Properties

Label 33.4.a.c.1.2
Level $33$
Weight $4$
Character 33.1
Self dual yes
Analytic conductor $1.947$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [33,4,Mod(1,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, 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("33.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 33.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: \(1.94706303019\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.42443\) of defining polynomial
Character \(\chi\) \(=\) 33.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.42443 q^{2} -3.00000 q^{3} +21.4244 q^{4} -16.8489 q^{5} -16.2733 q^{6} -7.69772 q^{7} +72.8199 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+5.42443 q^{2} -3.00000 q^{3} +21.4244 q^{4} -16.8489 q^{5} -16.2733 q^{6} -7.69772 q^{7} +72.8199 q^{8} +9.00000 q^{9} -91.3954 q^{10} -11.0000 q^{11} -64.2733 q^{12} +24.8489 q^{13} -41.7557 q^{14} +50.5466 q^{15} +223.611 q^{16} -15.9420 q^{17} +48.8199 q^{18} +15.1511 q^{19} -360.977 q^{20} +23.0931 q^{21} -59.6687 q^{22} +17.7557 q^{23} -218.460 q^{24} +158.884 q^{25} +134.791 q^{26} -27.0000 q^{27} -164.919 q^{28} -128.547 q^{29} +274.186 q^{30} +219.395 q^{31} +630.402 q^{32} +33.0000 q^{33} -86.4763 q^{34} +129.698 q^{35} +192.820 q^{36} +92.0703 q^{37} +82.1863 q^{38} -74.5466 q^{39} -1226.93 q^{40} -459.942 q^{41} +125.267 q^{42} +64.9648 q^{43} -235.669 q^{44} -151.640 q^{45} +96.3146 q^{46} +497.408 q^{47} -670.832 q^{48} -283.745 q^{49} +861.855 q^{50} +47.8260 q^{51} +532.373 q^{52} -526.919 q^{53} -146.460 q^{54} +185.337 q^{55} -560.547 q^{56} -45.4534 q^{57} -697.292 q^{58} -578.443 q^{59} +1082.93 q^{60} -221.569 q^{61} +1190.09 q^{62} -69.2794 q^{63} +1630.68 q^{64} -418.675 q^{65} +179.006 q^{66} -860.745 q^{67} -341.548 q^{68} -53.2671 q^{69} +703.536 q^{70} +580.919 q^{71} +655.379 q^{72} +510.116 q^{73} +499.429 q^{74} -476.652 q^{75} +324.605 q^{76} +84.6749 q^{77} -404.373 q^{78} +1035.12 q^{79} -3767.59 q^{80} +81.0000 q^{81} -2494.92 q^{82} +606.211 q^{83} +494.757 q^{84} +268.605 q^{85} +352.397 q^{86} +385.640 q^{87} -801.018 q^{88} -23.4411 q^{89} -822.559 q^{90} -191.279 q^{91} +380.406 q^{92} -658.186 q^{93} +2698.15 q^{94} -255.279 q^{95} -1891.20 q^{96} +719.490 q^{97} -1539.16 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} - 3 q^{6} + 24 q^{7} + 57 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} - 3 q^{6} + 24 q^{7} + 57 q^{8} + 18 q^{9} - 104 q^{10} - 22 q^{11} - 99 q^{12} + 30 q^{13} - 182 q^{14} + 42 q^{15} + 201 q^{16} + 106 q^{17} + 9 q^{18} + 50 q^{19} - 328 q^{20} - 72 q^{21} - 11 q^{22} + 134 q^{23} - 171 q^{24} + 42 q^{25} + 112 q^{26} - 54 q^{27} + 202 q^{28} - 198 q^{29} + 312 q^{30} + 360 q^{31} + 857 q^{32} + 66 q^{33} - 626 q^{34} + 220 q^{35} + 297 q^{36} - 328 q^{37} - 72 q^{38} - 90 q^{39} - 1272 q^{40} - 782 q^{41} + 546 q^{42} + 386 q^{43} - 363 q^{44} - 126 q^{45} - 418 q^{46} + 266 q^{47} - 603 q^{48} + 378 q^{49} + 1379 q^{50} - 318 q^{51} + 592 q^{52} - 522 q^{53} - 27 q^{54} + 154 q^{55} - 1062 q^{56} - 150 q^{57} - 390 q^{58} - 172 q^{59} + 984 q^{60} - 778 q^{61} + 568 q^{62} + 216 q^{63} + 809 q^{64} - 404 q^{65} + 33 q^{66} - 776 q^{67} + 1070 q^{68} - 402 q^{69} + 304 q^{70} + 630 q^{71} + 513 q^{72} + 1296 q^{73} + 2358 q^{74} - 126 q^{75} + 728 q^{76} - 264 q^{77} - 336 q^{78} + 652 q^{79} - 3832 q^{80} + 162 q^{81} - 1070 q^{82} - 324 q^{83} - 606 q^{84} + 616 q^{85} - 1068 q^{86} + 594 q^{87} - 627 q^{88} - 756 q^{89} - 936 q^{90} - 28 q^{91} + 1726 q^{92} - 1080 q^{93} + 3722 q^{94} - 156 q^{95} - 2571 q^{96} - 452 q^{97} - 4467 q^{98} - 198 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.42443 1.91783 0.958913 0.283702i \(-0.0915625\pi\)
0.958913 + 0.283702i \(0.0915625\pi\)
\(3\) −3.00000 −0.577350
\(4\) 21.4244 2.67805
\(5\) −16.8489 −1.50701 −0.753504 0.657444i \(-0.771638\pi\)
−0.753504 + 0.657444i \(0.771638\pi\)
\(6\) −16.2733 −1.10726
\(7\) −7.69772 −0.415638 −0.207819 0.978167i \(-0.566636\pi\)
−0.207819 + 0.978167i \(0.566636\pi\)
\(8\) 72.8199 3.21821
\(9\) 9.00000 0.333333
\(10\) −91.3954 −2.89018
\(11\) −11.0000 −0.301511
\(12\) −64.2733 −1.54617
\(13\) 24.8489 0.530141 0.265071 0.964229i \(-0.414605\pi\)
0.265071 + 0.964229i \(0.414605\pi\)
\(14\) −41.7557 −0.797120
\(15\) 50.5466 0.870071
\(16\) 223.611 3.49392
\(17\) −15.9420 −0.227441 −0.113721 0.993513i \(-0.536277\pi\)
−0.113721 + 0.993513i \(0.536277\pi\)
\(18\) 48.8199 0.639275
\(19\) 15.1511 0.182943 0.0914713 0.995808i \(-0.470843\pi\)
0.0914713 + 0.995808i \(0.470843\pi\)
\(20\) −360.977 −4.03585
\(21\) 23.0931 0.239968
\(22\) −59.6687 −0.578246
\(23\) 17.7557 0.160971 0.0804853 0.996756i \(-0.474353\pi\)
0.0804853 + 0.996756i \(0.474353\pi\)
\(24\) −218.460 −1.85804
\(25\) 158.884 1.27107
\(26\) 134.791 1.01672
\(27\) −27.0000 −0.192450
\(28\) −164.919 −1.11310
\(29\) −128.547 −0.823121 −0.411560 0.911383i \(-0.635016\pi\)
−0.411560 + 0.911383i \(0.635016\pi\)
\(30\) 274.186 1.66864
\(31\) 219.395 1.27112 0.635558 0.772053i \(-0.280770\pi\)
0.635558 + 0.772053i \(0.280770\pi\)
\(32\) 630.402 3.48251
\(33\) 33.0000 0.174078
\(34\) −86.4763 −0.436193
\(35\) 129.698 0.626369
\(36\) 192.820 0.892685
\(37\) 92.0703 0.409088 0.204544 0.978857i \(-0.434429\pi\)
0.204544 + 0.978857i \(0.434429\pi\)
\(38\) 82.1863 0.350852
