Properties

Label 2205.2.a.q.1.1
Level $2205$
Weight $2$
Character 2205.1
Self dual yes
Analytic conductor $17.607$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,2,Mod(1,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

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: yes
Analytic conductor: \(17.6070136457\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -4.41421 q^{8} +O(q^{10})\) \(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -4.41421 q^{8} -2.41421 q^{10} +0.828427 q^{11} +4.82843 q^{13} +3.00000 q^{16} +4.82843 q^{17} -2.82843 q^{19} +3.82843 q^{20} -2.00000 q^{22} -0.414214 q^{23} +1.00000 q^{25} -11.6569 q^{26} +1.00000 q^{29} +6.00000 q^{31} +1.58579 q^{32} -11.6569 q^{34} +6.82843 q^{38} -4.41421 q^{40} -7.82843 q^{41} +3.58579 q^{43} +3.17157 q^{44} +1.00000 q^{46} +2.00000 q^{47} -2.41421 q^{50} +18.4853 q^{52} +1.17157 q^{53} +0.828427 q^{55} -2.41421 q^{58} +4.48528 q^{59} -5.48528 q^{61} -14.4853 q^{62} -9.82843 q^{64} +4.82843 q^{65} +9.58579 q^{67} +18.4853 q^{68} -4.48528 q^{71} +0.828427 q^{73} -10.8284 q^{76} +14.8284 q^{79} +3.00000 q^{80} +18.8995 q^{82} +13.7279 q^{83} +4.82843 q^{85} -8.65685 q^{86} -3.65685 q^{88} -8.65685 q^{89} -1.58579 q^{92} -4.82843 q^{94} -2.82843 q^{95} -11.6569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8} - 2 q^{10} - 4 q^{11} + 4 q^{13} + 6 q^{16} + 4 q^{17} + 2 q^{20} - 4 q^{22} + 2 q^{23} + 2 q^{25} - 12 q^{26} + 2 q^{29} + 12 q^{31} + 6 q^{32} - 12 q^{34} + 8 q^{38} - 6 q^{40} - 10 q^{41} + 10 q^{43} + 12 q^{44} + 2 q^{46} + 4 q^{47} - 2 q^{50} + 20 q^{52} + 8 q^{53} - 4 q^{55} - 2 q^{58} - 8 q^{59} + 6 q^{61} - 12 q^{62} - 14 q^{64} + 4 q^{65} + 22 q^{67} + 20 q^{68} + 8 q^{71} - 4 q^{73} - 16 q^{76} + 24 q^{79} + 6 q^{80} + 18 q^{82} + 2 q^{83} + 4 q^{85} - 6 q^{86} + 4 q^{88} - 6 q^{89} - 6 q^{92} - 4 q^{94} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −4.41421 −1.56066
\(9\) 0 0
\(10\) −2.41421 −0.763441
\(11\) 0.828427 0.249780 0.124890 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 0 0
\(13\) 4.82843 1.33916 0.669582 0.742738i \(-0.266473\pi\)
0.669582 + 0.742738i \(0.266473\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 4.82843 1.17107 0.585533 0.810649i \(-0.300885\pi\)
0.585533 + 0.810649i \(0.300885\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 3.82843 0.856062
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −0.414214 −0.0863695 −0.0431847 0.999067i \(-0.513750\pi\)
−0.0431847 + 0.999067i \(0.513750\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −11.6569 −2.28610
\(27\) 0 0
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.58579 0.280330
\(33\) 0 0
\(34\) −11.6569 −1.99913
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 6.82843 1.10772
\(39\) 0 0
\(40\) −4.41421 −0.697948
\(41\) −7.82843 −1.22259 −0.611297 0.791401i \(-0.709352\pi\)
−0.611297 + 0.791401i \(0.709352\pi\)
\(42\) 0 0
\(43\) 3.58579 0.546827 0.273414 0.961897i \(-0.411847\pi\)
0.273414 + 0.961897i \(0.411847\pi\)
\(44\) 3.17157 0.478133
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.41421 −0.341421
\(51\) 0 0
\(52\) 18.4853 2.56345
\(53\) 1.17157 0.160928 0.0804640 0.996758i \(-0.474360\pi\)
0.0804640 + 0.996758i \(0.474360\pi\)
\(54\) 0 0
\(55\) 0.828427 0.111705
\(56\) 0 0
\(57\) 0 0
\(58\) −2.41421 −0.317002
\(59\) 4.48528 0.583934 0.291967 0.956428i \(-0.405690\pi\)
0.291967 + 0.956428i \(0.405690\pi\)
\(60\) 0 0
\(61\) −5.48528 −0.702318 −0.351159 0.936316i \(-0.614212\pi\)
−0.351159 + 0.936316i \(0.614212\pi\)
\(62\) −14.4853 −1.83963
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) 4.82843 0.598893
\(66\) 0 0
\(67\) 9.58579 1.17109 0.585545 0.810640i \(-0.300881\pi\)
0.585545 + 0.810640i \(0.300881\pi\)
\(68\) 18.4853 2.24167
\(69\) 0 0
\(70\) 0 0
\(71\) −4.48528 −0.532305 −0.266152 0.963931i \(-0.585752\pi\)
−0.266152 + 0.963931i \(0.585752\pi\)
\(72\) 0 0
\(73\) 0.828427 0.0969601 0.0484800 0.998824i \(-0.484562\pi\)
0.0484800 + 0.998824i \(0.484562\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −10.8284 −1.24211
\(77\) 0 0
\(78\) 0 0
\(79\) 14.8284 1.66833 0.834164 0.551516i \(-0.185951\pi\)
0.834164 + 0.551516i \(0.185951\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) 18.8995 2.08710
\(83\) 13.7279 1.50684 0.753418 0.657542i \(-0.228404\pi\)
0.753418 + 0.657542i \(0.228404\pi\)
\(84\) 0 0
