Properties

Label 4840.2.a.y.1.4
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $1$
Dimension $4$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.737640\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.19353 q^{3} +1.00000 q^{5} -1.83785 q^{7} -1.57549 q^{9} +O(q^{10})\) \(q+1.19353 q^{3} +1.00000 q^{5} -1.83785 q^{7} -1.57549 q^{9} -1.25546 q^{13} +1.19353 q^{15} +6.44899 q^{17} +1.25120 q^{19} -2.19353 q^{21} -4.35567 q^{23} +1.00000 q^{25} -5.46097 q^{27} -0.726479 q^{29} -11.0790 q^{31} -1.83785 q^{35} -11.5086 q^{37} -1.49843 q^{39} +7.92427 q^{41} +0.606873 q^{43} -1.57549 q^{45} -3.98194 q^{47} -3.62230 q^{49} +7.69704 q^{51} +6.83290 q^{53} +1.49334 q^{57} -1.41201 q^{59} -9.75208 q^{61} +2.89552 q^{63} -1.25546 q^{65} +12.5899 q^{67} -5.19862 q^{69} -6.79665 q^{71} -10.0652 q^{73} +1.19353 q^{75} +6.51977 q^{79} -1.79134 q^{81} +0.644326 q^{83} +6.44899 q^{85} -0.867073 q^{87} +9.39723 q^{89} +2.30735 q^{91} -13.2231 q^{93} +1.25120 q^{95} -4.66191 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9} + 3 q^{13} - 2 q^{15} + 11 q^{17} + 2 q^{19} - 2 q^{21} - 11 q^{23} + 4 q^{25} + q^{27} + 4 q^{29} - 17 q^{31} - 7 q^{35} - 3 q^{37} + q^{39} + 13 q^{41} - 7 q^{43} - 4 q^{45} - q^{47} - 5 q^{49} - q^{51} - 15 q^{53} + 17 q^{57} - 17 q^{59} + 4 q^{61} + 15 q^{63} + 3 q^{65} + 7 q^{67} + 4 q^{69} - 15 q^{71} + 7 q^{73} - 2 q^{75} - 12 q^{79} - 8 q^{81} + 9 q^{83} + 11 q^{85} - 23 q^{87} - 12 q^{89} - 24 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97}+O(q^{100}) \) 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
\(3\) 1.19353 0.689083 0.344542 0.938771i \(-0.388034\pi\)
0.344542 + 0.938771i \(0.388034\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.83785 −0.694643 −0.347322 0.937746i \(-0.612909\pi\)
−0.347322 + 0.937746i \(0.612909\pi\)
\(8\) 0 0
\(9\) −1.57549 −0.525164
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −1.25546 −0.348202 −0.174101 0.984728i \(-0.555702\pi\)
−0.174101 + 0.984728i \(0.555702\pi\)
\(14\) 0 0
\(15\) 1.19353 0.308167
\(16\) 0 0
\(17\) 6.44899 1.56411 0.782055 0.623210i \(-0.214172\pi\)
0.782055 + 0.623210i \(0.214172\pi\)
\(18\) 0 0
\(19\) 1.25120 0.287045 0.143522 0.989647i \(-0.454157\pi\)
0.143522 + 0.989647i \(0.454157\pi\)
\(20\) 0 0
\(21\) −2.19353 −0.478667
\(22\) 0 0
\(23\) −4.35567 −0.908221 −0.454110 0.890945i \(-0.650043\pi\)
−0.454110 + 0.890945i \(0.650043\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.46097 −1.05097
\(28\) 0 0
\(29\) −0.726479 −0.134904 −0.0674519 0.997723i \(-0.521487\pi\)
−0.0674519 + 0.997723i \(0.521487\pi\)
\(30\) 0 0
\(31\) −11.0790 −1.98985 −0.994924 0.100626i \(-0.967916\pi\)
−0.994924 + 0.100626i \(0.967916\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.83785 −0.310654
\(36\) 0 0
\(37\) −11.5086 −1.89200 −0.946001 0.324162i \(-0.894918\pi\)
−0.946001 + 0.324162i \(0.894918\pi\)
\(38\) 0 0
\(39\) −1.49843 −0.239940
\(40\) 0 0
\(41\) 7.92427 1.23756 0.618781 0.785563i \(-0.287627\pi\)
0.618781 + 0.785563i \(0.287627\pi\)
\(42\) 0 0
\(43\) 0.606873 0.0925473 0.0462736 0.998929i \(-0.485265\pi\)
0.0462736 + 0.998929i \(0.485265\pi\)
\(44\) 0 0
\(45\) −1.57549 −0.234861
\(46\) 0 0
\(47\) −3.98194 −0.580826 −0.290413 0.956901i \(-0.593793\pi\)
−0.290413 + 0.956901i \(0.593793\pi\)
\(48\) 0 0
\(49\) −3.62230 −0.517471
\(50\) 0 0
\(51\) 7.69704 1.07780
\(52\) 0 0
\(53\) 6.83290 0.938571 0.469285 0.883047i \(-0.344512\pi\)
0.469285 + 0.883047i \(0.344512\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.49334 0.197798
\(58\) 0 0
\(59\) −1.41201 −0.183828 −0.0919141 0.995767i \(-0.529299\pi\)
−0.0919141 + 0.995767i \(0.529299\pi\)
\(60\) 0 0
\(61\) −9.75208 −1.24863 −0.624313 0.781174i \(-0.714621\pi\)
−0.624313 + 0.781174i \(0.714621\pi\)
\(62\) 0 0
\(63\) 2.89552 0.364802
\(64\) 0 0
\(65\) −1.25546 −0.155721
\(66\) 0 0
\(67\) 12.5899 1.53811 0.769053 0.639186i \(-0.220728\pi\)
0.769053 + 0.639186i \(0.220728\pi\)
\(68\) 0 0
\(69\) −5.19862 −0.625840
\(70\) 0 0
\(71\) −6.79665 −0.806614 −0.403307 0.915065i \(-0.632139\pi\)
−0.403307 + 0.915065i \(0.632139\pi\)
\(72\) 0 0
\(73\) −10.0652 −1.17804 −0.589022 0.808117i \(-0.700487\pi\)
