Properties

Label 4600.2.e.q.4049.5
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4600,2,Mod(4049,4600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4600.4049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (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: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.3356224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} + 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.5
Root \(-0.254102i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.q.4049.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.93543i q^{3} +4.93543i q^{7} -0.745898 q^{9} +O(q^{10})\) \(q+1.93543i q^{3} +4.93543i q^{7} -0.745898 q^{9} +0.745898 q^{11} -1.74590i q^{13} -6.10856i q^{17} -5.44364 q^{19} -9.55220 q^{21} +1.00000i q^{23} +4.36266i q^{27} +1.66492 q^{29} -1.61676 q^{31} +1.44364i q^{33} +4.34625i q^{37} +3.37907 q^{39} -6.95184 q^{41} -5.01641i q^{43} +2.68133i q^{47} -17.3585 q^{49} +11.8227 q^{51} -13.7417i q^{53} -10.5358i q^{57} -12.2171 q^{59} -13.9794 q^{61} -3.68133i q^{63} +13.1044i q^{67} -1.93543 q^{69} -9.67716 q^{71} -5.69774i q^{73} +3.68133i q^{77} -10.3791 q^{79} -10.6813 q^{81} -0.637339i q^{83} +3.22235i q^{87} -2.72532 q^{89} +8.61676 q^{91} -3.12914i q^{93} +7.12497i q^{97} -0.556364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 6 q^{11} - 14 q^{19} - 12 q^{21} + 2 q^{29} + 20 q^{31} - 14 q^{39} - 20 q^{41} - 12 q^{49} + 18 q^{51} - 20 q^{59} - 26 q^{61} + 4 q^{69} + 20 q^{71} - 28 q^{79} - 50 q^{81} + 40 q^{89} + 22 q^{91} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.93543i 1.11742i 0.829362 + 0.558711i \(0.188704\pi\)
−0.829362 + 0.558711i \(0.811296\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.93543i 1.86542i 0.360630 + 0.932709i \(0.382562\pi\)
−0.360630 + 0.932709i \(0.617438\pi\)
\(8\) 0 0
\(9\) −0.745898 −0.248633
\(10\) 0 0
\(11\) 0.745898 0.224897 0.112448 0.993658i \(-0.464131\pi\)
0.112448 + 0.993658i \(0.464131\pi\)
\(12\) 0 0
\(13\) − 1.74590i − 0.484225i −0.970248 0.242113i \(-0.922160\pi\)
0.970248 0.242113i \(-0.0778403\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.10856i − 1.48154i −0.671757 0.740772i \(-0.734460\pi\)
0.671757 0.740772i \(-0.265540\pi\)
\(18\) 0 0
\(19\) −5.44364 −1.24886 −0.624428 0.781083i \(-0.714668\pi\)
−0.624428 + 0.781083i \(0.714668\pi\)
\(20\) 0 0
\(21\) −9.55220 −2.08446
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.36266i 0.839595i
\(28\) 0 0
\(29\) 1.66492 0.309169 0.154584 0.987980i \(-0.450596\pi\)
0.154584 + 0.987980i \(0.450596\pi\)
\(30\) 0 0
\(31\) −1.61676 −0.290379 −0.145190 0.989404i \(-0.546379\pi\)
−0.145190 + 0.989404i \(0.546379\pi\)
\(32\) 0 0
\(33\) 1.44364i 0.251305i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.34625i 0.714520i 0.934005 + 0.357260i \(0.116289\pi\)
−0.934005 + 0.357260i \(0.883711\pi\)
\(38\) 0 0
\(39\) 3.37907 0.541084
\(40\) 0 0
\(41\) −6.95184 −1.08569 −0.542847 0.839831i \(-0.682654\pi\)
−0.542847 + 0.839831i \(0.682654\pi\)
\(42\) 0 0
\(43\) − 5.01641i − 0.764995i −0.923957 0.382497i \(-0.875064\pi\)
0.923957 0.382497i \(-0.124936\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.68133i 0.391112i 0.980693 + 0.195556i \(0.0626512\pi\)
−0.980693 + 0.195556i \(0.937349\pi\)
\(48\) 0 0
\(49\) −17.3585 −2.47978
\(50\) 0 0
\(51\) 11.8227 1.65551
\(52\) 0 0
\(53\) − 13.7417i − 1.88757i −0.330558 0.943786i \(-0.607237\pi\)
0.330558 0.943786i \(-0.392763\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 10.5358i − 1.39550i
\(58\) 0 0
\(59\) −12.2171 −1.59053 −0.795267 0.606260i \(-0.792669\pi\)
−0.795267 + 0.606260i \(0.792669\pi\)
\(60\) 0 0
\(61\) −13.9794 −1.78988 −0.894941 0.446185i \(-0.852782\pi\)
−0.894941 + 0.446185i \(0.852782\pi\)
\(62\) 0 0
\(63\) − 3.68133i − 0.463804i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.1044i 1.60096i 0.599362 + 0.800478i \(0.295421\pi\)
−0.599362 + 0.800478i \(0.704579\pi\)
\(68\) 0 0
\(69\) −1.93543 −0.232999
\(70\) 0 0
\(71\) −9.67716 −1.14847 −0.574234 0.818691i \(-0.694700\pi\)
−0.574234 + 0.818691i \(0.694700\pi\)
\(72\) 0 0
\(73\) − 5.69774i − 0.666870i −0.942773 0.333435i \(-0.891792\pi\)
