Properties

Label 363.4.a.l.1.2
Level $363$
Weight $4$
Character 363.1
Self dual yes
Analytic conductor $21.418$
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] = [363,4,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, 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("363.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.19615 q^{2} -3.00000 q^{3} +19.0000 q^{4} +9.00000 q^{5} -15.5885 q^{6} +24.2487 q^{7} +57.1577 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+5.19615 q^{2} -3.00000 q^{3} +19.0000 q^{4} +9.00000 q^{5} -15.5885 q^{6} +24.2487 q^{7} +57.1577 q^{8} +9.00000 q^{9} +46.7654 q^{10} -57.0000 q^{12} -71.0141 q^{13} +126.000 q^{14} -27.0000 q^{15} +145.000 q^{16} -88.3346 q^{17} +46.7654 q^{18} +145.492 q^{19} +171.000 q^{20} -72.7461 q^{21} +90.0000 q^{23} -171.473 q^{24} -44.0000 q^{25} -369.000 q^{26} -27.0000 q^{27} +460.726 q^{28} +88.3346 q^{29} -140.296 q^{30} -188.000 q^{31} +296.181 q^{32} -459.000 q^{34} +218.238 q^{35} +171.000 q^{36} +133.000 q^{37} +756.000 q^{38} +213.042 q^{39} +514.419 q^{40} -36.3731 q^{41} -378.000 q^{42} +72.7461 q^{43} +81.0000 q^{45} +467.654 q^{46} +72.0000 q^{47} -435.000 q^{48} +245.000 q^{49} -228.631 q^{50} +265.004 q^{51} -1349.27 q^{52} -45.0000 q^{53} -140.296 q^{54} +1386.00 q^{56} -436.477 q^{57} +459.000 q^{58} +378.000 q^{59} -513.000 q^{60} -623.538 q^{61} -976.877 q^{62} +218.238 q^{63} +379.000 q^{64} -639.127 q^{65} -386.000 q^{67} -1678.36 q^{68} -270.000 q^{69} +1134.00 q^{70} -198.000 q^{71} +514.419 q^{72} -76.2102 q^{73} +691.088 q^{74} +132.000 q^{75} +2764.35 q^{76} +1107.00 q^{78} +152.420 q^{79} +1305.00 q^{80} +81.0000 q^{81} -189.000 q^{82} -1247.08 q^{83} -1382.18 q^{84} -795.011 q^{85} +378.000 q^{86} -265.004 q^{87} +45.0000 q^{89} +420.888 q^{90} -1722.00 q^{91} +1710.00 q^{92} +564.000 q^{93} +374.123 q^{94} +1309.43 q^{95} -888.542 q^{96} +89.0000 q^{97} +1273.06 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 38 q^{4} + 18 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 38 q^{4} + 18 q^{5} + 18 q^{9} - 114 q^{12} + 252 q^{14} - 54 q^{15} + 290 q^{16} + 342 q^{20} + 180 q^{23} - 88 q^{25} - 738 q^{26} - 54 q^{27} - 376 q^{31} - 918 q^{34} + 342 q^{36} + 266 q^{37} + 1512 q^{38} - 756 q^{42} + 162 q^{45} + 144 q^{47} - 870 q^{48} + 490 q^{49} - 90 q^{53} + 2772 q^{56} + 918 q^{58} + 756 q^{59} - 1026 q^{60} + 758 q^{64} - 772 q^{67} - 540 q^{69} + 2268 q^{70} - 396 q^{71} + 264 q^{75} + 2214 q^{78} + 2610 q^{80} + 162 q^{81} - 378 q^{82} + 756 q^{86} + 90 q^{89} - 3444 q^{91} + 3420 q^{92} + 1128 q^{93} + 178 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.19615 1.83712 0.918559 0.395285i \(-0.129354\pi\)
0.918559 + 0.395285i \(0.129354\pi\)
\(3\) −3.00000 −0.577350
\(4\) 19.0000 2.37500
\(5\) 9.00000 0.804984 0.402492 0.915423i \(-0.368144\pi\)
0.402492 + 0.915423i \(0.368144\pi\)
\(6\) −15.5885 −1.06066
\(7\) 24.2487 1.30931 0.654654 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 57.1577 2.52604
\(9\) 9.00000 0.333333
\(10\) 46.7654 1.47885
\(11\) 0 0
\(12\) −57.0000 −1.37121
\(13\) −71.0141 −1.51506 −0.757529 0.652801i \(-0.773594\pi\)
−0.757529 + 0.652801i \(0.773594\pi\)
\(14\) 126.000 2.40535
\(15\) −27.0000 −0.464758
\(16\) 145.000 2.26562
\(17\) −88.3346 −1.26025 −0.630126 0.776493i \(-0.716997\pi\)
−0.630126 + 0.776493i \(0.716997\pi\)
\(18\) 46.7654 0.612372
\(19\) 145.492 1.75675 0.878374 0.477974i \(-0.158629\pi\)
0.878374 + 0.477974i \(0.158629\pi\)
\(20\) 171.000 1.91184
\(21\) −72.7461 −0.755929
\(22\) 0 0
\(23\) 90.0000 0.815926 0.407963 0.912998i \(-0.366239\pi\)
0.407963 + 0.912998i \(0.366239\pi\)
\(24\) −171.473 −1.45841
\(25\) −44.0000 −0.352000
\(26\) −369.000 −2.78334
\(27\) −27.0000 −0.192450
\(28\) 460.726 3.10960
\(29\) 88.3346 0.565632 0.282816 0.959174i \(-0.408731\pi\)
0.282816 + 0.959174i \(0.408731\pi\)
\(30\) −140.296 −0.853815
\(31\) −188.000 −1.08922 −0.544610 0.838690i \(-0.683322\pi\)
−0.544610 + 0.838690i \(0.683322\pi\)
\(32\) 296.181 1.63618
\(33\) 0 0
\(34\) −459.000 −2.31523
\(35\) 218.238 1.05397
\(36\) 171.000 0.791667
\(37\) 133.000 0.590948 0.295474 0.955351i \(-0.404522\pi\)
0.295474 + 0.955351i \(0.404522\pi\)
\(38\) 756.000 3.22735
\(39\) 213.042 0.874720
\(40\) 514.419 2.03342
\(41\) −36.3731 −0.138549 −0.0692746 0.997598i \(-0.522068\pi\)
−0.0692746 + 0.997598i \(0.522068\pi\)
\(42\) −378.000 −1.38873
\(43\) 72.7461 0.257993 0.128996 0.991645i \(-0.458824\pi\)
0.128996 + 0.991645i \(0.458824\pi\)
\(44\) 0 0
\(45\) 81.0000 0.268328
\(46\) 467.654 1.49895
\(47\) 72.0000 0.223453 0.111726 0.993739i \(-0.464362\pi\)
0.111726 + 0.993739i \(0.464362\pi\)
\(48\) −435.000 −1.30806
\(49\) 245.000 0.714286
\(50\) −228.631 −0.646665
\(51\) 265.004 0.727607
\(52\) −1349.27 −3.59826
\(53\) −45.0000 −0.116627 −0.0583134 0.998298i \(-0.518572\pi\)
−0.0583134 + 0.998298i \(0.518572\pi\)
\(54\) −140.296 −0.353553
\(55\) 0 0
\(56\) 1386.00 3.30736
\(57\) −436.477 −1.01426
