Properties

Label 6900.2.f.b.6349.1
Level $6900$
Weight $2$
Character 6900.6349
Analytic conductor $55.097$
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] = [6900,2,Mod(6349,6900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6900.6349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6900.f (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: \(55.0967773947\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 6349.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 6900.6349
Dual form 6900.2.f.b.6349.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +2.00000i q^{13} -7.00000i q^{17} +6.00000 q^{19} -3.00000 q^{21} -1.00000i q^{23} +1.00000i q^{27} +9.00000 q^{29} +9.00000 q^{31} +2.00000i q^{33} -7.00000i q^{37} +2.00000 q^{39} +5.00000 q^{41} +8.00000i q^{47} -2.00000 q^{49} -7.00000 q^{51} +11.0000i q^{53} -6.00000i q^{57} -9.00000 q^{59} +3.00000i q^{63} -3.00000i q^{67} -1.00000 q^{69} +3.00000 q^{71} +6.00000i q^{73} +6.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} -5.00000i q^{83} -9.00000i q^{87} +6.00000 q^{91} -9.00000i q^{93} -10.0000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} - 4 q^{11} + 12 q^{19} - 6 q^{21} + 18 q^{29} + 18 q^{31} + 4 q^{39} + 10 q^{41} - 4 q^{49} - 14 q^{51} - 18 q^{59} - 2 q^{69} + 6 q^{71} + 16 q^{79} + 2 q^{81} + 12 q^{91} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1201\) \(3451\) \(4601\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.00000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7.00000i − 1.69775i −0.528594 0.848875i \(-0.677281\pi\)
0.528594 0.848875i \(-0.322719\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 9.00000 1.61645 0.808224 0.588875i \(-0.200429\pi\)
0.808224 + 0.588875i \(0.200429\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 7.00000i − 1.15079i −0.817875 0.575396i \(-0.804848\pi\)
0.817875 0.575396i \(-0.195152\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −7.00000 −0.980196
\(52\) 0 0
\(53\) 11.0000i 1.51097i 0.655168 + 0.755483i \(0.272598\pi\)
−0.655168 + 0.755483i \(0.727402\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.00000i − 0.794719i
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 3.00000i 0.377964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.00000i − 0.366508i −0.983066 0.183254i \(-0.941337\pi\)
0.983066 0.183254i \(-0.0586631\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 5.00000i − 0.548821i −0.961613 0.274411i \(-0.911517\pi\)
0.961613 0.274411i \(-0.0884828\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 9.00000i − 0.964901i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) − 9.00000i − 0.933257i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.00000i 0.483368i 0.970355 + 0.241684i \(0.0776998\pi\)
−0.970355 + 0.241684i \(0.922300\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) 0 0
\(113\) − 21.0000i − 1.97551i −0.156001 0.987757i \(-0.549860\pi\)
0.156001 0.987757i \(-0.450140\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2.00000i − 0.184900i
\(118\) 0 0
\(119\) −21.0000 −1.92507
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) − 5.00000i − 0.450835i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) − 18.0000i − 1.56080i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 14.0000i − 1.19610i −0.801459 0.598050i \(-0.795942\pi\)
0.801459 0.598050i \(-0.204058\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) − 4.00000i − 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.00000i 0.164957i
\(148\) 0 0
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 7.00000i 0.565916i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 5.00000i − 0.399043i −0.979893 0.199522i \(-0.936061\pi\)
0.979893 0.199522i \(-0.0639388\pi\)
\(158\) 0 0
\(159\) 11.0000 0.872357
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 14.0000i 1.09656i 0.836293 + 0.548282i \(0.184718\pi\)
−0.836293 + 0.548282i \(0.815282\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 4.00000i − 0.309529i −0.987951 0.154765i \(-0.950538\pi\)
