Properties

Label 4004.2.m.c.2157.3
Level $4004$
Weight $2$
Character 4004.2157
Analytic conductor $31.972$
Analytic rank $0$
Dimension $36$
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] = [4004,2,Mod(2157,4004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4004.2157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.m (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2157.3
Character \(\chi\) \(=\) 4004.2157
Dual form 4004.2.m.c.2157.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.10315 q^{3} +1.71558i q^{5} +1.00000i q^{7} +6.62954 q^{9} +O(q^{10})\) \(q-3.10315 q^{3} +1.71558i q^{5} +1.00000i q^{7} +6.62954 q^{9} -1.00000i q^{11} +(1.66202 - 3.19964i) q^{13} -5.32371i q^{15} +5.08192 q^{17} -7.32222i q^{19} -3.10315i q^{21} -3.22025 q^{23} +2.05677 q^{25} -11.2630 q^{27} -9.04384 q^{29} -4.98361i q^{31} +3.10315i q^{33} -1.71558 q^{35} +1.36062i q^{37} +(-5.15750 + 9.92896i) q^{39} +12.1147i q^{41} +8.57894 q^{43} +11.3735i q^{45} +0.265550i q^{47} -1.00000 q^{49} -15.7700 q^{51} -9.73208 q^{53} +1.71558 q^{55} +22.7219i q^{57} +0.610665i q^{59} -1.47181 q^{61} +6.62954i q^{63} +(5.48925 + 2.85134i) q^{65} -12.4920i q^{67} +9.99293 q^{69} +1.07609i q^{71} -7.37610i q^{73} -6.38248 q^{75} +1.00000 q^{77} -10.6681 q^{79} +15.0622 q^{81} +1.00686i q^{83} +8.71846i q^{85} +28.0644 q^{87} +2.76495i q^{89} +(3.19964 + 1.66202i) q^{91} +15.4649i q^{93} +12.5619 q^{95} +1.77103i q^{97} -6.62954i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(925\) \(2003\) \(3433\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) −3.10315 −1.79160 −0.895802 0.444453i \(-0.853398\pi\)
−0.895802 + 0.444453i \(0.853398\pi\)
\(4\) 0 0
\(5\) 1.71558i 0.767232i 0.923493 + 0.383616i \(0.125321\pi\)
−0.923493 + 0.383616i \(0.874679\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 6.62954 2.20985
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) 1.66202 3.19964i 0.460962 0.887420i
\(14\) 0 0
\(15\) 5.32371i 1.37458i
\(16\) 0 0
\(17\) 5.08192 1.23255 0.616273 0.787532i \(-0.288642\pi\)
0.616273 + 0.787532i \(0.288642\pi\)
\(18\) 0 0
\(19\) 7.32222i 1.67983i −0.542717 0.839916i \(-0.682604\pi\)
0.542717 0.839916i \(-0.317396\pi\)
\(20\) 0 0
\(21\) 3.10315i 0.677163i
\(22\) 0 0
\(23\) −3.22025 −0.671469 −0.335735 0.941957i \(-0.608985\pi\)
−0.335735 + 0.941957i \(0.608985\pi\)
\(24\) 0 0
\(25\) 2.05677 0.411355
\(26\) 0 0
\(27\) −11.2630 −2.16757
\(28\) 0 0
\(29\) −9.04384 −1.67940 −0.839700 0.543051i \(-0.817269\pi\)
−0.839700 + 0.543051i \(0.817269\pi\)
\(30\) 0 0
\(31\) 4.98361i 0.895082i −0.894263 0.447541i \(-0.852300\pi\)
0.894263 0.447541i \(-0.147700\pi\)
\(32\) 0 0
\(33\) 3.10315i 0.540189i
\(34\) 0 0
\(35\) −1.71558 −0.289986
\(36\) 0 0
\(37\) 1.36062i 0.223685i 0.993726 + 0.111843i \(0.0356753\pi\)
−0.993726 + 0.111843i \(0.964325\pi\)
\(38\) 0 0
\(39\) −5.15750 + 9.92896i −0.825861 + 1.58991i
\(40\) 0 0
\(41\) 12.1147i 1.89199i 0.324174 + 0.945997i \(0.394914\pi\)
−0.324174 + 0.945997i \(0.605086\pi\)
\(42\) 0 0
\(43\) 8.57894 1.30828 0.654138 0.756375i \(-0.273032\pi\)
0.654138 + 0.756375i \(0.273032\pi\)
\(44\) 0 0
\(45\) 11.3735i 1.69547i
\(46\) 0 0
\(47\) 0.265550i 0.0387344i 0.999812 + 0.0193672i \(0.00616516\pi\)
−0.999812 + 0.0193672i \(0.993835\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −15.7700 −2.20824
\(52\) 0 0
\(53\) −9.73208 −1.33680 −0.668402 0.743800i \(-0.733022\pi\)
−0.668402 + 0.743800i \(0.733022\pi\)
\(54\) 0 0
\(55\) 1.71558 0.231329
\(56\) 0 0
\(57\) 22.7219i 3.00959i
\(58\) 0 0
\(59\) 0.610665i 0.0795018i 0.999210 + 0.0397509i \(0.0126564\pi\)
−0.999210 + 0.0397509i \(0.987344\pi\)
\(60\) 0 0
\(61\) −1.47181 −0.188446 −0.0942229 0.995551i \(-0.530037\pi\)
−0.0942229 + 0.995551i \(0.530037\pi\)
\(62\) 0 0
\(63\) 6.62954i 0.835243i
\(64\) 0 0
\(65\) 5.48925 + 2.85134i 0.680857 + 0.353665i
\(66\) 0 0
\(67\) 12.4920i 1.52614i −0.646313 0.763072i \(-0.723690\pi\)
0.646313 0.763072i \(-0.276310\pi\)
\(68\) 0 0
\(69\) 9.99293 1.20301
\(70\) 0 0
\(71\) 1.07609i 0.127708i 0.997959 + 0.0638541i \(0.0203392\pi\)
−0.997959 + 0.0638541i \(0.979661\pi\)
\(72\) 0 0
\(73\) 7.37610i 0.863307i −0.902039 0.431654i \(-0.857930\pi\)
0.902039 0.431654i \(-0.142070\pi\)
\(74\) 0 0
\(75\) −6.38248 −0.736985
