Properties

Label 1089.3.c.g.604.1
Level $1089$
Weight $3$
Character 1089.604
Analytic conductor $29.673$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,3,Mod(604,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.604");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.c (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: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 363)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 604.1
Root \(-1.93185i\) of defining polynomial
Character \(\chi\) \(=\) 1089.604
Dual form 1089.3.c.g.604.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.93185i q^{2} +0.267949 q^{4} -3.19615 q^{5} +13.7632i q^{7} -8.24504i q^{8} +O(q^{10})\) \(q-1.93185i q^{2} +0.267949 q^{4} -3.19615 q^{5} +13.7632i q^{7} -8.24504i q^{8} +6.17449i q^{10} -13.6245i q^{13} +26.5885 q^{14} -14.8564 q^{16} +19.6603i q^{17} -3.58630i q^{19} -0.856406 q^{20} -10.0526 q^{23} -14.7846 q^{25} -26.3205 q^{26} +3.68784i q^{28} -47.6400i q^{29} -2.67949 q^{31} -4.27981i q^{32} +37.9808 q^{34} -43.9893i q^{35} +1.87564 q^{37} -6.92820 q^{38} +26.3524i q^{40} +62.2626i q^{41} -2.92996i q^{43} +19.4201i q^{46} -62.8897 q^{47} -140.426 q^{49} +28.5617i q^{50} -3.65067i q^{52} -14.2154 q^{53} +113.478 q^{56} -92.0333 q^{58} -103.909 q^{59} +51.6424i q^{61} +5.17638i q^{62} -67.6936 q^{64} +43.5460i q^{65} -16.0910 q^{67} +5.26796i q^{68} -84.9808 q^{70} -54.6936 q^{71} -85.4348i q^{73} -3.62347i q^{74} -0.960947i q^{76} -8.83701i q^{79} +47.4833 q^{80} +120.282 q^{82} +56.0137i q^{83} -62.8373i q^{85} -5.66025 q^{86} -44.1628 q^{89} +187.517 q^{91} -2.69358 q^{92} +121.494i q^{94} +11.4624i q^{95} +78.2295 q^{97} +271.281i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 8 q^{5} + 44 q^{14} - 4 q^{16} + 52 q^{20} + 36 q^{23} + 24 q^{25} - 36 q^{26} - 80 q^{31} + 48 q^{34} + 56 q^{37} - 16 q^{47} - 340 q^{49} - 140 q^{53} + 156 q^{56} - 188 q^{58} - 284 q^{59} - 56 q^{64} - 196 q^{67} - 236 q^{70} - 4 q^{71} + 280 q^{80} + 204 q^{82} + 12 q^{86} - 336 q^{89} + 660 q^{91} + 204 q^{92} + 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.93185i − 0.965926i −0.875641 0.482963i \(-0.839561\pi\)
0.875641 0.482963i \(-0.160439\pi\)
\(3\) 0 0
\(4\) 0.267949 0.0669873
\(5\) −3.19615 −0.639230 −0.319615 0.947547i \(-0.603554\pi\)
−0.319615 + 0.947547i \(0.603554\pi\)
\(6\) 0 0
\(7\) 13.7632i 1.96617i 0.183147 + 0.983086i \(0.441372\pi\)
−0.183147 + 0.983086i \(0.558628\pi\)
\(8\) − 8.24504i − 1.03063i
\(9\) 0 0
\(10\) 6.17449i 0.617449i
\(11\) 0 0
\(12\) 0 0
\(13\) − 13.6245i − 1.04804i −0.851706 0.524019i \(-0.824432\pi\)
0.851706 0.524019i \(-0.175568\pi\)
\(14\) 26.5885 1.89918
\(15\) 0 0
\(16\) −14.8564 −0.928525
\(17\) 19.6603i 1.15649i 0.815864 + 0.578244i \(0.196262\pi\)
−0.815864 + 0.578244i \(0.803738\pi\)
\(18\) 0 0
\(19\) − 3.58630i − 0.188753i −0.995537 0.0943764i \(-0.969914\pi\)
0.995537 0.0943764i \(-0.0300857\pi\)
\(20\) −0.856406 −0.0428203
\(21\) 0 0
\(22\) 0 0
\(23\) −10.0526 −0.437068 −0.218534 0.975829i \(-0.570127\pi\)
−0.218534 + 0.975829i \(0.570127\pi\)
\(24\) 0 0
\(25\) −14.7846 −0.591384
\(26\) −26.3205 −1.01233
\(27\) 0 0
\(28\) 3.68784i 0.131708i
\(29\) − 47.6400i − 1.64276i −0.570384 0.821378i \(-0.693205\pi\)
0.570384 0.821378i \(-0.306795\pi\)
\(30\) 0 0
\(31\) −2.67949 −0.0864352 −0.0432176 0.999066i \(-0.513761\pi\)
−0.0432176 + 0.999066i \(0.513761\pi\)
\(32\) − 4.27981i − 0.133744i
\(33\) 0 0
\(34\) 37.9808 1.11708
\(35\) − 43.9893i − 1.25684i
\(36\) 0 0
\(37\) 1.87564 0.0506931 0.0253465 0.999679i \(-0.491931\pi\)
0.0253465 + 0.999679i \(0.491931\pi\)
\(38\) −6.92820 −0.182321
\(39\) 0 0
\(40\) 26.3524i 0.658810i
\(41\) 62.2626i 1.51860i 0.650741 + 0.759300i \(0.274458\pi\)
−0.650741 + 0.759300i \(0.725542\pi\)
\(42\) 0 0
\(43\) − 2.92996i − 0.0681387i −0.999419 0.0340693i \(-0.989153\pi\)
0.999419 0.0340693i \(-0.0108467\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 19.4201i 0.422175i
\(47\) −62.8897 −1.33808 −0.669040 0.743227i \(-0.733294\pi\)
−0.669040 + 0.743227i \(0.733294\pi\)
\(48\) 0 0
\(49\) −140.426 −2.86583
\(50\) 28.5617i 0.571233i
\(51\) 0 0
\(52\) − 3.65067i − 0.0702053i
\(53\) −14.2154 −0.268215 −0.134107 0.990967i \(-0.542817\pi\)
−0.134107 + 0.990967i \(0.542817\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 113.478 2.02640
\(57\) 0 0
\(58\) −92.0333 −1.58678
\(59\) −103.909 −1.76117 −0.880584 0.473889i \(-0.842850\pi\)
−0.880584 + 0.473889i \(0.842850\pi\)
\(60\) 0 0
\(61\) 51.6424i 0.846596i 0.905991 + 0.423298i \(0.139128\pi\)
−0.905991 + 0.423298i \(0.860872\pi\)
\(62\) 5.17638i 0.0834900i
\(63\) 0 0
\(64\) −67.6936 −1.05771
