Properties

Label 2432.2.c.a.1217.1
Level $2432$
Weight $2$
Character 2432.1217
Analytic conductor $19.420$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2432,2,Mod(1217,2432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2432.1217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2432 = 2^{7} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2432.c (of order \(2\), degree \(1\), 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: \(19.4196177716\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1217.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2432.1217
Dual form 2432.2.c.a.1217.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -2.00000i q^{5} -1.00000 q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -2.00000i q^{5} -1.00000 q^{7} +2.00000 q^{9} +4.00000i q^{11} +1.00000i q^{13} -2.00000 q^{15} -5.00000 q^{17} -1.00000i q^{19} +1.00000i q^{21} -9.00000 q^{23} +1.00000 q^{25} -5.00000i q^{27} +5.00000i q^{29} -2.00000 q^{31} +4.00000 q^{33} +2.00000i q^{35} -10.0000i q^{37} +1.00000 q^{39} -12.0000 q^{41} +10.0000i q^{43} -4.00000i q^{45} -12.0000 q^{47} -6.00000 q^{49} +5.00000i q^{51} -9.00000i q^{53} +8.00000 q^{55} -1.00000 q^{57} +9.00000i q^{59} -2.00000i q^{61} -2.00000 q^{63} +2.00000 q^{65} -13.0000i q^{67} +9.00000i q^{69} +10.0000 q^{71} +7.00000 q^{73} -1.00000i q^{75} -4.00000i q^{77} -10.0000 q^{79} +1.00000 q^{81} +8.00000i q^{83} +10.0000i q^{85} +5.00000 q^{87} +6.00000 q^{89} -1.00000i q^{91} +2.00000i q^{93} -2.00000 q^{95} -6.00000 q^{97} +8.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} + 4 q^{9} - 4 q^{15} - 10 q^{17} - 18 q^{23} + 2 q^{25} - 4 q^{31} + 8 q^{33} + 2 q^{39} - 24 q^{41} - 24 q^{47} - 12 q^{49} + 16 q^{55} - 2 q^{57} - 4 q^{63} + 4 q^{65} + 20 q^{71} + 14 q^{73} - 20 q^{79} + 2 q^{81} + 10 q^{87} + 12 q^{89} - 4 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times\).

\(n\) \(1407\) \(1921\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.00000i − 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) − 2.00000i − 0.894427i −0.894427 0.447214i \(-0.852416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) −9.00000 −1.87663 −0.938315 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) 5.00000i 0.928477i 0.885710 + 0.464238i \(0.153672\pi\)
−0.885710 + 0.464238i \(0.846328\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 2.00000i 0.338062i
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) − 4.00000i − 0.596285i
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 5.00000i 0.700140i
\(52\) 0 0
\(53\) − 9.00000i − 1.23625i −0.786082 0.618123i \(-0.787894\pi\)
0.786082 0.618123i \(-0.212106\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 9.00000i 1.17170i 0.810419 + 0.585850i \(0.199239\pi\)
−0.810419 + 0.585850i \(0.800761\pi\)
\(60\) 0 0
\(61\) − 2.00000i − 0.256074i −0.991769 0.128037i \(-0.959132\pi\)
0.991769 0.128037i \(-0.0408676\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) − 13.0000i − 1.58820i −0.607785 0.794101i \(-0.707942\pi\)
0.607785 0.794101i \(-0.292058\pi\)
\(68\) 0 0
\(69\) 9.00000i 1.08347i
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) − 1.00000i − 0.115470i
\(76\) 0 0
\(77\) − 4.00000i − 0.455842i
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 0 0
\(85\) 10.0000i 1.08465i
\(86\) 0 0
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) − 1.00000i − 0.104828i
\(92\) 0 0
\(93\) 2.00000i 0.207390i
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 8.00000i 0.804030i
\(100\) 0 0
\(101\) 4.00000i 0.398015i 0.979998 + 0.199007i \(0.0637718\pi\)
