Properties

Label 338.4.b.a.337.1
Level $338$
Weight $4$
Character 338.337
Analytic conductor $19.943$
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] = [338,4,Mod(337,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.b (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: \(19.9426455819\)
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 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 338.337
Dual form 338.4.b.a.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +2.00000i q^{5} +6.00000i q^{6} +5.00000i q^{7} +8.00000i q^{8} -18.0000 q^{9} +O(q^{10})\) \(q-2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +2.00000i q^{5} +6.00000i q^{6} +5.00000i q^{7} +8.00000i q^{8} -18.0000 q^{9} +4.00000 q^{10} -13.0000i q^{11} +12.0000 q^{12} +10.0000 q^{14} -6.00000i q^{15} +16.0000 q^{16} -27.0000 q^{17} +36.0000i q^{18} +75.0000i q^{19} -8.00000i q^{20} -15.0000i q^{21} -26.0000 q^{22} +187.000 q^{23} -24.0000i q^{24} +121.000 q^{25} +135.000 q^{27} -20.0000i q^{28} -13.0000 q^{29} -12.0000 q^{30} -104.000i q^{31} -32.0000i q^{32} +39.0000i q^{33} +54.0000i q^{34} -10.0000 q^{35} +72.0000 q^{36} -423.000i q^{37} +150.000 q^{38} -16.0000 q^{40} +195.000i q^{41} -30.0000 q^{42} -199.000 q^{43} +52.0000i q^{44} -36.0000i q^{45} -374.000i q^{46} -388.000i q^{47} -48.0000 q^{48} +318.000 q^{49} -242.000i q^{50} +81.0000 q^{51} +618.000 q^{53} -270.000i q^{54} +26.0000 q^{55} -40.0000 q^{56} -225.000i q^{57} +26.0000i q^{58} -491.000i q^{59} +24.0000i q^{60} +175.000 q^{61} -208.000 q^{62} -90.0000i q^{63} -64.0000 q^{64} +78.0000 q^{66} +817.000i q^{67} +108.000 q^{68} -561.000 q^{69} +20.0000i q^{70} +79.0000i q^{71} -144.000i q^{72} -230.000i q^{73} -846.000 q^{74} -363.000 q^{75} -300.000i q^{76} +65.0000 q^{77} +764.000 q^{79} +32.0000i q^{80} +81.0000 q^{81} +390.000 q^{82} -732.000i q^{83} +60.0000i q^{84} -54.0000i q^{85} +398.000i q^{86} +39.0000 q^{87} +104.000 q^{88} +1041.00i q^{89} -72.0000 q^{90} -748.000 q^{92} +312.000i q^{93} -776.000 q^{94} -150.000 q^{95} +96.0000i q^{96} -97.0000i q^{97} -636.000i q^{98} +234.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 8 q^{4} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 8 q^{4} - 36 q^{9} + 8 q^{10} + 24 q^{12} + 20 q^{14} + 32 q^{16} - 54 q^{17} - 52 q^{22} + 374 q^{23} + 242 q^{25} + 270 q^{27} - 26 q^{29} - 24 q^{30} - 20 q^{35} + 144 q^{36} + 300 q^{38} - 32 q^{40} - 60 q^{42} - 398 q^{43} - 96 q^{48} + 636 q^{49} + 162 q^{51} + 1236 q^{53} + 52 q^{55} - 80 q^{56} + 350 q^{61} - 416 q^{62} - 128 q^{64} + 156 q^{66} + 216 q^{68} - 1122 q^{69} - 1692 q^{74} - 726 q^{75} + 130 q^{77} + 1528 q^{79} + 162 q^{81} + 780 q^{82} + 78 q^{87} + 208 q^{88} - 144 q^{90} - 1496 q^{92} - 1552 q^{94} - 300 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) −3.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) −4.00000 −0.500000
\(5\) 2.00000i 0.178885i 0.995992 + 0.0894427i \(0.0285086\pi\)
−0.995992 + 0.0894427i \(0.971491\pi\)
\(6\) 6.00000i 0.408248i
\(7\) 5.00000i 0.269975i 0.990847 + 0.134987i \(0.0430994\pi\)
−0.990847 + 0.134987i \(0.956901\pi\)
\(8\) 8.00000i 0.353553i
\(9\) −18.0000 −0.666667
\(10\) 4.00000 0.126491
\(11\) − 13.0000i − 0.356332i −0.984000 0.178166i \(-0.942984\pi\)
0.984000 0.178166i \(-0.0570163\pi\)
\(12\) 12.0000 0.288675
\(13\) 0 0
\(14\) 10.0000 0.190901
\(15\) − 6.00000i − 0.103280i
\(16\) 16.0000 0.250000
\(17\) −27.0000 −0.385204 −0.192602 0.981277i \(-0.561693\pi\)
−0.192602 + 0.981277i \(0.561693\pi\)
\(18\) 36.0000i 0.471405i
\(19\) 75.0000i 0.905588i 0.891615 + 0.452794i \(0.149573\pi\)
−0.891615 + 0.452794i \(0.850427\pi\)
\(20\) − 8.00000i − 0.0894427i
\(21\) − 15.0000i − 0.155870i
\(22\) −26.0000 −0.251964
\(23\) 187.000 1.69531 0.847656 0.530546i \(-0.178013\pi\)
0.847656 + 0.530546i \(0.178013\pi\)
\(24\) − 24.0000i − 0.204124i
\(25\) 121.000 0.968000
\(26\) 0 0
\(27\) 135.000 0.962250
\(28\) − 20.0000i − 0.134987i
\(29\) −13.0000 −0.0832427 −0.0416214 0.999133i \(-0.513252\pi\)
−0.0416214 + 0.999133i \(0.513252\pi\)
\(30\) −12.0000 −0.0730297
\(31\) − 104.000i − 0.602547i −0.953538 0.301273i \(-0.902588\pi\)
0.953538 0.301273i \(-0.0974117\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 39.0000i 0.205728i
\(34\) 54.0000i 0.272380i
\(35\) −10.0000 −0.0482945
\(36\) 72.0000 0.333333
\(37\) − 423.000i − 1.87948i −0.341890 0.939740i \(-0.611067\pi\)
0.341890 0.939740i \(-0.388933\pi\)
\(38\) 150.000 0.640348
\(39\) 0 0
\(40\) −16.0000 −0.0632456
\(41\) 195.000i 0.742778i 0.928477 + 0.371389i \(0.121118\pi\)
−0.928477 + 0.371389i \(0.878882\pi\)
\(42\) −30.0000 −0.110217
\(43\) −199.000 −0.705749 −0.352875 0.935671i \(-0.614796\pi\)
−0.352875 + 0.935671i \(0.614796\pi\)
\(44\) 52.0000i 0.178166i
\(45\) − 36.0000i − 0.119257i
\(46\) − 374.000i − 1.19877i
\(47\) − 388.000i − 1.20416i −0.798435 0.602081i \(-0.794338\pi\)
0.798435 0.602081i \(-0.205662\pi\)
\(48\) −48.0000 −0.144338
\(49\) 318.000 0.927114
\(50\) − 242.000i − 0.684479i
\(51\) 81.0000 0.222397
\(52\) 0 0
\(53\) 618.000 1.60168 0.800838 0.598881i \(-0.204388\pi\)
0.800838 + 0.598881i \(0.204388\pi\)
\(54\) − 270.000i − 0.680414i
\(55\) 26.0000 0.0637425
\(56\) −40.0000 −0.0954504
\(57\) − 225.000i − 0.522842i
\(58\) 26.0000i 0.0588615i
\(59\) − 491.000i − 1.08344i −0.840560 0.541718i \(-0.817774\pi\)
