Properties

Label 5040.2.t.v.1009.2
Level $5040$
Weight $2$
Character 5040.1009
Analytic conductor $40.245$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5040,2,Mod(1009,5040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5040.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.t (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.2
Root \(-0.854638 - 0.854638i\) of defining polynomial
Character \(\chi\) \(=\) 5040.1009
Dual form 5040.2.t.v.1009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.17009 + 0.539189i) q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+(-2.17009 + 0.539189i) q^{5} -1.00000i q^{7} +2.00000 q^{11} -0.921622i q^{13} -1.07838i q^{17} +3.07838 q^{19} -2.34017i q^{23} +(4.41855 - 2.34017i) q^{25} -6.68035 q^{29} +7.75872 q^{31} +(0.539189 + 2.17009i) q^{35} +10.8371i q^{37} -6.49693 q^{41} -6.52359i q^{43} +4.68035i q^{47} -1.00000 q^{49} +3.75872i q^{53} +(-4.34017 + 1.07838i) q^{55} -10.5236 q^{59} -4.15676 q^{61} +(0.496928 + 2.00000i) q^{65} -4.68035i q^{67} +2.00000 q^{71} -7.07838i q^{73} -2.00000i q^{77} +6.15676 q^{79} -6.83710i q^{83} +(0.581449 + 2.34017i) q^{85} +8.34017 q^{89} -0.921622 q^{91} +(-6.68035 + 1.65983i) q^{95} -8.43907i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} + 12 q^{11} + 12 q^{19} - 2 q^{25} + 4 q^{29} - 4 q^{31} - 4 q^{41} - 6 q^{49} - 4 q^{55} - 32 q^{59} - 12 q^{61} - 32 q^{65} + 12 q^{71} + 24 q^{79} + 32 q^{85} + 28 q^{89} - 12 q^{91} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times\).

\(n\) \(2017\) \(2801\) \(3151\) \(3601\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2.17009 + 0.539189i −0.970492 + 0.241133i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 0.921622i 0.255612i −0.991799 0.127806i \(-0.959207\pi\)
0.991799 0.127806i \(-0.0407935\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.07838i 0.261545i −0.991412 0.130773i \(-0.958254\pi\)
0.991412 0.130773i \(-0.0417457\pi\)
\(18\) 0 0
\(19\) 3.07838 0.706228 0.353114 0.935580i \(-0.385123\pi\)
0.353114 + 0.935580i \(0.385123\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.34017i 0.487960i −0.969780 0.243980i \(-0.921547\pi\)
0.969780 0.243980i \(-0.0784531\pi\)
\(24\) 0 0
\(25\) 4.41855 2.34017i 0.883710 0.468035i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.68035 −1.24051 −0.620255 0.784401i \(-0.712971\pi\)
−0.620255 + 0.784401i \(0.712971\pi\)
\(30\) 0 0
\(31\) 7.75872 1.39351 0.696754 0.717310i \(-0.254627\pi\)
0.696754 + 0.717310i \(0.254627\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.539189 + 2.17009i 0.0911396 + 0.366812i
\(36\) 0 0
\(37\) 10.8371i 1.78161i 0.454387 + 0.890804i \(0.349858\pi\)
−0.454387 + 0.890804i \(0.650142\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.49693 −1.01465 −0.507325 0.861755i \(-0.669366\pi\)
−0.507325 + 0.861755i \(0.669366\pi\)
\(42\) 0 0
\(43\) 6.52359i 0.994838i −0.867510 0.497419i \(-0.834281\pi\)
0.867510 0.497419i \(-0.165719\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.68035i 0.682699i 0.939937 + 0.341349i \(0.110884\pi\)
−0.939937 + 0.341349i \(0.889116\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.75872i 0.516300i 0.966105 + 0.258150i \(0.0831129\pi\)
−0.966105 + 0.258150i \(0.916887\pi\)
\(54\) 0 0
\(55\) −4.34017 + 1.07838i −0.585229 + 0.145408i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.5236 −1.37005 −0.685027 0.728517i \(-0.740210\pi\)
−0.685027 + 0.728517i \(0.740210\pi\)
\(60\) 0 0
\(61\) −4.15676 −0.532218 −0.266109 0.963943i \(-0.585738\pi\)
−0.266109 + 0.963943i \(0.585738\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.496928 + 2.00000i 0.0616364 + 0.248069i
\(66\) 0 0
\(67\) 4.68035i 0.571795i −0.958260 0.285898i \(-0.907708\pi\)
0.958260 0.285898i \(-0.0922917\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 7.07838i 0.828461i −0.910172 0.414231i \(-0.864051\pi\)
0.910172 0.414231i \(-0.135949\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) 6.15676 0.692689 0.346345 0.938107i \(-0.387423\pi\)
0.346345 + 0.938107i \(0.387423\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.83710i 0.750469i −0.926930 0.375235i \(-0.877562\pi\)
0.926930 0.375235i \(-0.122438\pi\)
\(84\) 0 0