\(65\) 43.5460i 0.669938i
\(66\) 0 0
\(67\) −16.0910 −0.240165 −0.120082 0.992764i \(-0.538316\pi\)
−0.120082 + 0.992764i \(0.538316\pi\)
\(68\) 5.26796i 0.0774700i
\(69\) 0 0
\(70\) −84.9808 −1.21401
\(71\) −54.6936 −0.770332 −0.385166 0.922847i \(-0.625856\pi\)
−0.385166 + 0.922847i \(0.625856\pi\)
\(72\) 0 0
\(73\) − 85.4348i − 1.17034i −0.810911 0.585170i \(-0.801028\pi\)
0.810911 0.585170i \(-0.198972\pi\)
\(74\) − 3.62347i − 0.0489658i
\(75\) 0 0
\(76\) − 0.960947i − 0.0126440i
\(77\) 0 0
\(78\) 0 0
\(79\) − 8.83701i − 0.111861i −0.998435 0.0559305i \(-0.982187\pi\)
0.998435 0.0559305i \(-0.0178125\pi\)
\(80\) 47.4833 0.593542
\(81\) 0 0
\(82\) 120.282 1.46685
\(83\) 56.0137i 0.674864i 0.941350 + 0.337432i \(0.109558\pi\)
−0.941350 + 0.337432i \(0.890442\pi\)
\(84\) 0 0
\(85\) − 62.8373i − 0.739262i
\(86\) −5.66025 −0.0658169
\(87\) 0 0
\(88\) 0 0
\(89\) −44.1628 −0.496212 −0.248106 0.968733i \(-0.579808\pi\)
−0.248106 + 0.968733i \(0.579808\pi\)
\(90\) 0 0
\(91\) 187.517 2.06062
\(92\) −2.69358 −0.0292780
\(93\) 0 0
\(94\) 121.494i 1.29249i
\(95\) 11.4624i 0.120656i
\(96\) 0 0
\(97\) 78.2295 0.806489 0.403245 0.915092i \(-0.367882\pi\)
0.403245 + 0.915092i \(0.367882\pi\)
\(98\) 271.281i 2.76818i
\(99\) 0 0
\(100\) −3.96152 −0.0396152
\(101\) 20.6783i 0.204736i 0.994747 + 0.102368i \(0.0326419\pi\)
−0.994747 + 0.102368i \(0.967358\pi\)
\(102\) 0 0
\(103\) −122.799 −1.19222 −0.596110 0.802903i \(-0.703288\pi\)
−0.596110 + 0.802903i \(0.703288\pi\)
\(104\) −112.335 −1.08014
\(105\) 0 0
\(106\) 27.4620i 0.259076i
\(107\) − 14.4467i − 0.135016i −0.997719 0.0675081i \(-0.978495\pi\)
0.997719 0.0675081i \(-0.0215049\pi\)
\(108\) 0 0
\(109\) 107.490i 0.986149i 0.869987 + 0.493074i \(0.164127\pi\)
−0.869987 + 0.493074i \(0.835873\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 204.472i − 1.82564i
\(113\) 127.785 1.13084 0.565419 0.824804i \(-0.308715\pi\)
0.565419 + 0.824804i \(0.308715\pi\)
\(114\) 0 0
\(115\) 32.1295 0.279387
\(116\) − 12.7651i − 0.110044i
\(117\) 0 0
\(118\) 200.737i 1.70116i
\(119\) −270.588 −2.27385
\(120\) 0 0
\(121\) 0 0
\(122\) 99.7654 0.817749
\(123\) 0 0
\(124\) −0.717968 −0.00579006
\(125\) 127.158 1.01726
\(126\) 0 0
\(127\) − 162.553i − 1.27994i −0.768398 0.639972i \(-0.778946\pi\)
0.768398 0.639972i \(-0.221054\pi\)
\(128\) 113.655i 0.887928i
\(129\) 0 0
\(130\) 84.1244 0.647110
\(131\) 29.0893i 0.222055i 0.993817 + 0.111028i \(0.0354142\pi\)
−0.993817 + 0.111028i \(0.964586\pi\)
\(132\) 0 0
\(133\) 49.3590 0.371120
\(134\) 31.0855i 0.231981i
\(135\) 0 0
\(136\) 162.100 1.19191
\(137\) −52.0666 −0.380048 −0.190024 0.981779i \(-0.560857\pi\)
−0.190024 + 0.981779i \(0.560857\pi\)
\(138\) 0 0
\(139\) − 163.913i − 1.17923i −0.807685 0.589614i \(-0.799280\pi\)
0.807685 0.589614i \(-0.200720\pi\)
\(140\) − 11.7869i − 0.0841921i
\(141\) 0 0
\(142\) 105.660i 0.744084i
\(143\) 0 0
\(144\) 0 0
\(145\) 152.265i 1.05010i
\(146\) −165.047 −1.13046
\(147\) 0 0
\(148\) 0.502577 0.00339579
\(149\) 170.602i 1.14498i 0.819911 + 0.572491i \(0.194023\pi\)
−0.819911 + 0.572491i \(0.805977\pi\)
\(150\) 0 0
\(151\) 40.1528i 0.265912i 0.991122 + 0.132956i \(0.0424470\pi\)
−0.991122 + 0.132956i \(0.957553\pi\)
\(152\) −29.5692 −0.194534
\(153\) 0 0
\(154\) 0 0
\(155\) 8.56406 0.0552520
\(156\) 0 0
\(157\) 97.9615 0.623959 0.311979 0.950089i \(-0.399008\pi\)
0.311979 + 0.950089i \(0.399008\pi\)
\(158\) −17.0718 −0.108049
\(159\) 0 0
\(160\) 13.6789i 0.0854932i
\(161\) − 138.355i − 0.859350i
\(162\) 0 0
\(163\) −225.478 −1.38330 −0.691651 0.722232i \(-0.743116\pi\)
−0.691651 + 0.722232i \(0.743116\pi\)
\(164\) 16.6832i 0.101727i
\(165\) 0 0
\(166\) 108.210 0.651869
\(167\) 36.7768i 0.220221i 0.993919 + 0.110110i \(0.0351204\pi\)
−0.993919 + 0.110110i \(0.964880\pi\)
\(168\) 0 0
\(169\) −16.6269 −0.0983842
\(170\) −121.392 −0.714072
\(171\) 0 0
\(172\) − 0.785081i − 0.00456443i
\(173\) 242.371i 1.40099i 0.713659 + 0.700493i \(0.247037\pi\)
−0.713659 + 0.700493i \(0.752963\pi\)
\(174\) 0 0
\(175\) − 203.484i − 1.16276i
\(176\) 0 0
\(177\) 0 0
\(178\) 85.3160i 0.479304i
\(179\) −9.32051 −0.0520699 −0.0260349 0.999661i \(-0.508288\pi\)
−0.0260349 + 0.999661i \(0.508288\pi\)
\(180\) 0 0
\(181\) 231.186 1.27727 0.638635 0.769510i \(-0.279499\pi\)
0.638635 + 0.769510i \(0.279499\pi\)
\(182\) − 362.254i − 1.99041i
\(183\) 0 0
\(184\) 82.8838i 0.450455i
\(185\) −5.99485 −0.0324046
\(186\) 0 0
\(187\) 0 0
\(188\) −16.8513 −0.0896343
\(189\) 0 0
\(190\) 22.1436 0.116545
\(191\) −244.133 −1.27818 −0.639092 0.769130i \(-0.720690\pi\)
−0.639092 + 0.769130i \(0.720690\pi\)
\(192\) 0 0