0.942773 0.333435i \(-0.108208\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.68133i 0.419527i
\(78\) 0 0
\(79\) −10.3791 −1.16774 −0.583868 0.811848i \(-0.698462\pi\)
−0.583868 + 0.811848i \(0.698462\pi\)
\(80\) 0 0
\(81\) −10.6813 −1.18681
\(82\) 0 0
\(83\) − 0.637339i − 0.0699570i −0.999388 0.0349785i \(-0.988864\pi\)
0.999388 0.0349785i \(-0.0111363\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.22235i 0.345472i
\(88\) 0 0
\(89\) −2.72532 −0.288884 −0.144442 0.989513i \(-0.546139\pi\)
−0.144442 + 0.989513i \(0.546139\pi\)
\(90\) 0 0
\(91\) 8.61676 0.903282
\(92\) 0 0
\(93\) − 3.12914i − 0.324476i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.12497i 0.723431i 0.932289 + 0.361715i \(0.117809\pi\)
−0.932289 + 0.361715i \(0.882191\pi\)
\(98\) 0 0
\(99\) −0.556364 −0.0559167
\(100\) 0 0
\(101\) 14.9753 1.49009 0.745047 0.667012i \(-0.232427\pi\)
0.745047 + 0.667012i \(0.232427\pi\)
\(102\) 0 0
\(103\) 10.8667i 1.07073i 0.844622 + 0.535364i \(0.179825\pi\)
−0.844622 + 0.535364i \(0.820175\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.49180i − 0.724259i −0.932128 0.362130i \(-0.882050\pi\)
0.932128 0.362130i \(-0.117950\pi\)
\(108\) 0 0
\(109\) 18.3309 1.75578 0.877891 0.478860i \(-0.158950\pi\)
0.877891 + 0.478860i \(0.158950\pi\)
\(110\) 0 0
\(111\) −8.41188 −0.798420
\(112\) 0 0
\(113\) − 0.379068i − 0.0356597i −0.999841 0.0178299i \(-0.994324\pi\)
0.999841 0.0178299i \(-0.00567572\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.30226i 0.120394i
\(118\) 0 0
\(119\) 30.1484 2.76370
\(120\) 0 0
\(121\) −10.4436 −0.949421
\(122\) 0 0
\(123\) − 13.4548i − 1.21318i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.64852i 0.235018i 0.993072 + 0.117509i \(0.0374909\pi\)
−0.993072 + 0.117509i \(0.962509\pi\)
\(128\) 0 0
\(129\) 9.70892 0.854822
\(130\) 0 0
\(131\) −5.53579 −0.483664 −0.241832 0.970318i \(-0.577748\pi\)
−0.241832 + 0.970318i \(0.577748\pi\)
\(132\) 0 0
\(133\) − 26.8667i − 2.32964i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.16896i 0.527050i 0.964653 + 0.263525i \(0.0848851\pi\)
−0.964653 + 0.263525i \(0.915115\pi\)
\(138\) 0 0
\(139\) 20.7693 1.76163 0.880815 0.473460i \(-0.156995\pi\)
0.880815 + 0.473460i \(0.156995\pi\)
\(140\) 0 0
\(141\) −5.18953 −0.437038
\(142\) 0 0
\(143\) − 1.30226i − 0.108901i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 33.5962i − 2.77097i
\(148\) 0 0
\(149\) 1.06457 0.0872128 0.0436064 0.999049i \(-0.486115\pi\)
0.0436064 + 0.999049i \(0.486115\pi\)
\(150\) 0 0
\(151\) −12.6608 −1.03032 −0.515159 0.857095i \(-0.672267\pi\)
−0.515159 + 0.857095i \(0.672267\pi\)
\(152\) 0 0
\(153\) 4.55636i 0.368360i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.8873i 1.18813i 0.804416 + 0.594067i \(0.202479\pi\)
−0.804416 + 0.594067i \(0.797521\pi\)
\(158\) 0 0
\(159\) 26.5962 2.10921
\(160\) 0 0
\(161\) −4.93543 −0.388967
\(162\) 0 0
\(163\) − 6.72949i − 0.527094i −0.964646 0.263547i \(-0.915108\pi\)
0.964646 0.263547i \(-0.0848925\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 2.98359i − 0.230877i −0.993315 0.115439i \(-0.963173\pi\)
0.993315 0.115439i \(-0.0368274\pi\)
\(168\) 0 0
\(169\) 9.95184 0.765526
\(170\) 0 0
\(171\) 4.06040 0.310506
\(172\) 0 0
\(173\) 15.8503i 1.20508i 0.798091 + 0.602538i \(0.205844\pi\)
−0.798091 + 0.602538i \(0.794156\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 23.6454i − 1.77730i
\(178\) 0 0
\(179\) −3.02759 −0.226292 −0.113146 0.993578i \(-0.536093\pi\)
−0.113146 + 0.993578i \(0.536093\pi\)
\(180\) 0 0
\(181\) 1.82270 0.135481 0.0677403 0.997703i \(-0.478421\pi\)
0.0677403 + 0.997703i \(0.478421\pi\)
\(182\) 0 0
\(183\) − 27.0562i − 2.00005i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 4.55636i − 0.333194i
\(188\) 0 0
\(189\) −21.5316 −1.56619
\(190\) 0 0
\(191\) 19.6126 1.41912 0.709559 0.704646i \(-0.248894\pi\)
0.709559 + 0.704646i \(0.248894\pi\)
\(192\) 0 0
\(193\) 0.335076i 0.0241193i 0.999927 + 0.0120597i \(0.00383880\pi\)
−0.999927 + 0.0120597i \(0.996161\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 20.5316i − 1.46282i −0.681939 0.731409i \(-0.738863\pi\)
0.681939 0.731409i \(-0.261137\pi\)
\(198\) 0 0