\(58\) 459.000 1.03913
\(59\) 378.000 0.834092 0.417046 0.908885i \(-0.363065\pi\)
0.417046 + 0.908885i \(0.363065\pi\)
\(60\) −513.000 −1.10380
\(61\) −623.538 −1.30879 −0.654393 0.756155i \(-0.727076\pi\)
−0.654393 + 0.756155i \(0.727076\pi\)
\(62\) −976.877 −2.00102
\(63\) 218.238 0.436436
\(64\) 379.000 0.740234
\(65\) −639.127 −1.21960
\(66\) 0 0
\(67\) −386.000 −0.703842 −0.351921 0.936030i \(-0.614471\pi\)
−0.351921 + 0.936030i \(0.614471\pi\)
\(68\) −1678.36 −2.99310
\(69\) −270.000 −0.471075
\(70\) 1134.00 1.93627
\(71\) −198.000 −0.330962 −0.165481 0.986213i \(-0.552918\pi\)
−0.165481 + 0.986213i \(0.552918\pi\)
\(72\) 514.419 0.842012
\(73\) −76.2102 −0.122188 −0.0610941 0.998132i \(-0.519459\pi\)
−0.0610941 + 0.998132i \(0.519459\pi\)
\(74\) 691.088 1.08564
\(75\) 132.000 0.203227
\(76\) 2764.35 4.17228
\(77\) 0 0
\(78\) 1107.00 1.60696
\(79\) 152.420 0.217071 0.108536 0.994093i \(-0.465384\pi\)
0.108536 + 0.994093i \(0.465384\pi\)
\(80\) 1305.00 1.82379
\(81\) 81.0000 0.111111
\(82\) −189.000 −0.254531
\(83\) −1247.08 −1.64921 −0.824605 0.565709i \(-0.808603\pi\)
−0.824605 + 0.565709i \(0.808603\pi\)
\(84\) −1382.18 −1.79533
\(85\) −795.011 −1.01448
\(86\) 378.000 0.473963
\(87\) −265.004 −0.326568
\(88\) 0 0
\(89\) 45.0000 0.0535954 0.0267977 0.999641i \(-0.491469\pi\)
0.0267977 + 0.999641i \(0.491469\pi\)
\(90\) 420.888 0.492950
\(91\) −1722.00 −1.98368
\(92\) 1710.00 1.93782
\(93\) 564.000 0.628861
\(94\) 374.123 0.410509
\(95\) 1309.43 1.41416
\(96\) −888.542 −0.944650
\(97\) 89.0000 0.0931606 0.0465803 0.998915i \(-0.485168\pi\)
0.0465803 + 0.998915i \(0.485168\pi\)
\(98\) 1273.06 1.31223
\(99\) 0 0
\(100\) −836.000 −0.836000
\(101\) −1288.65 −1.26955 −0.634777 0.772695i \(-0.718908\pi\)
−0.634777 + 0.772695i \(0.718908\pi\)
\(102\) 1377.00 1.33670
\(103\) 1358.00 1.29910 0.649552 0.760317i \(-0.274956\pi\)
0.649552 + 0.760317i \(0.274956\pi\)
\(104\) −4059.00 −3.82709
\(105\) −654.715 −0.608511
\(106\) −233.827 −0.214257
\(107\) −1299.04 −1.17367 −0.586835 0.809706i \(-0.699626\pi\)
−0.586835 + 0.809706i \(0.699626\pi\)
\(108\) −513.000 −0.457069
\(109\) −1203.78 −1.05781 −0.528903 0.848683i \(-0.677396\pi\)
−0.528903 + 0.848683i \(0.677396\pi\)
\(110\) 0 0
\(111\) −399.000 −0.341184
\(112\) 3516.06 2.96640
\(113\) 711.000 0.591905 0.295952 0.955203i \(-0.404363\pi\)
0.295952 + 0.955203i \(0.404363\pi\)
\(114\) −2268.00 −1.86331
\(115\) 810.000 0.656808
\(116\) 1678.36 1.34338
\(117\) −639.127 −0.505020
\(118\) 1964.15 1.53232
\(119\) −2142.00 −1.65006
\(120\) −1543.26 −1.17400
\(121\) 0 0
\(122\) −3240.00 −2.40439
\(123\) 109.119 0.0799914
\(124\) −3572.00 −2.58690
\(125\) −1521.00 −1.08834
\(126\) 1134.00 0.801784
\(127\) −651.251 −0.455033 −0.227516 0.973774i \(-0.573061\pi\)
−0.227516 + 0.973774i \(0.573061\pi\)
\(128\) −400.104 −0.276285
\(129\) −218.238 −0.148952
\(130\) −3321.00 −2.24055
\(131\) −2608.47 −1.73972 −0.869859 0.493301i \(-0.835790\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(132\) 0 0
\(133\) 3528.00 2.30012
\(134\) −2005.71 −1.29304
\(135\) −243.000 −0.154919
\(136\) −5049.00 −3.18344
\(137\) 2394.00 1.49294 0.746472 0.665417i \(-0.231746\pi\)
0.746472 + 0.665417i \(0.231746\pi\)
\(138\) −1402.96 −0.865420
\(139\) −183.597 −0.112033 −0.0560163 0.998430i \(-0.517840\pi\)
−0.0560163 + 0.998430i \(0.517840\pi\)
\(140\) 4146.53 2.50318
\(141\) −216.000 −0.129011
\(142\) −1028.84 −0.608015
\(143\) 0 0
\(144\) 1305.00 0.755208
\(145\) 795.011 0.455325
\(146\) −396.000 −0.224474
\(147\) −735.000 −0.412393
\(148\) 2527.00 1.40350
\(149\) 3611.33 1.98558 0.992790 0.119869i \(-0.0382473\pi\)
0.992790 + 0.119869i \(0.0382473\pi\)
\(150\) 685.892 0.373352
\(151\) −1357.93 −0.731832 −0.365916 0.930648i \(-0.619244\pi\)
−0.365916 + 0.930648i \(0.619244\pi\)
\(152\) 8316.00 4.43761
\(153\) −795.011 −0.420084
\(154\) 0 0
\(155\) −1692.00 −0.876805
\(156\) 4047.80 2.07746
\(157\) 1306.00 0.663886 0.331943 0.943299i \(-0.392296\pi\)
0.331943 + 0.943299i \(0.392296\pi\)
\(158\) 792.000 0.398786
\(159\) 135.000 0.0673346
\(160\) 2665.63 1.31710
\(161\) 2182.38 1.06830
\(162\) 420.888 0.204124
\(163\) 2546.00 1.22342 0.611712 0.791081i \(-0.290481\pi\)
0.611712 + 0.791081i \(0.290481\pi\)
\(164\) −691.088 −0.329054
\(165\) 0 0
\(166\) −6480.00 −3.02979
\(167\) 675.500 0.313004 0.156502 0.987678i \(-0.449978\pi\)
0.156502 + 0.987678i \(0.449978\pi\)
\(168\) −4158.00 −1.90950
\(169\) 2846.00 1.29540
\(170\) −4131.00 −1.86372
\(171\) 1309.43 0.585583
\(172\) 1382.18 0.612732
\(173\) 3180.05 1.39754 0.698770 0.715347i \(-0.253731\pi\)
0.698770 + 0.715347i \(0.253731\pi\)
\(174\) −1377.00 −0.599943
\(175\) −1066.94 −0.460876
\(176\) 0 0
\(177\) −1134.00 −0.481563
\(178\) 233.827 0.0984610
\(179\) −2556.00 −1.06729 −0.533644 0.845709i \(-0.679178\pi\)
−0.533644 + 0.845709i \(0.679178\pi\)
\(180\) 1539.00 0.637279
\(181\) −775.000 −0.318261 −0.159131 0.987258i \(-0.550869\pi\)
−0.159131 + 0.987258i \(0.550869\pi\)
