Properties

Label 1150.2.b.a.599.2
Level $1150$
Weight $2$
Character 1150.599
Analytic conductor $9.183$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1150,2,Mod(599,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, 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("1150.599");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.b (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: \(9.18279623245\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.a.599.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -6.00000 q^{9} +3.00000 q^{11} -3.00000i q^{12} -6.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -5.00000i q^{17} -6.00000i q^{18} +1.00000 q^{19} +12.0000 q^{21} +3.00000i q^{22} +1.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} -9.00000i q^{27} +4.00000i q^{28} +8.00000 q^{29} -8.00000 q^{31} +1.00000i q^{32} +9.00000i q^{33} +5.00000 q^{34} +6.00000 q^{36} -2.00000i q^{37} +1.00000i q^{38} +18.0000 q^{39} -7.00000 q^{41} +12.0000i q^{42} +4.00000i q^{43} -3.00000 q^{44} -1.00000 q^{46} -10.0000i q^{47} +3.00000i q^{48} -9.00000 q^{49} +15.0000 q^{51} +6.00000i q^{52} -12.0000i q^{53} +9.00000 q^{54} -4.00000 q^{56} +3.00000i q^{57} +8.00000i q^{58} -4.00000 q^{59} -8.00000 q^{61} -8.00000i q^{62} +24.0000i q^{63} -1.00000 q^{64} -9.00000 q^{66} -3.00000i q^{67} +5.00000i q^{68} -3.00000 q^{69} +4.00000 q^{71} +6.00000i q^{72} -7.00000i q^{73} +2.00000 q^{74} -1.00000 q^{76} -12.0000i q^{77} +18.0000i q^{78} +6.00000 q^{79} +9.00000 q^{81} -7.00000i q^{82} +11.0000i q^{83} -12.0000 q^{84} -4.00000 q^{86} +24.0000i q^{87} -3.00000i q^{88} +3.00000 q^{89} -24.0000 q^{91} -1.00000i q^{92} -24.0000i q^{93} +10.0000 q^{94} -3.00000 q^{96} +14.0000i q^{97} -9.00000i q^{98} -18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9} + 6 q^{11} + 8 q^{14} + 2 q^{16} + 2 q^{19} + 24 q^{21} + 6 q^{24} + 12 q^{26} + 16 q^{29} - 16 q^{31} + 10 q^{34} + 12 q^{36} + 36 q^{39} - 14 q^{41} - 6 q^{44} - 2 q^{46}+ \cdots - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 3.00000i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) − 3.00000i − 0.866025i
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 5.00000i − 1.21268i −0.795206 0.606339i \(-0.792637\pi\)
0.795206 0.606339i \(-0.207363\pi\)
\(18\) − 6.00000i − 1.41421i
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 12.0000 2.61861
\(22\) 3.00000i 0.639602i
\(23\) 1.00000i 0.208514i
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) − 9.00000i − 1.73205i
\(28\) 4.00000i 0.755929i
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 9.00000i 1.56670i
\(34\) 5.00000 0.857493
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 1.00000i 0.162221i
\(39\) 18.0000 2.88231
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 12.0000i 1.85164i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 10.0000i − 1.45865i −0.684167 0.729325i \(-0.739834\pi\)
0.684167 0.729325i \(-0.260166\pi\)
\(48\) 3.00000i 0.433013i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 15.0000 2.10042
\(52\) 6.00000i 0.832050i
\(53\) − 12.0000i − 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 3.00000i 0.397360i
\(58\) 8.00000i 1.05045i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) − 8.00000i − 1.01600i
\(63\) 24.0000i 3.02372i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −9.00000 −1.10782
\(67\) − 3.00000i − 0.366508i −0.983066 0.183254i \(-0.941337\pi\)
0.983066 0.183254i \(-0.0586631\pi\)
\(68\) 5.00000i 0.606339i
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 6.00000i 0.707107i
\(73\) − 7.00000i − 0.819288i −0.912245 0.409644i \(-0.865653\pi\)
0.912245 0.409644i \(-0.134347\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) − 12.0000i − 1.36753i
\(78\) 18.0000i 2.03810i
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) − 7.00000i − 0.773021i
\(83\) 11.0000i 1.20741i 0.797209 + 0.603703i \(0.206309\pi\)
−0.797209 + 0.603703i \(0.793691\pi\)
\(84\) −12.0000 −1.30931
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 24.0000i 2.57307i
\(88\) − 3.00000i − 0.319801i
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −24.0000 −2.51588
