Properties

Label 2394.2.e.b.1063.5
Level $2394$
Weight $2$
Character 2394.1063
Analytic conductor $19.116$
Analytic rank $0$
Dimension $12$
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] = [2394,2,Mod(1063,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2394.1063");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.e (of order \(2\), degree \(1\), 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: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + 2x^{10} + 54x^{8} - 114x^{7} + 120x^{6} + 46x^{5} + 9x^{4} - 4x^{3} + 8x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 798)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1063.5
Root \(1.25342 - 1.25342i\) of defining polynomial
Character \(\chi\) \(=\) 2394.1063
Dual form 2394.2.e.b.1063.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +2.50684i q^{5} +(-2.53963 - 0.741811i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +2.50684i q^{5} +(-2.53963 - 0.741811i) q^{7} +1.00000i q^{8} +2.50684 q^{10} +4.07545 q^{11} -0.506836 q^{13} +(-0.741811 + 2.53963i) q^{14} +1.00000 q^{16} -2.30760i q^{17} +(-4.30085 - 0.709018i) q^{19} -2.50684i q^{20} -4.07545i q^{22} +6.34027 q^{23} -1.28423 q^{25} +0.506836i q^{26} +(2.53963 + 0.741811i) q^{28} +8.66728i q^{29} -1.39640 q^{31} -1.00000i q^{32} -2.30760 q^{34} +(1.85960 - 6.36643i) q^{35} -1.68285i q^{37} +(-0.709018 + 4.30085i) q^{38} -2.50684 q^{40} +0.402139 q^{41} -6.73509 q^{43} -4.07545 q^{44} -6.34027i q^{46} +5.33646i q^{47} +(5.89943 + 3.76785i) q^{49} +1.28423i q^{50} +0.506836 q^{52} +1.66122i q^{53} +10.2165i q^{55} +(0.741811 - 2.53963i) q^{56} +8.66728 q^{58} -9.42907 q^{59} +5.31706i q^{61} +1.39640i q^{62} -1.00000 q^{64} -1.27056i q^{65} -3.06939i q^{67} +2.30760i q^{68} +(-6.36643 - 1.85960i) q^{70} +8.08532i q^{71} +13.5144i q^{73} -1.68285 q^{74} +(4.30085 + 0.709018i) q^{76} +(-10.3501 - 3.02321i) q^{77} -3.37893i q^{79} +2.50684i q^{80} -0.402139i q^{82} +10.5106i q^{83} +5.78478 q^{85} +6.73509i q^{86} +4.07545i q^{88} -11.7389 q^{89} +(1.28718 + 0.375977i) q^{91} -6.34027 q^{92} +5.33646 q^{94} +(1.77739 - 10.7815i) q^{95} +4.87048 q^{97} +(3.76785 - 5.89943i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} + 4 q^{10} - 12 q^{11} + 20 q^{13} + 4 q^{14} + 12 q^{16} - 8 q^{19} + 12 q^{23} - 12 q^{25} + 24 q^{31} + 4 q^{34} - 4 q^{35} - 4 q^{40} + 16 q^{41} + 12 q^{44} + 4 q^{49} - 20 q^{52} - 4 q^{56} + 8 q^{58} - 40 q^{59} - 12 q^{64} - 24 q^{70} + 8 q^{76} - 8 q^{77} + 8 q^{85} + 16 q^{89} + 24 q^{91} - 12 q^{92} + 44 q^{95} - 60 q^{97} + 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 2.50684i 1.12109i 0.828124 + 0.560546i \(0.189409\pi\)
−0.828124 + 0.560546i \(0.810591\pi\)
\(6\) 0 0
\(7\) −2.53963 0.741811i −0.959890 0.280378i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 2.50684 0.792731
\(11\) 4.07545 1.22880 0.614398 0.788997i \(-0.289399\pi\)
0.614398 + 0.788997i \(0.289399\pi\)
\(12\) 0 0
\(13\) −0.506836 −0.140571 −0.0702855 0.997527i \(-0.522391\pi\)
−0.0702855 + 0.997527i \(0.522391\pi\)
\(14\) −0.741811 + 2.53963i −0.198257 + 0.678744i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.30760i 0.559676i −0.960047 0.279838i \(-0.909719\pi\)
0.960047 0.279838i \(-0.0902807\pi\)
\(18\) 0 0
\(19\) −4.30085 0.709018i −0.986682 0.162660i
\(20\) 2.50684i 0.560546i
\(21\) 0 0
\(22\) 4.07545i 0.868889i
\(23\) 6.34027 1.32204 0.661019 0.750369i \(-0.270124\pi\)
0.661019 + 0.750369i \(0.270124\pi\)
\(24\) 0 0
\(25\) −1.28423 −0.256845
\(26\) 0.506836i 0.0993987i
\(27\) 0 0
\(28\) 2.53963 + 0.741811i 0.479945 + 0.140189i
\(29\) 8.66728i 1.60947i 0.593632 + 0.804737i \(0.297694\pi\)
−0.593632 + 0.804737i \(0.702306\pi\)
\(30\) 0 0
\(31\) −1.39640 −0.250802 −0.125401 0.992106i \(-0.540022\pi\)
−0.125401 + 0.992106i \(0.540022\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −2.30760 −0.395751
\(35\) 1.85960 6.36643i 0.314329 1.07612i
\(36\) 0 0
\(37\) 1.68285i 0.276660i −0.990386 0.138330i \(-0.955827\pi\)
0.990386 0.138330i \(-0.0441734\pi\)
\(38\) −0.709018 + 4.30085i −0.115018 + 0.697690i
\(39\) 0 0
\(40\) −2.50684 −0.396366
\(41\) 0.402139 0.0628036 0.0314018 0.999507i \(-0.490003\pi\)
0.0314018 + 0.999507i \(0.490003\pi\)
\(42\) 0 0
\(43\) −6.73509 −1.02709 −0.513546 0.858062i \(-0.671668\pi\)
−0.513546 + 0.858062i \(0.671668\pi\)
\(44\) −4.07545 −0.614398
\(45\) 0 0
\(46\) 6.34027i 0.934822i
\(47\) 5.33646i 0.778403i 0.921153 + 0.389202i \(0.127249\pi\)
−0.921153 + 0.389202i \(0.872751\pi\)
\(48\) 0 0
\(49\) 5.89943 + 3.76785i 0.842776 + 0.538264i
\(50\) 1.28423i 0.181617i
\(51\) 0 0
\(52\) 0.506836 0.0702855
\(53\) 1.66122i 0.228187i 0.993470 + 0.114093i \(0.0363963\pi\)
−0.993470 + 0.114093i \(0.963604\pi\)
\(54\) 0 0
\(55\) 10.2165i 1.37759i
\(56\) 0.741811 2.53963i 0.0991286 0.339372i
\(57\) 0 0
\(58\) 8.66728 1.13807
\(59\) −9.42907 −1.22756 −0.613780 0.789477i \(-0.710352\pi\)
−0.613780 + 0.789477i \(0.710352\pi\)
\(60\) 0 0
\(61\) 5.31706i 0.680779i 0.940284 + 0.340390i \(0.110559\pi\)
