Properties

Label 1450.2.b.l.349.3
Level $1450$
Weight $2$
Character 1450.349
Analytic conductor $11.578$
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] = [1450,2,Mod(349,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1450.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.b (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: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.24681024.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 290)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.3
Root \(2.14510i\) of defining polynomial
Character \(\chi\) \(=\) 1450.349
Dual form 1450.2.b.l.349.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +3.14510i q^{3} -1.00000 q^{4} +3.14510 q^{6} +3.89167i q^{7} +1.00000i q^{8} -6.89167 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +3.14510i q^{3} -1.00000 q^{4} +3.14510 q^{6} +3.89167i q^{7} +1.00000i q^{8} -6.89167 q^{9} -4.29021 q^{11} -3.14510i q^{12} +4.34803i q^{13} +3.89167 q^{14} +1.00000 q^{16} -1.60147i q^{17} +6.89167i q^{18} +1.20293 q^{19} -12.2397 q^{21} +4.29021i q^{22} -8.34803i q^{23} -3.14510 q^{24} +4.34803 q^{26} -12.2397i q^{27} -3.89167i q^{28} +1.00000 q^{29} -2.39853 q^{31} -1.00000i q^{32} -13.4931i q^{33} -1.60147 q^{34} +6.89167 q^{36} +9.78334i q^{37} -1.20293i q^{38} -13.6750 q^{39} +5.78334 q^{41} +12.2397i q^{42} +3.60147i q^{43} +4.29021 q^{44} -8.34803 q^{46} -8.58041i q^{47} +3.14510i q^{48} -8.14510 q^{49} +5.03677 q^{51} -4.34803i q^{52} +4.80440i q^{53} -12.2397 q^{54} -3.89167 q^{56} +3.78334i q^{57} -1.00000i q^{58} +4.05783 q^{59} +11.4353 q^{61} +2.39853i q^{62} -26.8201i q^{63} -1.00000 q^{64} -13.4931 q^{66} -9.08727i q^{67} +1.60147i q^{68} +26.2554 q^{69} -12.0000 q^{71} -6.89167i q^{72} -2.39853i q^{73} +9.78334 q^{74} -1.20293 q^{76} -16.6961i q^{77} +13.6750i q^{78} -7.55096 q^{79} +17.8201 q^{81} -5.78334i q^{82} +2.79707i q^{83} +12.2397 q^{84} +3.60147 q^{86} +3.14510i q^{87} -4.29021i q^{88} +4.29021 q^{89} -16.9211 q^{91} +8.34803i q^{92} -7.54364i q^{93} -8.58041 q^{94} +3.14510 q^{96} +0.348034i q^{97} +8.14510i q^{98} +29.5667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 6 q^{6} - 12 q^{9} - 6 q^{14} + 6 q^{16} - 24 q^{21} - 6 q^{24} + 6 q^{26} + 6 q^{29} - 18 q^{31} - 6 q^{34} + 12 q^{36} + 6 q^{39} - 24 q^{41} - 30 q^{46} - 36 q^{49} - 12 q^{51} - 24 q^{54} + 6 q^{56} + 30 q^{59} + 30 q^{61} - 6 q^{64} - 48 q^{66} + 18 q^{69} - 72 q^{71} - 18 q^{79} + 6 q^{81} + 24 q^{84} + 18 q^{86} - 48 q^{91} + 6 q^{96} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 3.14510i 1.81583i 0.419159 + 0.907913i \(0.362325\pi\)
−0.419159 + 0.907913i \(0.637675\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 3.14510 1.28398
\(7\) 3.89167i 1.47091i 0.677572 + 0.735457i \(0.263032\pi\)
−0.677572 + 0.735457i \(0.736968\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −6.89167 −2.29722
\(10\) 0 0
\(11\) −4.29021 −1.29355 −0.646773 0.762683i \(-0.723882\pi\)
−0.646773 + 0.762683i \(0.723882\pi\)
\(12\) − 3.14510i − 0.907913i
\(13\) 4.34803i 1.20593i 0.797769 + 0.602964i \(0.206014\pi\)
−0.797769 + 0.602964i \(0.793986\pi\)
\(14\) 3.89167 1.04009
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.60147i − 0.388412i −0.980961 0.194206i \(-0.937787\pi\)
0.980961 0.194206i \(-0.0622131\pi\)
\(18\) 6.89167i 1.62438i
\(19\) 1.20293 0.275971 0.137986 0.990434i \(-0.455937\pi\)
0.137986 + 0.990434i \(0.455937\pi\)
\(20\) 0 0
\(21\) −12.2397 −2.67092
\(22\) 4.29021i 0.914675i
\(23\) − 8.34803i − 1.74069i −0.492447 0.870343i \(-0.663897\pi\)
0.492447 0.870343i \(-0.336103\pi\)
\(24\) −3.14510 −0.641991
\(25\) 0 0
\(26\) 4.34803 0.852720
\(27\) − 12.2397i − 2.35553i
\(28\) − 3.89167i − 0.735457i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −2.39853 −0.430790 −0.215395 0.976527i \(-0.569104\pi\)
−0.215395 + 0.976527i \(0.569104\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 13.4931i − 2.34885i
\(34\) −1.60147 −0.274649
\(35\) 0 0
\(36\) 6.89167 1.14861
\(37\) 9.78334i 1.60837i 0.594378 + 0.804186i \(0.297398\pi\)
−0.594378 + 0.804186i \(0.702602\pi\)
\(38\) − 1.20293i − 0.195141i
\(39\) −13.6750 −2.18975
\(40\) 0 0
\(41\) 5.78334 0.903206 0.451603 0.892219i \(-0.350852\pi\)
0.451603 + 0.892219i \(0.350852\pi\)
\(42\) 12.2397i 1.88863i
\(43\) 3.60147i 0.549218i 0.961556 + 0.274609i \(0.0885485\pi\)
−0.961556 + 0.274609i \(0.911452\pi\)
\(44\) 4.29021 0.646773
\(45\) 0 0
\(46\) −8.34803 −1.23085
\(47\) − 8.58041i − 1.25158i −0.779991 0.625791i \(-0.784776\pi\)
0.779991 0.625791i \(-0.215224\pi\)
\(48\) 3.14510i 0.453956i
\(49\) −8.14510 −1.16359
\(50\) 0 0
\(51\) 5.03677 0.705289
\(52\) − 4.34803i − 0.602964i
\(53\) 4.80440i 0.659935i 0.943992 + 0.329967i \(0.107038\pi\)
−0.943992 + 0.329967i \(0.892962\pi\)
\(54\) −12.2397 −1.66561
\(55\) 0 0
\(56\) −3.89167 −0.520046
\(57\) 3.78334i 0.501116i
\(58\) − 1.00000i − 0.131306i
\(59\) 4.05783 0.528284 0.264142 0.964484i \(-0.414911\pi\)
