Properties

Label 150.4.a.c.1.1
Level $150$
Weight $4$
Character 150.1
Self dual yes
Analytic conductor $8.850$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,4,Mod(1,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 150.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: \(8.85028650086\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 150.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} -23.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} -23.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -30.0000 q^{11} +12.0000 q^{12} -29.0000 q^{13} +46.0000 q^{14} +16.0000 q^{16} -78.0000 q^{17} -18.0000 q^{18} +149.000 q^{19} -69.0000 q^{21} +60.0000 q^{22} -150.000 q^{23} -24.0000 q^{24} +58.0000 q^{26} +27.0000 q^{27} -92.0000 q^{28} -234.000 q^{29} -217.000 q^{31} -32.0000 q^{32} -90.0000 q^{33} +156.000 q^{34} +36.0000 q^{36} -146.000 q^{37} -298.000 q^{38} -87.0000 q^{39} -156.000 q^{41} +138.000 q^{42} +433.000 q^{43} -120.000 q^{44} +300.000 q^{46} -30.0000 q^{47} +48.0000 q^{48} +186.000 q^{49} -234.000 q^{51} -116.000 q^{52} +552.000 q^{53} -54.0000 q^{54} +184.000 q^{56} +447.000 q^{57} +468.000 q^{58} -270.000 q^{59} +275.000 q^{61} +434.000 q^{62} -207.000 q^{63} +64.0000 q^{64} +180.000 q^{66} -803.000 q^{67} -312.000 q^{68} -450.000 q^{69} +660.000 q^{71} -72.0000 q^{72} +646.000 q^{73} +292.000 q^{74} +596.000 q^{76} +690.000 q^{77} +174.000 q^{78} +992.000 q^{79} +81.0000 q^{81} +312.000 q^{82} +846.000 q^{83} -276.000 q^{84} -866.000 q^{86} -702.000 q^{87} +240.000 q^{88} -1488.00 q^{89} +667.000 q^{91} -600.000 q^{92} -651.000 q^{93} +60.0000 q^{94} -96.0000 q^{96} +319.000 q^{97} -372.000 q^{98} -270.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −6.00000 −0.408248
\(7\) −23.0000 −1.24188 −0.620942 0.783857i \(-0.713250\pi\)
−0.620942 + 0.783857i \(0.713250\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −30.0000 −0.822304 −0.411152 0.911567i \(-0.634873\pi\)
−0.411152 + 0.911567i \(0.634873\pi\)
\(12\) 12.0000 0.288675
\(13\) −29.0000 −0.618704 −0.309352 0.950948i \(-0.600112\pi\)
−0.309352 + 0.950948i \(0.600112\pi\)
\(14\) 46.0000 0.878144
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −78.0000 −1.11281 −0.556405 0.830911i \(-0.687820\pi\)
−0.556405 + 0.830911i \(0.687820\pi\)
\(18\) −18.0000 −0.235702
\(19\) 149.000 1.79910 0.899551 0.436815i \(-0.143894\pi\)
0.899551 + 0.436815i \(0.143894\pi\)
\(20\) 0 0
\(21\) −69.0000 −0.717002
\(22\) 60.0000 0.581456
\(23\) −150.000 −1.35988 −0.679938 0.733269i \(-0.737993\pi\)
−0.679938 + 0.733269i \(0.737993\pi\)
\(24\) −24.0000 −0.204124
\(25\) 0 0
\(26\) 58.0000 0.437490
\(27\) 27.0000 0.192450
\(28\) −92.0000 −0.620942
\(29\) −234.000 −1.49837 −0.749185 0.662361i \(-0.769554\pi\)
−0.749185 + 0.662361i \(0.769554\pi\)
\(30\) 0 0
\(31\) −217.000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) −32.0000 −0.176777
\(33\) −90.0000 −0.474757
\(34\) 156.000 0.786876
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) −146.000 −0.648710 −0.324355 0.945936i \(-0.605147\pi\)
−0.324355 + 0.945936i \(0.605147\pi\)
\(38\) −298.000 −1.27216
\(39\) −87.0000 −0.357209
\(40\) 0 0
\(41\) −156.000 −0.594222 −0.297111 0.954843i \(-0.596023\pi\)
−0.297111 + 0.954843i \(0.596023\pi\)
\(42\) 138.000 0.506997
\(43\) 433.000 1.53563 0.767813 0.640675i \(-0.221345\pi\)
0.767813 + 0.640675i \(0.221345\pi\)
\(44\) −120.000 −0.411152
\(45\) 0 0
\(46\) 300.000 0.961578
\(47\) −30.0000 −0.0931053 −0.0465527 0.998916i \(-0.514824\pi\)
−0.0465527 + 0.998916i \(0.514824\pi\)
\(48\) 48.0000 0.144338
\(49\) 186.000 0.542274
\(50\) 0 0
\(51\) −234.000 −0.642481
\(52\) −116.000 −0.309352
\(53\) 552.000 1.43062 0.715312 0.698806i \(-0.246285\pi\)
0.715312 + 0.698806i \(0.246285\pi\)
\(54\) −54.0000 −0.136083
\(55\) 0 0
\(56\) 184.000 0.439072
\(57\) 447.000 1.03871
\(58\) 468.000 1.05951
\(59\) −270.000 −0.595780 −0.297890 0.954600i \(-0.596283\pi\)
−0.297890 + 0.954600i \(0.596283\pi\)
\(60\) 0 0
\(61\) 275.000 0.577215 0.288608 0.957447i \(-0.406808\pi\)
0.288608 + 0.957447i \(0.406808\pi\)
\(62\) 434.000 0.889001
\(63\) −207.000 −0.413961
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 180.000 0.335704
\(67\) −803.000 −1.46421 −0.732105 0.681192i \(-0.761462\pi\)
−0.732105 + 0.681192i \(0.761462\pi\)
\(68\) −312.000 −0.556405
\(69\) −450.000 −0.785125
\(70\) 0 0
\(71\) 660.000 1.10321 0.551603 0.834107i \(-0.314016\pi\)
0.551603 + 0.834107i \(0.314016\pi\)
\(72\) −72.0000 −0.117851
\(73\) 646.000 1.03573 0.517867 0.855461i \(-0.326726\pi\)
0.517867 + 0.855461i \(0.326726\pi\)
\(74\) 292.000 0.458707
\(75\) 0 0
\(76\) 596.000 0.899551
\(77\) 690.000 1.02121
\(78\) 174.000 0.252585
\(79\) 992.000 1.41277 0.706384 0.707829i \(-0.250325\pi\)
