Properties

Label 3450.2.d.k.2899.2
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
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] = [3450,2,Mod(2899,3450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3450, 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("3450.2899");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (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: \(27.5483886973\)
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 2899.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3450.2899
Dual form 3450.2.d.k.2899.1

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) − 1.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 3.00000i 0.832050i 0.909353 + 0.416025i \(0.136577\pi\)
−0.909353 + 0.416025i \(0.863423\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) − 5.00000i − 1.06600i
\(23\) 1.00000i 0.208514i
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) 1.00000i 0.192450i
\(28\) 1.00000i 0.188982i
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 5.00000i 0.870388i
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) 3.00000i 0.486664i
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) − 1.00000i − 0.154303i
\(43\) 5.00000i 0.762493i 0.924473 + 0.381246i \(0.124505\pi\)
−0.924473 + 0.381246i \(0.875495\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) − 3.00000i − 0.416025i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) − 3.00000i − 0.397360i
\(58\) 1.00000i 0.131306i
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 9.00000i 1.05337i 0.850060 + 0.526685i \(0.176565\pi\)
−0.850060 + 0.526685i \(0.823435\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) 5.00000i 0.569803i
\(78\) 3.00000i 0.339683i
\(79\) 7.00000 0.787562 0.393781 0.919204i \(-0.371167\pi\)
0.393781 + 0.919204i \(0.371167\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 3.00000i − 0.331295i
\(83\) − 1.00000i − 0.109764i −0.998493 0.0548821i \(-0.982522\pi\)
0.998493 0.0548821i \(-0.0174783\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −5.00000 −0.539164
\(87\) − 1.00000i − 0.107211i
\(88\) 5.00000i 0.533002i
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) − 1.00000i − 0.104257i
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 6.00000i 0.606092i
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) 7.00000i 0.689730i 0.938652 + 0.344865i \(0.112075\pi\)
−0.938652 + 0.344865i \(0.887925\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) − 1.00000i − 0.0944911i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 3.00000 0.280976
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) − 3.00000i − 0.277350i
\(118\) 6.00000i 0.552345i
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 4.00000i 0.362143i
\(123\) 3.00000i 0.270501i
\(124\) 0 0
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 18.0000i 1.59724i 0.601834 + 0.798621i \(0.294437\pi\)
−0.601834 + 0.798621i \(0.705563\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 5.00000 0.440225
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) − 5.00000i − 0.435194i
\(133\) − 3.00000i − 0.260133i
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) − 16.0000i − 1.36697i −0.729964 0.683486i \(-0.760463\pi\)
0.729964 0.683486i \(-0.239537\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) − 2.00000i − 0.167836i
\(143\) − 15.0000i − 1.25436i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −9.00000 −0.744845
\(147\) − 6.00000i − 0.494872i
\(148\) 4.00000i 0.328798i
\(149\) 20.0000 1.63846 0.819232 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) − 3.00000i − 0.243332i
\(153\) 2.00000i 0.161690i
\(154\) −5.00000 −0.402911
\(155\) 0 0
\(156\) −3.00000 −0.240192
