662 −1.02249 −0.511247 0.859434i \(-0.670816\pi\)
−0.511247 + 0.859434i \(0.670816\pi\)
\(798\) 0 0
\(799\) −24.0907 −0.852266
\(800\) 24.6760 0.872428
\(801\) 0 0
\(802\) 52.5567 1.85584
\(803\) 1.31616 0.0464462
\(804\) 0 0
\(805\) −2.42467 −0.0854584
\(806\) 71.8171 2.52965
\(807\) 0 0
\(808\) 127.021 4.46858
\(809\) −19.0426 −0.669501 −0.334750 0.942307i \(-0.608652\pi\)
−0.334750 + 0.942307i \(0.608652\pi\)
\(810\) 0 0
\(811\) −20.8783 −0.733137 −0.366569 0.930391i \(-0.619467\pi\)
−0.366569 + 0.930391i \(0.619467\pi\)
\(812\) 2.20500 0.0773804
\(813\) 0 0
\(814\) −7.97066 −0.279372
\(815\) −0.855118 −0.0299535
\(816\) 0 0
\(817\) −9.90381 −0.346491
\(818\) −3.81060 −0.133234
\(819\) 0 0
\(820\) 9.27979 0.324064
\(821\) −3.60466 −0.125804 −0.0629018 0.998020i \(-0.520035\pi\)
−0.0629018 + 0.998020i \(0.520035\pi\)
\(822\) 0 0
\(823\) 27.2288 0.949135 0.474567 0.880219i \(-0.342605\pi\)
0.474567 + 0.880219i \(0.342605\pi\)
\(824\) 48.3109 1.68299
\(825\) 0 0
\(826\) 10.3572 0.360374
\(827\) −47.3936 −1.64804 −0.824019 0.566562i \(-0.808273\pi\)
−0.824019 + 0.566562i \(0.808273\pi\)
\(828\) 0 0
\(829\) −41.8388 −1.45312 −0.726560 0.687103i \(-0.758882\pi\)
−0.726560 + 0.687103i \(0.758882\pi\)
\(830\) −17.6698 −0.613328
\(831\) 0 0
\(832\) −91.6339 −3.17683
\(833\) 14.2258 0.492894
\(834\) 0 0
\(835\) −7.73194 −0.267575
\(836\) −2.11446 −0.0731302
\(837\) 0 0
\(838\) −7.91085 −0.273276
\(839\) 31.6385 1.09228 0.546141 0.837693i \(-0.316096\pi\)
0.546141 + 0.837693i \(0.316096\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 37.6111 1.29617
\(843\) 0 0
\(844\) 11.1977 0.385440
\(845\) −6.42023 −0.220862
\(846\) 0 0
\(847\) −4.27107 −0.146756
\(848\) 199.086 6.83665
\(849\) 0 0
\(850\) −5.72905 −0.196505
\(851\) 45.2013 1.54948
\(852\) 0 0
\(853\) 44.2000 1.51338 0.756690 0.653773i \(-0.226815\pi\)
0.756690 + 0.653773i \(0.226815\pi\)
\(854\) −6.78892 −0.232312
\(855\) 0 0
\(856\) 23.2394 0.794305
\(857\) −14.4775 −0.494541 −0.247270 0.968947i \(-0.579534\pi\)
−0.247270 + 0.968947i \(0.579534\pi\)
\(858\) 0 0
\(859\) 22.5883 0.770703 0.385352 0.922770i \(-0.374080\pi\)
0.385352 + 0.922770i \(0.374080\pi\)
\(860\) 57.8241 1.97179
\(861\) 0 0
\(862\) 32.9983 1.12393
\(863\) 22.9900 0.782590 0.391295 0.920265i \(-0.372027\pi\)
0.391295 + 0.920265i \(0.372027\pi\)
\(864\) 0 0
\(865\) −8.07449 −0.274541
