Properties

Label 624.4.a.l.1.1
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
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] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{43}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.55744\) of defining polynomial
Character \(\chi\) \(=\) 624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -7.11488 q^{5} -8.88512 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -7.11488 q^{5} -8.88512 q^{7} +9.00000 q^{9} -26.0000 q^{11} -13.0000 q^{13} +21.3446 q^{15} +16.2298 q^{17} -35.5744 q^{19} +26.6554 q^{21} -153.379 q^{23} -74.3785 q^{25} -27.0000 q^{27} +223.379 q^{29} -126.723 q^{31} +78.0000 q^{33} +63.2166 q^{35} +217.608 q^{37} +39.0000 q^{39} +105.804 q^{41} +183.608 q^{43} -64.0339 q^{45} -96.6893 q^{47} -264.055 q^{49} -48.6893 q^{51} +386.460 q^{53} +184.987 q^{55} +106.723 q^{57} -34.5273 q^{59} +274.919 q^{61} -79.9661 q^{63} +92.4934 q^{65} -93.9397 q^{67} +460.136 q^{69} +741.812 q^{71} +640.689 q^{73} +223.136 q^{75} +231.013 q^{77} +182.945 q^{79} +81.0000 q^{81} +288.825 q^{83} -115.473 q^{85} -670.136 q^{87} +963.115 q^{89} +115.507 q^{91} +380.169 q^{93} +253.107 q^{95} -481.932 q^{97} -234.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 12 q^{5} - 44 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 12 q^{5} - 44 q^{7} + 18 q^{9} - 52 q^{11} - 26 q^{13} - 36 q^{15} - 20 q^{17} + 60 q^{19} + 132 q^{21} + 8 q^{23} + 166 q^{25} - 54 q^{27} + 132 q^{29} + 140 q^{31} + 156 q^{33} - 608 q^{35} + 68 q^{37} + 78 q^{39} + 28 q^{41} + 108 q^{45} - 36 q^{47} + 626 q^{49} + 60 q^{51} + 668 q^{53} - 312 q^{55} - 180 q^{57} + 508 q^{59} + 340 q^{61} - 396 q^{63} - 156 q^{65} + 940 q^{67} - 24 q^{69} - 300 q^{71} + 1124 q^{73} - 498 q^{75} + 1144 q^{77} + 1520 q^{79} + 162 q^{81} - 524 q^{83} - 808 q^{85} - 396 q^{87} + 1900 q^{89} + 572 q^{91} - 420 q^{93} + 2080 q^{95} - 1436 q^{97} - 468 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −7.11488 −0.636374 −0.318187 0.948028i \(-0.603074\pi\)
−0.318187 + 0.948028i \(0.603074\pi\)
\(6\) 0 0
\(7\) −8.88512 −0.479752 −0.239876 0.970804i \(-0.577107\pi\)
−0.239876 + 0.970804i \(0.577107\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −26.0000 −0.712663 −0.356332 0.934360i \(-0.615973\pi\)
−0.356332 + 0.934360i \(0.615973\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 21.3446 0.367411
\(16\) 0 0
\(17\) 16.2298 0.231547 0.115773 0.993276i \(-0.463065\pi\)
0.115773 + 0.993276i \(0.463065\pi\)
\(18\) 0 0
\(19\) −35.5744 −0.429543 −0.214772 0.976664i \(-0.568901\pi\)
−0.214772 + 0.976664i \(0.568901\pi\)
\(20\) 0 0
\(21\) 26.6554 0.276985
\(22\) 0 0
\(23\) −153.379 −1.39051 −0.695253 0.718765i \(-0.744708\pi\)
−0.695253 + 0.718765i \(0.744708\pi\)
\(24\) 0 0
\(25\) −74.3785 −0.595028
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 223.379 1.43036 0.715178 0.698942i \(-0.246345\pi\)
0.715178 + 0.698942i \(0.246345\pi\)
\(30\) 0 0
\(31\) −126.723 −0.734198 −0.367099 0.930182i \(-0.619649\pi\)
−0.367099 + 0.930182i \(0.619649\pi\)
\(32\) 0 0
\(33\) 78.0000 0.411456
\(34\) 0 0
\(35\) 63.2166 0.305301
\(36\) 0 0
\(37\) 217.608 0.966881 0.483440 0.875377i \(-0.339387\pi\)
0.483440 + 0.875377i \(0.339387\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) 105.804 0.403020 0.201510 0.979486i \(-0.435415\pi\)
0.201510 + 0.979486i \(0.435415\pi\)
\(42\) 0 0
\(43\) 183.608 0.651163 0.325581 0.945514i \(-0.394440\pi\)
0.325581 + 0.945514i \(0.394440\pi\)
\(44\) 0 0
\(45\) −64.0339 −0.212125
\(46\) 0 0
