Properties

Label 1620.4.a.j.1.2
Level $1620$
Weight $4$
Character 1620.1
Self dual yes
Analytic conductor $95.583$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,4,Mod(1,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.a (trivial)

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: yes
Analytic conductor: \(95.5830942093\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 151x^{4} - 212x^{3} + 4412x^{2} + 8656x - 19148 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-6.01416\) of defining polynomial
Character \(\chi\) \(=\) 1620.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.00000 q^{5} -10.4813 q^{7} +O(q^{10})\) \(q+5.00000 q^{5} -10.4813 q^{7} -33.3429 q^{11} -23.9383 q^{13} +72.5536 q^{17} -45.9922 q^{19} +163.189 q^{23} +25.0000 q^{25} +30.9773 q^{29} +304.799 q^{31} -52.4065 q^{35} +150.155 q^{37} -409.065 q^{41} -249.174 q^{43} -155.899 q^{47} -233.142 q^{49} +263.062 q^{53} -166.714 q^{55} -245.829 q^{59} -466.473 q^{61} -119.691 q^{65} -177.016 q^{67} +45.2138 q^{71} -949.331 q^{73} +349.476 q^{77} +496.917 q^{79} +354.636 q^{83} +362.768 q^{85} -1193.20 q^{89} +250.904 q^{91} -229.961 q^{95} -408.798 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 30 q^{5} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 30 q^{5} + 12 q^{7} - 84 q^{13} - 12 q^{17} - 114 q^{19} - 30 q^{23} + 150 q^{25} - 168 q^{29} - 324 q^{31} + 60 q^{35} - 492 q^{37} - 312 q^{41} - 156 q^{43} - 462 q^{47} - 588 q^{49} - 1014 q^{53} - 1008 q^{59} + 36 q^{61} - 420 q^{65} + 144 q^{67} - 1212 q^{71} - 900 q^{73} - 672 q^{77} - 936 q^{79} - 288 q^{83} - 60 q^{85} + 120 q^{89} + 2286 q^{91} - 570 q^{95} - 1188 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −10.4813 −0.565937 −0.282968 0.959129i \(-0.591319\pi\)
−0.282968 + 0.959129i \(0.591319\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −33.3429 −0.913932 −0.456966 0.889484i \(-0.651064\pi\)
−0.456966 + 0.889484i \(0.651064\pi\)
\(12\) 0 0
\(13\) −23.9383 −0.510714 −0.255357 0.966847i \(-0.582193\pi\)
−0.255357 + 0.966847i \(0.582193\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 72.5536 1.03511 0.517554 0.855651i \(-0.326843\pi\)
0.517554 + 0.855651i \(0.326843\pi\)
\(18\) 0 0
\(19\) −45.9922 −0.555333 −0.277667 0.960678i \(-0.589561\pi\)
−0.277667 + 0.960678i \(0.589561\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 163.189 1.47944 0.739721 0.672914i \(-0.234957\pi\)
0.739721 + 0.672914i \(0.234957\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 30.9773 0.198357 0.0991783 0.995070i \(-0.468379\pi\)
0.0991783 + 0.995070i \(0.468379\pi\)
\(30\) 0 0
\(31\) 304.799 1.76592 0.882960 0.469448i \(-0.155547\pi\)
0.882960 + 0.469448i \(0.155547\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −52.4065 −0.253095
\(36\) 0 0
\(37\) 150.155 0.667171 0.333586 0.942720i \(-0.391741\pi\)
0.333586 + 0.942720i \(0.391741\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −409.065 −1.55818 −0.779088 0.626915i \(-0.784317\pi\)
−0.779088 + 0.626915i \(0.784317\pi\)
\(42\) 0 0
\(43\) −249.174 −0.883689 −0.441845 0.897092i \(-0.645676\pi\)
−0.441845 + 0.897092i \(0.645676\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −155.899 −0.483835 −0.241917 0.970297i \(-0.577776\pi\)
−0.241917 + 0.970297i \(0.577776\pi\)
\(48\) 0 0
\(49\) −233.142 −0.679716
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 263.062 0.681780 0.340890 0.940103i \(-0.389272\pi\)
0.340890 + 0.940103i \(0.389272\pi\)
\(54\) 0 0
\(55\) −166.714 −0.408723
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −245.829 −0.542445 −0.271223 0.962517i \(-0.587428\pi\)
−0.271223 + 0.962517i \(0.587428\pi\)
\(60\) 0 0
\(61\) −466.473 −0.979110 −0.489555 0.871972i \(-0.662841\pi\)
−0.489555 + 0.871972i \(0.662841\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −119.691 −0.228398
\(66\) 0 0
\(67\) −177.016 −0.322775 −0.161387 0.986891i \(-0.551597\pi\)
−0.161387 + 0.986891i \(0.551597\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 45.2138 0.0755759 0.0377879 0.999286i \(-0.487969\pi\)
