Properties

Label 2175.2.c.n.349.1
Level $2175$
Weight $2$
Character 2175.349
Analytic conductor $17.367$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,2,Mod(349,2175)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1267360000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 37x^{4} + 44x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 435)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.1
Root \(-1.75660i\) of defining polynomial
Character \(\chi\) \(=\) 2175.349
Dual form 2175.2.c.n.349.8

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.75660i − 1.94921i −0.223932 0.974605i \(-0.571890\pi\)
0.223932 0.974605i \(-0.428110\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −5.59883 −2.79942
\(5\) 0 0
\(6\) 2.75660 1.12538
\(7\) 0.393832i 0.148855i 0.997226 + 0.0744273i \(0.0237129\pi\)
−0.997226 + 0.0744273i \(0.976287\pi\)
\(8\) 9.92054i 3.50744i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −0.393832 −0.118745 −0.0593725 0.998236i \(-0.518910\pi\)
−0.0593725 + 0.998236i \(0.518910\pi\)
\(12\) − 5.59883i − 1.61624i
\(13\) 2.56511i 0.711433i 0.934594 + 0.355716i \(0.115763\pi\)
−0.934594 + 0.355716i \(0.884237\pi\)
\(14\) 1.08564 0.290149
\(15\) 0 0
\(16\) 16.1493 4.03732
\(17\) 2.07830i 0.504063i 0.967719 + 0.252031i \(0.0810986\pi\)
−0.967719 + 0.252031i \(0.918901\pi\)
\(18\) 2.75660i 0.649736i
\(19\) 0.958939 0.219996 0.109998 0.993932i \(-0.464916\pi\)
0.109998 + 0.993932i \(0.464916\pi\)
\(20\) 0 0
\(21\) −0.393832 −0.0859413
\(22\) 1.08564i 0.231459i
\(23\) − 6.15661i − 1.28374i −0.766813 0.641871i \(-0.778159\pi\)
0.766813 0.641871i \(-0.221841\pi\)
\(24\) −9.92054 −2.02502
\(25\) 0 0
\(26\) 7.07097 1.38673
\(27\) − 1.00000i − 0.192450i
\(28\) − 2.20500i − 0.416706i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −10.1566 −1.82418 −0.912090 0.409989i \(-0.865532\pi\)
−0.912090 + 0.409989i \(0.865532\pi\)
\(32\) − 24.6760i − 4.36214i
\(33\) − 0.393832i − 0.0685574i
\(34\) 5.72905 0.982524
\(35\) 0 0
\(36\) 5.59883 0.933139
\(37\) − 7.34192i − 1.20700i −0.797361 0.603502i \(-0.793771\pi\)
0.797361 0.603502i \(-0.206229\pi\)
\(38\) − 2.64341i − 0.428818i
\(39\) −2.56511 −0.410746
\(40\) 0 0
\(41\) −1.65745 −0.258850 −0.129425 0.991589i \(-0.541313\pi\)
−0.129425 + 0.991589i \(0.541313\pi\)
\(42\) 1.08564i 0.167517i
\(43\) − 10.3279i − 1.57499i −0.616323 0.787494i \(-0.711378\pi\)
0.616323 0.787494i \(-0.288622\pi\)
\(44\) 2.20500 0.332417
\(45\) 0 0
\(46\) −16.9713 −2.50228
\(47\) − 11.5915i − 1.69079i −0.534138 0.845397i \(-0.679364\pi\)
0.534138 0.845397i \(-0.320636\pi\)
\(48\) 16.1493i 2.33095i
\(49\) 6.84490 0.977842
\(50\) 0 0
\(51\) −2.07830 −0.291021
\(52\) − 14.3616i − 1.99160i
\(53\) 12.3279i 1.69336i 0.532099 + 0.846682i \(0.321404\pi\)
−0.532099 + 0.846682i \(0.678596\pi\)
\(54\) −2.75660 −0.375126
\(55\) 0 0
\(56\) −3.90703 −0.522099
\(57\) 0.958939i 0.127015i
\(58\) − 2.75660i − 0.361959i
\(59\) 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 \(0.631105\pi\)
\(62\) 27.9977i 3.55571i
\(63\) − 0.393832i − 0.0496182i
\(64\) −35.7232 −4.46540
\(65\) 0 0
\(66\) −1.08564 −0.133633
\(67\) − 7.42023i − 0.906525i −0.891377 0.453262i \(-0.850260\pi\)
0.891377 0.453262i \(-0.149740\pi\)
\(68\) − 11.6361i − 1.41108i
\(69\) 6.15661 0.741168
\(70\) 0 0
\(71\) 5.98533 0.710328 0.355164 0.934804i \(-0.384425\pi\)
0.355164 + 0.934804i \(0.384425\pi\)
\(72\) − 9.92054i − 1.16915i
\(73\) − 3.34192i − 0.391142i −0.980690 0.195571i \(-0.937344\pi\)
0.980690 0.195571i \(-0.0626561\pi\)
\(74\) −20.2387 −2.35270
\(75\) 0 0
\(76\) −5.36894 −0.615860
\(77\) − 0.155104i − 0.0176757i
\(78\) 7.07097i 0.800630i
\(79\) 2.06745 0.232607 0.116303 0.993214i \(-0.462896\pi\)
0.116303 + 0.993214i \(0.462896\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 4.56892i 0.504553i
\(83\) − 6.41000i − 0.703589i −0.936077 0.351795i \(-0.885572\pi\)
0.936077 0.351795i \(-0.114428\pi\)
\(84\) 2.20500 0.240585
\(85\) 0 0
\(86\) −28.4698 −3.06998
\(87\) 1.00000i 0.107211i
\(88\) − 3.90703i − 0.416491i
\(89\) −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
\(92\) 34.4698i 3.59373i
\(93\) − 10.1566i − 1.05319i
\(94\) −31.9531 −3.29571
\(95\) 0 0
\(96\) 24.6760 2.51848
\(97\) − 18.4575i − 1.87407i −0.349233 0.937036i \(-0.613558\pi\)
0.349233 0.937036i \(-0.386442\pi\)
\(98\) − 18.8686i − 1.90602i
\(99\) 0.393832 0.0395816
\(100\) 0 0
\(101\) −12.8038 −1.27403 −0.637015 0.770852i \(-0.719831\pi\)
