Properties

Label 2240.2.g.l.449.2
Level $2240$
Weight $2$
Character 2240.449
Analytic conductor $17.886$
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] = [2240,2,Mod(449,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2240.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.g (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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(0.432320 - 0.432320i\) of defining polynomial
Character \(\chi\) \(=\) 2240.449
Dual form 2240.2.g.l.449.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.76156i q^{3} +(-0.432320 - 2.19388i) q^{5} -1.00000i q^{7} -0.103084 q^{9} +O(q^{10})\) \(q-1.76156i q^{3} +(-0.432320 - 2.19388i) q^{5} -1.00000i q^{7} -0.103084 q^{9} +0.626198 q^{11} -5.49084i q^{13} +(-3.86464 + 0.761557i) q^{15} -0.896916i q^{17} +6.38776 q^{19} -1.76156 q^{21} -3.72928i q^{23} +(-4.62620 + 1.89692i) q^{25} -5.10308i q^{27} -7.87859 q^{29} +7.52311 q^{31} -1.10308i q^{33} +(-2.19388 + 0.432320i) q^{35} +6.00000i q^{37} -9.67243 q^{39} +7.72928 q^{41} -1.72928i q^{43} +(0.0445652 + 0.226153i) q^{45} -5.87859i q^{47} -1.00000 q^{49} -1.57997 q^{51} +6.77551i q^{53} +(-0.270718 - 1.37380i) q^{55} -11.2524i q^{57} -0.593923 q^{59} -7.13536 q^{61} +0.103084i q^{63} +(-12.0462 + 2.37380i) q^{65} -5.79383i q^{67} -6.56934 q^{69} +5.52311 q^{71} -3.72928i q^{73} +(3.34153 + 8.14931i) q^{75} -0.626198i q^{77} +5.67243 q^{79} -9.29862 q^{81} +17.4340i q^{83} +(-1.96772 + 0.387755i) q^{85} +13.8786i q^{87} -14.2986 q^{89} -5.49084 q^{91} -13.2524i q^{93} +(-2.76156 - 14.0140i) q^{95} +10.1493i q^{97} -0.0645508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{9} - 14 q^{11} - 18 q^{15} + 8 q^{19} + 2 q^{21} - 10 q^{25} + 6 q^{29} + 20 q^{31} + 2 q^{35} + 10 q^{39} + 36 q^{41} + 28 q^{45} - 6 q^{49} - 42 q^{51} - 12 q^{55} + 12 q^{59} - 48 q^{61} - 22 q^{65} + 36 q^{69} + 8 q^{71} + 40 q^{75} - 34 q^{79} + 30 q^{81} - 14 q^{85} - 10 q^{91} - 4 q^{95} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.76156i 1.01704i −0.861052 0.508518i \(-0.830194\pi\)
0.861052 0.508518i \(-0.169806\pi\)
\(4\) 0 0
\(5\) −0.432320 2.19388i −0.193340 0.981132i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −0.103084 −0.0343612
\(10\) 0 0
\(11\) 0.626198 0.188806 0.0944029 0.995534i \(-0.469906\pi\)
0.0944029 + 0.995534i \(0.469906\pi\)
\(12\) 0 0
\(13\) 5.49084i 1.52288i −0.648233 0.761442i \(-0.724492\pi\)
0.648233 0.761442i \(-0.275508\pi\)
\(14\) 0 0
\(15\) −3.86464 + 0.761557i −0.997846 + 0.196633i
\(16\) 0 0
\(17\) 0.896916i 0.217534i −0.994067 0.108767i \(-0.965310\pi\)
0.994067 0.108767i \(-0.0346903\pi\)
\(18\) 0 0
\(19\) 6.38776 1.46545 0.732726 0.680524i \(-0.238248\pi\)
0.732726 + 0.680524i \(0.238248\pi\)
\(20\) 0 0
\(21\) −1.76156 −0.384403
\(22\) 0 0
\(23\) 3.72928i 0.777609i −0.921320 0.388805i \(-0.872888\pi\)
0.921320 0.388805i \(-0.127112\pi\)
\(24\) 0 0
\(25\) −4.62620 + 1.89692i −0.925240 + 0.379383i
\(26\) 0 0
\(27\) 5.10308i 0.982089i
\(28\) 0 0
\(29\) −7.87859 −1.46302 −0.731509 0.681832i \(-0.761184\pi\)
−0.731509 + 0.681832i \(0.761184\pi\)
\(30\) 0 0
\(31\) 7.52311 1.35119 0.675596 0.737272i \(-0.263887\pi\)
0.675596 + 0.737272i \(0.263887\pi\)
\(32\) 0 0
\(33\) 1.10308i 0.192022i
\(34\) 0 0
\(35\) −2.19388 + 0.432320i −0.370833 + 0.0730755i
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) −9.67243 −1.54883
\(40\) 0 0
\(41\) 7.72928 1.20711 0.603556 0.797321i \(-0.293750\pi\)
0.603556 + 0.797321i \(0.293750\pi\)
\(42\) 0 0
\(43\) 1.72928i 0.263713i −0.991269 0.131856i \(-0.957906\pi\)
0.991269 0.131856i \(-0.0420938\pi\)
\(44\) 0 0
\(45\) 0.0445652 + 0.226153i 0.00664339 + 0.0337129i
\(46\) 0 0
\(47\) 5.87859i 0.857481i −0.903428 0.428741i \(-0.858957\pi\)
0.903428 0.428741i \(-0.141043\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −1.57997 −0.221240
\(52\) 0 0
\(53\) 6.77551i 0.930688i 0.885130 + 0.465344i \(0.154069\pi\)
−0.885130 + 0.465344i \(0.845931\pi\)
\(54\) 0 0
\(55\) −0.270718 1.37380i −0.0365036 0.185243i
\(56\) 0 0
\(57\) 11.2524i 1.49042i
\(58\) 0 0
\(59\) −0.593923 −0.0773221 −0.0386611 0.999252i \(-0.512309\pi\)
−0.0386611 + 0.999252i \(0.512309\pi\)
\(60\) 0 0
\(61\) −7.13536 −0.913589 −0.456795 0.889572i \(-0.651003\pi\)
−0.456795 + 0.889572i \(0.651003\pi\)
\(62\) 0 0
\(63\) 0.103084i 0.0129873i
\(64\) 0 0
\(65\) −12.0462 + 2.37380i −1.49415 + 0.294434i
\(66\) 0 0
\(67\) 5.79383i 0.707829i −0.935278 0.353915i \(-0.884850\pi\)
0.935278 0.353915i \(-0.115150\pi\)
\(68\) 0 0
\(69\) −6.56934 −0.790856
\(70\) 0 0
\(71\) 5.52311 0.655473 0.327737 0.944769i \(-0.393714\pi\)
0.327737 + 0.944769i \(0.393714\pi\)