\(47\) −96.6893 −0.300076 −0.150038 0.988680i \(-0.547940\pi\)
−0.150038 + 0.988680i \(0.547940\pi\)
\(48\) 0 0
\(49\) −264.055 −0.769838
\(50\) 0 0
\(51\) −48.6893 −0.133684
\(52\) 0 0
\(53\) 386.460 1.00159 0.500795 0.865566i \(-0.333041\pi\)
0.500795 + 0.865566i \(0.333041\pi\)
\(54\) 0 0
\(55\) 184.987 0.453520
\(56\) 0 0
\(57\) 106.723 0.247997
\(58\) 0 0
\(59\) −34.5273 −0.0761876 −0.0380938 0.999274i \(-0.512129\pi\)
−0.0380938 + 0.999274i \(0.512129\pi\)
\(60\) 0 0
\(61\) 274.919 0.577045 0.288523 0.957473i \(-0.406836\pi\)
0.288523 + 0.957473i \(0.406836\pi\)
\(62\) 0 0
\(63\) −79.9661 −0.159917
\(64\) 0 0
\(65\) 92.4934 0.176498
\(66\) 0 0
\(67\) −93.9397 −0.171292 −0.0856460 0.996326i \(-0.527295\pi\)
−0.0856460 + 0.996326i \(0.527295\pi\)
\(68\) 0 0
\(69\) 460.136 0.802809
\(70\) 0 0
\(71\) 741.812 1.23996 0.619978 0.784619i \(-0.287142\pi\)
0.619978 + 0.784619i \(0.287142\pi\)
\(72\) 0 0
\(73\) 640.689 1.02722 0.513610 0.858024i \(-0.328308\pi\)
0.513610 + 0.858024i \(0.328308\pi\)
\(74\) 0 0
\(75\) 223.136 0.343540
\(76\) 0 0
\(77\) 231.013 0.341901
\(78\) 0 0
\(79\) 182.945 0.260544 0.130272 0.991478i \(-0.458415\pi\)
0.130272 + 0.991478i \(0.458415\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 288.825 0.381960 0.190980 0.981594i \(-0.438834\pi\)
0.190980 + 0.981594i \(0.438834\pi\)
\(84\) 0 0
\(85\) −115.473 −0.147350
\(86\) 0 0
\(87\) −670.136 −0.825817
\(88\) 0 0
\(89\) 963.115 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(90\) 0 0
\(91\) 115.507 0.133059
\(92\) 0 0
\(93\) 380.169 0.423890
\(94\) 0 0
\(95\) 253.107 0.273350
\(96\) 0 0
\(97\) −481.932 −0.504462 −0.252231 0.967667i \(-0.581164\pi\)
−0.252231 + 0.967667i \(0.581164\pi\)
\(98\) 0 0
\(99\) −234.000 −0.237554
\(100\) 0 0
\(101\) 1138.85 1.12198 0.560990 0.827823i \(-0.310421\pi\)
0.560990 + 0.827823i \(0.310421\pi\)
\(102\) 0 0
\(103\) 1090.96 1.04365 0.521823 0.853054i \(-0.325252\pi\)
0.521823 + 0.853054i \(0.325252\pi\)
\(104\) 0 0
\(105\) −189.650 −0.176266
\(106\) 0 0
\(107\) −124.595 −0.112571 −0.0562853 0.998415i \(-0.517926\pi\)
−0.0562853 + 0.998415i \(0.517926\pi\)
\(108\) 0 0
\(109\) −1400.89 −1.23102 −0.615510 0.788129i \(-0.711050\pi\)
−0.615510 + 0.788129i \(0.711050\pi\)
\(110\) 0 0
\(111\) −652.825 −0.558229
\(112\) 0 0
\(113\) −541.812 −0.451056 −0.225528 0.974237i \(-0.572411\pi\)
−0.225528 + 0.974237i \(0.572411\pi\)
\(114\) 0 0
\(115\) 1091.27 0.884882
\(116\) 0 0
\(117\) −117.000 −0.0924500
\(118\) 0 0
\(119\) −144.203 −0.111085
\(120\) 0 0
\(121\) −655.000 −0.492111
\(122\) 0 0
\(123\) −317.412 −0.232684
\(124\) 0 0
\(125\) 1418.55 1.01503
\(126\) 0 0
\(127\) 1552.07 1.08444 0.542219 0.840237i \(-0.317584\pi\)
0.542219 + 0.840237i \(0.317584\pi\)
\(128\) 0 0
\(129\) −550.825 −0.375949
\(130\) 0 0
\(131\) 2281.97 1.52196 0.760981 0.648774i \(-0.224718\pi\)
0.760981 + 0.648774i \(0.224718\pi\)
\(132\) 0 0
\(133\) 316.083 0.206074
\(134\) 0 0
\(135\) 192.102 0.122470
\(136\) 0 0
\(137\) −1443.06 −0.899921 −0.449960 0.893049i \(-0.648562\pi\)
−0.449960 + 0.893049i \(0.648562\pi\)
\(138\) 0 0
\(139\) 3060.41 1.86748 0.933742 0.357946i \(-0.116523\pi\)
0.933742 + 0.357946i \(0.116523\pi\)
\(140\) 0 0
\(141\) 290.068 0.173249
\(142\) 0 0
\(143\) 338.000 0.197657
\(144\) 0 0
\(145\) −1589.31 −0.910242
\(146\) 0 0
\(147\) 792.164 0.444466
\(148\) 0 0
\(149\) 302.520 0.166331 0.0831657 0.996536i \(-0.473497\pi\)
0.0831657 + 0.996536i \(0.473497\pi\)
\(150\) 0 0
\(151\) −383.574 −0.206721 −0.103360 0.994644i \(-0.532960\pi\)
−0.103360 + 0.994644i \(0.532960\pi\)
