Properties

Label 60.3.b.a.29.3
Level $60$
Weight $3$
Character 60.29
Analytic conductor $1.635$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,3,Mod(29,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 60.b (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: \(1.63488158616\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 29.3
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 60.29
Dual form 60.3.b.a.29.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.23607 - 2.00000i) q^{3} +(2.23607 + 4.47214i) q^{5} -8.00000i q^{7} +(1.00000 - 8.94427i) q^{9} +O(q^{10})\) \(q+(2.23607 - 2.00000i) q^{3} +(2.23607 + 4.47214i) q^{5} -8.00000i q^{7} +(1.00000 - 8.94427i) q^{9} +8.94427i q^{11} +12.0000i q^{13} +(13.9443 + 5.52786i) q^{15} -31.3050 q^{17} -6.00000 q^{19} +(-16.0000 - 17.8885i) q^{21} +4.47214 q^{23} +(-15.0000 + 20.0000i) q^{25} +(-15.6525 - 22.0000i) q^{27} +26.8328i q^{29} +34.0000 q^{31} +(17.8885 + 20.0000i) q^{33} +(35.7771 - 17.8885i) q^{35} -44.0000i q^{37} +(24.0000 + 26.8328i) q^{39} -17.8885i q^{41} -28.0000i q^{43} +(42.2361 - 15.5279i) q^{45} +4.47214 q^{47} -15.0000 q^{49} +(-70.0000 + 62.6099i) q^{51} +40.2492 q^{53} +(-40.0000 + 20.0000i) q^{55} +(-13.4164 + 12.0000i) q^{57} -98.3870i q^{59} +74.0000 q^{61} +(-71.5542 - 8.00000i) q^{63} +(-53.6656 + 26.8328i) q^{65} +92.0000i q^{67} +(10.0000 - 8.94427i) q^{69} -53.6656i q^{71} +56.0000i q^{73} +(6.45898 + 74.7214i) q^{75} +71.5542 q^{77} -78.0000 q^{79} +(-79.0000 - 17.8885i) q^{81} -102.859 q^{83} +(-70.0000 - 140.000i) q^{85} +(53.6656 + 60.0000i) q^{87} +17.8885i q^{89} +96.0000 q^{91} +(76.0263 - 68.0000i) q^{93} +(-13.4164 - 26.8328i) q^{95} +32.0000i q^{97} +(80.0000 + 8.94427i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} + 20 q^{15} - 24 q^{19} - 64 q^{21} - 60 q^{25} + 136 q^{31} + 96 q^{39} + 160 q^{45} - 60 q^{49} - 280 q^{51} - 160 q^{55} + 296 q^{61} + 40 q^{69} + 160 q^{75} - 312 q^{79} - 316 q^{81} - 280 q^{85} + 384 q^{91} + 320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.23607 2.00000i 0.745356 0.666667i
\(4\) 0 0
\(5\) 2.23607 + 4.47214i 0.447214 + 0.894427i
\(6\) 0 0
\(7\) 8.00000i 1.14286i −0.820652 0.571429i \(-0.806389\pi\)
0.820652 0.571429i \(-0.193611\pi\)
\(8\) 0 0
\(9\) 1.00000 8.94427i 0.111111 0.993808i
\(10\) 0 0
\(11\) 8.94427i 0.813116i 0.913625 + 0.406558i \(0.133271\pi\)
−0.913625 + 0.406558i \(0.866729\pi\)
\(12\) 0 0
\(13\) 12.0000i 0.923077i 0.887120 + 0.461538i \(0.152702\pi\)
−0.887120 + 0.461538i \(0.847298\pi\)
\(14\) 0 0
\(15\) 13.9443 + 5.52786i 0.929618 + 0.368524i
\(16\) 0 0
\(17\) −31.3050 −1.84147 −0.920734 0.390191i \(-0.872409\pi\)
−0.920734 + 0.390191i \(0.872409\pi\)
\(18\) 0 0
\(19\) −6.00000 −0.315789 −0.157895 0.987456i \(-0.550471\pi\)
−0.157895 + 0.987456i \(0.550471\pi\)
\(20\) 0 0
\(21\) −16.0000 17.8885i −0.761905 0.851835i
\(22\) 0 0
\(23\) 4.47214 0.194441 0.0972203 0.995263i \(-0.469005\pi\)
0.0972203 + 0.995263i \(0.469005\pi\)
\(24\) 0 0
\(25\) −15.0000 + 20.0000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) −15.6525 22.0000i −0.579721 0.814815i
\(28\) 0 0
\(29\) 26.8328i 0.925270i 0.886549 + 0.462635i \(0.153096\pi\)
−0.886549 + 0.462635i \(0.846904\pi\)
\(30\) 0 0
\(31\) 34.0000 1.09677 0.548387 0.836225i \(-0.315242\pi\)
0.548387 + 0.836225i \(0.315242\pi\)
\(32\) 0 0
\(33\) 17.8885 + 20.0000i 0.542077 + 0.606061i
\(34\) 0 0
\(35\) 35.7771 17.8885i 1.02220 0.511101i
\(36\) 0 0
\(37\) 44.0000i 1.18919i −0.804026 0.594595i \(-0.797313\pi\)
0.804026 0.594595i \(-0.202687\pi\)
\(38\) 0 0
\(39\) 24.0000 + 26.8328i 0.615385 + 0.688021i
\(40\) 0 0
\(41\) 17.8885i 0.436306i −0.975915 0.218153i \(-0.929997\pi\)
0.975915 0.218153i \(-0.0700032\pi\)
\(42\) 0 0
\(43\) 28.0000i 0.651163i −0.945514 0.325581i \(-0.894440\pi\)
0.945514 0.325581i \(-0.105560\pi\)
\(44\) 0 0
\(45\) 42.2361 15.5279i 0.938579 0.345064i
\(46\) 0 0
\(47\) 4.47214 0.0951518 0.0475759 0.998868i \(-0.484850\pi\)
0.0475759 + 0.998868i \(0.484850\pi\)
\(48\) 0 0
\(49\) −15.0000 −0.306122
\(50\) 0 0
\(51\) −70.0000 + 62.6099i −1.37255 + 1.22765i
\(52\) 0 0
\(53\) 40.2492 0.759419 0.379710 0.925106i \(-0.376024\pi\)
0.379710 + 0.925106i \(0.376024\pi\)
\(54\) 0 0
\(55\) −40.0000 + 20.0000i −0.727273 + 0.363636i
\(56\) 0 0
\(57\) −13.4164 + 12.0000i −0.235376 + 0.210526i
\(58\) 0 0
\(59\) 98.3870i 1.66758i −0.552085 0.833788i \(-0.686167\pi\)
0.552085 0.833788i \(-0.313833\pi\)
\(60\) 0 0
\(61\) 74.0000 1.21311 0.606557 0.795040i \(-0.292550\pi\)
0.606557 + 0.795040i \(0.292550\pi\)
\(62\) 0 0
\(63\) −71.5542 8.00000i −1.13578 0.126984i
\(64\) 0 0
\(65\) −53.6656 + 26.8328i −0.825625 + 0.412813i
\(66\) 0 0
\(67\) 92.0000i 1.37313i 0.727066 + 0.686567i \(0.240883\pi\)
−0.727066 + 0.686567i \(0.759117\pi\)
\(68\) 0 0
\(69\) 10.0000 8.94427i 0.144928 0.129627i
\(70\) 0 0
\(71\) 53.6656i 0.755854i −0.925835 0.377927i \(-0.876637\pi\)
0.925835 0.377927i \(-0.123363\pi\)
\(72\) 0 0
\(73\) 56.0000i 0.767123i 0.923515 + 0.383562i \(0.125303\pi\)
−0.923515 + 0.383562i \(0.874697\pi\)
\(74\) 0 0