0.840560 0.541718i \(-0.182226\pi\)
\(60\) 24.0000i 0.0516398i
\(61\) 175.000 0.367319 0.183659 0.982990i \(-0.441206\pi\)
0.183659 + 0.982990i \(0.441206\pi\)
\(62\) −208.000 −0.426065
\(63\) − 90.0000i − 0.179983i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 78.0000 0.145472
\(67\) 817.000i 1.48974i 0.667211 + 0.744869i \(0.267488\pi\)
−0.667211 + 0.744869i \(0.732512\pi\)
\(68\) 108.000 0.192602
\(69\) −561.000 −0.978789
\(70\) 20.0000i 0.0341494i
\(71\) 79.0000i 0.132050i 0.997818 + 0.0660252i \(0.0210318\pi\)
−0.997818 + 0.0660252i \(0.978968\pi\)
\(72\) − 144.000i − 0.235702i
\(73\) − 230.000i − 0.368760i −0.982855 0.184380i \(-0.940972\pi\)
0.982855 0.184380i \(-0.0590277\pi\)
\(74\) −846.000 −1.32899
\(75\) −363.000 −0.558875
\(76\) − 300.000i − 0.452794i
\(77\) 65.0000 0.0962005
\(78\) 0 0
\(79\) 764.000 1.08806 0.544030 0.839066i \(-0.316898\pi\)
0.544030 + 0.839066i \(0.316898\pi\)
\(80\) 32.0000i 0.0447214i
\(81\) 81.0000 0.111111
\(82\) 390.000 0.525223
\(83\) − 732.000i − 0.968041i −0.875057 0.484021i \(-0.839176\pi\)
0.875057 0.484021i \(-0.160824\pi\)
\(84\) 60.0000i 0.0779350i
\(85\) − 54.0000i − 0.0689073i
\(86\) 398.000i 0.499040i
\(87\) 39.0000 0.0480602
\(88\) 104.000 0.125982
\(89\) 1041.00i 1.23984i 0.784665 + 0.619920i \(0.212835\pi\)
−0.784665 + 0.619920i \(0.787165\pi\)
\(90\) −72.0000 −0.0843274
\(91\) 0 0
\(92\) −748.000 −0.847656
\(93\) 312.000i 0.347881i
\(94\) −776.000 −0.851471
\(95\) −150.000 −0.161997
\(96\) 96.0000i 0.102062i
\(97\) − 97.0000i − 0.101535i −0.998711 0.0507673i \(-0.983833\pi\)
0.998711 0.0507673i \(-0.0161667\pi\)
\(98\) − 636.000i − 0.655568i
\(99\) 234.000i 0.237554i
\(100\) −484.000 −0.484000
\(101\) 809.000 0.797015 0.398507 0.917165i \(-0.369528\pi\)
0.398507 + 0.917165i \(0.369528\pi\)
\(102\) − 162.000i − 0.157259i
\(103\) −1288.00 −1.23214 −0.616070 0.787691i \(-0.711276\pi\)
−0.616070 + 0.787691i \(0.711276\pi\)
\(104\) 0 0
\(105\) 30.0000 0.0278829
\(106\) − 1236.00i − 1.13256i
\(107\) 1277.00 1.15376 0.576880 0.816829i \(-0.304270\pi\)
0.576880 + 0.816829i \(0.304270\pi\)
\(108\) −540.000 −0.481125
\(109\) 826.000i 0.725839i 0.931820 + 0.362920i \(0.118220\pi\)
−0.931820 + 0.362920i \(0.881780\pi\)
\(110\) − 52.0000i − 0.0450728i
\(111\) 1269.00i 1.08512i
\(112\) 80.0000i 0.0674937i
\(113\) 947.000 0.788374 0.394187 0.919030i \(-0.371026\pi\)
0.394187 + 0.919030i \(0.371026\pi\)
\(114\) −450.000 −0.369705
\(115\) 374.000i 0.303267i
\(116\) 52.0000 0.0416214
\(117\) 0 0
\(118\) −982.000 −0.766105
\(119\) − 135.000i − 0.103995i
\(120\) 48.0000 0.0365148
\(121\) 1162.00 0.873028
\(122\) − 350.000i − 0.259734i
\(123\) − 585.000i − 0.428843i
\(124\) 416.000i 0.301273i
\(125\) 492.000i 0.352047i
\(126\) −180.000 −0.127267
\(127\) −1177.00 −0.822377 −0.411188 0.911550i \(-0.634886\pi\)
−0.411188 + 0.911550i \(0.634886\pi\)
\(128\) 128.000i 0.0883883i
\(129\) 597.000 0.407464
\(130\) 0 0
\(131\) −1420.00 −0.947069 −0.473534 0.880775i \(-0.657022\pi\)
−0.473534 + 0.880775i \(0.657022\pi\)
\(132\) − 156.000i − 0.102864i
\(133\) −375.000 −0.244486
\(134\) 1634.00 1.05340
\(135\) 270.000i 0.172133i
\(136\) − 216.000i − 0.136190i
\(137\) 2409.00i 1.50230i 0.660133 + 0.751149i \(0.270500\pi\)
−0.660133 + 0.751149i \(0.729500\pi\)
\(138\) 1122.00i 0.692109i
\(139\) 2827.00 1.72506 0.862529 0.506008i \(-0.168879\pi\)
0.862529 + 0.506008i \(0.168879\pi\)
\(140\) 40.0000 0.0241473
\(141\) 1164.00i 0.695223i
\(142\) 158.000 0.0933737
\(143\) 0 0
\(144\) −288.000 −0.166667
\(145\) − 26.0000i − 0.0148909i
\(146\) −460.000 −0.260753
\(147\) −954.000 −0.535269
\(148\) 1692.00i 0.939740i
\(149\) 855.000i 0.470096i 0.971984 + 0.235048i \(0.0755248\pi\)
−0.971984 + 0.235048i \(0.924475\pi\)
\(150\) 726.000i 0.395184i
\(151\) − 2064.00i − 1.11236i −0.831063 0.556179i \(-0.812267\pi\)
0.831063 0.556179i \(-0.187733\pi\)
\(152\) −600.000 −0.320174
\(153\) 486.000 0.256802
\(154\) − 130.000i − 0.0680240i
\(155\) 208.000 0.107787
\(156\) 0 0
\(157\) −1894.00 −0.962788 −0.481394 0.876504i \(-0.659869\pi\)
−0.481394 + 0.876504i \(0.659869\pi\)
\(158\) − 1528.00i − 0.769374i
\(159\) −1854.00 −0.924728
\(160\) 64.0000 0.0316228
\(161\) 935.000i 0.457691i
\(162\) − 162.000i − 0.0785674i
\(163\) 985.000i 0.473320i 0.971593 + 0.236660i \(0.0760527\pi\)
−0.971593 + 0.236660i \(0.923947\pi\)
\(164\) − 780.000i − 0.371389i
\(165\) −78.0000 −0.0368018
\(166\) −1464.00 −0.684509
\(167\) 2355.00i 1.09123i 0.838036 + 0.545615i \(0.183704\pi\)
−0.838036 + 0.545615i \(0.816296\pi\)
\(168\) 120.000 0.0551083
\(169\) 0 0
\(170\) −108.000 −0.0487248
\(171\) − 1350.00i − 0.603726i
\(172\) 796.000 0.352875
\(173\) 3889.00 1.70911 0.854553 0.519365i \(-0.173831\pi\)
0.854553 + 0.519365i \(0.173831\pi\)
\(174\) − 78.0000i − 0.0339837i
\(175\) 605.000i 0.261335i
\(176\) − 208.000i − 0.0890829i
\(177\) 1473.00i 0.625522i
\(178\) 2082.00 0.876699
\(179\) −2229.00 −0.930745 −0.465372 0.885115i \(-0.654080\pi\)
−0.465372 + 0.885115i \(0.654080\pi\)
\(180\) 144.000i 0.0596285i
\(181\) 1038.00 0.426265 0.213132 0.977023i \(-0.431633\pi\)
0.213132 + 0.977023i \(0.431633\pi\)
\(182\) 0 0
\(183\) −525.000 −0.212072
\(184\) 1496.00i 0.599384i
\(185\) 846.000 0.336212
\(186\) 624.000 0.245989
\(187\) 351.000i 0.137260i
\(188\) 1552.00i 0.602081i
\(189\) 675.000i 0.259783i
\(190\) 300.000i 0.114549i