−0.940284 + 0.340390i \(0.889441\pi\)
\(62\) 1.39640i 0.177343i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 1.27056i 0.157593i
\(66\) 0 0
\(67\) 3.06939i 0.374986i −0.982266 0.187493i \(-0.939964\pi\)
0.982266 0.187493i \(-0.0600362\pi\)
\(68\) 2.30760i 0.279838i
\(69\) 0 0
\(70\) −6.36643 1.85960i −0.760934 0.222264i
\(71\) 8.08532i 0.959551i 0.877391 + 0.479775i \(0.159282\pi\)
−0.877391 + 0.479775i \(0.840718\pi\)
\(72\) 0 0
\(73\) 13.5144i 1.58174i 0.611984 + 0.790870i \(0.290372\pi\)
−0.611984 + 0.790870i \(0.709628\pi\)
\(74\) −1.68285 −0.195628
\(75\) 0 0
\(76\) 4.30085 + 0.709018i 0.493341 + 0.0813299i
\(77\) −10.3501 3.02321i −1.17951 0.344527i
\(78\) 0 0
\(79\) 3.37893i 0.380159i −0.981769 0.190079i \(-0.939125\pi\)
0.981769 0.190079i \(-0.0608745\pi\)
\(80\) 2.50684i 0.280273i
\(81\) 0 0
\(82\) 0.402139i 0.0444089i
\(83\) 10.5106i 1.15368i 0.816856 + 0.576842i \(0.195715\pi\)
−0.816856 + 0.576842i \(0.804285\pi\)
\(84\) 0 0
\(85\) 5.78478 0.627448
\(86\) 6.73509i 0.726264i
\(87\) 0 0
\(88\) 4.07545i 0.434445i
\(89\) −11.7389 −1.24432 −0.622162 0.782889i \(-0.713745\pi\)
−0.622162 + 0.782889i \(0.713745\pi\)
\(90\) 0 0
\(91\) 1.28718 + 0.375977i 0.134933 + 0.0394130i
\(92\) −6.34027 −0.661019
\(93\) 0 0
\(94\) 5.33646 0.550414
\(95\) 1.77739 10.7815i 0.182356 1.10616i
\(96\) 0 0
\(97\) 4.87048 0.494523 0.247261 0.968949i \(-0.420469\pi\)
0.247261 + 0.968949i \(0.420469\pi\)
\(98\) 3.76785 5.89943i 0.380610 0.595933i
\(99\) 0 0
\(100\) 1.28423 0.128423
\(101\) 10.8277i 1.07740i 0.842498 + 0.538700i \(0.181084\pi\)
−0.842498 + 0.538700i \(0.818916\pi\)
\(102\) 0 0
\(103\) 5.51407 0.543317 0.271659 0.962394i \(-0.412428\pi\)
0.271659 + 0.962394i \(0.412428\pi\)
\(104\) 0.506836i 0.0496994i
\(105\) 0 0
\(106\) 1.66122 0.161352
\(107\) 1.64796i 0.159315i 0.996822 + 0.0796573i \(0.0253826\pi\)
−0.996822 + 0.0796573i \(0.974617\pi\)
\(108\) 0 0
\(109\) 18.0910i 1.73281i 0.499345 + 0.866403i \(0.333574\pi\)
−0.499345 + 0.866403i \(0.666426\pi\)
\(110\) 10.2165 0.974104
\(111\) 0 0
\(112\) −2.53963 0.741811i −0.239972 0.0700945i
\(113\) 2.33655i 0.219804i 0.993942 + 0.109902i \(0.0350538\pi\)
−0.993942 + 0.109902i \(0.964946\pi\)
\(114\) 0 0
\(115\) 15.8940i 1.48213i
\(116\) 8.66728i 0.804737i
\(117\) 0 0
\(118\) 9.42907i 0.868016i
\(119\) −1.71180 + 5.86046i −0.156921 + 0.537227i
\(120\) 0 0
\(121\) 5.60931 0.509937
\(122\) 5.31706 0.481384
\(123\) 0 0
\(124\) 1.39640 0.125401
\(125\) 9.31483i 0.833144i
\(126\) 0 0
\(127\) 1.29418i 0.114840i −0.998350 0.0574200i \(-0.981713\pi\)
0.998350 0.0574200i \(-0.0182874\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) −1.27056 −0.111435
\(131\) 4.38528i 0.383144i 0.981479 + 0.191572i \(0.0613585\pi\)
−0.981479 + 0.191572i \(0.938642\pi\)
\(132\) 0 0
\(133\) 10.3966 + 4.99106i 0.901500 + 0.432780i
\(134\) −3.06939 −0.265155
\(135\) 0 0
\(136\) 2.30760 0.197875
\(137\) 17.3945 1.48611 0.743056 0.669229i \(-0.233375\pi\)
0.743056 + 0.669229i \(0.233375\pi\)
\(138\) 0 0
\(139\) 4.29012i 0.363884i −0.983309 0.181942i \(-0.941762\pi\)
0.983309 0.181942i \(-0.0582382\pi\)
\(140\) −1.85960 + 6.36643i −0.157165 + 0.538062i
\(141\) 0 0
\(142\) 8.08532 0.678505
\(143\) −2.06559 −0.172733
\(144\) 0 0
\(145\) −21.7275 −1.80437
\(146\) 13.5144 1.11846
\(147\) 0 0
\(148\) 1.68285i 0.138330i
\(149\) 9.33082 0.764410 0.382205 0.924077i \(-0.375165\pi\)
0.382205 + 0.924077i \(0.375165\pi\)
\(150\) 0 0
\(151\) 7.62713i 0.620687i 0.950624 + 0.310344i \(0.100444\pi\)
−0.950624 + 0.310344i \(0.899556\pi\)
\(152\) 0.709018 4.30085i 0.0575089 0.348845i
\(153\) 0 0
\(154\) −3.02321 + 10.3501i −0.243618 + 0.834038i
\(155\) 3.50056i 0.281171i
\(156\) 0 0
\(157\) 13.0191i 1.03904i −0.854458 0.519520i \(-0.826111\pi\)
0.854458 0.519520i \(-0.173889\pi\)
\(158\) −3.37893 −0.268813
\(159\) 0 0
\(160\) 2.50684 0.198183
\(161\) −16.1019 4.70328i −1.26901 0.370671i
\(162\) 0 0
\(163\) −4.37938 −0.343020 −0.171510 0.985182i \(-0.554865\pi\)
−0.171510 + 0.985182i \(0.554865\pi\)
\(164\) −0.402139 −0.0314018
\(165\) 0 0
\(166\) 10.5106 0.815777
\(167\) 2.29406 0.177520 0.0887600 0.996053i \(-0.471710\pi\)
0.0887600 + 0.996053i \(0.471710\pi\)
\(168\) 0 0
\(169\) −12.7431 −0.980240
\(170\) 5.78478i 0.443673i
\(171\) 0 0
\(172\) 6.73509 0.513546
\(173\) −22.4568 −1.70736 −0.853681 0.520797i \(-0.825635\pi\)
−0.853681 + 0.520797i \(0.825635\pi\)
\(174\) 0 0
\(175\) 3.26146 + 0.952654i 0.246543 + 0.0720139i
\(176\) 4.07545 0.307199
\(177\) 0 0
\(178\) 11.7389i 0.879870i
\(179\) 12.0887i 0.903555i −0.892131 0.451777i \(-0.850790\pi\)
0.892131 0.451777i \(-0.149210\pi\)
\(180\) 0 0
\(181\) −14.6113 −1.08605 −0.543025 0.839716i \(-0.682721\pi\)
−0.543025 + 0.839716i \(0.682721\pi\)
\(182\) 0.375977 1.28718i 0.0278692 0.0954118i
\(183\) 0 0
\(184\) 6.34027i 0.467411i
\(185\) 4.21864 0.310161
\(186\) 0 0
\(187\) 9.40453i 0.687727i
\(188\) 5.33646i 0.389202i
\(189\) 0 0
\(190\) −10.7815 1.77739i −0.782174 0.128946i