\(92\) − 1.00000i − 0.104257i
\(93\) − 24.0000i − 2.48868i
\(94\) 10.0000 1.03142
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) −18.0000 −1.80907
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 15.0000i 1.48522i
\(103\) 10.0000i 0.985329i 0.870219 + 0.492665i \(0.163977\pi\)
−0.870219 + 0.492665i \(0.836023\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) − 5.00000i − 0.483368i −0.970355 0.241684i \(-0.922300\pi\)
0.970355 0.241684i \(-0.0776998\pi\)
\(108\) 9.00000i 0.866025i
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) − 4.00000i − 0.377964i
\(113\) 15.0000i 1.41108i 0.708669 + 0.705541i \(0.249296\pi\)
−0.708669 + 0.705541i \(0.750704\pi\)
\(114\) −3.00000 −0.280976
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) 36.0000i 3.32820i
\(118\) − 4.00000i − 0.368230i
\(119\) −20.0000 −1.83340
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) − 8.00000i − 0.724286i
\(123\) − 21.0000i − 1.89351i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) −24.0000 −2.13809
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) − 9.00000i − 0.783349i
\(133\) − 4.00000i − 0.346844i
\(134\) 3.00000 0.259161
\(135\) 0 0
\(136\) −5.00000 −0.428746
\(137\) 3.00000i 0.256307i 0.991754 + 0.128154i \(0.0409051\pi\)
−0.991754 + 0.128154i \(0.959095\pi\)
\(138\) − 3.00000i − 0.255377i
\(139\) 19.0000 1.61156 0.805779 0.592216i \(-0.201747\pi\)
0.805779 + 0.592216i \(0.201747\pi\)
\(140\) 0 0
\(141\) 30.0000 2.52646
\(142\) 4.00000i 0.335673i
\(143\) − 18.0000i − 1.50524i
\(144\) −6.00000 −0.500000
\(145\) 0 0
\(146\) 7.00000 0.579324
\(147\) − 27.0000i − 2.22692i
\(148\) 2.00000i 0.164399i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) − 1.00000i − 0.0811107i
\(153\) 30.0000i 2.42536i
\(154\) 12.0000 0.966988
\(155\) 0 0
\(156\) −18.0000 −1.44115
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 6.00000i 0.477334i
\(159\) 36.0000 2.85499
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 9.00000i 0.707107i
\(163\) 5.00000i 0.391630i 0.980641 + 0.195815i \(0.0627352\pi\)
−0.980641 + 0.195815i \(0.937265\pi\)
\(164\) 7.00000 0.546608
\(165\) 0 0
\(166\) −11.0000 −0.853766
\(167\) − 22.0000i − 1.70241i −0.524832 0.851206i \(-0.675872\pi\)
0.524832 0.851206i \(-0.324128\pi\)
\(168\) − 12.0000i − 0.925820i
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) − 4.00000i − 0.304997i
\(173\) 4.00000i 0.304114i 0.988372 + 0.152057i \(0.0485898\pi\)
−0.988372 + 0.152057i \(0.951410\pi\)
\(174\) −24.0000 −1.81944
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) − 12.0000i − 0.901975i
\(178\) 3.00000i 0.224860i
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) − 24.0000i − 1.77900i
\(183\) − 24.0000i − 1.77413i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 24.0000 1.75977
\(187\) − 15.0000i − 1.09691i
\(188\) 10.0000i 0.729325i
\(189\) −36.0000 −2.61861
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) − 3.00000i − 0.216506i
\(193\) − 9.00000i − 0.647834i −0.946085 0.323917i \(-0.895000\pi\)
0.946085 0.323917i \(-0.105000\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 8.00000i 0.569976i 0.958531 + 0.284988i \(0.0919897\pi\)
−0.958531 + 0.284988i \(0.908010\pi\)
\(198\) − 18.0000i − 1.27920i
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) 9.00000 0.634811
\(202\) 4.00000i 0.281439i
\(203\) − 32.0000i − 2.24596i
\(204\) −15.0000 −1.05021
\(205\) 0 0
\(206\) −10.0000 −0.696733
\(207\) − 6.00000i − 0.417029i
\(208\) − 6.00000i − 0.416025i
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 12.0000i 0.824163i
\(213\) 12.0000i 0.822226i
\(214\) 5.00000 0.341793
\(215\) 0 0
\(216\) −9.00000 −0.612372
\(217\) 32.0000i 2.17230i
\(218\) 10.0000i 0.677285i
\(219\) 21.0000 1.41905
\(220\) 0 0
\(221\) −30.0000 −2.01802
\(222\) 6.00000i 0.402694i
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −15.0000 −0.997785
\(227\) 8.00000i 0.530979i 0.964114 + 0.265489i \(0.0855335\pi\)
−0.964114 + 0.265489i \(0.914466\pi\)
\(228\) − 3.00000i − 0.198680i
\(229\) 30.0000 1.98246 0.991228 0.132164i \(-0.0421925\pi\)
0.991228 + 0.132164i \(0.0421925\pi\)
\(230\) 0 0
\(231\) 36.0000 2.36863