\(157\) 4.00000i 0.319235i 0.987179 + 0.159617i \(0.0510260\pi\)
−0.987179 + 0.159617i \(0.948974\pi\)
\(158\) 7.00000i 0.556890i
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 1.00000i 0.0785674i
\(163\) − 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
\(166\) 1.00000 0.0776151
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 1.00000i 0.0771517i
\(169\) 4.00000 0.307692
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) − 5.00000i − 0.381246i
\(173\) − 9.00000i − 0.684257i −0.939653 0.342129i \(-0.888852\pi\)
0.939653 0.342129i \(-0.111148\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) − 6.00000i − 0.450988i
\(178\) 2.00000i 0.149906i
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 3.00000i 0.222375i
\(183\) − 4.00000i − 0.295689i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 10.0000i 0.731272i
\(188\) − 6.00000i − 0.437595i
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) − 15.0000i − 1.06871i −0.845262 0.534353i \(-0.820555\pi\)
0.845262 0.534353i \(-0.179445\pi\)
\(198\) 5.00000i 0.355335i
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 6.00000i 0.422159i
\(203\) − 1.00000i − 0.0701862i
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) −7.00000 −0.487713
\(207\) − 1.00000i − 0.0695048i
\(208\) 3.00000i 0.208013i
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 0 0
\(213\) 2.00000i 0.137038i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 4.00000i 0.270914i
\(219\) 9.00000 0.608164
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) − 4.00000i − 0.268462i
\(223\) 2.00000i 0.133930i 0.997755 + 0.0669650i \(0.0213316\pi\)
−0.997755 + 0.0669650i \(0.978668\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 24.0000i 1.59294i 0.604681 + 0.796468i \(0.293301\pi\)
−0.604681 + 0.796468i \(0.706699\pi\)
\(228\) 3.00000i 0.198680i
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 0 0
\(231\) 5.00000 0.328976
\(232\) − 1.00000i − 0.0656532i
\(233\) − 11.0000i − 0.720634i −0.932830 0.360317i \(-0.882669\pi\)
0.932830 0.360317i \(-0.117331\pi\)
\(234\) 3.00000 0.196116
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) − 7.00000i − 0.454699i
\(238\) − 2.00000i − 0.129641i
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 14.0000i 0.899954i
\(243\) − 1.00000i − 0.0641500i
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) −3.00000 −0.191273
\(247\) 9.00000i 0.572656i
\(248\) 0 0
\(249\) −1.00000 −0.0633724
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) − 5.00000i − 0.314347i
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 22.0000i 1.37232i 0.727450 + 0.686161i \(0.240706\pi\)
−0.727450 + 0.686161i \(0.759294\pi\)
\(258\) 5.00000i 0.311286i
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 12.0000i 0.741362i
\(263\) 28.0000i 1.72655i 0.504730 + 0.863277i \(0.331592\pi\)
−0.504730 + 0.863277i \(0.668408\pi\)
\(264\) 5.00000 0.307729
\(265\) 0 0
\(266\) 3.00000 0.183942
\(267\) − 2.00000i − 0.122398i
\(268\) − 4.00000i − 0.244339i
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) − 3.00000i − 0.181568i
\(274\) 16.0000 0.966595
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 31.0000i 1.86261i 0.364241 + 0.931305i \(0.381328\pi\)
−0.364241 + 0.931305i \(0.618672\pi\)
\(278\) 10.0000i 0.599760i
\(279\) 0 0
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 6.00000i 0.357295i
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 15.0000 0.886969
\(287\) 3.00000i 0.177084i
\(288\) − 1.00000i − 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) − 9.00000i − 0.526685i
\(293\) − 28.0000i − 1.63578i −0.575376 0.817889i \(-0.695144\pi\)
0.575376 0.817889i \(-0.304856\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) − 5.00000i − 0.290129i