\(191\) −0.142938 −0.0103427 −0.00517133 0.999987i \(-0.501646\pi\)
−0.00517133 + 0.999987i \(0.501646\pi\)
\(192\) 0 0
\(193\) 5.94006i 0.427575i 0.976880 + 0.213787i \(0.0685800\pi\)
−0.976880 + 0.213787i \(0.931420\pi\)
\(194\) 4.87048i 0.349680i
\(195\) 0 0
\(196\) −5.89943 3.76785i −0.421388 0.269132i
\(197\) −9.50252 −0.677027 −0.338513 0.940962i \(-0.609924\pi\)
−0.338513 + 0.940962i \(0.609924\pi\)
\(198\) 0 0
\(199\) 23.3287i 1.65373i 0.562404 + 0.826863i \(0.309877\pi\)
−0.562404 + 0.826863i \(0.690123\pi\)
\(200\) 1.28423i 0.0908086i
\(201\) 0 0
\(202\) 10.8277 0.761836
\(203\) 6.42948 22.0117i 0.451261 1.54492i
\(204\) 0 0
\(205\) 1.00810i 0.0704086i
\(206\) 5.51407i 0.384183i
\(207\) 0 0
\(208\) −0.506836 −0.0351428
\(209\) −17.5279 2.88957i −1.21243 0.199876i
\(210\) 0 0
\(211\) 23.1860i 1.59619i −0.602532 0.798095i \(-0.705841\pi\)
0.602532 0.798095i \(-0.294159\pi\)
\(212\) 1.66122i 0.114093i
\(213\) 0 0
\(214\) 1.64796 0.112652
\(215\) 16.8838i 1.15146i
\(216\) 0 0
\(217\) 3.54635 + 1.03587i 0.240742 + 0.0703193i
\(218\) 18.0910 1.22528
\(219\) 0 0
\(220\) 10.2165i 0.688796i
\(221\) 1.16958i 0.0786742i
\(222\) 0 0
\(223\) 23.0366 1.54264 0.771322 0.636446i \(-0.219596\pi\)
0.771322 + 0.636446i \(0.219596\pi\)
\(224\) −0.741811 + 2.53963i −0.0495643 + 0.169686i
\(225\) 0 0
\(226\) 2.33655 0.155425
\(227\) 17.6866 1.17390 0.586951 0.809623i \(-0.300328\pi\)
0.586951 + 0.809623i \(0.300328\pi\)
\(228\) 0 0
\(229\) 3.87815i 0.256275i −0.991756 0.128138i \(-0.959100\pi\)
0.991756 0.128138i \(-0.0408999\pi\)
\(230\) 15.8940 1.04802
\(231\) 0 0
\(232\) −8.66728 −0.569035
\(233\) 21.9618 1.43877 0.719384 0.694613i \(-0.244424\pi\)
0.719384 + 0.694613i \(0.244424\pi\)
\(234\) 0 0
\(235\) −13.3776 −0.872661
\(236\) 9.42907 0.613780
\(237\) 0 0
\(238\) 5.86046 + 1.71180i 0.379877 + 0.110960i
\(239\) 5.77182 0.373348 0.186674 0.982422i \(-0.440229\pi\)
0.186674 + 0.982422i \(0.440229\pi\)
\(240\) 0 0
\(241\) −10.8837 −0.701079 −0.350539 0.936548i \(-0.614002\pi\)
−0.350539 + 0.936548i \(0.614002\pi\)
\(242\) 5.60931i 0.360580i
\(243\) 0 0
\(244\) 5.31706i 0.340390i
\(245\) −9.44538 + 14.7889i −0.603443 + 0.944829i
\(246\) 0 0
\(247\) 2.17983 + 0.359356i 0.138699 + 0.0228653i
\(248\) 1.39640i 0.0886717i
\(249\) 0 0
\(250\) 9.31483 0.589122
\(251\) 23.2434i 1.46711i −0.679629 0.733556i \(-0.737859\pi\)
0.679629 0.733556i \(-0.262141\pi\)
\(252\) 0 0
\(253\) 25.8395 1.62451
\(254\) −1.29418 −0.0812042
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −29.4331 −1.83599 −0.917995 0.396593i \(-0.870192\pi\)
−0.917995 + 0.396593i \(0.870192\pi\)
\(258\) 0 0
\(259\) −1.24836 + 4.27383i −0.0775693 + 0.265563i
\(260\) 1.27056i 0.0787965i
\(261\) 0 0
\(262\) 4.38528 0.270923
\(263\) 9.57036 0.590134 0.295067 0.955477i \(-0.404658\pi\)
0.295067 + 0.955477i \(0.404658\pi\)
\(264\) 0 0
\(265\) −4.16441 −0.255818
\(266\) 4.99106 10.3966i 0.306021 0.637457i
\(267\) 0 0
\(268\) 3.06939i 0.187493i
\(269\) 20.3251 1.23924 0.619621 0.784901i \(-0.287286\pi\)
0.619621 + 0.784901i \(0.287286\pi\)
\(270\) 0 0
\(271\) 27.0034i 1.64034i −0.572120 0.820170i \(-0.693879\pi\)
0.572120 0.820170i \(-0.306121\pi\)
\(272\) 2.30760i 0.139919i
\(273\) 0 0
\(274\) 17.3945i 1.05084i
\(275\) −5.23381 −0.315610
\(276\) 0 0
\(277\) 1.93416 0.116213 0.0581063 0.998310i \(-0.481494\pi\)
0.0581063 + 0.998310i \(0.481494\pi\)
\(278\) −4.29012 −0.257305
\(279\) 0 0
\(280\) 6.36643 + 1.85960i 0.380467 + 0.111132i
\(281\) 26.0581i 1.55450i 0.629194 + 0.777248i \(0.283385\pi\)
−0.629194 + 0.777248i \(0.716615\pi\)
\(282\) 0 0
\(283\) 20.4608i 1.21627i 0.793835 + 0.608133i \(0.208081\pi\)
−0.793835 + 0.608133i \(0.791919\pi\)
\(284\) 8.08532i 0.479775i
\(285\) 0 0
\(286\) 2.06559i 0.122141i
\(287\) −1.02129 0.298311i −0.0602846 0.0176088i
\(288\) 0 0
\(289\) 11.6750 0.686763
\(290\) 21.7275i 1.27588i
\(291\) 0 0
\(292\) 13.5144i 0.790870i
\(293\) 16.4410 0.960495 0.480248 0.877133i \(-0.340547\pi\)
0.480248 + 0.877133i \(0.340547\pi\)
\(294\) 0 0
\(295\) 23.6371i 1.37621i
\(296\) 1.68285 0.0978139
\(297\) 0 0
\(298\) 9.33082i 0.540520i
\(299\) −3.21348 −0.185840
\(300\) 0 0
\(301\) 17.1046 + 4.99616i 0.985895 + 0.287974i
\(302\) 7.62713 0.438892
\(303\) 0 0
\(304\) −4.30085 0.709018i −0.246671 0.0406650i
\(305\) −13.3290 −0.763216
\(306\) 0 0
\(307\) −8.09858 −0.462210 −0.231105 0.972929i \(-0.574234\pi\)
−0.231105 + 0.972929i \(0.574234\pi\)
\(308\) 10.3501 + 3.02321i 0.589754 + 0.172264i
\(309\) 0 0
\(310\) −3.50056 −0.198818
\(311\) 8.87438i 0.503220i 0.967829 + 0.251610i \(0.0809600\pi\)
−0.967829 + 0.251610i \(0.919040\pi\)
\(312\) 0 0
\(313\) 22.9535i 1.29741i 0.761039 + 0.648706i \(0.224689\pi\)
−0.761039 + 0.648706i \(0.775311\pi\)
\(314\) −13.0191 −0.734712
\(315\) 0 0
\(316\) 3.37893i 0.190079i
\(317\) 19.9885i 1.12267i −0.827589 0.561334i \(-0.810288\pi\)
0.827589 0.561334i \(-0.189712\pi\)
\(318\) 0 0
\(319\) 35.3231i 1.97771i
\(320\) 2.50684i 0.140136i
\(321\) 0 0