0.706384 + 0.707829i \(0.250325\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 312.000 0.420178
\(83\) 846.000 1.11880 0.559401 0.828897i \(-0.311031\pi\)
0.559401 + 0.828897i \(0.311031\pi\)
\(84\) −276.000 −0.358501
\(85\) 0 0
\(86\) −866.000 −1.08585
\(87\) −702.000 −0.865084
\(88\) 240.000 0.290728
\(89\) −1488.00 −1.77222 −0.886111 0.463474i \(-0.846603\pi\)
−0.886111 + 0.463474i \(0.846603\pi\)
\(90\) 0 0
\(91\) 667.000 0.768358
\(92\) −600.000 −0.679938
\(93\) −651.000 −0.725866
\(94\) 60.0000 0.0658354
\(95\) 0 0
\(96\) −96.0000 −0.102062
\(97\) 319.000 0.333913 0.166956 0.985964i \(-0.446606\pi\)
0.166956 + 0.985964i \(0.446606\pi\)
\(98\) −372.000 −0.383446
\(99\) −270.000 −0.274101
\(100\) 0 0
\(101\) −792.000 −0.780267 −0.390133 0.920758i \(-0.627571\pi\)
−0.390133 + 0.920758i \(0.627571\pi\)
\(102\) 468.000 0.454303
\(103\) −812.000 −0.776784 −0.388392 0.921494i \(-0.626969\pi\)
−0.388392 + 0.921494i \(0.626969\pi\)
\(104\) 232.000 0.218745
\(105\) 0 0
\(106\) −1104.00 −1.01160
\(107\) 1416.00 1.27934 0.639672 0.768648i \(-0.279070\pi\)
0.639672 + 0.768648i \(0.279070\pi\)
\(108\) 108.000 0.0962250
\(109\) −55.0000 −0.0483307 −0.0241653 0.999708i \(-0.507693\pi\)
−0.0241653 + 0.999708i \(0.507693\pi\)
\(110\) 0 0
\(111\) −438.000 −0.374533
\(112\) −368.000 −0.310471
\(113\) −1404.00 −1.16882 −0.584412 0.811457i \(-0.698675\pi\)
−0.584412 + 0.811457i \(0.698675\pi\)
\(114\) −894.000 −0.734480
\(115\) 0 0
\(116\) −936.000 −0.749185
\(117\) −261.000 −0.206235
\(118\) 540.000 0.421280
\(119\) 1794.00 1.38198
\(120\) 0 0
\(121\) −431.000 −0.323817
\(122\) −550.000 −0.408153
\(123\) −468.000 −0.343074
\(124\) −868.000 −0.628619
\(125\) 0 0
\(126\) 414.000 0.292715
\(127\) −1280.00 −0.894344 −0.447172 0.894448i \(-0.647569\pi\)
−0.447172 + 0.894448i \(0.647569\pi\)
\(128\) −128.000 −0.0883883
\(129\) 1299.00 0.886594
\(130\) 0 0
\(131\) 480.000 0.320136 0.160068 0.987106i \(-0.448829\pi\)
0.160068 + 0.987106i \(0.448829\pi\)
\(132\) −360.000 −0.237379
\(133\) −3427.00 −2.23428
\(134\) 1606.00 1.03535
\(135\) 0 0
\(136\) 624.000 0.393438
\(137\) 282.000 0.175860 0.0879302 0.996127i \(-0.471975\pi\)
0.0879302 + 0.996127i \(0.471975\pi\)
\(138\) 900.000 0.555167
\(139\) 1604.00 0.978773 0.489387 0.872067i \(-0.337221\pi\)
0.489387 + 0.872067i \(0.337221\pi\)
\(140\) 0 0
\(141\) −90.0000 −0.0537544
\(142\) −1320.00 −0.780084
\(143\) 870.000 0.508763
\(144\) 144.000 0.0833333
\(145\) 0 0
\(146\) −1292.00 −0.732375
\(147\) 558.000 0.313082
\(148\) −584.000 −0.324355
\(149\) −774.000 −0.425561 −0.212780 0.977100i \(-0.568252\pi\)
−0.212780 + 0.977100i \(0.568252\pi\)
\(150\) 0 0
\(151\) 293.000 0.157907 0.0789536 0.996878i \(-0.474842\pi\)
0.0789536 + 0.996878i \(0.474842\pi\)
\(152\) −1192.00 −0.636079
\(153\) −702.000 −0.370937
\(154\) −1380.00 −0.722101
\(155\) 0 0
\(156\) −348.000 −0.178604
\(157\) 1729.00 0.878912 0.439456 0.898264i \(-0.355171\pi\)
0.439456 + 0.898264i \(0.355171\pi\)
\(158\) −1984.00 −0.998978
\(159\) 1656.00 0.825971
\(160\) 0 0
\(161\) 3450.00 1.68881
\(162\) −162.000 −0.0785674
\(163\) 1123.00 0.539633 0.269816 0.962912i \(-0.413037\pi\)
0.269816 + 0.962912i \(0.413037\pi\)
\(164\) −624.000 −0.297111
\(165\) 0 0
\(166\) −1692.00 −0.791112
\(167\) −1200.00 −0.556041 −0.278020 0.960575i \(-0.589678\pi\)
−0.278020 + 0.960575i \(0.589678\pi\)
\(168\) 552.000 0.253498
\(169\) −1356.00 −0.617205
\(170\) 0 0
\(171\) 1341.00 0.599701
\(172\) 1732.00 0.767813
\(173\) 1734.00 0.762044 0.381022 0.924566i \(-0.375572\pi\)
0.381022 + 0.924566i \(0.375572\pi\)
\(174\) 1404.00 0.611707
\(175\) 0 0
\(176\) −480.000 −0.205576
\(177\) −810.000 −0.343974
\(178\) 2976.00 1.25315
\(179\) 2586.00 1.07981 0.539907 0.841725i \(-0.318459\pi\)
0.539907 + 0.841725i \(0.318459\pi\)
\(180\) 0 0
\(181\) −3931.00 −1.61430 −0.807152 0.590344i \(-0.798992\pi\)
−0.807152 + 0.590344i \(0.798992\pi\)
\(182\) −1334.00 −0.543311
\(183\) 825.000 0.333255
\(184\) 1200.00 0.480789
\(185\) 0 0
\(186\) 1302.00 0.513265
\(187\) 2340.00 0.915068
\(188\) −120.000 −0.0465527
\(189\) −621.000 −0.239001
\(190\) 0 0
\(191\) 1566.00 0.593255 0.296628 0.954993i \(-0.404138\pi\)
0.296628 + 0.954993i \(0.404138\pi\)
\(192\) 192.000 0.0721688
\(193\) −2291.00 −0.854455 −0.427227 0.904144i \(-0.640510\pi\)
−0.427227 + 0.904144i \(0.640510\pi\)
\(194\) −638.000 −0.236112
\(195\) 0 0
\(196\) 744.000 0.271137
\(197\) −2142.00 −0.774676 −0.387338 0.921938i \(-0.626605\pi\)
−0.387338 + 0.921938i \(0.626605\pi\)
\(198\) 540.000 0.193819
\(199\) −4903.00 −1.74656 −0.873278 0.487223i \(-0.838010\pi\)
−0.873278 + 0.487223i \(0.838010\pi\)
\(200\) 0 0
\(201\) −2409.00 −0.845362
\(202\) 1584.00 0.551732
\(203\) 5382.00 1.86080
\(204\) −936.000 −0.321241
