Properties

Label 2175.2.a.v.1.4
Level $2175$
Weight $2$
Character 2175.1
Self dual yes
Analytic conductor $17.367$
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] = [2175,2,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, 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("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
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 \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.75660 q^{2} +1.00000 q^{3} +5.59883 q^{4} +2.75660 q^{6} -0.393832 q^{7} +9.92054 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.75660 q^{2} +1.00000 q^{3} +5.59883 q^{4} +2.75660 q^{6} -0.393832 q^{7} +9.92054 q^{8} +1.00000 q^{9} -0.393832 q^{11} +5.59883 q^{12} +2.56511 q^{13} -1.08564 q^{14} +16.1493 q^{16} -2.07830 q^{17} +2.75660 q^{18} -0.958939 q^{19} -0.393832 q^{21} -1.08564 q^{22} -6.15661 q^{23} +9.92054 q^{24} +7.07097 q^{26} +1.00000 q^{27} -2.20500 q^{28} -1.00000 q^{29} -10.1566 q^{31} +24.6760 q^{32} -0.393832 q^{33} -5.72905 q^{34} +5.59883 q^{36} +7.34192 q^{37} -2.64341 q^{38} +2.56511 q^{39} -1.65745 q^{41} -1.08564 q^{42} -10.3279 q^{43} -2.20500 q^{44} -16.9713 q^{46} +11.5915 q^{47} +16.1493 q^{48} -6.84490 q^{49} -2.07830 q^{51} +14.3616 q^{52} +12.3279 q^{53} +2.75660 q^{54} -3.90703 q^{56} -0.958939 q^{57} -2.75660 q^{58} -9.54022 q^{59} -6.25340 q^{61} -27.9977 q^{62} -0.393832 q^{63} +35.7232 q^{64} -1.08564 q^{66} +7.42023 q^{67} -11.6361 q^{68} -6.15661 q^{69} +5.98533 q^{71} +9.92054 q^{72} -3.34192 q^{73} +20.2387 q^{74} -5.36894 q^{76} +0.155104 q^{77} +7.07097 q^{78} -2.06745 q^{79} +1.00000 q^{81} -4.56892 q^{82} -6.41000 q^{83} -2.20500 q^{84} -28.4698 q^{86} -1.00000 q^{87} -3.90703 q^{88} +15.8302 q^{89} -1.01022 q^{91} -34.4698 q^{92} -10.1566 q^{93} +31.9531 q^{94} +24.6760 q^{96} +18.4575 q^{97} -18.8686 q^{98} -0.393832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 4 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 4 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} + 12 q^{8} + 4 q^{9} - 2 q^{11} + 5 q^{12} + 8 q^{13} - 3 q^{14} + 11 q^{16} + 10 q^{17} + 3 q^{18} - 2 q^{19} - 2 q^{21} - 3 q^{22} + 12 q^{23} + 12 q^{24} - 7 q^{26} + 4 q^{27} + 9 q^{28} - 4 q^{29} - 4 q^{31} + 17 q^{32} - 2 q^{33} - q^{34} + 5 q^{36} + 16 q^{37} + 10 q^{38} + 8 q^{39} - 12 q^{41} - 3 q^{42} - 2 q^{43} + 9 q^{44} - 8 q^{46} + 12 q^{47} + 11 q^{48} + 6 q^{49} + 10 q^{51} + 3 q^{52} + 10 q^{53} + 3 q^{54} - 2 q^{57} - 3 q^{58} + 2 q^{59} - 26 q^{61} - 20 q^{62} - 2 q^{63} + 34 q^{64} - 3 q^{66} - 2 q^{67} - 9 q^{68} + 12 q^{69} - 10 q^{71} + 12 q^{72} + 48 q^{74} + 16 q^{76} + 34 q^{77} - 7 q^{78} + 22 q^{79} + 4 q^{81} - 38 q^{82} + 10 q^{83} + 9 q^{84} - 4 q^{86} - 4 q^{87} - 4 q^{89} - 8 q^{91} - 28 q^{92} - 4 q^{93} + 39 q^{94} + 17 q^{96} + 22 q^{97} - 34 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.75660 1.94921 0.974605 0.223932i \(-0.0718895\pi\)
0.974605 + 0.223932i \(0.0718895\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.59883 2.79942
\(5\) 0 0
\(6\) 2.75660 1.12538
\(7\) −0.393832 −0.148855 −0.0744273 0.997226i \(-0.523713\pi\)
−0.0744273 + 0.997226i \(0.523713\pi\)
\(8\) 9.92054 3.50744
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.393832 −0.118745 −0.0593725 0.998236i \(-0.518910\pi\)
−0.0593725 + 0.998236i \(0.518910\pi\)
\(12\) 5.59883 1.61624
\(13\) 2.56511 0.711433 0.355716 0.934594i \(-0.384237\pi\)
0.355716 + 0.934594i \(0.384237\pi\)
\(14\) −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 \(0.581099\pi\)
\(18\) 2.75660 0.649736
\(19\) −0.958939 −0.219996 −0.109998 0.993932i \(-0.535084\pi\)
−0.109998 + 0.993932i \(0.535084\pi\)
\(20\) 0 0
\(21\) −0.393832 −0.0859413
\(22\) −1.08564 −0.231459
\(23\) −6.15661 −1.28374 −0.641871 0.766813i \(-0.721841\pi\)
−0.641871 + 0.766813i \(0.721841\pi\)
\(24\) 9.92054 2.02502
\(25\) 0 0
\(26\) 7.07097 1.38673
\(27\) 1.00000 0.192450
\(28\) −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
\(33\) −0.393832 −0.0685574
\(34\) −5.72905 −0.982524
\(35\) 0 0
\(36\) 5.59883 0.933139
\(37\) 7.34192 1.20700 0.603502 0.797361i \(-0.293771\pi\)
0.603502 + 0.797361i \(0.293771\pi\)
\(38\) −2.64341 −0.428818
\(39\) 2.56511 0.410746
\(40\) 0 0
\(41\) −1.65745 −0.258850 −0.129425 0.991589i \(-0.541313\pi\)
−0.129425 + 0.991589i \(0.541313\pi\)
\(42\) −1.08564 −0.167517