−0.979998 + 0.199007i \(0.936228\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) 11.0000i 1.06341i 0.846930 + 0.531705i \(0.178449\pi\)
−0.846930 + 0.531705i \(0.821551\pi\)
\(108\) 0 0
\(109\) 15.0000i 1.43674i 0.695662 + 0.718370i \(0.255111\pi\)
−0.695662 + 0.718370i \(0.744889\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 18.0000i 1.67851i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 12.0000i 1.08200i
\(124\) 0 0
\(125\) − 12.0000i − 1.07331i
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 0 0
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) 14.0000i 1.22319i 0.791173 + 0.611593i \(0.209471\pi\)
−0.791173 + 0.611593i \(0.790529\pi\)
\(132\) 0 0
\(133\) 1.00000i 0.0867110i
\(134\) 0 0
\(135\) −10.0000 −0.860663
\(136\) 0 0
\(137\) 13.0000 1.11066 0.555332 0.831628i \(-0.312591\pi\)
0.555332 + 0.831628i \(0.312591\pi\)
\(138\) 0 0
\(139\) − 2.00000i − 0.169638i −0.996396 0.0848189i \(-0.972969\pi\)
0.996396 0.0848189i \(-0.0270312\pi\)
\(140\) 0 0
\(141\) 12.0000i 1.01058i
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 10.0000 0.830455
\(146\) 0 0
\(147\) 6.00000i 0.494872i
\(148\) 0 0
\(149\) − 8.00000i − 0.655386i −0.944784 0.327693i \(-0.893729\pi\)
0.944784 0.327693i \(-0.106271\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) −10.0000 −0.808452
\(154\) 0 0
\(155\) 4.00000i 0.321288i
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 0 0
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 9.00000 0.709299
\(162\) 0 0
\(163\) − 10.0000i − 0.783260i −0.920123 0.391630i \(-0.871911\pi\)
0.920123 0.391630i \(-0.128089\pi\)
\(164\) 0 0
\(165\) − 8.00000i − 0.622799i
\(166\) 0 0
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) − 2.00000i − 0.152944i
\(172\) 0 0
\(173\) − 14.0000i − 1.06440i −0.846619 0.532200i \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 9.00000 0.676481
\(178\) 0 0
\(179\) 12.0000i 0.896922i 0.893802 + 0.448461i \(0.148028\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(180\) 0 0
\(181\) − 2.00000i − 0.148659i −0.997234 0.0743294i \(-0.976318\pi\)
0.997234 0.0743294i \(-0.0236816\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −20.0000 −1.47043
\(186\) 0 0
\(187\) − 20.0000i − 1.46254i
\(188\) 0 0
\(189\) 5.00000i 0.363696i
\(190\) 0 0
\(191\) −23.0000 −1.66422 −0.832111 0.554609i \(-0.812868\pi\)
−0.832111 + 0.554609i \(0.812868\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) − 2.00000i − 0.143223i
\(196\) 0 0
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 0 0
\(199\) −5.00000 −0.354441 −0.177220 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) 0 0
\(201\) −13.0000 −0.916949
\(202\) 0 0
\(203\) − 5.00000i − 0.350931i
\(204\) 0 0
\(205\) 24.0000i 1.67623i
\(206\) 0 0
\(207\) −18.0000 −1.25109
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) − 13.0000i − 0.894957i −0.894295 0.447478i \(-0.852322\pi\)
0.894295 0.447478i \(-0.147678\pi\)
\(212\) 0 0
\(213\) − 10.0000i − 0.685189i
\(214\) 0 0
\(215\) 20.0000 1.36399
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 0 0
\(219\) − 7.00000i − 0.473016i
\(220\) 0 0
\(221\) − 5.00000i − 0.336336i
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 0 0
\(227\) − 1.00000i − 0.0663723i −0.999449 0.0331862i \(-0.989435\pi\)
0.999449 0.0331862i \(-0.0105654\pi\)
\(228\) 0 0
\(229\) − 14.0000i − 0.925146i −0.886581 0.462573i \(-0.846926\pi\)
0.886581 0.462573i \(-0.153074\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 24.0000i 1.56559i
\(236\) 0 0
\(237\) 10.0000i 0.649570i
\(238\) 0 0
\(239\) −19.0000 −1.22901 −0.614504 0.788914i \(-0.710644\pi\)
