Properties

Label 3450.2.d.ba.2899.5
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3450,2,Mod(2899,3450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3450, 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("3450.2899");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (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: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.181494784.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 12x^{3} + 49x^{2} - 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.5
Root \(0.138157 - 0.138157i\) of defining polynomial
Character \(\chi\) \(=\) 3450.2899
Dual form 3450.2.d.ba.2899.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -0.723686i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -0.723686i q^{7} -1.00000i q^{8} -1.00000 q^{9} +5.51445 q^{11} -1.00000i q^{12} +4.96183i q^{13} +0.723686 q^{14} +1.00000 q^{16} -4.23814i q^{17} -1.00000i q^{18} -4.96183 q^{19} +0.723686 q^{21} +5.51445i q^{22} -1.00000i q^{23} +1.00000 q^{24} -4.96183 q^{26} -1.00000i q^{27} +0.723686i q^{28} +1.00000 q^{29} +6.00000 q^{31} +1.00000i q^{32} +5.51445i q^{33} +4.23814 q^{34} +1.00000 q^{36} +4.23814i q^{37} -4.96183i q^{38} -4.96183 q^{39} +1.03817 q^{41} +0.723686i q^{42} -0.961825i q^{43} -5.51445 q^{44} +1.00000 q^{46} +7.96183i q^{47} +1.00000i q^{48} +6.47628 q^{49} +4.23814 q^{51} -4.96183i q^{52} -6.47628i q^{53} +1.00000 q^{54} -0.723686 q^{56} -4.96183i q^{57} +1.00000i q^{58} +9.02891 q^{59} +7.44737 q^{61} +6.00000i q^{62} +0.723686i q^{63} -1.00000 q^{64} -5.51445 q^{66} +11.9237i q^{67} +4.23814i q^{68} +1.00000 q^{69} -0.514453 q^{71} +1.00000i q^{72} -0.447372i q^{73} -4.23814 q^{74} +4.96183 q^{76} -3.99073i q^{77} -4.96183i q^{78} +5.51445 q^{79} +1.00000 q^{81} +1.03817i q^{82} +6.17106i q^{83} -0.723686 q^{84} +0.961825 q^{86} +1.00000i q^{87} -5.51445i q^{88} +2.23814 q^{89} +3.59080 q^{91} +1.00000i q^{92} +6.00000i q^{93} -7.96183 q^{94} -1.00000 q^{96} -2.55263i q^{97} +6.47628i q^{98} -5.51445 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 6 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} - 6 q^{6} - 6 q^{9} + 6 q^{11} + 2 q^{14} + 6 q^{16} + 2 q^{19} + 2 q^{21} + 6 q^{24} + 2 q^{26} + 6 q^{29} + 36 q^{31} - 4 q^{34} + 6 q^{36} + 2 q^{39} + 38 q^{41} - 6 q^{44} + 6 q^{46} - 20 q^{49} - 4 q^{51} + 6 q^{54} - 2 q^{56} + 40 q^{61} - 6 q^{64} - 6 q^{66} + 6 q^{69} + 24 q^{71} + 4 q^{74} - 2 q^{76} + 6 q^{79} + 6 q^{81} - 2 q^{84} - 26 q^{86} - 16 q^{89} + 58 q^{91} - 16 q^{94} - 6 q^{96} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) − 0.723686i − 0.273528i −0.990604 0.136764i \(-0.956330\pi\)
0.990604 0.136764i \(-0.0436701\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.51445 1.66267 0.831335 0.555771i \(-0.187577\pi\)
0.831335 + 0.555771i \(0.187577\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 4.96183i 1.37616i 0.725634 + 0.688081i \(0.241547\pi\)
−0.725634 + 0.688081i \(0.758453\pi\)
\(14\) 0.723686 0.193413
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 4.23814i − 1.02790i −0.857820 0.513950i \(-0.828182\pi\)
0.857820 0.513950i \(-0.171818\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −4.96183 −1.13832 −0.569160 0.822227i \(-0.692732\pi\)
−0.569160 + 0.822227i \(0.692732\pi\)
\(20\) 0 0
\(21\) 0.723686 0.157921
\(22\) 5.51445i 1.17569i
\(23\) − 1.00000i − 0.208514i
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −4.96183 −0.973094
\(27\) − 1.00000i − 0.192450i
\(28\) 0.723686i 0.136764i
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 5.51445i 0.959943i
\(34\) 4.23814 0.726835
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.23814i 0.696746i 0.937356 + 0.348373i \(0.113266\pi\)
−0.937356 + 0.348373i \(0.886734\pi\)
\(38\) − 4.96183i − 0.804914i
\(39\) −4.96183 −0.794528
\(40\) 0 0
\(41\) 1.03817 0.162136 0.0810678 0.996709i \(-0.474167\pi\)
0.0810678 + 0.996709i \(0.474167\pi\)
\(42\) 0.723686i 0.111667i
\(43\) − 0.961825i − 0.146677i −0.997307 0.0733385i \(-0.976635\pi\)
0.997307 0.0733385i \(-0.0233653\pi\)
\(44\) −5.51445 −0.831335
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 7.96183i 1.16135i 0.814135 + 0.580676i \(0.197212\pi\)
−0.814135 + 0.580676i \(0.802788\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 6.47628 0.925183
\(50\) 0 0
\(51\) 4.23814 0.593458
\(52\) − 4.96183i − 0.688081i
\(53\) − 6.47628i − 0.889585i −0.895634 0.444793i \(-0.853277\pi\)
0.895634 0.444793i \(-0.146723\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −0.723686 −0.0967066
\(57\) − 4.96183i − 0.657210i
\(58\) 1.00000i 0.131306i
\(59\) 9.02891 1.17546 0.587732 0.809056i \(-0.300021\pi\)
0.587732 + 0.809056i \(0.300021\pi\)
\(60\) 0 0
\(61\) 7.44737 0.953538 0.476769 0.879029i \(-0.341808\pi\)
0.476769 + 0.879029i \(0.341808\pi\)
\(62\) 6.00000i 0.762001i
\(63\) 0.723686i 0.0911759i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −5.51445 −0.678782
\(67\) 11.9237i 1.45671i 0.685202 + 0.728353i \(0.259714\pi\)
