Properties

Label 1216.3.g.e.417.37
Level $1216$
Weight $3$
Character 1216.417
Analytic conductor $33.134$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,3,Mod(417,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, 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("1216.417");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.g (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: \(33.1336001462\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 417.37
Character \(\chi\) \(=\) 1216.417
Dual form 1216.3.g.e.417.40

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.58612 q^{3} -0.323607i q^{5} +8.09640 q^{7} +12.0325 q^{9} +O(q^{10})\) \(q+4.58612 q^{3} -0.323607i q^{5} +8.09640 q^{7} +12.0325 q^{9} +15.7627i q^{11} +2.70240 q^{13} -1.48410i q^{15} -23.8912 q^{17} +(-12.2592 + 14.5160i) q^{19} +37.1310 q^{21} +28.7327 q^{23} +24.8953 q^{25} +13.9072 q^{27} +51.7916 q^{29} +53.3446i q^{31} +72.2897i q^{33} -2.62006i q^{35} -30.8820 q^{37} +12.3935 q^{39} -41.0208i q^{41} +1.24753i q^{43} -3.89379i q^{45} +1.82806 q^{47} +16.5517 q^{49} -109.568 q^{51} -10.5509 q^{53} +5.10094 q^{55} +(-56.2219 + 66.5719i) q^{57} +13.8763 q^{59} -96.8096i q^{61} +97.4197 q^{63} -0.874518i q^{65} +22.4624 q^{67} +131.772 q^{69} -88.9010i q^{71} -34.3165 q^{73} +114.173 q^{75} +127.621i q^{77} -113.440i q^{79} -44.5120 q^{81} +65.8732i q^{83} +7.73138i q^{85} +237.522 q^{87} +120.371i q^{89} +21.8797 q^{91} +244.645i q^{93} +(4.69747 + 3.96715i) q^{95} -88.2932i q^{97} +189.665i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 264 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 264 q^{9} + 24 q^{17} - 160 q^{25} + 328 q^{49} + 8 q^{57} - 472 q^{73} - 736 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 4.58612 1.52871 0.764353 0.644798i \(-0.223059\pi\)
0.764353 + 0.644798i \(0.223059\pi\)
\(4\) 0 0
\(5\) 0.323607i 0.0647215i −0.999476 0.0323607i \(-0.989697\pi\)
0.999476 0.0323607i \(-0.0103025\pi\)
\(6\) 0 0
\(7\) 8.09640 1.15663 0.578314 0.815814i \(-0.303711\pi\)
0.578314 + 0.815814i \(0.303711\pi\)
\(8\) 0 0
\(9\) 12.0325 1.33694
\(10\) 0 0
\(11\) 15.7627i 1.43298i 0.697600 + 0.716488i \(0.254251\pi\)
−0.697600 + 0.716488i \(0.745749\pi\)
\(12\) 0 0
\(13\) 2.70240 0.207877 0.103939 0.994584i \(-0.466855\pi\)
0.103939 + 0.994584i \(0.466855\pi\)
\(14\) 0 0
\(15\) 1.48410i 0.0989401i
\(16\) 0 0
\(17\) −23.8912 −1.40537 −0.702684 0.711502i \(-0.748015\pi\)
−0.702684 + 0.711502i \(0.748015\pi\)
\(18\) 0 0
\(19\) −12.2592 + 14.5160i −0.645219 + 0.763998i
\(20\) 0 0
\(21\) 37.1310 1.76814
\(22\) 0 0
\(23\) 28.7327 1.24925 0.624624 0.780925i \(-0.285252\pi\)
0.624624 + 0.780925i \(0.285252\pi\)
\(24\) 0 0
\(25\) 24.8953 0.995811
\(26\) 0 0
\(27\) 13.9072 0.515083
\(28\) 0 0
\(29\) 51.7916 1.78592 0.892959 0.450139i \(-0.148626\pi\)
0.892959 + 0.450139i \(0.148626\pi\)
\(30\) 0 0
\(31\) 53.3446i 1.72079i 0.509625 + 0.860397i \(0.329784\pi\)
−0.509625 + 0.860397i \(0.670216\pi\)
\(32\) 0 0
\(33\) 72.2897i 2.19060i
\(34\) 0 0
\(35\) 2.62006i 0.0748587i
\(36\) 0 0
\(37\) −30.8820 −0.834649 −0.417325 0.908757i \(-0.637032\pi\)
−0.417325 + 0.908757i \(0.637032\pi\)
\(38\) 0 0
\(39\) 12.3935 0.317783
\(40\) 0 0
\(41\) 41.0208i 1.00051i −0.865879 0.500254i \(-0.833240\pi\)
0.865879 0.500254i \(-0.166760\pi\)
\(42\) 0 0
\(43\) 1.24753i 0.0290123i 0.999895 + 0.0145062i \(0.00461761\pi\)
−0.999895 + 0.0145062i \(0.995382\pi\)
\(44\) 0 0
\(45\) 3.89379i 0.0865288i
\(46\) 0 0
\(47\) 1.82806 0.0388949 0.0194474 0.999811i \(-0.493809\pi\)
0.0194474 + 0.999811i \(0.493809\pi\)
\(48\) 0 0
\(49\) 16.5517 0.337790
\(50\) 0 0
\(51\) −109.568 −2.14839
\(52\) 0 0
\(53\) −10.5509 −0.199073 −0.0995365 0.995034i \(-0.531736\pi\)
−0.0995365 + 0.995034i \(0.531736\pi\)
\(54\) 0 0
\(55\) 5.10094 0.0927443
\(56\) 0 0
\(57\) −56.2219 + 66.5719i −0.986349 + 1.16793i
\(58\) 0 0
\(59\) 13.8763 0.235192 0.117596 0.993062i \(-0.462481\pi\)
0.117596 + 0.993062i \(0.462481\pi\)
\(60\) 0 0
\(61\) 96.8096i 1.58704i −0.608542 0.793522i \(-0.708245\pi\)
0.608542 0.793522i \(-0.291755\pi\)
\(62\) 0 0
\(63\) 97.4197 1.54634
\(64\) 0 0
\(65\) 0.874518i 0.0134541i
\(66\) 0 0
\(67\) 22.4624 0.335260 0.167630 0.985850i \(-0.446389\pi\)
0.167630 + 0.985850i \(0.446389\pi\)
\(68\) 0 0
\(69\) 131.772 1.90973
\(70\) 0 0
\(71\) 88.9010i 1.25213i −0.779772 0.626064i \(-0.784665\pi\)