\(205\) 0 0
\(206\) 1624.00 0.549269
\(207\) −1350.00 −0.453292
\(208\) −464.000 −0.154676
\(209\) −4470.00 −1.47941
\(210\) 0 0
\(211\) 605.000 0.197393 0.0986965 0.995118i \(-0.468533\pi\)
0.0986965 + 0.995118i \(0.468533\pi\)
\(212\) 2208.00 0.715312
\(213\) 1980.00 0.636936
\(214\) −2832.00 −0.904633
\(215\) 0 0
\(216\) −216.000 −0.0680414
\(217\) 4991.00 1.56134
\(218\) 110.000 0.0341750
\(219\) 1938.00 0.597981
\(220\) 0 0
\(221\) 2262.00 0.688500
\(222\) 876.000 0.264835
\(223\) 145.000 0.0435422 0.0217711 0.999763i \(-0.493069\pi\)
0.0217711 + 0.999763i \(0.493069\pi\)
\(224\) 736.000 0.219536
\(225\) 0 0
\(226\) 2808.00 0.826484
\(227\) −2964.00 −0.866641 −0.433321 0.901240i \(-0.642658\pi\)
−0.433321 + 0.901240i \(0.642658\pi\)
\(228\) 1788.00 0.519356
\(229\) −5635.00 −1.62608 −0.813038 0.582211i \(-0.802188\pi\)
−0.813038 + 0.582211i \(0.802188\pi\)
\(230\) 0 0
\(231\) 2070.00 0.589593
\(232\) 1872.00 0.529754
\(233\) −4164.00 −1.17078 −0.585392 0.810750i \(-0.699059\pi\)
−0.585392 + 0.810750i \(0.699059\pi\)
\(234\) 522.000 0.145830
\(235\) 0 0
\(236\) −1080.00 −0.297890
\(237\) 2976.00 0.815662
\(238\) −3588.00 −0.977208
\(239\) −1944.00 −0.526138 −0.263069 0.964777i \(-0.584735\pi\)
−0.263069 + 0.964777i \(0.584735\pi\)
\(240\) 0 0
\(241\) 857.000 0.229063 0.114532 0.993420i \(-0.463463\pi\)
0.114532 + 0.993420i \(0.463463\pi\)
\(242\) 862.000 0.228973
\(243\) 243.000 0.0641500
\(244\) 1100.00 0.288608
\(245\) 0 0
\(246\) 936.000 0.242590
\(247\) −4321.00 −1.11311
\(248\) 1736.00 0.444500
\(249\) 2538.00 0.645941
\(250\) 0 0
\(251\) −3924.00 −0.986776 −0.493388 0.869809i \(-0.664242\pi\)
−0.493388 + 0.869809i \(0.664242\pi\)
\(252\) −828.000 −0.206981
\(253\) 4500.00 1.11823
\(254\) 2560.00 0.632396
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 2844.00 0.690287 0.345144 0.938550i \(-0.387830\pi\)
0.345144 + 0.938550i \(0.387830\pi\)
\(258\) −2598.00 −0.626916
\(259\) 3358.00 0.805621
\(260\) 0 0
\(261\) −2106.00 −0.499456
\(262\) −960.000 −0.226370
\(263\) 6060.00 1.42082 0.710410 0.703788i \(-0.248510\pi\)
0.710410 + 0.703788i \(0.248510\pi\)
\(264\) 720.000 0.167852
\(265\) 0 0
\(266\) 6854.00 1.57987
\(267\) −4464.00 −1.02319
\(268\) −3212.00 −0.732105
\(269\) 3906.00 0.885327 0.442664 0.896688i \(-0.354034\pi\)
0.442664 + 0.896688i \(0.354034\pi\)
\(270\) 0 0
\(271\) 2144.00 0.480586 0.240293 0.970700i \(-0.422757\pi\)
0.240293 + 0.970700i \(0.422757\pi\)
\(272\) −1248.00 −0.278203
\(273\) 2001.00 0.443612
\(274\) −564.000 −0.124352
\(275\) 0 0
\(276\) −1800.00 −0.392563
\(277\) −2321.00 −0.503449 −0.251725 0.967799i \(-0.580998\pi\)
−0.251725 + 0.967799i \(0.580998\pi\)
\(278\) −3208.00 −0.692097
\(279\) −1953.00 −0.419079
\(280\) 0 0
\(281\) −6822.00 −1.44828 −0.724140 0.689654i \(-0.757763\pi\)
−0.724140 + 0.689654i \(0.757763\pi\)
\(282\) 180.000 0.0380101
\(283\) −4049.00 −0.850488 −0.425244 0.905079i \(-0.639812\pi\)
−0.425244 + 0.905079i \(0.639812\pi\)
\(284\) 2640.00 0.551603
\(285\) 0 0
\(286\) −1740.00 −0.359749
\(287\) 3588.00 0.737955
\(288\) −288.000 −0.0589256
\(289\) 1171.00 0.238347
\(290\) 0 0
\(291\) 957.000 0.192785
\(292\) 2584.00 0.517867
\(293\) −2238.00 −0.446230 −0.223115 0.974792i \(-0.571623\pi\)
−0.223115 + 0.974792i \(0.571623\pi\)
\(294\) −1116.00 −0.221382
\(295\) 0 0
\(296\) 1168.00 0.229353
\(297\) −810.000 −0.158252
\(298\) 1548.00 0.300917
\(299\) 4350.00 0.841361
\(300\) 0 0
\(301\) −9959.00 −1.90707
\(302\) −586.000 −0.111657
\(303\) −2376.00 −0.450487
\(304\) 2384.00 0.449776
\(305\) 0 0
\(306\) 1404.00 0.262292
\(307\) −1385.00 −0.257479 −0.128740 0.991678i \(-0.541093\pi\)
−0.128740 + 0.991678i \(0.541093\pi\)
\(308\) 2760.00 0.510603
\(309\) −2436.00 −0.448476
\(310\) 0 0
\(311\) −5670.00 −1.03381 −0.516907 0.856042i \(-0.672917\pi\)
−0.516907 + 0.856042i \(0.672917\pi\)
\(312\) 696.000 0.126292
\(313\) 421.000 0.0760266 0.0380133 0.999277i \(-0.487897\pi\)
0.0380133 + 0.999277i \(0.487897\pi\)
\(314\) −3458.00 −0.621485
\(315\) 0 0
\(316\) 3968.00 0.706384
\(317\) −9984.00 −1.76895 −0.884475 0.466587i \(-0.845483\pi\)
−0.884475 + 0.466587i \(0.845483\pi\)
\(318\) −3312.00 −0.584049
\(319\) 7020.00 1.23211
\(320\) 0 0
\(321\) 4248.00 0.738630
\(322\) −6900.00 −1.19417
\(323\) −11622.0 −2.00206
\(324\) 324.000 0.0555556
\(325\) 0 0
\(326\) −2246.00 −0.381578
\(327\) −165.000 −0.0279037
\(328\) 1248.00 0.210089
\(329\) 690.000 0.115626
\(330\) 0 0
\(331\) −4228.00 −0.702090 −0.351045 0.936359i \(-0.614174\pi\)
−0.351045 + 0.936359i \(0.614174\pi\)
\(332\) 3384.00 0.559401
\(333\) −1314.00 −0.216237
\(334\) 2400.00 0.393180
\(335\) 0 0
\(336\) −1104.00 −0.179250
\(337\) −5393.00 −0.871737 −0.435869 0.900010i \(-0.643559\pi\)
−0.435869 + 0.900010i \(0.643559\pi\)