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −10.6681 −1.20026 −0.600130 0.799902i \(-0.704885\pi\)
−0.600130 + 0.799902i \(0.704885\pi\)
\(80\) 0 0
\(81\) 15.0622 1.67357
\(82\) 0 0
\(83\) 1.00686i 0.110517i 0.998472 + 0.0552584i \(0.0175983\pi\)
−0.998472 + 0.0552584i \(0.982402\pi\)
\(84\) 0 0
\(85\) 8.71846i 0.945649i
\(86\) 0 0
\(87\) 28.0644 3.00882
\(88\) 0 0
\(89\) 2.76495i 0.293084i 0.989204 + 0.146542i \(0.0468144\pi\)
−0.989204 + 0.146542i \(0.953186\pi\)
\(90\) 0 0
\(91\) 3.19964 + 1.66202i 0.335413 + 0.174227i
\(92\) 0 0
\(93\) 15.4649i 1.60363i
\(94\) 0 0
\(95\) 12.5619 1.28882
\(96\) 0 0
\(97\) 1.77103i 0.179821i 0.995950 + 0.0899107i \(0.0286582\pi\)
−0.995950 + 0.0899107i \(0.971342\pi\)
\(98\) 0 0
\(99\) 6.62954i 0.666294i
\(100\) 0 0
\(101\) 16.5672 1.64850 0.824250 0.566226i \(-0.191597\pi\)
0.824250 + 0.566226i \(0.191597\pi\)
\(102\) 0 0
\(103\) 12.8945 1.27053 0.635266 0.772294i \(-0.280891\pi\)
0.635266 + 0.772294i \(0.280891\pi\)
\(104\) 0 0
\(105\) 5.32371 0.519541
\(106\) 0 0
\(107\) −15.5121 −1.49961 −0.749806 0.661658i \(-0.769853\pi\)
−0.749806 + 0.661658i \(0.769853\pi\)
\(108\) 0 0
\(109\) 2.76650i 0.264982i 0.991184 + 0.132491i \(0.0422976\pi\)
−0.991184 + 0.132491i \(0.957702\pi\)
\(110\) 0 0
\(111\) 4.22222i 0.400756i
\(112\) 0 0
\(113\) −7.63951 −0.718665 −0.359332 0.933210i \(-0.616996\pi\)
−0.359332 + 0.933210i \(0.616996\pi\)
\(114\) 0 0
\(115\) 5.52461i 0.515173i
\(116\) 0 0
\(117\) 11.0184 21.2121i 1.01865 1.96106i
\(118\) 0 0
\(119\) 5.08192i 0.465859i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 37.5937i 3.38971i
\(124\) 0 0
\(125\) 12.1065i 1.08284i
\(126\) 0 0
\(127\) −20.4565 −1.81522 −0.907612 0.419809i \(-0.862097\pi\)
−0.907612 + 0.419809i \(0.862097\pi\)
\(128\) 0 0
\(129\) −26.6217 −2.34391
\(130\) 0 0
\(131\) −15.1564 −1.32422 −0.662109 0.749408i \(-0.730338\pi\)
−0.662109 + 0.749408i \(0.730338\pi\)
\(132\) 0 0
\(133\) 7.32222 0.634917
\(134\) 0 0
\(135\) 19.3226i 1.66303i
\(136\) 0 0
\(137\) 7.89611i 0.674610i −0.941395 0.337305i \(-0.890485\pi\)
0.941395 0.337305i \(-0.109515\pi\)
\(138\) 0 0
\(139\) −9.11731 −0.773320 −0.386660 0.922222i \(-0.626371\pi\)
−0.386660 + 0.922222i \(0.626371\pi\)
\(140\) 0 0
\(141\) 0.824040i 0.0693967i
\(142\) 0 0
\(143\) −3.19964 1.66202i −0.267567 0.138985i
\(144\) 0 0
\(145\) 15.5155i 1.28849i
\(146\) 0 0
\(147\) 3.10315 0.255943
\(148\) 0 0
\(149\) 8.08924i 0.662697i −0.943509 0.331348i \(-0.892496\pi\)
0.943509 0.331348i \(-0.107504\pi\)
\(150\) 0 0
\(151\) 20.1840i 1.64255i 0.570535 + 0.821273i \(0.306736\pi\)
−0.570535 + 0.821273i \(0.693264\pi\)
\(152\) 0 0
\(153\) 33.6908 2.72374
\(154\) 0 0
\(155\) 8.54979 0.686736
\(156\) 0 0
\(157\) −14.6475 −1.16900 −0.584498 0.811395i \(-0.698709\pi\)
−0.584498 + 0.811395i \(0.698709\pi\)
\(158\) 0 0
\(159\) 30.2001 2.39502
\(160\) 0 0
\(161\) 3.22025i 0.253791i
\(162\) 0 0
\(163\) 12.8860i 1.00931i 0.863322 + 0.504653i \(0.168380\pi\)
−0.863322 + 0.504653i \(0.831620\pi\)
\(164\) 0 0
\(165\) −5.32371 −0.414450
\(166\) 0 0
\(167\) 17.2959i 1.33840i −0.743083 0.669200i \(-0.766637\pi\)
0.743083 0.669200i \(-0.233363\pi\)
\(168\) 0 0
\(169\) −7.47537 10.6357i −0.575029 0.818133i
\(170\) 0 0
\(171\) 48.5429i 3.71217i
\(172\) 0 0
\(173\) −12.4223 −0.944450 −0.472225 0.881478i \(-0.656549\pi\)
−0.472225 + 0.881478i \(0.656549\pi\)
\(174\) 0 0
\(175\) 2.05677i 0.155478i
\(176\) 0 0
\(177\) 1.89498i 0.142436i
\(178\) 0 0
\(179\) −11.1096 −0.830370 −0.415185 0.909737i \(-0.636283\pi\)
−0.415185 + 0.909737i \(0.636283\pi\)
\(180\) 0 0
\(181\) 18.5362 1.37779 0.688894 0.724862i \(-0.258097\pi\)
0.688894 + 0.724862i \(0.258097\pi\)
\(182\) 0 0
\(183\) 4.56724 0.337620
\(184\) 0 0
\(185\) −2.33426 −0.171619
\(186\) 0 0
\(187\) 5.08192i 0.371627i
\(188\) 0 0
\(189\) 11.2630i 0.819263i
\(190\) 0 0
\(191\) −18.5932 −1.34535 −0.672677 0.739936i \(-0.734856\pi\)
−0.672677 + 0.739936i \(0.734856\pi\)
\(192\) 0 0
\(193\) 3.28669i 0.236581i −0.992979 0.118291i \(-0.962259\pi\)
0.992979 0.118291i \(-0.0377415\pi\)
\(194\) 0 0
\(195\) −17.0340 8.84812i −1.21983 0.633627i
\(196\) 0 0
\(197\) 17.4262i 1.24156i −0.783984 0.620781i \(-0.786815\pi\)
0.783984 0.620781i \(-0.213185\pi\)
\(198\) 0 0