\(39\) −74.5466 −0.306077
\(40\) −1226.93 −4.84987
\(41\) −459.942 −1.75197 −0.875986 0.482336i \(-0.839788\pi\)
−0.875986 + 0.482336i \(0.839788\pi\)
\(42\) 125.267 0.460218
\(43\) 64.9648 0.230396 0.115198 0.993343i \(-0.463250\pi\)
0.115198 + 0.993343i \(0.463250\pi\)
\(44\) −235.669 −0.807464
\(45\) −151.640 −0.502336
\(46\) 96.3146 0.308713
\(47\) 497.408 1.54371 0.771855 0.635799i \(-0.219329\pi\)
0.771855 + 0.635799i \(0.219329\pi\)
\(48\) −670.832 −2.01721
\(49\) −283.745 −0.827245
\(50\) 861.855 2.43769
\(51\) 47.8260 0.131313
\(52\) 532.373 1.41975
\(53\) −526.919 −1.36562 −0.682811 0.730596i \(-0.739243\pi\)
−0.682811 + 0.730596i \(0.739243\pi\)
\(54\) −146.460 −0.369086
\(55\) 185.337 0.454380
\(56\) −560.547 −1.33761
\(57\) −45.4534 −0.105622
\(58\) −697.292 −1.57860
\(59\) −578.443 −1.27639 −0.638194 0.769876i \(-0.720318\pi\)
−0.638194 + 0.769876i \(0.720318\pi\)
\(60\) 1082.93 2.33010
\(61\) −221.569 −0.465067 −0.232533 0.972588i \(-0.574701\pi\)
−0.232533 + 0.972588i \(0.574701\pi\)
\(62\) 1190.09 2.43778
\(63\) −69.2794 −0.138546
\(64\) 1630.68 3.18493
\(65\) −418.675 −0.798927
\(66\) 179.006 0.333851
\(67\) −860.745 −1.56950 −0.784752 0.619810i \(-0.787210\pi\)
−0.784752 + 0.619810i \(0.787210\pi\)
\(68\) −341.548 −0.609100
\(69\) −53.2671 −0.0929364
\(70\) 703.536 1.20127
\(71\) 580.919 0.971020 0.485510 0.874231i \(-0.338634\pi\)
0.485510 + 0.874231i \(0.338634\pi\)
\(72\) 655.379 1.07274
\(73\) 510.116 0.817871 0.408935 0.912563i \(-0.365900\pi\)
0.408935 + 0.912563i \(0.365900\pi\)
\(74\) 499.429 0.784560
\(75\) −476.652 −0.733854
\(76\) 324.605 0.489930
\(77\) 84.6749 0.125319
\(78\) −404.373 −0.587002
\(79\) 1035.12 1.47418 0.737088 0.675797i \(-0.236200\pi\)
0.737088 + 0.675797i \(0.236200\pi\)
\(80\) −3767.59 −5.26536
\(81\) 81.0000 0.111111
\(82\) −2494.92 −3.35998
\(83\) 606.211 0.801690 0.400845 0.916146i \(-0.368717\pi\)
0.400845 + 0.916146i \(0.368717\pi\)
\(84\) 494.757 0.642648
\(85\) 268.605 0.342756
\(86\) 352.397 0.441860
\(87\) 385.640 0.475229
\(88\) −801.018 −0.970328
\(89\) −23.4411 −0.0279186 −0.0139593 0.999903i \(-0.504444\pi\)
−0.0139593 + 0.999903i \(0.504444\pi\)
\(90\) −822.559 −0.963392
\(91\) −191.279 −0.220347
\(92\) 380.406 0.431088
\(93\) −658.186 −0.733879
\(94\) 2698.15 2.96057
\(95\) −255.279 −0.275696
\(96\) −1891.20 −2.01063
\(97\) 719.490 0.753126 0.376563 0.926391i \(-0.377106\pi\)
0.376563 + 0.926391i \(0.377106\pi\)
\(98\) −1539.16 −1.58651
\(99\) −99.0000 −0.100504
\(100\) 3404.00 3.40400
\(101\) 1871.27 1.84355 0.921774 0.387727i \(-0.126740\pi\)
0.921774 + 0.387727i \(0.126740\pi\)
\(102\) 259.429 0.251836
\(103\) −428.745 −0.410151 −0.205075 0.978746i \(-0.565744\pi\)
−0.205075 + 0.978746i \(0.565744\pi\)
\(104\) 1809.49 1.70611
\(105\) −389.093 −0.361634
\(106\) −2858.24 −2.61902
\(107\) 1148.02 1.03723 0.518616 0.855008i \(-0.326448\pi\)
0.518616 + 0.855008i \(0.326448\pi\)
\(108\) −578.460 −0.515392
\(109\) −1828.32 −1.60662 −0.803308 0.595564i \(-0.796929\pi\)
−0.803308 + 0.595564i \(0.796929\pi\)
\(110\) 1005.35 0.871421
\(111\) −276.211 −0.236187
\(112\) −1721.29 −1.45220
\(113\) 1126.40 0.937722 0.468861 0.883272i \(-0.344665\pi\)
0.468861 + 0.883272i \(0.344665\pi\)
\(114\) −246.559 −0.202565
\(115\) −299.163 −0.242584
\(116\) −2754.04 −2.20436
\(117\) 223.640 0.176714
\(118\) −3137.72 −2.44789
\(119\) 122.717 0.0945332
\(120\) 3680.79 2.80008
\(121\) 121.000 0.0909091
\(122\) −1201.89 −0.891916
\(123\) 1379.83 1.01150
\(124\) 4700.42 3.40412
\(125\) −570.907 −0.408508
\(126\) −375.801 −0.265707
\(127\) 661.304 0.462057 0.231029 0.972947i \(-0.425791\pi\)
0.231029 + 0.972947i \(0.425791\pi\)
\(128\) 3802.31 2.62562
\(129\) −194.895 −0.133019
\(130\) −2271.07 −1.53220
\(131\) 622.186 0.414967 0.207483 0.978239i \(-0.433473\pi\)
0.207483 + 0.978239i \(0.433473\pi\)
\(132\) 707.006 0.466189
\(133\) −116.629 −0.0760378
\(134\) −4669.05 −3.01003
\(135\) 454.919 0.290024
\(136\) −1160.89 −0.731955
\(137\) −1872.84 −1.16794 −0.583969 0.811776i \(-0.698501\pi\)
−0.583969 + 0.811776i \(0.698501\pi\)
\(138\) −288.944 −0.178236
\(139\) −954.058 −0.582174 −0.291087 0.956697i \(-0.594017\pi\)
−0.291087 + 0.956697i \(0.594017\pi\)
\(140\) 2778.70 1.67745
\(141\) −1492.22 −0.891261
\(142\) 3151.15 1.86225
\(143\) −273.337 −0.159844
\(144\) 2012.50 1.16464
\(145\) 2165.86 1.24045
\(146\) 2767.09 1.56853
\(147\) 851.236 0.477610
\(148\) 1972.55 1.09556
\(149\) −2047.01 −1.12549 −0.562745 0.826631i \(-0.690255\pi\)
−0.562745 + 0.826631i \(0.690255\pi\)
\(150\) −2585.57 −1.40740
\(151\) 475.863 0.256458 0.128229 0.991745i \(-0.459071\pi\)
0.128229 + 0.991745i \(0.459071\pi\)
\(152\) 1103.30 0.588749
\(153\) −143.478 −0.0758138
\(154\) 459.313 0.240341
\(155\) −3696.56 −1.91558
\(156\) −1597.12 −0.819691
\(157\) −647.466 −0.329130 −0.164565 0.986366i \(-0.552622\pi\)
−0.164565 + 0.986366i \(0.552622\pi\)
\(158\) 5614.92 2.82721
\(159\) 1580.76 0.788442
\(160\) −10621.5 −5.24817
\(161\) −136.678 −0.0669054
\(162\) 439.379 0.213092
\(163\) 1093.23 0.525329 0.262665 0.964887i \(-0.415399\pi\)
0.262665 + 0.964887i \(0.415399\pi\)
\(164\) −9853.99 −4.69188
\(165\) −556.012 −0.262336
\(166\) 3288.35 1.53750
\(167\) 1123.25 0.520479 0.260240 0.965544i \(-0.416198\pi\)
0.260240 + 0.965544i \(0.416198\pi\)
\(168\) 1681.64 0.772270
\(169\) −1579.53 −0.718951