0.264142 + 0.964484i \(0.414911\pi\)
\(60\) 0 0
\(61\) 11.4353 1.46414 0.732071 0.681229i \(-0.238554\pi\)
0.732071 + 0.681229i \(0.238554\pi\)
\(62\) 2.39853i 0.304614i
\(63\) − 26.8201i − 3.37902i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −13.4931 −1.66089
\(67\) − 9.08727i − 1.11019i −0.831788 0.555094i \(-0.812682\pi\)
0.831788 0.555094i \(-0.187318\pi\)
\(68\) 1.60147i 0.194206i
\(69\) 26.2554 3.16078
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) − 6.89167i − 0.812191i
\(73\) − 2.39853i − 0.280727i −0.990100 0.140364i \(-0.955173\pi\)
0.990100 0.140364i \(-0.0448272\pi\)
\(74\) 9.78334 1.13729
\(75\) 0 0
\(76\) −1.20293 −0.137986
\(77\) − 16.6961i − 1.90269i
\(78\) 13.6750i 1.54839i
\(79\) −7.55096 −0.849550 −0.424775 0.905299i \(-0.639647\pi\)
−0.424775 + 0.905299i \(0.639647\pi\)
\(80\) 0 0
\(81\) 17.8201 1.98001
\(82\) − 5.78334i − 0.638663i
\(83\) 2.79707i 0.307018i 0.988147 + 0.153509i \(0.0490574\pi\)
−0.988147 + 0.153509i \(0.950943\pi\)
\(84\) 12.2397 1.33546
\(85\) 0 0
\(86\) 3.60147 0.388356
\(87\) 3.14510i 0.337190i
\(88\) − 4.29021i − 0.457337i
\(89\) 4.29021 0.454761 0.227380 0.973806i \(-0.426984\pi\)
0.227380 + 0.973806i \(0.426984\pi\)
\(90\) 0 0
\(91\) −16.9211 −1.77382
\(92\) 8.34803i 0.870343i
\(93\) − 7.54364i − 0.782239i
\(94\) −8.58041 −0.885002
\(95\) 0 0
\(96\) 3.14510 0.320996
\(97\) 0.348034i 0.0353375i 0.999844 + 0.0176687i \(0.00562443\pi\)
−0.999844 + 0.0176687i \(0.994376\pi\)
\(98\) 8.14510i 0.822780i
\(99\) 29.5667 2.97156
\(100\) 0 0
\(101\) −18.4216 −1.83302 −0.916508 0.400017i \(-0.869004\pi\)
−0.916508 + 0.400017i \(0.869004\pi\)
\(102\) − 5.03677i − 0.498715i
\(103\) 12.0735i 1.18964i 0.803858 + 0.594821i \(0.202777\pi\)
−0.803858 + 0.594821i \(0.797223\pi\)
\(104\) −4.34803 −0.426360
\(105\) 0 0
\(106\) 4.80440 0.466644
\(107\) 5.78334i 0.559097i 0.960132 + 0.279548i \(0.0901847\pi\)
−0.960132 + 0.279548i \(0.909815\pi\)
\(108\) 12.2397i 1.17777i
\(109\) −10.7971 −1.03417 −0.517086 0.855934i \(-0.672983\pi\)
−0.517086 + 0.855934i \(0.672983\pi\)
\(110\) 0 0
\(111\) −30.7696 −2.92052
\(112\) 3.89167i 0.367728i
\(113\) 12.2324i 1.15073i 0.817898 + 0.575363i \(0.195139\pi\)
−0.817898 + 0.575363i \(0.804861\pi\)
\(114\) 3.78334 0.354342
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) − 29.9652i − 2.77029i
\(118\) − 4.05783i − 0.373553i
\(119\) 6.23238 0.571321
\(120\) 0 0
\(121\) 7.40586 0.673260
\(122\) − 11.4353i − 1.03530i
\(123\) 18.1892i 1.64007i
\(124\) 2.39853 0.215395
\(125\) 0 0
\(126\) −26.8201 −2.38933
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −11.3270 −0.997285
\(130\) 0 0
\(131\) −20.9863 −1.83358 −0.916790 0.399371i \(-0.869229\pi\)
−0.916790 + 0.399371i \(0.869229\pi\)
\(132\) 13.4931i 1.17443i
\(133\) 4.68141i 0.405930i
\(134\) −9.08727 −0.785021
\(135\) 0 0
\(136\) 1.60147 0.137325
\(137\) − 3.65197i − 0.312009i −0.987756 0.156004i \(-0.950139\pi\)
0.987756 0.156004i \(-0.0498614\pi\)
\(138\) − 26.2554i − 2.23501i
\(139\) −13.6750 −1.15990 −0.579950 0.814652i \(-0.696928\pi\)
−0.579950 + 0.814652i \(0.696928\pi\)
\(140\) 0 0
\(141\) 26.9863 2.27265
\(142\) 12.0000i 1.00702i
\(143\) − 18.6540i − 1.55992i
\(144\) −6.89167 −0.574306
\(145\) 0 0
\(146\) −2.39853 −0.198504
\(147\) − 25.6172i − 2.11287i
\(148\) − 9.78334i − 0.804186i
\(149\) −7.27648 −0.596112 −0.298056 0.954548i \(-0.596338\pi\)
−0.298056 + 0.954548i \(0.596338\pi\)
\(150\) 0 0
\(151\) −8.69607 −0.707676 −0.353838 0.935307i \(-0.615124\pi\)
−0.353838 + 0.935307i \(0.615124\pi\)
\(152\) 1.20293i 0.0975706i
\(153\) 11.0368i 0.892270i
\(154\) −16.6961 −1.34541
\(155\) 0 0
\(156\) 13.6750 1.09488
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 7.55096i 0.600723i
\(159\) −15.1103 −1.19833
\(160\) 0 0
\(161\) 32.4878 2.56040
\(162\) − 17.8201i − 1.40008i
\(163\) − 0.580411i − 0.0454613i −0.999742 0.0227306i \(-0.992764\pi\)
0.999742 0.0227306i \(-0.00723601\pi\)
\(164\) −5.78334 −0.451603
\(165\) 0 0
\(166\) 2.79707 0.217095
\(167\) − 20.2554i − 1.56741i −0.621132 0.783706i \(-0.713327\pi\)
0.621132 0.783706i \(-0.286673\pi\)
\(168\) − 12.2397i − 0.944314i
\(169\) −5.90540 −0.454261
\(170\) 0 0
\(171\) −8.29021 −0.633968
\(172\) − 3.60147i − 0.274609i
\(173\) − 1.94217i − 0.147661i −0.997271 0.0738303i \(-0.976478\pi\)
0.997271 0.0738303i \(-0.0235223\pi\)
\(174\) 3.14510 0.238430
\(175\) 0 0
\(176\) −4.29021 −0.323386
\(177\) 12.7623i 0.959272i
\(178\) − 4.29021i − 0.321564i
\(179\) 10.9284 0.816830 0.408415 0.912796i \(-0.366082\pi\)
0.408415 + 0.912796i \(0.366082\pi\)
\(180\) 0 0
\(181\) −1.20293 −0.0894132 −0.0447066 0.999000i \(-0.514235\pi\)
−0.0447066 + 0.999000i \(0.514235\pi\)
\(182\) 16.9211i 1.25428i
\(183\) 35.9652i 2.65863i
\(184\) 8.34803 0.615425
\(185\) 0 0
\(186\) −7.54364 −0.553126
\(187\) 6.87062i 0.502429i
\(188\) 8.58041i 0.625791i
\(189\) 47.6329 3.46478
\(190\) 0 0
\(191\) −15.7760 −1.14151 −0.570756 0.821120i \(-0.693350\pi\)