−0.589022 + 0.808117i \(0.700487\pi\)
\(74\) 0 0
\(75\) 1.19353 0.137817
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.51977 0.733531 0.366765 0.930313i \(-0.380465\pi\)
0.366765 + 0.930313i \(0.380465\pi\)
\(80\) 0 0
\(81\) −1.79134 −0.199038
\(82\) 0 0
\(83\) 0.644326 0.0707239 0.0353620 0.999375i \(-0.488742\pi\)
0.0353620 + 0.999375i \(0.488742\pi\)
\(84\) 0 0
\(85\) 6.44899 0.699491
\(86\) 0 0
\(87\) −0.867073 −0.0929600
\(88\) 0 0
\(89\) 9.39723 0.996104 0.498052 0.867147i \(-0.334049\pi\)
0.498052 + 0.867147i \(0.334049\pi\)
\(90\) 0 0
\(91\) 2.30735 0.241876
\(92\) 0 0
\(93\) −13.2231 −1.37117
\(94\) 0 0
\(95\) 1.25120 0.128370
\(96\) 0 0
\(97\) −4.66191 −0.473345 −0.236673 0.971589i \(-0.576057\pi\)
−0.236673 + 0.971589i \(0.576057\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.1078 1.00576 0.502880 0.864356i \(-0.332274\pi\)
0.502880 + 0.864356i \(0.332274\pi\)
\(102\) 0 0
\(103\) −3.71135 −0.365690 −0.182845 0.983142i \(-0.558531\pi\)
−0.182845 + 0.983142i \(0.558531\pi\)
\(104\) 0 0
\(105\) −2.19353 −0.214066
\(106\) 0 0
\(107\) −7.69278 −0.743689 −0.371845 0.928295i \(-0.621275\pi\)
−0.371845 + 0.928295i \(0.621275\pi\)
\(108\) 0 0
\(109\) −3.06569 −0.293640 −0.146820 0.989163i \(-0.546904\pi\)
−0.146820 + 0.989163i \(0.546904\pi\)
\(110\) 0 0
\(111\) −13.7358 −1.30375
\(112\) 0 0
\(113\) −10.5168 −0.989341 −0.494670 0.869081i \(-0.664711\pi\)
−0.494670 + 0.869081i \(0.664711\pi\)
\(114\) 0 0
\(115\) −4.35567 −0.406169
\(116\) 0 0
\(117\) 1.97797 0.182864
\(118\) 0 0
\(119\) −11.8523 −1.08650
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 9.45783 0.852784
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −5.97978 −0.530620 −0.265310 0.964163i \(-0.585474\pi\)
−0.265310 + 0.964163i \(0.585474\pi\)
\(128\) 0 0
\(129\) 0.724319 0.0637728
\(130\) 0 0
\(131\) −9.13159 −0.797831 −0.398915 0.916988i \(-0.630613\pi\)
−0.398915 + 0.916988i \(0.630613\pi\)
\(132\) 0 0
\(133\) −2.29952 −0.199394
\(134\) 0 0
\(135\) −5.46097 −0.470006
\(136\) 0 0
\(137\) 5.48597 0.468698 0.234349 0.972153i \(-0.424704\pi\)
0.234349 + 0.972153i \(0.424704\pi\)
\(138\) 0 0
\(139\) −17.7510 −1.50562 −0.752808 0.658240i \(-0.771301\pi\)
−0.752808 + 0.658240i \(0.771301\pi\)
\(140\) 0 0
\(141\) −4.75255 −0.400237
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −0.726479 −0.0603308
\(146\) 0 0
\(147\) −4.32331 −0.356581
\(148\) 0 0
\(149\) −15.4005 −1.26166 −0.630829 0.775922i \(-0.717285\pi\)
−0.630829 + 0.775922i \(0.717285\pi\)
\(150\) 0 0
\(151\) −14.4892 −1.17912 −0.589558 0.807726i \(-0.700698\pi\)
−0.589558 + 0.807726i \(0.700698\pi\)
\(152\) 0 0
\(153\) −10.1603 −0.821415
\(154\) 0 0
\(155\) −11.0790 −0.889887
\(156\) 0 0
\(157\) −4.88487 −0.389855 −0.194928 0.980818i \(-0.562447\pi\)
−0.194928 + 0.980818i \(0.562447\pi\)
\(158\) 0 0
\(159\) 8.15525 0.646753
\(160\) 0 0
\(161\) 8.00509 0.630889
\(162\) 0 0
\(163\) −23.0578 −1.80603 −0.903013 0.429612i \(-0.858650\pi\)
−0.903013 + 0.429612i \(0.858650\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.24094 0.250791 0.125396 0.992107i \(-0.459980\pi\)
0.125396 + 0.992107i \(0.459980\pi\)
\(168\) 0 0
\(169\) −11.4238 −0.878755
\(170\) 0 0
\(171\) −1.97125 −0.150746
\(172\) 0 0
\(173\) 17.6970 1.34548 0.672741 0.739878i \(-0.265117\pi\)
0.672741 + 0.739878i \(0.265117\pi\)
\(174\) 0 0
\(175\) −1.83785 −0.138929
\(176\) 0 0
\(177\) −1.68527 −0.126673
\(178\) 0 0
\(179\) 17.6078 1.31607 0.658036 0.752987i \(-0.271388\pi\)
0.658036 + 0.752987i \(0.271388\pi\)
\(180\) 0 0
\(181\) −4.79712 −0.356567 −0.178284 0.983979i \(-0.557054\pi\)
−0.178284 + 0.983979i \(0.557054\pi\)
\(182\) 0 0
\(183\) −11.6394 −0.860407
\(184\) 0 0
\(185\) −11.5086 −0.846129
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 10.0365 0.730046
\(190\) 0 0
\(191\) −11.6292 −0.841459 −0.420730 0.907186i \(-0.638226\pi\)
−0.420730 + 0.907186i \(0.638226\pi\)
\(192\) 0 0
\(193\) 4.05323 0.291758 0.145879 0.989302i \(-0.453399\pi\)
0.145879 + 0.989302i \(0.453399\pi\)
\(194\) 0 0
\(195\) −1.49843 −0.107305
\(196\) 0 0
\(197\) −17.4066 −1.24017 −0.620084 0.784536i \(-0.712901\pi\)