\(85\) 4.82843 0.523716
\(86\) −8.65685 −0.933493
\(87\) 0 0
\(88\) −3.65685 −0.389822
\(89\) −8.65685 −0.917625 −0.458812 0.888533i \(-0.651725\pi\)
−0.458812 + 0.888533i \(0.651725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.58579 −0.165330
\(93\) 0 0
\(94\) −4.82843 −0.498014
\(95\) −2.82843 −0.290191
\(96\) 0 0
\(97\) −11.6569 −1.18357 −0.591787 0.806094i \(-0.701577\pi\)
−0.591787 + 0.806094i \(0.701577\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 3.82843 0.382843
\(101\) −10.3137 −1.02625 −0.513126 0.858313i \(-0.671513\pi\)
−0.513126 + 0.858313i \(0.671513\pi\)
\(102\) 0 0
\(103\) 2.41421 0.237880 0.118940 0.992901i \(-0.462050\pi\)
0.118940 + 0.992901i \(0.462050\pi\)
\(104\) −21.3137 −2.08998
\(105\) 0 0
\(106\) −2.82843 −0.274721
\(107\) −11.2426 −1.08687 −0.543434 0.839452i \(-0.682876\pi\)
−0.543434 + 0.839452i \(0.682876\pi\)
\(108\) 0 0
\(109\) −13.4853 −1.29166 −0.645828 0.763483i \(-0.723488\pi\)
−0.645828 + 0.763483i \(0.723488\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) 0 0
\(113\) 4.48528 0.421940 0.210970 0.977493i \(-0.432338\pi\)
0.210970 + 0.977493i \(0.432338\pi\)
\(114\) 0 0
\(115\) −0.414214 −0.0386256
\(116\) 3.82843 0.355461
\(117\) 0 0
\(118\) −10.8284 −0.996838
\(119\) 0 0
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 13.2426 1.19893
\(123\) 0 0
\(124\) 22.9706 2.06282
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −9.31371 −0.826458 −0.413229 0.910627i \(-0.635599\pi\)
−0.413229 + 0.910627i \(0.635599\pi\)
\(128\) 20.5563 1.81694
\(129\) 0 0
\(130\) −11.6569 −1.02237
\(131\) −19.3137 −1.68745 −0.843723 0.536778i \(-0.819641\pi\)
−0.843723 + 0.536778i \(0.819641\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −23.1421 −1.99918
\(135\) 0 0
\(136\) −21.3137 −1.82764
\(137\) 9.65685 0.825041 0.412520 0.910948i \(-0.364649\pi\)
0.412520 + 0.910948i \(0.364649\pi\)
\(138\) 0 0
\(139\) 16.1421 1.36916 0.684579 0.728939i \(-0.259986\pi\)
0.684579 + 0.728939i \(0.259986\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.8284 0.908701
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 0 0
\(149\) 2.17157 0.177902 0.0889511 0.996036i \(-0.471649\pi\)
0.0889511 + 0.996036i \(0.471649\pi\)
\(150\) 0 0
\(151\) 11.6569 0.948621 0.474311 0.880358i \(-0.342697\pi\)
0.474311 + 0.880358i \(0.342697\pi\)
\(152\) 12.4853 1.01269
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −17.3137 −1.38178 −0.690892 0.722958i \(-0.742782\pi\)
−0.690892 + 0.722958i \(0.742782\pi\)
\(158\) −35.7990 −2.84801
\(159\) 0 0
\(160\) 1.58579 0.125367
\(161\) 0 0
\(162\) 0 0
\(163\) 12.3431 0.966790 0.483395 0.875402i \(-0.339404\pi\)
0.483395 + 0.875402i \(0.339404\pi\)
\(164\) −29.9706 −2.34031
\(165\) 0 0
\(166\) −33.1421 −2.57233
\(167\) 22.4142 1.73446 0.867232 0.497904i \(-0.165897\pi\)
0.867232 + 0.497904i \(0.165897\pi\)
\(168\) 0 0
\(169\) 10.3137 0.793362
\(170\) −11.6569 −0.894040
\(171\) 0 0
\(172\) 13.7279 1.04674
\(173\) 3.31371 0.251937 0.125968 0.992034i \(-0.459796\pi\)
0.125968 + 0.992034i \(0.459796\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.48528 0.187335
\(177\) 0 0
\(178\) 20.8995 1.56648
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −2.65685 −0.197482 −0.0987412 0.995113i \(-0.531482\pi\)
−0.0987412 + 0.995113i \(0.531482\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.82843 0.134793
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 7.65685 0.558433
\(189\) 0 0
\(190\) 6.82843 0.495386
\(191\) 12.8284 0.928232 0.464116 0.885774i \(-0.346372\pi\)
0.464116 + 0.885774i \(0.346372\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 28.1421 2.02049
\(195\) 0 0
\(196\) 0 0
\(197\) 12.3431 0.879413 0.439706 0.898142i \(-0.355083\pi\)
0.439706 + 0.898142i \(0.355083\pi\)
\(198\) 0 0
\(199\) 9.65685 0.684556 0.342278 0.939599i \(-0.388801\pi\)
0.342278 + 0.939599i \(0.388801\pi\)
\(200\) −4.41421 −0.312132
\(201\) 0 0
\(202\) 24.8995 1.75192
\(203\) 0 0
\(204\) 0 0
\(205\) −7.82843 −0.546761
\(206\) −5.82843 −0.406086
\(207\) 0 0
\(208\) 14.4853 1.00437
\(209\) −2.34315 −0.162079
\(210\) 0 0
\(211\) 20.4853 1.41026 0.705132 0.709076i \(-0.250888\pi\)
0.705132 + 0.709076i \(0.250888\pi\)
\(212\) 4.48528 0.308050
\(213\) 0 0
\(214\) 27.1421 1.85540
\(215\) 3.58579 0.244549
\(216\) 0 0
\(217\) 0 0
\(218\) 32.5563 2.20499
\(219\) 0 0
\(220\) 3.17157 0.213827