\(152\) 0 0
\(153\) 146.068 0.0771822
\(154\) 0 0
\(155\) 901.620 0.467225
\(156\) 0 0
\(157\) −2583.38 −1.31322 −0.656612 0.754229i \(-0.728011\pi\)
−0.656612 + 0.754229i \(0.728011\pi\)
\(158\) 0 0
\(159\) −1159.38 −0.578269
\(160\) 0 0
\(161\) 1362.79 0.667097
\(162\) 0 0
\(163\) −1303.45 −0.626346 −0.313173 0.949696i \(-0.601392\pi\)
−0.313173 + 0.949696i \(0.601392\pi\)
\(164\) 0 0
\(165\) −554.960 −0.261840
\(166\) 0 0
\(167\) −1040.04 −0.481921 −0.240961 0.970535i \(-0.577462\pi\)
−0.240961 + 0.970535i \(0.577462\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −320.169 −0.143181
\(172\) 0 0
\(173\) 2989.10 1.31362 0.656811 0.754055i \(-0.271905\pi\)
0.656811 + 0.754055i \(0.271905\pi\)
\(174\) 0 0
\(175\) 660.862 0.285466
\(176\) 0 0
\(177\) 103.582 0.0439870
\(178\) 0 0
\(179\) 1786.06 0.745788 0.372894 0.927874i \(-0.378365\pi\)
0.372894 + 0.927874i \(0.378365\pi\)
\(180\) 0 0
\(181\) −1320.84 −0.542416 −0.271208 0.962521i \(-0.587423\pi\)
−0.271208 + 0.962521i \(0.587423\pi\)
\(182\) 0 0
\(183\) −824.757 −0.333157
\(184\) 0 0
\(185\) −1548.26 −0.615298
\(186\) 0 0
\(187\) −421.974 −0.165015
\(188\) 0 0
\(189\) 239.898 0.0923282
\(190\) 0 0
\(191\) 436.595 0.165397 0.0826987 0.996575i \(-0.473646\pi\)
0.0826987 + 0.996575i \(0.473646\pi\)
\(192\) 0 0
\(193\) −2402.19 −0.895924 −0.447962 0.894053i \(-0.647850\pi\)
−0.447962 + 0.894053i \(0.647850\pi\)
\(194\) 0 0
\(195\) −277.480 −0.101901
\(196\) 0 0
\(197\) 1129.94 0.408654 0.204327 0.978903i \(-0.434499\pi\)
0.204327 + 0.978903i \(0.434499\pi\)
\(198\) 0 0
\(199\) −1859.61 −0.662433 −0.331216 0.943555i \(-0.607459\pi\)
−0.331216 + 0.943555i \(0.607459\pi\)
\(200\) 0 0
\(201\) 281.819 0.0988955
\(202\) 0 0
\(203\) −1984.75 −0.686216
\(204\) 0 0
\(205\) −752.783 −0.256472
\(206\) 0 0
\(207\) −1380.41 −0.463502
\(208\) 0 0
\(209\) 924.934 0.306120
\(210\) 0 0
\(211\) −859.729 −0.280503 −0.140252 0.990116i \(-0.544791\pi\)
−0.140252 + 0.990116i \(0.544791\pi\)
\(212\) 0 0
\(213\) −2225.43 −0.715889
\(214\) 0 0
\(215\) −1306.35 −0.414383
\(216\) 0 0
\(217\) 1125.95 0.352233
\(218\) 0 0
\(219\) −1922.07 −0.593065
\(220\) 0 0
\(221\) −210.987 −0.0642195
\(222\) 0 0
\(223\) −265.345 −0.0796807 −0.0398403 0.999206i \(-0.512685\pi\)
−0.0398403 + 0.999206i \(0.512685\pi\)
\(224\) 0 0
\(225\) −669.407 −0.198343
\(226\) 0 0
\(227\) −1519.43 −0.444265 −0.222133 0.975016i \(-0.571302\pi\)
−0.222133 + 0.975016i \(0.571302\pi\)
\(228\) 0 0
\(229\) 2069.00 0.597045 0.298522 0.954403i \(-0.403506\pi\)
0.298522 + 0.954403i \(0.403506\pi\)
\(230\) 0 0
\(231\) −693.040 −0.197397
\(232\) 0 0
\(233\) −4624.94 −1.30039 −0.650193 0.759769i \(-0.725312\pi\)
−0.650193 + 0.759769i \(0.725312\pi\)
\(234\) 0 0
\(235\) 687.932 0.190961
\(236\) 0 0
\(237\) −548.836 −0.150425
\(238\) 0 0
\(239\) −2098.35 −0.567913 −0.283957 0.958837i \(-0.591647\pi\)
−0.283957 + 0.958837i \(0.591647\pi\)
\(240\) 0 0
\(241\) −1345.96 −0.359755 −0.179878 0.983689i \(-0.557570\pi\)
−0.179878 + 0.983689i \(0.557570\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 1878.72 0.489905
\(246\) 0 0
\(247\) 462.467 0.119134
\(248\) 0 0
\(249\) −866.475 −0.220524
\(250\) 0 0
\(251\) −4117.24 −1.03537 −0.517685 0.855571i \(-0.673206\pi\)
−0.517685 + 0.855571i \(0.673206\pi\)
\(252\) 0 0
\(253\) 3987.84 0.990962
\(254\) 0 0
\(255\) 346.418 0.0850727
\(256\) 0 0
\(257\) 3393.64 0.823694 0.411847 0.911253i \(-0.364884\pi\)
0.411847 + 0.911253i \(0.364884\pi\)
\(258\) 0 0
\(259\) −1933.48 −0.463862
\(260\) 0 0