\(182\) −8947.77 −3.64425
\(183\) 1870.61 0.755627
\(184\) 5144.19 2.06106
\(185\) 1197.00 0.475704
\(186\) 2930.63 1.15529
\(187\) 0 0
\(188\) 1368.00 0.530700
\(189\) −654.715 −0.251976
\(190\) 6804.00 2.59797
\(191\) 2628.00 0.995578 0.497789 0.867298i \(-0.334145\pi\)
0.497789 + 0.867298i \(0.334145\pi\)
\(192\) −1137.00 −0.427375
\(193\) −2790.33 −1.04069 −0.520344 0.853957i \(-0.674196\pi\)
−0.520344 + 0.853957i \(0.674196\pi\)
\(194\) 462.458 0.171147
\(195\) 1917.38 0.704136
\(196\) 4655.00 1.69643
\(197\) 3590.54 1.29856 0.649278 0.760551i \(-0.275071\pi\)
0.649278 + 0.760551i \(0.275071\pi\)
\(198\) 0 0
\(199\) 1618.00 0.576367 0.288183 0.957575i \(-0.406949\pi\)
0.288183 + 0.957575i \(0.406949\pi\)
\(200\) −2514.94 −0.889165
\(201\) 1158.00 0.406363
\(202\) −6696.00 −2.33232
\(203\) 2142.00 0.740586
\(204\) 5035.07 1.72807
\(205\) −327.358 −0.111530
\(206\) 7056.37 2.38661
\(207\) 810.000 0.271975
\(208\) −10297.0 −3.43255
\(209\) 0 0
\(210\) −3402.00 −1.11791
\(211\) −1860.22 −0.606934 −0.303467 0.952842i \(-0.598144\pi\)
−0.303467 + 0.952842i \(0.598144\pi\)
\(212\) −855.000 −0.276989
\(213\) 594.000 0.191081
\(214\) −6750.00 −2.15617
\(215\) 654.715 0.207680
\(216\) −1543.26 −0.486136
\(217\) −4558.76 −1.42612
\(218\) −6255.00 −1.94331
\(219\) 228.631 0.0705453
\(220\) 0 0
\(221\) 6273.00 1.90936
\(222\) −2073.26 −0.626795
\(223\) −964.000 −0.289481 −0.144740 0.989470i \(-0.546235\pi\)
−0.144740 + 0.989470i \(0.546235\pi\)
\(224\) 7182.00 2.14227
\(225\) −396.000 −0.117333
\(226\) 3694.46 1.08740
\(227\) 2857.88 0.835614 0.417807 0.908536i \(-0.362799\pi\)
0.417807 + 0.908536i \(0.362799\pi\)
\(228\) −8293.06 −2.40887
\(229\) −5519.00 −1.59260 −0.796301 0.604901i \(-0.793213\pi\)
−0.796301 + 0.604901i \(0.793213\pi\)
\(230\) 4208.88 1.20663
\(231\) 0 0
\(232\) 5049.00 1.42881
\(233\) 3611.33 1.01539 0.507695 0.861537i \(-0.330498\pi\)
0.507695 + 0.861537i \(0.330498\pi\)
\(234\) −3321.00 −0.927780
\(235\) 648.000 0.179876
\(236\) 7182.00 1.98097
\(237\) −457.261 −0.125326
\(238\) −11130.2 −3.03135
\(239\) −3263.18 −0.883171 −0.441585 0.897219i \(-0.645584\pi\)
−0.441585 + 0.897219i \(0.645584\pi\)
\(240\) −3915.00 −1.05297
\(241\) −3457.17 −0.924050 −0.462025 0.886867i \(-0.652877\pi\)
−0.462025 + 0.886867i \(0.652877\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) −11847.2 −3.10836
\(245\) 2205.00 0.574989
\(246\) 567.000 0.146954
\(247\) −10332.0 −2.66158
\(248\) −10745.6 −2.75141
\(249\) 3741.23 0.952172
\(250\) −7903.35 −1.99941
\(251\) 6714.00 1.68838 0.844191 0.536042i \(-0.180081\pi\)
0.844191 + 0.536042i \(0.180081\pi\)
\(252\) 4146.53 1.03653
\(253\) 0 0
\(254\) −3384.00 −0.835949
\(255\) 2385.03 0.585712
\(256\) −5111.00 −1.24780
\(257\) 1341.00 0.325484 0.162742 0.986669i \(-0.447966\pi\)
0.162742 + 0.986669i \(0.447966\pi\)
\(258\) −1134.00 −0.273642
\(259\) 3225.08 0.773732
\(260\) −12143.4 −2.89655
\(261\) 795.011 0.188544
\(262\) −13554.0 −3.19606
\(263\) −956.092 −0.224164 −0.112082 0.993699i \(-0.535752\pi\)
−0.112082 + 0.993699i \(0.535752\pi\)
\(264\) 0 0
\(265\) −405.000 −0.0938828
\(266\) 18332.0 4.22560
\(267\) −135.000 −0.0309433
\(268\) −7334.00 −1.67162
\(269\) 3087.00 0.699694 0.349847 0.936807i \(-0.386234\pi\)
0.349847 + 0.936807i \(0.386234\pi\)
\(270\) −1262.67 −0.284605
\(271\) −4035.68 −0.904613 −0.452306 0.891863i \(-0.649399\pi\)
−0.452306 + 0.891863i \(0.649399\pi\)
\(272\) −12808.5 −2.85526
\(273\) 5166.00 1.14528
\(274\) 12439.6 2.74271
\(275\) 0 0
\(276\) −5130.00 −1.11880
\(277\) 3791.46 0.822407 0.411203 0.911544i \(-0.365109\pi\)
0.411203 + 0.911544i \(0.365109\pi\)
\(278\) −954.000 −0.205817
\(279\) −1692.00 −0.363073
\(280\) 12474.0 2.66237
\(281\) 2722.78 0.578034 0.289017 0.957324i \(-0.406672\pi\)
0.289017 + 0.957324i \(0.406672\pi\)
\(282\) −1122.37 −0.237007
\(283\) 3238.94 0.680335 0.340167 0.940365i \(-0.389516\pi\)
0.340167 + 0.940365i \(0.389516\pi\)
\(284\) −3762.00 −0.786034
\(285\) −3928.29 −0.816463
\(286\) 0 0
\(287\) −882.000 −0.181404
\(288\) 2665.63 0.545394
\(289\) 2890.00 0.588235
\(290\) 4131.00 0.836485
\(291\) −267.000 −0.0537863
\(292\) −1447.99 −0.290197
\(293\) −4338.79 −0.865101 −0.432551 0.901610i \(-0.642386\pi\)
−0.432551 + 0.901610i \(0.642386\pi\)
\(294\) −3819.17 −0.757614
\(295\) 3402.00 0.671431
\(296\) 7601.97 1.49276
\(297\) 0 0
\(298\) 18765.0 3.64774
\(299\) −6391.27 −1.23618
\(300\) 2508.00 0.482665
\(301\) 1764.00 0.337792
\(302\) −7056.00 −1.34446
\(303\) 3865.94 0.732978
\(304\) 21096.4 3.98013
\(305\) −5611.84 −1.05355
\(306\) −4131.00 −0.771744
\(307\) 5781.59 1.07483 0.537415 0.843318i \(-0.319401\pi\)
0.537415 + 0.843318i \(0.319401\pi\)
\(308\) 0 0
\(309\) −4074.00 −0.750038
\(310\) −8791.89 −1.61079
\(311\) 5220.00 0.951765 0.475883 0.879509i \(-0.342129\pi\)
0.475883 + 0.879509i \(0.342129\pi\)
\(312\) 12177.0 2.20957
\(313\) −3977.00 −0.718190 −0.359095 0.933301i \(-0.616915\pi\)
−0.359095 + 0.933301i \(0.616915\pi\)
\(314\) 6786.18 1.21964