\(221\) 23.3137 1.56825
\(222\) 0 0
\(223\) −0.343146 −0.0229787 −0.0114894 0.999934i \(-0.503657\pi\)
−0.0114894 + 0.999934i \(0.503657\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −10.8284 −0.720296
\(227\) −6.97056 −0.462652 −0.231326 0.972876i \(-0.574306\pi\)
−0.231326 + 0.972876i \(0.574306\pi\)
\(228\) 0 0
\(229\) 11.6569 0.770307 0.385153 0.922853i \(-0.374149\pi\)
0.385153 + 0.922853i \(0.374149\pi\)
\(230\) 1.00000 0.0659380
\(231\) 0 0
\(232\) −4.41421 −0.289807
\(233\) 16.8284 1.10247 0.551233 0.834351i \(-0.314157\pi\)
0.551233 + 0.834351i \(0.314157\pi\)
\(234\) 0 0
\(235\) 2.00000 0.130466
\(236\) 17.1716 1.11777
\(237\) 0 0
\(238\) 0 0
\(239\) 21.3137 1.37867 0.689335 0.724443i \(-0.257903\pi\)
0.689335 + 0.724443i \(0.257903\pi\)
\(240\) 0 0
\(241\) −27.6569 −1.78153 −0.890767 0.454460i \(-0.849832\pi\)
−0.890767 + 0.454460i \(0.849832\pi\)
\(242\) 24.8995 1.60060
\(243\) 0 0
\(244\) −21.0000 −1.34439
\(245\) 0 0
\(246\) 0 0
\(247\) −13.6569 −0.868965
\(248\) −26.4853 −1.68182
\(249\) 0 0
\(250\) −2.41421 −0.152688
\(251\) −9.31371 −0.587876 −0.293938 0.955824i \(-0.594966\pi\)
−0.293938 + 0.955824i \(0.594966\pi\)
\(252\) 0 0
\(253\) −0.343146 −0.0215734
\(254\) 22.4853 1.41085
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 6.34315 0.395675 0.197837 0.980235i \(-0.436608\pi\)
0.197837 + 0.980235i \(0.436608\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 18.4853 1.14641
\(261\) 0 0
\(262\) 46.6274 2.88065
\(263\) 29.0416 1.79078 0.895392 0.445279i \(-0.146896\pi\)
0.895392 + 0.445279i \(0.146896\pi\)
\(264\) 0 0
\(265\) 1.17157 0.0719691
\(266\) 0 0
\(267\) 0 0
\(268\) 36.6985 2.24172
\(269\) 20.4558 1.24721 0.623607 0.781738i \(-0.285666\pi\)
0.623607 + 0.781738i \(0.285666\pi\)
\(270\) 0 0
\(271\) −16.4853 −1.00141 −0.500705 0.865618i \(-0.666926\pi\)
−0.500705 + 0.865618i \(0.666926\pi\)
\(272\) 14.4853 0.878299
\(273\) 0 0
\(274\) −23.3137 −1.40843
\(275\) 0.828427 0.0499560
\(276\) 0 0
\(277\) 16.1421 0.969887 0.484943 0.874546i \(-0.338840\pi\)
0.484943 + 0.874546i \(0.338840\pi\)
\(278\) −38.9706 −2.33730
\(279\) 0 0
\(280\) 0 0
\(281\) 30.2843 1.80661 0.903304 0.429001i \(-0.141134\pi\)
0.903304 + 0.429001i \(0.141134\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −17.1716 −1.01895
\(285\) 0 0
\(286\) −9.65685 −0.571022
\(287\) 0 0
\(288\) 0 0
\(289\) 6.31371 0.371395
\(290\) −2.41421 −0.141768
\(291\) 0 0
\(292\) 3.17157 0.185602
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) 0 0
\(295\) 4.48528 0.261143
\(296\) 0 0
\(297\) 0 0
\(298\) −5.24264 −0.303698
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) −28.1421 −1.61940
\(303\) 0 0
\(304\) −8.48528 −0.486664
\(305\) −5.48528 −0.314086
\(306\) 0 0
\(307\) 4.75736 0.271517 0.135758 0.990742i \(-0.456653\pi\)
0.135758 + 0.990742i \(0.456653\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −14.4853 −0.822709
\(311\) −13.1716 −0.746891 −0.373446 0.927652i \(-0.621824\pi\)
−0.373446 + 0.927652i \(0.621824\pi\)
\(312\) 0 0
\(313\) 6.34315 0.358536 0.179268 0.983800i \(-0.442627\pi\)
0.179268 + 0.983800i \(0.442627\pi\)
\(314\) 41.7990 2.35885
\(315\) 0 0
\(316\) 56.7696 3.19354
\(317\) 13.7990 0.775028 0.387514 0.921864i \(-0.373334\pi\)
0.387514 + 0.921864i \(0.373334\pi\)
\(318\) 0 0
\(319\) 0.828427 0.0463830
\(320\) −9.82843 −0.549426
\(321\) 0 0
\(322\) 0 0
\(323\) −13.6569 −0.759888
\(324\) 0 0
\(325\) 4.82843 0.267833
\(326\) −29.7990 −1.65041
\(327\) 0 0
\(328\) 34.5563 1.90806
\(329\) 0 0
\(330\) 0 0
\(331\) 22.9706 1.26258 0.631288 0.775548i \(-0.282527\pi\)
0.631288 + 0.775548i \(0.282527\pi\)
\(332\) 52.5563 2.88440
\(333\) 0 0
\(334\) −54.1127 −2.96092
\(335\) 9.58579 0.523727
\(336\) 0 0
\(337\) 9.17157 0.499607 0.249804 0.968296i \(-0.419634\pi\)
0.249804 + 0.968296i \(0.419634\pi\)
\(338\) −24.8995 −1.35435
\(339\) 0 0
\(340\) 18.4853 1.00251
\(341\) 4.97056 0.269171
\(342\) 0 0
\(343\) 0 0
\(344\) −15.8284 −0.853412
\(345\) 0 0
\(346\) −8.00000 −0.430083
\(347\) −7.92893 −0.425647 −0.212824 0.977091i \(-0.568266\pi\)
−0.212824 + 0.977091i \(0.568266\pi\)
\(348\) 0 0
\(349\) 15.3431 0.821300 0.410650 0.911793i \(-0.365302\pi\)
0.410650 + 0.911793i \(0.365302\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.31371 0.0700209
\(353\) −26.8284 −1.42793 −0.713967 0.700180i \(-0.753103\pi\)
−0.713967 + 0.700180i \(0.753103\pi\)
