Properties

Label 2450.2.c.p.99.1
Level $2450$
Weight $2$
Character 2450.99
Analytic conductor $19.563$
Analytic rank $0$
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] = [2450,2,Mod(99,2450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2450.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.c (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: \(19.5633484952\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2450.99
Dual form 2450.2.c.p.99.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} -2.00000i q^{12} -5.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} -3.00000i q^{22} +3.00000i q^{23} -2.00000 q^{24} -5.00000 q^{26} +4.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} +6.00000 q^{34} +1.00000 q^{36} -11.0000i q^{37} +1.00000i q^{38} +10.0000 q^{39} -3.00000 q^{41} -10.0000i q^{43} -3.00000 q^{44} +3.00000 q^{46} +3.00000i q^{47} +2.00000i q^{48} -12.0000 q^{51} +5.00000i q^{52} +3.00000i q^{53} +4.00000 q^{54} -2.00000i q^{57} -6.00000i q^{58} +4.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +6.00000 q^{66} +4.00000i q^{67} -6.00000i q^{68} -6.00000 q^{69} +12.0000 q^{71} -1.00000i q^{72} +4.00000i q^{73} -11.0000 q^{74} +1.00000 q^{76} -10.0000i q^{78} +10.0000 q^{79} -11.0000 q^{81} +3.00000i q^{82} +12.0000i q^{83} -10.0000 q^{86} +12.0000i q^{87} +3.00000i q^{88} +6.00000 q^{89} -3.00000i q^{92} +8.00000i q^{93} +3.00000 q^{94} +2.00000 q^{96} +14.0000i q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 6 q^{11} + 2 q^{16} - 2 q^{19} - 4 q^{24} - 10 q^{26} + 12 q^{29} + 8 q^{31} + 12 q^{34} + 2 q^{36} + 20 q^{39} - 6 q^{41} - 6 q^{44} + 6 q^{46} - 24 q^{51} + 8 q^{54} + 8 q^{61} - 2 q^{64} + 12 q^{66} - 12 q^{69} + 24 q^{71} - 22 q^{74} + 2 q^{76} + 20 q^{79} - 22 q^{81} - 20 q^{86} + 12 q^{89} + 6 q^{94} + 4 q^{96} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) − 2.00000i − 0.577350i
\(13\) − 5.00000i − 1.38675i −0.720577 0.693375i \(-0.756123\pi\)
0.720577 0.693375i \(-0.243877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 3.00000i − 0.639602i
\(23\) 3.00000i 0.625543i 0.949828 + 0.312772i \(0.101257\pi\)
−0.949828 + 0.312772i \(0.898743\pi\)
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) −5.00000 −0.980581
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 6.00000i 1.04447i
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 11.0000i − 1.80839i −0.427121 0.904194i \(-0.640472\pi\)
0.427121 0.904194i \(-0.359528\pi\)
\(38\) 1.00000i 0.162221i
\(39\) 10.0000 1.60128
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) − 10.0000i − 1.52499i −0.646997 0.762493i \(-0.723975\pi\)
0.646997 0.762493i \(-0.276025\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 2.00000i 0.288675i
\(49\) 0 0
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 5.00000i 0.693375i
\(53\) 3.00000i 0.412082i 0.978543 + 0.206041i \(0.0660580\pi\)
−0.978543 + 0.206041i \(0.933942\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.00000i − 0.264906i
\(58\) − 6.00000i − 0.787839i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) −11.0000 −1.27872
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) − 10.0000i − 1.13228i
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 3.00000i 0.331295i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 12.0000i 1.28654i
\(88\) 3.00000i 0.319801i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 3.00000i − 0.312772i
\(93\) 8.00000i 0.829561i
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 12.0000i 1.18818i
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 5.00000 0.490290
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) − 4.00000i − 0.384900i