\(43\) −10.3279 −1.57499 −0.787494 0.616323i \(-0.788622\pi\)
−0.787494 + 0.616323i \(0.788622\pi\)
\(44\) −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 \(0.179364\pi\)
\(48\) 16.1493 2.33095
\(49\) −6.84490 −0.977842
\(50\) 0 0
\(51\) −2.07830 −0.291021
\(52\) 14.3616 1.99160
\(53\) 12.3279 1.69336 0.846682 0.532099i \(-0.178596\pi\)
0.846682 + 0.532099i \(0.178596\pi\)
\(54\) 2.75660 0.375126
\(55\) 0 0
\(56\) −3.90703 −0.522099
\(57\) −0.958939 −0.127015
\(58\) −2.75660 −0.361959
\(59\) −9.54022 −1.24203 −0.621015 0.783798i \(-0.713280\pi\)
−0.621015 + 0.783798i \(0.713280\pi\)
\(60\) 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
\(63\) −0.393832 −0.0496182
\(64\) 35.7232 4.46540
\(65\) 0 0
\(66\) −1.08564 −0.133633
\(67\) 7.42023 0.906525 0.453262 0.891377i \(-0.350260\pi\)
0.453262 + 0.891377i \(0.350260\pi\)
\(68\) −11.6361 −1.41108
\(69\) −6.15661 −0.741168
\(70\) 0 0
\(71\) 5.98533 0.710328 0.355164 0.934804i \(-0.384425\pi\)
0.355164 + 0.934804i \(0.384425\pi\)
\(72\) 9.92054 1.16915
\(73\) −3.34192 −0.391142 −0.195571 0.980690i \(-0.562656\pi\)
−0.195571 + 0.980690i \(0.562656\pi\)
\(74\) 20.2387 2.35270
\(75\) 0 0
\(76\) −5.36894 −0.615860
\(77\) 0.155104 0.0176757
\(78\) 7.07097 0.800630
\(79\) −2.06745 −0.232607 −0.116303 0.993214i \(-0.537104\pi\)
−0.116303 + 0.993214i \(0.537104\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −4.56892 −0.504553
\(83\) −6.41000 −0.703589 −0.351795 0.936077i \(-0.614428\pi\)
−0.351795 + 0.936077i \(0.614428\pi\)
\(84\) −2.20500 −0.240585
\(85\) 0 0
\(86\) −28.4698 −3.06998
\(87\) −1.00000 −0.107211
\(88\) −3.90703 −0.416491
\(89\) 15.8302 1.67800 0.839000 0.544131i \(-0.183140\pi\)
0.839000 + 0.544131i \(0.183140\pi\)
\(90\) 0 0
\(91\) −1.01022 −0.105900
\(92\) −34.4698 −3.59373
\(93\) −10.1566 −1.05319
\(94\) 31.9531 3.29571
\(95\) 0 0
\(96\) 24.6760 2.51848
\(97\) 18.4575 1.87407 0.937036 0.349233i \(-0.113558\pi\)
0.937036 + 0.349233i \(0.113558\pi\)
\(98\) −18.8686 −1.90602
\(99\) −0.393832 −0.0395816
\(100\) 0 0
\(101\) −12.8038 −1.27403 −0.637015 0.770852i \(-0.719831\pi\)
−0.637015 + 0.770852i \(0.719831\pi\)
\(102\) −5.72905 −0.567260
\(103\) −4.86979 −0.479834 −0.239917 0.970793i \(-0.577120\pi\)
−0.239917 + 0.970793i \(0.577120\pi\)
\(104\) 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 \(0.463880\pi\)
\(108\) 5.59883 0.538748
\(109\) 8.55044 0.818984 0.409492 0.912314i \(-0.365706\pi\)
0.409492 + 0.912314i \(0.365706\pi\)
\(110\) 0 0
\(111\) 7.34192 0.696864
\(112\) −6.36011 −0.600973
\(113\) 11.2085 1.05441 0.527204 0.849739i \(-0.323240\pi\)
0.527204 + 0.849739i \(0.323240\pi\)
\(114\) −2.64341 −0.247578
\(115\) 0 0
\(116\) −5.59883 −0.519839
\(117\) 2.56511 0.237144
\(118\) −26.2985 −2.42098
\(119\) 0.818503 0.0750321
\(120\) 0 0
\(121\) −10.8449 −0.985900
\(122\) −17.2381 −1.56066
\(123\) −1.65745 −0.149447
\(124\) −56.8652 −5.10664
\(125\) 0 0
\(126\) −1.08564 −0.0967163
\(127\) −20.1566 −1.78861 −0.894305 0.447458i \(-0.852329\pi\)
−0.894305 + 0.447458i \(0.852329\pi\)
\(128\) 49.1226 4.34187
\(129\) −10.3279 −0.909319
\(130\) 0 0
\(131\) −5.50235 −0.480742 −0.240371 0.970681i \(-0.577269\pi\)
−0.240371 + 0.970681i \(0.577269\pi\)
\(132\) −2.20500 −0.191921
\(133\) 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 \(0.603791\pi\)
\(138\) −16.9713 −1.44469
\(139\) −9.35277 −0.793292 −0.396646 0.917972i \(-0.629826\pi\)
−0.396646 + 0.917972i \(0.629826\pi\)
\(140\) 0 0
\(141\) 11.5915 0.976180
\(142\) 16.4992 1.38458
\(143\) −1.01022 −0.0844790
\(144\) 16.1493 1.34577
\(145\) 0 0
\(146\) −9.21234 −0.762418
\(147\) −6.84490 −0.564558
\(148\) 41.1062 3.37891
\(149\) −11.5402 −0.945411 −0.472706 0.881220i \(-0.656723\pi\)
−0.472706 + 0.881220i \(0.656723\pi\)
\(150\) 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
\(153\) −2.07830 −0.168021
\(154\) 0.427559 0.0344537
\(155\) 0 0
\(156\) 14.3616 1.14985
\(157\) 11.5953 0.925407 0.462704 0.886513i \(-0.346879\pi\)
0.462704 + 0.886513i \(0.346879\pi\)
\(158\) −5.69914 −0.453399
\(159\) 12.3279 0.977665
\(160\) 0 0
\(161\) 2.42467 0.191091
\(162\) 2.75660 0.216579
\(163\) 0.855118 0.0669780 0.0334890 0.999439i \(-0.489338\pi\)
0.0334890 + 0.999439i \(0.489338\pi\)
\(164\) −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 \(0.596706\pi\)
\(168\) −3.90703 −0.301434
\(169\) −6.42023 −0.493863
\(170\) 0 0
\(171\) −0.958939 −0.0733319