\(232\) − 8.00000i − 0.525226i
\(233\) − 14.0000i − 0.917170i −0.888650 0.458585i \(-0.848356\pi\)
0.888650 0.458585i \(-0.151644\pi\)
\(234\) −36.0000 −2.35339
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 18.0000i 1.16923i
\(238\) − 20.0000i − 1.29641i
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 23.0000 1.48156 0.740780 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(242\) − 2.00000i − 0.128565i
\(243\) 0 0
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 21.0000 1.33891
\(247\) − 6.00000i − 0.381771i
\(248\) 8.00000i 0.508001i
\(249\) −33.0000 −2.09129
\(250\) 0 0
\(251\) −7.00000 −0.441836 −0.220918 0.975292i \(-0.570905\pi\)
−0.220918 + 0.975292i \(0.570905\pi\)
\(252\) − 24.0000i − 1.51186i
\(253\) 3.00000i 0.188608i
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) − 12.0000i − 0.747087i
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −48.0000 −2.97113
\(262\) 12.0000i 0.741362i
\(263\) − 4.00000i − 0.246651i −0.992366 0.123325i \(-0.960644\pi\)
0.992366 0.123325i \(-0.0393559\pi\)
\(264\) 9.00000 0.553912
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 9.00000i 0.550791i
\(268\) 3.00000i 0.183254i
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 10.0000 0.607457 0.303728 0.952759i \(-0.401768\pi\)
0.303728 + 0.952759i \(0.401768\pi\)
\(272\) − 5.00000i − 0.303170i
\(273\) − 72.0000i − 4.35764i
\(274\) −3.00000 −0.181237
\(275\) 0 0
\(276\) 3.00000 0.180579
\(277\) 28.0000i 1.68236i 0.540758 + 0.841178i \(0.318138\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 19.0000i 1.13954i
\(279\) 48.0000 2.87368
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 30.0000i 1.78647i
\(283\) − 33.0000i − 1.96165i −0.194900 0.980823i \(-0.562438\pi\)
0.194900 0.980823i \(-0.437562\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 18.0000 1.06436
\(287\) 28.0000i 1.65279i
\(288\) − 6.00000i − 0.353553i
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −42.0000 −2.46208
\(292\) 7.00000i 0.409644i
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) 27.0000 1.57467
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) − 27.0000i − 1.56670i
\(298\) − 6.00000i − 0.347571i
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 20.0000i 1.15087i
\(303\) 12.0000i 0.689382i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −30.0000 −1.71499
\(307\) 9.00000i 0.513657i 0.966457 + 0.256829i \(0.0826776\pi\)
−0.966457 + 0.256829i \(0.917322\pi\)
\(308\) 12.0000i 0.683763i
\(309\) −30.0000 −1.70664
\(310\) 0 0
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) − 18.0000i − 1.01905i
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) − 12.0000i − 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) 36.0000i 2.01878i
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 15.0000 0.837218
\(322\) 4.00000i 0.222911i
\(323\) − 5.00000i − 0.278207i
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) −5.00000 −0.276924
\(327\) 30.0000i 1.65900i
\(328\) 7.00000i 0.386510i
\(329\) −40.0000 −2.20527
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) − 11.0000i − 0.603703i
\(333\) 12.0000i 0.657596i
\(334\) 22.0000 1.20379
\(335\) 0 0
\(336\) 12.0000 0.654654
\(337\) − 3.00000i − 0.163420i −0.996656 0.0817102i \(-0.973962\pi\)
0.996656 0.0817102i \(-0.0260382\pi\)
\(338\) − 23.0000i − 1.25104i
\(339\) −45.0000 −2.44406
\(340\) 0 0
\(341\) −24.0000 −1.29967
\(342\) − 6.00000i − 0.324443i
\(343\) 8.00000i 0.431959i
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −4.00000 −0.215041
\(347\) − 7.00000i − 0.375780i −0.982190 0.187890i \(-0.939835\pi\)
0.982190 0.187890i \(-0.0601648\pi\)
\(348\) − 24.0000i − 1.28654i
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −54.0000 −2.88231
\(352\) 3.00000i 0.159901i
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −3.00000 −0.159000
\(357\) − 60.0000i − 3.17554i
\(358\) 3.00000i 0.158555i
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) − 22.0000i − 1.15629i
\(363\) − 6.00000i − 0.314918i
\(364\) 24.0000 1.25794
\(365\) 0 0
\(366\) 24.0000 1.25450
\(367\) − 2.00000i − 0.104399i −0.998637 0.0521996i \(-0.983377\pi\)