\(298\) 20.0000i 1.15857i
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) 5.00000 0.288195
\(302\) 18.0000i 1.03578i
\(303\) − 6.00000i − 0.344691i
\(304\) 3.00000 0.172062
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) − 5.00000i − 0.284901i
\(309\) 7.00000 0.398216
\(310\) 0 0
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) − 3.00000i − 0.169842i
\(313\) 32.0000i 1.80875i 0.426742 + 0.904373i \(0.359661\pi\)
−0.426742 + 0.904373i \(0.640339\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) −7.00000 −0.393781
\(317\) 13.0000i 0.730153i 0.930978 + 0.365076i \(0.118957\pi\)
−0.930978 + 0.365076i \(0.881043\pi\)
\(318\) 0 0
\(319\) −5.00000 −0.279946
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 1.00000i 0.0557278i
\(323\) − 6.00000i − 0.333849i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) − 4.00000i − 0.221201i
\(328\) 3.00000i 0.165647i
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 1.00000i 0.0548821i
\(333\) 4.00000i 0.219199i
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) − 8.00000i − 0.435788i −0.975972 0.217894i \(-0.930081\pi\)
0.975972 0.217894i \(-0.0699187\pi\)
\(338\) 4.00000i 0.217571i
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) − 3.00000i − 0.162221i
\(343\) − 13.0000i − 0.701934i
\(344\) 5.00000 0.269582
\(345\) 0 0
\(346\) 9.00000 0.483843
\(347\) 20.0000i 1.07366i 0.843692 + 0.536828i \(0.180378\pi\)
−0.843692 + 0.536828i \(0.819622\pi\)
\(348\) 1.00000i 0.0536056i
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) 0 0
\(351\) −3.00000 −0.160128
\(352\) − 5.00000i − 0.266501i
\(353\) 11.0000i 0.585471i 0.956193 + 0.292735i \(0.0945655\pi\)
−0.956193 + 0.292735i \(0.905434\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 2.00000i 0.105851i
\(358\) − 10.0000i − 0.528516i
\(359\) 9.00000 0.475002 0.237501 0.971387i \(-0.423672\pi\)
0.237501 + 0.971387i \(0.423672\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) − 6.00000i − 0.315353i
\(363\) − 14.0000i − 0.734809i
\(364\) −3.00000 −0.157243
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) − 37.0000i − 1.93138i −0.259690 0.965692i \(-0.583620\pi\)
0.259690 0.965692i \(-0.416380\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 22.0000i − 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) −10.0000 −0.517088
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 3.00000i 0.154508i
\(378\) 1.00000i 0.0514344i
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) − 3.00000i − 0.153493i
\(383\) − 9.00000i − 0.459879i −0.973205 0.229939i \(-0.926147\pi\)
0.973205 0.229939i \(-0.0738528\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) − 5.00000i − 0.254164i
\(388\) 0 0
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) − 6.00000i − 0.303046i
\(393\) − 12.0000i − 0.605320i
\(394\) 15.0000 0.755689
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) − 2.00000i − 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 11.0000i 0.551380i
\(399\) −3.00000 −0.150188
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 4.00000i 0.199502i
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 1.00000 0.0496292
\(407\) 20.0000i 0.991363i
\(408\) 2.00000i 0.0990148i
\(409\) −17.0000 −0.840596 −0.420298 0.907386i \(-0.638074\pi\)
−0.420298 + 0.907386i \(0.638074\pi\)
\(410\) 0 0
\(411\) −16.0000 −0.789222
\(412\) − 7.00000i − 0.344865i
\(413\) − 6.00000i − 0.295241i
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −3.00000 −0.147087
\(417\) − 10.0000i − 0.489702i
\(418\) − 15.0000i − 0.733674i
\(419\) −25.0000 −1.22133 −0.610665 0.791889i \(-0.709098\pi\)
−0.610665 + 0.791889i \(0.709098\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) − 2.00000i − 0.0973585i