0.987951 0.154765i \(-0.0494619\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 0 0
\(173\) − 4.00000i − 0.304114i −0.988372 0.152057i \(-0.951410\pi\)
0.988372 0.152057i \(-0.0485898\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 9.00000i 0.676481i
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 14.0000i 1.02378i
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) 24.0000i 1.72756i 0.503871 + 0.863779i \(0.331909\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.0000i 0.997459i 0.866758 + 0.498729i \(0.166200\pi\)
−0.866758 + 0.498729i \(0.833800\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) −3.00000 −0.211604
\(202\) 0 0
\(203\) − 27.0000i − 1.89503i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000i 0.0695048i
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −23.0000 −1.58339 −0.791693 0.610920i \(-0.790800\pi\)
−0.791693 + 0.610920i \(0.790800\pi\)
\(212\) 0 0
\(213\) − 3.00000i − 0.205557i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 27.0000i − 1.83288i
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 14.0000 0.941742
\(222\) 0 0
\(223\) 2.00000i 0.133930i 0.997755 + 0.0669650i \(0.0213316\pi\)
−0.997755 + 0.0669650i \(0.978668\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) − 16.0000i − 1.04819i −0.851658 0.524097i \(-0.824403\pi\)
0.851658 0.524097i \(-0.175597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 8.00000i − 0.519656i
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 12.0000i 0.763542i
\(248\) 0 0
\(249\) −5.00000 −0.316862
\(250\) 0 0
\(251\) −30.0000 −1.89358 −0.946792 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) 0 0
\(253\) 2.00000i 0.125739i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 8.00000i − 0.499026i −0.968371 0.249513i \(-0.919729\pi\)
0.968371 0.249513i \(-0.0802706\pi\)
\(258\) 0 0
\(259\) −21.0000 −1.30488
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) 0 0
\(263\) − 11.0000i − 0.678289i −0.940734 0.339145i \(-0.889862\pi\)
0.940734 0.339145i \(-0.110138\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.00000 0.182913 0.0914566 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 0 0
\(271\) 31.0000 1.88312 0.941558 0.336851i \(-0.109362\pi\)
0.941558 + 0.336851i \(0.109362\pi\)
\(272\) 0 0
\(273\) − 6.00000i − 0.363137i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 26.0000i 1.56219i 0.624413 + 0.781094i \(0.285338\pi\)
−0.624413 + 0.781094i \(0.714662\pi\)
\(278\) 0 0
\(279\) −9.00000 −0.538816
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 5.00000i 0.297219i 0.988896 + 0.148610i \(0.0474798\pi\)
−0.988896 + 0.148610i \(0.952520\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 15.0000i − 0.885422i
\(288\) 0 0
\(289\) −32.0000 −1.88235
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) 9.00000i 0.525786i 0.964825 + 0.262893i \(0.0846766\pi\)
−0.964825 + 0.262893i \(0.915323\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 2.00000i − 0.116052i
\(298\) 0 0
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 3.00000i − 0.172345i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 30.0000i − 1.71219i −0.516818 0.856095i \(-0.672884\pi\)
0.516818 0.856095i \(-0.327116\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 29.0000i 1.63918i 0.572953 + 0.819588i \(0.305798\pi\)
−0.572953 + 0.819588i \(0.694202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) 5.00000 0.279073
\(322\) 0 0
\(323\) − 42.0000i − 2.33694i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 21.0000 1.15426 0.577132 0.816651i \(-0.304172\pi\)
0.577132 + 0.816651i \(0.304172\pi\)
\(332\) 0 0
\(333\) 7.00000i 0.383598i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 26.0000i − 1.41631i −0.706057 0.708155i \(-0.749528\pi\)
0.706057 0.708155i \(-0.250472\pi\)
\(338\) 0 0
\(339\) −21.0000 −1.14056
\(340\) 0 0
\(341\) −18.0000 −0.974755
\(342\) 0 0
\(343\) − 15.0000i − 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 0 0
\(349\) −31.0000 −1.65939 −0.829696 0.558216i \(-0.811486\pi\)
