Properties

Label 1200.3.bg.j.1057.1
Level $1200$
Weight $3$
Character 1200.1057
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,3,Mod(193,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.193");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.bg (of order \(4\), degree \(2\), 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: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1057.1
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1200.1057
Dual form 1200.3.bg.j.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.22474 + 1.22474i) q^{3} +(-7.34847 - 7.34847i) q^{7} -3.00000i q^{9} +18.0000 q^{11} +(-7.34847 + 7.34847i) q^{13} +(-4.89898 - 4.89898i) q^{17} +10.0000i q^{19} +18.0000 q^{21} +(19.5959 - 19.5959i) q^{23} +(3.67423 + 3.67423i) q^{27} -22.0000 q^{31} +(-22.0454 + 22.0454i) q^{33} +(7.34847 + 7.34847i) q^{37} -18.0000i q^{39} -18.0000 q^{41} +(-29.3939 + 29.3939i) q^{43} +(-44.0908 - 44.0908i) q^{47} +59.0000i q^{49} +12.0000 q^{51} +(4.89898 - 4.89898i) q^{53} +(-12.2474 - 12.2474i) q^{57} +90.0000i q^{59} +2.00000 q^{61} +(-22.0454 + 22.0454i) q^{63} +(-44.0908 - 44.0908i) q^{67} +48.0000i q^{69} -72.0000 q^{71} +(-44.0908 + 44.0908i) q^{73} +(-132.272 - 132.272i) q^{77} -70.0000i q^{79} -9.00000 q^{81} +(-53.8888 + 53.8888i) q^{83} -90.0000i q^{89} +108.000 q^{91} +(26.9444 - 26.9444i) q^{93} +(-102.879 - 102.879i) q^{97} -54.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 72 q^{11} + 72 q^{21} - 88 q^{31} - 72 q^{41} + 48 q^{51} + 8 q^{61} - 288 q^{71} - 36 q^{81} + 432 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

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
\(3\) −1.22474 + 1.22474i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −7.34847 7.34847i −1.04978 1.04978i −0.998694 0.0510871i \(-0.983731\pi\)
−0.0510871 0.998694i \(-0.516269\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) 18.0000 1.63636 0.818182 0.574960i \(-0.194982\pi\)
0.818182 + 0.574960i \(0.194982\pi\)
\(12\) 0 0
\(13\) −7.34847 + 7.34847i −0.565267 + 0.565267i −0.930799 0.365532i \(-0.880887\pi\)
0.365532 + 0.930799i \(0.380887\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.89898 4.89898i −0.288175 0.288175i 0.548183 0.836358i \(-0.315320\pi\)
−0.836358 + 0.548183i \(0.815320\pi\)
\(18\) 0 0
\(19\) 10.0000i 0.526316i 0.964753 + 0.263158i \(0.0847640\pi\)
−0.964753 + 0.263158i \(0.915236\pi\)
\(20\) 0 0
\(21\) 18.0000 0.857143
\(22\) 0 0
\(23\) 19.5959 19.5959i 0.851996 0.851996i −0.138382 0.990379i \(-0.544190\pi\)
0.990379 + 0.138382i \(0.0441903\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.67423 + 3.67423i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −22.0000 −0.709677 −0.354839 0.934928i \(-0.615464\pi\)
−0.354839 + 0.934928i \(0.615464\pi\)
\(32\) 0 0
\(33\) −22.0454 + 22.0454i −0.668043 + 0.668043i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.34847 + 7.34847i 0.198607 + 0.198607i 0.799403 0.600795i \(-0.205149\pi\)
−0.600795 + 0.799403i \(0.705149\pi\)
\(38\) 0 0
\(39\) 18.0000i 0.461538i
\(40\) 0 0
\(41\) −18.0000 −0.439024 −0.219512 0.975610i \(-0.570447\pi\)
−0.219512 + 0.975610i \(0.570447\pi\)
\(42\) 0 0
\(43\) −29.3939 + 29.3939i −0.683579 + 0.683579i −0.960805 0.277226i \(-0.910585\pi\)
0.277226 + 0.960805i \(0.410585\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −44.0908 44.0908i −0.938102 0.938102i 0.0600905 0.998193i \(-0.480861\pi\)
−0.998193 + 0.0600905i \(0.980861\pi\)
\(48\) 0 0
\(49\) 59.0000i 1.20408i
\(50\) 0 0
\(51\) 12.0000 0.235294
\(52\) 0 0
\(53\) 4.89898 4.89898i 0.0924336 0.0924336i −0.659378 0.751812i \(-0.729180\pi\)
0.751812 + 0.659378i \(0.229180\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −12.2474 12.2474i −0.214868 0.214868i
\(58\) 0 0
\(59\) 90.0000i 1.52542i 0.646738 + 0.762712i \(0.276133\pi\)
−0.646738 + 0.762712i \(0.723867\pi\)
\(60\) 0 0
\(61\) 2.00000 0.0327869 0.0163934 0.999866i \(-0.494782\pi\)
0.0163934 + 0.999866i \(0.494782\pi\)
\(62\) 0 0
\(63\) −22.0454 + 22.0454i −0.349927 + 0.349927i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −44.0908 44.0908i −0.658072 0.658072i 0.296852 0.954924i \(-0.404063\pi\)
−0.954924 + 0.296852i \(0.904063\pi\)
\(68\) 0 0
\(69\) 48.0000i 0.695652i
\(70\) 0 0
\(71\) −72.0000 −1.01408 −0.507042 0.861921i \(-0.669261\pi\)
−0.507042 + 0.861921i \(0.669261\pi\)
\(72\) 0 0
\(73\) −44.0908 + 44.0908i −0.603984 + 0.603984i −0.941367 0.337384i \(-0.890458\pi\)
0.337384 + 0.941367i \(0.390458\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −132.272 132.272i −1.71782 1.71782i
\(78\) 0 0