\(423\) − 6.00000i − 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) −2.00000 −0.0969003
\(427\) − 4.00000i − 0.193574i
\(428\) − 12.0000i − 0.580042i
\(429\) −15.0000 −0.724207
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 12.0000i − 0.576683i −0.957528 0.288342i \(-0.906896\pi\)
0.957528 0.288342i \(-0.0931039\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 3.00000i 0.143509i
\(438\) 9.00000i 0.430037i
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 6.00000i 0.285391i
\(443\) − 20.0000i − 0.950229i −0.879924 0.475114i \(-0.842407\pi\)
0.879924 0.475114i \(-0.157593\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) − 20.0000i − 0.945968i
\(448\) 1.00000i 0.0472456i
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 15.0000 0.706322
\(452\) 6.00000i 0.282216i
\(453\) − 18.0000i − 0.845714i
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) − 18.0000i − 0.842004i −0.907060 0.421002i \(-0.861678\pi\)
0.907060 0.421002i \(-0.138322\pi\)
\(458\) 8.00000i 0.373815i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −37.0000 −1.72326 −0.861631 0.507535i \(-0.830557\pi\)
−0.861631 + 0.507535i \(0.830557\pi\)
\(462\) 5.00000i 0.232621i
\(463\) 12.0000i 0.557687i 0.960337 + 0.278844i \(0.0899511\pi\)
−0.960337 + 0.278844i \(0.910049\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) 11.0000 0.509565
\(467\) 27.0000i 1.24941i 0.780860 + 0.624705i \(0.214781\pi\)
−0.780860 + 0.624705i \(0.785219\pi\)
\(468\) 3.00000i 0.138675i
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) − 6.00000i − 0.276172i
\(473\) − 25.0000i − 1.14950i
\(474\) 7.00000 0.321521
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 8.00000i 0.365911i
\(479\) −17.0000 −0.776750 −0.388375 0.921501i \(-0.626963\pi\)
−0.388375 + 0.921501i \(0.626963\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) − 2.00000i − 0.0910975i
\(483\) − 1.00000i − 0.0455016i
\(484\) −14.0000 −0.636364
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 18.0000i − 0.815658i −0.913058 0.407829i \(-0.866286\pi\)
0.913058 0.407829i \(-0.133714\pi\)
\(488\) − 4.00000i − 0.181071i
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) − 3.00000i − 0.135250i
\(493\) − 2.00000i − 0.0900755i
\(494\) −9.00000 −0.404929
\(495\) 0 0
\(496\) 0 0
\(497\) 2.00000i 0.0897123i
\(498\) − 1.00000i − 0.0448111i
\(499\) 34.0000 1.52205 0.761025 0.648723i \(-0.224697\pi\)
0.761025 + 0.648723i \(0.224697\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) − 12.0000i − 0.535586i
\(503\) − 11.0000i − 0.490466i −0.969464 0.245233i \(-0.921136\pi\)
0.969464 0.245233i \(-0.0788644\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 5.00000 0.222277
\(507\) − 4.00000i − 0.177646i
\(508\) − 18.0000i − 0.798621i
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) 9.00000 0.398137
\(512\) 1.00000i 0.0441942i
\(513\) 3.00000i 0.132453i
\(514\) −22.0000 −0.970378
\(515\) 0 0
\(516\) −5.00000 −0.220113
\(517\) − 30.0000i − 1.31940i
\(518\) − 4.00000i − 0.175750i
\(519\) −9.00000 −0.395056
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) − 1.00000i − 0.0437688i
\(523\) 19.0000i 0.830812i 0.909636 + 0.415406i \(0.136360\pi\)
−0.909636 + 0.415406i \(0.863640\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) 0 0
\(528\) 5.00000i 0.217597i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 3.00000i 0.130066i
\(533\) − 9.00000i − 0.389833i
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 10.0000i 0.431532i
\(538\) 9.00000i 0.388018i
\(539\) −30.0000 −1.29219
\(540\) 0 0
\(541\) −31.0000 −1.33279 −0.666397 0.745597i \(-0.732164\pi\)
−0.666397 + 0.745597i \(0.732164\pi\)