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 22.0000 2.08815
\(112\) 0 0
\(113\) 12.0000i 1.12887i 0.825479 + 0.564433i \(0.190905\pi\)
−0.825479 + 0.564433i \(0.809095\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 5.00000i 0.462250i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) − 4.00000i − 0.362143i
\(123\) − 6.00000i − 0.541002i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 19.0000i 1.68598i 0.537931 + 0.842989i \(0.319206\pi\)
−0.537931 + 0.842989i \(0.680794\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 20.0000 1.76090
\(130\) 0 0
\(131\) −3.00000 −0.262111 −0.131056 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) − 6.00000i − 0.522233i
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 6.00000i 0.510754i
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) − 12.0000i − 1.00702i
\(143\) − 15.0000i − 1.25436i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) 11.0000i 0.904194i
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) − 1.00000i − 0.0811107i
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) −10.0000 −0.800641
\(157\) 5.00000i 0.399043i 0.979893 + 0.199522i \(0.0639388\pi\)
−0.979893 + 0.199522i \(0.936061\pi\)
\(158\) − 10.0000i − 0.795557i
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 11.0000i 0.864242i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 9.00000i 0.696441i 0.937413 + 0.348220i \(0.113214\pi\)
−0.937413 + 0.348220i \(0.886786\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 10.0000i 0.762493i
\(173\) − 3.00000i − 0.228086i −0.993476 0.114043i \(-0.963620\pi\)
0.993476 0.114043i \(-0.0363801\pi\)
\(174\) 12.0000 0.909718
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) − 6.00000i − 0.449719i
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 8.00000i 0.591377i
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 18.0000i 1.31629i
\(188\) − 3.00000i − 0.218797i
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) − 2.00000i − 0.144338i
\(193\) − 4.00000i − 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000i 0.213741i 0.994273 + 0.106871i \(0.0340831\pi\)
−0.994273 + 0.106871i \(0.965917\pi\)
\(198\) 3.00000i 0.213201i
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) − 12.0000i − 0.844317i
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) − 3.00000i − 0.208514i
\(208\) − 5.00000i − 0.346688i
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) − 3.00000i − 0.206041i
\(213\) 24.0000i 1.64445i
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) − 4.00000i − 0.270914i
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 30.0000 2.01802
\(222\) − 22.0000i − 1.47654i
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) − 24.0000i − 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) 2.00000i 0.132453i
\(229\) −28.0000 −1.85029 −0.925146 0.379611i \(-0.876058\pi\)
−0.925146 + 0.379611i \(0.876058\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 5.00000 0.326860
\(235\) 0 0
\(236\) 0 0
\(237\) 20.0000i 1.29914i
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 25.0000 1.61039 0.805196 0.593009i \(-0.202060\pi\)
0.805196 + 0.593009i \(0.202060\pi\)
\(242\) 2.00000i 0.128565i
\(243\) − 10.0000i − 0.641500i
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 5.00000i 0.318142i
\(248\) 4.00000i 0.254000i
\(249\) −24.0000 −1.52094
\(250\) 0 0
\(251\) 15.0000 0.946792 0.473396 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 0 0
\(253\) 9.00000i 0.565825i
\(254\) 19.0000 1.19217
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 12.0000i − 0.748539i −0.927320 0.374270i \(-0.877893\pi\)
0.927320 0.374270i \(-0.122107\pi\)
\(258\) − 20.0000i − 1.24515i
