Properties

Label 476.2.a.a.1.1
Level $476$
Weight $2$
Character 476.1
Self dual yes
Analytic conductor $3.801$
Analytic rank $1$
Dimension $2$
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] = [476,2,Mod(1,476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(476, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("476.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 476.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: \(3.80087913621\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 476.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.30278 q^{3} +0.302776 q^{5} +1.00000 q^{7} +7.90833 q^{9} +O(q^{10})\) \(q-3.30278 q^{3} +0.302776 q^{5} +1.00000 q^{7} +7.90833 q^{9} -2.60555 q^{11} +0.605551 q^{13} -1.00000 q^{15} -1.00000 q^{17} -6.00000 q^{19} -3.30278 q^{21} +4.60555 q^{23} -4.90833 q^{25} -16.2111 q^{27} -1.39445 q^{29} -10.3028 q^{31} +8.60555 q^{33} +0.302776 q^{35} -7.21110 q^{37} -2.00000 q^{39} -8.51388 q^{41} -0.697224 q^{43} +2.39445 q^{45} +10.0000 q^{47} +1.00000 q^{49} +3.30278 q^{51} +4.30278 q^{53} -0.788897 q^{55} +19.8167 q^{57} -9.21110 q^{59} -0.697224 q^{61} +7.90833 q^{63} +0.183346 q^{65} +6.30278 q^{67} -15.2111 q^{69} -2.00000 q^{71} +4.51388 q^{73} +16.2111 q^{75} -2.60555 q^{77} -6.00000 q^{79} +29.8167 q^{81} -5.21110 q^{83} -0.302776 q^{85} +4.60555 q^{87} -2.00000 q^{89} +0.605551 q^{91} +34.0278 q^{93} -1.81665 q^{95} -12.9083 q^{97} -20.6056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} + 2 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{17} - 12 q^{19} - 3 q^{21} + 2 q^{23} + q^{25} - 18 q^{27} - 10 q^{29} - 17 q^{31} + 10 q^{33} - 3 q^{35} - 4 q^{39} + q^{41} - 5 q^{43} + 12 q^{45} + 20 q^{47} + 2 q^{49} + 3 q^{51} + 5 q^{53} - 16 q^{55} + 18 q^{57} - 4 q^{59} - 5 q^{61} + 5 q^{63} + 22 q^{65} + 9 q^{67} - 16 q^{69} - 4 q^{71} - 9 q^{73} + 18 q^{75} + 2 q^{77} - 12 q^{79} + 38 q^{81} + 4 q^{83} + 3 q^{85} + 2 q^{87} - 4 q^{89} - 6 q^{91} + 32 q^{93} + 18 q^{95} - 15 q^{97} - 34 q^{99}+O(q^{100}) \) 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
\(3\) −3.30278 −1.90686 −0.953429 0.301617i \(-0.902474\pi\)
−0.953429 + 0.301617i \(0.902474\pi\)
\(4\) 0 0
\(5\) 0.302776 0.135405 0.0677027 0.997706i \(-0.478433\pi\)
0.0677027 + 0.997706i \(0.478433\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 7.90833 2.63611
\(10\) 0 0
\(11\) −2.60555 −0.785603 −0.392802 0.919623i \(-0.628494\pi\)
−0.392802 + 0.919623i \(0.628494\pi\)
\(12\) 0 0
\(13\) 0.605551 0.167950 0.0839749 0.996468i \(-0.473238\pi\)
0.0839749 + 0.996468i \(0.473238\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −3.30278 −0.720725
\(22\) 0 0
\(23\) 4.60555 0.960324 0.480162 0.877180i \(-0.340578\pi\)
0.480162 + 0.877180i \(0.340578\pi\)
\(24\) 0 0
\(25\) −4.90833 −0.981665
\(26\) 0 0
\(27\) −16.2111 −3.11983
\(28\) 0 0
\(29\) −1.39445 −0.258943 −0.129471 0.991583i \(-0.541328\pi\)
−0.129471 + 0.991583i \(0.541328\pi\)
\(30\) 0 0
\(31\) −10.3028 −1.85043 −0.925217 0.379439i \(-0.876117\pi\)
−0.925217 + 0.379439i \(0.876117\pi\)
\(32\) 0 0
\(33\) 8.60555 1.49803
\(34\) 0 0
\(35\) 0.302776 0.0511784
\(36\) 0 0
\(37\) −7.21110 −1.18550 −0.592749 0.805387i \(-0.701957\pi\)
−0.592749 + 0.805387i \(0.701957\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −8.51388 −1.32964 −0.664822 0.747002i \(-0.731493\pi\)
−0.664822 + 0.747002i \(0.731493\pi\)
\(42\) 0 0
\(43\) −0.697224 −0.106326 −0.0531629 0.998586i \(-0.516930\pi\)
−0.0531629 + 0.998586i \(0.516930\pi\)
\(44\) 0 0
\(45\) 2.39445 0.356943
\(46\) 0 0
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.30278 0.462481
\(52\) 0 0
\(53\) 4.30278 0.591032 0.295516 0.955338i \(-0.404508\pi\)
0.295516 + 0.955338i \(0.404508\pi\)
\(54\) 0 0
\(55\) −0.788897 −0.106375
\(56\) 0 0
\(57\) 19.8167 2.62478
\(58\) 0 0
\(59\) −9.21110 −1.19918 −0.599592 0.800306i \(-0.704670\pi\)
−0.599592 + 0.800306i \(0.704670\pi\)
\(60\) 0 0
\(61\) −0.697224 −0.0892704 −0.0446352 0.999003i \(-0.514213\pi\)
−0.0446352 + 0.999003i \(0.514213\pi\)
\(62\) 0 0
\(63\) 7.90833 0.996356
\(64\) 0 0
\(65\) 0.183346 0.0227413
\(66\) 0 0