\(866\) 49.5930 1.68524
\(867\) 0 0
\(868\) −22.3953 −0.760147
\(869\) −0.814230 −0.0276209
\(870\) 0 0
\(871\) 19.0337 0.644931
\(872\) 84.8250 2.87254
\(873\) 0 0
\(874\) 16.2744 0.550491
\(875\) 0.393832 0.0133140
\(876\) 0 0
\(877\) 13.5741 0.458364 0.229182 0.973384i \(-0.426395\pi\)
0.229182 + 0.973384i \(0.426395\pi\)
\(878\) −44.7191 −1.50920
\(879\) 0 0
\(880\) 6.36011 0.214399
\(881\) −9.91235 −0.333956 −0.166978 0.985961i \(-0.553401\pi\)
−0.166978 + 0.985961i \(0.553401\pi\)
\(882\) 0 0
\(883\) 39.3192 1.32320 0.661598 0.749859i \(-0.269879\pi\)
0.661598 + 0.749859i \(0.269879\pi\)
\(884\) 29.8478 1.00389
\(885\) 0 0
\(886\) 97.5180 3.27618
\(887\) 59.2520 1.98949 0.994743 0.102403i \(-0.0326531\pi\)
0.994743 + 0.102403i \(0.0326531\pi\)
\(888\) 0 0
\(889\) 7.93832 0.266243
\(890\) −43.6376 −1.46274
\(891\) 0 0
\(892\) 70.2678 2.35274
\(893\) −11.1155 −0.371968
\(894\) 0 0
\(895\) −12.4100 −0.414821
\(896\) 19.3461 0.646307
\(897\) 0 0
\(898\) −114.942 −3.83567
\(899\) −10.1566 −0.338742
\(900\) 0 0
\(901\) −25.6211 −0.853562
\(902\) 1.79939 0.0599131
\(903\) 0 0
\(904\) 111.195 3.69828
\(905\) −2.49765 −0.0830248
\(906\) 0 0
\(907\) −5.91210 −0.196308 −0.0981541 0.995171i \(-0.531294\pi\)
−0.0981541 + 0.995171i \(0.531294\pi\)
\(908\) −7.28682 −0.241822
\(909\) 0 0
\(910\) −2.78478 −0.0923144
\(911\) 12.4185 0.411445 0.205722 0.978610i \(-0.434046\pi\)
0.205722 + 0.978610i \(0.434046\pi\)
\(912\) 0 0
\(913\) −2.52447 −0.0835476
\(914\) −44.4118 −1.46901
\(915\) 0 0
\(916\) 18.4024 0.608031
\(917\) 2.16700 0.0715607
\(918\) 0 0
\(919\) 16.4332 0.542081 0.271041 0.962568i \(-0.412632\pi\)
0.271041 + 0.962568i \(0.412632\pi\)
\(920\) −61.0769 −2.01364
\(921\) 0 0
\(922\) −47.7983 −1.57415
\(923\) 15.3530 0.505351
\(924\) 0 0
\(925\) −7.34192 −0.241401
\(926\) 53.2510 1.74994
\(927\) 0 0
\(928\) 24.6760 0.810029
\(929\) −13.8358 −0.453936 −0.226968 0.973902i \(-0.572881\pi\)
−0.226968 + 0.973902i \(0.572881\pi\)
\(930\) 0 0
\(931\) 6.56384 0.215121
\(932\) −99.1637 −3.24822
\(933\) 0 0
\(934\) 22.0528 0.721589
\(935\) −0.818503 −0.0267679
\(936\) 0 0
\(937\) 20.7528 0.677964 0.338982 0.940793i \(-0.389917\pi\)
0.338982 + 0.940793i \(0.389917\pi\)
\(938\) −8.05567 −0.263027
\(939\) 0 0
\(940\) 64.8989 2.11677
\(941\) −40.1666 −1.30940 −0.654698 0.755891i \(-0.727204\pi\)
−0.654698 + 0.755891i \(0.727204\pi\)