\(354\) 0 0
\(355\) −4.48528 −0.238054
\(356\) −33.1421 −1.75653
\(357\) 0 0
\(358\) −24.1421 −1.27595
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 6.41421 0.337124
\(363\) 0 0
\(364\) 0 0
\(365\) 0.828427 0.0433619
\(366\) 0 0
\(367\) 2.75736 0.143933 0.0719665 0.997407i \(-0.477073\pi\)
0.0719665 + 0.997407i \(0.477073\pi\)
\(368\) −1.24264 −0.0647771
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −20.9706 −1.08581 −0.542907 0.839793i \(-0.682677\pi\)
−0.542907 + 0.839793i \(0.682677\pi\)
\(374\) −9.65685 −0.499344
\(375\) 0 0
\(376\) −8.82843 −0.455291
\(377\) 4.82843 0.248677
\(378\) 0 0
\(379\) 26.8284 1.37808 0.689042 0.724722i \(-0.258032\pi\)
0.689042 + 0.724722i \(0.258032\pi\)
\(380\) −10.8284 −0.555487
\(381\) 0 0
\(382\) −30.9706 −1.58459
\(383\) −2.89949 −0.148157 −0.0740786 0.997252i \(-0.523602\pi\)
−0.0740786 + 0.997252i \(0.523602\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.82843 0.245760
\(387\) 0 0
\(388\) −44.6274 −2.26561
\(389\) −23.6569 −1.19945 −0.599725 0.800206i \(-0.704723\pi\)
−0.599725 + 0.800206i \(0.704723\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) 0 0
\(393\) 0 0
\(394\) −29.7990 −1.50125
\(395\) 14.8284 0.746099
\(396\) 0 0
\(397\) 16.6274 0.834506 0.417253 0.908790i \(-0.362993\pi\)
0.417253 + 0.908790i \(0.362993\pi\)
\(398\) −23.3137 −1.16861
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) −30.3137 −1.51379 −0.756897 0.653534i \(-0.773286\pi\)
−0.756897 + 0.653534i \(0.773286\pi\)
\(402\) 0 0
\(403\) 28.9706 1.44313
\(404\) −39.4853 −1.96447
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14.7990 0.731763 0.365881 0.930661i \(-0.380768\pi\)
0.365881 + 0.930661i \(0.380768\pi\)
\(410\) 18.8995 0.933380
\(411\) 0 0
\(412\) 9.24264 0.455352
\(413\) 0 0
\(414\) 0 0
\(415\) 13.7279 0.673877
\(416\) 7.65685 0.375408
\(417\) 0 0
\(418\) 5.65685 0.276686
\(419\) −0.686292 −0.0335275 −0.0167638 0.999859i \(-0.505336\pi\)
−0.0167638 + 0.999859i \(0.505336\pi\)
\(420\) 0 0
\(421\) 13.4853 0.657232 0.328616 0.944464i \(-0.393418\pi\)
0.328616 + 0.944464i \(0.393418\pi\)
\(422\) −49.4558 −2.40747
\(423\) 0 0
\(424\) −5.17157 −0.251154
\(425\) 4.82843 0.234213
\(426\) 0 0
\(427\) 0 0
\(428\) −43.0416 −2.08050
\(429\) 0 0
\(430\) −8.65685 −0.417471
\(431\) −17.7990 −0.857347 −0.428674 0.903459i \(-0.641019\pi\)
−0.428674 + 0.903459i \(0.641019\pi\)
\(432\) 0 0
\(433\) −7.79899 −0.374796 −0.187398 0.982284i \(-0.560005\pi\)
−0.187398 + 0.982284i \(0.560005\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −51.6274 −2.47250
\(437\) 1.17157 0.0560439
\(438\) 0 0
\(439\) 33.9411 1.61992 0.809961 0.586484i \(-0.199488\pi\)
0.809961 + 0.586484i \(0.199488\pi\)
\(440\) −3.65685 −0.174334
\(441\) 0 0
\(442\) −56.2843 −2.67717
\(443\) −30.2132 −1.43547 −0.717736 0.696315i \(-0.754822\pi\)
−0.717736 + 0.696315i \(0.754822\pi\)
\(444\) 0 0
\(445\) −8.65685 −0.410374
\(446\) 0.828427 0.0392272
\(447\) 0 0
\(448\) 0 0
\(449\) −3.82843 −0.180675 −0.0903373 0.995911i \(-0.528795\pi\)
−0.0903373 + 0.995911i \(0.528795\pi\)
\(450\) 0 0
\(451\) −6.48528 −0.305380
\(452\) 17.1716 0.807683
\(453\) 0 0
\(454\) 16.8284 0.789797
\(455\) 0 0
\(456\) 0 0
\(457\) 24.2843 1.13597 0.567985 0.823039i \(-0.307723\pi\)
0.567985 + 0.823039i \(0.307723\pi\)
\(458\) −28.1421 −1.31500
\(459\) 0 0
\(460\) −1.58579 −0.0739377
\(461\) 41.3137 1.92417 0.962086 0.272748i \(-0.0879324\pi\)
0.962086 + 0.272748i \(0.0879324\pi\)
\(462\) 0 0
\(463\) −37.0416 −1.72147 −0.860735 0.509053i \(-0.829996\pi\)
−0.860735 + 0.509053i \(0.829996\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −40.6274 −1.88203
\(467\) −3.10051 −0.143474 −0.0717371 0.997424i \(-0.522854\pi\)
−0.0717371 + 0.997424i \(0.522854\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4.82843 −0.222719
\(471\) 0 0
\(472\) −19.7990 −0.911322
\(473\) 2.97056 0.136587
\(474\) 0 0
\(475\) −2.82843 −0.129777
\(476\) 0 0
\(477\) 0 0
\(478\) −51.4558 −2.35354
\(479\) −35.6569 −1.62920 −0.814602 0.580021i \(-0.803044\pi\)
−0.814602 + 0.580021i \(0.803044\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 66.7696 3.04127
\(483\) 0 0
\(484\) −39.4853 −1.79479
\(485\) −11.6569 −0.529310
\(486\) 0 0
\(487\) 4.34315 0.196807 0.0984034 0.995147i \(-0.468626\pi\)
0.0984034 + 0.995147i \(0.468626\pi\)
\(488\) 24.2132 1.09608
\(489\) 0 0
\(490\) 0 0