0.779772 0.626064i \(-0.215335\pi\)
\(72\) 0 0
\(73\) −34.3165 −0.470090 −0.235045 0.971985i \(-0.575524\pi\)
−0.235045 + 0.971985i \(0.575524\pi\)
\(74\) 0 0
\(75\) 114.173 1.52230
\(76\) 0 0
\(77\) 127.621i 1.65742i
\(78\) 0 0
\(79\) 113.440i 1.43595i −0.696067 0.717977i \(-0.745068\pi\)
0.696067 0.717977i \(-0.254932\pi\)
\(80\) 0 0
\(81\) −44.5120 −0.549531
\(82\) 0 0
\(83\) 65.8732i 0.793653i 0.917894 + 0.396826i \(0.129888\pi\)
−0.917894 + 0.396826i \(0.870112\pi\)
\(84\) 0 0
\(85\) 7.73138i 0.0909574i
\(86\) 0 0
\(87\) 237.522 2.73014
\(88\) 0 0
\(89\) 120.371i 1.35249i 0.736679 + 0.676243i \(0.236393\pi\)
−0.736679 + 0.676243i \(0.763607\pi\)
\(90\) 0 0
\(91\) 21.8797 0.240437
\(92\) 0 0
\(93\) 244.645i 2.63059i
\(94\) 0 0
\(95\) 4.69747 + 3.96715i 0.0494471 + 0.0417595i
\(96\) 0 0
\(97\) 88.2932i 0.910239i −0.890430 0.455120i \(-0.849597\pi\)
0.890430 0.455120i \(-0.150403\pi\)
\(98\) 0 0
\(99\) 189.665i 1.91580i
\(100\) 0 0
\(101\) 98.1162i 0.971447i 0.874112 + 0.485724i \(0.161444\pi\)
−0.874112 + 0.485724i \(0.838556\pi\)
\(102\) 0 0
\(103\) 136.279i 1.32309i −0.749904 0.661547i \(-0.769900\pi\)
0.749904 0.661547i \(-0.230100\pi\)
\(104\) 0 0
\(105\) 12.0159i 0.114437i
\(106\) 0 0
\(107\) 195.523 1.82732 0.913659 0.406482i \(-0.133244\pi\)
0.913659 + 0.406482i \(0.133244\pi\)
\(108\) 0 0
\(109\) 11.6177 0.106584 0.0532922 0.998579i \(-0.483029\pi\)
0.0532922 + 0.998579i \(0.483029\pi\)
\(110\) 0 0
\(111\) −141.629 −1.27593
\(112\) 0 0
\(113\) 97.8220i 0.865681i −0.901470 0.432841i \(-0.857511\pi\)
0.901470 0.432841i \(-0.142489\pi\)
\(114\) 0 0
\(115\) 9.29812i 0.0808532i
\(116\) 0 0
\(117\) 32.5166 0.277919
\(118\) 0 0
\(119\) −193.433 −1.62549
\(120\) 0 0
\(121\) −127.464 −1.05342
\(122\) 0 0
\(123\) 188.126i 1.52948i
\(124\) 0 0
\(125\) 16.1465i 0.129172i
\(126\) 0 0
\(127\) 190.390i 1.49913i −0.661930 0.749566i \(-0.730262\pi\)
0.661930 0.749566i \(-0.269738\pi\)
\(128\) 0 0
\(129\) 5.72131i 0.0443513i
\(130\) 0 0
\(131\) 5.14461i 0.0392718i 0.999807 + 0.0196359i \(0.00625070\pi\)
−0.999807 + 0.0196359i \(0.993749\pi\)
\(132\) 0 0
\(133\) −99.2550 + 117.527i −0.746278 + 0.883662i
\(134\) 0 0
\(135\) 4.50048i 0.0333369i
\(136\) 0 0
\(137\) 50.3123 0.367243 0.183621 0.982997i \(-0.441218\pi\)
0.183621 + 0.982997i \(0.441218\pi\)
\(138\) 0 0
\(139\) 114.207i 0.821636i −0.911717 0.410818i \(-0.865243\pi\)
0.911717 0.410818i \(-0.134757\pi\)
\(140\) 0 0
\(141\) 8.38370 0.0594588
\(142\) 0 0
\(143\) 42.5973i 0.297883i
\(144\) 0 0
\(145\) 16.7601i 0.115587i
\(146\) 0 0
\(147\) 75.9081 0.516382
\(148\) 0 0
\(149\) 204.095i 1.36976i 0.728654 + 0.684882i \(0.240146\pi\)
−0.728654 + 0.684882i \(0.759854\pi\)
\(150\) 0 0
\(151\) 130.732i 0.865773i 0.901448 + 0.432887i \(0.142505\pi\)
−0.901448 + 0.432887i \(0.857495\pi\)
\(152\) 0 0
\(153\) −287.471 −1.87889
\(154\) 0 0
\(155\) 17.2627 0.111372
\(156\) 0 0
\(157\) 29.8681i 0.190243i −0.995466 0.0951213i \(-0.969676\pi\)
0.995466 0.0951213i \(-0.0303239\pi\)
\(158\) 0 0
\(159\) −48.3875 −0.304324
\(160\) 0 0
\(161\) 232.632 1.44492
\(162\) 0 0
\(163\) 266.286i 1.63366i 0.576881 + 0.816828i \(0.304270\pi\)
−0.576881 + 0.816828i \(0.695730\pi\)
\(164\) 0 0
\(165\) 23.3935 0.141779
\(166\) 0 0
\(167\) 17.3715i 0.104021i −0.998647 0.0520104i \(-0.983437\pi\)
0.998647 0.0520104i \(-0.0165629\pi\)
\(168\) 0 0
\(169\) −161.697 −0.956787
\(170\) 0 0
\(171\) −147.508 + 174.663i −0.862619 + 1.02142i
\(172\) 0 0
\(173\) 49.2803 0.284857 0.142429 0.989805i \(-0.454509\pi\)
0.142429 + 0.989805i \(0.454509\pi\)
\(174\) 0 0
\(175\) 201.562 1.15178
\(176\) 0 0
\(177\) 63.6384 0.359539
\(178\) 0 0
\(179\) −56.7394 −0.316980 −0.158490 0.987361i \(-0.550663\pi\)
−0.158490 + 0.987361i \(0.550663\pi\)
\(180\) 0 0
\(181\) −230.448 −1.27319 −0.636597 0.771196i \(-0.719659\pi\)
−0.636597 + 0.771196i \(0.719659\pi\)
\(182\) 0 0
\(183\) 443.980i 2.42612i
\(184\) 0 0
\(185\) 9.99365i 0.0540197i
\(186\) 0 0
\(187\) 376.591i 2.01386i
\(188\) 0 0
\(189\) 112.599 0.595760
\(190\) 0 0
\(191\) −80.7648 −0.422852 −0.211426 0.977394i \(-0.567811\pi\)
−0.211426 + 0.977394i \(0.567811\pi\)
\(192\) 0 0
\(193\) 113.655i 0.588885i 0.955669 + 0.294443i \(0.0951340\pi\)
−0.955669 + 0.294443i \(0.904866\pi\)
\(194\) 0 0
\(195\) 4.01064i 0.0205674i
\(196\) 0 0