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 3.00000i 0.185341i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) − 4.00000i − 0.244339i
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 6.00000i 0.363803i
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 4.00000i 0.239904i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −3.00000 −0.178965 −0.0894825 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) 6.00000i 0.357295i
\(283\) − 26.0000i − 1.54554i −0.634686 0.772770i \(-0.718871\pi\)
0.634686 0.772770i \(-0.281129\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) −15.0000 −0.886969
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −28.0000 −1.64139
\(292\) − 4.00000i − 0.234082i
\(293\) 27.0000i 1.57736i 0.614806 + 0.788678i \(0.289234\pi\)
−0.614806 + 0.788678i \(0.710766\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 11.0000 0.639362
\(297\) 12.0000i 0.696311i
\(298\) 18.0000i 1.04271i
\(299\) 15.0000 0.867472
\(300\) 0 0
\(301\) 0 0
\(302\) − 14.0000i − 0.805609i
\(303\) 24.0000i 1.37876i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 10.0000i 0.566139i
\(313\) − 8.00000i − 0.452187i −0.974106 0.226093i \(-0.927405\pi\)
0.974106 0.226093i \(-0.0725954\pi\)
\(314\) 5.00000 0.282166
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 6.00000i 0.336463i
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) − 6.00000i − 0.333849i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 8.00000i 0.442401i
\(328\) − 3.00000i − 0.165647i
\(329\) 0 0
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) 11.0000i 0.602796i
\(334\) 9.00000 0.492458
\(335\) 0 0
\(336\) 0 0
\(337\) − 14.0000i − 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) 12.0000i 0.652714i
\(339\) −24.0000 −1.30350
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) − 1.00000i − 0.0540738i
\(343\) 0 0
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) −3.00000 −0.161281
\(347\) − 24.0000i − 1.28839i −0.764862 0.644194i \(-0.777193\pi\)
0.764862 0.644194i \(-0.222807\pi\)
\(348\) − 12.0000i − 0.643268i
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 20.0000 1.06752
\(352\) − 3.00000i − 0.159901i
\(353\) − 12.0000i − 0.638696i −0.947638 0.319348i \(-0.896536\pi\)
0.947638 0.319348i \(-0.103464\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) − 3.00000i − 0.158555i
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 2.00000i 0.105118i
\(363\) − 4.00000i − 0.209946i
\(364\) 0 0
\(365\) 0 0
\(366\) 8.00000 0.418167
\(367\) − 1.00000i − 0.0521996i −0.999659 0.0260998i \(-0.991691\pi\)
0.999659 0.0260998i \(-0.00830876\pi\)
\(368\) 3.00000i 0.156386i
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) 0 0
\(372\) − 8.00000i − 0.414781i
\(373\) − 34.0000i − 1.76045i −0.474554 0.880227i \(-0.657390\pi\)
0.474554 0.880227i \(-0.342610\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) − 30.0000i − 1.54508i
\(378\) 0 0
\(379\) 25.0000 1.28416 0.642082 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(380\) 0 0
\(381\) −38.0000 −1.94680
\(382\) − 12.0000i − 0.613973i
\(383\) − 15.0000i − 0.766464i −0.923652 0.383232i \(-0.874811\pi\)
0.923652 0.383232i \(-0.125189\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) 10.0000i 0.508329i
\(388\) − 14.0000i − 0.710742i
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) − 6.00000i − 0.302660i
\(394\) 3.00000 0.151138
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 4.00000i 0.200502i
\(399\) 0 0
\(400\) 0 0
\(401\) 21.0000 1.04869 0.524345 0.851506i \(-0.324310\pi\)
0.524345 + 0.851506i \(0.324310\pi\)
\(402\) 8.00000i 0.399004i
\(403\) − 20.0000i − 0.996271i