\(199\) 6.64221 0.470854 0.235427 0.971892i \(-0.424351\pi\)
0.235427 + 0.971892i \(0.424351\pi\)
\(200\) 0 0
\(201\) 38.7646i 2.73425i
\(202\) 0 0
\(203\) 9.04384i 0.634753i
\(204\) 0 0
\(205\) −20.7837 −1.45160
\(206\) 0 0
\(207\) −21.3488 −1.48384
\(208\) 0 0
\(209\) −7.32222 −0.506488
\(210\) 0 0
\(211\) 6.13924 0.422642 0.211321 0.977417i \(-0.432223\pi\)
0.211321 + 0.977417i \(0.432223\pi\)
\(212\) 0 0
\(213\) 3.33926i 0.228803i
\(214\) 0 0
\(215\) 14.7179i 1.00375i
\(216\) 0 0
\(217\) 4.98361 0.338309
\(218\) 0 0
\(219\) 22.8891i 1.54670i
\(220\) 0 0
\(221\) 8.44626 16.2603i 0.568157 1.09379i
\(222\) 0 0
\(223\) 8.90531i 0.596344i −0.954512 0.298172i \(-0.903623\pi\)
0.954512 0.298172i \(-0.0963769\pi\)
\(224\) 0 0
\(225\) 13.6355 0.909031
\(226\) 0 0
\(227\) 14.1194i 0.937138i −0.883427 0.468569i \(-0.844770\pi\)
0.883427 0.468569i \(-0.155230\pi\)
\(228\) 0 0
\(229\) 2.92634i 0.193378i 0.995315 + 0.0966890i \(0.0308252\pi\)
−0.995315 + 0.0966890i \(0.969175\pi\)
\(230\) 0 0
\(231\) −3.10315 −0.204172
\(232\) 0 0
\(233\) −9.49861 −0.622275 −0.311137 0.950365i \(-0.600710\pi\)
−0.311137 + 0.950365i \(0.600710\pi\)
\(234\) 0 0
\(235\) −0.455573 −0.0297183
\(236\) 0 0
\(237\) 33.1049 2.15039
\(238\) 0 0
\(239\) 16.9333i 1.09533i 0.836699 + 0.547663i \(0.184482\pi\)
−0.836699 + 0.547663i \(0.815518\pi\)
\(240\) 0 0
\(241\) 5.70871i 0.367730i 0.982951 + 0.183865i \(0.0588609\pi\)
−0.982951 + 0.183865i \(0.941139\pi\)
\(242\) 0 0
\(243\) −12.9512 −0.830818
\(244\) 0 0
\(245\) 1.71558i 0.109605i
\(246\) 0 0
\(247\) −23.4284 12.1697i −1.49072 0.774338i
\(248\) 0 0
\(249\) 3.12443i 0.198002i
\(250\) 0 0
\(251\) 17.0766 1.07787 0.538934 0.842348i \(-0.318827\pi\)
0.538934 + 0.842348i \(0.318827\pi\)
\(252\) 0 0
\(253\) 3.22025i 0.202456i
\(254\) 0 0
\(255\) 27.0547i 1.69423i
\(256\) 0 0
\(257\) −17.6369 −1.10016 −0.550080 0.835112i \(-0.685403\pi\)
−0.550080 + 0.835112i \(0.685403\pi\)
\(258\) 0 0
\(259\) −1.36062 −0.0845451
\(260\) 0 0
\(261\) −59.9565 −3.71121
\(262\) 0 0
\(263\) 22.4613 1.38502 0.692510 0.721408i \(-0.256505\pi\)
0.692510 + 0.721408i \(0.256505\pi\)
\(264\) 0 0
\(265\) 16.6962i 1.02564i
\(266\) 0 0
\(267\) 8.58006i 0.525091i
\(268\) 0 0
\(269\) 6.52344 0.397741 0.198871 0.980026i \(-0.436273\pi\)
0.198871 + 0.980026i \(0.436273\pi\)
\(270\) 0 0
\(271\) 2.68382i 0.163030i 0.996672 + 0.0815151i \(0.0259759\pi\)
−0.996672 + 0.0815151i \(0.974024\pi\)
\(272\) 0 0
\(273\) −9.92896 5.15750i −0.600928 0.312146i
\(274\) 0 0
\(275\) 2.05677i 0.124028i
\(276\) 0 0
\(277\) 17.1046 1.02771 0.513856 0.857876i \(-0.328216\pi\)
0.513856 + 0.857876i \(0.328216\pi\)
\(278\) 0 0
\(279\) 33.0390i 1.97799i
\(280\) 0 0
\(281\) 32.5978i 1.94462i −0.233696 0.972310i \(-0.575082\pi\)
0.233696 0.972310i \(-0.424918\pi\)
\(282\) 0 0
\(283\) −1.68236 −0.100006 −0.0500030 0.998749i \(-0.515923\pi\)
−0.0500030 + 0.998749i \(0.515923\pi\)
\(284\) 0 0
\(285\) −38.9814 −2.30906
\(286\) 0 0
\(287\) −12.1147 −0.715107
\(288\) 0 0
\(289\) 8.82590 0.519171
\(290\) 0 0
\(291\) 5.49579i 0.322169i
\(292\) 0 0
\(293\) 10.9268i 0.638353i −0.947695 0.319176i \(-0.896594\pi\)
0.947695 0.319176i \(-0.103406\pi\)
\(294\) 0 0
\(295\) −1.04765 −0.0609963
\(296\) 0 0
\(297\) 11.2630i 0.653546i
\(298\) 0 0
\(299\) −5.35213 + 10.3036i −0.309522 + 0.595875i
\(300\) 0 0
\(301\) 8.57894i 0.494482i
\(302\) 0 0
\(303\) −51.4106 −2.95346
\(304\) 0 0
\(305\) 2.52501i 0.144582i
\(306\) 0 0
\(307\) 28.4746i 1.62513i −0.582872 0.812564i \(-0.698071\pi\)
0.582872 0.812564i \(-0.301929\pi\)
\(308\) 0 0
\(309\) −40.0135 −2.27629
\(310\) 0 0
\(311\) −4.49257 −0.254750 −0.127375 0.991855i \(-0.540655\pi\)
−0.127375 + 0.991855i \(0.540655\pi\)
\(312\) 0 0
\(313\) −0.176055 −0.00995119 −0.00497560 0.999988i \(-0.501584\pi\)
−0.00497560 + 0.999988i \(0.501584\pi\)
\(314\) 0 0
\(315\) −11.3735 −0.640826
\(316\) 0 0
\(317\) 5.56199i 0.312392i −0.987726 0.156196i \(-0.950077\pi\)
0.987726 0.156196i \(-0.0499232\pi\)
\(318\) 0 0
\(319\) 9.04384i 0.506358i
\(320\) 0 0
\(321\) 48.1364 2.68671
\(322\) 0 0
\(323\) 37.2109i 2.07047i
\(324\) 0 0
\(325\) 3.41840 6.58093i 0.189619 0.365045i
\(326\) 0 0
\(327\) 8.58486i 0.474744i
\(328\) 0 0
\(329\) −0.265550 −0.0146402
\(330\) 0 0
\(331\) 20.3149i 1.11661i −0.829636 0.558305i \(-0.811452\pi\)