\(85\) 0.581449 + 2.34017i 0.0630670 + 0.253827i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.34017 0.884057 0.442028 0.897001i \(-0.354259\pi\)
0.442028 + 0.897001i \(0.354259\pi\)
\(90\) 0 0
\(91\) −0.921622 −0.0966123
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.68035 + 1.65983i −0.685389 + 0.170295i
\(96\) 0 0
\(97\) 8.43907i 0.856858i −0.903576 0.428429i \(-0.859067\pi\)
0.903576 0.428429i \(-0.140933\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.81658 0.578772 0.289386 0.957213i \(-0.406549\pi\)
0.289386 + 0.957213i \(0.406549\pi\)
\(102\) 0 0
\(103\) 2.15676i 0.212511i −0.994339 0.106256i \(-0.966114\pi\)
0.994339 0.106256i \(-0.0338862\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.4969i 1.59482i −0.603439 0.797409i \(-0.706203\pi\)
0.603439 0.797409i \(-0.293797\pi\)
\(108\) 0 0
\(109\) 12.8371 1.22957 0.614786 0.788694i \(-0.289243\pi\)
0.614786 + 0.788694i \(0.289243\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.23513i 0.492480i −0.969209 0.246240i \(-0.920805\pi\)
0.969209 0.246240i \(-0.0791951\pi\)
\(114\) 0 0
\(115\) 1.26180 + 5.07838i 0.117663 + 0.473561i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.07838 −0.0988547
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −8.32684 + 7.46081i −0.744775 + 0.667315i
\(126\) 0 0
\(127\) 1.84324i 0.163562i −0.996650 0.0817808i \(-0.973939\pi\)
0.996650 0.0817808i \(-0.0260607\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.47641 0.128995 0.0644973 0.997918i \(-0.479456\pi\)
0.0644973 + 0.997918i \(0.479456\pi\)
\(132\) 0 0
\(133\) 3.07838i 0.266929i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.43907i 0.379255i −0.981856 0.189628i \(-0.939272\pi\)
0.981856 0.189628i \(-0.0607281\pi\)
\(138\) 0 0
\(139\) 13.6020 1.15370 0.576852 0.816849i \(-0.304281\pi\)
0.576852 + 0.816849i \(0.304281\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.84324i 0.154140i
\(144\) 0 0
\(145\) 14.4969 3.60197i 1.20390 0.299127i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.6742 1.28408 0.642040 0.766671i \(-0.278088\pi\)
0.642040 + 0.766671i \(0.278088\pi\)
\(150\) 0 0
\(151\) −5.84324 −0.475516 −0.237758 0.971324i \(-0.576413\pi\)
−0.237758 + 0.971324i \(0.576413\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −16.8371 + 4.18342i −1.35239 + 0.336020i
\(156\) 0 0
\(157\) 4.92162i 0.392788i 0.980525 + 0.196394i \(0.0629232\pi\)
−0.980525 + 0.196394i \(0.937077\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.34017 −0.184431
\(162\) 0 0
\(163\) 9.84324i 0.770982i −0.922712 0.385491i \(-0.874032\pi\)
0.922712 0.385491i \(-0.125968\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.2039i 1.48605i −0.669266 0.743023i \(-0.733391\pi\)
0.669266 0.743023i \(-0.266609\pi\)
\(168\) 0 0
\(169\) 12.1506 0.934662
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 22.4391i 1.70601i −0.521902 0.853005i \(-0.674777\pi\)
0.521902 0.853005i \(-0.325223\pi\)
\(174\) 0 0
\(175\) −2.34017 4.41855i −0.176900 0.334011i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −8.52359 −0.633553 −0.316777 0.948500i \(-0.602601\pi\)
−0.316777 + 0.948500i \(0.602601\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.84324 23.5174i −0.429604 1.72904i
\(186\) 0 0
\(187\) 2.15676i 0.157718i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.3607 1.11146 0.555730 0.831363i \(-0.312439\pi\)
0.555730 + 0.831363i \(0.312439\pi\)
\(192\) 0 0
\(193\) 8.36683i 0.602258i −0.953583 0.301129i \(-0.902637\pi\)
0.953583 0.301129i \(-0.0973635\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.7587i 0.837774i 0.908038 + 0.418887i \(0.137580\pi\)
−0.908038 + 0.418887i \(0.862420\pi\)
\(198\) 0 0
\(199\) −22.5958 −1.60178 −0.800888 0.598814i \(-0.795639\pi\)
−0.800888 + 0.598814i \(0.795639\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.68035i 0.468868i
\(204\) 0 0
\(205\) 14.0989 3.50307i 0.984710 0.244665i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.15676 0.425872
\(210\) 0 0
\(211\) 13.6742 0.941371 0.470685 0.882301i \(-0.344007\pi\)
0.470685 + 0.882301i \(0.344007\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.51745 + 14.1568i 0.239888 + 0.965483i
\(216\) 0 0
\(217\) 7.75872i 0.526696i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.993857 −0.0668541
\(222\) 0 0