−0.570756 + 0.821120i \(0.693350\pi\)
\(192\) − 3.14510i − 0.226978i
\(193\) 9.94217i 0.715653i 0.933788 + 0.357827i \(0.116482\pi\)
−0.933788 + 0.357827i \(0.883518\pi\)
\(194\) 0.348034 0.0249874
\(195\) 0 0
\(196\) 8.14510 0.581793
\(197\) 25.1681i 1.79316i 0.442885 + 0.896578i \(0.353955\pi\)
−0.442885 + 0.896578i \(0.646045\pi\)
\(198\) − 29.5667i − 2.10121i
\(199\) 18.7696 1.33054 0.665271 0.746602i \(-0.268316\pi\)
0.665271 + 0.746602i \(0.268316\pi\)
\(200\) 0 0
\(201\) 28.5804 2.01591
\(202\) 18.4216i 1.29614i
\(203\) 3.89167i 0.273142i
\(204\) −5.03677 −0.352645
\(205\) 0 0
\(206\) 12.0735 0.841204
\(207\) 57.5319i 3.99874i
\(208\) 4.34803i 0.301482i
\(209\) −5.16082 −0.356981
\(210\) 0 0
\(211\) 12.2902 0.846093 0.423046 0.906108i \(-0.360961\pi\)
0.423046 + 0.906108i \(0.360961\pi\)
\(212\) − 4.80440i − 0.329967i
\(213\) − 37.7412i − 2.58599i
\(214\) 5.78334 0.395341
\(215\) 0 0
\(216\) 12.2397 0.832806
\(217\) − 9.33431i − 0.633654i
\(218\) 10.7971i 0.731270i
\(219\) 7.54364 0.509752
\(220\) 0 0
\(221\) 6.96323 0.468397
\(222\) 30.7696i 2.06512i
\(223\) 20.2324i 1.35486i 0.735587 + 0.677430i \(0.236906\pi\)
−0.735587 + 0.677430i \(0.763094\pi\)
\(224\) 3.89167 0.260023
\(225\) 0 0
\(226\) 12.2324 0.813686
\(227\) − 8.98627i − 0.596440i −0.954497 0.298220i \(-0.903607\pi\)
0.954497 0.298220i \(-0.0963929\pi\)
\(228\) − 3.78334i − 0.250558i
\(229\) 21.2692 1.40551 0.702753 0.711434i \(-0.251954\pi\)
0.702753 + 0.711434i \(0.251954\pi\)
\(230\) 0 0
\(231\) 52.5108 3.45496
\(232\) 1.00000i 0.0656532i
\(233\) 11.1608i 0.731170i 0.930778 + 0.365585i \(0.119131\pi\)
−0.930778 + 0.365585i \(0.880869\pi\)
\(234\) −29.9652 −1.95889
\(235\) 0 0
\(236\) −4.05783 −0.264142
\(237\) − 23.7486i − 1.54263i
\(238\) − 6.23238i − 0.403985i
\(239\) 1.92645 0.124612 0.0623059 0.998057i \(-0.480155\pi\)
0.0623059 + 0.998057i \(0.480155\pi\)
\(240\) 0 0
\(241\) −17.7255 −1.14180 −0.570900 0.821019i \(-0.693406\pi\)
−0.570900 + 0.821019i \(0.693406\pi\)
\(242\) − 7.40586i − 0.476067i
\(243\) 19.3270i 1.23983i
\(244\) −11.4353 −0.732071
\(245\) 0 0
\(246\) 18.1892 1.15970
\(247\) 5.23039i 0.332801i
\(248\) − 2.39853i − 0.152307i
\(249\) −8.79707 −0.557492
\(250\) 0 0
\(251\) −14.3638 −0.906632 −0.453316 0.891350i \(-0.649759\pi\)
−0.453316 + 0.891350i \(0.649759\pi\)
\(252\) 26.8201i 1.68951i
\(253\) 35.8148i 2.25166i
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.3638i 0.895986i 0.894037 + 0.447993i \(0.147861\pi\)
−0.894037 + 0.447993i \(0.852139\pi\)
\(258\) 11.3270i 0.705187i
\(259\) −38.0735 −2.36578
\(260\) 0 0
\(261\) −6.89167 −0.426584
\(262\) 20.9863i 1.29654i
\(263\) − 12.0000i − 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) 13.4931 0.830445
\(265\) 0 0
\(266\) 4.68141 0.287036
\(267\) 13.4931i 0.825767i
\(268\) 9.08727i 0.555094i
\(269\) 12.1083 0.738258 0.369129 0.929378i \(-0.379656\pi\)
0.369129 + 0.929378i \(0.379656\pi\)
\(270\) 0 0
\(271\) 3.49314 0.212193 0.106096 0.994356i \(-0.466165\pi\)
0.106096 + 0.994356i \(0.466165\pi\)
\(272\) − 1.60147i − 0.0971031i
\(273\) − 53.2186i − 3.22094i
\(274\) −3.65197 −0.220623
\(275\) 0 0
\(276\) −26.2554 −1.58039
\(277\) − 22.1471i − 1.33069i −0.746536 0.665345i \(-0.768284\pi\)
0.746536 0.665345i \(-0.231716\pi\)
\(278\) 13.6750i 0.820173i
\(279\) 16.5299 0.989620
\(280\) 0 0
\(281\) −14.8779 −0.887544 −0.443772 0.896140i \(-0.646360\pi\)
−0.443772 + 0.896140i \(0.646360\pi\)
\(282\) − 26.9863i − 1.60701i
\(283\) 5.20293i 0.309282i 0.987971 + 0.154641i \(0.0494221\pi\)
−0.987971 + 0.154641i \(0.950578\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −18.6540 −1.10303
\(287\) 22.5069i 1.32854i
\(288\) 6.89167i 0.406096i
\(289\) 14.4353 0.849136
\(290\) 0 0
\(291\) −1.09460 −0.0641667
\(292\) 2.39853i 0.140364i
\(293\) 17.7833i 1.03891i 0.854497 + 0.519457i \(0.173866\pi\)
−0.854497 + 0.519457i \(0.826134\pi\)
\(294\) −25.6172 −1.49402
\(295\) 0 0
\(296\) −9.78334 −0.568645
\(297\) 52.5108i 3.04699i
\(298\) 7.27648i 0.421515i
\(299\) 36.2975 2.09914
\(300\) 0 0
\(301\) −14.0157 −0.807853
\(302\) 8.69607i 0.500402i
\(303\) − 57.9378i − 3.32844i
\(304\) 1.20293 0.0689928
\(305\) 0 0
\(306\) 11.0368 0.630930
\(307\) − 2.40586i − 0.137310i −0.997640 0.0686549i \(-0.978129\pi\)
0.997640 0.0686549i \(-0.0218707\pi\)
\(308\) 16.6961i 0.951347i
\(309\) −37.9725 −2.16018
\(310\) 0 0
\(311\) 16.2470 0.921285 0.460642 0.887586i \(-0.347619\pi\)
0.460642 + 0.887586i \(0.347619\pi\)
\(312\) − 13.6750i − 0.774195i
\(313\) − 12.1745i − 0.688146i −0.938943 0.344073i \(-0.888193\pi\)
0.938943 0.344073i \(-0.111807\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) 7.55096 0.424775
\(317\) 13.6823i 0.768477i 0.923234 + 0.384238i \(0.125536\pi\)
−0.923234 + 0.384238i \(0.874464\pi\)
\(318\) 15.1103i 0.847345i
\(319\) −4.29021 −0.240205
\(320\) 0 0
\(321\) −18.1892 −1.01522
\(322\) − 32.4878i − 1.81047i
\(323\) − 1.92645i − 0.107191i
\(324\) −17.8201 −0.990006
\(325\) 0 0