\(193\) 10.7888i 0.0559004i 0.999609 + 0.0279502i \(0.00889799\pi\)
−0.999609 + 0.0279502i \(0.991102\pi\)
\(194\) − 151.128i − 0.779009i
\(195\) 0 0
\(196\) −37.6269 −0.191974
\(197\) − 109.967i − 0.558208i −0.960261 0.279104i \(-0.909963\pi\)
0.960261 0.279104i \(-0.0900373\pi\)
\(198\) 0 0
\(199\) −28.5218 −0.143326 −0.0716629 0.997429i \(-0.522831\pi\)
−0.0716629 + 0.997429i \(0.522831\pi\)
\(200\) 121.900i 0.609499i
\(201\) 0 0
\(202\) 39.9474 0.197760
\(203\) 655.678 3.22994
\(204\) 0 0
\(205\) − 199.001i − 0.970735i
\(206\) 237.229i 1.15160i
\(207\) 0 0
\(208\) 202.411i 0.973130i
\(209\) 0 0
\(210\) 0 0
\(211\) 172.101i 0.815643i 0.913062 + 0.407822i \(0.133711\pi\)
−0.913062 + 0.407822i \(0.866289\pi\)
\(212\) −3.80900 −0.0179670
\(213\) 0 0
\(214\) −27.9090 −0.130416
\(215\) 9.36461i 0.0435563i
\(216\) 0 0
\(217\) − 36.8784i − 0.169946i
\(218\) 207.655 0.952546
\(219\) 0 0
\(220\) 0 0
\(221\) 267.862 1.21204
\(222\) 0 0
\(223\) −146.603 −0.657410 −0.328705 0.944433i \(-0.606612\pi\)
−0.328705 + 0.944433i \(0.606612\pi\)
\(224\) 58.9038 0.262963
\(225\) 0 0
\(226\) − 246.861i − 1.09230i
\(227\) 68.7616i 0.302914i 0.988464 + 0.151457i \(0.0483966\pi\)
−0.988464 + 0.151457i \(0.951603\pi\)
\(228\) 0 0
\(229\) −252.755 −1.10373 −0.551867 0.833932i \(-0.686084\pi\)
−0.551867 + 0.833932i \(0.686084\pi\)
\(230\) − 62.0694i − 0.269867i
\(231\) 0 0
\(232\) −392.794 −1.69308
\(233\) − 282.853i − 1.21396i −0.794717 0.606980i \(-0.792381\pi\)
0.794717 0.606980i \(-0.207619\pi\)
\(234\) 0 0
\(235\) 201.005 0.855341
\(236\) −27.8423 −0.117976
\(237\) 0 0
\(238\) 522.737i 2.19637i
\(239\) − 360.434i − 1.50809i −0.656822 0.754046i \(-0.728100\pi\)
0.656822 0.754046i \(-0.271900\pi\)
\(240\) 0 0
\(241\) 111.282i 0.461752i 0.972983 + 0.230876i \(0.0741592\pi\)
−0.972983 + 0.230876i \(0.925841\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 13.8375i 0.0567112i
\(245\) 448.822 1.83193
\(246\) 0 0
\(247\) −48.8616 −0.197820
\(248\) 22.0925i 0.0890828i
\(249\) 0 0
\(250\) − 245.650i − 0.982599i
\(251\) 366.406 1.45979 0.729893 0.683561i \(-0.239570\pi\)
0.729893 + 0.683561i \(0.239570\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −314.028 −1.23633
\(255\) 0 0
\(256\) −51.2102 −0.200040
\(257\) 47.5500 0.185019 0.0925097 0.995712i \(-0.470511\pi\)
0.0925097 + 0.995712i \(0.470511\pi\)
\(258\) 0 0
\(259\) 25.8149i 0.0996713i
\(260\) 11.6681i 0.0448773i
\(261\) 0 0
\(262\) 56.1962 0.214489
\(263\) − 63.4637i − 0.241307i −0.992695 0.120654i \(-0.961501\pi\)
0.992695 0.120654i \(-0.0384990\pi\)
\(264\) 0 0
\(265\) 45.4346 0.171451
\(266\) − 95.3542i − 0.358475i
\(267\) 0 0
\(268\) −4.31158 −0.0160880
\(269\) −145.172 −0.539672 −0.269836 0.962906i \(-0.586969\pi\)
−0.269836 + 0.962906i \(0.586969\pi\)
\(270\) 0 0
\(271\) 315.890i 1.16565i 0.812599 + 0.582823i \(0.198052\pi\)
−0.812599 + 0.582823i \(0.801948\pi\)
\(272\) − 292.081i − 1.07383i
\(273\) 0 0
\(274\) 100.585i 0.367099i
\(275\) 0 0
\(276\) 0 0
\(277\) 319.073i 1.15189i 0.817489 + 0.575944i \(0.195365\pi\)
−0.817489 + 0.575944i \(0.804635\pi\)
\(278\) −316.655 −1.13905
\(279\) 0 0
\(280\) −362.694 −1.29533
\(281\) 365.935i 1.30226i 0.758966 + 0.651130i \(0.225705\pi\)
−0.758966 + 0.651130i \(0.774295\pi\)
\(282\) 0 0
\(283\) − 231.978i − 0.819711i −0.912151 0.409855i \(-0.865579\pi\)
0.912151 0.409855i \(-0.134421\pi\)
\(284\) −14.6551 −0.0516025
\(285\) 0 0
\(286\) 0 0
\(287\) −856.932 −2.98583
\(288\) 0 0
\(289\) −97.5270 −0.337464
\(290\) 294.153 1.01432
\(291\) 0 0
\(292\) − 22.8922i − 0.0783979i
\(293\) − 151.128i − 0.515794i −0.966172 0.257897i \(-0.916970\pi\)
0.966172 0.257897i \(-0.0830296\pi\)
\(294\) 0 0
\(295\) 332.109 1.12579
\(296\) − 15.4648i − 0.0522458i
\(297\) 0 0
\(298\) 329.578 1.10597
\(299\) 136.961i 0.458064i
\(300\) 0 0
\(301\) 40.3257 0.133972
\(302\) 77.5692 0.256852
\(303\) 0 0
\(304\) 53.2796i 0.175262i
\(305\) − 165.057i − 0.541170i
\(306\) 0 0
\(307\) − 396.582i − 1.29180i −0.763423 0.645899i \(-0.776483\pi\)
0.763423 0.645899i \(-0.223517\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) − 16.5445i − 0.0533694i
\(311\) 219.951 0.707239 0.353619 0.935389i \(-0.384951\pi\)
0.353619 + 0.935389i \(0.384951\pi\)
\(312\) 0 0
\(313\) −31.6410 −0.101090 −0.0505448 0.998722i \(-0.516096\pi\)
−0.0505448 + 0.998722i \(0.516096\pi\)
\(314\) − 189.247i − 0.602698i
\(315\) 0 0
\(316\) − 2.36787i − 0.00749326i
\(317\) 307.990 0.971576 0.485788 0.874077i \(-0.338533\pi\)
0.485788 + 0.874077i \(0.338533\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 216.359 0.676122
\(321\) 0 0
\(322\) −267.282 −0.830068
\(323\) 70.5077 0.218290
\(324\) 0 0
\(325\) 201.433i 0.619793i