−0.685202 + 0.728353i \(0.740286\pi\)
\(68\) 4.23814i 0.513950i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −0.514453 −0.0610544 −0.0305272 0.999534i \(-0.509719\pi\)
−0.0305272 + 0.999534i \(0.509719\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 0.447372i − 0.0523609i −0.999657 0.0261805i \(-0.991666\pi\)
0.999657 0.0261805i \(-0.00833445\pi\)
\(74\) −4.23814 −0.492674
\(75\) 0 0
\(76\) 4.96183 0.569160
\(77\) − 3.99073i − 0.454786i
\(78\) − 4.96183i − 0.561816i
\(79\) 5.51445 0.620424 0.310212 0.950667i \(-0.399600\pi\)
0.310212 + 0.950667i \(0.399600\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.03817i 0.114647i
\(83\) 6.17106i 0.677362i 0.940901 + 0.338681i \(0.109981\pi\)
−0.940901 + 0.338681i \(0.890019\pi\)
\(84\) −0.723686 −0.0789606
\(85\) 0 0
\(86\) 0.961825 0.103716
\(87\) 1.00000i 0.107211i
\(88\) − 5.51445i − 0.587843i
\(89\) 2.23814 0.237242 0.118621 0.992940i \(-0.462153\pi\)
0.118621 + 0.992940i \(0.462153\pi\)
\(90\) 0 0
\(91\) 3.59080 0.376418
\(92\) 1.00000i 0.104257i
\(93\) 6.00000i 0.622171i
\(94\) −7.96183 −0.821200
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 2.55263i − 0.259180i −0.991568 0.129590i \(-0.958634\pi\)
0.991568 0.129590i \(-0.0413661\pi\)
\(98\) 6.47628i 0.654203i
\(99\) −5.51445 −0.554223
\(100\) 0 0
\(101\) −1.06708 −0.106179 −0.0530893 0.998590i \(-0.516907\pi\)
−0.0530893 + 0.998590i \(0.516907\pi\)
\(102\) 4.23814i 0.419638i
\(103\) − 5.75259i − 0.566820i −0.958999 0.283410i \(-0.908534\pi\)
0.958999 0.283410i \(-0.0914657\pi\)
\(104\) 4.96183 0.486547
\(105\) 0 0
\(106\) 6.47628 0.629032
\(107\) 15.9237i 1.53940i 0.638407 + 0.769699i \(0.279594\pi\)
−0.638407 + 0.769699i \(0.720406\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −17.2670 −1.65388 −0.826942 0.562288i \(-0.809921\pi\)
−0.826942 + 0.562288i \(0.809921\pi\)
\(110\) 0 0
\(111\) −4.23814 −0.402266
\(112\) − 0.723686i − 0.0683819i
\(113\) 6.23814i 0.586835i 0.955984 + 0.293417i \(0.0947926\pi\)
−0.955984 + 0.293417i \(0.905207\pi\)
\(114\) 4.96183 0.464718
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) − 4.96183i − 0.458721i
\(118\) 9.02891i 0.831178i
\(119\) −3.06708 −0.281159
\(120\) 0 0
\(121\) 19.4092 1.76447
\(122\) 7.44737i 0.674253i
\(123\) 1.03817i 0.0936091i
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) −0.723686 −0.0644711
\(127\) − 12.9526i − 1.14935i −0.818380 0.574677i \(-0.805128\pi\)
0.818380 0.574677i \(-0.194872\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0.961825 0.0846840
\(130\) 0 0
\(131\) 11.5815 1.01188 0.505942 0.862568i \(-0.331145\pi\)
0.505942 + 0.862568i \(0.331145\pi\)
\(132\) − 5.51445i − 0.479972i
\(133\) 3.59080i 0.311362i
\(134\) −11.9237 −1.03005
\(135\) 0 0
\(136\) −4.23814 −0.363417
\(137\) 22.2381i 1.89993i 0.312353 + 0.949966i \(0.398883\pi\)
−0.312353 + 0.949966i \(0.601117\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) −18.9907 −1.61077 −0.805386 0.592750i \(-0.798042\pi\)
−0.805386 + 0.592750i \(0.798042\pi\)
\(140\) 0 0
\(141\) −7.96183 −0.670507
\(142\) − 0.514453i − 0.0431720i
\(143\) 27.3618i 2.28810i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 0.447372 0.0370248
\(147\) 6.47628i 0.534154i
\(148\) − 4.23814i − 0.348373i
\(149\) 5.92365 0.485284 0.242642 0.970116i \(-0.421986\pi\)
0.242642 + 0.970116i \(0.421986\pi\)
\(150\) 0 0
\(151\) 6.47628 0.527032 0.263516 0.964655i \(-0.415118\pi\)
0.263516 + 0.964655i \(0.415118\pi\)
\(152\) 4.96183i 0.402457i
\(153\) 4.23814i 0.342633i
\(154\) 3.99073 0.321582
\(155\) 0 0
\(156\) 4.96183 0.397264
\(157\) 9.37102i 0.747889i 0.927451 + 0.373944i \(0.121995\pi\)
−0.927451 + 0.373944i \(0.878005\pi\)
\(158\) 5.51445i 0.438706i
\(159\) 6.47628 0.513602
\(160\) 0 0
\(161\) −0.723686 −0.0570344
\(162\) 1.00000i 0.0785674i
\(163\) 13.9237i 1.09058i 0.838246 + 0.545292i \(0.183581\pi\)
−0.838246 + 0.545292i \(0.816419\pi\)
\(164\) −1.03817 −0.0810678
\(165\) 0 0
\(166\) −6.17106 −0.478967
\(167\) 9.61971i 0.744396i 0.928153 + 0.372198i \(0.121396\pi\)
−0.928153 + 0.372198i \(0.878604\pi\)
\(168\) − 0.723686i − 0.0558336i
\(169\) −11.6197 −0.893824
\(170\) 0 0
\(171\) 4.96183 0.379440
\(172\) 0.961825i 0.0733385i
\(173\) − 1.51445i − 0.115142i −0.998341 0.0575709i \(-0.981664\pi\)
0.998341 0.0575709i \(-0.0183355\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) 5.51445 0.415668
\(177\) 9.02891i 0.678654i
\(178\) 2.23814i 0.167756i
\(179\) −10.4763 −0.783034 −0.391517 0.920171i \(-0.628050\pi\)
−0.391517 + 0.920171i \(0.628050\pi\)
\(180\) 0 0
\(181\) −21.7433 −1.61617 −0.808084 0.589067i \(-0.799495\pi\)
−0.808084 + 0.589067i \(0.799495\pi\)
\(182\) 3.59080i 0.266168i
\(183\) 7.44737i 0.550526i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) − 23.3710i − 1.70906i
\(188\) − 7.96183i − 0.580676i
\(189\) −0.723686 −0.0526404
\(190\) 0 0
\(191\) 8.40920 0.608468 0.304234 0.952597i \(-0.401600\pi\)