\(199\) −16.7253 −1.18563 −0.592813 0.805340i \(-0.701983\pi\)
−0.592813 + 0.805340i \(0.701983\pi\)
\(200\) 0 0
\(201\) −25.3627 −1.78894
\(202\) 0 0
\(203\) 8.21712i 0.576729i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 0.745898i − 0.0518435i
\(208\) 0 0
\(209\) −4.06040 −0.280864
\(210\) 0 0
\(211\) −13.3955 −0.922183 −0.461091 0.887353i \(-0.652542\pi\)
−0.461091 + 0.887353i \(0.652542\pi\)
\(212\) 0 0
\(213\) − 18.7295i − 1.28332i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 7.97942i − 0.541679i
\(218\) 0 0
\(219\) 11.0276 0.745175
\(220\) 0 0
\(221\) −10.6649 −0.717400
\(222\) 0 0
\(223\) − 1.27468i − 0.0853587i −0.999089 0.0426794i \(-0.986411\pi\)
0.999089 0.0426794i \(-0.0135894\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.57978i 0.237598i 0.992918 + 0.118799i \(0.0379045\pi\)
−0.992918 + 0.118799i \(0.962096\pi\)
\(228\) 0 0
\(229\) −10.4119 −0.688036 −0.344018 0.938963i \(-0.611788\pi\)
−0.344018 + 0.938963i \(0.611788\pi\)
\(230\) 0 0
\(231\) −7.12497 −0.468788
\(232\) 0 0
\(233\) − 3.06040i − 0.200493i −0.994963 0.100247i \(-0.968037\pi\)
0.994963 0.100247i \(-0.0319632\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 20.0880i − 1.30485i
\(238\) 0 0
\(239\) 19.1895 1.24127 0.620634 0.784100i \(-0.286875\pi\)
0.620634 + 0.784100i \(0.286875\pi\)
\(240\) 0 0
\(241\) 0.346255 0.0223042 0.0111521 0.999938i \(-0.496450\pi\)
0.0111521 + 0.999938i \(0.496450\pi\)
\(242\) 0 0
\(243\) − 7.58501i − 0.486579i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 9.50403i 0.604727i
\(248\) 0 0
\(249\) 1.23353 0.0781715
\(250\) 0 0
\(251\) 14.3861 0.908041 0.454021 0.890991i \(-0.349989\pi\)
0.454021 + 0.890991i \(0.349989\pi\)
\(252\) 0 0
\(253\) 0.745898i 0.0468942i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.28586i 0.0802095i 0.999195 + 0.0401047i \(0.0127692\pi\)
−0.999195 + 0.0401047i \(0.987231\pi\)
\(258\) 0 0
\(259\) −21.4506 −1.33288
\(260\) 0 0
\(261\) −1.24186 −0.0768694
\(262\) 0 0
\(263\) − 28.2294i − 1.74070i −0.492437 0.870348i \(-0.663894\pi\)
0.492437 0.870348i \(-0.336106\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 5.27468i − 0.322805i
\(268\) 0 0
\(269\) 19.4395 1.18525 0.592623 0.805480i \(-0.298093\pi\)
0.592623 + 0.805480i \(0.298093\pi\)
\(270\) 0 0
\(271\) −6.42723 −0.390426 −0.195213 0.980761i \(-0.562540\pi\)
−0.195213 + 0.980761i \(0.562540\pi\)
\(272\) 0 0
\(273\) 16.6772i 1.00935i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.08514i 0.245452i 0.992441 + 0.122726i \(0.0391637\pi\)
−0.992441 + 0.122726i \(0.960836\pi\)
\(278\) 0 0
\(279\) 1.20594 0.0721978
\(280\) 0 0
\(281\) −6.38741 −0.381041 −0.190520 0.981683i \(-0.561018\pi\)
−0.190520 + 0.981683i \(0.561018\pi\)
\(282\) 0 0
\(283\) 21.4835i 1.27706i 0.769597 + 0.638530i \(0.220457\pi\)
−0.769597 + 0.638530i \(0.779543\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 34.3103i − 2.02527i
\(288\) 0 0
\(289\) −20.3145 −1.19497
\(290\) 0 0
\(291\) −13.7899 −0.808378
\(292\) 0 0
\(293\) 16.1208i 0.941787i 0.882190 + 0.470894i \(0.156068\pi\)
−0.882190 + 0.470894i \(0.843932\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.25410i 0.188822i
\(298\) 0 0
\(299\) 1.74590 0.100968
\(300\) 0 0
\(301\) 24.7581 1.42704
\(302\) 0 0
\(303\) 28.9836i 1.66506i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 6.96302i − 0.397400i −0.980060 0.198700i \(-0.936328\pi\)
0.980060 0.198700i \(-0.0636720\pi\)
\(308\) 0 0
\(309\) −21.0318 −1.19645
\(310\) 0 0
\(311\) −4.14031 −0.234776 −0.117388 0.993086i \(-0.537452\pi\)
−0.117388 + 0.993086i \(0.537452\pi\)
\(312\) 0 0
\(313\) − 6.26528i − 0.354135i −0.984199 0.177067i \(-0.943339\pi\)
0.984199 0.177067i \(-0.0566610\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.53996i 0.535817i 0.963444 + 0.267909i \(0.0863326\pi\)
−0.963444 + 0.267909i \(0.913667\pi\)
\(318\) 0 0
\(319\) 1.24186 0.0695310
\(320\) 0 0
\(321\) 14.4999 0.809304
\(322\) 0 0
\(323\) 33.2528i 1.85023i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 35.4782i 1.96195i
\(328\) 0 0
\(329\) −13.2335 −0.729588
\(330\) 0 0
\(331\) −14.3023 −0.786123 −0.393062 0.919512i \(-0.628584\pi\)
−0.393062 + 0.919512i \(0.628584\pi\)