\(338\) 2712.00 0.436430
\(339\) −4212.00 −0.674821
\(340\) 0 0
\(341\) 6510.00 1.03383
\(342\) −2682.00 −0.424052
\(343\) 3611.00 0.568442
\(344\) −3464.00 −0.542925
\(345\) 0 0
\(346\) −3468.00 −0.538846
\(347\) 7914.00 1.22434 0.612170 0.790726i \(-0.290297\pi\)
0.612170 + 0.790726i \(0.290297\pi\)
\(348\) −2808.00 −0.432542
\(349\) 1010.00 0.154911 0.0774557 0.996996i \(-0.475320\pi\)
0.0774557 + 0.996996i \(0.475320\pi\)
\(350\) 0 0
\(351\) −783.000 −0.119070
\(352\) 960.000 0.145364
\(353\) −4722.00 −0.711974 −0.355987 0.934491i \(-0.615855\pi\)
−0.355987 + 0.934491i \(0.615855\pi\)
\(354\) 1620.00 0.243226
\(355\) 0 0
\(356\) −5952.00 −0.886111
\(357\) 5382.00 0.797887
\(358\) −5172.00 −0.763544
\(359\) 6204.00 0.912074 0.456037 0.889961i \(-0.349268\pi\)
0.456037 + 0.889961i \(0.349268\pi\)
\(360\) 0 0
\(361\) 15342.0 2.23677
\(362\) 7862.00 1.14148
\(363\) −1293.00 −0.186956
\(364\) 2668.00 0.384179
\(365\) 0 0
\(366\) −1650.00 −0.235647
\(367\) −1361.00 −0.193579 −0.0967897 0.995305i \(-0.530857\pi\)
−0.0967897 + 0.995305i \(0.530857\pi\)
\(368\) −2400.00 −0.339969
\(369\) −1404.00 −0.198074
\(370\) 0 0
\(371\) −12696.0 −1.77667
\(372\) −2604.00 −0.362933
\(373\) 913.000 0.126738 0.0633691 0.997990i \(-0.479815\pi\)
0.0633691 + 0.997990i \(0.479815\pi\)
\(374\) −4680.00 −0.647051
\(375\) 0 0
\(376\) 240.000 0.0329177
\(377\) 6786.00 0.927047
\(378\) 1242.00 0.168999
\(379\) −8881.00 −1.20366 −0.601829 0.798625i \(-0.705561\pi\)
−0.601829 + 0.798625i \(0.705561\pi\)
\(380\) 0 0
\(381\) −3840.00 −0.516350
\(382\) −3132.00 −0.419495
\(383\) −5460.00 −0.728441 −0.364221 0.931313i \(-0.618665\pi\)
−0.364221 + 0.931313i \(0.618665\pi\)
\(384\) −384.000 −0.0510310
\(385\) 0 0
\(386\) 4582.00 0.604191
\(387\) 3897.00 0.511875
\(388\) 1276.00 0.166956
\(389\) −13884.0 −1.80963 −0.904816 0.425803i \(-0.859992\pi\)
−0.904816 + 0.425803i \(0.859992\pi\)
\(390\) 0 0
\(391\) 11700.0 1.51328
\(392\) −1488.00 −0.191723
\(393\) 1440.00 0.184831
\(394\) 4284.00 0.547779
\(395\) 0 0
\(396\) −1080.00 −0.137051
\(397\) 3781.00 0.477992 0.238996 0.971021i \(-0.423182\pi\)
0.238996 + 0.971021i \(0.423182\pi\)
\(398\) 9806.00 1.23500
\(399\) −10281.0 −1.28996
\(400\) 0 0
\(401\) 9024.00 1.12378 0.561892 0.827211i \(-0.310074\pi\)
0.561892 + 0.827211i \(0.310074\pi\)
\(402\) 4818.00 0.597761
\(403\) 6293.00 0.777858
\(404\) −3168.00 −0.390133
\(405\) 0 0
\(406\) −10764.0 −1.31578
\(407\) 4380.00 0.533436
\(408\) 1872.00 0.227151
\(409\) 14789.0 1.78794 0.893972 0.448123i \(-0.147907\pi\)
0.893972 + 0.448123i \(0.147907\pi\)
\(410\) 0 0
\(411\) 846.000 0.101533
\(412\) −3248.00 −0.388392
\(413\) 6210.00 0.739889
\(414\) 2700.00 0.320526
\(415\) 0 0
\(416\) 928.000 0.109372
\(417\) 4812.00 0.565095
\(418\) 8940.00 1.04610
\(419\) 9840.00 1.14729 0.573646 0.819103i \(-0.305528\pi\)
0.573646 + 0.819103i \(0.305528\pi\)
\(420\) 0 0
\(421\) 5510.00 0.637865 0.318932 0.947778i \(-0.396676\pi\)
0.318932 + 0.947778i \(0.396676\pi\)
\(422\) −1210.00 −0.139578
\(423\) −270.000 −0.0310351
\(424\) −4416.00 −0.505802
\(425\) 0 0
\(426\) −3960.00 −0.450382
\(427\) −6325.00 −0.716834
\(428\) 5664.00 0.639672
\(429\) 2610.00 0.293734
\(430\) 0 0
\(431\) 11070.0 1.23718 0.618588 0.785715i \(-0.287705\pi\)
0.618588 + 0.785715i \(0.287705\pi\)
\(432\) 432.000 0.0481125
\(433\) 12133.0 1.34659 0.673297 0.739373i \(-0.264878\pi\)
0.673297 + 0.739373i \(0.264878\pi\)
\(434\) −9982.00 −1.10404
\(435\) 0 0
\(436\) −220.000 −0.0241653
\(437\) −22350.0 −2.44656
\(438\) −3876.00 −0.422837
\(439\) −1873.00 −0.203630 −0.101815 0.994803i \(-0.532465\pi\)
−0.101815 + 0.994803i \(0.532465\pi\)
\(440\) 0 0
\(441\) 1674.00 0.180758
\(442\) −4524.00 −0.486843
\(443\) 576.000 0.0617756 0.0308878 0.999523i \(-0.490167\pi\)
0.0308878 + 0.999523i \(0.490167\pi\)
\(444\) −1752.00 −0.187266
\(445\) 0 0
\(446\) −290.000 −0.0307890
\(447\) −2322.00 −0.245698
\(448\) −1472.00 −0.155235
\(449\) −4884.00 −0.513341 −0.256671 0.966499i \(-0.582626\pi\)
−0.256671 + 0.966499i \(0.582626\pi\)
\(450\) 0 0
\(451\) 4680.00 0.488631
\(452\) −5616.00 −0.584412
\(453\) 879.000 0.0911678
\(454\) 5928.00 0.612808
\(455\) 0 0
\(456\) −3576.00 −0.367240
\(457\) 15802.0 1.61748 0.808738 0.588169i \(-0.200151\pi\)
0.808738 + 0.588169i \(0.200151\pi\)
\(458\) 11270.0 1.14981
\(459\) −2106.00 −0.214160
\(460\) 0 0
\(461\) −15360.0 −1.55181 −0.775907 0.630847i \(-0.782708\pi\)
−0.775907 + 0.630847i \(0.782708\pi\)
\(462\) −4140.00 −0.416905
\(463\) −1712.00 −0.171843 −0.0859216 0.996302i \(-0.527383\pi\)
−0.0859216 + 0.996302i \(0.527383\pi\)
\(464\) −3744.00 −0.374592
\(465\) 0 0
\(466\) 8328.00 0.827869
\(467\) 16278.0 1.61297 0.806484 0.591256i \(-0.201368\pi\)
0.806484 + 0.591256i \(0.201368\pi\)