\(79\) 70.0000i 0.886076i −0.896503 0.443038i \(-0.853901\pi\)
0.896503 0.443038i \(-0.146099\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) −53.8888 + 53.8888i −0.649262 + 0.649262i −0.952815 0.303552i \(-0.901827\pi\)
0.303552 + 0.952815i \(0.401827\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 90.0000i 1.01124i −0.862757 0.505618i \(-0.831265\pi\)
0.862757 0.505618i \(-0.168735\pi\)
\(90\) 0 0
\(91\) 108.000 1.18681
\(92\) 0 0
\(93\) 26.9444 26.9444i 0.289725 0.289725i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −102.879 102.879i −1.06060 1.06060i −0.998041 0.0625628i \(-0.980073\pi\)
−0.0625628 0.998041i \(-0.519927\pi\)
\(98\) 0 0
\(99\) 54.0000i 0.545455i
\(100\) 0 0
\(101\) −108.000 −1.06931 −0.534653 0.845071i \(-0.679558\pi\)
−0.534653 + 0.845071i \(0.679558\pi\)
\(102\) 0 0
\(103\) −66.1362 + 66.1362i −0.642099 + 0.642099i −0.951071 0.308972i \(-0.900015\pi\)
0.308972 + 0.951071i \(0.400015\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −44.0908 44.0908i −0.412064 0.412064i 0.470393 0.882457i \(-0.344112\pi\)
−0.882457 + 0.470393i \(0.844112\pi\)
\(108\) 0 0
\(109\) 170.000i 1.55963i −0.626008 0.779817i \(-0.715312\pi\)
0.626008 0.779817i \(-0.284688\pi\)
\(110\) 0 0
\(111\) −18.0000 −0.162162
\(112\) 0 0
\(113\) −44.0908 + 44.0908i −0.390184 + 0.390184i −0.874753 0.484569i \(-0.838976\pi\)
0.484569 + 0.874753i \(0.338976\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 22.0454 + 22.0454i 0.188422 + 0.188422i
\(118\) 0 0
\(119\) 72.0000i 0.605042i
\(120\) 0 0
\(121\) 203.000 1.67769
\(122\) 0 0
\(123\) 22.0454 22.0454i 0.179231 0.179231i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −7.34847 7.34847i −0.0578620 0.0578620i 0.677584 0.735446i \(-0.263027\pi\)
−0.735446 + 0.677584i \(0.763027\pi\)
\(128\) 0 0
\(129\) 72.0000i 0.558140i
\(130\) 0 0
\(131\) 18.0000 0.137405 0.0687023 0.997637i \(-0.478114\pi\)
0.0687023 + 0.997637i \(0.478114\pi\)
\(132\) 0 0
\(133\) 73.4847 73.4847i 0.552516 0.552516i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 142.070 + 142.070i 1.03701 + 1.03701i 0.999288 + 0.0377220i \(0.0120101\pi\)
0.0377220 + 0.999288i \(0.487990\pi\)
\(138\) 0 0
\(139\) 170.000i 1.22302i −0.791236 0.611511i \(-0.790562\pi\)
0.791236 0.611511i \(-0.209438\pi\)
\(140\) 0 0
\(141\) 108.000 0.765957
\(142\) 0 0
\(143\) −132.272 + 132.272i −0.924982 + 0.924982i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −72.2599 72.2599i −0.491564 0.491564i
\(148\) 0 0
\(149\) 180.000i 1.20805i 0.796964 + 0.604027i \(0.206438\pi\)
−0.796964 + 0.604027i \(0.793562\pi\)
\(150\) 0 0
\(151\) −22.0000 −0.145695 −0.0728477 0.997343i \(-0.523209\pi\)
−0.0728477 + 0.997343i \(0.523209\pi\)
\(152\) 0 0
\(153\) −14.6969 + 14.6969i −0.0960584 + 0.0960584i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −139.621 139.621i −0.889305 0.889305i 0.105151 0.994456i \(-0.466467\pi\)
−0.994456 + 0.105151i \(0.966467\pi\)
\(158\) 0 0
\(159\) 12.0000i 0.0754717i
\(160\) 0 0
\(161\) −288.000 −1.78882
\(162\) 0 0
\(163\) 117.576 117.576i 0.721322 0.721322i −0.247552 0.968875i \(-0.579626\pi\)
0.968875 + 0.247552i \(0.0796262\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.5959 19.5959i −0.117341 0.117341i 0.645998 0.763339i \(-0.276441\pi\)
−0.763339 + 0.645998i \(0.776441\pi\)
\(168\) 0 0
\(169\) 61.0000i 0.360947i
\(170\) 0 0
\(171\) 30.0000 0.175439
\(172\) 0 0
\(173\) −142.070 + 142.070i −0.821216 + 0.821216i −0.986282 0.165066i \(-0.947216\pi\)
0.165066 + 0.986282i \(0.447216\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −110.227 110.227i −0.622752 0.622752i
\(178\) 0 0
\(179\) 90.0000i 0.502793i −0.967884 0.251397i \(-0.919110\pi\)
0.967884 0.251397i \(-0.0808899\pi\)
\(180\) 0 0
\(181\) −98.0000 −0.541436 −0.270718 0.962659i \(-0.587261\pi\)
−0.270718 + 0.962659i \(0.587261\pi\)
\(182\) 0 0
\(183\) −2.44949 + 2.44949i −0.0133852 + 0.0133852i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −88.1816 88.1816i −0.471560 0.471560i
\(188\) 0 0
\(189\) 54.0000i 0.285714i
\(190\) 0 0
\(191\) −252.000 −1.31937 −0.659686 0.751541i \(-0.729311\pi\)
−0.659686 + 0.751541i \(0.729311\pi\)
\(192\) 0 0
\(193\) −264.545 + 264.545i −1.37070 + 1.37070i −0.511292 + 0.859407i \(0.670833\pi\)
−0.859407 + 0.511292i \(0.829167\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −127.373 127.373i −0.646566 0.646566i 0.305596 0.952161i \(-0.401144\pi\)
−0.952161 + 0.305596i \(0.901144\pi\)
\(198\) 0 0
\(199\) 290.000i 1.45729i 0.684893 + 0.728643i \(0.259849\pi\)
−0.684893 + 0.728643i \(0.740151\pi\)
\(200\) 0 0
\(201\) 108.000 0.537313