\(326\) −0.580411 −0.0321460
\(327\) − 33.9579i − 1.87788i
\(328\) 5.78334i 0.319332i
\(329\) 33.3921 1.84097
\(330\) 0 0
\(331\) 8.87062 0.487573 0.243787 0.969829i \(-0.421610\pi\)
0.243787 + 0.969829i \(0.421610\pi\)
\(332\) − 2.79707i − 0.153509i
\(333\) − 67.4236i − 3.69479i
\(334\) −20.2554 −1.10833
\(335\) 0 0
\(336\) −12.2397 −0.667731
\(337\) 14.1819i 0.772536i 0.922387 + 0.386268i \(0.126236\pi\)
−0.922387 + 0.386268i \(0.873764\pi\)
\(338\) 5.90540i 0.321211i
\(339\) −38.4721 −2.08952
\(340\) 0 0
\(341\) 10.2902 0.557246
\(342\) 8.29021i 0.448283i
\(343\) − 4.45636i − 0.240621i
\(344\) −3.60147 −0.194178
\(345\) 0 0
\(346\) −1.94217 −0.104412
\(347\) 27.1755i 1.45886i 0.684058 + 0.729428i \(0.260214\pi\)
−0.684058 + 0.729428i \(0.739786\pi\)
\(348\) − 3.14510i − 0.168595i
\(349\) −1.31859 −0.0705824 −0.0352912 0.999377i \(-0.511236\pi\)
−0.0352912 + 0.999377i \(0.511236\pi\)
\(350\) 0 0
\(351\) 53.2186 2.84060
\(352\) 4.29021i 0.228669i
\(353\) 13.8990i 0.739769i 0.929078 + 0.369885i \(0.120603\pi\)
−0.929078 + 0.369885i \(0.879397\pi\)
\(354\) 12.7623 0.678308
\(355\) 0 0
\(356\) −4.29021 −0.227380
\(357\) 19.6015i 1.03742i
\(358\) − 10.9284i − 0.577586i
\(359\) −1.26076 −0.0665403 −0.0332702 0.999446i \(-0.510592\pi\)
−0.0332702 + 0.999446i \(0.510592\pi\)
\(360\) 0 0
\(361\) −17.5530 −0.923840
\(362\) 1.20293i 0.0632247i
\(363\) 23.2922i 1.22252i
\(364\) 16.9211 0.886908
\(365\) 0 0
\(366\) 35.9652 1.87993
\(367\) 17.2765i 0.901825i 0.892568 + 0.450912i \(0.148901\pi\)
−0.892568 + 0.450912i \(0.851099\pi\)
\(368\) − 8.34803i − 0.435171i
\(369\) −39.8569 −2.07487
\(370\) 0 0
\(371\) −18.6971 −0.970707
\(372\) 7.54364i 0.391119i
\(373\) 32.9515i 1.70616i 0.521777 + 0.853082i \(0.325269\pi\)
−0.521777 + 0.853082i \(0.674731\pi\)
\(374\) 6.87062 0.355271
\(375\) 0 0
\(376\) 8.58041 0.442501
\(377\) 4.34803i 0.223935i
\(378\) − 47.6329i − 2.44997i
\(379\) 4.87062 0.250187 0.125093 0.992145i \(-0.460077\pi\)
0.125093 + 0.992145i \(0.460077\pi\)
\(380\) 0 0
\(381\) 25.1608 1.28903
\(382\) 15.7760i 0.807171i
\(383\) − 2.06622i − 0.105579i −0.998606 0.0527894i \(-0.983189\pi\)
0.998606 0.0527894i \(-0.0168112\pi\)
\(384\) −3.14510 −0.160498
\(385\) 0 0
\(386\) 9.94217 0.506043
\(387\) − 24.8201i − 1.26168i
\(388\) − 0.348034i − 0.0176687i
\(389\) −23.9725 −1.21546 −0.607728 0.794145i \(-0.707919\pi\)
−0.607728 + 0.794145i \(0.707919\pi\)
\(390\) 0 0
\(391\) −13.3691 −0.676104
\(392\) − 8.14510i − 0.411390i
\(393\) − 66.0040i − 3.32946i
\(394\) 25.1681 1.26795
\(395\) 0 0
\(396\) −29.5667 −1.48578
\(397\) − 24.1544i − 1.21228i −0.795360 0.606138i \(-0.792718\pi\)
0.795360 0.606138i \(-0.207282\pi\)
\(398\) − 18.7696i − 0.940836i
\(399\) −14.7235 −0.737098
\(400\) 0 0
\(401\) 16.2470 0.811338 0.405669 0.914020i \(-0.367039\pi\)
0.405669 + 0.914020i \(0.367039\pi\)
\(402\) − 28.5804i − 1.42546i
\(403\) − 10.4289i − 0.519501i
\(404\) 18.4216 0.916508
\(405\) 0 0
\(406\) 3.89167 0.193140
\(407\) − 41.9725i − 2.08050i
\(408\) 5.03677i 0.249357i
\(409\) −9.08727 −0.449337 −0.224668 0.974435i \(-0.572130\pi\)
−0.224668 + 0.974435i \(0.572130\pi\)
\(410\) 0 0
\(411\) 11.4858 0.566553
\(412\) − 12.0735i − 0.594821i
\(413\) 15.7917i 0.777060i
\(414\) 57.5319 2.82754
\(415\) 0 0
\(416\) 4.34803 0.213180
\(417\) − 43.0093i − 2.10618i
\(418\) 5.16082i 0.252424i
\(419\) −32.4446 −1.58502 −0.792512 0.609856i \(-0.791227\pi\)
−0.792512 + 0.609856i \(0.791227\pi\)
\(420\) 0 0
\(421\) 25.5667 1.24604 0.623022 0.782204i \(-0.285905\pi\)
0.623022 + 0.782204i \(0.285905\pi\)
\(422\) − 12.2902i − 0.598278i
\(423\) 59.1334i 2.87516i
\(424\) −4.80440 −0.233322
\(425\) 0 0
\(426\) −37.7412 −1.82857
\(427\) 44.5025i 2.15362i
\(428\) − 5.78334i − 0.279548i
\(429\) 58.6686 2.83255
\(430\) 0 0
\(431\) −1.70979 −0.0823579 −0.0411790 0.999152i \(-0.513111\pi\)
−0.0411790 + 0.999152i \(0.513111\pi\)
\(432\) − 12.2397i − 0.588883i
\(433\) − 23.7412i − 1.14093i −0.821322 0.570465i \(-0.806763\pi\)
0.821322 0.570465i \(-0.193237\pi\)
\(434\) −9.33431 −0.448061
\(435\) 0 0
\(436\) 10.7971 0.517086
\(437\) − 10.0421i − 0.480379i
\(438\) − 7.54364i − 0.360449i
\(439\) 8.47941 0.404700 0.202350 0.979313i \(-0.435142\pi\)
0.202350 + 0.979313i \(0.435142\pi\)
\(440\) 0 0
\(441\) 56.1334 2.67302
\(442\) − 6.96323i − 0.331207i
\(443\) 18.4490i 0.876540i 0.898843 + 0.438270i \(0.144409\pi\)
−0.898843 + 0.438270i \(0.855591\pi\)
\(444\) 30.7696 1.46026
\(445\) 0 0
\(446\) 20.2324 0.958031
\(447\) − 22.8853i − 1.08244i
\(448\) − 3.89167i − 0.183864i
\(449\) 11.3775 0.536936 0.268468 0.963289i \(-0.413483\pi\)
0.268468 + 0.963289i \(0.413483\pi\)
\(450\) 0 0
\(451\) −24.8117 −1.16834
\(452\) − 12.2324i − 0.575363i
\(453\) − 27.3500i − 1.28502i
\(454\) −8.98627 −0.421747
\(455\) 0 0
\(456\) −3.78334 −0.177171
\(457\) 4.40586i 0.206098i 0.994676 + 0.103049i \(0.0328598\pi\)
−0.994676 + 0.103049i \(0.967140\pi\)
\(458\) − 21.2692i − 0.993842i
\(459\) −19.6015 −0.914918