0.998637 0.0521996i \(-0.0166232\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 42.0000 2.18643
\(370\) 0 0
\(371\) −48.0000 −2.49204
\(372\) 24.0000i 1.24434i
\(373\) − 12.0000i − 0.621336i −0.950518 0.310668i \(-0.899447\pi\)
0.950518 0.310668i \(-0.100553\pi\)
\(374\) 15.0000 0.775632
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) − 48.0000i − 2.47213i
\(378\) − 36.0000i − 1.85164i
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) −24.0000 −1.22956
\(382\) 12.0000i 0.613973i
\(383\) − 6.00000i − 0.306586i −0.988181 0.153293i \(-0.951012\pi\)
0.988181 0.153293i \(-0.0489878\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 9.00000 0.458088
\(387\) − 24.0000i − 1.21999i
\(388\) − 14.0000i − 0.710742i
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) 9.00000i 0.454569i
\(393\) 36.0000i 1.81596i
\(394\) −8.00000 −0.403034
\(395\) 0 0
\(396\) 18.0000 0.904534
\(397\) 12.0000i 0.602263i 0.953583 + 0.301131i \(0.0973643\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) − 18.0000i − 0.902258i
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) −5.00000 −0.249688 −0.124844 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) 9.00000i 0.448879i
\(403\) 48.0000i 2.39105i
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) 32.0000 1.58813
\(407\) − 6.00000i − 0.297409i
\(408\) − 15.0000i − 0.742611i
\(409\) −21.0000 −1.03838 −0.519192 0.854658i \(-0.673767\pi\)
−0.519192 + 0.854658i \(0.673767\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) − 10.0000i − 0.492665i
\(413\) 16.0000i 0.787309i
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 57.0000i 2.79130i
\(418\) 3.00000i 0.146735i
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) − 1.00000i − 0.0486792i
\(423\) 60.0000i 2.91730i
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) 32.0000i 1.54859i
\(428\) 5.00000i 0.241684i
\(429\) 54.0000 2.60714
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) − 9.00000i − 0.433013i
\(433\) 29.0000i 1.39365i 0.717241 + 0.696826i \(0.245405\pi\)
−0.717241 + 0.696826i \(0.754595\pi\)
\(434\) −32.0000 −1.53605
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 1.00000i 0.0478365i
\(438\) 21.0000i 1.00342i
\(439\) 6.00000 0.286364 0.143182 0.989696i \(-0.454267\pi\)
0.143182 + 0.989696i \(0.454267\pi\)
\(440\) 0 0
\(441\) 54.0000 2.57143
\(442\) − 30.0000i − 1.42695i
\(443\) − 39.0000i − 1.85295i −0.376361 0.926473i \(-0.622825\pi\)
0.376361 0.926473i \(-0.377175\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) 0 0
\(447\) − 18.0000i − 0.851371i
\(448\) 4.00000i 0.188982i
\(449\) −5.00000 −0.235965 −0.117982 0.993016i \(-0.537643\pi\)
−0.117982 + 0.993016i \(0.537643\pi\)
\(450\) 0 0
\(451\) −21.0000 −0.988851
\(452\) − 15.0000i − 0.705541i
\(453\) 60.0000i 2.81905i
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) 7.00000i 0.327446i 0.986506 + 0.163723i \(0.0523504\pi\)
−0.986506 + 0.163723i \(0.947650\pi\)
\(458\) 30.0000i 1.40181i
\(459\) −45.0000 −2.10042
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 36.0000i 1.67487i
\(463\) 26.0000i 1.20832i 0.796862 + 0.604161i \(0.206492\pi\)
−0.796862 + 0.604161i \(0.793508\pi\)
\(464\) 8.00000 0.371391
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) − 28.0000i − 1.29569i −0.761774 0.647843i \(-0.775671\pi\)
0.761774 0.647843i \(-0.224329\pi\)
\(468\) − 36.0000i − 1.66410i
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 42.0000 1.93526
\(472\) 4.00000i 0.184115i
\(473\) 12.0000i 0.551761i
\(474\) −18.0000 −0.826767
\(475\) 0 0
\(476\) 20.0000 0.916698
\(477\) 72.0000i 3.29665i
\(478\) 6.00000i 0.274434i
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 23.0000i 1.04762i
\(483\) 12.0000i 0.546019i
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) 30.0000i 1.35943i 0.733476 + 0.679715i \(0.237896\pi\)
−0.733476 + 0.679715i \(0.762104\pi\)
\(488\) 8.00000i 0.362143i
\(489\) −15.0000 −0.678323
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 21.0000i 0.946753i
\(493\) − 40.0000i − 1.80151i
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) − 16.0000i − 0.717698i
\(498\) − 33.0000i − 1.47877i