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −58.7878 58.7878i −0.283999 0.283999i
\(208\) 0 0
\(209\) 180.000i 0.861244i
\(210\) 0 0
\(211\) −122.000 −0.578199 −0.289100 0.957299i \(-0.593356\pi\)
−0.289100 + 0.957299i \(0.593356\pi\)
\(212\) 0 0
\(213\) 88.1816 88.1816i 0.413998 0.413998i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 161.666 + 161.666i 0.745006 + 0.745006i
\(218\) 0 0
\(219\) 108.000i 0.493151i
\(220\) 0 0
\(221\) 72.0000 0.325792
\(222\) 0 0
\(223\) 80.8332 80.8332i 0.362481 0.362481i −0.502245 0.864725i \(-0.667492\pi\)
0.864725 + 0.502245i \(0.167492\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 53.8888 + 53.8888i 0.237395 + 0.237395i 0.815771 0.578375i \(-0.196313\pi\)
−0.578375 + 0.815771i \(0.696313\pi\)
\(228\) 0 0
\(229\) 50.0000i 0.218341i 0.994023 + 0.109170i \(0.0348194\pi\)
−0.994023 + 0.109170i \(0.965181\pi\)
\(230\) 0 0
\(231\) 324.000 1.40260
\(232\) 0 0
\(233\) −44.0908 + 44.0908i −0.189231 + 0.189231i −0.795364 0.606133i \(-0.792720\pi\)
0.606133 + 0.795364i \(0.292720\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 85.7321 + 85.7321i 0.361739 + 0.361739i
\(238\) 0 0
\(239\) 180.000i 0.753138i 0.926389 + 0.376569i \(0.122896\pi\)
−0.926389 + 0.376569i \(0.877104\pi\)
\(240\) 0 0
\(241\) −178.000 −0.738589 −0.369295 0.929312i \(-0.620401\pi\)
−0.369295 + 0.929312i \(0.620401\pi\)
\(242\) 0 0
\(243\) 11.0227 11.0227i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −73.4847 73.4847i −0.297509 0.297509i
\(248\) 0 0
\(249\) 132.000i 0.530120i
\(250\) 0 0
\(251\) −342.000 −1.36255 −0.681275 0.732028i \(-0.738574\pi\)
−0.681275 + 0.732028i \(0.738574\pi\)
\(252\) 0 0
\(253\) 352.727 352.727i 1.39418 1.39418i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 289.040 + 289.040i 1.12467 + 1.12467i 0.991030 + 0.133638i \(0.0426660\pi\)
0.133638 + 0.991030i \(0.457334\pi\)
\(258\) 0 0
\(259\) 108.000i 0.416988i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 264.545 264.545i 1.00587 1.00587i 0.00589147 0.999983i \(-0.498125\pi\)
0.999983 0.00589147i \(-0.00187532\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 110.227 + 110.227i 0.412835 + 0.412835i
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 478.000 1.76384 0.881919 0.471401i \(-0.156252\pi\)
0.881919 + 0.471401i \(0.156252\pi\)
\(272\) 0 0
\(273\) −132.272 + 132.272i −0.484514 + 0.484514i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 80.8332 + 80.8332i 0.291816 + 0.291816i 0.837798 0.545981i \(-0.183843\pi\)
−0.545981 + 0.837798i \(0.683843\pi\)
\(278\) 0 0
\(279\) 66.0000i 0.236559i
\(280\) 0 0
\(281\) 162.000 0.576512 0.288256 0.957553i \(-0.406925\pi\)
0.288256 + 0.957553i \(0.406925\pi\)
\(282\) 0 0
\(283\) −249.848 + 249.848i −0.882855 + 0.882855i −0.993824 0.110969i \(-0.964605\pi\)
0.110969 + 0.993824i \(0.464605\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 132.272 + 132.272i 0.460880 + 0.460880i
\(288\) 0 0
\(289\) 241.000i 0.833910i
\(290\) 0 0
\(291\) 252.000 0.865979
\(292\) 0 0
\(293\) 372.322 372.322i 1.27073 1.27073i 0.325017 0.945708i \(-0.394630\pi\)
0.945708 0.325017i \(-0.105370\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 66.1362 + 66.1362i 0.222681 + 0.222681i
\(298\) 0 0
\(299\) 288.000i 0.963211i
\(300\) 0 0
\(301\) 432.000 1.43522
\(302\) 0 0
\(303\) 132.272 132.272i 0.436543 0.436543i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 396.817 + 396.817i 1.29256 + 1.29256i 0.933195 + 0.359369i \(0.117008\pi\)
0.359369 + 0.933195i \(0.382992\pi\)
\(308\) 0 0
\(309\) 162.000i 0.524272i
\(310\) 0 0
\(311\) −252.000 −0.810289 −0.405145 0.914253i \(-0.632779\pi\)
−0.405145 + 0.914253i \(0.632779\pi\)
\(312\) 0 0
\(313\) −117.576 + 117.576i −0.375641 + 0.375641i −0.869527 0.493886i \(-0.835576\pi\)
0.493886 + 0.869527i \(0.335576\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −274.343 274.343i −0.865435 0.865435i 0.126528 0.991963i \(-0.459617\pi\)
−0.991963 + 0.126528i \(0.959617\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 108.000 0.336449
\(322\) 0 0
\(323\) 48.9898 48.9898i 0.151671 0.151671i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 208.207 + 208.207i 0.636718 + 0.636718i
\(328\) 0 0
\(329\) 648.000i 1.96960i
\(330\) 0 0
\(331\) 418.000 1.26284 0.631420 0.775441i \(-0.282472\pi\)
0.631420 + 0.775441i \(0.282472\pi\)
\(332\) 0 0
\(333\) 22.0454 22.0454i 0.0662024 0.0662024i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 191.060 + 191.060i 0.566944 + 0.566944i 0.931271 0.364327i \(-0.118701\pi\)
−0.364327 + 0.931271i \(0.618701\pi\)
\(338\) 0 0
\(339\) 108.000i 0.318584i
