Properties

Label 3344.2.o.b.1519.19
Level $3344$
Weight $2$
Character 3344.1519
Analytic conductor $26.702$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3344,2,Mod(1519,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3344.1519");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.o (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: \(26.7019744359\)
Analytic rank: \(0\)
Dimension: \(64\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1519.19
Character \(\chi\) \(=\) 3344.1519
Dual form 3344.2.o.b.1519.20

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41745 q^{3} +2.36069 q^{5} +2.85642i q^{7} -0.990838 q^{9} +O(q^{10})\) \(q-1.41745 q^{3} +2.36069 q^{5} +2.85642i q^{7} -0.990838 q^{9} +1.00000i q^{11} -5.49937i q^{13} -3.34616 q^{15} -0.747022 q^{17} +(-1.90515 - 3.92051i) q^{19} -4.04883i q^{21} +6.90067i q^{23} +0.572849 q^{25} +5.65681 q^{27} +2.31965i q^{29} -1.19493 q^{31} -1.41745i q^{33} +6.74312i q^{35} -4.00399i q^{37} +7.79507i q^{39} +9.90538i q^{41} +2.59611i q^{43} -2.33906 q^{45} +8.06027i q^{47} -1.15915 q^{49} +1.05887 q^{51} +1.81166i q^{53} +2.36069i q^{55} +(2.70046 + 5.55712i) q^{57} +2.29532 q^{59} -9.40218 q^{61} -2.83025i q^{63} -12.9823i q^{65} -12.0326 q^{67} -9.78135i q^{69} -0.121811 q^{71} -5.12024 q^{73} -0.811984 q^{75} -2.85642 q^{77} +4.46660 q^{79} -5.04572 q^{81} +10.5652i q^{83} -1.76349 q^{85} -3.28799i q^{87} -5.36644i q^{89} +15.7085 q^{91} +1.69376 q^{93} +(-4.49747 - 9.25510i) q^{95} -10.3695i q^{97} -0.990838i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 64 q + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 64 q + 56 q^{9} + 16 q^{17} + 64 q^{25} + 32 q^{45} - 88 q^{49} + 32 q^{57} + 64 q^{61} + 40 q^{73} - 48 q^{81} - 24 q^{85} + 80 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3344\mathbb{Z}\right)^\times\).

\(n\) \(705\) \(837\) \(2433\) \(2927\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(3\) −1.41745 −0.818365 −0.409182 0.912453i \(-0.634186\pi\)
−0.409182 + 0.912453i \(0.634186\pi\)
\(4\) 0 0
\(5\) 2.36069 1.05573 0.527866 0.849328i \(-0.322992\pi\)
0.527866 + 0.849328i \(0.322992\pi\)
\(6\) 0 0
\(7\) 2.85642i 1.07963i 0.841785 + 0.539813i \(0.181505\pi\)
−0.841785 + 0.539813i \(0.818495\pi\)
\(8\) 0 0
\(9\) −0.990838 −0.330279
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) 5.49937i 1.52525i −0.646841 0.762625i \(-0.723910\pi\)
0.646841 0.762625i \(-0.276090\pi\)
\(14\) 0 0
\(15\) −3.34616 −0.863974
\(16\) 0 0
\(17\) −0.747022 −0.181179 −0.0905897 0.995888i \(-0.528875\pi\)
−0.0905897 + 0.995888i \(0.528875\pi\)
\(18\) 0 0
\(19\) −1.90515 3.92051i −0.437072 0.899426i
\(20\) 0 0
\(21\) 4.04883i 0.883528i
\(22\) 0 0
\(23\) 6.90067i 1.43889i 0.694549 + 0.719445i \(0.255604\pi\)
−0.694549 + 0.719445i \(0.744396\pi\)
\(24\) 0 0
\(25\) 0.572849 0.114570
\(26\) 0 0
\(27\) 5.65681 1.08865
\(28\) 0 0
\(29\) 2.31965i 0.430749i 0.976532 + 0.215374i \(0.0690972\pi\)
−0.976532 + 0.215374i \(0.930903\pi\)
\(30\) 0 0
\(31\) −1.19493 −0.214616 −0.107308 0.994226i \(-0.534223\pi\)
−0.107308 + 0.994226i \(0.534223\pi\)
\(32\) 0 0
\(33\) 1.41745i 0.246746i
\(34\) 0 0
\(35\) 6.74312i 1.13980i
\(36\) 0 0
\(37\) 4.00399i 0.658252i −0.944286 0.329126i \(-0.893246\pi\)
0.944286 0.329126i \(-0.106754\pi\)
\(38\) 0 0
\(39\) 7.79507i 1.24821i
\(40\) 0 0
\(41\) 9.90538i 1.54696i 0.633821 + 0.773480i \(0.281486\pi\)
−0.633821 + 0.773480i \(0.718514\pi\)
\(42\) 0 0
\(43\) 2.59611i 0.395904i 0.980212 + 0.197952i \(0.0634290\pi\)
−0.980212 + 0.197952i \(0.936571\pi\)
\(44\) 0 0
\(45\) −2.33906 −0.348687
\(46\) 0 0
\(47\) 8.06027i 1.17571i 0.808966 + 0.587856i \(0.200028\pi\)
−0.808966 + 0.587856i \(0.799972\pi\)
\(48\) 0 0
\(49\) −1.15915 −0.165593
\(50\) 0 0
\(51\) 1.05887 0.148271
\(52\) 0 0
\(53\) 1.81166i 0.248851i 0.992229 + 0.124426i \(0.0397088\pi\)
−0.992229 + 0.124426i \(0.960291\pi\)
\(54\) 0 0
\(55\) 2.36069i 0.318315i
\(56\) 0 0
\(57\) 2.70046 + 5.55712i 0.357684 + 0.736059i
\(58\) 0 0
\(59\) 2.29532 0.298825 0.149413 0.988775i \(-0.452262\pi\)
0.149413 + 0.988775i \(0.452262\pi\)
\(60\) 0 0
\(61\) −9.40218 −1.20383 −0.601913 0.798562i \(-0.705595\pi\)
−0.601913 + 0.798562i \(0.705595\pi\)
\(62\) 0 0
\(63\) 2.83025i 0.356578i
\(64\) 0 0
\(65\) 12.9823i 1.61026i
\(66\) 0 0
\(67\) −12.0326 −1.47002 −0.735008 0.678058i \(-0.762822\pi\)