0.0377879 + 0.999286i \(0.487969\pi\)
\(72\) 0 0
\(73\) −949.331 −1.52207 −0.761033 0.648713i \(-0.775307\pi\)
−0.761033 + 0.648713i \(0.775307\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 349.476 0.517228
\(78\) 0 0
\(79\) 496.917 0.707690 0.353845 0.935304i \(-0.384874\pi\)
0.353845 + 0.935304i \(0.384874\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 354.636 0.468992 0.234496 0.972117i \(-0.424656\pi\)
0.234496 + 0.972117i \(0.424656\pi\)
\(84\) 0 0
\(85\) 362.768 0.462914
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1193.20 −1.42111 −0.710554 0.703643i \(-0.751556\pi\)
−0.710554 + 0.703643i \(0.751556\pi\)
\(90\) 0 0
\(91\) 250.904 0.289032
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −229.961 −0.248353
\(96\) 0 0
\(97\) −408.798 −0.427908 −0.213954 0.976844i \(-0.568634\pi\)
−0.213954 + 0.976844i \(0.568634\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −890.427 −0.877235 −0.438618 0.898674i \(-0.644532\pi\)
−0.438618 + 0.898674i \(0.644532\pi\)
\(102\) 0 0
\(103\) 1531.17 1.46476 0.732381 0.680895i \(-0.238409\pi\)
0.732381 + 0.680895i \(0.238409\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −989.489 −0.893996 −0.446998 0.894535i \(-0.647507\pi\)
−0.446998 + 0.894535i \(0.647507\pi\)
\(108\) 0 0
\(109\) −296.637 −0.260667 −0.130334 0.991470i \(-0.541605\pi\)
−0.130334 + 0.991470i \(0.541605\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1036.15 −0.862588 −0.431294 0.902211i \(-0.641943\pi\)
−0.431294 + 0.902211i \(0.641943\pi\)
\(114\) 0 0
\(115\) 815.943 0.661627
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −760.456 −0.585806
\(120\) 0 0
\(121\) −219.254 −0.164729
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1040.10 −0.726726 −0.363363 0.931648i \(-0.618372\pi\)
−0.363363 + 0.931648i \(0.618372\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −316.994 −0.211419 −0.105709 0.994397i \(-0.533711\pi\)
−0.105709 + 0.994397i \(0.533711\pi\)
\(132\) 0 0
\(133\) 482.058 0.314283
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1701.23 −1.06092 −0.530459 0.847711i \(-0.677980\pi\)
−0.530459 + 0.847711i \(0.677980\pi\)
\(138\) 0 0
\(139\) 2756.36 1.68196 0.840978 0.541070i \(-0.181981\pi\)
0.840978 + 0.541070i \(0.181981\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 798.170 0.466757
\(144\) 0 0
\(145\) 154.887 0.0887078
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 328.707 0.180730 0.0903648 0.995909i \(-0.471197\pi\)
0.0903648 + 0.995909i \(0.471197\pi\)
\(150\) 0 0
\(151\) −316.280 −0.170454 −0.0852269 0.996362i \(-0.527162\pi\)
−0.0852269 + 0.996362i \(0.527162\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1524.00 0.789744
\(156\) 0 0
\(157\) 3521.19 1.78995 0.894973 0.446120i \(-0.147194\pi\)
0.894973 + 0.446120i \(0.147194\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1710.43 −0.837271
\(162\) 0 0
\(163\) −1988.65 −0.955602 −0.477801 0.878468i \(-0.658566\pi\)
−0.477801 + 0.878468i \(0.658566\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2632.95 −1.22002 −0.610011 0.792393i \(-0.708835\pi\)
−0.610011 + 0.792393i \(0.708835\pi\)
\(168\) 0 0
\(169\) −1623.96 −0.739172
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1683.23 −0.739732 −0.369866 0.929085i \(-0.620596\pi\)
−0.369866 + 0.929085i \(0.620596\pi\)
\(174\) 0 0
\(175\) −262.032 −0.113187
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2341.61 −0.977767 −0.488884 0.872349i \(-0.662596\pi\)
−0.488884 + 0.872349i \(0.662596\pi\)
\(180\) 0 0
\(181\) 469.689 0.192882 0.0964412 0.995339i \(-0.469254\pi\)
0.0964412 + 0.995339i \(0.469254\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 750.775 0.298368
\(186\) 0 0
\(187\) −2419.14 −0.946018
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1276.72 −0.483666 −0.241833 0.970318i \(-0.577749\pi\)
−0.241833 + 0.970318i \(0.577749\pi\)
\(192\) 0 0
\(193\) −1106.88 −0.412822 −0.206411 0.978465i \(-0.566178\pi\)