\(542\) − 28.0000i − 1.20270i
\(543\) 6.00000i 0.257485i
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 3.00000 0.128388
\(547\) − 6.00000i − 0.256541i −0.991739 0.128271i \(-0.959057\pi\)
0.991739 0.128271i \(-0.0409426\pi\)
\(548\) 16.0000i 0.683486i
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) 3.00000 0.127804
\(552\) − 1.00000i − 0.0425628i
\(553\) − 7.00000i − 0.297670i
\(554\) −31.0000 −1.31706
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) − 2.00000i − 0.0847427i −0.999102 0.0423714i \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) 0 0
\(559\) −15.0000 −0.634432
\(560\) 0 0
\(561\) 10.0000 0.422200
\(562\) − 22.0000i − 0.928014i
\(563\) 11.0000i 0.463595i 0.972764 + 0.231797i \(0.0744606\pi\)
−0.972764 + 0.231797i \(0.925539\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) − 1.00000i − 0.0419961i
\(568\) 2.00000i 0.0839181i
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 15.0000i 0.627182i
\(573\) 3.00000i 0.125327i
\(574\) −3.00000 −0.125218
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 17.0000i − 0.707719i −0.935299 0.353860i \(-0.884869\pi\)
0.935299 0.353860i \(-0.115131\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) −1.00000 −0.0414870
\(582\) 0 0
\(583\) 0 0
\(584\) 9.00000 0.372423
\(585\) 0 0
\(586\) 28.0000 1.15667
\(587\) − 2.00000i − 0.0825488i −0.999148 0.0412744i \(-0.986858\pi\)
0.999148 0.0412744i \(-0.0131418\pi\)
\(588\) 6.00000i 0.247436i
\(589\) 0 0
\(590\) 0 0
\(591\) −15.0000 −0.617018
\(592\) − 4.00000i − 0.164399i
\(593\) 9.00000i 0.369586i 0.982777 + 0.184793i \(0.0591614\pi\)
−0.982777 + 0.184793i \(0.940839\pi\)
\(594\) 5.00000 0.205152
\(595\) 0 0
\(596\) −20.0000 −0.819232
\(597\) − 11.0000i − 0.450200i
\(598\) − 3.00000i − 0.122679i
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 5.00000i 0.203785i
\(603\) − 4.00000i − 0.162893i
\(604\) −18.0000 −0.732410
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) 14.0000i 0.568242i 0.958788 + 0.284121i \(0.0917018\pi\)
−0.958788 + 0.284121i \(0.908298\pi\)
\(608\) 3.00000i 0.121666i
\(609\) −1.00000 −0.0405220
\(610\) 0 0
\(611\) −18.0000 −0.728202
\(612\) − 2.00000i − 0.0808452i
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 5.00000 0.201456
\(617\) − 10.0000i − 0.402585i −0.979531 0.201292i \(-0.935486\pi\)
0.979531 0.201292i \(-0.0645141\pi\)
\(618\) 7.00000i 0.281581i
\(619\) 36.0000 1.44696 0.723481 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) − 20.0000i − 0.801927i
\(623\) − 2.00000i − 0.0801283i
\(624\) 3.00000 0.120096
\(625\) 0 0
\(626\) −32.0000 −1.27898
\(627\) 15.0000i 0.599042i
\(628\) − 4.00000i − 0.159617i
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) − 7.00000i − 0.278445i
\(633\) 2.00000i 0.0794929i
\(634\) −13.0000 −0.516296
\(635\) 0 0
\(636\) 0 0
\(637\) 18.0000i 0.713186i
\(638\) − 5.00000i − 0.197952i
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 12.0000i 0.473602i
\(643\) − 19.0000i − 0.749287i −0.927169 0.374643i \(-0.877765\pi\)
0.927169 0.374643i \(-0.122235\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −30.0000 −1.17760
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) 1.00000i 0.0391330i 0.999809 + 0.0195665i \(0.00622861\pi\)
−0.999809 + 0.0195665i \(0.993771\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) − 9.00000i − 0.351123i
\(658\) 6.00000i 0.233904i
\(659\) −45.0000 −1.75295 −0.876476 0.481446i \(-0.840112\pi\)
−0.876476 + 0.481446i \(0.840112\pi\)
\(660\) 0 0
\(661\) 24.0000 0.933492 0.466746 0.884391i \(-0.345426\pi\)
0.466746 + 0.884391i \(0.345426\pi\)
\(662\) 20.0000i 0.777322i
\(663\) − 6.00000i − 0.233021i
\(664\) −1.00000 −0.0388075
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 1.00000i 0.0387202i