\(315\) 1964.15 0.351324
\(316\) 2895.99 0.515545
\(317\) 9918.00 1.75726 0.878628 0.477506i \(-0.158459\pi\)
0.878628 + 0.477506i \(0.158459\pi\)
\(318\) 701.481 0.123702
\(319\) 0 0
\(320\) 3411.00 0.595877
\(321\) 3897.11 0.677619
\(322\) 11340.0 1.96259
\(323\) −12852.0 −2.21395
\(324\) 1539.00 0.263889
\(325\) 3124.62 0.533301
\(326\) 13229.4 2.24757
\(327\) 3611.33 0.610724
\(328\) −2079.00 −0.349980
\(329\) 1745.91 0.292568
\(330\) 0 0
\(331\) 8756.00 1.45400 0.726999 0.686639i \(-0.240915\pi\)
0.726999 + 0.686639i \(0.240915\pi\)
\(332\) −23694.5 −3.91687
\(333\) 1197.00 0.196983
\(334\) 3510.00 0.575026
\(335\) −3474.00 −0.566582
\(336\) −10548.2 −1.71265
\(337\) −9919.45 −1.60340 −0.801702 0.597724i \(-0.796072\pi\)
−0.801702 + 0.597724i \(0.796072\pi\)
\(338\) 14788.2 2.37981
\(339\) −2133.00 −0.341736
\(340\) −15105.2 −2.40940
\(341\) 0 0
\(342\) 6804.00 1.07578
\(343\) −2376.37 −0.374088
\(344\) 4158.00 0.651699
\(345\) −2430.00 −0.379208
\(346\) 16524.0 2.56744
\(347\) 1330.22 0.205792 0.102896 0.994692i \(-0.467189\pi\)
0.102896 + 0.994692i \(0.467189\pi\)
\(348\) −5035.07 −0.775598
\(349\) −2842.30 −0.435944 −0.217972 0.975955i \(-0.569944\pi\)
−0.217972 + 0.975955i \(0.569944\pi\)
\(350\) −5544.00 −0.846684
\(351\) 1917.38 0.291573
\(352\) 0 0
\(353\) −1431.00 −0.215763 −0.107882 0.994164i \(-0.534407\pi\)
−0.107882 + 0.994164i \(0.534407\pi\)
\(354\) −5892.44 −0.884688
\(355\) −1782.00 −0.266419
\(356\) 855.000 0.127289
\(357\) 6426.00 0.952661
\(358\) −13281.4 −1.96073
\(359\) 5393.61 0.792935 0.396467 0.918049i \(-0.370236\pi\)
0.396467 + 0.918049i \(0.370236\pi\)
\(360\) 4629.77 0.677807
\(361\) 14309.0 2.08616
\(362\) −4027.02 −0.584683
\(363\) 0 0
\(364\) −32718.0 −4.71123
\(365\) −685.892 −0.0983595
\(366\) 9720.00 1.38818
\(367\) 3554.00 0.505497 0.252748 0.967532i \(-0.418666\pi\)
0.252748 + 0.967532i \(0.418666\pi\)
\(368\) 13050.0 1.84858
\(369\) −327.358 −0.0461831
\(370\) 6219.79 0.873924
\(371\) −1091.19 −0.152700
\(372\) 10716.0 1.49354
\(373\) 5577.20 0.774200 0.387100 0.922038i \(-0.373477\pi\)
0.387100 + 0.922038i \(0.373477\pi\)
\(374\) 0 0
\(375\) 4563.00 0.628353
\(376\) 4115.35 0.564450
\(377\) −6273.00 −0.856965
\(378\) −3402.00 −0.462910
\(379\) −13186.0 −1.78712 −0.893561 0.448942i \(-0.851801\pi\)
−0.893561 + 0.448942i \(0.851801\pi\)
\(380\) 24879.2 3.35862
\(381\) 1953.75 0.262713
\(382\) 13655.5 1.82899
\(383\) −3330.00 −0.444269 −0.222135 0.975016i \(-0.571302\pi\)
−0.222135 + 0.975016i \(0.571302\pi\)
\(384\) 1200.31 0.159513
\(385\) 0 0
\(386\) −14499.0 −1.91186
\(387\) 654.715 0.0859975
\(388\) 1691.00 0.221256
\(389\) 3537.00 0.461010 0.230505 0.973071i \(-0.425962\pi\)
0.230505 + 0.973071i \(0.425962\pi\)
\(390\) 9963.00 1.29358
\(391\) −7950.11 −1.02827
\(392\) 14003.6 1.80431
\(393\) 7825.41 1.00443
\(394\) 18657.0 2.38560
\(395\) 1371.78 0.174739
\(396\) 0 0
\(397\) −4501.00 −0.569014 −0.284507 0.958674i \(-0.591830\pi\)
−0.284507 + 0.958674i \(0.591830\pi\)
\(398\) 8407.37 1.05885
\(399\) −10584.0 −1.32798
\(400\) −6380.00 −0.797500
\(401\) 3303.00 0.411332 0.205666 0.978622i \(-0.434064\pi\)
0.205666 + 0.978622i \(0.434064\pi\)
\(402\) 6017.14 0.746537
\(403\) 13350.6 1.65023
\(404\) −24484.3 −3.01519
\(405\) 729.000 0.0894427
\(406\) 11130.2 1.36054
\(407\) 0 0
\(408\) 15147.0 1.83796
\(409\) 9576.51 1.15777 0.578885 0.815409i \(-0.303488\pi\)
0.578885 + 0.815409i \(0.303488\pi\)
\(410\) −1701.00 −0.204894
\(411\) −7182.00 −0.861951
\(412\) 25802.0 3.08537
\(413\) 9166.01 1.09208
\(414\) 4208.88 0.499651
\(415\) −11223.7 −1.32759
\(416\) −21033.0 −2.47891
\(417\) 550.792 0.0646820
\(418\) 0 0
\(419\) −6750.00 −0.787015 −0.393507 0.919322i \(-0.628738\pi\)
−0.393507 + 0.919322i \(0.628738\pi\)
\(420\) −12439.6 −1.44521
\(421\) −13481.0 −1.56063 −0.780313 0.625389i \(-0.784940\pi\)
−0.780313 + 0.625389i \(0.784940\pi\)
\(422\) −9666.00 −1.11501
\(423\) 648.000 0.0744843
\(424\) −2572.10 −0.294604
\(425\) 3886.72 0.443609
\(426\) 3086.51 0.351038
\(427\) −15120.0 −1.71360
\(428\) −24681.7 −2.78747
\(429\) 0 0
\(430\) 3402.00 0.381533
\(431\) 1382.18 0.154471 0.0772356 0.997013i \(-0.475391\pi\)
0.0772356 + 0.997013i \(0.475391\pi\)
\(432\) −3915.00 −0.436020
\(433\) −1531.00 −0.169920 −0.0849598 0.996384i \(-0.527076\pi\)
−0.0849598 + 0.996384i \(0.527076\pi\)
\(434\) −23688.0 −2.61995
\(435\) −2385.03 −0.262882
\(436\) −22871.7 −2.51229
\(437\) 13094.3 1.43338
\(438\) 1188.00 0.129600
\(439\) −1919.11 −0.208643 −0.104321 0.994544i \(-0.533267\pi\)
−0.104321 + 0.994544i \(0.533267\pi\)
\(440\) 0 0
\(441\) 2205.00 0.238095
\(442\) 32595.5 3.50771
\(443\) 14598.0 1.56563 0.782813 0.622258i \(-0.213784\pi\)
0.782813 + 0.622258i \(0.213784\pi\)
\(444\) −7581.00 −0.810312
\(445\) 405.000 0.0431435
\(446\) −5009.09 −0.531810
\(447\) −10834.0 −1.14637
\(448\) 9190.26 0.969194
\(449\) −12591.0 −1.32340 −0.661699 0.749769i \(-0.730164\pi\)
−0.661699 + 0.749769i \(0.730164\pi\)