\(942\) 0 0
\(943\) −10.2043 −0.332297
\(944\) 154.068 5.01447
\(945\) 0 0
\(946\) 11.2123 0.364544
\(947\) 17.4144 0.565894 0.282947 0.959136i \(-0.408688\pi\)
0.282947 + 0.959136i \(0.408688\pi\)
\(948\) 0 0
\(949\) −8.57239 −0.278271
\(950\) −2.64341 −0.0857636
\(951\) 0 0
\(952\) −8.11999 −0.263170
\(953\) −13.5121 −0.437701 −0.218851 0.975758i \(-0.570231\pi\)
−0.218851 + 0.975758i \(0.570231\pi\)
\(954\) 0 0
\(955\) 6.32085 0.204538
\(956\) −119.060 −3.85067
\(957\) 0 0
\(958\) 111.333 3.59700
\(959\) −2.95316 −0.0953626
\(960\) 0 0
\(961\) 72.1567 2.32763
\(962\) 51.9145 1.67379
\(963\) 0 0
\(964\) 85.7610 2.76218
\(965\) 25.2921 0.814183
\(966\) 0 0
\(967\) −24.0598 −0.773712 −0.386856 0.922140i \(-0.626439\pi\)
−0.386856 + 0.922140i \(0.626439\pi\)
\(968\) −107.587 −3.45798
\(969\) 0 0
\(970\) −50.8798 −1.63365
\(971\) −34.7877 −1.11639 −0.558195 0.829710i \(-0.688506\pi\)
−0.558195 + 0.829710i \(0.688506\pi\)
\(972\) 0 0
\(973\) −3.68342 −0.118085
\(974\) 7.83344 0.251000
\(975\) 0 0
\(976\) −100.988 −3.23254
\(977\) −35.7566 −1.14396 −0.571978 0.820269i \(-0.693824\pi\)
−0.571978 + 0.820269i \(0.693824\pi\)
\(978\) 0 0
\(979\) −6.23446 −0.199254
\(980\) −38.3234 −1.22420
\(981\) 0 0
\(982\) −0.435166 −0.0138867
\(983\) 7.19064 0.229346 0.114673 0.993403i \(-0.463418\pi\)
0.114673 + 0.993403i \(0.463418\pi\)
\(984\) 0 0
\(985\) 21.2047 0.675638
\(986\) −5.72905 −0.182450
\(987\) 0 0
\(988\) 13.7719 0.438143
\(989\) −63.5847 −2.02188
\(990\) 0 0
\(991\) −19.9123 −0.632537 −0.316268 0.948670i \(-0.602430\pi\)
−0.316268 + 0.948670i \(0.602430\pi\)
\(992\) −250.624 −7.95733
\(993\) 0 0
\(994\) −6.49790 −0.206101
\(995\) −2.64723 −0.0839228
\(996\) 0 0
\(997\) −1.57302 −0.0498179 −0.0249089 0.999690i \(-0.507930\pi\)
−0.0249089 + 0.999690i \(0.507930\pi\)
\(998\) 7.29734 0.230993
\(999\) 0 0
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 1305.2.a.r.1.4 4
3.2 odd 2 435.2.a.j.1.1 4
5.4 even 2 6525.2.a.bi.1.1 4
12.11 even 2 6960.2.a.co.1.2 4
15.2 even 4 2175.2.c.n.349.1 8
15.8 even 4 2175.2.c.n.349.8 8
15.14 odd 2 2175.2.a.v.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.1 4 3.2 odd 2
1305.2.a.r.1.4 4 1.1 even 1 trivial
2175.2.a.v.1.4 4 15.14 odd 2
2175.2.c.n.349.1 8 15.2 even 4
2175.2.c.n.349.8 8 15.8 even 4
6525.2.a.bi.1.1 4 5.4 even 2
6960.2.a.co.1.2 4 12.11 even 2