−0.637015 + 0.770852i \(0.719831\pi\)
\(102\) 5.72905i 0.567260i
\(103\) − 4.86979i − 0.479834i −0.970793 0.239917i \(-0.922880\pi\)
0.970793 0.239917i \(-0.0771203\pi\)
\(104\) −25.4472 −2.49531
\(105\) 0 0
\(106\) 33.9830 3.30072
\(107\) − 2.34255i − 0.226463i −0.993569 0.113231i \(-0.963880\pi\)
0.993569 0.113231i \(-0.0361201\pi\)
\(108\) 5.59883i 0.538748i
\(109\) −8.55044 −0.818984 −0.409492 0.912314i \(-0.634294\pi\)
−0.409492 + 0.912314i \(0.634294\pi\)
\(110\) 0 0
\(111\) 7.34192 0.696864
\(112\) 6.36011i 0.600973i
\(113\) 11.2085i 1.05441i 0.849739 + 0.527204i \(0.176760\pi\)
−0.849739 + 0.527204i \(0.823240\pi\)
\(114\) 2.64341 0.247578
\(115\) 0 0
\(116\) −5.59883 −0.519839
\(117\) − 2.56511i − 0.237144i
\(118\) − 26.2985i − 2.42098i
\(119\) −0.818503 −0.0750321
\(120\) 0 0
\(121\) −10.8449 −0.985900
\(122\) 17.2381i 1.56066i
\(123\) − 1.65745i − 0.149447i
\(124\) 56.8652 5.10664
\(125\) 0 0
\(126\) −1.08564 −0.0967163
\(127\) 20.1566i 1.78861i 0.447458 + 0.894305i \(0.352329\pi\)
−0.447458 + 0.894305i \(0.647671\pi\)
\(128\) 49.1226i 4.34187i
\(129\) 10.3279 0.909319
\(130\) 0 0
\(131\) −5.50235 −0.480742 −0.240371 0.970681i \(-0.577269\pi\)
−0.240371 + 0.970681i \(0.577269\pi\)
\(132\) 2.20500i 0.191921i
\(133\) 0.377661i 0.0327474i
\(134\) −20.4546 −1.76701
\(135\) 0 0
\(136\) −20.6179 −1.76797
\(137\) 7.49853i 0.640643i 0.947309 + 0.320321i \(0.103791\pi\)
−0.947309 + 0.320321i \(0.896209\pi\)
\(138\) − 16.9713i − 1.44469i
\(139\) 9.35277 0.793292 0.396646 0.917972i \(-0.370174\pi\)
0.396646 + 0.917972i \(0.370174\pi\)
\(140\) 0 0
\(141\) 11.5915 0.976180
\(142\) − 16.4992i − 1.38458i
\(143\) − 1.01022i − 0.0844790i
\(144\) −16.1493 −1.34577
\(145\) 0 0
\(146\) −9.21234 −0.762418
\(147\) 6.84490i 0.564558i
\(148\) 41.1062i 3.37891i
\(149\) 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 \(0.420398\pi\)
\(152\) 9.51320i 0.771622i
\(153\) − 2.07830i − 0.168021i
\(154\) −0.427559 −0.0344537
\(155\) 0 0
\(156\) 14.3616 1.14985
\(157\) − 11.5953i − 0.925407i −0.886513 0.462704i \(-0.846879\pi\)
0.886513 0.462704i \(-0.153121\pi\)
\(158\) − 5.69914i − 0.453399i
\(159\) −12.3279 −0.977665
\(160\) 0 0
\(161\) 2.42467 0.191091
\(162\) − 2.75660i − 0.216579i
\(163\) 0.855118i 0.0669780i 0.999439 + 0.0334890i \(0.0106619\pi\)
−0.999439 + 0.0334890i \(0.989338\pi\)
\(164\) 9.27979 0.724630
\(165\) 0 0
\(166\) −17.6698 −1.37144
\(167\) 7.73194i 0.598315i 0.954204 + 0.299158i \(0.0967056\pi\)
−0.954204 + 0.299158i \(0.903294\pi\)
\(168\) − 3.90703i − 0.301434i
\(169\) 6.42023 0.493863
\(170\) 0 0
\(171\) −0.958939 −0.0733319
\(172\) 57.8241i 4.40905i
\(173\) − 8.07449i − 0.613892i −0.951727 0.306946i \(-0.900693\pi\)
0.951727 0.306946i \(-0.0993071\pi\)
\(174\) 2.75660 0.208977
\(175\) 0 0
\(176\) −6.36011 −0.479411
\(177\) 9.54022i 0.717087i
\(178\) 43.6376i 3.27077i
\(179\) −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 \(0.529590\pi\)
\(182\) 2.78478i 0.206421i
\(183\) − 6.25340i − 0.462264i
\(184\) 61.0769 4.50265
\(185\) 0 0
\(186\) −27.9977 −2.05289
\(187\) − 0.818503i − 0.0598549i
\(188\) 64.8989i 4.73324i
\(189\) 0.393832 0.0286471
\(190\) 0 0
\(191\) −6.32085 −0.457361 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(192\) − 35.7232i − 2.57810i
\(193\) − 25.2921i − 1.82057i −0.413984 0.910284i \(-0.635863\pi\)
0.413984 0.910284i \(-0.364137\pi\)
\(194\) −50.8798 −3.65296
\(195\) 0 0
\(196\) −38.3234 −2.73739
\(197\) − 21.2047i − 1.51077i −0.655280 0.755386i \(-0.727449\pi\)
0.655280 0.755386i \(-0.272551\pi\)
\(198\) − 1.08564i − 0.0771529i
\(199\) 2.64723 0.187657 0.0938285 0.995588i \(-0.470089\pi\)
0.0938285 + 0.995588i \(0.470089\pi\)
\(200\) 0 0
\(201\) 7.42023 0.523382
\(202\) 35.2950i 2.48335i
\(203\) 0.393832i 0.0276416i
\(204\) 11.6361 0.814688
\(205\) 0 0
\(206\) −13.4240 −0.935297
\(207\) 6.15661i 0.427914i
\(208\) 41.4246i 2.87228i
\(209\) −0.377661 −0.0261234
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) − 69.0218i − 4.74043i
\(213\) 5.98533i 0.410108i
\(214\) −6.45747 −0.441423
\(215\) 0 0
\(216\) 9.92054 0.675007
\(217\) − 4.00000i − 0.271538i
\(218\) 23.5701i 1.59637i
\(219\) 3.34192 0.225826
\(220\) 0 0
\(221\) −5.33107 −0.358607
\(222\) − 20.2387i − 1.35833i
\(223\) − 12.5504i − 0.840440i −0.907422 0.420220i \(-0.861953\pi\)
0.907422 0.420220i \(-0.138047\pi\)
\(224\) 9.71820 0.649324
\(225\) 0 0
\(226\) 30.8974 2.05526
\(227\) 1.30149i 0.0863829i 0.999067 + 0.0431914i \(0.0137525\pi\)