\(223\) 21.6742i 1.45141i −0.688005 0.725706i \(-0.741513\pi\)
0.688005 0.725706i \(-0.258487\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.5174i 0.764440i −0.924071 0.382220i \(-0.875160\pi\)
0.924071 0.382220i \(-0.124840\pi\)
\(228\) 0 0
\(229\) −12.8371 −0.848300 −0.424150 0.905592i \(-0.639427\pi\)
−0.424150 + 0.905592i \(0.639427\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.76487i 0.443181i −0.975140 0.221591i \(-0.928875\pi\)
0.975140 0.221591i \(-0.0711248\pi\)
\(234\) 0 0
\(235\) −2.52359 10.1568i −0.164621 0.662554i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 23.3607 1.51108 0.755539 0.655104i \(-0.227375\pi\)
0.755539 + 0.655104i \(0.227375\pi\)
\(240\) 0 0
\(241\) −14.6803 −0.945644 −0.472822 0.881158i \(-0.656765\pi\)
−0.472822 + 0.881158i \(0.656765\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.17009 0.539189i 0.138642 0.0344475i
\(246\) 0 0
\(247\) 2.83710i 0.180520i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.16290 0.578357 0.289179 0.957275i \(-0.406618\pi\)
0.289179 + 0.957275i \(0.406618\pi\)
\(252\) 0 0
\(253\) 4.68035i 0.294251i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.07838i 0.316781i 0.987377 + 0.158390i \(0.0506304\pi\)
−0.987377 + 0.158390i \(0.949370\pi\)
\(258\) 0 0
\(259\) 10.8371 0.673385
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.65983i 0.349000i −0.984657 0.174500i \(-0.944169\pi\)
0.984657 0.174500i \(-0.0558309\pi\)
\(264\) 0 0
\(265\) −2.02666 8.15676i −0.124497 0.501066i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −27.8576 −1.69851 −0.849255 0.527984i \(-0.822948\pi\)
−0.849255 + 0.527984i \(0.822948\pi\)
\(270\) 0 0
\(271\) −25.1194 −1.52590 −0.762948 0.646460i \(-0.776249\pi\)
−0.762948 + 0.646460i \(0.776249\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.83710 4.68035i 0.532897 0.282235i
\(276\) 0 0
\(277\) 28.1978i 1.69424i −0.531401 0.847121i \(-0.678334\pi\)
0.531401 0.847121i \(-0.321666\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.3545 1.21425 0.607125 0.794606i \(-0.292323\pi\)
0.607125 + 0.794606i \(0.292323\pi\)
\(282\) 0 0
\(283\) 23.5174i 1.39797i 0.715138 + 0.698984i \(0.246364\pi\)
−0.715138 + 0.698984i \(0.753636\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.49693i 0.383502i
\(288\) 0 0
\(289\) 15.8371 0.931594
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.92162i 0.170683i −0.996352 0.0853415i \(-0.972802\pi\)
0.996352 0.0853415i \(-0.0271981\pi\)
\(294\) 0 0
\(295\) 22.8371 5.67420i 1.32963 0.330365i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.15676 −0.124728
\(300\) 0 0
\(301\) −6.52359 −0.376014
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.02052 2.24128i 0.516513 0.128335i
\(306\) 0 0
\(307\) 10.4703i 0.597570i 0.954321 + 0.298785i \(0.0965813\pi\)
−0.954321 + 0.298785i \(0.903419\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 23.8310 1.35133 0.675665 0.737209i \(-0.263857\pi\)
0.675665 + 0.737209i \(0.263857\pi\)
\(312\) 0 0
\(313\) 32.7526i 1.85129i −0.378399 0.925643i \(-0.623525\pi\)
0.378399 0.925643i \(-0.376475\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.9155i 1.00623i 0.864218 + 0.503117i \(0.167813\pi\)
−0.864218 + 0.503117i \(0.832187\pi\)
\(318\) 0 0
\(319\) −13.3607 −0.748055
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.31965i 0.184710i
\(324\) 0 0
\(325\) −2.15676 4.07223i −0.119635 0.225887i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.68035 0.258036
\(330\) 0 0
\(331\) 1.36069 0.0747904 0.0373952 0.999301i \(-0.488094\pi\)
0.0373952 + 0.999301i \(0.488094\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.52359 + 10.1568i 0.137878 + 0.554923i
\(336\) 0 0
\(337\) 25.3607i 1.38148i −0.723101 0.690742i \(-0.757284\pi\)
0.723101 0.690742i \(-0.242716\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 15.5174 0.840317
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.8638i 0.905294i 0.891690 + 0.452647i \(0.149520\pi\)
−0.891690 + 0.452647i \(0.850480\pi\)
\(348\) 0 0
\(349\) −9.51745 −0.509457 −0.254729 0.967013i \(-0.581986\pi\)
−0.254729 + 0.967013i \(0.581986\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 35.7998i 1.90543i −0.303867 0.952715i \(-0.598278\pi\)
0.303867 0.952715i \(-0.401722\pi\)
\(354\) 0 0