0.829636 0.558305i \(-0.188548\pi\)
\(332\) 0 0
\(333\) 9.02031i 0.494310i
\(334\) 0 0
\(335\) 21.4311 1.17091
\(336\) 0 0
\(337\) −29.0083 −1.58019 −0.790093 0.612987i \(-0.789968\pi\)
−0.790093 + 0.612987i \(0.789968\pi\)
\(338\) 0 0
\(339\) 23.7065 1.28756
\(340\) 0 0
\(341\) −4.98361 −0.269877
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 17.1437i 0.922986i
\(346\) 0 0
\(347\) −23.8427 −1.27995 −0.639973 0.768397i \(-0.721054\pi\)
−0.639973 + 0.768397i \(0.721054\pi\)
\(348\) 0 0
\(349\) 29.8451i 1.59757i −0.601615 0.798786i \(-0.705476\pi\)
0.601615 0.798786i \(-0.294524\pi\)
\(350\) 0 0
\(351\) −18.7194 + 36.0375i −0.999165 + 1.92354i
\(352\) 0 0
\(353\) 15.4542i 0.822544i 0.911513 + 0.411272i \(0.134915\pi\)
−0.911513 + 0.411272i \(0.865085\pi\)
\(354\) 0 0
\(355\) −1.84612 −0.0979819
\(356\) 0 0
\(357\) 15.7700i 0.834635i
\(358\) 0 0
\(359\) 8.37938i 0.442247i 0.975246 + 0.221123i \(0.0709723\pi\)
−0.975246 + 0.221123i \(0.929028\pi\)
\(360\) 0 0
\(361\) −34.6149 −1.82183
\(362\) 0 0
\(363\) 3.10315 0.162873
\(364\) 0 0
\(365\) 12.6543 0.662357
\(366\) 0 0
\(367\) 5.78081 0.301756 0.150878 0.988552i \(-0.451790\pi\)
0.150878 + 0.988552i \(0.451790\pi\)
\(368\) 0 0
\(369\) 80.3147i 4.18102i
\(370\) 0 0
\(371\) 9.73208i 0.505265i
\(372\) 0 0
\(373\) 4.26380 0.220771 0.110386 0.993889i \(-0.464791\pi\)
0.110386 + 0.993889i \(0.464791\pi\)
\(374\) 0 0
\(375\) 37.5682i 1.94002i
\(376\) 0 0
\(377\) −15.0311 + 28.9370i −0.774139 + 1.49033i
\(378\) 0 0
\(379\) 17.0712i 0.876890i −0.898758 0.438445i \(-0.855529\pi\)
0.898758 0.438445i \(-0.144471\pi\)
\(380\) 0 0
\(381\) 63.4797 3.25216
\(382\) 0 0
\(383\) 1.28626i 0.0657250i −0.999460 0.0328625i \(-0.989538\pi\)
0.999460 0.0328625i \(-0.0104623\pi\)
\(384\) 0 0
\(385\) 1.71558i 0.0874342i
\(386\) 0 0
\(387\) 56.8744 2.89109
\(388\) 0 0
\(389\) 22.8254 1.15729 0.578646 0.815579i \(-0.303581\pi\)
0.578646 + 0.815579i \(0.303581\pi\)
\(390\) 0 0
\(391\) −16.3651 −0.827617
\(392\) 0 0
\(393\) 47.0325 2.37247
\(394\) 0 0
\(395\) 18.3021i 0.920878i
\(396\) 0 0
\(397\) 9.90885i 0.497311i −0.968592 0.248656i \(-0.920011\pi\)
0.968592 0.248656i \(-0.0799887\pi\)
\(398\) 0 0
\(399\) −22.7219 −1.13752
\(400\) 0 0
\(401\) 8.17088i 0.408035i −0.978967 0.204017i \(-0.934600\pi\)
0.978967 0.204017i \(-0.0653999\pi\)
\(402\) 0 0
\(403\) −15.9457 8.28286i −0.794314 0.412599i
\(404\) 0 0
\(405\) 25.8404i 1.28402i
\(406\) 0 0
\(407\) 1.36062 0.0674436
\(408\) 0 0
\(409\) 14.9797i 0.740700i −0.928892 0.370350i \(-0.879238\pi\)
0.928892 0.370350i \(-0.120762\pi\)
\(410\) 0 0
\(411\) 24.5028i 1.20863i
\(412\) 0 0
\(413\) −0.610665 −0.0300489
\(414\) 0 0
\(415\) −1.72735 −0.0847920
\(416\) 0 0
\(417\) 28.2924 1.38548
\(418\) 0 0
\(419\) 17.2674 0.843566 0.421783 0.906697i \(-0.361404\pi\)
0.421783 + 0.906697i \(0.361404\pi\)
\(420\) 0 0
\(421\) 30.4363i 1.48338i 0.670746 + 0.741688i \(0.265974\pi\)
−0.670746 + 0.741688i \(0.734026\pi\)
\(422\) 0 0
\(423\) 1.76047i 0.0855971i
\(424\) 0 0
\(425\) 10.4524 0.507014
\(426\) 0 0
\(427\) 1.47181i 0.0712258i
\(428\) 0 0
\(429\) 9.92896 + 5.15750i 0.479375 + 0.249007i
\(430\) 0 0
\(431\) 22.2055i 1.06960i 0.844978 + 0.534801i \(0.179613\pi\)
−0.844978 + 0.534801i \(0.820387\pi\)
\(432\) 0 0
\(433\) 21.2497 1.02119 0.510597 0.859820i \(-0.329425\pi\)
0.510597 + 0.859820i \(0.329425\pi\)
\(434\) 0 0
\(435\) 48.1468i 2.30846i
\(436\) 0 0
\(437\) 23.5794i 1.12796i
\(438\) 0 0
\(439\) −4.51477 −0.215478 −0.107739 0.994179i \(-0.534361\pi\)
−0.107739 + 0.994179i \(0.534361\pi\)
\(440\) 0 0
\(441\) −6.62954 −0.315692
\(442\) 0 0
\(443\) −19.5619 −0.929416 −0.464708 0.885464i \(-0.653841\pi\)
−0.464708 + 0.885464i \(0.653841\pi\)
\(444\) 0 0
\(445\) −4.74351 −0.224864
\(446\) 0 0
\(447\) 25.1021i 1.18729i
\(448\) 0 0
\(449\) 13.2952i 0.627439i 0.949516 + 0.313720i \(0.101575\pi\)
−0.949516 + 0.313720i \(0.898425\pi\)
\(450\) 0 0
\(451\) 12.1147 0.570458
\(452\) 0 0
\(453\) 62.6338i 2.94279i
\(454\) 0 0
\(455\) −2.85134 + 5.48925i −0.133673 + 0.257340i
\(456\) 0 0
\(457\) 34.9067i 1.63287i −0.577439 0.816434i \(-0.695948\pi\)
0.577439 0.816434i \(-0.304052\pi\)
\(458\) 0 0
\(459\) −57.2377 −2.67163
\(460\) 0 0
\(461\) 34.2750i 1.59635i −0.602428 0.798173i \(-0.705800\pi\)