\(172\) −57.8241 −4.40905
\(173\) −8.07449 −0.613892 −0.306946 0.951727i \(-0.599307\pi\)
−0.306946 + 0.951727i \(0.599307\pi\)
\(174\) −2.75660 −0.208977
\(175\) 0 0
\(176\) −6.36011 −0.479411
\(177\) −9.54022 −0.717087
\(178\) 43.6376 3.27077
\(179\) 12.4100 0.927567 0.463784 0.885949i \(-0.346492\pi\)
0.463784 + 0.885949i \(0.346492\pi\)
\(180\) 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
\(183\) −6.25340 −0.462264
\(184\) −61.0769 −4.50265
\(185\) 0 0
\(186\) −27.9977 −2.05289
\(187\) 0.818503 0.0598549
\(188\) 64.8989 4.73324
\(189\) −0.393832 −0.0286471
\(190\) 0 0
\(191\) −6.32085 −0.457361 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(192\) 35.7232 2.57810
\(193\) −25.2921 −1.82057 −0.910284 0.413984i \(-0.864137\pi\)
−0.910284 + 0.413984i \(0.864137\pi\)
\(194\) 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 \(0.227449\pi\)
\(198\) −1.08564 −0.0771529
\(199\) −2.64723 −0.187657 −0.0938285 0.995588i \(-0.529911\pi\)
−0.0938285 + 0.995588i \(0.529911\pi\)
\(200\) 0 0
\(201\) 7.42023 0.523382
\(202\) −35.2950 −2.48335
\(203\) 0.393832 0.0276416
\(204\) −11.6361 −0.814688
\(205\) 0 0
\(206\) −13.4240 −0.935297
\(207\) −6.15661 −0.427914
\(208\) 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
\(213\) 5.98533 0.410108
\(214\) 6.45747 0.441423
\(215\) 0 0
\(216\) 9.92054 0.675007
\(217\) 4.00000 0.271538
\(218\) 23.5701 1.59637
\(219\) −3.34192 −0.225826
\(220\) 0 0
\(221\) −5.33107 −0.358607
\(222\) 20.2387 1.35833
\(223\) −12.5504 −0.840440 −0.420220 0.907422i \(-0.638047\pi\)
−0.420220 + 0.907422i \(0.638047\pi\)
\(224\) −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 \(0.513753\pi\)
\(228\) −5.36894 −0.355567
\(229\) 3.28682 0.217199 0.108600 0.994086i \(-0.465363\pi\)
0.108600 + 0.994086i \(0.465363\pi\)
\(230\) 0 0
\(231\) 0.155104 0.0102051
\(232\) −9.92054 −0.651315
\(233\) −17.7115 −1.16032 −0.580159 0.814503i \(-0.697010\pi\)
−0.580159 + 0.814503i \(0.697010\pi\)
\(234\) 7.07097 0.462244
\(235\) 0 0
\(236\) −53.4141 −3.47696
\(237\) −2.06745 −0.134296
\(238\) 2.25628 0.146253
\(239\) 21.2651 1.37553 0.687763 0.725935i \(-0.258593\pi\)
0.687763 + 0.725935i \(0.258593\pi\)
\(240\) 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
\(243\) 1.00000 0.0641500
\(244\) −35.0117 −2.24140
\(245\) 0 0
\(246\) −4.56892 −0.291304
\(247\) −2.45978 −0.156512
\(248\) −100.759 −6.39820
\(249\) −6.41000 −0.406217
\(250\) 0 0
\(251\) 26.8917 1.69739 0.848696 0.528882i \(-0.177388\pi\)
0.848696 + 0.528882i \(0.177388\pi\)
\(252\) −2.20500 −0.138902
\(253\) 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 \(0.431414\pi\)
\(258\) −28.4698 −1.77245
\(259\) −2.89149 −0.179668
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) −15.1678 −0.937067
\(263\) 13.6294 0.840423 0.420212 0.907426i \(-0.361956\pi\)
0.420212 + 0.907426i \(0.361956\pi\)
\(264\) −3.90703 −0.240461
\(265\) 0 0
\(266\) 1.04106 0.0638315
\(267\) 15.8302 0.968794
\(268\) 41.5446 2.53774
\(269\) −8.46129 −0.515894 −0.257947 0.966159i \(-0.583046\pi\)
−0.257947 + 0.966159i \(0.583046\pi\)
\(270\) 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
\(273\) −1.01022 −0.0611414
\(274\) −20.6704 −1.24875
\(275\) 0 0
\(276\) −34.4698 −2.07484
\(277\) 17.1464 1.03023 0.515113 0.857122i \(-0.327750\pi\)
0.515113 + 0.857122i \(0.327750\pi\)
\(278\) −25.7818 −1.54629
\(279\) −10.1566 −0.608060
\(280\) 0 0
\(281\) −12.2985 −0.733670 −0.366835 0.930286i \(-0.619559\pi\)
−0.366835 + 0.930286i \(0.619559\pi\)
\(282\) 31.9531 1.90278
\(283\) −25.1830 −1.49697 −0.748487 0.663149i \(-0.769219\pi\)
−0.748487 + 0.663149i \(0.769219\pi\)
\(284\) 33.5109 1.98851
\(285\) 0 0
\(286\) −2.78478 −0.164667
\(287\) 0.652757 0.0385311
\(288\) 24.6760 1.45405
\(289\) −12.6807 −0.745921
\(290\) 0 0
\(291\) 18.4575 1.08200
\(292\) −18.7109 −1.09497
\(293\) −12.3170 −0.719569 −0.359784 0.933035i \(-0.617150\pi\)
−0.359784 + 0.933035i \(0.617150\pi\)
\(294\) −18.8686 −1.10044
\(295\) 0 0
\(296\) 72.8358 4.23350
\(297\) −0.393832 −0.0228525
\(298\) −31.8117 −1.84280
\(299\) −15.7924 −0.913296
\(300\) 0 0
\(301\) 4.06745 0.234444
\(302\) 16.7660 0.964773
\(303\) −12.8038 −0.735561
\(304\) −15.4862 −0.888193
\(305\) 0 0
\(306\) −5.72905 −0.327508
\(307\) 22.7379 1.29772 0.648860 0.760908i \(-0.275246\pi\)
0.648860 + 0.760908i \(0.275246\pi\)
\(308\) 0.868401 0.0494817
\(309\) −4.86979 −0.277032
\(310\) 0 0