−0.735008 + 0.678058i \(0.762822\pi\)
\(68\) 0 0
\(69\) 9.78135i 1.17754i
\(70\) 0 0
\(71\) −0.121811 −0.0144563 −0.00722815 0.999974i \(-0.502301\pi\)
−0.00722815 + 0.999974i \(0.502301\pi\)
\(72\) 0 0
\(73\) −5.12024 −0.599279 −0.299640 0.954052i \(-0.596866\pi\)
−0.299640 + 0.954052i \(0.596866\pi\)
\(74\) 0 0
\(75\) −0.811984 −0.0937599
\(76\) 0 0
\(77\) −2.85642 −0.325520
\(78\) 0 0
\(79\) 4.46660 0.502532 0.251266 0.967918i \(-0.419153\pi\)
0.251266 + 0.967918i \(0.419153\pi\)
\(80\) 0 0
\(81\) −5.04572 −0.560636
\(82\) 0 0
\(83\) 10.5652i 1.15969i 0.814729 + 0.579843i \(0.196886\pi\)
−0.814729 + 0.579843i \(0.803114\pi\)
\(84\) 0 0
\(85\) −1.76349 −0.191277
\(86\) 0 0
\(87\) 3.28799i 0.352510i
\(88\) 0 0
\(89\) 5.36644i 0.568841i −0.958700 0.284421i \(-0.908199\pi\)
0.958700 0.284421i \(-0.0918012\pi\)
\(90\) 0 0
\(91\) 15.7085 1.64670
\(92\) 0 0
\(93\) 1.69376 0.175634
\(94\) 0 0
\(95\) −4.49747 9.25510i −0.461431 0.949553i
\(96\) 0 0
\(97\) 10.3695i 1.05287i −0.850217 0.526433i \(-0.823529\pi\)
0.850217 0.526433i \(-0.176471\pi\)
\(98\) 0 0
\(99\) 0.990838i 0.0995830i
\(100\) 0 0
\(101\) −5.49048 −0.546323 −0.273161 0.961968i \(-0.588069\pi\)
−0.273161 + 0.961968i \(0.588069\pi\)
\(102\) 0 0
\(103\) −16.2513 −1.60129 −0.800643 0.599141i \(-0.795509\pi\)
−0.800643 + 0.599141i \(0.795509\pi\)
\(104\) 0 0
\(105\) 9.55803i 0.932768i
\(106\) 0 0
\(107\) 8.82047 0.852707 0.426354 0.904557i \(-0.359798\pi\)
0.426354 + 0.904557i \(0.359798\pi\)
\(108\) 0 0
\(109\) 4.68875i 0.449101i −0.974462 0.224551i \(-0.927909\pi\)
0.974462 0.224551i \(-0.0720914\pi\)
\(110\) 0 0
\(111\) 5.67546i 0.538690i
\(112\) 0 0
\(113\) 11.2668i 1.05989i 0.848032 + 0.529945i \(0.177787\pi\)
−0.848032 + 0.529945i \(0.822213\pi\)
\(114\) 0 0
\(115\) 16.2903i 1.51908i
\(116\) 0 0
\(117\) 5.44898i 0.503759i
\(118\) 0 0
\(119\) 2.13381i 0.195606i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 14.0404i 1.26598i
\(124\) 0 0
\(125\) −10.4511 −0.934777
\(126\) 0 0
\(127\) −15.8349 −1.40512 −0.702559 0.711625i \(-0.747959\pi\)
−0.702559 + 0.711625i \(0.747959\pi\)
\(128\) 0 0
\(129\) 3.67986i 0.323994i
\(130\) 0 0
\(131\) 0.303222i 0.0264926i 0.999912 + 0.0132463i \(0.00421655\pi\)
−0.999912 + 0.0132463i \(0.995783\pi\)
\(132\) 0 0
\(133\) 11.1986 5.44192i 0.971044 0.471875i
\(134\) 0 0
\(135\) 13.3540 1.14933
\(136\) 0 0
\(137\) 12.9530 1.10665 0.553323 0.832967i \(-0.313360\pi\)
0.553323 + 0.832967i \(0.313360\pi\)
\(138\) 0 0
\(139\) 1.77585i 0.150626i 0.997160 + 0.0753130i \(0.0239956\pi\)
−0.997160 + 0.0753130i \(0.976004\pi\)
\(140\) 0 0
\(141\) 11.4250i 0.962160i
\(142\) 0 0
\(143\) 5.49937 0.459880
\(144\) 0 0
\(145\) 5.47598i 0.454755i
\(146\) 0 0
\(147\) 1.64303 0.135515
\(148\) 0 0
\(149\) −8.89791 −0.728946 −0.364473 0.931214i \(-0.618751\pi\)
−0.364473 + 0.931214i \(0.618751\pi\)
\(150\) 0 0
\(151\) −15.8154 −1.28704 −0.643519 0.765430i \(-0.722526\pi\)
−0.643519 + 0.765430i \(0.722526\pi\)
\(152\) 0 0
\(153\) 0.740178 0.0598398
\(154\) 0 0
\(155\) −2.82086 −0.226577
\(156\) 0 0
\(157\) 20.4342 1.63082 0.815412 0.578881i \(-0.196511\pi\)
0.815412 + 0.578881i \(0.196511\pi\)
\(158\) 0 0
\(159\) 2.56794i 0.203651i
\(160\) 0 0
\(161\) −19.7112 −1.55346
\(162\) 0 0
\(163\) 6.90104i 0.540531i −0.962786 0.270266i \(-0.912888\pi\)
0.962786 0.270266i \(-0.0871115\pi\)
\(164\) 0 0
\(165\) 3.34616i 0.260498i
\(166\) 0 0
\(167\) −16.6670 −1.28973 −0.644866 0.764296i \(-0.723087\pi\)
−0.644866 + 0.764296i \(0.723087\pi\)
\(168\) 0 0
\(169\) −17.2430 −1.32639
\(170\) 0 0
\(171\) 1.88770 + 3.88459i 0.144356 + 0.297062i
\(172\) 0 0
\(173\) 6.35319i 0.483024i 0.970398 + 0.241512i \(0.0776434\pi\)
−0.970398 + 0.241512i \(0.922357\pi\)
\(174\) 0 0
\(175\) 1.63630i 0.123693i
\(176\) 0 0
\(177\) −3.25350 −0.244548
\(178\) 0 0
\(179\) −17.6854 −1.32187 −0.660936 0.750442i \(-0.729840\pi\)
−0.660936 + 0.750442i \(0.729840\pi\)
\(180\) 0 0
\(181\) 24.3156i 1.80736i 0.428205 + 0.903682i \(0.359146\pi\)
−0.428205 + 0.903682i \(0.640854\pi\)
\(182\) 0 0
\(183\) 13.3271 0.985168
\(184\) 0 0
\(185\) 9.45218i 0.694938i
\(186\) 0 0
\(187\) 0.747022i 0.0546276i
\(188\) 0 0
\(189\) 16.1582i 1.17534i
\(190\) 0 0
\(191\) 4.41500i 0.319458i 0.987161 + 0.159729i \(0.0510621\pi\)
−0.987161 + 0.159729i \(0.948938\pi\)