\(170\) 1457.03 0.657346
\(171\) 136.360 0.0609809
\(172\) 1391.83 0.617014
\(173\) 46.0123 0.0202211 0.0101106 0.999949i \(-0.496782\pi\)
0.0101106 + 0.999949i \(0.496782\pi\)
\(174\) 2091.88 0.911406
\(175\) −1223.04 −0.528305
\(176\) −2459.72 −1.05346
\(177\) 1735.33 0.736923
\(178\) −127.155 −0.0535430
\(179\) −831.975 −0.347401 −0.173700 0.984799i \(-0.555572\pi\)
−0.173700 + 0.984799i \(0.555572\pi\)
\(180\) −3248.79 −1.34528
\(181\) −1810.63 −0.743553 −0.371776 0.928322i \(-0.621251\pi\)
−0.371776 + 0.928322i \(0.621251\pi\)
\(182\) −1037.58 −0.422586
\(183\) 664.708 0.268506
\(184\) 1292.97 0.518037
\(185\) −1551.28 −0.616499
\(186\) −3570.28 −1.40745
\(187\) 175.362 0.0685762
\(188\) 10656.7 4.13414
\(189\) 207.838 0.0799895
\(190\) −1384.75 −0.528737
\(191\) 458.898 0.173847 0.0869233 0.996215i \(-0.472296\pi\)
0.0869233 + 0.996215i \(0.472296\pi\)
\(192\) −4892.05 −1.83882
\(193\) 1778.91 0.663465 0.331733 0.943373i \(-0.392367\pi\)
0.331733 + 0.943373i \(0.392367\pi\)
\(194\) 3902.82 1.44436
\(195\) 1256.02 0.461260
\(196\) −6079.08 −2.21541
\(197\) 5304.53 1.91844 0.959218 0.282666i \(-0.0912188\pi\)
0.959218 + 0.282666i \(0.0912188\pi\)
\(198\) −537.018 −0.192749
\(199\) −5138.40 −1.83041 −0.915205 0.402989i \(-0.867971\pi\)
−0.915205 + 0.402989i \(0.867971\pi\)
\(200\) 11569.9 4.09058
\(201\) 2582.24 0.906153
\(202\) 10150.6 3.53560
\(203\) 989.515 0.342120
\(204\) 1024.65 0.351664
\(205\) 7749.50 2.64024
\(206\) −2325.70 −0.786597
\(207\) 159.801 0.0536568
\(208\) 5556.47 1.85227
\(209\) −166.663 −0.0551593
\(210\) −2110.61 −0.693551
\(211\) −4262.36 −1.39068 −0.695339 0.718682i \(-0.744746\pi\)
−0.695339 + 0.718682i \(0.744746\pi\)
\(212\) −11288.9 −3.65721
\(213\) −1742.76 −0.560619
\(214\) 6227.38 1.98923
\(215\) −1094.58 −0.347209
\(216\) −1966.14 −0.619345
\(217\) −1688.84 −0.528323
\(218\) −9917.58 −3.08121
\(219\) −1530.35 −0.472198
\(220\) 3970.75 1.21685
\(221\) −396.141 −0.120576
\(222\) −1498.29 −0.452966
\(223\) −1377.80 −0.413740 −0.206870 0.978368i \(-0.566328\pi\)
−0.206870 + 0.978368i \(0.566328\pi\)
\(224\) −4852.65 −1.44746
\(225\) 1429.96 0.423691
\(226\) 6110.06 1.79839
\(227\) −1227.28 −0.358843 −0.179422 0.983772i \(-0.557423\pi\)
−0.179422 + 0.983772i \(0.557423\pi\)
\(228\) −973.814 −0.282861
\(229\) 3890.28 1.12261 0.561304 0.827610i \(-0.310300\pi\)
0.561304 + 0.827610i \(0.310300\pi\)
\(230\) −1622.79 −0.465233
\(231\) −254.025 −0.0723532
\(232\) −9360.74 −2.64898
\(233\) −3218.14 −0.904837 −0.452419 0.891806i \(-0.649439\pi\)
−0.452419 + 0.891806i \(0.649439\pi\)
\(234\) 1213.12 0.338906
\(235\) −8380.75 −2.32638
\(236\) −12392.8 −3.41823
\(237\) −3105.35 −0.851115
\(238\) 665.670 0.181298
\(239\) −428.098 −0.115864 −0.0579318 0.998321i \(-0.518451\pi\)
−0.0579318 + 0.998321i \(0.518451\pi\)
\(240\) 11302.8 3.03996
\(241\) 1231.16 0.329070 0.164535 0.986371i \(-0.447388\pi\)
0.164535 + 0.986371i \(0.447388\pi\)
\(242\) 656.356 0.174348
\(243\) −243.000 −0.0641500
\(244\) −4747.00 −1.24547
\(245\) 4780.78 1.24667
\(246\) 7484.77 1.93988
\(247\) 376.489 0.0969854
\(248\) 15976.3 4.09072
\(249\) −1818.63 −0.462856
\(250\) −3096.84 −0.783446
\(251\) 2838.22 0.713732 0.356866 0.934156i \(-0.383845\pi\)
0.356866 + 0.934156i \(0.383845\pi\)
\(252\) −1484.27 −0.371033
\(253\) −195.313 −0.0485344
\(254\) 3587.20 0.886145
\(255\) −805.814 −0.197890
\(256\) 7579.90 1.85056
\(257\) 342.007 0.0830110 0.0415055 0.999138i \(-0.486785\pi\)
0.0415055 + 0.999138i \(0.486785\pi\)
\(258\) −1057.19 −0.255108
\(259\) −708.731 −0.170032
\(260\) −8969.87 −2.13957
\(261\) −1156.92 −0.274374
\(262\) 3375.01 0.795834
\(263\) 5895.00 1.38213 0.691067 0.722791i \(-0.257141\pi\)
0.691067 + 0.722791i \(0.257141\pi\)
\(264\) 2403.06 0.560219
\(265\) 8877.99 2.05800
\(266\) −632.647 −0.145827
\(267\) 70.3234 0.0161188
\(268\) −18441.0 −4.20322
\(269\) −2496.18 −0.565779 −0.282890 0.959152i \(-0.591293\pi\)
−0.282890 + 0.959152i \(0.591293\pi\)
\(270\) 2467.68 0.556215
\(271\) −2249.68 −0.504274 −0.252137 0.967692i \(-0.581133\pi\)
−0.252137 + 0.967692i \(0.581133\pi\)
\(272\) −3564.80 −0.794662
\(273\) 573.838 0.127217
\(274\) −10159.1 −2.23990
\(275\) −1747.72 −0.383243
\(276\) −1141.22 −0.248889
\(277\) 4082.59 0.885556 0.442778 0.896631i \(-0.353993\pi\)
0.442778 + 0.896631i \(0.353993\pi\)
\(278\) −5175.22 −1.11651
\(279\) 1974.56 0.423705
\(280\) 9444.57 2.01579
\(281\) −1033.79 −0.219468 −0.109734 0.993961i \(-0.535000\pi\)
−0.109734 + 0.993961i \(0.535000\pi\)
\(282\) −8094.46 −1.70928
\(283\) −7809.14 −1.64030 −0.820150 0.572148i \(-0.806110\pi\)
−0.820150 + 0.572148i \(0.806110\pi\)
\(284\) 12445.9 2.60044
\(285\) 765.838 0.159173
\(286\) −1482.70 −0.306552
\(287\) 3540.50 0.728186
\(288\) 5673.61 1.16084
\(289\) −4658.85 −0.948270
\(290\) 11748.6 2.37896
\(291\) −2158.47 −0.434817
\(292\) 10928.9 2.19030
\(293\) −1949.19 −0.388645 −0.194323 0.980938i \(-0.562251\pi\)
−0.194323 + 0.980938i \(0.562251\pi\)
\(294\) 4617.47 0.915973
\(295\) 9746.10 1.92353
\(296\) 6704.55 1.31653
\(297\) 297.000 0.0580259
\(298\) −11103.9 −2.15849
\(299\) 441.209 0.0853371
\(300\) −10212.0 −1.96530
\(301\) −500.081 −0.0957614
\(302\) 2581.28 0.491842
\(303\) −5613.81 −1.06437
\(304\) 3387.96 0.639187
\(305\) 3733.19 0.700859
\(306\) −778.286 −0.145398
\(307\) 2364.09 0.439497 0.219748 0.975557i \(-0.429476\pi\)