\(460\) 0 0
\(461\) 15.3113 0.713116 0.356558 0.934273i \(-0.383950\pi\)
0.356558 + 0.934273i \(0.383950\pi\)
\(462\) − 52.5108i − 2.44303i
\(463\) − 17.0873i − 0.794113i −0.917794 0.397056i \(-0.870032\pi\)
0.917794 0.397056i \(-0.129968\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) 11.1608 0.517015
\(467\) 22.3711i 1.03521i 0.855620 + 0.517605i \(0.173176\pi\)
−0.855620 + 0.517605i \(0.826824\pi\)
\(468\) 29.9652i 1.38514i
\(469\) 35.3647 1.63299
\(470\) 0 0
\(471\) −31.4510 −1.44919
\(472\) 4.05783i 0.186777i
\(473\) − 15.4510i − 0.710439i
\(474\) −23.7486 −1.09081
\(475\) 0 0
\(476\) −6.23238 −0.285661
\(477\) − 33.1103i − 1.51602i
\(478\) − 1.92645i − 0.0881139i
\(479\) 8.00733 0.365864 0.182932 0.983126i \(-0.441441\pi\)
0.182932 + 0.983126i \(0.441441\pi\)
\(480\) 0 0
\(481\) −42.5383 −1.93958
\(482\) 17.7255i 0.807375i
\(483\) 102.177i 4.64924i
\(484\) −7.40586 −0.336630
\(485\) 0 0
\(486\) 19.3270 0.876690
\(487\) 17.0441i 0.772342i 0.922427 + 0.386171i \(0.126203\pi\)
−0.922427 + 0.386171i \(0.873797\pi\)
\(488\) 11.4353i 0.517652i
\(489\) 1.82545 0.0825498
\(490\) 0 0
\(491\) 37.0873 1.67373 0.836863 0.547413i \(-0.184387\pi\)
0.836863 + 0.547413i \(0.184387\pi\)
\(492\) − 18.1892i − 0.820033i
\(493\) − 1.60147i − 0.0721264i
\(494\) 5.23039 0.235326
\(495\) 0 0
\(496\) −2.39853 −0.107697
\(497\) − 46.7001i − 2.09478i
\(498\) 8.79707i 0.394206i
\(499\) 23.5089 1.05240 0.526200 0.850361i \(-0.323616\pi\)
0.526200 + 0.850361i \(0.323616\pi\)
\(500\) 0 0
\(501\) 63.7054 2.84615
\(502\) 14.3638i 0.641086i
\(503\) 17.1608i 0.765163i 0.923922 + 0.382582i \(0.124965\pi\)
−0.923922 + 0.382582i \(0.875035\pi\)
\(504\) 26.8201 1.19466
\(505\) 0 0
\(506\) 35.8148 1.59216
\(507\) − 18.5731i − 0.824860i
\(508\) 8.00000i 0.354943i
\(509\) 14.3638 0.636662 0.318331 0.947980i \(-0.396878\pi\)
0.318331 + 0.947980i \(0.396878\pi\)
\(510\) 0 0
\(511\) 9.33431 0.412925
\(512\) − 1.00000i − 0.0441942i
\(513\) − 14.7235i − 0.650059i
\(514\) 14.3638 0.633558
\(515\) 0 0
\(516\) 11.3270 0.498642
\(517\) 36.8117i 1.61898i
\(518\) 38.0735i 1.67286i
\(519\) 6.10833 0.268126
\(520\) 0 0
\(521\) −32.0388 −1.40364 −0.701822 0.712352i \(-0.747630\pi\)
−0.701822 + 0.712352i \(0.747630\pi\)
\(522\) 6.89167i 0.301640i
\(523\) − 26.9442i − 1.17819i −0.808065 0.589093i \(-0.799485\pi\)
0.808065 0.589093i \(-0.200515\pi\)
\(524\) 20.9863 0.916790
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 3.84117i 0.167324i
\(528\) − 13.4931i − 0.587213i
\(529\) −46.6897 −2.02999
\(530\) 0 0
\(531\) −27.9652 −1.21359
\(532\) − 4.68141i − 0.202965i
\(533\) 25.1462i 1.08920i
\(534\) 13.4931 0.583905
\(535\) 0 0
\(536\) 9.08727 0.392510
\(537\) 34.3711i 1.48322i
\(538\) − 12.1083i − 0.522027i
\(539\) 34.9442 1.50515
\(540\) 0 0
\(541\) −13.5005 −0.580430 −0.290215 0.956961i \(-0.593727\pi\)
−0.290215 + 0.956961i \(0.593727\pi\)
\(542\) − 3.49314i − 0.150043i
\(543\) − 3.78334i − 0.162359i
\(544\) −1.60147 −0.0686623
\(545\) 0 0
\(546\) −53.2186 −2.27755
\(547\) − 35.1755i − 1.50399i −0.659166 0.751997i \(-0.729091\pi\)
0.659166 0.751997i \(-0.270909\pi\)
\(548\) 3.65197i 0.156004i
\(549\) −78.8084 −3.36346
\(550\) 0 0
\(551\) 1.20293 0.0512466
\(552\) 26.2554i 1.11751i
\(553\) − 29.3859i − 1.24961i
\(554\) −22.1471 −0.940940
\(555\) 0 0
\(556\) 13.6750 0.579950
\(557\) − 18.6382i − 0.789728i −0.918740 0.394864i \(-0.870792\pi\)
0.918740 0.394864i \(-0.129208\pi\)
\(558\) − 16.5299i − 0.699767i
\(559\) −15.6593 −0.662318
\(560\) 0 0
\(561\) −21.6088 −0.912324
\(562\) 14.8779i 0.627588i
\(563\) − 0.421581i − 0.0177675i −0.999961 0.00888376i \(-0.997172\pi\)
0.999961 0.00888376i \(-0.00282783\pi\)
\(564\) −26.9863 −1.13633
\(565\) 0 0
\(566\) 5.20293 0.218696
\(567\) 69.3500i 2.91243i
\(568\) − 12.0000i − 0.503509i
\(569\) −7.49314 −0.314129 −0.157064 0.987588i \(-0.550203\pi\)
−0.157064 + 0.987588i \(0.550203\pi\)
\(570\) 0 0
\(571\) 6.46369 0.270497 0.135249 0.990812i \(-0.456817\pi\)
0.135249 + 0.990812i \(0.456817\pi\)
\(572\) 18.6540i 0.779961i
\(573\) − 49.6172i − 2.07279i
\(574\) 22.5069 0.939418
\(575\) 0 0
\(576\) 6.89167 0.287153
\(577\) 31.5594i 1.31383i 0.753963 + 0.656917i \(0.228140\pi\)
−0.753963 + 0.656917i \(0.771860\pi\)
\(578\) − 14.4353i − 0.600430i
\(579\) −31.2692 −1.29950
\(580\) 0 0
\(581\) −10.8853 −0.451597
\(582\) 1.09460i 0.0453727i
\(583\) − 20.6118i − 0.853656i
\(584\) 2.39853 0.0992521
\(585\) 0 0
\(586\) 17.7833 0.734623
\(587\) 11.5941i 0.478541i 0.970953 + 0.239271i \(0.0769083\pi\)
−0.970953 + 0.239271i \(0.923092\pi\)
\(588\) 25.6172i 1.05643i
\(589\) −2.88527 −0.118886
\(590\) 0 0
\(591\) −79.1564 −3.25606
\(592\) 9.78334i 0.402093i
\(593\) − 18.4648i − 0.758257i −0.925344 0.379128i \(-0.876224\pi\)
0.925344 0.379128i \(-0.123776\pi\)
\(594\) 52.5108 2.15455
\(595\) 0 0
\(596\) 7.27648 0.298056
\(597\) 59.0324i 2.41603i
\(598\) − 36.2975i − 1.48432i
\(599\) 28.3711 1.15921 0.579605 0.814897i \(-0.303207\pi\)