\(450\) −2057.68 −0.215555
\(451\) 0 0
\(452\) 13509.0 1.40577
\(453\) 4073.78 0.422523
\(454\) 14850.0 1.53512
\(455\) −15498.0 −1.59683
\(456\) −24948.0 −2.56206
\(457\) 16778.4 1.71742 0.858708 0.512465i \(-0.171267\pi\)
0.858708 + 0.512465i \(0.171267\pi\)
\(458\) −28677.6 −2.92580
\(459\) 2385.03 0.242536
\(460\) 15390.0 1.55992
\(461\) 14138.7 1.42843 0.714215 0.699926i \(-0.246784\pi\)
0.714215 + 0.699926i \(0.246784\pi\)
\(462\) 0 0
\(463\) 6902.00 0.692793 0.346396 0.938088i \(-0.387405\pi\)
0.346396 + 0.938088i \(0.387405\pi\)
\(464\) 12808.5 1.28151
\(465\) 5076.00 0.506223
\(466\) 18765.0 1.86539
\(467\) 15894.0 1.57492 0.787459 0.616367i \(-0.211396\pi\)
0.787459 + 0.616367i \(0.211396\pi\)
\(468\) −12143.4 −1.19942
\(469\) −9360.00 −0.921545
\(470\) 3367.11 0.330453
\(471\) −3918.00 −0.383295
\(472\) 21605.6 2.10695
\(473\) 0 0
\(474\) −2376.00 −0.230239
\(475\) −6401.66 −0.618375
\(476\) −40698.0 −3.91889
\(477\) −405.000 −0.0388756
\(478\) −16956.0 −1.62249
\(479\) −9716.81 −0.926873 −0.463436 0.886130i \(-0.653384\pi\)
−0.463436 + 0.886130i \(0.653384\pi\)
\(480\) −7996.88 −0.760429
\(481\) −9444.87 −0.895320
\(482\) −17964.0 −1.69759
\(483\) −6547.15 −0.616782
\(484\) 0 0
\(485\) 801.000 0.0749929
\(486\) −1262.67 −0.117851
\(487\) −7622.00 −0.709211 −0.354606 0.935016i \(-0.615385\pi\)
−0.354606 + 0.935016i \(0.615385\pi\)
\(488\) −35640.0 −3.30604
\(489\) −7638.00 −0.706344
\(490\) 11457.5 1.05632
\(491\) 14902.6 1.36974 0.684871 0.728664i \(-0.259859\pi\)
0.684871 + 0.728664i \(0.259859\pi\)
\(492\) 2073.26 0.189980
\(493\) −7803.00 −0.712839
\(494\) −53686.6 −4.88963
\(495\) 0 0
\(496\) −27260.0 −2.46776
\(497\) −4801.24 −0.433331
\(498\) 19440.0 1.74925
\(499\) 13006.0 1.16679 0.583395 0.812188i \(-0.301724\pi\)
0.583395 + 0.812188i \(0.301724\pi\)
\(500\) −28899.0 −2.58481
\(501\) −2026.50 −0.180713
\(502\) 34887.0 3.10176
\(503\) −14185.5 −1.25746 −0.628728 0.777626i \(-0.716424\pi\)
−0.628728 + 0.777626i \(0.716424\pi\)
\(504\) 12474.0 1.10245
\(505\) −11597.8 −1.02197
\(506\) 0 0
\(507\) −8538.00 −0.747901
\(508\) −12373.8 −1.08070
\(509\) 18234.0 1.58783 0.793917 0.608026i \(-0.208038\pi\)
0.793917 + 0.608026i \(0.208038\pi\)
\(510\) 12393.0 1.07602
\(511\) −1848.00 −0.159982
\(512\) −23356.7 −2.01607
\(513\) −3928.29 −0.338086
\(514\) 6968.04 0.597952
\(515\) 12222.0 1.04576
\(516\) −4146.53 −0.353761
\(517\) 0 0
\(518\) 16758.0 1.42144
\(519\) −9540.14 −0.806870
\(520\) −36531.0 −3.08075
\(521\) 20502.0 1.72401 0.862005 0.506900i \(-0.169209\pi\)
0.862005 + 0.506900i \(0.169209\pi\)
\(522\) 4131.00 0.346377
\(523\) −10600.2 −0.886257 −0.443128 0.896458i \(-0.646131\pi\)
−0.443128 + 0.896458i \(0.646131\pi\)
\(524\) −49560.9 −4.13183
\(525\) 3200.83 0.266087
\(526\) −4968.00 −0.411816
\(527\) 16606.9 1.37269
\(528\) 0 0
\(529\) −4067.00 −0.334265
\(530\) −2104.44 −0.172474
\(531\) 3402.00 0.278031
\(532\) 67032.0 5.46279
\(533\) 2583.00 0.209910
\(534\) −701.481 −0.0568465
\(535\) −11691.3 −0.944787
\(536\) −22062.9 −1.77793
\(537\) 7668.00 0.616199
\(538\) 16040.5 1.28542
\(539\) 0 0
\(540\) −4617.00 −0.367933
\(541\) 6082.96 0.483414 0.241707 0.970349i \(-0.422293\pi\)
0.241707 + 0.970349i \(0.422293\pi\)
\(542\) −20970.0 −1.66188
\(543\) 2325.00 0.183748
\(544\) −26163.0 −2.06200
\(545\) −10834.0 −0.851517
\(546\) 26843.3 2.10401
\(547\) 19378.2 1.51472 0.757360 0.652998i \(-0.226489\pi\)
0.757360 + 0.652998i \(0.226489\pi\)
\(548\) 45486.0 3.54574
\(549\) −5611.84 −0.436262
\(550\) 0 0
\(551\) 12852.0 0.993673
\(552\) −15432.6 −1.18995
\(553\) 3696.00 0.284213
\(554\) 19701.0 1.51086
\(555\) −3591.00 −0.274648
\(556\) −3488.35 −0.266077
\(557\) −14216.7 −1.08147 −0.540736 0.841192i \(-0.681854\pi\)
−0.540736 + 0.841192i \(0.681854\pi\)
\(558\) −8791.89 −0.667008
\(559\) −5166.00 −0.390874
\(560\) 31644.6 2.38791
\(561\) 0 0
\(562\) 14148.0 1.06192
\(563\) −2130.42 −0.159479 −0.0797394 0.996816i \(-0.525409\pi\)
−0.0797394 + 0.996816i \(0.525409\pi\)
\(564\) −4104.00 −0.306400
\(565\) 6399.00 0.476474
\(566\) 16830.0 1.24985
\(567\) 1964.15 0.145479
\(568\) −11317.2 −0.836021
\(569\) 17251.2 1.27102 0.635509 0.772094i \(-0.280790\pi\)
0.635509 + 0.772094i \(0.280790\pi\)
\(570\) −20412.0 −1.49994
\(571\) 9079.41 0.665432 0.332716 0.943027i \(-0.392035\pi\)
0.332716 + 0.943027i \(0.392035\pi\)
\(572\) 0 0
\(573\) −7884.00 −0.574797
\(574\) −4583.01 −0.333260
\(575\) −3960.00 −0.287206
\(576\) 3411.00 0.246745
\(577\) −7427.00 −0.535858 −0.267929 0.963439i \(-0.586339\pi\)
−0.267929 + 0.963439i \(0.586339\pi\)
\(578\) 15016.9 1.08066
\(579\) 8371.00 0.600841
\(580\) 15105.2 1.08140
\(581\) −30240.0 −2.15932
\(582\) −1387.37 −0.0988118
\(583\) 0 0
\(584\) −4356.00 −0.308652
\(585\) −5752.14 −0.406533
\(586\) −22545.0 −1.58929
\(587\) −9972.00 −0.701173 −0.350586 0.936530i \(-0.614018\pi\)
−0.350586 + 0.936530i \(0.614018\pi\)
\(588\) −13965.0 −0.979433
\(589\) −27352.5 −1.91348
\(590\) 17677.3 1.23350
\(591\) −10771.6 −0.749721