\(67\) 6.30278 0.770007 0.385003 0.922915i \(-0.374200\pi\)
0.385003 + 0.922915i \(0.374200\pi\)
\(68\) 0 0
\(69\) −15.2111 −1.83120
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 4.51388 0.528309 0.264155 0.964480i \(-0.414907\pi\)
0.264155 + 0.964480i \(0.414907\pi\)
\(74\) 0 0
\(75\) 16.2111 1.87190
\(76\) 0 0
\(77\) −2.60555 −0.296930
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 29.8167 3.31296
\(82\) 0 0
\(83\) −5.21110 −0.571993 −0.285996 0.958231i \(-0.592325\pi\)
−0.285996 + 0.958231i \(0.592325\pi\)
\(84\) 0 0
\(85\) −0.302776 −0.0328406
\(86\) 0 0
\(87\) 4.60555 0.493767
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0.605551 0.0634790
\(92\) 0 0
\(93\) 34.0278 3.52851
\(94\) 0 0
\(95\) −1.81665 −0.186385
\(96\) 0 0
\(97\) −12.9083 −1.31064 −0.655321 0.755350i \(-0.727467\pi\)
−0.655321 + 0.755350i \(0.727467\pi\)
\(98\) 0 0
\(99\) −20.6056 −2.07094
\(100\) 0 0
\(101\) 16.4222 1.63407 0.817035 0.576588i \(-0.195616\pi\)
0.817035 + 0.576588i \(0.195616\pi\)
\(102\) 0 0
\(103\) −1.21110 −0.119333 −0.0596667 0.998218i \(-0.519004\pi\)
−0.0596667 + 0.998218i \(0.519004\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) −16.6056 −1.60532 −0.802660 0.596437i \(-0.796582\pi\)
−0.802660 + 0.596437i \(0.796582\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 23.8167 2.26058
\(112\) 0 0
\(113\) 4.78890 0.450502 0.225251 0.974301i \(-0.427680\pi\)
0.225251 + 0.974301i \(0.427680\pi\)
\(114\) 0 0
\(115\) 1.39445 0.130033
\(116\) 0 0
\(117\) 4.78890 0.442734
\(118\) 0 0
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) −4.21110 −0.382828
\(122\) 0 0
\(123\) 28.1194 2.53544
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 20.1194 1.78531 0.892655 0.450740i \(-0.148840\pi\)
0.892655 + 0.450740i \(0.148840\pi\)
\(128\) 0 0
\(129\) 2.30278 0.202748
\(130\) 0 0
\(131\) 18.4222 1.60956 0.804778 0.593576i \(-0.202284\pi\)
0.804778 + 0.593576i \(0.202284\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) −4.90833 −0.422442
\(136\) 0 0
\(137\) −13.9083 −1.18827 −0.594134 0.804366i \(-0.702505\pi\)
−0.594134 + 0.804366i \(0.702505\pi\)
\(138\) 0 0
\(139\) −19.5139 −1.65515 −0.827573 0.561358i \(-0.810279\pi\)
−0.827573 + 0.561358i \(0.810279\pi\)
\(140\) 0 0
\(141\) −33.0278 −2.78144
\(142\) 0 0
\(143\) −1.57779 −0.131942
\(144\) 0 0
\(145\) −0.422205 −0.0350622
\(146\) 0 0
\(147\) −3.30278 −0.272408
\(148\) 0 0
\(149\) −5.90833 −0.484029 −0.242015 0.970273i \(-0.577808\pi\)
−0.242015 + 0.970273i \(0.577808\pi\)
\(150\) 0 0
\(151\) −11.1194 −0.904886 −0.452443 0.891793i \(-0.649447\pi\)
−0.452443 + 0.891793i \(0.649447\pi\)
\(152\) 0 0
\(153\) −7.90833 −0.639350
\(154\) 0 0
\(155\) −3.11943 −0.250559
\(156\) 0 0
\(157\) −21.8167 −1.74116 −0.870579 0.492028i \(-0.836256\pi\)
−0.870579 + 0.492028i \(0.836256\pi\)
\(158\) 0 0
\(159\) −14.2111 −1.12701
\(160\) 0 0
\(161\) 4.60555 0.362968
\(162\) 0 0
\(163\) −0.183346 −0.0143608 −0.00718039 0.999974i \(-0.502286\pi\)
−0.00718039 + 0.999974i \(0.502286\pi\)
\(164\) 0 0
\(165\) 2.60555 0.202842
\(166\) 0 0
\(167\) 23.1194 1.78904 0.894518 0.447033i \(-0.147519\pi\)
0.894518 + 0.447033i \(0.147519\pi\)
\(168\) 0 0
\(169\) −12.6333 −0.971793
\(170\) 0 0
\(171\) −47.4500 −3.62859
\(172\) 0 0
\(173\) −5.90833 −0.449202 −0.224601 0.974451i \(-0.572108\pi\)
−0.224601 + 0.974451i \(0.572108\pi\)
\(174\) 0 0
\(175\) −4.90833 −0.371035
\(176\) 0 0
\(177\) 30.4222 2.28667
\(178\) 0 0
\(179\) 21.9083 1.63751 0.818753 0.574146i \(-0.194666\pi\)
0.818753 + 0.574146i \(0.194666\pi\)
\(180\) 0 0
\(181\) 21.6333 1.60799 0.803996 0.594635i \(-0.202704\pi\)
0.803996 + 0.594635i \(0.202704\pi\)
\(182\) 0 0
\(183\) 2.30278 0.170226
\(184\) 0 0
\(185\) −2.18335 −0.160523
\(186\) 0 0
\(187\) 2.60555 0.190537
\(188\) 0 0
\(189\) −16.2111 −1.17918
\(190\) 0 0
\(191\) 8.72498 0.631317 0.315659 0.948873i \(-0.397775\pi\)
0.315659 + 0.948873i \(0.397775\pi\)
\(192\) 0 0
\(193\) −12.6056 −0.907367 −0.453684 0.891163i \(-0.649890\pi\)
−0.453684 + 0.891163i \(0.649890\pi\)
\(194\) 0 0
\(195\) −0.605551 −0.0433644
\(196\) 0 0