\(332\) 0 0
\(333\) − 3.24186i − 0.177653i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 3.34731i − 0.182340i −0.995835 0.0911699i \(-0.970939\pi\)
0.995835 0.0911699i \(-0.0290606\pi\)
\(338\) 0 0
\(339\) 0.733661 0.0398470
\(340\) 0 0
\(341\) −1.20594 −0.0653054
\(342\) 0 0
\(343\) − 51.1236i − 2.76042i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 23.9383i − 1.28507i −0.766255 0.642537i \(-0.777882\pi\)
0.766255 0.642537i \(-0.222118\pi\)
\(348\) 0 0
\(349\) 11.9149 0.637788 0.318894 0.947790i \(-0.396689\pi\)
0.318894 + 0.947790i \(0.396689\pi\)
\(350\) 0 0
\(351\) 7.61676 0.406553
\(352\) 0 0
\(353\) 25.2447i 1.34364i 0.740714 + 0.671820i \(0.234487\pi\)
−0.740714 + 0.671820i \(0.765513\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 58.3502i 3.08822i
\(358\) 0 0
\(359\) −20.4671 −1.08021 −0.540105 0.841598i \(-0.681615\pi\)
−0.540105 + 0.841598i \(0.681615\pi\)
\(360\) 0 0
\(361\) 10.6332 0.559641
\(362\) 0 0
\(363\) − 20.2130i − 1.06090i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 36.4999i − 1.90528i −0.304104 0.952639i \(-0.598357\pi\)
0.304104 0.952639i \(-0.401643\pi\)
\(368\) 0 0
\(369\) 5.18537 0.269939
\(370\) 0 0
\(371\) 67.8214 3.52111
\(372\) 0 0
\(373\) 5.57978i 0.288910i 0.989511 + 0.144455i \(0.0461429\pi\)
−0.989511 + 0.144455i \(0.953857\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 2.90679i − 0.149707i
\(378\) 0 0
\(379\) −36.6168 −1.88088 −0.940438 0.339964i \(-0.889585\pi\)
−0.940438 + 0.339964i \(0.889585\pi\)
\(380\) 0 0
\(381\) −5.12603 −0.262614
\(382\) 0 0
\(383\) 28.7581i 1.46947i 0.678353 + 0.734736i \(0.262694\pi\)
−0.678353 + 0.734736i \(0.737306\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.74173i 0.190203i
\(388\) 0 0
\(389\) −26.3913 −1.33809 −0.669046 0.743221i \(-0.733297\pi\)
−0.669046 + 0.743221i \(0.733297\pi\)
\(390\) 0 0
\(391\) 6.10856 0.308923
\(392\) 0 0
\(393\) − 10.7141i − 0.540457i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.08931i 0.0546710i 0.999626 + 0.0273355i \(0.00870225\pi\)
−0.999626 + 0.0273355i \(0.991298\pi\)
\(398\) 0 0
\(399\) 51.9987 2.60319
\(400\) 0 0
\(401\) −22.5962 −1.12840 −0.564200 0.825638i \(-0.690815\pi\)
−0.564200 + 0.825638i \(0.690815\pi\)
\(402\) 0 0
\(403\) 2.82270i 0.140609i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.24186i 0.160693i
\(408\) 0 0
\(409\) 16.0510 0.793671 0.396835 0.917890i \(-0.370108\pi\)
0.396835 + 0.917890i \(0.370108\pi\)
\(410\) 0 0
\(411\) −11.9396 −0.588937
\(412\) 0 0
\(413\) − 60.2968i − 2.96701i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 40.1976i 1.96849i
\(418\) 0 0
\(419\) 24.4342 1.19369 0.596845 0.802356i \(-0.296421\pi\)
0.596845 + 0.802356i \(0.296421\pi\)
\(420\) 0 0
\(421\) −11.3473 −0.553034 −0.276517 0.961009i \(-0.589180\pi\)
−0.276517 + 0.961009i \(0.589180\pi\)
\(422\) 0 0
\(423\) − 2.00000i − 0.0972433i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 68.9945i − 3.33888i
\(428\) 0 0
\(429\) 2.52044 0.121688
\(430\) 0 0
\(431\) 7.83805 0.377546 0.188773 0.982021i \(-0.439549\pi\)
0.188773 + 0.982021i \(0.439549\pi\)
\(432\) 0 0
\(433\) 37.9023i 1.82147i 0.412990 + 0.910735i \(0.364484\pi\)
−0.412990 + 0.910735i \(0.635516\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 5.44364i − 0.260404i
\(438\) 0 0
\(439\) 17.7704 0.848134 0.424067 0.905631i \(-0.360602\pi\)
0.424067 + 0.905631i \(0.360602\pi\)
\(440\) 0 0
\(441\) 12.9477 0.616556
\(442\) 0 0
\(443\) − 16.9466i − 0.805158i −0.915385 0.402579i \(-0.868114\pi\)
0.915385 0.402579i \(-0.131886\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.06040i 0.0974535i
\(448\) 0 0
\(449\) 21.0890 0.995253 0.497627 0.867391i \(-0.334205\pi\)
0.497627 + 0.867391i \(0.334205\pi\)
\(450\) 0 0
\(451\) −5.18537 −0.244169
\(452\) 0 0
\(453\) − 24.5040i − 1.15130i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13.1372i 0.614532i 0.951624 + 0.307266i \(0.0994142\pi\)
−0.951624 + 0.307266i \(0.900586\pi\)
\(458\) 0 0
\(459\) 26.6496 1.24390
\(460\) 0 0
\(461\) −14.5439 −0.677375 −0.338687 0.940899i \(-0.609983\pi\)
−0.338687 + 0.940899i \(0.609983\pi\)
\(462\) 0 0
\(463\) − 19.4283i − 0.902909i −0.892294 0.451455i \(-0.850905\pi\)