\(197\) 19.9354i 0.101195i 0.998719 + 0.0505974i \(0.0161125\pi\)
−0.998719 + 0.0505974i \(0.983887\pi\)
\(198\) 0 0
\(199\) 12.3811 0.0622164 0.0311082 0.999516i \(-0.490096\pi\)
0.0311082 + 0.999516i \(0.490096\pi\)
\(200\) 0 0
\(201\) 103.015 0.512514
\(202\) 0 0
\(203\) 419.326 2.06564
\(204\) 0 0
\(205\) −13.2746 −0.0647544
\(206\) 0 0
\(207\) 345.725 1.67017
\(208\) 0 0
\(209\) −228.811 193.238i −1.09479 0.924582i
\(210\) 0 0
\(211\) 265.822 1.25982 0.629909 0.776669i \(-0.283092\pi\)
0.629909 + 0.776669i \(0.283092\pi\)
\(212\) 0 0
\(213\) 407.711i 1.91413i
\(214\) 0 0
\(215\) 0.403710 0.00187772
\(216\) 0 0
\(217\) 431.899i 1.99032i
\(218\) 0 0
\(219\) −157.380 −0.718629
\(220\) 0 0
\(221\) −64.5638 −0.292144
\(222\) 0 0
\(223\) 181.609i 0.814390i −0.913341 0.407195i \(-0.866507\pi\)
0.913341 0.407195i \(-0.133493\pi\)
\(224\) 0 0
\(225\) 299.552 1.33134
\(226\) 0 0
\(227\) −413.152 −1.82005 −0.910025 0.414553i \(-0.863938\pi\)
−0.910025 + 0.414553i \(0.863938\pi\)
\(228\) 0 0
\(229\) 142.290i 0.621356i 0.950515 + 0.310678i \(0.100556\pi\)
−0.950515 + 0.310678i \(0.899444\pi\)
\(230\) 0 0
\(231\) 585.287i 2.53371i
\(232\) 0 0
\(233\) −222.260 −0.953905 −0.476952 0.878929i \(-0.658259\pi\)
−0.476952 + 0.878929i \(0.658259\pi\)
\(234\) 0 0
\(235\) 0.591574i 0.00251733i
\(236\) 0 0
\(237\) 520.250i 2.19515i
\(238\) 0 0
\(239\) 391.140 1.63657 0.818286 0.574812i \(-0.194925\pi\)
0.818286 + 0.574812i \(0.194925\pi\)
\(240\) 0 0
\(241\) 300.859i 1.24838i −0.781274 0.624188i \(-0.785430\pi\)
0.781274 0.624188i \(-0.214570\pi\)
\(242\) 0 0
\(243\) −329.302 −1.35515
\(244\) 0 0
\(245\) 5.35626i 0.0218623i
\(246\) 0 0
\(247\) −33.1292 + 39.2280i −0.134126 + 0.158818i
\(248\) 0 0
\(249\) 302.102i 1.21326i
\(250\) 0 0
\(251\) 120.770i 0.481154i −0.970630 0.240577i \(-0.922663\pi\)
0.970630 0.240577i \(-0.0773367\pi\)
\(252\) 0 0
\(253\) 452.906i 1.79014i
\(254\) 0 0
\(255\) 35.4570i 0.139047i
\(256\) 0 0
\(257\) 1.62541i 0.00632457i 0.999995 + 0.00316228i \(0.00100659\pi\)
−0.999995 + 0.00316228i \(0.998993\pi\)
\(258\) 0 0
\(259\) −250.033 −0.965379
\(260\) 0 0
\(261\) 623.181 2.38766
\(262\) 0 0
\(263\) −136.060 −0.517337 −0.258669 0.965966i \(-0.583284\pi\)
−0.258669 + 0.965966i \(0.583284\pi\)
\(264\) 0 0
\(265\) 3.41434i 0.0128843i
\(266\) 0 0
\(267\) 552.037i 2.06755i
\(268\) 0 0
\(269\) 0.387264 0.00143964 0.000719822 1.00000i \(-0.499771\pi\)
0.000719822 1.00000i \(0.499771\pi\)
\(270\) 0 0
\(271\) −431.503 −1.59226 −0.796130 0.605125i \(-0.793123\pi\)
−0.796130 + 0.605125i \(0.793123\pi\)
\(272\) 0 0
\(273\) 100.343 0.367557
\(274\) 0 0
\(275\) 392.418i 1.42697i
\(276\) 0 0
\(277\) 143.616i 0.518470i −0.965814 0.259235i \(-0.916530\pi\)
0.965814 0.259235i \(-0.0834704\pi\)
\(278\) 0 0
\(279\) 641.867i 2.30060i
\(280\) 0 0
\(281\) 197.212i 0.701822i −0.936409 0.350911i \(-0.885872\pi\)
0.936409 0.350911i \(-0.114128\pi\)
\(282\) 0 0
\(283\) 47.0745i 0.166341i −0.996535 0.0831705i \(-0.973495\pi\)
0.996535 0.0831705i \(-0.0265046\pi\)
\(284\) 0 0
\(285\) 21.5432 + 18.1938i 0.0755900 + 0.0638380i
\(286\) 0 0
\(287\) 332.121i 1.15722i
\(288\) 0 0
\(289\) 281.792 0.975057
\(290\) 0 0
\(291\) 404.923i 1.39149i
\(292\) 0 0
\(293\) 544.484 1.85831 0.929154 0.369692i \(-0.120537\pi\)
0.929154 + 0.369692i \(0.120537\pi\)
\(294\) 0 0
\(295\) 4.49048i 0.0152220i
\(296\) 0 0
\(297\) 219.216i 0.738101i
\(298\) 0 0
\(299\) 77.6474 0.259690
\(300\) 0 0
\(301\) 10.1005i 0.0335565i
\(302\) 0 0
\(303\) 449.972i 1.48506i
\(304\) 0 0
\(305\) −31.3283 −0.102716
\(306\) 0 0
\(307\) −80.0883 −0.260874 −0.130437 0.991457i \(-0.541638\pi\)
−0.130437 + 0.991457i \(0.541638\pi\)
\(308\) 0 0
\(309\) 624.990i 2.02262i
\(310\) 0 0
\(311\) −149.245 −0.479889 −0.239944 0.970787i \(-0.577129\pi\)
−0.239944 + 0.970787i \(0.577129\pi\)
\(312\) 0 0
\(313\) 104.148 0.332743 0.166371 0.986063i \(-0.446795\pi\)
0.166371 + 0.986063i \(0.446795\pi\)
\(314\) 0 0
\(315\) 31.5257i 0.100082i
\(316\) 0 0
\(317\) −32.7397 −0.103280 −0.0516399 0.998666i \(-0.516445\pi\)
−0.0516399 + 0.998666i \(0.516445\pi\)
\(318\) 0 0
\(319\) 816.377i 2.55918i
\(320\) 0 0
\(321\) 896.691 2.79343
\(322\) 0 0
\(323\) 292.886 346.804i 0.906769 1.07370i
\(324\) 0 0
\(325\) 67.2771 0.207006
\(326\) 0 0
\(327\) 53.2801 0.162936
\(328\) 0 0
\(329\) 14.8007 0.0449870
\(330\) 0 0