\(75\) 6.45898 + 74.7214i 0.0861197 + 0.996285i
\(76\) 0 0
\(77\) 71.5542 0.929275
\(78\) 0 0
\(79\) −78.0000 −0.987342 −0.493671 0.869649i \(-0.664345\pi\)
−0.493671 + 0.869649i \(0.664345\pi\)
\(80\) 0 0
\(81\) −79.0000 17.8885i −0.975309 0.220846i
\(82\) 0 0
\(83\) −102.859 −1.23927 −0.619633 0.784891i \(-0.712719\pi\)
−0.619633 + 0.784891i \(0.712719\pi\)
\(84\) 0 0
\(85\) −70.0000 140.000i −0.823529 1.64706i
\(86\) 0 0
\(87\) 53.6656 + 60.0000i 0.616846 + 0.689655i
\(88\) 0 0
\(89\) 17.8885i 0.200995i 0.994937 + 0.100497i \(0.0320434\pi\)
−0.994937 + 0.100497i \(0.967957\pi\)
\(90\) 0 0
\(91\) 96.0000 1.05495
\(92\) 0 0
\(93\) 76.0263 68.0000i 0.817487 0.731183i
\(94\) 0 0
\(95\) −13.4164 26.8328i −0.141225 0.282451i
\(96\) 0 0
\(97\) 32.0000i 0.329897i 0.986302 + 0.164948i \(0.0527458\pi\)
−0.986302 + 0.164948i \(0.947254\pi\)
\(98\) 0 0
\(99\) 80.0000 + 8.94427i 0.808081 + 0.0903462i
\(100\) 0 0
\(101\) 152.053i 1.50547i 0.658323 + 0.752736i \(0.271266\pi\)
−0.658323 + 0.752736i \(0.728734\pi\)
\(102\) 0 0
\(103\) 104.000i 1.00971i −0.863205 0.504854i \(-0.831546\pi\)
0.863205 0.504854i \(-0.168454\pi\)
\(104\) 0 0
\(105\) 44.2229 111.554i 0.421171 1.06242i
\(106\) 0 0
\(107\) 147.580 1.37926 0.689628 0.724163i \(-0.257774\pi\)
0.689628 + 0.724163i \(0.257774\pi\)
\(108\) 0 0
\(109\) 74.0000 0.678899 0.339450 0.940624i \(-0.389759\pi\)
0.339450 + 0.940624i \(0.389759\pi\)
\(110\) 0 0
\(111\) −88.0000 98.3870i −0.792793 0.886369i
\(112\) 0 0
\(113\) 40.2492 0.356188 0.178094 0.984013i \(-0.443007\pi\)
0.178094 + 0.984013i \(0.443007\pi\)
\(114\) 0 0
\(115\) 10.0000 + 20.0000i 0.0869565 + 0.173913i
\(116\) 0 0
\(117\) 107.331 + 12.0000i 0.917361 + 0.102564i
\(118\) 0 0
\(119\) 250.440i 2.10453i
\(120\) 0 0
\(121\) 41.0000 0.338843
\(122\) 0 0
\(123\) −35.7771 40.0000i −0.290871 0.325203i
\(124\) 0 0
\(125\) −122.984 22.3607i −0.983870 0.178885i
\(126\) 0 0
\(127\) 16.0000i 0.125984i 0.998014 + 0.0629921i \(0.0200643\pi\)
−0.998014 + 0.0629921i \(0.979936\pi\)
\(128\) 0 0
\(129\) −56.0000 62.6099i −0.434109 0.485348i
\(130\) 0 0
\(131\) 80.4984i 0.614492i −0.951630 0.307246i \(-0.900593\pi\)
0.951630 0.307246i \(-0.0994074\pi\)
\(132\) 0 0
\(133\) 48.0000i 0.360902i
\(134\) 0 0
\(135\) 63.3870 119.193i 0.469533 0.882915i
\(136\) 0 0
\(137\) −174.413 −1.27309 −0.636545 0.771240i \(-0.719637\pi\)
−0.636545 + 0.771240i \(0.719637\pi\)
\(138\) 0 0
\(139\) −118.000 −0.848921 −0.424460 0.905446i \(-0.639536\pi\)
−0.424460 + 0.905446i \(0.639536\pi\)
\(140\) 0 0
\(141\) 10.0000 8.94427i 0.0709220 0.0634346i
\(142\) 0 0
\(143\) −107.331 −0.750568
\(144\) 0 0
\(145\) −120.000 + 60.0000i −0.827586 + 0.413793i
\(146\) 0 0
\(147\) −33.5410 + 30.0000i −0.228170 + 0.204082i
\(148\) 0 0
\(149\) 98.3870i 0.660315i −0.943926 0.330158i \(-0.892898\pi\)
0.943926 0.330158i \(-0.107102\pi\)
\(150\) 0 0
\(151\) 34.0000 0.225166 0.112583 0.993642i \(-0.464088\pi\)
0.112583 + 0.993642i \(0.464088\pi\)
\(152\) 0 0
\(153\) −31.3050 + 280.000i −0.204608 + 1.83007i
\(154\) 0 0
\(155\) 76.0263 + 152.053i 0.490492 + 0.980985i
\(156\) 0 0
\(157\) 92.0000i 0.585987i 0.956114 + 0.292994i \(0.0946515\pi\)
−0.956114 + 0.292994i \(0.905349\pi\)
\(158\) 0 0
\(159\) 90.0000 80.4984i 0.566038 0.506280i
\(160\) 0 0
\(161\) 35.7771i 0.222218i
\(162\) 0 0
\(163\) 68.0000i 0.417178i −0.978003 0.208589i \(-0.933113\pi\)
0.978003 0.208589i \(-0.0668871\pi\)
\(164\) 0 0
\(165\) −49.4427 + 124.721i −0.299653 + 0.755887i
\(166\) 0 0
\(167\) −67.0820 −0.401689 −0.200844 0.979623i \(-0.564369\pi\)
−0.200844 + 0.979623i \(0.564369\pi\)
\(168\) 0 0
\(169\) 25.0000 0.147929
\(170\) 0 0
\(171\) −6.00000 + 53.6656i −0.0350877 + 0.313834i
\(172\) 0 0
\(173\) 76.0263 0.439458 0.219729 0.975561i \(-0.429483\pi\)
0.219729 + 0.975561i \(0.429483\pi\)
\(174\) 0 0
\(175\) 160.000 + 120.000i 0.914286 + 0.685714i
\(176\) 0 0
\(177\) −196.774 220.000i −1.11172 1.24294i
\(178\) 0 0
\(179\) 259.384i 1.44907i −0.689237 0.724536i \(-0.742054\pi\)
0.689237 0.724536i \(-0.257946\pi\)
\(180\) 0 0
\(181\) −166.000 −0.917127 −0.458564 0.888662i \(-0.651636\pi\)
−0.458564 + 0.888662i \(0.651636\pi\)
\(182\) 0 0
\(183\) 165.469 148.000i 0.904202 0.808743i
\(184\) 0 0
\(185\) 196.774 98.3870i 1.06364 0.531822i
\(186\) 0 0
\(187\) 280.000i 1.49733i
\(188\) 0 0
\(189\) −176.000 + 125.220i −0.931217 + 0.662539i
\(190\) 0 0
\(191\) 214.663i 1.12389i 0.827175 + 0.561944i \(0.189946\pi\)
−0.827175 + 0.561944i \(0.810054\pi\)
\(192\) 0 0
\(193\) 32.0000i 0.165803i 0.996558 + 0.0829016i \(0.0264187\pi\)
−0.996558 + 0.0829016i \(0.973581\pi\)
\(194\) 0 0
\(195\) −66.3344 + 167.331i −0.340176 + 0.858109i
\(196\) 0 0
\(197\) 4.47214 0.0227012 0.0113506 0.999936i \(-0.496387\pi\)
0.0113506 + 0.999936i \(0.496387\pi\)
\(198\) 0 0
\(199\) 114.000 0.572864 0.286432 0.958101i \(-0.407531\pi\)
0.286432 + 0.958101i \(0.407531\pi\)
\(200\) 0 0
\(201\) 184.000 + 205.718i 0.915423 + 1.02347i
\(202\) 0 0
\(203\) 214.663 1.05745
\(204\) 0 0
\(205\) 80.0000 40.0000i 0.390244 0.195122i
\(206\) 0 0
\(207\) 4.47214 40.0000i 0.0216045 0.193237i