0.304234 + 0.952597i \(0.401600\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 12.9907i 0.935093i 0.883968 + 0.467547i \(0.154862\pi\)
−0.883968 + 0.467547i \(0.845138\pi\)
\(194\) 2.55263 0.183268
\(195\) 0 0
\(196\) −6.47628 −0.462591
\(197\) 20.5815i 1.46637i 0.680027 + 0.733187i \(0.261968\pi\)
−0.680027 + 0.733187i \(0.738032\pi\)
\(198\) − 5.51445i − 0.391895i
\(199\) 4.64734 0.329441 0.164720 0.986340i \(-0.447328\pi\)
0.164720 + 0.986340i \(0.447328\pi\)
\(200\) 0 0
\(201\) −11.9237 −0.841029
\(202\) − 1.06708i − 0.0750796i
\(203\) − 0.723686i − 0.0507928i
\(204\) −4.23814 −0.296729
\(205\) 0 0
\(206\) 5.75259 0.400802
\(207\) 1.00000i 0.0695048i
\(208\) 4.96183i 0.344041i
\(209\) −27.3618 −1.89265
\(210\) 0 0
\(211\) −23.3328 −1.60630 −0.803150 0.595777i \(-0.796844\pi\)
−0.803150 + 0.595777i \(0.796844\pi\)
\(212\) 6.47628i 0.444793i
\(213\) − 0.514453i − 0.0352498i
\(214\) −15.9237 −1.08852
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) − 4.34212i − 0.294762i
\(218\) − 17.2670i − 1.16947i
\(219\) 0.447372 0.0302306
\(220\) 0 0
\(221\) 21.0289 1.41456
\(222\) − 4.23814i − 0.284445i
\(223\) − 25.2947i − 1.69386i −0.531707 0.846928i \(-0.678449\pi\)
0.531707 0.846928i \(-0.321551\pi\)
\(224\) 0.723686 0.0483533
\(225\) 0 0
\(226\) −6.23814 −0.414955
\(227\) 7.13288i 0.473426i 0.971580 + 0.236713i \(0.0760701\pi\)
−0.971580 + 0.236713i \(0.923930\pi\)
\(228\) 4.96183i 0.328605i
\(229\) 7.58154 0.501002 0.250501 0.968116i \(-0.419405\pi\)
0.250501 + 0.968116i \(0.419405\pi\)
\(230\) 0 0
\(231\) 3.99073 0.262571
\(232\) − 1.00000i − 0.0656532i
\(233\) − 24.8855i − 1.63030i −0.579249 0.815151i \(-0.696654\pi\)
0.579249 0.815151i \(-0.303346\pi\)
\(234\) 4.96183 0.324365
\(235\) 0 0
\(236\) −9.02891 −0.587732
\(237\) 5.51445i 0.358202i
\(238\) − 3.06708i − 0.198809i
\(239\) 18.0960 1.17053 0.585266 0.810841i \(-0.300990\pi\)
0.585266 + 0.810841i \(0.300990\pi\)
\(240\) 0 0
\(241\) 26.8762 1.73125 0.865624 0.500694i \(-0.166922\pi\)
0.865624 + 0.500694i \(0.166922\pi\)
\(242\) 19.4092i 1.24767i
\(243\) 1.00000i 0.0641500i
\(244\) −7.44737 −0.476769
\(245\) 0 0
\(246\) −1.03817 −0.0661916
\(247\) − 24.6197i − 1.56651i
\(248\) − 6.00000i − 0.381000i
\(249\) −6.17106 −0.391075
\(250\) 0 0
\(251\) −8.79077 −0.554868 −0.277434 0.960745i \(-0.589484\pi\)
−0.277434 + 0.960745i \(0.589484\pi\)
\(252\) − 0.723686i − 0.0455879i
\(253\) − 5.51445i − 0.346691i
\(254\) 12.9526 0.812716
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 10.9526i − 0.683202i −0.939845 0.341601i \(-0.889031\pi\)
0.939845 0.341601i \(-0.110969\pi\)
\(258\) 0.961825i 0.0598806i
\(259\) 3.06708 0.190579
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 11.5815i 0.715510i
\(263\) − 20.8762i − 1.28728i −0.765327 0.643641i \(-0.777423\pi\)
0.765327 0.643641i \(-0.222577\pi\)
\(264\) 5.51445 0.339391
\(265\) 0 0
\(266\) −3.59080 −0.220166
\(267\) 2.23814i 0.136972i
\(268\) − 11.9237i − 0.728353i
\(269\) −21.3618 −1.30245 −0.651225 0.758885i \(-0.725744\pi\)
−0.651225 + 0.758885i \(0.725744\pi\)
\(270\) 0 0
\(271\) 11.0289 0.669958 0.334979 0.942226i \(-0.391271\pi\)
0.334979 + 0.942226i \(0.391271\pi\)
\(272\) − 4.23814i − 0.256975i
\(273\) 3.59080i 0.217325i
\(274\) −22.2381 −1.34346
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) − 15.4381i − 0.927586i −0.885944 0.463793i \(-0.846488\pi\)
0.885944 0.463793i \(-0.153512\pi\)
\(278\) − 18.9907i − 1.13899i
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −2.16179 −0.128962 −0.0644808 0.997919i \(-0.520539\pi\)
−0.0644808 + 0.997919i \(0.520539\pi\)
\(282\) − 7.96183i − 0.474120i
\(283\) 16.9526i 1.00772i 0.863784 + 0.503862i \(0.168088\pi\)
−0.863784 + 0.503862i \(0.831912\pi\)
\(284\) 0.514453 0.0305272
\(285\) 0 0
\(286\) −27.3618 −1.61793
\(287\) − 0.751312i − 0.0443486i
\(288\) − 1.00000i − 0.0589256i
\(289\) −0.961825 −0.0565780
\(290\) 0 0
\(291\) 2.55263 0.149638
\(292\) 0.447372i 0.0261805i
\(293\) 13.0289i 0.761157i 0.924749 + 0.380578i \(0.124275\pi\)
−0.924749 + 0.380578i \(0.875725\pi\)
\(294\) −6.47628 −0.377704
\(295\) 0 0
\(296\) 4.23814 0.246337
\(297\) − 5.51445i − 0.319981i
\(298\) 5.92365i 0.343148i
\(299\) 4.96183 0.286950
\(300\) 0 0
\(301\) −0.696059 −0.0401202
\(302\) 6.47628i 0.372668i
\(303\) − 1.06708i − 0.0613022i
\(304\) −4.96183 −0.284580
\(305\) 0 0
\(306\) −4.23814 −0.242278
\(307\) − 27.0671i − 1.54480i −0.635136 0.772400i \(-0.719056\pi\)
0.635136 0.772400i \(-0.280944\pi\)
\(308\) 3.99073i 0.227393i
\(309\) 5.75259 0.327254
\(310\) 0 0
\(311\) −31.4670 −1.78433 −0.892165 0.451709i \(-0.850814\pi\)
−0.892165 + 0.451709i \(0.850814\pi\)
\(312\) 4.96183i 0.280908i
\(313\) 27.2947i 1.54279i 0.636359 + 0.771393i \(0.280440\pi\)
−0.636359 + 0.771393i \(0.719560\pi\)
\(314\) −9.37102 −0.528837
\(315\) 0 0
\(316\) −5.51445 −0.310212
\(317\) 11.0000i 0.617822i 0.951091 + 0.308911i \(0.0999645\pi\)
−0.951091 + 0.308911i \(0.900036\pi\)
\(318\) 6.47628i 0.363172i
\(319\) 5.51445 0.308750
\(320\) 0 0
\(321\) −15.9237 −0.888772