−0.620084 + 0.784536i \(0.712901\pi\)
\(198\) 0 0
\(199\) 22.0409 1.56244 0.781218 0.624259i \(-0.214599\pi\)
0.781218 + 0.624259i \(0.214599\pi\)
\(200\) 0 0
\(201\) 15.0264 1.05988
\(202\) 0 0
\(203\) 1.33516 0.0937100
\(204\) 0 0
\(205\) 7.92427 0.553455
\(206\) 0 0
\(207\) 6.86233 0.476965
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.27193 −0.569463 −0.284731 0.958607i \(-0.591904\pi\)
−0.284731 + 0.958607i \(0.591904\pi\)
\(212\) 0 0
\(213\) −8.11198 −0.555824
\(214\) 0 0
\(215\) 0.606873 0.0413884
\(216\) 0 0
\(217\) 20.3616 1.38223
\(218\) 0 0
\(219\) −12.0131 −0.811770
\(220\) 0 0
\(221\) −8.09646 −0.544627
\(222\) 0 0
\(223\) 10.8470 0.726372 0.363186 0.931717i \(-0.381689\pi\)
0.363186 + 0.931717i \(0.381689\pi\)
\(224\) 0 0
\(225\) −1.57549 −0.105033
\(226\) 0 0
\(227\) −5.43321 −0.360615 −0.180308 0.983610i \(-0.557709\pi\)
−0.180308 + 0.983610i \(0.557709\pi\)
\(228\) 0 0
\(229\) 20.8976 1.38095 0.690476 0.723355i \(-0.257401\pi\)
0.690476 + 0.723355i \(0.257401\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26.9714 1.76696 0.883478 0.468472i \(-0.155195\pi\)
0.883478 + 0.468472i \(0.155195\pi\)
\(234\) 0 0
\(235\) −3.98194 −0.259753
\(236\) 0 0
\(237\) 7.78152 0.505464
\(238\) 0 0
\(239\) 23.2655 1.50492 0.752459 0.658639i \(-0.228867\pi\)
0.752459 + 0.658639i \(0.228867\pi\)
\(240\) 0 0
\(241\) 9.93604 0.640037 0.320018 0.947411i \(-0.396311\pi\)
0.320018 + 0.947411i \(0.396311\pi\)
\(242\) 0 0
\(243\) 14.2449 0.913811
\(244\) 0 0
\(245\) −3.62230 −0.231420
\(246\) 0 0
\(247\) −1.57083 −0.0999497
\(248\) 0 0
\(249\) 0.769020 0.0487347
\(250\) 0 0
\(251\) −17.8421 −1.12618 −0.563092 0.826394i \(-0.690388\pi\)
−0.563092 + 0.826394i \(0.690388\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 7.69704 0.482008
\(256\) 0 0
\(257\) −27.9332 −1.74243 −0.871214 0.490903i \(-0.836667\pi\)
−0.871214 + 0.490903i \(0.836667\pi\)
\(258\) 0 0
\(259\) 21.1511 1.31427
\(260\) 0 0
\(261\) 1.14456 0.0708467
\(262\) 0 0
\(263\) −14.8691 −0.916871 −0.458436 0.888728i \(-0.651590\pi\)
−0.458436 + 0.888728i \(0.651590\pi\)
\(264\) 0 0
\(265\) 6.83290 0.419742
\(266\) 0 0
\(267\) 11.2158 0.686399
\(268\) 0 0
\(269\) −7.98061 −0.486586 −0.243293 0.969953i \(-0.578228\pi\)
−0.243293 + 0.969953i \(0.578228\pi\)
\(270\) 0 0
\(271\) −18.8092 −1.14258 −0.571290 0.820748i \(-0.693557\pi\)
−0.571290 + 0.820748i \(0.693557\pi\)
\(272\) 0 0
\(273\) 2.75389 0.166673
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −10.8607 −0.652554 −0.326277 0.945274i \(-0.605794\pi\)
−0.326277 + 0.945274i \(0.605794\pi\)
\(278\) 0 0
\(279\) 17.4549 1.04500
\(280\) 0 0
\(281\) −24.1095 −1.43825 −0.719126 0.694880i \(-0.755457\pi\)
−0.719126 + 0.694880i \(0.755457\pi\)
\(282\) 0 0
\(283\) −9.11974 −0.542112 −0.271056 0.962564i \(-0.587373\pi\)
−0.271056 + 0.962564i \(0.587373\pi\)
\(284\) 0 0
\(285\) 1.49334 0.0884578
\(286\) 0 0
\(287\) −14.5636 −0.859665
\(288\) 0 0
\(289\) 24.5895 1.44644
\(290\) 0 0
\(291\) −5.56412 −0.326174
\(292\) 0 0
\(293\) 26.2288 1.53230 0.766151 0.642660i \(-0.222169\pi\)
0.766151 + 0.642660i \(0.222169\pi\)
\(294\) 0 0
\(295\) −1.41201 −0.0822105
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.46838 0.316245
\(300\) 0 0
\(301\) −1.11534 −0.0642873
\(302\) 0 0
\(303\) 12.0639 0.693052
\(304\) 0 0
\(305\) −9.75208 −0.558402
\(306\) 0 0
\(307\) −8.16034 −0.465735 −0.232868 0.972508i \(-0.574811\pi\)
−0.232868 + 0.972508i \(0.574811\pi\)
\(308\) 0 0
\(309\) −4.42960 −0.251991
\(310\) 0 0
\(311\) −13.4933 −0.765137 −0.382569 0.923927i \(-0.624960\pi\)
−0.382569 + 0.923927i \(0.624960\pi\)
\(312\) 0 0
\(313\) 7.83820 0.443041 0.221521 0.975156i \(-0.428898\pi\)
0.221521 + 0.975156i \(0.428898\pi\)
\(314\) 0 0
\(315\) 2.89552 0.163144
\(316\) 0 0
\(317\) 6.32808 0.355421 0.177710 0.984083i \(-0.443131\pi\)
0.177710 + 0.984083i \(0.443131\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −9.18154 −0.512464
\(322\) 0 0
\(323\) 8.06897 0.448969
\(324\) 0 0
\(325\) −1.25546 −0.0696405
\(326\) 0 0
\(327\) −3.65898 −0.202342
\(328\) 0 0
\(329\) 7.31822 0.403467
\(330\) 0 0
\(331\) −2.92392 −0.160713 −0.0803566 0.996766i \(-0.525606\pi\)