0.219748 + 0.975557i \(0.429476\pi\)
\(308\) 1814.11 0.335612
\(309\) 1286.24 0.236801
\(310\) −20051.7 −3.67375
\(311\) −1989.17 −0.362686 −0.181343 0.983420i \(-0.558044\pi\)
−0.181343 + 0.983420i \(0.558044\pi\)
\(312\) −5428.47 −0.985021
\(313\) −3878.67 −0.700433 −0.350216 0.936669i \(-0.613892\pi\)
−0.350216 + 0.936669i \(0.613892\pi\)
\(314\) −3512.13 −0.631214
\(315\) 1167.28 0.208790
\(316\) 22176.8 3.94792
\(317\) 2913.73 0.516251 0.258126 0.966111i \(-0.416895\pi\)
0.258126 + 0.966111i \(0.416895\pi\)
\(318\) 8574.71 1.51209
\(319\) 1414.01 0.248180
\(320\) −27475.1 −4.79971
\(321\) −3444.07 −0.598846
\(322\) −741.402 −0.128313
\(323\) −241.540 −0.0416087
\(324\) 1735.38 0.297562
\(325\) 3948.09 0.673847
\(326\) 5930.17 1.00749
\(327\) 5484.95 0.927580
\(328\) −33492.9 −5.63822
\(329\) −3828.90 −0.641624
\(330\) −3016.05 −0.503115
\(331\) 8104.46 1.34580 0.672902 0.739731i \(-0.265047\pi\)
0.672902 + 0.739731i \(0.265047\pi\)
\(332\) 12987.7 2.14697
\(333\) 828.633 0.136363
\(334\) 6093.02 0.998189
\(335\) 14502.6 2.36525
\(336\) 5163.88 0.838430
\(337\) 5919.19 0.956792 0.478396 0.878144i \(-0.341218\pi\)
0.478396 + 0.878144i \(0.341218\pi\)
\(338\) −8568.07 −1.37882
\(339\) −3379.19 −0.541394
\(340\) 5754.70 0.917919
\(341\) −2413.35 −0.383256
\(342\) 739.677 0.116951
\(343\) 4824.51 0.759472
\(344\) 4730.73 0.741465
\(345\) 897.490 0.140056
\(346\) 249.590 0.0387805
\(347\) −8540.59 −1.32128 −0.660638 0.750705i \(-0.729714\pi\)
−0.660638 + 0.750705i \(0.729714\pi\)
\(348\) 8262.11 1.27269
\(349\) 937.337 0.143767 0.0718833 0.997413i \(-0.477099\pi\)
0.0718833 + 0.997413i \(0.477099\pi\)
\(350\) −6634.31 −1.01320
\(351\) −670.919 −0.102026
\(352\) −6934.42 −1.05002
\(353\) −211.118 −0.0318319 −0.0159160 0.999873i \(-0.505066\pi\)
−0.0159160 + 0.999873i \(0.505066\pi\)
\(354\) 9413.17 1.41329
\(355\) −9787.82 −1.46333
\(356\) −502.213 −0.0747675
\(357\) −368.151 −0.0545788
\(358\) −4512.99 −0.666254
\(359\) −1376.31 −0.202337 −0.101169 0.994869i \(-0.532258\pi\)
−0.101169 + 0.994869i \(0.532258\pi\)
\(360\) −11042.4 −1.61662
\(361\) −6629.44 −0.966532
\(362\) −9821.63 −1.42600
\(363\) −363.000 −0.0524864
\(364\) −4098.05 −0.590100
\(365\) −8594.87 −1.23254
\(366\) 3605.66 0.514948
\(367\) 1030.45 0.146564 0.0732821 0.997311i \(-0.476653\pi\)
0.0732821 + 0.997311i \(0.476653\pi\)
\(368\) 3970.37 0.562418
\(369\) −4139.48 −0.583991
\(370\) −8414.81 −1.18234
\(371\) 4056.07 0.567603
\(372\) −14101.3 −1.96537
\(373\) 9365.39 1.30006 0.650029 0.759909i \(-0.274757\pi\)
0.650029 + 0.759909i \(0.274757\pi\)
\(374\) 951.239 0.131517
\(375\) 1712.72 0.235852
\(376\) 36221.2 4.96799
\(377\) −3194.24 −0.436370
\(378\) 1127.40 0.153406
\(379\) −7120.23 −0.965017 −0.482509 0.875891i \(-0.660274\pi\)
−0.482509 + 0.875891i \(0.660274\pi\)
\(380\) −5469.22 −0.738329
\(381\) −1983.91 −0.266769
\(382\) 2489.26 0.333407
\(383\) −1163.56 −0.155235 −0.0776176 0.996983i \(-0.524731\pi\)
−0.0776176 + 0.996983i \(0.524731\pi\)
\(384\) −11406.9 −1.51590
\(385\) −1426.67 −0.188857
\(386\) 9649.57 1.27241
\(387\) 584.684 0.0767988
\(388\) 15414.7 2.01691
\(389\) −10958.9 −1.42838 −0.714188 0.699954i \(-0.753204\pi\)
−0.714188 + 0.699954i \(0.753204\pi\)
\(390\) 6813.22 0.884617
\(391\) −283.062 −0.0366114
\(392\) −20662.3 −2.66225
\(393\) −1866.56 −0.239581
\(394\) 28774.0 3.67923
\(395\) −17440.6 −2.22159
\(396\) −2121.02 −0.269155
\(397\) −2172.09 −0.274595 −0.137298 0.990530i \(-0.543842\pi\)
−0.137298 + 0.990530i \(0.543842\pi\)
\(398\) −27872.9 −3.51041
\(399\) 349.888 0.0439005
\(400\) 35528.2 4.44102
\(401\) 7830.71 0.975180 0.487590 0.873073i \(-0.337876\pi\)
0.487590 + 0.873073i \(0.337876\pi\)
\(402\) 14007.2 1.73784
\(403\) 5451.73 0.673870
\(404\) 40090.9 4.93712
\(405\) −1364.76 −0.167445
\(406\) 5367.55 0.656126
\(407\) −1012.77 −0.123345
\(408\) 3482.68 0.422594
\(409\) −10731.2 −1.29736 −0.648682 0.761060i \(-0.724679\pi\)
−0.648682 + 0.761060i \(0.724679\pi\)
\(410\) 42036.6 5.06351
\(411\) 5618.52 0.674309
\(412\) −9185.62 −1.09841
\(413\) 4452.69 0.530515
\(414\) 866.831 0.102904
\(415\) −10214.0 −1.20815
\(416\) 15664.8 1.84622
\(417\) 2862.17 0.336118
\(418\) −904.049 −0.105786
\(419\) −7315.88 −0.852994 −0.426497 0.904489i \(-0.640252\pi\)
−0.426497 + 0.904489i \(0.640252\pi\)
\(420\) −8336.10 −0.968476
\(421\) −12495.7 −1.44657 −0.723284 0.690551i \(-0.757368\pi\)
−0.723284 + 0.690551i \(0.757368\pi\)
\(422\) −23120.9 −2.66708
\(423\) 4476.67 0.514570
\(424\) −38370.2 −4.39486
\(425\) −2532.93 −0.289094
\(426\) −9453.46 −1.07517
\(427\) 1705.58 0.193299
\(428\) 24595.8 2.77776
\(429\) 820.012 0.0922857
\(430\) −5937.49 −0.665887
\(431\) 6075.01 0.678939 0.339470 0.940617i \(-0.389752\pi\)
0.339470 + 0.940617i \(0.389752\pi\)
\(432\) −6037.49 −0.672405
\(433\) 5641.79 0.626160 0.313080 0.949727i \(-0.398639\pi\)
0.313080 + 0.949727i \(0.398639\pi\)
\(434\) −9161.01 −1.01323
\(435\) −6497.59 −0.716174
\(436\) −39170.7 −4.30260
\(437\) 269.019 0.0294484
\(438\) −8301.26 −0.905593
\(439\) 10897.0 1.18470 0.592351 0.805680i \(-0.298200\pi\)
0.592351 + 0.805680i \(0.298200\pi\)
\(440\) 13496.2 1.46229
\(441\) −2553.71 −0.275748
\(442\) −2148.84 −0.231244
\(443\) −7720.83 −0.828054 −0.414027 0.910265i \(-0.635878\pi\)
−0.414027 + 0.910265i \(0.635878\pi\)
\(444\) −5917.66 −0.632522