\(261\) 2010.41 0.476786
\(262\) 0 0
\(263\) 6335.19 1.48534 0.742671 0.669657i \(-0.233559\pi\)
0.742671 + 0.669657i \(0.233559\pi\)
\(264\) 0 0
\(265\) −2749.61 −0.637386
\(266\) 0 0
\(267\) −2889.34 −0.662266
\(268\) 0 0
\(269\) 619.702 0.140461 0.0702303 0.997531i \(-0.477627\pi\)
0.0702303 + 0.997531i \(0.477627\pi\)
\(270\) 0 0
\(271\) −5165.75 −1.15792 −0.578961 0.815355i \(-0.696542\pi\)
−0.578961 + 0.815355i \(0.696542\pi\)
\(272\) 0 0
\(273\) −346.520 −0.0768217
\(274\) 0 0
\(275\) 1933.84 0.424055
\(276\) 0 0
\(277\) 3534.38 0.766643 0.383322 0.923615i \(-0.374780\pi\)
0.383322 + 0.923615i \(0.374780\pi\)
\(278\) 0 0
\(279\) −1140.51 −0.244733
\(280\) 0 0
\(281\) 4115.81 0.873767 0.436883 0.899518i \(-0.356082\pi\)
0.436883 + 0.899518i \(0.356082\pi\)
\(282\) 0 0
\(283\) −3944.56 −0.828550 −0.414275 0.910152i \(-0.635965\pi\)
−0.414275 + 0.910152i \(0.635965\pi\)
\(284\) 0 0
\(285\) −759.322 −0.157819
\(286\) 0 0
\(287\) −940.083 −0.193350
\(288\) 0 0
\(289\) −4649.60 −0.946386
\(290\) 0 0
\(291\) 1445.80 0.291251
\(292\) 0 0
\(293\) 772.441 0.154015 0.0770076 0.997031i \(-0.475463\pi\)
0.0770076 + 0.997031i \(0.475463\pi\)
\(294\) 0 0
\(295\) 245.657 0.0484838
\(296\) 0 0
\(297\) 702.000 0.137152
\(298\) 0 0
\(299\) 1993.92 0.385657
\(300\) 0 0
\(301\) −1631.38 −0.312396
\(302\) 0 0
\(303\) −3416.55 −0.647775
\(304\) 0 0
\(305\) −1956.01 −0.367217
\(306\) 0 0
\(307\) −4885.74 −0.908286 −0.454143 0.890929i \(-0.650054\pi\)
−0.454143 + 0.890929i \(0.650054\pi\)
\(308\) 0 0
\(309\) −3272.88 −0.602549
\(310\) 0 0
\(311\) −177.544 −0.0323717 −0.0161859 0.999869i \(-0.505152\pi\)
−0.0161859 + 0.999869i \(0.505152\pi\)
\(312\) 0 0
\(313\) −6677.03 −1.20578 −0.602888 0.797826i \(-0.705983\pi\)
−0.602888 + 0.797826i \(0.705983\pi\)
\(314\) 0 0
\(315\) 568.949 0.101767
\(316\) 0 0
\(317\) 1806.53 0.320079 0.160040 0.987111i \(-0.448838\pi\)
0.160040 + 0.987111i \(0.448838\pi\)
\(318\) 0 0
\(319\) −5807.84 −1.01936
\(320\) 0 0
\(321\) 373.785 0.0649927
\(322\) 0 0
\(323\) −577.364 −0.0994593
\(324\) 0 0
\(325\) 966.921 0.165031
\(326\) 0 0
\(327\) 4202.68 0.710730
\(328\) 0 0
\(329\) 859.096 0.143962
\(330\) 0 0
\(331\) −11447.2 −1.90088 −0.950442 0.310901i \(-0.899369\pi\)
−0.950442 + 0.310901i \(0.899369\pi\)
\(332\) 0 0
\(333\) 1958.47 0.322294
\(334\) 0 0
\(335\) 668.370 0.109006
\(336\) 0 0
\(337\) −821.752 −0.132830 −0.0664149 0.997792i \(-0.521156\pi\)
−0.0664149 + 0.997792i \(0.521156\pi\)
\(338\) 0 0
\(339\) 1625.43 0.260417
\(340\) 0 0
\(341\) 3294.80 0.523236
\(342\) 0 0
\(343\) 5393.75 0.849083
\(344\) 0 0
\(345\) −3273.81 −0.510887
\(346\) 0 0
\(347\) 4819.55 0.745611 0.372805 0.927910i \(-0.378396\pi\)
0.372805 + 0.927910i \(0.378396\pi\)
\(348\) 0 0
\(349\) 7922.72 1.21517 0.607584 0.794255i \(-0.292139\pi\)
0.607584 + 0.794255i \(0.292139\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) 0 0
\(353\) −2860.73 −0.431336 −0.215668 0.976467i \(-0.569193\pi\)
−0.215668 + 0.976467i \(0.569193\pi\)
\(354\) 0 0
\(355\) −5277.90 −0.789075
\(356\) 0 0
\(357\) 432.610 0.0641349
\(358\) 0 0
\(359\) 11785.8 1.73267 0.866334 0.499464i \(-0.166470\pi\)
0.866334 + 0.499464i \(0.166470\pi\)
\(360\) 0 0
\(361\) −5593.46 −0.815493
\(362\) 0 0
\(363\) 1965.00 0.284121
\(364\) 0 0
\(365\) −4558.43 −0.653696
\(366\) 0 0
\(367\) −2017.19 −0.286911 −0.143456 0.989657i \(-0.545821\pi\)
−0.143456 + 0.989657i \(0.545821\pi\)
\(368\) 0 0
\(369\) 952.237 0.134340
\(370\) 0 0
\(371\) −3433.74 −0.480515
\(372\) 0 0
\(373\) 5148.21 0.714650 0.357325 0.933980i \(-0.383689\pi\)