\(331\) 171.550 0.518277 0.259139 0.965840i \(-0.416561\pi\)
0.259139 + 0.965840i \(0.416561\pi\)
\(332\) 0 0
\(333\) −371.587 −1.11588
\(334\) 0 0
\(335\) 7.26901i 0.0216985i
\(336\) 0 0
\(337\) 182.851i 0.542585i −0.962497 0.271292i \(-0.912549\pi\)
0.962497 0.271292i \(-0.0874510\pi\)
\(338\) 0 0
\(339\) 448.623i 1.32337i
\(340\) 0 0
\(341\) −840.857 −2.46585
\(342\) 0 0
\(343\) −262.714 −0.765931
\(344\) 0 0
\(345\) 42.6423i 0.123601i
\(346\) 0 0
\(347\) 459.359i 1.32380i 0.749591 + 0.661901i \(0.230250\pi\)
−0.749591 + 0.661901i \(0.769750\pi\)
\(348\) 0 0
\(349\) 308.948i 0.885239i −0.896710 0.442619i \(-0.854049\pi\)
0.896710 0.442619i \(-0.145951\pi\)
\(350\) 0 0
\(351\) 37.5830 0.107074
\(352\) 0 0
\(353\) −138.347 −0.391917 −0.195958 0.980612i \(-0.562782\pi\)
−0.195958 + 0.980612i \(0.562782\pi\)
\(354\) 0 0
\(355\) −28.7690 −0.0810395
\(356\) 0 0
\(357\) −887.107 −2.48489
\(358\) 0 0
\(359\) 572.384 1.59439 0.797193 0.603725i \(-0.206317\pi\)
0.797193 + 0.603725i \(0.206317\pi\)
\(360\) 0 0
\(361\) −60.4264 355.907i −0.167386 0.985891i
\(362\) 0 0
\(363\) −584.563 −1.61037
\(364\) 0 0
\(365\) 11.1051i 0.0304249i
\(366\) 0 0
\(367\) −569.204 −1.55096 −0.775482 0.631370i \(-0.782493\pi\)
−0.775482 + 0.631370i \(0.782493\pi\)
\(368\) 0 0
\(369\) 493.582i 1.33762i
\(370\) 0 0
\(371\) −85.4241 −0.230254
\(372\) 0 0
\(373\) 426.308 1.14292 0.571458 0.820631i \(-0.306378\pi\)
0.571458 + 0.820631i \(0.306378\pi\)
\(374\) 0 0
\(375\) 74.0496i 0.197466i
\(376\) 0 0
\(377\) 139.962 0.371251
\(378\) 0 0
\(379\) −411.263 −1.08513 −0.542563 0.840015i \(-0.682546\pi\)
−0.542563 + 0.840015i \(0.682546\pi\)
\(380\) 0 0
\(381\) 873.150i 2.29173i
\(382\) 0 0
\(383\) 485.648i 1.26801i −0.773329 0.634005i \(-0.781410\pi\)
0.773329 0.634005i \(-0.218590\pi\)
\(384\) 0 0
\(385\) 41.2992 0.107271
\(386\) 0 0
\(387\) 15.0109i 0.0387877i
\(388\) 0 0
\(389\) 151.447i 0.389324i −0.980870 0.194662i \(-0.937639\pi\)
0.980870 0.194662i \(-0.0623610\pi\)
\(390\) 0 0
\(391\) −686.460 −1.75565
\(392\) 0 0
\(393\) 23.5938i 0.0600350i
\(394\) 0 0
\(395\) −36.7101 −0.0929370
\(396\) 0 0
\(397\) 391.307i 0.985661i −0.870125 0.492830i \(-0.835962\pi\)
0.870125 0.492830i \(-0.164038\pi\)
\(398\) 0 0
\(399\) −455.195 + 538.993i −1.14084 + 1.35086i
\(400\) 0 0
\(401\) 531.100i 1.32444i 0.749310 + 0.662220i \(0.230385\pi\)
−0.749310 + 0.662220i \(0.769615\pi\)
\(402\) 0 0
\(403\) 144.159i 0.357714i
\(404\) 0 0
\(405\) 14.4044i 0.0355664i
\(406\) 0 0
\(407\) 486.785i 1.19603i
\(408\) 0 0
\(409\) 6.97313i 0.0170492i −0.999964 0.00852461i \(-0.997286\pi\)
0.999964 0.00852461i \(-0.00271350\pi\)
\(410\) 0 0
\(411\) 230.738 0.561406
\(412\) 0 0
\(413\) 112.348 0.272030
\(414\) 0 0
\(415\) 21.3170 0.0513664
\(416\) 0 0
\(417\) 523.769i 1.25604i
\(418\) 0 0
\(419\) 244.894i 0.584473i −0.956346 0.292236i \(-0.905601\pi\)
0.956346 0.292236i \(-0.0943994\pi\)
\(420\) 0 0
\(421\) −669.982 −1.59141 −0.795703 0.605687i \(-0.792898\pi\)
−0.795703 + 0.605687i \(0.792898\pi\)
\(422\) 0 0
\(423\) 21.9961 0.0520002
\(424\) 0 0
\(425\) −594.779 −1.39948
\(426\) 0 0
\(427\) 783.810i 1.83562i
\(428\) 0 0
\(429\) 195.356i 0.455375i
\(430\) 0 0
\(431\) 603.062i 1.39922i −0.714526 0.699608i \(-0.753358\pi\)
0.714526 0.699608i \(-0.246642\pi\)
\(432\) 0 0
\(433\) 614.205i 1.41849i −0.704964 0.709243i \(-0.749037\pi\)
0.704964 0.709243i \(-0.250963\pi\)
\(434\) 0 0
\(435\) 76.8640i 0.176699i
\(436\) 0 0
\(437\) −352.239 + 417.083i −0.806038 + 0.954424i
\(438\) 0 0
\(439\) 111.058i 0.252980i −0.991968 0.126490i \(-0.959629\pi\)
0.991968 0.126490i \(-0.0403711\pi\)
\(440\) 0 0
\(441\) 199.158 0.451605
\(442\) 0 0
\(443\) 254.595i 0.574708i −0.957825 0.287354i \(-0.907224\pi\)
0.957825 0.287354i \(-0.0927756\pi\)
\(444\) 0 0
\(445\) 38.9530 0.0875349
\(446\) 0 0
\(447\) 936.002i 2.09397i
\(448\) 0 0
\(449\) 637.170i 1.41909i 0.704662 + 0.709543i \(0.251099\pi\)
−0.704662 + 0.709543i \(0.748901\pi\)
\(450\) 0 0
\(451\) 646.600 1.43370
\(452\) 0 0
\(453\) 599.551i 1.32351i
\(454\) 0 0
\(455\) 7.08045i 0.0155614i
\(456\) 0 0
\(457\) −619.257 −1.35505 −0.677524 0.735500i \(-0.736947\pi\)
−0.677524 + 0.735500i \(0.736947\pi\)
\(458\) 0 0
\(459\) −332.261 −0.723881
\(460\) 0 0
\(461\) 693.390i 1.50410i 0.659106 + 0.752050i \(0.270935\pi\)
−0.659106 + 0.752050i \(0.729065\pi\)
\(462\) 0 0