−0.614504 + 0.788914i \(0.710644\pi\)
\(240\) 0 0
\(241\) −16.0000 −1.03065 −0.515325 0.856995i \(-0.672329\pi\)
−0.515325 + 0.856995i \(0.672329\pi\)
\(242\) 0 0
\(243\) − 16.0000i − 1.02640i
\(244\) 0 0
\(245\) 12.0000i 0.766652i
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) − 6.00000i − 0.378717i −0.981908 0.189358i \(-0.939359\pi\)
0.981908 0.189358i \(-0.0606408\pi\)
\(252\) 0 0
\(253\) − 36.0000i − 2.26330i
\(254\) 0 0
\(255\) 10.0000 0.626224
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 10.0000i 0.621370i
\(260\) 0 0
\(261\) 10.0000i 0.618984i
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) −18.0000 −1.10573
\(266\) 0 0
\(267\) − 6.00000i − 0.367194i
\(268\) 0 0
\(269\) − 6.00000i − 0.365826i −0.983129 0.182913i \(-0.941447\pi\)
0.983129 0.182913i \(-0.0585527\pi\)
\(270\) 0 0
\(271\) −3.00000 −0.182237 −0.0911185 0.995840i \(-0.529044\pi\)
−0.0911185 + 0.995840i \(0.529044\pi\)
\(272\) 0 0
\(273\) −1.00000 −0.0605228
\(274\) 0 0
\(275\) 4.00000i 0.241209i
\(276\) 0 0
\(277\) − 6.00000i − 0.360505i −0.983620 0.180253i \(-0.942309\pi\)
0.983620 0.180253i \(-0.0576915\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) 0 0
\(283\) − 24.0000i − 1.42665i −0.700832 0.713326i \(-0.747188\pi\)
0.700832 0.713326i \(-0.252812\pi\)
\(284\) 0 0
\(285\) 2.00000i 0.118470i
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 6.00000i 0.351726i
\(292\) 0 0
\(293\) 21.0000i 1.22683i 0.789760 + 0.613417i \(0.210205\pi\)
−0.789760 + 0.613417i \(0.789795\pi\)
\(294\) 0 0
\(295\) 18.0000 1.04800
\(296\) 0 0
\(297\) 20.0000 1.16052
\(298\) 0 0
\(299\) − 9.00000i − 0.520483i
\(300\) 0 0
\(301\) − 10.0000i − 0.576390i
\(302\) 0 0
\(303\) 4.00000 0.229794
\(304\) 0 0
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 8.00000i 0.455104i
\(310\) 0 0
\(311\) −33.0000 −1.87126 −0.935629 0.352985i \(-0.885167\pi\)
−0.935629 + 0.352985i \(0.885167\pi\)
\(312\) 0 0
\(313\) −21.0000 −1.18699 −0.593495 0.804838i \(-0.702252\pi\)
−0.593495 + 0.804838i \(0.702252\pi\)
\(314\) 0 0
\(315\) 4.00000i 0.225374i
\(316\) 0 0
\(317\) 31.0000i 1.74113i 0.492050 + 0.870567i \(0.336248\pi\)
−0.492050 + 0.870567i \(0.663752\pi\)
\(318\) 0 0
\(319\) −20.0000 −1.11979
\(320\) 0 0
\(321\) 11.0000 0.613960
\(322\) 0 0
\(323\) 5.00000i 0.278207i
\(324\) 0 0
\(325\) 1.00000i 0.0554700i
\(326\) 0 0
\(327\) 15.0000 0.829502
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) − 1.00000i − 0.0549650i −0.999622 0.0274825i \(-0.991251\pi\)
0.999622 0.0274825i \(-0.00874905\pi\)
\(332\) 0 0
\(333\) − 20.0000i − 1.09599i
\(334\) 0 0
\(335\) −26.0000 −1.42053
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) 6.00000i 0.325875i
\(340\) 0 0
\(341\) − 8.00000i − 0.433224i
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 18.0000 0.969087
\(346\) 0 0
\(347\) − 22.0000i − 1.18102i −0.807030 0.590511i \(-0.798926\pi\)
0.807030 0.590511i \(-0.201074\pi\)
\(348\) 0 0
\(349\) 26.0000i 1.39175i 0.718164 + 0.695874i \(0.244983\pi\)
−0.718164 + 0.695874i \(0.755017\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) −35.0000 −1.86286 −0.931431 0.363918i \(-0.881439\pi\)
−0.931431 + 0.363918i \(0.881439\pi\)
\(354\) 0 0
\(355\) − 20.0000i − 1.06149i
\(356\) 0 0
\(357\) − 5.00000i − 0.264628i
\(358\) 0 0
\(359\) −3.00000 −0.158334 −0.0791670 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) − 14.0000i − 0.732793i
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) −24.0000 −1.24939
\(370\) 0 0
\(371\) 9.00000i 0.467257i
\(372\) 0 0
\(373\) 13.0000i 0.673114i 0.941663 + 0.336557i \(0.109263\pi\)
