Properties

Label 300.4.d.d.49.1
Level $300$
Weight $4$
Character 300.49
Analytic conductor $17.701$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,4,Mod(49,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 300.d (of order \(2\), degree \(1\), 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: \(17.7005730017\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 300.49
Dual form 300.4.d.d.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} -32.0000i q^{7} -9.00000 q^{9} +36.0000 q^{11} -10.0000i q^{13} +78.0000i q^{17} -140.000 q^{19} -96.0000 q^{21} -192.000i q^{23} +27.0000i q^{27} -6.00000 q^{29} -16.0000 q^{31} -108.000i q^{33} +34.0000i q^{37} -30.0000 q^{39} -390.000 q^{41} -52.0000i q^{43} -408.000i q^{47} -681.000 q^{49} +234.000 q^{51} -114.000i q^{53} +420.000i q^{57} -516.000 q^{59} -58.0000 q^{61} +288.000i q^{63} +892.000i q^{67} -576.000 q^{69} -120.000 q^{71} -646.000i q^{73} -1152.00i q^{77} +1168.00 q^{79} +81.0000 q^{81} -732.000i q^{83} +18.0000i q^{87} +1590.00 q^{89} -320.000 q^{91} +48.0000i q^{93} -194.000i q^{97} -324.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} + 72 q^{11} - 280 q^{19} - 192 q^{21} - 12 q^{29} - 32 q^{31} - 60 q^{39} - 780 q^{41} - 1362 q^{49} + 468 q^{51} - 1032 q^{59} - 116 q^{61} - 1152 q^{69} - 240 q^{71} + 2336 q^{79} + 162 q^{81}+ \cdots - 648 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 3.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 32.0000i − 1.72784i −0.503631 0.863919i \(-0.668003\pi\)
0.503631 0.863919i \(-0.331997\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 36.0000 0.986764 0.493382 0.869813i \(-0.335760\pi\)
0.493382 + 0.869813i \(0.335760\pi\)
\(12\) 0 0
\(13\) − 10.0000i − 0.213346i −0.994294 0.106673i \(-0.965980\pi\)
0.994294 0.106673i \(-0.0340198\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 78.0000i 1.11281i 0.830911 + 0.556405i \(0.187820\pi\)
−0.830911 + 0.556405i \(0.812180\pi\)
\(18\) 0 0
\(19\) −140.000 −1.69043 −0.845216 0.534425i \(-0.820528\pi\)
−0.845216 + 0.534425i \(0.820528\pi\)
\(20\) 0 0
\(21\) −96.0000 −0.997567
\(22\) 0 0
\(23\) − 192.000i − 1.74064i −0.492485 0.870321i \(-0.663911\pi\)
0.492485 0.870321i \(-0.336089\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) −6.00000 −0.0384197 −0.0192099 0.999815i \(-0.506115\pi\)
−0.0192099 + 0.999815i \(0.506115\pi\)
\(30\) 0 0
\(31\) −16.0000 −0.0926995 −0.0463498 0.998925i \(-0.514759\pi\)
−0.0463498 + 0.998925i \(0.514759\pi\)
\(32\) 0 0
\(33\) − 108.000i − 0.569709i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 34.0000i 0.151069i 0.997143 + 0.0755347i \(0.0240664\pi\)
−0.997143 + 0.0755347i \(0.975934\pi\)
\(38\) 0 0
\(39\) −30.0000 −0.123176
\(40\) 0 0
\(41\) −390.000 −1.48556 −0.742778 0.669538i \(-0.766492\pi\)
−0.742778 + 0.669538i \(0.766492\pi\)
\(42\) 0 0
\(43\) − 52.0000i − 0.184417i −0.995740 0.0922084i \(-0.970607\pi\)
0.995740 0.0922084i \(-0.0293926\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 408.000i − 1.26623i −0.774057 0.633116i \(-0.781776\pi\)
0.774057 0.633116i \(-0.218224\pi\)
\(48\) 0 0
\(49\) −681.000 −1.98542
\(50\) 0 0
\(51\) 234.000 0.642481
\(52\) 0 0
\(53\) − 114.000i − 0.295455i −0.989028 0.147727i \(-0.952804\pi\)
0.989028 0.147727i \(-0.0471958\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 420.000i 0.975971i
\(58\) 0 0
\(59\) −516.000 −1.13860 −0.569301 0.822129i \(-0.692786\pi\)
−0.569301 + 0.822129i \(0.692786\pi\)
\(60\) 0 0
\(61\) −58.0000 −0.121740 −0.0608700 0.998146i \(-0.519388\pi\)
−0.0608700 + 0.998146i \(0.519388\pi\)
\(62\) 0 0
\(63\) 288.000i 0.575946i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 892.000i 1.62649i 0.581918 + 0.813247i \(0.302302\pi\)
−0.581918 + 0.813247i \(0.697698\pi\)
\(68\) 0 0
\(69\) −576.000 −1.00496
\(70\) 0 0
\(71\) −120.000 −0.200583 −0.100291 0.994958i \(-0.531978\pi\)
−0.100291 + 0.994958i \(0.531978\pi\)
\(72\) 0 0
\(73\) − 646.000i − 1.03573i −0.855461 0.517867i \(-0.826726\pi\)
0.855461 0.517867i \(-0.173274\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1152.00i − 1.70497i
\(78\) 0 0
\(79\) 1168.00 1.66342 0.831711 0.555209i \(-0.187362\pi\)