\(468\) −1044.00 −0.103117
\(469\) 18469.0 1.81838
\(470\) 0 0
\(471\) 5187.00 0.507440
\(472\) 2160.00 0.210640
\(473\) −12990.0 −1.26275
\(474\) −5952.00 −0.576760
\(475\) 0 0
\(476\) 7176.00 0.690990
\(477\) 4968.00 0.476874
\(478\) 3888.00 0.372036
\(479\) −14766.0 −1.40851 −0.704254 0.709948i \(-0.748719\pi\)
−0.704254 + 0.709948i \(0.748719\pi\)
\(480\) 0 0
\(481\) 4234.00 0.401359
\(482\) −1714.00 −0.161972
\(483\) 10350.0 0.975034
\(484\) −1724.00 −0.161908
\(485\) 0 0
\(486\) −486.000 −0.0453609
\(487\) 3319.00 0.308826 0.154413 0.988006i \(-0.450651\pi\)
0.154413 + 0.988006i \(0.450651\pi\)
\(488\) −2200.00 −0.204076
\(489\) 3369.00 0.311557
\(490\) 0 0
\(491\) 11064.0 1.01693 0.508464 0.861083i \(-0.330214\pi\)
0.508464 + 0.861083i \(0.330214\pi\)
\(492\) −1872.00 −0.171537
\(493\) 18252.0 1.66740
\(494\) 8642.00 0.787089
\(495\) 0 0
\(496\) −3472.00 −0.314309
\(497\) −15180.0 −1.37005
\(498\) −5076.00 −0.456749
\(499\) −14131.0 −1.26772 −0.633858 0.773449i \(-0.718530\pi\)
−0.633858 + 0.773449i \(0.718530\pi\)
\(500\) 0 0
\(501\) −3600.00 −0.321030
\(502\) 7848.00 0.697756
\(503\) −11988.0 −1.06266 −0.531331 0.847165i \(-0.678308\pi\)
−0.531331 + 0.847165i \(0.678308\pi\)
\(504\) 1656.00 0.146357
\(505\) 0 0
\(506\) −9000.00 −0.790709
\(507\) −4068.00 −0.356344
\(508\) −5120.00 −0.447172
\(509\) 10806.0 0.940997 0.470499 0.882401i \(-0.344074\pi\)
0.470499 + 0.882401i \(0.344074\pi\)
\(510\) 0 0
\(511\) −14858.0 −1.28626
\(512\) −512.000 −0.0441942
\(513\) 4023.00 0.346237
\(514\) −5688.00 −0.488107
\(515\) 0 0
\(516\) 5196.00 0.443297
\(517\) 900.000 0.0765608
\(518\) −6716.00 −0.569660
\(519\) 5202.00 0.439966
\(520\) 0 0
\(521\) 22578.0 1.89858 0.949290 0.314402i \(-0.101804\pi\)
0.949290 + 0.314402i \(0.101804\pi\)
\(522\) 4212.00 0.353169
\(523\) −12065.0 −1.00873 −0.504365 0.863491i \(-0.668273\pi\)
−0.504365 + 0.863491i \(0.668273\pi\)
\(524\) 1920.00 0.160068
\(525\) 0 0
\(526\) −12120.0 −1.00467
\(527\) 16926.0 1.39907
\(528\) −1440.00 −0.118689
\(529\) 10333.0 0.849264
\(530\) 0 0
\(531\) −2430.00 −0.198593
\(532\) −13708.0 −1.11714
\(533\) 4524.00 0.367648
\(534\) 8928.00 0.723506
\(535\) 0 0
\(536\) 6424.00 0.517676
\(537\) 7758.00 0.623431
\(538\) −7812.00 −0.626021
\(539\) −5580.00 −0.445914
\(540\) 0 0
\(541\) −12055.0 −0.958013 −0.479006 0.877811i \(-0.659003\pi\)
−0.479006 + 0.877811i \(0.659003\pi\)
\(542\) −4288.00 −0.339825
\(543\) −11793.0 −0.932019
\(544\) 2496.00 0.196719
\(545\) 0 0
\(546\) −4002.00 −0.313681
\(547\) −6176.00 −0.482754 −0.241377 0.970431i \(-0.577599\pi\)
−0.241377 + 0.970431i \(0.577599\pi\)
\(548\) 1128.00 0.0879302
\(549\) 2475.00 0.192405
\(550\) 0 0
\(551\) −34866.0 −2.69572
\(552\) 3600.00 0.277584
\(553\) −22816.0 −1.75449
\(554\) 4642.00 0.355992
\(555\) 0 0
\(556\) 6416.00 0.489387
\(557\) −8274.00 −0.629409 −0.314704 0.949190i \(-0.601905\pi\)
−0.314704 + 0.949190i \(0.601905\pi\)
\(558\) 3906.00 0.296334
\(559\) −12557.0 −0.950098
\(560\) 0 0
\(561\) 7020.00 0.528315
\(562\) 13644.0 1.02409
\(563\) −966.000 −0.0723127 −0.0361563 0.999346i \(-0.511511\pi\)
−0.0361563 + 0.999346i \(0.511511\pi\)
\(564\) −360.000 −0.0268772
\(565\) 0 0
\(566\) 8098.00 0.601386
\(567\) −1863.00 −0.137987
\(568\) −5280.00 −0.390042
\(569\) 19002.0 1.40001 0.700005 0.714138i \(-0.253181\pi\)
0.700005 + 0.714138i \(0.253181\pi\)
\(570\) 0 0
\(571\) 8645.00 0.633594 0.316797 0.948493i \(-0.397393\pi\)
0.316797 + 0.948493i \(0.397393\pi\)
\(572\) 3480.00 0.254381
\(573\) 4698.00 0.342516
\(574\) −7176.00 −0.521813
\(575\) 0 0
\(576\) 576.000 0.0416667
\(577\) −10931.0 −0.788672 −0.394336 0.918966i \(-0.629025\pi\)
−0.394336 + 0.918966i \(0.629025\pi\)
\(578\) −2342.00 −0.168537
\(579\) −6873.00 −0.493320
\(580\) 0 0
\(581\) −19458.0 −1.38942
\(582\) −1914.00 −0.136319
\(583\) −16560.0 −1.17641
\(584\) −5168.00 −0.366187
\(585\) 0 0
\(586\) 4476.00 0.315532
\(587\) −8904.00 −0.626077 −0.313039 0.949740i \(-0.601347\pi\)
−0.313039 + 0.949740i \(0.601347\pi\)
\(588\) 2232.00 0.156541
\(589\) −32333.0 −2.26190
\(590\) 0 0
\(591\) −6426.00 −0.447259
\(592\) −2336.00 −0.162177
\(593\) −8820.00 −0.610782 −0.305391 0.952227i \(-0.598787\pi\)
−0.305391 + 0.952227i \(0.598787\pi\)
\(594\) 1620.00 0.111901
\(595\) 0 0
\(596\) −3096.00 −0.212780
\(597\) −14709.0 −1.00837
\(598\) −8700.00 −0.594932
\(599\) 9804.00 0.668749 0.334374 0.942440i \(-0.391475\pi\)
0.334374 + 0.942440i \(0.391475\pi\)
\(600\) 0 0
\(601\) −23437.0 −1.59071 −0.795354 0.606146i \(-0.792715\pi\)
−0.795354 + 0.606146i \(0.792715\pi\)
\(602\) 19918.0 1.34850
\(603\) −7227.00 −0.488070
\(604\) 1172.00 0.0789536
\(605\) 0 0
\(606\) 4752.00 0.318543
\(607\) −2648.00 −0.177066 −0.0885330 0.996073i \(-0.528218\pi\)