0.579605 + 0.814897i \(0.303207\pi\)
\(600\) 0 0
\(601\) −34.4648 −1.40585 −0.702923 0.711266i \(-0.748122\pi\)
−0.702923 + 0.711266i \(0.748122\pi\)
\(602\) 14.0157i 0.571238i
\(603\) 62.6265i 2.55035i
\(604\) 8.69607 0.353838
\(605\) 0 0
\(606\) −57.9378 −2.35356
\(607\) − 36.7275i − 1.49072i −0.666660 0.745362i \(-0.732277\pi\)
0.666660 0.745362i \(-0.267723\pi\)
\(608\) − 1.20293i − 0.0487853i
\(609\) −12.2397 −0.495978
\(610\) 0 0
\(611\) 37.3079 1.50932
\(612\) − 11.0368i − 0.446135i
\(613\) 8.95149i 0.361547i 0.983525 + 0.180774i \(0.0578601\pi\)
−0.983525 + 0.180774i \(0.942140\pi\)
\(614\) −2.40586 −0.0970927
\(615\) 0 0
\(616\) 16.6961 0.672704
\(617\) − 22.3985i − 0.901731i −0.892592 0.450866i \(-0.851115\pi\)
0.892592 0.450866i \(-0.148885\pi\)
\(618\) 37.9725i 1.52748i
\(619\) −31.5392 −1.26767 −0.633834 0.773469i \(-0.718520\pi\)
−0.633834 + 0.773469i \(0.718520\pi\)
\(620\) 0 0
\(621\) −102.177 −4.10024
\(622\) − 16.2470i − 0.651447i
\(623\) 16.6961i 0.668914i
\(624\) −13.6750 −0.547439
\(625\) 0 0
\(626\) −12.1745 −0.486593
\(627\) − 16.2313i − 0.648216i
\(628\) − 10.0000i − 0.399043i
\(629\) 15.6677 0.624712
\(630\) 0 0
\(631\) −41.0598 −1.63457 −0.817283 0.576237i \(-0.804521\pi\)
−0.817283 + 0.576237i \(0.804521\pi\)
\(632\) − 7.55096i − 0.300361i
\(633\) 38.6540i 1.53636i
\(634\) 13.6823 0.543395
\(635\) 0 0
\(636\) 15.1103 0.599163
\(637\) − 35.4152i − 1.40320i
\(638\) 4.29021i 0.169851i
\(639\) 82.7001 3.27156
\(640\) 0 0
\(641\) 40.5383 1.60117 0.800583 0.599221i \(-0.204523\pi\)
0.800583 + 0.599221i \(0.204523\pi\)
\(642\) 18.1892i 0.717871i
\(643\) − 35.2765i − 1.39117i −0.718445 0.695584i \(-0.755146\pi\)
0.718445 0.695584i \(-0.244854\pi\)
\(644\) −32.4878 −1.28020
\(645\) 0 0
\(646\) −1.92645 −0.0757953
\(647\) 3.69514i 0.145271i 0.997359 + 0.0726355i \(0.0231410\pi\)
−0.997359 + 0.0726355i \(0.976859\pi\)
\(648\) 17.8201i 0.700040i
\(649\) −17.4089 −0.683360
\(650\) 0 0
\(651\) 29.3574 1.15061
\(652\) 0.580411i 0.0227306i
\(653\) − 8.52152i − 0.333473i −0.986002 0.166736i \(-0.946677\pi\)
0.986002 0.166736i \(-0.0533229\pi\)
\(654\) −33.9579 −1.32786
\(655\) 0 0
\(656\) 5.78334 0.225802
\(657\) 16.5299i 0.644893i
\(658\) − 33.3921i − 1.30176i
\(659\) 5.10193 0.198743 0.0993715 0.995050i \(-0.468317\pi\)
0.0993715 + 0.995050i \(0.468317\pi\)
\(660\) 0 0
\(661\) −14.6961 −0.571611 −0.285805 0.958288i \(-0.592261\pi\)
−0.285805 + 0.958288i \(0.592261\pi\)
\(662\) − 8.87062i − 0.344766i
\(663\) 21.9001i 0.850528i
\(664\) −2.79707 −0.108547
\(665\) 0 0
\(666\) −67.4236 −2.61261
\(667\) − 8.34803i − 0.323237i
\(668\) 20.2554i 0.783706i
\(669\) −63.6329 −2.46019
\(670\) 0 0
\(671\) −49.0598 −1.89393
\(672\) 12.2397i 0.472157i
\(673\) − 25.5813i − 0.986088i −0.870004 0.493044i \(-0.835884\pi\)
0.870004 0.493044i \(-0.164116\pi\)
\(674\) 14.1819 0.546265
\(675\) 0 0
\(676\) 5.90540 0.227131
\(677\) 21.2344i 0.816103i 0.912959 + 0.408052i \(0.133792\pi\)
−0.912959 + 0.408052i \(0.866208\pi\)
\(678\) 38.4721i 1.47751i
\(679\) −1.35443 −0.0519784
\(680\) 0 0
\(681\) 28.2628 1.08303
\(682\) − 10.2902i − 0.394032i
\(683\) − 22.1324i − 0.846874i −0.905925 0.423437i \(-0.860823\pi\)
0.905925 0.423437i \(-0.139177\pi\)
\(684\) 8.29021 0.316984
\(685\) 0 0
\(686\) −4.45636 −0.170145
\(687\) 66.8937i 2.55215i
\(688\) 3.60147i 0.137305i
\(689\) −20.8897 −0.795833
\(690\) 0 0
\(691\) 8.08088 0.307411 0.153705 0.988117i \(-0.450879\pi\)
0.153705 + 0.988117i \(0.450879\pi\)
\(692\) 1.94217i 0.0738303i
\(693\) 115.064i 4.37091i
\(694\) 27.1755 1.03157
\(695\) 0 0
\(696\) −3.14510 −0.119215
\(697\) − 9.26182i − 0.350817i
\(698\) 1.31859i 0.0499093i
\(699\) −35.1019 −1.32768
\(700\) 0 0
\(701\) 28.2902 1.06851 0.534253 0.845325i \(-0.320593\pi\)
0.534253 + 0.845325i \(0.320593\pi\)
\(702\) − 53.2186i − 2.00861i
\(703\) 11.7687i 0.443864i
\(704\) 4.29021 0.161693
\(705\) 0 0
\(706\) 13.8990 0.523096
\(707\) − 71.6907i − 2.69621i
\(708\) − 12.7623i − 0.479636i
\(709\) −52.2060 −1.96064 −0.980318 0.197423i \(-0.936743\pi\)
−0.980318 + 0.197423i \(0.936743\pi\)
\(710\) 0 0
\(711\) 52.0388 1.95161
\(712\) 4.29021i 0.160782i
\(713\) 20.0230i 0.749869i
\(714\) 19.6015 0.733566
\(715\) 0 0
\(716\) −10.9284 −0.408415
\(717\) 6.05889i 0.226273i
\(718\) 1.26076i 0.0470511i
\(719\) −27.0177 −1.00759 −0.503795 0.863823i \(-0.668063\pi\)
−0.503795 + 0.863823i \(0.668063\pi\)
\(720\) 0 0
\(721\) −46.9863 −1.74986
\(722\) 17.5530i 0.653253i
\(723\) − 55.7486i − 2.07331i
\(724\) 1.20293 0.0447066
\(725\) 0 0
\(726\) 23.2922 0.864455
\(727\) 33.7559i 1.25194i 0.779849 + 0.625968i \(0.215296\pi\)
−0.779849 + 0.625968i \(0.784704\pi\)
\(728\) − 16.9211i − 0.627138i
\(729\) −7.32499 −0.271296
\(730\) 0 0
\(731\) 5.76762 0.213323
\(732\) − 35.9652i − 1.32931i
\(733\) 9.52059i 0.351651i 0.984421 + 0.175826i \(0.0562595\pi\)
−0.984421 + 0.175826i \(0.943741\pi\)
\(734\) 17.2765 0.637686
\(735\) 0 0
\(736\) −8.34803 −0.307713