\(355\) −4.34017 + 1.07838i −0.230352 + 0.0572343i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 22.3135 1.17766 0.588831 0.808256i \(-0.299588\pi\)
0.588831 + 0.808256i \(0.299588\pi\)
\(360\) 0 0
\(361\) −9.52359 −0.501242
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.81658 + 15.3607i 0.199769 + 0.804015i
\(366\) 0 0
\(367\) 20.3135i 1.06036i −0.847886 0.530178i \(-0.822125\pi\)
0.847886 0.530178i \(-0.177875\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.75872 0.195143
\(372\) 0 0
\(373\) 16.0000i 0.828449i 0.910175 + 0.414224i \(0.135947\pi\)
−0.910175 + 0.414224i \(0.864053\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.15676i 0.317089i
\(378\) 0 0
\(379\) 6.15676 0.316251 0.158126 0.987419i \(-0.449455\pi\)
0.158126 + 0.987419i \(0.449455\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 26.8371i 1.37131i −0.727926 0.685656i \(-0.759515\pi\)
0.727926 0.685656i \(-0.240485\pi\)
\(384\) 0 0
\(385\) 1.07838 + 4.34017i 0.0549592 + 0.221196i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5.63317 −0.285613 −0.142806 0.989751i \(-0.545613\pi\)
−0.142806 + 0.989751i \(0.545613\pi\)
\(390\) 0 0
\(391\) −2.52359 −0.127623
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −13.3607 + 3.31965i −0.672249 + 0.167030i
\(396\) 0 0
\(397\) 37.7998i 1.89712i 0.316604 + 0.948558i \(0.397457\pi\)
−0.316604 + 0.948558i \(0.602543\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.6332 0.680808 0.340404 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(402\) 0 0
\(403\) 7.15061i 0.356197i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.6742i 1.07435i
\(408\) 0 0
\(409\) −12.3545 −0.610893 −0.305447 0.952209i \(-0.598806\pi\)
−0.305447 + 0.952209i \(0.598806\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.5236i 0.517832i
\(414\) 0 0
\(415\) 3.68649 + 14.8371i 0.180963 + 0.728325i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −28.9939 −1.41644 −0.708221 0.705991i \(-0.750502\pi\)
−0.708221 + 0.705991i \(0.750502\pi\)
\(420\) 0 0
\(421\) −15.1629 −0.738994 −0.369497 0.929232i \(-0.620470\pi\)
−0.369497 + 0.929232i \(0.620470\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.52359 4.76487i −0.122412 0.231130i
\(426\) 0 0
\(427\) 4.15676i 0.201159i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10.3135 −0.496784 −0.248392 0.968660i \(-0.579902\pi\)
−0.248392 + 0.968660i \(0.579902\pi\)
\(432\) 0 0
\(433\) 20.4391i 0.982239i −0.871092 0.491120i \(-0.836588\pi\)
0.871092 0.491120i \(-0.163412\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.20394i 0.344611i
\(438\) 0 0
\(439\) −16.9216 −0.807625 −0.403812 0.914842i \(-0.632315\pi\)
−0.403812 + 0.914842i \(0.632315\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.8104i 0.608642i 0.952569 + 0.304321i \(0.0984296\pi\)
−0.952569 + 0.304321i \(0.901570\pi\)
\(444\) 0 0
\(445\) −18.0989 + 4.49693i −0.857970 + 0.213175i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −14.6270 −0.690292 −0.345146 0.938549i \(-0.612171\pi\)
−0.345146 + 0.938549i \(0.612171\pi\)
\(450\) 0 0
\(451\) −12.9939 −0.611857
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.00000 0.496928i 0.0937614 0.0232964i
\(456\) 0 0
\(457\) 14.1568i 0.662225i 0.943591 + 0.331113i \(0.107424\pi\)
−0.943591 + 0.331113i \(0.892576\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.340173 −0.0158434 −0.00792172 0.999969i \(-0.502522\pi\)
−0.00792172 + 0.999969i \(0.502522\pi\)
\(462\) 0 0
\(463\) 9.84324i 0.457454i −0.973491 0.228727i \(-0.926544\pi\)
0.973491 0.228727i \(-0.0734564\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.5174i 0.532964i −0.963840 0.266482i \(-0.914139\pi\)
0.963840 0.266482i \(-0.0858613\pi\)
\(468\) 0 0
\(469\) −4.68035 −0.216118
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 13.0472i 0.599910i
\(474\) 0 0
\(475\) 13.6020 7.20394i 0.624101 0.330539i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 19.5174 0.891775 0.445887 0.895089i \(-0.352888\pi\)
0.445887 + 0.895089i \(0.352888\pi\)
\(480\) 0 0
\(481\) 9.98771 0.455401
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.55025 + 18.3135i 0.206616 + 0.831574i
\(486\) 0 0
\(487\) 23.1506i 1.04905i −0.851394 0.524527i \(-0.824242\pi\)