\(322\) − 0.723686i − 0.0403294i
\(323\) 21.0289i 1.17008i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −13.9237 −0.771160
\(327\) − 17.2670i − 0.954870i
\(328\) − 1.03817i − 0.0573236i
\(329\) 5.76186 0.317662
\(330\) 0 0
\(331\) −8.51445 −0.467997 −0.233998 0.972237i \(-0.575181\pi\)
−0.233998 + 0.972237i \(0.575181\pi\)
\(332\) − 6.17106i − 0.338681i
\(333\) − 4.23814i − 0.232249i
\(334\) −9.61971 −0.526367
\(335\) 0 0
\(336\) 0.723686 0.0394803
\(337\) 1.52372i 0.0830024i 0.999138 + 0.0415012i \(0.0132140\pi\)
−0.999138 + 0.0415012i \(0.986786\pi\)
\(338\) − 11.6197i − 0.632029i
\(339\) −6.23814 −0.338809
\(340\) 0 0
\(341\) 33.0867 1.79175
\(342\) 4.96183i 0.268305i
\(343\) − 9.75259i − 0.526591i
\(344\) −0.961825 −0.0518581
\(345\) 0 0
\(346\) 1.51445 0.0814175
\(347\) − 25.5815i − 1.37329i −0.726993 0.686644i \(-0.759083\pi\)
0.726993 0.686644i \(-0.240917\pi\)
\(348\) − 1.00000i − 0.0536056i
\(349\) 26.8855 1.43915 0.719573 0.694417i \(-0.244337\pi\)
0.719573 + 0.694417i \(0.244337\pi\)
\(350\) 0 0
\(351\) 4.96183 0.264843
\(352\) 5.51445i 0.293921i
\(353\) 10.6197i 0.565230i 0.959233 + 0.282615i \(0.0912019\pi\)
−0.959233 + 0.282615i \(0.908798\pi\)
\(354\) −9.02891 −0.479881
\(355\) 0 0
\(356\) −2.23814 −0.118621
\(357\) − 3.06708i − 0.162327i
\(358\) − 10.4763i − 0.553689i
\(359\) 5.43810 0.287012 0.143506 0.989649i \(-0.454162\pi\)
0.143506 + 0.989649i \(0.454162\pi\)
\(360\) 0 0
\(361\) 5.61971 0.295774
\(362\) − 21.7433i − 1.14280i
\(363\) 19.4092i 1.01872i
\(364\) −3.59080 −0.188209
\(365\) 0 0
\(366\) −7.44737 −0.389280
\(367\) − 8.56190i − 0.446927i −0.974712 0.223464i \(-0.928264\pi\)
0.974712 0.223464i \(-0.0717364\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) −1.03817 −0.0540452
\(370\) 0 0
\(371\) −4.68679 −0.243326
\(372\) − 6.00000i − 0.311086i
\(373\) − 26.2960i − 1.36155i −0.732491 0.680776i \(-0.761643\pi\)
0.732491 0.680776i \(-0.238357\pi\)
\(374\) 23.3710 1.20849
\(375\) 0 0
\(376\) 7.96183 0.410600
\(377\) 4.96183i 0.255547i
\(378\) − 0.723686i − 0.0372224i
\(379\) 6.34212 0.325773 0.162886 0.986645i \(-0.447920\pi\)
0.162886 + 0.986645i \(0.447920\pi\)
\(380\) 0 0
\(381\) 12.9526 0.663580
\(382\) 8.40920i 0.430252i
\(383\) 35.4959i 1.81376i 0.421393 + 0.906878i \(0.361541\pi\)
−0.421393 + 0.906878i \(0.638459\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −12.9907 −0.661211
\(387\) 0.961825i 0.0488923i
\(388\) 2.55263i 0.129590i
\(389\) 9.84730 0.499278 0.249639 0.968339i \(-0.419688\pi\)
0.249639 + 0.968339i \(0.419688\pi\)
\(390\) 0 0
\(391\) −4.23814 −0.214332
\(392\) − 6.47628i − 0.327101i
\(393\) 11.5815i 0.584211i
\(394\) −20.5815 −1.03688
\(395\) 0 0
\(396\) 5.51445 0.277112
\(397\) 18.8762i 0.947370i 0.880694 + 0.473685i \(0.157076\pi\)
−0.880694 + 0.473685i \(0.842924\pi\)
\(398\) 4.64734i 0.232950i
\(399\) −3.59080 −0.179765
\(400\) 0 0
\(401\) 33.5052 1.67317 0.836585 0.547838i \(-0.184549\pi\)
0.836585 + 0.547838i \(0.184549\pi\)
\(402\) − 11.9237i − 0.594698i
\(403\) 29.7710i 1.48300i
\(404\) 1.06708 0.0530893
\(405\) 0 0
\(406\) 0.723686 0.0359159
\(407\) 23.3710i 1.15846i
\(408\) − 4.23814i − 0.209819i
\(409\) 25.5341 1.26258 0.631290 0.775547i \(-0.282526\pi\)
0.631290 + 0.775547i \(0.282526\pi\)
\(410\) 0 0
\(411\) −22.2381 −1.09693
\(412\) 5.75259i 0.283410i
\(413\) − 6.53409i − 0.321522i
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −4.96183 −0.243273
\(417\) − 18.9907i − 0.929980i
\(418\) − 27.3618i − 1.33831i
\(419\) 37.2000 1.81734 0.908669 0.417518i \(-0.137100\pi\)
0.908669 + 0.417518i \(0.137100\pi\)
\(420\) 0 0
\(421\) −11.5815 −0.564449 −0.282225 0.959348i \(-0.591072\pi\)
−0.282225 + 0.959348i \(0.591072\pi\)
\(422\) − 23.3328i − 1.13583i
\(423\) − 7.96183i − 0.387117i
\(424\) −6.47628 −0.314516
\(425\) 0 0
\(426\) 0.514453 0.0249254
\(427\) − 5.38956i − 0.260819i
\(428\) − 15.9237i − 0.769699i
\(429\) −27.3618 −1.32104
\(430\) 0 0
\(431\) 7.37102 0.355050 0.177525 0.984116i \(-0.443191\pi\)
0.177525 + 0.984116i \(0.443191\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 19.3710i 0.930912i 0.885071 + 0.465456i \(0.154110\pi\)
−0.885071 + 0.465456i \(0.845890\pi\)
\(434\) 4.34212 0.208428
\(435\) 0 0
\(436\) 17.2670 0.826942
\(437\) 4.96183i 0.237356i
\(438\) 0.447372i 0.0213763i
\(439\) −17.5052 −0.835477 −0.417738 0.908567i \(-0.637177\pi\)
−0.417738 + 0.908567i \(0.637177\pi\)
\(440\) 0 0
\(441\) −6.47628 −0.308394
\(442\) 21.0289i 1.00024i
\(443\) − 17.3710i − 0.825322i −0.910885 0.412661i \(-0.864599\pi\)
0.910885 0.412661i \(-0.135401\pi\)
\(444\) 4.23814 0.201133
\(445\) 0 0
\(446\) 25.2947 1.19774
\(447\) 5.92365i 0.280179i
\(448\) 0.723686i 0.0341909i
\(449\) 16.8947 0.797312 0.398656 0.917100i \(-0.369477\pi\)
0.398656 + 0.917100i \(0.369477\pi\)
\(450\) 0 0
\(451\) 5.72497 0.269578
\(452\) − 6.23814i − 0.293417i
\(453\) 6.47628i 0.304282i
\(454\) −7.13288 −0.334763
\(455\) 0 0
\(456\) −4.96183 −0.232359
\(457\) − 21.9237i − 1.02555i −0.858524 0.512773i \(-0.828618\pi\)
