Properties

Label 1200.3.bg.j.193.2
Level $1200$
Weight $3$
Character 1200.193
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 193.2
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1200.193
Dual form 1200.3.bg.j.1057.2

$q$-expansion

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

Character values

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

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

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 0.0510871 0.998694i \(-0.483731\pi\)
0.998694 0.0510871i \(-0.0162686\pi\)
\(8\) 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 \(-0.119113\pi\)
−0.365532 + 0.930799i \(0.619113\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.89898 4.89898i 0.288175 0.288175i −0.548183 0.836358i \(-0.684680\pi\)
0.836358 + 0.548183i \(0.184680\pi\)
\(18\) 0 0
\(19\) 10.0000i 0.526316i −0.964753 0.263158i \(-0.915236\pi\)
0.964753 0.263158i \(-0.0847640\pi\)
\(20\) 0 0
\(21\) 18.0000 0.857143
\(22\) 0 0
\(23\) −19.5959 19.5959i −0.851996 0.851996i 0.138382 0.990379i \(-0.455810\pi\)
−0.990379 + 0.138382i \(0.955810\pi\)
\(24\) 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 \(-0.794851\pi\)
0.600795 + 0.799403i \(0.294851\pi\)
\(38\) 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 \(-0.0894151\pi\)
−0.277226 + 0.960805i \(0.589415\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 44.0908 44.0908i 0.938102 0.938102i −0.0600905 0.998193i \(-0.519139\pi\)
0.998193 + 0.0600905i \(0.0191389\pi\)
\(48\) 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 \(-0.270820\pi\)
−0.751812 + 0.659378i \(0.770820\pi\)
\(54\) 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 \(-0.723867\pi\)
0.646738 0.762712i \(-0.276133\pi\)
\(60\) 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 \(-0.595937\pi\)
0.954924 + 0.296852i \(0.0959367\pi\)
\(68\) 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 \(-0.109542\pi\)
−0.337384 + 0.941367i \(0.609542\pi\)
\(74\) 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 \(0.146099\pi\)
−0.896503 + 0.443038i \(0.853901\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) 53.8888 + 53.8888i 0.649262 + 0.649262i 0.952815 0.303552i \(-0.0981727\pi\)
−0.303552 + 0.952815i \(0.598173\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 90.0000i 1.01124i 0.862757 + 0.505618i \(0.168735\pi\)
−0.862757 + 0.505618i \(0.831265\pi\)
\(90\) 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 0.0625628 0.998041i \(-0.480073\pi\)
0.998041 0.0625628i \(-0.0199274\pi\)
\(98\) 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 \(-0.0999849\pi\)
−0.308972 + 0.951071i \(0.599985\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 44.0908 44.0908i 0.412064 0.412064i −0.470393 0.882457i \(-0.655888\pi\)
0.882457 + 0.470393i \(0.155888\pi\)
\(108\) 0 0
\(109\) 170.000i 1.55963i 0.626008 + 0.779817i \(0.284688\pi\)
−0.626008 + 0.779817i \(0.715312\pi\)
\(110\) 0 0
\(111\) −18.0000 −0.162162
\(112\) 0 0
\(113\) 44.0908 + 44.0908i 0.390184 + 0.390184i 0.874753 0.484569i \(-0.161024\pi\)
−0.484569 + 0.874753i \(0.661024\pi\)
\(114\) 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 \(-0.736973\pi\)
0.735446 + 0.677584i \(0.236973\pi\)
\(128\) 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 −0.0377220 + 0.999288i \(0.512010\pi\)
−0.999288 + 0.0377220i \(0.987990\pi\)
\(138\) 0 0
\(139\) 170.000i 1.22302i 0.791236 + 0.611511i \(0.209438\pi\)