\(72\) 0 0
\(73\) 3.72928i 0.436479i −0.975895 0.218240i \(-0.929969\pi\)
0.975895 0.218240i \(-0.0700315\pi\)
\(74\) 0 0
\(75\) 3.34153 + 8.14931i 0.385846 + 0.941002i
\(76\) 0 0
\(77\) 0.626198i 0.0713619i
\(78\) 0 0
\(79\) 5.67243 0.638198 0.319099 0.947721i \(-0.396620\pi\)
0.319099 + 0.947721i \(0.396620\pi\)
\(80\) 0 0
\(81\) −9.29862 −1.03318
\(82\) 0 0
\(83\) 17.4340i 1.91363i 0.290700 + 0.956814i \(0.406112\pi\)
−0.290700 + 0.956814i \(0.593888\pi\)
\(84\) 0 0
\(85\) −1.96772 + 0.387755i −0.213430 + 0.0420580i
\(86\) 0 0
\(87\) 13.8786i 1.48794i
\(88\) 0 0
\(89\) −14.2986 −1.51565 −0.757826 0.652457i \(-0.773738\pi\)
−0.757826 + 0.652457i \(0.773738\pi\)
\(90\) 0 0
\(91\) −5.49084 −0.575596
\(92\) 0 0
\(93\) 13.2524i 1.37421i
\(94\) 0 0
\(95\) −2.76156 14.0140i −0.283330 1.43780i
\(96\) 0 0
\(97\) 10.1493i 1.03051i 0.857038 + 0.515253i \(0.172302\pi\)
−0.857038 + 0.515253i \(0.827698\pi\)
\(98\) 0 0
\(99\) −0.0645508 −0.00648760
\(100\) 0 0
\(101\) −9.64015 −0.959231 −0.479615 0.877479i \(-0.659224\pi\)
−0.479615 + 0.877479i \(0.659224\pi\)
\(102\) 0 0
\(103\) 0.626198i 0.0617011i −0.999524 0.0308506i \(-0.990178\pi\)
0.999524 0.0308506i \(-0.00982160\pi\)
\(104\) 0 0
\(105\) 0.761557 + 3.86464i 0.0743204 + 0.377150i
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 18.9248 1.81267 0.906335 0.422561i \(-0.138869\pi\)
0.906335 + 0.422561i \(0.138869\pi\)
\(110\) 0 0
\(111\) 10.5693 1.00320
\(112\) 0 0
\(113\) 1.04623i 0.0984209i −0.998788 0.0492105i \(-0.984329\pi\)
0.998788 0.0492105i \(-0.0156705\pi\)
\(114\) 0 0
\(115\) −8.18159 + 1.61224i −0.762937 + 0.150343i
\(116\) 0 0
\(117\) 0.566016i 0.0523282i
\(118\) 0 0
\(119\) −0.896916 −0.0822202
\(120\) 0 0
\(121\) −10.6079 −0.964352
\(122\) 0 0
\(123\) 13.6156i 1.22767i
\(124\) 0 0
\(125\) 6.16160 + 9.32924i 0.551110 + 0.834432i
\(126\) 0 0
\(127\) 21.2803i 1.88832i 0.329485 + 0.944161i \(0.393125\pi\)
−0.329485 + 0.944161i \(0.606875\pi\)
\(128\) 0 0
\(129\) −3.04623 −0.268205
\(130\) 0 0
\(131\) 9.91087 0.865917 0.432958 0.901414i \(-0.357470\pi\)
0.432958 + 0.901414i \(0.357470\pi\)
\(132\) 0 0
\(133\) 6.38776i 0.553889i
\(134\) 0 0
\(135\) −11.1955 + 2.20617i −0.963559 + 0.189877i
\(136\) 0 0
\(137\) 3.45856i 0.295485i −0.989026 0.147743i \(-0.952799\pi\)
0.989026 0.147743i \(-0.0472007\pi\)
\(138\) 0 0
\(139\) −20.4157 −1.73163 −0.865817 0.500361i \(-0.833201\pi\)
−0.865817 + 0.500361i \(0.833201\pi\)
\(140\) 0 0
\(141\) −10.3555 −0.872089
\(142\) 0 0
\(143\) 3.43835i 0.287530i
\(144\) 0 0
\(145\) 3.40608 + 17.2847i 0.282859 + 1.43541i
\(146\) 0 0
\(147\) 1.76156i 0.145291i
\(148\) 0 0
\(149\) −17.0462 −1.39648 −0.698241 0.715863i \(-0.746033\pi\)
−0.698241 + 0.715863i \(0.746033\pi\)
\(150\) 0 0
\(151\) 2.89692 0.235748 0.117874 0.993029i \(-0.462392\pi\)
0.117874 + 0.993029i \(0.462392\pi\)
\(152\) 0 0
\(153\) 0.0924575i 0.00747474i
\(154\) 0 0
\(155\) −3.25240 16.5048i −0.261239 1.32570i
\(156\) 0 0
\(157\) 10.1170i 0.807427i 0.914885 + 0.403714i \(0.132281\pi\)
−0.914885 + 0.403714i \(0.867719\pi\)
\(158\) 0 0
\(159\) 11.9354 0.946543
\(160\) 0 0
\(161\) −3.72928 −0.293909
\(162\) 0 0
\(163\) 0.476886i 0.0373526i −0.999826 0.0186763i \(-0.994055\pi\)
0.999826 0.0186763i \(-0.00594519\pi\)
\(164\) 0 0
\(165\) −2.42003 + 0.476886i −0.188399 + 0.0371255i
\(166\) 0 0
\(167\) 13.1676i 1.01894i 0.860488 + 0.509471i \(0.170159\pi\)
−0.860488 + 0.509471i \(0.829841\pi\)
\(168\) 0 0
\(169\) −17.1493 −1.31918
\(170\) 0 0
\(171\) −0.658473 −0.0503547
\(172\) 0 0
\(173\) 13.9677i 1.06195i −0.847389 0.530973i \(-0.821826\pi\)
0.847389 0.530973i \(-0.178174\pi\)
\(174\) 0 0
\(175\) 1.89692 + 4.62620i 0.143393 + 0.349708i
\(176\) 0 0
\(177\) 1.04623i 0.0786394i
\(178\) 0 0
\(179\) 7.45856 0.557479 0.278740 0.960367i \(-0.410083\pi\)
0.278740 + 0.960367i \(0.410083\pi\)
\(180\) 0 0
\(181\) 14.1170 1.04931 0.524656 0.851315i \(-0.324194\pi\)
0.524656 + 0.851315i \(0.324194\pi\)
\(182\) 0 0
\(183\) 12.5693i 0.929153i
\(184\) 0 0
\(185\) 13.1633 2.59392i 0.967783 0.190709i
\(186\) 0 0
\(187\) 0.561647i 0.0410717i
\(188\) 0 0
\(189\) −5.10308 −0.371195
\(190\) 0 0
\(191\) −8.42003 −0.609252 −0.304626 0.952472i \(-0.598531\pi\)
−0.304626 + 0.952472i \(0.598531\pi\)
\(192\) 0 0
\(193\) 22.2986i 1.60509i −0.596591 0.802545i \(-0.703479\pi\)
0.596591 0.802545i \(-0.296521\pi\)
\(194\) 0 0
\(195\) 4.18159 + 21.2201i 0.299450 + 1.51960i
\(196\) 0 0
\(197\) 26.3632i 1.87830i 0.343509 + 0.939149i \(0.388384\pi\)
−0.343509 + 0.939149i \(0.611616\pi\)
\(198\) 0 0
\(199\) −22.4402 −1.59075 −0.795373 0.606120i \(-0.792725\pi\)
−0.795373 + 0.606120i \(0.792725\pi\)