−0.999067 + 0.0431914i \(0.986247\pi\)
\(228\) − 5.36894i − 0.355567i
\(229\) −3.28682 −0.217199 −0.108600 0.994086i \(-0.534637\pi\)
−0.108600 + 0.994086i \(0.534637\pi\)
\(230\) 0 0
\(231\) 0.155104 0.0102051
\(232\) 9.92054i 0.651315i
\(233\) − 17.7115i − 1.16032i −0.814503 0.580159i \(-0.802990\pi\)
0.814503 0.580159i \(-0.197010\pi\)
\(234\) −7.07097 −0.462244
\(235\) 0 0
\(236\) −53.4141 −3.47696
\(237\) 2.06745i 0.134296i
\(238\) 2.25628i 0.146253i
\(239\) −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 \(0.335773\pi\)
\(242\) 29.8950i 1.92172i
\(243\) 1.00000i 0.0641500i
\(244\) 35.0117 2.24140
\(245\) 0 0
\(246\) −4.56892 −0.291304
\(247\) 2.45978i 0.156512i
\(248\) − 100.759i − 6.39820i
\(249\) 6.41000 0.406217
\(250\) 0 0
\(251\) 26.8917 1.69739 0.848696 0.528882i \(-0.177388\pi\)
0.848696 + 0.528882i \(0.177388\pi\)
\(252\) 2.20500i 0.138902i
\(253\) 2.42467i 0.152438i
\(254\) 55.5637 3.48637
\(255\) 0 0
\(256\) 63.9648 3.99780
\(257\) − 6.85512i − 0.427611i −0.976876 0.213805i \(-0.931414\pi\)
0.976876 0.213805i \(-0.0685858\pi\)
\(258\) − 28.4698i − 1.77245i
\(259\) 2.89149 0.179668
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 15.1678i 0.937067i
\(263\) 13.6294i 0.840423i 0.907426 + 0.420212i \(0.138044\pi\)
−0.907426 + 0.420212i \(0.861956\pi\)
\(264\) 3.90703 0.240461
\(265\) 0 0
\(266\) 1.04106 0.0638315
\(267\) − 15.8302i − 0.968794i
\(268\) 41.5446i 2.53774i
\(269\) 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 \(0.438295\pi\)
\(272\) 33.5631i 2.03506i
\(273\) − 1.01022i − 0.0611414i
\(274\) 20.6704 1.24875
\(275\) 0 0
\(276\) −34.4698 −2.07484
\(277\) − 17.1464i − 1.03023i −0.857122 0.515113i \(-0.827750\pi\)
0.857122 0.515113i \(-0.172250\pi\)
\(278\) − 25.7818i − 1.54629i
\(279\) 10.1566 0.608060
\(280\) 0 0
\(281\) −12.2985 −0.733670 −0.366835 0.930286i \(-0.619559\pi\)
−0.366835 + 0.930286i \(0.619559\pi\)
\(282\) − 31.9531i − 1.90278i
\(283\) − 25.1830i − 1.49697i −0.663149 0.748487i \(-0.730781\pi\)
0.663149 0.748487i \(-0.269219\pi\)
\(284\) −33.5109 −1.98851
\(285\) 0 0
\(286\) −2.78478 −0.164667
\(287\) − 0.652757i − 0.0385311i
\(288\) 24.6760i 1.45405i
\(289\) 12.6807 0.745921
\(290\) 0 0
\(291\) 18.4575 1.08200
\(292\) 18.7109i 1.09497i
\(293\) − 12.3170i − 0.719569i −0.933035 0.359784i \(-0.882850\pi\)
0.933035 0.359784i \(-0.117150\pi\)
\(294\) 18.8686 1.10044
\(295\) 0 0
\(296\) 72.8358 4.23350
\(297\) 0.393832i 0.0228525i
\(298\) − 31.8117i − 1.84280i
\(299\) 15.7924 0.913296
\(300\) 0 0
\(301\) 4.06745 0.234444
\(302\) − 16.7660i − 0.964773i
\(303\) − 12.8038i − 0.735561i
\(304\) 15.4862 0.888193
\(305\) 0 0
\(306\) −5.72905 −0.327508
\(307\) − 22.7379i − 1.29772i −0.760908 0.648860i \(-0.775246\pi\)
0.760908 0.648860i \(-0.224754\pi\)
\(308\) 0.868401i 0.0494817i
\(309\) 4.86979 0.277032
\(310\) 0 0
\(311\) −5.33810 −0.302696 −0.151348 0.988481i \(-0.548361\pi\)
−0.151348 + 0.988481i \(0.548361\pi\)
\(312\) − 25.4472i − 1.44067i
\(313\) 1.96338i 0.110977i 0.998459 + 0.0554885i \(0.0176716\pi\)
−0.998459 + 0.0554885i \(0.982328\pi\)
\(314\) −31.9636 −1.80381
\(315\) 0 0
\(316\) −11.5753 −0.651163
\(317\) 1.29064i 0.0724895i 0.999343 + 0.0362448i \(0.0115396\pi\)
−0.999343 + 0.0362448i \(0.988460\pi\)
\(318\) 33.9830i 1.90567i
\(319\) −0.393832 −0.0220504
\(320\) 0 0
\(321\) 2.34255 0.130748
\(322\) − 6.68384i − 0.372476i
\(323\) 1.99297i 0.110892i
\(324\) −5.59883 −0.311046
\(325\) 0 0
\(326\) 2.35722 0.130554
\(327\) − 8.55044i − 0.472840i
\(328\) − 16.4428i − 0.907902i
\(329\) 4.56511 0.251683
\(330\) 0 0
\(331\) 28.9971 1.59382 0.796911 0.604096i \(-0.206466\pi\)
0.796911 + 0.604096i \(0.206466\pi\)
\(332\) 35.8885i 1.96964i
\(333\) 7.34192i 0.402335i
\(334\) 21.3138 1.16624
\(335\) 0 0
\(336\) −6.36011 −0.346972
\(337\) − 16.7854i − 0.914356i −0.889375 0.457178i \(-0.848860\pi\)
0.889375 0.457178i \(-0.151140\pi\)
\(338\) − 17.6980i − 0.962643i
\(339\) −11.2085 −0.608763
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 2.64341i 0.142939i
\(343\) 5.45257i 0.294411i
\(344\) 102.458 5.52417
\(345\) 0 0
\(346\) −22.2581 −1.19660
\(347\) − 17.8681i − 0.959210i −0.877485 0.479605i \(-0.840780\pi\)
0.877485 0.479605i \(-0.159220\pi\)
\(348\) − 5.59883i − 0.300129i
\(349\) −8.73789 −0.467728 −0.233864 0.972269i \(-0.575137\pi\)
−0.233864 + 0.972269i \(0.575137\pi\)
\(350\) 0 0
\(351\) 2.56511 0.136915
\(352\) 9.71820i 0.517982i
\(353\) 22.6017i 1.20297i 0.798885 + 0.601484i \(0.205424\pi\)