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) − 33.0000i − 1.63575i
\(408\) − 12.0000i − 0.594089i
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) 24.0000 1.18383
\(412\) − 4.00000i − 0.197066i
\(413\) 0 0
\(414\) −3.00000 −0.147442
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) − 8.00000i − 0.391762i
\(418\) 3.00000i 0.146735i
\(419\) 15.0000 0.732798 0.366399 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 1.00000i 0.0486792i
\(423\) − 3.00000i − 0.145865i
\(424\) −3.00000 −0.145693
\(425\) 0 0
\(426\) 24.0000 1.16280
\(427\) 0 0
\(428\) − 12.0000i − 0.580042i
\(429\) 30.0000 1.44841
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 4.00000i 0.192450i
\(433\) 16.0000i 0.768911i 0.923144 + 0.384455i \(0.125611\pi\)
−0.923144 + 0.384455i \(0.874389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) − 3.00000i − 0.143509i
\(438\) 8.00000i 0.382255i
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 30.0000i − 1.42695i
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) −22.0000 −1.04407
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) − 36.0000i − 1.70274i
\(448\) 0 0
\(449\) 3.00000 0.141579 0.0707894 0.997491i \(-0.477448\pi\)
0.0707894 + 0.997491i \(0.477448\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) − 12.0000i − 0.564433i
\(453\) 28.0000i 1.31555i
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) 22.0000i 1.02912i 0.857455 + 0.514558i \(0.172044\pi\)
−0.857455 + 0.514558i \(0.827956\pi\)
\(458\) 28.0000i 1.30835i
\(459\) −24.0000 −1.12022
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) − 19.0000i − 0.883005i −0.897260 0.441502i \(-0.854446\pi\)
0.897260 0.441502i \(-0.145554\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) − 18.0000i − 0.832941i −0.909149 0.416470i \(-0.863267\pi\)
0.909149 0.416470i \(-0.136733\pi\)
\(468\) − 5.00000i − 0.231125i
\(469\) 0 0
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) − 30.0000i − 1.37940i
\(474\) 20.0000 0.918630
\(475\) 0 0
\(476\) 0 0
\(477\) − 3.00000i − 0.137361i
\(478\) 6.00000i 0.274434i
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −55.0000 −2.50778
\(482\) − 25.0000i − 1.13872i
\(483\) 0 0
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 16.0000i 0.725029i 0.931978 + 0.362515i \(0.118082\pi\)
−0.931978 + 0.362515i \(0.881918\pi\)
\(488\) 4.00000i 0.181071i
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 36.0000i 1.62136i
\(494\) 5.00000 0.224961
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 24.0000i 1.07547i
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) −18.0000 −0.804181
\(502\) − 15.0000i − 0.669483i
\(503\) − 24.0000i − 1.07011i −0.844818 0.535054i \(-0.820291\pi\)
0.844818 0.535054i \(-0.179709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 9.00000 0.400099
\(507\) − 24.0000i − 1.06588i
\(508\) − 19.0000i − 0.842989i
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) − 4.00000i − 0.176604i
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) −20.0000 −0.880451
\(517\) 9.00000i 0.395820i
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −33.0000 −1.44576 −0.722878 0.690976i \(-0.757181\pi\)
−0.722878 + 0.690976i \(0.757181\pi\)
\(522\) 6.00000i 0.262613i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) 3.00000 0.131056
\(525\) 0 0
\(526\) 0 0
\(527\) 24.0000i 1.04546i
\(528\) 6.00000i 0.261116i
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 15.0000i 0.649722i
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 6.00000i 0.258919i
\(538\) 12.0000i 0.517357i
\(539\) 0 0
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) − 16.0000i − 0.687259i
\(543\) − 4.00000i − 0.171656i