−0.0885330 + 0.996073i \(0.528218\pi\)
\(608\) −4768.00 −0.318039
\(609\) 16146.0 1.07433
\(610\) 0 0
\(611\) 870.000 0.0576046
\(612\) −2808.00 −0.185468
\(613\) −794.000 −0.0523154 −0.0261577 0.999658i \(-0.508327\pi\)
−0.0261577 + 0.999658i \(0.508327\pi\)
\(614\) 2770.00 0.182065
\(615\) 0 0
\(616\) −5520.00 −0.361051
\(617\) −18720.0 −1.22146 −0.610728 0.791840i \(-0.709123\pi\)
−0.610728 + 0.791840i \(0.709123\pi\)
\(618\) 4872.00 0.317121
\(619\) −8959.00 −0.581733 −0.290866 0.956764i \(-0.593944\pi\)
−0.290866 + 0.956764i \(0.593944\pi\)
\(620\) 0 0
\(621\) −4050.00 −0.261708
\(622\) 11340.0 0.731017
\(623\) 34224.0 2.20089
\(624\) −1392.00 −0.0893022
\(625\) 0 0
\(626\) −842.000 −0.0537589
\(627\) −13410.0 −0.854137
\(628\) 6916.00 0.439456
\(629\) 11388.0 0.721891
\(630\) 0 0
\(631\) −12373.0 −0.780604 −0.390302 0.920687i \(-0.627629\pi\)
−0.390302 + 0.920687i \(0.627629\pi\)
\(632\) −7936.00 −0.499489
\(633\) 1815.00 0.113965
\(634\) 19968.0 1.25084
\(635\) 0 0
\(636\) 6624.00 0.412985
\(637\) −5394.00 −0.335507
\(638\) −14040.0 −0.871237
\(639\) 5940.00 0.367735
\(640\) 0 0
\(641\) 24900.0 1.53431 0.767154 0.641463i \(-0.221672\pi\)
0.767154 + 0.641463i \(0.221672\pi\)
\(642\) −8496.00 −0.522290
\(643\) 14668.0 0.899610 0.449805 0.893127i \(-0.351493\pi\)
0.449805 + 0.893127i \(0.351493\pi\)
\(644\) 13800.0 0.844404
\(645\) 0 0
\(646\) 23244.0 1.41567
\(647\) 10788.0 0.655518 0.327759 0.944761i \(-0.393707\pi\)
0.327759 + 0.944761i \(0.393707\pi\)
\(648\) −648.000 −0.0392837
\(649\) 8100.00 0.489912
\(650\) 0 0
\(651\) 14973.0 0.901441
\(652\) 4492.00 0.269816
\(653\) 14214.0 0.851817 0.425909 0.904766i \(-0.359954\pi\)
0.425909 + 0.904766i \(0.359954\pi\)
\(654\) 330.000 0.0197309
\(655\) 0 0
\(656\) −2496.00 −0.148556
\(657\) 5814.00 0.345245
\(658\) −1380.00 −0.0817599
\(659\) −588.000 −0.0347576 −0.0173788 0.999849i \(-0.505532\pi\)
−0.0173788 + 0.999849i \(0.505532\pi\)
\(660\) 0 0
\(661\) −3166.00 −0.186298 −0.0931491 0.995652i \(-0.529693\pi\)
−0.0931491 + 0.995652i \(0.529693\pi\)
\(662\) 8456.00 0.496453
\(663\) 6786.00 0.397506
\(664\) −6768.00 −0.395556
\(665\) 0 0
\(666\) 2628.00 0.152902
\(667\) 35100.0 2.03760
\(668\) −4800.00 −0.278020
\(669\) 435.000 0.0251391
\(670\) 0 0
\(671\) −8250.00 −0.474646
\(672\) 2208.00 0.126749
\(673\) −9182.00 −0.525914 −0.262957 0.964808i \(-0.584698\pi\)
−0.262957 + 0.964808i \(0.584698\pi\)
\(674\) 10786.0 0.616411
\(675\) 0 0
\(676\) −5424.00 −0.308603
\(677\) −11742.0 −0.666590 −0.333295 0.942823i \(-0.608161\pi\)
−0.333295 + 0.942823i \(0.608161\pi\)
\(678\) 8424.00 0.477171
\(679\) −7337.00 −0.414681
\(680\) 0 0
\(681\) −8892.00 −0.500356
\(682\) −13020.0 −0.731029
\(683\) 6024.00 0.337485 0.168742 0.985660i \(-0.446029\pi\)
0.168742 + 0.985660i \(0.446029\pi\)
\(684\) 5364.00 0.299850
\(685\) 0 0
\(686\) −7222.00 −0.401949
\(687\) −16905.0 −0.938815
\(688\) 6928.00 0.383906
\(689\) −16008.0 −0.885132
\(690\) 0 0
\(691\) 9344.00 0.514418 0.257209 0.966356i \(-0.417197\pi\)
0.257209 + 0.966356i \(0.417197\pi\)
\(692\) 6936.00 0.381022
\(693\) 6210.00 0.340402
\(694\) −15828.0 −0.865739
\(695\) 0 0
\(696\) 5616.00 0.305853
\(697\) 12168.0 0.661257
\(698\) −2020.00 −0.109539
\(699\) −12492.0 −0.675953
\(700\) 0 0
\(701\) 21234.0 1.14408 0.572038 0.820227i \(-0.306153\pi\)
0.572038 + 0.820227i \(0.306153\pi\)
\(702\) 1566.00 0.0841950
\(703\) −21754.0 −1.16709
\(704\) −1920.00 −0.102788
\(705\) 0 0
\(706\) 9444.00 0.503441
\(707\) 18216.0 0.969000
\(708\) −3240.00 −0.171987
\(709\) −1723.00 −0.0912675 −0.0456337 0.998958i \(-0.514531\pi\)
−0.0456337 + 0.998958i \(0.514531\pi\)
\(710\) 0 0
\(711\) 8928.00 0.470923
\(712\) 11904.0 0.626575
\(713\) 32550.0 1.70969
\(714\) −10764.0 −0.564191
\(715\) 0 0
\(716\) 10344.0 0.539907
\(717\) −5832.00 −0.303766
\(718\) −12408.0 −0.644934
\(719\) 18510.0 0.960093 0.480046 0.877243i \(-0.340620\pi\)
0.480046 + 0.877243i \(0.340620\pi\)
\(720\) 0 0
\(721\) 18676.0 0.964675
\(722\) −30684.0 −1.58163
\(723\) 2571.00 0.132250
\(724\) −15724.0 −0.807152
\(725\) 0 0
\(726\) 2586.00 0.132198
\(727\) 1009.00 0.0514742 0.0257371 0.999669i \(-0.491807\pi\)
0.0257371 + 0.999669i \(0.491807\pi\)
\(728\) −5336.00 −0.271656
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −33774.0 −1.70886
\(732\) 3300.00 0.166628
\(733\) 21994.0 1.10828 0.554138 0.832425i \(-0.313048\pi\)
0.554138 + 0.832425i \(0.313048\pi\)
\(734\) 2722.00 0.136881
\(735\) 0 0
\(736\) 4800.00 0.240394
\(737\) 24090.0 1.20403
\(738\) 2808.00 0.140059
\(739\) −13948.0 −0.694297 −0.347148 0.937810i \(-0.612850\pi\)
−0.347148 + 0.937810i \(0.612850\pi\)
\(740\) 0 0
\(741\) −12963.0 −0.642655
\(742\) 25392.0 1.25629
\(743\) 26508.0 1.30886 0.654431 0.756122i \(-0.272908\pi\)