0.831711 + 0.555209i \(0.187362\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 732.000i − 0.968041i −0.875057 0.484021i \(-0.839176\pi\)
0.875057 0.484021i \(-0.160824\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 18.0000i 0.0221816i
\(88\) 0 0
\(89\) 1590.00 1.89370 0.946852 0.321669i \(-0.104244\pi\)
0.946852 + 0.321669i \(0.104244\pi\)
\(90\) 0 0
\(91\) −320.000 −0.368628
\(92\) 0 0
\(93\) 48.0000i 0.0535201i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 194.000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 0 0
\(99\) −324.000 −0.328921
\(100\) 0 0
\(101\) 798.000 0.786178 0.393089 0.919500i \(-0.371406\pi\)
0.393089 + 0.919500i \(0.371406\pi\)
\(102\) 0 0
\(103\) 272.000i 0.260203i 0.991501 + 0.130102i \(0.0415304\pi\)
−0.991501 + 0.130102i \(0.958470\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 156.000i − 0.140945i −0.997514 0.0704724i \(-0.977549\pi\)
0.997514 0.0704724i \(-0.0224507\pi\)
\(108\) 0 0
\(109\) −1622.00 −1.42532 −0.712658 0.701512i \(-0.752509\pi\)
−0.712658 + 0.701512i \(0.752509\pi\)
\(110\) 0 0
\(111\) 102.000 0.0872199
\(112\) 0 0
\(113\) 1074.00i 0.894101i 0.894509 + 0.447051i \(0.147526\pi\)
−0.894509 + 0.447051i \(0.852474\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 90.0000i 0.0711154i
\(118\) 0 0
\(119\) 2496.00 1.92276
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) 1170.00i 0.857686i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1528.00i 1.06762i 0.845604 + 0.533811i \(0.179241\pi\)
−0.845604 + 0.533811i \(0.820759\pi\)
\(128\) 0 0
\(129\) −156.000 −0.106473
\(130\) 0 0
\(131\) 2412.00 1.60868 0.804341 0.594168i \(-0.202518\pi\)
0.804341 + 0.594168i \(0.202518\pi\)
\(132\) 0 0
\(133\) 4480.00i 2.92079i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2106.00i − 1.31334i −0.754178 0.656671i \(-0.771964\pi\)
0.754178 0.656671i \(-0.228036\pi\)
\(138\) 0 0
\(139\) 556.000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −1224.00 −0.731060
\(142\) 0 0
\(143\) − 360.000i − 0.210522i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2043.00i 1.14628i
\(148\) 0 0
\(149\) 2418.00 1.32946 0.664732 0.747081i \(-0.268546\pi\)
0.664732 + 0.747081i \(0.268546\pi\)
\(150\) 0 0
\(151\) 2840.00 1.53057 0.765285 0.643692i \(-0.222598\pi\)
0.765285 + 0.643692i \(0.222598\pi\)
\(152\) 0 0
\(153\) − 702.000i − 0.370937i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 2054.00i − 1.04412i −0.852908 0.522061i \(-0.825163\pi\)
0.852908 0.522061i \(-0.174837\pi\)
\(158\) 0 0
\(159\) −342.000 −0.170581
\(160\) 0 0
\(161\) −6144.00 −3.00755
\(162\) 0 0
\(163\) − 460.000i − 0.221043i −0.993874 0.110521i \(-0.964748\pi\)
0.993874 0.110521i \(-0.0352521\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 2016.00i − 0.934148i −0.884218 0.467074i \(-0.845308\pi\)
0.884218 0.467074i \(-0.154692\pi\)
\(168\) 0 0
\(169\) 2097.00 0.954483
\(170\) 0 0
\(171\) 1260.00 0.563477
\(172\) 0 0
\(173\) − 618.000i − 0.271593i −0.990737 0.135797i \(-0.956641\pi\)
0.990737 0.135797i \(-0.0433594\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1548.00i 0.657372i
\(178\) 0 0
\(179\) 2964.00 1.23765 0.618826 0.785528i \(-0.287609\pi\)
0.618826 + 0.785528i \(0.287609\pi\)
\(180\) 0 0
\(181\) −370.000 −0.151944 −0.0759721 0.997110i \(-0.524206\pi\)
−0.0759721 + 0.997110i \(0.524206\pi\)
\(182\) 0 0
\(183\) 174.000i 0.0702866i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2808.00i 1.09808i
\(188\) 0 0
\(189\) 864.000 0.332522
\(190\) 0 0
\(191\) 1104.00 0.418234 0.209117 0.977891i \(-0.432941\pi\)
0.209117 + 0.977891i \(0.432941\pi\)
\(192\) 0 0
\(193\) − 2398.00i − 0.894362i −0.894444 0.447181i \(-0.852428\pi\)
0.894444 0.447181i \(-0.147572\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 1278.00i − 0.462202i −0.972930 0.231101i \(-0.925767\pi\)
0.972930 0.231101i \(-0.0742327\pi\)
\(198\) 0 0
\(199\) −4472.00 −1.59302 −0.796512 0.604623i \(-0.793324\pi\)
−0.796512 + 0.604623i \(0.793324\pi\)
\(200\) 0 0
\(201\) 2676.00 0.939057
\(202\) 0 0
\(203\) 192.000i 0.0663830i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1728.00i 0.580214i
\(208\) 0 0
\(209\) −5040.00 −1.66806
\(210\) 0 0
\(211\) 1340.00 0.437201 0.218600 0.975814i \(-0.429851\pi\)