−0.798885 + 0.601484i \(0.794576\pi\)
\(354\) 26.2985 1.39775
\(355\) 0 0
\(356\) 88.6308 4.69742
\(357\) − 0.818503i − 0.0433198i
\(358\) 34.2094i 1.80802i
\(359\) 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
\(362\) 6.88503i 0.361869i
\(363\) − 10.8449i − 0.569209i
\(364\) 5.65607 0.296458
\(365\) 0 0
\(366\) −17.2381 −0.901050
\(367\) 17.4511i 0.910938i 0.890252 + 0.455469i \(0.150528\pi\)
−0.890252 + 0.455469i \(0.849472\pi\)
\(368\) − 99.4247i − 5.18287i
\(369\) 1.65745 0.0862834
\(370\) 0 0
\(371\) −4.85512 −0.252065
\(372\) 56.8652i 2.94832i
\(373\) 32.2018i 1.66734i 0.552260 + 0.833672i \(0.313766\pi\)
−0.552260 + 0.833672i \(0.686234\pi\)
\(374\) −2.25628 −0.116670
\(375\) 0 0
\(376\) 114.994 5.93036
\(377\) 2.56511i 0.132110i
\(378\) − 1.08564i − 0.0558392i
\(379\) −32.3660 −1.66253 −0.831265 0.555876i \(-0.812383\pi\)
−0.831265 + 0.555876i \(0.812383\pi\)
\(380\) 0 0
\(381\) −20.1566 −1.03265
\(382\) 17.4240i 0.891492i
\(383\) 25.8739i 1.32209i 0.750345 + 0.661047i \(0.229887\pi\)
−0.750345 + 0.661047i \(0.770113\pi\)
\(384\) −49.1226 −2.50678
\(385\) 0 0
\(386\) −69.7203 −3.54867
\(387\) 10.3279i 0.524996i
\(388\) 103.340i 5.24631i
\(389\) −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
\(392\) 67.9051i 3.42972i
\(393\) − 5.50235i − 0.277557i
\(394\) −58.4528 −2.94481
\(395\) 0 0
\(396\) −2.20500 −0.110806
\(397\) 4.39534i 0.220596i 0.993899 + 0.110298i \(0.0351805\pi\)
−0.993899 + 0.110298i \(0.964820\pi\)
\(398\) − 7.29734i − 0.365783i
\(399\) −0.377661 −0.0189067
\(400\) 0 0
\(401\) −19.0658 −0.952099 −0.476049 0.879418i \(-0.657932\pi\)
−0.476049 + 0.879418i \(0.657932\pi\)
\(402\) − 20.4546i − 1.02018i
\(403\) − 26.0528i − 1.29778i
\(404\) 71.6865 3.56654
\(405\) 0 0
\(406\) 1.08564 0.0538793
\(407\) 2.89149i 0.143326i
\(408\) − 20.6179i − 1.02074i
\(409\) 1.38235 0.0683530 0.0341765 0.999416i \(-0.489119\pi\)
0.0341765 + 0.999416i \(0.489119\pi\)
\(410\) 0 0
\(411\) −7.49853 −0.369875
\(412\) 27.2651i 1.34326i
\(413\) 3.75725i 0.184882i
\(414\) 16.9713 0.834094
\(415\) 0 0
\(416\) 63.2965 3.10337
\(417\) 9.35277i 0.458007i
\(418\) 1.04106i 0.0509199i
\(419\) −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 \(0.392113\pi\)
\(422\) − 5.51320i − 0.268378i
\(423\) 11.5915i 0.563598i
\(424\) −122.299 −5.93938
\(425\) 0 0
\(426\) 16.4992 0.799387
\(427\) − 2.46279i − 0.119183i
\(428\) 13.1155i 0.633964i
\(429\) 1.01022 0.0487740
\(430\) 0 0
\(431\) −11.9707 −0.576607 −0.288303 0.957539i \(-0.593091\pi\)
−0.288303 + 0.957539i \(0.593091\pi\)
\(432\) − 16.1493i − 0.776982i
\(433\) − 17.9907i − 0.864576i −0.901736 0.432288i \(-0.857706\pi\)
0.901736 0.432288i \(-0.142294\pi\)
\(434\) −11.0264 −0.529284
\(435\) 0 0
\(436\) 47.8725 2.29268
\(437\) − 5.90381i − 0.282418i
\(438\) − 9.21234i − 0.440182i
\(439\) 16.2226 0.774260 0.387130 0.922025i \(-0.373466\pi\)
0.387130 + 0.922025i \(0.373466\pi\)
\(440\) 0 0
\(441\) −6.84490 −0.325947
\(442\) 14.6956i 0.698999i
\(443\) 35.3762i 1.68078i 0.541986 + 0.840388i \(0.317673\pi\)
−0.541986 + 0.840388i \(0.682327\pi\)
\(444\) −41.1062 −1.95081
\(445\) 0 0
\(446\) −34.5965 −1.63819
\(447\) 11.5402i 0.545834i
\(448\) − 14.0690i − 0.664696i
\(449\) −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
\(452\) − 62.7546i − 2.95173i
\(453\) 6.08212i 0.285763i
\(454\) 3.58768 0.168378
\(455\) 0 0
\(456\) −9.51320 −0.445496
\(457\) − 16.1111i − 0.753646i −0.926285 0.376823i \(-0.877017\pi\)
0.926285 0.376823i \(-0.122983\pi\)
\(458\) 9.06045i 0.423367i
\(459\) 2.07830 0.0970069
\(460\) 0 0
\(461\) 17.3396 0.807586 0.403793 0.914850i \(-0.367692\pi\)
0.403793 + 0.914850i \(0.367692\pi\)
\(462\) − 0.427559i − 0.0198918i
\(463\) − 19.3177i − 0.897768i −0.893590 0.448884i \(-0.851822\pi\)
0.893590 0.448884i \(-0.148178\pi\)
\(464\) 16.1493 0.749711
\(465\) 0 0
\(466\) −48.8235 −2.26170
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 14.3616i 0.663866i
\(469\) 2.92232 0.134940
\(470\) 0 0
\(471\) 11.5953 0.534284
\(472\) 94.6441i 4.35635i
\(473\) 4.06745i 0.187022i
\(474\) 5.69914 0.261770
\(475\) 0 0
\(476\) 4.58266 0.210046
\(477\) − 12.3279i − 0.564455i
\(478\) 58.6194i 2.68119i
\(479\) 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
\(482\) − 42.2246i − 1.92328i
\(483\) 2.42467i 0.110326i
\(484\) 60.7188 2.75994
\(485\) 0 0
\(486\) 2.75660 0.125042
\(487\) 2.84171i 0.128770i 0.997925 + 0.0643850i \(0.0205086\pi\)
−0.997925 + 0.0643850i \(0.979491\pi\)