\(197\) 13.0278 0.928189 0.464095 0.885786i \(-0.346380\pi\)
0.464095 + 0.885786i \(0.346380\pi\)
\(198\) 0 0
\(199\) −4.51388 −0.319980 −0.159990 0.987119i \(-0.551146\pi\)
−0.159990 + 0.987119i \(0.551146\pi\)
\(200\) 0 0
\(201\) −20.8167 −1.46829
\(202\) 0 0
\(203\) −1.39445 −0.0978711
\(204\) 0 0
\(205\) −2.57779 −0.180041
\(206\) 0 0
\(207\) 36.4222 2.53152
\(208\) 0 0
\(209\) 15.6333 1.08138
\(210\) 0 0
\(211\) 9.02776 0.621496 0.310748 0.950492i \(-0.399420\pi\)
0.310748 + 0.950492i \(0.399420\pi\)
\(212\) 0 0
\(213\) 6.60555 0.452605
\(214\) 0 0
\(215\) −0.211103 −0.0143971
\(216\) 0 0
\(217\) −10.3028 −0.699398
\(218\) 0 0
\(219\) −14.9083 −1.00741
\(220\) 0 0
\(221\) −0.605551 −0.0407338
\(222\) 0 0
\(223\) 21.6333 1.44867 0.724337 0.689446i \(-0.242146\pi\)
0.724337 + 0.689446i \(0.242146\pi\)
\(224\) 0 0
\(225\) −38.8167 −2.58778
\(226\) 0 0
\(227\) 10.9083 0.724011 0.362006 0.932176i \(-0.382092\pi\)
0.362006 + 0.932176i \(0.382092\pi\)
\(228\) 0 0
\(229\) 15.8167 1.04519 0.522597 0.852580i \(-0.324963\pi\)
0.522597 + 0.852580i \(0.324963\pi\)
\(230\) 0 0
\(231\) 8.60555 0.566204
\(232\) 0 0
\(233\) −1.21110 −0.0793420 −0.0396710 0.999213i \(-0.512631\pi\)
−0.0396710 + 0.999213i \(0.512631\pi\)
\(234\) 0 0
\(235\) 3.02776 0.197509
\(236\) 0 0
\(237\) 19.8167 1.28723
\(238\) 0 0
\(239\) −8.69722 −0.562577 −0.281288 0.959623i \(-0.590762\pi\)
−0.281288 + 0.959623i \(0.590762\pi\)
\(240\) 0 0
\(241\) 3.09167 0.199152 0.0995761 0.995030i \(-0.468251\pi\)
0.0995761 + 0.995030i \(0.468251\pi\)
\(242\) 0 0
\(243\) −49.8444 −3.19752
\(244\) 0 0
\(245\) 0.302776 0.0193436
\(246\) 0 0
\(247\) −3.63331 −0.231182
\(248\) 0 0
\(249\) 17.2111 1.09071
\(250\) 0 0
\(251\) 23.2111 1.46507 0.732536 0.680728i \(-0.238337\pi\)
0.732536 + 0.680728i \(0.238337\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 0 0
\(255\) 1.00000 0.0626224
\(256\) 0 0
\(257\) 7.81665 0.487589 0.243795 0.969827i \(-0.421608\pi\)
0.243795 + 0.969827i \(0.421608\pi\)
\(258\) 0 0
\(259\) −7.21110 −0.448076
\(260\) 0 0
\(261\) −11.0278 −0.682601
\(262\) 0 0
\(263\) −6.42221 −0.396010 −0.198005 0.980201i \(-0.563446\pi\)
−0.198005 + 0.980201i \(0.563446\pi\)
\(264\) 0 0
\(265\) 1.30278 0.0800289
\(266\) 0 0
\(267\) 6.60555 0.404253
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −10.1833 −0.618594 −0.309297 0.950965i \(-0.600094\pi\)
−0.309297 + 0.950965i \(0.600094\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 12.7889 0.771200
\(276\) 0 0
\(277\) −11.0278 −0.662594 −0.331297 0.943527i \(-0.607486\pi\)
−0.331297 + 0.943527i \(0.607486\pi\)
\(278\) 0 0
\(279\) −81.4777 −4.87794
\(280\) 0 0
\(281\) −20.3028 −1.21116 −0.605581 0.795784i \(-0.707059\pi\)
−0.605581 + 0.795784i \(0.707059\pi\)
\(282\) 0 0
\(283\) 10.3028 0.612436 0.306218 0.951961i \(-0.400936\pi\)
0.306218 + 0.951961i \(0.400936\pi\)
\(284\) 0 0
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) −8.51388 −0.502558
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 42.6333 2.49921
\(292\) 0 0
\(293\) −19.8167 −1.15770 −0.578851 0.815434i \(-0.696499\pi\)
−0.578851 + 0.815434i \(0.696499\pi\)
\(294\) 0 0
\(295\) −2.78890 −0.162376
\(296\) 0 0
\(297\) 42.2389 2.45095
\(298\) 0 0
\(299\) 2.78890 0.161286
\(300\) 0 0
\(301\) −0.697224 −0.0401873
\(302\) 0 0
\(303\) −54.2389 −3.11594
\(304\) 0 0
\(305\) −0.211103 −0.0120877
\(306\) 0 0
\(307\) −26.0000 −1.48390 −0.741949 0.670456i \(-0.766098\pi\)
−0.741949 + 0.670456i \(0.766098\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 16.7250 0.948387 0.474193 0.880421i \(-0.342740\pi\)
0.474193 + 0.880421i \(0.342740\pi\)
\(312\) 0 0
\(313\) 26.7250 1.51059 0.755293 0.655388i \(-0.227495\pi\)
0.755293 + 0.655388i \(0.227495\pi\)
\(314\) 0 0
\(315\) 2.39445 0.134912
\(316\) 0 0
\(317\) −5.39445 −0.302982 −0.151491 0.988459i \(-0.548408\pi\)
−0.151491 + 0.988459i \(0.548408\pi\)
\(318\) 0 0
\(319\) 3.63331 0.203426
\(320\) 0 0
\(321\) 54.8444 3.06112
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) −2.97224 −0.164870
\(326\) 0 0
\(327\) 6.60555 0.365288
\(328\) 0 0
\(329\) 10.0000 0.551318
\(330\) 0 0