−0.829696 + 0.558216i \(0.811486\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 28.0000i 1.49029i 0.666903 + 0.745145i \(0.267620\pi\)
−0.666903 + 0.745145i \(0.732380\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 21.0000i 1.11144i
\(358\) 0 0
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7.00000i 0.365397i 0.983169 + 0.182699i \(0.0584832\pi\)
−0.983169 + 0.182699i \(0.941517\pi\)
\(368\) 0 0
\(369\) −5.00000 −0.260290
\(370\) 0 0
\(371\) 33.0000 1.71327
\(372\) 0 0
\(373\) − 34.0000i − 1.76045i −0.474554 0.880227i \(-0.657390\pi\)
0.474554 0.880227i \(-0.342610\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.0000i 0.927047i
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) − 27.0000i − 1.37964i −0.723983 0.689818i \(-0.757691\pi\)
0.723983 0.689818i \(-0.242309\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −7.00000 −0.354005
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 28.0000i − 1.40528i −0.711546 0.702640i \(-0.752005\pi\)
0.711546 0.702640i \(-0.247995\pi\)
\(398\) 0 0
\(399\) −18.0000 −0.901127
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 18.0000i 0.896644i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 14.0000i 0.693954i
\(408\) 0 0
\(409\) 35.0000 1.73064 0.865319 0.501221i \(-0.167116\pi\)
0.865319 + 0.501221i \(0.167116\pi\)
\(410\) 0 0
\(411\) −14.0000 −0.690569
\(412\) 0 0
\(413\) 27.0000i 1.32858i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 5.00000i − 0.244851i
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 0 0
\(423\) − 8.00000i − 0.388973i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) 0 0
\(433\) − 1.00000i − 0.0480569i −0.999711 0.0240285i \(-0.992351\pi\)
0.999711 0.0240285i \(-0.00764923\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 6.00000i − 0.287019i
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 4.00000i − 0.189194i
\(448\) 0 0
\(449\) 17.0000 0.802280 0.401140 0.916017i \(-0.368614\pi\)
0.401140 + 0.916017i \(0.368614\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) 0 0
\(453\) 4.00000i 0.187936i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 7.00000i 0.327446i 0.986506 + 0.163723i \(0.0523504\pi\)
−0.986506 + 0.163723i \(0.947650\pi\)
\(458\) 0 0
\(459\) 7.00000 0.326732
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 10.0000i 0.464739i 0.972628 + 0.232370i \(0.0746479\pi\)
−0.972628 + 0.232370i \(0.925352\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.0000i 0.971764i 0.874024 + 0.485882i \(0.161502\pi\)
−0.874024 + 0.485882i \(0.838498\pi\)
\(468\) 0 0
\(469\) −9.00000 −0.415581
\(470\) 0 0
\(471\) −5.00000 −0.230388
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 11.0000i − 0.503655i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 14.0000 0.638345
\(482\) 0 0
\(483\) 3.00000i 0.136505i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 8.00000i − 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) 0 0
\(489\) 14.0000 0.633102
\(490\) 0 0
\(491\) −5.00000 −0.225647 −0.112823 0.993615i \(-0.535989\pi\)
−0.112823 + 0.993615i \(0.535989\pi\)
\(492\) 0 0
\(493\) − 63.0000i − 2.83738i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 9.00000i − 0.403705i
\(498\) 0 0
\(499\) −15.0000 −0.671492 −0.335746 0.941953i \(-0.608988\pi\)
−0.335746 + 0.941953i \(0.608988\pi\)
\(500\) 0 0
\(501\) −4.00000 −0.178707
\(502\) 0 0
\(503\) 33.0000i 1.47140i 0.677309 + 0.735699i \(0.263146\pi\)
−0.677309 + 0.735699i \(0.736854\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) 0 0
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 18.0000 0.796273
\(512\) 0 0
\(513\) 6.00000i 0.264906i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 16.0000i − 0.703679i
\(518\) 0 0
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 0 0
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 63.0000i − 2.74432i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 9.00000 0.390567
\(532\) 0 0
\(533\) 10.0000i 0.433148i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 16.0000i 0.690451i