0.602428 0.798173i \(-0.294200\pi\)
\(462\) 0 0
\(463\) 33.9899i 1.57964i 0.613337 + 0.789822i \(0.289827\pi\)
−0.613337 + 0.789822i \(0.710173\pi\)
\(464\) 0 0
\(465\) −26.5313 −1.23036
\(466\) 0 0
\(467\) −34.5307 −1.59789 −0.798946 0.601403i \(-0.794609\pi\)
−0.798946 + 0.601403i \(0.794609\pi\)
\(468\) 0 0
\(469\) 12.4920 0.576828
\(470\) 0 0
\(471\) 45.4533 2.09438
\(472\) 0 0
\(473\) 8.57894i 0.394460i
\(474\) 0 0
\(475\) 15.0601i 0.691007i
\(476\) 0 0
\(477\) −64.5192 −2.95413
\(478\) 0 0
\(479\) 15.1076i 0.690284i 0.938551 + 0.345142i \(0.112169\pi\)
−0.938551 + 0.345142i \(0.887831\pi\)
\(480\) 0 0
\(481\) 4.35351 + 2.26139i 0.198503 + 0.103110i
\(482\) 0 0
\(483\) 9.99293i 0.454694i
\(484\) 0 0
\(485\) −3.03836 −0.137965
\(486\) 0 0
\(487\) 38.6163i 1.74987i −0.484239 0.874936i \(-0.660903\pi\)
0.484239 0.874936i \(-0.339097\pi\)
\(488\) 0 0
\(489\) 39.9871i 1.80828i
\(490\) 0 0
\(491\) −11.7053 −0.528254 −0.264127 0.964488i \(-0.585084\pi\)
−0.264127 + 0.964488i \(0.585084\pi\)
\(492\) 0 0
\(493\) −45.9601 −2.06994
\(494\) 0 0
\(495\) 11.3735 0.511202
\(496\) 0 0
\(497\) −1.07609 −0.0482692
\(498\) 0 0
\(499\) 1.85956i 0.0832455i 0.999133 + 0.0416227i \(0.0132528\pi\)
−0.999133 + 0.0416227i \(0.986747\pi\)
\(500\) 0 0
\(501\) 53.6719i 2.39788i
\(502\) 0 0
\(503\) 3.45118 0.153881 0.0769403 0.997036i \(-0.475485\pi\)
0.0769403 + 0.997036i \(0.475485\pi\)
\(504\) 0 0
\(505\) 28.4224i 1.26478i
\(506\) 0 0
\(507\) 23.1972 + 33.0043i 1.03022 + 1.46577i
\(508\) 0 0
\(509\) 30.5030i 1.35202i 0.736891 + 0.676011i \(0.236293\pi\)
−0.736891 + 0.676011i \(0.763707\pi\)
\(510\) 0 0
\(511\) 7.37610 0.326299
\(512\) 0 0
\(513\) 82.4702i 3.64115i
\(514\) 0 0
\(515\) 22.1216i 0.974792i
\(516\) 0 0
\(517\) 0.265550 0.0116789
\(518\) 0 0
\(519\) 38.5483 1.69208
\(520\) 0 0
\(521\) 22.3987 0.981307 0.490653 0.871355i \(-0.336758\pi\)
0.490653 + 0.871355i \(0.336758\pi\)
\(522\) 0 0
\(523\) 4.09703 0.179151 0.0895753 0.995980i \(-0.471449\pi\)
0.0895753 + 0.995980i \(0.471449\pi\)
\(524\) 0 0
\(525\) 6.38248i 0.278554i
\(526\) 0 0
\(527\) 25.3263i 1.10323i
\(528\) 0 0
\(529\) −12.6300 −0.549129
\(530\) 0 0
\(531\) 4.04843i 0.175687i
\(532\) 0 0
\(533\) 38.7626 + 20.1349i 1.67899 + 0.872137i
\(534\) 0 0
\(535\) 26.6123i 1.15055i
\(536\) 0 0
\(537\) 34.4747 1.48770
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 4.60972i 0.198187i 0.995078 + 0.0990936i \(0.0315943\pi\)
−0.995078 + 0.0990936i \(0.968406\pi\)
\(542\) 0 0
\(543\) −57.5207 −2.46845
\(544\) 0 0
\(545\) −4.74616 −0.203303
\(546\) 0 0
\(547\) −28.7361 −1.22867 −0.614333 0.789047i \(-0.710575\pi\)
−0.614333 + 0.789047i \(0.710575\pi\)
\(548\) 0 0
\(549\) −9.75742 −0.416436
\(550\) 0 0
\(551\) 66.2210i 2.82111i
\(552\) 0 0
\(553\) 10.6681i 0.453656i
\(554\) 0 0
\(555\) 7.24357 0.307472
\(556\) 0 0
\(557\) 10.8600i 0.460152i −0.973173 0.230076i \(-0.926102\pi\)
0.973173 0.230076i \(-0.0738976\pi\)
\(558\) 0 0
\(559\) 14.2584 27.4495i 0.603065 1.16099i
\(560\) 0 0
\(561\) 15.7700i 0.665808i
\(562\) 0 0
\(563\) 23.4139 0.986777 0.493388 0.869809i \(-0.335758\pi\)
0.493388 + 0.869809i \(0.335758\pi\)
\(564\) 0 0
\(565\) 13.1062i 0.551383i
\(566\) 0 0
\(567\) 15.0622i 0.632552i
\(568\) 0 0
\(569\) 26.5389 1.11257 0.556285 0.830991i \(-0.312226\pi\)
0.556285 + 0.830991i \(0.312226\pi\)
\(570\) 0 0
\(571\) 31.0785 1.30060 0.650298 0.759679i \(-0.274644\pi\)
0.650298 + 0.759679i \(0.274644\pi\)
\(572\) 0 0
\(573\) 57.6974 2.41034
\(574\) 0 0
\(575\) −6.62333 −0.276212
\(576\) 0 0
\(577\) 44.7936i 1.86478i −0.361451 0.932391i \(-0.617719\pi\)
0.361451 0.932391i \(-0.382281\pi\)
\(578\) 0 0
\(579\) 10.1991i 0.423860i
\(580\) 0 0
\(581\) −1.00686 −0.0417714
\(582\) 0 0
\(583\) 9.73208i 0.403062i
\(584\) 0 0
\(585\) 36.3912 + 18.9030i 1.50459 + 0.781545i
\(586\) 0 0
\(587\) 31.7357i 1.30987i −0.755685 0.654935i \(-0.772696\pi\)
0.755685 0.654935i \(-0.227304\pi\)
\(588\) 0 0
\(589\) −36.4910 −1.50359
\(590\) 0 0
\(591\) 54.0760i 2.22439i
\(592\) 0 0
\(593\) 28.3865i 1.16569i −0.812582 0.582847i \(-0.801939\pi\)
0.812582 0.582847i \(-0.198061\pi\)
\(594\) 0 0
\(595\) −8.71846 −0.357422
\(596\) 0 0
\(597\) −20.6118 −0.843583
\(598\) 0 0
\(599\) 21.3055 0.870518 0.435259 0.900305i \(-0.356657\pi\)