\(326\) 435.590i 1.33617i
\(327\) 0 0
\(328\) 513.358 1.56511
\(329\) − 865.564i − 2.63089i
\(330\) 0 0
\(331\) −152.515 −0.460771 −0.230386 0.973099i \(-0.573999\pi\)
−0.230386 + 0.973099i \(0.573999\pi\)
\(332\) 15.0088i 0.0452073i
\(333\) 0 0
\(334\) 71.0474 0.212717
\(335\) 51.4294 0.153521
\(336\) 0 0
\(337\) − 404.069i − 1.19902i −0.800368 0.599509i \(-0.795363\pi\)
0.800368 0.599509i \(-0.204637\pi\)
\(338\) 32.1208i 0.0950319i
\(339\) 0 0
\(340\) − 16.8372i − 0.0495212i
\(341\) 0 0
\(342\) 0 0
\(343\) − 1258.31i − 3.66854i
\(344\) −24.1577 −0.0702258
\(345\) 0 0
\(346\) 468.224 1.35325
\(347\) − 66.6240i − 0.192000i −0.995381 0.0960000i \(-0.969395\pi\)
0.995381 0.0960000i \(-0.0306049\pi\)
\(348\) 0 0
\(349\) 36.5147i 0.104627i 0.998631 + 0.0523134i \(0.0166595\pi\)
−0.998631 + 0.0523134i \(0.983341\pi\)
\(350\) −393.100 −1.12314
\(351\) 0 0
\(352\) 0 0
\(353\) 329.196 0.932567 0.466284 0.884635i \(-0.345593\pi\)
0.466284 + 0.884635i \(0.345593\pi\)
\(354\) 0 0
\(355\) 174.809 0.492420
\(356\) −11.8334 −0.0332399
\(357\) 0 0
\(358\) 18.0058i 0.0502956i
\(359\) − 618.017i − 1.72149i −0.509032 0.860747i \(-0.669997\pi\)
0.509032 0.860747i \(-0.330003\pi\)
\(360\) 0 0
\(361\) 348.138 0.964372
\(362\) − 446.617i − 1.23375i
\(363\) 0 0
\(364\) 50.2449 0.138036
\(365\) 273.063i 0.748117i
\(366\) 0 0
\(367\) −180.809 −0.492668 −0.246334 0.969185i \(-0.579226\pi\)
−0.246334 + 0.969185i \(0.579226\pi\)
\(368\) 149.345 0.405829
\(369\) 0 0
\(370\) 11.5812i 0.0313004i
\(371\) − 195.649i − 0.527356i
\(372\) 0 0
\(373\) − 343.842i − 0.921829i −0.887444 0.460915i \(-0.847521\pi\)
0.887444 0.460915i \(-0.152479\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 518.529i 1.37907i
\(377\) −649.070 −1.72167
\(378\) 0 0
\(379\) −353.611 −0.933012 −0.466506 0.884518i \(-0.654487\pi\)
−0.466506 + 0.884518i \(0.654487\pi\)
\(380\) 3.07133i 0.00808245i
\(381\) 0 0
\(382\) 471.629i 1.23463i
\(383\) 158.070 0.412716 0.206358 0.978477i \(-0.433839\pi\)
0.206358 + 0.978477i \(0.433839\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 20.8423 0.0539957
\(387\) 0 0
\(388\) 20.9615 0.0540245
\(389\) 407.242 1.04690 0.523448 0.852058i \(-0.324646\pi\)
0.523448 + 0.852058i \(0.324646\pi\)
\(390\) 0 0
\(391\) − 197.636i − 0.505463i
\(392\) 1157.82i 2.95361i
\(393\) 0 0
\(394\) −212.440 −0.539187
\(395\) 28.2444i 0.0715049i
\(396\) 0 0
\(397\) 4.07180 0.0102564 0.00512821 0.999987i \(-0.498368\pi\)
0.00512821 + 0.999987i \(0.498368\pi\)
\(398\) 55.0999i 0.138442i
\(399\) 0 0
\(400\) 219.646 0.549115
\(401\) −433.862 −1.08195 −0.540975 0.841039i \(-0.681944\pi\)
−0.540975 + 0.841039i \(0.681944\pi\)
\(402\) 0 0
\(403\) 36.5067i 0.0905874i
\(404\) 5.54074i 0.0137147i
\(405\) 0 0
\(406\) − 1266.67i − 3.11988i
\(407\) 0 0
\(408\) 0 0
\(409\) 105.088i 0.256938i 0.991713 + 0.128469i \(0.0410064\pi\)
−0.991713 + 0.128469i \(0.958994\pi\)
\(410\) −384.440 −0.937658
\(411\) 0 0
\(412\) −32.9038 −0.0798636
\(413\) − 1430.12i − 3.46276i
\(414\) 0 0
\(415\) − 179.028i − 0.431394i
\(416\) −58.3102 −0.140169
\(417\) 0 0
\(418\) 0 0
\(419\) −72.6757 −0.173450 −0.0867252 0.996232i \(-0.527640\pi\)
−0.0867252 + 0.996232i \(0.527640\pi\)
\(420\) 0 0
\(421\) −343.172 −0.815135 −0.407567 0.913175i \(-0.633623\pi\)
−0.407567 + 0.913175i \(0.633623\pi\)
\(422\) 332.473 0.787851
\(423\) 0 0
\(424\) 117.207i 0.276430i
\(425\) − 290.670i − 0.683929i
\(426\) 0 0
\(427\) −710.764 −1.66455
\(428\) − 3.87099i − 0.00904438i
\(429\) 0 0
\(430\) 18.0910 0.0420722
\(431\) − 409.946i − 0.951151i −0.879675 0.475575i \(-0.842240\pi\)
0.879675 0.475575i \(-0.157760\pi\)
\(432\) 0 0
\(433\) −573.158 −1.32369 −0.661845 0.749641i \(-0.730226\pi\)
−0.661845 + 0.749641i \(0.730226\pi\)
\(434\) −71.2436 −0.164156
\(435\) 0 0
\(436\) 28.8019i 0.0660594i
\(437\) 36.0515i 0.0824977i
\(438\) 0 0
\(439\) − 146.143i − 0.332899i −0.986050 0.166449i \(-0.946770\pi\)
0.986050 0.166449i \(-0.0532302\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 517.469i − 1.17074i
\(443\) −470.788 −1.06273 −0.531364 0.847144i \(-0.678320\pi\)
−0.531364 + 0.847144i \(0.678320\pi\)
\(444\) 0 0
\(445\) 141.151 0.317194
\(446\) 283.214i 0.635010i
\(447\) 0 0
\(448\) − 931.680i − 2.07964i
\(449\) 271.722 0.605171 0.302585 0.953122i \(-0.402150\pi\)
0.302585 + 0.953122i \(0.402150\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 34.2398 0.0757517
\(453\) 0 0
\(454\) 132.837 0.292593
\(455\) −599.332 −1.31721
\(456\) 0 0
\(457\) 731.035i 1.59964i 0.600241 + 0.799819i \(0.295071\pi\)
−0.600241 + 0.799819i \(0.704929\pi\)
\(458\) 488.285i 1.06613i
\(459\) 0 0
\(460\) 8.60908 0.0187154