−0.0803566 + 0.996766i \(0.525606\pi\)
\(332\) 0 0
\(333\) 18.1317 0.993612
\(334\) 0 0
\(335\) 12.5899 0.687861
\(336\) 0 0
\(337\) 22.3306 1.21642 0.608211 0.793775i \(-0.291887\pi\)
0.608211 + 0.793775i \(0.291887\pi\)
\(338\) 0 0
\(339\) −12.5521 −0.681738
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 19.5222 1.05410
\(344\) 0 0
\(345\) −5.19862 −0.279884
\(346\) 0 0
\(347\) 5.62309 0.301863 0.150932 0.988544i \(-0.451773\pi\)
0.150932 + 0.988544i \(0.451773\pi\)
\(348\) 0 0
\(349\) −4.13639 −0.221416 −0.110708 0.993853i \(-0.535312\pi\)
−0.110708 + 0.993853i \(0.535312\pi\)
\(350\) 0 0
\(351\) 6.85605 0.365949
\(352\) 0 0
\(353\) 17.3112 0.921380 0.460690 0.887561i \(-0.347602\pi\)
0.460690 + 0.887561i \(0.347602\pi\)
\(354\) 0 0
\(355\) −6.79665 −0.360729
\(356\) 0 0
\(357\) −14.1460 −0.748687
\(358\) 0 0
\(359\) 15.0666 0.795187 0.397594 0.917562i \(-0.369845\pi\)
0.397594 + 0.917562i \(0.369845\pi\)
\(360\) 0 0
\(361\) −17.4345 −0.917605
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −10.0652 −0.526837
\(366\) 0 0
\(367\) −5.13586 −0.268089 −0.134045 0.990975i \(-0.542797\pi\)
−0.134045 + 0.990975i \(0.542797\pi\)
\(368\) 0 0
\(369\) −12.4846 −0.649924
\(370\) 0 0
\(371\) −12.5579 −0.651972
\(372\) 0 0
\(373\) −4.24462 −0.219778 −0.109889 0.993944i \(-0.535050\pi\)
−0.109889 + 0.993944i \(0.535050\pi\)
\(374\) 0 0
\(375\) 1.19353 0.0616335
\(376\) 0 0
\(377\) 0.912067 0.0469738
\(378\) 0 0
\(379\) −19.9850 −1.02656 −0.513280 0.858221i \(-0.671570\pi\)
−0.513280 + 0.858221i \(0.671570\pi\)
\(380\) 0 0
\(381\) −7.13703 −0.365641
\(382\) 0 0
\(383\) 7.47296 0.381850 0.190925 0.981605i \(-0.438851\pi\)
0.190925 + 0.981605i \(0.438851\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.956124 −0.0486025
\(388\) 0 0
\(389\) −23.4196 −1.18742 −0.593709 0.804680i \(-0.702337\pi\)
−0.593709 + 0.804680i \(0.702337\pi\)
\(390\) 0 0
\(391\) −28.0897 −1.42056
\(392\) 0 0
\(393\) −10.8988 −0.549772
\(394\) 0 0
\(395\) 6.51977 0.328045
\(396\) 0 0
\(397\) 13.2416 0.664578 0.332289 0.943178i \(-0.392179\pi\)
0.332289 + 0.943178i \(0.392179\pi\)
\(398\) 0 0
\(399\) −2.74454 −0.137399
\(400\) 0 0
\(401\) 8.32901 0.415931 0.207965 0.978136i \(-0.433316\pi\)
0.207965 + 0.978136i \(0.433316\pi\)
\(402\) 0 0
\(403\) 13.9093 0.692870
\(404\) 0 0
\(405\) −1.79134 −0.0890125
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 22.7399 1.12442 0.562209 0.826995i \(-0.309952\pi\)
0.562209 + 0.826995i \(0.309952\pi\)
\(410\) 0 0
\(411\) 6.54765 0.322972
\(412\) 0 0
\(413\) 2.59507 0.127695
\(414\) 0 0
\(415\) 0.644326 0.0316287
\(416\) 0 0
\(417\) −21.1863 −1.03750
\(418\) 0 0
\(419\) −15.9446 −0.778946 −0.389473 0.921038i \(-0.627343\pi\)
−0.389473 + 0.921038i \(0.627343\pi\)
\(420\) 0 0
\(421\) −33.0997 −1.61318 −0.806589 0.591112i \(-0.798689\pi\)
−0.806589 + 0.591112i \(0.798689\pi\)
\(422\) 0 0
\(423\) 6.27352 0.305029
\(424\) 0 0
\(425\) 6.44899 0.312822
\(426\) 0 0
\(427\) 17.9229 0.867349
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.1286 0.728718 0.364359 0.931259i \(-0.381288\pi\)
0.364359 + 0.931259i \(0.381288\pi\)
\(432\) 0 0
\(433\) 10.7199 0.515165 0.257583 0.966256i \(-0.417074\pi\)
0.257583 + 0.966256i \(0.417074\pi\)
\(434\) 0 0
\(435\) −0.867073 −0.0415730
\(436\) 0 0
\(437\) −5.44981 −0.260700
\(438\) 0 0
\(439\) −35.9820 −1.71733 −0.858663 0.512540i \(-0.828705\pi\)
−0.858663 + 0.512540i \(0.828705\pi\)
\(440\) 0 0
\(441\) 5.70690 0.271757
\(442\) 0 0
\(443\) 3.65928 0.173857 0.0869287 0.996215i \(-0.472295\pi\)
0.0869287 + 0.996215i \(0.472295\pi\)
\(444\) 0 0
\(445\) 9.39723 0.445471
\(446\) 0 0
\(447\) −18.3809 −0.869388
\(448\) 0 0
\(449\) 7.35426 0.347069 0.173534 0.984828i \(-0.444481\pi\)
0.173534 + 0.984828i \(0.444481\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −17.2933 −0.812508
\(454\) 0 0
\(455\) 2.30735 0.108170
\(456\) 0 0
\(457\) 31.4949 1.47327 0.736635 0.676291i \(-0.236414\pi\)
0.736635 + 0.676291i \(0.236414\pi\)
\(458\) 0 0
\(459\) −35.2178 −1.64382
\(460\) 0 0
\(461\) 39.7866 1.85305 0.926523 0.376239i \(-0.122783\pi\)
0.926523 + 0.376239i \(0.122783\pi\)
\(462\) 0 0