0.851394 0.524527i \(-0.175758\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) 0 0
\(493\) 7.20394i 0.324449i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.00000i 0.0897123i
\(498\) 0 0
\(499\) 27.2039 1.21782 0.608908 0.793241i \(-0.291608\pi\)
0.608908 + 0.793241i \(0.291608\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.8371i 0.839905i 0.907546 + 0.419952i \(0.137953\pi\)
−0.907546 + 0.419952i \(0.862047\pi\)
\(504\) 0 0
\(505\) −12.6225 + 3.13624i −0.561693 + 0.139561i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.81044 0.301867 0.150934 0.988544i \(-0.451772\pi\)
0.150934 + 0.988544i \(0.451772\pi\)
\(510\) 0 0
\(511\) −7.07838 −0.313129
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.16290 + 4.68035i 0.0512434 + 0.206241i
\(516\) 0 0
\(517\) 9.36069i 0.411683i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 25.8166 1.13105 0.565523 0.824733i \(-0.308675\pi\)
0.565523 + 0.824733i \(0.308675\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.36683i 0.364465i
\(528\) 0 0
\(529\) 17.5236 0.761895
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.98771i 0.259357i
\(534\) 0 0
\(535\) 8.89496 + 35.7998i 0.384563 + 1.54776i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 25.8843 1.11285 0.556426 0.830897i \(-0.312172\pi\)
0.556426 + 0.830897i \(0.312172\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −27.8576 + 6.92162i −1.19329 + 0.296490i
\(546\) 0 0
\(547\) 11.3197i 0.483993i −0.970277 0.241997i \(-0.922198\pi\)
0.970277 0.241997i \(-0.0778023\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −20.5646 −0.876083
\(552\) 0 0
\(553\) 6.15676i 0.261812i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 26.6491i 1.12916i 0.825378 + 0.564580i \(0.190962\pi\)
−0.825378 + 0.564580i \(0.809038\pi\)
\(558\) 0 0
\(559\) −6.01229 −0.254293
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 46.3545i 1.95361i −0.214128 0.976806i \(-0.568691\pi\)
0.214128 0.976806i \(-0.431309\pi\)
\(564\) 0 0
\(565\) 2.82273 + 11.3607i 0.118753 + 0.477948i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −14.3668 −0.602289 −0.301145 0.953579i \(-0.597369\pi\)
−0.301145 + 0.953579i \(0.597369\pi\)
\(570\) 0 0
\(571\) −38.7214 −1.62044 −0.810220 0.586126i \(-0.800652\pi\)
−0.810220 + 0.586126i \(0.800652\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.47641 10.3402i −0.228382 0.431215i
\(576\) 0 0
\(577\) 43.4740i 1.80984i 0.425577 + 0.904922i \(0.360071\pi\)
−0.425577 + 0.904922i \(0.639929\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.83710 −0.283651
\(582\) 0 0
\(583\) 7.51745i 0.311341i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0288i 1.48707i 0.668699 + 0.743533i \(0.266851\pi\)
−0.668699 + 0.743533i \(0.733149\pi\)
\(588\) 0 0
\(589\) 23.8843 0.984135
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 31.4863i 1.29299i 0.762920 + 0.646493i \(0.223765\pi\)
−0.762920 + 0.646493i \(0.776235\pi\)
\(594\) 0 0
\(595\) 2.34017 0.581449i 0.0959377 0.0238371i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −29.0349 −1.18633 −0.593167 0.805080i \(-0.702123\pi\)
−0.593167 + 0.805080i \(0.702123\pi\)
\(600\) 0 0
\(601\) 15.3607 0.626576 0.313288 0.949658i \(-0.398570\pi\)
0.313288 + 0.949658i \(0.398570\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 15.1906 3.77432i 0.617586 0.153448i
\(606\) 0 0
\(607\) 13.0472i 0.529569i 0.964308 + 0.264784i \(0.0853008\pi\)
−0.964308 + 0.264784i \(0.914699\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.31351 0.174506
\(612\) 0 0
\(613\) 15.5174i 0.626744i 0.949630 + 0.313372i \(0.101459\pi\)
−0.949630 + 0.313372i \(0.898541\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.7649i 0.916479i 0.888829 + 0.458240i \(0.151520\pi\)
−0.888829 + 0.458240i \(0.848480\pi\)
\(618\) 0 0
\(619\) 7.92777 0.318644 0.159322 0.987227i \(-0.449069\pi\)
0.159322 + 0.987227i \(0.449069\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.34017i 0.334142i
\(624\) 0 0
\(625\) 14.0472 20.6803i 0.561887 0.827214i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.6865 0.465971
\(630\) 0 0
\(631\) −19.2039 −0.764497 −0.382248 0.924060i \(-0.624850\pi\)
−0.382248 + 0.924060i \(0.624850\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.993857 + 4.00000i 0.0394400 + 0.158735i