\(461\) 271.180i 0.588243i 0.955768 + 0.294121i \(0.0950270\pi\)
−0.955768 + 0.294121i \(0.904973\pi\)
\(462\) 0 0
\(463\) −755.381 −1.63149 −0.815746 0.578411i \(-0.803673\pi\)
−0.815746 + 0.578411i \(0.803673\pi\)
\(464\) 707.758i 1.52534i
\(465\) 0 0
\(466\) −546.429 −1.17260
\(467\) −173.919 −0.372418 −0.186209 0.982510i \(-0.559620\pi\)
−0.186209 + 0.982510i \(0.559620\pi\)
\(468\) 0 0
\(469\) − 221.464i − 0.472205i
\(470\) − 388.312i − 0.826196i
\(471\) 0 0
\(472\) 856.734i 1.81511i
\(473\) 0 0
\(474\) 0 0
\(475\) 53.0221i 0.111625i
\(476\) −72.5040 −0.152319
\(477\) 0 0
\(478\) −696.305 −1.45671
\(479\) 458.787i 0.957802i 0.877869 + 0.478901i \(0.158965\pi\)
−0.877869 + 0.478901i \(0.841035\pi\)
\(480\) 0 0
\(481\) − 25.5547i − 0.0531283i
\(482\) 214.981 0.446018
\(483\) 0 0
\(484\) 0 0
\(485\) −250.033 −0.515533
\(486\) 0 0
\(487\) 483.734 0.993295 0.496647 0.867952i \(-0.334564\pi\)
0.496647 + 0.867952i \(0.334564\pi\)
\(488\) 425.794 0.872528
\(489\) 0 0
\(490\) − 867.057i − 1.76950i
\(491\) 676.366i 1.37753i 0.724986 + 0.688763i \(0.241846\pi\)
−0.724986 + 0.688763i \(0.758154\pi\)
\(492\) 0 0
\(493\) 936.615 1.89983
\(494\) 94.3933i 0.191080i
\(495\) 0 0
\(496\) 39.8076 0.0802573
\(497\) − 752.759i − 1.51460i
\(498\) 0 0
\(499\) 851.486 1.70638 0.853192 0.521597i \(-0.174663\pi\)
0.853192 + 0.521597i \(0.174663\pi\)
\(500\) 34.0718 0.0681436
\(501\) 0 0
\(502\) − 707.843i − 1.41005i
\(503\) 224.969i 0.447254i 0.974675 + 0.223627i \(0.0717898\pi\)
−0.974675 + 0.223627i \(0.928210\pi\)
\(504\) 0 0
\(505\) − 66.0911i − 0.130873i
\(506\) 0 0
\(507\) 0 0
\(508\) − 43.5559i − 0.0857400i
\(509\) 845.520 1.66114 0.830570 0.556914i \(-0.188015\pi\)
0.830570 + 0.556914i \(0.188015\pi\)
\(510\) 0 0
\(511\) 1175.86 2.30109
\(512\) 553.549i 1.08115i
\(513\) 0 0
\(514\) − 91.8595i − 0.178715i
\(515\) 392.483 0.762104
\(516\) 0 0
\(517\) 0 0
\(518\) 49.8705 0.0962751
\(519\) 0 0
\(520\) 359.038 0.690459
\(521\) −272.841 −0.523687 −0.261844 0.965110i \(-0.584330\pi\)
−0.261844 + 0.965110i \(0.584330\pi\)
\(522\) 0 0
\(523\) 371.713i 0.710733i 0.934727 + 0.355366i \(0.115644\pi\)
−0.934727 + 0.355366i \(0.884356\pi\)
\(524\) 7.79445i 0.0148749i
\(525\) 0 0
\(526\) −122.603 −0.233085
\(527\) − 52.6796i − 0.0999613i
\(528\) 0 0
\(529\) −427.946 −0.808972
\(530\) − 87.7728i − 0.165609i
\(531\) 0 0
\(532\) 13.2257 0.0248603
\(533\) 848.296 1.59155
\(534\) 0 0
\(535\) 46.1740i 0.0863065i
\(536\) 132.671i 0.247521i
\(537\) 0 0
\(538\) 280.450i 0.521283i
\(539\) 0 0
\(540\) 0 0
\(541\) − 202.644i − 0.374573i −0.982305 0.187287i \(-0.940031\pi\)
0.982305 0.187287i \(-0.0599693\pi\)
\(542\) 610.252 1.12593
\(543\) 0 0
\(544\) 84.1422 0.154673
\(545\) − 343.555i − 0.630376i
\(546\) 0 0
\(547\) 835.947i 1.52824i 0.645075 + 0.764120i \(0.276826\pi\)
−0.645075 + 0.764120i \(0.723174\pi\)
\(548\) −13.9512 −0.0254584
\(549\) 0 0
\(550\) 0 0
\(551\) −170.851 −0.310075
\(552\) 0 0
\(553\) 121.626 0.219938
\(554\) 616.401 1.11264
\(555\) 0 0
\(556\) − 43.9203i − 0.0789933i
\(557\) − 840.516i − 1.50901i −0.656297 0.754503i \(-0.727878\pi\)
0.656297 0.754503i \(-0.272122\pi\)
\(558\) 0 0
\(559\) −39.9193 −0.0714119
\(560\) 653.523i 1.16700i
\(561\) 0 0
\(562\) 706.932 1.25789
\(563\) 519.631i 0.922968i 0.887149 + 0.461484i \(0.152683\pi\)
−0.887149 + 0.461484i \(0.847317\pi\)
\(564\) 0 0
\(565\) −408.419 −0.722866
\(566\) −448.147 −0.791780
\(567\) 0 0
\(568\) 450.951i 0.793928i
\(569\) − 46.7089i − 0.0820894i −0.999157 0.0410447i \(-0.986931\pi\)
0.999157 0.0410447i \(-0.0130686\pi\)
\(570\) 0 0
\(571\) 318.501i 0.557795i 0.960321 + 0.278897i \(0.0899689\pi\)
−0.960321 + 0.278897i \(0.910031\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1655.47i 2.88409i
\(575\) 148.623 0.258475
\(576\) 0 0
\(577\) −219.963 −0.381218 −0.190609 0.981666i \(-0.561046\pi\)
−0.190609 + 0.981666i \(0.561046\pi\)
\(578\) 188.408i 0.325965i
\(579\) 0 0
\(580\) 40.7992i 0.0703434i
\(581\) −770.928 −1.32690
\(582\) 0 0
\(583\) 0 0
\(584\) −704.414 −1.20619
\(585\) 0 0
\(586\) −291.956 −0.498219
\(587\) −147.674 −0.251575 −0.125787 0.992057i \(-0.540146\pi\)
−0.125787 + 0.992057i \(0.540146\pi\)
\(588\) 0 0
\(589\) 9.60947i 0.0163149i
\(590\) − 641.585i − 1.08743i
\(591\) 0 0
\(592\) −27.8653 −0.0470698
\(593\) 446.800i 0.753457i 0.926324 + 0.376728i \(0.122951\pi\)
−0.926324 + 0.376728i \(0.877049\pi\)
\(594\) 0 0
\(595\) 864.842 1.45352
\(596\) 45.7127i 0.0766992i
\(597\) 0 0
\(598\) 264.588 0.442456
\(599\) 353.205 0.589658 0.294829 0.955550i \(-0.404737\pi\)
0.294829 + 0.955550i \(0.404737\pi\)
\(600\) 0 0
\(601\) 1152.51i 1.91765i 0.284005 + 0.958823i \(0.408337\pi\)