−0.941663 + 0.336557i \(0.890737\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) −5.00000 −0.257513
\(378\) 0 0
\(379\) 17.0000i 0.873231i 0.899648 + 0.436616i \(0.143823\pi\)
−0.899648 + 0.436616i \(0.856177\pi\)
\(380\) 0 0
\(381\) 6.00000i 0.307389i
\(382\) 0 0
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 0 0
\(387\) 20.0000i 1.01666i
\(388\) 0 0
\(389\) − 12.0000i − 0.608424i −0.952604 0.304212i \(-0.901607\pi\)
0.952604 0.304212i \(-0.0983931\pi\)
\(390\) 0 0
\(391\) 45.0000 2.27575
\(392\) 0 0
\(393\) 14.0000 0.706207
\(394\) 0 0
\(395\) 20.0000i 1.00631i
\(396\) 0 0
\(397\) − 26.0000i − 1.30490i −0.757831 0.652451i \(-0.773741\pi\)
0.757831 0.652451i \(-0.226259\pi\)
\(398\) 0 0
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 0 0
\(403\) − 2.00000i − 0.0996271i
\(404\) 0 0
\(405\) − 2.00000i − 0.0993808i
\(406\) 0 0
\(407\) 40.0000 1.98273
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) − 13.0000i − 0.641243i
\(412\) 0 0
\(413\) − 9.00000i − 0.442861i
\(414\) 0 0
\(415\) 16.0000 0.785409
\(416\) 0 0
\(417\) −2.00000 −0.0979404
\(418\) 0 0
\(419\) − 34.0000i − 1.66101i −0.557012 0.830504i \(-0.688052\pi\)
0.557012 0.830504i \(-0.311948\pi\)
\(420\) 0 0
\(421\) 15.0000i 0.731055i 0.930800 + 0.365528i \(0.119111\pi\)
−0.930800 + 0.365528i \(0.880889\pi\)
\(422\) 0 0
\(423\) −24.0000 −1.16692
\(424\) 0 0
\(425\) −5.00000 −0.242536
\(426\) 0 0
\(427\) 2.00000i 0.0967868i
\(428\) 0 0
\(429\) 4.00000i 0.193122i
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 8.00000 0.384455 0.192228 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) 0 0
\(435\) − 10.0000i − 0.479463i
\(436\) 0 0
\(437\) 9.00000i 0.430528i
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −12.0000 −0.571429
\(442\) 0 0
\(443\) − 20.0000i − 0.950229i −0.879924 0.475114i \(-0.842407\pi\)
0.879924 0.475114i \(-0.157593\pi\)
\(444\) 0 0
\(445\) − 12.0000i − 0.568855i
\(446\) 0 0
\(447\) −8.00000 −0.378387
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) − 48.0000i − 2.26023i
\(452\) 0 0
\(453\) 8.00000i 0.375873i
\(454\) 0 0
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 0 0
\(459\) 25.0000i 1.16690i
\(460\) 0 0
\(461\) − 24.0000i − 1.11779i −0.829238 0.558896i \(-0.811225\pi\)
0.829238 0.558896i \(-0.188775\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) 6.00000i 0.277647i 0.990317 + 0.138823i \(0.0443321\pi\)
−0.990317 + 0.138823i \(0.955668\pi\)
\(468\) 0 0
\(469\) 13.0000i 0.600284i
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) −40.0000 −1.83920
\(474\) 0 0
\(475\) − 1.00000i − 0.0458831i
\(476\) 0 0
\(477\) − 18.0000i − 0.824163i
\(478\) 0 0
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) − 9.00000i − 0.409514i
\(484\) 0 0
\(485\) 12.0000i 0.544892i
\(486\) 0 0
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 0 0
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) 20.0000i 0.902587i 0.892375 + 0.451294i \(0.149037\pi\)
−0.892375 + 0.451294i \(0.850963\pi\)
\(492\) 0 0
\(493\) − 25.0000i − 1.12594i
\(494\) 0 0
\(495\) 16.0000 0.719147
\(496\) 0 0
\(497\) −10.0000 −0.448561
\(498\) 0 0
\(499\) − 10.0000i − 0.447661i −0.974628 0.223831i \(-0.928144\pi\)
0.974628 0.223831i \(-0.0718563\pi\)
\(500\) 0 0
\(501\) − 24.0000i − 1.07224i
\(502\) 0 0
\(503\) 11.0000 0.490466 0.245233 0.969464i \(-0.421136\pi\)
0.245233 + 0.969464i \(0.421136\pi\)
\(504\) 0 0
\(505\) 8.00000 0.355995
\(506\) 0 0
\(507\) − 12.0000i − 0.532939i
\(508\) 0 0
\(509\) − 6.00000i − 0.265945i −0.991120 0.132973i \(-0.957548\pi\)
0.991120 0.132973i \(-0.0424523\pi\)