\(208\) 0 0
\(209\) 53.6656i 0.256773i
\(210\) 0 0
\(211\) −6.00000 −0.0284360 −0.0142180 0.999899i \(-0.504526\pi\)
−0.0142180 + 0.999899i \(0.504526\pi\)
\(212\) 0 0
\(213\) −107.331 120.000i −0.503903 0.563380i
\(214\) 0 0
\(215\) 125.220 62.6099i 0.582418 0.291209i
\(216\) 0 0
\(217\) 272.000i 1.25346i
\(218\) 0 0
\(219\) 112.000 + 125.220i 0.511416 + 0.571780i
\(220\) 0 0
\(221\) 375.659i 1.69982i
\(222\) 0 0
\(223\) 272.000i 1.21973i 0.792505 + 0.609865i \(0.208777\pi\)
−0.792505 + 0.609865i \(0.791223\pi\)
\(224\) 0 0
\(225\) 163.885 + 154.164i 0.728380 + 0.685174i
\(226\) 0 0
\(227\) −245.967 −1.08356 −0.541779 0.840521i \(-0.682249\pi\)
−0.541779 + 0.840521i \(0.682249\pi\)
\(228\) 0 0
\(229\) 154.000 0.672489 0.336245 0.941775i \(-0.390843\pi\)
0.336245 + 0.941775i \(0.390843\pi\)
\(230\) 0 0
\(231\) 160.000 143.108i 0.692641 0.619517i
\(232\) 0 0
\(233\) 183.358 0.786942 0.393471 0.919337i \(-0.371274\pi\)
0.393471 + 0.919337i \(0.371274\pi\)
\(234\) 0 0
\(235\) 10.0000 + 20.0000i 0.0425532 + 0.0851064i
\(236\) 0 0
\(237\) −174.413 + 156.000i −0.735921 + 0.658228i
\(238\) 0 0
\(239\) 178.885i 0.748475i 0.927333 + 0.374237i \(0.122095\pi\)
−0.927333 + 0.374237i \(0.877905\pi\)
\(240\) 0 0
\(241\) −206.000 −0.854772 −0.427386 0.904069i \(-0.640565\pi\)
−0.427386 + 0.904069i \(0.640565\pi\)
\(242\) 0 0
\(243\) −212.426 + 118.000i −0.874183 + 0.485597i
\(244\) 0 0
\(245\) −33.5410 67.0820i −0.136902 0.273804i
\(246\) 0 0
\(247\) 72.0000i 0.291498i
\(248\) 0 0
\(249\) −230.000 + 205.718i −0.923695 + 0.826178i
\(250\) 0 0
\(251\) 26.8328i 0.106904i −0.998570 0.0534518i \(-0.982978\pi\)
0.998570 0.0534518i \(-0.0170224\pi\)
\(252\) 0 0
\(253\) 40.0000i 0.158103i
\(254\) 0 0
\(255\) −436.525 173.050i −1.71186 0.678626i
\(256\) 0 0
\(257\) 40.2492 0.156612 0.0783059 0.996929i \(-0.475049\pi\)
0.0783059 + 0.996929i \(0.475049\pi\)
\(258\) 0 0
\(259\) −352.000 −1.35907
\(260\) 0 0
\(261\) 240.000 + 26.8328i 0.919540 + 0.102808i
\(262\) 0 0
\(263\) −210.190 −0.799203 −0.399602 0.916689i \(-0.630851\pi\)
−0.399602 + 0.916689i \(0.630851\pi\)
\(264\) 0 0
\(265\) 90.0000 + 180.000i 0.339623 + 0.679245i
\(266\) 0 0
\(267\) 35.7771 + 40.0000i 0.133997 + 0.149813i
\(268\) 0 0
\(269\) 134.164i 0.498751i 0.968407 + 0.249376i \(0.0802254\pi\)
−0.968407 + 0.249376i \(0.919775\pi\)
\(270\) 0 0
\(271\) −398.000 −1.46863 −0.734317 0.678806i \(-0.762498\pi\)
−0.734317 + 0.678806i \(0.762498\pi\)
\(272\) 0 0
\(273\) 214.663 192.000i 0.786310 0.703297i
\(274\) 0 0
\(275\) −178.885 134.164i −0.650493 0.487869i
\(276\) 0 0
\(277\) 292.000i 1.05415i 0.849818 + 0.527076i \(0.176712\pi\)
−0.849818 + 0.527076i \(0.823288\pi\)
\(278\) 0 0
\(279\) 34.0000 304.105i 0.121864 1.08998i
\(280\) 0 0
\(281\) 53.6656i 0.190981i −0.995430 0.0954904i \(-0.969558\pi\)
0.995430 0.0954904i \(-0.0304419\pi\)
\(282\) 0 0
\(283\) 52.0000i 0.183746i 0.995771 + 0.0918728i \(0.0292853\pi\)
−0.995771 + 0.0918728i \(0.970715\pi\)
\(284\) 0 0
\(285\) −83.6656 33.1672i −0.293564 0.116376i
\(286\) 0 0
\(287\) −143.108 −0.498635
\(288\) 0 0
\(289\) 691.000 2.39100
\(290\) 0 0
\(291\) 64.0000 + 71.5542i 0.219931 + 0.245891i
\(292\) 0 0
\(293\) −389.076 −1.32790 −0.663952 0.747775i \(-0.731122\pi\)
−0.663952 + 0.747775i \(0.731122\pi\)
\(294\) 0 0
\(295\) 440.000 220.000i 1.49153 0.745763i
\(296\) 0 0
\(297\) 196.774 140.000i 0.662539 0.471380i
\(298\) 0 0
\(299\) 53.6656i 0.179484i
\(300\) 0 0
\(301\) −224.000 −0.744186
\(302\) 0 0
\(303\) 304.105 + 340.000i 1.00365 + 1.12211i
\(304\) 0 0
\(305\) 165.469 + 330.938i 0.542521 + 1.08504i
\(306\) 0 0
\(307\) 492.000i 1.60261i 0.598259 + 0.801303i \(0.295859\pi\)
−0.598259 + 0.801303i \(0.704141\pi\)
\(308\) 0 0
\(309\) −208.000 232.551i −0.673139 0.752592i
\(310\) 0 0
\(311\) 482.991i 1.55302i 0.630102 + 0.776512i \(0.283013\pi\)
−0.630102 + 0.776512i \(0.716987\pi\)
\(312\) 0 0
\(313\) 568.000i 1.81470i −0.420380 0.907348i \(-0.638103\pi\)
0.420380 0.907348i \(-0.361897\pi\)
\(314\) 0 0
\(315\) −124.223 337.889i −0.394358 1.07266i
\(316\) 0 0
\(317\) 612.683 1.93275 0.966376 0.257132i \(-0.0827774\pi\)
0.966376 + 0.257132i \(0.0827774\pi\)
\(318\) 0 0
\(319\) −240.000 −0.752351
\(320\) 0 0
\(321\) 330.000 295.161i 1.02804 0.919505i
\(322\) 0 0
\(323\) 187.830 0.581516
\(324\) 0 0
\(325\) −240.000 180.000i −0.738462 0.553846i
\(326\) 0 0
\(327\) 165.469 148.000i 0.506021 0.452599i
\(328\) 0 0
\(329\) 35.7771i 0.108745i
\(330\) 0 0
\(331\) 202.000 0.610272 0.305136 0.952309i \(-0.401298\pi\)
0.305136 + 0.952309i \(0.401298\pi\)
\(332\) 0 0
\(333\) −393.548 44.0000i −1.18183 0.132132i
\(334\) 0 0
\(335\) −411.437 + 205.718i −1.22817 + 0.614084i
\(336\) 0 0
\(337\) 368.000i 1.09199i −0.837789 0.545994i \(-0.816152\pi\)
0.837789 0.545994i \(-0.183848\pi\)
\(338\) 0 0
\(339\) 90.0000 80.4984i 0.265487 0.237459i
\(340\) 0 0
\(341\) 304.105i 0.891804i
\(342\) 0 0
\(343\) 272.000i 0.793003i
\(344\) 0 0
\(345\) 62.3607 + 24.7214i 0.180756 + 0.0716561i
\(346\) 0 0
\(347\) 254.912 0.734616 0.367308 0.930099i \(-0.380280\pi\)