−0.284005 + 0.958823i \(0.591663\pi\)
\(602\) − 77.9032i − 0.129407i
\(603\) 0 0
\(604\) 10.7589i 0.0178128i
\(605\) 0 0
\(606\) 0 0
\(607\) 249.462i 0.410975i 0.978660 + 0.205487i \(0.0658780\pi\)
−0.978660 + 0.205487i \(0.934122\pi\)
\(608\) −15.3487 −0.0252445
\(609\) 0 0
\(610\) −318.865 −0.522730
\(611\) 856.841i 1.40236i
\(612\) 0 0
\(613\) 320.244i 0.522421i 0.965282 + 0.261210i \(0.0841217\pi\)
−0.965282 + 0.261210i \(0.915878\pi\)
\(614\) −766.137 −1.24778
\(615\) 0 0
\(616\) 0 0
\(617\) 1036.56 1.67999 0.839997 0.542591i \(-0.182557\pi\)
0.839997 + 0.542591i \(0.182557\pi\)
\(618\) 0 0
\(619\) −701.542 −1.13335 −0.566674 0.823942i \(-0.691770\pi\)
−0.566674 + 0.823942i \(0.691770\pi\)
\(620\) 2.29473 0.00370118
\(621\) 0 0
\(622\) − 424.913i − 0.683140i
\(623\) − 607.822i − 0.975637i
\(624\) 0 0
\(625\) −36.8001 −0.0588801
\(626\) 61.1257i 0.0976450i
\(627\) 0 0
\(628\) 26.2487 0.0417973
\(629\) 36.8757i 0.0586259i
\(630\) 0 0
\(631\) 1199.17 1.90043 0.950214 0.311597i \(-0.100864\pi\)
0.950214 + 0.311597i \(0.100864\pi\)
\(632\) −72.8616 −0.115287
\(633\) 0 0
\(634\) − 594.990i − 0.938471i
\(635\) 519.544i 0.818179i
\(636\) 0 0
\(637\) 1913.23i 3.00350i
\(638\) 0 0
\(639\) 0 0
\(640\) − 363.258i − 0.567590i
\(641\) 739.327 1.15340 0.576698 0.816958i \(-0.304341\pi\)
0.576698 + 0.816958i \(0.304341\pi\)
\(642\) 0 0
\(643\) 410.087 0.637772 0.318886 0.947793i \(-0.396691\pi\)
0.318886 + 0.947793i \(0.396691\pi\)
\(644\) − 37.0722i − 0.0575655i
\(645\) 0 0
\(646\) − 136.210i − 0.210852i
\(647\) 352.115 0.544228 0.272114 0.962265i \(-0.412277\pi\)
0.272114 + 0.962265i \(0.412277\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 389.138 0.598675
\(651\) 0 0
\(652\) −60.4167 −0.0926636
\(653\) −864.574 −1.32400 −0.662002 0.749502i \(-0.730293\pi\)
−0.662002 + 0.749502i \(0.730293\pi\)
\(654\) 0 0
\(655\) − 92.9737i − 0.141945i
\(656\) − 924.998i − 1.41006i
\(657\) 0 0
\(658\) −1672.14 −2.54125
\(659\) − 985.279i − 1.49511i −0.664199 0.747556i \(-0.731227\pi\)
0.664199 0.747556i \(-0.268773\pi\)
\(660\) 0 0
\(661\) −529.651 −0.801288 −0.400644 0.916234i \(-0.631214\pi\)
−0.400644 + 0.916234i \(0.631214\pi\)
\(662\) 294.637i 0.445071i
\(663\) 0 0
\(664\) 461.836 0.695536
\(665\) −157.759 −0.237231
\(666\) 0 0
\(667\) 478.903i 0.717996i
\(668\) 9.85433i 0.0147520i
\(669\) 0 0
\(670\) − 99.3540i − 0.148290i
\(671\) 0 0
\(672\) 0 0
\(673\) 535.126i 0.795135i 0.917573 + 0.397567i \(0.130145\pi\)
−0.917573 + 0.397567i \(0.869855\pi\)
\(674\) −780.601 −1.15816
\(675\) 0 0
\(676\) −4.45517 −0.00659049
\(677\) − 740.764i − 1.09419i −0.837072 0.547093i \(-0.815734\pi\)
0.837072 0.547093i \(-0.184266\pi\)
\(678\) 0 0
\(679\) 1076.69i 1.58570i
\(680\) −518.096 −0.761906
\(681\) 0 0
\(682\) 0 0
\(683\) 376.722 0.551569 0.275785 0.961219i \(-0.411062\pi\)
0.275785 + 0.961219i \(0.411062\pi\)
\(684\) 0 0
\(685\) 166.413 0.242939
\(686\) −2430.87 −3.54354
\(687\) 0 0
\(688\) 43.5287i 0.0632685i
\(689\) 193.678i 0.281099i
\(690\) 0 0
\(691\) 336.404 0.486836 0.243418 0.969921i \(-0.421731\pi\)
0.243418 + 0.969921i \(0.421731\pi\)
\(692\) 64.9430i 0.0938483i
\(693\) 0 0
\(694\) −128.708 −0.185458
\(695\) 523.890i 0.753799i
\(696\) 0 0
\(697\) −1224.10 −1.75624
\(698\) 70.5411 0.101062
\(699\) 0 0
\(700\) − 54.5232i − 0.0778903i
\(701\) 842.500i 1.20185i 0.799304 + 0.600927i \(0.205202\pi\)
−0.799304 + 0.600927i \(0.794798\pi\)
\(702\) 0 0
\(703\) − 6.72663i − 0.00956846i
\(704\) 0 0
\(705\) 0 0
\(706\) − 635.958i − 0.900791i
\(707\) −284.600 −0.402546
\(708\) 0 0
\(709\) 318.882 0.449763 0.224882 0.974386i \(-0.427800\pi\)
0.224882 + 0.974386i \(0.427800\pi\)
\(710\) − 337.705i − 0.475641i
\(711\) 0 0
\(712\) 364.125i 0.511411i
\(713\) 26.9358 0.0377781
\(714\) 0 0
\(715\) 0 0
\(716\) −2.49742 −0.00348802
\(717\) 0 0
\(718\) −1193.92 −1.66284
\(719\) 548.788 0.763266 0.381633 0.924314i \(-0.375362\pi\)
0.381633 + 0.924314i \(0.375362\pi\)
\(720\) 0 0
\(721\) − 1690.10i − 2.34411i
\(722\) − 672.552i − 0.931512i
\(723\) 0 0
\(724\) 61.9461 0.0855609
\(725\) 704.338i 0.971501i
\(726\) 0 0
\(727\) 897.699 1.23480 0.617399 0.786650i \(-0.288186\pi\)
0.617399 + 0.786650i \(0.288186\pi\)
\(728\) − 1546.08i − 2.12374i
\(729\) 0 0
\(730\) 527.517 0.722626
\(731\) 57.6039 0.0788015
\(732\) 0 0
\(733\) − 198.000i − 0.270123i −0.990837 0.135061i \(-0.956877\pi\)
0.990837 0.135061i \(-0.0431231\pi\)
\(734\) 349.296i 0.475880i
\(735\) 0 0
\(736\) 43.0230i 0.0584552i
\(737\) 0 0
\(738\) 0 0
\(739\) − 560.286i − 0.758168i −0.925362 0.379084i \(-0.876239\pi\)
0.925362 0.379084i \(-0.123761\pi\)
\(740\) −1.60631 −0.00217069