\(331\) −32.7250 −1.79873 −0.899364 0.437201i \(-0.855970\pi\)
−0.899364 + 0.437201i \(0.855970\pi\)
\(332\) 0 0
\(333\) −57.0278 −3.12510
\(334\) 0 0
\(335\) 1.90833 0.104263
\(336\) 0 0
\(337\) 25.2111 1.37334 0.686668 0.726971i \(-0.259073\pi\)
0.686668 + 0.726971i \(0.259073\pi\)
\(338\) 0 0
\(339\) −15.8167 −0.859043
\(340\) 0 0
\(341\) 26.8444 1.45371
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −4.60555 −0.247955
\(346\) 0 0
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) 4.78890 0.256344 0.128172 0.991752i \(-0.459089\pi\)
0.128172 + 0.991752i \(0.459089\pi\)
\(350\) 0 0
\(351\) −9.81665 −0.523974
\(352\) 0 0
\(353\) −2.60555 −0.138680 −0.0693398 0.997593i \(-0.522089\pi\)
−0.0693398 + 0.997593i \(0.522089\pi\)
\(354\) 0 0
\(355\) −0.605551 −0.0321393
\(356\) 0 0
\(357\) 3.30278 0.174801
\(358\) 0 0
\(359\) −35.5139 −1.87435 −0.937175 0.348859i \(-0.886569\pi\)
−0.937175 + 0.348859i \(0.886569\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 13.9083 0.729998
\(364\) 0 0
\(365\) 1.36669 0.0715359
\(366\) 0 0
\(367\) −5.72498 −0.298842 −0.149421 0.988774i \(-0.547741\pi\)
−0.149421 + 0.988774i \(0.547741\pi\)
\(368\) 0 0
\(369\) −67.3305 −3.50509
\(370\) 0 0
\(371\) 4.30278 0.223389
\(372\) 0 0
\(373\) 8.69722 0.450325 0.225163 0.974321i \(-0.427709\pi\)
0.225163 + 0.974321i \(0.427709\pi\)
\(374\) 0 0
\(375\) 9.90833 0.511664
\(376\) 0 0
\(377\) −0.844410 −0.0434893
\(378\) 0 0
\(379\) 22.2389 1.14233 0.571167 0.820834i \(-0.306491\pi\)
0.571167 + 0.820834i \(0.306491\pi\)
\(380\) 0 0
\(381\) −66.4500 −3.40433
\(382\) 0 0
\(383\) 33.0278 1.68764 0.843820 0.536627i \(-0.180302\pi\)
0.843820 + 0.536627i \(0.180302\pi\)
\(384\) 0 0
\(385\) −0.788897 −0.0402059
\(386\) 0 0
\(387\) −5.51388 −0.280286
\(388\) 0 0
\(389\) −24.9083 −1.26290 −0.631451 0.775416i \(-0.717540\pi\)
−0.631451 + 0.775416i \(0.717540\pi\)
\(390\) 0 0
\(391\) −4.60555 −0.232913
\(392\) 0 0
\(393\) −60.8444 −3.06919
\(394\) 0 0
\(395\) −1.81665 −0.0914058
\(396\) 0 0
\(397\) −16.9083 −0.848605 −0.424302 0.905521i \(-0.639481\pi\)
−0.424302 + 0.905521i \(0.639481\pi\)
\(398\) 0 0
\(399\) 19.8167 0.992074
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) −6.23886 −0.310780
\(404\) 0 0
\(405\) 9.02776 0.448593
\(406\) 0 0
\(407\) 18.7889 0.931331
\(408\) 0 0
\(409\) 33.0278 1.63312 0.816559 0.577262i \(-0.195879\pi\)
0.816559 + 0.577262i \(0.195879\pi\)
\(410\) 0 0
\(411\) 45.9361 2.26586
\(412\) 0 0
\(413\) −9.21110 −0.453249
\(414\) 0 0
\(415\) −1.57779 −0.0774509
\(416\) 0 0
\(417\) 64.4500 3.15613
\(418\) 0 0
\(419\) 2.30278 0.112498 0.0562490 0.998417i \(-0.482086\pi\)
0.0562490 + 0.998417i \(0.482086\pi\)
\(420\) 0 0
\(421\) 9.90833 0.482902 0.241451 0.970413i \(-0.422377\pi\)
0.241451 + 0.970413i \(0.422377\pi\)
\(422\) 0 0
\(423\) 79.0833 3.84516
\(424\) 0 0
\(425\) 4.90833 0.238089
\(426\) 0 0
\(427\) −0.697224 −0.0337411
\(428\) 0 0
\(429\) 5.21110 0.251594
\(430\) 0 0
\(431\) −38.8444 −1.87107 −0.935535 0.353235i \(-0.885082\pi\)
−0.935535 + 0.353235i \(0.885082\pi\)
\(432\) 0 0
\(433\) −31.6333 −1.52020 −0.760100 0.649806i \(-0.774850\pi\)
−0.760100 + 0.649806i \(0.774850\pi\)
\(434\) 0 0
\(435\) 1.39445 0.0668587
\(436\) 0 0
\(437\) −27.6333 −1.32188
\(438\) 0 0
\(439\) 3.30278 0.157633 0.0788164 0.996889i \(-0.474886\pi\)
0.0788164 + 0.996889i \(0.474886\pi\)
\(440\) 0 0
\(441\) 7.90833 0.376587
\(442\) 0 0
\(443\) 18.4222 0.875265 0.437633 0.899154i \(-0.355817\pi\)
0.437633 + 0.899154i \(0.355817\pi\)
\(444\) 0 0
\(445\) −0.605551 −0.0287059
\(446\) 0 0
\(447\) 19.5139 0.922975
\(448\) 0 0
\(449\) 18.2389 0.860745 0.430372 0.902651i \(-0.358382\pi\)
0.430372 + 0.902651i \(0.358382\pi\)
\(450\) 0 0
\(451\) 22.1833 1.04457
\(452\) 0 0
\(453\) 36.7250 1.72549
\(454\) 0 0
\(455\) 0.183346 0.00859540
\(456\) 0 0
\(457\) 16.9083 0.790938 0.395469 0.918479i \(-0.370582\pi\)
0.395469 + 0.918479i \(0.370582\pi\)
\(458\) 0 0
\(459\) 16.2111 0.756669
\(460\) 0 0
\(461\) −5.21110 −0.242705 −0.121353 0.992609i \(-0.538723\pi\)
−0.121353 + 0.992609i \(0.538723\pi\)
\(462\) 0 0
\(463\) −17.5416 −0.815229 −0.407614 0.913154i \(-0.633639\pi\)