\(491\) −9.31371 −0.420322 −0.210161 0.977667i \(-0.567399\pi\)
−0.210161 + 0.977667i \(0.567399\pi\)
\(492\) 0 0
\(493\) 4.82843 0.217461
\(494\) 32.9706 1.48342
\(495\) 0 0
\(496\) 18.0000 0.808224
\(497\) 0 0
\(498\) 0 0
\(499\) −0.828427 −0.0370855 −0.0185427 0.999828i \(-0.505903\pi\)
−0.0185427 + 0.999828i \(0.505903\pi\)
\(500\) 3.82843 0.171212
\(501\) 0 0
\(502\) 22.4853 1.00357
\(503\) −15.8701 −0.707611 −0.353805 0.935319i \(-0.615113\pi\)
−0.353805 + 0.935319i \(0.615113\pi\)
\(504\) 0 0
\(505\) −10.3137 −0.458954
\(506\) 0.828427 0.0368281
\(507\) 0 0
\(508\) −35.6569 −1.58202
\(509\) 13.3431 0.591425 0.295712 0.955277i \(-0.404443\pi\)
0.295712 + 0.955277i \(0.404443\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 31.2426 1.38074
\(513\) 0 0
\(514\) −15.3137 −0.675459
\(515\) 2.41421 0.106383
\(516\) 0 0
\(517\) 1.65685 0.0728684
\(518\) 0 0
\(519\) 0 0
\(520\) −21.3137 −0.934668
\(521\) 14.9706 0.655872 0.327936 0.944700i \(-0.393647\pi\)
0.327936 + 0.944700i \(0.393647\pi\)
\(522\) 0 0
\(523\) −35.6569 −1.55917 −0.779583 0.626299i \(-0.784569\pi\)
−0.779583 + 0.626299i \(0.784569\pi\)
\(524\) −73.9411 −3.23013
\(525\) 0 0
\(526\) −70.1127 −3.05706
\(527\) 28.9706 1.26198
\(528\) 0 0
\(529\) −22.8284 −0.992540
\(530\) −2.82843 −0.122859
\(531\) 0 0
\(532\) 0 0
\(533\) −37.7990 −1.63726
\(534\) 0 0
\(535\) −11.2426 −0.486062
\(536\) −42.3137 −1.82767
\(537\) 0 0
\(538\) −49.3848 −2.12913
\(539\) 0 0
\(540\) 0 0
\(541\) 7.34315 0.315706 0.157853 0.987463i \(-0.449543\pi\)
0.157853 + 0.987463i \(0.449543\pi\)
\(542\) 39.7990 1.70951
\(543\) 0 0
\(544\) 7.65685 0.328285
\(545\) −13.4853 −0.577646
\(546\) 0 0
\(547\) 24.8995 1.06463 0.532313 0.846548i \(-0.321323\pi\)
0.532313 + 0.846548i \(0.321323\pi\)
\(548\) 36.9706 1.57930
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) −2.82843 −0.120495
\(552\) 0 0
\(553\) 0 0
\(554\) −38.9706 −1.65570
\(555\) 0 0
\(556\) 61.7990 2.62086
\(557\) −22.2843 −0.944215 −0.472107 0.881541i \(-0.656507\pi\)
−0.472107 + 0.881541i \(0.656507\pi\)
\(558\) 0 0
\(559\) 17.3137 0.732292
\(560\) 0 0
\(561\) 0 0
\(562\) −73.1127 −3.08407
\(563\) 41.7279 1.75862 0.879311 0.476248i \(-0.158003\pi\)
0.879311 + 0.476248i \(0.158003\pi\)
\(564\) 0 0
\(565\) 4.48528 0.188697
\(566\) 33.7990 1.42068
\(567\) 0 0
\(568\) 19.7990 0.830747
\(569\) −7.65685 −0.320992 −0.160496 0.987036i \(-0.551309\pi\)
−0.160496 + 0.987036i \(0.551309\pi\)
\(570\) 0 0
\(571\) 9.17157 0.383818 0.191909 0.981413i \(-0.438532\pi\)
0.191909 + 0.981413i \(0.438532\pi\)
\(572\) 15.3137 0.640298
\(573\) 0 0
\(574\) 0 0
\(575\) −0.414214 −0.0172739
\(576\) 0 0
\(577\) 43.9411 1.82929 0.914646 0.404255i \(-0.132469\pi\)
0.914646 + 0.404255i \(0.132469\pi\)
\(578\) −15.2426 −0.634010
\(579\) 0 0
\(580\) 3.82843 0.158967
\(581\) 0 0
\(582\) 0 0
\(583\) 0.970563 0.0401966
\(584\) −3.65685 −0.151322
\(585\) 0 0
\(586\) 38.6274 1.59568
\(587\) −34.2843 −1.41506 −0.707532 0.706682i \(-0.750191\pi\)
−0.707532 + 0.706682i \(0.750191\pi\)
\(588\) 0 0
\(589\) −16.9706 −0.699260
\(590\) −10.8284 −0.445799
\(591\) 0 0
\(592\) 0 0
\(593\) 4.20101 0.172515 0.0862574 0.996273i \(-0.472509\pi\)
0.0862574 + 0.996273i \(0.472509\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.31371 0.340543
\(597\) 0 0
\(598\) 4.82843 0.197449
\(599\) −6.34315 −0.259174 −0.129587 0.991568i \(-0.541365\pi\)
−0.129587 + 0.991568i \(0.541365\pi\)
\(600\) 0 0
\(601\) −19.6569 −0.801820 −0.400910 0.916117i \(-0.631306\pi\)
−0.400910 + 0.916117i \(0.631306\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 44.6274 1.81586
\(605\) −10.3137 −0.419312
\(606\) 0 0
\(607\) −38.2132 −1.55103 −0.775513 0.631332i \(-0.782509\pi\)
−0.775513 + 0.631332i \(0.782509\pi\)
\(608\) −4.48528 −0.181902
\(609\) 0 0
\(610\) 13.2426 0.536179
\(611\) 9.65685 0.390675
\(612\) 0 0
\(613\) 35.4558 1.43205 0.716024 0.698076i \(-0.245960\pi\)
0.716024 + 0.698076i \(0.245960\pi\)
\(614\) −11.4853 −0.463508
\(615\) 0 0
\(616\) 0 0
\(617\) −11.3137 −0.455473 −0.227736 0.973723i \(-0.573132\pi\)
−0.227736 + 0.973723i \(0.573132\pi\)
\(618\) 0 0
\(619\) −25.5147 −1.02552 −0.512762 0.858531i \(-0.671378\pi\)
−0.512762 + 0.858531i \(0.671378\pi\)
\(620\) 22.9706 0.922520
\(621\) 0 0
\(622\) 31.7990 1.27502
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −15.3137 −0.612059
\(627\) 0 0
\(628\) −66.2843 −2.64503