\(463\) −221.733 −0.478905 −0.239453 0.970908i \(-0.576968\pi\)
−0.239453 + 0.970908i \(0.576968\pi\)
\(464\) 0 0
\(465\) 79.1688 0.170255
\(466\) 0 0
\(467\) 386.716i 0.828085i −0.910258 0.414042i \(-0.864117\pi\)
0.910258 0.414042i \(-0.135883\pi\)
\(468\) 0 0
\(469\) 181.865 0.387772
\(470\) 0 0
\(471\) 136.979i 0.290825i
\(472\) 0 0
\(473\) −19.6645 −0.0415739
\(474\) 0 0
\(475\) −305.195 + 361.379i −0.642516 + 0.760798i
\(476\) 0 0
\(477\) −126.953 −0.266149
\(478\) 0 0
\(479\) −263.608 −0.550330 −0.275165 0.961397i \(-0.588733\pi\)
−0.275165 + 0.961397i \(0.588733\pi\)
\(480\) 0 0
\(481\) −83.4557 −0.173505
\(482\) 0 0
\(483\) 1066.88 2.20885
\(484\) 0 0
\(485\) −28.5723 −0.0589120
\(486\) 0 0
\(487\) 312.880i 0.642464i −0.947000 0.321232i \(-0.895903\pi\)
0.947000 0.321232i \(-0.104097\pi\)
\(488\) 0 0
\(489\) 1221.22i 2.49738i
\(490\) 0 0
\(491\) 67.1921i 0.136847i −0.997656 0.0684237i \(-0.978203\pi\)
0.997656 0.0684237i \(-0.0217970\pi\)
\(492\) 0 0
\(493\) −1237.37 −2.50987
\(494\) 0 0
\(495\) 61.3768 0.123994
\(496\) 0 0
\(497\) 719.779i 1.44825i
\(498\) 0 0
\(499\) 638.944i 1.28045i −0.768188 0.640224i \(-0.778841\pi\)
0.768188 0.640224i \(-0.221159\pi\)
\(500\) 0 0
\(501\) 79.6676i 0.159017i
\(502\) 0 0
\(503\) 47.2084 0.0938537 0.0469269 0.998898i \(-0.485057\pi\)
0.0469269 + 0.998898i \(0.485057\pi\)
\(504\) 0 0
\(505\) 31.7511 0.0628735
\(506\) 0 0
\(507\) −741.561 −1.46265
\(508\) 0 0
\(509\) −152.130 −0.298879 −0.149440 0.988771i \(-0.547747\pi\)
−0.149440 + 0.988771i \(0.547747\pi\)
\(510\) 0 0
\(511\) −277.840 −0.543719
\(512\) 0 0
\(513\) −170.491 + 201.877i −0.332341 + 0.393522i
\(514\) 0 0
\(515\) −44.1008 −0.0856326
\(516\) 0 0
\(517\) 28.8152i 0.0557354i
\(518\) 0 0
\(519\) 226.005 0.435463
\(520\) 0 0
\(521\) 273.833i 0.525591i −0.964852 0.262795i \(-0.915356\pi\)
0.964852 0.262795i \(-0.0846445\pi\)
\(522\) 0 0
\(523\) −598.807 −1.14495 −0.572473 0.819923i \(-0.694016\pi\)
−0.572473 + 0.819923i \(0.694016\pi\)
\(524\) 0 0
\(525\) 924.388 1.76074
\(526\) 0 0
\(527\) 1274.47i 2.41835i
\(528\) 0 0
\(529\) 296.569 0.560622
\(530\) 0 0
\(531\) 166.966 0.314437
\(532\) 0 0
\(533\) 110.855i 0.207983i
\(534\) 0 0
\(535\) 63.2727i 0.118267i
\(536\) 0 0
\(537\) −260.213 −0.484569
\(538\) 0 0
\(539\) 260.900i 0.484045i
\(540\) 0 0
\(541\) 1004.48i 1.85671i 0.371694 + 0.928355i \(0.378777\pi\)
−0.371694 + 0.928355i \(0.621223\pi\)
\(542\) 0 0
\(543\) −1056.86 −1.94634
\(544\) 0 0
\(545\) 3.75957i 0.00689830i
\(546\) 0 0
\(547\) 221.020 0.404059 0.202029 0.979379i \(-0.435246\pi\)
0.202029 + 0.979379i \(0.435246\pi\)
\(548\) 0 0
\(549\) 1164.86i 2.12178i
\(550\) 0 0
\(551\) −634.921 + 751.805i −1.15231 + 1.36444i
\(552\) 0 0
\(553\) 918.458i 1.66086i
\(554\) 0 0
\(555\) 45.8320i 0.0825803i
\(556\) 0 0
\(557\) 403.065i 0.723636i −0.932249 0.361818i \(-0.882156\pi\)
0.932249 0.361818i \(-0.117844\pi\)
\(558\) 0 0
\(559\) 3.37133i 0.00603100i
\(560\) 0 0
\(561\) 1727.09i 3.07859i
\(562\) 0 0
\(563\) 892.295 1.58489 0.792447 0.609941i \(-0.208807\pi\)
0.792447 + 0.609941i \(0.208807\pi\)
\(564\) 0 0
\(565\) −31.6559 −0.0560282
\(566\) 0 0
\(567\) −360.387 −0.635603
\(568\) 0 0
\(569\) 276.824i 0.486510i 0.969962 + 0.243255i \(0.0782152\pi\)
−0.969962 + 0.243255i \(0.921785\pi\)
\(570\) 0 0
\(571\) 606.699i 1.06252i 0.847209 + 0.531260i \(0.178281\pi\)
−0.847209 + 0.531260i \(0.821719\pi\)
\(572\) 0 0
\(573\) −370.397 −0.646417
\(574\) 0 0
\(575\) 715.309 1.24402
\(576\) 0 0
\(577\) 804.482 1.39425 0.697125 0.716950i \(-0.254462\pi\)
0.697125 + 0.716950i \(0.254462\pi\)
\(578\) 0 0
\(579\) 521.234i 0.900232i
\(580\) 0 0
\(581\) 533.336i 0.917961i
\(582\) 0 0
\(583\) 166.310i 0.285267i
\(584\) 0 0
\(585\) 10.5226i 0.0179874i
\(586\) 0 0
\(587\) 138.544i 0.236020i −0.993012 0.118010i \(-0.962348\pi\)
0.993012 0.118010i \(-0.0376515\pi\)
\(588\) 0 0
\(589\) −774.348 653.960i −1.31468 1.11029i
\(590\) 0 0
\(591\) 91.4259i 0.154697i
\(592\) 0 0
\(593\) −680.758 −1.14799 −0.573995 0.818859i \(-0.694607\pi\)
−0.573995 + 0.818859i \(0.694607\pi\)
\(594\) 0 0
\(595\) 62.5964i 0.105204i
\(596\) 0 0
\(597\) 56.7810 0.0951105
\(598\) 0 0
\(599\) 529.164i 0.883413i 0.897160 + 0.441706i \(0.145627\pi\)
−0.897160 + 0.441706i \(0.854373\pi\)
\(600\) 0 0
\(601\) 448.349i 0.746004i 0.927830 + 0.373002i \(0.121672\pi\)
−0.927830 + 0.373002i \(0.878328\pi\)