\(510\) 0 0
\(511\) −7.00000 −0.309662
\(512\) 0 0
\(513\) −5.00000 −0.220755
\(514\) 0 0
\(515\) 16.0000i 0.705044i
\(516\) 0 0
\(517\) − 48.0000i − 2.11104i
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 32.0000 1.40195 0.700973 0.713188i \(-0.252749\pi\)
0.700973 + 0.713188i \(0.252749\pi\)
\(522\) 0 0
\(523\) − 17.0000i − 0.743358i −0.928361 0.371679i \(-0.878782\pi\)
0.928361 0.371679i \(-0.121218\pi\)
\(524\) 0 0
\(525\) 1.00000i 0.0436436i
\(526\) 0 0
\(527\) 10.0000 0.435607
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) 18.0000i 0.781133i
\(532\) 0 0
\(533\) − 12.0000i − 0.519778i
\(534\) 0 0
\(535\) 22.0000 0.951143
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) − 24.0000i − 1.03375i
\(540\) 0 0
\(541\) 4.00000i 0.171973i 0.996296 + 0.0859867i \(0.0274043\pi\)
−0.996296 + 0.0859867i \(0.972596\pi\)
\(542\) 0 0
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) 30.0000 1.28506
\(546\) 0 0
\(547\) − 4.00000i − 0.171028i −0.996337 0.0855138i \(-0.972747\pi\)
0.996337 0.0855138i \(-0.0272532\pi\)
\(548\) 0 0
\(549\) − 4.00000i − 0.170716i
\(550\) 0 0
\(551\) 5.00000 0.213007
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 0 0
\(555\) 20.0000i 0.848953i
\(556\) 0 0
\(557\) 22.0000i 0.932170i 0.884740 + 0.466085i \(0.154336\pi\)
−0.884740 + 0.466085i \(0.845664\pi\)
\(558\) 0 0
\(559\) −10.0000 −0.422955
\(560\) 0 0
\(561\) −20.0000 −0.844401
\(562\) 0 0
\(563\) − 20.0000i − 0.842900i −0.906852 0.421450i \(-0.861521\pi\)
0.906852 0.421450i \(-0.138479\pi\)
\(564\) 0 0
\(565\) 12.0000i 0.504844i
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) − 20.0000i − 0.836974i −0.908223 0.418487i \(-0.862561\pi\)
0.908223 0.418487i \(-0.137439\pi\)
\(572\) 0 0
\(573\) 23.0000i 0.960839i
\(574\) 0 0
\(575\) −9.00000 −0.375326
\(576\) 0 0
\(577\) 3.00000 0.124892 0.0624458 0.998048i \(-0.480110\pi\)
0.0624458 + 0.998048i \(0.480110\pi\)
\(578\) 0 0
\(579\) 4.00000i 0.166234i
\(580\) 0 0
\(581\) − 8.00000i − 0.331896i
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) 0 0
\(585\) 4.00000 0.165380
\(586\) 0 0
\(587\) 48.0000i 1.98117i 0.136892 + 0.990586i \(0.456289\pi\)
−0.136892 + 0.990586i \(0.543711\pi\)
\(588\) 0 0
\(589\) 2.00000i 0.0824086i
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) − 10.0000i − 0.409960i
\(596\) 0 0
\(597\) 5.00000i 0.204636i
\(598\) 0 0
\(599\) 28.0000 1.14405 0.572024 0.820237i \(-0.306158\pi\)
0.572024 + 0.820237i \(0.306158\pi\)
\(600\) 0 0
\(601\) 34.0000 1.38689 0.693444 0.720510i \(-0.256092\pi\)
0.693444 + 0.720510i \(0.256092\pi\)
\(602\) 0 0
\(603\) − 26.0000i − 1.05880i
\(604\) 0 0
\(605\) 10.0000i 0.406558i
\(606\) 0 0
\(607\) 10.0000 0.405887 0.202944 0.979190i \(-0.434949\pi\)
0.202944 + 0.979190i \(0.434949\pi\)
\(608\) 0 0
\(609\) −5.00000 −0.202610
\(610\) 0 0
\(611\) − 12.0000i − 0.485468i
\(612\) 0 0
\(613\) 12.0000i 0.484675i 0.970192 + 0.242338i \(0.0779142\pi\)
−0.970192 + 0.242338i \(0.922086\pi\)
\(614\) 0 0
\(615\) 24.0000 0.967773
\(616\) 0 0
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 0 0
\(619\) − 10.0000i − 0.401934i −0.979598 0.200967i \(-0.935592\pi\)
0.979598 0.200967i \(-0.0644084\pi\)
\(620\) 0 0
\(621\) 45.0000i 1.80579i
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) − 4.00000i − 0.159745i
\(628\) 0 0
\(629\) 50.0000i 1.99363i
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) −13.0000 −0.516704
\(634\) 0 0
\(635\) 12.0000i 0.476205i
\(636\) 0 0
\(637\) − 6.00000i − 0.237729i
\(638\) 0 0
\(639\) 20.0000 0.791188