0.367308 + 0.930099i \(0.380280\pi\)
\(348\) 0 0
\(349\) −118.000 −0.338109 −0.169054 0.985607i \(-0.554071\pi\)
−0.169054 + 0.985607i \(0.554071\pi\)
\(350\) 0 0
\(351\) 264.000 187.830i 0.752137 0.535127i
\(352\) 0 0
\(353\) −31.3050 −0.0886826 −0.0443413 0.999016i \(-0.514119\pi\)
−0.0443413 + 0.999016i \(0.514119\pi\)
\(354\) 0 0
\(355\) 240.000 120.000i 0.676056 0.338028i
\(356\) 0 0
\(357\) 500.879 + 560.000i 1.40302 + 1.56863i
\(358\) 0 0
\(359\) 53.6656i 0.149486i −0.997203 0.0747432i \(-0.976186\pi\)
0.997203 0.0747432i \(-0.0238137\pi\)
\(360\) 0 0
\(361\) −325.000 −0.900277
\(362\) 0 0
\(363\) 91.6788 82.0000i 0.252559 0.225895i
\(364\) 0 0
\(365\) −250.440 + 125.220i −0.686136 + 0.343068i
\(366\) 0 0
\(367\) 352.000i 0.959128i 0.877507 + 0.479564i \(0.159205\pi\)
−0.877507 + 0.479564i \(0.840795\pi\)
\(368\) 0 0
\(369\) −160.000 17.8885i −0.433604 0.0484784i
\(370\) 0 0
\(371\) 321.994i 0.867908i
\(372\) 0 0
\(373\) 132.000i 0.353887i 0.984221 + 0.176944i \(0.0566211\pi\)
−0.984221 + 0.176944i \(0.943379\pi\)
\(374\) 0 0
\(375\) −319.721 + 195.967i −0.852590 + 0.522580i
\(376\) 0 0
\(377\) −321.994 −0.854095
\(378\) 0 0
\(379\) 394.000 1.03958 0.519789 0.854295i \(-0.326011\pi\)
0.519789 + 0.854295i \(0.326011\pi\)
\(380\) 0 0
\(381\) 32.0000 + 35.7771i 0.0839895 + 0.0939031i
\(382\) 0 0
\(383\) 76.0263 0.198502 0.0992511 0.995062i \(-0.468355\pi\)
0.0992511 + 0.995062i \(0.468355\pi\)
\(384\) 0 0
\(385\) 160.000 + 320.000i 0.415584 + 0.831169i
\(386\) 0 0
\(387\) −250.440 28.0000i −0.647131 0.0723514i
\(388\) 0 0
\(389\) 277.272i 0.712783i −0.934337 0.356391i \(-0.884007\pi\)
0.934337 0.356391i \(-0.115993\pi\)
\(390\) 0 0
\(391\) −140.000 −0.358056
\(392\) 0 0
\(393\) −160.997 180.000i −0.409661 0.458015i
\(394\) 0 0
\(395\) −174.413 348.827i −0.441553 0.883105i
\(396\) 0 0
\(397\) 652.000i 1.64232i 0.570700 + 0.821159i \(0.306672\pi\)
−0.570700 + 0.821159i \(0.693328\pi\)
\(398\) 0 0
\(399\) 96.0000 + 107.331i 0.240602 + 0.269001i
\(400\) 0 0
\(401\) 178.885i 0.446098i 0.974807 + 0.223049i \(0.0716011\pi\)
−0.974807 + 0.223049i \(0.928399\pi\)
\(402\) 0 0
\(403\) 408.000i 1.01241i
\(404\) 0 0
\(405\) −96.6494 393.299i −0.238640 0.971108i
\(406\) 0 0
\(407\) 393.548 0.966948
\(408\) 0 0
\(409\) −206.000 −0.503667 −0.251834 0.967771i \(-0.581034\pi\)
−0.251834 + 0.967771i \(0.581034\pi\)
\(410\) 0 0
\(411\) −390.000 + 348.827i −0.948905 + 0.848727i
\(412\) 0 0
\(413\) −787.096 −1.90580
\(414\) 0 0
\(415\) −230.000 460.000i −0.554217 1.10843i
\(416\) 0 0
\(417\) −263.856 + 236.000i −0.632748 + 0.565947i
\(418\) 0 0
\(419\) 205.718i 0.490974i 0.969400 + 0.245487i \(0.0789479\pi\)
−0.969400 + 0.245487i \(0.921052\pi\)
\(420\) 0 0
\(421\) −38.0000 −0.0902613 −0.0451306 0.998981i \(-0.514370\pi\)
−0.0451306 + 0.998981i \(0.514370\pi\)
\(422\) 0 0
\(423\) 4.47214 40.0000i 0.0105724 0.0945626i
\(424\) 0 0
\(425\) 469.574 626.099i 1.10488 1.47317i
\(426\) 0 0
\(427\) 592.000i 1.38642i
\(428\) 0 0
\(429\) −240.000 + 214.663i −0.559441 + 0.500379i
\(430\) 0 0
\(431\) 608.210i 1.41116i −0.708630 0.705581i \(-0.750686\pi\)
0.708630 0.705581i \(-0.249314\pi\)
\(432\) 0 0
\(433\) 272.000i 0.628176i 0.949394 + 0.314088i \(0.101699\pi\)
−0.949394 + 0.314088i \(0.898301\pi\)
\(434\) 0 0
\(435\) −148.328 + 374.164i −0.340984 + 0.860147i
\(436\) 0 0
\(437\) −26.8328 −0.0614023
\(438\) 0 0
\(439\) −366.000 −0.833713 −0.416856 0.908972i \(-0.636868\pi\)
−0.416856 + 0.908972i \(0.636868\pi\)
\(440\) 0 0
\(441\) −15.0000 + 134.164i −0.0340136 + 0.304227i
\(442\) 0 0
\(443\) 576.906 1.30227 0.651135 0.758962i \(-0.274293\pi\)
0.651135 + 0.758962i \(0.274293\pi\)
\(444\) 0 0
\(445\) −80.0000 + 40.0000i −0.179775 + 0.0898876i
\(446\) 0 0
\(447\) −196.774 220.000i −0.440210 0.492170i
\(448\) 0 0
\(449\) 429.325i 0.956181i 0.878311 + 0.478090i \(0.158671\pi\)
−0.878311 + 0.478090i \(0.841329\pi\)
\(450\) 0 0
\(451\) 160.000 0.354767
\(452\) 0 0
\(453\) 76.0263 68.0000i 0.167829 0.150110i
\(454\) 0 0
\(455\) 214.663 + 429.325i 0.471786 + 0.943572i
\(456\) 0 0
\(457\) 104.000i 0.227571i −0.993505 0.113786i \(-0.963702\pi\)
0.993505 0.113786i \(-0.0362977\pi\)
\(458\) 0 0
\(459\) 490.000 + 688.709i 1.06754 + 1.50046i
\(460\) 0 0
\(461\) 509.823i 1.10591i −0.833212 0.552954i \(-0.813501\pi\)
0.833212 0.552954i \(-0.186499\pi\)
\(462\) 0 0
\(463\) 96.0000i 0.207343i 0.994612 + 0.103672i \(0.0330591\pi\)
−0.994612 + 0.103672i \(0.966941\pi\)
\(464\) 0 0
\(465\) 474.105 + 187.947i 1.01958 + 0.404188i
\(466\) 0 0
\(467\) 147.580 0.316018 0.158009 0.987438i \(-0.449492\pi\)
0.158009 + 0.987438i \(0.449492\pi\)
\(468\) 0 0
\(469\) 736.000 1.56930
\(470\) 0 0
\(471\) 184.000 + 205.718i 0.390658 + 0.436769i
\(472\) 0 0
\(473\) 250.440 0.529471
\(474\) 0 0
\(475\) 90.0000 120.000i 0.189474 0.252632i
\(476\) 0 0
\(477\) 40.2492 360.000i 0.0843799 0.754717i
\(478\) 0 0
\(479\) 572.433i 1.19506i −0.801847 0.597530i \(-0.796149\pi\)
0.801847 0.597530i \(-0.203851\pi\)
\(480\) 0 0
\(481\) 528.000 1.09771
\(482\) 0 0
\(483\) −71.5542 80.0000i −0.148145 0.165631i