0.218600 + 0.975814i \(0.429851\pi\)
\(212\) 0 0
\(213\) 360.000i 0.115807i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 512.000i 0.160170i
\(218\) 0 0
\(219\) −1938.00 −0.597981
\(220\) 0 0
\(221\) 780.000 0.237414
\(222\) 0 0
\(223\) 2360.00i 0.708687i 0.935115 + 0.354344i \(0.115296\pi\)
−0.935115 + 0.354344i \(0.884704\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1380.00i − 0.403497i −0.979437 0.201748i \(-0.935338\pi\)
0.979437 0.201748i \(-0.0646624\pi\)
\(228\) 0 0
\(229\) −1694.00 −0.488833 −0.244416 0.969670i \(-0.578596\pi\)
−0.244416 + 0.969670i \(0.578596\pi\)
\(230\) 0 0
\(231\) −3456.00 −0.984364
\(232\) 0 0
\(233\) − 5190.00i − 1.45926i −0.683841 0.729631i \(-0.739692\pi\)
0.683841 0.729631i \(-0.260308\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 3504.00i − 0.960377i
\(238\) 0 0
\(239\) −2352.00 −0.636562 −0.318281 0.947996i \(-0.603105\pi\)
−0.318281 + 0.947996i \(0.603105\pi\)
\(240\) 0 0
\(241\) −3502.00 −0.936032 −0.468016 0.883720i \(-0.655031\pi\)
−0.468016 + 0.883720i \(0.655031\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1400.00i 0.360647i
\(248\) 0 0
\(249\) −2196.00 −0.558899
\(250\) 0 0
\(251\) 4788.00 1.20405 0.602024 0.798478i \(-0.294361\pi\)
0.602024 + 0.798478i \(0.294361\pi\)
\(252\) 0 0
\(253\) − 6912.00i − 1.71760i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 1506.00i − 0.365532i −0.983156 0.182766i \(-0.941495\pi\)
0.983156 0.182766i \(-0.0585051\pi\)
\(258\) 0 0
\(259\) 1088.00 0.261023
\(260\) 0 0
\(261\) 54.0000 0.0128066
\(262\) 0 0
\(263\) − 432.000i − 0.101286i −0.998717 0.0506431i \(-0.983873\pi\)
0.998717 0.0506431i \(-0.0161271\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 4770.00i − 1.09333i
\(268\) 0 0
\(269\) −54.0000 −0.0122395 −0.00611977 0.999981i \(-0.501948\pi\)
−0.00611977 + 0.999981i \(0.501948\pi\)
\(270\) 0 0
\(271\) −6496.00 −1.45610 −0.728051 0.685522i \(-0.759574\pi\)
−0.728051 + 0.685522i \(0.759574\pi\)
\(272\) 0 0
\(273\) 960.000i 0.212827i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 466.000i 0.101080i 0.998722 + 0.0505401i \(0.0160943\pi\)
−0.998722 + 0.0505401i \(0.983906\pi\)
\(278\) 0 0
\(279\) 144.000 0.0308998
\(280\) 0 0
\(281\) −4854.00 −1.03048 −0.515241 0.857045i \(-0.672298\pi\)
−0.515241 + 0.857045i \(0.672298\pi\)
\(282\) 0 0
\(283\) − 4516.00i − 0.948581i −0.880368 0.474290i \(-0.842705\pi\)
0.880368 0.474290i \(-0.157295\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12480.0i 2.56680i
\(288\) 0 0
\(289\) −1171.00 −0.238347
\(290\) 0 0
\(291\) −582.000 −0.117242
\(292\) 0 0
\(293\) 8574.00i 1.70955i 0.518998 + 0.854775i \(0.326305\pi\)
−0.518998 + 0.854775i \(0.673695\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 972.000i 0.189903i
\(298\) 0 0
\(299\) −1920.00 −0.371359
\(300\) 0 0
\(301\) −1664.00 −0.318642
\(302\) 0 0
\(303\) − 2394.00i − 0.453900i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 3476.00i − 0.646208i −0.946363 0.323104i \(-0.895274\pi\)
0.946363 0.323104i \(-0.104726\pi\)
\(308\) 0 0
\(309\) 816.000 0.150229
\(310\) 0 0
\(311\) 2424.00 0.441969 0.220985 0.975277i \(-0.429073\pi\)
0.220985 + 0.975277i \(0.429073\pi\)
\(312\) 0 0
\(313\) − 1558.00i − 0.281353i −0.990056 0.140676i \(-0.955072\pi\)
0.990056 0.140676i \(-0.0449277\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8538.00i 1.51275i 0.654138 + 0.756375i \(0.273032\pi\)
−0.654138 + 0.756375i \(0.726968\pi\)
\(318\) 0 0
\(319\) −216.000 −0.0379112
\(320\) 0 0
\(321\) −468.000 −0.0813745
\(322\) 0 0
\(323\) − 10920.0i − 1.88113i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4866.00i 0.822906i
\(328\) 0 0
\(329\) −13056.0 −2.18784
\(330\) 0 0
\(331\) −988.000 −0.164065 −0.0820323 0.996630i \(-0.526141\pi\)
−0.0820323 + 0.996630i \(0.526141\pi\)
\(332\) 0 0
\(333\) − 306.000i − 0.0503564i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 2546.00i − 0.411541i −0.978600 0.205771i \(-0.934030\pi\)
0.978600 0.205771i \(-0.0659701\pi\)
\(338\) 0 0
\(339\) 3222.00 0.516209
\(340\) 0 0
\(341\) −576.000 −0.0914726
\(342\) 0 0
\(343\) 10816.0i 1.70265i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 8556.00i − 1.32366i −0.749654 0.661830i \(-0.769780\pi\)