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) − 34.0000i − 1.34083i −0.741987 0.670415i \(-0.766116\pi\)
0.741987 0.670415i \(-0.233884\pi\)
\(644\) 0 0
\(645\) − 20.0000i − 0.787499i
\(646\) 0 0
\(647\) −37.0000 −1.45462 −0.727310 0.686309i \(-0.759230\pi\)
−0.727310 + 0.686309i \(0.759230\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) − 2.00000i − 0.0783862i
\(652\) 0 0
\(653\) 46.0000i 1.80012i 0.435767 + 0.900060i \(0.356477\pi\)
−0.435767 + 0.900060i \(0.643523\pi\)
\(654\) 0 0
\(655\) 28.0000 1.09405
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) − 19.0000i − 0.740135i −0.929005 0.370067i \(-0.879335\pi\)
0.929005 0.370067i \(-0.120665\pi\)
\(660\) 0 0
\(661\) − 15.0000i − 0.583432i −0.956505 0.291716i \(-0.905774\pi\)
0.956505 0.291716i \(-0.0942263\pi\)
\(662\) 0 0
\(663\) −5.00000 −0.194184
\(664\) 0 0
\(665\) 2.00000 0.0775567
\(666\) 0 0
\(667\) − 45.0000i − 1.74241i
\(668\) 0 0
\(669\) − 16.0000i − 0.618596i
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) −20.0000 −0.770943 −0.385472 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) 0 0
\(675\) − 5.00000i − 0.192450i
\(676\) 0 0
\(677\) − 15.0000i − 0.576497i −0.957556 0.288248i \(-0.906927\pi\)
0.957556 0.288248i \(-0.0930729\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) −1.00000 −0.0383201
\(682\) 0 0
\(683\) − 4.00000i − 0.153056i −0.997067 0.0765279i \(-0.975617\pi\)
0.997067 0.0765279i \(-0.0243834\pi\)
\(684\) 0 0
\(685\) − 26.0000i − 0.993409i
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 0 0
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) − 30.0000i − 1.14125i −0.821209 0.570627i \(-0.806700\pi\)
0.821209 0.570627i \(-0.193300\pi\)
\(692\) 0 0
\(693\) − 8.00000i − 0.303895i
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 60.0000 2.27266
\(698\) 0 0
\(699\) − 10.0000i − 0.378235i
\(700\) 0 0
\(701\) 20.0000i 0.755390i 0.925930 + 0.377695i \(0.123283\pi\)
−0.925930 + 0.377695i \(0.876717\pi\)
\(702\) 0 0
\(703\) −10.0000 −0.377157
\(704\) 0 0
\(705\) 24.0000 0.903892
\(706\) 0 0
\(707\) − 4.00000i − 0.150435i
\(708\) 0 0
\(709\) 10.0000i 0.375558i 0.982211 + 0.187779i \(0.0601289\pi\)
−0.982211 + 0.187779i \(0.939871\pi\)
\(710\) 0 0
\(711\) −20.0000 −0.750059
\(712\) 0 0
\(713\) 18.0000 0.674105
\(714\) 0 0
\(715\) 8.00000i 0.299183i
\(716\) 0 0
\(717\) 19.0000i 0.709568i
\(718\) 0 0
\(719\) −3.00000 −0.111881 −0.0559406 0.998434i \(-0.517816\pi\)
−0.0559406 + 0.998434i \(0.517816\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 16.0000i 0.595046i
\(724\) 0 0
\(725\) 5.00000i 0.185695i
\(726\) 0 0
\(727\) −29.0000 −1.07555 −0.537775 0.843088i \(-0.680735\pi\)
−0.537775 + 0.843088i \(0.680735\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) − 50.0000i − 1.84932i
\(732\) 0 0
\(733\) − 6.00000i − 0.221615i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353437\pi\)
\(734\) 0 0
\(735\) 12.0000 0.442627
\(736\) 0 0
\(737\) 52.0000 1.91544
\(738\) 0 0
\(739\) 20.0000i 0.735712i 0.929883 + 0.367856i \(0.119908\pi\)
−0.929883 + 0.367856i \(0.880092\pi\)
\(740\) 0 0
\(741\) − 1.00000i − 0.0367359i
\(742\) 0 0
\(743\) −50.0000 −1.83432 −0.917161 0.398517i \(-0.869525\pi\)
−0.917161 + 0.398517i \(0.869525\pi\)
\(744\) 0 0
\(745\) −16.0000 −0.586195
\(746\) 0 0
\(747\) 16.0000i 0.585409i
\(748\) 0 0
\(749\) − 11.0000i − 0.401931i
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) −6.00000 −0.218652
\(754\) 0 0
\(755\) 16.0000i 0.582300i
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 0 0
\(759\) −36.0000 −1.30672
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) − 15.0000i − 0.543036i