−0.407614 + 0.913154i \(0.633639\pi\)
\(464\) 0 0
\(465\) 10.3028 0.477780
\(466\) 0 0
\(467\) 2.78890 0.129055 0.0645274 0.997916i \(-0.479446\pi\)
0.0645274 + 0.997916i \(0.479446\pi\)
\(468\) 0 0
\(469\) 6.30278 0.291035
\(470\) 0 0
\(471\) 72.0555 3.32014
\(472\) 0 0
\(473\) 1.81665 0.0835298
\(474\) 0 0
\(475\) 29.4500 1.35126
\(476\) 0 0
\(477\) 34.0278 1.55802
\(478\) 0 0
\(479\) −20.6972 −0.945680 −0.472840 0.881148i \(-0.656771\pi\)
−0.472840 + 0.881148i \(0.656771\pi\)
\(480\) 0 0
\(481\) −4.36669 −0.199104
\(482\) 0 0
\(483\) −15.2111 −0.692129
\(484\) 0 0
\(485\) −3.90833 −0.177468
\(486\) 0 0
\(487\) −29.6333 −1.34281 −0.671407 0.741089i \(-0.734310\pi\)
−0.671407 + 0.741089i \(0.734310\pi\)
\(488\) 0 0
\(489\) 0.605551 0.0273840
\(490\) 0 0
\(491\) 34.3028 1.54806 0.774031 0.633147i \(-0.218237\pi\)
0.774031 + 0.633147i \(0.218237\pi\)
\(492\) 0 0
\(493\) 1.39445 0.0628028
\(494\) 0 0
\(495\) −6.23886 −0.280416
\(496\) 0 0
\(497\) −2.00000 −0.0897123
\(498\) 0 0
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 0 0
\(501\) −76.3583 −3.41144
\(502\) 0 0
\(503\) 36.1194 1.61049 0.805243 0.592945i \(-0.202035\pi\)
0.805243 + 0.592945i \(0.202035\pi\)
\(504\) 0 0
\(505\) 4.97224 0.221262
\(506\) 0 0
\(507\) 41.7250 1.85307
\(508\) 0 0
\(509\) 10.6056 0.470083 0.235041 0.971985i \(-0.424477\pi\)
0.235041 + 0.971985i \(0.424477\pi\)
\(510\) 0 0
\(511\) 4.51388 0.199682
\(512\) 0 0
\(513\) 97.2666 4.29443
\(514\) 0 0
\(515\) −0.366692 −0.0161584
\(516\) 0 0
\(517\) −26.0555 −1.14592
\(518\) 0 0
\(519\) 19.5139 0.856564
\(520\) 0 0
\(521\) −32.3305 −1.41643 −0.708213 0.705999i \(-0.750498\pi\)
−0.708213 + 0.705999i \(0.750498\pi\)
\(522\) 0 0
\(523\) −19.0278 −0.832026 −0.416013 0.909359i \(-0.636573\pi\)
−0.416013 + 0.909359i \(0.636573\pi\)
\(524\) 0 0
\(525\) 16.2111 0.707511
\(526\) 0 0
\(527\) 10.3028 0.448796
\(528\) 0 0
\(529\) −1.78890 −0.0777781
\(530\) 0 0
\(531\) −72.8444 −3.16118
\(532\) 0 0
\(533\) −5.15559 −0.223313
\(534\) 0 0
\(535\) −5.02776 −0.217369
\(536\) 0 0
\(537\) −72.3583 −3.12249
\(538\) 0 0
\(539\) −2.60555 −0.112229
\(540\) 0 0
\(541\) −17.8167 −0.765998 −0.382999 0.923749i \(-0.625109\pi\)
−0.382999 + 0.923749i \(0.625109\pi\)
\(542\) 0 0
\(543\) −71.4500 −3.06621
\(544\) 0 0
\(545\) −0.605551 −0.0259390
\(546\) 0 0
\(547\) 25.4500 1.08816 0.544081 0.839033i \(-0.316878\pi\)
0.544081 + 0.839033i \(0.316878\pi\)
\(548\) 0 0
\(549\) −5.51388 −0.235327
\(550\) 0 0
\(551\) 8.36669 0.356433
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) 0 0
\(555\) 7.21110 0.306094
\(556\) 0 0
\(557\) −3.21110 −0.136059 −0.0680294 0.997683i \(-0.521671\pi\)
−0.0680294 + 0.997683i \(0.521671\pi\)
\(558\) 0 0
\(559\) −0.422205 −0.0178574
\(560\) 0 0
\(561\) −8.60555 −0.363327
\(562\) 0 0
\(563\) −18.2389 −0.768676 −0.384338 0.923192i \(-0.625570\pi\)
−0.384338 + 0.923192i \(0.625570\pi\)
\(564\) 0 0
\(565\) 1.44996 0.0610003
\(566\) 0 0
\(567\) 29.8167 1.25218
\(568\) 0 0
\(569\) −23.4861 −0.984589 −0.492295 0.870429i \(-0.663842\pi\)
−0.492295 + 0.870429i \(0.663842\pi\)
\(570\) 0 0
\(571\) 9.63331 0.403141 0.201571 0.979474i \(-0.435395\pi\)
0.201571 + 0.979474i \(0.435395\pi\)
\(572\) 0 0
\(573\) −28.8167 −1.20383
\(574\) 0 0
\(575\) −22.6056 −0.942717
\(576\) 0 0
\(577\) 2.97224 0.123736 0.0618681 0.998084i \(-0.480294\pi\)
0.0618681 + 0.998084i \(0.480294\pi\)
\(578\) 0 0
\(579\) 41.6333 1.73022
\(580\) 0 0
\(581\) −5.21110 −0.216193
\(582\) 0 0
\(583\) −11.2111 −0.464316
\(584\) 0 0
\(585\) 1.44996 0.0599485
\(586\) 0 0
\(587\) −2.78890 −0.115110 −0.0575551 0.998342i \(-0.518330\pi\)
−0.0575551 + 0.998342i \(0.518330\pi\)
\(588\) 0 0
\(589\) 61.8167 2.54711
\(590\) 0 0
\(591\) −43.0278 −1.76993
\(592\) 0 0
\(593\) −33.6333 −1.38115 −0.690577 0.723259i \(-0.742643\pi\)
−0.690577 + 0.723259i \(0.742643\pi\)
\(594\) 0 0
\(595\) −0.302776 −0.0124126
\(596\) 0 0
\(597\) 14.9083 0.610157
\(598\) 0 0
\(599\) 21.5416 0.880167 0.440084 0.897957i \(-0.354949\pi\)
0.440084 + 0.897957i \(0.354949\pi\)
\(600\) 0 0
\(601\) 4.78890 0.195343 0.0976716 0.995219i \(-0.468861\pi\)