0.749654 0.661830i \(-0.230220\pi\)
\(348\) 0 0
\(349\) 3706.00 0.568417 0.284209 0.958762i \(-0.408269\pi\)
0.284209 + 0.958762i \(0.408269\pi\)
\(350\) 0 0
\(351\) 270.000 0.0410585
\(352\) 0 0
\(353\) 11394.0i 1.71796i 0.512005 + 0.858982i \(0.328903\pi\)
−0.512005 + 0.858982i \(0.671097\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 7488.00i − 1.11010i
\(358\) 0 0
\(359\) 264.000 0.0388117 0.0194058 0.999812i \(-0.493823\pi\)
0.0194058 + 0.999812i \(0.493823\pi\)
\(360\) 0 0
\(361\) 12741.0 1.85756
\(362\) 0 0
\(363\) 105.000i 0.0151820i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 10232.0i − 1.45533i −0.685933 0.727665i \(-0.740606\pi\)
0.685933 0.727665i \(-0.259394\pi\)
\(368\) 0 0
\(369\) 3510.00 0.495185
\(370\) 0 0
\(371\) −3648.00 −0.510498
\(372\) 0 0
\(373\) − 562.000i − 0.0780141i −0.999239 0.0390070i \(-0.987581\pi\)
0.999239 0.0390070i \(-0.0124195\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 60.0000i 0.00819670i
\(378\) 0 0
\(379\) 7228.00 0.979624 0.489812 0.871828i \(-0.337065\pi\)
0.489812 + 0.871828i \(0.337065\pi\)
\(380\) 0 0
\(381\) 4584.00 0.616392
\(382\) 0 0
\(383\) − 5736.00i − 0.765263i −0.923901 0.382632i \(-0.875018\pi\)
0.923901 0.382632i \(-0.124982\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 468.000i 0.0614723i
\(388\) 0 0
\(389\) 9186.00 1.19730 0.598649 0.801012i \(-0.295705\pi\)
0.598649 + 0.801012i \(0.295705\pi\)
\(390\) 0 0
\(391\) 14976.0 1.93700
\(392\) 0 0
\(393\) − 7236.00i − 0.928773i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 394.000i 0.0498093i 0.999690 + 0.0249047i \(0.00792822\pi\)
−0.999690 + 0.0249047i \(0.992072\pi\)
\(398\) 0 0
\(399\) 13440.0 1.68632
\(400\) 0 0
\(401\) −1614.00 −0.200996 −0.100498 0.994937i \(-0.532044\pi\)
−0.100498 + 0.994937i \(0.532044\pi\)
\(402\) 0 0
\(403\) 160.000i 0.0197771i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1224.00i 0.149070i
\(408\) 0 0
\(409\) −1034.00 −0.125007 −0.0625037 0.998045i \(-0.519909\pi\)
−0.0625037 + 0.998045i \(0.519909\pi\)
\(410\) 0 0
\(411\) −6318.00 −0.758258
\(412\) 0 0
\(413\) 16512.0i 1.96732i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 1668.00i − 0.195881i
\(418\) 0 0
\(419\) −3708.00 −0.432333 −0.216167 0.976356i \(-0.569355\pi\)
−0.216167 + 0.976356i \(0.569355\pi\)
\(420\) 0 0
\(421\) −4930.00 −0.570721 −0.285360 0.958420i \(-0.592113\pi\)
−0.285360 + 0.958420i \(0.592113\pi\)
\(422\) 0 0
\(423\) 3672.00i 0.422077i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1856.00i 0.210347i
\(428\) 0 0
\(429\) −1080.00 −0.121545
\(430\) 0 0
\(431\) −2592.00 −0.289680 −0.144840 0.989455i \(-0.546267\pi\)
−0.144840 + 0.989455i \(0.546267\pi\)
\(432\) 0 0
\(433\) 2162.00i 0.239952i 0.992777 + 0.119976i \(0.0382817\pi\)
−0.992777 + 0.119976i \(0.961718\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 26880.0i 2.94244i
\(438\) 0 0
\(439\) −1352.00 −0.146987 −0.0734937 0.997296i \(-0.523415\pi\)
−0.0734937 + 0.997296i \(0.523415\pi\)
\(440\) 0 0
\(441\) 6129.00 0.661808
\(442\) 0 0
\(443\) 5532.00i 0.593303i 0.954986 + 0.296652i \(0.0958700\pi\)
−0.954986 + 0.296652i \(0.904130\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 7254.00i − 0.767567i
\(448\) 0 0
\(449\) 3198.00 0.336131 0.168066 0.985776i \(-0.446248\pi\)
0.168066 + 0.985776i \(0.446248\pi\)
\(450\) 0 0
\(451\) −14040.0 −1.46589
\(452\) 0 0
\(453\) − 8520.00i − 0.883674i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1510.00i 0.154562i 0.997009 + 0.0772810i \(0.0246239\pi\)
−0.997009 + 0.0772810i \(0.975376\pi\)
\(458\) 0 0
\(459\) −2106.00 −0.214160
\(460\) 0 0
\(461\) 16086.0 1.62516 0.812581 0.582848i \(-0.198062\pi\)
0.812581 + 0.582848i \(0.198062\pi\)
\(462\) 0 0
\(463\) 5384.00i 0.540423i 0.962801 + 0.270211i \(0.0870936\pi\)
−0.962801 + 0.270211i \(0.912906\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2604.00i 0.258027i 0.991643 + 0.129014i \(0.0411811\pi\)
−0.991643 + 0.129014i \(0.958819\pi\)
\(468\) 0 0
\(469\) 28544.0 2.81032
\(470\) 0 0
\(471\) −6162.00 −0.602824
\(472\) 0 0
\(473\) − 1872.00i − 0.181976i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1026.00i 0.0984849i
\(478\) 0 0
\(479\) −11136.0 −1.06225 −0.531124 0.847294i \(-0.678230\pi\)