\(192\) 0 0
\(193\) 19.3885i 1.39561i 0.716287 + 0.697806i \(0.245840\pi\)
−0.716287 + 0.697806i \(0.754160\pi\)
\(194\) 0 0
\(195\) 18.4017i 1.31778i
\(196\) 0 0
\(197\) −18.6160 −1.32634 −0.663169 0.748470i \(-0.730789\pi\)
−0.663169 + 0.748470i \(0.730789\pi\)
\(198\) 0 0
\(199\) 4.43761i 0.314574i 0.987553 + 0.157287i \(0.0502748\pi\)
−0.987553 + 0.157287i \(0.949725\pi\)
\(200\) 0 0
\(201\) 17.0556 1.20301
\(202\) 0 0
\(203\) −6.62591 −0.465048
\(204\) 0 0
\(205\) 23.3835i 1.63318i
\(206\) 0 0
\(207\) 6.83745i 0.475236i
\(208\) 0 0
\(209\) 3.92051 1.90515i 0.271187 0.131782i
\(210\) 0 0
\(211\) 8.71830 0.600193 0.300096 0.953909i \(-0.402981\pi\)
0.300096 + 0.953909i \(0.402981\pi\)
\(212\) 0 0
\(213\) 0.172661 0.0118305
\(214\) 0 0
\(215\) 6.12862i 0.417968i
\(216\) 0 0
\(217\) 3.41323i 0.231705i
\(218\) 0 0
\(219\) 7.25768 0.490429
\(220\) 0 0
\(221\) 4.10815i 0.276344i
\(222\) 0 0
\(223\) 1.12773 0.0755184 0.0377592 0.999287i \(-0.487978\pi\)
0.0377592 + 0.999287i \(0.487978\pi\)
\(224\) 0 0
\(225\) −0.567601 −0.0378401
\(226\) 0 0
\(227\) −13.7541 −0.912895 −0.456447 0.889750i \(-0.650878\pi\)
−0.456447 + 0.889750i \(0.650878\pi\)
\(228\) 0 0
\(229\) 15.1290 0.999751 0.499876 0.866097i \(-0.333379\pi\)
0.499876 + 0.866097i \(0.333379\pi\)
\(230\) 0 0
\(231\) 4.04883 0.266394
\(232\) 0 0
\(233\) 9.86111 0.646023 0.323012 0.946395i \(-0.395305\pi\)
0.323012 + 0.946395i \(0.395305\pi\)
\(234\) 0 0
\(235\) 19.0278i 1.24124i
\(236\) 0 0
\(237\) −6.33118 −0.411254
\(238\) 0 0
\(239\) 4.98810i 0.322654i 0.986901 + 0.161327i \(0.0515773\pi\)
−0.986901 + 0.161327i \(0.948423\pi\)
\(240\) 0 0
\(241\) 5.56526i 0.358490i 0.983804 + 0.179245i \(0.0573655\pi\)
−0.983804 + 0.179245i \(0.942634\pi\)
\(242\) 0 0
\(243\) −9.81837 −0.629849
\(244\) 0 0
\(245\) −2.73639 −0.174821
\(246\) 0 0
\(247\) −21.5603 + 10.4771i −1.37185 + 0.666645i
\(248\) 0 0
\(249\) 14.9757i 0.949045i
\(250\) 0 0
\(251\) 1.63885i 0.103443i −0.998662 0.0517215i \(-0.983529\pi\)
0.998662 0.0517215i \(-0.0164708\pi\)
\(252\) 0 0
\(253\) −6.90067 −0.433842
\(254\) 0 0
\(255\) 2.49965 0.156534
\(256\) 0 0
\(257\) 3.23102i 0.201546i 0.994909 + 0.100773i \(0.0321315\pi\)
−0.994909 + 0.100773i \(0.967868\pi\)
\(258\) 0 0
\(259\) 11.4371 0.710667
\(260\) 0 0
\(261\) 2.29840i 0.142268i
\(262\) 0 0
\(263\) 16.6565i 1.02709i −0.858064 0.513543i \(-0.828333\pi\)
0.858064 0.513543i \(-0.171667\pi\)
\(264\) 0 0
\(265\) 4.27677i 0.262720i
\(266\) 0 0
\(267\) 7.60665i 0.465519i
\(268\) 0 0
\(269\) 26.1317i 1.59328i −0.604454 0.796640i \(-0.706609\pi\)
0.604454 0.796640i \(-0.293391\pi\)
\(270\) 0 0
\(271\) 1.63390i 0.0992523i −0.998768 0.0496262i \(-0.984197\pi\)
0.998768 0.0496262i \(-0.0158030\pi\)
\(272\) 0 0
\(273\) −22.2660 −1.34760
\(274\) 0 0
\(275\) 0.572849i 0.0345441i
\(276\) 0 0
\(277\) 4.17565 0.250890 0.125445 0.992101i \(-0.459964\pi\)
0.125445 + 0.992101i \(0.459964\pi\)
\(278\) 0 0
\(279\) 1.18399 0.0708834
\(280\) 0 0
\(281\) 14.9726i 0.893192i −0.894736 0.446596i \(-0.852636\pi\)
0.894736 0.446596i \(-0.147364\pi\)
\(282\) 0 0
\(283\) 0.171136i 0.0101730i −0.999987 0.00508650i \(-0.998381\pi\)
0.999987 0.00508650i \(-0.00161909\pi\)
\(284\) 0 0
\(285\) 6.37494 + 13.1186i 0.377619 + 0.777081i
\(286\) 0 0
\(287\) −28.2939 −1.67014
\(288\) 0 0
\(289\) −16.4420 −0.967174
\(290\) 0 0
\(291\) 14.6983i 0.861629i
\(292\) 0 0
\(293\) 21.0522i 1.22988i 0.788573 + 0.614941i \(0.210820\pi\)
−0.788573 + 0.614941i \(0.789180\pi\)
\(294\) 0 0
\(295\) 5.41853 0.315479
\(296\) 0 0
\(297\) 5.65681i 0.328241i
\(298\) 0 0
\(299\) 37.9493 2.19467
\(300\) 0 0
\(301\) −7.41560 −0.427428
\(302\) 0 0
\(303\) 7.78247 0.447091
\(304\) 0 0
\(305\) −22.1956 −1.27092
\(306\) 0 0
\(307\) −6.42551 −0.366723 −0.183361 0.983046i \(-0.558698\pi\)
−0.183361 + 0.983046i \(0.558698\pi\)
\(308\) 0 0
\(309\) 23.0354 1.31044
\(310\) 0 0
\(311\) 30.6582i 1.73847i 0.494401 + 0.869234i \(0.335388\pi\)
−0.494401 + 0.869234i \(0.664612\pi\)
\(312\) 0 0
\(313\) 20.0875 1.13541 0.567707 0.823231i \(-0.307831\pi\)
0.567707 + 0.823231i \(0.307831\pi\)
\(314\) 0 0
\(315\) 6.68134i 0.376451i
\(316\) 0 0
\(317\) 11.2341i 0.630971i 0.948930 + 0.315485i \(0.102167\pi\)
−0.948930 + 0.315485i \(0.897833\pi\)
\(318\) 0 0
\(319\) −2.31965 −0.129876
\(320\) 0 0
\(321\) −12.5026 −0.697826