−0.791236 + 0.611511i \(0.790562\pi\)
\(140\) 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 \(-0.793562\pi\)
0.796964 0.604027i \(-0.206438\pi\)
\(150\) 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 \(-0.533533\pi\)
0.994456 + 0.105151i \(0.0335326\pi\)
\(158\) 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 \(-0.420374\pi\)
−0.968875 + 0.247552i \(0.920374\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.5959 19.5959i 0.117341 0.117341i −0.645998 0.763339i \(-0.723559\pi\)
0.763339 + 0.645998i \(0.223559\pi\)
\(168\) 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 \(-0.0527838\pi\)
−0.165066 + 0.986282i \(0.552784\pi\)
\(174\) 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 \(0.0808899\pi\)
−0.967884 + 0.251397i \(0.919110\pi\)
\(180\) 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 0.859407 + 0.511292i \(0.170833\pi\)
0.511292 + 0.859407i \(0.329167\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 127.373 127.373i 0.646566 0.646566i −0.305596 0.952161i \(-0.598856\pi\)
0.952161 + 0.305596i \(0.0988556\pi\)
\(198\) 0 0
\(199\) 290.000i 1.45729i −0.684893 0.728643i \(-0.740151\pi\)
0.684893 0.728643i \(-0.259849\pi\)
\(200\) 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 \(-0.332508\pi\)
−0.864725 + 0.502245i \(0.832508\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −53.8888 + 53.8888i −0.237395 + 0.237395i −0.815771 0.578375i \(-0.803687\pi\)
0.578375 + 0.815771i \(0.303687\pi\)
\(228\) 0 0
\(229\) 50.0000i 0.218341i −0.994023 0.109170i \(-0.965181\pi\)
0.994023 0.109170i \(-0.0348194\pi\)
\(230\) 0 0
\(231\) 324.000 1.40260
\(232\) 0 0
\(233\) 44.0908 + 44.0908i 0.189231 + 0.189231i 0.795364 0.606133i \(-0.207280\pi\)
−0.606133 + 0.795364i \(0.707280\pi\)
\(234\) 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 \(-0.877104\pi\)
0.926389 0.376569i \(-0.122896\pi\)
\(240\) 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 −0.133638 + 0.991030i \(0.542666\pi\)
−0.991030 + 0.133638i \(0.957334\pi\)
\(258\) 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 −0.999983 0.00589147i \(-0.998125\pi\)
−0.00589147 0.999983i \(-0.501875\pi\)
\(264\) 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 \(-0.816157\pi\)
0.545981 + 0.837798i \(0.316157\pi\)
\(278\) 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 \(-0.0353954\pi\)
−0.110969 + 0.993824i \(0.535395\pi\)
\(284\) 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 −0.945708 0.325017i \(-0.894630\pi\)
−0.325017 0.945708i \(-0.605370\pi\)
\(294\) 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 −0.359369 + 0.933195i \(0.617008\pi\)
−0.933195 + 0.359369i \(0.882992\pi\)
\(308\) 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 \(-0.164424\pi\)
−0.493886 + 0.869527i \(0.664424\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 274.343 274.343i 0.865435 0.865435i −0.126528 0.991963i \(-0.540383\pi\)
0.991963 + 0.126528i \(0.0403834\pi\)
\(318\) 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 \(-0.881299\pi\)
0.364327 + 0.931271i \(0.381299\pi\)
\(338\) 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 \(-0.721362\pi\)
0.767778 + 0.640716i \(0.221362\pi\)
\(348\) 0 0
\(349\) 70.0000i 0.200573i 0.994959 + 0.100287i \(0.0319759\pi\)
−0.994959 + 0.100287i \(0.968024\pi\)
\(350\) 0 0
\(351\) −54.0000 −0.153846
\(352\) 0 0
\(353\) 44.0908 + 44.0908i 0.124903 + 0.124903i 0.766795 0.641892i \(-0.221850\pi\)