0.357325 + 0.933980i \(0.383689\pi\)
\(374\) 0 0
\(375\) −4255.66 −0.586030
\(376\) 0 0
\(377\) −2903.92 −0.396710
\(378\) 0 0
\(379\) 3419.10 0.463397 0.231698 0.972788i \(-0.425572\pi\)
0.231698 + 0.972788i \(0.425572\pi\)
\(380\) 0 0
\(381\) −4656.20 −0.626101
\(382\) 0 0
\(383\) 5427.04 0.724044 0.362022 0.932170i \(-0.382087\pi\)
0.362022 + 0.932170i \(0.382087\pi\)
\(384\) 0 0
\(385\) −1643.63 −0.217577
\(386\) 0 0
\(387\) 1652.47 0.217054
\(388\) 0 0
\(389\) 6843.88 0.892027 0.446014 0.895026i \(-0.352843\pi\)
0.446014 + 0.895026i \(0.352843\pi\)
\(390\) 0 0
\(391\) −2489.30 −0.321967
\(392\) 0 0
\(393\) −6845.92 −0.878705
\(394\) 0 0
\(395\) −1301.63 −0.165803
\(396\) 0 0
\(397\) −8382.62 −1.05973 −0.529863 0.848083i \(-0.677757\pi\)
−0.529863 + 0.848083i \(0.677757\pi\)
\(398\) 0 0
\(399\) −948.248 −0.118977
\(400\) 0 0
\(401\) 3525.30 0.439015 0.219508 0.975611i \(-0.429555\pi\)
0.219508 + 0.975611i \(0.429555\pi\)
\(402\) 0 0
\(403\) 1647.40 0.203630
\(404\) 0 0
\(405\) −576.305 −0.0707082
\(406\) 0 0
\(407\) −5657.82 −0.689060
\(408\) 0 0
\(409\) 940.772 0.113736 0.0568682 0.998382i \(-0.481889\pi\)
0.0568682 + 0.998382i \(0.481889\pi\)
\(410\) 0 0
\(411\) 4329.19 0.519569
\(412\) 0 0
\(413\) 306.779 0.0365511
\(414\) 0 0
\(415\) −2054.95 −0.243069
\(416\) 0 0
\(417\) −9181.22 −1.07819
\(418\) 0 0
\(419\) −532.791 −0.0621206 −0.0310603 0.999518i \(-0.509888\pi\)
−0.0310603 + 0.999518i \(0.509888\pi\)
\(420\) 0 0
\(421\) 14425.3 1.66994 0.834970 0.550295i \(-0.185485\pi\)
0.834970 + 0.550295i \(0.185485\pi\)
\(422\) 0 0
\(423\) −870.203 −0.100025
\(424\) 0 0
\(425\) −1207.15 −0.137777
\(426\) 0 0
\(427\) −2442.69 −0.276838
\(428\) 0 0
\(429\) −1014.00 −0.114117
\(430\) 0 0
\(431\) 15224.9 1.70152 0.850762 0.525551i \(-0.176141\pi\)
0.850762 + 0.525551i \(0.176141\pi\)
\(432\) 0 0
\(433\) 10115.5 1.12268 0.561338 0.827587i \(-0.310287\pi\)
0.561338 + 0.827587i \(0.310287\pi\)
\(434\) 0 0
\(435\) 4767.93 0.525528
\(436\) 0 0
\(437\) 5456.35 0.597283
\(438\) 0 0
\(439\) 15925.0 1.73134 0.865669 0.500617i \(-0.166894\pi\)
0.865669 + 0.500617i \(0.166894\pi\)
\(440\) 0 0
\(441\) −2376.49 −0.256613
\(442\) 0 0
\(443\) −2387.09 −0.256014 −0.128007 0.991773i \(-0.540858\pi\)
−0.128007 + 0.991773i \(0.540858\pi\)
\(444\) 0 0
\(445\) −6852.44 −0.729971
\(446\) 0 0
\(447\) −907.559 −0.0960315
\(448\) 0 0
\(449\) −13985.3 −1.46995 −0.734973 0.678096i \(-0.762805\pi\)
−0.734973 + 0.678096i \(0.762805\pi\)
\(450\) 0 0
\(451\) −2750.91 −0.287218
\(452\) 0 0
\(453\) 1150.72 0.119350
\(454\) 0 0
\(455\) −821.815 −0.0846754
\(456\) 0 0
\(457\) −6520.34 −0.667414 −0.333707 0.942677i \(-0.608300\pi\)
−0.333707 + 0.942677i \(0.608300\pi\)
\(458\) 0 0
\(459\) −438.203 −0.0445612
\(460\) 0 0
\(461\) 6543.86 0.661124 0.330562 0.943784i \(-0.392762\pi\)
0.330562 + 0.943784i \(0.392762\pi\)
\(462\) 0 0
\(463\) −13172.7 −1.32221 −0.661107 0.750292i \(-0.729913\pi\)
−0.661107 + 0.750292i \(0.729913\pi\)
\(464\) 0 0
\(465\) −2704.86 −0.269752
\(466\) 0 0
\(467\) 3166.27 0.313742 0.156871 0.987619i \(-0.449859\pi\)
0.156871 + 0.987619i \(0.449859\pi\)
\(468\) 0 0
\(469\) 834.666 0.0821776
\(470\) 0 0
\(471\) 7750.14 0.758190
\(472\) 0 0
\(473\) −4773.82 −0.464060
\(474\) 0 0
\(475\) 2645.97 0.255590
\(476\) 0 0
\(477\) 3478.14 0.333863
\(478\) 0 0
\(479\) 6470.71 0.617232 0.308616 0.951187i \(-0.400134\pi\)
0.308616 + 0.951187i \(0.400134\pi\)
\(480\) 0 0
\(481\) −2828.91 −0.268164
\(482\) 0 0
\(483\) −4088.36 −0.385149
\(484\) 0 0
\(485\) 3428.89 0.321026