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000i 1.19719i 0.801050 + 0.598597i \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) 12.0000i 0.512615i
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) − 6.00000i − 0.255377i
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 27.0000i 1.14403i 0.820244 + 0.572013i \(0.193837\pi\)
−0.820244 + 0.572013i \(0.806163\pi\)
\(558\) 4.00000i 0.169334i
\(559\) −50.0000 −2.11477
\(560\) 0 0
\(561\) −36.0000 −1.51992
\(562\) 3.00000i 0.126547i
\(563\) 18.0000i 0.758610i 0.925272 + 0.379305i \(0.123837\pi\)
−0.925272 + 0.379305i \(0.876163\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) −26.0000 −1.09286
\(567\) 0 0
\(568\) 12.0000i 0.503509i
\(569\) 3.00000 0.125767 0.0628833 0.998021i \(-0.479970\pi\)
0.0628833 + 0.998021i \(0.479970\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 15.0000i 0.627182i
\(573\) 24.0000i 1.00261i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 20.0000i 0.832611i 0.909225 + 0.416305i \(0.136675\pi\)
−0.909225 + 0.416305i \(0.863325\pi\)
\(578\) 19.0000i 0.790296i
\(579\) 8.00000 0.332469
\(580\) 0 0
\(581\) 0 0
\(582\) 28.0000i 1.16064i
\(583\) 9.00000i 0.372742i
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 27.0000 1.11536
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) − 11.0000i − 0.452097i
\(593\) 36.0000i 1.47834i 0.673517 + 0.739171i \(0.264783\pi\)
−0.673517 + 0.739171i \(0.735217\pi\)
\(594\) 12.0000 0.492366
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) − 8.00000i − 0.327418i
\(598\) − 15.0000i − 0.613396i
\(599\) −42.0000 −1.71607 −0.858037 0.513588i \(-0.828316\pi\)
−0.858037 + 0.513588i \(0.828316\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) 24.0000 0.974933
\(607\) − 19.0000i − 0.771186i −0.922669 0.385593i \(-0.873997\pi\)
0.922669 0.385593i \(-0.126003\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) 0 0
\(610\) 0 0
\(611\) 15.0000 0.606835
\(612\) 6.00000i 0.242536i
\(613\) 47.0000i 1.89831i 0.314806 + 0.949156i \(0.398061\pi\)
−0.314806 + 0.949156i \(0.601939\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 8.00000i 0.321807i
\(619\) −1.00000 −0.0401934 −0.0200967 0.999798i \(-0.506397\pi\)
−0.0200967 + 0.999798i \(0.506397\pi\)
\(620\) 0 0
\(621\) −12.0000 −0.481543
\(622\) 12.0000i 0.481156i
\(623\) 0 0
\(624\) 10.0000 0.400320
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) − 6.00000i − 0.239617i
\(628\) − 5.00000i − 0.199522i
\(629\) 66.0000 2.63159
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 10.0000i 0.397779i
\(633\) − 2.00000i − 0.0794929i
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) − 18.0000i − 0.712627i
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −45.0000 −1.77739 −0.888697 0.458496i \(-0.848388\pi\)
−0.888697 + 0.458496i \(0.848388\pi\)
\(642\) 24.0000i 0.947204i
\(643\) − 38.0000i − 1.49857i −0.662246 0.749287i \(-0.730396\pi\)
0.662246 0.749287i \(-0.269604\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) − 21.0000i − 0.825595i −0.910823 0.412798i \(-0.864552\pi\)
0.910823 0.412798i \(-0.135448\pi\)
\(648\) − 11.0000i − 0.432121i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) 21.0000i 0.821794i 0.911682 + 0.410897i \(0.134784\pi\)
−0.911682 + 0.410897i \(0.865216\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) − 4.00000i − 0.156055i
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −44.0000 −1.71140 −0.855701 0.517471i \(-0.826874\pi\)
−0.855701 + 0.517471i \(0.826874\pi\)
\(662\) 7.00000i 0.272063i
\(663\) 60.0000i 2.33021i
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 11.0000 0.426241
\(667\) 18.0000i 0.696963i
\(668\) − 9.00000i − 0.348220i