\(737\) 38.9863i 1.43608i
\(738\) 39.8569i 1.46715i
\(739\) −0.0735473 −0.00270548 −0.00135274 0.999999i \(-0.500431\pi\)
−0.00135274 + 0.999999i \(0.500431\pi\)
\(740\) 0 0
\(741\) −16.4501 −0.604309
\(742\) 18.6971i 0.686393i
\(743\) 27.4196i 1.00593i 0.864308 + 0.502964i \(0.167757\pi\)
−0.864308 + 0.502964i \(0.832243\pi\)
\(744\) 7.54364 0.276563
\(745\) 0 0
\(746\) 32.9515 1.20644
\(747\) − 19.2765i − 0.705289i
\(748\) − 6.87062i − 0.251215i
\(749\) −22.5069 −0.822383
\(750\) 0 0
\(751\) −17.0873 −0.623523 −0.311762 0.950160i \(-0.600919\pi\)
−0.311762 + 0.950160i \(0.600919\pi\)
\(752\) − 8.58041i − 0.312895i
\(753\) − 45.1755i − 1.64629i
\(754\) 4.34803 0.158346
\(755\) 0 0
\(756\) −47.6329 −1.73239
\(757\) − 25.7833i − 0.937111i −0.883434 0.468556i \(-0.844775\pi\)
0.883434 0.468556i \(-0.155225\pi\)
\(758\) − 4.87062i − 0.176909i
\(759\) −112.641 −4.08862
\(760\) 0 0
\(761\) 9.28381 0.336538 0.168269 0.985741i \(-0.446182\pi\)
0.168269 + 0.985741i \(0.446182\pi\)
\(762\) − 25.1608i − 0.911480i
\(763\) − 42.0186i − 1.52118i
\(764\) 15.7760 0.570756
\(765\) 0 0
\(766\) −2.06622 −0.0746555
\(767\) 17.6436i 0.637073i
\(768\) 3.14510i 0.113489i
\(769\) 33.1019 1.19369 0.596843 0.802358i \(-0.296421\pi\)
0.596843 + 0.802358i \(0.296421\pi\)
\(770\) 0 0
\(771\) −45.1755 −1.62696
\(772\) − 9.94217i − 0.357827i
\(773\) 25.7687i 0.926835i 0.886140 + 0.463418i \(0.153377\pi\)
−0.886140 + 0.463418i \(0.846623\pi\)
\(774\) −24.8201 −0.892141
\(775\) 0 0
\(776\) −0.348034 −0.0124937
\(777\) − 119.745i − 4.29584i
\(778\) 23.9725i 0.859457i
\(779\) 6.95696 0.249259
\(780\) 0 0
\(781\) 51.4825 1.84219
\(782\) 13.3691i 0.478078i
\(783\) − 12.2397i − 0.437411i
\(784\) −8.14510 −0.290897
\(785\) 0 0
\(786\) −66.0040 −2.35428
\(787\) 33.3500i 1.18880i 0.804170 + 0.594400i \(0.202610\pi\)
−0.804170 + 0.594400i \(0.797390\pi\)
\(788\) − 25.1681i − 0.896578i
\(789\) 37.7412 1.34362
\(790\) 0 0
\(791\) −47.6044 −1.69262
\(792\) 29.5667i 1.05061i
\(793\) 49.7211i 1.76565i
\(794\) −24.1544 −0.857208
\(795\) 0 0
\(796\) −18.7696 −0.665271
\(797\) 7.05982i 0.250072i 0.992152 + 0.125036i \(0.0399046\pi\)
−0.992152 + 0.125036i \(0.960095\pi\)
\(798\) 14.7235i 0.521207i
\(799\) −13.7412 −0.486130
\(800\) 0 0
\(801\) −29.5667 −1.04469
\(802\) − 16.2470i − 0.573703i
\(803\) 10.2902i 0.363133i
\(804\) −28.5804 −1.00795
\(805\) 0 0
\(806\) −10.4289 −0.367343
\(807\) 38.0819i 1.34055i
\(808\) − 18.4216i − 0.648069i
\(809\) −18.6814 −0.656803 −0.328402 0.944538i \(-0.606510\pi\)
−0.328402 + 0.944538i \(0.606510\pi\)
\(810\) 0 0
\(811\) 36.2133 1.27162 0.635811 0.771845i \(-0.280666\pi\)
0.635811 + 0.771845i \(0.280666\pi\)
\(812\) − 3.89167i − 0.136571i
\(813\) 10.9863i 0.385305i
\(814\) −41.9725 −1.47114
\(815\) 0 0
\(816\) 5.03677 0.176322
\(817\) 4.33231i 0.151569i
\(818\) 9.08727i 0.317729i
\(819\) 116.615 4.07485
\(820\) 0 0
\(821\) −36.8706 −1.28679 −0.643397 0.765533i \(-0.722475\pi\)
−0.643397 + 0.765533i \(0.722475\pi\)
\(822\) − 11.4858i − 0.400614i
\(823\) 39.0324i 1.36058i 0.732942 + 0.680291i \(0.238147\pi\)
−0.732942 + 0.680291i \(0.761853\pi\)
\(824\) −12.0735 −0.420602
\(825\) 0 0
\(826\) 15.7917 0.549465
\(827\) − 17.2103i − 0.598459i −0.954181 0.299230i \(-0.903270\pi\)
0.954181 0.299230i \(-0.0967297\pi\)
\(828\) − 57.5319i − 1.99937i
\(829\) −32.7034 −1.13584 −0.567918 0.823085i \(-0.692251\pi\)
−0.567918 + 0.823085i \(0.692251\pi\)
\(830\) 0 0
\(831\) 69.6549 2.41630
\(832\) − 4.34803i − 0.150741i
\(833\) 13.0441i 0.451951i
\(834\) −43.0093 −1.48929
\(835\) 0 0
\(836\) 5.16082 0.178491
\(837\) 29.3574i 1.01474i
\(838\) 32.4446i 1.12078i
\(839\) 32.3049 1.11529 0.557644 0.830080i \(-0.311706\pi\)
0.557644 + 0.830080i \(0.311706\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) − 25.5667i − 0.881086i
\(843\) − 46.7927i − 1.61162i
\(844\) −12.2902 −0.423046
\(845\) 0 0
\(846\) 59.1334 2.03305
\(847\) 28.8212i 0.990307i
\(848\) 4.80440i 0.164984i
\(849\) −16.3638 −0.561603
\(850\) 0 0
\(851\) 81.6717 2.79967
\(852\) 37.7412i 1.29299i
\(853\) 46.7422i 1.60042i 0.599719 + 0.800211i \(0.295279\pi\)
−0.599719 + 0.800211i \(0.704721\pi\)
\(854\) 44.5025 1.52284
\(855\) 0 0
\(856\) −5.78334 −0.197671
\(857\) − 30.4648i − 1.04066i −0.853966 0.520328i \(-0.825810\pi\)
0.853966 0.520328i \(-0.174190\pi\)
\(858\) − 58.6686i − 2.00291i
\(859\) 30.7971 1.05078 0.525391 0.850861i \(-0.323919\pi\)
0.525391 + 0.850861i \(0.323919\pi\)
\(860\) 0 0
\(861\) −70.7864 −2.41239
\(862\) 1.70979i 0.0582358i
\(863\) 34.9010i 1.18804i 0.804449 + 0.594022i \(0.202461\pi\)
−0.804449 + 0.594022i \(0.797539\pi\)
\(864\) −12.2397 −0.416403
\(865\) 0 0
\(866\) −23.7412 −0.806760
\(867\) 45.4005i 1.54188i
\(868\) 9.33431i 0.316827i
\(869\) 32.3952 1.09893
\(870\) 0 0
\(871\) 39.5118 1.33881
\(872\) − 10.7971i − 0.365635i
\(873\) − 2.39853i − 0.0811781i
\(874\) −10.0421 −0.339679
\(875\) 0 0
\(876\) −7.54364 −0.254876