\(191\) −2141.00 −0.811085 −0.405543 0.914076i \(-0.632917\pi\)
−0.405543 + 0.914076i \(0.632917\pi\)
\(192\) 192.000 0.0721688
\(193\) − 2627.00i − 0.979770i −0.871787 0.489885i \(-0.837039\pi\)
0.871787 0.489885i \(-0.162961\pi\)
\(194\) −194.000 −0.0717958
\(195\) 0 0
\(196\) −1272.00 −0.463557
\(197\) 1203.00i 0.435077i 0.976052 + 0.217539i \(0.0698028\pi\)
−0.976052 + 0.217539i \(0.930197\pi\)
\(198\) 468.000 0.167976
\(199\) −743.000 −0.264673 −0.132336 0.991205i \(-0.542248\pi\)
−0.132336 + 0.991205i \(0.542248\pi\)
\(200\) 968.000i 0.342240i
\(201\) − 2451.00i − 0.860101i
\(202\) − 1618.00i − 0.563575i
\(203\) − 65.0000i − 0.0224734i
\(204\) −324.000 −0.111199
\(205\) −390.000 −0.132872
\(206\) 2576.00i 0.871254i
\(207\) −3366.00 −1.13021
\(208\) 0 0
\(209\) 975.000 0.322690
\(210\) − 60.0000i − 0.0197162i
\(211\) −355.000 −0.115826 −0.0579128 0.998322i \(-0.518445\pi\)
−0.0579128 + 0.998322i \(0.518445\pi\)
\(212\) −2472.00 −0.800838
\(213\) − 237.000i − 0.0762393i
\(214\) − 2554.00i − 0.815831i
\(215\) − 398.000i − 0.126248i
\(216\) 1080.00i 0.340207i
\(217\) 520.000 0.162672
\(218\) 1652.00 0.513246
\(219\) 690.000i 0.212904i
\(220\) −104.000 −0.0318713
\(221\) 0 0
\(222\) 2538.00 0.767295
\(223\) − 2283.00i − 0.685565i −0.939415 0.342782i \(-0.888631\pi\)
0.939415 0.342782i \(-0.111369\pi\)
\(224\) 160.000 0.0477252
\(225\) −2178.00 −0.645333
\(226\) − 1894.00i − 0.557465i
\(227\) 2451.00i 0.716646i 0.933598 + 0.358323i \(0.116651\pi\)
−0.933598 + 0.358323i \(0.883349\pi\)
\(228\) 900.000i 0.261421i
\(229\) 1878.00i 0.541929i 0.962589 + 0.270964i \(0.0873426\pi\)
−0.962589 + 0.270964i \(0.912657\pi\)
\(230\) 748.000 0.214442
\(231\) −195.000 −0.0555414
\(232\) − 104.000i − 0.0294308i
\(233\) −1630.00 −0.458304 −0.229152 0.973391i \(-0.573595\pi\)
−0.229152 + 0.973391i \(0.573595\pi\)
\(234\) 0 0
\(235\) 776.000 0.215407
\(236\) 1964.00i 0.541718i
\(237\) −2292.00 −0.628192
\(238\) −270.000 −0.0735357
\(239\) − 5544.00i − 1.50047i −0.661173 0.750233i \(-0.729941\pi\)
0.661173 0.750233i \(-0.270059\pi\)
\(240\) − 96.0000i − 0.0258199i
\(241\) − 5523.00i − 1.47621i −0.674683 0.738107i \(-0.735720\pi\)
0.674683 0.738107i \(-0.264280\pi\)
\(242\) − 2324.00i − 0.617324i
\(243\) −3888.00 −1.02640
\(244\) −700.000 −0.183659
\(245\) 636.000i 0.165847i
\(246\) −1170.00 −0.303238
\(247\) 0 0
\(248\) 832.000 0.213032
\(249\) 2196.00i 0.558899i
\(250\) 984.000 0.248934
\(251\) −2175.00 −0.546951 −0.273476 0.961879i \(-0.588173\pi\)
−0.273476 + 0.961879i \(0.588173\pi\)
\(252\) 360.000i 0.0899915i
\(253\) − 2431.00i − 0.604094i
\(254\) 2354.00i 0.581508i
\(255\) 162.000i 0.0397837i
\(256\) 256.000 0.0625000
\(257\) 5685.00 1.37985 0.689923 0.723883i \(-0.257644\pi\)
0.689923 + 0.723883i \(0.257644\pi\)
\(258\) − 1194.00i − 0.288121i
\(259\) 2115.00 0.507412
\(260\) 0 0
\(261\) 234.000 0.0554952
\(262\) 2840.00i 0.669679i
\(263\) 6117.00 1.43418 0.717092 0.696979i \(-0.245473\pi\)
0.717092 + 0.696979i \(0.245473\pi\)
\(264\) −312.000 −0.0727359
\(265\) 1236.00i 0.286517i
\(266\) 750.000i 0.172878i
\(267\) − 3123.00i − 0.715822i
\(268\) − 3268.00i − 0.744869i
\(269\) −5109.00 −1.15800 −0.578999 0.815329i \(-0.696556\pi\)
−0.578999 + 0.815329i \(0.696556\pi\)
\(270\) 540.000 0.121716
\(271\) − 7549.00i − 1.69214i −0.533074 0.846068i \(-0.678963\pi\)
0.533074 0.846068i \(-0.321037\pi\)
\(272\) −432.000 −0.0963009
\(273\) 0 0
\(274\) 4818.00 1.06228
\(275\) − 1573.00i − 0.344929i
\(276\) 2244.00 0.489395
\(277\) 981.000 0.212789 0.106395 0.994324i \(-0.466069\pi\)
0.106395 + 0.994324i \(0.466069\pi\)
\(278\) − 5654.00i − 1.21980i
\(279\) 1872.00i 0.401698i
\(280\) − 80.0000i − 0.0170747i
\(281\) 2762.00i 0.586360i 0.956057 + 0.293180i \(0.0947135\pi\)
−0.956057 + 0.293180i \(0.905287\pi\)
\(282\) 2328.00 0.491597
\(283\) −3925.00 −0.824442 −0.412221 0.911084i \(-0.635247\pi\)
−0.412221 + 0.911084i \(0.635247\pi\)
\(284\) − 316.000i − 0.0660252i
\(285\) 450.000 0.0935288
\(286\) 0 0
\(287\) −975.000 −0.200531
\(288\) 576.000i 0.117851i
\(289\) −4184.00 −0.851618
\(290\) −52.0000 −0.0105295
\(291\) 291.000i 0.0586210i
\(292\) 920.000i 0.184380i
\(293\) − 7711.00i − 1.53748i −0.639562 0.768740i \(-0.720884\pi\)
0.639562 0.768740i \(-0.279116\pi\)
\(294\) 1908.00i 0.378493i
\(295\) 982.000 0.193811
\(296\) 3384.00 0.664497
\(297\) − 1755.00i − 0.342880i
\(298\) 1710.00 0.332408
\(299\) 0 0
\(300\) 1452.00 0.279438
\(301\) − 995.000i − 0.190534i
\(302\) −4128.00 −0.786555
\(303\) −2427.00 −0.460157
\(304\) 1200.00i 0.226397i
\(305\) 350.000i 0.0657080i
\(306\) − 972.000i − 0.181587i
\(307\) − 10388.0i − 1.93119i −0.260056 0.965594i \(-0.583741\pi\)
0.260056 0.965594i \(-0.416259\pi\)
\(308\) −260.000 −0.0481002
\(309\) 3864.00 0.711376
\(310\) − 416.000i − 0.0762168i
\(311\) 7272.00 1.32591 0.662954 0.748660i \(-0.269303\pi\)
0.662954 + 0.748660i \(0.269303\pi\)
\(312\) 0 0
\(313\) 7910.00 1.42843 0.714217 0.699925i \(-0.246783\pi\)
0.714217 + 0.699925i \(0.246783\pi\)
\(314\) 3788.00i 0.680794i
\(315\) 180.000 0.0321964
\(316\) −3056.00 −0.544030
\(317\) 7398.00i 1.31077i 0.755296 + 0.655383i \(0.227493\pi\)
−0.755296 + 0.655383i \(0.772507\pi\)
\(318\) 3708.00i 0.653881i
\(319\) 169.000i 0.0296620i
\(320\) − 128.000i − 0.0223607i
\(321\) −3831.00 −0.666123
\(322\) 1870.00 0.323637
\(323\) − 2025.00i − 0.348836i
\(324\) −324.000 −0.0555556