\(488\) − 62.0371i − 2.80829i
\(489\) −0.855118 −0.0386698
\(490\) 0 0
\(491\) 0.157863 0.00712428 0.00356214 0.999994i \(-0.498866\pi\)
0.00356214 + 0.999994i \(0.498866\pi\)
\(492\) 9.27979i 0.418365i
\(493\) 2.07830i 0.0936021i
\(494\) 6.78063 0.305075
\(495\) 0 0
\(496\) −164.022 −7.36480
\(497\) 2.35722i 0.105736i
\(498\) − 17.6698i − 0.791803i
\(499\) −2.64723 −0.118506 −0.0592531 0.998243i \(-0.518872\pi\)
−0.0592531 + 0.998243i \(0.518872\pi\)
\(500\) 0 0
\(501\) −7.73194 −0.345437
\(502\) − 74.1297i − 3.30857i
\(503\) − 13.9545i − 0.622200i −0.950377 0.311100i \(-0.899303\pi\)
0.950377 0.311100i \(-0.100697\pi\)
\(504\) 3.90703 0.174033
\(505\) 0 0
\(506\) 6.68384 0.297133
\(507\) 6.42023i 0.285132i
\(508\) − 112.853i − 5.00706i
\(509\) 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
\(512\) − 78.0802i − 3.45069i
\(513\) − 0.958939i − 0.0423382i
\(514\) −18.8968 −0.833502
\(515\) 0 0
\(516\) −57.8241 −2.54556
\(517\) 4.56511i 0.200773i
\(518\) − 7.97066i − 0.350211i
\(519\) 8.07449 0.354431
\(520\) 0 0
\(521\) −12.2253 −0.535601 −0.267800 0.963474i \(-0.586297\pi\)
−0.267800 + 0.963474i \(0.586297\pi\)
\(522\) 2.75660i 0.120653i
\(523\) − 1.66659i − 0.0728748i −0.999336 0.0364374i \(-0.988399\pi\)
0.999336 0.0364374i \(-0.0116010\pi\)
\(524\) 30.8067 1.34580
\(525\) 0 0
\(526\) 37.5707 1.63816
\(527\) − 21.1085i − 0.919501i
\(528\) − 6.36011i − 0.276788i
\(529\) −14.9038 −0.647992
\(530\) 0 0
\(531\) −9.54022 −0.414010
\(532\) − 2.11446i − 0.0916736i
\(533\) − 4.25154i − 0.184155i
\(534\) −43.6376 −1.88838
\(535\) 0 0
\(536\) 73.6126 3.17958
\(537\) − 12.4100i − 0.535531i
\(538\) − 23.3244i − 1.00558i
\(539\) −2.69574 −0.116114
\(540\) 0 0
\(541\) −34.6558 −1.48997 −0.744984 0.667083i \(-0.767543\pi\)
−0.744984 + 0.667083i \(0.767543\pi\)
\(542\) − 17.4839i − 0.750996i
\(543\) − 2.49765i − 0.107185i
\(544\) 51.2842 2.19879
\(545\) 0 0
\(546\) −2.78478 −0.119177
\(547\) 38.2530i 1.63558i 0.575515 + 0.817791i \(0.304801\pi\)
−0.575515 + 0.817791i \(0.695199\pi\)
\(548\) − 41.9830i − 1.79343i
\(549\) 6.25340 0.266888
\(550\) 0 0
\(551\) 0.958939 0.0408522
\(552\) 61.0769i 2.59960i
\(553\) 0.814230i 0.0346246i
\(554\) −47.2657 −2.00813
\(555\) 0 0
\(556\) −52.3646 −2.22075
\(557\) − 28.7760i − 1.21928i −0.792679 0.609639i \(-0.791314\pi\)
0.792679 0.609639i \(-0.208686\pi\)
\(558\) − 27.9977i − 1.18524i
\(559\) 26.4921 1.12050
\(560\) 0 0
\(561\) 0.818503 0.0345572
\(562\) 33.9022i 1.43008i
\(563\) 32.9809i 1.38998i 0.719020 + 0.694989i \(0.244591\pi\)
−0.719020 + 0.694989i \(0.755409\pi\)
\(564\) −64.8989 −2.73274
\(565\) 0 0
\(566\) −69.4194 −2.91792
\(567\) 0.393832i 0.0165394i
\(568\) 59.3777i 2.49143i
\(569\) −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 \(0.541526\pi\)
\(572\) 5.65607i 0.236492i
\(573\) − 6.32085i − 0.264057i
\(574\) −1.79939 −0.0751051
\(575\) 0 0
\(576\) 35.7232 1.48847
\(577\) 11.9977i 0.499470i 0.968314 + 0.249735i \(0.0803435\pi\)
−0.968314 + 0.249735i \(0.919656\pi\)
\(578\) − 34.9555i − 1.45396i
\(579\) 25.2921 1.05111
\(580\) 0 0
\(581\) 2.52447 0.104733
\(582\) − 50.8798i − 2.10904i
\(583\) − 4.85512i − 0.201078i
\(584\) 33.1537 1.37191
\(585\) 0 0
\(586\) −33.9531 −1.40259
\(587\) − 11.2194i − 0.463073i −0.972826 0.231536i \(-0.925625\pi\)
0.972826 0.231536i \(-0.0743753\pi\)
\(588\) − 38.3234i − 1.58043i
\(589\) −9.73957 −0.401312
\(590\) 0 0
\(591\) 21.2047 0.872245
\(592\) − 118.567i − 4.87306i
\(593\) 7.69682i 0.316071i 0.987433 + 0.158035i \(0.0505160\pi\)
−0.987433 + 0.158035i \(0.949484\pi\)
\(594\) 1.08564 0.0445442
\(595\) 0 0
\(596\) −64.6118 −2.64660
\(597\) 2.64723i 0.108344i
\(598\) − 43.5332i − 1.78020i
\(599\) −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 \(0.479806\pi\)
\(602\) − 11.2123i − 0.456981i
\(603\) 7.42023i 0.302175i
\(604\) −34.0528 −1.38559
\(605\) 0 0
\(606\) −35.2950 −1.43376
\(607\) 37.0441i 1.50357i 0.659407 + 0.751786i \(0.270807\pi\)
−0.659407 + 0.751786i \(0.729193\pi\)
\(608\) − 23.6628i − 0.959652i
\(609\) −0.393832 −0.0159589
\(610\) 0 0
\(611\) 29.7334 1.20289
\(612\) 11.6361i 0.470361i
\(613\) 13.3839i 0.540569i 0.962781 + 0.270284i \(0.0871177\pi\)
−0.962781 + 0.270284i \(0.912882\pi\)
\(614\) −62.6792 −2.52953
\(615\) 0 0
\(616\) 1.53871 0.0619966
\(617\) − 12.6364i − 0.508721i −0.967109 0.254361i \(-0.918135\pi\)
0.967109 0.254361i \(-0.0818650\pi\)
\(618\) − 13.4240i − 0.539994i
\(619\) 41.2447 1.65776 0.828882 0.559424i \(-0.188978\pi\)