\(340\) 0 0
\(341\) −396.000 −1.16129
\(342\) 0 0
\(343\) 73.4847 73.4847i 0.214241 0.214241i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −44.0908 44.0908i −0.127063 0.127063i 0.640716 0.767778i \(-0.278638\pi\)
−0.767778 + 0.640716i \(0.778638\pi\)
\(348\) 0 0
\(349\) 70.0000i 0.200573i −0.994959 0.100287i \(-0.968024\pi\)
0.994959 0.100287i \(-0.0319759\pi\)
\(350\) 0 0
\(351\) −54.0000 −0.153846
\(352\) 0 0
\(353\) −44.0908 + 44.0908i −0.124903 + 0.124903i −0.766795 0.641892i \(-0.778150\pi\)
0.641892 + 0.766795i \(0.278150\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −88.1816 88.1816i −0.247007 0.247007i
\(358\) 0 0
\(359\) 540.000i 1.50418i −0.659061 0.752089i \(-0.729046\pi\)
0.659061 0.752089i \(-0.270954\pi\)
\(360\) 0 0
\(361\) 261.000 0.722992
\(362\) 0 0
\(363\) −248.623 + 248.623i −0.684912 + 0.684912i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 66.1362 + 66.1362i 0.180208 + 0.180208i 0.791446 0.611239i \(-0.209328\pi\)
−0.611239 + 0.791446i \(0.709328\pi\)
\(368\) 0 0
\(369\) 54.0000i 0.146341i
\(370\) 0 0
\(371\) −72.0000 −0.194070
\(372\) 0 0
\(373\) 213.106 213.106i 0.571329 0.571329i −0.361171 0.932500i \(-0.617623\pi\)
0.932500 + 0.361171i \(0.117623\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 170.000i 0.448549i −0.974526 0.224274i \(-0.927999\pi\)
0.974526 0.224274i \(-0.0720012\pi\)
\(380\) 0 0
\(381\) 18.0000 0.0472441
\(382\) 0 0
\(383\) −151.868 + 151.868i −0.396523 + 0.396523i −0.877005 0.480482i \(-0.840462\pi\)
0.480482 + 0.877005i \(0.340462\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 88.1816 + 88.1816i 0.227860 + 0.227860i
\(388\) 0 0
\(389\) 360.000i 0.925450i 0.886502 + 0.462725i \(0.153128\pi\)
−0.886502 + 0.462725i \(0.846872\pi\)
\(390\) 0 0
\(391\) −192.000 −0.491049
\(392\) 0 0
\(393\) −22.0454 + 22.0454i −0.0560952 + 0.0560952i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 301.287 + 301.287i 0.758910 + 0.758910i 0.976124 0.217214i \(-0.0696970\pi\)
−0.217214 + 0.976124i \(0.569697\pi\)
\(398\) 0 0
\(399\) 180.000i 0.451128i
\(400\) 0 0
\(401\) −558.000 −1.39152 −0.695761 0.718274i \(-0.744933\pi\)
−0.695761 + 0.718274i \(0.744933\pi\)
\(402\) 0 0
\(403\) 161.666 161.666i 0.401157 0.401157i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 132.272 + 132.272i 0.324994 + 0.324994i
\(408\) 0 0
\(409\) 670.000i 1.63814i −0.573692 0.819071i \(-0.694489\pi\)
0.573692 0.819071i \(-0.305511\pi\)
\(410\) 0 0
\(411\) −348.000 −0.846715
\(412\) 0 0
\(413\) 661.362 661.362i 1.60136 1.60136i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 208.207 + 208.207i 0.499296 + 0.499296i
\(418\) 0 0
\(419\) 630.000i 1.50358i 0.659403 + 0.751790i \(0.270809\pi\)
−0.659403 + 0.751790i \(0.729191\pi\)
\(420\) 0 0
\(421\) 142.000 0.337292 0.168646 0.985677i \(-0.446061\pi\)
0.168646 + 0.985677i \(0.446061\pi\)
\(422\) 0 0
\(423\) −132.272 + 132.272i −0.312701 + 0.312701i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −14.6969 14.6969i −0.0344191 0.0344191i
\(428\) 0 0
\(429\) 324.000i 0.755245i
\(430\) 0 0
\(431\) −612.000 −1.41995 −0.709977 0.704225i \(-0.751295\pi\)
−0.709977 + 0.704225i \(0.751295\pi\)
\(432\) 0 0
\(433\) 102.879 102.879i 0.237595 0.237595i −0.578259 0.815853i \(-0.696268\pi\)
0.815853 + 0.578259i \(0.196268\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 195.959 + 195.959i 0.448419 + 0.448419i
\(438\) 0 0
\(439\) 430.000i 0.979499i −0.871863 0.489749i \(-0.837088\pi\)
0.871863 0.489749i \(-0.162912\pi\)
\(440\) 0 0
\(441\) 177.000 0.401361
\(442\) 0 0
\(443\) 582.979 582.979i 1.31598 1.31598i 0.399049 0.916930i \(-0.369340\pi\)
0.916930 0.399049i \(-0.130660\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −220.454 220.454i −0.493186 0.493186i
\(448\) 0 0
\(449\) 90.0000i 0.200445i −0.994965 0.100223i \(-0.968044\pi\)
0.994965 0.100223i \(-0.0319555\pi\)
\(450\) 0 0
\(451\) −324.000 −0.718404
\(452\) 0 0
\(453\) 26.9444 26.9444i 0.0594799 0.0594799i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 191.060 + 191.060i 0.418075 + 0.418075i 0.884540 0.466465i \(-0.154473\pi\)
−0.466465 + 0.884540i \(0.654473\pi\)
\(458\) 0 0
\(459\) 36.0000i 0.0784314i
\(460\) 0 0
\(461\) −828.000 −1.79610 −0.898048 0.439898i \(-0.855015\pi\)
−0.898048 + 0.439898i \(0.855015\pi\)
\(462\) 0 0
\(463\) 7.34847 7.34847i 0.0158714 0.0158714i −0.699127 0.714998i \(-0.746428\pi\)
0.714998 + 0.699127i \(0.246428\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 347.828 + 347.828i 0.744813 + 0.744813i 0.973500 0.228687i \(-0.0734433\pi\)
−0.228687 + 0.973500i \(0.573443\pi\)
\(468\) 0 0