\(602\) 0 0
\(603\) 270.279 0.448223
\(604\) 0 0
\(605\) 41.2482i 0.0681788i
\(606\) 0 0
\(607\) 1053.81i 1.73609i 0.496482 + 0.868047i \(0.334625\pi\)
−0.496482 + 0.868047i \(0.665375\pi\)
\(608\) 0 0
\(609\) 1923.08 3.15776
\(610\) 0 0
\(611\) 4.94016 0.00808536
\(612\) 0 0
\(613\) 673.805i 1.09919i −0.835430 0.549596i \(-0.814782\pi\)
0.835430 0.549596i \(-0.185218\pi\)
\(614\) 0 0
\(615\) −60.8791 −0.0989904
\(616\) 0 0
\(617\) −938.504 −1.52108 −0.760538 0.649293i \(-0.775065\pi\)
−0.760538 + 0.649293i \(0.775065\pi\)
\(618\) 0 0
\(619\) 873.689i 1.41145i −0.708484 0.705726i \(-0.750621\pi\)
0.708484 0.705726i \(-0.249379\pi\)
\(620\) 0 0
\(621\) 399.593 0.643467
\(622\) 0 0
\(623\) 974.574i 1.56432i
\(624\) 0 0
\(625\) 617.157 0.987451
\(626\) 0 0
\(627\) −1049.35 886.211i −1.67361 1.41341i
\(628\) 0 0
\(629\) 737.810 1.17299
\(630\) 0 0
\(631\) 888.811 1.40858 0.704288 0.709915i \(-0.251267\pi\)
0.704288 + 0.709915i \(0.251267\pi\)
\(632\) 0 0
\(633\) 1219.09 1.92589
\(634\) 0 0
\(635\) −61.6115 −0.0970260
\(636\) 0 0
\(637\) 44.7294 0.0702189
\(638\) 0 0
\(639\) 1069.70i 1.67402i
\(640\) 0 0
\(641\) 491.699i 0.767081i 0.923524 + 0.383540i \(0.125295\pi\)
−0.923524 + 0.383540i \(0.874705\pi\)
\(642\) 0 0
\(643\) 852.032i 1.32509i 0.749023 + 0.662544i \(0.230523\pi\)
−0.749023 + 0.662544i \(0.769477\pi\)
\(644\) 0 0
\(645\) 1.85146 0.00287048
\(646\) 0 0
\(647\) 510.498 0.789023 0.394511 0.918891i \(-0.370914\pi\)
0.394511 + 0.918891i \(0.370914\pi\)
\(648\) 0 0
\(649\) 218.729i 0.337024i
\(650\) 0 0
\(651\) 1980.74i 3.04261i
\(652\) 0 0
\(653\) 1068.09i 1.63567i 0.575452 + 0.817836i \(0.304826\pi\)
−0.575452 + 0.817836i \(0.695174\pi\)
\(654\) 0 0
\(655\) 1.66483 0.00254173
\(656\) 0 0
\(657\) −412.913 −0.628482
\(658\) 0 0
\(659\) −42.0110 −0.0637496 −0.0318748 0.999492i \(-0.510148\pi\)
−0.0318748 + 0.999492i \(0.510148\pi\)
\(660\) 0 0
\(661\) 141.464 0.214014 0.107007 0.994258i \(-0.465873\pi\)
0.107007 + 0.994258i \(0.465873\pi\)
\(662\) 0 0
\(663\) −296.097 −0.446602
\(664\) 0 0
\(665\) 38.0326 + 32.1197i 0.0571919 + 0.0483002i
\(666\) 0 0
\(667\) 1488.11 2.23105
\(668\) 0 0
\(669\) 832.880i 1.24496i
\(670\) 0 0
\(671\) 1525.98 2.27419
\(672\) 0 0
\(673\) 1032.50i 1.53417i 0.641546 + 0.767085i \(0.278294\pi\)
−0.641546 + 0.767085i \(0.721706\pi\)
\(674\) 0 0
\(675\) 346.225 0.512925
\(676\) 0 0
\(677\) 662.404 0.978440 0.489220 0.872160i \(-0.337282\pi\)
0.489220 + 0.872160i \(0.337282\pi\)
\(678\) 0 0
\(679\) 714.857i 1.05281i
\(680\) 0 0
\(681\) −1894.76 −2.78232
\(682\) 0 0
\(683\) −828.996 −1.21376 −0.606879 0.794795i \(-0.707579\pi\)
−0.606879 + 0.794795i \(0.707579\pi\)
\(684\) 0 0
\(685\) 16.2814i 0.0237685i
\(686\) 0 0
\(687\) 652.561i 0.949870i
\(688\) 0 0
\(689\) −28.5127 −0.0413827
\(690\) 0 0
\(691\) 685.276i 0.991716i −0.868404 0.495858i \(-0.834854\pi\)
0.868404 0.495858i \(-0.165146\pi\)
\(692\) 0 0
\(693\) 1535.60i 2.21587i
\(694\) 0 0
\(695\) −36.9584 −0.0531775
\(696\) 0 0
\(697\) 980.039i 1.40608i
\(698\) 0 0
\(699\) −1019.31 −1.45824
\(700\) 0 0
\(701\) 76.5531i 0.109206i 0.998508 + 0.0546028i \(0.0173893\pi\)
−0.998508 + 0.0546028i \(0.982611\pi\)
\(702\) 0 0
\(703\) 378.587 448.282i 0.538531 0.637671i
\(704\) 0 0
\(705\) 2.71303i 0.00384826i
\(706\) 0 0
\(707\) 794.388i 1.12360i
\(708\) 0 0
\(709\) 893.900i 1.26079i −0.776275 0.630395i \(-0.782893\pi\)
0.776275 0.630395i \(-0.217107\pi\)
\(710\) 0 0
\(711\) 1364.97i 1.91978i
\(712\) 0 0
\(713\) 1532.74i 2.14970i
\(714\) 0 0
\(715\) 13.7848 0.0192794
\(716\) 0 0
\(717\) 1793.82 2.50184
\(718\) 0 0
\(719\) 735.713 1.02325 0.511623 0.859210i \(-0.329045\pi\)
0.511623 + 0.859210i \(0.329045\pi\)
\(720\) 0 0
\(721\) 1103.37i 1.53033i
\(722\) 0 0
\(723\) 1379.77i 1.90840i
\(724\) 0 0
\(725\) 1289.37 1.77844
\(726\) 0 0
\(727\) 1366.64 1.87984 0.939919 0.341396i \(-0.110900\pi\)
0.939919 + 0.341396i \(0.110900\pi\)
\(728\) 0 0
\(729\) −1109.61 −1.52210
\(730\) 0 0
\(731\) 29.8050i 0.0407729i
\(732\) 0 0
\(733\) 231.571i 0.315923i 0.987445 + 0.157961i \(0.0504921\pi\)
−0.987445 + 0.157961i \(0.949508\pi\)
\(734\) 0 0
\(735\) 24.5644i 0.0334210i
\(736\) 0 0
\(737\) 354.069i 0.480420i
\(738\) 0 0
\(739\) 1118.87i 1.51404i 0.653394 + 0.757018i \(0.273345\pi\)
−0.653394 + 0.757018i \(0.726655\pi\)
\(740\) 0 0
\(741\) −151.934 + 179.904i −0.205040 + 0.242786i