\(322\) 0 0
\(323\) 1.42319 + 2.92871i 0.0791885 + 0.162958i
\(324\) 0 0
\(325\) 3.15031i 0.174748i
\(326\) 0 0
\(327\) 6.64607i 0.367528i
\(328\) 0 0
\(329\) −23.0235 −1.26933
\(330\) 0 0
\(331\) −12.7640 −0.701571 −0.350785 0.936456i \(-0.614085\pi\)
−0.350785 + 0.936456i \(0.614085\pi\)
\(332\) 0 0
\(333\) 3.96731i 0.217407i
\(334\) 0 0
\(335\) −28.4052 −1.55194
\(336\) 0 0
\(337\) 34.0522i 1.85494i −0.373893 0.927472i \(-0.621977\pi\)
0.373893 0.927472i \(-0.378023\pi\)
\(338\) 0 0
\(339\) 15.9701i 0.867376i
\(340\) 0 0
\(341\) 1.19493i 0.0647093i
\(342\) 0 0
\(343\) 16.6839i 0.900848i
\(344\) 0 0
\(345\) 23.0907i 1.24316i
\(346\) 0 0
\(347\) 3.89622i 0.209160i 0.994516 + 0.104580i \(0.0333498\pi\)
−0.994516 + 0.104580i \(0.966650\pi\)
\(348\) 0 0
\(349\) 11.7263 0.627697 0.313849 0.949473i \(-0.398382\pi\)
0.313849 + 0.949473i \(0.398382\pi\)
\(350\) 0 0
\(351\) 31.1089i 1.66047i
\(352\) 0 0
\(353\) −18.8518 −1.00338 −0.501691 0.865047i \(-0.667288\pi\)
−0.501691 + 0.865047i \(0.667288\pi\)
\(354\) 0 0
\(355\) −0.287558 −0.0152620
\(356\) 0 0
\(357\) 3.02457i 0.160077i
\(358\) 0 0
\(359\) 24.2241i 1.27850i −0.768999 0.639250i \(-0.779245\pi\)
0.768999 0.639250i \(-0.220755\pi\)
\(360\) 0 0
\(361\) −11.7408 + 14.9383i −0.617936 + 0.786229i
\(362\) 0 0
\(363\) 1.41745 0.0743968
\(364\) 0 0
\(365\) −12.0873 −0.632678
\(366\) 0 0
\(367\) 26.8522i 1.40167i 0.713322 + 0.700837i \(0.247190\pi\)
−0.713322 + 0.700837i \(0.752810\pi\)
\(368\) 0 0
\(369\) 9.81463i 0.510929i
\(370\) 0 0
\(371\) −5.17488 −0.268666
\(372\) 0 0
\(373\) 22.6582i 1.17320i −0.809877 0.586600i \(-0.800466\pi\)
0.809877 0.586600i \(-0.199534\pi\)
\(374\) 0 0
\(375\) 14.8139 0.764988
\(376\) 0 0
\(377\) 12.7566 0.657000
\(378\) 0 0
\(379\) −22.6082 −1.16131 −0.580654 0.814151i \(-0.697203\pi\)
−0.580654 + 0.814151i \(0.697203\pi\)
\(380\) 0 0
\(381\) 22.4451 1.14990
\(382\) 0 0
\(383\) 12.6404 0.645893 0.322947 0.946417i \(-0.395327\pi\)
0.322947 + 0.946417i \(0.395327\pi\)
\(384\) 0 0
\(385\) −6.74312 −0.343661
\(386\) 0 0
\(387\) 2.57233i 0.130759i
\(388\) 0 0
\(389\) −12.6466 −0.641206 −0.320603 0.947214i \(-0.603886\pi\)
−0.320603 + 0.947214i \(0.603886\pi\)
\(390\) 0 0
\(391\) 5.15495i 0.260697i
\(392\) 0 0
\(393\) 0.429801i 0.0216806i
\(394\) 0 0
\(395\) 10.5443 0.530539
\(396\) 0 0
\(397\) 27.2631 1.36830 0.684149 0.729342i \(-0.260174\pi\)
0.684149 + 0.729342i \(0.260174\pi\)
\(398\) 0 0
\(399\) −15.8735 + 7.71365i −0.794668 + 0.386165i
\(400\) 0 0
\(401\) 15.9672i 0.797363i 0.917089 + 0.398682i \(0.130532\pi\)
−0.917089 + 0.398682i \(0.869468\pi\)
\(402\) 0 0
\(403\) 6.57138i 0.327344i
\(404\) 0 0
\(405\) −11.9114 −0.591881
\(406\) 0 0
\(407\) 4.00399 0.198471
\(408\) 0 0
\(409\) 26.6851i 1.31949i 0.751488 + 0.659747i \(0.229337\pi\)
−0.751488 + 0.659747i \(0.770663\pi\)
\(410\) 0 0
\(411\) −18.3602 −0.905640
\(412\) 0 0
\(413\) 6.55640i 0.322619i
\(414\) 0 0
\(415\) 24.9412i 1.22432i
\(416\) 0 0
\(417\) 2.51718i 0.123267i
\(418\) 0 0
\(419\) 5.48659i 0.268038i 0.990979 + 0.134019i \(0.0427883\pi\)
−0.990979 + 0.134019i \(0.957212\pi\)
\(420\) 0 0
\(421\) 33.1252i 1.61442i 0.590264 + 0.807211i \(0.299024\pi\)
−0.590264 + 0.807211i \(0.700976\pi\)
\(422\) 0 0
\(423\) 7.98643i 0.388313i
\(424\) 0 0
\(425\) −0.427931 −0.0207577
\(426\) 0 0
\(427\) 26.8566i 1.29968i
\(428\) 0 0
\(429\) −7.79507 −0.376350
\(430\) 0 0
\(431\) 7.09401 0.341706 0.170853 0.985296i \(-0.445348\pi\)
0.170853 + 0.985296i \(0.445348\pi\)
\(432\) 0 0
\(433\) 0.765804i 0.0368022i 0.999831 + 0.0184011i \(0.00585759\pi\)
−0.999831 + 0.0184011i \(0.994142\pi\)
\(434\) 0 0
\(435\) 7.76192i 0.372156i
\(436\) 0 0
\(437\) 27.0542 13.1468i 1.29418 0.628899i
\(438\) 0 0
\(439\) −22.7411 −1.08537 −0.542687 0.839935i \(-0.682593\pi\)
−0.542687 + 0.839935i \(0.682593\pi\)
\(440\) 0 0
\(441\) 1.14853 0.0546918
\(442\) 0 0
\(443\) 14.0871i 0.669300i 0.942343 + 0.334650i \(0.108618\pi\)
−0.942343 + 0.334650i \(0.891382\pi\)
\(444\) 0 0
\(445\) 12.6685i 0.600544i
\(446\) 0 0
\(447\) 12.6123 0.596543
\(448\) 0 0
\(449\) 18.2937i 0.863332i −0.902034 0.431666i \(-0.857926\pi\)
0.902034 0.431666i \(-0.142074\pi\)
\(450\) 0 0
\(451\) −9.90538 −0.466426
\(452\) 0 0
\(453\) 22.4175 1.05327
\(454\) 0 0
\(455\) 37.0829 1.73847
\(456\) 0 0
\(457\) −17.8553 −0.835238 −0.417619 0.908622i \(-0.637135\pi\)