\(484\) 0 0
\(485\) −143.108 + 71.5542i −0.295069 + 0.147534i
\(486\) 0 0
\(487\) 648.000i 1.33060i −0.746578 0.665298i \(-0.768305\pi\)
0.746578 0.665298i \(-0.231695\pi\)
\(488\) 0 0
\(489\) −136.000 152.053i −0.278119 0.310946i
\(490\) 0 0
\(491\) 134.164i 0.273247i −0.990623 0.136623i \(-0.956375\pi\)
0.990623 0.136623i \(-0.0436250\pi\)
\(492\) 0 0
\(493\) 840.000i 1.70385i
\(494\) 0 0
\(495\) 138.885 + 377.771i 0.280577 + 0.763173i
\(496\) 0 0
\(497\) −429.325 −0.863833
\(498\) 0 0
\(499\) −486.000 −0.973948 −0.486974 0.873416i \(-0.661899\pi\)
−0.486974 + 0.873416i \(0.661899\pi\)
\(500\) 0 0
\(501\) −150.000 + 134.164i −0.299401 + 0.267793i
\(502\) 0 0
\(503\) 791.568 1.57369 0.786847 0.617148i \(-0.211712\pi\)
0.786847 + 0.617148i \(0.211712\pi\)
\(504\) 0 0
\(505\) −680.000 + 340.000i −1.34653 + 0.673267i
\(506\) 0 0
\(507\) 55.9017 50.0000i 0.110260 0.0986193i
\(508\) 0 0
\(509\) 26.8328i 0.0527167i 0.999653 + 0.0263584i \(0.00839110\pi\)
−0.999653 + 0.0263584i \(0.991609\pi\)
\(510\) 0 0
\(511\) 448.000 0.876712
\(512\) 0 0
\(513\) 93.9149 + 132.000i 0.183070 + 0.257310i
\(514\) 0 0
\(515\) 465.102 232.551i 0.903111 0.451555i
\(516\) 0 0
\(517\) 40.0000i 0.0773694i
\(518\) 0 0
\(519\) 170.000 152.053i 0.327553 0.292972i
\(520\) 0 0
\(521\) 983.870i 1.88843i 0.329336 + 0.944213i \(0.393175\pi\)
−0.329336 + 0.944213i \(0.606825\pi\)
\(522\) 0 0
\(523\) 292.000i 0.558317i 0.960245 + 0.279159i \(0.0900556\pi\)
−0.960245 + 0.279159i \(0.909944\pi\)
\(524\) 0 0
\(525\) 597.771 51.6718i 1.13861 0.0984226i
\(526\) 0 0
\(527\) −1064.37 −2.01967
\(528\) 0 0
\(529\) −509.000 −0.962193
\(530\) 0 0
\(531\) −880.000 98.3870i −1.65725 0.185286i
\(532\) 0 0
\(533\) 214.663 0.402744
\(534\) 0 0
\(535\) 330.000 + 660.000i 0.616822 + 1.23364i
\(536\) 0 0
\(537\) −518.768 580.000i −0.966048 1.08007i
\(538\) 0 0
\(539\) 134.164i 0.248913i
\(540\) 0 0
\(541\) −86.0000 −0.158965 −0.0794824 0.996836i \(-0.525327\pi\)
−0.0794824 + 0.996836i \(0.525327\pi\)
\(542\) 0 0
\(543\) −371.187 + 332.000i −0.683586 + 0.611418i
\(544\) 0 0
\(545\) 165.469 + 330.938i 0.303613 + 0.607226i
\(546\) 0 0
\(547\) 164.000i 0.299817i −0.988700 0.149909i \(-0.952102\pi\)
0.988700 0.149909i \(-0.0478979\pi\)
\(548\) 0 0
\(549\) 74.0000 661.876i 0.134791 1.20560i
\(550\) 0 0
\(551\) 160.997i 0.292190i
\(552\) 0 0
\(553\) 624.000i 1.12839i
\(554\) 0 0
\(555\) 243.226 613.548i 0.438245 1.10549i
\(556\) 0 0
\(557\) 362.243 0.650347 0.325173 0.945654i \(-0.394577\pi\)
0.325173 + 0.945654i \(0.394577\pi\)
\(558\) 0 0
\(559\) 336.000 0.601073
\(560\) 0 0
\(561\) −560.000 626.099i −0.998217 1.11604i
\(562\) 0 0
\(563\) −997.286 −1.77138 −0.885689 0.464278i \(-0.846314\pi\)
−0.885689 + 0.464278i \(0.846314\pi\)
\(564\) 0 0
\(565\) 90.0000 + 180.000i 0.159292 + 0.318584i
\(566\) 0 0
\(567\) −143.108 + 632.000i −0.252396 + 1.11464i
\(568\) 0 0
\(569\) 626.099i 1.10035i −0.835049 0.550175i \(-0.814561\pi\)
0.835049 0.550175i \(-0.185439\pi\)
\(570\) 0 0
\(571\) 394.000 0.690018 0.345009 0.938599i \(-0.387876\pi\)
0.345009 + 0.938599i \(0.387876\pi\)
\(572\) 0 0
\(573\) 429.325 + 480.000i 0.749258 + 0.837696i
\(574\) 0 0
\(575\) −67.0820 + 89.4427i −0.116664 + 0.155553i
\(576\) 0 0
\(577\) 608.000i 1.05373i −0.849950 0.526863i \(-0.823368\pi\)
0.849950 0.526863i \(-0.176632\pi\)
\(578\) 0 0
\(579\) 64.0000 + 71.5542i 0.110535 + 0.123582i
\(580\) 0 0
\(581\) 822.873i 1.41630i
\(582\) 0 0
\(583\) 360.000i 0.617496i
\(584\) 0 0
\(585\) 186.334 + 506.833i 0.318520 + 0.866381i
\(586\) 0 0
\(587\) −711.070 −1.21136 −0.605681 0.795707i \(-0.707099\pi\)
−0.605681 + 0.795707i \(0.707099\pi\)
\(588\) 0 0
\(589\) −204.000 −0.346350
\(590\) 0 0
\(591\) 10.0000 8.94427i 0.0169205 0.0151341i
\(592\) 0 0
\(593\) −603.738 −1.01811 −0.509054 0.860734i \(-0.670005\pi\)
−0.509054 + 0.860734i \(0.670005\pi\)
\(594\) 0 0
\(595\) −1120.00 + 560.000i −1.88235 + 0.941176i
\(596\) 0 0
\(597\) 254.912 228.000i 0.426988 0.381910i
\(598\) 0 0
\(599\) 53.6656i 0.0895920i 0.998996 + 0.0447960i \(0.0142638\pi\)
−0.998996 + 0.0447960i \(0.985736\pi\)
\(600\) 0 0
\(601\) 434.000 0.722130 0.361065 0.932541i \(-0.382413\pi\)
0.361065 + 0.932541i \(0.382413\pi\)
\(602\) 0 0
\(603\) 822.873 + 92.0000i 1.36463 + 0.152570i
\(604\) 0 0
\(605\) 91.6788 + 183.358i 0.151535 + 0.303070i
\(606\) 0 0
\(607\) 656.000i 1.08072i 0.841432 + 0.540362i \(0.181713\pi\)
−0.841432 + 0.540362i \(0.818287\pi\)
\(608\) 0 0
\(609\) 480.000 429.325i 0.788177 0.704967i
\(610\) 0 0
\(611\) 53.6656i 0.0878325i
\(612\) 0 0
\(613\) 844.000i 1.37684i −0.725315 0.688418i \(-0.758306\pi\)
0.725315 0.688418i \(-0.241694\pi\)
\(614\) 0 0
\(615\) 98.8854 249.443i 0.160789 0.405598i
\(616\) 0 0
\(617\) −460.630 −0.746564 −0.373282 0.927718i \(-0.621768\pi\)
−0.373282 + 0.927718i \(0.621768\pi\)
\(618\) 0 0
\(619\) −1046.00 −1.68982 −0.844911 0.534907i \(-0.820347\pi\)
−0.844911 + 0.534907i \(0.820347\pi\)
\(620\) 0 0
\(621\) −70.0000 98.3870i −0.112721 0.158433i
\(622\) 0 0
\(623\) 143.108 0.229708
\(624\) 0 0
\(625\) −175.000 600.000i −0.280000 0.960000i