\(325\) 0 0
\(326\) 1970.00 0.334688
\(327\) − 2478.00i − 0.419063i
\(328\) −1560.00 −0.262612
\(329\) 1940.00 0.325093
\(330\) 156.000i 0.0260228i
\(331\) − 2377.00i − 0.394718i −0.980331 0.197359i \(-0.936764\pi\)
0.980331 0.197359i \(-0.0632365\pi\)
\(332\) 2928.00i 0.484021i
\(333\) 7614.00i 1.25299i
\(334\) 4710.00 0.771616
\(335\) −1634.00 −0.266492
\(336\) − 240.000i − 0.0389675i
\(337\) 7618.00 1.23139 0.615696 0.787984i \(-0.288875\pi\)
0.615696 + 0.787984i \(0.288875\pi\)
\(338\) 0 0
\(339\) −2841.00 −0.455168
\(340\) 216.000i 0.0344537i
\(341\) −1352.00 −0.214706
\(342\) −2700.00 −0.426898
\(343\) 3305.00i 0.520272i
\(344\) − 1592.00i − 0.249520i
\(345\) − 1122.00i − 0.175091i
\(346\) − 7778.00i − 1.20852i
\(347\) 375.000 0.0580146 0.0290073 0.999579i \(-0.490765\pi\)
0.0290073 + 0.999579i \(0.490765\pi\)
\(348\) −156.000 −0.0240301
\(349\) − 9727.00i − 1.49190i −0.666000 0.745952i \(-0.731995\pi\)
0.666000 0.745952i \(-0.268005\pi\)
\(350\) 1210.00 0.184792
\(351\) 0 0
\(352\) −416.000 −0.0629911
\(353\) 2263.00i 0.341211i 0.985339 + 0.170605i \(0.0545723\pi\)
−0.985339 + 0.170605i \(0.945428\pi\)
\(354\) 2946.00 0.442311
\(355\) −158.000 −0.0236219
\(356\) − 4164.00i − 0.619920i
\(357\) 405.000i 0.0600417i
\(358\) 4458.00i 0.658136i
\(359\) 4488.00i 0.659798i 0.944016 + 0.329899i \(0.107015\pi\)
−0.944016 + 0.329899i \(0.892985\pi\)
\(360\) 288.000 0.0421637
\(361\) 1234.00 0.179910
\(362\) − 2076.00i − 0.301415i
\(363\) −3486.00 −0.504043
\(364\) 0 0
\(365\) 460.000 0.0659658
\(366\) 1050.00i 0.149957i
\(367\) −1627.00 −0.231413 −0.115707 0.993283i \(-0.536913\pi\)
−0.115707 + 0.993283i \(0.536913\pi\)
\(368\) 2992.00 0.423828
\(369\) − 3510.00i − 0.495185i
\(370\) − 1692.00i − 0.237738i
\(371\) 3090.00i 0.432412i
\(372\) − 1248.00i − 0.173940i
\(373\) 2987.00 0.414641 0.207320 0.978273i \(-0.433526\pi\)
0.207320 + 0.978273i \(0.433526\pi\)
\(374\) 702.000 0.0970576
\(375\) − 1476.00i − 0.203254i
\(376\) 3104.00 0.425736
\(377\) 0 0
\(378\) 1350.00 0.183694
\(379\) − 8867.00i − 1.20176i −0.799339 0.600880i \(-0.794817\pi\)
0.799339 0.600880i \(-0.205183\pi\)
\(380\) 600.000 0.0809983
\(381\) 3531.00 0.474800
\(382\) 4282.00i 0.573524i
\(383\) 11403.0i 1.52132i 0.649150 + 0.760661i \(0.275125\pi\)
−0.649150 + 0.760661i \(0.724875\pi\)
\(384\) − 384.000i − 0.0510310i
\(385\) 130.000i 0.0172089i
\(386\) −5254.00 −0.692802
\(387\) 3582.00 0.470499
\(388\) 388.000i 0.0507673i
\(389\) −2622.00 −0.341750 −0.170875 0.985293i \(-0.554659\pi\)
−0.170875 + 0.985293i \(0.554659\pi\)
\(390\) 0 0
\(391\) −5049.00 −0.653041
\(392\) 2544.00i 0.327784i
\(393\) 4260.00 0.546790
\(394\) 2406.00 0.307646
\(395\) 1528.00i 0.194638i
\(396\) − 936.000i − 0.118777i
\(397\) − 659.000i − 0.0833105i −0.999132 0.0416552i \(-0.986737\pi\)
0.999132 0.0416552i \(-0.0132631\pi\)
\(398\) 1486.00i 0.187152i
\(399\) 1125.00 0.141154
\(400\) 1936.00 0.242000
\(401\) 14685.0i 1.82876i 0.404854 + 0.914381i \(0.367322\pi\)
−0.404854 + 0.914381i \(0.632678\pi\)
\(402\) −4902.00 −0.608183
\(403\) 0 0
\(404\) −3236.00 −0.398507
\(405\) 162.000i 0.0198762i
\(406\) −130.000 −0.0158911
\(407\) −5499.00 −0.669718
\(408\) 648.000i 0.0786294i
\(409\) − 7829.00i − 0.946502i −0.880928 0.473251i \(-0.843080\pi\)
0.880928 0.473251i \(-0.156920\pi\)
\(410\) 780.000i 0.0939548i
\(411\) − 7227.00i − 0.867352i
\(412\) 5152.00 0.616070
\(413\) 2455.00 0.292500
\(414\) 6732.00i 0.799178i
\(415\) 1464.00 0.173169
\(416\) 0 0
\(417\) −8481.00 −0.995962
\(418\) − 1950.00i − 0.228176i
\(419\) −2919.00 −0.340340 −0.170170 0.985415i \(-0.554432\pi\)
−0.170170 + 0.985415i \(0.554432\pi\)
\(420\) −120.000 −0.0139414
\(421\) − 3110.00i − 0.360029i −0.983664 0.180014i \(-0.942386\pi\)
0.983664 0.180014i \(-0.0576144\pi\)
\(422\) 710.000i 0.0819011i
\(423\) 6984.00i 0.802775i
\(424\) 4944.00i 0.566278i
\(425\) −3267.00 −0.372877
\(426\) −474.000 −0.0539093
\(427\) 875.000i 0.0991668i
\(428\) −5108.00 −0.576880
\(429\) 0 0
\(430\) −796.000 −0.0892710
\(431\) − 9135.00i − 1.02092i −0.859901 0.510461i \(-0.829475\pi\)
0.859901 0.510461i \(-0.170525\pi\)
\(432\) 2160.00 0.240563
\(433\) 11669.0 1.29510 0.647548 0.762025i \(-0.275795\pi\)
0.647548 + 0.762025i \(0.275795\pi\)
\(434\) − 1040.00i − 0.115027i
\(435\) 78.0000i 0.00859727i
\(436\) − 3304.00i − 0.362920i
\(437\) 14025.0i 1.53526i
\(438\) 1380.00 0.150546
\(439\) −13529.0 −1.47085 −0.735426 0.677605i \(-0.763018\pi\)
−0.735426 + 0.677605i \(0.763018\pi\)
\(440\) 208.000i 0.0225364i
\(441\) −5724.00 −0.618076
\(442\) 0 0
\(443\) −1932.00 −0.207206 −0.103603 0.994619i \(-0.533037\pi\)
−0.103603 + 0.994619i \(0.533037\pi\)
\(444\) − 5076.00i − 0.542559i
\(445\) −2082.00 −0.221789
\(446\) −4566.00 −0.484768
\(447\) − 2565.00i − 0.271410i
\(448\) − 320.000i − 0.0337468i
\(449\) 5357.00i 0.563057i 0.959553 + 0.281528i \(0.0908413\pi\)
−0.959553 + 0.281528i \(0.909159\pi\)
\(450\) 4356.00i 0.456320i
\(451\) 2535.00 0.264675
\(452\) −3788.00 −0.394187
\(453\) 6192.00i 0.642220i
\(454\) 4902.00 0.506745
\(455\) 0 0
\(456\) 1800.00 0.184852
\(457\) 19399.0i 1.98566i 0.119532 + 0.992830i \(0.461861\pi\)
−0.119532 + 0.992830i \(0.538139\pi\)
\(458\) 3756.00 0.383202
\(459\) −3645.00 −0.370662
\(460\) − 1496.00i − 0.151633i
\(461\) − 15549.0i − 1.57091i −0.618919 0.785455i \(-0.712429\pi\)
0.618919 0.785455i \(-0.287571\pi\)