\(486\) 0 0
\(487\) −7620.36 −0.709058 −0.354529 0.935045i \(-0.615359\pi\)
−0.354529 + 0.935045i \(0.615359\pi\)
\(488\) 0 0
\(489\) 3910.36 0.361621
\(490\) 0 0
\(491\) 14472.0 1.33017 0.665084 0.746769i \(-0.268396\pi\)
0.665084 + 0.746769i \(0.268396\pi\)
\(492\) 0 0
\(493\) 3625.38 0.331194
\(494\) 0 0
\(495\) 1664.88 0.151173
\(496\) 0 0
\(497\) −6591.09 −0.594871
\(498\) 0 0
\(499\) 12338.1 1.10687 0.553436 0.832892i \(-0.313316\pi\)
0.553436 + 0.832892i \(0.313316\pi\)
\(500\) 0 0
\(501\) 3120.12 0.278237
\(502\) 0 0
\(503\) 12491.1 1.10725 0.553627 0.832765i \(-0.313243\pi\)
0.553627 + 0.832765i \(0.313243\pi\)
\(504\) 0 0
\(505\) −8102.79 −0.713999
\(506\) 0 0
\(507\) −507.000 −0.0444116
\(508\) 0 0
\(509\) −21986.9 −1.91464 −0.957318 0.289036i \(-0.906665\pi\)
−0.957318 + 0.289036i \(0.906665\pi\)
\(510\) 0 0
\(511\) −5692.60 −0.492810
\(512\) 0 0
\(513\) 960.508 0.0826657
\(514\) 0 0
\(515\) −7762.05 −0.664149
\(516\) 0 0
\(517\) 2513.92 0.213853
\(518\) 0 0
\(519\) −8967.29 −0.758421
\(520\) 0 0
\(521\) −6385.80 −0.536981 −0.268491 0.963282i \(-0.586525\pi\)
−0.268491 + 0.963282i \(0.586525\pi\)
\(522\) 0 0
\(523\) 11858.5 0.991467 0.495733 0.868475i \(-0.334899\pi\)
0.495733 + 0.868475i \(0.334899\pi\)
\(524\) 0 0
\(525\) −1982.59 −0.164814
\(526\) 0 0
\(527\) −2056.69 −0.170001
\(528\) 0 0
\(529\) 11358.0 0.933506
\(530\) 0 0
\(531\) −310.746 −0.0253959
\(532\) 0 0
\(533\) −1375.45 −0.111778
\(534\) 0 0
\(535\) 886.479 0.0716370
\(536\) 0 0
\(537\) −5358.17 −0.430581
\(538\) 0 0
\(539\) 6865.42 0.548636
\(540\) 0 0
\(541\) 19726.4 1.56766 0.783829 0.620976i \(-0.213264\pi\)
0.783829 + 0.620976i \(0.213264\pi\)
\(542\) 0 0
\(543\) 3962.52 0.313164
\(544\) 0 0
\(545\) 9967.18 0.783389
\(546\) 0 0
\(547\) 9070.36 0.708996 0.354498 0.935057i \(-0.384652\pi\)
0.354498 + 0.935057i \(0.384652\pi\)
\(548\) 0 0
\(549\) 2474.27 0.192348
\(550\) 0 0
\(551\) −7946.55 −0.614400
\(552\) 0 0
\(553\) −1625.49 −0.124996
\(554\) 0 0
\(555\) 4644.77 0.355242
\(556\) 0 0
\(557\) 21557.9 1.63992 0.819961 0.572419i \(-0.193995\pi\)
0.819961 + 0.572419i \(0.193995\pi\)
\(558\) 0 0
\(559\) −2386.91 −0.180600
\(560\) 0 0
\(561\) 1265.92 0.0952713
\(562\) 0 0
\(563\) −22899.8 −1.71423 −0.857113 0.515128i \(-0.827745\pi\)
−0.857113 + 0.515128i \(0.827745\pi\)
\(564\) 0 0
\(565\) 3854.92 0.287040
\(566\) 0 0
\(567\) −719.695 −0.0533057
\(568\) 0 0
\(569\) −14009.0 −1.03214 −0.516071 0.856546i \(-0.672606\pi\)
−0.516071 + 0.856546i \(0.672606\pi\)
\(570\) 0 0
\(571\) 22800.2 1.67103 0.835517 0.549465i \(-0.185168\pi\)
0.835517 + 0.549465i \(0.185168\pi\)
\(572\) 0 0
\(573\) −1309.79 −0.0954923
\(574\) 0 0
\(575\) 11408.1 0.827390
\(576\) 0 0
\(577\) 16722.7 1.20655 0.603273 0.797535i \(-0.293863\pi\)
0.603273 + 0.797535i \(0.293863\pi\)
\(578\) 0 0
\(579\) 7206.57 0.517262
\(580\) 0 0
\(581\) −2566.24 −0.183246
\(582\) 0 0
\(583\) −10047.9 −0.713797
\(584\) 0 0
\(585\) 832.441 0.0588328
\(586\) 0 0
\(587\) −4728.47 −0.332479 −0.166239 0.986085i \(-0.553162\pi\)
−0.166239 + 0.986085i \(0.553162\pi\)
\(588\) 0 0
\(589\) 4508.10 0.315370
\(590\) 0 0
\(591\) −3389.82 −0.235937
\(592\) 0 0
\(593\) 15903.8 1.10133 0.550667 0.834725i \(-0.314374\pi\)
0.550667 + 0.834725i \(0.314374\pi\)
\(594\) 0 0
\(595\) 1025.99 0.0706915
\(596\) 0 0
\(597\) 5578.82 0.382456
\(598\) 0 0
\(599\) 7515.22 0.512627 0.256313 0.966594i \(-0.417492\pi\)
0.256313 + 0.966594i \(0.417492\pi\)
\(600\) 0 0
\(601\) 26704.5 1.81247 0.906237 0.422769i \(-0.138942\pi\)
0.906237 + 0.422769i \(0.138942\pi\)