\(469\) 648.000i 1.38166i
\(470\) 0 0
\(471\) 342.000 0.726115
\(472\) 0 0
\(473\) −529.090 + 529.090i −1.11858 + 1.11858i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −14.6969 14.6969i −0.0308112 0.0308112i
\(478\) 0 0
\(479\) 360.000i 0.751566i −0.926708 0.375783i \(-0.877374\pi\)
0.926708 0.375783i \(-0.122626\pi\)
\(480\) 0 0
\(481\) −108.000 −0.224532
\(482\) 0 0
\(483\) 352.727 352.727i 0.730283 0.730283i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 580.529 + 580.529i 1.19205 + 1.19205i 0.976490 + 0.215561i \(0.0691581\pi\)
0.215561 + 0.976490i \(0.430842\pi\)
\(488\) 0 0
\(489\) 288.000i 0.588957i
\(490\) 0 0
\(491\) 18.0000 0.0366599 0.0183299 0.999832i \(-0.494165\pi\)
0.0183299 + 0.999832i \(0.494165\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 529.090 + 529.090i 1.06457 + 1.06457i
\(498\) 0 0
\(499\) 590.000i 1.18236i 0.806538 + 0.591182i \(0.201339\pi\)
−0.806538 + 0.591182i \(0.798661\pi\)
\(500\) 0 0
\(501\) 48.0000 0.0958084
\(502\) 0 0
\(503\) 93.0806 93.0806i 0.185051 0.185051i −0.608502 0.793553i \(-0.708229\pi\)
0.793553 + 0.608502i \(0.208229\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −74.7094 74.7094i −0.147356 0.147356i
\(508\) 0 0
\(509\) 540.000i 1.06090i −0.847715 0.530452i \(-0.822022\pi\)
0.847715 0.530452i \(-0.177978\pi\)
\(510\) 0 0
\(511\) 648.000 1.26810
\(512\) 0 0
\(513\) −36.7423 + 36.7423i −0.0716225 + 0.0716225i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −793.635 793.635i −1.53508 1.53508i
\(518\) 0 0
\(519\) 348.000i 0.670520i
\(520\) 0 0
\(521\) 342.000 0.656430 0.328215 0.944603i \(-0.393553\pi\)
0.328215 + 0.944603i \(0.393553\pi\)
\(522\) 0 0
\(523\) 264.545 264.545i 0.505822 0.505822i −0.407419 0.913241i \(-0.633571\pi\)
0.913241 + 0.407419i \(0.133571\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 107.778 + 107.778i 0.204511 + 0.204511i
\(528\) 0 0
\(529\) 239.000i 0.451796i
\(530\) 0 0
\(531\) 270.000 0.508475
\(532\) 0 0
\(533\) 132.272 132.272i 0.248166 0.248166i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 110.227 + 110.227i 0.205265 + 0.205265i
\(538\) 0 0
\(539\) 1062.00i 1.97032i
\(540\) 0 0
\(541\) 2.00000 0.00369686 0.00184843 0.999998i \(-0.499412\pi\)
0.00184843 + 0.999998i \(0.499412\pi\)
\(542\) 0 0
\(543\) 120.025 120.025i 0.221041 0.221041i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −264.545 264.545i −0.483629 0.483629i 0.422660 0.906288i \(-0.361097\pi\)
−0.906288 + 0.422660i \(0.861097\pi\)
\(548\) 0 0
\(549\) 6.00000i 0.0109290i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −514.393 + 514.393i −0.930186 + 0.930186i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 484.999 + 484.999i 0.870734 + 0.870734i 0.992552 0.121818i \(-0.0388725\pi\)
−0.121818 + 0.992552i \(0.538872\pi\)
\(558\) 0 0
\(559\) 432.000i 0.772809i
\(560\) 0 0
\(561\) 216.000 0.385027
\(562\) 0 0
\(563\) −396.817 + 396.817i −0.704827 + 0.704827i −0.965443 0.260616i \(-0.916074\pi\)
0.260616 + 0.965443i \(0.416074\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 66.1362 + 66.1362i 0.116642 + 0.116642i
\(568\) 0 0
\(569\) 630.000i 1.10721i −0.832781 0.553603i \(-0.813253\pi\)
0.832781 0.553603i \(-0.186747\pi\)
\(570\) 0 0
\(571\) −302.000 −0.528897 −0.264448 0.964400i \(-0.585190\pi\)
−0.264448 + 0.964400i \(0.585190\pi\)
\(572\) 0 0
\(573\) 308.636 308.636i 0.538631 0.538631i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 484.999 + 484.999i 0.840553 + 0.840553i 0.988931 0.148378i \(-0.0474052\pi\)
−0.148378 + 0.988931i \(0.547405\pi\)
\(578\) 0 0
\(579\) 648.000i 1.11917i
\(580\) 0 0
\(581\) 792.000 1.36317
\(582\) 0 0
\(583\) 88.1816 88.1816i 0.151255 0.151255i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −582.979 582.979i −0.993149 0.993149i 0.00682753 0.999977i \(-0.497827\pi\)
−0.999977 + 0.00682753i \(0.997827\pi\)
\(588\) 0 0
\(589\) 220.000i 0.373514i
\(590\) 0 0
\(591\) 312.000 0.527919
\(592\) 0 0
\(593\) −533.989 + 533.989i −0.900487 + 0.900487i −0.995478 0.0949912i \(-0.969718\pi\)
0.0949912 + 0.995478i \(0.469718\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −355.176 355.176i −0.594935 0.594935i
\(598\) 0 0
\(599\) 540.000i 0.901503i −0.892650 0.450751i \(-0.851156\pi\)
0.892650 0.450751i \(-0.148844\pi\)
\(600\) 0 0
\(601\) −758.000 −1.26123 −0.630616 0.776095i \(-0.717198\pi\)
−0.630616 + 0.776095i \(0.717198\pi\)
\(602\) 0 0
\(603\) −132.272 + 132.272i −0.219357 + 0.219357i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −668.711 668.711i −1.10167 1.10167i −0.994210 0.107455i \(-0.965730\pi\)