0.0976716 + 0.995219i \(0.468861\pi\)
\(602\) 0 0
\(603\) 49.8444 2.02982
\(604\) 0 0
\(605\) −1.27502 −0.0518369
\(606\) 0 0
\(607\) −26.9083 −1.09218 −0.546088 0.837728i \(-0.683883\pi\)
−0.546088 + 0.837728i \(0.683883\pi\)
\(608\) 0 0
\(609\) 4.60555 0.186626
\(610\) 0 0
\(611\) 6.05551 0.244980
\(612\) 0 0
\(613\) 15.9361 0.643652 0.321826 0.946799i \(-0.395703\pi\)
0.321826 + 0.946799i \(0.395703\pi\)
\(614\) 0 0
\(615\) 8.51388 0.343313
\(616\) 0 0
\(617\) −2.60555 −0.104896 −0.0524478 0.998624i \(-0.516702\pi\)
−0.0524478 + 0.998624i \(0.516702\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) −74.6611 −2.99605
\(622\) 0 0
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) 23.6333 0.945332
\(626\) 0 0
\(627\) −51.6333 −2.06204
\(628\) 0 0
\(629\) 7.21110 0.287525
\(630\) 0 0
\(631\) −12.1194 −0.482467 −0.241233 0.970467i \(-0.577552\pi\)
−0.241233 + 0.970467i \(0.577552\pi\)
\(632\) 0 0
\(633\) −29.8167 −1.18511
\(634\) 0 0
\(635\) 6.09167 0.241741
\(636\) 0 0
\(637\) 0.605551 0.0239928
\(638\) 0 0
\(639\) −15.8167 −0.625697
\(640\) 0 0
\(641\) −36.4222 −1.43859 −0.719295 0.694704i \(-0.755535\pi\)
−0.719295 + 0.694704i \(0.755535\pi\)
\(642\) 0 0
\(643\) −12.3305 −0.486269 −0.243134 0.969993i \(-0.578176\pi\)
−0.243134 + 0.969993i \(0.578176\pi\)
\(644\) 0 0
\(645\) 0.697224 0.0274532
\(646\) 0 0
\(647\) −34.6056 −1.36048 −0.680242 0.732987i \(-0.738125\pi\)
−0.680242 + 0.732987i \(0.738125\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 34.0278 1.33365
\(652\) 0 0
\(653\) 2.00000 0.0782660 0.0391330 0.999234i \(-0.487540\pi\)
0.0391330 + 0.999234i \(0.487540\pi\)
\(654\) 0 0
\(655\) 5.57779 0.217942
\(656\) 0 0
\(657\) 35.6972 1.39268
\(658\) 0 0
\(659\) −46.5416 −1.81300 −0.906502 0.422201i \(-0.861258\pi\)
−0.906502 + 0.422201i \(0.861258\pi\)
\(660\) 0 0
\(661\) −27.6333 −1.07481 −0.537406 0.843324i \(-0.680596\pi\)
−0.537406 + 0.843324i \(0.680596\pi\)
\(662\) 0 0
\(663\) 2.00000 0.0776736
\(664\) 0 0
\(665\) −1.81665 −0.0704468
\(666\) 0 0
\(667\) −6.42221 −0.248669
\(668\) 0 0
\(669\) −71.4500 −2.76242
\(670\) 0 0
\(671\) 1.81665 0.0701311
\(672\) 0 0
\(673\) 36.8444 1.42025 0.710124 0.704077i \(-0.248639\pi\)
0.710124 + 0.704077i \(0.248639\pi\)
\(674\) 0 0
\(675\) 79.5694 3.06263
\(676\) 0 0
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) −12.9083 −0.495376
\(680\) 0 0
\(681\) −36.0278 −1.38059
\(682\) 0 0
\(683\) 25.0278 0.957660 0.478830 0.877908i \(-0.341061\pi\)
0.478830 + 0.877908i \(0.341061\pi\)
\(684\) 0 0
\(685\) −4.21110 −0.160898
\(686\) 0 0
\(687\) −52.2389 −1.99304
\(688\) 0 0
\(689\) 2.60555 0.0992636
\(690\) 0 0
\(691\) 0.697224 0.0265237 0.0132618 0.999912i \(-0.495779\pi\)
0.0132618 + 0.999912i \(0.495779\pi\)
\(692\) 0 0
\(693\) −20.6056 −0.782740
\(694\) 0 0
\(695\) −5.90833 −0.224116
\(696\) 0 0
\(697\) 8.51388 0.322486
\(698\) 0 0
\(699\) 4.00000 0.151294
\(700\) 0 0
\(701\) 3.21110 0.121282 0.0606408 0.998160i \(-0.480686\pi\)
0.0606408 + 0.998160i \(0.480686\pi\)
\(702\) 0 0
\(703\) 43.2666 1.63183
\(704\) 0 0
\(705\) −10.0000 −0.376622
\(706\) 0 0
\(707\) 16.4222 0.617621
\(708\) 0 0
\(709\) −45.8167 −1.72068 −0.860340 0.509720i \(-0.829749\pi\)
−0.860340 + 0.509720i \(0.829749\pi\)
\(710\) 0 0
\(711\) −47.4500 −1.77951
\(712\) 0 0
\(713\) −47.4500 −1.77702
\(714\) 0 0
\(715\) −0.477718 −0.0178656
\(716\) 0 0
\(717\) 28.7250 1.07275
\(718\) 0 0
\(719\) −20.5139 −0.765039 −0.382519 0.923948i \(-0.624943\pi\)
−0.382519 + 0.923948i \(0.624943\pi\)
\(720\) 0 0
\(721\) −1.21110 −0.0451038
\(722\) 0 0
\(723\) −10.2111 −0.379755
\(724\) 0 0
\(725\) 6.84441 0.254195
\(726\) 0 0
\(727\) −37.2666 −1.38214 −0.691071 0.722787i \(-0.742861\pi\)
−0.691071 + 0.722787i \(0.742861\pi\)
\(728\) 0 0
\(729\) 75.1749 2.78426
\(730\) 0 0
\(731\) 0.697224 0.0257878
\(732\) 0 0
\(733\) 27.6333 1.02066 0.510330 0.859979i \(-0.329523\pi\)
0.510330 + 0.859979i \(0.329523\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) −16.4222 −0.604920
\(738\) 0 0
\(739\) −14.0917 −0.518371 −0.259185 0.965828i \(-0.583454\pi\)
−0.259185 + 0.965828i \(0.583454\pi\)