\(463\) −14.1053 −0.655529 −0.327764 0.944759i \(-0.606295\pi\)
−0.327764 + 0.944759i \(0.606295\pi\)
\(464\) 0 0
\(465\) −13.2231 −0.613206
\(466\) 0 0
\(467\) 5.15219 0.238415 0.119207 0.992869i \(-0.461965\pi\)
0.119207 + 0.992869i \(0.461965\pi\)
\(468\) 0 0
\(469\) −23.1384 −1.06843
\(470\) 0 0
\(471\) −5.83023 −0.268643
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.25120 0.0574089
\(476\) 0 0
\(477\) −10.7652 −0.492904
\(478\) 0 0
\(479\) 16.1277 0.736891 0.368446 0.929649i \(-0.379890\pi\)
0.368446 + 0.929649i \(0.379890\pi\)
\(480\) 0 0
\(481\) 14.4486 0.658800
\(482\) 0 0
\(483\) 9.55429 0.434735
\(484\) 0 0
\(485\) −4.66191 −0.211686
\(486\) 0 0
\(487\) 0.406098 0.0184020 0.00920102 0.999958i \(-0.497071\pi\)
0.00920102 + 0.999958i \(0.497071\pi\)
\(488\) 0 0
\(489\) −27.5201 −1.24450
\(490\) 0 0
\(491\) 11.8359 0.534147 0.267074 0.963676i \(-0.413943\pi\)
0.267074 + 0.963676i \(0.413943\pi\)
\(492\) 0 0
\(493\) −4.68506 −0.211004
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.4912 0.560309
\(498\) 0 0
\(499\) 27.3527 1.22448 0.612238 0.790674i \(-0.290269\pi\)
0.612238 + 0.790674i \(0.290269\pi\)
\(500\) 0 0
\(501\) 3.86815 0.172816
\(502\) 0 0
\(503\) −18.7444 −0.835771 −0.417885 0.908500i \(-0.637229\pi\)
−0.417885 + 0.908500i \(0.637229\pi\)
\(504\) 0 0
\(505\) 10.1078 0.449789
\(506\) 0 0
\(507\) −13.6346 −0.605535
\(508\) 0 0
\(509\) 35.4484 1.57122 0.785610 0.618722i \(-0.212349\pi\)
0.785610 + 0.618722i \(0.212349\pi\)
\(510\) 0 0
\(511\) 18.4984 0.818320
\(512\) 0 0
\(513\) −6.83276 −0.301674
\(514\) 0 0
\(515\) −3.71135 −0.163542
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 21.1219 0.927149
\(520\) 0 0
\(521\) −20.4748 −0.897016 −0.448508 0.893779i \(-0.648044\pi\)
−0.448508 + 0.893779i \(0.648044\pi\)
\(522\) 0 0
\(523\) −30.2594 −1.32315 −0.661576 0.749878i \(-0.730112\pi\)
−0.661576 + 0.749878i \(0.730112\pi\)
\(524\) 0 0
\(525\) −2.19353 −0.0957334
\(526\) 0 0
\(527\) −71.4484 −3.11234
\(528\) 0 0
\(529\) −4.02810 −0.175135
\(530\) 0 0
\(531\) 2.22461 0.0965400
\(532\) 0 0
\(533\) −9.94862 −0.430922
\(534\) 0 0
\(535\) −7.69278 −0.332588
\(536\) 0 0
\(537\) 21.0154 0.906882
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.49204 0.365101 0.182551 0.983196i \(-0.441565\pi\)
0.182551 + 0.983196i \(0.441565\pi\)
\(542\) 0 0
\(543\) −5.72549 −0.245704
\(544\) 0 0
\(545\) −3.06569 −0.131320
\(546\) 0 0
\(547\) 26.0488 1.11377 0.556883 0.830591i \(-0.311997\pi\)
0.556883 + 0.830591i \(0.311997\pi\)
\(548\) 0 0
\(549\) 15.3643 0.655734
\(550\) 0 0
\(551\) −0.908970 −0.0387234
\(552\) 0 0
\(553\) −11.9824 −0.509542
\(554\) 0 0
\(555\) −13.7358 −0.583054
\(556\) 0 0
\(557\) 11.8872 0.503677 0.251839 0.967769i \(-0.418965\pi\)
0.251839 + 0.967769i \(0.418965\pi\)
\(558\) 0 0
\(559\) −0.761906 −0.0322252
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −38.1919 −1.60959 −0.804797 0.593550i \(-0.797726\pi\)
−0.804797 + 0.593550i \(0.797726\pi\)
\(564\) 0 0
\(565\) −10.5168 −0.442447
\(566\) 0 0
\(567\) 3.29222 0.138260
\(568\) 0 0
\(569\) 29.9184 1.25425 0.627123 0.778920i \(-0.284232\pi\)
0.627123 + 0.778920i \(0.284232\pi\)
\(570\) 0 0
\(571\) 34.1271 1.42817 0.714087 0.700057i \(-0.246842\pi\)
0.714087 + 0.700057i \(0.246842\pi\)
\(572\) 0 0
\(573\) −13.8798 −0.579835
\(574\) 0 0
\(575\) −4.35567 −0.181644
\(576\) 0 0
\(577\) −5.81087 −0.241910 −0.120955 0.992658i \(-0.538596\pi\)
−0.120955 + 0.992658i \(0.538596\pi\)
\(578\) 0 0
\(579\) 4.83764 0.201045
\(580\) 0 0
\(581\) −1.18418 −0.0491279
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 1.97797 0.0817791
\(586\) 0 0
\(587\) 7.05828 0.291326 0.145663 0.989334i \(-0.453468\pi\)
0.145663 + 0.989334i \(0.453468\pi\)
\(588\) 0 0
\(589\) −13.8620 −0.571175
\(590\) 0 0
\(591\) −20.7752 −0.854579
\(592\) 0 0
\(593\) 35.1649 1.44405 0.722024 0.691868i \(-0.243212\pi\)
0.722024 + 0.691868i \(0.243212\pi\)
\(594\) 0 0
\(595\) −11.8523 −0.485897
\(596\) 0 0
\(597\) 26.3064 1.07665
\(598\) 0 0
\(599\) 45.3634 1.85350 0.926750 0.375678i \(-0.122590\pi\)
0.926750 + 0.375678i \(0.122590\pi\)
\(600\) 0 0
\(601\) 42.7574 1.74411 0.872055 0.489408i \(-0.162787\pi\)