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) − 34.0000i − 1.31060i −0.755367 0.655302i \(-0.772541\pi\)
0.755367 0.655302i \(-0.227459\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) − 3.00000i − 0.115299i −0.998337 0.0576497i \(-0.981639\pi\)
0.998337 0.0576497i \(-0.0183606\pi\)
\(678\) 24.0000i 0.921714i
\(679\) 0 0
\(680\) 0 0
\(681\) 48.0000 1.83936
\(682\) − 12.0000i − 0.459504i
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) − 56.0000i − 2.13653i
\(688\) − 10.0000i − 0.381246i
\(689\) 15.0000 0.571454
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 3.00000i 0.114043i
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) −12.0000 −0.454859
\(697\) − 18.0000i − 0.681799i
\(698\) 10.0000i 0.378506i
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) − 20.0000i − 0.754851i
\(703\) 11.0000i 0.414873i
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) 0 0
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 6.00000i 0.224860i
\(713\) 12.0000i 0.449404i
\(714\) 0 0
\(715\) 0 0
\(716\) −3.00000 −0.112115
\(717\) − 12.0000i − 0.448148i
\(718\) 6.00000i 0.223918i
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18.0000i 0.669891i
\(723\) 50.0000i 1.85952i
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −4.00000 −0.148454
\(727\) 29.0000i 1.07555i 0.843088 + 0.537775i \(0.180735\pi\)
−0.843088 + 0.537775i \(0.819265\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 60.0000 2.21918
\(732\) − 8.00000i − 0.295689i
\(733\) − 47.0000i − 1.73598i −0.496578 0.867992i \(-0.665410\pi\)
0.496578 0.867992i \(-0.334590\pi\)
\(734\) −1.00000 −0.0369107
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) 12.0000i 0.442026i
\(738\) − 3.00000i − 0.110432i
\(739\) 37.0000 1.36107 0.680534 0.732717i \(-0.261748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) −10.0000 −0.367359
\(742\) 0 0
\(743\) 9.00000i 0.330178i 0.986279 + 0.165089i \(0.0527911\pi\)
−0.986279 + 0.165089i \(0.947209\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) −34.0000 −1.24483
\(747\) − 12.0000i − 0.439057i
\(748\) − 18.0000i − 0.658145i
\(749\) 0 0
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) 3.00000i 0.109399i
\(753\) 30.0000i 1.09326i
\(754\) −30.0000 −1.09254
\(755\) 0 0
\(756\) 0 0
\(757\) − 26.0000i − 0.944986i −0.881334 0.472493i \(-0.843354\pi\)
0.881334 0.472493i \(-0.156646\pi\)
\(758\) − 25.0000i − 0.908041i
\(759\) −18.0000 −0.653359
\(760\) 0 0
\(761\) 51.0000 1.84875 0.924374 0.381487i \(-0.124588\pi\)
0.924374 + 0.381487i \(0.124588\pi\)
\(762\) 38.0000i 1.37659i
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −15.0000 −0.541972
\(767\) 0 0
\(768\) 2.00000i 0.0721688i
\(769\) −49.0000 −1.76699 −0.883493 0.468445i \(-0.844814\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) 4.00000i 0.143963i
\(773\) − 39.0000i − 1.40273i −0.712801 0.701366i \(-0.752574\pi\)
0.712801 0.701366i \(-0.247426\pi\)
\(774\) 10.0000 0.359443
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) − 24.0000i − 0.860442i
\(779\) 3.00000 0.107486
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 18.0000i 0.643679i
\(783\) 24.0000i 0.857690i
\(784\) 0 0
\(785\) 0 0
\(786\) −6.00000 −0.214013
\(787\) − 34.0000i − 1.21197i −0.795476 0.605985i \(-0.792779\pi\)
0.795476 0.605985i \(-0.207221\pi\)
\(788\) − 3.00000i − 0.106871i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) − 3.00000i − 0.106600i
\(793\) − 20.0000i − 0.710221i
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) − 21.0000i − 0.741536i
\(803\) 12.0000i 0.423471i
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −20.0000 −0.704470
\(807\) − 24.0000i − 0.844840i
\(808\) 12.0000i 0.422159i