−0.107455 0.994210i \(-0.534270\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 648.000 1.06056
\(612\) 0 0
\(613\) 66.1362 66.1362i 0.107889 0.107889i −0.651101 0.758991i \(-0.725693\pi\)
0.758991 + 0.651101i \(0.225693\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −592.777 592.777i −0.960740 0.960740i 0.0385180 0.999258i \(-0.487736\pi\)
−0.999258 + 0.0385180i \(0.987736\pi\)
\(618\) 0 0
\(619\) 1030.00i 1.66397i −0.554795 0.831987i \(-0.687203\pi\)
0.554795 0.831987i \(-0.312797\pi\)
\(620\) 0 0
\(621\) 144.000 0.231884
\(622\) 0 0
\(623\) −661.362 + 661.362i −1.06158 + 1.06158i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −220.454 220.454i −0.351601 0.351601i
\(628\) 0 0
\(629\) 72.0000i 0.114467i
\(630\) 0 0
\(631\) −242.000 −0.383518 −0.191759 0.981442i \(-0.561419\pi\)
−0.191759 + 0.981442i \(0.561419\pi\)
\(632\) 0 0
\(633\) 149.419 149.419i 0.236049 0.236049i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −433.560 433.560i −0.680627 0.680627i
\(638\) 0 0
\(639\) 216.000i 0.338028i
\(640\) 0 0
\(641\) 162.000 0.252730 0.126365 0.991984i \(-0.459669\pi\)
0.126365 + 0.991984i \(0.459669\pi\)
\(642\) 0 0
\(643\) 264.545 264.545i 0.411423 0.411423i −0.470811 0.882234i \(-0.656039\pi\)
0.882234 + 0.470811i \(0.156039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −93.0806 93.0806i −0.143865 0.143865i 0.631506 0.775371i \(-0.282437\pi\)
−0.775371 + 0.631506i \(0.782437\pi\)
\(648\) 0 0
\(649\) 1620.00i 2.49615i
\(650\) 0 0
\(651\) −396.000 −0.608295
\(652\) 0 0
\(653\) −68.5857 + 68.5857i −0.105032 + 0.105032i −0.757670 0.652638i \(-0.773662\pi\)
0.652638 + 0.757670i \(0.273662\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 132.272 + 132.272i 0.201328 + 0.201328i
\(658\) 0 0
\(659\) 630.000i 0.955994i 0.878361 + 0.477997i \(0.158637\pi\)
−0.878361 + 0.477997i \(0.841363\pi\)
\(660\) 0 0
\(661\) 622.000 0.940998 0.470499 0.882400i \(-0.344074\pi\)
0.470499 + 0.882400i \(0.344074\pi\)
\(662\) 0 0
\(663\) −88.1816 + 88.1816i −0.133004 + 0.133004i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 198.000i 0.295964i
\(670\) 0 0
\(671\) 36.0000 0.0536513
\(672\) 0 0
\(673\) 102.879 102.879i 0.152866 0.152866i −0.626531 0.779397i \(-0.715526\pi\)
0.779397 + 0.626531i \(0.215526\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −176.363 176.363i −0.260507 0.260507i 0.564753 0.825260i \(-0.308971\pi\)
−0.825260 + 0.564753i \(0.808971\pi\)
\(678\) 0 0
\(679\) 1512.00i 2.22680i
\(680\) 0 0
\(681\) −132.000 −0.193833
\(682\) 0 0
\(683\) −445.807 + 445.807i −0.652719 + 0.652719i −0.953647 0.300928i \(-0.902704\pi\)
0.300928 + 0.953647i \(0.402704\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −61.2372 61.2372i −0.0891372 0.0891372i
\(688\) 0 0
\(689\) 72.0000i 0.104499i
\(690\) 0 0
\(691\) −682.000 −0.986975 −0.493488 0.869753i \(-0.664278\pi\)
−0.493488 + 0.869753i \(0.664278\pi\)
\(692\) 0 0
\(693\) −396.817 + 396.817i −0.572608 + 0.572608i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 88.1816 + 88.1816i 0.126516 + 0.126516i
\(698\) 0 0
\(699\) 108.000i 0.154506i
\(700\) 0 0
\(701\) −468.000 −0.667618 −0.333809 0.942641i \(-0.608334\pi\)
−0.333809 + 0.942641i \(0.608334\pi\)
\(702\) 0 0
\(703\) −73.4847 + 73.4847i −0.104530 + 0.104530i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 793.635 + 793.635i 1.12254 + 1.12254i
\(708\) 0 0
\(709\) 310.000i 0.437236i −0.975811 0.218618i \(-0.929845\pi\)
0.975811 0.218618i \(-0.0701548\pi\)
\(710\) 0 0
\(711\) −210.000 −0.295359
\(712\) 0 0
\(713\) −431.110 + 431.110i −0.604643 + 0.604643i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −220.454 220.454i −0.307467 0.307467i
\(718\) 0 0
\(719\) 180.000i 0.250348i 0.992135 + 0.125174i \(0.0399489\pi\)
−0.992135 + 0.125174i \(0.960051\pi\)
\(720\) 0 0
\(721\) 972.000 1.34813
\(722\) 0 0
\(723\) 218.005 218.005i 0.301528 0.301528i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 507.044 + 507.044i 0.697448 + 0.697448i 0.963859 0.266412i \(-0.0858381\pi\)
−0.266412 + 0.963859i \(0.585838\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) 288.000 0.393981
\(732\) 0 0
\(733\) 800.983 800.983i 1.09275 1.09275i 0.0975121 0.995234i \(-0.468912\pi\)
0.995234 0.0975121i \(-0.0310885\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −793.635 793.635i −1.07684 1.07684i
\(738\) 0 0
\(739\) 350.000i 0.473613i −0.971557 0.236806i \(-0.923899\pi\)
0.971557 0.236806i \(-0.0761007\pi\)
\(740\) 0 0
\(741\) 180.000 0.242915
\(742\) 0 0
\(743\) 925.907 925.907i 1.24617 1.24617i 0.288778 0.957396i \(-0.406751\pi\)