0.828882 + 0.559424i \(0.188978\pi\)
\(620\) 0 0
\(621\) −6.15661 −0.247056
\(622\) 14.7150i 0.590018i
\(623\) − 6.23446i − 0.249778i
\(624\) −41.4246 −1.65831
\(625\) 0 0
\(626\) 5.41226 0.216318
\(627\) − 0.377661i − 0.0150823i
\(628\) 64.9203i 2.59060i
\(629\) 15.2587 0.608406
\(630\) 0 0
\(631\) 29.8009 1.18635 0.593177 0.805072i \(-0.297873\pi\)
0.593177 + 0.805072i \(0.297873\pi\)
\(632\) 20.5103i 0.815854i
\(633\) 2.00000i 0.0794929i
\(634\) 3.55777 0.141297
\(635\) 0 0
\(636\) 69.0218 2.73689
\(637\) 17.5579i 0.695669i
\(638\) 1.08564i 0.0429808i
\(639\) −5.98533 −0.236776
\(640\) 0 0
\(641\) 17.7774 0.702167 0.351083 0.936344i \(-0.385813\pi\)
0.351083 + 0.936344i \(0.385813\pi\)
\(642\) − 6.45747i − 0.254856i
\(643\) 44.5534i 1.75702i 0.477727 + 0.878508i \(0.341461\pi\)
−0.477727 + 0.878508i \(0.658539\pi\)
\(644\) −13.5753 −0.534943
\(645\) 0 0
\(646\) 5.49381 0.216151
\(647\) 42.9339i 1.68790i 0.536418 + 0.843952i \(0.319777\pi\)
−0.536418 + 0.843952i \(0.680223\pi\)
\(648\) 9.92054i 0.389716i
\(649\) −3.75725 −0.147485
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) − 4.78766i − 0.187499i
\(653\) 19.8426i 0.776500i 0.921554 + 0.388250i \(0.126920\pi\)
−0.921554 + 0.388250i \(0.873080\pi\)
\(654\) −23.5701 −0.921665
\(655\) 0 0
\(656\) −26.7666 −1.04506
\(657\) 3.34192i 0.130381i
\(658\) − 12.5842i − 0.490582i
\(659\) −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 \(0.748677\pi\)
\(662\) − 79.9332i − 3.10669i
\(663\) − 5.33107i − 0.207042i
\(664\) 63.5907 2.46780
\(665\) 0 0
\(666\) 20.2387 0.784235
\(667\) − 6.15661i − 0.238385i
\(668\) − 43.2898i − 1.67493i
\(669\) 12.5504 0.485228
\(670\) 0 0
\(671\) 2.46279 0.0950749
\(672\) 9.71820i 0.374888i
\(673\) − 22.9000i − 0.882731i −0.897327 0.441365i \(-0.854494\pi\)
0.897327 0.441365i \(-0.145506\pi\)
\(674\) −46.2705 −1.78227
\(675\) 0 0
\(676\) −35.9458 −1.38253
\(677\) 36.0068i 1.38385i 0.721967 + 0.691927i \(0.243238\pi\)
−0.721967 + 0.691927i \(0.756762\pi\)
\(678\) 30.8974i 1.18661i
\(679\) 7.26915 0.278964
\(680\) 0 0
\(681\) −1.30149 −0.0498732
\(682\) − 11.0264i − 0.422222i
\(683\) − 9.12896i − 0.349310i −0.984630 0.174655i \(-0.944119\pi\)
0.984630 0.174655i \(-0.0558810\pi\)
\(684\) 5.36894 0.205287
\(685\) 0 0
\(686\) 15.0305 0.573869
\(687\) − 3.28682i − 0.125400i
\(688\) − 166.788i − 6.35873i
\(689\) −31.6223 −1.20472
\(690\) 0 0
\(691\) 5.85956 0.222908 0.111454 0.993770i \(-0.464449\pi\)
0.111454 + 0.993770i \(0.464449\pi\)
\(692\) 45.2077i 1.71854i
\(693\) 0.155104i 0.00589191i
\(694\) −49.2552 −1.86970
\(695\) 0 0
\(696\) −9.92054 −0.376037
\(697\) − 3.44469i − 0.130477i
\(698\) 24.0868i 0.911700i
\(699\) 17.7115 0.669910
\(700\) 0 0
\(701\) −7.56955 −0.285898 −0.142949 0.989730i \(-0.545658\pi\)
−0.142949 + 0.989730i \(0.545658\pi\)
\(702\) − 7.07097i − 0.266877i
\(703\) − 7.04046i − 0.265536i
\(704\) 14.0690 0.530244
\(705\) 0 0
\(706\) 62.3039 2.34484
\(707\) − 5.04256i − 0.189645i
\(708\) − 53.4141i − 2.00742i
\(709\) −14.5209 −0.545342 −0.272671 0.962107i \(-0.587907\pi\)
−0.272671 + 0.962107i \(0.587907\pi\)
\(710\) 0 0
\(711\) −2.06745 −0.0775356
\(712\) − 157.044i − 5.88549i
\(713\) 62.5302i 2.34178i
\(714\) −2.25628 −0.0844393
\(715\) 0 0
\(716\) 69.4815 2.59665
\(717\) − 21.2651i − 0.794161i
\(718\) − 36.3402i − 1.35621i
\(719\) −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
\(722\) 49.8405i 1.85487i
\(723\) 15.3177i 0.569670i
\(724\) 13.9839 0.519709
\(725\) 0 0
\(726\) −29.8950 −1.10951
\(727\) − 40.8856i − 1.51636i −0.652044 0.758182i \(-0.726088\pi\)
0.652044 0.758182i \(-0.273912\pi\)
\(728\) − 10.0219i − 0.371438i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 21.4645 0.793892
\(732\) 35.0117i 1.29407i
\(733\) − 25.4915i − 0.941550i −0.882253 0.470775i \(-0.843974\pi\)
0.882253 0.470775i \(-0.156026\pi\)
\(734\) 48.1056 1.77561
\(735\) 0 0
\(736\) −151.920 −5.59986
\(737\) 2.92232i 0.107645i
\(738\) − 4.56892i − 0.168184i
\(739\) −9.56830 −0.351975 −0.175988 0.984392i \(-0.556312\pi\)
−0.175988 + 0.984392i \(0.556312\pi\)
\(740\) 0 0
\(741\) −2.45978 −0.0903624
\(742\) 13.3836i 0.491328i
\(743\) 18.0455i 0.662025i 0.943626 + 0.331013i \(0.107390\pi\)
−0.943626 + 0.331013i \(0.892610\pi\)
\(744\) 100.759 3.69400
\(745\) 0 0
\(746\) 88.7673 3.25000
\(747\) 6.41000i 0.234530i
\(748\) 4.58266i 0.167559i
\(749\) 0.922572 0.0337100
\(750\) 0 0
\(751\) 11.9255 0.435168 0.217584 0.976042i \(-0.430182\pi\)