0.858524 0.512773i \(-0.171382\pi\)
\(458\) 7.58154i 0.354262i
\(459\) −4.23814 −0.197819
\(460\) 0 0
\(461\) −2.65788 −0.123790 −0.0618950 0.998083i \(-0.519714\pi\)
−0.0618950 + 0.998083i \(0.519714\pi\)
\(462\) 3.99073i 0.185666i
\(463\) − 12.0000i − 0.557687i −0.960337 0.278844i \(-0.910049\pi\)
0.960337 0.278844i \(-0.0899511\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) 24.8855 1.15280
\(467\) 15.8289i 0.732476i 0.930521 + 0.366238i \(0.119354\pi\)
−0.930521 + 0.366238i \(0.880646\pi\)
\(468\) 4.96183i 0.229360i
\(469\) 8.62898 0.398449
\(470\) 0 0
\(471\) −9.37102 −0.431794
\(472\) − 9.02891i − 0.415589i
\(473\) − 5.30394i − 0.243875i
\(474\) −5.51445 −0.253287
\(475\) 0 0
\(476\) 3.06708 0.140579
\(477\) 6.47628i 0.296528i
\(478\) 18.0960i 0.827691i
\(479\) −27.5723 −1.25981 −0.629905 0.776673i \(-0.716906\pi\)
−0.629905 + 0.776673i \(0.716906\pi\)
\(480\) 0 0
\(481\) −21.0289 −0.958836
\(482\) 26.8762i 1.22418i
\(483\) − 0.723686i − 0.0329288i
\(484\) −19.4092 −0.882236
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 16.6868i − 0.756151i −0.925775 0.378075i \(-0.876586\pi\)
0.925775 0.378075i \(-0.123414\pi\)
\(488\) − 7.44737i − 0.337127i
\(489\) −13.9237 −0.629649
\(490\) 0 0
\(491\) −13.4288 −0.606035 −0.303017 0.952985i \(-0.597994\pi\)
−0.303017 + 0.952985i \(0.597994\pi\)
\(492\) − 1.03817i − 0.0468045i
\(493\) − 4.23814i − 0.190876i
\(494\) 24.6197 1.10769
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 0.372303i 0.0167001i
\(498\) − 6.17106i − 0.276532i
\(499\) 8.99073 0.402480 0.201240 0.979542i \(-0.435503\pi\)
0.201240 + 0.979542i \(0.435503\pi\)
\(500\) 0 0
\(501\) −9.61971 −0.429777
\(502\) − 8.79077i − 0.392351i
\(503\) 18.9618i 0.845466i 0.906254 + 0.422733i \(0.138929\pi\)
−0.906254 + 0.422733i \(0.861071\pi\)
\(504\) 0.723686 0.0322355
\(505\) 0 0
\(506\) 5.51445 0.245147
\(507\) − 11.6197i − 0.516049i
\(508\) 12.9526i 0.574677i
\(509\) −26.2276 −1.16252 −0.581259 0.813719i \(-0.697440\pi\)
−0.581259 + 0.813719i \(0.697440\pi\)
\(510\) 0 0
\(511\) −0.323757 −0.0143222
\(512\) 1.00000i 0.0441942i
\(513\) 4.96183i 0.219070i
\(514\) 10.9526 0.483097
\(515\) 0 0
\(516\) −0.961825 −0.0423420
\(517\) 43.9051i 1.93094i
\(518\) 3.06708i 0.134760i
\(519\) 1.51445 0.0664771
\(520\) 0 0
\(521\) 32.2381 1.41238 0.706189 0.708023i \(-0.250413\pi\)
0.706189 + 0.708023i \(0.250413\pi\)
\(522\) − 1.00000i − 0.0437688i
\(523\) 5.59080i 0.244469i 0.992501 + 0.122234i \(0.0390060\pi\)
−0.992501 + 0.122234i \(0.960994\pi\)
\(524\) −11.5815 −0.505942
\(525\) 0 0
\(526\) 20.8762 0.910246
\(527\) − 25.4288i − 1.10770i
\(528\) 5.51445i 0.239986i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −9.02891 −0.391821
\(532\) − 3.59080i − 0.155681i
\(533\) 5.15124i 0.223125i
\(534\) −2.23814 −0.0968538
\(535\) 0 0
\(536\) 11.9237 0.515023
\(537\) − 10.4763i − 0.452085i
\(538\) − 21.3618i − 0.920971i
\(539\) 35.7131 1.53827
\(540\) 0 0
\(541\) 23.6486 1.01673 0.508367 0.861141i \(-0.330249\pi\)
0.508367 + 0.861141i \(0.330249\pi\)
\(542\) 11.0289i 0.473732i
\(543\) − 21.7433i − 0.933095i
\(544\) 4.23814 0.181709
\(545\) 0 0
\(546\) −3.59080 −0.153672
\(547\) − 39.3328i − 1.68175i −0.541229 0.840876i \(-0.682041\pi\)
0.541229 0.840876i \(-0.317959\pi\)
\(548\) − 22.2381i − 0.949966i
\(549\) −7.44737 −0.317846
\(550\) 0 0
\(551\) −4.96183 −0.211381
\(552\) − 1.00000i − 0.0425628i
\(553\) − 3.99073i − 0.169703i
\(554\) 15.4381 0.655902
\(555\) 0 0
\(556\) 18.9907 0.805386
\(557\) 34.2658i 1.45189i 0.687754 + 0.725944i \(0.258597\pi\)
−0.687754 + 0.725944i \(0.741403\pi\)
\(558\) − 6.00000i − 0.254000i
\(559\) 4.77241 0.201851
\(560\) 0 0
\(561\) 23.3710 0.986725
\(562\) − 2.16179i − 0.0911896i
\(563\) 20.5434i 0.865799i 0.901442 + 0.432900i \(0.142510\pi\)
−0.901442 + 0.432900i \(0.857490\pi\)
\(564\) 7.96183 0.335253
\(565\) 0 0
\(566\) −16.9526 −0.712569
\(567\) − 0.723686i − 0.0303920i
\(568\) 0.514453i 0.0215860i
\(569\) 27.4474 1.15065 0.575327 0.817924i \(-0.304875\pi\)
0.575327 + 0.817924i \(0.304875\pi\)
\(570\) 0 0
\(571\) −5.58154 −0.233580 −0.116790 0.993157i \(-0.537260\pi\)
−0.116790 + 0.993157i \(0.537260\pi\)
\(572\) − 27.3618i − 1.14405i
\(573\) 8.40920i 0.351299i
\(574\) 0.751312 0.0313592
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 4.56190i − 0.189914i −0.995481 0.0949571i \(-0.969729\pi\)
0.995481 0.0949571i \(-0.0302714\pi\)
\(578\) − 0.961825i − 0.0400067i
\(579\) −12.9907 −0.539876
\(580\) 0 0
\(581\) 4.46591 0.185277
\(582\) 2.55263i 0.105810i
\(583\) − 35.7131i − 1.47909i
\(584\) −0.447372 −0.0185124
\(585\) 0 0
\(586\) −13.0289 −0.538219
\(587\) 23.4474i 0.967777i 0.875130 + 0.483888i \(0.160776\pi\)
−0.875130 + 0.483888i \(0.839224\pi\)
\(588\) − 6.47628i − 0.267077i
\(589\) −29.7710 −1.22669
\(590\) 0 0
\(591\) −20.5815 −0.846611
\(592\) 4.23814i 0.174186i
\(593\) 14.4092i 0.591715i 0.955232 + 0.295857i \(0.0956053\pi\)
−0.955232 + 0.295857i \(0.904395\pi\)
\(594\) 5.51445 0.226261