0.957396 0.288778i \(-0.0932488\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 161.666 + 161.666i 0.216421 + 0.216421i
\(748\) 0 0
\(749\) 648.000i 0.865154i
\(750\) 0 0
\(751\) 338.000 0.450067 0.225033 0.974351i \(-0.427751\pi\)
0.225033 + 0.974351i \(0.427751\pi\)
\(752\) 0 0
\(753\) 418.863 418.863i 0.556259 0.556259i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −286.590 286.590i −0.378587 0.378587i 0.492005 0.870592i \(-0.336264\pi\)
−0.870592 + 0.492005i \(0.836264\pi\)
\(758\) 0 0
\(759\) 864.000i 1.13834i
\(760\) 0 0
\(761\) −1278.00 −1.67937 −0.839685 0.543074i \(-0.817260\pi\)
−0.839685 + 0.543074i \(0.817260\pi\)
\(762\) 0 0
\(763\) −1249.24 + 1249.24i −1.63727 + 1.63727i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −661.362 661.362i −0.862271 0.862271i
\(768\) 0 0
\(769\) 590.000i 0.767230i −0.923493 0.383615i \(-0.874679\pi\)
0.923493 0.383615i \(-0.125321\pi\)
\(770\) 0 0
\(771\) −708.000 −0.918288
\(772\) 0 0
\(773\) 127.373 127.373i 0.164778 0.164778i −0.619902 0.784680i \(-0.712827\pi\)
0.784680 + 0.619902i \(0.212827\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 132.272 + 132.272i 0.170235 + 0.170235i
\(778\) 0 0
\(779\) 180.000i 0.231065i
\(780\) 0 0
\(781\) −1296.00 −1.65941
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −264.545 264.545i −0.336143 0.336143i 0.518770 0.854914i \(-0.326390\pi\)
−0.854914 + 0.518770i \(0.826390\pi\)
\(788\) 0 0
\(789\) 648.000i 0.821293i
\(790\) 0 0
\(791\) 648.000 0.819216
\(792\) 0 0
\(793\) −14.6969 + 14.6969i −0.0185333 + 0.0185333i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −200.858 200.858i −0.252018 0.252018i 0.569780 0.821797i \(-0.307029\pi\)
−0.821797 + 0.569780i \(0.807029\pi\)
\(798\) 0 0
\(799\) 432.000i 0.540676i
\(800\) 0 0
\(801\) −270.000 −0.337079
\(802\) 0 0
\(803\) −793.635 + 793.635i −0.988337 + 0.988337i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 630.000i 0.778739i 0.921082 + 0.389370i \(0.127307\pi\)
−0.921082 + 0.389370i \(0.872693\pi\)
\(810\) 0 0
\(811\) 218.000 0.268804 0.134402 0.990927i \(-0.457089\pi\)
0.134402 + 0.990927i \(0.457089\pi\)
\(812\) 0 0
\(813\) −585.428 + 585.428i −0.720084 + 0.720084i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −293.939 293.939i −0.359778 0.359778i
\(818\) 0 0
\(819\) 324.000i 0.395604i
\(820\) 0 0
\(821\) 432.000 0.526188 0.263094 0.964770i \(-0.415257\pi\)
0.263094 + 0.964770i \(0.415257\pi\)
\(822\) 0 0
\(823\) −360.075 + 360.075i −0.437515 + 0.437515i −0.891175 0.453660i \(-0.850118\pi\)
0.453660 + 0.891175i \(0.350118\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −876.917 876.917i −1.06036 1.06036i −0.998057 0.0623022i \(-0.980156\pi\)
−0.0623022 0.998057i \(-0.519844\pi\)
\(828\) 0 0
\(829\) 70.0000i 0.0844391i −0.999108 0.0422195i \(-0.986557\pi\)
0.999108 0.0422195i \(-0.0134429\pi\)
\(830\) 0 0
\(831\) −198.000 −0.238267
\(832\) 0 0
\(833\) 289.040 289.040i 0.346987 0.346987i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −80.8332 80.8332i −0.0965749 0.0965749i
\(838\) 0 0
\(839\) 540.000i 0.643623i −0.946804 0.321812i \(-0.895708\pi\)
0.946804 0.321812i \(-0.104292\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) −198.409 + 198.409i −0.235360 + 0.235360i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1491.74 1491.74i −1.76120 1.76120i
\(848\) 0 0
\(849\) 612.000i 0.720848i
\(850\) 0 0
\(851\) 288.000 0.338425
\(852\) 0 0
\(853\) −815.680 + 815.680i −0.956249 + 0.956249i −0.999082 0.0428336i \(-0.986361\pi\)
0.0428336 + 0.999082i \(0.486361\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −298.838 298.838i −0.348702 0.348702i 0.510924 0.859626i \(-0.329303\pi\)
−0.859626 + 0.510924i \(0.829303\pi\)
\(858\) 0 0
\(859\) 310.000i 0.360885i −0.983586 0.180442i \(-0.942247\pi\)
0.983586 0.180442i \(-0.0577529\pi\)
\(860\) 0 0
\(861\) −324.000 −0.376307
\(862\) 0 0
\(863\) −225.353 + 225.353i −0.261128 + 0.261128i −0.825512 0.564385i \(-0.809114\pi\)
0.564385 + 0.825512i \(0.309114\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 295.164 + 295.164i 0.340442 + 0.340442i
\(868\) 0 0
\(869\) 1260.00i 1.44994i
\(870\) 0 0
\(871\) 648.000 0.743972
\(872\) 0 0
\(873\) −308.636 + 308.636i −0.353535 + 0.353535i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −213.106 213.106i −0.242994 0.242994i 0.575094 0.818088i \(-0.304966\pi\)
−0.818088 + 0.575094i \(0.804966\pi\)
\(878\) 0 0
\(879\) 912.000i 1.03754i
\(880\) 0 0
\(881\) 1062.00 1.20545 0.602724 0.797950i \(-0.294082\pi\)
0.602724 + 0.797950i \(0.294082\pi\)
\(882\) 0 0
\(883\) 117.576 117.576i 0.133155 0.133155i −0.637388 0.770543i \(-0.719985\pi\)