0.892294 0.451455i \(-0.149095\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 3.61259i − 0.167171i −0.996501 0.0835855i \(-0.973363\pi\)
0.996501 0.0835855i \(-0.0266372\pi\)
\(468\) 0 0
\(469\) −64.6758 −2.98645
\(470\) 0 0
\(471\) −28.8133 −1.32765
\(472\) 0 0
\(473\) − 3.74173i − 0.172045i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 10.2499i 0.469312i
\(478\) 0 0
\(479\) −5.13720 −0.234725 −0.117362 0.993089i \(-0.537444\pi\)
−0.117362 + 0.993089i \(0.537444\pi\)
\(480\) 0 0
\(481\) 7.58812 0.345988
\(482\) 0 0
\(483\) − 9.55220i − 0.434640i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 24.7693i − 1.12240i −0.827679 0.561202i \(-0.810339\pi\)
0.827679 0.561202i \(-0.189661\pi\)
\(488\) 0 0
\(489\) 13.0245 0.588987
\(490\) 0 0
\(491\) 3.54413 0.159944 0.0799721 0.996797i \(-0.474517\pi\)
0.0799721 + 0.996797i \(0.474517\pi\)
\(492\) 0 0
\(493\) − 10.1703i − 0.458047i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 47.7610i − 2.14237i
\(498\) 0 0
\(499\) −31.5358 −1.41174 −0.705868 0.708344i \(-0.749443\pi\)
−0.705868 + 0.708344i \(0.749443\pi\)
\(500\) 0 0
\(501\) 5.77454 0.257988
\(502\) 0 0
\(503\) 32.2346i 1.43727i 0.695388 + 0.718635i \(0.255233\pi\)
−0.695388 + 0.718635i \(0.744767\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 19.2611i 0.855416i
\(508\) 0 0
\(509\) −0.302263 −0.0133976 −0.00669878 0.999978i \(-0.502132\pi\)
−0.00669878 + 0.999978i \(0.502132\pi\)
\(510\) 0 0
\(511\) 28.1208 1.24399
\(512\) 0 0
\(513\) − 23.7487i − 1.04853i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.00000i 0.0879599i
\(518\) 0 0
\(519\) −30.6772 −1.34658
\(520\) 0 0
\(521\) −30.3463 −1.32949 −0.664747 0.747069i \(-0.731461\pi\)
−0.664747 + 0.747069i \(0.731461\pi\)
\(522\) 0 0
\(523\) 42.5878i 1.86224i 0.364717 + 0.931118i \(0.381166\pi\)
−0.364717 + 0.931118i \(0.618834\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.87609i 0.430209i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 9.11273 0.395459
\(532\) 0 0
\(533\) 12.1372i 0.525721i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 5.85969i − 0.252864i
\(538\) 0 0
\(539\) −12.9477 −0.557696
\(540\) 0 0
\(541\) 38.9313 1.67379 0.836893 0.547367i \(-0.184370\pi\)
0.836893 + 0.547367i \(0.184370\pi\)
\(542\) 0 0
\(543\) 3.52772i 0.151389i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 16.6004i 0.709780i 0.934908 + 0.354890i \(0.115482\pi\)
−0.934908 + 0.354890i \(0.884518\pi\)
\(548\) 0 0
\(549\) 10.4272 0.445023
\(550\) 0 0
\(551\) −9.06324 −0.386107
\(552\) 0 0
\(553\) − 51.2252i − 2.17832i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17.4506i 0.739408i 0.929150 + 0.369704i \(0.120541\pi\)
−0.929150 + 0.369704i \(0.879459\pi\)
\(558\) 0 0
\(559\) −8.75814 −0.370430
\(560\) 0 0
\(561\) 8.81853 0.372319
\(562\) 0 0
\(563\) − 8.25827i − 0.348045i −0.984742 0.174022i \(-0.944324\pi\)
0.984742 0.174022i \(-0.0556765\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 52.7170i − 2.21391i
\(568\) 0 0
\(569\) 41.4178 1.73633 0.868163 0.496279i \(-0.165301\pi\)
0.868163 + 0.496279i \(0.165301\pi\)
\(570\) 0 0
\(571\) 7.19059 0.300917 0.150458 0.988616i \(-0.451925\pi\)
0.150458 + 0.988616i \(0.451925\pi\)
\(572\) 0 0
\(573\) 37.9588i 1.58575i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 39.3103i 1.63651i 0.574855 + 0.818255i \(0.305058\pi\)
−0.574855 + 0.818255i \(0.694942\pi\)
\(578\) 0 0
\(579\) −0.648517 −0.0269515
\(580\) 0 0
\(581\) 3.14554 0.130499
\(582\) 0 0
\(583\) − 10.2499i − 0.424509i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.7899i 1.18829i 0.804359 + 0.594143i \(0.202509\pi\)
−0.804359 + 0.594143i \(0.797491\pi\)
\(588\) 0 0
\(589\) 8.80107 0.362642
\(590\) 0 0
\(591\) 39.7376 1.63458
\(592\) 0 0
\(593\) 40.8873i 1.67904i 0.543330 + 0.839519i \(0.317163\pi\)
−0.543330 + 0.839519i \(0.682837\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 32.3707i − 1.32485i
\(598\) 0 0
\(599\) −19.1801 −0.783679 −0.391840 0.920034i \(-0.628161\pi\)
−0.391840 + 0.920034i \(0.628161\pi\)
\(600\) 0 0
\(601\) 3.36683 0.137336 0.0686679 0.997640i \(-0.478125\pi\)
0.0686679 + 0.997640i \(0.478125\pi\)
\(602\) 0 0