−0.206411 + 0.978465i \(0.566178\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1945.23 −0.703513 −0.351757 0.936092i \(-0.614416\pi\)
−0.351757 + 0.936092i \(0.614416\pi\)
\(198\) 0 0
\(199\) −426.899 −0.152071 −0.0760353 0.997105i \(-0.524226\pi\)
−0.0760353 + 0.997105i \(0.524226\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −324.682 −0.112257
\(204\) 0 0
\(205\) −2045.32 −0.696837
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1533.51 0.507537
\(210\) 0 0
\(211\) 4343.21 1.41706 0.708529 0.705682i \(-0.249359\pi\)
0.708529 + 0.705682i \(0.249359\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1245.87 −0.395198
\(216\) 0 0
\(217\) −3194.69 −0.999399
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1736.81 −0.528644
\(222\) 0 0
\(223\) −2854.90 −0.857302 −0.428651 0.903470i \(-0.641011\pi\)
−0.428651 + 0.903470i \(0.641011\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 377.834 0.110475 0.0552373 0.998473i \(-0.482408\pi\)
0.0552373 + 0.998473i \(0.482408\pi\)
\(228\) 0 0
\(229\) 6143.68 1.77286 0.886432 0.462858i \(-0.153176\pi\)
0.886432 + 0.462858i \(0.153176\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4323.86 −1.21573 −0.607866 0.794040i \(-0.707974\pi\)
−0.607866 + 0.794040i \(0.707974\pi\)
\(234\) 0 0
\(235\) −779.496 −0.216378
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1827.21 −0.494528 −0.247264 0.968948i \(-0.579531\pi\)
−0.247264 + 0.968948i \(0.579531\pi\)
\(240\) 0 0
\(241\) 447.748 0.119676 0.0598381 0.998208i \(-0.480942\pi\)
0.0598381 + 0.998208i \(0.480942\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1165.71 −0.303978
\(246\) 0 0
\(247\) 1100.97 0.283616
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4046.98 1.01770 0.508851 0.860855i \(-0.330071\pi\)
0.508851 + 0.860855i \(0.330071\pi\)
\(252\) 0 0
\(253\) −5441.17 −1.35211
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2283.13 −0.554155 −0.277077 0.960848i \(-0.589366\pi\)
−0.277077 + 0.960848i \(0.589366\pi\)
\(258\) 0 0
\(259\) −1573.82 −0.377577
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3423.31 −0.802625 −0.401312 0.915941i \(-0.631446\pi\)
−0.401312 + 0.915941i \(0.631446\pi\)
\(264\) 0 0
\(265\) 1315.31 0.304901
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2257.48 0.511676 0.255838 0.966720i \(-0.417649\pi\)
0.255838 + 0.966720i \(0.417649\pi\)
\(270\) 0 0
\(271\) −8085.45 −1.81239 −0.906193 0.422865i \(-0.861024\pi\)
−0.906193 + 0.422865i \(0.861024\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −833.571 −0.182786
\(276\) 0 0
\(277\) 4840.74 1.05001 0.525004 0.851100i \(-0.324064\pi\)
0.525004 + 0.851100i \(0.324064\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6555.12 −1.39162 −0.695811 0.718225i \(-0.744955\pi\)
−0.695811 + 0.718225i \(0.744955\pi\)
\(282\) 0 0
\(283\) 4103.01 0.861832 0.430916 0.902392i \(-0.358191\pi\)
0.430916 + 0.902392i \(0.358191\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4287.53 0.881829
\(288\) 0 0
\(289\) 351.026 0.0714485
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8500.10 −1.69482 −0.847408 0.530942i \(-0.821838\pi\)
−0.847408 + 0.530942i \(0.821838\pi\)
\(294\) 0 0
\(295\) −1229.15 −0.242589
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3906.45 −0.755571
\(300\) 0 0
\(301\) 2611.66 0.500112
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2332.36 −0.437871
\(306\) 0 0
\(307\) 5386.01 1.00129 0.500645 0.865653i \(-0.333096\pi\)
0.500645 + 0.865653i \(0.333096\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1489.18 0.271522 0.135761 0.990742i \(-0.456652\pi\)
0.135761 + 0.990742i \(0.456652\pi\)
\(312\) 0 0
\(313\) −4425.29 −0.799144 −0.399572 0.916702i \(-0.630841\pi\)
−0.399572 + 0.916702i \(0.630841\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3235.94 −0.573340 −0.286670 0.958029i \(-0.592548\pi\)
−0.286670 + 0.958029i \(0.592548\pi\)
\(318\) 0 0
\(319\) −1032.87 −0.181284
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3336.90 −0.574830
\(324\) 0 0
\(325\) −598.456 −0.102143