−0.417619 + 0.908622i \(0.637135\pi\)
\(458\) 0 0
\(459\) −4.22576 −0.197242
\(460\) 0 0
\(461\) 22.0096 1.02509 0.512545 0.858660i \(-0.328703\pi\)
0.512545 + 0.858660i \(0.328703\pi\)
\(462\) 0 0
\(463\) 21.2764i 0.988799i −0.869235 0.494399i \(-0.835388\pi\)
0.869235 0.494399i \(-0.164612\pi\)
\(464\) 0 0
\(465\) 3.99843 0.185423
\(466\) 0 0
\(467\) 4.06408i 0.188063i 0.995569 + 0.0940316i \(0.0299755\pi\)
−0.995569 + 0.0940316i \(0.970025\pi\)
\(468\) 0 0
\(469\) 34.3702i 1.58707i
\(470\) 0 0
\(471\) −28.9644 −1.33461
\(472\) 0 0
\(473\) −2.59611 −0.119369
\(474\) 0 0
\(475\) −1.09137 2.24586i −0.0500753 0.103047i
\(476\) 0 0
\(477\) 1.79507i 0.0821904i
\(478\) 0 0
\(479\) 28.4469i 1.29977i −0.760032 0.649886i \(-0.774817\pi\)
0.760032 0.649886i \(-0.225183\pi\)
\(480\) 0 0
\(481\) −22.0194 −1.00400
\(482\) 0 0
\(483\) 27.9397 1.27130
\(484\) 0 0
\(485\) 24.4792i 1.11154i
\(486\) 0 0
\(487\) 20.5512 0.931265 0.465632 0.884978i \(-0.345827\pi\)
0.465632 + 0.884978i \(0.345827\pi\)
\(488\) 0 0
\(489\) 9.78188i 0.442352i
\(490\) 0 0
\(491\) 2.32477i 0.104915i −0.998623 0.0524577i \(-0.983295\pi\)
0.998623 0.0524577i \(-0.0167055\pi\)
\(492\) 0 0
\(493\) 1.73283i 0.0780428i
\(494\) 0 0
\(495\) 2.33906i 0.105133i
\(496\) 0 0
\(497\) 0.347943i 0.0156074i
\(498\) 0 0
\(499\) 3.39830i 0.152129i 0.997103 + 0.0760645i \(0.0242355\pi\)
−0.997103 + 0.0760645i \(0.975765\pi\)
\(500\) 0 0
\(501\) 23.6246 1.05547
\(502\) 0 0
\(503\) 30.1892i 1.34607i −0.739610 0.673035i \(-0.764990\pi\)
0.739610 0.673035i \(-0.235010\pi\)
\(504\) 0 0
\(505\) −12.9613 −0.576770
\(506\) 0 0
\(507\) 24.4411 1.08547
\(508\) 0 0
\(509\) 13.0179i 0.577010i 0.957478 + 0.288505i \(0.0931582\pi\)
−0.957478 + 0.288505i \(0.906842\pi\)
\(510\) 0 0
\(511\) 14.6256i 0.646997i
\(512\) 0 0
\(513\) −10.7771 22.1776i −0.475820 0.979164i
\(514\) 0 0
\(515\) −38.3642 −1.69053
\(516\) 0 0
\(517\) −8.06027 −0.354490
\(518\) 0 0
\(519\) 9.00533i 0.395290i
\(520\) 0 0
\(521\) 3.20269i 0.140312i 0.997536 + 0.0701562i \(0.0223498\pi\)
−0.997536 + 0.0701562i \(0.977650\pi\)
\(522\) 0 0
\(523\) 24.6892 1.07958 0.539791 0.841799i \(-0.318503\pi\)
0.539791 + 0.841799i \(0.318503\pi\)
\(524\) 0 0
\(525\) 2.31937i 0.101226i
\(526\) 0 0
\(527\) 0.892641 0.0388841
\(528\) 0 0
\(529\) −24.6193 −1.07040
\(530\) 0 0
\(531\) −2.27429 −0.0986958
\(532\) 0 0
\(533\) 54.4733 2.35950
\(534\) 0 0
\(535\) 20.8224 0.900230
\(536\) 0 0
\(537\) 25.0682 1.08177
\(538\) 0 0
\(539\) 1.15915i 0.0499280i
\(540\) 0 0
\(541\) −10.1904 −0.438120 −0.219060 0.975711i \(-0.570299\pi\)
−0.219060 + 0.975711i \(0.570299\pi\)
\(542\) 0 0
\(543\) 34.4661i 1.47908i
\(544\) 0 0
\(545\) 11.0687i 0.474130i
\(546\) 0 0
\(547\) 30.2689 1.29421 0.647103 0.762403i \(-0.275980\pi\)
0.647103 + 0.762403i \(0.275980\pi\)
\(548\) 0 0
\(549\) 9.31604 0.397599
\(550\) 0 0
\(551\) 9.09422 4.41930i 0.387427 0.188268i
\(552\) 0 0
\(553\) 12.7585i 0.542546i
\(554\) 0 0
\(555\) 13.3980i 0.568713i
\(556\) 0 0
\(557\) 0.0246473 0.00104434 0.000522169 1.00000i \(-0.499834\pi\)
0.000522169 1.00000i \(0.499834\pi\)
\(558\) 0 0
\(559\) 14.2770 0.603852
\(560\) 0 0
\(561\) 1.05887i 0.0447053i
\(562\) 0 0
\(563\) 24.7494 1.04306 0.521531 0.853233i \(-0.325361\pi\)
0.521531 + 0.853233i \(0.325361\pi\)
\(564\) 0 0
\(565\) 26.5974i 1.11896i
\(566\) 0 0
\(567\) 14.4127i 0.605277i
\(568\) 0 0
\(569\) 2.57504i 0.107951i 0.998542 + 0.0539756i \(0.0171893\pi\)
−0.998542 + 0.0539756i \(0.982811\pi\)
\(570\) 0 0
\(571\) 3.52277i 0.147423i 0.997280 + 0.0737117i \(0.0234845\pi\)
−0.997280 + 0.0737117i \(0.976516\pi\)
\(572\) 0 0
\(573\) 6.25804i 0.261433i
\(574\) 0 0
\(575\) 3.95304i 0.164853i
\(576\) 0 0
\(577\) −13.0816 −0.544594 −0.272297 0.962213i \(-0.587783\pi\)
−0.272297 + 0.962213i \(0.587783\pi\)
\(578\) 0 0
\(579\) 27.4821i 1.14212i
\(580\) 0 0
\(581\) −30.1788 −1.25203
\(582\) 0 0
\(583\) −1.81166 −0.0750315
\(584\) 0 0
\(585\) 12.8634i 0.531834i
\(586\) 0 0
\(587\) 26.5279i 1.09492i −0.836831 0.547461i \(-0.815595\pi\)
0.836831 0.547461i \(-0.184405\pi\)
\(588\) 0 0
\(589\) 2.27653 + 4.68475i 0.0938028 + 0.193032i
\(590\) 0 0
\(591\) 26.3873 1.08543
\(592\) 0 0
\(593\) 7.60973 0.312494 0.156247 0.987718i \(-0.450060\pi\)
0.156247 + 0.987718i \(0.450060\pi\)
\(594\) 0 0
\(595\) 5.03726i 0.206507i