\(595\) 0 0
\(596\) −5.92365 −0.242642
\(597\) 4.64734i 0.190203i
\(598\) 4.96183i 0.202904i
\(599\) −27.9815 −1.14329 −0.571646 0.820500i \(-0.693695\pi\)
−0.571646 + 0.820500i \(0.693695\pi\)
\(600\) 0 0
\(601\) −6.38029 −0.260257 −0.130129 0.991497i \(-0.541539\pi\)
−0.130129 + 0.991497i \(0.541539\pi\)
\(602\) − 0.696059i − 0.0283693i
\(603\) − 11.9237i − 0.485569i
\(604\) −6.47628 −0.263516
\(605\) 0 0
\(606\) 1.06708 0.0433472
\(607\) − 41.6946i − 1.69233i −0.532920 0.846166i \(-0.678905\pi\)
0.532920 0.846166i \(-0.321095\pi\)
\(608\) − 4.96183i − 0.201229i
\(609\) 0.723686 0.0293252
\(610\) 0 0
\(611\) −39.5052 −1.59821
\(612\) − 4.23814i − 0.171317i
\(613\) − 10.8486i − 0.438170i −0.975706 0.219085i \(-0.929693\pi\)
0.975706 0.219085i \(-0.0703072\pi\)
\(614\) 27.0671 1.09234
\(615\) 0 0
\(616\) −3.99073 −0.160791
\(617\) − 17.5052i − 0.704732i −0.935862 0.352366i \(-0.885377\pi\)
0.935862 0.352366i \(-0.114623\pi\)
\(618\) 5.75259i 0.231403i
\(619\) −22.3421 −0.898005 −0.449003 0.893530i \(-0.648221\pi\)
−0.449003 + 0.893530i \(0.648221\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) − 31.4670i − 1.26171i
\(623\) − 1.61971i − 0.0648923i
\(624\) −4.96183 −0.198632
\(625\) 0 0
\(626\) −27.2947 −1.09091
\(627\) − 27.3618i − 1.09272i
\(628\) − 9.37102i − 0.373944i
\(629\) 17.9618 0.716185
\(630\) 0 0
\(631\) −13.4105 −0.533863 −0.266931 0.963716i \(-0.586010\pi\)
−0.266931 + 0.963716i \(0.586010\pi\)
\(632\) − 5.51445i − 0.219353i
\(633\) − 23.3328i − 0.927397i
\(634\) −11.0000 −0.436866
\(635\) 0 0
\(636\) −6.47628 −0.256801
\(637\) 32.1342i 1.27320i
\(638\) 5.51445i 0.218319i
\(639\) 0.514453 0.0203515
\(640\) 0 0
\(641\) 35.9538 1.42009 0.710046 0.704156i \(-0.248674\pi\)
0.710046 + 0.704156i \(0.248674\pi\)
\(642\) − 15.9237i − 0.628456i
\(643\) 2.48555i 0.0980204i 0.998798 + 0.0490102i \(0.0156067\pi\)
−0.998798 + 0.0490102i \(0.984393\pi\)
\(644\) 0.723686 0.0285172
\(645\) 0 0
\(646\) −21.0289 −0.827371
\(647\) − 44.4381i − 1.74704i −0.486786 0.873521i \(-0.661831\pi\)
0.486786 0.873521i \(-0.338169\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 49.7895 1.95441
\(650\) 0 0
\(651\) 4.34212 0.170181
\(652\) − 13.9237i − 0.545292i
\(653\) − 21.5526i − 0.843420i −0.906731 0.421710i \(-0.861430\pi\)
0.906731 0.421710i \(-0.138570\pi\)
\(654\) 17.2670 0.675195
\(655\) 0 0
\(656\) 1.03817 0.0405339
\(657\) 0.447372i 0.0174536i
\(658\) 5.76186i 0.224621i
\(659\) 7.33413 0.285697 0.142849 0.989745i \(-0.454374\pi\)
0.142849 + 0.989745i \(0.454374\pi\)
\(660\) 0 0
\(661\) −46.1618 −1.79549 −0.897743 0.440520i \(-0.854794\pi\)
−0.897743 + 0.440520i \(0.854794\pi\)
\(662\) − 8.51445i − 0.330924i
\(663\) 21.0289i 0.816695i
\(664\) 6.17106 0.239483
\(665\) 0 0
\(666\) 4.23814 0.164225
\(667\) − 1.00000i − 0.0387202i
\(668\) − 9.61971i − 0.372198i
\(669\) 25.2947 0.977949
\(670\) 0 0
\(671\) 41.0682 1.58542
\(672\) 0.723686i 0.0279168i
\(673\) − 37.8184i − 1.45779i −0.684624 0.728896i \(-0.740034\pi\)
0.684624 0.728896i \(-0.259966\pi\)
\(674\) −1.52372 −0.0586916
\(675\) 0 0
\(676\) 11.6197 0.446912
\(677\) − 2.21051i − 0.0849569i −0.999097 0.0424785i \(-0.986475\pi\)
0.999097 0.0424785i \(-0.0135254\pi\)
\(678\) − 6.23814i − 0.239574i
\(679\) −1.84730 −0.0708929
\(680\) 0 0
\(681\) −7.13288 −0.273333
\(682\) 33.0867i 1.26696i
\(683\) − 29.6393i − 1.13412i −0.823677 0.567059i \(-0.808081\pi\)
0.823677 0.567059i \(-0.191919\pi\)
\(684\) −4.96183 −0.189720
\(685\) 0 0
\(686\) 9.75259 0.372356
\(687\) 7.58154i 0.289254i
\(688\) − 0.961825i − 0.0366692i
\(689\) 32.1342 1.22421
\(690\) 0 0
\(691\) −40.1512 −1.52743 −0.763713 0.645556i \(-0.776626\pi\)
−0.763713 + 0.645556i \(0.776626\pi\)
\(692\) 1.51445i 0.0575709i
\(693\) 3.99073i 0.151595i
\(694\) 25.5815 0.971062
\(695\) 0 0
\(696\) 1.00000 0.0379049
\(697\) − 4.39993i − 0.166659i
\(698\) 26.8855i 1.01763i
\(699\) 24.8855 0.941255
\(700\) 0 0
\(701\) −40.8762 −1.54387 −0.771937 0.635700i \(-0.780712\pi\)
−0.771937 + 0.635700i \(0.780712\pi\)
\(702\) 4.96183i 0.187272i
\(703\) − 21.0289i − 0.793120i
\(704\) −5.51445 −0.207834
\(705\) 0 0
\(706\) −10.6197 −0.399678
\(707\) 0.772232i 0.0290428i
\(708\) − 9.02891i − 0.339327i
\(709\) 25.0565 0.941018 0.470509 0.882395i \(-0.344070\pi\)
0.470509 + 0.882395i \(0.344070\pi\)
\(710\) 0 0
\(711\) −5.51445 −0.206808
\(712\) − 2.23814i − 0.0838778i
\(713\) − 6.00000i − 0.224702i
\(714\) 3.06708 0.114783
\(715\) 0 0
\(716\) 10.4763 0.391517
\(717\) 18.0960i 0.675807i
\(718\) 5.43810i 0.202948i
\(719\) 34.5145 1.28717 0.643586 0.765374i \(-0.277446\pi\)
0.643586 + 0.765374i \(0.277446\pi\)
\(720\) 0 0
\(721\) −4.16307 −0.155041
\(722\) 5.61971i 0.209144i
\(723\) 26.8762i 0.999537i
\(724\) 21.7433 0.808084
\(725\) 0 0
\(726\) −19.4092 −0.720343
\(727\) − 37.0289i − 1.37333i −0.726976 0.686663i \(-0.759075\pi\)
0.726976 0.686663i \(-0.240925\pi\)
\(728\) − 3.59080i − 0.133084i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −4.07635 −0.150769