\(322\) −4.70328 + 16.1019i −0.262104 + 0.897326i
\(323\) −1.63613 + 9.92465i −0.0910368 + 0.552222i
\(324\) 0 0
\(325\) 0.650893 0.0361050
\(326\) 4.37938i 0.242552i
\(327\) 0 0
\(328\) 0.402139i 0.0222044i
\(329\) 3.95865 13.5526i 0.218247 0.747181i
\(330\) 0 0
\(331\) 19.4117i 1.06697i −0.845811 0.533483i \(-0.820883\pi\)
0.845811 0.533483i \(-0.179117\pi\)
\(332\) 10.5106i 0.576842i
\(333\) 0 0
\(334\) 2.29406i 0.125526i
\(335\) 7.69446 0.420394
\(336\) 0 0
\(337\) 26.3229i 1.43390i 0.697124 + 0.716951i \(0.254463\pi\)
−0.697124 + 0.716951i \(0.745537\pi\)
\(338\) 12.7431i 0.693134i
\(339\) 0 0
\(340\) −5.78478 −0.313724
\(341\) −5.69098 −0.308184
\(342\) 0 0
\(343\) −12.1873 13.9452i −0.658055 0.752970i
\(344\) 6.73509i 0.363132i
\(345\) 0 0
\(346\) 22.4568i 1.20729i
\(347\) 6.55866 0.352087 0.176044 0.984382i \(-0.443670\pi\)
0.176044 + 0.984382i \(0.443670\pi\)
\(348\) 0 0
\(349\) 28.8644i 1.54508i −0.634968 0.772539i \(-0.718987\pi\)
0.634968 0.772539i \(-0.281013\pi\)
\(350\) 0.952654 3.26146i 0.0509215 0.174332i
\(351\) 0 0
\(352\) 4.07545i 0.217222i
\(353\) 34.6309i 1.84321i 0.388124 + 0.921607i \(0.373123\pi\)
−0.388124 + 0.921607i \(0.626877\pi\)
\(354\) 0 0
\(355\) −20.2686 −1.07574
\(356\) 11.7389 0.622162
\(357\) 0 0
\(358\) −12.0887 −0.638910
\(359\) −11.1407 −0.587983 −0.293992 0.955808i \(-0.594984\pi\)
−0.293992 + 0.955808i \(0.594984\pi\)
\(360\) 0 0
\(361\) 17.9946 + 6.09875i 0.947084 + 0.320987i
\(362\) 14.6113i 0.767953i
\(363\) 0 0
\(364\) −1.28718 0.375977i −0.0674663 0.0197065i
\(365\) −33.8784 −1.77327
\(366\) 0 0
\(367\) 11.0195i 0.575213i 0.957749 + 0.287607i \(0.0928595\pi\)
−0.957749 + 0.287607i \(0.907140\pi\)
\(368\) 6.34027 0.330509
\(369\) 0 0
\(370\) 4.21864i 0.219317i
\(371\) 1.23231 4.21889i 0.0639785 0.219034i
\(372\) 0 0
\(373\) 27.0191i 1.39900i −0.714634 0.699499i \(-0.753407\pi\)
0.714634 0.699499i \(-0.246593\pi\)
\(374\) −9.40453 −0.486296
\(375\) 0 0
\(376\) −5.33646 −0.275207
\(377\) 4.39289i 0.226245i
\(378\) 0 0
\(379\) 24.9769i 1.28298i −0.767133 0.641488i \(-0.778318\pi\)
0.767133 0.641488i \(-0.221682\pi\)
\(380\) −1.77739 + 10.7815i −0.0911782 + 0.553080i
\(381\) 0 0
\(382\) 0.142938i 0.00731337i
\(383\) 12.4870 0.638057 0.319029 0.947745i \(-0.396643\pi\)
0.319029 + 0.947745i \(0.396643\pi\)
\(384\) 0 0
\(385\) 7.57870 25.9461i 0.386246 1.32234i
\(386\) 5.94006 0.302341
\(387\) 0 0
\(388\) −4.87048 −0.247261
\(389\) −31.4241 −1.59327 −0.796634 0.604462i \(-0.793388\pi\)
−0.796634 + 0.604462i \(0.793388\pi\)
\(390\) 0 0
\(391\) 14.6308i 0.739913i
\(392\) −3.76785 + 5.89943i −0.190305 + 0.297966i
\(393\) 0 0
\(394\) 9.50252i 0.478730i
\(395\) 8.47041 0.426193
\(396\) 0 0
\(397\) 37.1197i 1.86299i −0.363759 0.931493i \(-0.618507\pi\)
0.363759 0.931493i \(-0.381493\pi\)
\(398\) 23.3287 1.16936
\(399\) 0 0
\(400\) −1.28423 −0.0642114
\(401\) 38.8566i 1.94041i 0.242289 + 0.970204i \(0.422102\pi\)
−0.242289 + 0.970204i \(0.577898\pi\)
\(402\) 0 0
\(403\) 0.707748 0.0352554
\(404\) 10.8277i 0.538700i
\(405\) 0 0
\(406\) −22.0117 6.42948i −1.09242 0.319090i
\(407\) 6.85839i 0.339958i
\(408\) 0 0
\(409\) 9.77207 0.483198 0.241599 0.970376i \(-0.422328\pi\)
0.241599 + 0.970376i \(0.422328\pi\)
\(410\) 1.00810 0.0497864
\(411\) 0 0
\(412\) −5.51407 −0.271659
\(413\) 23.9463 + 6.99459i 1.17832 + 0.344181i
\(414\) 0 0
\(415\) −26.3482 −1.29338
\(416\) 0.506836i 0.0248497i
\(417\) 0 0
\(418\) −2.88957 + 17.5279i −0.141333 + 0.857318i
\(419\) 30.2647i 1.47853i −0.673416 0.739264i \(-0.735174\pi\)
0.673416 0.739264i \(-0.264826\pi\)
\(420\) 0 0
\(421\) 21.7967i 1.06231i −0.847276 0.531153i \(-0.821759\pi\)
0.847276 0.531153i \(-0.178241\pi\)
\(422\) −23.1860 −1.12868
\(423\) 0 0
\(424\) −1.66122 −0.0806761
\(425\) 2.96349i 0.143750i
\(426\) 0 0
\(427\) 3.94425 13.5034i 0.190876 0.653473i
\(428\) 1.64796i 0.0796573i
\(429\) 0 0
\(430\) −16.8838 −0.814208
\(431\) 31.4075i 1.51285i 0.654082 + 0.756423i \(0.273055\pi\)
−0.654082 + 0.756423i \(0.726945\pi\)
\(432\) 0 0
\(433\) −13.5121 −0.649348 −0.324674 0.945826i \(-0.605255\pi\)
−0.324674 + 0.945826i \(0.605255\pi\)
\(434\) 1.03587 3.54635i 0.0497232 0.170230i
\(435\) 0 0
\(436\) 18.0910i 0.866403i
\(437\) −27.2685 4.49536i −1.30443 0.215042i
\(438\) 0 0
\(439\) 26.4310 1.26148 0.630742 0.775993i \(-0.282751\pi\)
0.630742 + 0.775993i \(0.282751\pi\)
\(440\) −10.2165 −0.487052
\(441\) 0 0
\(442\) 1.16958 0.0556311
\(443\) 27.7520 1.31854 0.659269 0.751907i \(-0.270866\pi\)
0.659269 + 0.751907i \(0.270866\pi\)
\(444\) 0 0
\(445\) 29.4276i 1.39500i
\(446\) 23.0366i 1.09081i
\(447\) 0 0
\(448\) 2.53963 + 0.741811i 0.119986 + 0.0350473i
\(449\) 15.1061i 0.712900i 0.934314 + 0.356450i \(0.116013\pi\)
−0.934314 + 0.356450i \(0.883987\pi\)
\(450\) 0 0
\(451\) 1.63890 0.0771728
\(452\) 2.33655i 0.109902i
\(453\) 0 0
\(454\) 17.6866i 0.830073i
\(455\) −0.942512 + 3.22674i −0.0441856 + 0.151272i
\(456\) 0 0
\(457\) −20.9766 −0.981245 −0.490623 0.871372i \(-0.663231\pi\)