\(596\) 0 0
\(597\) 6.29009i 0.257436i
\(598\) 0 0
\(599\) 36.6027 1.49555 0.747773 0.663954i \(-0.231123\pi\)
0.747773 + 0.663954i \(0.231123\pi\)
\(600\) 0 0
\(601\) 16.3639i 0.667497i −0.942662 0.333748i \(-0.891686\pi\)
0.942662 0.333748i \(-0.108314\pi\)
\(602\) 0 0
\(603\) 11.9224 0.485516
\(604\) 0 0
\(605\) −2.36069 −0.0959756
\(606\) 0 0
\(607\) 46.9997 1.90766 0.953830 0.300347i \(-0.0971023\pi\)
0.953830 + 0.300347i \(0.0971023\pi\)
\(608\) 0 0
\(609\) 9.39189 0.380579
\(610\) 0 0
\(611\) 44.3264 1.79325
\(612\) 0 0
\(613\) 47.3632 1.91298 0.956490 0.291764i \(-0.0942422\pi\)
0.956490 + 0.291764i \(0.0942422\pi\)
\(614\) 0 0
\(615\) 33.1449i 1.33653i
\(616\) 0 0
\(617\) 20.8329 0.838701 0.419350 0.907825i \(-0.362258\pi\)
0.419350 + 0.907825i \(0.362258\pi\)
\(618\) 0 0
\(619\) 23.1478i 0.930389i 0.885209 + 0.465194i \(0.154016\pi\)
−0.885209 + 0.465194i \(0.845984\pi\)
\(620\) 0 0
\(621\) 39.0358i 1.56645i
\(622\) 0 0
\(623\) 15.3288 0.614136
\(624\) 0 0
\(625\) −27.5361 −1.10144
\(626\) 0 0
\(627\) −5.55712 + 2.70046i −0.221930 + 0.107846i
\(628\) 0 0
\(629\) 2.99107i 0.119262i
\(630\) 0 0
\(631\) 15.2532i 0.607219i 0.952797 + 0.303609i \(0.0981918\pi\)
−0.952797 + 0.303609i \(0.901808\pi\)
\(632\) 0 0
\(633\) −12.3578 −0.491177
\(634\) 0 0
\(635\) −37.3812 −1.48343
\(636\) 0 0
\(637\) 6.37458i 0.252570i
\(638\) 0 0
\(639\) 0.120695 0.00477462
\(640\) 0 0
\(641\) 37.6876i 1.48857i 0.667862 + 0.744285i \(0.267210\pi\)
−0.667862 + 0.744285i \(0.732790\pi\)
\(642\) 0 0
\(643\) 7.12184i 0.280858i −0.990091 0.140429i \(-0.955152\pi\)
0.990091 0.140429i \(-0.0448482\pi\)
\(644\) 0 0
\(645\) 8.68700i 0.342050i
\(646\) 0 0
\(647\) 0.654870i 0.0257456i −0.999917 0.0128728i \(-0.995902\pi\)
0.999917 0.0128728i \(-0.00409765\pi\)
\(648\) 0 0
\(649\) 2.29532i 0.0900992i
\(650\) 0 0
\(651\) 4.83808i 0.189619i
\(652\) 0 0
\(653\) 43.6710 1.70898 0.854489 0.519469i \(-0.173870\pi\)
0.854489 + 0.519469i \(0.173870\pi\)
\(654\) 0 0
\(655\) 0.715812i 0.0279691i
\(656\) 0 0
\(657\) 5.07333 0.197930
\(658\) 0 0
\(659\) −36.2529 −1.41221 −0.706106 0.708106i \(-0.749550\pi\)
−0.706106 + 0.708106i \(0.749550\pi\)
\(660\) 0 0
\(661\) 12.0067i 0.467008i 0.972356 + 0.233504i \(0.0750192\pi\)
−0.972356 + 0.233504i \(0.924981\pi\)
\(662\) 0 0
\(663\) 5.82309i 0.226150i
\(664\) 0 0
\(665\) 26.4365 12.8467i 1.02516 0.498173i
\(666\) 0 0
\(667\) −16.0072 −0.619800
\(668\) 0 0
\(669\) −1.59850 −0.0618016
\(670\) 0 0
\(671\) 9.40218i 0.362967i
\(672\) 0 0
\(673\) 37.3402i 1.43936i −0.694307 0.719679i \(-0.744289\pi\)
0.694307 0.719679i \(-0.255711\pi\)
\(674\) 0 0
\(675\) 3.24050 0.124727
\(676\) 0 0
\(677\) 41.2570i 1.58563i 0.609460 + 0.792817i \(0.291386\pi\)
−0.609460 + 0.792817i \(0.708614\pi\)
\(678\) 0 0
\(679\) 29.6198 1.13670
\(680\) 0 0
\(681\) 19.4958 0.747081
\(682\) 0 0
\(683\) −40.7405 −1.55889 −0.779447 0.626469i \(-0.784500\pi\)
−0.779447 + 0.626469i \(0.784500\pi\)
\(684\) 0 0
\(685\) 30.5779 1.16832
\(686\) 0 0
\(687\) −21.4446 −0.818161
\(688\) 0 0
\(689\) 9.96301 0.379560
\(690\) 0 0
\(691\) 3.47735i 0.132285i 0.997810 + 0.0661424i \(0.0210692\pi\)
−0.997810 + 0.0661424i \(0.978931\pi\)
\(692\) 0 0
\(693\) 2.83025 0.107512
\(694\) 0 0
\(695\) 4.19224i 0.159021i
\(696\) 0 0
\(697\) 7.39953i 0.280277i
\(698\) 0 0
\(699\) −13.9776 −0.528682
\(700\) 0 0
\(701\) 39.9299 1.50813 0.754066 0.656799i \(-0.228090\pi\)
0.754066 + 0.656799i \(0.228090\pi\)
\(702\) 0 0
\(703\) −15.6977 + 7.62822i −0.592050 + 0.287704i
\(704\) 0 0
\(705\) 26.9709i 1.01578i
\(706\) 0 0
\(707\) 15.6831i 0.589824i
\(708\) 0 0
\(709\) 31.1202 1.16874 0.584371 0.811486i \(-0.301341\pi\)
0.584371 + 0.811486i \(0.301341\pi\)
\(710\) 0 0
\(711\) −4.42568 −0.165976
\(712\) 0 0
\(713\) 8.24584i 0.308809i
\(714\) 0 0
\(715\) 12.9823 0.485510
\(716\) 0 0
\(717\) 7.07038i 0.264048i
\(718\) 0 0
\(719\) 40.6566i 1.51624i −0.652118 0.758118i \(-0.726119\pi\)
0.652118 0.758118i \(-0.273881\pi\)
\(720\) 0 0
\(721\) 46.4205i 1.72879i
\(722\) 0 0
\(723\) 7.88848i 0.293376i
\(724\) 0 0
\(725\) 1.32881i 0.0493508i
\(726\) 0 0
\(727\) 24.1161i 0.894417i 0.894430 + 0.447208i \(0.147582\pi\)
−0.894430 + 0.447208i \(0.852418\pi\)
\(728\) 0 0
\(729\) 29.0542 1.07608
\(730\) 0 0
\(731\) 1.93935i 0.0717296i
\(732\) 0 0
\(733\) 28.1403 1.03939 0.519693 0.854353i \(-0.326046\pi\)