\(626\) 0 0
\(627\) −107.331 120.000i −0.171182 0.191388i
\(628\) 0 0
\(629\) 1377.42i 2.18985i
\(630\) 0 0
\(631\) −46.0000 −0.0729002 −0.0364501 0.999335i \(-0.511605\pi\)
−0.0364501 + 0.999335i \(0.511605\pi\)
\(632\) 0 0
\(633\) −13.4164 + 12.0000i −0.0211950 + 0.0189573i
\(634\) 0 0
\(635\) −71.5542 + 35.7771i −0.112684 + 0.0563419i
\(636\) 0 0
\(637\) 180.000i 0.282575i
\(638\) 0 0
\(639\) −480.000 53.6656i −0.751174 0.0839838i
\(640\) 0 0
\(641\) 1144.87i 1.78606i −0.449994 0.893032i \(-0.648574\pi\)
0.449994 0.893032i \(-0.351426\pi\)
\(642\) 0 0
\(643\) 804.000i 1.25039i −0.780469 0.625194i \(-0.785020\pi\)
0.780469 0.625194i \(-0.214980\pi\)
\(644\) 0 0
\(645\) 154.780 390.440i 0.239969 0.605333i
\(646\) 0 0
\(647\) 576.906 0.891662 0.445831 0.895117i \(-0.352908\pi\)
0.445831 + 0.895117i \(0.352908\pi\)
\(648\) 0 0
\(649\) 880.000 1.35593
\(650\) 0 0
\(651\) −544.000 608.210i −0.835637 0.934271i
\(652\) 0 0
\(653\) 1077.78 1.65051 0.825256 0.564758i \(-0.191031\pi\)
0.825256 + 0.564758i \(0.191031\pi\)
\(654\) 0 0
\(655\) 360.000 180.000i 0.549618 0.274809i
\(656\) 0 0
\(657\) 500.879 + 56.0000i 0.762373 + 0.0852359i
\(658\) 0 0
\(659\) 813.929i 1.23510i 0.786533 + 0.617548i \(0.211874\pi\)
−0.786533 + 0.617548i \(0.788126\pi\)
\(660\) 0 0
\(661\) 1082.00 1.63691 0.818457 0.574568i \(-0.194830\pi\)
0.818457 + 0.574568i \(0.194830\pi\)
\(662\) 0 0
\(663\) −751.319 840.000i −1.13321 1.26697i
\(664\) 0 0
\(665\) −214.663 + 107.331i −0.322801 + 0.161400i
\(666\) 0 0
\(667\) 120.000i 0.179910i
\(668\) 0 0
\(669\) 544.000 + 608.210i 0.813154 + 0.909134i
\(670\) 0 0
\(671\) 661.876i 0.986403i
\(672\) 0 0
\(673\) 1056.00i 1.56909i 0.620070 + 0.784547i \(0.287104\pi\)
−0.620070 + 0.784547i \(0.712896\pi\)
\(674\) 0 0
\(675\) 674.787 + 16.9505i 0.999685 + 0.0251118i
\(676\) 0 0
\(677\) 612.683 0.904996 0.452498 0.891765i \(-0.350533\pi\)
0.452498 + 0.891765i \(0.350533\pi\)
\(678\) 0 0
\(679\) 256.000 0.377025
\(680\) 0 0
\(681\) −550.000 + 491.935i −0.807636 + 0.722371i
\(682\) 0 0
\(683\) −317.522 −0.464893 −0.232446 0.972609i \(-0.574673\pi\)
−0.232446 + 0.972609i \(0.574673\pi\)
\(684\) 0 0
\(685\) −390.000 780.000i −0.569343 1.13869i
\(686\) 0 0
\(687\) 344.354 308.000i 0.501244 0.448326i
\(688\) 0 0
\(689\) 482.991i 0.701002i
\(690\) 0 0
\(691\) 922.000 1.33430 0.667149 0.744924i \(-0.267514\pi\)
0.667149 + 0.744924i \(0.267514\pi\)
\(692\) 0 0
\(693\) 71.5542 640.000i 0.103253 0.923521i
\(694\) 0 0
\(695\) −263.856 527.712i −0.379649 0.759298i
\(696\) 0 0
\(697\) 560.000i 0.803443i
\(698\) 0 0
\(699\) 410.000 366.715i 0.586552 0.524628i
\(700\) 0 0
\(701\) 474.046i 0.676243i −0.941102 0.338122i \(-0.890209\pi\)
0.941102 0.338122i \(-0.109791\pi\)
\(702\) 0 0
\(703\) 264.000i 0.375533i
\(704\) 0 0
\(705\) 62.3607 + 24.7214i 0.0884549 + 0.0350658i
\(706\) 0 0
\(707\) 1216.42 1.72054
\(708\) 0 0
\(709\) −966.000 −1.36248 −0.681241 0.732059i \(-0.738559\pi\)
−0.681241 + 0.732059i \(0.738559\pi\)
\(710\) 0 0
\(711\) −78.0000 + 697.653i −0.109705 + 0.981228i
\(712\) 0 0
\(713\) 152.053 0.213258
\(714\) 0 0
\(715\) −240.000 480.000i −0.335664 0.671329i
\(716\) 0 0
\(717\) 357.771 + 400.000i 0.498983 + 0.557880i
\(718\) 0 0
\(719\) 1109.09i 1.54254i −0.636505 0.771272i \(-0.719621\pi\)
0.636505 0.771272i \(-0.280379\pi\)
\(720\) 0 0
\(721\) −832.000 −1.15395
\(722\) 0 0
\(723\) −460.630 + 412.000i −0.637109 + 0.569848i
\(724\) 0 0
\(725\) −536.656 402.492i −0.740216 0.555162i
\(726\) 0 0
\(727\) 408.000i 0.561210i −0.959823 0.280605i \(-0.909465\pi\)
0.959823 0.280605i \(-0.0905352\pi\)
\(728\) 0 0
\(729\) −239.000 + 688.709i −0.327846 + 0.944731i
\(730\) 0 0
\(731\) 876.539i 1.19910i
\(732\) 0 0
\(733\) 164.000i 0.223738i −0.993723 0.111869i \(-0.964316\pi\)
0.993723 0.111869i \(-0.0356837\pi\)
\(734\) 0 0
\(735\) −209.164 82.9180i −0.284577 0.112814i
\(736\) 0 0
\(737\) −822.873 −1.11652
\(738\) 0 0
\(739\) 1082.00 1.46414 0.732070 0.681229i \(-0.238554\pi\)
0.732070 + 0.681229i \(0.238554\pi\)
\(740\) 0 0
\(741\) −144.000 160.997i −0.194332 0.217270i
\(742\) 0 0
\(743\) −1140.39 −1.53485 −0.767426 0.641138i \(-0.778463\pi\)
−0.767426 + 0.641138i \(0.778463\pi\)
\(744\) 0 0
\(745\) 440.000 220.000i 0.590604 0.295302i
\(746\) 0 0
\(747\) −102.859 + 920.000i −0.137696 + 1.23159i
\(748\) 0 0
\(749\) 1180.64i 1.57629i
\(750\) 0 0
\(751\) −958.000 −1.27563 −0.637816 0.770189i \(-0.720162\pi\)
−0.637816 + 0.770189i \(0.720162\pi\)
\(752\) 0 0
\(753\) −53.6656 60.0000i −0.0712691 0.0796813i
\(754\) 0 0
\(755\) 76.0263 + 152.053i 0.100697 + 0.201394i
\(756\) 0 0
\(757\) 772.000i 1.01982i 0.860229 + 0.509908i \(0.170320\pi\)
−0.860229 + 0.509908i \(0.829680\pi\)
\(758\) 0 0
\(759\) 80.0000 + 89.4427i 0.105402 + 0.117843i
\(760\) 0 0
\(761\) 1126.98i 1.48092i −0.672102 0.740459i \(-0.734608\pi\)
0.672102 0.740459i \(-0.265392\pi\)
\(762\) 0 0
\(763\) 592.000i 0.775885i
\(764\) 0 0
\(765\) −1322.20 + 486.099i −1.72836 + 0.635424i
\(766\) 0 0
\(767\) 1180.64 1.53930
\(768\) 0 0
\(769\) −1326.00 −1.72432 −0.862159 0.506638i \(-0.830888\pi\)