\(311\) −5.33810 −0.302696 −0.151348 0.988481i \(-0.548361\pi\)
−0.151348 + 0.988481i \(0.548361\pi\)
\(312\) 25.4472 1.44067
\(313\) 1.96338 0.110977 0.0554885 0.998459i \(-0.482328\pi\)
0.0554885 + 0.998459i \(0.482328\pi\)
\(314\) 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 \(0.511540\pi\)
\(318\) 33.9830 1.90567
\(319\) 0.393832 0.0220504
\(320\) 0 0
\(321\) 2.34255 0.130748
\(322\) 6.68384 0.372476
\(323\) 1.99297 0.110892
\(324\) 5.59883 0.311046
\(325\) 0 0
\(326\) 2.35722 0.130554
\(327\) 8.55044 0.472840
\(328\) −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
\(333\) 7.34192 0.402335
\(334\) −21.3138 −1.16624
\(335\) 0 0
\(336\) −6.36011 −0.346972
\(337\) 16.7854 0.914356 0.457178 0.889375i \(-0.348860\pi\)
0.457178 + 0.889375i \(0.348860\pi\)
\(338\) −17.6980 −0.962643
\(339\) 11.2085 0.608763
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) −2.64341 −0.142939
\(343\) 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 \(0.340780\pi\)
\(348\) −5.59883 −0.300129
\(349\) 8.73789 0.467728 0.233864 0.972269i \(-0.424863\pi\)
0.233864 + 0.972269i \(0.424863\pi\)
\(350\) 0 0
\(351\) 2.56511 0.136915
\(352\) −9.71820 −0.517982
\(353\) 22.6017 1.20297 0.601484 0.798885i \(-0.294576\pi\)
0.601484 + 0.798885i \(0.294576\pi\)
\(354\) −26.2985 −1.39775
\(355\) 0 0
\(356\) 88.6308 4.69742
\(357\) 0.818503 0.0433198
\(358\) 34.2094 1.80802
\(359\) −13.1830 −0.695772 −0.347886 0.937537i \(-0.613100\pi\)
−0.347886 + 0.937537i \(0.613100\pi\)
\(360\) 0 0
\(361\) −18.0804 −0.951602
\(362\) −6.88503 −0.361869
\(363\) −10.8449 −0.569209
\(364\) −5.65607 −0.296458
\(365\) 0 0
\(366\) −17.2381 −0.901050
\(367\) −17.4511 −0.910938 −0.455469 0.890252i \(-0.650528\pi\)
−0.455469 + 0.890252i \(0.650528\pi\)
\(368\) −99.4247 −5.18287
\(369\) −1.65745 −0.0862834
\(370\) 0 0
\(371\) −4.85512 −0.252065
\(372\) −56.8652 −2.94832
\(373\) 32.2018 1.66734 0.833672 0.552260i \(-0.186234\pi\)
0.833672 + 0.552260i \(0.186234\pi\)
\(374\) 2.25628 0.116670
\(375\) 0 0
\(376\) 114.994 5.93036
\(377\) −2.56511 −0.132110
\(378\) −1.08564 −0.0558392
\(379\) 32.3660 1.66253 0.831265 0.555876i \(-0.187617\pi\)
0.831265 + 0.555876i \(0.187617\pi\)
\(380\) 0 0
\(381\) −20.1566 −1.03265
\(382\) −17.4240 −0.891492
\(383\) 25.8739 1.32209 0.661047 0.750345i \(-0.270113\pi\)
0.661047 + 0.750345i \(0.270113\pi\)
\(384\) 49.1226 2.50678
\(385\) 0 0
\(386\) −69.7203 −3.54867
\(387\) −10.3279 −0.524996
\(388\) 103.340 5.24631
\(389\) 32.6103 1.65341 0.826703 0.562639i \(-0.190214\pi\)
0.826703 + 0.562639i \(0.190214\pi\)
\(390\) 0 0
\(391\) 12.7953 0.647086
\(392\) −67.9051 −3.42972
\(393\) −5.50235 −0.277557
\(394\) 58.4528 2.94481
\(395\) 0 0
\(396\) −2.20500 −0.110806
\(397\) −4.39534 −0.220596 −0.110298 0.993899i \(-0.535180\pi\)
−0.110298 + 0.993899i \(0.535180\pi\)
\(398\) −7.29734 −0.365783
\(399\) 0.377661 0.0189067
\(400\) 0 0
\(401\) −19.0658 −0.952099 −0.476049 0.879418i \(-0.657932\pi\)
−0.476049 + 0.879418i \(0.657932\pi\)
\(402\) 20.4546 1.02018
\(403\) −26.0528 −1.29778
\(404\) −71.6865 −3.56654
\(405\) 0 0
\(406\) 1.08564 0.0538793
\(407\) −2.89149 −0.143326
\(408\) −20.6179 −1.02074
\(409\) −1.38235 −0.0683530 −0.0341765 0.999416i \(-0.510881\pi\)
−0.0341765 + 0.999416i \(0.510881\pi\)
\(410\) 0 0
\(411\) −7.49853 −0.369875
\(412\) −27.2651 −1.34326
\(413\) 3.75725 0.184882
\(414\) −16.9713 −0.834094
\(415\) 0 0
\(416\) 63.2965 3.10337
\(417\) −9.35277 −0.458007
\(418\) 1.04106 0.0509199
\(419\) 2.86979 0.140198 0.0700991 0.997540i \(-0.477668\pi\)
0.0700991 + 0.997540i \(0.477668\pi\)
\(420\) 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
\(423\) 11.5915 0.563598
\(424\) 122.299 5.93938
\(425\) 0 0
\(426\) 16.4992 0.799387
\(427\) 2.46279 0.119183
\(428\) 13.1155 0.633964
\(429\) −1.01022 −0.0487740
\(430\) 0 0
\(431\) −11.9707 −0.576607 −0.288303 0.957539i \(-0.593091\pi\)
−0.288303 + 0.957539i \(0.593091\pi\)
\(432\) 16.1493 0.776982
\(433\) −17.9907 −0.864576 −0.432288 0.901736i \(-0.642294\pi\)
−0.432288 + 0.901736i \(0.642294\pi\)
\(434\) 11.0264 0.529284
\(435\) 0 0
\(436\) 47.8725 2.29268
\(437\) 5.90381 0.282418
\(438\) −9.21234 −0.440182
\(439\) −16.2226 −0.774260 −0.387130 0.922025i \(-0.626534\pi\)
−0.387130 + 0.922025i \(0.626534\pi\)
\(440\) 0 0
\(441\) −6.84490 −0.325947
\(442\) −14.6956 −0.698999
\(443\) 35.3762 1.68078 0.840388 0.541986i \(-0.182327\pi\)