\(538\) 0 0
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) − 12.0000i − 0.514969i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 24.0000i − 1.02617i −0.858339 0.513083i \(-0.828503\pi\)
0.858339 0.513083i \(-0.171497\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 54.0000 2.30048
\(552\) 0 0
\(553\) − 24.0000i − 1.02058i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 11.0000i − 0.466085i −0.972467 0.233042i \(-0.925132\pi\)
0.972467 0.233042i \(-0.0748681\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 14.0000 0.591080
\(562\) 0 0
\(563\) 5.00000i 0.210725i 0.994434 + 0.105362i \(0.0336003\pi\)
−0.994434 + 0.105362i \(0.966400\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 3.00000i − 0.125988i
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 0 0
\(573\) − 4.00000i − 0.167102i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 32.0000i − 1.33218i −0.745873 0.666089i \(-0.767967\pi\)
0.745873 0.666089i \(-0.232033\pi\)
\(578\) 0 0
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) −15.0000 −0.622305
\(582\) 0 0
\(583\) − 22.0000i − 0.911147i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 2.00000i − 0.0825488i −0.999148 0.0412744i \(-0.986858\pi\)
0.999148 0.0412744i \(-0.0131418\pi\)
\(588\) 0 0
\(589\) 54.0000 2.22503
\(590\) 0 0
\(591\) 14.0000 0.575883
\(592\) 0 0
\(593\) 36.0000i 1.47834i 0.673517 + 0.739171i \(0.264783\pi\)
−0.673517 + 0.739171i \(0.735217\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 14.0000i 0.572982i
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 0 0
\(603\) 3.00000i 0.122169i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) 0 0
\(609\) −27.0000 −1.09410
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 9.00000i − 0.362326i −0.983453 0.181163i \(-0.942014\pi\)
0.983453 0.181163i \(-0.0579862\pi\)
\(618\) 0 0
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 12.0000i 0.479234i
\(628\) 0 0
\(629\) −49.0000 −1.95376
\(630\) 0 0
\(631\) −44.0000 −1.75161 −0.875806 0.482663i \(-0.839670\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) 0 0
\(633\) 23.0000i 0.914168i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 4.00000i − 0.158486i
\(638\) 0 0
\(639\) −3.00000 −0.118678
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 23.0000i 0.907031i 0.891248 + 0.453516i \(0.149830\pi\)
−0.891248 + 0.453516i \(0.850170\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.0000i 0.393141i 0.980490 + 0.196570i \(0.0629804\pi\)
−0.980490 + 0.196570i \(0.937020\pi\)
\(648\) 0 0
\(649\) 18.0000 0.706562
\(650\) 0 0
\(651\) −27.0000 −1.05821
\(652\) 0 0
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 6.00000i − 0.234082i
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) 6.00000 0.233373 0.116686 0.993169i \(-0.462773\pi\)
0.116686 + 0.993169i \(0.462773\pi\)
\(662\) 0 0
\(663\) − 14.0000i − 0.543715i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 9.00000i − 0.348481i
\(668\) 0 0
\(669\) 2.00000 0.0773245
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 16.0000i − 0.616755i −0.951264 0.308377i \(-0.900214\pi\)
0.951264 0.308377i \(-0.0997859\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 21.0000i − 0.807096i −0.914959 0.403548i \(-0.867777\pi\)
0.914959 0.403548i \(-0.132223\pi\)
\(678\) 0 0
\(679\) −30.0000 −1.15129
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 4.00000i 0.153056i 0.997067 + 0.0765279i \(0.0243834\pi\)
−0.997067 + 0.0765279i \(0.975617\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 16.0000i − 0.610438i
\(688\) 0 0
\(689\) −22.0000 −0.838133
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 0 0
\(693\) − 6.00000i − 0.227921i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 35.0000i − 1.32572i
\(698\) 0 0
\(699\) −16.0000 −0.605176
\(700\) 0 0
\(701\) −20.0000 −0.755390 −0.377695 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) 0 0
\(703\) − 42.0000i − 1.58406i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 9.00000i − 0.338480i