0.872055 + 0.489408i \(0.162787\pi\)
\(602\) 0 0
\(603\) −19.8354 −0.807758
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −39.7376 −1.61290 −0.806450 0.591302i \(-0.798614\pi\)
−0.806450 + 0.591302i \(0.798614\pi\)
\(608\) 0 0
\(609\) 1.59355 0.0645740
\(610\) 0 0
\(611\) 4.99917 0.202245
\(612\) 0 0
\(613\) 47.4880 1.91802 0.959011 0.283367i \(-0.0914515\pi\)
0.959011 + 0.283367i \(0.0914515\pi\)
\(614\) 0 0
\(615\) 9.45783 0.381377
\(616\) 0 0
\(617\) 16.6834 0.671649 0.335824 0.941925i \(-0.390985\pi\)
0.335824 + 0.941925i \(0.390985\pi\)
\(618\) 0 0
\(619\) −28.9511 −1.16364 −0.581822 0.813316i \(-0.697660\pi\)
−0.581822 + 0.813316i \(0.697660\pi\)
\(620\) 0 0
\(621\) 23.7862 0.954508
\(622\) 0 0
\(623\) −17.2707 −0.691937
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −74.2189 −2.95930
\(630\) 0 0
\(631\) 9.97512 0.397103 0.198552 0.980090i \(-0.436376\pi\)
0.198552 + 0.980090i \(0.436376\pi\)
\(632\) 0 0
\(633\) −9.87277 −0.392407
\(634\) 0 0
\(635\) −5.97978 −0.237300
\(636\) 0 0
\(637\) 4.54766 0.180185
\(638\) 0 0
\(639\) 10.7081 0.423605
\(640\) 0 0
\(641\) −43.2582 −1.70860 −0.854298 0.519784i \(-0.826012\pi\)
−0.854298 + 0.519784i \(0.826012\pi\)
\(642\) 0 0
\(643\) 10.3729 0.409069 0.204534 0.978859i \(-0.434432\pi\)
0.204534 + 0.978859i \(0.434432\pi\)
\(644\) 0 0
\(645\) 0.724319 0.0285200
\(646\) 0 0
\(647\) 8.97743 0.352939 0.176470 0.984306i \(-0.443532\pi\)
0.176470 + 0.984306i \(0.443532\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 24.3021 0.952475
\(652\) 0 0
\(653\) −8.15796 −0.319246 −0.159623 0.987178i \(-0.551028\pi\)
−0.159623 + 0.987178i \(0.551028\pi\)
\(654\) 0 0
\(655\) −9.13159 −0.356801
\(656\) 0 0
\(657\) 15.8577 0.618667
\(658\) 0 0
\(659\) 12.2222 0.476111 0.238055 0.971252i \(-0.423490\pi\)
0.238055 + 0.971252i \(0.423490\pi\)
\(660\) 0 0
\(661\) −22.7382 −0.884412 −0.442206 0.896914i \(-0.645804\pi\)
−0.442206 + 0.896914i \(0.645804\pi\)
\(662\) 0 0
\(663\) −9.66334 −0.375293
\(664\) 0 0
\(665\) −2.29952 −0.0891715
\(666\) 0 0
\(667\) 3.16431 0.122522
\(668\) 0 0
\(669\) 12.9462 0.500531
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −18.0804 −0.696949 −0.348475 0.937318i \(-0.613300\pi\)
−0.348475 + 0.937318i \(0.613300\pi\)
\(674\) 0 0
\(675\) −5.46097 −0.210193
\(676\) 0 0
\(677\) 19.2431 0.739572 0.369786 0.929117i \(-0.379431\pi\)
0.369786 + 0.929117i \(0.379431\pi\)
\(678\) 0 0
\(679\) 8.56790 0.328806
\(680\) 0 0
\(681\) −6.48469 −0.248494
\(682\) 0 0
\(683\) 33.1235 1.26743 0.633717 0.773565i \(-0.281528\pi\)
0.633717 + 0.773565i \(0.281528\pi\)
\(684\) 0 0
\(685\) 5.48597 0.209608
\(686\) 0 0
\(687\) 24.9419 0.951591
\(688\) 0 0
\(689\) −8.57844 −0.326813
\(690\) 0 0
\(691\) −43.3858 −1.65047 −0.825237 0.564787i \(-0.808958\pi\)
−0.825237 + 0.564787i \(0.808958\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −17.7510 −0.673332
\(696\) 0 0
\(697\) 51.1035 1.93568
\(698\) 0 0
\(699\) 32.1911 1.21758
\(700\) 0 0
\(701\) −8.37727 −0.316405 −0.158203 0.987407i \(-0.550570\pi\)
−0.158203 + 0.987407i \(0.550570\pi\)
\(702\) 0 0
\(703\) −14.3996 −0.543089
\(704\) 0 0
\(705\) −4.75255 −0.178992
\(706\) 0 0
\(707\) −18.5766 −0.698644
\(708\) 0 0
\(709\) 0.475440 0.0178555 0.00892777 0.999960i \(-0.497158\pi\)
0.00892777 + 0.999960i \(0.497158\pi\)
\(710\) 0 0
\(711\) −10.2718 −0.385224
\(712\) 0 0
\(713\) 48.2566 1.80722
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 27.7680 1.03701
\(718\) 0 0
\(719\) −1.66515 −0.0620997 −0.0310499 0.999518i \(-0.509885\pi\)
−0.0310499 + 0.999518i \(0.509885\pi\)
\(720\) 0 0
\(721\) 6.82091 0.254024
\(722\) 0 0
\(723\) 11.8589 0.441038
\(724\) 0 0
\(725\) −0.726479 −0.0269808
\(726\) 0 0
\(727\) 14.5667 0.540247 0.270124 0.962826i \(-0.412935\pi\)
0.270124 + 0.962826i \(0.412935\pi\)
\(728\) 0 0
\(729\) 22.3757 0.828730
\(730\) 0 0
\(731\) 3.91372 0.144754
\(732\) 0 0
\(733\) −42.7581 −1.57931 −0.789653 0.613554i \(-0.789739\pi\)
−0.789653 + 0.613554i \(0.789739\pi\)
\(734\) 0 0
\(735\) −4.32331 −0.159468
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 45.0404 1.65684 0.828419 0.560109i \(-0.189241\pi\)