\(636\) 0 0
\(637\) 0.921622i 0.0365160i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.94668 0.234880 0.117440 0.993080i \(-0.462531\pi\)
0.117440 + 0.993080i \(0.462531\pi\)
\(642\) 0 0
\(643\) 30.8904i 1.21820i −0.793094 0.609100i \(-0.791531\pi\)
0.793094 0.609100i \(-0.208469\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.2039i 0.754985i −0.926013 0.377492i \(-0.876786\pi\)
0.926013 0.377492i \(-0.123214\pi\)
\(648\) 0 0
\(649\) −21.0472 −0.826174
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 28.5548i 1.11744i −0.829358 0.558718i \(-0.811294\pi\)
0.829358 0.558718i \(-0.188706\pi\)
\(654\) 0 0
\(655\) −3.20394 + 0.796064i −0.125188 + 0.0311048i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 27.9877 1.09025 0.545123 0.838356i \(-0.316483\pi\)
0.545123 + 0.838356i \(0.316483\pi\)
\(660\) 0 0
\(661\) −22.1445 −0.861320 −0.430660 0.902514i \(-0.641719\pi\)
−0.430660 + 0.902514i \(0.641719\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.65983 + 6.68035i 0.0643653 + 0.259053i
\(666\) 0 0
\(667\) 15.6332i 0.605319i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −8.31351 −0.320940
\(672\) 0 0
\(673\) 2.21008i 0.0851923i −0.999092 0.0425962i \(-0.986437\pi\)
0.999092 0.0425962i \(-0.0135629\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.5486i 0.751315i −0.926758 0.375658i \(-0.877417\pi\)
0.926758 0.375658i \(-0.122583\pi\)
\(678\) 0 0
\(679\) −8.43907 −0.323862
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 11.8166i 0.452149i −0.974110 0.226074i \(-0.927411\pi\)
0.974110 0.226074i \(-0.0725893\pi\)
\(684\) 0 0
\(685\) 2.39350 + 9.63317i 0.0914508 + 0.368064i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.46412 0.131973
\(690\) 0 0
\(691\) −11.7587 −0.447323 −0.223661 0.974667i \(-0.571801\pi\)
−0.223661 + 0.974667i \(0.571801\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −29.5174 + 7.33403i −1.11966 + 0.278196i
\(696\) 0 0
\(697\) 7.00614i 0.265377i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −9.94668 −0.375681 −0.187840 0.982200i \(-0.560149\pi\)
−0.187840 + 0.982200i \(0.560149\pi\)
\(702\) 0 0
\(703\) 33.3607i 1.25822i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.81658i 0.218755i
\(708\) 0 0
\(709\) 11.0472 0.414886 0.207443 0.978247i \(-0.433486\pi\)
0.207443 + 0.978247i \(0.433486\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 18.1568i 0.679976i
\(714\) 0 0
\(715\) 0.993857 + 4.00000i 0.0371681 + 0.149592i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.15676 0.229608 0.114804 0.993388i \(-0.463376\pi\)
0.114804 + 0.993388i \(0.463376\pi\)
\(720\) 0 0
\(721\) −2.15676 −0.0803218
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −29.5174 + 15.6332i −1.09625 + 0.580601i
\(726\) 0 0
\(727\) 2.89043i 0.107200i −0.998562 0.0536000i \(-0.982930\pi\)
0.998562 0.0536000i \(-0.0170696\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7.03489 −0.260195
\(732\) 0 0
\(733\) 25.7998i 0.952936i 0.879192 + 0.476468i \(0.158083\pi\)
−0.879192 + 0.476468i \(0.841917\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.36069i 0.344806i
\(738\) 0 0
\(739\) −1.04718 −0.0385212 −0.0192606 0.999814i \(-0.506131\pi\)
−0.0192606 + 0.999814i \(0.506131\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.97334i 0.365886i 0.983123 + 0.182943i \(0.0585624\pi\)
−0.983123 + 0.182943i \(0.941438\pi\)
\(744\) 0 0
\(745\) −34.0144 + 8.45136i −1.24619 + 0.309634i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −16.4969 −0.602785
\(750\) 0 0
\(751\) −3.26633 −0.119190 −0.0595950 0.998223i \(-0.518981\pi\)
−0.0595950 + 0.998223i \(0.518981\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.6803 3.15061i 0.461485 0.114663i
\(756\) 0 0
\(757\) 49.9877i 1.81683i −0.418065 0.908417i \(-0.637292\pi\)
0.418065 0.908417i \(-0.362708\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.61265 −0.0947083 −0.0473542 0.998878i \(-0.515079\pi\)
−0.0473542 + 0.998878i \(0.515079\pi\)
\(762\) 0 0
\(763\) 12.8371i 0.464734i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.69878i 0.350202i
\(768\) 0 0
\(769\) 15.6742 0.565226 0.282613 0.959234i \(-0.408799\pi\)
0.282613 + 0.959234i \(0.408799\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 5.81205i 0.209045i 0.994523 + 0.104522i \(0.0333314\pi\)