\(809\) −39.0000 −1.37117 −0.685583 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) −47.0000 −1.65039 −0.825197 0.564846i \(-0.808936\pi\)
−0.825197 + 0.564846i \(0.808936\pi\)
\(812\) 0 0
\(813\) 32.0000i 1.12229i
\(814\) −33.0000 −1.15665
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) 10.0000i 0.349856i
\(818\) 22.0000i 0.769212i
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) − 24.0000i − 0.837096i
\(823\) 44.0000i 1.53374i 0.641800 + 0.766872i \(0.278188\pi\)
−0.641800 + 0.766872i \(0.721812\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 54.0000i 1.87776i 0.344239 + 0.938882i \(0.388137\pi\)
−0.344239 + 0.938882i \(0.611863\pi\)
\(828\) 3.00000i 0.104257i
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 4.00000 0.138758
\(832\) 5.00000i 0.173344i
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 3.00000 0.103757
\(837\) 16.0000i 0.553041i
\(838\) − 15.0000i − 0.518166i
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 34.0000i 1.17172i
\(843\) − 6.00000i − 0.206651i
\(844\) 1.00000 0.0344214
\(845\) 0 0
\(846\) −3.00000 −0.103142
\(847\) 0 0
\(848\) 3.00000i 0.103020i
\(849\) 52.0000 1.78464
\(850\) 0 0
\(851\) 33.0000 1.13123
\(852\) − 24.0000i − 0.822226i
\(853\) 1.00000i 0.0342393i 0.999853 + 0.0171197i \(0.00544963\pi\)
−0.999853 + 0.0171197i \(0.994550\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) − 18.0000i − 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) − 30.0000i − 1.02418i
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 39.0000i 1.32758i 0.747921 + 0.663788i \(0.231052\pi\)
−0.747921 + 0.663788i \(0.768948\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) − 38.0000i − 1.29055i
\(868\) 0 0
\(869\) 30.0000 1.01768
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 4.00000i 0.135457i
\(873\) − 14.0000i − 0.473828i
\(874\) −3.00000 −0.101477
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) 7.00000i 0.236373i 0.992991 + 0.118187i \(0.0377081\pi\)
−0.992991 + 0.118187i \(0.962292\pi\)
\(878\) 10.0000i 0.337484i
\(879\) −54.0000 −1.82137
\(880\) 0 0
\(881\) −33.0000 −1.11180 −0.555899 0.831250i \(-0.687626\pi\)
−0.555899 + 0.831250i \(0.687626\pi\)
\(882\) 0 0
\(883\) 8.00000i 0.269221i 0.990899 + 0.134611i \(0.0429784\pi\)
−0.990899 + 0.134611i \(0.957022\pi\)
\(884\) −30.0000 −1.00901
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 24.0000i 0.805841i 0.915235 + 0.402921i \(0.132005\pi\)
−0.915235 + 0.402921i \(0.867995\pi\)
\(888\) 22.0000i 0.738272i
\(889\) 0 0
\(890\) 0 0
\(891\) −33.0000 −1.10554
\(892\) 8.00000i 0.267860i
\(893\) − 3.00000i − 0.100391i
\(894\) −36.0000 −1.20402
\(895\) 0 0
\(896\) 0 0
\(897\) 30.0000i 1.00167i
\(898\) − 3.00000i − 0.100111i
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 9.00000i 0.299667i
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) 28.0000 0.930238
\(907\) 10.0000i 0.332045i 0.986122 + 0.166022i \(0.0530924\pi\)
−0.986122 + 0.166022i \(0.946908\pi\)
\(908\) 24.0000i 0.796468i
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) − 2.00000i − 0.0662266i
\(913\) 36.0000i 1.19143i
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) 28.0000 0.925146
\(917\) 0 0
\(918\) 24.0000i 0.792118i
\(919\) −38.0000 −1.25350 −0.626752 0.779219i \(-0.715616\pi\)
−0.626752 + 0.779219i \(0.715616\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 6.00000i 0.197599i
\(923\) − 60.0000i − 1.97492i
\(924\) 0 0
\(925\) 0 0
\(926\) −19.0000 −0.624379
\(927\) − 4.00000i − 0.131377i
\(928\) − 6.00000i − 0.196960i
\(929\) 33.0000 1.08269 0.541347 0.840799i \(-0.317914\pi\)
0.541347 + 0.840799i \(0.317914\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000i 0.196537i