\(603\) − 9.77454i − 0.398050i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 13.1924i 0.535462i 0.963494 + 0.267731i \(0.0862738\pi\)
−0.963494 + 0.267731i \(0.913726\pi\)
\(608\) 0 0
\(609\) −15.9037 −0.644450
\(610\) 0 0
\(611\) 4.68133 0.189386
\(612\) 0 0
\(613\) − 32.6207i − 1.31754i −0.752346 0.658768i \(-0.771078\pi\)
0.752346 0.658768i \(-0.228922\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 41.7212i 1.67963i 0.542872 + 0.839815i \(0.317337\pi\)
−0.542872 + 0.839815i \(0.682663\pi\)
\(618\) 0 0
\(619\) 0.339245 0.0136354 0.00681771 0.999977i \(-0.497830\pi\)
0.00681771 + 0.999977i \(0.497830\pi\)
\(620\) 0 0
\(621\) −4.36266 −0.175068
\(622\) 0 0
\(623\) − 13.4506i − 0.538889i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 7.85863i − 0.313843i
\(628\) 0 0
\(629\) 26.5494 1.05859
\(630\) 0 0
\(631\) 3.71725 0.147982 0.0739908 0.997259i \(-0.476426\pi\)
0.0739908 + 0.997259i \(0.476426\pi\)
\(632\) 0 0
\(633\) − 25.9260i − 1.03047i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 30.3062i 1.20077i
\(638\) 0 0
\(639\) 7.21818 0.285547
\(640\) 0 0
\(641\) −22.0328 −0.870244 −0.435122 0.900372i \(-0.643295\pi\)
−0.435122 + 0.900372i \(0.643295\pi\)
\(642\) 0 0
\(643\) 24.6842i 0.973449i 0.873556 + 0.486724i \(0.161808\pi\)
−0.873556 + 0.486724i \(0.838192\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.1731i 1.26486i 0.774619 + 0.632428i \(0.217942\pi\)
−0.774619 + 0.632428i \(0.782058\pi\)
\(648\) 0 0
\(649\) −9.11273 −0.357706
\(650\) 0 0
\(651\) 15.4436 0.605284
\(652\) 0 0
\(653\) 26.9466i 1.05450i 0.849709 + 0.527251i \(0.176777\pi\)
−0.849709 + 0.527251i \(0.823223\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 4.24993i 0.165806i
\(658\) 0 0
\(659\) 19.7417 0.769029 0.384514 0.923119i \(-0.374369\pi\)
0.384514 + 0.923119i \(0.374369\pi\)
\(660\) 0 0
\(661\) −1.28692 −0.0500552 −0.0250276 0.999687i \(-0.507967\pi\)
−0.0250276 + 0.999687i \(0.507967\pi\)
\(662\) 0 0
\(663\) − 20.6412i − 0.801639i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.66492i 0.0644661i
\(668\) 0 0
\(669\) 2.46705 0.0953817
\(670\) 0 0
\(671\) −10.4272 −0.402539
\(672\) 0 0
\(673\) − 12.4970i − 0.481725i −0.970559 0.240862i \(-0.922570\pi\)
0.970559 0.240862i \(-0.0774303\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 2.69251i − 0.103482i −0.998661 0.0517408i \(-0.983523\pi\)
0.998661 0.0517408i \(-0.0164770\pi\)
\(678\) 0 0
\(679\) −35.1648 −1.34950
\(680\) 0 0
\(681\) −6.92842 −0.265498
\(682\) 0 0
\(683\) 2.72115i 0.104122i 0.998644 + 0.0520610i \(0.0165790\pi\)
−0.998644 + 0.0520610i \(0.983421\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 20.1515i − 0.768827i
\(688\) 0 0
\(689\) −23.9917 −0.914010
\(690\) 0 0
\(691\) 38.3051 1.45719 0.728597 0.684942i \(-0.240173\pi\)
0.728597 + 0.684942i \(0.240173\pi\)
\(692\) 0 0
\(693\) − 2.74590i − 0.104308i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 42.4657i 1.60850i
\(698\) 0 0
\(699\) 5.92319 0.224036
\(700\) 0 0
\(701\) 39.8472 1.50501 0.752504 0.658588i \(-0.228846\pi\)
0.752504 + 0.658588i \(0.228846\pi\)
\(702\) 0 0
\(703\) − 23.6594i − 0.892332i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 73.9094i 2.77965i
\(708\) 0 0
\(709\) 44.8995 1.68624 0.843118 0.537728i \(-0.180717\pi\)
0.843118 + 0.537728i \(0.180717\pi\)
\(710\) 0 0
\(711\) 7.74173 0.290338
\(712\) 0 0
\(713\) − 1.61676i − 0.0605482i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 37.1400i 1.38702i
\(718\) 0 0
\(719\) 31.5400 1.17624 0.588121 0.808773i \(-0.299868\pi\)
0.588121 + 0.808773i \(0.299868\pi\)
\(720\) 0 0
\(721\) −53.6318 −1.99735
\(722\) 0 0
\(723\) 0.670152i 0.0249232i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 48.9219i − 1.81441i −0.420687 0.907206i \(-0.638211\pi\)
0.420687 0.907206i \(-0.361789\pi\)
\(728\) 0 0
\(729\) −17.3637 −0.643101
\(730\) 0 0
\(731\) −30.6430 −1.13337
\(732\) 0 0
\(733\) 14.1536i 0.522776i 0.965234 + 0.261388i \(0.0841801\pi\)
−0.965234 + 0.261388i \(0.915820\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.77454i 0.360050i
\(738\) 0 0
\(739\) −4.46421 −0.164219 −0.0821093 0.996623i \(-0.526166\pi\)
−0.0821093 + 0.996623i \(0.526166\pi\)
\(740\) 0 0