0.435259 + 0.900305i \(0.356657\pi\)
\(600\) 0 0
\(601\) −16.0139 −0.653219 −0.326610 0.945159i \(-0.605906\pi\)
−0.326610 + 0.945159i \(0.605906\pi\)
\(602\) 0 0
\(603\) 82.8164i 3.37255i
\(604\) 0 0
\(605\) 1.71558i 0.0697484i
\(606\) 0 0
\(607\) −17.8074 −0.722782 −0.361391 0.932414i \(-0.617698\pi\)
−0.361391 + 0.932414i \(0.617698\pi\)
\(608\) 0 0
\(609\) 28.0644i 1.13723i
\(610\) 0 0
\(611\) 0.849663 + 0.441349i 0.0343737 + 0.0178551i
\(612\) 0 0
\(613\) 32.7247i 1.32174i 0.750500 + 0.660870i \(0.229813\pi\)
−0.750500 + 0.660870i \(0.770187\pi\)
\(614\) 0 0
\(615\) 64.4950 2.60069
\(616\) 0 0
\(617\) 10.4929i 0.422430i 0.977440 + 0.211215i \(0.0677420\pi\)
−0.977440 + 0.211215i \(0.932258\pi\)
\(618\) 0 0
\(619\) 40.5207i 1.62867i −0.580398 0.814333i \(-0.697103\pi\)
0.580398 0.814333i \(-0.302897\pi\)
\(620\) 0 0
\(621\) 36.2697 1.45545
\(622\) 0 0
\(623\) −2.76495 −0.110775
\(624\) 0 0
\(625\) −10.4858 −0.419432
\(626\) 0 0
\(627\) 22.7219 0.907427
\(628\) 0 0
\(629\) 6.91458i 0.275702i
\(630\) 0 0
\(631\) 8.34947i 0.332387i −0.986093 0.166193i \(-0.946852\pi\)
0.986093 0.166193i \(-0.0531476\pi\)
\(632\) 0 0
\(633\) −19.0510 −0.757208
\(634\) 0 0
\(635\) 35.0949i 1.39270i
\(636\) 0 0
\(637\) −1.66202 + 3.19964i −0.0658517 + 0.126774i
\(638\) 0 0
\(639\) 7.13397i 0.282216i
\(640\) 0 0
\(641\) 32.6733 1.29052 0.645258 0.763964i \(-0.276750\pi\)
0.645258 + 0.763964i \(0.276750\pi\)
\(642\) 0 0
\(643\) 32.3767i 1.27681i 0.769699 + 0.638407i \(0.220406\pi\)
−0.769699 + 0.638407i \(0.779594\pi\)
\(644\) 0 0
\(645\) 45.6718i 1.79833i
\(646\) 0 0
\(647\) 22.3589 0.879021 0.439510 0.898238i \(-0.355152\pi\)
0.439510 + 0.898238i \(0.355152\pi\)
\(648\) 0 0
\(649\) 0.610665 0.0239707
\(650\) 0 0
\(651\) −15.4649 −0.606116
\(652\) 0 0
\(653\) 21.3370 0.834982 0.417491 0.908681i \(-0.362910\pi\)
0.417491 + 0.908681i \(0.362910\pi\)
\(654\) 0 0
\(655\) 26.0020i 1.01598i
\(656\) 0 0
\(657\) 48.9001i 1.90778i
\(658\) 0 0
\(659\) −38.7806 −1.51068 −0.755339 0.655334i \(-0.772528\pi\)
−0.755339 + 0.655334i \(0.772528\pi\)
\(660\) 0 0
\(661\) 28.3898i 1.10423i 0.833767 + 0.552117i \(0.186180\pi\)
−0.833767 + 0.552117i \(0.813820\pi\)
\(662\) 0 0
\(663\) −26.2100 + 50.4582i −1.01791 + 1.95963i
\(664\) 0 0
\(665\) 12.5619i 0.487128i
\(666\) 0 0
\(667\) 29.1235 1.12766
\(668\) 0 0
\(669\) 27.6345i 1.06841i
\(670\) 0 0
\(671\) 1.47181i 0.0568186i
\(672\) 0 0
\(673\) 26.7052 1.02941 0.514705 0.857367i \(-0.327901\pi\)
0.514705 + 0.857367i \(0.327901\pi\)
\(674\) 0 0
\(675\) −23.1655 −0.891639
\(676\) 0 0
\(677\) −40.2714 −1.54775 −0.773877 0.633336i \(-0.781685\pi\)
−0.773877 + 0.633336i \(0.781685\pi\)
\(678\) 0 0
\(679\) −1.77103 −0.0679661
\(680\) 0 0
\(681\) 43.8146i 1.67898i
\(682\) 0 0
\(683\) 14.8735i 0.569118i −0.958658 0.284559i \(-0.908153\pi\)
0.958658 0.284559i \(-0.0918472\pi\)
\(684\) 0 0
\(685\) 13.5464 0.517583
\(686\) 0 0
\(687\) 9.08087i 0.346457i
\(688\) 0 0
\(689\) −16.1749 + 31.1391i −0.616216 + 1.18631i
\(690\) 0 0
\(691\) 20.5711i 0.782562i 0.920271 + 0.391281i \(0.127968\pi\)
−0.920271 + 0.391281i \(0.872032\pi\)
\(692\) 0 0
\(693\) 6.62954 0.251835
\(694\) 0 0
\(695\) 15.6415i 0.593316i
\(696\) 0 0
\(697\) 61.5658i 2.33197i
\(698\) 0 0
\(699\) 29.4756 1.11487
\(700\) 0 0
\(701\) −21.1759 −0.799804 −0.399902 0.916558i \(-0.630956\pi\)
−0.399902 + 0.916558i \(0.630956\pi\)
\(702\) 0 0
\(703\) 9.96279 0.375754
\(704\) 0 0
\(705\) 1.41371 0.0532434
\(706\) 0 0
\(707\) 16.5672i 0.623074i
\(708\) 0 0
\(709\) 32.4743i 1.21960i −0.792557 0.609798i \(-0.791250\pi\)
0.792557 0.609798i \(-0.208750\pi\)
\(710\) 0 0
\(711\) −70.7249 −2.65239
\(712\) 0 0
\(713\) 16.0485i 0.601020i
\(714\) 0 0
\(715\) 2.85134 5.48925i 0.106634 0.205286i
\(716\) 0 0
\(717\) 52.5466i 1.96239i
\(718\) 0 0
\(719\) −21.7515 −0.811195 −0.405598 0.914052i \(-0.632937\pi\)
−0.405598 + 0.914052i \(0.632937\pi\)
\(720\) 0 0
\(721\) 12.8945i 0.480216i
\(722\) 0 0
\(723\) 17.7150i 0.658827i
\(724\) 0 0
\(725\) −18.6011 −0.690829
\(726\) 0 0
\(727\) −50.4648 −1.87163 −0.935817 0.352485i \(-0.885337\pi\)
−0.935817 + 0.352485i \(0.885337\pi\)
\(728\) 0 0
\(729\) −4.99710 −0.185078
\(730\) 0 0
\(731\) 43.5975 1.61251
\(732\) 0 0
\(733\) 12.3476i 0.456068i 0.973653 + 0.228034i \(0.0732298\pi\)