−0.641892 + 0.766795i \(0.721850\pi\)
\(354\) 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 \(0.270954\pi\)
−0.659061 + 0.752089i \(0.729046\pi\)
\(360\) 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 \(-0.790672\pi\)
0.611239 + 0.791446i \(0.290672\pi\)
\(368\) 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 \(-0.382377\pi\)
−0.932500 + 0.361171i \(0.882377\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 170.000i 0.448549i 0.974526 + 0.224274i \(0.0720012\pi\)
−0.974526 + 0.224274i \(0.927999\pi\)
\(380\) 0 0
\(381\) 18.0000 0.0472441
\(382\) 0 0
\(383\) 151.868 + 151.868i 0.396523 + 0.396523i 0.877005 0.480482i \(-0.159538\pi\)
−0.480482 + 0.877005i \(0.659538\pi\)
\(384\) 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 \(-0.846872\pi\)
0.886502 0.462725i \(-0.153128\pi\)
\(390\) 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 \(-0.930303\pi\)
0.217214 + 0.976124i \(0.430303\pi\)
\(398\) 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 \(0.305511\pi\)
−0.573692 + 0.819071i \(0.694489\pi\)
\(410\) 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 \(-0.729191\pi\)
0.659403 0.751790i \(-0.270809\pi\)
\(420\) 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 \(-0.303732\pi\)
−0.815853 + 0.578259i \(0.803732\pi\)
\(434\) 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 \(0.162912\pi\)
−0.871863 + 0.489749i \(0.837088\pi\)
\(440\) 0 0
\(441\) 177.000 0.401361
\(442\) 0 0
\(443\) −582.979 582.979i −1.31598 1.31598i −0.916930 0.399049i \(-0.869340\pi\)
−0.399049 0.916930i \(-0.630660\pi\)
\(444\) 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 \(0.0319555\pi\)
−0.994965 + 0.100223i \(0.968044\pi\)
\(450\) 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 \(-0.845527\pi\)
0.466465 + 0.884540i \(0.345527\pi\)
\(458\) 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 \(-0.253572\pi\)
−0.714998 + 0.699127i \(0.753572\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −347.828 + 347.828i −0.744813 + 0.744813i −0.973500 0.228687i \(-0.926557\pi\)
0.228687 + 0.973500i \(0.426557\pi\)
\(468\) 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 \(0.122626\pi\)
−0.926708 + 0.375783i \(0.877374\pi\)
\(480\) 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 −0.215561 + 0.976490i \(0.569158\pi\)
−0.976490 + 0.215561i \(0.930842\pi\)
\(488\) 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 \(-0.798661\pi\)
0.806538 0.591182i \(-0.201339\pi\)
\(500\) 0 0
\(501\) 48.0000 0.0958084
\(502\) 0 0
\(503\) −93.0806 93.0806i −0.185051 0.185051i 0.608502 0.793553i \(-0.291771\pi\)
−0.793553 + 0.608502i \(0.791771\pi\)
\(504\) 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 \(0.177978\pi\)
−0.847715 + 0.530452i \(0.822022\pi\)
\(510\) 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 \(-0.366429\pi\)
−0.913241 + 0.407419i \(0.866429\pi\)
\(524\) 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 \(-0.638903\pi\)
0.906288 + 0.422660i \(0.138903\pi\)
\(548\) 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 \(-0.961128\pi\)
0.121818 + 0.992552i \(0.461128\pi\)
\(558\) 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 \(-0.0839256\pi\)
−0.260616 + 0.965443i \(0.583926\pi\)
\(564\) 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 \(0.186747\pi\)
−0.832781 + 0.553603i \(0.813253\pi\)
\(570\) 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 \(-0.952595\pi\)