\(741\) −18.3944 −0.675736
\(742\) 0 0
\(743\) 34.2898i 1.25797i 0.777418 + 0.628985i \(0.216529\pi\)
−0.777418 + 0.628985i \(0.783471\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.475390i 0.0173936i
\(748\) 0 0
\(749\) 36.9753 1.35105
\(750\) 0 0
\(751\) −8.28275 −0.302242 −0.151121 0.988515i \(-0.548288\pi\)
−0.151121 + 0.988515i \(0.548288\pi\)
\(752\) 0 0
\(753\) 27.8433i 1.01467i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.32985i 0.121025i 0.998167 + 0.0605127i \(0.0192736\pi\)
−0.998167 + 0.0605127i \(0.980726\pi\)
\(758\) 0 0
\(759\) −1.44364 −0.0524007
\(760\) 0 0
\(761\) −18.4077 −0.667279 −0.333640 0.942701i \(-0.608277\pi\)
−0.333640 + 0.942701i \(0.608277\pi\)
\(762\) 0 0
\(763\) 90.4710i 3.27527i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 21.3298i 0.770176i
\(768\) 0 0
\(769\) −51.4506 −1.85536 −0.927679 0.373379i \(-0.878199\pi\)
−0.927679 + 0.373379i \(0.878199\pi\)
\(770\) 0 0
\(771\) −2.48869 −0.0896279
\(772\) 0 0
\(773\) − 17.4506i − 0.627656i −0.949480 0.313828i \(-0.898388\pi\)
0.949480 0.313828i \(-0.101612\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 41.5163i − 1.48939i
\(778\) 0 0
\(779\) 37.8433 1.35588
\(780\) 0 0
\(781\) −7.21818 −0.258287
\(782\) 0 0
\(783\) 7.26350i 0.259576i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 21.2335i − 0.756893i −0.925623 0.378447i \(-0.876458\pi\)
0.925623 0.378447i \(-0.123542\pi\)
\(788\) 0 0
\(789\) 54.6360 1.94509
\(790\) 0 0
\(791\) 1.87086 0.0665203
\(792\) 0 0
\(793\) 24.4067i 0.866706i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 25.3215i − 0.896934i −0.893799 0.448467i \(-0.851970\pi\)
0.893799 0.448467i \(-0.148030\pi\)
\(798\) 0 0
\(799\) 16.3791 0.579450
\(800\) 0 0
\(801\) 2.03281 0.0718259
\(802\) 0 0
\(803\) − 4.24993i − 0.149977i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 37.6238i 1.32442i
\(808\) 0 0
\(809\) −6.14137 −0.215919 −0.107960 0.994155i \(-0.534432\pi\)
−0.107960 + 0.994155i \(0.534432\pi\)
\(810\) 0 0
\(811\) 15.5051 0.544457 0.272229 0.962233i \(-0.412239\pi\)
0.272229 + 0.962233i \(0.412239\pi\)
\(812\) 0 0
\(813\) − 12.4395i − 0.436271i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 27.3075i 0.955368i
\(818\) 0 0
\(819\) −6.42723 −0.224586
\(820\) 0 0
\(821\) −21.4835 −0.749778 −0.374889 0.927070i \(-0.622319\pi\)
−0.374889 + 0.927070i \(0.622319\pi\)
\(822\) 0 0
\(823\) − 40.1565i − 1.39977i −0.714258 0.699883i \(-0.753235\pi\)
0.714258 0.699883i \(-0.246765\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38.0573i 1.32338i 0.749777 + 0.661691i \(0.230161\pi\)
−0.749777 + 0.661691i \(0.769839\pi\)
\(828\) 0 0
\(829\) 51.2580 1.78026 0.890132 0.455703i \(-0.150612\pi\)
0.890132 + 0.455703i \(0.150612\pi\)
\(830\) 0 0
\(831\) −7.90652 −0.274274
\(832\) 0 0
\(833\) 106.035i 3.67391i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 7.05339i − 0.243801i
\(838\) 0 0
\(839\) −45.4200 −1.56807 −0.784035 0.620716i \(-0.786842\pi\)
−0.784035 + 0.620716i \(0.786842\pi\)
\(840\) 0 0
\(841\) −26.2280 −0.904415
\(842\) 0 0
\(843\) − 12.3624i − 0.425783i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 51.5439i − 1.77107i
\(848\) 0 0
\(849\) −41.5798 −1.42701
\(850\) 0 0
\(851\) −4.34625 −0.148988
\(852\) 0 0
\(853\) − 40.0867i − 1.37254i −0.727346 0.686270i \(-0.759247\pi\)
0.727346 0.686270i \(-0.240753\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 3.63423i − 0.124143i −0.998072 0.0620715i \(-0.980229\pi\)
0.998072 0.0620715i \(-0.0197707\pi\)
\(858\) 0 0
\(859\) 37.7529 1.28811 0.644056 0.764978i \(-0.277250\pi\)
0.644056 + 0.764978i \(0.277250\pi\)
\(860\) 0 0
\(861\) 66.4053 2.26309
\(862\) 0 0
\(863\) 3.94555i 0.134308i 0.997743 + 0.0671541i \(0.0213919\pi\)
−0.997743 + 0.0671541i \(0.978608\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 39.3173i − 1.33529i
\(868\) 0 0
\(869\) −7.74173 −0.262620
\(870\) 0 0
\(871\) 22.8789 0.775223
\(872\) 0 0
\(873\) − 5.31450i − 0.179869i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 8.64958i − 0.292075i −0.989279 0.146038i \(-0.953348\pi\)
0.989279 0.146038i \(-0.0466521\pi\)
\(878\) 0 0
\(879\) −31.2007 −1.05237
\(880\) 0 0