\(668\) − 12.0000i − 0.464294i
\(669\) 2.00000 0.0773245
\(670\) 0 0
\(671\) −20.0000 −0.772091
\(672\) − 1.00000i − 0.0385758i
\(673\) 27.0000i 1.04077i 0.853931 + 0.520387i \(0.174212\pi\)
−0.853931 + 0.520387i \(0.825788\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) − 8.00000i − 0.307465i −0.988113 0.153732i \(-0.950871\pi\)
0.988113 0.153732i \(-0.0491294\pi\)
\(678\) − 6.00000i − 0.230429i
\(679\) 0 0
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) − 18.0000i − 0.688751i −0.938832 0.344375i \(-0.888091\pi\)
0.938832 0.344375i \(-0.111909\pi\)
\(684\) 3.00000 0.114708
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) − 8.00000i − 0.305219i
\(688\) 5.00000i 0.190623i
\(689\) 0 0
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) 9.00000i 0.342129i
\(693\) − 5.00000i − 0.189934i
\(694\) −20.0000 −0.759190
\(695\) 0 0
\(696\) −1.00000 −0.0379049
\(697\) 6.00000i 0.227266i
\(698\) 15.0000i 0.567758i
\(699\) −11.0000 −0.416058
\(700\) 0 0
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) − 3.00000i − 0.113228i
\(703\) − 12.0000i − 0.452589i
\(704\) 5.00000 0.188445
\(705\) 0 0
\(706\) −11.0000 −0.413990
\(707\) − 6.00000i − 0.225653i
\(708\) 6.00000i 0.225494i
\(709\) 44.0000 1.65245 0.826227 0.563337i \(-0.190483\pi\)
0.826227 + 0.563337i \(0.190483\pi\)
\(710\) 0 0
\(711\) −7.00000 −0.262521
\(712\) − 2.00000i − 0.0749532i
\(713\) 0 0
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) − 8.00000i − 0.298765i
\(718\) 9.00000i 0.335877i
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 7.00000 0.260694
\(722\) − 10.0000i − 0.372161i
\(723\) 2.00000i 0.0743808i
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) − 8.00000i − 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) − 3.00000i − 0.111187i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 10.0000 0.369863
\(732\) 4.00000i 0.147844i
\(733\) − 24.0000i − 0.886460i −0.896408 0.443230i \(-0.853832\pi\)
0.896408 0.443230i \(-0.146168\pi\)
\(734\) 37.0000 1.36569
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) − 20.0000i − 0.736709i
\(738\) 3.00000i 0.110432i
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) 0 0
\(741\) 9.00000 0.330623
\(742\) 0 0
\(743\) − 5.00000i − 0.183432i −0.995785 0.0917161i \(-0.970765\pi\)
0.995785 0.0917161i \(-0.0292352\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) 1.00000i 0.0365881i
\(748\) − 10.0000i − 0.365636i
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 7.00000 0.255434 0.127717 0.991811i \(-0.459235\pi\)
0.127717 + 0.991811i \(0.459235\pi\)
\(752\) 6.00000i 0.218797i
\(753\) 12.0000i 0.437304i
\(754\) −3.00000 −0.109254
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) − 28.0000i − 1.01768i −0.860862 0.508839i \(-0.830075\pi\)
0.860862 0.508839i \(-0.169925\pi\)
\(758\) 20.0000i 0.726433i
\(759\) −5.00000 −0.181489
\(760\) 0 0
\(761\) 23.0000 0.833749 0.416875 0.908964i \(-0.363125\pi\)
0.416875 + 0.908964i \(0.363125\pi\)
\(762\) 18.0000i 0.652071i
\(763\) − 4.00000i − 0.144810i
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) 9.00000 0.325183
\(767\) 18.0000i 0.649942i
\(768\) − 1.00000i − 0.0360844i
\(769\) −6.00000 −0.216366 −0.108183 0.994131i \(-0.534503\pi\)
−0.108183 + 0.994131i \(0.534503\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) − 10.0000i − 0.359908i
\(773\) 12.0000i 0.431610i 0.976436 + 0.215805i \(0.0692376\pi\)
−0.976436 + 0.215805i \(0.930762\pi\)
\(774\) 5.00000 0.179721
\(775\) 0 0
\(776\) 0 0
\(777\) 4.00000i 0.143499i
\(778\) − 10.0000i − 0.358517i
\(779\) −9.00000 −0.322458
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) 2.00000i 0.0715199i
\(783\) 1.00000i 0.0357371i