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 0 0
\(501\) 66.0000 2.94866
\(502\) − 7.00000i − 0.312425i
\(503\) − 24.0000i − 1.07011i −0.844818 0.535054i \(-0.820291\pi\)
0.844818 0.535054i \(-0.179709\pi\)
\(504\) 24.0000 1.06904
\(505\) 0 0
\(506\) −3.00000 −0.133366
\(507\) − 69.0000i − 3.06440i
\(508\) − 8.00000i − 0.354943i
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) 1.00000i 0.0441942i
\(513\) − 9.00000i − 0.397360i
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) − 30.0000i − 1.31940i
\(518\) − 8.00000i − 0.351500i
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 31.0000 1.35813 0.679067 0.734076i \(-0.262384\pi\)
0.679067 + 0.734076i \(0.262384\pi\)
\(522\) − 48.0000i − 2.10090i
\(523\) 37.0000i 1.61790i 0.587879 + 0.808949i \(0.299963\pi\)
−0.587879 + 0.808949i \(0.700037\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) 40.0000i 1.74243i
\(528\) 9.00000i 0.391675i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 4.00000i 0.173422i
\(533\) 42.0000i 1.81922i
\(534\) −9.00000 −0.389468
\(535\) 0 0
\(536\) −3.00000 −0.129580
\(537\) 9.00000i 0.388379i
\(538\) 6.00000i 0.258678i
\(539\) −27.0000 −1.16297
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 10.0000i 0.429537i
\(543\) − 66.0000i − 2.83233i
\(544\) 5.00000 0.214373
\(545\) 0 0
\(546\) 72.0000 3.08132
\(547\) 17.0000i 0.726868i 0.931620 + 0.363434i \(0.118396\pi\)
−0.931620 + 0.363434i \(0.881604\pi\)
\(548\) − 3.00000i − 0.128154i
\(549\) 48.0000 2.04859
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 3.00000i 0.127688i
\(553\) − 24.0000i − 1.02058i
\(554\) −28.0000 −1.18961
\(555\) 0 0
\(556\) −19.0000 −0.805779
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 48.0000i 2.03200i
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 45.0000 1.89990
\(562\) − 30.0000i − 1.26547i
\(563\) − 4.00000i − 0.168580i −0.996441 0.0842900i \(-0.973138\pi\)
0.996441 0.0842900i \(-0.0268622\pi\)
\(564\) −30.0000 −1.26323
\(565\) 0 0
\(566\) 33.0000 1.38709
\(567\) − 36.0000i − 1.51186i
\(568\) − 4.00000i − 0.167836i
\(569\) 9.00000 0.377300 0.188650 0.982044i \(-0.439589\pi\)
0.188650 + 0.982044i \(0.439589\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 18.0000i 0.752618i
\(573\) 36.0000i 1.50392i
\(574\) −28.0000 −1.16870
\(575\) 0 0
\(576\) 6.00000 0.250000
\(577\) − 17.0000i − 0.707719i −0.935299 0.353860i \(-0.884869\pi\)
0.935299 0.353860i \(-0.115131\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) 27.0000 1.12208
\(580\) 0 0
\(581\) 44.0000 1.82543
\(582\) − 42.0000i − 1.74096i
\(583\) − 36.0000i − 1.49097i
\(584\) −7.00000 −0.289662
\(585\) 0 0
\(586\) −24.0000 −0.991431
\(587\) − 17.0000i − 0.701665i −0.936438 0.350833i \(-0.885899\pi\)
0.936438 0.350833i \(-0.114101\pi\)
\(588\) 27.0000i 1.11346i
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) − 2.00000i − 0.0821995i
\(593\) − 35.0000i − 1.43728i −0.695383 0.718639i \(-0.744765\pi\)
0.695383 0.718639i \(-0.255235\pi\)
\(594\) 27.0000 1.10782
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) − 54.0000i − 2.21007i
\(598\) 6.00000i 0.245358i
\(599\) 20.0000 0.817178 0.408589 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) 0 0
\(601\) 31.0000 1.26452 0.632258 0.774758i \(-0.282128\pi\)
0.632258 + 0.774758i \(0.282128\pi\)
\(602\) 16.0000i 0.652111i
\(603\) 18.0000i 0.733017i
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) − 10.0000i − 0.405887i −0.979190 0.202944i \(-0.934949\pi\)
0.979190 0.202944i \(-0.0650509\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) 96.0000 3.89012
\(610\) 0 0
\(611\) −60.0000 −2.42734
\(612\) − 30.0000i − 1.21268i
\(613\) − 8.00000i − 0.323117i −0.986863 0.161558i \(-0.948348\pi\)
0.986863 0.161558i \(-0.0516520\pi\)
\(614\) −9.00000 −0.363210
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) − 30.0000i − 1.20775i −0.797077 0.603877i \(-0.793622\pi\)
0.797077 0.603877i \(-0.206378\pi\)
\(618\) − 30.0000i − 1.20678i
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 9.00000 0.361158
\(622\) 2.00000i 0.0801927i
\(623\) − 12.0000i − 0.480770i
\(624\) 18.0000 0.720577