\(881\) −56.1760 −1.89262 −0.946308 0.323266i \(-0.895219\pi\)
−0.946308 + 0.323266i \(0.895219\pi\)
\(882\) 0 0
\(883\) 12.1054i 0.407381i 0.979035 + 0.203690i \(0.0652936\pi\)
−0.979035 + 0.203690i \(0.934706\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 24.7058i − 0.829540i −0.909926 0.414770i \(-0.863862\pi\)
0.909926 0.414770i \(-0.136138\pi\)
\(888\) 0 0
\(889\) −13.0716 −0.438407
\(890\) 0 0
\(891\) −7.96719 −0.266911
\(892\) 0 0
\(893\) − 14.5962i − 0.488443i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.37907i 0.112824i
\(898\) 0 0
\(899\) −2.69179 −0.0897761
\(900\) 0 0
\(901\) −83.9422 −2.79652
\(902\) 0 0
\(903\) 47.9177i 1.59460i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 22.7170i 0.754305i 0.926151 + 0.377153i \(0.123097\pi\)
−0.926151 + 0.377153i \(0.876903\pi\)
\(908\) 0 0
\(909\) −11.1700 −0.370486
\(910\) 0 0
\(911\) −12.3156 −0.408033 −0.204016 0.978967i \(-0.565400\pi\)
−0.204016 + 0.978967i \(0.565400\pi\)
\(912\) 0 0
\(913\) − 0.475390i − 0.0157331i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 27.3215i − 0.902236i
\(918\) 0 0
\(919\) −46.2088 −1.52429 −0.762144 0.647408i \(-0.775853\pi\)
−0.762144 + 0.647408i \(0.775853\pi\)
\(920\) 0 0
\(921\) 13.4764 0.444064
\(922\) 0 0
\(923\) 16.8953i 0.556117i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 8.10545i − 0.266218i
\(928\) 0 0
\(929\) −9.12391 −0.299346 −0.149673 0.988736i \(-0.547822\pi\)
−0.149673 + 0.988736i \(0.547822\pi\)
\(930\) 0 0
\(931\) 94.4933 3.09689
\(932\) 0 0
\(933\) − 8.01330i − 0.262344i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 33.9190i − 1.10809i −0.832488 0.554043i \(-0.813084\pi\)
0.832488 0.554043i \(-0.186916\pi\)
\(938\) 0 0
\(939\) 12.1260 0.395718
\(940\) 0 0
\(941\) −37.2926 −1.21570 −0.607852 0.794050i \(-0.707969\pi\)
−0.607852 + 0.794050i \(0.707969\pi\)
\(942\) 0 0
\(943\) − 6.95184i − 0.226383i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33.7047i 1.09526i 0.836722 + 0.547629i \(0.184469\pi\)
−0.836722 + 0.547629i \(0.815531\pi\)
\(948\) 0 0
\(949\) −9.94767 −0.322915
\(950\) 0 0
\(951\) −18.4639 −0.598734
\(952\) 0 0
\(953\) 48.4465i 1.56934i 0.619917 + 0.784668i \(0.287166\pi\)
−0.619917 + 0.784668i \(0.712834\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.40354i 0.0776955i
\(958\) 0 0
\(959\) −30.4465 −0.983168
\(960\) 0 0
\(961\) −28.3861 −0.915680
\(962\) 0 0
\(963\) 5.58812i 0.180075i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 28.2528i − 0.908548i −0.890862 0.454274i \(-0.849899\pi\)
0.890862 0.454274i \(-0.150101\pi\)
\(968\) 0 0
\(969\) −64.3585 −2.06749
\(970\) 0 0
\(971\) 1.65553 0.0531284 0.0265642 0.999647i \(-0.491543\pi\)
0.0265642 + 0.999647i \(0.491543\pi\)
\(972\) 0 0
\(973\) 102.506i 3.28618i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 11.1630i − 0.357136i −0.983928 0.178568i \(-0.942854\pi\)
0.983928 0.178568i \(-0.0571465\pi\)
\(978\) 0 0
\(979\) −2.03281 −0.0649690
\(980\) 0 0
\(981\) −13.6730 −0.436545
\(982\) 0 0
\(983\) − 15.0615i − 0.480386i −0.970725 0.240193i \(-0.922789\pi\)
0.970725 0.240193i \(-0.0772107\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 25.6126i − 0.815258i
\(988\) 0 0
\(989\) 5.01641 0.159512
\(990\) 0 0
\(991\) −43.8971 −1.39444 −0.697219 0.716858i \(-0.745579\pi\)
−0.697219 + 0.716858i \(0.745579\pi\)
\(992\) 0 0
\(993\) − 27.6811i − 0.878432i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 38.5655i 1.22138i 0.791869 + 0.610691i \(0.209108\pi\)
−0.791869 + 0.610691i \(0.790892\pi\)
\(998\) 0 0
\(999\) −18.9612 −0.599907
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 4600.2.e.q.4049.5 6
5.2 odd 4 4600.2.a.v.1.3 3
5.3 odd 4 920.2.a.i.1.1 3
5.4 even 2 inner 4600.2.e.q.4049.2 6
15.8 even 4 8280.2.a.bl.1.3 3
20.3 even 4 1840.2.a.q.1.3 3
20.7 even 4 9200.2.a.ci.1.1 3
40.3 even 4 7360.2.a.cf.1.1 3
40.13 odd 4 7360.2.a.bw.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.1 3 5.3 odd 4
1840.2.a.q.1.3 3 20.3 even 4
4600.2.a.v.1.3 3 5.2 odd 4
4600.2.e.q.4049.2 6 5.4 even 2 inner
4600.2.e.q.4049.5 6 1.1 even 1 trivial
7360.2.a.bw.1.3 3 40.13 odd 4
7360.2.a.cf.1.1 3 40.3 even 4
8280.2.a.bl.1.3 3 15.8 even 4
9200.2.a.ci.1.1 3 20.7 even 4