\(740\) 0 0
\(741\) 12.0000 0.440831
\(742\) 0 0
\(743\) −40.6056 −1.48967 −0.744837 0.667247i \(-0.767473\pi\)
−0.744837 + 0.667247i \(0.767473\pi\)
\(744\) 0 0
\(745\) −1.78890 −0.0655401
\(746\) 0 0
\(747\) −41.2111 −1.50784
\(748\) 0 0
\(749\) −16.6056 −0.606754
\(750\) 0 0
\(751\) −36.7889 −1.34245 −0.671223 0.741256i \(-0.734231\pi\)
−0.671223 + 0.741256i \(0.734231\pi\)
\(752\) 0 0
\(753\) −76.6611 −2.79368
\(754\) 0 0
\(755\) −3.36669 −0.122526
\(756\) 0 0
\(757\) 33.9083 1.23242 0.616210 0.787582i \(-0.288667\pi\)
0.616210 + 0.787582i \(0.288667\pi\)
\(758\) 0 0
\(759\) 39.6333 1.43860
\(760\) 0 0
\(761\) 39.3944 1.42805 0.714024 0.700121i \(-0.246871\pi\)
0.714024 + 0.700121i \(0.246871\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 0 0
\(765\) −2.39445 −0.0865715
\(766\) 0 0
\(767\) −5.57779 −0.201403
\(768\) 0 0
\(769\) −9.81665 −0.353998 −0.176999 0.984211i \(-0.556639\pi\)
−0.176999 + 0.984211i \(0.556639\pi\)
\(770\) 0 0
\(771\) −25.8167 −0.929764
\(772\) 0 0
\(773\) 50.8444 1.82875 0.914373 0.404872i \(-0.132684\pi\)
0.914373 + 0.404872i \(0.132684\pi\)
\(774\) 0 0
\(775\) 50.5694 1.81651
\(776\) 0 0
\(777\) 23.8167 0.854418
\(778\) 0 0
\(779\) 51.0833 1.83025
\(780\) 0 0
\(781\) 5.21110 0.186468
\(782\) 0 0
\(783\) 22.6056 0.807856
\(784\) 0 0
\(785\) −6.60555 −0.235762
\(786\) 0 0
\(787\) 14.4222 0.514096 0.257048 0.966399i \(-0.417250\pi\)
0.257048 + 0.966399i \(0.417250\pi\)
\(788\) 0 0
\(789\) 21.2111 0.755135
\(790\) 0 0
\(791\) 4.78890 0.170274
\(792\) 0 0
\(793\) −0.422205 −0.0149929
\(794\) 0 0
\(795\) −4.30278 −0.152604
\(796\) 0 0
\(797\) 29.2111 1.03471 0.517355 0.855771i \(-0.326917\pi\)
0.517355 + 0.855771i \(0.326917\pi\)
\(798\) 0 0
\(799\) −10.0000 −0.353775
\(800\) 0 0
\(801\) −15.8167 −0.558854
\(802\) 0 0
\(803\) −11.7611 −0.415042
\(804\) 0 0
\(805\) 1.39445 0.0491479
\(806\) 0 0
\(807\) 19.8167 0.697579
\(808\) 0 0
\(809\) −19.4500 −0.683824 −0.341912 0.939732i \(-0.611075\pi\)
−0.341912 + 0.939732i \(0.611075\pi\)
\(810\) 0 0
\(811\) 37.3583 1.31183 0.655913 0.754836i \(-0.272284\pi\)
0.655913 + 0.754836i \(0.272284\pi\)
\(812\) 0 0
\(813\) 33.6333 1.17957
\(814\) 0 0
\(815\) −0.0555128 −0.00194453
\(816\) 0 0
\(817\) 4.18335 0.146357
\(818\) 0 0
\(819\) 4.78890 0.167338
\(820\) 0 0
\(821\) 11.5778 0.404068 0.202034 0.979379i \(-0.435245\pi\)
0.202034 + 0.979379i \(0.435245\pi\)
\(822\) 0 0
\(823\) 31.0278 1.08156 0.540780 0.841164i \(-0.318129\pi\)
0.540780 + 0.841164i \(0.318129\pi\)
\(824\) 0 0
\(825\) −42.2389 −1.47057
\(826\) 0 0
\(827\) −38.8444 −1.35075 −0.675376 0.737473i \(-0.736019\pi\)
−0.675376 + 0.737473i \(0.736019\pi\)
\(828\) 0 0
\(829\) −36.6056 −1.27136 −0.635682 0.771951i \(-0.719281\pi\)
−0.635682 + 0.771951i \(0.719281\pi\)
\(830\) 0 0
\(831\) 36.4222 1.26347
\(832\) 0 0
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) 7.00000 0.242245
\(836\) 0 0
\(837\) 167.019 5.77303
\(838\) 0 0
\(839\) 6.78890 0.234379 0.117189 0.993110i \(-0.462612\pi\)
0.117189 + 0.993110i \(0.462612\pi\)
\(840\) 0 0
\(841\) −27.0555 −0.932949
\(842\) 0 0
\(843\) 67.0555 2.30951
\(844\) 0 0
\(845\) −3.82506 −0.131586
\(846\) 0 0
\(847\) −4.21110 −0.144695
\(848\) 0 0
\(849\) −34.0278 −1.16783
\(850\) 0 0
\(851\) −33.2111 −1.13846
\(852\) 0 0
\(853\) −4.78890 −0.163969 −0.0819844 0.996634i \(-0.526126\pi\)
−0.0819844 + 0.996634i \(0.526126\pi\)
\(854\) 0 0
\(855\) −14.3667 −0.491331
\(856\) 0 0
\(857\) −5.51388 −0.188350 −0.0941752 0.995556i \(-0.530021\pi\)
−0.0941752 + 0.995556i \(0.530021\pi\)
\(858\) 0 0
\(859\) −2.36669 −0.0807505 −0.0403753 0.999185i \(-0.512855\pi\)
−0.0403753 + 0.999185i \(0.512855\pi\)
\(860\) 0 0
\(861\) 28.1194 0.958308
\(862\) 0 0
\(863\) −37.9361 −1.29136 −0.645680 0.763608i \(-0.723426\pi\)
−0.645680 + 0.763608i \(0.723426\pi\)
\(864\) 0 0
\(865\) −1.78890 −0.0608243
\(866\) 0 0
\(867\) −3.30278 −0.112168
\(868\) 0 0
\(869\) 15.6333 0.530324
\(870\) 0 0
\(871\) 3.81665 0.129322
\(872\) 0 0
\(873\) −102.083 −3.45500
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −27.8167 −0.939302 −0.469651 0.882852i \(-0.655620\pi\)