\(732\) − 7.44737i − 0.275263i
\(733\) 8.02763i 0.296507i 0.988949 + 0.148254i \(0.0473652\pi\)
−0.988949 + 0.148254i \(0.952635\pi\)
\(734\) 8.56190 0.316025
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 65.7524i 2.42202i
\(738\) − 1.03817i − 0.0382157i
\(739\) 34.3432 1.26334 0.631668 0.775239i \(-0.282371\pi\)
0.631668 + 0.775239i \(0.282371\pi\)
\(740\) 0 0
\(741\) 24.6197 0.904428
\(742\) − 4.68679i − 0.172058i
\(743\) − 30.8855i − 1.13308i −0.824035 0.566539i \(-0.808282\pi\)
0.824035 0.566539i \(-0.191718\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) 26.2960 0.962763
\(747\) − 6.17106i − 0.225787i
\(748\) 23.3710i 0.854529i
\(749\) 11.5237 0.421068
\(750\) 0 0
\(751\) 19.0658 0.695721 0.347860 0.937546i \(-0.386908\pi\)
0.347860 + 0.937546i \(0.386908\pi\)
\(752\) 7.96183i 0.290338i
\(753\) − 8.79077i − 0.320353i
\(754\) −4.96183 −0.180699
\(755\) 0 0
\(756\) 0.723686 0.0263202
\(757\) 26.1040i 0.948765i 0.880319 + 0.474383i \(0.157329\pi\)
−0.880319 + 0.474383i \(0.842671\pi\)
\(758\) 6.34212i 0.230356i
\(759\) 5.51445 0.200162
\(760\) 0 0
\(761\) 16.9433 0.614194 0.307097 0.951678i \(-0.400642\pi\)
0.307097 + 0.951678i \(0.400642\pi\)
\(762\) 12.9526i 0.469222i
\(763\) 12.4959i 0.452383i
\(764\) −8.40920 −0.304234
\(765\) 0 0
\(766\) −35.4959 −1.28252
\(767\) 44.7999i 1.61763i
\(768\) 1.00000i 0.0360844i
\(769\) 41.5815 1.49947 0.749734 0.661739i \(-0.230181\pi\)
0.749734 + 0.661739i \(0.230181\pi\)
\(770\) 0 0
\(771\) 10.9526 0.394447
\(772\) − 12.9907i − 0.467547i
\(773\) 39.2183i 1.41059i 0.708916 + 0.705293i \(0.249184\pi\)
−0.708916 + 0.705293i \(0.750816\pi\)
\(774\) −0.961825 −0.0345721
\(775\) 0 0
\(776\) −2.55263 −0.0916340
\(777\) 3.06708i 0.110031i
\(778\) 9.84730i 0.353043i
\(779\) −5.15124 −0.184562
\(780\) 0 0
\(781\) −2.83693 −0.101513
\(782\) − 4.23814i − 0.151556i
\(783\) − 1.00000i − 0.0357371i
\(784\) 6.47628 0.231296
\(785\) 0 0
\(786\) −11.5815 −0.413100
\(787\) 17.3039i 0.616819i 0.951254 + 0.308409i \(0.0997967\pi\)
−0.951254 + 0.308409i \(0.900203\pi\)
\(788\) − 20.5815i − 0.733187i
\(789\) 20.8762 0.743213
\(790\) 0 0
\(791\) 4.51445 0.160515
\(792\) 5.51445i 0.195948i
\(793\) 36.9526i 1.31222i
\(794\) −18.8762 −0.669892
\(795\) 0 0
\(796\) −4.64734 −0.164720
\(797\) − 12.3236i − 0.436524i −0.975890 0.218262i \(-0.929961\pi\)
0.975890 0.218262i \(-0.0700387\pi\)
\(798\) − 3.59080i − 0.127113i
\(799\) 33.7433 1.19375
\(800\) 0 0
\(801\) −2.23814 −0.0790808
\(802\) 33.5052i 1.18311i
\(803\) − 2.46701i − 0.0870589i
\(804\) 11.9237 0.420515
\(805\) 0 0
\(806\) −29.7710 −1.04864
\(807\) − 21.3618i − 0.751969i
\(808\) 1.06708i 0.0375398i
\(809\) 21.3618 0.751039 0.375520 0.926814i \(-0.377464\pi\)
0.375520 + 0.926814i \(0.377464\pi\)
\(810\) 0 0
\(811\) 47.0104 1.65076 0.825379 0.564579i \(-0.190962\pi\)
0.825379 + 0.564579i \(0.190962\pi\)
\(812\) 0.723686i 0.0253964i
\(813\) 11.0289i 0.386801i
\(814\) −23.3710 −0.819154
\(815\) 0 0
\(816\) 4.23814 0.148365
\(817\) 4.77241i 0.166965i
\(818\) 25.5341i 0.892779i
\(819\) −3.59080 −0.125473
\(820\) 0 0
\(821\) 30.4855 1.06395 0.531976 0.846759i \(-0.321449\pi\)
0.531976 + 0.846759i \(0.321449\pi\)
\(822\) − 22.2381i − 0.775644i
\(823\) − 6.34212i − 0.221072i −0.993872 0.110536i \(-0.964743\pi\)
0.993872 0.110536i \(-0.0352568\pi\)
\(824\) −5.75259 −0.200401
\(825\) 0 0
\(826\) 6.53409 0.227350
\(827\) − 56.1340i − 1.95197i −0.217838 0.975985i \(-0.569900\pi\)
0.217838 0.975985i \(-0.430100\pi\)
\(828\) − 1.00000i − 0.0347524i
\(829\) 9.93292 0.344985 0.172492 0.985011i \(-0.444818\pi\)
0.172492 + 0.985011i \(0.444818\pi\)
\(830\) 0 0
\(831\) 15.4381 0.535542
\(832\) − 4.96183i − 0.172020i
\(833\) − 27.4474i − 0.950995i
\(834\) 18.9907 0.657595
\(835\) 0 0
\(836\) 27.3618 0.946326
\(837\) − 6.00000i − 0.207390i
\(838\) 37.2000i 1.28505i
\(839\) −31.7802 −1.09718 −0.548588 0.836093i \(-0.684834\pi\)
−0.548588 + 0.836093i \(0.684834\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) − 11.5815i − 0.399126i
\(843\) − 2.16179i − 0.0744560i
\(844\) 23.3328 0.803150
\(845\) 0 0
\(846\) 7.96183 0.273733
\(847\) − 14.0462i − 0.482632i
\(848\) − 6.47628i − 0.222396i
\(849\) −16.9526 −0.581810
\(850\) 0 0
\(851\) 4.23814 0.145282
\(852\) 0.514453i 0.0176249i
\(853\) − 5.30394i − 0.181603i −0.995869 0.0908017i \(-0.971057\pi\)
0.995869 0.0908017i \(-0.0289430\pi\)
\(854\) 5.38956 0.184427
\(855\) 0 0
\(856\) 15.9237 0.544259
\(857\) 19.6579i 0.671501i 0.941951 + 0.335750i \(0.108990\pi\)
−0.941951 + 0.335750i \(0.891010\pi\)
\(858\) − 27.3618i − 0.934115i
\(859\) −57.7524 −1.97049 −0.985244 0.171159i \(-0.945249\pi\)
−0.985244 + 0.171159i \(0.945249\pi\)
\(860\) 0 0
\(861\) 0.751312 0.0256047
\(862\) 7.37102i 0.251058i
\(863\) − 27.3328i − 0.930421i −0.885200 0.465210i \(-0.845979\pi\)
0.885200 0.465210i \(-0.154021\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −19.3710 −0.658254
\(867\) − 0.961825i − 0.0326653i
\(868\) 4.34212i 0.147381i