0.828419 + 0.560109i \(0.189241\pi\)
\(740\) 0 0
\(741\) −1.87483 −0.0688736
\(742\) 0 0
\(743\) −21.6833 −0.795482 −0.397741 0.917498i \(-0.630206\pi\)
−0.397741 + 0.917498i \(0.630206\pi\)
\(744\) 0 0
\(745\) −15.4005 −0.564231
\(746\) 0 0
\(747\) −1.01513 −0.0371417
\(748\) 0 0
\(749\) 14.1382 0.516598
\(750\) 0 0
\(751\) −3.62118 −0.132139 −0.0660693 0.997815i \(-0.521046\pi\)
−0.0660693 + 0.997815i \(0.521046\pi\)
\(752\) 0 0
\(753\) −21.2950 −0.776035
\(754\) 0 0
\(755\) −14.4892 −0.527316
\(756\) 0 0
\(757\) 32.3595 1.17612 0.588062 0.808816i \(-0.299891\pi\)
0.588062 + 0.808816i \(0.299891\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −26.4350 −0.958268 −0.479134 0.877742i \(-0.659049\pi\)
−0.479134 + 0.877742i \(0.659049\pi\)
\(762\) 0 0
\(763\) 5.63428 0.203975
\(764\) 0 0
\(765\) −10.1603 −0.367348
\(766\) 0 0
\(767\) 1.77273 0.0640094
\(768\) 0 0
\(769\) −31.6062 −1.13975 −0.569874 0.821732i \(-0.693008\pi\)
−0.569874 + 0.821732i \(0.693008\pi\)
\(770\) 0 0
\(771\) −33.3391 −1.20068
\(772\) 0 0
\(773\) 28.3808 1.02079 0.510393 0.859941i \(-0.329500\pi\)
0.510393 + 0.859941i \(0.329500\pi\)
\(774\) 0 0
\(775\) −11.0790 −0.397970
\(776\) 0 0
\(777\) 25.2444 0.905639
\(778\) 0 0
\(779\) 9.91484 0.355236
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 3.96729 0.141779
\(784\) 0 0
\(785\) −4.88487 −0.174349
\(786\) 0 0
\(787\) 29.1912 1.04056 0.520278 0.853997i \(-0.325828\pi\)
0.520278 + 0.853997i \(0.325828\pi\)
\(788\) 0 0
\(789\) −17.7467 −0.631800
\(790\) 0 0
\(791\) 19.3284 0.687239
\(792\) 0 0
\(793\) 12.2434 0.434775
\(794\) 0 0
\(795\) 8.15525 0.289237
\(796\) 0 0
\(797\) 4.47951 0.158672 0.0793362 0.996848i \(-0.474720\pi\)
0.0793362 + 0.996848i \(0.474720\pi\)
\(798\) 0 0
\(799\) −25.6795 −0.908475
\(800\) 0 0
\(801\) −14.8053 −0.523119
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 8.00509 0.282142
\(806\) 0 0
\(807\) −9.52507 −0.335298
\(808\) 0 0
\(809\) 41.6062 1.46280 0.731398 0.681951i \(-0.238868\pi\)
0.731398 + 0.681951i \(0.238868\pi\)
\(810\) 0 0
\(811\) −18.0341 −0.633262 −0.316631 0.948549i \(-0.602552\pi\)
−0.316631 + 0.948549i \(0.602552\pi\)
\(812\) 0 0
\(813\) −22.4493 −0.787333
\(814\) 0 0
\(815\) −23.0578 −0.807680
\(816\) 0 0
\(817\) 0.759319 0.0265652
\(818\) 0 0
\(819\) −3.63522 −0.127025
\(820\) 0 0
\(821\) 7.32591 0.255676 0.127838 0.991795i \(-0.459196\pi\)
0.127838 + 0.991795i \(0.459196\pi\)
\(822\) 0 0
\(823\) 15.1762 0.529008 0.264504 0.964385i \(-0.414792\pi\)
0.264504 + 0.964385i \(0.414792\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −52.0554 −1.81014 −0.905072 0.425257i \(-0.860184\pi\)
−0.905072 + 0.425257i \(0.860184\pi\)
\(828\) 0 0
\(829\) 29.1386 1.01203 0.506013 0.862526i \(-0.331119\pi\)
0.506013 + 0.862526i \(0.331119\pi\)
\(830\) 0 0
\(831\) −12.9625 −0.449664
\(832\) 0 0
\(833\) −23.3602 −0.809381
\(834\) 0 0
\(835\) 3.24094 0.112157
\(836\) 0 0
\(837\) 60.5022 2.09126
\(838\) 0 0
\(839\) −42.4131 −1.46426 −0.732131 0.681163i \(-0.761474\pi\)
−0.732131 + 0.681163i \(0.761474\pi\)
\(840\) 0 0
\(841\) −28.4722 −0.981801
\(842\) 0 0
\(843\) −28.7753 −0.991075
\(844\) 0 0
\(845\) −11.4238 −0.392991
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −10.8847 −0.373560
\(850\) 0 0
\(851\) 50.1277 1.71836
\(852\) 0 0
\(853\) 5.09345 0.174396 0.0871982 0.996191i \(-0.472209\pi\)
0.0871982 + 0.996191i \(0.472209\pi\)
\(854\) 0 0
\(855\) −1.97125 −0.0674155
\(856\) 0 0
\(857\) 27.4535 0.937795 0.468897 0.883253i \(-0.344651\pi\)
0.468897 + 0.883253i \(0.344651\pi\)
\(858\) 0 0
\(859\) 27.9580 0.953914 0.476957 0.878927i \(-0.341740\pi\)
0.476957 + 0.878927i \(0.341740\pi\)
\(860\) 0 0
\(861\) −17.3821 −0.592380
\(862\) 0 0
\(863\) −1.45954 −0.0496834 −0.0248417 0.999691i \(-0.507908\pi\)
−0.0248417 + 0.999691i \(0.507908\pi\)
\(864\) 0 0
\(865\) 17.6970 0.601718
\(866\) 0 0
\(867\) 29.3482 0.996717
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −15.8062 −0.535572
\(872\) 0 0
\(873\) 7.34481 0.248584
\(874\) 0 0
\(875\) −1.83785 −0.0621308
\(876\) 0 0
\(877\) 3.94038 0.133057 0.0665287 0.997785i \(-0.478808\pi\)
0.0665287 + 0.997785i \(0.478808\pi\)
\(878\) 0 0