−0.490623 + 0.871372i \(0.663231\pi\)
\(458\) −3.87815 −0.181214
\(459\) 0 0
\(460\) 15.8940i 0.741063i
\(461\) 7.85663i 0.365920i 0.983120 + 0.182960i \(0.0585678\pi\)
−0.983120 + 0.182960i \(0.941432\pi\)
\(462\) 0 0
\(463\) 12.6622 0.588463 0.294231 0.955734i \(-0.404936\pi\)
0.294231 + 0.955734i \(0.404936\pi\)
\(464\) 8.66728i 0.402368i
\(465\) 0 0
\(466\) 21.9618i 1.01736i
\(467\) 8.29420i 0.383810i −0.981414 0.191905i \(-0.938534\pi\)
0.981414 0.191905i \(-0.0614665\pi\)
\(468\) 0 0
\(469\) −2.27691 + 7.79512i −0.105138 + 0.359945i
\(470\) 13.3776i 0.617065i
\(471\) 0 0
\(472\) 9.42907i 0.434008i
\(473\) −27.4485 −1.26209
\(474\) 0 0
\(475\) 5.52327 + 0.910540i 0.253425 + 0.0417784i
\(476\) 1.71180 5.86046i 0.0784605 0.268614i
\(477\) 0 0
\(478\) 5.77182i 0.263997i
\(479\) 3.93055i 0.179591i 0.995960 + 0.0897957i \(0.0286214\pi\)
−0.995960 + 0.0897957i \(0.971379\pi\)
\(480\) 0 0
\(481\) 0.852932i 0.0388903i
\(482\) 10.8837i 0.495738i
\(483\) 0 0
\(484\) −5.60931 −0.254969
\(485\) 12.2095i 0.554405i
\(486\) 0 0
\(487\) 19.0464i 0.863077i 0.902095 + 0.431538i \(0.142029\pi\)
−0.902095 + 0.431538i \(0.857971\pi\)
\(488\) −5.31706 −0.240692
\(489\) 0 0
\(490\) 14.7889 + 9.44538i 0.668095 + 0.426699i
\(491\) −19.5088 −0.880419 −0.440209 0.897895i \(-0.645096\pi\)
−0.440209 + 0.897895i \(0.645096\pi\)
\(492\) 0 0
\(493\) 20.0006 0.900784
\(494\) 0.359356 2.17983i 0.0161682 0.0980750i
\(495\) 0 0
\(496\) −1.39640 −0.0627004
\(497\) 5.99778 20.5337i 0.269037 0.921063i
\(498\) 0 0
\(499\) −14.9906 −0.671074 −0.335537 0.942027i \(-0.608918\pi\)
−0.335537 + 0.942027i \(0.608918\pi\)
\(500\) 9.31483i 0.416572i
\(501\) 0 0
\(502\) −23.2434 −1.03740
\(503\) 3.32055i 0.148056i −0.997256 0.0740279i \(-0.976415\pi\)
0.997256 0.0740279i \(-0.0235854\pi\)
\(504\) 0 0
\(505\) −27.1433 −1.20786
\(506\) 25.8395i 1.14870i
\(507\) 0 0
\(508\) 1.29418i 0.0574200i
\(509\) 7.68285 0.340537 0.170268 0.985398i \(-0.445537\pi\)
0.170268 + 0.985398i \(0.445537\pi\)
\(510\) 0 0
\(511\) 10.0251 34.3215i 0.443485 1.51830i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 29.4331i 1.29824i
\(515\) 13.8229i 0.609108i
\(516\) 0 0
\(517\) 21.7485i 0.956498i
\(518\) 4.27383 + 1.24836i 0.187781 + 0.0548498i
\(519\) 0 0
\(520\) 1.27056 0.0557175
\(521\) −37.1281 −1.62661 −0.813306 0.581836i \(-0.802335\pi\)
−0.813306 + 0.581836i \(0.802335\pi\)
\(522\) 0 0
\(523\) −24.7335 −1.08152 −0.540761 0.841177i \(-0.681863\pi\)
−0.540761 + 0.841177i \(0.681863\pi\)
\(524\) 4.38528i 0.191572i
\(525\) 0 0
\(526\) 9.57036i 0.417288i
\(527\) 3.22235i 0.140368i
\(528\) 0 0
\(529\) 17.1990 0.747784
\(530\) 4.16441i 0.180891i
\(531\) 0 0
\(532\) −10.3966 4.99106i −0.450750 0.216390i
\(533\) −0.203819 −0.00882837
\(534\) 0 0
\(535\) −4.13117 −0.178606
\(536\) 3.06939 0.132578
\(537\) 0 0
\(538\) 20.3251i 0.876276i
\(539\) 24.0429 + 15.3557i 1.03560 + 0.661416i
\(540\) 0 0
\(541\) −32.9110 −1.41496 −0.707478 0.706735i \(-0.750167\pi\)
−0.707478 + 0.706735i \(0.750167\pi\)
\(542\) −27.0034 −1.15990
\(543\) 0 0
\(544\) −2.30760 −0.0989377
\(545\) −45.3512 −1.94263
\(546\) 0 0
\(547\) 41.6086i 1.77905i 0.456883 + 0.889527i \(0.348966\pi\)
−0.456883 + 0.889527i \(0.651034\pi\)
\(548\) −17.3945 −0.743056
\(549\) 0 0
\(550\) 5.23381i 0.223170i
\(551\) 6.14526 37.2767i 0.261797 1.58804i
\(552\) 0 0
\(553\) −2.50652 + 8.58122i −0.106588 + 0.364910i
\(554\) 1.93416i 0.0821748i
\(555\) 0 0
\(556\) 4.29012i 0.181942i
\(557\) 8.31329 0.352245 0.176123 0.984368i \(-0.443644\pi\)
0.176123 + 0.984368i \(0.443644\pi\)
\(558\) 0 0
\(559\) 3.41359 0.144379
\(560\) 1.85960 6.36643i 0.0785824 0.269031i
\(561\) 0 0
\(562\) 26.0581 1.09920
\(563\) −8.97656 −0.378317 −0.189159 0.981947i \(-0.560576\pi\)
−0.189159 + 0.981947i \(0.560576\pi\)
\(564\) 0 0
\(565\) −5.85736 −0.246421
\(566\) 20.4608 0.860030
\(567\) 0 0
\(568\) −8.08532 −0.339252
\(569\) 33.0391i 1.38507i −0.721383 0.692536i \(-0.756493\pi\)
0.721383 0.692536i \(-0.243507\pi\)
\(570\) 0 0
\(571\) −21.4576 −0.897973 −0.448987 0.893538i \(-0.648215\pi\)
−0.448987 + 0.893538i \(0.648215\pi\)
\(572\) 2.06559 0.0863665
\(573\) 0 0
\(574\) −0.298311 + 1.02129i −0.0124513 + 0.0426276i
\(575\) −8.14235 −0.339559
\(576\) 0 0
\(577\) 22.0341i 0.917293i −0.888619 0.458646i \(-0.848334\pi\)
0.888619 0.458646i \(-0.151666\pi\)
\(578\) 11.6750i 0.485615i
\(579\) 0 0
\(580\) 21.7275 0.902184
\(581\) 7.79684 26.6929i 0.323468 1.10741i
\(582\) 0 0
\(583\) 6.77023i 0.280394i
\(584\) −13.5144 −0.559229
\(585\) 0 0
\(586\) 16.4410i 0.679173i
\(587\) 47.5397i 1.96217i 0.193573 + 0.981086i \(0.437992\pi\)
−0.193573 + 0.981086i \(0.562008\pi\)
\(588\) 0 0
\(589\) 6.00572 + 0.990075i 0.247461 + 0.0407953i
\(590\) −23.6371 −0.973126
\(591\) 0 0
\(592\) 1.68285i 0.0691649i
\(593\) 36.0330i 1.47970i −0.672774 0.739848i \(-0.734897\pi\)
0.672774 0.739848i \(-0.265103\pi\)
\(594\) 0 0
\(595\) −14.6912 4.29121i −0.602281 0.175923i
\(596\) −9.33082 −0.382205
\(597\) 0 0
\(598\) 3.21348i 0.131409i