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 9.00000i 0.359425i
\(628\) 14.0000i 0.558661i
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) 30.0000 1.19428 0.597141 0.802137i \(-0.296303\pi\)
0.597141 + 0.802137i \(0.296303\pi\)
\(632\) − 6.00000i − 0.238667i
\(633\) − 3.00000i − 0.119239i
\(634\) 12.0000 0.476581
\(635\) 0 0
\(636\) −36.0000 −1.42749
\(637\) 54.0000i 2.13956i
\(638\) 24.0000i 0.950169i
\(639\) −24.0000 −0.949425
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 15.0000i 0.592003i
\(643\) 32.0000i 1.26196i 0.775800 + 0.630978i \(0.217346\pi\)
−0.775800 + 0.630978i \(0.782654\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 5.00000 0.196722
\(647\) − 28.0000i − 1.10079i −0.834903 0.550397i \(-0.814476\pi\)
0.834903 0.550397i \(-0.185524\pi\)
\(648\) − 9.00000i − 0.353553i
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) −96.0000 −3.76254
\(652\) − 5.00000i − 0.195815i
\(653\) 14.0000i 0.547862i 0.961749 + 0.273931i \(0.0883240\pi\)
−0.961749 + 0.273931i \(0.911676\pi\)
\(654\) −30.0000 −1.17309
\(655\) 0 0
\(656\) −7.00000 −0.273304
\(657\) 42.0000i 1.63858i
\(658\) − 40.0000i − 1.55936i
\(659\) −5.00000 −0.194772 −0.0973862 0.995247i \(-0.531048\pi\)
−0.0973862 + 0.995247i \(0.531048\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 25.0000i 0.971653i
\(663\) − 90.0000i − 3.49531i
\(664\) 11.0000 0.426883
\(665\) 0 0
\(666\) −12.0000 −0.464991
\(667\) 8.00000i 0.309761i
\(668\) 22.0000i 0.851206i
\(669\) 0 0
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 12.0000i 0.462910i
\(673\) 34.0000i 1.31060i 0.755367 + 0.655302i \(0.227459\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) 3.00000 0.115556
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) − 45.0000i − 1.72821i
\(679\) 56.0000 2.14908
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) − 24.0000i − 0.919007i
\(683\) 35.0000i 1.33924i 0.742705 + 0.669619i \(0.233543\pi\)
−0.742705 + 0.669619i \(0.766457\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 90.0000i 3.43371i
\(688\) 4.00000i 0.152499i
\(689\) −72.0000 −2.74298
\(690\) 0 0
\(691\) 29.0000 1.10321 0.551606 0.834105i \(-0.314015\pi\)
0.551606 + 0.834105i \(0.314015\pi\)
\(692\) − 4.00000i − 0.152057i
\(693\) 72.0000i 2.73505i
\(694\) 7.00000 0.265716
\(695\) 0 0
\(696\) 24.0000 0.909718
\(697\) 35.0000i 1.32572i
\(698\) − 26.0000i − 0.984115i
\(699\) 42.0000 1.58859
\(700\) 0 0
\(701\) −8.00000 −0.302156 −0.151078 0.988522i \(-0.548274\pi\)
−0.151078 + 0.988522i \(0.548274\pi\)
\(702\) − 54.0000i − 2.03810i
\(703\) − 2.00000i − 0.0754314i
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) − 16.0000i − 0.601742i
\(708\) 12.0000i 0.450988i
\(709\) −16.0000 −0.600893 −0.300446 0.953799i \(-0.597136\pi\)
−0.300446 + 0.953799i \(0.597136\pi\)
\(710\) 0 0
\(711\) −36.0000 −1.35011
\(712\) − 3.00000i − 0.112430i
\(713\) − 8.00000i − 0.299602i
\(714\) 60.0000 2.24544
\(715\) 0 0
\(716\) −3.00000 −0.112115
\(717\) 18.0000i 0.672222i
\(718\) 20.0000i 0.746393i
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) 40.0000 1.48968
\(722\) − 18.0000i − 0.669891i
\(723\) 69.0000i 2.56614i
\(724\) 22.0000 0.817624
\(725\) 0 0
\(726\) 6.00000 0.222681
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 24.0000i 0.889499i
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 20.0000 0.739727
\(732\) 24.0000i 0.887066i
\(733\) 38.0000i 1.40356i 0.712393 + 0.701781i \(0.247612\pi\)
−0.712393 + 0.701781i \(0.752388\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) − 9.00000i − 0.331519i
\(738\) 42.0000i 1.54604i
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 18.0000 0.661247
\(742\) − 48.0000i − 1.76214i
\(743\) 26.0000i 0.953847i 0.878945 + 0.476924i \(0.158248\pi\)
−0.878945 + 0.476924i \(0.841752\pi\)
\(744\) −24.0000 −0.879883
\(745\) 0 0
\(746\) 12.0000 0.439351
\(747\) − 66.0000i − 2.41481i
\(748\) 15.0000i 0.548454i
\(749\) −20.0000 −0.730784
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) − 10.0000i − 0.364662i
\(753\) − 21.0000i − 0.765283i
\(754\) 48.0000 1.74806
\(755\) 0 0
\(756\) 36.0000 1.30931
\(757\) − 16.0000i − 0.581530i −0.956795 0.290765i \(-0.906090\pi\)