0.148378 + 0.988931i \(0.452595\pi\)
\(578\) 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 \(-0.502173\pi\)
0.999977 + 0.00682753i \(0.00217329\pi\)
\(588\) 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 \(-0.0302823\pi\)
−0.0949912 + 0.995478i \(0.530282\pi\)
\(594\) 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 \(0.148844\pi\)
−0.892650 + 0.450751i \(0.851156\pi\)
\(600\) 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 0.107455 0.994210i \(-0.465730\pi\)
0.994210 0.107455i \(-0.0342702\pi\)
\(608\) 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 \(-0.274307\pi\)
−0.758991 + 0.651101i \(0.774307\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 592.777 592.777i 0.960740 0.960740i −0.0385180 0.999258i \(-0.512264\pi\)
0.999258 + 0.0385180i \(0.0122637\pi\)
\(618\) 0 0
\(619\) 1030.00i 1.66397i 0.554795 + 0.831987i \(0.312797\pi\)
−0.554795 + 0.831987i \(0.687203\pi\)
\(620\) 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 \(-0.343961\pi\)
−0.882234 + 0.470811i \(0.843961\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 93.0806 93.0806i 0.143865 0.143865i −0.631506 0.775371i \(-0.717563\pi\)
0.775371 + 0.631506i \(0.217563\pi\)
\(648\) 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 \(-0.226338\pi\)
−0.652638 + 0.757670i \(0.726338\pi\)
\(654\) 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 \(-0.841363\pi\)
0.878361 0.477997i \(-0.158637\pi\)
\(660\) 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 \(-0.284474\pi\)
−0.779397 + 0.626531i \(0.784474\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 176.363 176.363i 0.260507 0.260507i −0.564753 0.825260i \(-0.691029\pi\)
0.825260 + 0.564753i \(0.191029\pi\)
\(678\) 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 \(-0.0972963\pi\)
−0.300928 + 0.953647i \(0.597296\pi\)
\(684\) 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 \(0.0701548\pi\)
−0.975811 + 0.218618i \(0.929845\pi\)
\(710\) 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 \(-0.960051\pi\)
0.992135 0.125174i \(-0.0399489\pi\)
\(720\) 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 \(-0.914162\pi\)
0.266412 + 0.963859i \(0.414162\pi\)
\(728\) 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 −0.995234 0.0975121i \(-0.968912\pi\)
−0.0975121 0.995234i \(-0.531088\pi\)
\(734\) 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 \(0.0761007\pi\)
−0.971557 + 0.236806i \(0.923899\pi\)
\(740\) 0 0
\(741\) 180.000 0.242915
\(742\) 0 0
\(743\) −925.907 925.907i −1.24617 1.24617i −0.957396 0.288778i \(-0.906751\pi\)
−0.288778 0.957396i \(-0.593249\pi\)
\(744\) 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 \(-0.663736\pi\)
0.870592 + 0.492005i \(0.163736\pi\)
\(758\) 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 \(0.125321\pi\)
−0.923493 + 0.383615i \(0.874679\pi\)
\(770\) 0 0
\(771\) −708.000 −0.918288
\(772\) 0 0
\(773\) −127.373 127.373i −0.164778 0.164778i 0.619902 0.784680i \(-0.287173\pi\)
−0.784680 + 0.619902i \(0.787173\pi\)
\(774\) 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 \(-0.673610\pi\)
0.854914 + 0.518770i \(0.173610\pi\)
\(788\) 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 \(-0.692971\pi\)
0.821797 + 0.569780i \(0.192971\pi\)
\(798\) 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 \(-0.872693\pi\)
0.921082 0.389370i \(-0.127307\pi\)
\(810\) 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 \(-0.149882\pi\)