0.840388 + 0.541986i \(0.182327\pi\)
\(444\) 41.1062 1.95081
\(445\) 0 0
\(446\) −34.5965 −1.63819
\(447\) −11.5402 −0.545834
\(448\) −14.0690 −0.664696
\(449\) 41.6971 1.96781 0.983903 0.178702i \(-0.0571898\pi\)
0.983903 + 0.178702i \(0.0571898\pi\)
\(450\) 0 0
\(451\) 0.652757 0.0307371
\(452\) 62.7546 2.95173
\(453\) 6.08212 0.285763
\(454\) −3.58768 −0.168378
\(455\) 0 0
\(456\) −9.51320 −0.445496
\(457\) 16.1111 0.753646 0.376823 0.926285i \(-0.377017\pi\)
0.376823 + 0.926285i \(0.377017\pi\)
\(458\) 9.06045 0.423367
\(459\) −2.07830 −0.0970069
\(460\) 0 0
\(461\) 17.3396 0.807586 0.403793 0.914850i \(-0.367692\pi\)
0.403793 + 0.914850i \(0.367692\pi\)
\(462\) 0.427559 0.0198918
\(463\) −19.3177 −0.897768 −0.448884 0.893590i \(-0.648178\pi\)
−0.448884 + 0.893590i \(0.648178\pi\)
\(464\) −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 \(0.440740\pi\)
\(468\) 14.3616 0.663866
\(469\) −2.92232 −0.134940
\(470\) 0 0
\(471\) 11.5953 0.534284
\(472\) −94.6441 −4.35635
\(473\) 4.06745 0.187022
\(474\) −5.69914 −0.261770
\(475\) 0 0
\(476\) 4.58266 0.210046
\(477\) 12.3279 0.564455
\(478\) 58.6194 2.68119
\(479\) −40.3877 −1.84536 −0.922681 0.385565i \(-0.874006\pi\)
−0.922681 + 0.385565i \(0.874006\pi\)
\(480\) 0 0
\(481\) 18.8328 0.858702
\(482\) 42.2246 1.92328
\(483\) 2.42467 0.110326
\(484\) −60.7188 −2.75994
\(485\) 0 0
\(486\) 2.75660 0.125042
\(487\) −2.84171 −0.128770 −0.0643850 0.997925i \(-0.520509\pi\)
−0.0643850 + 0.997925i \(0.520509\pi\)
\(488\) −62.0371 −2.80829
\(489\) 0.855118 0.0386698
\(490\) 0 0
\(491\) 0.157863 0.00712428 0.00356214 0.999994i \(-0.498866\pi\)
0.00356214 + 0.999994i \(0.498866\pi\)
\(492\) −9.27979 −0.418365
\(493\) 2.07830 0.0936021
\(494\) −6.78063 −0.305075
\(495\) 0 0
\(496\) −164.022 −7.36480
\(497\) −2.35722 −0.105736
\(498\) −17.6698 −0.791803
\(499\) 2.64723 0.118506 0.0592531 0.998243i \(-0.481128\pi\)
0.0592531 + 0.998243i \(0.481128\pi\)
\(500\) 0 0
\(501\) −7.73194 −0.345437
\(502\) 74.1297 3.30857
\(503\) −13.9545 −0.622200 −0.311100 0.950377i \(-0.600697\pi\)
−0.311100 + 0.950377i \(0.600697\pi\)
\(504\) −3.90703 −0.174033
\(505\) 0 0
\(506\) 6.68384 0.297133
\(507\) −6.42023 −0.285132
\(508\) −112.853 −5.00706
\(509\) −16.4921 −0.731001 −0.365500 0.930811i \(-0.619102\pi\)
−0.365500 + 0.930811i \(0.619102\pi\)
\(510\) 0 0
\(511\) 1.31616 0.0582233
\(512\) 78.0802 3.45069
\(513\) −0.958939 −0.0423382
\(514\) 18.8968 0.833502
\(515\) 0 0
\(516\) −57.8241 −2.54556
\(517\) −4.56511 −0.200773
\(518\) −7.97066 −0.350211
\(519\) −8.07449 −0.354431
\(520\) 0 0
\(521\) −12.2253 −0.535601 −0.267800 0.963474i \(-0.586297\pi\)
−0.267800 + 0.963474i \(0.586297\pi\)
\(522\) −2.75660 −0.120653
\(523\) −1.66659 −0.0728748 −0.0364374 0.999336i \(-0.511601\pi\)
−0.0364374 + 0.999336i \(0.511601\pi\)
\(524\) −30.8067 −1.34580
\(525\) 0 0
\(526\) 37.5707 1.63816
\(527\) 21.1085 0.919501
\(528\) −6.36011 −0.276788
\(529\) 14.9038 0.647992
\(530\) 0 0
\(531\) −9.54022 −0.414010
\(532\) 2.11446 0.0916736
\(533\) −4.25154 −0.184155
\(534\) 43.6376 1.88838
\(535\) 0 0
\(536\) 73.6126 3.17958
\(537\) 12.4100 0.535531
\(538\) −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
\(543\) −2.49765 −0.107185
\(544\) −51.2842 −2.19879
\(545\) 0 0
\(546\) −2.78478 −0.119177
\(547\) −38.2530 −1.63558 −0.817791 0.575515i \(-0.804801\pi\)
−0.817791 + 0.575515i \(0.804801\pi\)
\(548\) −41.9830 −1.79343
\(549\) −6.25340 −0.266888
\(550\) 0 0
\(551\) 0.958939 0.0408522
\(552\) −61.0769 −2.59960
\(553\) 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 \(0.291314\pi\)
\(558\) −27.9977 −1.18524
\(559\) −26.4921 −1.12050
\(560\) 0 0
\(561\) 0.818503 0.0345572
\(562\) −33.9022 −1.43008
\(563\) 32.9809 1.38998 0.694989 0.719020i \(-0.255409\pi\)
0.694989 + 0.719020i \(0.255409\pi\)
\(564\) 64.8989 2.73274
\(565\) 0 0
\(566\) −69.4194 −2.91792
\(567\) −0.393832 −0.0165394
\(568\) 59.3777 2.49143
\(569\) 16.8038 0.704453 0.352227 0.935915i \(-0.385425\pi\)
0.352227 + 0.935915i \(0.385425\pi\)
\(570\) 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
\(573\) −6.32085 −0.264057
\(574\) 1.79939 0.0751051
\(575\) 0 0
\(576\) 35.7232 1.48847
\(577\) −11.9977 −0.499470 −0.249735 0.968314i \(-0.580344\pi\)
−0.249735 + 0.968314i \(0.580344\pi\)
\(578\) −34.9555 −1.45396
\(579\) −25.2921 −1.05111
\(580\) 0 0
\(581\) 2.52447 0.104733