0.956795 0.290765i \(-0.0939098\pi\)
\(758\) − 5.00000i − 0.181608i
\(759\) −9.00000 −0.326679
\(760\) 0 0
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) − 24.0000i − 0.869428i
\(763\) − 40.0000i − 1.44810i
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) 24.0000i 0.866590i
\(768\) 3.00000i 0.108253i
\(769\) 21.0000 0.757279 0.378640 0.925544i \(-0.376392\pi\)
0.378640 + 0.925544i \(0.376392\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 9.00000i 0.323917i
\(773\) 4.00000i 0.143870i 0.997409 + 0.0719350i \(0.0229174\pi\)
−0.997409 + 0.0719350i \(0.977083\pi\)
\(774\) 24.0000 0.862662
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) − 24.0000i − 0.860995i
\(778\) 36.0000i 1.29066i
\(779\) −7.00000 −0.250801
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 5.00000i 0.178800i
\(783\) − 72.0000i − 2.57307i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) −36.0000 −1.28408
\(787\) 52.0000i 1.85360i 0.375555 + 0.926800i \(0.377452\pi\)
−0.375555 + 0.926800i \(0.622548\pi\)
\(788\) − 8.00000i − 0.284988i
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 60.0000 2.13335
\(792\) 18.0000i 0.639602i
\(793\) 48.0000i 1.70453i
\(794\) −12.0000 −0.425864
\(795\) 0 0
\(796\) 18.0000 0.637993
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 12.0000i 0.424795i
\(799\) −50.0000 −1.76887
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) − 5.00000i − 0.176556i
\(803\) − 21.0000i − 0.741074i
\(804\) −9.00000 −0.317406
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) 18.0000i 0.633630i
\(808\) − 4.00000i − 0.140720i
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 32.0000i 1.12298i
\(813\) 30.0000i 1.05215i
\(814\) 6.00000 0.210300
\(815\) 0 0
\(816\) 15.0000 0.525105
\(817\) 4.00000i 0.139942i
\(818\) − 21.0000i − 0.734248i
\(819\) 144.000 5.03177
\(820\) 0 0
\(821\) 16.0000 0.558404 0.279202 0.960232i \(-0.409930\pi\)
0.279202 + 0.960232i \(0.409930\pi\)
\(822\) − 9.00000i − 0.313911i
\(823\) − 26.0000i − 0.906303i −0.891434 0.453152i \(-0.850300\pi\)
0.891434 0.453152i \(-0.149700\pi\)
\(824\) 10.0000 0.348367
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) 17.0000i 0.591148i 0.955320 + 0.295574i \(0.0955109\pi\)
−0.955320 + 0.295574i \(0.904489\pi\)
\(828\) 6.00000i 0.208514i
\(829\) −24.0000 −0.833554 −0.416777 0.909009i \(-0.636840\pi\)
−0.416777 + 0.909009i \(0.636840\pi\)
\(830\) 0 0
\(831\) −84.0000 −2.91393
\(832\) 6.00000i 0.208013i
\(833\) 45.0000i 1.55916i
\(834\) −57.0000 −1.97375
\(835\) 0 0
\(836\) −3.00000 −0.103757
\(837\) 72.0000i 2.48868i
\(838\) − 3.00000i − 0.103633i
\(839\) −26.0000 −0.897620 −0.448810 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) − 14.0000i − 0.482472i
\(843\) − 90.0000i − 3.09976i
\(844\) 1.00000 0.0344214
\(845\) 0 0
\(846\) −60.0000 −2.06284
\(847\) 8.00000i 0.274883i
\(848\) − 12.0000i − 0.412082i
\(849\) 99.0000 3.39767
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) − 12.0000i − 0.411113i
\(853\) 12.0000i 0.410872i 0.978671 + 0.205436i \(0.0658613\pi\)
−0.978671 + 0.205436i \(0.934139\pi\)
\(854\) −32.0000 −1.09502
\(855\) 0 0
\(856\) −5.00000 −0.170896
\(857\) 21.0000i 0.717346i 0.933463 + 0.358673i \(0.116771\pi\)
−0.933463 + 0.358673i \(0.883229\pi\)
\(858\) 54.0000i 1.84353i
\(859\) 11.0000 0.375315 0.187658 0.982235i \(-0.439910\pi\)
0.187658 + 0.982235i \(0.439910\pi\)
\(860\) 0 0
\(861\) −84.0000 −2.86271
\(862\) − 4.00000i − 0.136241i
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) 9.00000 0.306186
\(865\) 0 0
\(866\) −29.0000 −0.985460
\(867\) − 24.0000i − 0.815083i
\(868\) − 32.0000i − 1.08615i
\(869\) 18.0000 0.610608
\(870\) 0 0
\(871\) −18.0000 −0.609907
\(872\) − 10.0000i − 0.338643i
\(873\) − 84.0000i − 2.84297i
\(874\) −1.00000 −0.0338255
\(875\) 0 0
\(876\) −21.0000 −0.709524
\(877\) − 38.0000i − 1.28317i −0.767052 0.641584i \(-0.778277\pi\)
0.767052 0.641584i \(-0.221723\pi\)
\(878\) 6.00000i 0.202490i
\(879\) −72.0000 −2.42850
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 54.0000i 1.81827i
\(883\) − 37.0000i − 1.24515i −0.782560 0.622575i \(-0.786087\pi\)
0.782560 0.622575i \(-0.213913\pi\)