0.519693 + 0.854353i \(0.326046\pi\)
\(734\) 0 0
\(735\) 3.87869 0.143068
\(736\) 0 0
\(737\) 12.0326i 0.443227i
\(738\) 0 0
\(739\) 45.6353i 1.67872i −0.543575 0.839361i \(-0.682930\pi\)
0.543575 0.839361i \(-0.317070\pi\)
\(740\) 0 0
\(741\) 30.5607 14.8508i 1.12267 0.545558i
\(742\) 0 0
\(743\) −8.05299 −0.295436 −0.147718 0.989030i \(-0.547193\pi\)
−0.147718 + 0.989030i \(0.547193\pi\)
\(744\) 0 0
\(745\) −21.0052 −0.769571
\(746\) 0 0
\(747\) 10.4684i 0.383020i
\(748\) 0 0
\(749\) 25.1950i 0.920605i
\(750\) 0 0
\(751\) 4.35756 0.159010 0.0795048 0.996834i \(-0.474666\pi\)
0.0795048 + 0.996834i \(0.474666\pi\)
\(752\) 0 0
\(753\) 2.32298i 0.0846541i
\(754\) 0 0
\(755\) −37.3352 −1.35877
\(756\) 0 0
\(757\) 5.01051 0.182110 0.0910550 0.995846i \(-0.470976\pi\)
0.0910550 + 0.995846i \(0.470976\pi\)
\(758\) 0 0
\(759\) 9.78135 0.355041
\(760\) 0 0
\(761\) 37.9954 1.37733 0.688666 0.725079i \(-0.258197\pi\)
0.688666 + 0.725079i \(0.258197\pi\)
\(762\) 0 0
\(763\) 13.3931 0.484861
\(764\) 0 0
\(765\) 1.74733 0.0631748
\(766\) 0 0
\(767\) 12.6228i 0.455783i
\(768\) 0 0
\(769\) −36.2100 −1.30577 −0.652883 0.757459i \(-0.726441\pi\)
−0.652883 + 0.757459i \(0.726441\pi\)
\(770\) 0 0
\(771\) 4.57981i 0.164938i
\(772\) 0 0
\(773\) 33.0530i 1.18883i 0.804158 + 0.594416i \(0.202617\pi\)
−0.804158 + 0.594416i \(0.797383\pi\)
\(774\) 0 0
\(775\) −0.684516 −0.0245886
\(776\) 0 0
\(777\) −16.2115 −0.581584
\(778\) 0 0
\(779\) 38.8341 18.8713i 1.39138 0.676133i
\(780\) 0 0
\(781\) 0.121811i 0.00435874i
\(782\) 0 0
\(783\) 13.1218i 0.468936i
\(784\) 0 0
\(785\) 48.2387 1.72171
\(786\) 0 0
\(787\) 12.2269 0.435842 0.217921 0.975966i \(-0.430073\pi\)
0.217921 + 0.975966i \(0.430073\pi\)
\(788\) 0 0
\(789\) 23.6098i 0.840531i
\(790\) 0 0
\(791\) −32.1827 −1.14428
\(792\) 0 0
\(793\) 51.7060i 1.83614i
\(794\) 0 0
\(795\) 6.06211i 0.215001i
\(796\) 0 0
\(797\) 38.6821i 1.37019i −0.728454 0.685094i \(-0.759761\pi\)
0.728454 0.685094i \(-0.240239\pi\)
\(798\) 0 0
\(799\) 6.02120i 0.213015i
\(800\) 0 0
\(801\) 5.31727i 0.187876i
\(802\) 0 0
\(803\) 5.12024i 0.180689i
\(804\) 0 0
\(805\) −46.5321 −1.64004
\(806\) 0 0
\(807\) 37.0404i 1.30388i
\(808\) 0 0
\(809\) −52.6496 −1.85106 −0.925531 0.378671i \(-0.876381\pi\)
−0.925531 + 0.378671i \(0.876381\pi\)
\(810\) 0 0
\(811\) 4.02553 0.141355 0.0706777 0.997499i \(-0.477484\pi\)
0.0706777 + 0.997499i \(0.477484\pi\)
\(812\) 0 0
\(813\) 2.31597i 0.0812246i
\(814\) 0 0
\(815\) 16.2912i 0.570656i
\(816\) 0 0
\(817\) 10.1781 4.94600i 0.356086 0.173038i
\(818\) 0 0
\(819\) −15.5646 −0.543871
\(820\) 0 0
\(821\) −30.5358 −1.06571 −0.532854 0.846207i \(-0.678880\pi\)
−0.532854 + 0.846207i \(0.678880\pi\)
\(822\) 0 0
\(823\) 8.16634i 0.284661i 0.989819 + 0.142330i \(0.0454596\pi\)
−0.989819 + 0.142330i \(0.954540\pi\)
\(824\) 0 0
\(825\) 0.811984i 0.0282697i
\(826\) 0 0
\(827\) 12.6480 0.439813 0.219907 0.975521i \(-0.429425\pi\)
0.219907 + 0.975521i \(0.429425\pi\)
\(828\) 0 0
\(829\) 13.4740i 0.467970i 0.972240 + 0.233985i \(0.0751766\pi\)
−0.972240 + 0.233985i \(0.924823\pi\)
\(830\) 0 0
\(831\) −5.91876 −0.205320
\(832\) 0 0
\(833\) 0.865909 0.0300020
\(834\) 0 0
\(835\) −39.3456 −1.36161
\(836\) 0 0
\(837\) −6.75951 −0.233643
\(838\) 0 0
\(839\) −18.3291 −0.632791 −0.316396 0.948627i \(-0.602473\pi\)
−0.316396 + 0.948627i \(0.602473\pi\)
\(840\) 0 0
\(841\) 23.6192 0.814455
\(842\) 0 0
\(843\) 21.2229i 0.730956i
\(844\) 0 0
\(845\) −40.7055 −1.40031
\(846\) 0 0
\(847\) 2.85642i 0.0981478i
\(848\) 0 0
\(849\) 0.242577i 0.00832522i
\(850\) 0 0
\(851\) 27.6303 0.947153
\(852\) 0 0
\(853\) −40.2794 −1.37914 −0.689569 0.724220i \(-0.742200\pi\)
−0.689569 + 0.724220i \(0.742200\pi\)
\(854\) 0 0
\(855\) 4.45627 + 9.17031i 0.152401 + 0.313618i
\(856\) 0 0
\(857\) 13.5459i 0.462719i −0.972868 0.231359i \(-0.925683\pi\)
0.972868 0.231359i \(-0.0743173\pi\)
\(858\) 0 0
\(859\) 32.8595i 1.12115i −0.828103 0.560575i \(-0.810580\pi\)
0.828103 0.560575i \(-0.189420\pi\)
\(860\) 0 0
\(861\) 40.1052 1.36678
\(862\) 0 0
\(863\) 30.4320 1.03592 0.517958 0.855406i \(-0.326692\pi\)
0.517958 + 0.855406i \(0.326692\pi\)
\(864\) 0 0
\(865\) 14.9979i 0.509944i
\(866\) 0 0
\(867\) 23.3056 0.791501
\(868\) 0 0
\(869\) 4.46660i 0.151519i
\(870\) 0 0
\(871\) 66.1717i 2.24214i