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1634.03 0.273820
\(330\) 0 0
\(331\) −871.620 −0.144739 −0.0723694 0.997378i \(-0.523056\pi\)
−0.0723694 + 0.997378i \(0.523056\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −885.079 −0.144349
\(336\) 0 0
\(337\) 3114.15 0.503378 0.251689 0.967808i \(-0.419014\pi\)
0.251689 + 0.967808i \(0.419014\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10162.9 −1.61393
\(342\) 0 0
\(343\) 6038.72 0.950613
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7418.63 1.14770 0.573851 0.818960i \(-0.305449\pi\)
0.573851 + 0.818960i \(0.305449\pi\)
\(348\) 0 0
\(349\) 6016.68 0.922824 0.461412 0.887186i \(-0.347343\pi\)
0.461412 + 0.887186i \(0.347343\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5745.76 0.866334 0.433167 0.901314i \(-0.357396\pi\)
0.433167 + 0.901314i \(0.357396\pi\)
\(354\) 0 0
\(355\) 226.069 0.0337986
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5133.25 −0.754659 −0.377329 0.926079i \(-0.623158\pi\)
−0.377329 + 0.926079i \(0.623158\pi\)
\(360\) 0 0
\(361\) −4743.72 −0.691605
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4746.66 −0.680689
\(366\) 0 0
\(367\) −1679.48 −0.238878 −0.119439 0.992842i \(-0.538110\pi\)
−0.119439 + 0.992842i \(0.538110\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2757.23 −0.385844
\(372\) 0 0
\(373\) −4910.54 −0.681658 −0.340829 0.940125i \(-0.610708\pi\)
−0.340829 + 0.940125i \(0.610708\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −741.543 −0.101303
\(378\) 0 0
\(379\) −9065.83 −1.22871 −0.614354 0.789030i \(-0.710583\pi\)
−0.614354 + 0.789030i \(0.710583\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8273.28 −1.10377 −0.551886 0.833919i \(-0.686092\pi\)
−0.551886 + 0.833919i \(0.686092\pi\)
\(384\) 0 0
\(385\) 1747.38 0.231311
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3008.56 −0.392134 −0.196067 0.980591i \(-0.562817\pi\)
−0.196067 + 0.980591i \(0.562817\pi\)
\(390\) 0 0
\(391\) 11839.9 1.53138
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2484.59 0.316489
\(396\) 0 0
\(397\) −11495.3 −1.45324 −0.726618 0.687042i \(-0.758909\pi\)
−0.726618 + 0.687042i \(0.758909\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10883.5 1.35535 0.677674 0.735362i \(-0.262988\pi\)
0.677674 + 0.735362i \(0.262988\pi\)
\(402\) 0 0
\(403\) −7296.36 −0.901880
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5006.60 −0.609749
\(408\) 0 0
\(409\) −6255.00 −0.756210 −0.378105 0.925763i \(-0.623424\pi\)
−0.378105 + 0.925763i \(0.623424\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2576.61 0.306990
\(414\) 0 0
\(415\) 1773.18 0.209740
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −10131.2 −1.18125 −0.590623 0.806947i \(-0.701118\pi\)
−0.590623 + 0.806947i \(0.701118\pi\)
\(420\) 0 0
\(421\) 2585.12 0.299266 0.149633 0.988742i \(-0.452191\pi\)
0.149633 + 0.988742i \(0.452191\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1813.84 0.207022
\(426\) 0 0
\(427\) 4889.24 0.554114
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5184.08 0.579370 0.289685 0.957122i \(-0.406449\pi\)
0.289685 + 0.957122i \(0.406449\pi\)
\(432\) 0 0
\(433\) 4384.88 0.486660 0.243330 0.969944i \(-0.421760\pi\)
0.243330 + 0.969944i \(0.421760\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7505.40 −0.821583
\(438\) 0 0
\(439\) −8649.17 −0.940324 −0.470162 0.882580i \(-0.655805\pi\)
−0.470162 + 0.882580i \(0.655805\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17120.9 1.83620 0.918100 0.396349i \(-0.129723\pi\)
0.918100 + 0.396349i \(0.129723\pi\)
\(444\) 0 0
\(445\) −5965.98 −0.635539
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2688.80 0.282611 0.141306 0.989966i \(-0.454870\pi\)
0.141306 + 0.989966i \(0.454870\pi\)
\(450\) 0 0
\(451\) 13639.4 1.42407
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1254.52 0.129259
\(456\) 0 0
\(457\) 1679.09 0.171870 0.0859350 0.996301i \(-0.472612\pi\)