−0.453660 + 0.891175i \(0.649882\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 876.917 876.917i 1.06036 1.06036i 0.0623022 0.998057i \(-0.480156\pi\)
0.998057 0.0623022i \(-0.0198443\pi\)
\(828\) 0 0
\(829\) 70.0000i 0.0844391i 0.999108 + 0.0422195i \(0.0134429\pi\)
−0.999108 + 0.0422195i \(0.986557\pi\)
\(830\) 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 \(0.104292\pi\)
−0.946804 + 0.321812i \(0.895708\pi\)
\(840\) 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 \(-0.0136385\pi\)
−0.0428336 + 0.999082i \(0.513639\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 298.838 298.838i 0.348702 0.348702i −0.510924 0.859626i \(-0.670697\pi\)
0.859626 + 0.510924i \(0.170697\pi\)
\(858\) 0 0
\(859\) 310.000i 0.360885i 0.983586 + 0.180442i \(0.0577529\pi\)
−0.983586 + 0.180442i \(0.942247\pi\)
\(860\) 0 0
\(861\) −324.000 −0.376307
\(862\) 0 0
\(863\) 225.353 + 225.353i 0.261128 + 0.261128i 0.825512 0.564385i \(-0.190886\pi\)
−0.564385 + 0.825512i \(0.690886\pi\)
\(864\) 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 \(-0.695034\pi\)
0.818088 + 0.575094i \(0.195034\pi\)
\(878\) 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 \(-0.280015\pi\)
−0.770543 + 0.637388i \(0.780015\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 509.494 509.494i 0.574401 0.574401i −0.358954 0.933355i \(-0.616867\pi\)
0.933355 + 0.358954i \(0.116867\pi\)
\(888\) 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 \(-0.646041\pi\)
0.896584 + 0.442874i \(0.146041\pi\)
\(908\) 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 \(-0.924835\pi\)
0.972249 0.233950i \(-0.0751652\pi\)
\(920\) 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 \(0.307963\pi\)
−0.567365 + 0.823466i \(0.692037\pi\)
\(930\) 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 \(-0.671539\pi\)
0.858270 + 0.513198i \(0.171539\pi\)
\(938\) 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 \(-0.936270\pi\)
0.198878 + 0.980024i \(0.436270\pi\)
\(948\) 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 \(-0.239585\pi\)
−0.683596 + 0.729861i \(0.739585\pi\)
\(954\) 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 \(-0.928550\pi\)
0.222588 + 0.974913i \(0.428550\pi\)
\(968\) 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 \(-0.657276\pi\)
0.880397 + 0.474238i \(0.157276\pi\)
\(978\) 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 \(-0.453069\pi\)
−0.989151 + 0.146905i \(0.953069\pi\)
\(984\) 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 \(-0.946365\pi\)
0.167703 + 0.985838i \(0.446365\pi\)
\(998\) 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.193.2 4
4.3 odd 2 75.3.f.a.43.1 yes 4
5.2 odd 4 inner 1200.3.bg.j.1057.2 4
5.3 odd 4 inner 1200.3.bg.j.1057.1 4
5.4 even 2 inner 1200.3.bg.j.193.1 4
12.11 even 2 225.3.g.f.118.2 4
20.3 even 4 75.3.f.a.7.2 yes 4
20.7 even 4 75.3.f.a.7.1 4
20.19 odd 2 75.3.f.a.43.2 yes 4
60.23 odd 4 225.3.g.f.82.1 4
60.47 odd 4 225.3.g.f.82.2 4
60.59 even 2 225.3.g.f.118.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.3.f.a.7.1 4 20.7 even 4
75.3.f.a.7.2 yes 4 20.3 even 4
75.3.f.a.43.1 yes 4 4.3 odd 2
75.3.f.a.43.2 yes 4 20.19 odd 2
225.3.g.f.82.1 4 60.23 odd 4
225.3.g.f.82.2 4 60.47 odd 4
225.3.g.f.118.1 4 60.59 even 2
225.3.g.f.118.2 4 12.11 even 2
1200.3.bg.j.193.1 4 5.4 even 2 inner
1200.3.bg.j.193.2 4 1.1 even 1 trivial
1200.3.bg.j.1057.1 4 5.3 odd 4 inner
1200.3.bg.j.1057.2 4 5.2 odd 4 inner