−0.531124 + 0.847294i \(0.678230\pi\)
\(480\) 0 0
\(481\) 340.000 0.0322301
\(482\) 0 0
\(483\) 18432.0i 1.73641i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 14624.0i − 1.36073i −0.732872 0.680366i \(-0.761821\pi\)
0.732872 0.680366i \(-0.238179\pi\)
\(488\) 0 0
\(489\) −1380.00 −0.127619
\(490\) 0 0
\(491\) 11844.0 1.08862 0.544310 0.838884i \(-0.316792\pi\)
0.544310 + 0.838884i \(0.316792\pi\)
\(492\) 0 0
\(493\) − 468.000i − 0.0427539i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3840.00i 0.346575i
\(498\) 0 0
\(499\) 11284.0 1.01231 0.506154 0.862443i \(-0.331067\pi\)
0.506154 + 0.862443i \(0.331067\pi\)
\(500\) 0 0
\(501\) −6048.00 −0.539331
\(502\) 0 0
\(503\) 4032.00i 0.357412i 0.983903 + 0.178706i \(0.0571910\pi\)
−0.983903 + 0.178706i \(0.942809\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 6291.00i − 0.551071i
\(508\) 0 0
\(509\) 17562.0 1.52932 0.764658 0.644436i \(-0.222908\pi\)
0.764658 + 0.644436i \(0.222908\pi\)
\(510\) 0 0
\(511\) −20672.0 −1.78958
\(512\) 0 0
\(513\) − 3780.00i − 0.325324i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 14688.0i − 1.24947i
\(518\) 0 0
\(519\) −1854.00 −0.156805
\(520\) 0 0
\(521\) 3162.00 0.265892 0.132946 0.991123i \(-0.457556\pi\)
0.132946 + 0.991123i \(0.457556\pi\)
\(522\) 0 0
\(523\) 6764.00i 0.565524i 0.959190 + 0.282762i \(0.0912507\pi\)
−0.959190 + 0.282762i \(0.908749\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1248.00i − 0.103157i
\(528\) 0 0
\(529\) −24697.0 −2.02983
\(530\) 0 0
\(531\) 4644.00 0.379534
\(532\) 0 0
\(533\) 3900.00i 0.316938i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 8892.00i − 0.714559i
\(538\) 0 0
\(539\) −24516.0 −1.95914
\(540\) 0 0
\(541\) 17798.0 1.41441 0.707205 0.707009i \(-0.249956\pi\)
0.707205 + 0.707009i \(0.249956\pi\)
\(542\) 0 0
\(543\) 1110.00i 0.0877250i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 19996.0i 1.56301i 0.623898 + 0.781506i \(0.285548\pi\)
−0.623898 + 0.781506i \(0.714452\pi\)
\(548\) 0 0
\(549\) 522.000 0.0405800
\(550\) 0 0
\(551\) 840.000 0.0649459
\(552\) 0 0
\(553\) − 37376.0i − 2.87412i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 11094.0i − 0.843928i −0.906613 0.421964i \(-0.861341\pi\)
0.906613 0.421964i \(-0.138659\pi\)
\(558\) 0 0
\(559\) −520.000 −0.0393446
\(560\) 0 0
\(561\) 8424.00 0.633978
\(562\) 0 0
\(563\) 900.000i 0.0673721i 0.999432 + 0.0336860i \(0.0107246\pi\)
−0.999432 + 0.0336860i \(0.989275\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 2592.00i − 0.191982i
\(568\) 0 0
\(569\) −7914.00 −0.583079 −0.291540 0.956559i \(-0.594168\pi\)
−0.291540 + 0.956559i \(0.594168\pi\)
\(570\) 0 0
\(571\) −2380.00 −0.174431 −0.0872153 0.996189i \(-0.527797\pi\)
−0.0872153 + 0.996189i \(0.527797\pi\)
\(572\) 0 0
\(573\) − 3312.00i − 0.241467i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 25726.0i 1.85613i 0.372417 + 0.928065i \(0.378529\pi\)
−0.372417 + 0.928065i \(0.621471\pi\)
\(578\) 0 0
\(579\) −7194.00 −0.516360
\(580\) 0 0
\(581\) −23424.0 −1.67262
\(582\) 0 0
\(583\) − 4104.00i − 0.291544i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 3612.00i − 0.253975i −0.991904 0.126987i \(-0.959469\pi\)
0.991904 0.126987i \(-0.0405308\pi\)
\(588\) 0 0
\(589\) 2240.00 0.156702
\(590\) 0 0
\(591\) −3834.00 −0.266852
\(592\) 0 0
\(593\) 2898.00i 0.200686i 0.994953 + 0.100343i \(0.0319940\pi\)
−0.994953 + 0.100343i \(0.968006\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13416.0i 0.919733i
\(598\) 0 0
\(599\) −2664.00 −0.181716 −0.0908582 0.995864i \(-0.528961\pi\)
−0.0908582 + 0.995864i \(0.528961\pi\)
\(600\) 0 0
\(601\) −502.000 −0.0340716 −0.0170358 0.999855i \(-0.505423\pi\)
−0.0170358 + 0.999855i \(0.505423\pi\)
\(602\) 0 0
\(603\) − 8028.00i − 0.542165i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 7976.00i − 0.533337i −0.963788 0.266669i \(-0.914077\pi\)
0.963788 0.266669i \(-0.0859230\pi\)
\(608\) 0 0
\(609\) 576.000 0.0383263
\(610\) 0 0
\(611\) −4080.00 −0.270146
\(612\) 0 0
\(613\) 20414.0i 1.34505i 0.740076 + 0.672523i \(0.234790\pi\)
−0.740076 + 0.672523i \(0.765210\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6342.00i 0.413808i 0.978361 + 0.206904i \(0.0663387\pi\)