\(445\) 394.956 0.0420735
\(446\) −7473.76 −0.793481
\(447\) 6141.04 0.649802
\(448\) −12552.5 −1.32378
\(449\) −7473.86 −0.785553 −0.392776 0.919634i \(-0.628485\pi\)
−0.392776 + 0.919634i \(0.628485\pi\)
\(450\) 7756.70 0.812565
\(451\) 5059.36 0.528240
\(452\) 24132.4 2.51127
\(453\) −1427.59 −0.148066
\(454\) −6657.29 −0.688198
\(455\) 3222.84 0.332064
\(456\) −3309.91 −0.339914
\(457\) 11140.5 1.14033 0.570167 0.821529i \(-0.306879\pi\)
0.570167 + 0.821529i \(0.306879\pi\)
\(458\) 21102.6 2.15296
\(459\) 430.434 0.0437711
\(460\) −6409.41 −0.649652
\(461\) 14328.8 1.44763 0.723817 0.689992i \(-0.242386\pi\)
0.723817 + 0.689992i \(0.242386\pi\)
\(462\) −1377.94 −0.138761
\(463\) 11760.7 1.18049 0.590246 0.807223i \(-0.299031\pi\)
0.590246 + 0.807223i \(0.299031\pi\)
\(464\) −28744.4 −2.87592
\(465\) 11089.7 1.10596
\(466\) −17456.5 −1.73532
\(467\) 11854.9 1.17469 0.587343 0.809338i \(-0.300174\pi\)
0.587343 + 0.809338i \(0.300174\pi\)
\(468\) 4791.35 0.473249
\(469\) 6625.77 0.652345
\(470\) −45460.8 −4.46160
\(471\) 1942.40 0.190023
\(472\) −42122.1 −4.10769
\(473\) −714.613 −0.0694671
\(474\) −16844.8 −1.63229
\(475\) 2407.27 0.232533
\(476\) 2629.14 0.253165
\(477\) −4742.27 −0.455207
\(478\) −2322.19 −0.222206
\(479\) −1324.68 −0.126359 −0.0631796 0.998002i \(-0.520124\pi\)
−0.0631796 + 0.998002i \(0.520124\pi\)
\(480\) 31864.6 3.03003
\(481\) 2287.84 0.216874
\(482\) 6678.34 0.631100
\(483\) 410.035 0.0386278
\(484\) 2592.36 0.243459
\(485\) −12122.6 −1.13497
\(486\) −1318.14 −0.123029
\(487\) 18636.4 1.73408 0.867040 0.498239i \(-0.166020\pi\)
0.867040 + 0.498239i \(0.166020\pi\)
\(488\) −16134.7 −1.49668
\(489\) −3279.70 −0.303299
\(490\) 25933.0 2.39089
\(491\) 124.552 0.0114480 0.00572398 0.999984i \(-0.498178\pi\)
0.00572398 + 0.999984i \(0.498178\pi\)
\(492\) 29562.0 2.70886
\(493\) 2049.29 0.187212
\(494\) 2042.24 0.186001
\(495\) 1668.04 0.151460
\(496\) 49059.2 4.44117
\(497\) −4471.75 −0.403592
\(498\) −9865.04 −0.887677
\(499\) 10230.2 0.917768 0.458884 0.888496i \(-0.348249\pi\)
0.458884 + 0.888496i \(0.348249\pi\)
\(500\) −12231.4 −1.09401
\(501\) −3369.76 −0.300499
\(502\) 15395.7 1.36881
\(503\) 5150.81 0.456587 0.228294 0.973592i \(-0.426685\pi\)
0.228294 + 0.973592i \(0.426685\pi\)
\(504\) −5044.92 −0.445870
\(505\) −31528.8 −2.77824
\(506\) −1059.46 −0.0930806
\(507\) 4738.60 0.415086
\(508\) 14168.1 1.23741
\(509\) −22.7715 −0.00198296 −0.000991481 1.00000i \(-0.500316\pi\)
−0.000991481 1.00000i \(0.500316\pi\)
\(510\) −4371.08 −0.379519
\(511\) −3926.73 −0.339938
\(512\) 10698.1 0.923429
\(513\) −409.081 −0.0352073
\(514\) 1855.19 0.159201
\(515\) 7223.87 0.618100
\(516\) −4175.50 −0.356233
\(517\) −5471.49 −0.465446
\(518\) −3844.46 −0.326093
\(519\) −138.037 −0.0116747
\(520\) −30487.8 −2.57112
\(521\) 21521.7 1.80976 0.904879 0.425669i \(-0.139961\pi\)
0.904879 + 0.425669i \(0.139961\pi\)
\(522\) −6275.63 −0.526201
\(523\) 2923.36 0.244416 0.122208 0.992504i \(-0.461002\pi\)
0.122208 + 0.992504i \(0.461002\pi\)
\(524\) 13330.0 1.11130
\(525\) 3669.13 0.305017
\(526\) 31977.0 2.65069
\(527\) −3497.60 −0.289104
\(528\) 7379.15 0.608213
\(529\) −11851.7 −0.974088
\(530\) 48158.0 3.94689
\(531\) −5205.99 −0.425462
\(532\) −2498.71 −0.203633
\(533\) −11429.0 −0.928792
\(534\) 381.464 0.0309130
\(535\) −19342.9 −1.56312
\(536\) −62679.3 −5.05100
\(537\) 2495.93 0.200572
\(538\) −13540.3 −1.08507
\(539\) 3121.20 0.249424
\(540\) 9746.38 0.776699
\(541\) 21272.8 1.69056 0.845278 0.534327i \(-0.179435\pi\)
0.845278 + 0.534327i \(0.179435\pi\)
\(542\) −12203.2 −0.967108
\(543\) 5431.89 0.429290
\(544\) −10049.9 −0.792067
\(545\) 30805.1 2.42118
\(546\) 3112.75 0.243980
\(547\) −18730.5 −1.46409 −0.732046 0.681256i \(-0.761434\pi\)
−0.732046 + 0.681256i \(0.761434\pi\)
\(548\) −40124.5 −3.12780
\(549\) −1994.12 −0.155022
\(550\) −9480.41 −0.734992
\(551\) −1947.63 −0.150584
\(552\) −3878.91 −0.299089
\(553\) −7968.04 −0.612723
\(554\) 22145.7 1.69834
\(555\) 4653.84 0.355936
\(556\) −20440.1 −1.55909
\(557\) −18885.0 −1.43659 −0.718297 0.695736i \(-0.755078\pi\)
−0.718297 + 0.695736i \(0.755078\pi\)
\(558\) 10710.9 0.812593
\(559\) 1614.30 0.122143
\(560\) 29001.8 2.18848
\(561\) −526.086 −0.0395925
\(562\) −5607.71 −0.420902
\(563\) 10285.1 0.769922 0.384961 0.922933i \(-0.374215\pi\)
0.384961 + 0.922933i \(0.374215\pi\)
\(564\) −31970.0 −2.38685
\(565\) −18978.5 −1.41315
\(566\) −42360.1 −3.14581
\(567\) −623.515 −0.0461820
\(568\) 42302.5 3.12495
\(569\) 18008.8 1.32683 0.663415 0.748251i \(-0.269106\pi\)
0.663415 + 0.748251i \(0.269106\pi\)
\(570\) 4154.24 0.305266
\(571\) −7010.79 −0.513822 −0.256911 0.966435i \(-0.582705\pi\)
−0.256911 + 0.966435i \(0.582705\pi\)
\(572\) −5856.10 −0.428070
\(573\) −1376.69 −0.100370
\(574\) 19205.2 1.39653
\(575\) 2821.10 0.204605
\(576\) 14676.1 1.06164
\(577\) −16398.9 −1.18318 −0.591589 0.806240i \(-0.701499\pi\)
−0.591589 + 0.806240i \(0.701499\pi\)
\(578\) −25271.6 −1.81862
\(579\) −5336.73 −0.383052
\(580\) 46402.4 3.32199
\(581\) −4666.44 −0.333213
\(582\) −11708.5 −0.833903
\(583\) 5796.11 0.411750
\(584\) 37146.6 2.63208
\(585\) −3768.07 −0.266309
\(586\) −10573.3 −0.745354
\(587\) 12823.5 0.901671 0.450836 0.892607i \(-0.351126\pi\)
0.450836 + 0.892607i \(0.351126\pi\)
\(588\) 18237.2 1.27907
\(589\) 3324.09 0.232541