0.217584 + 0.976042i \(0.430182\pi\)
\(752\) − 187.194i − 6.82627i
\(753\) 26.8917i 0.979989i
\(754\) 7.07097 0.257510
\(755\) 0 0
\(756\) −2.20500 −0.0801951
\(757\) 5.86152i 0.213041i 0.994311 + 0.106520i \(0.0339709\pi\)
−0.994311 + 0.106520i \(0.966029\pi\)
\(758\) 89.2201i 3.24062i
\(759\) −2.42467 −0.0880100
\(760\) 0 0
\(761\) 49.8739 1.80793 0.903963 0.427610i \(-0.140644\pi\)
0.903963 + 0.427610i \(0.140644\pi\)
\(762\) 55.5637i 2.01286i
\(763\) − 3.36744i − 0.121909i
\(764\) 35.3894 1.28034
\(765\) 0 0
\(766\) 71.3239 2.57704
\(767\) 24.4717i 0.883621i
\(768\) 63.9648i 2.30813i
\(769\) −29.5121 −1.06423 −0.532117 0.846671i \(-0.678604\pi\)
−0.532117 + 0.846671i \(0.678604\pi\)
\(770\) 0 0
\(771\) 6.85512 0.246881
\(772\) 141.607i 5.09653i
\(773\) 2.41935i 0.0870180i 0.999053 + 0.0435090i \(0.0138537\pi\)
−0.999053 + 0.0435090i \(0.986146\pi\)
\(774\) 28.4698 1.02333
\(775\) 0 0
\(776\) 183.108 6.57320
\(777\) 2.89149i 0.103731i
\(778\) 89.8934i 3.22283i
\(779\) −1.58939 −0.0569460
\(780\) 0 0
\(781\) −2.35722 −0.0843479
\(782\) − 35.2715i − 1.26131i
\(783\) − 1.00000i − 0.0357371i
\(784\) 110.540 3.94786
\(785\) 0 0
\(786\) −15.1678 −0.541016
\(787\) − 51.1549i − 1.82348i −0.410772 0.911738i \(-0.634741\pi\)
0.410772 0.911738i \(-0.365259\pi\)
\(788\) 118.722i 4.22928i
\(789\) −13.6294 −0.485218
\(790\) 0 0
\(791\) −4.41428 −0.156954
\(792\) 3.90703i 0.138830i
\(793\) − 16.0406i − 0.569619i
\(794\) 12.1162 0.429987
\(795\) 0 0
\(796\) −14.8214 −0.525330
\(797\) 28.8662i 1.02249i 0.859434 + 0.511247i \(0.170816\pi\)
−0.859434 + 0.511247i \(0.829184\pi\)
\(798\) 1.04106i 0.0368531i
\(799\) 24.0907 0.852266
\(800\) 0 0
\(801\) 15.8302 0.559334
\(802\) 52.5567i 1.85584i
\(803\) 1.31616i 0.0464462i
\(804\) −41.5446 −1.46517
\(805\) 0 0
\(806\) −71.8171 −2.52965
\(807\) 8.46129i 0.297851i
\(808\) − 127.021i − 4.46858i
\(809\) −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 \(0.619467\pi\)
\(812\) − 2.20500i − 0.0773804i
\(813\) 6.34255i 0.222443i
\(814\) 7.97066 0.279372
\(815\) 0 0
\(816\) −33.5631 −1.17494
\(817\) − 9.90381i − 0.346491i
\(818\) − 3.81060i − 0.133234i
\(819\) 1.01022 0.0353000
\(820\) 0 0
\(821\) 3.60466 0.125804 0.0629018 0.998020i \(-0.479965\pi\)
0.0629018 + 0.998020i \(0.479965\pi\)
\(822\) 20.6704i 0.720964i
\(823\) − 27.2288i − 0.949135i −0.880219 0.474567i \(-0.842605\pi\)
0.880219 0.474567i \(-0.157395\pi\)
\(824\) 48.3109 1.68299
\(825\) 0 0
\(826\) 10.3572 0.360374
\(827\) 47.3936i 1.64804i 0.566562 + 0.824019i \(0.308273\pi\)
−0.566562 + 0.824019i \(0.691727\pi\)
\(828\) − 34.4698i − 1.19791i
\(829\) 41.8388 1.45312 0.726560 0.687103i \(-0.241118\pi\)
0.726560 + 0.687103i \(0.241118\pi\)
\(830\) 0 0
\(831\) 17.1464 0.594802
\(832\) − 91.6339i − 3.17683i
\(833\) 14.2258i 0.492894i
\(834\) 25.7818 0.892752
\(835\) 0 0
\(836\) 2.11446 0.0731302
\(837\) 10.1566i 0.351064i
\(838\) 7.91085i 0.273276i
\(839\) 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
\(842\) − 37.6111i − 1.29617i
\(843\) − 12.2985i − 0.423584i
\(844\) −11.1977 −0.385440
\(845\) 0 0
\(846\) 31.9531 1.09857
\(847\) − 4.27107i − 0.146756i
\(848\) 199.086i 6.83665i
\(849\) 25.1830 0.864278
\(850\) 0 0
\(851\) −45.2013 −1.54948
\(852\) − 33.5109i − 1.14806i
\(853\) − 44.2000i − 1.51338i −0.653773 0.756690i \(-0.726815\pi\)
0.653773 0.756690i \(-0.273185\pi\)
\(854\) −6.78892 −0.232312
\(855\) 0 0
\(856\) 23.2394 0.794305
\(857\) 14.4775i 0.494541i 0.968947 + 0.247270i \(0.0795335\pi\)
−0.968947 + 0.247270i \(0.920466\pi\)
\(858\) − 2.78478i − 0.0950707i
\(859\) −22.5883 −0.770703 −0.385352 0.922770i \(-0.625920\pi\)
−0.385352 + 0.922770i \(0.625920\pi\)
\(860\) 0 0
\(861\) 0.652757 0.0222459
\(862\) 32.9983i 1.12393i
\(863\) 22.9900i 0.782590i 0.920265 + 0.391295i \(0.127973\pi\)
−0.920265 + 0.391295i \(0.872027\pi\)
\(864\) −24.6760 −0.839494
\(865\) 0 0
\(866\) −49.5930 −1.68524
\(867\) 12.6807i 0.430658i
\(868\) 22.3953i 0.760147i
\(869\) −0.814230 −0.0276209
\(870\) 0 0
\(871\) 19.0337 0.644931
\(872\) − 84.8250i − 2.87254i
\(873\) 18.4575i 0.624691i
\(874\) −16.2744 −0.550491
\(875\) 0 0
\(876\) −18.7109 −0.632182
\(877\) 13.5741i 0.458364i 0.973384 + 0.229182i \(0.0736051\pi\)
−0.973384 + 0.229182i \(0.926395\pi\)
\(878\) − 44.7191i − 1.50920i
\(879\) 12.3170 0.415443
\(880\) 0 0
\(881\) 9.91235 0.333956 0.166978 0.985961i \(-0.446599\pi\)
0.166978 + 0.985961i \(0.446599\pi\)
\(882\) 18.8686i 0.635340i
\(883\) − 39.3192i − 1.32320i −0.749859 0.661598i \(-0.769879\pi\)