\(592\) 19285.0 1.33887
\(593\) −21922.6 −1.51813 −0.759066 0.651014i \(-0.774344\pi\)
−0.759066 + 0.651014i \(0.774344\pi\)
\(594\) 0 0
\(595\) −19278.0 −1.32827
\(596\) 68615.2 4.71575
\(597\) −4854.00 −0.332765
\(598\) −33210.0 −2.27100
\(599\) 576.000 0.0392900 0.0196450 0.999807i \(-0.493746\pi\)
0.0196450 + 0.999807i \(0.493746\pi\)
\(600\) 7544.81 0.513360
\(601\) −7740.54 −0.525363 −0.262681 0.964883i \(-0.584607\pi\)
−0.262681 + 0.964883i \(0.584607\pi\)
\(602\) 9166.01 0.620563
\(603\) −3474.00 −0.234614
\(604\) −25800.6 −1.73810
\(605\) 0 0
\(606\) 20088.0 1.34657
\(607\) −14500.7 −0.969632 −0.484816 0.874616i \(-0.661113\pi\)
−0.484816 + 0.874616i \(0.661113\pi\)
\(608\) 43092.0 2.87436
\(609\) −6426.00 −0.427577
\(610\) −29160.0 −1.93550
\(611\) −5113.01 −0.338544
\(612\) −15105.2 −0.997700
\(613\) 5298.34 0.349100 0.174550 0.984648i \(-0.444153\pi\)
0.174550 + 0.984648i \(0.444153\pi\)
\(614\) 30042.0 1.97459
\(615\) 982.073 0.0643919
\(616\) 0 0
\(617\) 6939.00 0.452761 0.226381 0.974039i \(-0.427311\pi\)
0.226381 + 0.974039i \(0.427311\pi\)
\(618\) −21169.1 −1.37791
\(619\) 1286.00 0.0835036 0.0417518 0.999128i \(-0.486706\pi\)
0.0417518 + 0.999128i \(0.486706\pi\)
\(620\) −32148.0 −2.08241
\(621\) −2430.00 −0.157025
\(622\) 27123.9 1.74850
\(623\) 1091.19 0.0701728
\(624\) 30891.1 1.98179
\(625\) −8189.00 −0.524096
\(626\) −20665.1 −1.31940
\(627\) 0 0
\(628\) 24814.0 1.57673
\(629\) −11748.5 −0.744743
\(630\) 10206.0 0.645423
\(631\) −15110.0 −0.953280 −0.476640 0.879099i \(-0.658145\pi\)
−0.476640 + 0.879099i \(0.658145\pi\)
\(632\) 8712.00 0.548330
\(633\) 5580.67 0.350413
\(634\) 51535.4 3.22829
\(635\) −5861.26 −0.366294
\(636\) 2565.00 0.159920
\(637\) −17398.5 −1.08218
\(638\) 0 0
\(639\) −1782.00 −0.110321
\(640\) −3600.93 −0.222405
\(641\) 13293.0 0.819098 0.409549 0.912288i \(-0.365686\pi\)
0.409549 + 0.912288i \(0.365686\pi\)
\(642\) 20250.0 1.24487
\(643\) 20528.0 1.25901 0.629506 0.776995i \(-0.283257\pi\)
0.629506 + 0.776995i \(0.283257\pi\)
\(644\) 41465.3 2.53721
\(645\) −1964.15 −0.119904
\(646\) −66781.0 −4.06728
\(647\) −11124.0 −0.675934 −0.337967 0.941158i \(-0.609739\pi\)
−0.337967 + 0.941158i \(0.609739\pi\)
\(648\) 4629.77 0.280671
\(649\) 0 0
\(650\) 16236.0 0.979736
\(651\) 13676.3 0.823372
\(652\) 48374.0 2.90563
\(653\) 5562.00 0.333320 0.166660 0.986014i \(-0.446702\pi\)
0.166660 + 0.986014i \(0.446702\pi\)
\(654\) 18765.0 1.12197
\(655\) −23476.2 −1.40045
\(656\) −5274.09 −0.313901
\(657\) −685.892 −0.0407294
\(658\) 9072.00 0.537482
\(659\) 16378.3 0.968144 0.484072 0.875028i \(-0.339157\pi\)
0.484072 + 0.875028i \(0.339157\pi\)
\(660\) 0 0
\(661\) −28385.0 −1.67027 −0.835135 0.550045i \(-0.814611\pi\)
−0.835135 + 0.550045i \(0.814611\pi\)
\(662\) 45497.5 2.67116
\(663\) −18819.0 −1.10237
\(664\) −71280.0 −4.16596
\(665\) 31752.0 1.85156
\(666\) 6219.79 0.361880
\(667\) 7950.11 0.461514
\(668\) 12834.5 0.743386
\(669\) 2892.00 0.167132
\(670\) −18051.4 −1.04088
\(671\) 0 0
\(672\) −21546.0 −1.23684
\(673\) −6948.99 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(674\) −51543.0 −2.94564
\(675\) 1188.00 0.0677424
\(676\) 54074.0 3.07658
\(677\) −3923.10 −0.222713 −0.111357 0.993781i \(-0.535520\pi\)
−0.111357 + 0.993781i \(0.535520\pi\)
\(678\) −11083.4 −0.627810
\(679\) 2158.14 0.121976
\(680\) −45441.0 −2.56262
\(681\) −8573.65 −0.482442
\(682\) 0 0
\(683\) 2034.00 0.113951 0.0569757 0.998376i \(-0.481854\pi\)
0.0569757 + 0.998376i \(0.481854\pi\)
\(684\) 24879.2 1.39076
\(685\) 21546.0 1.20180
\(686\) −12348.0 −0.687243
\(687\) 16557.0 0.919489
\(688\) 10548.2 0.584514
\(689\) 3195.63 0.176697
\(690\) −12626.7 −0.696650
\(691\) 4598.00 0.253135 0.126567 0.991958i \(-0.459604\pi\)
0.126567 + 0.991958i \(0.459604\pi\)
\(692\) 60420.9 3.31916
\(693\) 0 0
\(694\) 6912.00 0.378063
\(695\) −1652.38 −0.0901845
\(696\) −15147.0 −0.824922
\(697\) 3213.00 0.174607
\(698\) −14769.0 −0.800881
\(699\) −10834.0 −0.586236
\(700\) −20271.9 −1.09458
\(701\) −5949.59 −0.320561 −0.160280 0.987072i \(-0.551240\pi\)
−0.160280 + 0.987072i \(0.551240\pi\)
\(702\) 9963.00 0.535654
\(703\) 19350.5 1.03815
\(704\) 0 0
\(705\) −1944.00 −0.103851
\(706\) −7435.69 −0.396382
\(707\) −31248.0 −1.66224
\(708\) −21546.0 −1.14371
\(709\) 24814.0 1.31440 0.657200 0.753716i \(-0.271741\pi\)
0.657200 + 0.753716i \(0.271741\pi\)
\(710\) −9259.54 −0.489443
\(711\) 1371.78 0.0723571
\(712\) 2572.10 0.135384
\(713\) −16920.0 −0.888722
\(714\) 33390.5 1.75015
\(715\) 0 0
\(716\) −48564.0 −2.53481
\(717\) 9789.55 0.509899
\(718\) 28026.0 1.45671
\(719\) 26442.0 1.37152 0.685758 0.727829i \(-0.259471\pi\)
0.685758 + 0.727829i \(0.259471\pi\)
\(720\) 11745.0 0.607931
\(721\) 32929.7 1.70093
\(722\) 74351.7 3.83253
\(723\) 10371.5 0.533501
\(724\) −14725.0 −0.755871
\(725\) −3886.72 −0.199102
\(726\) 0 0
\(727\) −28370.0 −1.44730 −0.723649 0.690169i \(-0.757536\pi\)
−0.723649 + 0.690169i \(0.757536\pi\)
\(728\) −98425.5 −5.01084