−0.994523 + 0.104522i \(0.966669\pi\)
\(774\) 0 0
\(775\) 34.2823 18.1568i 1.23146 0.652210i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.65368 10.6803i −0.0947140 0.381198i
\(786\) 0 0
\(787\) 39.3484i 1.40262i −0.712857 0.701310i \(-0.752599\pi\)
0.712857 0.701310i \(-0.247401\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.23513 −0.186140
\(792\) 0 0
\(793\) 3.83096i 0.136041i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28.2823i 1.00181i −0.865502 0.500905i \(-0.833000\pi\)
0.865502 0.500905i \(-0.167000\pi\)
\(798\) 0 0
\(799\) 5.04718 0.178556
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 14.1568i 0.499581i
\(804\) 0 0
\(805\) 5.07838 1.26180i 0.178989 0.0444724i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −15.6742 −0.551076 −0.275538 0.961290i \(-0.588856\pi\)
−0.275538 + 0.961290i \(0.588856\pi\)
\(810\) 0 0
\(811\) 42.1666 1.48067 0.740335 0.672238i \(-0.234667\pi\)
0.740335 + 0.672238i \(0.234667\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.30737 + 21.3607i 0.185909 + 0.748232i
\(816\) 0 0
\(817\) 20.0821i 0.702583i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 39.0472 1.36276 0.681378 0.731932i \(-0.261381\pi\)
0.681378 + 0.731932i \(0.261381\pi\)
\(822\) 0 0
\(823\) 36.5646i 1.27456i −0.770631 0.637281i \(-0.780059\pi\)
0.770631 0.637281i \(-0.219941\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 50.2245i 1.74648i 0.487294 + 0.873238i \(0.337984\pi\)
−0.487294 + 0.873238i \(0.662016\pi\)
\(828\) 0 0
\(829\) −32.8371 −1.14048 −0.570240 0.821478i \(-0.693150\pi\)
−0.570240 + 0.821478i \(0.693150\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.07838i 0.0373636i
\(834\) 0 0
\(835\) 10.3545 + 41.6742i 0.358334 + 1.44220i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 13.3607 0.461262 0.230631 0.973041i \(-0.425921\pi\)
0.230631 + 0.973041i \(0.425921\pi\)
\(840\) 0 0
\(841\) 15.6270 0.538863
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −26.3679 + 6.55148i −0.907083 + 0.225378i
\(846\) 0 0
\(847\) 7.00000i 0.240523i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 25.3607 0.869353
\(852\) 0 0
\(853\) 39.6430i 1.35735i 0.734438 + 0.678675i \(0.237446\pi\)
−0.734438 + 0.678675i \(0.762554\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 29.7054i 1.01472i 0.861735 + 0.507359i \(0.169378\pi\)
−0.861735 + 0.507359i \(0.830622\pi\)
\(858\) 0 0
\(859\) −3.07838 −0.105033 −0.0525164 0.998620i \(-0.516724\pi\)
−0.0525164 + 0.998620i \(0.516724\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6.39350i 0.217637i 0.994062 + 0.108819i \(0.0347068\pi\)
−0.994062 + 0.108819i \(0.965293\pi\)
\(864\) 0 0
\(865\) 12.0989 + 48.6947i 0.411375 + 1.65567i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 12.3135 0.417707
\(870\) 0 0
\(871\) −4.31351 −0.146158
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 7.46081 + 8.32684i 0.252221 + 0.281499i
\(876\) 0 0
\(877\) 1.21622i 0.0410689i −0.999789 0.0205345i \(-0.993463\pi\)
0.999789 0.0205345i \(-0.00653678\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −15.9733 −0.538155 −0.269078 0.963118i \(-0.586719\pi\)
−0.269078 + 0.963118i \(0.586719\pi\)
\(882\) 0 0
\(883\) 11.6865i 0.393282i 0.980476 + 0.196641i \(0.0630033\pi\)
−0.980476 + 0.196641i \(0.936997\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 25.6209i 0.860265i −0.902766 0.430132i \(-0.858467\pi\)
0.902766 0.430132i \(-0.141533\pi\)
\(888\) 0 0
\(889\) −1.84324 −0.0618204
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 14.4079i 0.482141i
\(894\) 0 0
\(895\) 21.7009 5.39189i 0.725380 0.180231i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −51.8310 −1.72866
\(900\) 0 0
\(901\) 4.05332 0.135036
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.4969 4.59583i 0.614859 0.152770i
\(906\) 0 0
\(907\) 57.7563i 1.91777i 0.283802 + 0.958883i \(0.408404\pi\)
−0.283802 + 0.958883i \(0.591596\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −35.9877 −1.19233 −0.596163 0.802863i \(-0.703309\pi\)
−0.596163 + 0.802863i \(0.703309\pi\)
\(912\) 0 0
\(913\) 13.6742i 0.452550i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.47641i 0.0487553i
\(918\) 0 0
\(919\) 46.7214 1.54120 0.770598 0.637321i \(-0.219958\pi\)