0.749859 0.661598i \(-0.230121\pi\)
\(884\) 29.8478 1.00389
\(885\) 0 0
\(886\) 97.5180 3.27618
\(887\) − 59.2520i − 1.98949i −0.102403 0.994743i \(-0.532653\pi\)
0.102403 0.994743i \(-0.467347\pi\)
\(888\) 72.8358i 2.44421i
\(889\) −7.93832 −0.266243
\(890\) 0 0
\(891\) −0.393832 −0.0131939
\(892\) 70.2678i 2.35274i
\(893\) − 11.1155i − 0.371968i
\(894\) 31.8117 1.06394
\(895\) 0 0
\(896\) −19.3461 −0.646307
\(897\) 15.7924i 0.527291i
\(898\) 114.942i 3.83567i
\(899\) −10.1566 −0.338742
\(900\) 0 0
\(901\) −25.6211 −0.853562
\(902\) − 1.79939i − 0.0599131i
\(903\) 4.06745i 0.135356i
\(904\) −111.195 −3.69828
\(905\) 0 0
\(906\) 16.7660 0.557012
\(907\) − 5.91210i − 0.196308i −0.995171 0.0981541i \(-0.968706\pi\)
0.995171 0.0981541i \(-0.0312938\pi\)
\(908\) − 7.28682i − 0.241822i
\(909\) 12.8038 0.424676
\(910\) 0 0
\(911\) −12.4185 −0.411445 −0.205722 0.978610i \(-0.565954\pi\)
−0.205722 + 0.978610i \(0.565954\pi\)
\(912\) 15.4862i 0.512799i
\(913\) 2.52447i 0.0835476i
\(914\) −44.4118 −1.46901
\(915\) 0 0
\(916\) 18.4024 0.608031
\(917\) − 2.16700i − 0.0715607i
\(918\) − 5.72905i − 0.189087i
\(919\) −16.4332 −0.542081 −0.271041 0.962568i \(-0.587368\pi\)
−0.271041 + 0.962568i \(0.587368\pi\)
\(920\) 0 0
\(921\) 22.7379 0.749239
\(922\) − 47.7983i − 1.57415i
\(923\) 15.3530i 0.505351i
\(924\) −0.868401 −0.0285683
\(925\) 0 0
\(926\) −53.2510 −1.74994
\(927\) 4.86979i 0.159945i
\(928\) − 24.6760i − 0.810029i
\(929\) −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
\(932\) 99.1637i 3.24822i
\(933\) − 5.33810i − 0.174762i
\(934\) −22.0528 −0.721589
\(935\) 0 0
\(936\) 25.4472 0.831769
\(937\) 20.7528i 0.677964i 0.940793 + 0.338982i \(0.110083\pi\)
−0.940793 + 0.338982i \(0.889917\pi\)
\(938\) − 8.05567i − 0.263027i
\(939\) −1.96338 −0.0640726
\(940\) 0 0
\(941\) 40.1666 1.30940 0.654698 0.755891i \(-0.272796\pi\)
0.654698 + 0.755891i \(0.272796\pi\)
\(942\) − 31.9636i − 1.04143i
\(943\) 10.2043i 0.332297i
\(944\) 154.068 5.01447
\(945\) 0 0
\(946\) 11.2123 0.364544
\(947\) − 17.4144i − 0.565894i −0.959136 0.282947i \(-0.908688\pi\)
0.959136 0.282947i \(-0.0913120\pi\)
\(948\) − 11.5753i − 0.375949i
\(949\) 8.57239 0.278271
\(950\) 0 0
\(951\) −1.29064 −0.0418518
\(952\) − 8.11999i − 0.263170i
\(953\) − 13.5121i − 0.437701i −0.975758 0.218851i \(-0.929769\pi\)
0.975758 0.218851i \(-0.0702307\pi\)
\(954\) −33.9830 −1.10024
\(955\) 0 0
\(956\) 119.060 3.85067
\(957\) − 0.393832i − 0.0127308i
\(958\) − 111.333i − 3.59700i
\(959\) −2.95316 −0.0953626
\(960\) 0 0
\(961\) 72.1567 2.32763
\(962\) − 51.9145i − 1.67379i
\(963\) 2.34255i 0.0754876i
\(964\) −85.7610 −2.76218
\(965\) 0 0
\(966\) 6.68384 0.215049
\(967\) − 24.0598i − 0.773712i −0.922140 0.386856i \(-0.873561\pi\)
0.922140 0.386856i \(-0.126439\pi\)
\(968\) − 107.587i − 3.45798i
\(969\) −1.99297 −0.0640233
\(970\) 0 0
\(971\) 34.7877 1.11639 0.558195 0.829710i \(-0.311494\pi\)
0.558195 + 0.829710i \(0.311494\pi\)
\(972\) − 5.59883i − 0.179583i
\(973\) 3.68342i 0.118085i
\(974\) 7.83344 0.251000
\(975\) 0 0
\(976\) −100.988 −3.23254
\(977\) 35.7566i 1.14396i 0.820269 + 0.571978i \(0.193824\pi\)
−0.820269 + 0.571978i \(0.806176\pi\)
\(978\) 2.35722i 0.0753755i
\(979\) 6.23446 0.199254
\(980\) 0 0
\(981\) 8.55044 0.272995
\(982\) − 0.435166i − 0.0138867i
\(983\) 7.19064i 0.229346i 0.993403 + 0.114673i \(0.0365820\pi\)
−0.993403 + 0.114673i \(0.963418\pi\)
\(984\) 16.4428 0.524177
\(985\) 0 0
\(986\) 5.72905 0.182450
\(987\) 4.56511i 0.145309i
\(988\) − 13.7719i − 0.438143i
\(989\) −63.5847 −2.02188
\(990\) 0 0
\(991\) −19.9123 −0.632537 −0.316268 0.948670i \(-0.602430\pi\)
−0.316268 + 0.948670i \(0.602430\pi\)
\(992\) 250.624i 7.95733i
\(993\) 28.9971i 0.920194i
\(994\) 6.49790 0.206101
\(995\) 0 0
\(996\) −35.8885 −1.13717
\(997\) − 1.57302i − 0.0498179i −0.999690 0.0249089i \(-0.992070\pi\)
0.999690 0.0249089i \(-0.00792958\pi\)
\(998\) 7.29734i 0.230993i
\(999\) −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.c.n.349.1 8
5.2 odd 4 2175.2.a.v.1.4 4
5.3 odd 4 435.2.a.j.1.1 4
5.4 even 2 inner 2175.2.c.n.349.8 8
15.2 even 4 6525.2.a.bi.1.1 4
15.8 even 4 1305.2.a.r.1.4 4
20.3 even 4 6960.2.a.co.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.1 4 5.3 odd 4
1305.2.a.r.1.4 4 15.8 even 4
2175.2.a.v.1.4 4 5.2 odd 4
2175.2.c.n.349.1 8 1.1 even 1 trivial
2175.2.c.n.349.8 8 5.4 even 2 inner
6525.2.a.bi.1.1 4 15.2 even 4
6960.2.a.co.1.2 4 20.3 even 4