\(602\) 0 0
\(603\) −845.457 −0.0570973
\(604\) 0 0
\(605\) 4660.24 0.313167
\(606\) 0 0
\(607\) 1113.76 0.0744748 0.0372374 0.999306i \(-0.488144\pi\)
0.0372374 + 0.999306i \(0.488144\pi\)
\(608\) 0 0
\(609\) 5954.24 0.396187
\(610\) 0 0
\(611\) 1256.96 0.0832261
\(612\) 0 0
\(613\) −12195.8 −0.803559 −0.401780 0.915736i \(-0.631608\pi\)
−0.401780 + 0.915736i \(0.631608\pi\)
\(614\) 0 0
\(615\) 2258.35 0.148074
\(616\) 0 0
\(617\) −447.009 −0.0291668 −0.0145834 0.999894i \(-0.504642\pi\)
−0.0145834 + 0.999894i \(0.504642\pi\)
\(618\) 0 0
\(619\) 22476.3 1.45945 0.729725 0.683740i \(-0.239648\pi\)
0.729725 + 0.683740i \(0.239648\pi\)
\(620\) 0 0
\(621\) 4141.22 0.267603
\(622\) 0 0
\(623\) −8557.39 −0.550313
\(624\) 0 0
\(625\) −795.520 −0.0509132
\(626\) 0 0
\(627\) −2774.80 −0.176738
\(628\) 0 0
\(629\) 3531.73 0.223878
\(630\) 0 0
\(631\) 7165.04 0.452037 0.226019 0.974123i \(-0.427429\pi\)
0.226019 + 0.974123i \(0.427429\pi\)
\(632\) 0 0
\(633\) 2579.19 0.161949
\(634\) 0 0
\(635\) −11042.8 −0.690109
\(636\) 0 0
\(637\) 3432.71 0.213515
\(638\) 0 0
\(639\) 6676.30 0.413319
\(640\) 0 0
\(641\) 12463.4 0.767980 0.383990 0.923337i \(-0.374550\pi\)
0.383990 + 0.923337i \(0.374550\pi\)
\(642\) 0 0
\(643\) −11009.8 −0.675249 −0.337625 0.941281i \(-0.609623\pi\)
−0.337625 + 0.941281i \(0.609623\pi\)
\(644\) 0 0
\(645\) 3919.05 0.239244
\(646\) 0 0
\(647\) 30825.9 1.87309 0.936547 0.350541i \(-0.114002\pi\)
0.936547 + 0.350541i \(0.114002\pi\)
\(648\) 0 0
\(649\) 897.710 0.0542961
\(650\) 0 0
\(651\) −3377.85 −0.203362
\(652\) 0 0
\(653\) 15675.9 0.939426 0.469713 0.882819i \(-0.344357\pi\)
0.469713 + 0.882819i \(0.344357\pi\)
\(654\) 0 0
\(655\) −16236.0 −0.968537
\(656\) 0 0
\(657\) 5766.20 0.342406
\(658\) 0 0
\(659\) −22568.8 −1.33408 −0.667038 0.745024i \(-0.732438\pi\)
−0.667038 + 0.745024i \(0.732438\pi\)
\(660\) 0 0
\(661\) 31243.5 1.83847 0.919237 0.393705i \(-0.128807\pi\)
0.919237 + 0.393705i \(0.128807\pi\)
\(662\) 0 0
\(663\) 632.960 0.0370771
\(664\) 0 0
\(665\) −2248.89 −0.131140
\(666\) 0 0
\(667\) −34261.5 −1.98892
\(668\) 0 0
\(669\) 796.034 0.0460037
\(670\) 0 0
\(671\) −7147.89 −0.411239
\(672\) 0 0
\(673\) −19818.1 −1.13511 −0.567556 0.823335i \(-0.692111\pi\)
−0.567556 + 0.823335i \(0.692111\pi\)
\(674\) 0 0
\(675\) 2008.22 0.114513
\(676\) 0 0
\(677\) 3421.40 0.194232 0.0971159 0.995273i \(-0.469038\pi\)
0.0971159 + 0.995273i \(0.469038\pi\)
\(678\) 0 0
\(679\) 4282.03 0.242016
\(680\) 0 0
\(681\) 4558.29 0.256497
\(682\) 0 0
\(683\) −11002.1 −0.616373 −0.308186 0.951326i \(-0.599722\pi\)
−0.308186 + 0.951326i \(0.599722\pi\)
\(684\) 0 0
\(685\) 10267.2 0.572686
\(686\) 0 0
\(687\) −6206.99 −0.344704
\(688\) 0 0
\(689\) −5023.97 −0.277791
\(690\) 0 0
\(691\) 9817.44 0.540482 0.270241 0.962793i \(-0.412897\pi\)
0.270241 + 0.962793i \(0.412897\pi\)
\(692\) 0 0
\(693\) 2079.12 0.113967
\(694\) 0 0
\(695\) −21774.4 −1.18842
\(696\) 0 0
\(697\) 1717.18 0.0933180
\(698\) 0 0
\(699\) 13874.8 0.750778
\(700\) 0 0
\(701\) −5776.14 −0.311215 −0.155608 0.987819i \(-0.549734\pi\)
−0.155608 + 0.987819i \(0.549734\pi\)
\(702\) 0 0
\(703\) −7741.28 −0.415317
\(704\) 0 0
\(705\) −2063.80 −0.110251
\(706\) 0 0
\(707\) −10118.8 −0.538271
\(708\) 0 0
\(709\) −35654.9 −1.88864 −0.944322 0.329022i \(-0.893281\pi\)
−0.944322 + 0.329022i \(0.893281\pi\)
\(710\) 0 0
\(711\) 1646.51 0.0868480
\(712\) 0 0
\(713\) 19436.6 1.02091
\(714\) 0 0
\(715\) −2404.83 −0.125784
\(716\) 0 0
\(717\) 6295.06 0.327885
\(718\) 0 0