\(200\) 0 0
\(201\) −10.2062 −0.719888
\(202\) 0 0
\(203\) 7.87859i 0.552969i
\(204\) 0 0
\(205\) −3.34153 16.9571i −0.233382 1.18434i
\(206\) 0 0
\(207\) 0.384428i 0.0267196i
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 15.1955 1.04610 0.523052 0.852301i \(-0.324793\pi\)
0.523052 + 0.852301i \(0.324793\pi\)
\(212\) 0 0
\(213\) 9.72928i 0.666639i
\(214\) 0 0
\(215\) −3.79383 + 0.747604i −0.258737 + 0.0509862i
\(216\) 0 0
\(217\) 7.52311i 0.510702i
\(218\) 0 0
\(219\) −6.56934 −0.443915
\(220\) 0 0
\(221\) −4.92482 −0.331279
\(222\) 0 0
\(223\) 6.89692i 0.461852i −0.972971 0.230926i \(-0.925825\pi\)
0.972971 0.230926i \(-0.0741755\pi\)
\(224\) 0 0
\(225\) 0.476886 0.195541i 0.0317924 0.0130361i
\(226\) 0 0
\(227\) 2.77988i 0.184507i −0.995736 0.0922535i \(-0.970593\pi\)
0.995736 0.0922535i \(-0.0294070\pi\)
\(228\) 0 0
\(229\) 18.6585 1.23299 0.616493 0.787360i \(-0.288553\pi\)
0.616493 + 0.787360i \(0.288553\pi\)
\(230\) 0 0
\(231\) −1.10308 −0.0725776
\(232\) 0 0
\(233\) 11.0462i 0.723663i −0.932244 0.361831i \(-0.882152\pi\)
0.932244 0.361831i \(-0.117848\pi\)
\(234\) 0 0
\(235\) −12.8969 + 2.54144i −0.841302 + 0.165785i
\(236\) 0 0
\(237\) 9.99230i 0.649070i
\(238\) 0 0
\(239\) 20.1493 1.30335 0.651675 0.758498i \(-0.274066\pi\)
0.651675 + 0.758498i \(0.274066\pi\)
\(240\) 0 0
\(241\) 3.72928 0.240224 0.120112 0.992760i \(-0.461675\pi\)
0.120112 + 0.992760i \(0.461675\pi\)
\(242\) 0 0
\(243\) 1.07081i 0.0686924i
\(244\) 0 0
\(245\) 0.432320 + 2.19388i 0.0276199 + 0.140162i
\(246\) 0 0
\(247\) 35.0741i 2.23171i
\(248\) 0 0
\(249\) 30.7110 1.94623
\(250\) 0 0
\(251\) 21.4985 1.35698 0.678488 0.734612i \(-0.262636\pi\)
0.678488 + 0.734612i \(0.262636\pi\)
\(252\) 0 0
\(253\) 2.33527i 0.146817i
\(254\) 0 0
\(255\) 0.683053 + 3.46626i 0.0427744 + 0.217066i
\(256\) 0 0
\(257\) 4.27072i 0.266400i −0.991089 0.133200i \(-0.957475\pi\)
0.991089 0.133200i \(-0.0425253\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 0.812155 0.0502711
\(262\) 0 0
\(263\) 11.4586i 0.706565i −0.935517 0.353283i \(-0.885065\pi\)
0.935517 0.353283i \(-0.114935\pi\)
\(264\) 0 0
\(265\) 14.8646 2.92919i 0.913128 0.179939i
\(266\) 0 0
\(267\) 25.1878i 1.54147i
\(268\) 0 0
\(269\) −7.07081 −0.431115 −0.215557 0.976491i \(-0.569157\pi\)
−0.215557 + 0.976491i \(0.569157\pi\)
\(270\) 0 0
\(271\) 17.0096 1.03326 0.516629 0.856209i \(-0.327187\pi\)
0.516629 + 0.856209i \(0.327187\pi\)
\(272\) 0 0
\(273\) 9.67243i 0.585402i
\(274\) 0 0
\(275\) −2.89692 + 1.18785i −0.174691 + 0.0716298i
\(276\) 0 0
\(277\) 18.7755i 1.12811i −0.825737 0.564056i \(-0.809240\pi\)
0.825737 0.564056i \(-0.190760\pi\)
\(278\) 0 0
\(279\) −0.775511 −0.0464286
\(280\) 0 0
\(281\) −4.59829 −0.274311 −0.137156 0.990550i \(-0.543796\pi\)
−0.137156 + 0.990550i \(0.543796\pi\)
\(282\) 0 0
\(283\) 13.5833i 0.807443i 0.914882 + 0.403722i \(0.132284\pi\)
−0.914882 + 0.403722i \(0.867716\pi\)
\(284\) 0 0
\(285\) −24.6864 + 4.86464i −1.46229 + 0.288156i
\(286\) 0 0
\(287\) 7.72928i 0.456245i
\(288\) 0 0
\(289\) 16.1955 0.952679
\(290\) 0 0
\(291\) 17.8786 1.04806
\(292\) 0 0
\(293\) 11.8261i 0.690889i −0.938439 0.345444i \(-0.887728\pi\)
0.938439 0.345444i \(-0.112272\pi\)
\(294\) 0 0
\(295\) 0.256765 + 1.30299i 0.0149494 + 0.0758632i
\(296\) 0 0
\(297\) 3.19554i 0.185424i
\(298\) 0 0
\(299\) −20.4769 −1.18421
\(300\) 0 0
\(301\) −1.72928 −0.0996741
\(302\) 0 0
\(303\) 16.9817i 0.975572i
\(304\) 0 0
\(305\) 3.08476 + 15.6541i 0.176633 + 0.896351i
\(306\) 0 0
\(307\) 20.9860i 1.19774i −0.800847 0.598868i \(-0.795617\pi\)
0.800847 0.598868i \(-0.204383\pi\)
\(308\) 0 0
\(309\) −1.10308 −0.0627522
\(310\) 0 0
\(311\) −22.5048 −1.27613 −0.638065 0.769983i \(-0.720265\pi\)
−0.638065 + 0.769983i \(0.720265\pi\)
\(312\) 0 0
\(313\) 12.4846i 0.705670i −0.935686 0.352835i \(-0.885218\pi\)
0.935686 0.352835i \(-0.114782\pi\)
\(314\) 0 0
\(315\) 0.226153 0.0445652i 0.0127423 0.00251096i
\(316\) 0 0
\(317\) 25.5231i 1.43352i 0.697319 + 0.716760i \(0.254376\pi\)
−0.697319 + 0.716760i \(0.745624\pi\)
\(318\) 0 0
\(319\) −4.93356 −0.276226
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.72928i 0.318786i
\(324\) 0 0
\(325\) 10.4157 + 25.4017i 0.577757 + 1.40903i
\(326\) 0 0
\(327\) 33.3372i 1.84355i
\(328\) 0 0
\(329\) −5.87859 −0.324097
\(330\) 0 0
\(331\) 2.68305 0.147474 0.0737370 0.997278i \(-0.476507\pi\)
0.0737370 + 0.997278i \(0.476507\pi\)
\(332\) 0 0
\(333\) 0.618502i 0.0338937i
\(334\) 0 0
\(335\) −12.7110 + 2.50479i −0.694474 + 0.136851i
\(336\) 0 0
\(337\) 12.5048i 0.681179i −0.940212 0.340590i \(-0.889373\pi\)
0.940212 0.340590i \(-0.110627\pi\)