−0.978361 + 0.206904i \(0.933661\pi\)
\(618\) 0 0
\(619\) −22676.0 −1.47242 −0.736208 0.676755i \(-0.763385\pi\)
−0.736208 + 0.676755i \(0.763385\pi\)
\(620\) 0 0
\(621\) 5184.00 0.334987
\(622\) 0 0
\(623\) − 50880.0i − 3.27201i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 15120.0i 0.963054i
\(628\) 0 0
\(629\) −2652.00 −0.168112
\(630\) 0 0
\(631\) −7048.00 −0.444654 −0.222327 0.974972i \(-0.571365\pi\)
−0.222327 + 0.974972i \(0.571365\pi\)
\(632\) 0 0
\(633\) − 4020.00i − 0.252418i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6810.00i 0.423582i
\(638\) 0 0
\(639\) 1080.00 0.0668609
\(640\) 0 0
\(641\) −20286.0 −1.25000 −0.624999 0.780625i \(-0.714901\pi\)
−0.624999 + 0.780625i \(0.714901\pi\)
\(642\) 0 0
\(643\) − 16108.0i − 0.987928i −0.869482 0.493964i \(-0.835548\pi\)
0.869482 0.493964i \(-0.164452\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 27456.0i − 1.66833i −0.551518 0.834163i \(-0.685951\pi\)
0.551518 0.834163i \(-0.314049\pi\)
\(648\) 0 0
\(649\) −18576.0 −1.12353
\(650\) 0 0
\(651\) 1536.00 0.0924740
\(652\) 0 0
\(653\) − 12522.0i − 0.750419i −0.926940 0.375210i \(-0.877571\pi\)
0.926940 0.375210i \(-0.122429\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5814.00i 0.345245i
\(658\) 0 0
\(659\) 16308.0 0.963990 0.481995 0.876174i \(-0.339912\pi\)
0.481995 + 0.876174i \(0.339912\pi\)
\(660\) 0 0
\(661\) 32078.0 1.88758 0.943789 0.330547i \(-0.107233\pi\)
0.943789 + 0.330547i \(0.107233\pi\)
\(662\) 0 0
\(663\) − 2340.00i − 0.137071i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1152.00i 0.0668750i
\(668\) 0 0
\(669\) 7080.00 0.409161
\(670\) 0 0
\(671\) −2088.00 −0.120129
\(672\) 0 0
\(673\) 4610.00i 0.264045i 0.991247 + 0.132023i \(0.0421472\pi\)
−0.991247 + 0.132023i \(0.957853\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 10782.0i − 0.612091i −0.952017 0.306046i \(-0.900994\pi\)
0.952017 0.306046i \(-0.0990060\pi\)
\(678\) 0 0
\(679\) −6208.00 −0.350871
\(680\) 0 0
\(681\) −4140.00 −0.232959
\(682\) 0 0
\(683\) 2892.00i 0.162019i 0.996713 + 0.0810097i \(0.0258145\pi\)
−0.996713 + 0.0810097i \(0.974186\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 5082.00i 0.282228i
\(688\) 0 0
\(689\) −1140.00 −0.0630342
\(690\) 0 0
\(691\) −29572.0 −1.62803 −0.814017 0.580841i \(-0.802724\pi\)
−0.814017 + 0.580841i \(0.802724\pi\)
\(692\) 0 0
\(693\) 10368.0i 0.568323i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 30420.0i − 1.65314i
\(698\) 0 0
\(699\) −15570.0 −0.842506
\(700\) 0 0
\(701\) 5766.00 0.310669 0.155334 0.987862i \(-0.450354\pi\)
0.155334 + 0.987862i \(0.450354\pi\)
\(702\) 0 0
\(703\) − 4760.00i − 0.255372i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 25536.0i − 1.35839i
\(708\) 0 0
\(709\) −3326.00 −0.176178 −0.0880892 0.996113i \(-0.528076\pi\)
−0.0880892 + 0.996113i \(0.528076\pi\)
\(710\) 0 0
\(711\) −10512.0 −0.554474
\(712\) 0 0
\(713\) 3072.00i 0.161357i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7056.00i 0.367519i
\(718\) 0 0
\(719\) 7728.00 0.400843 0.200421 0.979710i \(-0.435769\pi\)
0.200421 + 0.979710i \(0.435769\pi\)
\(720\) 0 0
\(721\) 8704.00 0.449589
\(722\) 0 0
\(723\) 10506.0i 0.540418i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 21616.0i 1.10274i 0.834260 + 0.551371i \(0.185895\pi\)
−0.834260 + 0.551371i \(0.814105\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 4056.00 0.205221
\(732\) 0 0
\(733\) 10118.0i 0.509846i 0.966961 + 0.254923i \(0.0820501\pi\)
−0.966961 + 0.254923i \(0.917950\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 32112.0i 1.60497i
\(738\) 0 0
\(739\) −10460.0 −0.520673 −0.260336 0.965518i \(-0.583833\pi\)
−0.260336 + 0.965518i \(0.583833\pi\)
\(740\) 0 0
\(741\) 4200.00 0.208220
\(742\) 0 0
\(743\) 17232.0i 0.850849i 0.904994 + 0.425424i \(0.139875\pi\)
−0.904994 + 0.425424i \(0.860125\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6588.00i 0.322680i
\(748\) 0 0
\(749\) −4992.00 −0.243530
\(750\) 0 0
\(751\) 26912.0 1.30763 0.653817 0.756653i \(-0.273167\pi\)
0.653817 + 0.756653i \(0.273167\pi\)
\(752\) 0 0
\(753\) − 14364.0i − 0.695157i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 13838.0i − 0.664400i −0.943209 0.332200i \(-0.892209\pi\)