\(462\) 390.000i 0.0392737i
\(463\) − 4072.00i − 0.408730i −0.978895 0.204365i \(-0.934487\pi\)
0.978895 0.204365i \(-0.0655129\pi\)
\(464\) −208.000 −0.0208107
\(465\) −624.000 −0.0622308
\(466\) 3260.00i 0.324070i
\(467\) −15224.0 −1.50853 −0.754264 0.656571i \(-0.772006\pi\)
−0.754264 + 0.656571i \(0.772006\pi\)
\(468\) 0 0
\(469\) −4085.00 −0.402191
\(470\) − 1552.00i − 0.152316i
\(471\) 5682.00 0.555866
\(472\) 3928.00 0.383053
\(473\) 2587.00i 0.251481i
\(474\) 4584.00i 0.444199i
\(475\) 9075.00i 0.876610i
\(476\) 540.000i 0.0519976i
\(477\) −11124.0 −1.06778
\(478\) −11088.0 −1.06099
\(479\) 10335.0i 0.985842i 0.870074 + 0.492921i \(0.164071\pi\)
−0.870074 + 0.492921i \(0.835929\pi\)
\(480\) −192.000 −0.0182574
\(481\) 0 0
\(482\) −11046.0 −1.04384
\(483\) − 2805.00i − 0.264248i
\(484\) −4648.00 −0.436514
\(485\) 194.000 0.0181631
\(486\) 7776.00i 0.725775i
\(487\) 6455.00i 0.600624i 0.953841 + 0.300312i \(0.0970908\pi\)
−0.953841 + 0.300312i \(0.902909\pi\)
\(488\) 1400.00i 0.129867i
\(489\) − 2955.00i − 0.273271i
\(490\) 1272.00 0.117272
\(491\) −7777.00 −0.714809 −0.357404 0.933950i \(-0.616338\pi\)
−0.357404 + 0.933950i \(0.616338\pi\)
\(492\) 2340.00i 0.214421i
\(493\) 351.000 0.0320654
\(494\) 0 0
\(495\) −468.000 −0.0424950
\(496\) − 1664.00i − 0.150637i
\(497\) −395.000 −0.0356502
\(498\) 4392.00 0.395201
\(499\) − 3044.00i − 0.273082i −0.990634 0.136541i \(-0.956401\pi\)
0.990634 0.136541i \(-0.0435986\pi\)
\(500\) − 1968.00i − 0.176023i
\(501\) − 7065.00i − 0.630022i
\(502\) 4350.00i 0.386753i
\(503\) 11347.0 1.00584 0.502920 0.864333i \(-0.332259\pi\)
0.502920 + 0.864333i \(0.332259\pi\)
\(504\) 720.000 0.0636336
\(505\) 1618.00i 0.142574i
\(506\) −4862.00 −0.427159
\(507\) 0 0
\(508\) 4708.00 0.411188
\(509\) 727.000i 0.0633079i 0.999499 + 0.0316539i \(0.0100774\pi\)
−0.999499 + 0.0316539i \(0.989923\pi\)
\(510\) 324.000 0.0281313
\(511\) 1150.00 0.0995558
\(512\) − 512.000i − 0.0441942i
\(513\) 10125.0i 0.871403i
\(514\) − 11370.0i − 0.975699i
\(515\) − 2576.00i − 0.220412i
\(516\) −2388.00 −0.203732
\(517\) −5044.00 −0.429081
\(518\) − 4230.00i − 0.358794i
\(519\) −11667.0 −0.986752
\(520\) 0 0
\(521\) 9582.00 0.805749 0.402874 0.915255i \(-0.368011\pi\)
0.402874 + 0.915255i \(0.368011\pi\)
\(522\) − 468.000i − 0.0392410i
\(523\) −10383.0 −0.868101 −0.434051 0.900889i \(-0.642916\pi\)
−0.434051 + 0.900889i \(0.642916\pi\)
\(524\) 5680.00 0.473534
\(525\) − 1815.00i − 0.150882i
\(526\) − 12234.0i − 1.01412i
\(527\) 2808.00i 0.232103i
\(528\) 624.000i 0.0514320i
\(529\) 22802.0 1.87409
\(530\) 2472.00 0.202598
\(531\) 8838.00i 0.722291i
\(532\) 1500.00 0.122243
\(533\) 0 0
\(534\) −6246.00 −0.506163
\(535\) 2554.00i 0.206391i
\(536\) −6536.00 −0.526702
\(537\) 6687.00 0.537366
\(538\) 10218.0i 0.818828i
\(539\) − 4134.00i − 0.330360i
\(540\) − 1080.00i − 0.0860663i
\(541\) − 12230.0i − 0.971920i −0.873981 0.485960i \(-0.838470\pi\)
0.873981 0.485960i \(-0.161530\pi\)
\(542\) −15098.0 −1.19652
\(543\) −3114.00 −0.246104
\(544\) 864.000i 0.0680950i
\(545\) −1652.00 −0.129842
\(546\) 0 0
\(547\) −14636.0 −1.14404 −0.572020 0.820239i \(-0.693840\pi\)
−0.572020 + 0.820239i \(0.693840\pi\)
\(548\) − 9636.00i − 0.751149i
\(549\) −3150.00 −0.244879
\(550\) −3146.00 −0.243902
\(551\) − 975.000i − 0.0753837i
\(552\) − 4488.00i − 0.346054i
\(553\) 3820.00i 0.293749i
\(554\) − 1962.00i − 0.150465i
\(555\) −2538.00 −0.194112
\(556\) −11308.0 −0.862529
\(557\) 765.000i 0.0581941i 0.999577 + 0.0290970i \(0.00926318\pi\)
−0.999577 + 0.0290970i \(0.990737\pi\)
\(558\) 3744.00 0.284043
\(559\) 0 0
\(560\) −160.000 −0.0120736
\(561\) − 1053.00i − 0.0792472i
\(562\) 5524.00 0.414619
\(563\) −5915.00 −0.442784 −0.221392 0.975185i \(-0.571060\pi\)
−0.221392 + 0.975185i \(0.571060\pi\)
\(564\) − 4656.00i − 0.347612i
\(565\) 1894.00i 0.141029i
\(566\) 7850.00i 0.582968i
\(567\) 405.000i 0.0299972i
\(568\) −632.000 −0.0466869
\(569\) 1217.00 0.0896648 0.0448324 0.998995i \(-0.485725\pi\)
0.0448324 + 0.998995i \(0.485725\pi\)
\(570\) − 900.000i − 0.0661348i
\(571\) 23436.0 1.71763 0.858814 0.512287i \(-0.171202\pi\)
0.858814 + 0.512287i \(0.171202\pi\)
\(572\) 0 0
\(573\) 6423.00 0.468280
\(574\) 1950.00i 0.141797i
\(575\) 22627.0 1.64106
\(576\) 1152.00 0.0833333
\(577\) 7854.00i 0.566666i 0.959022 + 0.283333i \(0.0914402\pi\)
−0.959022 + 0.283333i \(0.908560\pi\)
\(578\) 8368.00i 0.602185i
\(579\) 7881.00i 0.565670i
\(580\) 104.000i 0.00744546i
\(581\) 3660.00 0.261347
\(582\) 582.000 0.0414513
\(583\) − 8034.00i − 0.570728i
\(584\) 1840.00 0.130376
\(585\) 0 0
\(586\) −15422.0 −1.08716
\(587\) 17033.0i 1.19766i 0.800876 + 0.598831i \(0.204368\pi\)
−0.800876 + 0.598831i \(0.795632\pi\)
\(588\) 3816.00 0.267635
\(589\) 7800.00 0.545659
\(590\) − 1964.00i − 0.137045i
\(591\) − 3609.00i − 0.251192i
\(592\) − 6768.00i − 0.469870i
\(593\) 14506.0i 1.00454i 0.864712 + 0.502268i \(0.167501\pi\)
−0.864712 + 0.502268i \(0.832499\pi\)
\(594\) −3510.00 −0.242453
\(595\) 270.000 0.0186032
\(596\) − 3420.00i − 0.235048i
\(597\) 2229.00 0.152809
\(598\) 0 0
\(599\) 15388.0 1.04964 0.524822 0.851212i \(-0.324132\pi\)
0.524822 + 0.851212i \(0.324132\pi\)
\(600\) − 2904.00i − 0.197592i
\(601\) −6077.00 −0.412456 −0.206228 0.978504i \(-0.566119\pi\)
−0.206228 + 0.978504i \(0.566119\pi\)
\(602\) −1990.00 −0.134728