\(338\) 0 0
\(339\) −1.84299 −0.100098
\(340\) 0 0
\(341\) 4.71096 0.255113
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 2.84006 + 14.4123i 0.152904 + 0.775934i
\(346\) 0 0
\(347\) 28.9450i 1.55385i −0.629593 0.776925i \(-0.716778\pi\)
0.629593 0.776925i \(-0.283222\pi\)
\(348\) 0 0
\(349\) 32.1449 1.72068 0.860340 0.509721i \(-0.170251\pi\)
0.860340 + 0.509721i \(0.170251\pi\)
\(350\) 0 0
\(351\) −28.0202 −1.49561
\(352\) 0 0
\(353\) 8.17722i 0.435229i 0.976035 + 0.217615i \(0.0698276\pi\)
−0.976035 + 0.217615i \(0.930172\pi\)
\(354\) 0 0
\(355\) −2.38776 12.1170i −0.126729 0.643106i
\(356\) 0 0
\(357\) 1.57997i 0.0836208i
\(358\) 0 0
\(359\) −18.5048 −0.976646 −0.488323 0.872663i \(-0.662391\pi\)
−0.488323 + 0.872663i \(0.662391\pi\)
\(360\) 0 0
\(361\) 21.8034 1.14755
\(362\) 0 0
\(363\) 18.6864i 0.980781i
\(364\) 0 0
\(365\) −8.18159 + 1.61224i −0.428244 + 0.0843887i
\(366\) 0 0
\(367\) 27.4942i 1.43518i 0.696464 + 0.717592i \(0.254756\pi\)
−0.696464 + 0.717592i \(0.745244\pi\)
\(368\) 0 0
\(369\) −0.796763 −0.0414778
\(370\) 0 0
\(371\) 6.77551 0.351767
\(372\) 0 0
\(373\) 6.06455i 0.314011i −0.987598 0.157005i \(-0.949816\pi\)
0.987598 0.157005i \(-0.0501840\pi\)
\(374\) 0 0
\(375\) 16.4340 10.8540i 0.848647 0.560499i
\(376\) 0 0
\(377\) 43.2601i 2.22801i
\(378\) 0 0
\(379\) −5.72928 −0.294293 −0.147147 0.989115i \(-0.547009\pi\)
−0.147147 + 0.989115i \(0.547009\pi\)
\(380\) 0 0
\(381\) 37.4865 1.92049
\(382\) 0 0
\(383\) 1.72928i 0.0883622i −0.999024 0.0441811i \(-0.985932\pi\)
0.999024 0.0441811i \(-0.0140679\pi\)
\(384\) 0 0
\(385\) −1.37380 + 0.270718i −0.0700154 + 0.0137971i
\(386\) 0 0
\(387\) 0.178261i 0.00906150i
\(388\) 0 0
\(389\) −36.7187 −1.86171 −0.930855 0.365389i \(-0.880936\pi\)
−0.930855 + 0.365389i \(0.880936\pi\)
\(390\) 0 0
\(391\) −3.34485 −0.169157
\(392\) 0 0
\(393\) 17.4586i 0.880668i
\(394\) 0 0
\(395\) −2.45231 12.4446i −0.123389 0.626156i
\(396\) 0 0
\(397\) 2.03228i 0.101997i −0.998699 0.0509985i \(-0.983760\pi\)
0.998699 0.0509985i \(-0.0162404\pi\)
\(398\) 0 0
\(399\) −11.2524 −0.563324
\(400\) 0 0
\(401\) −1.64452 −0.0821234 −0.0410617 0.999157i \(-0.513074\pi\)
−0.0410617 + 0.999157i \(0.513074\pi\)
\(402\) 0 0
\(403\) 41.3082i 2.05771i
\(404\) 0 0
\(405\) 4.01999 + 20.4000i 0.199755 + 1.01369i
\(406\) 0 0
\(407\) 3.75719i 0.186237i
\(408\) 0 0
\(409\) 4.98168 0.246328 0.123164 0.992386i \(-0.460696\pi\)
0.123164 + 0.992386i \(0.460696\pi\)
\(410\) 0 0
\(411\) −6.09246 −0.300519
\(412\) 0 0
\(413\) 0.593923i 0.0292250i
\(414\) 0 0
\(415\) 38.2480 7.53707i 1.87752 0.369980i
\(416\) 0 0
\(417\) 35.9634i 1.76113i
\(418\) 0 0
\(419\) 22.9205 1.11974 0.559869 0.828581i \(-0.310852\pi\)
0.559869 + 0.828581i \(0.310852\pi\)
\(420\) 0 0
\(421\) 20.9527 1.02117 0.510587 0.859826i \(-0.329428\pi\)
0.510587 + 0.859826i \(0.329428\pi\)
\(422\) 0 0
\(423\) 0.605987i 0.0294641i
\(424\) 0 0
\(425\) 1.70138 + 4.14931i 0.0825288 + 0.201271i
\(426\) 0 0
\(427\) 7.13536i 0.345304i
\(428\) 0 0
\(429\) −6.05685 −0.292428
\(430\) 0 0
\(431\) 33.1589 1.59721 0.798604 0.601857i \(-0.205572\pi\)
0.798604 + 0.601857i \(0.205572\pi\)
\(432\) 0 0
\(433\) 18.5414i 0.891045i −0.895271 0.445522i \(-0.853018\pi\)
0.895271 0.445522i \(-0.146982\pi\)
\(434\) 0 0
\(435\) 30.4479 6.00000i 1.45987 0.287678i
\(436\) 0 0
\(437\) 23.8217i 1.13955i
\(438\) 0 0
\(439\) 20.8401 0.994642 0.497321 0.867567i \(-0.334317\pi\)
0.497321 + 0.867567i \(0.334317\pi\)
\(440\) 0 0
\(441\) 0.103084 0.00490875
\(442\) 0 0
\(443\) 17.5510i 0.833874i −0.908935 0.416937i \(-0.863104\pi\)
0.908935 0.416937i \(-0.136896\pi\)
\(444\) 0 0
\(445\) 6.18159 + 31.3694i 0.293035 + 1.48705i
\(446\) 0 0
\(447\) 30.0279i 1.42027i
\(448\) 0 0
\(449\) 13.1955 0.622736 0.311368 0.950289i \(-0.399213\pi\)
0.311368 + 0.950289i \(0.399213\pi\)
\(450\) 0 0
\(451\) 4.84006 0.227910
\(452\) 0 0
\(453\) 5.10308i 0.239764i
\(454\) 0 0
\(455\) 2.37380 + 12.0462i 0.111286 + 0.564736i
\(456\) 0 0
\(457\) 29.1020i 1.36134i 0.732592 + 0.680668i \(0.238310\pi\)
−0.732592 + 0.680668i \(0.761690\pi\)
\(458\) 0 0
\(459\) −4.57704 −0.213638
\(460\) 0 0
\(461\) 18.8280 0.876907 0.438454 0.898754i \(-0.355526\pi\)
0.438454 + 0.898754i \(0.355526\pi\)
\(462\) 0 0
\(463\) 14.0925i 0.654932i −0.944863 0.327466i \(-0.893805\pi\)
0.944863 0.327466i \(-0.106195\pi\)
\(464\) 0 0
\(465\) −29.0741 + 5.72928i −1.34828 + 0.265689i
\(466\) 0 0
\(467\) 12.2663i 0.567619i −0.958881 0.283809i \(-0.908402\pi\)
0.958881 0.283809i \(-0.0915983\pi\)
\(468\) 0 0
\(469\) −5.79383 −0.267534
\(470\) 0 0
\(471\) 17.8217 0.821182
\(472\) 0 0
\(473\) 1.08287i 0.0497905i