0.0859350 + 0.996301i \(0.472612\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2403.50 0.242825 0.121412 0.992602i \(-0.461258\pi\)
0.121412 + 0.992602i \(0.461258\pi\)
\(462\) 0 0
\(463\) 3048.56 0.306002 0.153001 0.988226i \(-0.451106\pi\)
0.153001 + 0.988226i \(0.451106\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6256.23 −0.619923 −0.309961 0.950749i \(-0.600316\pi\)
−0.309961 + 0.950749i \(0.600316\pi\)
\(468\) 0 0
\(469\) 1855.35 0.182670
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8308.16 0.807632
\(474\) 0 0
\(475\) −1149.80 −0.111067
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4633.25 0.441960 0.220980 0.975278i \(-0.429075\pi\)
0.220980 + 0.975278i \(0.429075\pi\)
\(480\) 0 0
\(481\) −3594.45 −0.340733
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2043.99 −0.191366
\(486\) 0 0
\(487\) 9823.94 0.914097 0.457049 0.889442i \(-0.348907\pi\)
0.457049 + 0.889442i \(0.348907\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2269.91 0.208634 0.104317 0.994544i \(-0.466734\pi\)
0.104317 + 0.994544i \(0.466734\pi\)
\(492\) 0 0
\(493\) 2247.52 0.205321
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −473.899 −0.0427712
\(498\) 0 0
\(499\) 4652.31 0.417367 0.208684 0.977983i \(-0.433082\pi\)
0.208684 + 0.977983i \(0.433082\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −13783.4 −1.22181 −0.610906 0.791703i \(-0.709195\pi\)
−0.610906 + 0.791703i \(0.709195\pi\)
\(504\) 0 0
\(505\) −4452.13 −0.392312
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 649.278 0.0565397 0.0282699 0.999600i \(-0.491000\pi\)
0.0282699 + 0.999600i \(0.491000\pi\)
\(510\) 0 0
\(511\) 9950.22 0.861393
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7655.84 0.655062
\(516\) 0 0
\(517\) 5198.13 0.442192
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21010.9 1.76680 0.883402 0.468616i \(-0.155247\pi\)
0.883402 + 0.468616i \(0.155247\pi\)
\(522\) 0 0
\(523\) −16162.4 −1.35130 −0.675652 0.737221i \(-0.736138\pi\)
−0.675652 + 0.737221i \(0.736138\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22114.3 1.82792
\(528\) 0 0
\(529\) 14463.5 1.18875
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9792.30 0.795781
\(534\) 0 0
\(535\) −4947.45 −0.399807
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 7773.64 0.621214
\(540\) 0 0
\(541\) 20812.9 1.65400 0.827002 0.562199i \(-0.190044\pi\)
0.827002 + 0.562199i \(0.190044\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1483.19 −0.116574
\(546\) 0 0
\(547\) −1000.02 −0.0781675 −0.0390838 0.999236i \(-0.512444\pi\)
−0.0390838 + 0.999236i \(0.512444\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1424.71 −0.110154
\(552\) 0 0
\(553\) −5208.33 −0.400508
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4720.42 0.359085 0.179543 0.983750i \(-0.442538\pi\)
0.179543 + 0.983750i \(0.442538\pi\)
\(558\) 0 0
\(559\) 5964.78 0.451312
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7863.47 0.588643 0.294321 0.955707i \(-0.404906\pi\)
0.294321 + 0.955707i \(0.404906\pi\)
\(564\) 0 0
\(565\) −5180.73 −0.385761
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −22555.2 −1.66180 −0.830900 0.556422i \(-0.812174\pi\)
−0.830900 + 0.556422i \(0.812174\pi\)
\(570\) 0 0
\(571\) −12656.8 −0.927619 −0.463810 0.885935i \(-0.653518\pi\)
−0.463810 + 0.885935i \(0.653518\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4079.71 0.295888
\(576\) 0 0
\(577\) 11720.0 0.845601 0.422801 0.906223i \(-0.361047\pi\)
0.422801 + 0.906223i \(0.361047\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3717.05 −0.265420
\(582\) 0 0
\(583\) −8771.24 −0.623100
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −949.761 −0.0667817 −0.0333908 0.999442i \(-0.510631\pi\)
−0.0333908 + 0.999442i \(0.510631\pi\)
\(588\) 0 0
\(589\) −14018.4 −0.980674
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −11575.2 −0.801580 −0.400790 0.916170i \(-0.631264\pi\)
−0.400790 + 0.916170i \(0.631264\pi\)
\(594\) 0 0
\(595\) −3802.28 −0.261980
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 27076.3 1.84692 0.923461 0.383693i \(-0.125348\pi\)