\(877\) − 36.9599i − 1.24805i −0.781406 0.624023i \(-0.785497\pi\)
0.781406 0.624023i \(-0.214503\pi\)
\(878\) − 8.47941i − 0.286166i
\(879\) −55.9304 −1.88649
\(880\) 0 0
\(881\) 25.0009 0.842303 0.421151 0.906990i \(-0.361626\pi\)
0.421151 + 0.906990i \(0.361626\pi\)
\(882\) − 56.1334i − 1.89011i
\(883\) 52.7696i 1.77584i 0.459999 + 0.887919i \(0.347850\pi\)
−0.459999 + 0.887919i \(0.652150\pi\)
\(884\) −6.96323 −0.234199
\(885\) 0 0
\(886\) 18.4490 0.619807
\(887\) 23.5667i 0.791292i 0.918403 + 0.395646i \(0.129479\pi\)
−0.918403 + 0.395646i \(0.870521\pi\)
\(888\) − 30.7696i − 1.03256i
\(889\) 31.1334 1.04418
\(890\) 0 0
\(891\) −76.4520 −2.56124
\(892\) − 20.2324i − 0.677430i
\(893\) − 10.3216i − 0.345401i
\(894\) −22.8853 −0.765398
\(895\) 0 0
\(896\) −3.89167 −0.130012
\(897\) 114.159i 3.81167i
\(898\) − 11.3775i − 0.379671i
\(899\) −2.39853 −0.0799956
\(900\) 0 0
\(901\) 7.69408 0.256327
\(902\) 24.8117i 0.826140i
\(903\) − 44.0809i − 1.46692i
\(904\) −12.2324 −0.406843
\(905\) 0 0
\(906\) −27.3500 −0.908644
\(907\) − 25.2103i − 0.837093i −0.908195 0.418546i \(-0.862540\pi\)
0.908195 0.418546i \(-0.137460\pi\)
\(908\) 8.98627i 0.298220i
\(909\) 126.955 4.21085
\(910\) 0 0
\(911\) 32.5961 1.07996 0.539979 0.841679i \(-0.318432\pi\)
0.539979 + 0.841679i \(0.318432\pi\)
\(912\) 3.78334i 0.125279i
\(913\) − 12.0000i − 0.397142i
\(914\) 4.40586 0.145733
\(915\) 0 0
\(916\) −21.2692 −0.702753
\(917\) − 81.6717i − 2.69704i
\(918\) 19.6015i 0.646945i
\(919\) −44.0000 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(920\) 0 0
\(921\) 7.56668 0.249331
\(922\) − 15.3113i − 0.504249i
\(923\) − 52.1764i − 1.71741i
\(924\) −52.5108 −1.72748
\(925\) 0 0
\(926\) −17.0873 −0.561523
\(927\) − 83.2069i − 2.73287i
\(928\) − 1.00000i − 0.0328266i
\(929\) −5.05249 −0.165767 −0.0828835 0.996559i \(-0.526413\pi\)
−0.0828835 + 0.996559i \(0.526413\pi\)
\(930\) 0 0
\(931\) −9.79800 −0.321116
\(932\) − 11.1608i − 0.365585i
\(933\) 51.0986i 1.67289i
\(934\) 22.3711 0.732004
\(935\) 0 0
\(936\) 29.9652 0.979444
\(937\) 19.6088i 0.640591i 0.947318 + 0.320296i \(0.103782\pi\)
−0.947318 + 0.320296i \(0.896218\pi\)
\(938\) − 35.3647i − 1.15470i
\(939\) 38.2902 1.24955
\(940\) 0 0
\(941\) 6.68141 0.217808 0.108904 0.994052i \(-0.465266\pi\)
0.108904 + 0.994052i \(0.465266\pi\)
\(942\) 31.4510i 1.02473i
\(943\) − 48.2795i − 1.57220i
\(944\) 4.05783 0.132071
\(945\) 0 0
\(946\) −15.4510 −0.502356
\(947\) − 33.4667i − 1.08752i −0.839240 0.543762i \(-0.817000\pi\)
0.839240 0.543762i \(-0.183000\pi\)
\(948\) 23.7486i 0.771317i
\(949\) 10.4289 0.338537
\(950\) 0 0
\(951\) −43.0324 −1.39542
\(952\) 6.23238i 0.201992i
\(953\) − 43.7412i − 1.41692i −0.705752 0.708459i \(-0.749391\pi\)
0.705752 0.708459i \(-0.250609\pi\)
\(954\) −33.1103 −1.07199
\(955\) 0 0
\(956\) −1.92645 −0.0623059
\(957\) − 13.4931i − 0.436171i
\(958\) − 8.00733i − 0.258705i
\(959\) 14.2123 0.458938
\(960\) 0 0
\(961\) −25.2470 −0.814420
\(962\) 42.5383i 1.37149i
\(963\) − 39.8569i − 1.28437i
\(964\) 17.7255 0.570900
\(965\) 0 0
\(966\) 102.177 3.28751
\(967\) 35.9304i 1.15544i 0.816233 + 0.577722i \(0.196058\pi\)
−0.816233 + 0.577722i \(0.803942\pi\)
\(968\) 7.40586i 0.238033i
\(969\) 6.05889 0.194640
\(970\) 0 0
\(971\) 26.7971 0.859959 0.429979 0.902839i \(-0.358521\pi\)
0.429979 + 0.902839i \(0.358521\pi\)
\(972\) − 19.3270i − 0.619913i
\(973\) − 53.2186i − 1.70611i
\(974\) 17.0441 0.546128
\(975\) 0 0
\(976\) 11.4353 0.366035
\(977\) 21.4510i 0.686279i 0.939284 + 0.343140i \(0.111490\pi\)
−0.939284 + 0.343140i \(0.888510\pi\)
\(978\) − 1.82545i − 0.0583715i
\(979\) −18.4059 −0.588254
\(980\) 0 0
\(981\) 74.4098 2.37572
\(982\) − 37.0873i − 1.18350i
\(983\) 49.8715i 1.59066i 0.606180 + 0.795328i \(0.292701\pi\)
−0.606180 + 0.795328i \(0.707299\pi\)
\(984\) −18.1892 −0.579851
\(985\) 0 0
\(986\) −1.60147 −0.0510011
\(987\) 105.022i 3.34288i
\(988\) − 5.23039i − 0.166401i
\(989\) 30.0652 0.956016
\(990\) 0 0
\(991\) −16.2167 −0.515139 −0.257570 0.966260i \(-0.582922\pi\)
−0.257570 + 0.966260i \(0.582922\pi\)
\(992\) 2.39853i 0.0761535i
\(993\) 27.8990i 0.885348i
\(994\) −46.7001 −1.48124
\(995\) 0 0
\(996\) 8.79707 0.278746
\(997\) − 6.31766i − 0.200082i −0.994983 0.100041i \(-0.968103\pi\)
0.994983 0.100041i \(-0.0318974\pi\)
\(998\) − 23.5089i − 0.744160i
\(999\) 119.745 3.78857
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 1450.2.b.l.349.3 6
5.2 odd 4 290.2.a.e.1.3 3
5.3 odd 4 1450.2.a.p.1.1 3
5.4 even 2 inner 1450.2.b.l.349.4 6
15.2 even 4 2610.2.a.x.1.1 3
20.7 even 4 2320.2.a.l.1.1 3
40.27 even 4 9280.2.a.by.1.3 3
40.37 odd 4 9280.2.a.bf.1.1 3
145.57 odd 4 8410.2.a.v.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
290.2.a.e.1.3 3 5.2 odd 4
1450.2.a.p.1.1 3 5.3 odd 4
1450.2.b.l.349.3 6 1.1 even 1 trivial
1450.2.b.l.349.4 6 5.4 even 2 inner
2320.2.a.l.1.1 3 20.7 even 4
2610.2.a.x.1.1 3 15.2 even 4
8410.2.a.v.1.1 3 145.57 odd 4
9280.2.a.bf.1.1 3 40.37 odd 4
9280.2.a.by.1.3 3 40.27 even 4