\(729\) 729.000 0.0370370
\(730\) −3564.00 −0.180698
\(731\) −6426.00 −0.325136
\(732\) 35541.7 1.79462
\(733\) −20332.5 −1.02456 −0.512278 0.858820i \(-0.671198\pi\)
−0.512278 + 0.858820i \(0.671198\pi\)
\(734\) 18467.1 0.928657
\(735\) −6615.00 −0.331970
\(736\) 26656.3 1.33500
\(737\) 0 0
\(738\) −1701.00 −0.0848437
\(739\) 9048.23 0.450399 0.225199 0.974313i \(-0.427697\pi\)
0.225199 + 0.974313i \(0.427697\pi\)
\(740\) 22743.0 1.12980
\(741\) 30996.0 1.53666
\(742\) −5670.00 −0.280529
\(743\) 9758.37 0.481830 0.240915 0.970546i \(-0.422552\pi\)
0.240915 + 0.970546i \(0.422552\pi\)
\(744\) 32236.9 1.58853
\(745\) 32501.9 1.59836
\(746\) 28980.0 1.42230
\(747\) −11223.7 −0.549737
\(748\) 0 0
\(749\) −31500.0 −1.53670
\(750\) 23710.0 1.15436
\(751\) −34010.0 −1.65252 −0.826260 0.563289i \(-0.809536\pi\)
−0.826260 + 0.563289i \(0.809536\pi\)
\(752\) 10440.0 0.506260
\(753\) −20142.0 −0.974788
\(754\) −32595.5 −1.57435
\(755\) −12221.4 −0.589113
\(756\) −12439.6 −0.598444
\(757\) −5345.00 −0.256628 −0.128314 0.991734i \(-0.540957\pi\)
−0.128314 + 0.991734i \(0.540957\pi\)
\(758\) −68516.5 −3.28315
\(759\) 0 0
\(760\) 74844.0 3.57221
\(761\) −22629.2 −1.07794 −0.538968 0.842326i \(-0.681186\pi\)
−0.538968 + 0.842326i \(0.681186\pi\)
\(762\) 10152.0 0.482635
\(763\) −29190.0 −1.38499
\(764\) 49932.0 2.36450
\(765\) −7155.10 −0.338161
\(766\) −17303.2 −0.816174
\(767\) −26843.3 −1.26370
\(768\) 15333.0 0.720419
\(769\) −34961.4 −1.63946 −0.819728 0.572753i \(-0.805876\pi\)
−0.819728 + 0.572753i \(0.805876\pi\)
\(770\) 0 0
\(771\) −4023.00 −0.187918
\(772\) −53016.3 −2.47163
\(773\) 29610.0 1.37775 0.688873 0.724882i \(-0.258106\pi\)
0.688873 + 0.724882i \(0.258106\pi\)
\(774\) 3402.00 0.157988
\(775\) 8272.00 0.383405
\(776\) 5087.03 0.235327
\(777\) −9675.24 −0.446714
\(778\) 18378.8 0.846930
\(779\) −5292.00 −0.243396
\(780\) 36430.2 1.67232
\(781\) 0 0
\(782\) −41310.0 −1.88906
\(783\) −2385.03 −0.108856
\(784\) 35525.0 1.61830
\(785\) 11754.0 0.534418
\(786\) 40662.0 1.84525
\(787\) −5785.05 −0.262026 −0.131013 0.991381i \(-0.541823\pi\)
−0.131013 + 0.991381i \(0.541823\pi\)
\(788\) 68220.3 3.08407
\(789\) 2868.28 0.129421
\(790\) 7128.00 0.321016
\(791\) 17240.8 0.774985
\(792\) 0 0
\(793\) 44280.0 1.98289
\(794\) −23387.9 −1.04535
\(795\) 1215.00 0.0542033
\(796\) 30742.0 1.36887
\(797\) −4626.00 −0.205598 −0.102799 0.994702i \(-0.532780\pi\)
−0.102799 + 0.994702i \(0.532780\pi\)
\(798\) −54996.1 −2.43965
\(799\) −6360.09 −0.281607
\(800\) −13032.0 −0.575936
\(801\) 405.000 0.0178651
\(802\) 17162.9 0.755664
\(803\) 0 0
\(804\) 22002.0 0.965113
\(805\) 19641.5 0.859963
\(806\) 69372.0 3.03167
\(807\) −9261.00 −0.403969
\(808\) −73656.0 −3.20694
\(809\) 31239.3 1.35762 0.678810 0.734314i \(-0.262496\pi\)
0.678810 + 0.734314i \(0.262496\pi\)
\(810\) 3788.00 0.164317
\(811\) 44340.5 1.91986 0.959929 0.280242i \(-0.0904146\pi\)
0.959929 + 0.280242i \(0.0904146\pi\)
\(812\) 40698.0 1.75889
\(813\) 12107.0 0.522278
\(814\) 0 0
\(815\) 22914.0 0.984837
\(816\) 38425.5 1.64848
\(817\) 10584.0 0.453228
\(818\) 49761.0 2.12696
\(819\) −15498.0 −0.661226
\(820\) −6219.79 −0.264884
\(821\) −38493.1 −1.63632 −0.818160 0.574991i \(-0.805006\pi\)
−0.818160 + 0.574991i \(0.805006\pi\)
\(822\) −37318.8 −1.58351
\(823\) 8678.00 0.367553 0.183776 0.982968i \(-0.441168\pi\)
0.183776 + 0.982968i \(0.441168\pi\)
\(824\) 77620.1 3.28158
\(825\) 0 0
\(826\) 47628.0 2.00628
\(827\) −27238.2 −1.14530 −0.572652 0.819799i \(-0.694085\pi\)
−0.572652 + 0.819799i \(0.694085\pi\)
\(828\) 15390.0 0.645941
\(829\) −19789.0 −0.829072 −0.414536 0.910033i \(-0.636056\pi\)
−0.414536 + 0.910033i \(0.636056\pi\)
\(830\) −58320.0 −2.43894
\(831\) −11374.4 −0.474817
\(832\) −26914.3 −1.12150
\(833\) −21642.0 −0.900180
\(834\) 2862.00 0.118828
\(835\) 6079.50 0.251964
\(836\) 0 0
\(837\) 5076.00 0.209620
\(838\) −35074.0 −1.44584
\(839\) −1800.00 −0.0740678 −0.0370339 0.999314i \(-0.511791\pi\)
−0.0370339 + 0.999314i \(0.511791\pi\)
\(840\) −37422.0 −1.53712
\(841\) −16586.0 −0.680061
\(842\) −70049.3 −2.86705
\(843\) −8168.35 −0.333728
\(844\) −35344.2 −1.44147
\(845\) 25614.0 1.04278
\(846\) 3367.11 0.136836
\(847\) 0 0
\(848\) −6525.00 −0.264233
\(849\) −9716.81 −0.392791
\(850\) 20196.0 0.814961
\(851\) 11970.0 0.482170
\(852\) 11286.0 0.453817
\(853\) 33513.5 1.34523 0.672614 0.739994i \(-0.265172\pi\)
0.672614 + 0.739994i \(0.265172\pi\)
\(854\) −78565.8 −3.14809
\(855\) 11784.9 0.471385
\(856\) −74250.0 −2.96473
\(857\) 26687.4 1.06374 0.531870 0.846826i \(-0.321489\pi\)
0.531870 + 0.846826i \(0.321489\pi\)
\(858\) 0 0
\(859\) 46694.0 1.85469 0.927345 0.374207i \(-0.122085\pi\)
0.927345 + 0.374207i \(0.122085\pi\)
\(860\) 12439.6 0.493240
\(861\) 2646.00 0.104733
\(862\) 7182.00 0.283782
\(863\) −36018.0 −1.42070 −0.710352 0.703847i \(-0.751464\pi\)
−0.710352 + 0.703847i \(0.751464\pi\)
\(864\) −7996.88 −0.314883
\(865\) 28620.4 1.12500
\(866\) −7955.31 −0.312162