\(784\) 6.00000 0.214286
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 37.0000i 1.31891i 0.751745 + 0.659454i \(0.229212\pi\)
−0.751745 + 0.659454i \(0.770788\pi\)
\(788\) 15.0000i 0.534353i
\(789\) 28.0000 0.996826
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) − 5.00000i − 0.177667i
\(793\) 12.0000i 0.426132i
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −11.0000 −0.389885
\(797\) 10.0000i 0.354218i 0.984191 + 0.177109i \(0.0566745\pi\)
−0.984191 + 0.177109i \(0.943325\pi\)
\(798\) − 3.00000i − 0.106199i
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 12.0000i 0.423735i
\(803\) − 45.0000i − 1.58802i
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) − 9.00000i − 0.316815i
\(808\) − 6.00000i − 0.211079i
\(809\) −39.0000 −1.37117 −0.685583 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 1.00000i 0.0350931i
\(813\) 28.0000i 0.982003i
\(814\) −20.0000 −0.701000
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 15.0000i 0.524784i
\(818\) − 17.0000i − 0.594391i
\(819\) −3.00000 −0.104828
\(820\) 0 0
\(821\) 53.0000 1.84971 0.924856 0.380317i \(-0.124185\pi\)
0.924856 + 0.380317i \(0.124185\pi\)
\(822\) − 16.0000i − 0.558064i
\(823\) 2.00000i 0.0697156i 0.999392 + 0.0348578i \(0.0110978\pi\)
−0.999392 + 0.0348578i \(0.988902\pi\)
\(824\) 7.00000 0.243857
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) − 53.0000i − 1.84299i −0.388390 0.921495i \(-0.626968\pi\)
0.388390 0.921495i \(-0.373032\pi\)
\(828\) 1.00000i 0.0347524i
\(829\) 23.0000 0.798823 0.399412 0.916772i \(-0.369214\pi\)
0.399412 + 0.916772i \(0.369214\pi\)
\(830\) 0 0
\(831\) 31.0000 1.07538
\(832\) − 3.00000i − 0.104006i
\(833\) − 12.0000i − 0.415775i
\(834\) 10.0000 0.346272
\(835\) 0 0
\(836\) 15.0000 0.518786
\(837\) 0 0
\(838\) − 25.0000i − 0.863611i
\(839\) −25.0000 −0.863096 −0.431548 0.902090i \(-0.642032\pi\)
−0.431548 + 0.902090i \(0.642032\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 6.00000i 0.206774i
\(843\) 22.0000i 0.757720i
\(844\) 2.00000 0.0688428
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) − 14.0000i − 0.481046i
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) − 2.00000i − 0.0685189i
\(853\) − 49.0000i − 1.67773i −0.544341 0.838864i \(-0.683220\pi\)
0.544341 0.838864i \(-0.316780\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) − 6.00000i − 0.204956i −0.994735 0.102478i \(-0.967323\pi\)
0.994735 0.102478i \(-0.0326771\pi\)
\(858\) − 15.0000i − 0.512092i
\(859\) −6.00000 −0.204717 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) 0 0
\(861\) 3.00000 0.102240
\(862\) 0 0
\(863\) 18.0000i 0.612727i 0.951915 + 0.306364i \(0.0991123\pi\)
−0.951915 + 0.306364i \(0.900888\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 12.0000 0.407777
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) −35.0000 −1.18729
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) − 4.00000i − 0.135457i
\(873\) 0 0
\(874\) −3.00000 −0.101477
\(875\) 0 0
\(876\) −9.00000 −0.304082
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) − 20.0000i − 0.674967i
\(879\) −28.0000 −0.944417
\(880\) 0 0
\(881\) 40.0000 1.34763 0.673817 0.738898i \(-0.264654\pi\)
0.673817 + 0.738898i \(0.264654\pi\)
\(882\) − 6.00000i − 0.202031i
\(883\) 24.0000i 0.807664i 0.914833 + 0.403832i \(0.132322\pi\)
−0.914833 + 0.403832i \(0.867678\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) − 6.00000i − 0.201460i −0.994914 0.100730i \(-0.967882\pi\)
0.994914 0.100730i \(-0.0321179\pi\)
\(888\) 4.00000i 0.134231i
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) − 2.00000i − 0.0669650i
\(893\) 18.0000i 0.602347i
\(894\) 20.0000 0.668900
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 3.00000i 0.100167i