−0.862159 + 0.506638i \(0.830888\pi\)
\(770\) 0 0
\(771\) 90.0000 80.4984i 0.116732 0.104408i
\(772\) 0 0
\(773\) 147.580 0.190919 0.0954596 0.995433i \(-0.469568\pi\)
0.0954596 + 0.995433i \(0.469568\pi\)
\(774\) 0 0
\(775\) −510.000 + 680.000i −0.658065 + 0.877419i
\(776\) 0 0
\(777\) −787.096 + 704.000i −1.01299 + 0.906049i
\(778\) 0 0
\(779\) 107.331i 0.137781i
\(780\) 0 0
\(781\) 480.000 0.614597
\(782\) 0 0
\(783\) 590.322 420.000i 0.753923 0.536398i
\(784\) 0 0
\(785\) −411.437 + 205.718i −0.524123 + 0.262061i
\(786\) 0 0
\(787\) 1132.00i 1.43837i 0.694817 + 0.719187i \(0.255485\pi\)
−0.694817 + 0.719187i \(0.744515\pi\)
\(788\) 0 0
\(789\) −470.000 + 420.381i −0.595691 + 0.532802i
\(790\) 0 0
\(791\) 321.994i 0.407072i
\(792\) 0 0
\(793\) 888.000i 1.11980i
\(794\) 0 0
\(795\) 561.246 + 222.492i 0.705970 + 0.279864i
\(796\) 0 0
\(797\) 612.683 0.768736 0.384368 0.923180i \(-0.374419\pi\)
0.384368 + 0.923180i \(0.374419\pi\)
\(798\) 0 0
\(799\) −140.000 −0.175219
\(800\) 0 0
\(801\) 160.000 + 17.8885i 0.199750 + 0.0223328i
\(802\) 0 0
\(803\) −500.879 −0.623760
\(804\) 0 0
\(805\) 160.000 80.0000i 0.198758 0.0993789i
\(806\) 0 0
\(807\) 268.328 + 300.000i 0.332501 + 0.371747i
\(808\) 0 0
\(809\) 1234.31i 1.52572i −0.646562 0.762861i \(-0.723794\pi\)
0.646562 0.762861i \(-0.276206\pi\)
\(810\) 0 0
\(811\) 874.000 1.07768 0.538841 0.842408i \(-0.318862\pi\)
0.538841 + 0.842408i \(0.318862\pi\)
\(812\) 0 0
\(813\) −889.955 + 796.000i −1.09466 + 0.979090i
\(814\) 0 0
\(815\) 304.105 152.053i 0.373135 0.186568i
\(816\) 0 0
\(817\) 168.000i 0.205630i
\(818\) 0 0
\(819\) 96.0000 858.650i 0.117216 1.04841i
\(820\) 0 0
\(821\) 259.384i 0.315937i 0.987444 + 0.157968i \(0.0504944\pi\)
−0.987444 + 0.157968i \(0.949506\pi\)
\(822\) 0 0
\(823\) 1368.00i 1.66221i −0.556114 0.831106i \(-0.687708\pi\)
0.556114 0.831106i \(-0.312292\pi\)
\(824\) 0 0
\(825\) −668.328 + 57.7709i −0.810095 + 0.0700253i
\(826\) 0 0
\(827\) −460.630 −0.556989 −0.278495 0.960438i \(-0.589835\pi\)
−0.278495 + 0.960438i \(0.589835\pi\)
\(828\) 0 0
\(829\) 1002.00 1.20869 0.604343 0.796725i \(-0.293436\pi\)
0.604343 + 0.796725i \(0.293436\pi\)
\(830\) 0 0
\(831\) 584.000 + 652.932i 0.702768 + 0.785718i
\(832\) 0 0
\(833\) 469.574 0.563715
\(834\) 0 0
\(835\) −150.000 300.000i −0.179641 0.359281i
\(836\) 0 0
\(837\) −532.184 748.000i −0.635823 0.893668i
\(838\) 0 0
\(839\) 304.105i 0.362462i 0.983441 + 0.181231i \(0.0580081\pi\)
−0.983441 + 0.181231i \(0.941992\pi\)
\(840\) 0 0
\(841\) 121.000 0.143876
\(842\) 0 0
\(843\) −107.331 120.000i −0.127321 0.142349i
\(844\) 0 0
\(845\) 55.9017 + 111.803i 0.0661559 + 0.132312i
\(846\) 0 0
\(847\) 328.000i 0.387249i
\(848\) 0 0
\(849\) 104.000 + 116.276i 0.122497 + 0.136956i
\(850\) 0 0
\(851\) 196.774i 0.231227i
\(852\) 0 0
\(853\) 772.000i 0.905041i 0.891754 + 0.452521i \(0.149475\pi\)
−0.891754 + 0.452521i \(0.850525\pi\)
\(854\) 0 0
\(855\) −253.416 + 93.1672i −0.296393 + 0.108967i
\(856\) 0 0
\(857\) 898.899 1.04889 0.524445 0.851444i \(-0.324273\pi\)
0.524445 + 0.851444i \(0.324273\pi\)
\(858\) 0 0
\(859\) −278.000 −0.323632 −0.161816 0.986821i \(-0.551735\pi\)
−0.161816 + 0.986821i \(0.551735\pi\)
\(860\) 0 0
\(861\) −320.000 + 286.217i −0.371661 + 0.332424i
\(862\) 0 0
\(863\) 576.906 0.668488 0.334244 0.942486i \(-0.391519\pi\)
0.334244 + 0.942486i \(0.391519\pi\)
\(864\) 0 0
\(865\) 170.000 + 340.000i 0.196532 + 0.393064i
\(866\) 0 0
\(867\) 1545.12 1382.00i 1.78215 1.59400i
\(868\) 0 0
\(869\) 697.653i 0.802823i
\(870\) 0 0
\(871\) −1104.00 −1.26751
\(872\) 0 0
\(873\) 286.217 + 32.0000i 0.327854 + 0.0366552i
\(874\) 0 0
\(875\) −178.885 + 983.870i −0.204441 + 1.12442i
\(876\) 0 0
\(877\) 628.000i 0.716078i −0.933707 0.358039i \(-0.883446\pi\)
0.933707 0.358039i \(-0.116554\pi\)
\(878\) 0 0
\(879\) −870.000 + 778.152i −0.989761 + 0.885269i
\(880\) 0 0
\(881\) 536.656i 0.609145i 0.952489 + 0.304572i \(0.0985135\pi\)
−0.952489 + 0.304572i \(0.901486\pi\)
\(882\) 0 0
\(883\) 948.000i 1.07361i −0.843705 0.536806i \(-0.819631\pi\)
0.843705 0.536806i \(-0.180369\pi\)
\(884\) 0 0
\(885\) 543.870 1371.93i 0.614542 1.55021i
\(886\) 0 0
\(887\) −281.745 −0.317638 −0.158819 0.987308i \(-0.550769\pi\)
−0.158819 + 0.987308i \(0.550769\pi\)
\(888\) 0 0
\(889\) 128.000 0.143982
\(890\) 0 0
\(891\) 160.000 706.597i 0.179574 0.793039i
\(892\) 0 0
\(893\) −26.8328 −0.0300479
\(894\) 0 0
\(895\) 1160.00 580.000i 1.29609 0.648045i
\(896\) 0 0
\(897\) 107.331 + 120.000i 0.119656 + 0.133779i
\(898\) 0 0
\(899\) 912.316i 1.01481i
\(900\) 0 0
\(901\) −1260.00 −1.39845
\(902\) 0 0
\(903\) −500.879 + 448.000i −0.554684 + 0.496124i
\(904\) 0 0
\(905\) −371.187 742.375i −0.410152 0.820303i
\(906\) 0 0
\(907\) 516.000i 0.568908i 0.958690 + 0.284454i \(0.0918124\pi\)
−0.958690 + 0.284454i \(0.908188\pi\)
\(908\) 0 0
\(909\) 1360.00 + 152.053i 1.49615 + 0.167275i
\(910\) 0 0
\(911\) 321.994i 0.353451i 0.984260 + 0.176725i \(0.0565505\pi\)
−0.984260 + 0.176725i \(0.943450\pi\)
\(912\) 0 0
\(913\) 920.000i 1.00767i
\(914\) 0 0