\(474\) 0 0
\(475\) −29.5510 + 12.1170i −1.35589 + 0.555968i
\(476\) 0 0
\(477\) 0.698445i 0.0319796i
\(478\) 0 0
\(479\) 14.1570 0.646850 0.323425 0.946254i \(-0.395166\pi\)
0.323425 + 0.946254i \(0.395166\pi\)
\(480\) 0 0
\(481\) 32.9450 1.50216
\(482\) 0 0
\(483\) 6.56934i 0.298915i
\(484\) 0 0
\(485\) 22.2663 4.38776i 1.01106 0.199238i
\(486\) 0 0
\(487\) 30.2341i 1.37004i −0.728526 0.685018i \(-0.759794\pi\)
0.728526 0.685018i \(-0.240206\pi\)
\(488\) 0 0
\(489\) −0.840061 −0.0379889
\(490\) 0 0
\(491\) −39.9065 −1.80096 −0.900478 0.434902i \(-0.856783\pi\)
−0.900478 + 0.434902i \(0.856783\pi\)
\(492\) 0 0
\(493\) 7.06644i 0.318256i
\(494\) 0 0
\(495\) 0.0279066 + 0.141617i 0.00125431 + 0.00636519i
\(496\) 0 0
\(497\) 5.52311i 0.247746i
\(498\) 0 0
\(499\) 27.3246 1.22322 0.611610 0.791160i \(-0.290522\pi\)
0.611610 + 0.791160i \(0.290522\pi\)
\(500\) 0 0
\(501\) 23.1955 1.03630
\(502\) 0 0
\(503\) 1.40171i 0.0624991i 0.999512 + 0.0312495i \(0.00994866\pi\)
−0.999512 + 0.0312495i \(0.990051\pi\)
\(504\) 0 0
\(505\) 4.16763 + 21.1493i 0.185457 + 0.941132i
\(506\) 0 0
\(507\) 30.2095i 1.34165i
\(508\) 0 0
\(509\) 18.6585 0.827022 0.413511 0.910499i \(-0.364302\pi\)
0.413511 + 0.910499i \(0.364302\pi\)
\(510\) 0 0
\(511\) −3.72928 −0.164974
\(512\) 0 0
\(513\) 32.5972i 1.43920i
\(514\) 0 0
\(515\) −1.37380 + 0.270718i −0.0605369 + 0.0119293i
\(516\) 0 0
\(517\) 3.68116i 0.161897i
\(518\) 0 0
\(519\) −24.6049 −1.08004
\(520\) 0 0
\(521\) −4.02791 −0.176466 −0.0882329 0.996100i \(-0.528122\pi\)
−0.0882329 + 0.996100i \(0.528122\pi\)
\(522\) 0 0
\(523\) 38.4436i 1.68102i −0.541796 0.840510i \(-0.682256\pi\)
0.541796 0.840510i \(-0.317744\pi\)
\(524\) 0 0
\(525\) 8.14931 3.34153i 0.355665 0.145836i
\(526\) 0 0
\(527\) 6.74760i 0.293930i
\(528\) 0 0
\(529\) 9.09246 0.395324
\(530\) 0 0
\(531\) 0.0612237 0.00265688
\(532\) 0 0
\(533\) 42.4402i 1.83829i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 13.1387i 0.566976i
\(538\) 0 0
\(539\) −0.626198 −0.0269723
\(540\) 0 0
\(541\) −12.4846 −0.536754 −0.268377 0.963314i \(-0.586487\pi\)
−0.268377 + 0.963314i \(0.586487\pi\)
\(542\) 0 0
\(543\) 24.8680i 1.06719i
\(544\) 0 0
\(545\) −8.18159 41.5187i −0.350461 1.77847i
\(546\) 0 0
\(547\) 37.7851i 1.61557i 0.589474 + 0.807787i \(0.299335\pi\)
−0.589474 + 0.807787i \(0.700665\pi\)
\(548\) 0 0
\(549\) 0.735539 0.0313921
\(550\) 0 0
\(551\) −50.3265 −2.14398
\(552\) 0 0
\(553\) 5.67243i 0.241216i
\(554\) 0 0
\(555\) −4.56934 23.1878i −0.193958 0.984269i
\(556\) 0 0
\(557\) 1.93545i 0.0820076i 0.999159 + 0.0410038i \(0.0130556\pi\)
−0.999159 + 0.0410038i \(0.986944\pi\)
\(558\) 0 0
\(559\) −9.49521 −0.401604
\(560\) 0 0
\(561\) −0.989374 −0.0417714
\(562\) 0 0
\(563\) 29.8463i 1.25787i 0.777457 + 0.628936i \(0.216509\pi\)
−0.777457 + 0.628936i \(0.783491\pi\)
\(564\) 0 0
\(565\) −2.29530 + 0.452306i −0.0965639 + 0.0190287i
\(566\) 0 0
\(567\) 9.29862i 0.390506i
\(568\) 0 0
\(569\) −15.3449 −0.643290 −0.321645 0.946860i \(-0.604236\pi\)
−0.321645 + 0.946860i \(0.604236\pi\)
\(570\) 0 0
\(571\) −35.8217 −1.49909 −0.749547 0.661952i \(-0.769728\pi\)
−0.749547 + 0.661952i \(0.769728\pi\)
\(572\) 0 0
\(573\) 14.8324i 0.619631i
\(574\) 0 0
\(575\) 7.07414 + 17.2524i 0.295012 + 0.719475i
\(576\) 0 0
\(577\) 2.92482i 0.121762i −0.998145 0.0608810i \(-0.980609\pi\)
0.998145 0.0608810i \(-0.0193910\pi\)
\(578\) 0 0
\(579\) −39.2803 −1.63243
\(580\) 0 0
\(581\) 17.4340 0.723284
\(582\) 0 0
\(583\) 4.24281i 0.175719i
\(584\) 0 0
\(585\) 1.24177 0.244700i 0.0513409 0.0101171i
\(586\) 0 0
\(587\) 19.5756i 0.807972i 0.914765 + 0.403986i \(0.132375\pi\)
−0.914765 + 0.403986i \(0.867625\pi\)
\(588\) 0 0
\(589\) 48.0558 1.98011
\(590\) 0 0
\(591\) 46.4402 1.91030
\(592\) 0 0
\(593\) 8.00770i 0.328837i 0.986391 + 0.164418i \(0.0525747\pi\)
−0.986391 + 0.164418i \(0.947425\pi\)
\(594\) 0 0
\(595\) 0.387755 + 1.96772i 0.0158964 + 0.0806688i
\(596\) 0 0
\(597\) 39.5298i 1.61785i
\(598\) 0 0
\(599\) 29.3005 1.19719 0.598593 0.801053i \(-0.295727\pi\)
0.598593 + 0.801053i \(0.295727\pi\)
\(600\) 0 0
\(601\) 21.3449 0.870675 0.435337 0.900267i \(-0.356629\pi\)
0.435337 + 0.900267i \(0.356629\pi\)
\(602\) 0 0
\(603\) 0.597250i 0.0243219i
\(604\) 0 0
\(605\) 4.58600 + 23.2724i 0.186447 + 0.946157i
\(606\) 0 0
\(607\) 9.53081i 0.386844i −0.981116 0.193422i \(-0.938041\pi\)
0.981116 0.193422i \(-0.0619586\pi\)
\(608\) 0 0
\(609\) 13.8786 0.562389
\(610\) 0 0
\(611\) −32.2784 −1.30584
\(612\) 0 0
\(613\) 9.75719i 0.394089i 0.980395 + 0.197045i \(0.0631344\pi\)
−0.980395 + 0.197045i \(0.936866\pi\)
\(614\) 0 0
\(615\) −29.8709 + 5.88629i −1.20451 + 0.237358i