\(898\) − 14.0000i − 0.467186i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 15.0000i 0.499445i
\(903\) − 5.00000i − 0.166390i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 18.0000 0.598010
\(907\) 33.0000i 1.09575i 0.836561 + 0.547874i \(0.184562\pi\)
−0.836561 + 0.547874i \(0.815438\pi\)
\(908\) − 24.0000i − 0.796468i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −9.00000 −0.298183 −0.149092 0.988823i \(-0.547635\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(912\) − 3.00000i − 0.0993399i
\(913\) 5.00000i 0.165476i
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) −8.00000 −0.264327
\(917\) − 12.0000i − 0.396275i
\(918\) 2.00000i 0.0660098i
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) − 37.0000i − 1.21853i
\(923\) − 6.00000i − 0.197492i
\(924\) −5.00000 −0.164488
\(925\) 0 0
\(926\) −12.0000 −0.394344
\(927\) − 7.00000i − 0.229910i
\(928\) 1.00000i 0.0328266i
\(929\) −33.0000 −1.08269 −0.541347 0.840799i \(-0.682086\pi\)
−0.541347 + 0.840799i \(0.682086\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 11.0000i 0.360317i
\(933\) 20.0000i 0.654771i
\(934\) −27.0000 −0.883467
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) − 60.0000i − 1.96011i −0.198715 0.980057i \(-0.563677\pi\)
0.198715 0.980057i \(-0.436323\pi\)
\(938\) 4.00000i 0.130605i
\(939\) 32.0000 1.04428
\(940\) 0 0
\(941\) 32.0000 1.04317 0.521585 0.853199i \(-0.325341\pi\)
0.521585 + 0.853199i \(0.325341\pi\)
\(942\) 4.00000i 0.130327i
\(943\) − 3.00000i − 0.0976934i
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 25.0000 0.812820
\(947\) − 32.0000i − 1.03986i −0.854209 0.519930i \(-0.825958\pi\)
0.854209 0.519930i \(-0.174042\pi\)
\(948\) 7.00000i 0.227349i
\(949\) −27.0000 −0.876457
\(950\) 0 0
\(951\) 13.0000 0.421554
\(952\) 2.00000i 0.0648204i
\(953\) 20.0000i 0.647864i 0.946080 + 0.323932i \(0.105005\pi\)
−0.946080 + 0.323932i \(0.894995\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 5.00000i 0.161627i
\(958\) − 17.0000i − 0.549245i
\(959\) −16.0000 −0.516667
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 12.0000i 0.386896i
\(963\) − 12.0000i − 0.386695i
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) 1.00000 0.0321745
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) − 14.0000i − 0.449977i
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) −31.0000 −0.994837 −0.497419 0.867511i \(-0.665719\pi\)
−0.497419 + 0.867511i \(0.665719\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 10.0000i − 0.320585i
\(974\) 18.0000 0.576757
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) − 24.0000i − 0.767828i −0.923369 0.383914i \(-0.874576\pi\)
0.923369 0.383914i \(-0.125424\pi\)
\(978\) − 4.00000i − 0.127906i
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) 9.00000i 0.287055i 0.989646 + 0.143528i \(0.0458446\pi\)
−0.989646 + 0.143528i \(0.954155\pi\)
\(984\) 3.00000 0.0956365
\(985\) 0 0
\(986\) 2.00000 0.0636930
\(987\) − 6.00000i − 0.190982i
\(988\) − 9.00000i − 0.286328i
\(989\) −5.00000 −0.158991
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 0 0
\(993\) − 20.0000i − 0.634681i
\(994\) −2.00000 −0.0634361
\(995\) 0 0
\(996\) 1.00000 0.0316862
\(997\) − 49.0000i − 1.55185i −0.630828 0.775923i \(-0.717285\pi\)
0.630828 0.775923i \(-0.282715\pi\)
\(998\) 34.0000i 1.07625i
\(999\) 4.00000 0.126554
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 3450.2.d.k.2899.2 2
5.2 odd 4 3450.2.a.e.1.1 1
5.3 odd 4 3450.2.a.w.1.1 yes 1
5.4 even 2 inner 3450.2.d.k.2899.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.e.1.1 1 5.2 odd 4
3450.2.a.w.1.1 yes 1 5.3 odd 4
3450.2.d.k.2899.1 2 5.4 even 2 inner
3450.2.d.k.2899.2 2 1.1 even 1 trivial