0.943209 0.332200i \(-0.107791\pi\)
\(758\) 0 0
\(759\) −20736.0 −0.991659
\(760\) 0 0
\(761\) −17238.0 −0.821126 −0.410563 0.911832i \(-0.634668\pi\)
−0.410563 + 0.911832i \(0.634668\pi\)
\(762\) 0 0
\(763\) 51904.0i 2.46271i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5160.00i 0.242916i
\(768\) 0 0
\(769\) −21698.0 −1.01749 −0.508745 0.860917i \(-0.669890\pi\)
−0.508745 + 0.860917i \(0.669890\pi\)
\(770\) 0 0
\(771\) −4518.00 −0.211040
\(772\) 0 0
\(773\) 18366.0i 0.854565i 0.904118 + 0.427283i \(0.140529\pi\)
−0.904118 + 0.427283i \(0.859471\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 3264.00i − 0.150702i
\(778\) 0 0
\(779\) 54600.0 2.51123
\(780\) 0 0
\(781\) −4320.00 −0.197928
\(782\) 0 0
\(783\) − 162.000i − 0.00739388i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 30316.0i 1.37312i 0.727071 + 0.686562i \(0.240881\pi\)
−0.727071 + 0.686562i \(0.759119\pi\)
\(788\) 0 0
\(789\) −1296.00 −0.0584776
\(790\) 0 0
\(791\) 34368.0 1.54486
\(792\) 0 0
\(793\) 580.000i 0.0259728i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 7494.00i − 0.333063i −0.986036 0.166531i \(-0.946743\pi\)
0.986036 0.166531i \(-0.0532567\pi\)
\(798\) 0 0
\(799\) 31824.0 1.40908
\(800\) 0 0
\(801\) −14310.0 −0.631235
\(802\) 0 0
\(803\) − 23256.0i − 1.02203i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 162.000i 0.00706651i
\(808\) 0 0
\(809\) 11526.0 0.500906 0.250453 0.968129i \(-0.419421\pi\)
0.250453 + 0.968129i \(0.419421\pi\)
\(810\) 0 0
\(811\) −33820.0 −1.46434 −0.732171 0.681121i \(-0.761493\pi\)
−0.732171 + 0.681121i \(0.761493\pi\)
\(812\) 0 0
\(813\) 19488.0i 0.840681i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7280.00i 0.311744i
\(818\) 0 0
\(819\) 2880.00 0.122876
\(820\) 0 0
\(821\) 19566.0 0.831739 0.415870 0.909424i \(-0.363477\pi\)
0.415870 + 0.909424i \(0.363477\pi\)
\(822\) 0 0
\(823\) − 40096.0i − 1.69825i −0.528193 0.849124i \(-0.677130\pi\)
0.528193 0.849124i \(-0.322870\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 31884.0i − 1.34065i −0.742069 0.670324i \(-0.766155\pi\)
0.742069 0.670324i \(-0.233845\pi\)
\(828\) 0 0
\(829\) 24442.0 1.02401 0.512006 0.858982i \(-0.328903\pi\)
0.512006 + 0.858982i \(0.328903\pi\)
\(830\) 0 0
\(831\) 1398.00 0.0583587
\(832\) 0 0
\(833\) − 53118.0i − 2.20940i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 432.000i − 0.0178400i
\(838\) 0 0
\(839\) 31944.0 1.31446 0.657228 0.753691i \(-0.271729\pi\)
0.657228 + 0.753691i \(0.271729\pi\)
\(840\) 0 0
\(841\) −24353.0 −0.998524
\(842\) 0 0
\(843\) 14562.0i 0.594949i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1120.00i 0.0454352i
\(848\) 0 0
\(849\) −13548.0 −0.547663
\(850\) 0 0
\(851\) 6528.00 0.262958
\(852\) 0 0
\(853\) 17486.0i 0.701887i 0.936397 + 0.350943i \(0.114139\pi\)
−0.936397 + 0.350943i \(0.885861\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 43434.0i − 1.73125i −0.500697 0.865623i \(-0.666923\pi\)
0.500697 0.865623i \(-0.333077\pi\)
\(858\) 0 0
\(859\) −10820.0 −0.429771 −0.214886 0.976639i \(-0.568938\pi\)
−0.214886 + 0.976639i \(0.568938\pi\)
\(860\) 0 0
\(861\) 37440.0 1.48194
\(862\) 0 0
\(863\) 29976.0i 1.18238i 0.806532 + 0.591191i \(0.201342\pi\)
−0.806532 + 0.591191i \(0.798658\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 3513.00i 0.137610i
\(868\) 0 0
\(869\) 42048.0 1.64140
\(870\) 0 0
\(871\) 8920.00 0.347007
\(872\) 0 0
\(873\) 1746.00i 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 40522.0i 1.56024i 0.625630 + 0.780120i \(0.284842\pi\)
−0.625630 + 0.780120i \(0.715158\pi\)
\(878\) 0 0
\(879\) 25722.0 0.987010
\(880\) 0 0
\(881\) 15570.0 0.595422 0.297711 0.954656i \(-0.403777\pi\)
0.297711 + 0.954656i \(0.403777\pi\)
\(882\) 0 0
\(883\) − 1084.00i − 0.0413131i −0.999787 0.0206566i \(-0.993424\pi\)
0.999787 0.0206566i \(-0.00657566\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8208.00i 0.310708i 0.987859 + 0.155354i \(0.0496518\pi\)
−0.987859 + 0.155354i \(0.950348\pi\)
\(888\) 0 0
\(889\) 48896.0 1.84468
\(890\) 0 0
\(891\) 2916.00 0.109640
\(892\) 0 0
\(893\) 57120.0i 2.14048i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 5760.00i 0.214404i
\(898\) 0 0