\(884\) 30.0000 1.00901
\(885\) 0 0
\(886\) 39.0000 1.31023
\(887\) − 12.0000i − 0.402921i −0.979497 0.201460i \(-0.935431\pi\)
0.979497 0.201460i \(-0.0645687\pi\)
\(888\) − 6.00000i − 0.201347i
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 27.0000 0.904534
\(892\) 0 0
\(893\) − 10.0000i − 0.334637i
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 18.0000i 0.601003i
\(898\) − 5.00000i − 0.166852i
\(899\) −64.0000 −2.13452
\(900\) 0 0
\(901\) −60.0000 −1.99889
\(902\) − 21.0000i − 0.699224i
\(903\) 48.0000i 1.59734i
\(904\) 15.0000 0.498893
\(905\) 0 0
\(906\) −60.0000 −1.99337
\(907\) 28.0000i 0.929725i 0.885383 + 0.464862i \(0.153896\pi\)
−0.885383 + 0.464862i \(0.846104\pi\)
\(908\) − 8.00000i − 0.265489i
\(909\) −24.0000 −0.796030
\(910\) 0 0
\(911\) −10.0000 −0.331315 −0.165657 0.986183i \(-0.552975\pi\)
−0.165657 + 0.986183i \(0.552975\pi\)
\(912\) 3.00000i 0.0993399i
\(913\) 33.0000i 1.09214i
\(914\) −7.00000 −0.231539
\(915\) 0 0
\(916\) −30.0000 −0.991228
\(917\) − 48.0000i − 1.58510i
\(918\) − 45.0000i − 1.48522i
\(919\) −52.0000 −1.71532 −0.857661 0.514216i \(-0.828083\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(920\) 0 0
\(921\) −27.0000 −0.889680
\(922\) − 24.0000i − 0.790398i
\(923\) − 24.0000i − 0.789970i
\(924\) −36.0000 −1.18431
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) − 60.0000i − 1.97066i
\(928\) 8.00000i 0.262613i
\(929\) 22.0000 0.721797 0.360898 0.932605i \(-0.382470\pi\)
0.360898 + 0.932605i \(0.382470\pi\)
\(930\) 0 0
\(931\) −9.00000 −0.294963
\(932\) 14.0000i 0.458585i
\(933\) 6.00000i 0.196431i
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) 36.0000 1.17670
\(937\) 9.00000i 0.294017i 0.989135 + 0.147009i \(0.0469645\pi\)
−0.989135 + 0.147009i \(0.953036\pi\)
\(938\) − 12.0000i − 0.391814i
\(939\) 30.0000 0.979013
\(940\) 0 0
\(941\) 4.00000 0.130396 0.0651981 0.997872i \(-0.479232\pi\)
0.0651981 + 0.997872i \(0.479232\pi\)
\(942\) 42.0000i 1.36843i
\(943\) − 7.00000i − 0.227951i
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) − 18.0000i − 0.584613i
\(949\) −42.0000 −1.36338
\(950\) 0 0
\(951\) 36.0000 1.16738
\(952\) 20.0000i 0.648204i
\(953\) − 27.0000i − 0.874616i −0.899312 0.437308i \(-0.855932\pi\)
0.899312 0.437308i \(-0.144068\pi\)
\(954\) −72.0000 −2.33109
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 72.0000i 2.32743i
\(958\) 4.00000i 0.129234i
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 12.0000i − 0.386896i
\(963\) 30.0000i 0.966736i
\(964\) −23.0000 −0.740780
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 28.0000i 0.900419i 0.892923 + 0.450210i \(0.148651\pi\)
−0.892923 + 0.450210i \(0.851349\pi\)
\(968\) 2.00000i 0.0642824i
\(969\) 15.0000 0.481869
\(970\) 0 0
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 0 0
\(973\) − 76.0000i − 2.43645i
\(974\) −30.0000 −0.961262
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) − 17.0000i − 0.543878i −0.962314 0.271939i \(-0.912335\pi\)
0.962314 0.271939i \(-0.0876649\pi\)
\(978\) − 15.0000i − 0.479647i
\(979\) 9.00000 0.287641
\(980\) 0 0
\(981\) −60.0000 −1.91565
\(982\) 0 0
\(983\) 42.0000i 1.33959i 0.742545 + 0.669796i \(0.233618\pi\)
−0.742545 + 0.669796i \(0.766382\pi\)
\(984\) −21.0000 −0.669456
\(985\) 0 0
\(986\) 40.0000 1.27386
\(987\) − 120.000i − 3.81964i
\(988\) 6.00000i 0.190885i
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) − 8.00000i − 0.254000i
\(993\) 75.0000i 2.38005i
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 33.0000 1.04565
\(997\) − 22.0000i − 0.696747i −0.937356 0.348373i \(-0.886734\pi\)
0.937356 0.348373i \(-0.113266\pi\)
\(998\) 12.0000i 0.379853i
\(999\) −18.0000 −0.569495
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 1150.2.b.a.599.2 2
5.2 odd 4 1150.2.a.d.1.1 1
5.3 odd 4 1150.2.a.e.1.1 yes 1
5.4 even 2 inner 1150.2.b.a.599.1 2
20.3 even 4 9200.2.a.bl.1.1 1
20.7 even 4 9200.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.2.a.d.1.1 1 5.2 odd 4
1150.2.a.e.1.1 yes 1 5.3 odd 4
1150.2.b.a.599.1 2 5.4 even 2 inner
1150.2.b.a.599.2 2 1.1 even 1 trivial
9200.2.a.a.1.1 1 20.7 even 4
9200.2.a.bl.1.1 1 20.3 even 4