\(629\) 0 0
\(630\) 0 0
\(631\) −20.1421 −0.801846 −0.400923 0.916112i \(-0.631310\pi\)
−0.400923 + 0.916112i \(0.631310\pi\)
\(632\) −65.4558 −2.60369
\(633\) 0 0
\(634\) −33.3137 −1.32306
\(635\) −9.31371 −0.369603
\(636\) 0 0
\(637\) 0 0
\(638\) −2.00000 −0.0791808
\(639\) 0 0
\(640\) 20.5563 0.812561
\(641\) −31.4853 −1.24359 −0.621797 0.783179i \(-0.713597\pi\)
−0.621797 + 0.783179i \(0.713597\pi\)
\(642\) 0 0
\(643\) −26.2843 −1.03655 −0.518275 0.855214i \(-0.673426\pi\)
−0.518275 + 0.855214i \(0.673426\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 32.9706 1.29721
\(647\) −31.0416 −1.22037 −0.610186 0.792258i \(-0.708905\pi\)
−0.610186 + 0.792258i \(0.708905\pi\)
\(648\) 0 0
\(649\) 3.71573 0.145855
\(650\) −11.6569 −0.457219
\(651\) 0 0
\(652\) 47.2548 1.85064
\(653\) 19.1716 0.750242 0.375121 0.926976i \(-0.377601\pi\)
0.375121 + 0.926976i \(0.377601\pi\)
\(654\) 0 0
\(655\) −19.3137 −0.754649
\(656\) −23.4853 −0.916946
\(657\) 0 0
\(658\) 0 0
\(659\) −21.1716 −0.824727 −0.412364 0.911019i \(-0.635297\pi\)
−0.412364 + 0.911019i \(0.635297\pi\)
\(660\) 0 0
\(661\) −31.8284 −1.23798 −0.618991 0.785398i \(-0.712458\pi\)
−0.618991 + 0.785398i \(0.712458\pi\)
\(662\) −55.4558 −2.15535
\(663\) 0 0
\(664\) −60.5980 −2.35166
\(665\) 0 0
\(666\) 0 0
\(667\) −0.414214 −0.0160384
\(668\) 85.8112 3.32013
\(669\) 0 0
\(670\) −23.1421 −0.894059
\(671\) −4.54416 −0.175425
\(672\) 0 0
\(673\) 29.6569 1.14319 0.571594 0.820537i \(-0.306325\pi\)
0.571594 + 0.820537i \(0.306325\pi\)
\(674\) −22.1421 −0.852883
\(675\) 0 0
\(676\) 39.4853 1.51866
\(677\) −28.1421 −1.08159 −0.540795 0.841154i \(-0.681877\pi\)
−0.540795 + 0.841154i \(0.681877\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −21.3137 −0.817343
\(681\) 0 0
\(682\) −12.0000 −0.459504
\(683\) 34.7574 1.32995 0.664977 0.746864i \(-0.268441\pi\)
0.664977 + 0.746864i \(0.268441\pi\)
\(684\) 0 0
\(685\) 9.65685 0.368969
\(686\) 0 0
\(687\) 0 0
\(688\) 10.7574 0.410120
\(689\) 5.65685 0.215509
\(690\) 0 0
\(691\) −0.828427 −0.0315149 −0.0157574 0.999876i \(-0.505016\pi\)
−0.0157574 + 0.999876i \(0.505016\pi\)
\(692\) 12.6863 0.482260
\(693\) 0 0
\(694\) 19.1421 0.726626
\(695\) 16.1421 0.612306
\(696\) 0 0
\(697\) −37.7990 −1.43174
\(698\) −37.0416 −1.40205
\(699\) 0 0
\(700\) 0 0
\(701\) 3.20101 0.120900 0.0604502 0.998171i \(-0.480746\pi\)
0.0604502 + 0.998171i \(0.480746\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −8.14214 −0.306868
\(705\) 0 0
\(706\) 64.7696 2.43763
\(707\) 0 0
\(708\) 0 0
\(709\) 15.6863 0.589111 0.294556 0.955634i \(-0.404828\pi\)
0.294556 + 0.955634i \(0.404828\pi\)
\(710\) 10.8284 0.406384
\(711\) 0 0
\(712\) 38.2132 1.43210
\(713\) −2.48528 −0.0930745
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 38.2843 1.43075
\(717\) 0 0
\(718\) −24.1421 −0.900976
\(719\) 21.1127 0.787371 0.393685 0.919245i \(-0.371200\pi\)
0.393685 + 0.919245i \(0.371200\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 26.5563 0.988325
\(723\) 0 0
\(724\) −10.1716 −0.378024
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 37.5858 1.39398 0.696990 0.717081i \(-0.254522\pi\)
0.696990 + 0.717081i \(0.254522\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2.00000 −0.0740233
\(731\) 17.3137 0.640371
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) −6.65685 −0.245709
\(735\) 0 0
\(736\) −0.656854 −0.0242120
\(737\) 7.94113 0.292515
\(738\) 0 0
\(739\) −21.1127 −0.776643 −0.388322 0.921524i \(-0.626945\pi\)
−0.388322 + 0.921524i \(0.626945\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0711 0.589590 0.294795 0.955560i \(-0.404749\pi\)
0.294795 + 0.955560i \(0.404749\pi\)
\(744\) 0 0
\(745\) 2.17157 0.0795603
\(746\) 50.6274 1.85360
\(747\) 0 0
\(748\) 15.3137 0.559925
\(749\) 0 0
\(750\) 0 0
\(751\) −30.3431 −1.10724 −0.553619 0.832770i \(-0.686753\pi\)
−0.553619 + 0.832770i \(0.686753\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) −11.6569 −0.424518
\(755\) 11.6569 0.424236
\(756\) 0 0
\(757\) −31.4558 −1.14328 −0.571641 0.820504i \(-0.693693\pi\)
−0.571641 + 0.820504i \(0.693693\pi\)
\(758\) −64.7696 −2.35254
\(759\) 0 0
\(760\) 12.4853 0.452889
\(761\) 9.31371 0.337622 0.168811 0.985648i \(-0.446007\pi\)
0.168811 + 0.985648i \(0.446007\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 49.1127 1.77684
\(765\) 0 0
\(766\) 7.00000 0.252920