\(708\) 0 0
\(709\) 32.0000 1.20179 0.600893 0.799330i \(-0.294812\pi\)
0.600893 + 0.799330i \(0.294812\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) − 9.00000i − 0.337053i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 21.0000i 0.784259i
\(718\) 0 0
\(719\) 43.0000 1.60363 0.801815 0.597573i \(-0.203868\pi\)
0.801815 + 0.597573i \(0.203868\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) − 26.0000i − 0.966950i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 23.0000i 0.853023i 0.904482 + 0.426511i \(0.140258\pi\)
−0.904482 + 0.426511i \(0.859742\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 47.0000i 1.73598i 0.496578 + 0.867992i \(0.334590\pi\)
−0.496578 + 0.867992i \(0.665410\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.00000i 0.221013i
\(738\) 0 0
\(739\) 23.0000 0.846069 0.423034 0.906114i \(-0.360965\pi\)
0.423034 + 0.906114i \(0.360965\pi\)
\(740\) 0 0
\(741\) 12.0000 0.440831
\(742\) 0 0
\(743\) 32.0000i 1.17397i 0.809599 + 0.586983i \(0.199684\pi\)
−0.809599 + 0.586983i \(0.800316\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.00000i 0.182940i
\(748\) 0 0
\(749\) 15.0000 0.548088
\(750\) 0 0
\(751\) 10.0000 0.364905 0.182453 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) 0 0
\(753\) 30.0000i 1.09326i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 1.00000i − 0.0363456i −0.999835 0.0181728i \(-0.994215\pi\)
0.999835 0.0181728i \(-0.00578490\pi\)
\(758\) 0 0
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) −39.0000 −1.41375 −0.706874 0.707339i \(-0.749895\pi\)
−0.706874 + 0.707339i \(0.749895\pi\)
\(762\) 0 0
\(763\) 30.0000i 1.08607i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 18.0000i − 0.649942i
\(768\) 0 0
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) −8.00000 −0.288113
\(772\) 0 0
\(773\) 54.0000i 1.94225i 0.238581 + 0.971123i \(0.423318\pi\)
−0.238581 + 0.971123i \(0.576682\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 21.0000i 0.753371i
\(778\) 0 0
\(779\) 30.0000 1.07486
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 0 0
\(783\) 9.00000i 0.321634i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 31.0000i − 1.10503i −0.833503 0.552515i \(-0.813668\pi\)
0.833503 0.552515i \(-0.186332\pi\)
\(788\) 0 0
\(789\) −11.0000 −0.391610
\(790\) 0 0
\(791\) −63.0000 −2.24002
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.0000i 0.389640i 0.980839 + 0.194820i \(0.0624123\pi\)
−0.980839 + 0.194820i \(0.937588\pi\)
\(798\) 0 0
\(799\) 56.0000 1.98114
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 12.0000i − 0.423471i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 3.00000i − 0.105605i
\(808\) 0 0
\(809\) −3.00000 −0.105474 −0.0527372 0.998608i \(-0.516795\pi\)
−0.0527372 + 0.998608i \(0.516795\pi\)
\(810\) 0 0
\(811\) −7.00000 −0.245803 −0.122902 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(812\) 0 0
\(813\) − 31.0000i − 1.08722i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) − 52.0000i − 1.81261i −0.422628 0.906303i \(-0.638892\pi\)
0.422628 0.906303i \(-0.361108\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.0000i 1.07798i 0.842314 + 0.538988i \(0.181193\pi\)
−0.842314 + 0.538988i \(0.818807\pi\)
\(828\) 0 0
\(829\) −49.0000 −1.70184 −0.850920 0.525295i \(-0.823955\pi\)
−0.850920 + 0.525295i \(0.823955\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 0 0
\(833\) 14.0000i 0.485071i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 9.00000i 0.311086i
\(838\) 0 0
\(839\) −2.00000 −0.0690477 −0.0345238 0.999404i \(-0.510991\pi\)
−0.0345238 + 0.999404i \(0.510991\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) − 10.0000i − 0.344418i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 21.0000i 0.721569i
\(848\) 0 0
\(849\) 5.00000 0.171600
\(850\) 0 0
\(851\) −7.00000 −0.239957
\(852\) 0 0
\(853\) 30.0000i 1.02718i 0.858036 + 0.513590i \(0.171685\pi\)
−0.858036 + 0.513590i \(0.828315\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 54.0000i 1.84460i 0.386469 + 0.922302i \(0.373695\pi\)