\(603\) − 14706.0i − 0.993159i
\(604\) 8256.00i 0.556179i
\(605\) 2324.00i 0.156172i
\(606\) 4854.00i 0.325380i
\(607\) 10215.0 0.683054 0.341527 0.939872i \(-0.389056\pi\)
0.341527 + 0.939872i \(0.389056\pi\)
\(608\) 2400.00 0.160087
\(609\) 195.000i 0.0129750i
\(610\) 700.000 0.0464626
\(611\) 0 0
\(612\) −1944.00 −0.128401
\(613\) − 3457.00i − 0.227776i −0.993494 0.113888i \(-0.963669\pi\)
0.993494 0.113888i \(-0.0363306\pi\)
\(614\) −20776.0 −1.36556
\(615\) 1170.00 0.0767137
\(616\) 520.000i 0.0340120i
\(617\) − 7169.00i − 0.467768i −0.972264 0.233884i \(-0.924856\pi\)
0.972264 0.233884i \(-0.0751437\pi\)
\(618\) − 7728.00i − 0.503019i
\(619\) − 20212.0i − 1.31242i −0.754578 0.656211i \(-0.772158\pi\)
0.754578 0.656211i \(-0.227842\pi\)
\(620\) −832.000 −0.0538934
\(621\) 25245.0 1.63132
\(622\) − 14544.0i − 0.937558i
\(623\) −5205.00 −0.334725
\(624\) 0 0
\(625\) 14141.0 0.905024
\(626\) − 15820.0i − 1.01005i
\(627\) −2925.00 −0.186305
\(628\) 7576.00 0.481394
\(629\) 11421.0i 0.723983i
\(630\) − 360.000i − 0.0227663i
\(631\) 8945.00i 0.564334i 0.959365 + 0.282167i \(0.0910532\pi\)
−0.959365 + 0.282167i \(0.908947\pi\)
\(632\) 6112.00i 0.384687i
\(633\) 1065.00 0.0668720
\(634\) 14796.0 0.926852
\(635\) − 2354.00i − 0.147111i
\(636\) 7416.00 0.462364
\(637\) 0 0
\(638\) 338.000 0.0209742
\(639\) − 1422.00i − 0.0880336i
\(640\) −256.000 −0.0158114
\(641\) −28243.0 −1.74030 −0.870149 0.492788i \(-0.835978\pi\)
−0.870149 + 0.492788i \(0.835978\pi\)
\(642\) 7662.00i 0.471020i
\(643\) 5231.00i 0.320825i 0.987050 + 0.160413i \(0.0512825\pi\)
−0.987050 + 0.160413i \(0.948718\pi\)
\(644\) − 3740.00i − 0.228846i
\(645\) 1194.00i 0.0728895i
\(646\) −4050.00 −0.246664
\(647\) 4871.00 0.295980 0.147990 0.988989i \(-0.452720\pi\)
0.147990 + 0.988989i \(0.452720\pi\)
\(648\) 648.000i 0.0392837i
\(649\) −6383.00 −0.386063
\(650\) 0 0
\(651\) −1560.00 −0.0939189
\(652\) − 3940.00i − 0.236660i
\(653\) 12255.0 0.734418 0.367209 0.930138i \(-0.380313\pi\)
0.367209 + 0.930138i \(0.380313\pi\)
\(654\) −4956.00 −0.296323
\(655\) − 2840.00i − 0.169417i
\(656\) 3120.00i 0.185694i
\(657\) 4140.00i 0.245840i
\(658\) − 3880.00i − 0.229876i
\(659\) −2145.00 −0.126794 −0.0633971 0.997988i \(-0.520193\pi\)
−0.0633971 + 0.997988i \(0.520193\pi\)
\(660\) 312.000 0.0184009
\(661\) − 2111.00i − 0.124218i −0.998069 0.0621092i \(-0.980217\pi\)
0.998069 0.0621092i \(-0.0197827\pi\)
\(662\) −4754.00 −0.279108
\(663\) 0 0
\(664\) 5856.00 0.342254
\(665\) − 750.000i − 0.0437350i
\(666\) 15228.0 0.885996
\(667\) −2431.00 −0.141122
\(668\) − 9420.00i − 0.545615i
\(669\) 6849.00i 0.395811i
\(670\) 3268.00i 0.188439i
\(671\) − 2275.00i − 0.130887i
\(672\) −480.000 −0.0275542
\(673\) 23273.0 1.33300 0.666499 0.745506i \(-0.267792\pi\)
0.666499 + 0.745506i \(0.267792\pi\)
\(674\) − 15236.0i − 0.870725i
\(675\) 16335.0 0.931458
\(676\) 0 0
\(677\) −5910.00 −0.335509 −0.167755 0.985829i \(-0.553652\pi\)
−0.167755 + 0.985829i \(0.553652\pi\)
\(678\) 5682.00i 0.321852i
\(679\) 485.000 0.0274118
\(680\) 432.000 0.0243624
\(681\) − 7353.00i − 0.413756i
\(682\) 2704.00i 0.151820i
\(683\) − 16747.0i − 0.938223i −0.883139 0.469111i \(-0.844574\pi\)
0.883139 0.469111i \(-0.155426\pi\)
\(684\) 5400.00i 0.301863i
\(685\) −4818.00 −0.268739
\(686\) 6610.00 0.367888
\(687\) − 5634.00i − 0.312883i
\(688\) −3184.00 −0.176437
\(689\) 0 0
\(690\) −2244.00 −0.123808
\(691\) 10309.0i 0.567544i 0.958892 + 0.283772i \(0.0915859\pi\)
−0.958892 + 0.283772i \(0.908414\pi\)
\(692\) −15556.0 −0.854553
\(693\) −1170.00 −0.0641337
\(694\) − 750.000i − 0.0410225i
\(695\) 5654.00i 0.308588i
\(696\) 312.000i 0.0169919i
\(697\) − 5265.00i − 0.286121i
\(698\) −19454.0 −1.05494
\(699\) 4890.00 0.264602
\(700\) − 2420.00i − 0.130668i
\(701\) 24294.0 1.30895 0.654473 0.756085i \(-0.272890\pi\)
0.654473 + 0.756085i \(0.272890\pi\)
\(702\) 0 0
\(703\) 31725.0 1.70204
\(704\) 832.000i 0.0445414i
\(705\) −2328.00 −0.124365
\(706\) 4526.00 0.241272
\(707\) 4045.00i 0.215174i
\(708\) − 5892.00i − 0.312761i
\(709\) − 12659.0i − 0.670548i −0.942121 0.335274i \(-0.891171\pi\)
0.942121 0.335274i \(-0.108829\pi\)
\(710\) 316.000i 0.0167032i
\(711\) −13752.0 −0.725373
\(712\) −8328.00 −0.438350
\(713\) − 19448.0i − 1.02151i
\(714\) 810.000 0.0424559
\(715\) 0 0
\(716\) 8916.00 0.465372
\(717\) 16632.0i 0.866295i
\(718\) 8976.00 0.466548
\(719\) −13091.0 −0.679015 −0.339508 0.940603i \(-0.610260\pi\)
−0.339508 + 0.940603i \(0.610260\pi\)
\(720\) − 576.000i − 0.0298142i
\(721\) − 6440.00i − 0.332647i
\(722\) − 2468.00i − 0.127215i
\(723\) 16569.0i 0.852293i
\(724\) −4152.00 −0.213132
\(725\) −1573.00 −0.0805790
\(726\) 6972.00i 0.356412i
\(727\) −10792.0 −0.550555 −0.275277 0.961365i \(-0.588770\pi\)
−0.275277 + 0.961365i \(0.588770\pi\)
\(728\) 0 0
\(729\) 9477.00 0.481481
\(730\) − 920.000i − 0.0466448i
\(731\) 5373.00 0.271857
\(732\) 2100.00 0.106036
\(733\) − 2698.00i − 0.135952i −0.997687 0.0679761i \(-0.978346\pi\)
0.997687 0.0679761i \(-0.0216542\pi\)
\(734\) 3254.00i 0.163634i
\(735\) − 1908.00i − 0.0957519i
\(736\) − 5984.00i − 0.299692i
\(737\) 10621.0 0.530841
\(738\) −7020.00 −0.350149
\(739\) − 2841.00i − 0.141418i −0.997497 0.0707090i \(-0.977474\pi\)
0.997497 0.0707090i \(-0.0225262\pi\)
\(740\) −3384.00 −0.168106
\(741\) 0 0
\(742\) 6180.00 0.305761