\(599\) 19.8527i 0.811161i −0.914059 0.405580i \(-0.867069\pi\)
0.914059 0.405580i \(-0.132931\pi\)
\(600\) 0 0
\(601\) −32.8582 −1.34032 −0.670158 0.742219i \(-0.733774\pi\)
−0.670158 + 0.742219i \(0.733774\pi\)
\(602\) 4.99616 17.1046i 0.203628 0.697133i
\(603\) 0 0
\(604\) 7.62713i 0.310344i
\(605\) 14.0616i 0.571686i
\(606\) 0 0
\(607\) 16.9412 0.687621 0.343810 0.939039i \(-0.388282\pi\)
0.343810 + 0.939039i \(0.388282\pi\)
\(608\) −0.709018 + 4.30085i −0.0287545 + 0.174422i
\(609\) 0 0
\(610\) 13.3290i 0.539675i
\(611\) 2.70471i 0.109421i
\(612\) 0 0
\(613\) 15.2574 0.616242 0.308121 0.951347i \(-0.400300\pi\)
0.308121 + 0.951347i \(0.400300\pi\)
\(614\) 8.09858i 0.326832i
\(615\) 0 0
\(616\) 3.02321 10.3501i 0.121809 0.417019i
\(617\) 46.9250 1.88913 0.944564 0.328328i \(-0.106485\pi\)
0.944564 + 0.328328i \(0.106485\pi\)
\(618\) 0 0
\(619\) 8.50433i 0.341818i −0.985287 0.170909i \(-0.945330\pi\)
0.985287 0.170909i \(-0.0546704\pi\)
\(620\) 3.50056i 0.140586i
\(621\) 0 0
\(622\) 8.87438 0.355830
\(623\) 29.8125 + 8.70806i 1.19441 + 0.348881i
\(624\) 0 0
\(625\) −29.7719 −1.19088
\(626\) 22.9535 0.917408
\(627\) 0 0
\(628\) 13.0191i 0.519520i
\(629\) −3.88336 −0.154840
\(630\) 0 0
\(631\) 43.6369 1.73716 0.868579 0.495552i \(-0.165034\pi\)
0.868579 + 0.495552i \(0.165034\pi\)
\(632\) 3.37893 0.134406
\(633\) 0 0
\(634\) −19.9885 −0.793846
\(635\) 3.24430 0.128746
\(636\) 0 0
\(637\) −2.99005 1.90968i −0.118470 0.0756644i
\(638\) 35.3231 1.39845
\(639\) 0 0
\(640\) −2.50684 −0.0990914
\(641\) 16.0386i 0.633488i −0.948511 0.316744i \(-0.897410\pi\)
0.948511 0.316744i \(-0.102590\pi\)
\(642\) 0 0
\(643\) 36.3835i 1.43482i 0.696649 + 0.717412i \(0.254673\pi\)
−0.696649 + 0.717412i \(0.745327\pi\)
\(644\) 16.1019 + 4.70328i 0.634505 + 0.185335i
\(645\) 0 0
\(646\) 9.92465 + 1.63613i 0.390480 + 0.0643727i
\(647\) 31.7012i 1.24630i −0.782101 0.623151i \(-0.785852\pi\)
0.782101 0.623151i \(-0.214148\pi\)
\(648\) 0 0
\(649\) −38.4277 −1.50842
\(650\) 0.650893i 0.0255301i
\(651\) 0 0
\(652\) 4.37938 0.171510
\(653\) −4.26096 −0.166744 −0.0833721 0.996518i \(-0.526569\pi\)
−0.0833721 + 0.996518i \(0.526569\pi\)
\(654\) 0 0
\(655\) −10.9932 −0.429539
\(656\) 0.402139 0.0157009
\(657\) 0 0
\(658\) −13.5526 3.95865i −0.528337 0.154324i
\(659\) 0.0621626i 0.00242151i 0.999999 + 0.00121076i \(0.000385395\pi\)
−0.999999 + 0.00121076i \(0.999615\pi\)
\(660\) 0 0
\(661\) 16.2637 0.632585 0.316293 0.948662i \(-0.397562\pi\)
0.316293 + 0.948662i \(0.397562\pi\)
\(662\) −19.4117 −0.754458
\(663\) 0 0
\(664\) −10.5106 −0.407889
\(665\) −12.5118 + 26.0626i −0.485185 + 1.01066i
\(666\) 0 0
\(667\) 54.9529i 2.12779i
\(668\) −2.29406 −0.0887600
\(669\) 0 0
\(670\) 7.69446i 0.297263i
\(671\) 21.6694i 0.836538i
\(672\) 0 0
\(673\) 5.20897i 0.200791i 0.994948 + 0.100395i \(0.0320108\pi\)
−0.994948 + 0.100395i \(0.967989\pi\)
\(674\) 26.3229 1.01392
\(675\) 0 0
\(676\) 12.7431 0.490120
\(677\) −40.7978 −1.56799 −0.783993 0.620770i \(-0.786820\pi\)
−0.783993 + 0.620770i \(0.786820\pi\)
\(678\) 0 0
\(679\) −12.3692 3.61298i −0.474687 0.138653i
\(680\) 5.78478i 0.221836i
\(681\) 0 0
\(682\) 5.69098i 0.217919i
\(683\) 9.91494i 0.379385i 0.981844 + 0.189692i \(0.0607491\pi\)
−0.981844 + 0.189692i \(0.939251\pi\)
\(684\) 0 0
\(685\) 43.6052i 1.66607i
\(686\) −13.9452 + 12.1873i −0.532430 + 0.465315i
\(687\) 0 0
\(688\) −6.73509 −0.256773
\(689\) 0.841968i 0.0320764i
\(690\) 0 0
\(691\) 18.4742i 0.702790i −0.936227 0.351395i \(-0.885707\pi\)
0.936227 0.351395i \(-0.114293\pi\)
\(692\) 22.4568 0.853681
\(693\) 0 0
\(694\) 6.55866i 0.248963i
\(695\) 10.7546 0.407947
\(696\) 0 0
\(697\) 0.927978i 0.0351497i
\(698\) −28.8644 −1.09253
\(699\) 0 0
\(700\) −3.26146 0.952654i −0.123272 0.0360069i
\(701\) 3.23079 0.122025 0.0610126 0.998137i \(-0.480567\pi\)
0.0610126 + 0.998137i \(0.480567\pi\)
\(702\) 0 0
\(703\) −1.19317 + 7.23770i −0.0450014 + 0.272975i
\(704\) −4.07545 −0.153599
\(705\) 0 0
\(706\) 34.6309 1.30335
\(707\) 8.03213 27.4984i 0.302079 1.03418i
\(708\) 0 0
\(709\) 24.7361 0.928984 0.464492 0.885577i \(-0.346237\pi\)
0.464492 + 0.885577i \(0.346237\pi\)
\(710\) 20.2686i 0.760666i
\(711\) 0 0
\(712\) 11.7389i 0.439935i
\(713\) −8.85358 −0.331569
\(714\) 0 0
\(715\) 5.17809i 0.193649i
\(716\) 12.0887i 0.451777i
\(717\) 0 0
\(718\) 11.1407i 0.415767i
\(719\) 13.1215i 0.489349i 0.969605 + 0.244675i \(0.0786811\pi\)
−0.969605 + 0.244675i \(0.921319\pi\)
\(720\) 0 0
\(721\) −14.0037 4.09039i −0.521524 0.152334i
\(722\) 6.09875 17.9946i 0.226972 0.669689i
\(723\) 0 0
\(724\) 14.6113 0.543025
\(725\) 11.1308i 0.413386i
\(726\) 0 0
\(727\) 5.24285i 0.194446i −0.995263 0.0972232i \(-0.969004\pi\)
0.995263 0.0972232i \(-0.0309961\pi\)
\(728\) −0.375977 + 1.28718i −0.0139346 + 0.0477059i
\(729\) 0 0
\(730\) 33.8784i 1.25389i
\(731\) 15.5419i 0.574839i
\(732\) 0 0
\(733\) 28.0831i 1.03727i −0.854995 0.518636i \(-0.826440\pi\)
0.854995 0.518636i \(-0.173560\pi\)