\(616\) 0 0
\(617\) 37.9267i 1.52687i 0.645884 + 0.763436i \(0.276489\pi\)
−0.645884 + 0.763436i \(0.723511\pi\)
\(618\) 0 0
\(619\) 41.9109 1.68454 0.842270 0.539056i \(-0.181219\pi\)
0.842270 + 0.539056i \(0.181219\pi\)
\(620\) 0 0
\(621\) −19.0308 −0.763681
\(622\) 0 0
\(623\) 14.2986i 0.572862i
\(624\) 0 0
\(625\) 17.8034 17.5510i 0.712137 0.702041i
\(626\) 0 0
\(627\) 7.04623i 0.281399i
\(628\) 0 0
\(629\) 5.38150 0.214574
\(630\) 0 0
\(631\) −8.42003 −0.335196 −0.167598 0.985855i \(-0.553601\pi\)
−0.167598 + 0.985855i \(0.553601\pi\)
\(632\) 0 0
\(633\) 26.7678i 1.06393i
\(634\) 0 0
\(635\) 46.6864 9.19991i 1.85269 0.365087i
\(636\) 0 0
\(637\) 5.49084i 0.217555i
\(638\) 0 0
\(639\) −0.569343 −0.0225229
\(640\) 0 0
\(641\) −2.07707 −0.0820392 −0.0410196 0.999158i \(-0.513061\pi\)
−0.0410196 + 0.999158i \(0.513061\pi\)
\(642\) 0 0
\(643\) 27.2480i 1.07456i 0.843405 + 0.537279i \(0.180548\pi\)
−0.843405 + 0.537279i \(0.819452\pi\)
\(644\) 0 0
\(645\) 1.31695 + 6.68305i 0.0518547 + 0.263145i
\(646\) 0 0
\(647\) 12.3632i 0.486047i −0.970020 0.243023i \(-0.921861\pi\)
0.970020 0.243023i \(-0.0781391\pi\)
\(648\) 0 0
\(649\) −0.371913 −0.0145989
\(650\) 0 0
\(651\) −13.2524 −0.519402
\(652\) 0 0
\(653\) 37.3449i 1.46142i −0.682690 0.730709i \(-0.739190\pi\)
0.682690 0.730709i \(-0.260810\pi\)
\(654\) 0 0
\(655\) −4.28467 21.7432i −0.167416 0.849578i
\(656\) 0 0
\(657\) 0.384428i 0.0149980i
\(658\) 0 0
\(659\) −14.1772 −0.552266 −0.276133 0.961119i \(-0.589053\pi\)
−0.276133 + 0.961119i \(0.589053\pi\)
\(660\) 0 0
\(661\) 1.76925 0.0688160 0.0344080 0.999408i \(-0.489045\pi\)
0.0344080 + 0.999408i \(0.489045\pi\)
\(662\) 0 0
\(663\) 8.67536i 0.336923i
\(664\) 0 0
\(665\) −14.0140 + 2.76156i −0.543438 + 0.107089i
\(666\) 0 0
\(667\) 29.3815i 1.13766i
\(668\) 0 0
\(669\) −12.1493 −0.469720
\(670\) 0 0
\(671\) −4.46815 −0.172491
\(672\) 0 0
\(673\) 4.05581i 0.156340i 0.996940 + 0.0781701i \(0.0249077\pi\)
−0.996940 + 0.0781701i \(0.975092\pi\)
\(674\) 0 0
\(675\) 9.68012 + 23.6079i 0.372588 + 0.908668i
\(676\) 0 0
\(677\) 9.37713i 0.360392i −0.983631 0.180196i \(-0.942327\pi\)
0.983631 0.180196i \(-0.0576733\pi\)
\(678\) 0 0
\(679\) 10.1493 0.389495
\(680\) 0 0
\(681\) −4.89692 −0.187650
\(682\) 0 0
\(683\) 5.49521i 0.210268i 0.994458 + 0.105134i \(0.0335272\pi\)
−0.994458 + 0.105134i \(0.966473\pi\)
\(684\) 0 0
\(685\) −7.58767 + 1.49521i −0.289910 + 0.0571290i
\(686\) 0 0
\(687\) 32.8680i 1.25399i
\(688\) 0 0
\(689\) 37.2032 1.41733
\(690\) 0 0
\(691\) 3.03416 0.115425 0.0577125 0.998333i \(-0.481619\pi\)
0.0577125 + 0.998333i \(0.481619\pi\)
\(692\) 0 0
\(693\) 0.0645508i 0.00245208i
\(694\) 0 0
\(695\) 8.82611 + 44.7895i 0.334793 + 1.69896i
\(696\) 0 0
\(697\) 6.93252i 0.262588i
\(698\) 0 0
\(699\) −19.4586 −0.735990
\(700\) 0 0
\(701\) −35.2234 −1.33037 −0.665186 0.746678i \(-0.731648\pi\)
−0.665186 + 0.746678i \(0.731648\pi\)
\(702\) 0 0
\(703\) 38.3265i 1.44551i
\(704\) 0 0
\(705\) 4.47689 + 22.7187i 0.168609 + 0.855634i
\(706\) 0 0
\(707\) 9.64015i 0.362555i
\(708\) 0 0
\(709\) 28.7187 1.07855 0.539276 0.842129i \(-0.318698\pi\)
0.539276 + 0.842129i \(0.318698\pi\)
\(710\) 0 0
\(711\) −0.584735 −0.0219293
\(712\) 0 0
\(713\) 28.0558i 1.05070i
\(714\) 0 0
\(715\) −7.54333 + 1.48647i −0.282104 + 0.0555908i
\(716\) 0 0
\(717\) 35.4942i 1.32555i
\(718\) 0 0
\(719\) 8.60599 0.320949 0.160475 0.987040i \(-0.448698\pi\)
0.160475 + 0.987040i \(0.448698\pi\)
\(720\) 0 0
\(721\) −0.626198 −0.0233208
\(722\) 0 0
\(723\) 6.56934i 0.244316i
\(724\) 0 0
\(725\) 36.4479 14.9450i 1.35364 0.555045i
\(726\) 0 0
\(727\) 15.2803i 0.566715i 0.959014 + 0.283358i \(0.0914483\pi\)
−0.959014 + 0.283358i \(0.908552\pi\)
\(728\) 0 0
\(729\) −26.0096 −0.963318
\(730\) 0 0
\(731\) −1.55102 −0.0573666
\(732\) 0 0
\(733\) 46.0235i 1.69992i 0.526849 + 0.849959i \(0.323373\pi\)
−0.526849 + 0.849959i \(0.676627\pi\)
\(734\) 0 0
\(735\) 3.86464 0.761557i 0.142549 0.0280905i
\(736\) 0 0
\(737\) 3.62809i 0.133642i
\(738\) 0 0
\(739\) 46.3467 1.70489 0.852446 0.522815i \(-0.175118\pi\)
0.852446 + 0.522815i \(0.175118\pi\)
\(740\) 0 0
\(741\) −61.7851 −2.26973
\(742\) 0 0
\(743\) 10.7755i 0.395315i 0.980271 + 0.197658i \(0.0633334\pi\)
−0.980271 + 0.197658i \(0.936667\pi\)
\(744\) 0 0
\(745\) 7.36943 + 37.3973i 0.269995 + 1.37013i
\(746\) 0 0
\(747\) 1.79716i 0.0657546i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.76781 0.100999 0.0504995 0.998724i \(-0.483919\pi\)
0.0504995 + 0.998724i \(0.483919\pi\)
\(752\) 0 0
\(753\) 37.8709i 1.38009i
\(754\) 0 0
\(755\) −1.25240 6.35548i −0.0455794 0.231300i
\(756\) 0 0