−0.973653 + 0.228034i \(0.926770\pi\)
\(734\) 0 0
\(735\) 5.32371i 0.196368i
\(736\) 0 0
\(737\) −12.4920 −0.460150
\(738\) 0 0
\(739\) 31.3855i 1.15454i 0.816555 + 0.577268i \(0.195881\pi\)
−0.816555 + 0.577268i \(0.804119\pi\)
\(740\) 0 0
\(741\) 72.7020 + 37.7643i 2.67077 + 1.38731i
\(742\) 0 0
\(743\) 38.4254i 1.40969i 0.709360 + 0.704846i \(0.248984\pi\)
−0.709360 + 0.704846i \(0.751016\pi\)
\(744\) 0 0
\(745\) 13.8778 0.508442
\(746\) 0 0
\(747\) 6.67499i 0.244225i
\(748\) 0 0
\(749\) 15.5121i 0.566800i
\(750\) 0 0
\(751\) 40.3811 1.47353 0.736763 0.676151i \(-0.236353\pi\)
0.736763 + 0.676151i \(0.236353\pi\)
\(752\) 0 0
\(753\) −52.9913 −1.93111
\(754\) 0 0
\(755\) −34.6273 −1.26021
\(756\) 0 0
\(757\) 22.2675 0.809326 0.404663 0.914466i \(-0.367389\pi\)
0.404663 + 0.914466i \(0.367389\pi\)
\(758\) 0 0
\(759\) 9.99293i 0.362720i
\(760\) 0 0
\(761\) 19.9937i 0.724771i 0.932028 + 0.362386i \(0.118038\pi\)
−0.932028 + 0.362386i \(0.881962\pi\)
\(762\) 0 0
\(763\) −2.76650 −0.100154
\(764\) 0 0
\(765\) 57.7993i 2.08974i
\(766\) 0 0
\(767\) 1.95391 + 1.01494i 0.0705515 + 0.0366473i
\(768\) 0 0
\(769\) 36.9427i 1.33219i 0.745868 + 0.666093i \(0.232035\pi\)
−0.745868 + 0.666093i \(0.767965\pi\)
\(770\) 0 0
\(771\) 54.7299 1.97105
\(772\) 0 0
\(773\) 38.8776i 1.39833i 0.714960 + 0.699165i \(0.246445\pi\)
−0.714960 + 0.699165i \(0.753555\pi\)
\(774\) 0 0
\(775\) 10.2502i 0.368196i
\(776\) 0 0
\(777\) 4.22222 0.151471
\(778\) 0 0
\(779\) 88.7063 3.17823
\(780\) 0 0
\(781\) 1.07609 0.0385055
\(782\) 0 0
\(783\) 101.861 3.64021
\(784\) 0 0
\(785\) 25.1290i 0.896892i
\(786\) 0 0
\(787\) 19.8181i 0.706437i −0.935541 0.353219i \(-0.885087\pi\)
0.935541 0.353219i \(-0.114913\pi\)
\(788\) 0 0
\(789\) −69.7006 −2.48141
\(790\) 0 0
\(791\) 7.63951i 0.271630i
\(792\) 0 0
\(793\) −2.44618 + 4.70926i −0.0868663 + 0.167231i
\(794\) 0 0
\(795\) 51.8108i 1.83754i
\(796\) 0 0
\(797\) −31.2756 −1.10784 −0.553918 0.832571i \(-0.686868\pi\)
−0.553918 + 0.832571i \(0.686868\pi\)
\(798\) 0 0
\(799\) 1.34950i 0.0477420i
\(800\) 0 0
\(801\) 18.3304i 0.647671i
\(802\) 0 0
\(803\) −7.37610 −0.260297
\(804\) 0 0
\(805\) 5.52461 0.194717
\(806\) 0 0
\(807\) −20.2432 −0.712595
\(808\) 0 0
\(809\) 28.0574 0.986444 0.493222 0.869903i \(-0.335819\pi\)
0.493222 + 0.869903i \(0.335819\pi\)
\(810\) 0 0
\(811\) 1.50469i 0.0528367i −0.999651 0.0264183i \(-0.991590\pi\)
0.999651 0.0264183i \(-0.00841019\pi\)
\(812\) 0 0
\(813\) 8.32828i 0.292086i
\(814\) 0 0
\(815\) −22.1069 −0.774372
\(816\) 0 0
\(817\) 62.8168i 2.19768i
\(818\) 0 0
\(819\) 21.2121 + 11.0184i 0.741212 + 0.385015i
\(820\) 0 0
\(821\) 1.71207i 0.0597516i 0.999554 + 0.0298758i \(0.00951118\pi\)
−0.999554 + 0.0298758i \(0.990489\pi\)
\(822\) 0 0
\(823\) 15.5558 0.542240 0.271120 0.962546i \(-0.412606\pi\)
0.271120 + 0.962546i \(0.412606\pi\)
\(824\) 0 0
\(825\) 6.38248i 0.222209i
\(826\) 0 0
\(827\) 14.9162i 0.518688i 0.965785 + 0.259344i \(0.0835063\pi\)
−0.965785 + 0.259344i \(0.916494\pi\)
\(828\) 0 0
\(829\) −18.2952 −0.635419 −0.317709 0.948188i \(-0.602914\pi\)
−0.317709 + 0.948188i \(0.602914\pi\)
\(830\) 0 0
\(831\) −53.0780 −1.84126
\(832\) 0 0
\(833\) −5.08192 −0.176078
\(834\) 0 0
\(835\) 29.6726 1.02686
\(836\) 0 0
\(837\) 56.1304i 1.94015i
\(838\) 0 0
\(839\) 30.6594i 1.05848i −0.848472 0.529240i \(-0.822477\pi\)
0.848472 0.529240i \(-0.177523\pi\)
\(840\) 0 0
\(841\) 52.7911 1.82038
\(842\) 0 0
\(843\) 101.156i 3.48399i
\(844\) 0 0
\(845\) 18.2465 12.8246i 0.627698 0.441180i
\(846\) 0 0
\(847\) 1.00000i 0.0343604i
\(848\) 0 0
\(849\) 5.22062 0.179171
\(850\) 0 0
\(851\) 4.38155i 0.150198i
\(852\) 0 0
\(853\) 12.4539i 0.426414i −0.977007 0.213207i \(-0.931609\pi\)
0.977007 0.213207i \(-0.0683909\pi\)
\(854\) 0 0
\(855\) 83.2794 2.84810
\(856\) 0 0
\(857\) 7.11998 0.243214 0.121607 0.992578i \(-0.461195\pi\)
0.121607 + 0.992578i \(0.461195\pi\)
\(858\) 0 0
\(859\) −45.7909 −1.56237 −0.781183 0.624302i \(-0.785383\pi\)
−0.781183 + 0.624302i \(0.785383\pi\)
\(860\) 0 0
\(861\) 37.5937 1.28119
\(862\) 0 0
\(863\) 13.5172i 0.460132i −0.973175 0.230066i \(-0.926106\pi\)
0.973175 0.230066i \(-0.0738942\pi\)
\(864\) 0 0
\(865\) 21.3115i 0.724612i
\(866\) 0 0
\(867\) −27.3881 −0.930149