\(743\) − 9191.00i − 0.453816i −0.973916 0.226908i \(-0.927138\pi\)
0.973916 0.226908i \(-0.0728616\pi\)
\(744\) −2496.00 −0.122994
\(745\) −1710.00 −0.0840934
\(746\) − 5974.00i − 0.293195i
\(747\) 13176.0i 0.645361i
\(748\) − 1404.00i − 0.0686301i
\(749\) 6385.00i 0.311486i
\(750\) −2952.00 −0.143722
\(751\) 1659.00 0.0806095 0.0403048 0.999187i \(-0.487167\pi\)
0.0403048 + 0.999187i \(0.487167\pi\)
\(752\) − 6208.00i − 0.301041i
\(753\) 6525.00 0.315782
\(754\) 0 0
\(755\) 4128.00 0.198985
\(756\) − 2700.00i − 0.129892i
\(757\) −13929.0 −0.668769 −0.334384 0.942437i \(-0.608528\pi\)
−0.334384 + 0.942437i \(0.608528\pi\)
\(758\) −17734.0 −0.849773
\(759\) 7293.00i 0.348774i
\(760\) − 1200.00i − 0.0572744i
\(761\) − 4587.00i − 0.218500i −0.994014 0.109250i \(-0.965155\pi\)
0.994014 0.109250i \(-0.0348449\pi\)
\(762\) − 7062.00i − 0.335734i
\(763\) −4130.00 −0.195958
\(764\) 8564.00 0.405543
\(765\) 972.000i 0.0459382i
\(766\) 22806.0 1.07574
\(767\) 0 0
\(768\) −768.000 −0.0360844
\(769\) 14499.0i 0.679905i 0.940443 + 0.339953i \(0.110411\pi\)
−0.940443 + 0.339953i \(0.889589\pi\)
\(770\) 260.000 0.0121685
\(771\) −17055.0 −0.796655
\(772\) 10508.0i 0.489885i
\(773\) 3059.00i 0.142335i 0.997464 + 0.0711673i \(0.0226724\pi\)
−0.997464 + 0.0711673i \(0.977328\pi\)
\(774\) − 7164.00i − 0.332693i
\(775\) − 12584.0i − 0.583265i
\(776\) 776.000 0.0358979
\(777\) −6345.00 −0.292954
\(778\) 5244.00i 0.241654i
\(779\) −14625.0 −0.672651
\(780\) 0 0
\(781\) 1027.00 0.0470537
\(782\) 10098.0i 0.461769i
\(783\) −1755.00 −0.0801004
\(784\) 5088.00 0.231778
\(785\) − 3788.00i − 0.172229i
\(786\) − 8520.00i − 0.386639i
\(787\) − 36407.0i − 1.64901i −0.565856 0.824504i \(-0.691454\pi\)
0.565856 0.824504i \(-0.308546\pi\)
\(788\) − 4812.00i − 0.217539i
\(789\) −18351.0 −0.828026
\(790\) 3056.00 0.137630
\(791\) 4735.00i 0.212841i
\(792\) −1872.00 −0.0839882
\(793\) 0 0
\(794\) −1318.00 −0.0589094
\(795\) − 3708.00i − 0.165420i
\(796\) 2972.00 0.132336
\(797\) 13137.0 0.583860 0.291930 0.956440i \(-0.405703\pi\)
0.291930 + 0.956440i \(0.405703\pi\)
\(798\) − 2250.00i − 0.0998109i
\(799\) 10476.0i 0.463848i
\(800\) − 3872.00i − 0.171120i
\(801\) − 18738.0i − 0.826560i
\(802\) 29370.0 1.29313
\(803\) −2990.00 −0.131401
\(804\) 9804.00i 0.430050i
\(805\) −1870.00 −0.0818743
\(806\) 0 0
\(807\) 15327.0 0.668570
\(808\) 6472.00i 0.281787i
\(809\) 15411.0 0.669743 0.334871 0.942264i \(-0.391307\pi\)
0.334871 + 0.942264i \(0.391307\pi\)
\(810\) 324.000 0.0140546
\(811\) − 27664.0i − 1.19780i −0.800824 0.598899i \(-0.795605\pi\)
0.800824 0.598899i \(-0.204395\pi\)
\(812\) 260.000i 0.0112367i
\(813\) 22647.0i 0.976956i
\(814\) 10998.0i 0.473562i
\(815\) −1970.00 −0.0846700
\(816\) 1296.00 0.0555994
\(817\) − 14925.0i − 0.639118i
\(818\) −15658.0 −0.669278
\(819\) 0 0
\(820\) 1560.00 0.0664361
\(821\) − 21397.0i − 0.909574i −0.890600 0.454787i \(-0.849715\pi\)
0.890600 0.454787i \(-0.150285\pi\)
\(822\) −14454.0 −0.613310
\(823\) 24249.0 1.02706 0.513528 0.858073i \(-0.328338\pi\)
0.513528 + 0.858073i \(0.328338\pi\)
\(824\) − 10304.0i − 0.435627i
\(825\) 4719.00i 0.199145i
\(826\) − 4910.00i − 0.206829i
\(827\) − 14028.0i − 0.589844i −0.955521 0.294922i \(-0.904706\pi\)
0.955521 0.294922i \(-0.0952937\pi\)
\(828\) 13464.0 0.565104
\(829\) −30451.0 −1.27576 −0.637881 0.770135i \(-0.720189\pi\)
−0.637881 + 0.770135i \(0.720189\pi\)
\(830\) − 2928.00i − 0.122449i
\(831\) −2943.00 −0.122854
\(832\) 0 0
\(833\) −8586.00 −0.357128
\(834\) 16962.0i 0.704252i
\(835\) −4710.00 −0.195205
\(836\) −3900.00 −0.161345
\(837\) − 14040.0i − 0.579801i
\(838\) 5838.00i 0.240657i
\(839\) − 20591.0i − 0.847295i −0.905827 0.423647i \(-0.860750\pi\)
0.905827 0.423647i \(-0.139250\pi\)
\(840\) 240.000i 0.00985808i
\(841\) −24220.0 −0.993071
\(842\) −6220.00 −0.254579
\(843\) − 8286.00i − 0.338535i
\(844\) 1420.00 0.0579128
\(845\) 0 0
\(846\) 13968.0 0.567647
\(847\) 5810.00i 0.235695i
\(848\) 9888.00 0.400419
\(849\) 11775.0 0.475992
\(850\) 6534.00i 0.263664i
\(851\) − 79101.0i − 3.18631i
\(852\) 948.000i 0.0381197i
\(853\) 5798.00i 0.232731i 0.993206 + 0.116366i \(0.0371244\pi\)
−0.993206 + 0.116366i \(0.962876\pi\)
\(854\) 1750.00 0.0701215
\(855\) 2700.00 0.107998
\(856\) 10216.0i 0.407916i
\(857\) −5686.00 −0.226640 −0.113320 0.993559i \(-0.536148\pi\)
−0.113320 + 0.993559i \(0.536148\pi\)
\(858\) 0 0
\(859\) −46708.0 −1.85525 −0.927623 0.373518i \(-0.878152\pi\)
−0.927623 + 0.373518i \(0.878152\pi\)
\(860\) 1592.00i 0.0631241i
\(861\) 2925.00 0.115777
\(862\) −18270.0 −0.721901
\(863\) 25168.0i 0.992733i 0.868113 + 0.496367i \(0.165333\pi\)
−0.868113 + 0.496367i \(0.834667\pi\)
\(864\) − 4320.00i − 0.170103i
\(865\) 7778.00i 0.305734i
\(866\) − 23338.0i − 0.915771i
\(867\) 12552.0 0.491682
\(868\) −2080.00 −0.0813362
\(869\) − 9932.00i − 0.387710i
\(870\) 156.000 0.00607919
\(871\) 0 0
\(872\) −6608.00 −0.256623
\(873\) 1746.00i 0.0676897i
\(874\) 28050.0 1.08559
\(875\) −2460.00 −0.0950436
\(876\) − 2760.00i − 0.106452i
\(877\) 18663.0i 0.718591i 0.933224 + 0.359296i \(0.116983\pi\)
−0.933224 + 0.359296i \(0.883017\pi\)
\(878\) 27058.0i 1.04005i
\(879\) 23133.0i 0.887664i
\(880\) 416.000 0.0159356
\(881\) −4971.00 −0.190099 −0.0950495 0.995473i \(-0.530301\pi\)
−0.0950495 + 0.995473i \(0.530301\pi\)
\(882\) 11448.0i 0.437046i
\(883\) −6892.00 −0.262666 −0.131333 0.991338i \(-0.541926\pi\)