\(757\) 14.8401i 0.539371i 0.962948 + 0.269686i \(0.0869198\pi\)
−0.962948 + 0.269686i \(0.913080\pi\)
\(758\) 0 0
\(759\) −4.11371 −0.149318
\(760\) 0 0
\(761\) 31.3169 1.13524 0.567619 0.823291i \(-0.307865\pi\)
0.567619 + 0.823291i \(0.307865\pi\)
\(762\) 0 0
\(763\) 18.9248i 0.685125i
\(764\) 0 0
\(765\) 0.202840 0.0399712i 0.00733371 0.00144516i
\(766\) 0 0
\(767\) 3.26113i 0.117753i
\(768\) 0 0
\(769\) −8.92586 −0.321875 −0.160937 0.986965i \(-0.551452\pi\)
−0.160937 + 0.986965i \(0.551452\pi\)
\(770\) 0 0
\(771\) −7.52311 −0.270938
\(772\) 0 0
\(773\) 36.7711i 1.32257i −0.750136 0.661283i \(-0.770012\pi\)
0.750136 0.661283i \(-0.229988\pi\)
\(774\) 0 0
\(775\) −34.8034 + 14.2707i −1.25018 + 0.512619i
\(776\) 0 0
\(777\) 10.5693i 0.379173i
\(778\) 0 0
\(779\) 49.3728 1.76896
\(780\) 0 0
\(781\) 3.45856 0.123757
\(782\) 0 0
\(783\) 40.2051i 1.43681i
\(784\) 0 0
\(785\) 22.1955 4.37380i 0.792193 0.156108i
\(786\) 0 0
\(787\) 6.53707i 0.233021i 0.993189 + 0.116511i \(0.0371709\pi\)
−0.993189 + 0.116511i \(0.962829\pi\)
\(788\) 0 0
\(789\) −20.1849 −0.718602
\(790\) 0 0
\(791\) −1.04623 −0.0371996
\(792\) 0 0
\(793\) 39.1791i 1.39129i
\(794\) 0 0
\(795\) −5.15994 26.1849i −0.183004 0.928683i
\(796\) 0 0
\(797\) 7.82611i 0.277215i 0.990347 + 0.138607i \(0.0442626\pi\)
−0.990347 + 0.138607i \(0.955737\pi\)
\(798\) 0 0
\(799\) −5.27261 −0.186531
\(800\) 0 0
\(801\) 1.47396 0.0520796
\(802\) 0 0
\(803\) 2.33527i 0.0824099i
\(804\) 0 0
\(805\) 1.61224 + 8.18159i 0.0568242 + 0.288363i
\(806\) 0 0
\(807\) 12.4556i 0.438459i
\(808\) 0 0
\(809\) −38.4113 −1.35047 −0.675235 0.737603i \(-0.735958\pi\)
−0.675235 + 0.737603i \(0.735958\pi\)
\(810\) 0 0
\(811\) −7.64015 −0.268282 −0.134141 0.990962i \(-0.542828\pi\)
−0.134141 + 0.990962i \(0.542828\pi\)
\(812\) 0 0
\(813\) 29.9634i 1.05086i
\(814\) 0 0
\(815\) −1.04623 + 0.206167i −0.0366478 + 0.00722173i
\(816\) 0 0
\(817\) 11.0462i 0.386459i
\(818\) 0 0
\(819\) 0.566016 0.0197782
\(820\) 0 0
\(821\) −2.56165 −0.0894021 −0.0447011 0.999000i \(-0.514234\pi\)
−0.0447011 + 0.999000i \(0.514234\pi\)
\(822\) 0 0
\(823\) 0.784248i 0.0273372i 0.999907 + 0.0136686i \(0.00435098\pi\)
−0.999907 + 0.0136686i \(0.995649\pi\)
\(824\) 0 0
\(825\) 2.09246 + 5.10308i 0.0728500 + 0.177667i
\(826\) 0 0
\(827\) 7.28904i 0.253465i 0.991937 + 0.126732i \(0.0404489\pi\)
−0.991937 + 0.126732i \(0.959551\pi\)
\(828\) 0 0
\(829\) −11.0708 −0.384505 −0.192253 0.981345i \(-0.561579\pi\)
−0.192253 + 0.981345i \(0.561579\pi\)
\(830\) 0 0
\(831\) −33.0741 −1.14733
\(832\) 0 0
\(833\) 0.896916i 0.0310763i
\(834\) 0 0
\(835\) 28.8882 5.69264i 0.999717 0.197002i
\(836\) 0 0
\(837\) 38.3911i 1.32699i
\(838\) 0 0
\(839\) 5.55976 0.191944 0.0959721 0.995384i \(-0.469404\pi\)
0.0959721 + 0.995384i \(0.469404\pi\)
\(840\) 0 0
\(841\) 33.0722 1.14042
\(842\) 0 0
\(843\) 8.10015i 0.278984i
\(844\) 0 0
\(845\) 7.41400 + 37.6235i 0.255049 + 1.29429i
\(846\) 0 0
\(847\) 10.6079i 0.364491i
\(848\) 0 0
\(849\) 23.9278 0.821198
\(850\) 0 0
\(851\) 22.3757 0.767029
\(852\) 0 0
\(853\) 16.0804i 0.550582i −0.961361 0.275291i \(-0.911226\pi\)
0.961361 0.275291i \(-0.0887742\pi\)
\(854\) 0 0
\(855\) 0.284672 + 1.44461i 0.00973556 + 0.0494046i
\(856\) 0 0
\(857\) 1.58767i 0.0542336i −0.999632 0.0271168i \(-0.991367\pi\)
0.999632 0.0271168i \(-0.00863261\pi\)
\(858\) 0 0
\(859\) 4.94171 0.168609 0.0843044 0.996440i \(-0.473133\pi\)
0.0843044 + 0.996440i \(0.473133\pi\)
\(860\) 0 0
\(861\) −13.6156 −0.464017
\(862\) 0 0
\(863\) 37.6647i 1.28212i −0.767490 0.641061i \(-0.778494\pi\)
0.767490 0.641061i \(-0.221506\pi\)
\(864\) 0 0
\(865\) −30.6435 + 6.03853i −1.04191 + 0.205316i
\(866\) 0 0
\(867\) 28.5294i 0.968908i
\(868\) 0 0
\(869\) 3.55206 0.120495
\(870\) 0 0
\(871\) −31.8130 −1.07794
\(872\) 0 0
\(873\) 1.04623i 0.0354095i
\(874\) 0 0
\(875\) 9.32924 6.16160i 0.315386 0.208300i
\(876\) 0 0
\(877\) 3.66473i 0.123749i −0.998084 0.0618746i \(-0.980292\pi\)
0.998084 0.0618746i \(-0.0197079\pi\)
\(878\) 0 0
\(879\) −20.8324 −0.702658
\(880\) 0 0
\(881\) −43.4373 −1.46344 −0.731720 0.681605i \(-0.761282\pi\)
−0.731720 + 0.681605i \(0.761282\pi\)
\(882\) 0 0
\(883\) 47.3853i 1.59464i 0.603556 + 0.797321i \(0.293750\pi\)
−0.603556 + 0.797321i \(0.706250\pi\)
\(884\) 0 0
\(885\) 2.29530 0.452306i 0.0771556 0.0152041i
\(886\) 0 0
\(887\) 21.3169i 0.715753i −0.933769 0.357877i \(-0.883501\pi\)
0.933769 0.357877i \(-0.116499\pi\)
\(888\) 0 0
\(889\) 21.2803 0.713718
\(890\) 0 0
\(891\) −5.82278 −0.195071
\(892\) 0 0
\(893\) 37.5510i 1.25660i
\(894\) 0 0
\(895\) −3.22449 16.3632i −0.107783 0.546961i
\(896\) 0 0
\(897\) 36.0712i 1.20438i