\(915\) 1031.88 + 409.062i 1.12773 + 0.447062i
\(916\) 0 0
\(917\) −643.988 −0.702277
\(918\) 0 0
\(919\) 1314.00 1.42982 0.714908 0.699219i \(-0.246469\pi\)
0.714908 + 0.699219i \(0.246469\pi\)
\(920\) 0 0
\(921\) 984.000 + 1100.15i 1.06840 + 1.19451i
\(922\) 0 0
\(923\) 643.988 0.697711
\(924\) 0 0
\(925\) 880.000 + 660.000i 0.951351 + 0.713514i
\(926\) 0 0
\(927\) −930.204 104.000i −1.00346 0.112190i
\(928\) 0 0
\(929\) 1502.64i 1.61748i 0.588167 + 0.808739i \(0.299850\pi\)
−0.588167 + 0.808739i \(0.700150\pi\)
\(930\) 0 0
\(931\) 90.0000 0.0966702
\(932\) 0 0
\(933\) 965.981 + 1080.00i 1.03535 + 1.15756i
\(934\) 0 0
\(935\) 1252.20 626.099i 1.33925 0.669625i
\(936\) 0 0
\(937\) 1288.00i 1.37460i −0.726374 0.687300i \(-0.758796\pi\)
0.726374 0.687300i \(-0.241204\pi\)
\(938\) 0 0
\(939\) −1136.00 1270.09i −1.20980 1.35259i
\(940\) 0 0
\(941\) 366.715i 0.389708i −0.980832 0.194854i \(-0.937577\pi\)
0.980832 0.194854i \(-0.0624233\pi\)
\(942\) 0 0
\(943\) 80.0000i 0.0848356i
\(944\) 0 0
\(945\) −953.548 507.096i −1.00905 0.536609i
\(946\) 0 0
\(947\) −1677.05 −1.77091 −0.885455 0.464726i \(-0.846153\pi\)
−0.885455 + 0.464726i \(0.846153\pi\)
\(948\) 0 0
\(949\) −672.000 −0.708114
\(950\) 0 0
\(951\) 1370.00 1225.37i 1.44059 1.28850i
\(952\) 0 0
\(953\) 827.345 0.868148 0.434074 0.900877i \(-0.357076\pi\)
0.434074 + 0.900877i \(0.357076\pi\)
\(954\) 0 0
\(955\) −960.000 + 480.000i −1.00524 + 0.502618i
\(956\) 0 0
\(957\) −536.656 + 480.000i −0.560769 + 0.501567i
\(958\) 0 0
\(959\) 1395.31i 1.45496i
\(960\) 0 0
\(961\) 195.000 0.202914
\(962\) 0 0
\(963\) 147.580 1320.00i 0.153251 1.37072i
\(964\) 0 0
\(965\) −143.108 + 71.5542i −0.148299 + 0.0741494i
\(966\) 0 0
\(967\) 536.000i 0.554292i 0.960828 + 0.277146i \(0.0893885\pi\)
−0.960828 + 0.277146i \(0.910611\pi\)
\(968\) 0 0
\(969\) 420.000 375.659i 0.433437 0.387677i
\(970\) 0 0
\(971\) 134.164i 0.138171i −0.997611 0.0690855i \(-0.977992\pi\)
0.997611 0.0690855i \(-0.0220081\pi\)
\(972\) 0 0
\(973\) 944.000i 0.970195i
\(974\) 0 0
\(975\) −896.656 + 77.5078i −0.919648 + 0.0794951i
\(976\) 0 0
\(977\) −174.413 −0.178519 −0.0892596 0.996008i \(-0.528450\pi\)
−0.0892596 + 0.996008i \(0.528450\pi\)
\(978\) 0 0
\(979\) −160.000 −0.163432
\(980\) 0 0
\(981\) 74.0000 661.876i 0.0754332 0.674695i
\(982\) 0 0
\(983\) −1855.94 −1.88803 −0.944016 0.329898i \(-0.892986\pi\)
−0.944016 + 0.329898i \(0.892986\pi\)
\(984\) 0 0
\(985\) 10.0000 + 20.0000i 0.0101523 + 0.0203046i
\(986\) 0 0
\(987\) −71.5542 80.0000i −0.0724966 0.0810537i
\(988\) 0 0
\(989\) 125.220i 0.126613i
\(990\) 0 0
\(991\) −1646.00 −1.66095 −0.830474 0.557057i \(-0.811930\pi\)
−0.830474 + 0.557057i \(0.811930\pi\)
\(992\) 0 0
\(993\) 451.686 404.000i 0.454870 0.406848i
\(994\) 0 0
\(995\) 254.912 + 509.823i 0.256193 + 0.512385i
\(996\) 0 0
\(997\) 1164.00i 1.16750i −0.811933 0.583751i \(-0.801584\pi\)
0.811933 0.583751i \(-0.198416\pi\)
\(998\) 0 0
\(999\) −968.000 + 688.709i −0.968969 + 0.689398i
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 60.3.b.a.29.3 yes 4
3.2 odd 2 inner 60.3.b.a.29.1 4
4.3 odd 2 240.3.c.d.209.2 4
5.2 odd 4 300.3.g.e.101.1 2
5.3 odd 4 300.3.g.h.101.2 2
5.4 even 2 inner 60.3.b.a.29.2 yes 4
8.3 odd 2 960.3.c.g.449.3 4
8.5 even 2 960.3.c.h.449.2 4
9.2 odd 6 1620.3.t.b.1349.4 8
9.4 even 3 1620.3.t.b.269.3 8
9.5 odd 6 1620.3.t.b.269.2 8
9.7 even 3 1620.3.t.b.1349.1 8
12.11 even 2 240.3.c.d.209.4 4
15.2 even 4 300.3.g.e.101.2 2
15.8 even 4 300.3.g.h.101.1 2
15.14 odd 2 inner 60.3.b.a.29.4 yes 4
20.3 even 4 1200.3.l.h.401.1 2
20.7 even 4 1200.3.l.q.401.2 2
20.19 odd 2 240.3.c.d.209.3 4
24.5 odd 2 960.3.c.h.449.4 4
24.11 even 2 960.3.c.g.449.1 4
40.19 odd 2 960.3.c.g.449.2 4
40.29 even 2 960.3.c.h.449.3 4
45.4 even 6 1620.3.t.b.269.4 8
45.14 odd 6 1620.3.t.b.269.1 8
45.29 odd 6 1620.3.t.b.1349.3 8
45.34 even 6 1620.3.t.b.1349.2 8
60.23 odd 4 1200.3.l.h.401.2 2
60.47 odd 4 1200.3.l.q.401.1 2
60.59 even 2 240.3.c.d.209.1 4
120.29 odd 2 960.3.c.h.449.1 4
120.59 even 2 960.3.c.g.449.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.3.b.a.29.1 4 3.2 odd 2 inner
60.3.b.a.29.2 yes 4 5.4 even 2 inner
60.3.b.a.29.3 yes 4 1.1 even 1 trivial
60.3.b.a.29.4 yes 4 15.14 odd 2 inner
240.3.c.d.209.1 4 60.59 even 2
240.3.c.d.209.2 4 4.3 odd 2
240.3.c.d.209.3 4 20.19 odd 2
240.3.c.d.209.4 4 12.11 even 2
300.3.g.e.101.1 2 5.2 odd 4
300.3.g.e.101.2 2 15.2 even 4
300.3.g.h.101.1 2 15.8 even 4
300.3.g.h.101.2 2 5.3 odd 4
960.3.c.g.449.1 4 24.11 even 2
960.3.c.g.449.2 4 40.19 odd 2
960.3.c.g.449.3 4 8.3 odd 2
960.3.c.g.449.4 4 120.59 even 2
960.3.c.h.449.1 4 120.29 odd 2
960.3.c.h.449.2 4 8.5 even 2
960.3.c.h.449.3 4 40.29 even 2
960.3.c.h.449.4 4 24.5 odd 2
1200.3.l.h.401.1 2 20.3 even 4
1200.3.l.h.401.2 2 60.23 odd 4
1200.3.l.q.401.1 2 60.47 odd 4
1200.3.l.q.401.2 2 20.7 even 4
1620.3.t.b.269.1 8 45.14 odd 6
1620.3.t.b.269.2 8 9.5 odd 6
1620.3.t.b.269.3 8 9.4 even 3
1620.3.t.b.269.4 8 45.4 even 6
1620.3.t.b.1349.1 8 9.7 even 3
1620.3.t.b.1349.2 8 45.34 even 6
1620.3.t.b.1349.3 8 45.29 odd 6
1620.3.t.b.1349.4 8 9.2 odd 6