0.770598 + 0.637321i \(0.219958\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.84324i 0.0606711i
\(924\) 0 0
\(925\) 25.3607 + 47.8843i 0.833854 + 1.57443i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −53.0493 −1.74049 −0.870245 0.492619i \(-0.836040\pi\)
−0.870245 + 0.492619i \(0.836040\pi\)
\(930\) 0 0
\(931\) −3.07838 −0.100890
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.16290 + 4.68035i 0.0380308 + 0.153064i
\(936\) 0 0
\(937\) 16.1256i 0.526799i −0.964687 0.263400i \(-0.915156\pi\)
0.964687 0.263400i \(-0.0848437\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −24.7070 −0.805425 −0.402713 0.915326i \(-0.631933\pi\)
−0.402713 + 0.915326i \(0.631933\pi\)
\(942\) 0 0
\(943\) 15.2039i 0.495108i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.53797i 0.212455i −0.994342 0.106228i \(-0.966123\pi\)
0.994342 0.106228i \(-0.0338772\pi\)
\(948\) 0 0
\(949\) −6.52359 −0.211765
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.11327i 0.198028i 0.995086 + 0.0990142i \(0.0315689\pi\)
−0.995086 + 0.0990142i \(0.968431\pi\)
\(954\) 0 0
\(955\) −33.3340 + 8.28231i −1.07866 + 0.268009i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.43907 −0.143345
\(960\) 0 0
\(961\) 29.1978 0.941864
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.51130 + 18.1568i 0.145224 + 0.584487i
\(966\) 0 0
\(967\) 25.6209i 0.823912i 0.911204 + 0.411956i \(0.135154\pi\)
−0.911204 + 0.411956i \(0.864846\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.05332 0.130077 0.0650387 0.997883i \(-0.479283\pi\)
0.0650387 + 0.997883i \(0.479283\pi\)
\(972\) 0 0
\(973\) 13.6020i 0.436059i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.81205i 0.121958i 0.998139 + 0.0609791i \(0.0194223\pi\)
−0.998139 + 0.0609791i \(0.980578\pi\)
\(978\) 0 0
\(979\) 16.6803 0.533106
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0000i 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 0 0
\(985\) −6.34017 25.5174i −0.202015 0.813053i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −15.2663 −0.485441
\(990\) 0 0
\(991\) 42.4079 1.34713 0.673565 0.739128i \(-0.264762\pi\)
0.673565 + 0.739128i \(0.264762\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 49.0349 12.1834i 1.55451 0.386240i
\(996\) 0 0
\(997\) 43.4740i 1.37683i 0.725315 + 0.688417i \(0.241694\pi\)
−0.725315 + 0.688417i \(0.758306\pi\)
\(998\) 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 5040.2.t.v.1009.2 6
3.2 odd 2 1680.2.t.k.1009.6 6
4.3 odd 2 315.2.d.e.64.6 6
5.4 even 2 inner 5040.2.t.v.1009.1 6
12.11 even 2 105.2.d.b.64.1 6
15.2 even 4 8400.2.a.dj.1.2 3
15.8 even 4 8400.2.a.dg.1.2 3
15.14 odd 2 1680.2.t.k.1009.3 6
20.3 even 4 1575.2.a.x.1.3 3
20.7 even 4 1575.2.a.w.1.1 3
20.19 odd 2 315.2.d.e.64.1 6
28.27 even 2 2205.2.d.l.1324.6 6
60.23 odd 4 525.2.a.j.1.1 3
60.47 odd 4 525.2.a.k.1.3 3
60.59 even 2 105.2.d.b.64.6 yes 6
84.11 even 6 735.2.q.e.79.6 12
84.23 even 6 735.2.q.e.214.1 12
84.47 odd 6 735.2.q.f.214.1 12
84.59 odd 6 735.2.q.f.79.6 12
84.83 odd 2 735.2.d.b.589.1 6
140.139 even 2 2205.2.d.l.1324.1 6
420.59 odd 6 735.2.q.f.79.1 12
420.83 even 4 3675.2.a.bi.1.1 3
420.167 even 4 3675.2.a.bj.1.3 3
420.179 even 6 735.2.q.e.79.1 12
420.299 odd 6 735.2.q.f.214.6 12
420.359 even 6 735.2.q.e.214.6 12
420.419 odd 2 735.2.d.b.589.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.1 6 12.11 even 2
105.2.d.b.64.6 yes 6 60.59 even 2
315.2.d.e.64.1 6 20.19 odd 2
315.2.d.e.64.6 6 4.3 odd 2
525.2.a.j.1.1 3 60.23 odd 4
525.2.a.k.1.3 3 60.47 odd 4
735.2.d.b.589.1 6 84.83 odd 2
735.2.d.b.589.6 6 420.419 odd 2
735.2.q.e.79.1 12 420.179 even 6
735.2.q.e.79.6 12 84.11 even 6
735.2.q.e.214.1 12 84.23 even 6
735.2.q.e.214.6 12 420.359 even 6
735.2.q.f.79.1 12 420.59 odd 6
735.2.q.f.79.6 12 84.59 odd 6
735.2.q.f.214.1 12 84.47 odd 6
735.2.q.f.214.6 12 420.299 odd 6
1575.2.a.w.1.1 3 20.7 even 4
1575.2.a.x.1.3 3 20.3 even 4
1680.2.t.k.1009.3 6 15.14 odd 2
1680.2.t.k.1009.6 6 3.2 odd 2
2205.2.d.l.1324.1 6 140.139 even 2
2205.2.d.l.1324.6 6 28.27 even 2
3675.2.a.bi.1.1 3 420.83 even 4
3675.2.a.bj.1.3 3 420.167 even 4
5040.2.t.v.1009.1 6 5.4 even 2 inner
5040.2.t.v.1009.2 6 1.1 even 1 trivial
8400.2.a.dg.1.2 3 15.8 even 4
8400.2.a.dj.1.2 3 15.2 even 4