−0.386469 + 0.922302i \(0.626305\pi\)
\(858\) 0 0
\(859\) 43.0000 1.46714 0.733571 0.679613i \(-0.237852\pi\)
0.733571 + 0.679613i \(0.237852\pi\)
\(860\) 0 0
\(861\) −15.0000 −0.511199
\(862\) 0 0
\(863\) − 6.00000i − 0.204242i −0.994772 0.102121i \(-0.967437\pi\)
0.994772 0.102121i \(-0.0325630\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 32.0000i 1.08678i
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 6.00000 0.203302
\(872\) 0 0
\(873\) 10.0000i 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 34.0000i − 1.14810i −0.818821 0.574049i \(-0.805372\pi\)
0.818821 0.574049i \(-0.194628\pi\)
\(878\) 0 0
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) −16.0000 −0.539054 −0.269527 0.962993i \(-0.586867\pi\)
−0.269527 + 0.962993i \(0.586867\pi\)
\(882\) 0 0
\(883\) − 12.0000i − 0.403832i −0.979403 0.201916i \(-0.935283\pi\)
0.979403 0.201916i \(-0.0647168\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 38.0000i 1.27592i 0.770072 + 0.637958i \(0.220220\pi\)
−0.770072 + 0.637958i \(0.779780\pi\)
\(888\) 0 0
\(889\) −48.0000 −1.60987
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 0 0
\(893\) 48.0000i 1.60626i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 2.00000i − 0.0667781i
\(898\) 0 0
\(899\) 81.0000 2.70150
\(900\) 0 0
\(901\) 77.0000 2.56524
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 37.0000i 1.22856i 0.789086 + 0.614282i \(0.210554\pi\)
−0.789086 + 0.614282i \(0.789446\pi\)
\(908\) 0 0
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) −56.0000 −1.85536 −0.927681 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) 0 0
\(913\) 10.0000i 0.330952i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 36.0000i 1.18882i
\(918\) 0 0
\(919\) 12.0000 0.395843 0.197922 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 0 0
\(921\) −30.0000 −0.988534
\(922\) 0 0
\(923\) 6.00000i 0.197492i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 8.00000i 0.262754i
\(928\) 0 0
\(929\) 57.0000 1.87011 0.935055 0.354504i \(-0.115350\pi\)
0.935055 + 0.354504i \(0.115350\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 46.0000i 1.50275i 0.659873 + 0.751377i \(0.270610\pi\)
−0.659873 + 0.751377i \(0.729390\pi\)
\(938\) 0 0
\(939\) 29.0000 0.946379
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 0 0
\(943\) − 5.00000i − 0.162822i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 18.0000i 0.581857i
\(958\) 0 0
\(959\) −42.0000 −1.35625
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) − 5.00000i − 0.161123i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 4.00000i 0.128631i 0.997930 + 0.0643157i \(0.0204865\pi\)
−0.997930 + 0.0643157i \(0.979514\pi\)
\(968\) 0 0
\(969\) −42.0000 −1.34923
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 0 0
\(973\) − 15.0000i − 0.480878i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 37.0000i − 1.18373i −0.806035 0.591867i \(-0.798391\pi\)
0.806035 0.591867i \(-0.201609\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) − 41.0000i − 1.30770i −0.756626 0.653848i \(-0.773153\pi\)
0.756626 0.653848i \(-0.226847\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 24.0000i − 0.763928i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −59.0000 −1.87420 −0.937098 0.349065i \(-0.886499\pi\)
−0.937098 + 0.349065i \(0.886499\pi\)
\(992\) 0 0
\(993\) − 21.0000i − 0.666415i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 16.0000i 0.506725i 0.967371 + 0.253363i \(0.0815366\pi\)
−0.967371 + 0.253363i \(0.918463\pi\)
\(998\) 0 0
\(999\) 7.00000 0.221470
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 6900.2.f.b.6349.1 2
5.2 odd 4 6900.2.a.c.1.1 1
5.3 odd 4 1380.2.a.e.1.1 1
5.4 even 2 inner 6900.2.f.b.6349.2 2
15.8 even 4 4140.2.a.b.1.1 1
20.3 even 4 5520.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.e.1.1 1 5.3 odd 4
4140.2.a.b.1.1 1 15.8 even 4
5520.2.a.m.1.1 1 20.3 even 4
6900.2.a.c.1.1 1 5.2 odd 4
6900.2.f.b.6349.1 2 1.1 even 1 trivial
6900.2.f.b.6349.2 2 5.4 even 2 inner