\(741\) 0 0
\(742\) −377.965 −0.509387
\(743\) 1007.49i 1.35598i 0.735071 + 0.677990i \(0.237149\pi\)
−0.735071 + 0.677990i \(0.762851\pi\)
\(744\) 0 0
\(745\) − 545.271i − 0.731907i
\(746\) −664.252 −0.890419
\(747\) 0 0
\(748\) 0 0
\(749\) 198.833 0.265465
\(750\) 0 0
\(751\) 1181.32 1.57300 0.786501 0.617589i \(-0.211891\pi\)
0.786501 + 0.617589i \(0.211891\pi\)
\(752\) 934.315 1.24244
\(753\) 0 0
\(754\) 1253.91i 1.66301i
\(755\) − 128.334i − 0.169979i
\(756\) 0 0
\(757\) −1244.66 −1.64420 −0.822101 0.569341i \(-0.807198\pi\)
−0.822101 + 0.569341i \(0.807198\pi\)
\(758\) 683.125i 0.901220i
\(759\) 0 0
\(760\) 94.5077 0.124352
\(761\) − 1191.43i − 1.56562i −0.622263 0.782808i \(-0.713787\pi\)
0.622263 0.782808i \(-0.286213\pi\)
\(762\) 0 0
\(763\) −1479.41 −1.93894
\(764\) −65.4153 −0.0856221
\(765\) 0 0
\(766\) − 305.369i − 0.398654i
\(767\) 1415.71i 1.84577i
\(768\) 0 0
\(769\) − 823.778i − 1.07123i −0.844461 0.535617i \(-0.820079\pi\)
0.844461 0.535617i \(-0.179921\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.89085i 0.00374462i
\(773\) 1439.82 1.86264 0.931318 0.364207i \(-0.118660\pi\)
0.931318 + 0.364207i \(0.118660\pi\)
\(774\) 0 0
\(775\) 39.6152 0.0511164
\(776\) − 645.006i − 0.831193i
\(777\) 0 0
\(778\) − 786.731i − 1.01122i
\(779\) 223.292 0.286640
\(780\) 0 0
\(781\) 0 0
\(782\) −381.804 −0.488240
\(783\) 0 0
\(784\) 2086.22 2.66100
\(785\) −313.100 −0.398853
\(786\) 0 0
\(787\) 638.284i 0.811034i 0.914088 + 0.405517i \(0.132908\pi\)
−0.914088 + 0.405517i \(0.867092\pi\)
\(788\) − 29.4655i − 0.0373928i
\(789\) 0 0
\(790\) 54.5641 0.0690684
\(791\) 1758.72i 2.22342i
\(792\) 0 0
\(793\) 703.601 0.887265
\(794\) − 7.86611i − 0.00990694i
\(795\) 0 0
\(796\) −7.64240 −0.00960100
\(797\) 255.590 0.320690 0.160345 0.987061i \(-0.448739\pi\)
0.160345 + 0.987061i \(0.448739\pi\)
\(798\) 0 0
\(799\) − 1236.43i − 1.54747i
\(800\) 63.2753i 0.0790941i
\(801\) 0 0
\(802\) 838.156i 1.04508i
\(803\) 0 0
\(804\) 0 0
\(805\) 442.205i 0.549323i
\(806\) 70.5256 0.0875007
\(807\) 0 0
\(808\) 170.494 0.211007
\(809\) 356.384i 0.440524i 0.975441 + 0.220262i \(0.0706912\pi\)
−0.975441 + 0.220262i \(0.929309\pi\)
\(810\) 0 0
\(811\) 166.748i 0.205608i 0.994702 + 0.102804i \(0.0327815\pi\)
−0.994702 + 0.102804i \(0.967218\pi\)
\(812\) 175.688 0.216365
\(813\) 0 0
\(814\) 0 0
\(815\) 720.663 0.884249
\(816\) 0 0
\(817\) −10.5077 −0.0128614
\(818\) 203.014 0.248183
\(819\) 0 0
\(820\) − 53.3221i − 0.0650269i
\(821\) 1028.61i 1.25287i 0.779472 + 0.626437i \(0.215487\pi\)
−0.779472 + 0.626437i \(0.784513\pi\)
\(822\) 0 0
\(823\) −445.570 −0.541398 −0.270699 0.962664i \(-0.587255\pi\)
−0.270699 + 0.962664i \(0.587255\pi\)
\(824\) 1012.48i 1.22874i
\(825\) 0 0
\(826\) −2762.78 −3.34477
\(827\) 954.877i 1.15463i 0.816522 + 0.577314i \(0.195899\pi\)
−0.816522 + 0.577314i \(0.804101\pi\)
\(828\) 0 0
\(829\) 1098.95 1.32563 0.662817 0.748781i \(-0.269361\pi\)
0.662817 + 0.748781i \(0.269361\pi\)
\(830\) −345.856 −0.416694
\(831\) 0 0
\(832\) 922.291i 1.10852i
\(833\) − 2760.81i − 3.31430i
\(834\) 0 0
\(835\) − 117.544i − 0.140772i
\(836\) 0 0
\(837\) 0 0
\(838\) 140.399i 0.167540i
\(839\) −952.974 −1.13585 −0.567923 0.823082i \(-0.692253\pi\)
−0.567923 + 0.823082i \(0.692253\pi\)
\(840\) 0 0
\(841\) −1428.57 −1.69865
\(842\) 662.957i 0.787360i
\(843\) 0 0
\(844\) 46.1142i 0.0546377i
\(845\) 53.1422 0.0628902
\(846\) 0 0
\(847\) 0 0
\(848\) 211.190 0.249044
\(849\) 0 0
\(850\) −561.531 −0.660624
\(851\) −18.8550 −0.0221563
\(852\) 0 0
\(853\) − 376.647i − 0.441556i −0.975324 0.220778i \(-0.929140\pi\)
0.975324 0.220778i \(-0.0708597\pi\)
\(854\) 1373.09i 1.60783i
\(855\) 0 0
\(856\) −119.114 −0.139152
\(857\) − 552.323i − 0.644484i −0.946657 0.322242i \(-0.895563\pi\)
0.946657 0.322242i \(-0.104437\pi\)
\(858\) 0 0
\(859\) 1556.85 1.81240 0.906200 0.422849i \(-0.138970\pi\)
0.906200 + 0.422849i \(0.138970\pi\)
\(860\) 2.50924i 0.00291772i
\(861\) 0 0
\(862\) −791.955 −0.918741
\(863\) −201.173 −0.233109 −0.116555 0.993184i \(-0.537185\pi\)
−0.116555 + 0.993184i \(0.537185\pi\)
\(864\) 0 0
\(865\) − 774.654i − 0.895554i
\(866\) 1107.26i 1.27859i
\(867\) 0 0
\(868\) − 9.88153i − 0.0113843i
\(869\) 0 0
\(870\) 0 0
\(871\) 219.232i 0.251702i
\(872\) 886.261 1.01635
\(873\) 0 0
\(874\) 69.6462 0.0796867
\(875\) 1750.10i 2.00011i
\(876\) 0 0
\(877\) − 696.933i − 0.794678i −0.917672 0.397339i \(-0.869934\pi\)
0.917672 0.397339i \(-0.130066\pi\)
\(878\) −282.326 −0.321555
\(879\) 0 0
\(880\) 0 0
\(881\) 159.841 0.181431 0.0907156 0.995877i \(-0.471085\pi\)
0.0907156 + 0.995877i \(0.471085\pi\)
\(882\) 0 0