0.923461 + 0.383693i \(0.125348\pi\)
\(600\) 0 0
\(601\) 19863.3 1.34815 0.674077 0.738661i \(-0.264542\pi\)
0.674077 + 0.738661i \(0.264542\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1096.27 −0.0736689
\(606\) 0 0
\(607\) −15270.4 −1.02110 −0.510550 0.859848i \(-0.670558\pi\)
−0.510550 + 0.859848i \(0.670558\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3731.95 0.247101
\(612\) 0 0
\(613\) −13256.0 −0.873418 −0.436709 0.899603i \(-0.643856\pi\)
−0.436709 + 0.899603i \(0.643856\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −14348.4 −0.936214 −0.468107 0.883672i \(-0.655064\pi\)
−0.468107 + 0.883672i \(0.655064\pi\)
\(618\) 0 0
\(619\) −24004.8 −1.55870 −0.779348 0.626591i \(-0.784450\pi\)
−0.779348 + 0.626591i \(0.784450\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12506.2 0.804257
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10894.3 0.690594
\(630\) 0 0
\(631\) 28832.3 1.81901 0.909505 0.415693i \(-0.136461\pi\)
0.909505 + 0.415693i \(0.136461\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5200.52 −0.325002
\(636\) 0 0
\(637\) 5581.02 0.347140
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −25123.6 −1.54808 −0.774042 0.633134i \(-0.781768\pi\)
−0.774042 + 0.633134i \(0.781768\pi\)
\(642\) 0 0
\(643\) −31360.0 −1.92335 −0.961676 0.274187i \(-0.911591\pi\)
−0.961676 + 0.274187i \(0.911591\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7235.97 0.439684 0.219842 0.975536i \(-0.429446\pi\)
0.219842 + 0.975536i \(0.429446\pi\)
\(648\) 0 0
\(649\) 8196.66 0.495758
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −21525.6 −1.28999 −0.644995 0.764187i \(-0.723141\pi\)
−0.644995 + 0.764187i \(0.723141\pi\)
\(654\) 0 0
\(655\) −1584.97 −0.0945494
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 19452.1 1.14984 0.574920 0.818209i \(-0.305033\pi\)
0.574920 + 0.818209i \(0.305033\pi\)
\(660\) 0 0
\(661\) −716.749 −0.0421760 −0.0210880 0.999778i \(-0.506713\pi\)
−0.0210880 + 0.999778i \(0.506713\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2410.29 0.140552
\(666\) 0 0
\(667\) 5055.14 0.293457
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 15553.5 0.894840
\(672\) 0 0
\(673\) 22459.7 1.28641 0.643207 0.765692i \(-0.277603\pi\)
0.643207 + 0.765692i \(0.277603\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1671.06 −0.0948654 −0.0474327 0.998874i \(-0.515104\pi\)
−0.0474327 + 0.998874i \(0.515104\pi\)
\(678\) 0 0
\(679\) 4284.73 0.242169
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 22202.2 1.24384 0.621921 0.783080i \(-0.286353\pi\)
0.621921 + 0.783080i \(0.286353\pi\)
\(684\) 0 0
\(685\) −8506.14 −0.474457
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6297.24 −0.348194
\(690\) 0 0
\(691\) −3871.94 −0.213163 −0.106582 0.994304i \(-0.533991\pi\)
−0.106582 + 0.994304i \(0.533991\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13781.8 0.752193
\(696\) 0 0
\(697\) −29679.1 −1.61288
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −21918.3 −1.18095 −0.590474 0.807057i \(-0.701059\pi\)
−0.590474 + 0.807057i \(0.701059\pi\)
\(702\) 0 0
\(703\) −6905.96 −0.370502
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9332.83 0.496460
\(708\) 0 0
\(709\) 22333.8 1.18302 0.591512 0.806296i \(-0.298531\pi\)
0.591512 + 0.806296i \(0.298531\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 49739.7 2.61258
\(714\) 0 0
\(715\) 3990.85 0.208740
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 33133.5 1.71859 0.859297 0.511477i \(-0.170901\pi\)
0.859297 + 0.511477i \(0.170901\pi\)
\(720\) 0 0
\(721\) −16048.6 −0.828963
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 774.433 0.0396713
\(726\) 0 0
\(727\) 11182.1 0.570457 0.285228 0.958460i \(-0.407931\pi\)
0.285228 + 0.958460i \(0.407931\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −18078.5 −0.914714
\(732\) 0 0
\(733\) −32311.7 −1.62819 −0.814093 0.580735i \(-0.802765\pi\)