\(767\) 21.6569 0.781984
\(768\) 0 0
\(769\) 0.627417 0.0226252 0.0113126 0.999936i \(-0.496399\pi\)
0.0113126 + 0.999936i \(0.496399\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.65685 −0.275576
\(773\) −37.1127 −1.33485 −0.667425 0.744677i \(-0.732604\pi\)
−0.667425 + 0.744677i \(0.732604\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) 51.4558 1.84716
\(777\) 0 0
\(778\) 57.1127 2.04759
\(779\) 22.1421 0.793324
\(780\) 0 0
\(781\) −3.71573 −0.132959
\(782\) 4.82843 0.172664
\(783\) 0 0
\(784\) 0 0
\(785\) −17.3137 −0.617953
\(786\) 0 0
\(787\) −2.55635 −0.0911240 −0.0455620 0.998962i \(-0.514508\pi\)
−0.0455620 + 0.998962i \(0.514508\pi\)
\(788\) 47.2548 1.68338
\(789\) 0 0
\(790\) −35.7990 −1.27367
\(791\) 0 0
\(792\) 0 0
\(793\) −26.4853 −0.940520
\(794\) −40.1421 −1.42459
\(795\) 0 0
\(796\) 36.9706 1.31039
\(797\) −8.00000 −0.283375 −0.141687 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) 0 0
\(799\) 9.65685 0.341635
\(800\) 1.58579 0.0560660
\(801\) 0 0
\(802\) 73.1838 2.58421
\(803\) 0.686292 0.0242187
\(804\) 0 0
\(805\) 0 0
\(806\) −69.9411 −2.46357
\(807\) 0 0
\(808\) 45.5269 1.60163
\(809\) −35.6274 −1.25259 −0.626297 0.779585i \(-0.715430\pi\)
−0.626297 + 0.779585i \(0.715430\pi\)
\(810\) 0 0
\(811\) 20.6274 0.724327 0.362163 0.932115i \(-0.382038\pi\)
0.362163 + 0.932115i \(0.382038\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 12.3431 0.432362
\(816\) 0 0
\(817\) −10.1421 −0.354828
\(818\) −35.7279 −1.24920
\(819\) 0 0
\(820\) −29.9706 −1.04662
\(821\) −47.9411 −1.67316 −0.836578 0.547847i \(-0.815448\pi\)
−0.836578 + 0.547847i \(0.815448\pi\)
\(822\) 0 0
\(823\) 2.07107 0.0721929 0.0360964 0.999348i \(-0.488508\pi\)
0.0360964 + 0.999348i \(0.488508\pi\)
\(824\) −10.6569 −0.371249
\(825\) 0 0
\(826\) 0 0
\(827\) 26.2132 0.911522 0.455761 0.890102i \(-0.349367\pi\)
0.455761 + 0.890102i \(0.349367\pi\)
\(828\) 0 0
\(829\) −29.3137 −1.01811 −0.509054 0.860735i \(-0.670005\pi\)
−0.509054 + 0.860735i \(0.670005\pi\)
\(830\) −33.1421 −1.15038
\(831\) 0 0
\(832\) −47.4558 −1.64524
\(833\) 0 0
\(834\) 0 0
\(835\) 22.4142 0.775676
\(836\) −8.97056 −0.310253
\(837\) 0 0
\(838\) 1.65685 0.0572351
\(839\) 15.1716 0.523781 0.261890 0.965098i \(-0.415654\pi\)
0.261890 + 0.965098i \(0.415654\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −32.5563 −1.12197
\(843\) 0 0
\(844\) 78.4264 2.69955
\(845\) 10.3137 0.354802
\(846\) 0 0
\(847\) 0 0
\(848\) 3.51472 0.120696
\(849\) 0 0
\(850\) −11.6569 −0.399827
\(851\) 0 0
\(852\) 0 0
\(853\) −2.54416 −0.0871102 −0.0435551 0.999051i \(-0.513868\pi\)
−0.0435551 + 0.999051i \(0.513868\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 49.6274 1.69623
\(857\) 34.2843 1.17113 0.585564 0.810626i \(-0.300873\pi\)
0.585564 + 0.810626i \(0.300873\pi\)
\(858\) 0 0
\(859\) −1.37258 −0.0468319 −0.0234160 0.999726i \(-0.507454\pi\)
−0.0234160 + 0.999726i \(0.507454\pi\)
\(860\) 13.7279 0.468118
\(861\) 0 0
\(862\) 42.9706 1.46358
\(863\) −14.5563 −0.495504 −0.247752 0.968823i \(-0.579692\pi\)
−0.247752 + 0.968823i \(0.579692\pi\)
\(864\) 0 0
\(865\) 3.31371 0.112669
\(866\) 18.8284 0.639816
\(867\) 0 0
\(868\) 0 0
\(869\) 12.2843 0.416715
\(870\) 0 0
\(871\) 46.2843 1.56828
\(872\) 59.5269 2.01584
\(873\) 0 0
\(874\) −2.82843 −0.0956730
\(875\) 0 0
\(876\) 0 0
\(877\) 25.1716 0.849984 0.424992 0.905197i \(-0.360277\pi\)
0.424992 + 0.905197i \(0.360277\pi\)
\(878\) −81.9411 −2.76538
\(879\) 0 0
\(880\) 2.48528 0.0837788
\(881\) 1.82843 0.0616013 0.0308006 0.999526i \(-0.490194\pi\)
0.0308006 + 0.999526i \(0.490194\pi\)
\(882\) 0 0
\(883\) 18.2843 0.615315 0.307657 0.951497i \(-0.400455\pi\)
0.307657 + 0.951497i \(0.400455\pi\)
\(884\) 89.2548 3.00196
\(885\) 0 0
\(886\) 72.9411 2.45051
\(887\) 29.9289 1.00492 0.502458 0.864602i \(-0.332429\pi\)
0.502458 + 0.864602i \(0.332429\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 20.8995 0.700553
\(891\) 0 0
\(892\) −1.31371 −0.0439862
\(893\) −5.65685 −0.189299
\(894\) 0 0
\(895\) 10.0000 0.334263
\(896\) 0 0
\(897\) 0 0
\(898\) 9.24264 0.308431
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) 5.65685 0.188457
\(902\) 15.6569 0.521316
\(903\) 0 0
\(904\) −19.7990 −0.658505
\(905\) −2.65685 −0.0883168
\(906\) 0 0
\(907\) −14.2132 −0.471942 −0.235971 0.971760i \(-0.575827\pi\)
−0.235971 + 0.971760i \(0.575827\pi\)