\(742\) 0 0
\(743\) 541.335i 0.728580i −0.931286 0.364290i \(-0.881312\pi\)
0.931286 0.364290i \(-0.118688\pi\)
\(744\) 0 0
\(745\) 66.0466 0.0886531
\(746\) 0 0
\(747\) 792.617i 1.06107i
\(748\) 0 0
\(749\) 1583.03 2.11353
\(750\) 0 0
\(751\) 567.078i 0.755098i −0.925990 0.377549i \(-0.876767\pi\)
0.925990 0.377549i \(-0.123233\pi\)
\(752\) 0 0
\(753\) 553.864i 0.735543i
\(754\) 0 0
\(755\) 42.3058 0.0560341
\(756\) 0 0
\(757\) 360.370i 0.476050i 0.971259 + 0.238025i \(0.0765000\pi\)
−0.971259 + 0.238025i \(0.923500\pi\)
\(758\) 0 0
\(759\) 2077.08i 2.73660i
\(760\) 0 0
\(761\) 40.9265 0.0537798 0.0268899 0.999638i \(-0.491440\pi\)
0.0268899 + 0.999638i \(0.491440\pi\)
\(762\) 0 0
\(763\) 94.0615 0.123279
\(764\) 0 0
\(765\) 93.0276i 0.121605i
\(766\) 0 0
\(767\) 37.4994 0.0488910
\(768\) 0 0
\(769\) −170.819 −0.222132 −0.111066 0.993813i \(-0.535426\pi\)
−0.111066 + 0.993813i \(0.535426\pi\)
\(770\) 0 0
\(771\) 7.45433i 0.00966840i
\(772\) 0 0
\(773\) −1027.67 −1.32945 −0.664725 0.747088i \(-0.731451\pi\)
−0.664725 + 0.747088i \(0.731451\pi\)
\(774\) 0 0
\(775\) 1328.03i 1.71359i
\(776\) 0 0
\(777\) −1146.68 −1.47578
\(778\) 0 0
\(779\) 595.457 + 502.881i 0.764386 + 0.645546i
\(780\) 0 0
\(781\) 1401.32 1.79427
\(782\) 0 0
\(783\) 720.278 0.919895
\(784\) 0 0
\(785\) −9.66554 −0.0123128
\(786\) 0 0
\(787\) −1370.72 −1.74171 −0.870853 0.491544i \(-0.836433\pi\)
−0.870853 + 0.491544i \(0.836433\pi\)
\(788\) 0 0
\(789\) −623.985 −0.790856
\(790\) 0 0
\(791\) 792.006i 1.00127i
\(792\) 0 0
\(793\) 261.619i 0.329910i
\(794\) 0 0
\(795\) 15.6586i 0.0196963i
\(796\) 0 0
\(797\) 507.322 0.636539 0.318269 0.948000i \(-0.396898\pi\)
0.318269 + 0.948000i \(0.396898\pi\)
\(798\) 0 0
\(799\) −43.6746 −0.0546616
\(800\) 0 0
\(801\) 1448.36i 1.80819i
\(802\) 0 0
\(803\) 540.922i 0.673627i
\(804\) 0 0
\(805\) 75.2813i 0.0935172i
\(806\) 0 0
\(807\) 1.77604 0.00220079
\(808\) 0 0
\(809\) 808.551 0.999445 0.499722 0.866186i \(-0.333435\pi\)
0.499722 + 0.866186i \(0.333435\pi\)
\(810\) 0 0
\(811\) −96.4414 −0.118917 −0.0594583 0.998231i \(-0.518937\pi\)
−0.0594583 + 0.998231i \(0.518937\pi\)
\(812\) 0 0
\(813\) −1978.92 −2.43410
\(814\) 0 0
\(815\) 86.1721 0.105733
\(816\) 0 0
\(817\) −18.1091 15.2936i −0.0221653 0.0187193i
\(818\) 0 0
\(819\) 263.267 0.321450
\(820\) 0 0
\(821\) 1175.04i 1.43123i 0.698496 + 0.715614i \(0.253853\pi\)
−0.698496 + 0.715614i \(0.746147\pi\)
\(822\) 0 0
\(823\) 486.014 0.590539 0.295270 0.955414i \(-0.404591\pi\)
0.295270 + 0.955414i \(0.404591\pi\)
\(824\) 0 0
\(825\) 1799.67i 2.18142i
\(826\) 0 0
\(827\) 706.002 0.853690 0.426845 0.904325i \(-0.359625\pi\)
0.426845 + 0.904325i \(0.359625\pi\)
\(828\) 0 0
\(829\) 264.350 0.318878 0.159439 0.987208i \(-0.449032\pi\)
0.159439 + 0.987208i \(0.449032\pi\)
\(830\) 0 0
\(831\) 658.640i 0.792588i
\(832\) 0 0
\(833\) −395.441 −0.474719
\(834\) 0 0
\(835\) −5.62154 −0.00673238
\(836\) 0 0
\(837\) 741.876i 0.886351i
\(838\) 0 0
\(839\) 350.939i 0.418283i 0.977885 + 0.209141i \(0.0670669\pi\)
−0.977885 + 0.209141i \(0.932933\pi\)
\(840\) 0 0
\(841\) 1841.37 2.18950
\(842\) 0 0
\(843\) 904.437i 1.07288i
\(844\) 0 0
\(845\) 52.3263i 0.0619247i
\(846\) 0 0
\(847\) −1032.00 −1.21841
\(848\) 0 0
\(849\) 215.889i 0.254287i
\(850\) 0 0
\(851\) −887.325 −1.04268
\(852\) 0 0
\(853\) 918.300i 1.07655i −0.842768 0.538277i \(-0.819075\pi\)
0.842768 0.538277i \(-0.180925\pi\)
\(854\) 0 0
\(855\) 56.5222 + 47.7346i 0.0661078 + 0.0558300i
\(856\) 0 0
\(857\) 755.091i 0.881087i 0.897731 + 0.440543i \(0.145214\pi\)
−0.897731 + 0.440543i \(0.854786\pi\)
\(858\) 0 0
\(859\) 893.583i 1.04026i 0.854087 + 0.520130i \(0.174116\pi\)
−0.854087 + 0.520130i \(0.825884\pi\)
\(860\) 0 0
\(861\) 1523.15i 1.76904i
\(862\) 0 0
\(863\) 610.340i 0.707230i −0.935391 0.353615i \(-0.884952\pi\)
0.935391 0.353615i \(-0.115048\pi\)
\(864\) 0 0
\(865\) 15.9475i 0.0184364i
\(866\) 0 0
\(867\) 1292.33 1.49058
\(868\) 0 0
\(869\) 1788.13 2.05769
\(870\) 0 0
\(871\) 60.7026 0.0696930
\(872\) 0 0
\(873\) 1062.38i 1.21694i
\(874\) 0 0
\(875\) 130.728i 0.149404i
\(876\) 0 0
\(877\) −103.356 −0.117852 −0.0589259 0.998262i \(-0.518768\pi\)
−0.0589259 + 0.998262i \(0.518768\pi\)
\(878\) 0 0
\(879\) 2497.07 2.84081
\(880\) 0 0
\(881\) −875.630 −0.993904 −0.496952 0.867778i \(-0.665548\pi\)