\(899\) 96.0000 0.00356149
\(900\) 0 0
\(901\) 8892.00 0.328785
\(902\) 0 0
\(903\) 4992.00i 0.183968i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 34076.0i − 1.24749i −0.781627 0.623746i \(-0.785610\pi\)
0.781627 0.623746i \(-0.214390\pi\)
\(908\) 0 0
\(909\) −7182.00 −0.262059
\(910\) 0 0
\(911\) −15072.0 −0.548142 −0.274071 0.961709i \(-0.588370\pi\)
−0.274071 + 0.961709i \(0.588370\pi\)
\(912\) 0 0
\(913\) − 26352.0i − 0.955229i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 77184.0i − 2.77954i
\(918\) 0 0
\(919\) −24392.0 −0.875536 −0.437768 0.899088i \(-0.644231\pi\)
−0.437768 + 0.899088i \(0.644231\pi\)
\(920\) 0 0
\(921\) −10428.0 −0.373088
\(922\) 0 0
\(923\) 1200.00i 0.0427936i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 2448.00i − 0.0867345i
\(928\) 0 0
\(929\) −13602.0 −0.480374 −0.240187 0.970727i \(-0.577209\pi\)
−0.240187 + 0.970727i \(0.577209\pi\)
\(930\) 0 0
\(931\) 95340.0 3.35622
\(932\) 0 0
\(933\) − 7272.00i − 0.255171i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 47974.0i 1.67262i 0.548259 + 0.836309i \(0.315291\pi\)
−0.548259 + 0.836309i \(0.684709\pi\)
\(938\) 0 0
\(939\) −4674.00 −0.162439
\(940\) 0 0
\(941\) −48330.0 −1.67430 −0.837148 0.546976i \(-0.815779\pi\)
−0.837148 + 0.546976i \(0.815779\pi\)
\(942\) 0 0
\(943\) 74880.0i 2.58582i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9324.00i 0.319946i 0.987121 + 0.159973i \(0.0511408\pi\)
−0.987121 + 0.159973i \(0.948859\pi\)
\(948\) 0 0
\(949\) −6460.00 −0.220970
\(950\) 0 0
\(951\) 25614.0 0.873387
\(952\) 0 0
\(953\) − 14838.0i − 0.504355i −0.967681 0.252177i \(-0.918853\pi\)
0.967681 0.252177i \(-0.0811466\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 648.000i 0.0218881i
\(958\) 0 0
\(959\) −67392.0 −2.26924
\(960\) 0 0
\(961\) −29535.0 −0.991407
\(962\) 0 0
\(963\) 1404.00i 0.0469816i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 11360.0i − 0.377780i −0.981998 0.188890i \(-0.939511\pi\)
0.981998 0.188890i \(-0.0604889\pi\)
\(968\) 0 0
\(969\) −32760.0 −1.08607
\(970\) 0 0
\(971\) −6972.00 −0.230424 −0.115212 0.993341i \(-0.536755\pi\)
−0.115212 + 0.993341i \(0.536755\pi\)
\(972\) 0 0
\(973\) − 17792.0i − 0.586213i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 41166.0i 1.34802i 0.738722 + 0.674011i \(0.235430\pi\)
−0.738722 + 0.674011i \(0.764570\pi\)
\(978\) 0 0
\(979\) 57240.0 1.86864
\(980\) 0 0
\(981\) 14598.0 0.475105
\(982\) 0 0
\(983\) 22464.0i 0.728881i 0.931227 + 0.364441i \(0.118740\pi\)
−0.931227 + 0.364441i \(0.881260\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 39168.0i 1.26315i
\(988\) 0 0
\(989\) −9984.00 −0.321004
\(990\) 0 0
\(991\) −10192.0 −0.326700 −0.163350 0.986568i \(-0.552230\pi\)
−0.163350 + 0.986568i \(0.552230\pi\)
\(992\) 0 0
\(993\) 2964.00i 0.0947228i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 322.000i 0.0102285i 0.999987 + 0.00511426i \(0.00162793\pi\)
−0.999987 + 0.00511426i \(0.998372\pi\)
\(998\) 0 0
\(999\) −918.000 −0.0290733
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 300.4.d.d.49.1 2
3.2 odd 2 900.4.d.b.649.1 2
4.3 odd 2 1200.4.f.e.49.2 2
5.2 odd 4 60.4.a.b.1.1 1
5.3 odd 4 300.4.a.e.1.1 1
5.4 even 2 inner 300.4.d.d.49.2 2
15.2 even 4 180.4.a.c.1.1 1
15.8 even 4 900.4.a.b.1.1 1
15.14 odd 2 900.4.d.b.649.2 2
20.3 even 4 1200.4.a.s.1.1 1
20.7 even 4 240.4.a.j.1.1 1
20.19 odd 2 1200.4.f.e.49.1 2
40.27 even 4 960.4.a.a.1.1 1
40.37 odd 4 960.4.a.bb.1.1 1
45.2 even 12 1620.4.i.g.1081.1 2
45.7 odd 12 1620.4.i.a.1081.1 2
45.22 odd 12 1620.4.i.a.541.1 2
45.32 even 12 1620.4.i.g.541.1 2
60.47 odd 4 720.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.4.a.b.1.1 1 5.2 odd 4
180.4.a.c.1.1 1 15.2 even 4
240.4.a.j.1.1 1 20.7 even 4
300.4.a.e.1.1 1 5.3 odd 4
300.4.d.d.49.1 2 1.1 even 1 trivial
300.4.d.d.49.2 2 5.4 even 2 inner
720.4.a.c.1.1 1 60.47 odd 4
900.4.a.b.1.1 1 15.8 even 4
900.4.d.b.649.1 2 3.2 odd 2
900.4.d.b.649.2 2 15.14 odd 2
960.4.a.a.1.1 1 40.27 even 4
960.4.a.bb.1.1 1 40.37 odd 4
1200.4.a.s.1.1 1 20.3 even 4
1200.4.f.e.49.1 2 20.19 odd 2
1200.4.f.e.49.2 2 4.3 odd 2
1620.4.i.a.541.1 2 45.22 odd 12
1620.4.i.a.1081.1 2 45.7 odd 12
1620.4.i.g.541.1 2 45.32 even 12
1620.4.i.g.1081.1 2 45.2 even 12