\(908\) −26.6863 −0.885616
\(909\) 0 0
\(910\) 0 0
\(911\) 10.2010 0.337975 0.168987 0.985618i \(-0.445950\pi\)
0.168987 + 0.985618i \(0.445950\pi\)
\(912\) 0 0
\(913\) 11.3726 0.376378
\(914\) −58.6274 −1.93922
\(915\) 0 0
\(916\) 44.6274 1.47453
\(917\) 0 0
\(918\) 0 0
\(919\) −43.1127 −1.42216 −0.711078 0.703113i \(-0.751793\pi\)
−0.711078 + 0.703113i \(0.751793\pi\)
\(920\) 1.82843 0.0602815
\(921\) 0 0
\(922\) −99.7401 −3.28477
\(923\) −21.6569 −0.712844
\(924\) 0 0
\(925\) 0 0
\(926\) 89.4264 2.93873
\(927\) 0 0
\(928\) 1.58579 0.0520560
\(929\) 5.48528 0.179966 0.0899831 0.995943i \(-0.471319\pi\)
0.0899831 + 0.995943i \(0.471319\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 64.4264 2.11036
\(933\) 0 0
\(934\) 7.48528 0.244926
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −34.6274 −1.13123 −0.565614 0.824670i \(-0.691361\pi\)
−0.565614 + 0.824670i \(0.691361\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 7.65685 0.249739
\(941\) −46.2843 −1.50882 −0.754412 0.656401i \(-0.772078\pi\)
−0.754412 + 0.656401i \(0.772078\pi\)
\(942\) 0 0
\(943\) 3.24264 0.105595
\(944\) 13.4558 0.437950
\(945\) 0 0
\(946\) −7.17157 −0.233168
\(947\) −33.1838 −1.07833 −0.539164 0.842201i \(-0.681260\pi\)
−0.539164 + 0.842201i \(0.681260\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 6.82843 0.221543
\(951\) 0 0
\(952\) 0 0
\(953\) 13.6569 0.442389 0.221194 0.975230i \(-0.429004\pi\)
0.221194 + 0.975230i \(0.429004\pi\)
\(954\) 0 0
\(955\) 12.8284 0.415118
\(956\) 81.5980 2.63907
\(957\) 0 0
\(958\) 86.0833 2.78122
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) −105.882 −3.41024
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) 37.5269 1.20678 0.603392 0.797445i \(-0.293815\pi\)
0.603392 + 0.797445i \(0.293815\pi\)
\(968\) 45.5269 1.46329
\(969\) 0 0
\(970\) 28.1421 0.903590
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −10.4853 −0.335970
\(975\) 0 0
\(976\) −16.4558 −0.526739
\(977\) 1.31371 0.0420293 0.0210146 0.999779i \(-0.493310\pi\)
0.0210146 + 0.999779i \(0.493310\pi\)
\(978\) 0 0
\(979\) −7.17157 −0.229204
\(980\) 0 0
\(981\) 0 0
\(982\) 22.4853 0.717534
\(983\) 28.2132 0.899861 0.449931 0.893063i \(-0.351449\pi\)
0.449931 + 0.893063i \(0.351449\pi\)
\(984\) 0 0
\(985\) 12.3431 0.393285
\(986\) −11.6569 −0.371230
\(987\) 0 0
\(988\) −52.2843 −1.66338
\(989\) −1.48528 −0.0472292
\(990\) 0 0
\(991\) −4.34315 −0.137965 −0.0689823 0.997618i \(-0.521975\pi\)
−0.0689823 + 0.997618i \(0.521975\pi\)
\(992\) 9.51472 0.302093
\(993\) 0 0
\(994\) 0 0
\(995\) 9.65685 0.306143
\(996\) 0 0
\(997\) −33.4558 −1.05956 −0.529779 0.848136i \(-0.677725\pi\)
−0.529779 + 0.848136i \(0.677725\pi\)
\(998\) 2.00000 0.0633089
\(999\) 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 2205.2.a.q.1.1 2
3.2 odd 2 245.2.a.g.1.2 2
7.3 odd 6 315.2.j.e.226.2 4
7.5 odd 6 315.2.j.e.46.2 4
7.6 odd 2 2205.2.a.n.1.1 2
12.11 even 2 3920.2.a.bv.1.1 2
15.2 even 4 1225.2.b.h.99.4 4
15.8 even 4 1225.2.b.h.99.1 4
15.14 odd 2 1225.2.a.m.1.1 2
21.2 odd 6 245.2.e.e.116.1 4
21.5 even 6 35.2.e.a.11.1 4
21.11 odd 6 245.2.e.e.226.1 4
21.17 even 6 35.2.e.a.16.1 yes 4
21.20 even 2 245.2.a.h.1.2 2
84.47 odd 6 560.2.q.k.81.1 4
84.59 odd 6 560.2.q.k.401.1 4
84.83 odd 2 3920.2.a.bq.1.2 2
105.17 odd 12 175.2.k.a.149.1 8
105.38 odd 12 175.2.k.a.149.4 8
105.47 odd 12 175.2.k.a.74.4 8
105.59 even 6 175.2.e.c.51.2 4
105.62 odd 4 1225.2.b.g.99.4 4
105.68 odd 12 175.2.k.a.74.1 8
105.83 odd 4 1225.2.b.g.99.1 4
105.89 even 6 175.2.e.c.151.2 4
105.104 even 2 1225.2.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.e.a.11.1 4 21.5 even 6
35.2.e.a.16.1 yes 4 21.17 even 6
175.2.e.c.51.2 4 105.59 even 6
175.2.e.c.151.2 4 105.89 even 6
175.2.k.a.74.1 8 105.68 odd 12
175.2.k.a.74.4 8 105.47 odd 12
175.2.k.a.149.1 8 105.17 odd 12
175.2.k.a.149.4 8 105.38 odd 12
245.2.a.g.1.2 2 3.2 odd 2
245.2.a.h.1.2 2 21.20 even 2
245.2.e.e.116.1 4 21.2 odd 6
245.2.e.e.226.1 4 21.11 odd 6
315.2.j.e.46.2 4 7.5 odd 6
315.2.j.e.226.2 4 7.3 odd 6
560.2.q.k.81.1 4 84.47 odd 6
560.2.q.k.401.1 4 84.59 odd 6
1225.2.a.k.1.1 2 105.104 even 2
1225.2.a.m.1.1 2 15.14 odd 2
1225.2.b.g.99.1 4 105.83 odd 4
1225.2.b.g.99.4 4 105.62 odd 4
1225.2.b.h.99.1 4 15.8 even 4
1225.2.b.h.99.4 4 15.2 even 4
2205.2.a.n.1.1 2 7.6 odd 2
2205.2.a.q.1.1 2 1.1 even 1 trivial
3920.2.a.bq.1.2 2 84.83 odd 2
3920.2.a.bv.1.1 2 12.11 even 2