\(868\) 0 0
\(869\) 10.6681i 0.361892i
\(870\) 0 0
\(871\) −39.9700 20.7620i −1.35433 0.703494i
\(872\) 0 0
\(873\) 11.7411i 0.397378i
\(874\) 0 0
\(875\) −12.1065 −0.409274
\(876\) 0 0
\(877\) 31.7379i 1.07171i −0.844309 0.535857i \(-0.819989\pi\)
0.844309 0.535857i \(-0.180011\pi\)
\(878\) 0 0
\(879\) 33.9076i 1.14368i
\(880\) 0 0
\(881\) −29.6781 −0.999881 −0.499941 0.866060i \(-0.666645\pi\)
−0.499941 + 0.866060i \(0.666645\pi\)
\(882\) 0 0
\(883\) 30.0595 1.01158 0.505791 0.862656i \(-0.331201\pi\)
0.505791 + 0.862656i \(0.331201\pi\)
\(884\) 0 0
\(885\) 3.25100 0.109281
\(886\) 0 0
\(887\) −17.0959 −0.574023 −0.287012 0.957927i \(-0.592662\pi\)
−0.287012 + 0.957927i \(0.592662\pi\)
\(888\) 0 0
\(889\) 20.4565i 0.686090i
\(890\) 0 0
\(891\) 15.0622i 0.504602i
\(892\) 0 0
\(893\) 1.94441 0.0650673
\(894\) 0 0
\(895\) 19.0594i 0.637087i
\(896\) 0 0
\(897\) 16.6085 31.9738i 0.554540 1.06757i
\(898\) 0 0
\(899\) 45.0709i 1.50320i
\(900\) 0 0
\(901\) −49.4577 −1.64767
\(902\) 0 0
\(903\) 26.6217i 0.885916i
\(904\) 0 0
\(905\) 31.8005i 1.05708i
\(906\) 0 0
\(907\) 0.531933 0.0176625 0.00883127 0.999961i \(-0.497189\pi\)
0.00883127 + 0.999961i \(0.497189\pi\)
\(908\) 0 0
\(909\) 109.833 3.64293
\(910\) 0 0
\(911\) 11.1156 0.368276 0.184138 0.982900i \(-0.441051\pi\)
0.184138 + 0.982900i \(0.441051\pi\)
\(912\) 0 0
\(913\) 1.00686 0.0333221
\(914\) 0 0
\(915\) 7.83549i 0.259033i
\(916\) 0 0
\(917\) 15.1564i 0.500507i
\(918\) 0 0
\(919\) −17.8972 −0.590375 −0.295188 0.955439i \(-0.595382\pi\)
−0.295188 + 0.955439i \(0.595382\pi\)
\(920\) 0 0
\(921\) 88.3608i 2.91159i
\(922\) 0 0
\(923\) 3.44310 + 1.78848i 0.113331 + 0.0588686i
\(924\) 0 0
\(925\) 2.79850i 0.0920140i
\(926\) 0 0
\(927\) 85.4845 2.80768
\(928\) 0 0
\(929\) 18.7219i 0.614245i 0.951670 + 0.307123i \(0.0993662\pi\)
−0.951670 + 0.307123i \(0.900634\pi\)
\(930\) 0 0
\(931\) 7.32222i 0.239976i
\(932\) 0 0
\(933\) 13.9411 0.456411
\(934\) 0 0
\(935\) 8.71846 0.285124
\(936\) 0 0
\(937\) 26.6111 0.869348 0.434674 0.900588i \(-0.356864\pi\)
0.434674 + 0.900588i \(0.356864\pi\)
\(938\) 0 0
\(939\) 0.546324 0.0178286
\(940\) 0 0
\(941\) 14.1803i 0.462265i −0.972922 0.231133i \(-0.925757\pi\)
0.972922 0.231133i \(-0.0742431\pi\)
\(942\) 0 0
\(943\) 39.0123i 1.27042i
\(944\) 0 0
\(945\) 19.3226 0.628565
\(946\) 0 0
\(947\) 10.7724i 0.350057i −0.984563 0.175029i \(-0.943998\pi\)
0.984563 0.175029i \(-0.0560018\pi\)
\(948\) 0 0
\(949\) −23.6009 12.2592i −0.766116 0.397952i
\(950\) 0 0
\(951\) 17.2597i 0.559683i
\(952\) 0 0
\(953\) −6.62221 −0.214515 −0.107257 0.994231i \(-0.534207\pi\)
−0.107257 + 0.994231i \(0.534207\pi\)
\(954\) 0 0
\(955\) 31.8981i 1.03220i
\(956\) 0 0
\(957\) 28.0644i 0.907193i
\(958\) 0 0
\(959\) 7.89611 0.254979
\(960\) 0 0
\(961\) 6.16368 0.198828
\(962\) 0 0
\(963\) −102.838 −3.31391
\(964\) 0 0
\(965\) 5.63860 0.181513
\(966\) 0 0
\(967\) 17.2050i 0.553275i −0.960974 0.276638i \(-0.910780\pi\)
0.960974 0.276638i \(-0.0892202\pi\)
\(968\) 0 0
\(969\) 115.471i 3.70946i
\(970\) 0 0
\(971\) −9.76690 −0.313435 −0.156717 0.987643i \(-0.550091\pi\)
−0.156717 + 0.987643i \(0.550091\pi\)
\(972\) 0 0
\(973\) 9.11731i 0.292287i
\(974\) 0 0
\(975\) −10.6078 + 20.4216i −0.339722 + 0.654015i
\(976\) 0 0
\(977\) 46.7229i 1.49480i −0.664375 0.747399i \(-0.731302\pi\)
0.664375 0.747399i \(-0.268698\pi\)
\(978\) 0 0
\(979\) 2.76495 0.0883683
\(980\) 0 0
\(981\) 18.3406i 0.585571i
\(982\) 0 0
\(983\) 1.49831i 0.0477886i −0.999714 0.0238943i \(-0.992393\pi\)
0.999714 0.0238943i \(-0.00760652\pi\)
\(984\) 0 0
\(985\) 29.8960 0.952567
\(986\) 0 0
\(987\) 0.824040 0.0262295
\(988\) 0 0
\(989\) −27.6264 −0.878467
\(990\) 0 0
\(991\) −44.7378 −1.42114 −0.710571 0.703625i \(-0.751564\pi\)
−0.710571 + 0.703625i \(0.751564\pi\)
\(992\) 0 0
\(993\) 63.0403i 2.00052i
\(994\) 0 0
\(995\) 11.3953i 0.361254i
\(996\) 0 0
\(997\) −42.4348 −1.34392 −0.671961 0.740586i \(-0.734548\pi\)
−0.671961 + 0.740586i \(0.734548\pi\)
\(998\) 0 0
\(999\) 15.3247i 0.484853i
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 4004.2.m.c.2157.3 36
13.12 even 2 inner 4004.2.m.c.2157.4 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4004.2.m.c.2157.3 36 1.1 even 1 trivial
4004.2.m.c.2157.4 yes 36 13.12 even 2 inner