0.770543 + 0.637388i \(0.219985\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −509.494 509.494i −0.574401 0.574401i 0.358954 0.933355i \(-0.383133\pi\)
−0.933355 + 0.358954i \(0.883133\pi\)
\(888\) 0 0
\(889\) 108.000i 0.121485i
\(890\) 0 0
\(891\) −162.000 −0.181818
\(892\) 0 0
\(893\) 440.908 440.908i 0.493738 0.493738i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −352.727 352.727i −0.393229 0.393229i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −48.0000 −0.0532741
\(902\) 0 0
\(903\) −529.090 + 529.090i −0.585924 + 0.585924i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −411.514 411.514i −0.453709 0.453709i 0.442874 0.896584i \(-0.353959\pi\)
−0.896584 + 0.442874i \(0.853959\pi\)
\(908\) 0 0
\(909\) 324.000i 0.356436i
\(910\) 0 0
\(911\) −792.000 −0.869374 −0.434687 0.900582i \(-0.643141\pi\)
−0.434687 + 0.900582i \(0.643141\pi\)
\(912\) 0 0
\(913\) −969.998 + 969.998i −1.06243 + 1.06243i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −132.272 132.272i −0.144245 0.144245i
\(918\) 0 0
\(919\) 430.000i 0.467900i 0.972249 + 0.233950i \(0.0751652\pi\)
−0.972249 + 0.233950i \(0.924835\pi\)
\(920\) 0 0
\(921\) −972.000 −1.05537
\(922\) 0 0
\(923\) 529.090 529.090i 0.573228 0.573228i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 198.409 + 198.409i 0.214033 + 0.214033i
\(928\) 0 0
\(929\) 1530.00i 1.64693i −0.567365 0.823466i \(-0.692037\pi\)
0.567365 0.823466i \(-0.307963\pi\)
\(930\) 0 0
\(931\) −590.000 −0.633727
\(932\) 0 0
\(933\) 308.636 308.636i 0.330799 0.330799i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −323.333 323.333i −0.345072 0.345072i 0.513198 0.858270i \(-0.328461\pi\)
−0.858270 + 0.513198i \(0.828461\pi\)
\(938\) 0 0
\(939\) 288.000i 0.306709i
\(940\) 0 0
\(941\) 1152.00 1.22423 0.612115 0.790769i \(-0.290319\pi\)
0.612115 + 0.790769i \(0.290319\pi\)
\(942\) 0 0
\(943\) −352.727 + 352.727i −0.374047 + 0.374047i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 739.746 + 739.746i 0.781147 + 0.781147i 0.980024 0.198878i \(-0.0637296\pi\)
−0.198878 + 0.980024i \(0.563730\pi\)
\(948\) 0 0
\(949\) 648.000i 0.682824i
\(950\) 0 0
\(951\) 672.000 0.706625
\(952\) 0 0
\(953\) −44.0908 + 44.0908i −0.0462653 + 0.0462653i −0.729861 0.683596i \(-0.760415\pi\)
0.683596 + 0.729861i \(0.260415\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2088.00i 2.17727i
\(960\) 0 0
\(961\) −477.000 −0.496358
\(962\) 0 0
\(963\) −132.272 + 132.272i −0.137355 + 0.137355i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 727.498 + 727.498i 0.752325 + 0.752325i 0.974913 0.222588i \(-0.0714503\pi\)
−0.222588 + 0.974913i \(0.571450\pi\)
\(968\) 0 0
\(969\) 120.000i 0.123839i
\(970\) 0 0
\(971\) 1278.00 1.31617 0.658084 0.752944i \(-0.271367\pi\)
0.658084 + 0.752944i \(0.271367\pi\)
\(972\) 0 0
\(973\) −1249.24 + 1249.24i −1.28391 + 1.28391i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −396.817 396.817i −0.406159 0.406159i 0.474238 0.880397i \(-0.342724\pi\)
−0.880397 + 0.474238i \(0.842724\pi\)
\(978\) 0 0
\(979\) 1620.00i 1.65475i
\(980\) 0 0
\(981\) −510.000 −0.519878
\(982\) 0 0
\(983\) 827.928 827.928i 0.842246 0.842246i −0.146905 0.989151i \(-0.546931\pi\)
0.989151 + 0.146905i \(0.0469311\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −793.635 793.635i −0.804088 0.804088i
\(988\) 0 0
\(989\) 1152.00i 1.16481i
\(990\) 0 0
\(991\) 118.000 0.119072 0.0595358 0.998226i \(-0.481038\pi\)
0.0595358 + 0.998226i \(0.481038\pi\)
\(992\) 0 0
\(993\) −511.943 + 511.943i −0.515552 + 0.515552i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 815.680 + 815.680i 0.818134 + 0.818134i 0.985838 0.167703i \(-0.0536350\pi\)
−0.167703 + 0.985838i \(0.553635\pi\)
\(998\) 0 0
\(999\) 54.0000i 0.0540541i
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 1200.3.bg.j.1057.1 4
4.3 odd 2 75.3.f.a.7.2 yes 4
5.2 odd 4 inner 1200.3.bg.j.193.2 4
5.3 odd 4 inner 1200.3.bg.j.193.1 4
5.4 even 2 inner 1200.3.bg.j.1057.2 4
12.11 even 2 225.3.g.f.82.1 4
20.3 even 4 75.3.f.a.43.2 yes 4
20.7 even 4 75.3.f.a.43.1 yes 4
20.19 odd 2 75.3.f.a.7.1 4
60.23 odd 4 225.3.g.f.118.1 4
60.47 odd 4 225.3.g.f.118.2 4
60.59 even 2 225.3.g.f.82.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.3.f.a.7.1 4 20.19 odd 2
75.3.f.a.7.2 yes 4 4.3 odd 2
75.3.f.a.43.1 yes 4 20.7 even 4
75.3.f.a.43.2 yes 4 20.3 even 4
225.3.g.f.82.1 4 12.11 even 2
225.3.g.f.82.2 4 60.59 even 2
225.3.g.f.118.1 4 60.23 odd 4
225.3.g.f.118.2 4 60.47 odd 4
1200.3.bg.j.193.1 4 5.3 odd 4 inner
1200.3.bg.j.193.2 4 5.2 odd 4 inner
1200.3.bg.j.1057.1 4 1.1 even 1 trivial
1200.3.bg.j.1057.2 4 5.4 even 2 inner