\(883\) −185.864 −0.210491 −0.105246 0.994446i \(-0.533563\pi\)
−0.105246 + 0.994446i \(0.533563\pi\)
\(884\) 71.7733 0.0811915
\(885\) 0 0
\(886\) 909.493i 1.02652i
\(887\) 1061.32i 1.19652i 0.801301 + 0.598262i \(0.204142\pi\)
−0.801301 + 0.598262i \(0.795858\pi\)
\(888\) 0 0
\(889\) 2237.25 2.51659
\(890\) − 272.683i − 0.306385i
\(891\) 0 0
\(892\) −39.2820 −0.0440382
\(893\) 225.542i 0.252566i
\(894\) 0 0
\(895\) 29.7898 0.0332847
\(896\) −1564.25 −1.74582
\(897\) 0 0
\(898\) − 524.926i − 0.584550i
\(899\) 127.651i 0.141992i
\(900\) 0 0
\(901\) − 279.479i − 0.310187i
\(902\) 0 0
\(903\) 0 0
\(904\) − 1053.59i − 1.16548i
\(905\) −738.905 −0.816470
\(906\) 0 0
\(907\) −974.991 −1.07496 −0.537481 0.843276i \(-0.680624\pi\)
−0.537481 + 0.843276i \(0.680624\pi\)
\(908\) 18.4246i 0.0202914i
\(909\) 0 0
\(910\) 1157.82i 1.27233i
\(911\) −1106.62 −1.21473 −0.607366 0.794422i \(-0.707774\pi\)
−0.607366 + 0.794422i \(0.707774\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1412.25 1.54513
\(915\) 0 0
\(916\) −67.7255 −0.0739362
\(917\) −400.361 −0.436599
\(918\) 0 0
\(919\) 865.081i 0.941329i 0.882312 + 0.470665i \(0.155986\pi\)
−0.882312 + 0.470665i \(0.844014\pi\)
\(920\) − 264.909i − 0.287945i
\(921\) 0 0
\(922\) 523.879 0.568199
\(923\) 745.172i 0.807337i
\(924\) 0 0
\(925\) −27.7307 −0.0299791
\(926\) 1459.28i 1.57590i
\(927\) 0 0
\(928\) −203.890 −0.219709
\(929\) −1520.93 −1.63717 −0.818583 0.574388i \(-0.805240\pi\)
−0.818583 + 0.574388i \(0.805240\pi\)
\(930\) 0 0
\(931\) 503.609i 0.540933i
\(932\) − 75.7901i − 0.0813199i
\(933\) 0 0
\(934\) 335.986i 0.359728i
\(935\) 0 0
\(936\) 0 0
\(937\) − 835.522i − 0.891699i −0.895108 0.445849i \(-0.852902\pi\)
0.895108 0.445849i \(-0.147098\pi\)
\(938\) −427.836 −0.456115
\(939\) 0 0
\(940\) 53.8592 0.0572970
\(941\) − 860.626i − 0.914586i −0.889316 0.457293i \(-0.848819\pi\)
0.889316 0.457293i \(-0.151181\pi\)
\(942\) 0 0
\(943\) − 625.898i − 0.663731i
\(944\) 1543.71 1.63529
\(945\) 0 0
\(946\) 0 0
\(947\) −1268.54 −1.33954 −0.669768 0.742571i \(-0.733606\pi\)
−0.669768 + 0.742571i \(0.733606\pi\)
\(948\) 0 0
\(949\) −1164.01 −1.22656
\(950\) 102.431 0.107822
\(951\) 0 0
\(952\) 2231.01i 2.34350i
\(953\) 290.553i 0.304882i 0.988313 + 0.152441i \(0.0487135\pi\)
−0.988313 + 0.152441i \(0.951287\pi\)
\(954\) 0 0
\(955\) 780.287 0.817055
\(956\) − 96.5780i − 0.101023i
\(957\) 0 0
\(958\) 886.309 0.925166
\(959\) − 716.604i − 0.747240i
\(960\) 0 0
\(961\) −953.820 −0.992529
\(962\) −49.3679 −0.0513180
\(963\) 0 0
\(964\) 29.8180i 0.0309315i
\(965\) − 34.4826i − 0.0357333i
\(966\) 0 0
\(967\) − 1253.94i − 1.29673i −0.761330 0.648365i \(-0.775453\pi\)
0.761330 0.648365i \(-0.224547\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 483.027i 0.497966i
\(971\) −516.837 −0.532273 −0.266137 0.963935i \(-0.585747\pi\)
−0.266137 + 0.963935i \(0.585747\pi\)
\(972\) 0 0
\(973\) 2255.96 2.31856
\(974\) − 934.503i − 0.959449i
\(975\) 0 0
\(976\) − 767.220i − 0.786086i
\(977\) −106.061 −0.108558 −0.0542792 0.998526i \(-0.517286\pi\)
−0.0542792 + 0.998526i \(0.517286\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 120.261 0.122716
\(981\) 0 0
\(982\) 1306.64 1.33059
\(983\) 1625.88 1.65400 0.826998 0.562204i \(-0.190046\pi\)
0.826998 + 0.562204i \(0.190046\pi\)
\(984\) 0 0
\(985\) 351.471i 0.356823i
\(986\) − 1809.40i − 1.83509i
\(987\) 0 0
\(988\) −13.0924 −0.0132514
\(989\) 29.4536i 0.0297812i
\(990\) 0 0
\(991\) −458.638 −0.462803 −0.231402 0.972858i \(-0.574331\pi\)
−0.231402 + 0.972858i \(0.574331\pi\)
\(992\) 11.4677i 0.0115602i
\(993\) 0 0
\(994\) −1454.22 −1.46300
\(995\) 91.1601 0.0916182
\(996\) 0 0
\(997\) − 140.671i − 0.141095i −0.997508 0.0705474i \(-0.977525\pi\)
0.997508 0.0705474i \(-0.0224746\pi\)
\(998\) − 1644.94i − 1.64824i
\(999\) 0 0
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 1089.3.c.g.604.1 4
3.2 odd 2 363.3.c.c.241.4 yes 4
11.10 odd 2 inner 1089.3.c.g.604.4 4
33.2 even 10 363.3.g.b.40.1 16
33.5 odd 10 363.3.g.b.118.1 16
33.8 even 10 363.3.g.b.112.1 16
33.14 odd 10 363.3.g.b.112.4 16
33.17 even 10 363.3.g.b.118.4 16
33.20 odd 10 363.3.g.b.40.4 16
33.26 odd 10 363.3.g.b.94.1 16
33.29 even 10 363.3.g.b.94.4 16
33.32 even 2 363.3.c.c.241.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.3.c.c.241.1 4 33.32 even 2
363.3.c.c.241.4 yes 4 3.2 odd 2
363.3.g.b.40.1 16 33.2 even 10
363.3.g.b.40.4 16 33.20 odd 10
363.3.g.b.94.1 16 33.26 odd 10
363.3.g.b.94.4 16 33.29 even 10
363.3.g.b.112.1 16 33.8 even 10
363.3.g.b.112.4 16 33.14 odd 10
363.3.g.b.118.1 16 33.5 odd 10
363.3.g.b.118.4 16 33.17 even 10
1089.3.c.g.604.1 4 1.1 even 1 trivial
1089.3.c.g.604.4 4 11.10 odd 2 inner