−0.814093 + 0.580735i \(0.802765\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5902.21 0.294994
\(738\) 0 0
\(739\) 16348.4 0.813784 0.406892 0.913476i \(-0.366613\pi\)
0.406892 + 0.913476i \(0.366613\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −13984.2 −0.690486 −0.345243 0.938513i \(-0.612204\pi\)
−0.345243 + 0.938513i \(0.612204\pi\)
\(744\) 0 0
\(745\) 1643.53 0.0808248
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 10371.1 0.505945
\(750\) 0 0
\(751\) 24688.8 1.19961 0.599805 0.800146i \(-0.295245\pi\)
0.599805 + 0.800146i \(0.295245\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1581.40 −0.0762293
\(756\) 0 0
\(757\) 21515.4 1.03301 0.516507 0.856283i \(-0.327232\pi\)
0.516507 + 0.856283i \(0.327232\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19156.0 0.912487 0.456244 0.889855i \(-0.349195\pi\)
0.456244 + 0.889855i \(0.349195\pi\)
\(762\) 0 0
\(763\) 3109.14 0.147521
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5884.73 0.277034
\(768\) 0 0
\(769\) −39460.0 −1.85041 −0.925203 0.379473i \(-0.876105\pi\)
−0.925203 + 0.379473i \(0.876105\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2246.03 0.104507 0.0522537 0.998634i \(-0.483360\pi\)
0.0522537 + 0.998634i \(0.483360\pi\)
\(774\) 0 0
\(775\) 7619.98 0.353184
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 18813.8 0.865307
\(780\) 0 0
\(781\) −1507.56 −0.0690712
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 17606.0 0.800488
\(786\) 0 0
\(787\) −5801.11 −0.262754 −0.131377 0.991332i \(-0.541940\pi\)
−0.131377 + 0.991332i \(0.541940\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 10860.2 0.488170
\(792\) 0 0
\(793\) 11166.5 0.500045
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 40399.1 1.79549 0.897747 0.440512i \(-0.145203\pi\)
0.897747 + 0.440512i \(0.145203\pi\)
\(798\) 0 0
\(799\) −11311.1 −0.500821
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 31653.4 1.39106
\(804\) 0 0
\(805\) −8552.14 −0.374439
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30986.8 −1.34665 −0.673324 0.739348i \(-0.735134\pi\)
−0.673324 + 0.739348i \(0.735134\pi\)
\(810\) 0 0
\(811\) 8658.08 0.374878 0.187439 0.982276i \(-0.439981\pi\)
0.187439 + 0.982276i \(0.439981\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −9943.25 −0.427358
\(816\) 0 0
\(817\) 11460.0 0.490742
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −30463.1 −1.29497 −0.647484 0.762079i \(-0.724179\pi\)
−0.647484 + 0.762079i \(0.724179\pi\)
\(822\) 0 0
\(823\) −10357.9 −0.438704 −0.219352 0.975646i \(-0.570394\pi\)
−0.219352 + 0.975646i \(0.570394\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 30108.4 1.26599 0.632993 0.774157i \(-0.281826\pi\)
0.632993 + 0.774157i \(0.281826\pi\)
\(828\) 0 0
\(829\) 6248.62 0.261789 0.130895 0.991396i \(-0.458215\pi\)
0.130895 + 0.991396i \(0.458215\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −16915.3 −0.703579
\(834\) 0 0
\(835\) −13164.7 −0.545610
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −10579.6 −0.435339 −0.217669 0.976023i \(-0.569845\pi\)
−0.217669 + 0.976023i \(0.569845\pi\)
\(840\) 0 0
\(841\) −23429.4 −0.960655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8119.80 −0.330568
\(846\) 0 0
\(847\) 2298.07 0.0932260
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 24503.6 0.987041
\(852\) 0 0
\(853\) −20228.7 −0.811980 −0.405990 0.913878i \(-0.633073\pi\)
−0.405990 + 0.913878i \(0.633073\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −43333.4 −1.72724 −0.863618 0.504146i \(-0.831807\pi\)
−0.863618 + 0.504146i \(0.831807\pi\)
\(858\) 0 0
\(859\) 16584.0 0.658718 0.329359 0.944205i \(-0.393167\pi\)
0.329359 + 0.944205i \(0.393167\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3281.90 0.129452 0.0647261 0.997903i \(-0.479383\pi\)
0.0647261 + 0.997903i \(0.479383\pi\)
\(864\) 0 0
\(865\) −8416.15 −0.330818
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −16568.6 −0.646781
\(870\) 0 0
\(871\) 4237.45 0.164846
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1310.16 −0.0506189