0.654431 + 0.756122i \(0.272908\pi\)
\(744\) 5208.00 0.256632
\(745\) 0 0
\(746\) −1826.00 −0.0896174
\(747\) 7614.00 0.372934
\(748\) 9360.00 0.457534
\(749\) −32568.0 −1.58880
\(750\) 0 0
\(751\) −1600.00 −0.0777428 −0.0388714 0.999244i \(-0.512376\pi\)
−0.0388714 + 0.999244i \(0.512376\pi\)
\(752\) −480.000 −0.0232763
\(753\) −11772.0 −0.569715
\(754\) −13572.0 −0.655521
\(755\) 0 0
\(756\) −2484.00 −0.119500
\(757\) −30101.0 −1.44523 −0.722615 0.691250i \(-0.757060\pi\)
−0.722615 + 0.691250i \(0.757060\pi\)
\(758\) 17762.0 0.851115
\(759\) 13500.0 0.645611
\(760\) 0 0
\(761\) 35628.0 1.69713 0.848564 0.529093i \(-0.177468\pi\)
0.848564 + 0.529093i \(0.177468\pi\)
\(762\) 7680.00 0.365114
\(763\) 1265.00 0.0600211
\(764\) 6264.00 0.296628
\(765\) 0 0
\(766\) 10920.0 0.515086
\(767\) 7830.00 0.368611
\(768\) 768.000 0.0360844
\(769\) −12517.0 −0.586963 −0.293482 0.955965i \(-0.594814\pi\)
−0.293482 + 0.955965i \(0.594814\pi\)
\(770\) 0 0
\(771\) 8532.00 0.398538
\(772\) −9164.00 −0.427227
\(773\) −14124.0 −0.657186 −0.328593 0.944472i \(-0.606574\pi\)
−0.328593 + 0.944472i \(0.606574\pi\)
\(774\) −7794.00 −0.361950
\(775\) 0 0
\(776\) −2552.00 −0.118056
\(777\) 10074.0 0.465126
\(778\) 27768.0 1.27960
\(779\) −23244.0 −1.06907
\(780\) 0 0
\(781\) −19800.0 −0.907170
\(782\) −23400.0 −1.07005
\(783\) −6318.00 −0.288361
\(784\) 2976.00 0.135569
\(785\) 0 0
\(786\) −2880.00 −0.130695
\(787\) −40433.0 −1.83136 −0.915680 0.401907i \(-0.868347\pi\)
−0.915680 + 0.401907i \(0.868347\pi\)
\(788\) −8568.00 −0.387338
\(789\) 18180.0 0.820311
\(790\) 0 0
\(791\) 32292.0 1.45154
\(792\) 2160.00 0.0969094
\(793\) −7975.00 −0.357126
\(794\) −7562.00 −0.337992
\(795\) 0 0
\(796\) −19612.0 −0.873278
\(797\) 27300.0 1.21332 0.606660 0.794962i \(-0.292509\pi\)
0.606660 + 0.794962i \(0.292509\pi\)
\(798\) 20562.0 0.912139
\(799\) 2340.00 0.103609
\(800\) 0 0
\(801\) −13392.0 −0.590740
\(802\) −18048.0 −0.794635
\(803\) −19380.0 −0.851688
\(804\) −9636.00 −0.422681
\(805\) 0 0
\(806\) −12586.0 −0.550028
\(807\) 11718.0 0.511144
\(808\) 6336.00 0.275866
\(809\) −2856.00 −0.124118 −0.0620591 0.998072i \(-0.519767\pi\)
−0.0620591 + 0.998072i \(0.519767\pi\)
\(810\) 0 0
\(811\) −12619.0 −0.546379 −0.273189 0.961960i \(-0.588079\pi\)
−0.273189 + 0.961960i \(0.588079\pi\)
\(812\) 21528.0 0.930400
\(813\) 6432.00 0.277466
\(814\) −8760.00 −0.377196
\(815\) 0 0
\(816\) −3744.00 −0.160620
\(817\) 64517.0 2.76275
\(818\) −29578.0 −1.26427
\(819\) 6003.00 0.256119
\(820\) 0 0
\(821\) −29082.0 −1.23626 −0.618130 0.786076i \(-0.712109\pi\)
−0.618130 + 0.786076i \(0.712109\pi\)
\(822\) −1692.00 −0.0717947
\(823\) −10235.0 −0.433499 −0.216749 0.976227i \(-0.569545\pi\)
−0.216749 + 0.976227i \(0.569545\pi\)
\(824\) 6496.00 0.274635
\(825\) 0 0
\(826\) −12420.0 −0.523180
\(827\) −26976.0 −1.13428 −0.567139 0.823622i \(-0.691950\pi\)
−0.567139 + 0.823622i \(0.691950\pi\)
\(828\) −5400.00 −0.226646
\(829\) 37802.0 1.58374 0.791868 0.610692i \(-0.209109\pi\)
0.791868 + 0.610692i \(0.209109\pi\)
\(830\) 0 0
\(831\) −6963.00 −0.290666
\(832\) −1856.00 −0.0773380
\(833\) −14508.0 −0.603448
\(834\) −9624.00 −0.399583
\(835\) 0 0
\(836\) −17880.0 −0.739704
\(837\) −5859.00 −0.241955
\(838\) −19680.0 −0.811258
\(839\) −16974.0 −0.698460 −0.349230 0.937037i \(-0.613557\pi\)
−0.349230 + 0.937037i \(0.613557\pi\)
\(840\) 0 0
\(841\) 30367.0 1.24511
\(842\) −11020.0 −0.451038
\(843\) −20466.0 −0.836164
\(844\) 2420.00 0.0986965
\(845\) 0 0
\(846\) 540.000 0.0219451
\(847\) 9913.00 0.402143
\(848\) 8832.00 0.357656
\(849\) −12147.0 −0.491029
\(850\) 0 0
\(851\) 21900.0 0.882165
\(852\) 7920.00 0.318468
\(853\) 24937.0 1.00097 0.500485 0.865745i \(-0.333155\pi\)
0.500485 + 0.865745i \(0.333155\pi\)
\(854\) 12650.0 0.506878
\(855\) 0 0
\(856\) −11328.0 −0.452317
\(857\) −15756.0 −0.628022 −0.314011 0.949419i \(-0.601673\pi\)
−0.314011 + 0.949419i \(0.601673\pi\)
\(858\) −5220.00 −0.207701
\(859\) 38144.0 1.51508 0.757542 0.652787i \(-0.226400\pi\)
0.757542 + 0.652787i \(0.226400\pi\)
\(860\) 0 0
\(861\) 10764.0 0.426058
\(862\) −22140.0 −0.874816
\(863\) −5448.00 −0.214892 −0.107446 0.994211i \(-0.534267\pi\)
−0.107446 + 0.994211i \(0.534267\pi\)
\(864\) −864.000 −0.0340207
\(865\) 0 0
\(866\) −24266.0 −0.952185
\(867\) 3513.00 0.137610
\(868\) 19964.0 0.780671
\(869\) −29760.0 −1.16172
\(870\) 0 0
\(871\) 23287.0 0.905913
\(872\) 440.000 0.0170875
\(873\) 2871.00 0.111304
\(874\) 44700.0 1.72998
\(875\) 0 0
\(876\) 7752.00 0.298991
\(877\) −21191.0 −0.815928 −0.407964 0.912998i \(-0.633761\pi\)
−0.407964 + 0.912998i \(0.633761\pi\)
\(878\) 3746.00 0.143988
\(879\) −6714.00 −0.257631
\(880\) 0 0
\(881\) 18216.0 0.696609 0.348305 0.937381i \(-0.386758\pi\)