\(590\) 52867.0 3.68899
\(591\) −15913.6 −1.10761
\(592\) 20587.9 1.42932
\(593\) 16899.5 1.17029 0.585144 0.810929i \(-0.301038\pi\)
0.585144 + 0.810929i \(0.301038\pi\)
\(594\) 1611.06 0.111284
\(595\) −2067.64 −0.142462
\(596\) −43856.1 −3.01412
\(597\) 15415.2 1.05679
\(598\) 2393.31 0.163662
\(599\) −15074.9 −1.02829 −0.514143 0.857704i \(-0.671890\pi\)
−0.514143 + 0.857704i \(0.671890\pi\)
\(600\) −34709.7 −2.36170
\(601\) −11418.8 −0.775014 −0.387507 0.921867i \(-0.626664\pi\)
−0.387507 + 0.921867i \(0.626664\pi\)
\(602\) −2712.65 −0.183654
\(603\) −7746.71 −0.523168
\(604\) 10195.1 0.686809
\(605\) −2038.71 −0.137001
\(606\) −30451.7 −2.04128
\(607\) −17952.8 −1.20046 −0.600232 0.799826i \(-0.704925\pi\)
−0.600232 + 0.799826i \(0.704925\pi\)
\(608\) 9551.30 0.637100
\(609\) −2968.54 −0.197523
\(610\) 20250.4 1.34412
\(611\) 12360.0 0.818384
\(612\) −3073.94 −0.203033
\(613\) 12528.9 0.825507 0.412753 0.910843i \(-0.364567\pi\)
0.412753 + 0.910843i \(0.364567\pi\)
\(614\) 12823.8 0.842878
\(615\) −23248.5 −1.52434
\(616\) 6166.01 0.403305
\(617\) −8586.10 −0.560232 −0.280116 0.959966i \(-0.590373\pi\)
−0.280116 + 0.959966i \(0.590373\pi\)
\(618\) 6977.09 0.454142
\(619\) 18415.4 1.19576 0.597882 0.801584i \(-0.296009\pi\)
0.597882 + 0.801584i \(0.296009\pi\)
\(620\) −79196.7 −5.13003
\(621\) −479.404 −0.0309788
\(622\) −10790.1 −0.695569
\(623\) 180.443 0.0116040
\(624\) −16669.4 −1.06941
\(625\) −10241.4 −0.655448
\(626\) −21039.6 −1.34331
\(627\) 499.988 0.0318462
\(628\) −13871.6 −0.881427
\(629\) −1467.79 −0.0930436
\(630\) 6331.82 0.400422
\(631\) 2374.38 0.149798 0.0748989 0.997191i \(-0.476137\pi\)
0.0748989 + 0.997191i \(0.476137\pi\)
\(632\) 75377.1 4.74421
\(633\) 12787.1 0.802909
\(634\) 15805.3 0.990080
\(635\) −11142.2 −0.696324
\(636\) 33866.8 2.11149
\(637\) −7050.74 −0.438557
\(638\) 7670.21 0.475966
\(639\) 5228.27 0.323673
\(640\) −64064.6 −3.95684
\(641\) 11086.0 0.683104 0.341552 0.939863i \(-0.389047\pi\)
0.341552 + 0.939863i \(0.389047\pi\)
\(642\) −18682.1 −1.14848
\(643\) 19934.1 1.22259 0.611294 0.791403i \(-0.290649\pi\)
0.611294 + 0.791403i \(0.290649\pi\)
\(644\) −2928.26 −0.179176
\(645\) 3283.75 0.200461
\(646\) −1310.21 −0.0797983
\(647\) −30634.8 −1.86148 −0.930739 0.365684i \(-0.880835\pi\)
−0.930739 + 0.365684i \(0.880835\pi\)
\(648\) 5898.41 0.357579
\(649\) 6362.87 0.384845
\(650\) 21416.1 1.29232
\(651\) 5066.53 0.305028
\(652\) 23421.9 1.40686
\(653\) −9818.07 −0.588378 −0.294189 0.955747i \(-0.595049\pi\)
−0.294189 + 0.955747i \(0.595049\pi\)
\(654\) 29752.7 1.77894
\(655\) −10483.1 −0.625358
\(656\) −102848. −6.12125
\(657\) 4591.04 0.272624
\(658\) −20769.6 −1.23052
\(659\) 16478.5 0.974070 0.487035 0.873383i \(-0.338078\pi\)
0.487035 + 0.873383i \(0.338078\pi\)
\(660\) −11912.2 −0.702551
\(661\) 2958.12 0.174066 0.0870328 0.996205i \(-0.472262\pi\)
0.0870328 + 0.996205i \(0.472262\pi\)
\(662\) 43962.1 2.58102
\(663\) 1188.42 0.0696146
\(664\) 44144.2 2.58001
\(665\) 1965.07 0.114590
\(666\) 4494.86 0.261520
\(667\) −2282.44 −0.132498
\(668\) 24065.1 1.39387
\(669\) 4133.39 0.238873
\(670\) 78668.2 4.53614
\(671\) 2437.26 0.140223
\(672\) 14558.0 0.835692
\(673\) −29960.3 −1.71602 −0.858012 0.513630i \(-0.828300\pi\)
−0.858012 + 0.513630i \(0.828300\pi\)
\(674\) 32108.2 1.83496
\(675\) −4289.87 −0.244618
\(676\) −33840.6 −1.92539
\(677\) −4514.73 −0.256300 −0.128150 0.991755i \(-0.540904\pi\)
−0.128150 + 0.991755i \(0.540904\pi\)
\(678\) −18330.2 −1.03830
\(679\) −5538.43 −0.313027
\(680\) 19559.7 1.10306
\(681\) 3681.84 0.207178
\(682\) −13091.0 −0.735018
\(683\) −13555.7 −0.759438 −0.379719 0.925102i \(-0.623979\pi\)
−0.379719 + 0.925102i \(0.623979\pi\)
\(684\) 2921.44 0.163310
\(685\) 31555.2 1.76009
\(686\) 26170.2 1.45653
\(687\) −11670.8 −0.648137
\(688\) 14526.8 0.804986
\(689\) −13093.3 −0.723972
\(690\) 4868.37 0.268603
\(691\) −11471.3 −0.631535 −0.315768 0.948837i \(-0.602262\pi\)
−0.315768 + 0.948837i \(0.602262\pi\)
\(692\) 985.787 0.0541532
\(693\) 762.074 0.0417731
\(694\) −46327.8 −2.53398
\(695\) 16074.8 0.877340
\(696\) 28082.2 1.52939
\(697\) 7332.40 0.398471
\(698\) 5084.52 0.275719
\(699\) 9654.41 0.522408
\(700\) −26203.0 −1.41483
\(701\) 22229.0 1.19769 0.598843 0.800866i \(-0.295627\pi\)
0.598843 + 0.800866i \(0.295627\pi\)
\(702\) −3639.35 −0.195667
\(703\) 1394.97 0.0748397
\(704\) −17937.5 −0.960292
\(705\) 25142.3 1.34314
\(706\) −1145.19 −0.0610480
\(707\) −14404.5 −0.766248
\(708\) 37178.4 1.97352
\(709\) −15081.2 −0.798851 −0.399426 0.916766i \(-0.630790\pi\)
−0.399426 + 0.916766i \(0.630790\pi\)
\(710\) −53093.4 −2.80642
\(711\) 9316.06 0.491392
\(712\) −1706.98 −0.0898480
\(713\) 3895.52 0.204612
\(714\) −1997.01 −0.104673
\(715\) 4605.42 0.240885
\(716\) −17824.6 −0.930358
\(717\) 1284.30 0.0668938
\(718\) −7465.71 −0.388047
\(719\) −7399.80 −0.383819 −0.191910 0.981413i \(-0.561468\pi\)
−0.191910 + 0.981413i \(0.561468\pi\)
\(720\) −33908.3 −1.75512
\(721\) 3300.36 0.170474
\(722\) −35960.9 −1.85364
\(723\) −3693.48 −0.189989
\(724\) −38791.7 −1.99127
\(725\) −20424.0 −1.04625
\(726\) −1969.07 −0.100660
\(727\) 1705.77 0.0870202 0.0435101 0.999053i \(-0.486146\pi\)
0.0435101 + 0.999053i \(0.486146\pi\)
\(728\) −13928.9 −0.709122
\(729\) 729.000 0.0370370
\(730\) −46622.3 −2.36379
\(731\) −1035.67 −0.0524017