\(582\) 50.8798 2.10904
\(583\) −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 \(0.425625\pi\)
\(588\) −38.3234 −1.58043
\(589\) 9.73957 0.401312
\(590\) 0 0
\(591\) 21.2047 0.872245
\(592\) 118.567 4.87306
\(593\) 7.69682 0.316071 0.158035 0.987433i \(-0.449484\pi\)
0.158035 + 0.987433i \(0.449484\pi\)
\(594\) −1.08564 −0.0445442
\(595\) 0 0
\(596\) −64.6118 −2.64660
\(597\) −2.64723 −0.108344
\(598\) −43.5332 −1.78020
\(599\) 20.0543 0.819396 0.409698 0.912221i \(-0.365634\pi\)
0.409698 + 0.912221i \(0.365634\pi\)
\(600\) 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
\(603\) 7.42023 0.302175
\(604\) 34.0528 1.38559
\(605\) 0 0
\(606\) −35.2950 −1.43376
\(607\) −37.0441 −1.50357 −0.751786 0.659407i \(-0.770807\pi\)
−0.751786 + 0.659407i \(0.770807\pi\)
\(608\) −23.6628 −0.959652
\(609\) 0.393832 0.0159589
\(610\) 0 0
\(611\) 29.7334 1.20289
\(612\) −11.6361 −0.470361
\(613\) 13.3839 0.540569 0.270284 0.962781i \(-0.412882\pi\)
0.270284 + 0.962781i \(0.412882\pi\)
\(614\) 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 \(0.418135\pi\)
\(618\) −13.4240 −0.539994
\(619\) −41.2447 −1.65776 −0.828882 0.559424i \(-0.811022\pi\)
−0.828882 + 0.559424i \(0.811022\pi\)
\(620\) 0 0
\(621\) −6.15661 −0.247056
\(622\) −14.7150 −0.590018
\(623\) −6.23446 −0.249778
\(624\) 41.4246 1.65831
\(625\) 0 0
\(626\) 5.41226 0.216318
\(627\) 0.377661 0.0150823
\(628\) 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
\(633\) 2.00000 0.0794929
\(634\) −3.55777 −0.141297
\(635\) 0 0
\(636\) 69.0218 2.73689
\(637\) −17.5579 −0.695669
\(638\) 1.08564 0.0429808
\(639\) 5.98533 0.236776
\(640\) 0 0
\(641\) 17.7774 0.702167 0.351083 0.936344i \(-0.385813\pi\)
0.351083 + 0.936344i \(0.385813\pi\)
\(642\) 6.45747 0.254856
\(643\) 44.5534 1.75702 0.878508 0.477727i \(-0.158539\pi\)
0.878508 + 0.477727i \(0.158539\pi\)
\(644\) 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 \(0.819777\pi\)
\(648\) 9.92054 0.389716
\(649\) 3.75725 0.147485
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 4.78766 0.187499
\(653\) 19.8426 0.776500 0.388250 0.921554i \(-0.373080\pi\)
0.388250 + 0.921554i \(0.373080\pi\)
\(654\) 23.5701 0.921665
\(655\) 0 0
\(656\) −26.7666 −1.04506
\(657\) −3.34192 −0.130381
\(658\) −12.5842 −0.490582
\(659\) 23.0483 0.897836 0.448918 0.893573i \(-0.351810\pi\)
0.448918 + 0.893573i \(0.351810\pi\)
\(660\) 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
\(663\) −5.33107 −0.207042
\(664\) −63.5907 −2.46780
\(665\) 0 0
\(666\) 20.2387 0.784235
\(667\) 6.15661 0.238385
\(668\) −43.2898 −1.67493
\(669\) −12.5504 −0.485228
\(670\) 0 0
\(671\) 2.46279 0.0950749
\(672\) −9.71820 −0.374888
\(673\) −22.9000 −0.882731 −0.441365 0.897327i \(-0.645506\pi\)
−0.441365 + 0.897327i \(0.645506\pi\)
\(674\) 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 \(0.743238\pi\)
\(678\) 30.8974 1.18661
\(679\) −7.26915 −0.278964
\(680\) 0 0
\(681\) −1.30149 −0.0498732
\(682\) 11.0264 0.422222
\(683\) −9.12896 −0.349310 −0.174655 0.984630i \(-0.555881\pi\)
−0.174655 + 0.984630i \(0.555881\pi\)
\(684\) −5.36894 −0.205287
\(685\) 0 0
\(686\) 15.0305 0.573869
\(687\) 3.28682 0.125400
\(688\) −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
\(693\) 0.155104 0.00589191
\(694\) 49.2552 1.86970
\(695\) 0 0
\(696\) −9.92054 −0.376037
\(697\) 3.44469 0.130477
\(698\) 24.0868 0.911700
\(699\) −17.7115 −0.669910
\(700\) 0 0
\(701\) −7.56955 −0.285898 −0.142949 0.989730i \(-0.545658\pi\)
−0.142949 + 0.989730i \(0.545658\pi\)
\(702\) 7.07097 0.266877
\(703\) −7.04046 −0.265536
\(704\) −14.0690 −0.530244
\(705\) 0 0
\(706\) 62.3039 2.34484
\(707\) 5.04256 0.189645
\(708\) −53.4141 −2.00742
\(709\) 14.5209 0.545342 0.272671 0.962107i \(-0.412093\pi\)
0.272671 + 0.962107i \(0.412093\pi\)
\(710\) 0 0
\(711\) −2.06745 −0.0775356
\(712\) 157.044 5.88549
\(713\) 62.5302 2.34178
\(714\) 2.25628 0.0844393
\(715\) 0 0
\(716\) 69.4815 2.59665
\(717\) 21.2651 0.794161
\(718\) −36.3402 −1.35621
\(719\) 12.2164 0.455596 0.227798 0.973708i \(-0.426847\pi\)
0.227798 + 0.973708i \(0.426847\pi\)
\(720\) 0 0
\(721\) 1.91788 0.0714255
\(722\) −49.8405 −1.85487
\(723\) 15.3177 0.569670
\(724\) −13.9839 −0.519709
\(725\) 0 0
\(726\) −29.8950 −1.10951
\(727\) 40.8856 1.51636 0.758182 0.652044i \(-0.226088\pi\)
0.758182 + 0.652044i \(0.226088\pi\)
\(728\) −10.0219 −0.371438