−0.496952 + 0.867778i \(0.665548\pi\)
\(882\) 0 0
\(883\) 446.254i 0.505384i −0.967547 0.252692i \(-0.918684\pi\)
0.967547 0.252692i \(-0.0813159\pi\)
\(884\) 0 0
\(885\) 20.5938i 0.0232699i
\(886\) 0 0
\(887\) 680.106i 0.766748i −0.923593 0.383374i \(-0.874762\pi\)
0.923593 0.383374i \(-0.125238\pi\)
\(888\) 0 0
\(889\) 1541.47i 1.73394i
\(890\) 0 0
\(891\) 701.630i 0.787464i
\(892\) 0 0
\(893\) −22.4105 + 26.5361i −0.0250957 + 0.0297156i
\(894\) 0 0
\(895\) 18.3613i 0.0205154i
\(896\) 0 0
\(897\) 356.100 0.396990
\(898\) 0 0
\(899\) 2762.80i 3.07319i
\(900\) 0 0
\(901\) 252.073 0.279771
\(902\) 0 0
\(903\) 46.3221i 0.0512980i
\(904\) 0 0
\(905\) 74.5747i 0.0824030i
\(906\) 0 0
\(907\) 1241.68 1.36899 0.684496 0.729016i \(-0.260022\pi\)
0.684496 + 0.729016i \(0.260022\pi\)
\(908\) 0 0
\(909\) 1180.58i 1.29877i
\(910\) 0 0
\(911\) 673.760i 0.739583i −0.929115 0.369791i \(-0.879429\pi\)
0.929115 0.369791i \(-0.120571\pi\)
\(912\) 0 0
\(913\) −1038.34 −1.13728
\(914\) 0 0
\(915\) −143.675 −0.157022
\(916\) 0 0
\(917\) 41.6528i 0.0454229i
\(918\) 0 0
\(919\) −1399.96 −1.52335 −0.761677 0.647957i \(-0.775624\pi\)
−0.761677 + 0.647957i \(0.775624\pi\)
\(920\) 0 0
\(921\) −367.294 −0.398799
\(922\) 0 0
\(923\) 240.247i 0.260289i
\(924\) 0 0
\(925\) −768.817 −0.831153
\(926\) 0 0
\(927\) 1639.77i 1.76890i
\(928\) 0 0
\(929\) 479.876 0.516551 0.258275 0.966071i \(-0.416846\pi\)
0.258275 + 0.966071i \(0.416846\pi\)
\(930\) 0 0
\(931\) −202.910 + 240.264i −0.217949 + 0.258071i
\(932\) 0 0
\(933\) −684.457 −0.733609
\(934\) 0 0
\(935\) −121.868 −0.130340
\(936\) 0 0
\(937\) −482.124 −0.514540 −0.257270 0.966340i \(-0.582823\pi\)
−0.257270 + 0.966340i \(0.582823\pi\)
\(938\) 0 0
\(939\) 477.637 0.508665
\(940\) 0 0
\(941\) −1233.93 −1.31129 −0.655647 0.755068i \(-0.727604\pi\)
−0.655647 + 0.755068i \(0.727604\pi\)
\(942\) 0 0
\(943\) 1178.64i 1.24988i
\(944\) 0 0
\(945\) 36.4377i 0.0385584i
\(946\) 0 0
\(947\) 488.859i 0.516219i 0.966116 + 0.258109i \(0.0830995\pi\)
−0.966116 + 0.258109i \(0.916901\pi\)
\(948\) 0 0
\(949\) −92.7371 −0.0977209
\(950\) 0 0
\(951\) −150.148 −0.157885
\(952\) 0 0
\(953\) 1830.89i 1.92118i −0.277965 0.960591i \(-0.589660\pi\)
0.277965 0.960591i \(-0.410340\pi\)
\(954\) 0 0
\(955\) 26.1361i 0.0273676i
\(956\) 0 0
\(957\) 3744.00i 3.91223i
\(958\) 0 0
\(959\) 407.348 0.424764
\(960\) 0 0
\(961\) −1884.65 −1.96113
\(962\) 0 0
\(963\) 2352.62 2.44302
\(964\) 0 0
\(965\) 36.7796 0.0381135
\(966\) 0 0
\(967\) −529.813 −0.547893 −0.273947 0.961745i \(-0.588329\pi\)
−0.273947 + 0.961745i \(0.588329\pi\)
\(968\) 0 0
\(969\) 1343.21 1590.49i 1.38618 1.64137i
\(970\) 0 0
\(971\) 1335.82 1.37571 0.687856 0.725847i \(-0.258552\pi\)
0.687856 + 0.725847i \(0.258552\pi\)
\(972\) 0 0
\(973\) 924.670i 0.950328i
\(974\) 0 0
\(975\) 308.541 0.316452
\(976\) 0 0
\(977\) 803.318i 0.822229i 0.911584 + 0.411115i \(0.134860\pi\)
−0.911584 + 0.411115i \(0.865140\pi\)
\(978\) 0 0
\(979\) −1897.38 −1.93808
\(980\) 0 0
\(981\) 139.790 0.142497
\(982\) 0 0
\(983\) 784.639i 0.798209i −0.916906 0.399104i \(-0.869321\pi\)
0.916906 0.399104i \(-0.130679\pi\)
\(984\) 0 0
\(985\) 6.45123 0.00654947
\(986\) 0 0
\(987\) 67.8778 0.0687718
\(988\) 0 0
\(989\) 35.8449i 0.0362436i
\(990\) 0 0
\(991\) 1505.00i 1.51867i 0.650699 + 0.759336i \(0.274476\pi\)
−0.650699 + 0.759336i \(0.725524\pi\)
\(992\) 0 0
\(993\) 786.747 0.792293
\(994\) 0 0
\(995\) 4.00660i 0.00402674i
\(996\) 0 0
\(997\) 1376.43i 1.38057i 0.723539 + 0.690284i \(0.242514\pi\)
−0.723539 + 0.690284i \(0.757486\pi\)
\(998\) 0 0
\(999\) −429.484 −0.429914
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 1216.3.g.e.417.37 yes 48
4.3 odd 2 inner 1216.3.g.e.417.9 48
8.3 odd 2 inner 1216.3.g.e.417.39 yes 48
8.5 even 2 inner 1216.3.g.e.417.11 yes 48
19.18 odd 2 inner 1216.3.g.e.417.10 yes 48
76.75 even 2 inner 1216.3.g.e.417.38 yes 48
152.37 odd 2 inner 1216.3.g.e.417.40 yes 48
152.75 even 2 inner 1216.3.g.e.417.12 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.3.g.e.417.9 48 4.3 odd 2 inner
1216.3.g.e.417.10 yes 48 19.18 odd 2 inner
1216.3.g.e.417.11 yes 48 8.5 even 2 inner
1216.3.g.e.417.12 yes 48 152.75 even 2 inner
1216.3.g.e.417.37 yes 48 1.1 even 1 trivial
1216.3.g.e.417.38 yes 48 76.75 even 2 inner
1216.3.g.e.417.39 yes 48 8.3 odd 2 inner
1216.3.g.e.417.40 yes 48 152.37 odd 2 inner