\(898\) 0 0
\(899\) −59.2716 −1.97682
\(900\) 0 0
\(901\) 6.07707 0.202456
\(902\) 0 0
\(903\) 3.04623i 0.101372i
\(904\) 0 0
\(905\) −6.10308 30.9711i −0.202873 1.02951i
\(906\) 0 0
\(907\) 18.9325i 0.628644i −0.949316 0.314322i \(-0.898223\pi\)
0.949316 0.314322i \(-0.101777\pi\)
\(908\) 0 0
\(909\) 0.993743 0.0329604
\(910\) 0 0
\(911\) −12.6705 −0.419794 −0.209897 0.977724i \(-0.567313\pi\)
−0.209897 + 0.977724i \(0.567313\pi\)
\(912\) 0 0
\(913\) 10.9171i 0.361304i
\(914\) 0 0
\(915\) 27.5756 5.43398i 0.911621 0.179642i
\(916\) 0 0
\(917\) 9.91087i 0.327286i
\(918\) 0 0
\(919\) −22.8690 −0.754379 −0.377190 0.926136i \(-0.623109\pi\)
−0.377190 + 0.926136i \(0.623109\pi\)
\(920\) 0 0
\(921\) −36.9681 −1.21814
\(922\) 0 0
\(923\) 30.3265i 0.998210i
\(924\) 0 0
\(925\) −11.3815 27.7572i −0.374221 0.912651i
\(926\) 0 0
\(927\) 0.0645508i 0.00212013i
\(928\) 0 0
\(929\) −50.4190 −1.65419 −0.827097 0.562060i \(-0.810009\pi\)
−0.827097 + 0.562060i \(0.810009\pi\)
\(930\) 0 0
\(931\) −6.38776 −0.209350
\(932\) 0 0
\(933\) 39.6435i 1.29787i
\(934\) 0 0
\(935\) −1.23219 + 0.242812i −0.0402968 + 0.00794079i
\(936\) 0 0
\(937\) 47.9344i 1.56595i 0.622054 + 0.782974i \(0.286298\pi\)
−0.622054 + 0.782974i \(0.713702\pi\)
\(938\) 0 0
\(939\) −21.9923 −0.717692
\(940\) 0 0
\(941\) −59.9667 −1.95486 −0.977429 0.211264i \(-0.932242\pi\)
−0.977429 + 0.211264i \(0.932242\pi\)
\(942\) 0 0
\(943\) 28.8247i 0.938660i
\(944\) 0 0
\(945\) 2.20617 + 11.1955i 0.0717666 + 0.364191i
\(946\) 0 0
\(947\) 27.5231i 0.894381i −0.894439 0.447191i \(-0.852425\pi\)
0.894439 0.447191i \(-0.147575\pi\)
\(948\) 0 0
\(949\) −20.4769 −0.664708
\(950\) 0 0
\(951\) 44.9604 1.45794
\(952\) 0 0
\(953\) 0.840061i 0.0272123i 0.999907 + 0.0136061i \(0.00433110\pi\)
−0.999907 + 0.0136061i \(0.995669\pi\)
\(954\) 0 0
\(955\) 3.64015 + 18.4725i 0.117793 + 0.597757i
\(956\) 0 0
\(957\) 8.69075i 0.280932i
\(958\) 0 0
\(959\) −3.45856 −0.111683
\(960\) 0 0
\(961\) 25.5972 0.825718
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −48.9205 + 9.64015i −1.57481 + 0.310327i
\(966\) 0 0
\(967\) 31.8988i 1.02580i −0.858449 0.512898i \(-0.828572\pi\)
0.858449 0.512898i \(-0.171428\pi\)
\(968\) 0 0
\(969\) −10.0925 −0.324216
\(970\) 0 0
\(971\) −28.4157 −0.911902 −0.455951 0.890005i \(-0.650701\pi\)
−0.455951 + 0.890005i \(0.650701\pi\)
\(972\) 0 0
\(973\) 20.4157i 0.654496i
\(974\) 0 0
\(975\) 44.7466 18.3478i 1.43304 0.587599i
\(976\) 0 0
\(977\) 9.55102i 0.305564i −0.988260 0.152782i \(-0.951177\pi\)
0.988260 0.152782i \(-0.0488233\pi\)
\(978\) 0 0
\(979\) −8.95377 −0.286164
\(980\) 0 0
\(981\) −1.95084 −0.0622856
\(982\) 0 0
\(983\) 0.0443400i 0.00141423i −1.00000 0.000707113i \(-0.999775\pi\)
1.00000 0.000707113i \(-0.000225081\pi\)
\(984\) 0 0
\(985\) 57.8376 11.3973i 1.84286 0.363149i
\(986\) 0 0
\(987\) 10.3555i 0.329619i
\(988\) 0 0
\(989\) −6.44898 −0.205066
\(990\) 0 0
\(991\) −13.9913 −0.444447 −0.222224 0.974996i \(-0.571331\pi\)
−0.222224 + 0.974996i \(0.571331\pi\)
\(992\) 0 0
\(993\) 4.72635i 0.149986i
\(994\) 0 0
\(995\) 9.70138 + 49.2311i 0.307554 + 1.56073i
\(996\) 0 0
\(997\) 19.0419i 0.603062i −0.953456 0.301531i \(-0.902502\pi\)
0.953456 0.301531i \(-0.0974976\pi\)
\(998\) 0 0
\(999\) 30.6185 0.968727
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 2240.2.g.l.449.2 6
4.3 odd 2 2240.2.g.m.449.5 6
5.4 even 2 inner 2240.2.g.l.449.5 6
8.3 odd 2 560.2.g.f.449.2 6
8.5 even 2 280.2.g.b.169.5 yes 6
20.19 odd 2 2240.2.g.m.449.2 6
24.5 odd 2 2520.2.t.g.1009.3 6
24.11 even 2 5040.2.t.y.1009.3 6
40.3 even 4 2800.2.a.bq.1.3 3
40.13 odd 4 1400.2.a.t.1.1 3
40.19 odd 2 560.2.g.f.449.5 6
40.27 even 4 2800.2.a.br.1.1 3
40.29 even 2 280.2.g.b.169.2 6
40.37 odd 4 1400.2.a.s.1.3 3
56.13 odd 2 1960.2.g.c.1569.2 6
120.29 odd 2 2520.2.t.g.1009.4 6
120.59 even 2 5040.2.t.y.1009.4 6
280.13 even 4 9800.2.a.cd.1.3 3
280.69 odd 2 1960.2.g.c.1569.5 6
280.237 even 4 9800.2.a.cg.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.g.b.169.2 6 40.29 even 2
280.2.g.b.169.5 yes 6 8.5 even 2
560.2.g.f.449.2 6 8.3 odd 2
560.2.g.f.449.5 6 40.19 odd 2
1400.2.a.s.1.3 3 40.37 odd 4
1400.2.a.t.1.1 3 40.13 odd 4
1960.2.g.c.1569.2 6 56.13 odd 2
1960.2.g.c.1569.5 6 280.69 odd 2
2240.2.g.l.449.2 6 1.1 even 1 trivial
2240.2.g.l.449.5 6 5.4 even 2 inner
2240.2.g.m.449.2 6 20.19 odd 2
2240.2.g.m.449.5 6 4.3 odd 2
2520.2.t.g.1009.3 6 24.5 odd 2
2520.2.t.g.1009.4 6 120.29 odd 2
2800.2.a.bq.1.3 3 40.3 even 4
2800.2.a.br.1.1 3 40.27 even 4
5040.2.t.y.1009.3 6 24.11 even 2
5040.2.t.y.1009.4 6 120.59 even 2
9800.2.a.cd.1.3 3 280.13 even 4
9800.2.a.cg.1.1 3 280.237 even 4