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 21.4645 0.793892
\(732\) −35.0117 −1.29407
\(733\) −25.4915 −0.941550 −0.470775 0.882253i \(-0.656026\pi\)
−0.470775 + 0.882253i \(0.656026\pi\)
\(734\) −48.1056 −1.77561
\(735\) 0 0
\(736\) −151.920 −5.59986
\(737\) −2.92232 −0.107645
\(738\) −4.56892 −0.168184
\(739\) 9.56830 0.351975 0.175988 0.984392i \(-0.443688\pi\)
0.175988 + 0.984392i \(0.443688\pi\)
\(740\) 0 0
\(741\) −2.45978 −0.0903624
\(742\) −13.3836 −0.491328
\(743\) 18.0455 0.662025 0.331013 0.943626i \(-0.392610\pi\)
0.331013 + 0.943626i \(0.392610\pi\)
\(744\) −100.759 −3.69400
\(745\) 0 0
\(746\) 88.7673 3.25000
\(747\) −6.41000 −0.234530
\(748\) 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
\(753\) 26.8917 0.979989
\(754\) −7.07097 −0.257510
\(755\) 0 0
\(756\) −2.20500 −0.0801951
\(757\) −5.86152 −0.213041 −0.106520 0.994311i \(-0.533971\pi\)
−0.106520 + 0.994311i \(0.533971\pi\)
\(758\) 89.2201 3.24062
\(759\) 2.42467 0.0880100
\(760\) 0 0
\(761\) 49.8739 1.80793 0.903963 0.427610i \(-0.140644\pi\)
0.903963 + 0.427610i \(0.140644\pi\)
\(762\) −55.5637 −2.01286
\(763\) −3.36744 −0.121909
\(764\) −35.3894 −1.28034
\(765\) 0 0
\(766\) 71.3239 2.57704
\(767\) −24.4717 −0.883621
\(768\) 63.9648 2.30813
\(769\) 29.5121 1.06423 0.532117 0.846671i \(-0.321396\pi\)
0.532117 + 0.846671i \(0.321396\pi\)
\(770\) 0 0
\(771\) 6.85512 0.246881
\(772\) −141.607 −5.09653
\(773\) 2.41935 0.0870180 0.0435090 0.999053i \(-0.486146\pi\)
0.0435090 + 0.999053i \(0.486146\pi\)
\(774\) −28.4698 −1.02333
\(775\) 0 0
\(776\) 183.108 6.57320
\(777\) −2.89149 −0.103731
\(778\) 89.8934 3.22283
\(779\) 1.58939 0.0569460
\(780\) 0 0
\(781\) −2.35722 −0.0843479
\(782\) 35.2715 1.26131
\(783\) −1.00000 −0.0357371
\(784\) −110.540 −3.94786
\(785\) 0 0
\(786\) −15.1678 −0.541016
\(787\) 51.1549 1.82348 0.911738 0.410772i \(-0.134741\pi\)
0.911738 + 0.410772i \(0.134741\pi\)
\(788\) 118.722 4.22928
\(789\) 13.6294 0.485218
\(790\) 0 0
\(791\) −4.41428 −0.156954
\(792\) −3.90703 −0.138830
\(793\) −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 \(0.670816\pi\)
\(798\) 1.04106 0.0368531
\(799\) −24.0907 −0.852266
\(800\) 0 0
\(801\) 15.8302 0.559334
\(802\) −52.5567 −1.85584
\(803\) 1.31616 0.0464462
\(804\) 41.5446 1.46517
\(805\) 0 0
\(806\) −71.8171 −2.52965
\(807\) −8.46129 −0.297851
\(808\) −127.021 −4.46858
\(809\) 19.0426 0.669501 0.334750 0.942307i \(-0.391348\pi\)
0.334750 + 0.942307i \(0.391348\pi\)
\(810\) 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
\(813\) 6.34255 0.222443
\(814\) −7.97066 −0.279372
\(815\) 0 0
\(816\) −33.5631 −1.17494
\(817\) 9.90381 0.346491
\(818\) −3.81060 −0.133234
\(819\) −1.01022 −0.0353000
\(820\) 0 0
\(821\) 3.60466 0.125804 0.0629018 0.998020i \(-0.479965\pi\)
0.0629018 + 0.998020i \(0.479965\pi\)
\(822\) −20.6704 −0.720964
\(823\) −27.2288 −0.949135 −0.474567 0.880219i \(-0.657395\pi\)
−0.474567 + 0.880219i \(0.657395\pi\)
\(824\) −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 \(0.808273\pi\)
\(828\) −34.4698 −1.19791
\(829\) −41.8388 −1.45312 −0.726560 0.687103i \(-0.758882\pi\)
−0.726560 + 0.687103i \(0.758882\pi\)
\(830\) 0 0
\(831\) 17.1464 0.594802
\(832\) 91.6339 3.17683
\(833\) 14.2258 0.492894
\(834\) −25.7818 −0.892752
\(835\) 0 0
\(836\) 2.11446 0.0731302
\(837\) −10.1566 −0.351064
\(838\) 7.91085 0.273276
\(839\) −31.6385 −1.09228 −0.546141 0.837693i \(-0.683904\pi\)
−0.546141 + 0.837693i \(0.683904\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 37.6111 1.29617
\(843\) −12.2985 −0.423584
\(844\) 11.1977 0.385440
\(845\) 0 0
\(846\) 31.9531 1.09857
\(847\) 4.27107 0.146756
\(848\) 199.086 6.83665
\(849\) −25.1830 −0.864278
\(850\) 0 0
\(851\) −45.2013 −1.54948
\(852\) 33.5109 1.14806
\(853\) −44.2000 −1.51338 −0.756690 0.653773i \(-0.773185\pi\)
−0.756690 + 0.653773i \(0.773185\pi\)
\(854\) 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 \(0.579534\pi\)
\(858\) −2.78478 −0.0950707
\(859\) 22.5883 0.770703 0.385352 0.922770i \(-0.374080\pi\)
0.385352 + 0.922770i \(0.374080\pi\)
\(860\) 0 0
\(861\) 0.652757 0.0222459
\(862\) −32.9983 −1.12393
\(863\) 22.9900 0.782590 0.391295 0.920265i \(-0.372027\pi\)
0.391295 + 0.920265i \(0.372027\pi\)
\(864\) 24.6760 0.839494
\(865\) 0 0
\(866\) −49.5930 −1.68524
\(867\) −12.6807 −0.430658
\(868\) 22.3953 0.760147
\(869\) 0.814230 0.0276209
\(870\) 0 0
\(871\) 19.0337 0.644931