\(879\) 31.3048 1.05588
\(880\) 0 0
\(881\) −44.6771 −1.50521 −0.752604 0.658473i \(-0.771203\pi\)
−0.752604 + 0.658473i \(0.771203\pi\)
\(882\) 0 0
\(883\) 47.5197 1.59916 0.799582 0.600557i \(-0.205054\pi\)
0.799582 + 0.600557i \(0.205054\pi\)
\(884\) 0 0
\(885\) −1.68527 −0.0566498
\(886\) 0 0
\(887\) −45.7025 −1.53454 −0.767270 0.641325i \(-0.778385\pi\)
−0.767270 + 0.641325i \(0.778385\pi\)
\(888\) 0 0
\(889\) 10.9900 0.368591
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −4.98220 −0.166723
\(894\) 0 0
\(895\) 17.6078 0.588565
\(896\) 0 0
\(897\) 6.52666 0.217919
\(898\) 0 0
\(899\) 8.04867 0.268438
\(900\) 0 0
\(901\) 44.0653 1.46803
\(902\) 0 0
\(903\) −1.33119 −0.0442993
\(904\) 0 0
\(905\) −4.79712 −0.159462
\(906\) 0 0
\(907\) −36.5095 −1.21228 −0.606139 0.795359i \(-0.707282\pi\)
−0.606139 + 0.795359i \(0.707282\pi\)
\(908\) 0 0
\(909\) −15.9247 −0.528189
\(910\) 0 0
\(911\) −22.1432 −0.733637 −0.366819 0.930293i \(-0.619553\pi\)
−0.366819 + 0.930293i \(0.619553\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −11.6394 −0.384786
\(916\) 0 0
\(917\) 16.7825 0.554208
\(918\) 0 0
\(919\) −54.4511 −1.79618 −0.898088 0.439816i \(-0.855044\pi\)
−0.898088 + 0.439816i \(0.855044\pi\)
\(920\) 0 0
\(921\) −9.73958 −0.320930
\(922\) 0 0
\(923\) 8.53293 0.280865
\(924\) 0 0
\(925\) −11.5086 −0.378401
\(926\) 0 0
\(927\) 5.84720 0.192047
\(928\) 0 0
\(929\) −0.727990 −0.0238846 −0.0119423 0.999929i \(-0.503801\pi\)
−0.0119423 + 0.999929i \(0.503801\pi\)
\(930\) 0 0
\(931\) −4.53221 −0.148537
\(932\) 0 0
\(933\) −16.1047 −0.527243
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 54.3101 1.77423 0.887116 0.461546i \(-0.152705\pi\)
0.887116 + 0.461546i \(0.152705\pi\)
\(938\) 0 0
\(939\) 9.35511 0.305292
\(940\) 0 0
\(941\) −19.3033 −0.629270 −0.314635 0.949213i \(-0.601882\pi\)
−0.314635 + 0.949213i \(0.601882\pi\)
\(942\) 0 0
\(943\) −34.5155 −1.12398
\(944\) 0 0
\(945\) 10.0365 0.326486
\(946\) 0 0
\(947\) 8.04409 0.261398 0.130699 0.991422i \(-0.458278\pi\)
0.130699 + 0.991422i \(0.458278\pi\)
\(948\) 0 0
\(949\) 12.6365 0.410198
\(950\) 0 0
\(951\) 7.55274 0.244914
\(952\) 0 0
\(953\) 39.1448 1.26803 0.634013 0.773323i \(-0.281407\pi\)
0.634013 + 0.773323i \(0.281407\pi\)
\(954\) 0 0
\(955\) −11.6292 −0.376312
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.0824 −0.325578
\(960\) 0 0
\(961\) 91.7444 2.95950
\(962\) 0 0
\(963\) 12.1199 0.390559
\(964\) 0 0
\(965\) 4.05323 0.130478
\(966\) 0 0
\(967\) 38.4906 1.23777 0.618887 0.785480i \(-0.287584\pi\)
0.618887 + 0.785480i \(0.287584\pi\)
\(968\) 0 0
\(969\) 9.63053 0.309377
\(970\) 0 0
\(971\) −53.8570 −1.72835 −0.864176 0.503189i \(-0.832160\pi\)
−0.864176 + 0.503189i \(0.832160\pi\)
\(972\) 0 0
\(973\) 32.6237 1.04587
\(974\) 0 0
\(975\) −1.49843 −0.0479881
\(976\) 0 0
\(977\) −10.6010 −0.339156 −0.169578 0.985517i \(-0.554240\pi\)
−0.169578 + 0.985517i \(0.554240\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 4.82997 0.154209
\(982\) 0 0
\(983\) −17.3284 −0.552691 −0.276346 0.961058i \(-0.589123\pi\)
−0.276346 + 0.961058i \(0.589123\pi\)
\(984\) 0 0
\(985\) −17.4066 −0.554620
\(986\) 0 0
\(987\) 8.73450 0.278022
\(988\) 0 0
\(989\) −2.64334 −0.0840534
\(990\) 0 0
\(991\) −1.68386 −0.0534894 −0.0267447 0.999642i \(-0.508514\pi\)
−0.0267447 + 0.999642i \(0.508514\pi\)
\(992\) 0 0
\(993\) −3.48978 −0.110745
\(994\) 0 0
\(995\) 22.0409 0.698742
\(996\) 0 0
\(997\) −16.7562 −0.530674 −0.265337 0.964156i \(-0.585483\pi\)
−0.265337 + 0.964156i \(0.585483\pi\)
\(998\) 0 0
\(999\) 62.8482 1.98843
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 4840.2.a.y.1.4 4
4.3 odd 2 9680.2.a.cu.1.1 4
11.3 even 5 440.2.y.a.361.1 8
11.4 even 5 440.2.y.a.401.1 yes 8
11.10 odd 2 4840.2.a.z.1.4 4
44.3 odd 10 880.2.bo.d.801.2 8
44.15 odd 10 880.2.bo.d.401.2 8
44.43 even 2 9680.2.a.ct.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.a.361.1 8 11.3 even 5
440.2.y.a.401.1 yes 8 11.4 even 5
880.2.bo.d.401.2 8 44.15 odd 10
880.2.bo.d.801.2 8 44.3 odd 10
4840.2.a.y.1.4 4 1.1 even 1 trivial
4840.2.a.z.1.4 4 11.10 odd 2
9680.2.a.ct.1.1 4 44.43 even 2
9680.2.a.cu.1.1 4 4.3 odd 2