\(869\) 30.4092 1.03156
\(870\) 0 0
\(871\) −59.1631 −2.00466
\(872\) 17.2670i 0.584736i
\(873\) 2.55263i 0.0863934i
\(874\) −4.96183 −0.167836
\(875\) 0 0
\(876\) −0.447372 −0.0151153
\(877\) − 14.0000i − 0.472746i −0.971662 0.236373i \(-0.924041\pi\)
0.971662 0.236373i \(-0.0759588\pi\)
\(878\) − 17.5052i − 0.590771i
\(879\) −13.0289 −0.439454
\(880\) 0 0
\(881\) 19.4474 0.655199 0.327599 0.944817i \(-0.393760\pi\)
0.327599 + 0.944817i \(0.393760\pi\)
\(882\) − 6.47628i − 0.218068i
\(883\) − 34.2854i − 1.15380i −0.816816 0.576898i \(-0.804263\pi\)
0.816816 0.576898i \(-0.195737\pi\)
\(884\) −21.0289 −0.707279
\(885\) 0 0
\(886\) 17.3710 0.583591
\(887\) 35.9433i 1.20686i 0.797417 + 0.603429i \(0.206199\pi\)
−0.797417 + 0.603429i \(0.793801\pi\)
\(888\) 4.23814i 0.142223i
\(889\) −9.37358 −0.314380
\(890\) 0 0
\(891\) 5.51445 0.184741
\(892\) 25.2947i 0.846928i
\(893\) − 39.5052i − 1.32199i
\(894\) −5.92365 −0.198117
\(895\) 0 0
\(896\) −0.723686 −0.0241766
\(897\) 4.96183i 0.165671i
\(898\) 16.8947i 0.563785i
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) −27.4474 −0.914405
\(902\) 5.72497i 0.190621i
\(903\) − 0.696059i − 0.0231634i
\(904\) 6.23814 0.207477
\(905\) 0 0
\(906\) −6.47628 −0.215160
\(907\) − 34.0671i − 1.13118i −0.824687 0.565589i \(-0.808649\pi\)
0.824687 0.565589i \(-0.191351\pi\)
\(908\) − 7.13288i − 0.236713i
\(909\) 1.06708 0.0353929
\(910\) 0 0
\(911\) 16.0671 0.532326 0.266163 0.963928i \(-0.414244\pi\)
0.266163 + 0.963928i \(0.414244\pi\)
\(912\) − 4.96183i − 0.164302i
\(913\) 34.0300i 1.12623i
\(914\) 21.9237 0.725170
\(915\) 0 0
\(916\) −7.58154 −0.250501
\(917\) − 8.38139i − 0.276778i
\(918\) − 4.23814i − 0.139879i
\(919\) 19.7619 0.651884 0.325942 0.945390i \(-0.394319\pi\)
0.325942 + 0.945390i \(0.394319\pi\)
\(920\) 0 0
\(921\) 27.0671 0.891891
\(922\) − 2.65788i − 0.0875328i
\(923\) − 2.55263i − 0.0840208i
\(924\) −3.99073 −0.131285
\(925\) 0 0
\(926\) 12.0000 0.394344
\(927\) 5.75259i 0.188940i
\(928\) 1.00000i 0.0328266i
\(929\) 6.33285 0.207774 0.103887 0.994589i \(-0.466872\pi\)
0.103887 + 0.994589i \(0.466872\pi\)
\(930\) 0 0
\(931\) −32.1342 −1.05315
\(932\) 24.8855i 0.815151i
\(933\) − 31.4670i − 1.03018i
\(934\) −15.8289 −0.517939
\(935\) 0 0
\(936\) −4.96183 −0.162182
\(937\) − 25.8473i − 0.844395i −0.906504 0.422197i \(-0.861259\pi\)
0.906504 0.422197i \(-0.138741\pi\)
\(938\) 8.62898i 0.281746i
\(939\) −27.2947 −0.890728
\(940\) 0 0
\(941\) −9.04744 −0.294938 −0.147469 0.989067i \(-0.547113\pi\)
−0.147469 + 0.989067i \(0.547113\pi\)
\(942\) − 9.37102i − 0.305324i
\(943\) − 1.03817i − 0.0338076i
\(944\) 9.02891 0.293866
\(945\) 0 0
\(946\) 5.30394 0.172446
\(947\) − 0.971093i − 0.0315563i −0.999876 0.0157781i \(-0.994977\pi\)
0.999876 0.0157781i \(-0.00502255\pi\)
\(948\) − 5.51445i − 0.179101i
\(949\) 2.21978 0.0720571
\(950\) 0 0
\(951\) −11.0000 −0.356699
\(952\) 3.06708i 0.0994047i
\(953\) 5.19070i 0.168143i 0.996460 + 0.0840716i \(0.0267924\pi\)
−0.996460 + 0.0840716i \(0.973208\pi\)
\(954\) −6.47628 −0.209677
\(955\) 0 0
\(956\) −18.0960 −0.585266
\(957\) 5.51445i 0.178257i
\(958\) − 27.5723i − 0.890820i
\(959\) 16.0934 0.519684
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) − 21.0289i − 0.677999i
\(963\) − 15.9237i − 0.513132i
\(964\) −26.8762 −0.865624
\(965\) 0 0
\(966\) 0.723686 0.0232842
\(967\) − 17.0289i − 0.547613i −0.961785 0.273806i \(-0.911717\pi\)
0.961785 0.273806i \(-0.0882827\pi\)
\(968\) − 19.4092i − 0.623835i
\(969\) −21.0289 −0.675546
\(970\) 0 0
\(971\) −25.4683 −0.817316 −0.408658 0.912688i \(-0.634003\pi\)
−0.408658 + 0.912688i \(0.634003\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 13.7433i 0.440591i
\(974\) 16.6868 0.534679
\(975\) 0 0
\(976\) 7.44737 0.238385
\(977\) 23.9513i 0.766269i 0.923693 + 0.383135i \(0.125155\pi\)
−0.923693 + 0.383135i \(0.874845\pi\)
\(978\) − 13.9237i − 0.445229i
\(979\) 12.3421 0.394456
\(980\) 0 0
\(981\) 17.2670 0.551294
\(982\) − 13.4288i − 0.428531i
\(983\) 13.5723i 0.432888i 0.976295 + 0.216444i \(0.0694459\pi\)
−0.976295 + 0.216444i \(0.930554\pi\)
\(984\) 1.03817 0.0330958
\(985\) 0 0
\(986\) 4.23814 0.134970
\(987\) 5.76186i 0.183402i
\(988\) 24.6197i 0.783257i
\(989\) −0.961825 −0.0305843
\(990\) 0 0
\(991\) 25.9237 0.823492 0.411746 0.911299i \(-0.364919\pi\)
0.411746 + 0.911299i \(0.364919\pi\)
\(992\) 6.00000i 0.190500i
\(993\) − 8.51445i − 0.270198i
\(994\) −0.372303 −0.0118087
\(995\) 0 0
\(996\) 6.17106 0.195537
\(997\) 2.06708i 0.0654651i 0.999464 + 0.0327326i \(0.0104210\pi\)
−0.999464 + 0.0327326i \(0.989579\pi\)
\(998\) 8.99073i 0.284597i
\(999\) 4.23814 0.134089
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 3450.2.d.ba.2899.5 6
5.2 odd 4 3450.2.a.bp.1.2 3
5.3 odd 4 3450.2.a.bs.1.2 yes 3
5.4 even 2 inner 3450.2.d.ba.2899.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.bp.1.2 3 5.2 odd 4
3450.2.a.bs.1.2 yes 3 5.3 odd 4
3450.2.d.ba.2899.2 6 5.4 even 2 inner
3450.2.d.ba.2899.5 6 1.1 even 1 trivial