\(867\) −8670.00 −0.339618
\(868\) −86616.4 −3.38704
\(869\) 0 0
\(870\) −12393.0 −0.482945
\(871\) 27411.4 1.06636
\(872\) −68805.0 −2.67205
\(873\) 801.000 0.0310535
\(874\) 68040.0 2.63328
\(875\) −36882.3 −1.42497
\(876\) 4343.98 0.167545
\(877\) −9191.99 −0.353924 −0.176962 0.984218i \(-0.556627\pi\)
−0.176962 + 0.984218i \(0.556627\pi\)
\(878\) −9972.00 −0.383301
\(879\) 13016.4 0.499466
\(880\) 0 0
\(881\) −40005.0 −1.52986 −0.764928 0.644116i \(-0.777225\pi\)
−0.764928 + 0.644116i \(0.777225\pi\)
\(882\) 11457.5 0.437409
\(883\) 4492.00 0.171198 0.0855990 0.996330i \(-0.472720\pi\)
0.0855990 + 0.996330i \(0.472720\pi\)
\(884\) 119187. 4.53472
\(885\) −10206.0 −0.387651
\(886\) 75853.4 2.87624
\(887\) 43554.1 1.64871 0.824355 0.566074i \(-0.191538\pi\)
0.824355 + 0.566074i \(0.191538\pi\)
\(888\) −22805.9 −0.861843
\(889\) −15792.0 −0.595778
\(890\) 2104.44 0.0792596
\(891\) 0 0
\(892\) −18316.0 −0.687517
\(893\) 10475.4 0.392550
\(894\) −56295.0 −2.10603
\(895\) −23004.0 −0.859150
\(896\) −9702.00 −0.361742
\(897\) 19173.8 0.713706
\(898\) −65424.8 −2.43124
\(899\) −16606.9 −0.616097
\(900\) −7524.00 −0.278667
\(901\) 3975.06 0.146979
\(902\) 0 0
\(903\) −5292.00 −0.195024
\(904\) 40639.1 1.49517
\(905\) −6975.00 −0.256195
\(906\) 21168.0 0.776225
\(907\) 7634.00 0.279474 0.139737 0.990189i \(-0.455374\pi\)
0.139737 + 0.990189i \(0.455374\pi\)
\(908\) 54299.8 1.98458
\(909\) −11597.8 −0.423185
\(910\) −80530.0 −2.93356
\(911\) −43830.0 −1.59402 −0.797010 0.603966i \(-0.793586\pi\)
−0.797010 + 0.603966i \(0.793586\pi\)
\(912\) −63289.1 −2.29793
\(913\) 0 0
\(914\) 87183.0 3.15510
\(915\) 16835.5 0.608268
\(916\) −104861. −3.78243
\(917\) −63252.0 −2.27782
\(918\) 12393.0 0.445566
\(919\) −32531.4 −1.16769 −0.583847 0.811864i \(-0.698453\pi\)
−0.583847 + 0.811864i \(0.698453\pi\)
\(920\) 46297.7 1.65912
\(921\) −17344.8 −0.620553
\(922\) 73467.0 2.62419
\(923\) 14060.8 0.501426
\(924\) 0 0
\(925\) −5852.00 −0.208014
\(926\) 35863.8 1.27274
\(927\) 12222.0 0.433035
\(928\) 26163.0 0.925477
\(929\) 15651.0 0.552737 0.276368 0.961052i \(-0.410869\pi\)
0.276368 + 0.961052i \(0.410869\pi\)
\(930\) 26375.7 0.929992
\(931\) 35645.6 1.25482
\(932\) 68615.2 2.41155
\(933\) −15660.0 −0.549502
\(934\) 82587.6 2.89331
\(935\) 0 0
\(936\) −36531.0 −1.27570
\(937\) −6337.57 −0.220960 −0.110480 0.993878i \(-0.535239\pi\)
−0.110480 + 0.993878i \(0.535239\pi\)
\(938\) −48636.0 −1.69299
\(939\) 11931.0 0.414647
\(940\) 12312.0 0.427205
\(941\) 9846.71 0.341120 0.170560 0.985347i \(-0.445442\pi\)
0.170560 + 0.985347i \(0.445442\pi\)
\(942\) −20358.5 −0.704158
\(943\) −3273.58 −0.113046
\(944\) 54810.0 1.88974
\(945\) −5892.44 −0.202837
\(946\) 0 0
\(947\) −738.000 −0.0253239 −0.0126620 0.999920i \(-0.504031\pi\)
−0.0126620 + 0.999920i \(0.504031\pi\)
\(948\) −8687.97 −0.297650
\(949\) 5412.00 0.185122
\(950\) −33264.0 −1.13603
\(951\) −29754.0 −1.01455
\(952\) −122432. −4.16810
\(953\) −15541.7 −0.528274 −0.264137 0.964485i \(-0.585087\pi\)
−0.264137 + 0.964485i \(0.585087\pi\)
\(954\) −2104.44 −0.0714191
\(955\) 23652.0 0.801425
\(956\) −62000.5 −2.09753
\(957\) 0 0
\(958\) −50490.0 −1.70277
\(959\) 58051.4 1.95472
\(960\) −10233.0 −0.344030
\(961\) 5553.00 0.186399
\(962\) −49077.0 −1.64481
\(963\) −11691.3 −0.391224
\(964\) −65686.3 −2.19462
\(965\) −25113.0 −0.837737
\(966\) −34020.0 −1.13310
\(967\) −37692.9 −1.25349 −0.626743 0.779226i \(-0.715613\pi\)
−0.626743 + 0.779226i \(0.715613\pi\)
\(968\) 0 0
\(969\) 38556.0 1.27822
\(970\) 4162.12 0.137771
\(971\) −12402.0 −0.409886 −0.204943 0.978774i \(-0.565701\pi\)
−0.204943 + 0.978774i \(0.565701\pi\)
\(972\) −4617.00 −0.152356
\(973\) −4452.00 −0.146685
\(974\) −39605.1 −1.30290
\(975\) −9373.86 −0.307901
\(976\) −90413.1 −2.96522
\(977\) 31203.0 1.02177 0.510887 0.859648i \(-0.329317\pi\)
0.510887 + 0.859648i \(0.329317\pi\)
\(978\) −39688.2 −1.29764
\(979\) 0 0
\(980\) 41895.0 1.36560
\(981\) −10834.0 −0.352602
\(982\) 77436.0 2.51638
\(983\) −36540.0 −1.18560 −0.592800 0.805350i \(-0.701978\pi\)
−0.592800 + 0.805350i \(0.701978\pi\)
\(984\) 6237.00 0.202061
\(985\) 32314.9 1.04532
\(986\) −40545.6 −1.30957
\(987\) −5237.72 −0.168914
\(988\) −196308. −6.32124
\(989\) 6547.15 0.210503
\(990\) 0 0
\(991\) −56888.0 −1.82352 −0.911759 0.410725i \(-0.865276\pi\)
−0.911759 + 0.410725i \(0.865276\pi\)
\(992\) −55682.0 −1.78216
\(993\) −26268.0 −0.839466
\(994\) −24948.0 −0.796079
\(995\) 14562.0 0.463966
\(996\) 71083.4 2.26141
\(997\) 15711.4 0.499083 0.249542 0.968364i \(-0.419720\pi\)
0.249542 + 0.968364i \(0.419720\pi\)
\(998\) 67581.2 2.14353
\(999\) −3591.00 −0.113728
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 363.4.a.l.1.2 yes 2
3.2 odd 2 1089.4.a.s.1.1 2
11.10 odd 2 inner 363.4.a.l.1.1 2
33.32 even 2 1089.4.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.4.a.l.1.1 2 11.10 odd 2 inner
363.4.a.l.1.2 yes 2 1.1 even 1 trivial
1089.4.a.s.1.1 2 3.2 odd 2
1089.4.a.s.1.2 2 33.32 even 2