\(732\) 14241.0 0.719074
\(733\) 37122.6 1.87061 0.935303 0.353847i \(-0.115127\pi\)
0.935303 + 0.353847i \(0.115127\pi\)
\(734\) 5589.60 0.281084
\(735\) −14342.3 −0.719762
\(736\) 11193.2 0.560581
\(737\) 9468.20 0.473223
\(738\) −22454.3 −1.11999
\(739\) 34256.3 1.70520 0.852598 0.522568i \(-0.175026\pi\)
0.852598 + 0.522568i \(0.175026\pi\)
\(740\) −33235.3 −1.65102
\(741\) −1129.47 −0.0559945
\(742\) 22001.9 1.08856
\(743\) 1567.88 0.0774160 0.0387080 0.999251i \(-0.487676\pi\)
0.0387080 + 0.999251i \(0.487676\pi\)
\(744\) −47929.0 −2.36178
\(745\) 34489.8 1.69612
\(746\) 50801.9 2.49328
\(747\) 5455.90 0.267230
\(748\) 3757.03 0.183651
\(749\) −8837.17 −0.431112
\(750\) 9290.53 0.452323
\(751\) −955.613 −0.0464325 −0.0232163 0.999730i \(-0.507391\pi\)
−0.0232163 + 0.999730i \(0.507391\pi\)
\(752\) 111226. 5.39360
\(753\) −8514.65 −0.412073
\(754\) −17326.9 −0.836881
\(755\) −8017.75 −0.386484
\(756\) 4452.82 0.214216
\(757\) −14015.4 −0.672918 −0.336459 0.941698i \(-0.609229\pi\)
−0.336459 + 0.941698i \(0.609229\pi\)
\(758\) −38623.2 −1.85073
\(759\) 585.938 0.0280214
\(760\) −18589.4 −0.887249
\(761\) −36271.0 −1.72776 −0.863879 0.503699i \(-0.831972\pi\)
−0.863879 + 0.503699i \(0.831972\pi\)
\(762\) −10761.6 −0.511616
\(763\) 14073.9 0.667770
\(764\) 9831.63 0.465571
\(765\) 2417.44 0.114252
\(766\) −6311.64 −0.297714
\(767\) −14373.6 −0.676665
\(768\) −22739.7 −1.06842
\(769\) −18163.6 −0.851749 −0.425874 0.904782i \(-0.640033\pi\)
−0.425874 + 0.904782i \(0.640033\pi\)
\(770\) −7738.90 −0.362195
\(771\) −1026.02 −0.0479264
\(772\) 38112.1 1.77680
\(773\) −8345.65 −0.388321 −0.194160 0.980970i \(-0.562198\pi\)
−0.194160 + 0.980970i \(0.562198\pi\)
\(774\) 3171.57 0.147287
\(775\) 34858.4 1.61568
\(776\) 52393.2 2.42372
\(777\) 2126.19 0.0981683
\(778\) −59445.7 −2.73937
\(779\) −6968.65 −0.320510
\(780\) 26909.6 1.23528
\(781\) −6390.11 −0.292774
\(782\) −1535.45 −0.0702142
\(783\) 3470.76 0.158410
\(784\) −63448.5 −2.89033
\(785\) 10909.1 0.496001
\(786\) −10125.0 −0.459475
\(787\) −22996.2 −1.04158 −0.520791 0.853684i \(-0.674363\pi\)
−0.520791 + 0.853684i \(0.674363\pi\)
\(788\) 113647. 5.13768
\(789\) −17685.0 −0.797975
\(790\) −94605.0 −4.26063
\(791\) −8670.69 −0.389752
\(792\) −7209.17 −0.323443
\(793\) −5505.75 −0.246551
\(794\) −11782.4 −0.526626
\(795\) −26634.0 −1.18819
\(796\) −110087. −4.90194
\(797\) −2743.82 −0.121946 −0.0609730 0.998139i \(-0.519420\pi\)
−0.0609730 + 0.998139i \(0.519420\pi\)
\(798\) 1897.94 0.0841934
\(799\) −7929.68 −0.351104
\(800\) 100161. 4.42652
\(801\) −210.970 −0.00930619
\(802\) 42477.1 1.87022
\(803\) −5611.28 −0.246597
\(804\) 55322.9 2.42673
\(805\) 2302.88 0.100827
\(806\) 29572.5 1.29237
\(807\) 7488.53 0.326653
\(808\) 136266. 5.93293
\(809\) 41241.7 1.79231 0.896156 0.443738i \(-0.146348\pi\)
0.896156 + 0.443738i \(0.146348\pi\)
\(810\) −7403.03 −0.321131
\(811\) 12832.9 0.555641 0.277820 0.960633i \(-0.410388\pi\)
0.277820 + 0.960633i \(0.410388\pi\)
\(812\) 21199.8 0.916215
\(813\) 6749.03 0.291142
\(814\) −5493.72 −0.236554
\(815\) −18419.7 −0.791675
\(816\) 10694.4 0.458798
\(817\) 984.292 0.0421493
\(818\) −58210.4 −2.48812
\(819\) −1721.51 −0.0734488
\(820\) 166029. 7.07069
\(821\) 16368.5 0.695817 0.347908 0.937529i \(-0.386892\pi\)
0.347908 + 0.937529i \(0.386892\pi\)
\(822\) 30477.3 1.29321
\(823\) 3869.53 0.163892 0.0819461 0.996637i \(-0.473886\pi\)
0.0819461 + 0.996637i \(0.473886\pi\)
\(824\) −31221.2 −1.31995
\(825\) 5243.17 0.221265
\(826\) 24153.3 1.01743
\(827\) 7388.69 0.310677 0.155339 0.987861i \(-0.450353\pi\)
0.155339 + 0.987861i \(0.450353\pi\)
\(828\) 3423.65 0.143696
\(829\) 23990.1 1.00508 0.502539 0.864554i \(-0.332399\pi\)
0.502539 + 0.864554i \(0.332399\pi\)
\(830\) −55404.9 −2.31703
\(831\) −12247.8 −0.511276
\(832\) 40520.6 1.68846
\(833\) 4523.47 0.188150
\(834\) 15525.7 0.644616
\(835\) −18925.6 −0.784367
\(836\) −3570.65 −0.147720
\(837\) −5923.68 −0.244626
\(838\) −39684.5 −1.63589
\(839\) −18228.3 −0.750074 −0.375037 0.927010i \(-0.622370\pi\)
−0.375037 + 0.927010i \(0.622370\pi\)
\(840\) −28333.7 −1.16382
\(841\) −7864.78 −0.322472
\(842\) −67782.3 −2.77427
\(843\) 3101.36 0.126710
\(844\) −91318.7 −3.72431
\(845\) 26613.3 1.08346
\(846\) 24283.4 0.986855
\(847\) −931.424 −0.0377852
\(848\) −117825. −4.77137
\(849\) 23427.4 0.947028
\(850\) −13739.7 −0.554433
\(851\) 1634.77 0.0658511
\(852\) −37337.6 −1.50137
\(853\) −21737.3 −0.872534 −0.436267 0.899817i \(-0.643700\pi\)
−0.436267 + 0.899817i \(0.643700\pi\)
\(854\) 9251.79 0.370714
\(855\) −2297.51 −0.0918987
\(856\) 83599.0 3.33803
\(857\) −18712.2 −0.745852 −0.372926 0.927861i \(-0.621645\pi\)
−0.372926 + 0.927861i \(0.621645\pi\)
\(858\) 4448.10 0.176988
\(859\) 30527.6 1.21256 0.606279 0.795252i \(-0.292661\pi\)
0.606279 + 0.795252i \(0.292661\pi\)
\(860\) −23450.8 −0.929845
\(861\) −10621.5 −0.420418
\(862\) 32953.4 1.30209
\(863\) 10906.4 0.430196 0.215098 0.976592i \(-0.430993\pi\)
0.215098 + 0.976592i \(0.430993\pi\)
\(864\) −17020.8 −0.670209
\(865\) −775.255 −0.0304734
\(866\) 30603.5 1.20087
\(867\) 13976.6 0.547484
\(868\) −36182.5 −1.41488
\(869\) −11386.3 −0.444481
\(870\) −35245.7 −1.37350
\(871\) −21388.5 −0.832058
\(872\) −133138. −5.17043
\(873\) 6475.41 0.251042
\(874\) 1459.28 0.0564768
\(875\) 4394.68 0.169791
\(876\) −32786.8 −1.26457