\(872\) 0 0
\(873\) 10.2745i 0.347740i
\(874\) 0 0
\(875\) 29.8528i 1.00921i
\(876\) 0 0
\(877\) 27.9946i 0.945312i −0.881247 0.472656i \(-0.843295\pi\)
0.881247 0.472656i \(-0.156705\pi\)
\(878\) 0 0
\(879\) 29.8404i 1.00649i
\(880\) 0 0
\(881\) −9.94901 −0.335191 −0.167595 0.985856i \(-0.553600\pi\)
−0.167595 + 0.985856i \(0.553600\pi\)
\(882\) 0 0
\(883\) 45.8776i 1.54390i −0.635681 0.771952i \(-0.719281\pi\)
0.635681 0.771952i \(-0.280719\pi\)
\(884\) 0 0
\(885\) −7.68050 −0.258177
\(886\) 0 0
\(887\) 42.1643 1.41574 0.707869 0.706344i \(-0.249657\pi\)
0.707869 + 0.706344i \(0.249657\pi\)
\(888\) 0 0
\(889\) 45.2311i 1.51700i
\(890\) 0 0
\(891\) 5.04572i 0.169038i
\(892\) 0 0
\(893\) 31.6004 15.3561i 1.05747 0.513871i
\(894\) 0 0
\(895\) −41.7498 −1.39554
\(896\) 0 0
\(897\) −53.7913 −1.79604
\(898\) 0 0
\(899\) 2.77183i 0.0924458i
\(900\) 0 0
\(901\) 1.35335i 0.0450867i
\(902\) 0 0
\(903\) 10.5112 0.349792
\(904\) 0 0
\(905\) 57.4015i 1.90809i
\(906\) 0 0
\(907\) −5.36055 −0.177994 −0.0889971 0.996032i \(-0.528366\pi\)
−0.0889971 + 0.996032i \(0.528366\pi\)
\(908\) 0 0
\(909\) 5.44018 0.180439
\(910\) 0 0
\(911\) −36.0968 −1.19594 −0.597970 0.801519i \(-0.704026\pi\)
−0.597970 + 0.801519i \(0.704026\pi\)
\(912\) 0 0
\(913\) −10.5652 −0.349658
\(914\) 0 0
\(915\) 31.4611 1.04007
\(916\) 0 0
\(917\) −0.866129 −0.0286021
\(918\) 0 0
\(919\) 2.82205i 0.0930907i 0.998916 + 0.0465454i \(0.0148212\pi\)
−0.998916 + 0.0465454i \(0.985179\pi\)
\(920\) 0 0
\(921\) 9.10783 0.300113
\(922\) 0 0
\(923\) 0.669883i 0.0220495i
\(924\) 0 0
\(925\) 2.29368i 0.0754159i
\(926\) 0 0
\(927\) 16.1024 0.528872
\(928\) 0 0
\(929\) 21.0460 0.690498 0.345249 0.938511i \(-0.387794\pi\)
0.345249 + 0.938511i \(0.387794\pi\)
\(930\) 0 0
\(931\) 2.20836 + 4.54445i 0.0723759 + 0.148938i
\(932\) 0 0
\(933\) 43.4565i 1.42270i
\(934\) 0 0
\(935\) 1.76349i 0.0576721i
\(936\) 0 0
\(937\) −50.8139 −1.66002 −0.830008 0.557751i \(-0.811664\pi\)
−0.830008 + 0.557751i \(0.811664\pi\)
\(938\) 0 0
\(939\) −28.4730 −0.929182
\(940\) 0 0
\(941\) 36.6197i 1.19377i 0.802327 + 0.596885i \(0.203595\pi\)
−0.802327 + 0.596885i \(0.796405\pi\)
\(942\) 0 0
\(943\) −68.3538 −2.22591
\(944\) 0 0
\(945\) 38.1446i 1.24084i
\(946\) 0 0
\(947\) 40.3778i 1.31210i −0.754717 0.656051i \(-0.772226\pi\)
0.754717 0.656051i \(-0.227774\pi\)
\(948\) 0 0
\(949\) 28.1581i 0.914050i
\(950\) 0 0
\(951\) 15.9238i 0.516364i
\(952\) 0 0
\(953\) 20.9002i 0.677022i 0.940962 + 0.338511i \(0.109923\pi\)
−0.940962 + 0.338511i \(0.890077\pi\)
\(954\) 0 0
\(955\) 10.4224i 0.337262i
\(956\) 0 0
\(957\) 3.28799 0.106286
\(958\) 0 0
\(959\) 36.9991i 1.19476i
\(960\) 0 0
\(961\) −29.5721 −0.953940
\(962\) 0 0
\(963\) −8.73966 −0.281632
\(964\) 0 0
\(965\) 45.7701i 1.47339i
\(966\) 0 0
\(967\) 28.5238i 0.917265i 0.888626 + 0.458633i \(0.151661\pi\)
−0.888626 + 0.458633i \(0.848339\pi\)
\(968\) 0 0
\(969\) −2.01730 4.15129i −0.0648050 0.133359i
\(970\) 0 0
\(971\) 42.3137 1.35791 0.678955 0.734180i \(-0.262433\pi\)
0.678955 + 0.734180i \(0.262433\pi\)
\(972\) 0 0
\(973\) −5.07259 −0.162620
\(974\) 0 0
\(975\) 4.46540i 0.143007i
\(976\) 0 0
\(977\) 10.3560i 0.331317i −0.986183 0.165659i \(-0.947025\pi\)
0.986183 0.165659i \(-0.0529750\pi\)
\(978\) 0 0
\(979\) 5.36644 0.171512
\(980\) 0 0
\(981\) 4.64580i 0.148329i
\(982\) 0 0
\(983\) 32.6772 1.04224 0.521120 0.853483i \(-0.325514\pi\)
0.521120 + 0.853483i \(0.325514\pi\)
\(984\) 0 0
\(985\) −43.9467 −1.40026
\(986\) 0 0
\(987\) 32.6347 1.03877
\(988\) 0 0
\(989\) −17.9149 −0.569662
\(990\) 0 0
\(991\) −43.3073 −1.37570 −0.687851 0.725852i \(-0.741446\pi\)
−0.687851 + 0.725852i \(0.741446\pi\)
\(992\) 0 0
\(993\) 18.0923 0.574141
\(994\) 0 0
\(995\) 10.4758i 0.332106i
\(996\) 0 0
\(997\) −48.4845 −1.53552 −0.767760 0.640738i \(-0.778628\pi\)
−0.767760 + 0.640738i \(0.778628\pi\)
\(998\) 0 0
\(999\) 22.6498i 0.716609i
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 3344.2.o.b.1519.19 64
4.3 odd 2 inner 3344.2.o.b.1519.45 yes 64
19.18 odd 2 inner 3344.2.o.b.1519.46 yes 64
76.75 even 2 inner 3344.2.o.b.1519.20 yes 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3344.2.o.b.1519.19 64 1.1 even 1 trivial
3344.2.o.b.1519.20 yes 64 76.75 even 2 inner
3344.2.o.b.1519.45 yes 64 4.3 odd 2 inner
3344.2.o.b.1519.46 yes 64 19.18 odd 2 inner