\(764\) 0 0
\(765\) 20.0000i 0.723102i
\(766\) 0 0
\(767\) −9.00000 −0.324971
\(768\) 0 0
\(769\) 1.00000 0.0360609 0.0180305 0.999837i \(-0.494260\pi\)
0.0180305 + 0.999837i \(0.494260\pi\)
\(770\) 0 0
\(771\) 6.00000i 0.216085i
\(772\) 0 0
\(773\) 9.00000i 0.323708i 0.986815 + 0.161854i \(0.0517473\pi\)
−0.986815 + 0.161854i \(0.948253\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 10.0000 0.358748
\(778\) 0 0
\(779\) 12.0000i 0.429945i
\(780\) 0 0
\(781\) 40.0000i 1.43131i
\(782\) 0 0
\(783\) 25.0000 0.893427
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) 11.0000i 0.392108i 0.980593 + 0.196054i \(0.0628127\pi\)
−0.980593 + 0.196054i \(0.937187\pi\)
\(788\) 0 0
\(789\) − 16.0000i − 0.569615i
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 0 0
\(795\) 18.0000i 0.638394i
\(796\) 0 0
\(797\) 5.00000i 0.177109i 0.996071 + 0.0885545i \(0.0282248\pi\)
−0.996071 + 0.0885545i \(0.971775\pi\)
\(798\) 0 0
\(799\) 60.0000 2.12265
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 0 0
\(803\) 28.0000i 0.988099i
\(804\) 0 0
\(805\) − 18.0000i − 0.634417i
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) −5.00000 −0.175791 −0.0878953 0.996130i \(-0.528014\pi\)
−0.0878953 + 0.996130i \(0.528014\pi\)
\(810\) 0 0
\(811\) − 9.00000i − 0.316033i −0.987436 0.158016i \(-0.949490\pi\)
0.987436 0.158016i \(-0.0505099\pi\)
\(812\) 0 0
\(813\) 3.00000i 0.105215i
\(814\) 0 0
\(815\) −20.0000 −0.700569
\(816\) 0 0
\(817\) 10.0000 0.349856
\(818\) 0 0
\(819\) − 2.00000i − 0.0698857i
\(820\) 0 0
\(821\) − 50.0000i − 1.74501i −0.488603 0.872506i \(-0.662493\pi\)
0.488603 0.872506i \(-0.337507\pi\)
\(822\) 0 0
\(823\) −3.00000 −0.104573 −0.0522867 0.998632i \(-0.516651\pi\)
−0.0522867 + 0.998632i \(0.516651\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) − 19.0000i − 0.660695i −0.943859 0.330347i \(-0.892834\pi\)
0.943859 0.330347i \(-0.107166\pi\)
\(828\) 0 0
\(829\) 7.00000i 0.243120i 0.992584 + 0.121560i \(0.0387897\pi\)
−0.992584 + 0.121560i \(0.961210\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) 0 0
\(833\) 30.0000 1.03944
\(834\) 0 0
\(835\) − 48.0000i − 1.66111i
\(836\) 0 0
\(837\) 10.0000i 0.345651i
\(838\) 0 0
\(839\) −22.0000 −0.759524 −0.379762 0.925084i \(-0.623994\pi\)
−0.379762 + 0.925084i \(0.623994\pi\)
\(840\) 0 0
\(841\) 4.00000 0.137931
\(842\) 0 0
\(843\) − 20.0000i − 0.688837i
\(844\) 0 0
\(845\) − 24.0000i − 0.825625i
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 0 0
\(849\) −24.0000 −0.823678
\(850\) 0 0
\(851\) 90.0000i 3.08516i
\(852\) 0 0
\(853\) 40.0000i 1.36957i 0.728743 + 0.684787i \(0.240105\pi\)
−0.728743 + 0.684787i \(0.759895\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 0 0
\(859\) − 16.0000i − 0.545913i −0.962026 0.272956i \(-0.911998\pi\)
0.962026 0.272956i \(-0.0880015\pi\)
\(860\) 0 0
\(861\) − 12.0000i − 0.408959i
\(862\) 0 0
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) 0 0
\(865\) −28.0000 −0.952029
\(866\) 0 0
\(867\) − 8.00000i − 0.271694i
\(868\) 0 0
\(869\) − 40.0000i − 1.35691i
\(870\) 0 0
\(871\) 13.0000 0.440488
\(872\) 0 0
\(873\) −12.0000 −0.406138
\(874\) 0 0
\(875\) 12.0000i 0.405674i
\(876\) 0 0
\(877\) − 41.0000i − 1.38447i −0.721671 0.692236i \(-0.756626\pi\)
0.721671 0.692236i \(-0.243374\pi\)
\(878\) 0 0
\(879\) 21.0000 0.708312
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) 0 0
\(883\) 26.0000i 0.874970i 0.899226 + 0.437485i \(0.144131\pi\)
−0.899226 + 0.437485i \(0.855869\pi\)
\(884\) 0 0
\(885\) − 18.0000i − 0.605063i
\(886\) 0 0
\(887\) −18.0000 −0.604381 −0.302190 0.953248i \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) 0 0