\(877\) 21770.9 0.838256 0.419128 0.907927i \(-0.362336\pi\)
0.419128 + 0.907927i \(0.362336\pi\)
\(878\) 59109.8 2.27205
\(879\) 5847.58 0.224384
\(880\) 41443.4 1.58757
\(881\) 47206.9 1.80527 0.902634 0.430409i \(-0.141631\pi\)
0.902634 + 0.430409i \(0.141631\pi\)
\(882\) −13852.4 −0.528837
\(883\) −6059.68 −0.230945 −0.115473 0.993311i \(-0.536838\pi\)
−0.115473 + 0.993311i \(0.536838\pi\)
\(884\) −8487.09 −0.322909
\(885\) −29238.3 −1.11055
\(886\) −41881.1 −1.58806
\(887\) −37130.2 −1.40553 −0.702767 0.711420i \(-0.748052\pi\)
−0.702767 + 0.711420i \(0.748052\pi\)
\(888\) −20113.6 −0.760101
\(889\) −5090.53 −0.192048
\(890\) 2142.41 0.0806896
\(891\) −891.000 −0.0335013
\(892\) −29518.5 −1.10802
\(893\) 7536.30 0.282410
\(894\) 33311.6 1.24621
\(895\) 14017.8 0.523536
\(896\) −29269.1 −1.09131
\(897\) −1323.63 −0.0492694
\(898\) −40541.4 −1.50655
\(899\) −28202.5 −1.04628
\(900\) 30636.0 1.13467
\(901\) 8400.15 0.310599
\(902\) 27444.1 1.01307
\(903\) 1500.24 0.0552879
\(904\) 82024.1 3.01779
\(905\) 30507.0 1.12054
\(906\) −7743.85 −0.283965
\(907\) 1182.94 0.0433064 0.0216532 0.999766i \(-0.493107\pi\)
0.0216532 + 0.999766i \(0.493107\pi\)
\(908\) −26293.8 −0.961001
\(909\) 16841.4 0.614516
\(910\) 17482.1 0.636841
\(911\) 37676.4 1.37022 0.685112 0.728438i \(-0.259753\pi\)
0.685112 + 0.728438i \(0.259753\pi\)
\(912\) −10163.9 −0.369035
\(913\) −6668.32 −0.241719
\(914\) 60431.1 2.18696
\(915\) −11199.6 −0.404641
\(916\) 83347.1 3.00640
\(917\) −4789.41 −0.172476
\(918\) 2334.86 0.0839454
\(919\) 8697.82 0.312203 0.156101 0.987741i \(-0.450107\pi\)
0.156101 + 0.987741i \(0.450107\pi\)
\(920\) −21785.0 −0.780686
\(921\) −7092.26 −0.253744
\(922\) 77725.7 2.77631
\(923\) 14435.2 0.514778
\(924\) −5442.33 −0.193766
\(925\) 14628.5 0.519981
\(926\) 63795.3 2.26398
\(927\) −3858.71 −0.136717
\(928\) −81036.0 −2.86653
\(929\) −17247.5 −0.609119 −0.304559 0.952493i \(-0.598509\pi\)
−0.304559 + 0.952493i \(0.598509\pi\)
\(930\) 60155.2 2.12104
\(931\) −4299.06 −0.151338
\(932\) −68946.7 −2.42320
\(933\) 5967.51 0.209397
\(934\) 64306.0 2.25284
\(935\) −2954.65 −0.103345
\(936\) 16285.4 0.568702
\(937\) −41812.4 −1.45779 −0.728896 0.684624i \(-0.759966\pi\)
−0.728896 + 0.684624i \(0.759966\pi\)
\(938\) 35941.0 1.25108
\(939\) 11636.0 0.404395
\(940\) −179553. −6.23018
\(941\) 37655.9 1.30451 0.652257 0.757998i \(-0.273822\pi\)
0.652257 + 0.757998i \(0.273822\pi\)
\(942\) 10536.4 0.364431
\(943\) −8166.60 −0.282016
\(944\) −129346. −4.45959
\(945\) −3501.84 −0.120545
\(946\) −3876.37 −0.133226
\(947\) −21244.4 −0.728986 −0.364493 0.931206i \(-0.618758\pi\)
−0.364493 + 0.931206i \(0.618758\pi\)
\(948\) −66530.4 −2.27933
\(949\) 12675.8 0.433587
\(950\) 13058.1 0.445958
\(951\) −8741.20 −0.298058
\(952\) 8936.24 0.304228
\(953\) 1324.27 0.0450130 0.0225065 0.999747i \(-0.492835\pi\)
0.0225065 + 0.999747i \(0.492835\pi\)
\(954\) −25724.1 −0.873007
\(955\) −7731.91 −0.261988
\(956\) −9171.77 −0.310289
\(957\) −4242.04 −0.143287
\(958\) −7185.62 −0.242335
\(959\) 14416.6 0.485439
\(960\) 82425.4 2.77111
\(961\) 18343.4 0.615735
\(962\) 12410.2 0.415927
\(963\) 10332.2 0.345744
\(964\) 26376.9 0.881268
\(965\) −29972.6 −0.999847
\(966\) 2224.21 0.0740815
\(967\) 52267.1 1.73815 0.869077 0.494676i \(-0.164713\pi\)
0.869077 + 0.494676i \(0.164713\pi\)
\(968\) 8811.20 0.292565
\(969\) 724.619 0.0240228
\(970\) −65758.1 −2.17667
\(971\) −52489.8 −1.73479 −0.867394 0.497622i \(-0.834207\pi\)
−0.867394 + 0.497622i \(0.834207\pi\)
\(972\) −5206.14 −0.171797
\(973\) 7344.07 0.241973
\(974\) 101092. 3.32566
\(975\) −11844.3 −0.389046
\(976\) −49545.3 −1.62490
\(977\) 8324.11 0.272581 0.136291 0.990669i \(-0.456482\pi\)
0.136291 + 0.990669i \(0.456482\pi\)
\(978\) −17790.5 −0.581675
\(979\) 257.852 0.00841777
\(980\) 102426. 3.33864
\(981\) −16454.9 −0.535539
\(982\) 675.623 0.0219552
\(983\) −44407.1 −1.44086 −0.720431 0.693527i \(-0.756056\pi\)
−0.720431 + 0.693527i \(0.756056\pi\)
\(984\) 100479. 3.25523
\(985\) −89375.3 −2.89110
\(986\) 11116.2 0.359039
\(987\) 11486.7 0.370442
\(988\) 8066.05 0.259732
\(989\) 1153.50 0.0370870
\(990\) 9048.15 0.290474
\(991\) −45124.7 −1.44645 −0.723226 0.690612i \(-0.757341\pi\)
−0.723226 + 0.690612i \(0.757341\pi\)
\(992\) 138307. 4.42667
\(993\) −24313.4 −0.777001
\(994\) −24256.7 −0.774020
\(995\) 86576.2 2.75844
\(996\) −38963.2 −1.23955
\(997\) 5480.61 0.174095 0.0870474 0.996204i \(-0.472257\pi\)
0.0870474 + 0.996204i \(0.472257\pi\)
\(998\) 55492.9 1.76012
\(999\) −2485.90 −0.0787291
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 33.4.a.c.1.2 2
3.2 odd 2 99.4.a.f.1.1 2
4.3 odd 2 528.4.a.p.1.1 2
5.2 odd 4 825.4.c.h.199.4 4
5.3 odd 4 825.4.c.h.199.1 4
5.4 even 2 825.4.a.l.1.1 2
7.6 odd 2 1617.4.a.k.1.2 2
8.3 odd 2 2112.4.a.bg.1.2 2
8.5 even 2 2112.4.a.bn.1.2 2
11.10 odd 2 363.4.a.i.1.1 2
12.11 even 2 1584.4.a.bj.1.2 2
15.14 odd 2 2475.4.a.p.1.2 2
33.32 even 2 1089.4.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.2 2 1.1 even 1 trivial
99.4.a.f.1.1 2 3.2 odd 2
363.4.a.i.1.1 2 11.10 odd 2
528.4.a.p.1.1 2 4.3 odd 2
825.4.a.l.1.1 2 5.4 even 2
825.4.c.h.199.1 4 5.3 odd 4
825.4.c.h.199.4 4 5.2 odd 4
1089.4.a.u.1.2 2 33.32 even 2
1584.4.a.bj.1.2 2 12.11 even 2
1617.4.a.k.1.2 2 7.6 odd 2
2112.4.a.bg.1.2 2 8.3 odd 2
2112.4.a.bn.1.2 2 8.5 even 2
2475.4.a.p.1.2 2 15.14 odd 2