\(734\) 11.0195 0.406737
\(735\) 0 0
\(736\) 6.34027i 0.233706i
\(737\) 12.5092i 0.460781i
\(738\) 0 0
\(739\) 37.2496 1.37025 0.685125 0.728425i \(-0.259747\pi\)
0.685125 + 0.728425i \(0.259747\pi\)
\(740\) −4.21864 −0.155080
\(741\) 0 0
\(742\) −4.21889 1.23231i −0.154880 0.0452396i
\(743\) 22.4717i 0.824408i −0.911092 0.412204i \(-0.864759\pi\)
0.911092 0.412204i \(-0.135241\pi\)
\(744\) 0 0
\(745\) 23.3908i 0.856974i
\(746\) −27.0191 −0.989241
\(747\) 0 0
\(748\) 9.40453i 0.343864i
\(749\) 1.22248 4.18521i 0.0446683 0.152924i
\(750\) 0 0
\(751\) 43.1905i 1.57604i 0.615647 + 0.788022i \(0.288895\pi\)
−0.615647 + 0.788022i \(0.711105\pi\)
\(752\) 5.33646i 0.194601i
\(753\) 0 0
\(754\) −4.39289 −0.159980
\(755\) −19.1200 −0.695847
\(756\) 0 0
\(757\) 8.50229 0.309021 0.154511 0.987991i \(-0.450620\pi\)
0.154511 + 0.987991i \(0.450620\pi\)
\(758\) −24.9769 −0.907200
\(759\) 0 0
\(760\) 10.7815 + 1.77739i 0.391087 + 0.0644728i
\(761\) 4.95250i 0.179528i −0.995963 0.0897639i \(-0.971389\pi\)
0.995963 0.0897639i \(-0.0286113\pi\)
\(762\) 0 0
\(763\) 13.4201 45.9445i 0.485841 1.66330i
\(764\) 0.142938 0.00517133
\(765\) 0 0
\(766\) 12.4870i 0.451174i
\(767\) 4.77899 0.172559
\(768\) 0 0
\(769\) 5.92509i 0.213664i −0.994277 0.106832i \(-0.965929\pi\)
0.994277 0.106832i \(-0.0340708\pi\)
\(770\) −25.9461 7.57870i −0.935032 0.273118i
\(771\) 0 0
\(772\) 5.94006i 0.213787i
\(773\) 17.3774 0.625023 0.312512 0.949914i \(-0.398830\pi\)
0.312512 + 0.949914i \(0.398830\pi\)
\(774\) 0 0
\(775\) 1.79330 0.0644172
\(776\) 4.87048i 0.174840i
\(777\) 0 0
\(778\) 31.4241i 1.12661i
\(779\) −1.72954 0.285124i −0.0619672 0.0102156i
\(780\) 0 0
\(781\) 32.9513i 1.17909i
\(782\) −14.6308 −0.523197
\(783\) 0 0
\(784\) 5.89943 + 3.76785i 0.210694 + 0.134566i
\(785\) 32.6368 1.16486
\(786\) 0 0
\(787\) 11.7715 0.419609 0.209805 0.977743i \(-0.432717\pi\)
0.209805 + 0.977743i \(0.432717\pi\)
\(788\) 9.50252 0.338513
\(789\) 0 0
\(790\) 8.47041i 0.301364i
\(791\) 1.73328 5.93398i 0.0616284 0.210988i
\(792\) 0 0
\(793\) 2.69488i 0.0956979i
\(794\) −37.1197 −1.31733
\(795\) 0 0
\(796\) 23.3287i 0.826863i
\(797\) 1.38112 0.0489218 0.0244609 0.999701i \(-0.492213\pi\)
0.0244609 + 0.999701i \(0.492213\pi\)
\(798\) 0 0
\(799\) 12.3144 0.435654
\(800\) 1.28423i 0.0454043i
\(801\) 0 0
\(802\) 38.8566 1.37208
\(803\) 55.0772i 1.94363i
\(804\) 0 0
\(805\) 11.7904 40.3649i 0.415555 1.42268i
\(806\) 0.707748i 0.0249294i
\(807\) 0 0
\(808\) −10.8277 −0.380918
\(809\) 0.562056 0.0197608 0.00988042 0.999951i \(-0.496855\pi\)
0.00988042 + 0.999951i \(0.496855\pi\)
\(810\) 0 0
\(811\) 27.0061 0.948311 0.474156 0.880441i \(-0.342753\pi\)
0.474156 + 0.880441i \(0.342753\pi\)
\(812\) −6.42948 + 22.0117i −0.225631 + 0.772459i
\(813\) 0 0
\(814\) −6.85839 −0.240387
\(815\) 10.9784i 0.384556i
\(816\) 0 0
\(817\) 28.9666 + 4.77530i 1.01341 + 0.167067i
\(818\) 9.77207i 0.341672i
\(819\) 0 0
\(820\) 1.00810i 0.0352043i
\(821\) −5.06969 −0.176933 −0.0884667 0.996079i \(-0.528197\pi\)
−0.0884667 + 0.996079i \(0.528197\pi\)
\(822\) 0 0
\(823\) −42.8564 −1.49388 −0.746941 0.664890i \(-0.768478\pi\)
−0.746941 + 0.664890i \(0.768478\pi\)
\(824\) 5.51407i 0.192092i
\(825\) 0 0
\(826\) 6.99459 23.9463i 0.243373 0.833200i
\(827\) 35.3843i 1.23043i −0.788358 0.615217i \(-0.789069\pi\)
0.788358 0.615217i \(-0.210931\pi\)
\(828\) 0 0
\(829\) 28.4298 0.987407 0.493703 0.869630i \(-0.335643\pi\)
0.493703 + 0.869630i \(0.335643\pi\)
\(830\) 26.3482i 0.914561i
\(831\) 0 0
\(832\) 0.506836 0.0175714
\(833\) 8.69470 13.6136i 0.301253 0.471682i
\(834\) 0 0
\(835\) 5.75084i 0.199016i
\(836\) 17.5279 + 2.88957i 0.606215 + 0.0999378i
\(837\) 0 0
\(838\) −30.2647 −1.04548
\(839\) 15.7323 0.543139 0.271569 0.962419i \(-0.412457\pi\)
0.271569 + 0.962419i \(0.412457\pi\)
\(840\) 0 0
\(841\) −46.1218 −1.59041
\(842\) −21.7967 −0.751164
\(843\) 0 0
\(844\) 23.1860i 0.798095i
\(845\) 31.9449i 1.09894i
\(846\) 0 0
\(847\) −14.2456 4.16105i −0.489483 0.142975i
\(848\) 1.66122i 0.0570466i
\(849\) 0 0
\(850\) 2.96349 0.101647
\(851\) 10.6698i 0.365755i
\(852\) 0 0
\(853\) 28.5022i 0.975896i 0.872873 + 0.487948i \(0.162254\pi\)
−0.872873 + 0.487948i \(0.837746\pi\)
\(854\) −13.5034 3.94425i −0.462075 0.134969i
\(855\) 0 0
\(856\) −1.64796 −0.0563262
\(857\) 51.1600 1.74759 0.873796 0.486293i \(-0.161651\pi\)
0.873796 + 0.486293i \(0.161651\pi\)
\(858\) 0 0
\(859\) 44.1527i 1.50647i −0.657752 0.753235i \(-0.728492\pi\)
0.657752 0.753235i \(-0.271508\pi\)
\(860\) 16.8838i 0.575732i
\(861\) 0 0
\(862\) 31.4075 1.06974
\(863\) 17.8212i 0.606640i −0.952889 0.303320i \(-0.901905\pi\)
0.952889 0.303320i \(-0.0980952\pi\)
\(864\) 0 0
\(865\) 56.2956i 1.91411i
\(866\) 13.5121i 0.459159i
\(867\) 0 0
\(868\) −3.54635 1.03587i −0.120371 0.0351596i
\(869\) 13.7706i 0.467137i
\(870\) 0 0
\(871\) 1.55568i 0.0527122i
\(872\) −18.0910 −0.612640
\(873\) 0 0
\(874\) −4.49536 + 27.2685i −0.152058 + 0.922372i
\(875\) 6.90984 23.6562i 0.233595 0.799726i
\(876\) 0 0