0.348305 + 0.937381i \(0.386758\pi\)
\(882\) −3348.00 −0.127815
\(883\) −12767.0 −0.486573 −0.243286 0.969955i \(-0.578225\pi\)
−0.243286 + 0.969955i \(0.578225\pi\)
\(884\) 9048.00 0.344250
\(885\) 0 0
\(886\) −1152.00 −0.0436819
\(887\) −11010.0 −0.416775 −0.208388 0.978046i \(-0.566822\pi\)
−0.208388 + 0.978046i \(0.566822\pi\)
\(888\) 3504.00 0.132417
\(889\) 29440.0 1.11067
\(890\) 0 0
\(891\) −2430.00 −0.0913671
\(892\) 580.000 0.0217711
\(893\) −4470.00 −0.167506
\(894\) 4644.00 0.173734
\(895\) 0 0
\(896\) 2944.00 0.109768
\(897\) 13050.0 0.485760
\(898\) 9768.00 0.362987
\(899\) 50778.0 1.88381
\(900\) 0 0
\(901\) −43056.0 −1.59201
\(902\) −9360.00 −0.345514
\(903\) −29877.0 −1.10105
\(904\) 11232.0 0.413242
\(905\) 0 0
\(906\) −1758.00 −0.0644654
\(907\) −22772.0 −0.833662 −0.416831 0.908984i \(-0.636859\pi\)
−0.416831 + 0.908984i \(0.636859\pi\)
\(908\) −11856.0 −0.433321
\(909\) −7128.00 −0.260089
\(910\) 0 0
\(911\) 29802.0 1.08385 0.541923 0.840428i \(-0.317696\pi\)
0.541923 + 0.840428i \(0.317696\pi\)
\(912\) 7152.00 0.259678
\(913\) −25380.0 −0.919995
\(914\) −31604.0 −1.14373
\(915\) 0 0
\(916\) −22540.0 −0.813038
\(917\) −11040.0 −0.397571
\(918\) 4212.00 0.151434
\(919\) 48941.0 1.75671 0.878354 0.478011i \(-0.158642\pi\)
0.878354 + 0.478011i \(0.158642\pi\)
\(920\) 0 0
\(921\) −4155.00 −0.148656
\(922\) 30720.0 1.09730
\(923\) −19140.0 −0.682558
\(924\) 8280.00 0.294797
\(925\) 0 0
\(926\) 3424.00 0.121511
\(927\) −7308.00 −0.258928
\(928\) 7488.00 0.264877
\(929\) 31026.0 1.09573 0.547863 0.836568i \(-0.315441\pi\)
0.547863 + 0.836568i \(0.315441\pi\)
\(930\) 0 0
\(931\) 27714.0 0.975607
\(932\) −16656.0 −0.585392
\(933\) −17010.0 −0.596873
\(934\) −32556.0 −1.14054
\(935\) 0 0
\(936\) 2088.00 0.0729150
\(937\) −11183.0 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(938\) −36938.0 −1.28579
\(939\) 1263.00 0.0438940
\(940\) 0 0
\(941\) −2562.00 −0.0887554 −0.0443777 0.999015i \(-0.514130\pi\)
−0.0443777 + 0.999015i \(0.514130\pi\)
\(942\) −10374.0 −0.358814
\(943\) 23400.0 0.808069
\(944\) −4320.00 −0.148945
\(945\) 0 0
\(946\) 25980.0 0.892899
\(947\) −7638.00 −0.262093 −0.131046 0.991376i \(-0.541834\pi\)
−0.131046 + 0.991376i \(0.541834\pi\)
\(948\) 11904.0 0.407831
\(949\) −18734.0 −0.640813
\(950\) 0 0
\(951\) −29952.0 −1.02130
\(952\) −14352.0 −0.488604
\(953\) −51432.0 −1.74821 −0.874106 0.485735i \(-0.838552\pi\)
−0.874106 + 0.485735i \(0.838552\pi\)
\(954\) −9936.00 −0.337201
\(955\) 0 0
\(956\) −7776.00 −0.263069
\(957\) 21060.0 0.711362
\(958\) 29532.0 0.995966
\(959\) −6486.00 −0.218398
\(960\) 0 0
\(961\) 17298.0 0.580645
\(962\) −8468.00 −0.283804
\(963\) 12744.0 0.426448
\(964\) 3428.00 0.114532
\(965\) 0 0
\(966\) −20700.0 −0.689453
\(967\) −39728.0 −1.32116 −0.660582 0.750754i \(-0.729691\pi\)
−0.660582 + 0.750754i \(0.729691\pi\)
\(968\) 3448.00 0.114486
\(969\) −34866.0 −1.15589
\(970\) 0 0
\(971\) −47946.0 −1.58461 −0.792307 0.610123i \(-0.791120\pi\)
−0.792307 + 0.610123i \(0.791120\pi\)
\(972\) 972.000 0.0320750
\(973\) −36892.0 −1.21552
\(974\) −6638.00 −0.218373
\(975\) 0 0
\(976\) 4400.00 0.144304
\(977\) 22326.0 0.731087 0.365544 0.930794i \(-0.380883\pi\)
0.365544 + 0.930794i \(0.380883\pi\)
\(978\) −6738.00 −0.220304
\(979\) 44640.0 1.45730
\(980\) 0 0
\(981\) −495.000 −0.0161102
\(982\) −22128.0 −0.719076
\(983\) 48468.0 1.57262 0.786312 0.617830i \(-0.211988\pi\)
0.786312 + 0.617830i \(0.211988\pi\)
\(984\) 3744.00 0.121295
\(985\) 0 0
\(986\) −36504.0 −1.17903
\(987\) 2070.00 0.0667567
\(988\) −17284.0 −0.556556
\(989\) −64950.0 −2.08826
\(990\) 0 0
\(991\) −25141.0 −0.805883 −0.402942 0.915226i \(-0.632012\pi\)
−0.402942 + 0.915226i \(0.632012\pi\)
\(992\) 6944.00 0.222250
\(993\) −12684.0 −0.405352
\(994\) 30360.0 0.968773
\(995\) 0 0
\(996\) 10152.0 0.322970
\(997\) 35422.0 1.12520 0.562601 0.826729i \(-0.309801\pi\)
0.562601 + 0.826729i \(0.309801\pi\)
\(998\) 28262.0 0.896411
\(999\) −3942.00 −0.124844
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 150.4.a.c.1.1 1
3.2 odd 2 450.4.a.l.1.1 1
4.3 odd 2 1200.4.a.r.1.1 1
5.2 odd 4 150.4.c.b.49.1 2
5.3 odd 4 150.4.c.b.49.2 2
5.4 even 2 150.4.a.g.1.1 yes 1
15.2 even 4 450.4.c.h.199.2 2
15.8 even 4 450.4.c.h.199.1 2
15.14 odd 2 450.4.a.i.1.1 1
20.3 even 4 1200.4.f.q.49.1 2
20.7 even 4 1200.4.f.q.49.2 2
20.19 odd 2 1200.4.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.4.a.c.1.1 1 1.1 even 1 trivial
150.4.a.g.1.1 yes 1 5.4 even 2
150.4.c.b.49.1 2 5.2 odd 4
150.4.c.b.49.2 2 5.3 odd 4
450.4.a.i.1.1 1 15.14 odd 2
450.4.a.l.1.1 1 3.2 odd 2
450.4.c.h.199.1 2 15.8 even 4
450.4.c.h.199.2 2 15.2 even 4
1200.4.a.r.1.1 1 4.3 odd 2
1200.4.a.v.1.1 1 20.19 odd 2
1200.4.f.q.49.1 2 20.3 even 4
1200.4.f.q.49.2 2 20.7 even 4