\(889\) 6.00000 0.201234
\(890\) 0 0
\(891\) 4.00000i 0.134005i
\(892\) 0 0
\(893\) 12.0000i 0.401565i
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) −9.00000 −0.300501
\(898\) 0 0
\(899\) − 10.0000i − 0.333519i
\(900\) 0 0
\(901\) 45.0000i 1.49917i
\(902\) 0 0
\(903\) −10.0000 −0.332779
\(904\) 0 0
\(905\) −4.00000 −0.132964
\(906\) 0 0
\(907\) 51.0000i 1.69343i 0.532049 + 0.846714i \(0.321422\pi\)
−0.532049 + 0.846714i \(0.678578\pi\)
\(908\) 0 0
\(909\) 8.00000i 0.265343i
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) −32.0000 −1.05905
\(914\) 0 0
\(915\) 4.00000i 0.132236i
\(916\) 0 0
\(917\) − 14.0000i − 0.462321i
\(918\) 0 0
\(919\) −43.0000 −1.41844 −0.709220 0.704988i \(-0.750953\pi\)
−0.709220 + 0.704988i \(0.750953\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 0 0
\(923\) 10.0000i 0.329154i
\(924\) 0 0
\(925\) − 10.0000i − 0.328798i
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) 13.0000 0.426516 0.213258 0.976996i \(-0.431592\pi\)
0.213258 + 0.976996i \(0.431592\pi\)
\(930\) 0 0
\(931\) 6.00000i 0.196642i
\(932\) 0 0
\(933\) 33.0000i 1.08037i
\(934\) 0 0
\(935\) −40.0000 −1.30814
\(936\) 0 0
\(937\) −37.0000 −1.20874 −0.604369 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) 0 0
\(939\) 21.0000i 0.685309i
\(940\) 0 0
\(941\) − 3.00000i − 0.0977972i −0.998804 0.0488986i \(-0.984429\pi\)
0.998804 0.0488986i \(-0.0155711\pi\)
\(942\) 0 0
\(943\) 108.000 3.51696
\(944\) 0 0
\(945\) 10.0000 0.325300
\(946\) 0 0
\(947\) − 38.0000i − 1.23483i −0.786636 0.617417i \(-0.788179\pi\)
0.786636 0.617417i \(-0.211821\pi\)
\(948\) 0 0
\(949\) 7.00000i 0.227230i
\(950\) 0 0
\(951\) 31.0000 1.00524
\(952\) 0 0
\(953\) 8.00000 0.259145 0.129573 0.991570i \(-0.458639\pi\)
0.129573 + 0.991570i \(0.458639\pi\)
\(954\) 0 0
\(955\) 46.0000i 1.48853i
\(956\) 0 0
\(957\) 20.0000i 0.646508i
\(958\) 0 0
\(959\) −13.0000 −0.419792
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 22.0000i 0.708940i
\(964\) 0 0
\(965\) 8.00000i 0.257529i
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 5.00000 0.160623
\(970\) 0 0
\(971\) − 12.0000i − 0.385098i −0.981287 0.192549i \(-0.938325\pi\)
0.981287 0.192549i \(-0.0616755\pi\)
\(972\) 0 0
\(973\) 2.00000i 0.0641171i
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 24.0000i 0.767043i
\(980\) 0 0
\(981\) 30.0000i 0.957826i
\(982\) 0 0
\(983\) 2.00000 0.0637901 0.0318950 0.999491i \(-0.489846\pi\)
0.0318950 + 0.999491i \(0.489846\pi\)
\(984\) 0 0
\(985\) 24.0000 0.764704
\(986\) 0 0
\(987\) − 12.0000i − 0.381964i
\(988\) 0 0
\(989\) − 90.0000i − 2.86183i
\(990\) 0 0
\(991\) 10.0000 0.317660 0.158830 0.987306i \(-0.449228\pi\)
0.158830 + 0.987306i \(0.449228\pi\)
\(992\) 0 0
\(993\) −1.00000 −0.0317340
\(994\) 0 0
\(995\) 10.0000i 0.317021i
\(996\) 0 0
\(997\) 50.0000i 1.58352i 0.610835 + 0.791758i \(0.290834\pi\)
−0.610835 + 0.791758i \(0.709166\pi\)
\(998\) 0 0
\(999\) −50.0000 −1.58193
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 2432.2.c.a.1217.1 2
4.3 odd 2 2432.2.c.b.1217.2 yes 2
8.3 odd 2 2432.2.c.b.1217.1 yes 2
8.5 even 2 inner 2432.2.c.a.1217.2 yes 2
16.3 odd 4 4864.2.a.c.1.1 1
16.5 even 4 4864.2.a.f.1.1 1
16.11 odd 4 4864.2.a.n.1.1 1
16.13 even 4 4864.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.a.1217.1 2 1.1 even 1 trivial
2432.2.c.a.1217.2 yes 2 8.5 even 2 inner
2432.2.c.b.1217.1 yes 2 8.3 odd 2
2432.2.c.b.1217.2 yes 2 4.3 odd 2
4864.2.a.c.1.1 1 16.3 odd 4
4864.2.a.f.1.1 1 16.5 even 4
4864.2.a.k.1.1 1 16.13 even 4
4864.2.a.n.1.1 1 16.11 odd 4