\(876\) 0 0
\(877\) 18214.8 0.701333 0.350666 0.936500i \(-0.385955\pi\)
0.350666 + 0.936500i \(0.385955\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 29408.8 1.12464 0.562320 0.826920i \(-0.309909\pi\)
0.562320 + 0.826920i \(0.309909\pi\)
\(882\) 0 0
\(883\) −7442.63 −0.283652 −0.141826 0.989892i \(-0.545297\pi\)
−0.141826 + 0.989892i \(0.545297\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 20306.7 0.768696 0.384348 0.923188i \(-0.374426\pi\)
0.384348 + 0.923188i \(0.374426\pi\)
\(888\) 0 0
\(889\) 10901.6 0.411281
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7170.14 0.268690
\(894\) 0 0
\(895\) −11708.1 −0.437271
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9441.86 0.350282
\(900\) 0 0
\(901\) 19086.1 0.705716
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2348.45 0.0862597
\(906\) 0 0
\(907\) 6400.36 0.234311 0.117156 0.993114i \(-0.462622\pi\)
0.117156 + 0.993114i \(0.462622\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −9303.31 −0.338345 −0.169173 0.985586i \(-0.554110\pi\)
−0.169173 + 0.985586i \(0.554110\pi\)
\(912\) 0 0
\(913\) −11824.6 −0.428627
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3322.51 0.119650
\(918\) 0 0
\(919\) 15110.9 0.542397 0.271198 0.962523i \(-0.412580\pi\)
0.271198 + 0.962523i \(0.412580\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1082.34 −0.0385976
\(924\) 0 0
\(925\) 3753.87 0.133434
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 48972.2 1.72952 0.864761 0.502184i \(-0.167470\pi\)
0.864761 + 0.502184i \(0.167470\pi\)
\(930\) 0 0
\(931\) 10722.7 0.377469
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12095.7 −0.423072
\(936\) 0 0
\(937\) 41808.1 1.45764 0.728821 0.684705i \(-0.240069\pi\)
0.728821 + 0.684705i \(0.240069\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 54568.7 1.89042 0.945212 0.326456i \(-0.105855\pi\)
0.945212 + 0.326456i \(0.105855\pi\)
\(942\) 0 0
\(943\) −66754.7 −2.30523
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 38799.9 1.33139 0.665695 0.746224i \(-0.268135\pi\)
0.665695 + 0.746224i \(0.268135\pi\)
\(948\) 0 0
\(949\) 22725.3 0.777340
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −38007.3 −1.29190 −0.645948 0.763382i \(-0.723538\pi\)
−0.645948 + 0.763382i \(0.723538\pi\)
\(954\) 0 0
\(955\) −6383.59 −0.216302
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 17831.1 0.600412
\(960\) 0 0
\(961\) 63111.5 2.11848
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −5534.38 −0.184620
\(966\) 0 0
\(967\) 42906.7 1.42687 0.713437 0.700719i \(-0.247137\pi\)
0.713437 + 0.700719i \(0.247137\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 38855.2 1.28416 0.642082 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(972\) 0 0
\(973\) −28890.3 −0.951880
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −48239.0 −1.57963 −0.789817 0.613342i \(-0.789825\pi\)
−0.789817 + 0.613342i \(0.789825\pi\)
\(978\) 0 0
\(979\) 39784.6 1.29880
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 43167.5 1.40064 0.700320 0.713829i \(-0.253041\pi\)
0.700320 + 0.713829i \(0.253041\pi\)
\(984\) 0 0
\(985\) −9726.16 −0.314621
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −40662.3 −1.30737
\(990\) 0 0
\(991\) 57081.7 1.82973 0.914863 0.403764i \(-0.132298\pi\)
0.914863 + 0.403764i \(0.132298\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2134.49 −0.0680080
\(996\) 0 0
\(997\) 47673.1 1.51436 0.757182 0.653203i \(-0.226575\pi\)
0.757182 + 0.653203i \(0.226575\pi\)
\(998\) 0 0
\(999\) 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 1620.4.a.j.1.2 yes 6
3.2 odd 2 1620.4.a.i.1.2 6
9.2 odd 6 1620.4.i.x.1081.5 12
9.4 even 3 1620.4.i.w.541.5 12
9.5 odd 6 1620.4.i.x.541.5 12
9.7 even 3 1620.4.i.w.1081.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.4.a.i.1.2 6 3.2 odd 2
1620.4.a.j.1.2 yes 6 1.1 even 1 trivial
1620.4.i.w.541.5 12 9.4 even 3
1620.4.i.w.1081.5 12 9.7 even 3
1620.4.i.x.541.5 12 9.5 odd 6
1620.4.i.x.1081.5 12 9.2 odd 6