−0.469651 + 0.882852i \(0.655620\pi\)
\(878\) 0 0
\(879\) 65.4500 2.20757
\(880\) 0 0
\(881\) 18.1194 0.610459 0.305230 0.952279i \(-0.401267\pi\)
0.305230 + 0.952279i \(0.401267\pi\)
\(882\) 0 0
\(883\) 23.2750 0.783267 0.391633 0.920121i \(-0.371910\pi\)
0.391633 + 0.920121i \(0.371910\pi\)
\(884\) 0 0
\(885\) 9.21110 0.309628
\(886\) 0 0
\(887\) −44.7250 −1.50172 −0.750859 0.660463i \(-0.770360\pi\)
−0.750859 + 0.660463i \(0.770360\pi\)
\(888\) 0 0
\(889\) 20.1194 0.674784
\(890\) 0 0
\(891\) −77.6888 −2.60267
\(892\) 0 0
\(893\) −60.0000 −2.00782
\(894\) 0 0
\(895\) 6.63331 0.221727
\(896\) 0 0
\(897\) −9.21110 −0.307550
\(898\) 0 0
\(899\) 14.3667 0.479156
\(900\) 0 0
\(901\) −4.30278 −0.143346
\(902\) 0 0
\(903\) 2.30278 0.0766316
\(904\) 0 0
\(905\) 6.55004 0.217731
\(906\) 0 0
\(907\) −32.4222 −1.07656 −0.538281 0.842766i \(-0.680926\pi\)
−0.538281 + 0.842766i \(0.680926\pi\)
\(908\) 0 0
\(909\) 129.872 4.30759
\(910\) 0 0
\(911\) 18.4222 0.610355 0.305177 0.952296i \(-0.401284\pi\)
0.305177 + 0.952296i \(0.401284\pi\)
\(912\) 0 0
\(913\) 13.5778 0.449359
\(914\) 0 0
\(915\) 0.697224 0.0230495
\(916\) 0 0
\(917\) 18.4222 0.608355
\(918\) 0 0
\(919\) −28.5139 −0.940586 −0.470293 0.882510i \(-0.655852\pi\)
−0.470293 + 0.882510i \(0.655852\pi\)
\(920\) 0 0
\(921\) 85.8722 2.82958
\(922\) 0 0
\(923\) −1.21110 −0.0398639
\(924\) 0 0
\(925\) 35.3944 1.16376
\(926\) 0 0
\(927\) −9.57779 −0.314576
\(928\) 0 0
\(929\) −28.3028 −0.928584 −0.464292 0.885682i \(-0.653691\pi\)
−0.464292 + 0.885682i \(0.653691\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) −55.2389 −1.80844
\(934\) 0 0
\(935\) 0.788897 0.0257997
\(936\) 0 0
\(937\) −10.7889 −0.352458 −0.176229 0.984349i \(-0.556390\pi\)
−0.176229 + 0.984349i \(0.556390\pi\)
\(938\) 0 0
\(939\) −88.2666 −2.88047
\(940\) 0 0
\(941\) −40.3305 −1.31474 −0.657369 0.753569i \(-0.728331\pi\)
−0.657369 + 0.753569i \(0.728331\pi\)
\(942\) 0 0
\(943\) −39.2111 −1.27689
\(944\) 0 0
\(945\) −4.90833 −0.159668
\(946\) 0 0
\(947\) −30.2389 −0.982631 −0.491315 0.870982i \(-0.663484\pi\)
−0.491315 + 0.870982i \(0.663484\pi\)
\(948\) 0 0
\(949\) 2.73338 0.0887294
\(950\) 0 0
\(951\) 17.8167 0.577745
\(952\) 0 0
\(953\) −41.7527 −1.35250 −0.676252 0.736670i \(-0.736397\pi\)
−0.676252 + 0.736670i \(0.736397\pi\)
\(954\) 0 0
\(955\) 2.64171 0.0854838
\(956\) 0 0
\(957\) −12.0000 −0.387905
\(958\) 0 0
\(959\) −13.9083 −0.449123
\(960\) 0 0
\(961\) 75.1472 2.42410
\(962\) 0 0
\(963\) −131.322 −4.23180
\(964\) 0 0
\(965\) −3.81665 −0.122862
\(966\) 0 0
\(967\) −16.6972 −0.536947 −0.268473 0.963287i \(-0.586519\pi\)
−0.268473 + 0.963287i \(0.586519\pi\)
\(968\) 0 0
\(969\) −19.8167 −0.636603
\(970\) 0 0
\(971\) 30.2389 0.970411 0.485206 0.874400i \(-0.338745\pi\)
0.485206 + 0.874400i \(0.338745\pi\)
\(972\) 0 0
\(973\) −19.5139 −0.625586
\(974\) 0 0
\(975\) 9.81665 0.314385
\(976\) 0 0
\(977\) −5.51388 −0.176405 −0.0882023 0.996103i \(-0.528112\pi\)
−0.0882023 + 0.996103i \(0.528112\pi\)
\(978\) 0 0
\(979\) 5.21110 0.166548
\(980\) 0 0
\(981\) −15.8167 −0.504987
\(982\) 0 0
\(983\) −3.51388 −0.112075 −0.0560377 0.998429i \(-0.517847\pi\)
−0.0560377 + 0.998429i \(0.517847\pi\)
\(984\) 0 0
\(985\) 3.94449 0.125682
\(986\) 0 0
\(987\) −33.0278 −1.05129
\(988\) 0 0
\(989\) −3.21110 −0.102107
\(990\) 0 0
\(991\) 6.00000 0.190596 0.0952981 0.995449i \(-0.469620\pi\)
0.0952981 + 0.995449i \(0.469620\pi\)
\(992\) 0 0
\(993\) 108.083 3.42992
\(994\) 0 0
\(995\) −1.36669 −0.0433271
\(996\) 0 0
\(997\) 22.0917 0.699650 0.349825 0.936815i \(-0.386241\pi\)
0.349825 + 0.936815i \(0.386241\pi\)
\(998\) 0 0
\(999\) 116.900 3.69855
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 476.2.a.a.1.1 2
3.2 odd 2 4284.2.a.p.1.1 2
4.3 odd 2 1904.2.a.l.1.2 2
7.6 odd 2 3332.2.a.n.1.2 2
8.3 odd 2 7616.2.a.m.1.1 2
8.5 even 2 7616.2.a.z.1.2 2
17.16 even 2 8092.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.a.1.1 2 1.1 even 1 trivial
1904.2.a.l.1.2 2 4.3 odd 2
3332.2.a.n.1.2 2 7.6 odd 2
4284.2.a.p.1.1 2 3.2 odd 2
7616.2.a.m.1.1 2 8.3 odd 2
7616.2.a.z.1.2 2 8.5 even 2
8092.2.a.n.1.2 2 17.16 even 2