\(933\) − 24.0000i − 0.785725i
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) −5.00000 −0.163430
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) 0 0
\(939\) 16.0000 0.522140
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 10.0000i 0.325818i
\(943\) − 9.00000i − 0.293080i
\(944\) 0 0
\(945\) 0 0
\(946\) −30.0000 −0.975384
\(947\) − 30.0000i − 0.974869i −0.873160 0.487435i \(-0.837933\pi\)
0.873160 0.487435i \(-0.162067\pi\)
\(948\) − 20.0000i − 0.649570i
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) 36.0000 1.16738
\(952\) 0 0
\(953\) − 12.0000i − 0.388718i −0.980930 0.194359i \(-0.937737\pi\)
0.980930 0.194359i \(-0.0622627\pi\)
\(954\) −3.00000 −0.0971286
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) 36.0000i 1.16371i
\(958\) − 24.0000i − 0.775405i
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 55.0000i 1.77327i
\(963\) − 12.0000i − 0.386695i
\(964\) −25.0000 −0.805196
\(965\) 0 0
\(966\) 0 0
\(967\) − 32.0000i − 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) − 2.00000i − 0.0642824i
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −27.0000 −0.866471 −0.433236 0.901281i \(-0.642628\pi\)
−0.433236 + 0.901281i \(0.642628\pi\)
\(972\) 10.0000i 0.320750i
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) 30.0000i 0.959785i 0.877327 + 0.479893i \(0.159324\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(978\) − 8.00000i − 0.255812i
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 12.0000i 0.382935i
\(983\) − 57.0000i − 1.81802i −0.416777 0.909009i \(-0.636840\pi\)
0.416777 0.909009i \(-0.363160\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) − 5.00000i − 0.159071i
\(989\) 30.0000 0.953945
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) − 14.0000i − 0.444277i
\(994\) 0 0
\(995\) 0 0
\(996\) 24.0000 0.760469
\(997\) 14.0000i 0.443384i 0.975117 + 0.221692i \(0.0711580\pi\)
−0.975117 + 0.221692i \(0.928842\pi\)
\(998\) − 28.0000i − 0.886325i
\(999\) 44.0000 1.39210
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 2450.2.c.p.99.1 2
5.2 odd 4 490.2.a.j.1.1 1
5.3 odd 4 2450.2.a.f.1.1 1
5.4 even 2 inner 2450.2.c.p.99.2 2
7.3 odd 6 350.2.j.a.149.2 4
7.5 odd 6 350.2.j.a.249.1 4
7.6 odd 2 2450.2.c.f.99.1 2
15.2 even 4 4410.2.a.c.1.1 1
20.7 even 4 3920.2.a.g.1.1 1
35.2 odd 12 490.2.e.a.361.1 2
35.3 even 12 350.2.e.h.51.1 2
35.12 even 12 70.2.e.b.11.1 2
35.13 even 4 2450.2.a.p.1.1 1
35.17 even 12 70.2.e.b.51.1 yes 2
35.19 odd 6 350.2.j.a.249.2 4
35.24 odd 6 350.2.j.a.149.1 4
35.27 even 4 490.2.a.g.1.1 1
35.32 odd 12 490.2.e.a.471.1 2
35.33 even 12 350.2.e.h.151.1 2
35.34 odd 2 2450.2.c.f.99.2 2
105.17 odd 12 630.2.k.e.541.1 2
105.47 odd 12 630.2.k.e.361.1 2
105.62 odd 4 4410.2.a.m.1.1 1
140.27 odd 4 3920.2.a.be.1.1 1
140.47 odd 12 560.2.q.d.81.1 2
140.87 odd 12 560.2.q.d.401.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.b.11.1 2 35.12 even 12
70.2.e.b.51.1 yes 2 35.17 even 12
350.2.e.h.51.1 2 35.3 even 12
350.2.e.h.151.1 2 35.33 even 12
350.2.j.a.149.1 4 35.24 odd 6
350.2.j.a.149.2 4 7.3 odd 6
350.2.j.a.249.1 4 7.5 odd 6
350.2.j.a.249.2 4 35.19 odd 6
490.2.a.g.1.1 1 35.27 even 4
490.2.a.j.1.1 1 5.2 odd 4
490.2.e.a.361.1 2 35.2 odd 12
490.2.e.a.471.1 2 35.32 odd 12
560.2.q.d.81.1 2 140.47 odd 12
560.2.q.d.401.1 2 140.87 odd 12
630.2.k.e.361.1 2 105.47 odd 12
630.2.k.e.541.1 2 105.17 odd 12
2450.2.a.f.1.1 1 5.3 odd 4
2450.2.a.p.1.1 1 35.13 even 4
2450.2.c.f.99.1 2 7.6 odd 2
2450.2.c.f.99.2 2 35.34 odd 2
2450.2.c.p.99.1 2 1.1 even 1 trivial
2450.2.c.p.99.2 2 5.4 even 2 inner
3920.2.a.g.1.1 1 20.7 even 4
3920.2.a.be.1.1 1 140.27 odd 4
4410.2.a.c.1.1 1 15.2 even 4
4410.2.a.m.1.1 1 105.62 odd 4