\(877\) 6.47255i 0.218563i 0.994011 + 0.109281i \(0.0348549\pi\)
−0.994011 + 0.109281i \(0.965145\pi\)
\(878\) 26.4310i 0.892004i
\(879\) 0 0
\(880\) 10.2165i 0.344398i
\(881\) 4.78225i 0.161118i 0.996750 + 0.0805590i \(0.0256706\pi\)
−0.996750 + 0.0805590i \(0.974329\pi\)
\(882\) 0 0
\(883\) 17.8380 0.600297 0.300148 0.953892i \(-0.402964\pi\)
0.300148 + 0.953892i \(0.402964\pi\)
\(884\) 1.16958i 0.0393371i
\(885\) 0 0
\(886\) 27.7520i 0.932348i
\(887\) 57.0906 1.91692 0.958458 0.285234i \(-0.0920712\pi\)
0.958458 + 0.285234i \(0.0920712\pi\)
\(888\) 0 0
\(889\) −0.960038 + 3.28674i −0.0321986 + 0.110234i
\(890\) −29.4276 −0.986414
\(891\) 0 0
\(892\) −23.0366 −0.771322
\(893\) 3.78365 22.9513i 0.126615 0.768037i
\(894\) 0 0
\(895\) 30.3045 1.01297
\(896\) 0.741811 2.53963i 0.0247822 0.0848431i
\(897\) 0 0
\(898\) 15.1061 0.504097
\(899\) 12.1030i 0.403659i
\(900\) 0 0
\(901\) 3.83344 0.127711
\(902\) 1.63890i 0.0545694i
\(903\) 0 0
\(904\) −2.33655 −0.0777126
\(905\) 36.6282i 1.21756i
\(906\) 0 0
\(907\) 32.8683i 1.09137i −0.837989 0.545687i \(-0.816269\pi\)
0.837989 0.545687i \(-0.183731\pi\)
\(908\) −17.6866 −0.586951
\(909\) 0 0
\(910\) 3.22674 + 0.942512i 0.106965 + 0.0312440i
\(911\) 35.4594i 1.17482i 0.809289 + 0.587411i \(0.199853\pi\)
−0.809289 + 0.587411i \(0.800147\pi\)
\(912\) 0 0
\(913\) 42.8353i 1.41764i
\(914\) 20.9766i 0.693845i
\(915\) 0 0
\(916\) 3.87815i 0.128138i
\(917\) 3.25305 11.1370i 0.107425 0.367776i
\(918\) 0 0
\(919\) −17.4663 −0.576161 −0.288080 0.957606i \(-0.593017\pi\)
−0.288080 + 0.957606i \(0.593017\pi\)
\(920\) −15.8940 −0.524010
\(921\) 0 0
\(922\) 7.85663 0.258744
\(923\) 4.09793i 0.134885i
\(924\) 0 0
\(925\) 2.16117i 0.0710588i
\(926\) 12.6622i 0.416106i
\(927\) 0 0
\(928\) 8.66728 0.284517
\(929\) 20.8426i 0.683825i 0.939732 + 0.341912i \(0.111075\pi\)
−0.939732 + 0.341912i \(0.888925\pi\)
\(930\) 0 0
\(931\) −22.7011 20.3877i −0.743998 0.668181i
\(932\) −21.9618 −0.719384
\(933\) 0 0
\(934\) −8.29420 −0.271394
\(935\) 23.5756 0.771005
\(936\) 0 0
\(937\) 8.44662i 0.275939i −0.990436 0.137969i \(-0.955942\pi\)
0.990436 0.137969i \(-0.0440576\pi\)
\(938\) 7.79512 + 2.27691i 0.254520 + 0.0743437i
\(939\) 0 0
\(940\) 13.3776 0.436331
\(941\) 36.0231 1.17432 0.587160 0.809471i \(-0.300246\pi\)
0.587160 + 0.809471i \(0.300246\pi\)
\(942\) 0 0
\(943\) 2.54967 0.0830288
\(944\) −9.42907 −0.306890
\(945\) 0 0
\(946\) 27.4485i 0.892429i
\(947\) −34.9342 −1.13521 −0.567605 0.823301i \(-0.692130\pi\)
−0.567605 + 0.823301i \(0.692130\pi\)
\(948\) 0 0
\(949\) 6.84958i 0.222347i
\(950\) 0.910540 5.52327i 0.0295418 0.179198i
\(951\) 0 0
\(952\) −5.86046 1.71180i −0.189938 0.0554799i
\(953\) 44.7390i 1.44924i −0.689149 0.724620i \(-0.742015\pi\)
0.689149 0.724620i \(-0.257985\pi\)
\(954\) 0 0
\(955\) 0.358323i 0.0115951i
\(956\) −5.77182 −0.186674
\(957\) 0 0
\(958\) 3.93055 0.126990
\(959\) −44.1756 12.9034i −1.42650 0.416674i
\(960\) 0 0
\(961\) −29.0501 −0.937099
\(962\) 0.852932 0.0274996
\(963\) 0 0
\(964\) 10.8837 0.350539
\(965\) −14.8908 −0.479351
\(966\) 0 0
\(967\) −29.5592 −0.950559 −0.475280 0.879835i \(-0.657653\pi\)
−0.475280 + 0.879835i \(0.657653\pi\)
\(968\) 5.60931i 0.180290i
\(969\) 0 0
\(970\) 12.2095 0.392024
\(971\) −33.7683 −1.08368 −0.541838 0.840483i \(-0.682272\pi\)
−0.541838 + 0.840483i \(0.682272\pi\)
\(972\) 0 0
\(973\) −3.18246 + 10.8953i −0.102025 + 0.349288i
\(974\) 19.0464 0.610287
\(975\) 0 0
\(976\) 5.31706i 0.170195i
\(977\) 51.1745i 1.63722i 0.574351 + 0.818609i \(0.305255\pi\)
−0.574351 + 0.818609i \(0.694745\pi\)
\(978\) 0 0
\(979\) −47.8414 −1.52902
\(980\) 9.44538 14.7889i 0.301722 0.472414i
\(981\) 0 0
\(982\) 19.5088i 0.622550i
\(983\) 57.3236 1.82834 0.914170 0.405331i \(-0.132844\pi\)
0.914170 + 0.405331i \(0.132844\pi\)
\(984\) 0 0
\(985\) 23.8213i 0.759009i
\(986\) 20.0006i 0.636950i
\(987\) 0 0
\(988\) −2.17983 0.359356i −0.0693495 0.0114326i
\(989\) −42.7023 −1.35785
\(990\) 0 0
\(991\) 52.1178i 1.65558i −0.561041 0.827788i \(-0.689599\pi\)
0.561041 0.827788i \(-0.310401\pi\)
\(992\) 1.39640i 0.0443359i
\(993\) 0 0
\(994\) −20.5337 5.99778i −0.651290 0.190238i
\(995\) −58.4811 −1.85398
\(996\) 0 0
\(997\) 15.2597i 0.483281i 0.970366 + 0.241640i \(0.0776854\pi\)
−0.970366 + 0.241640i \(0.922315\pi\)
\(998\) 14.9906i 0.474521i
\(999\) 0 0
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 2394.2.e.b.1063.5 12
3.2 odd 2 798.2.e.b.265.8 yes 12
7.6 odd 2 2394.2.e.a.1063.2 12
19.18 odd 2 2394.2.e.a.1063.11 12
21.20 even 2 798.2.e.a.265.11 yes 12
57.56 even 2 798.2.e.a.265.2 12
133.132 even 2 inner 2394.2.e.b.1063.8 12
399.398 odd 2 798.2.e.b.265.5 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.e.a.265.2 12 57.56 even 2
798.2.e.a.265.11 yes 12 21.20 even 2
798.2.e.b.265.5 yes 12 399.398 odd 2
798.2.e.b.265.8 yes 12 3.2 odd 2
2394.2.e.a.1063.2 12 7.6 odd 2
2394.2.e.a.1063.11 12 19.18 odd 2
2394.2.e.b.1063.5 12 1.1 even 1 trivial
2394.2.e.b.1063.8 12 133.132 even 2 inner