−0.131333 + 0.991338i \(0.541926\pi\)
\(884\) 0 0
\(885\) −2946.00 −0.111897
\(886\) 3864.00i 0.146516i
\(887\) −24047.0 −0.910281 −0.455140 0.890420i \(-0.650411\pi\)
−0.455140 + 0.890420i \(0.650411\pi\)
\(888\) −10152.0 −0.383647
\(889\) − 5885.00i − 0.222021i
\(890\) 4164.00i 0.156829i
\(891\) − 1053.00i − 0.0395924i
\(892\) 9132.00i 0.342782i
\(893\) 29100.0 1.09048
\(894\) −5130.00 −0.191916
\(895\) − 4458.00i − 0.166497i
\(896\) −640.000 −0.0238626
\(897\) 0 0
\(898\) 10714.0 0.398141
\(899\) 1352.00i 0.0501576i
\(900\) 8712.00 0.322667
\(901\) −16686.0 −0.616971
\(902\) − 5070.00i − 0.187154i
\(903\) 2985.00i 0.110005i
\(904\) 7576.00i 0.278732i
\(905\) 2076.00i 0.0762526i
\(906\) 12384.0 0.454118
\(907\) 12843.0 0.470171 0.235085 0.971975i \(-0.424463\pi\)
0.235085 + 0.971975i \(0.424463\pi\)
\(908\) − 9804.00i − 0.358323i
\(909\) −14562.0 −0.531343
\(910\) 0 0
\(911\) −144.000 −0.00523703 −0.00261851 0.999997i \(-0.500833\pi\)
−0.00261851 + 0.999997i \(0.500833\pi\)
\(912\) − 3600.00i − 0.130710i
\(913\) −9516.00 −0.344944
\(914\) 38798.0 1.40407
\(915\) − 1050.00i − 0.0379365i
\(916\) − 7512.00i − 0.270964i
\(917\) − 7100.00i − 0.255684i
\(918\) 7290.00i 0.262098i
\(919\) −11061.0 −0.397028 −0.198514 0.980098i \(-0.563612\pi\)
−0.198514 + 0.980098i \(0.563612\pi\)
\(920\) −2992.00 −0.107221
\(921\) 31164.0i 1.11497i
\(922\) −31098.0 −1.11080
\(923\) 0 0
\(924\) 780.000 0.0277707
\(925\) − 51183.0i − 1.81934i
\(926\) −8144.00 −0.289016
\(927\) 23184.0 0.821427
\(928\) 416.000i 0.0147154i
\(929\) 26307.0i 0.929069i 0.885555 + 0.464534i \(0.153778\pi\)
−0.885555 + 0.464534i \(0.846222\pi\)
\(930\) 1248.00i 0.0440038i
\(931\) 23850.0i 0.839583i
\(932\) 6520.00 0.229152
\(933\) −21816.0 −0.765513
\(934\) 30448.0i 1.06669i
\(935\) −702.000 −0.0245539
\(936\) 0 0
\(937\) −46074.0 −1.60637 −0.803187 0.595727i \(-0.796864\pi\)
−0.803187 + 0.595727i \(0.796864\pi\)
\(938\) 8170.00i 0.284392i
\(939\) −23730.0 −0.824706
\(940\) −3104.00 −0.107704
\(941\) 36118.0i 1.25124i 0.780130 + 0.625618i \(0.215153\pi\)
−0.780130 + 0.625618i \(0.784847\pi\)
\(942\) − 11364.0i − 0.393056i
\(943\) 36465.0i 1.25924i
\(944\) − 7856.00i − 0.270859i
\(945\) −1350.00 −0.0464714
\(946\) 5174.00 0.177824
\(947\) 55515.0i 1.90496i 0.304604 + 0.952479i \(0.401476\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(948\) 9168.00 0.314096
\(949\) 0 0
\(950\) 18150.0 0.619857
\(951\) − 22194.0i − 0.756772i
\(952\) 1080.00 0.0367679
\(953\) 5353.00 0.181952 0.0909762 0.995853i \(-0.471001\pi\)
0.0909762 + 0.995853i \(0.471001\pi\)
\(954\) 22248.0i 0.755037i
\(955\) − 4282.00i − 0.145091i
\(956\) 22176.0i 0.750233i
\(957\) − 507.000i − 0.0171254i
\(958\) 20670.0 0.697095
\(959\) −12045.0 −0.405582
\(960\) 384.000i 0.0129099i
\(961\) 18975.0 0.636937
\(962\) 0 0
\(963\) −22986.0 −0.769173
\(964\) 22092.0i 0.738107i
\(965\) 5254.00 0.175267
\(966\) −5610.00 −0.186852
\(967\) − 5488.00i − 0.182505i −0.995828 0.0912524i \(-0.970913\pi\)
0.995828 0.0912524i \(-0.0290870\pi\)
\(968\) 9296.00i 0.308662i
\(969\) 6075.00i 0.201401i
\(970\) − 388.000i − 0.0128432i
\(971\) −37353.0 −1.23452 −0.617258 0.786761i \(-0.711756\pi\)
−0.617258 + 0.786761i \(0.711756\pi\)
\(972\) 15552.0 0.513200
\(973\) 14135.0i 0.465722i
\(974\) 12910.0 0.424705
\(975\) 0 0
\(976\) 2800.00 0.0918297
\(977\) − 12729.0i − 0.416824i −0.978041 0.208412i \(-0.933171\pi\)
0.978041 0.208412i \(-0.0668294\pi\)
\(978\) −5910.00 −0.193232
\(979\) 13533.0 0.441794
\(980\) − 2544.00i − 0.0829236i
\(981\) − 14868.0i − 0.483893i
\(982\) 15554.0i 0.505446i
\(983\) 56128.0i 1.82116i 0.413327 + 0.910582i \(0.364367\pi\)
−0.413327 + 0.910582i \(0.635633\pi\)
\(984\) 4680.00 0.151619
\(985\) −2406.00 −0.0778290
\(986\) − 702.000i − 0.0226737i
\(987\) −5820.00 −0.187693
\(988\) 0 0
\(989\) −37213.0 −1.19647
\(990\) 936.000i 0.0300485i
\(991\) 47001.0 1.50660 0.753298 0.657680i \(-0.228462\pi\)
0.753298 + 0.657680i \(0.228462\pi\)
\(992\) −3328.00 −0.106516
\(993\) 7131.00i 0.227891i
\(994\) 790.000i 0.0252085i
\(995\) − 1486.00i − 0.0473461i
\(996\) − 8784.00i − 0.279449i
\(997\) −24433.0 −0.776129 −0.388065 0.921632i \(-0.626856\pi\)
−0.388065 + 0.921632i \(0.626856\pi\)
\(998\) −6088.00 −0.193098
\(999\) − 57105.0i − 1.80853i
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 338.4.b.a.337.1 2
13.2 odd 12 26.4.c.a.9.1 yes 2
13.3 even 3 338.4.e.d.147.2 4
13.4 even 6 338.4.e.d.23.2 4
13.5 odd 4 338.4.a.a.1.1 1
13.6 odd 12 26.4.c.a.3.1 2
13.7 odd 12 338.4.c.d.315.1 2
13.8 odd 4 338.4.a.d.1.1 1
13.9 even 3 338.4.e.d.23.1 4
13.10 even 6 338.4.e.d.147.1 4
13.11 odd 12 338.4.c.d.191.1 2
13.12 even 2 inner 338.4.b.a.337.2 2
39.2 even 12 234.4.h.b.217.1 2
39.32 even 12 234.4.h.b.55.1 2
52.15 even 12 208.4.i.a.113.1 2
52.19 even 12 208.4.i.a.81.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.4.c.a.3.1 2 13.6 odd 12
26.4.c.a.9.1 yes 2 13.2 odd 12
208.4.i.a.81.1 2 52.19 even 12
208.4.i.a.113.1 2 52.15 even 12
234.4.h.b.55.1 2 39.32 even 12
234.4.h.b.217.1 2 39.2 even 12
338.4.a.a.1.1 1 13.5 odd 4
338.4.a.d.1.1 1 13.8 odd 4
338.4.b.a.337.1 2 1.1 even 1 trivial
338.4.b.a.337.2 2 13.12 even 2 inner
338.4.c.d.191.1 2 13.11 odd 12
338.4.c.d.315.1 2 13.7 odd 12
338.4.e.d.23.1 4 13.9 even 3
338.4.e.d.23.2 4 13.4 even 6
338.4.e.d.147.1 4 13.10 even 6
338.4.e.d.147.2 4 13.3 even 3