\(872\) 84.8250 2.87254
\(873\) 18.4575 0.624691
\(874\) 16.2744 0.550491
\(875\) 0 0
\(876\) −18.7109 −0.632182
\(877\) −13.5741 −0.458364 −0.229182 0.973384i \(-0.573605\pi\)
−0.229182 + 0.973384i \(0.573605\pi\)
\(878\) −44.7191 −1.50920
\(879\) −12.3170 −0.415443
\(880\) 0 0
\(881\) 9.91235 0.333956 0.166978 0.985961i \(-0.446599\pi\)
0.166978 + 0.985961i \(0.446599\pi\)
\(882\) −18.8686 −0.635340
\(883\) −39.3192 −1.32320 −0.661598 0.749859i \(-0.730121\pi\)
−0.661598 + 0.749859i \(0.730121\pi\)
\(884\) −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 \(0.0326531\pi\)
\(888\) 72.8358 2.44421
\(889\) 7.93832 0.266243
\(890\) 0 0
\(891\) −0.393832 −0.0131939
\(892\) −70.2678 −2.35274
\(893\) −11.1155 −0.371968
\(894\) −31.8117 −1.06394
\(895\) 0 0
\(896\) −19.3461 −0.646307
\(897\) −15.7924 −0.527291
\(898\) 114.942 3.83567
\(899\) 10.1566 0.338742
\(900\) 0 0
\(901\) −25.6211 −0.853562
\(902\) 1.79939 0.0599131
\(903\) 4.06745 0.135356
\(904\) 111.195 3.69828
\(905\) 0 0
\(906\) 16.7660 0.557012
\(907\) 5.91210 0.196308 0.0981541 0.995171i \(-0.468706\pi\)
0.0981541 + 0.995171i \(0.468706\pi\)
\(908\) −7.28682 −0.241822
\(909\) −12.8038 −0.424676
\(910\) 0 0
\(911\) −12.4185 −0.411445 −0.205722 0.978610i \(-0.565954\pi\)
−0.205722 + 0.978610i \(0.565954\pi\)
\(912\) −15.4862 −0.512799
\(913\) 2.52447 0.0835476
\(914\) 44.4118 1.46901
\(915\) 0 0
\(916\) 18.4024 0.608031
\(917\) 2.16700 0.0715607
\(918\) −5.72905 −0.189087
\(919\) 16.4332 0.542081 0.271041 0.962568i \(-0.412632\pi\)
0.271041 + 0.962568i \(0.412632\pi\)
\(920\) 0 0
\(921\) 22.7379 0.749239
\(922\) 47.7983 1.57415
\(923\) 15.3530 0.505351
\(924\) 0.868401 0.0285683
\(925\) 0 0
\(926\) −53.2510 −1.74994
\(927\) −4.86979 −0.159945
\(928\) −24.6760 −0.810029
\(929\) 13.8358 0.453936 0.226968 0.973902i \(-0.427119\pi\)
0.226968 + 0.973902i \(0.427119\pi\)
\(930\) 0 0
\(931\) 6.56384 0.215121
\(932\) −99.1637 −3.24822
\(933\) −5.33810 −0.174762
\(934\) 22.0528 0.721589
\(935\) 0 0
\(936\) 25.4472 0.831769
\(937\) −20.7528 −0.677964 −0.338982 0.940793i \(-0.610083\pi\)
−0.338982 + 0.940793i \(0.610083\pi\)
\(938\) −8.05567 −0.263027
\(939\) 1.96338 0.0640726
\(940\) 0 0
\(941\) 40.1666 1.30940 0.654698 0.755891i \(-0.272796\pi\)
0.654698 + 0.755891i \(0.272796\pi\)
\(942\) 31.9636 1.04143
\(943\) 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 \(0.408688\pi\)
\(948\) −11.5753 −0.375949
\(949\) −8.57239 −0.278271
\(950\) 0 0
\(951\) −1.29064 −0.0418518
\(952\) 8.11999 0.263170
\(953\) −13.5121 −0.437701 −0.218851 0.975758i \(-0.570231\pi\)
−0.218851 + 0.975758i \(0.570231\pi\)
\(954\) 33.9830 1.10024
\(955\) 0 0
\(956\) 119.060 3.85067
\(957\) 0.393832 0.0127308
\(958\) −111.333 −3.59700
\(959\) 2.95316 0.0953626
\(960\) 0 0
\(961\) 72.1567 2.32763
\(962\) 51.9145 1.67379
\(963\) 2.34255 0.0754876
\(964\) 85.7610 2.76218
\(965\) 0 0
\(966\) 6.68384 0.215049
\(967\) 24.0598 0.773712 0.386856 0.922140i \(-0.373561\pi\)
0.386856 + 0.922140i \(0.373561\pi\)
\(968\) −107.587 −3.45798
\(969\) 1.99297 0.0640233
\(970\) 0 0
\(971\) 34.7877 1.11639 0.558195 0.829710i \(-0.311494\pi\)
0.558195 + 0.829710i \(0.311494\pi\)
\(972\) 5.59883 0.179583
\(973\) 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 \(0.693824\pi\)
\(978\) 2.35722 0.0753755
\(979\) −6.23446 −0.199254
\(980\) 0 0
\(981\) 8.55044 0.272995
\(982\) 0.435166 0.0138867
\(983\) 7.19064 0.229346 0.114673 0.993403i \(-0.463418\pi\)
0.114673 + 0.993403i \(0.463418\pi\)
\(984\) −16.4428 −0.524177
\(985\) 0 0
\(986\) 5.72905 0.182450
\(987\) −4.56511 −0.145309
\(988\) −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
\(993\) 28.9971 0.920194
\(994\) −6.49790 −0.206101
\(995\) 0 0
\(996\) −35.8885 −1.13717
\(997\) 1.57302 0.0498179 0.0249089 0.999690i \(-0.492070\pi\)
0.0249089 + 0.999690i \(0.492070\pi\)
\(998\) 7.29734 0.230993
\(999\) 7.34192 0.232288
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 2175.2.a.v.1.4 4
3.2 odd 2 6525.2.a.bi.1.1 4
5.2 odd 4 2175.2.c.n.349.8 8
5.3 odd 4 2175.2.c.n.349.1 8
5.4 even 2 435.2.a.j.1.1 4
15.14 odd 2 1305.2.a.r.1.4 4
20.19 odd 2 6960.2.a.co.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.1 4 5.4 even 2
1305.2.a.r.1.4 4 15.14 odd 2
2175.2.a.v.1.4 4 1.1 even 1 trivial
2175.2.c.n.349.1 8 5.3 odd 4
2175.2.c.n.349.8 8 5.2 odd 4
6525.2.a.bi.1.1 4 3.2 odd 2
6960.2.a.co.1.2 4 20.19 odd 2