\(719\) −14353.1 −0.744478 −0.372239 0.928137i \(-0.621410\pi\)
−0.372239 + 0.928137i \(0.621410\pi\)
\(720\) 0 0
\(721\) −9693.32 −0.500691
\(722\) 0 0
\(723\) 4037.89 0.207705
\(724\) 0 0
\(725\) −16614.6 −0.851103
\(726\) 0 0
\(727\) 28167.8 1.43698 0.718491 0.695536i \(-0.244833\pi\)
0.718491 + 0.695536i \(0.244833\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 2979.92 0.150775
\(732\) 0 0
\(733\) −23945.2 −1.20660 −0.603299 0.797515i \(-0.706147\pi\)
−0.603299 + 0.797515i \(0.706147\pi\)
\(734\) 0 0
\(735\) −5636.15 −0.282847
\(736\) 0 0
\(737\) 2442.43 0.122073
\(738\) 0 0
\(739\) 4021.49 0.200180 0.100090 0.994978i \(-0.468087\pi\)
0.100090 + 0.994978i \(0.468087\pi\)
\(740\) 0 0
\(741\) −1387.40 −0.0687820
\(742\) 0 0
\(743\) −13584.9 −0.670770 −0.335385 0.942081i \(-0.608866\pi\)
−0.335385 + 0.942081i \(0.608866\pi\)
\(744\) 0 0
\(745\) −2152.39 −0.105849
\(746\) 0 0
\(747\) 2599.42 0.127320
\(748\) 0 0
\(749\) 1107.04 0.0540060
\(750\) 0 0
\(751\) −3537.89 −0.171903 −0.0859517 0.996299i \(-0.527393\pi\)
−0.0859517 + 0.996299i \(0.527393\pi\)
\(752\) 0 0
\(753\) 12351.7 0.597772
\(754\) 0 0
\(755\) 2729.08 0.131552
\(756\) 0 0
\(757\) −4421.46 −0.212286 −0.106143 0.994351i \(-0.533850\pi\)
−0.106143 + 0.994351i \(0.533850\pi\)
\(758\) 0 0
\(759\) −11963.5 −0.572132
\(760\) 0 0
\(761\) 1093.63 0.0520947 0.0260473 0.999661i \(-0.491708\pi\)
0.0260473 + 0.999661i \(0.491708\pi\)
\(762\) 0 0
\(763\) 12447.1 0.590584
\(764\) 0 0
\(765\) −1039.25 −0.0491168
\(766\) 0 0
\(767\) 448.855 0.0211307
\(768\) 0 0
\(769\) −15814.2 −0.741580 −0.370790 0.928717i \(-0.620913\pi\)
−0.370790 + 0.928717i \(0.620913\pi\)
\(770\) 0 0
\(771\) −10180.9 −0.475560
\(772\) 0 0
\(773\) −37380.8 −1.73932 −0.869659 0.493653i \(-0.835661\pi\)
−0.869659 + 0.493653i \(0.835661\pi\)
\(774\) 0 0
\(775\) 9425.48 0.436869
\(776\) 0 0
\(777\) 5800.43 0.267811
\(778\) 0 0
\(779\) −3763.92 −0.173115
\(780\) 0 0
\(781\) −19287.1 −0.883671
\(782\) 0 0
\(783\) −6031.22 −0.275272
\(784\) 0 0
\(785\) 18380.4 0.835701
\(786\) 0 0
\(787\) −1979.23 −0.0896469 −0.0448234 0.998995i \(-0.514273\pi\)
−0.0448234 + 0.998995i \(0.514273\pi\)
\(788\) 0 0
\(789\) −19005.6 −0.857562
\(790\) 0 0
\(791\) 4814.06 0.216395
\(792\) 0 0
\(793\) −3573.95 −0.160044
\(794\) 0 0
\(795\) 8248.84 0.367995
\(796\) 0 0
\(797\) −23575.5 −1.04779 −0.523894 0.851783i \(-0.675521\pi\)
−0.523894 + 0.851783i \(0.675521\pi\)
\(798\) 0 0
\(799\) −1569.24 −0.0694816
\(800\) 0 0
\(801\) 8668.03 0.382359
\(802\) 0 0
\(803\) −16657.9 −0.732061
\(804\) 0 0
\(805\) −9696.06 −0.424523
\(806\) 0 0
\(807\) −1859.11 −0.0810950
\(808\) 0 0
\(809\) −39145.7 −1.70122 −0.850611 0.525796i \(-0.823768\pi\)
−0.850611 + 0.525796i \(0.823768\pi\)
\(810\) 0 0
\(811\) 39342.9 1.70347 0.851735 0.523973i \(-0.175551\pi\)
0.851735 + 0.523973i \(0.175551\pi\)
\(812\) 0 0
\(813\) 15497.3 0.668527
\(814\) 0 0
\(815\) 9273.91 0.398590
\(816\) 0 0
\(817\) −6531.75 −0.279703
\(818\) 0 0
\(819\) 1039.56 0.0443530
\(820\) 0 0
\(821\) 216.569 0.00920622 0.00460311 0.999989i \(-0.498535\pi\)
0.00460311 + 0.999989i \(0.498535\pi\)
\(822\) 0 0
\(823\) −45061.7 −1.90857 −0.954284 0.298901i \(-0.903380\pi\)
−0.954284 + 0.298901i \(0.903380\pi\)
\(824\) 0 0
\(825\) −5801.52 −0.244828
\(826\) 0 0
\(827\) −1228.67 −0.0516625 −0.0258313 0.999666i \(-0.508223\pi\)
−0.0258313 + 0.999666i \(0.508223\pi\)
\(828\) 0 0
\(829\) −29560.3 −1.23845 −0.619224 0.785215i \(-0.712553\pi\)
−0.619224 + 0.785215i \(0.712553\pi\)
\(830\) 0 0
\(831\) −10603.1 −0.442622
\(832\) 0 0
\(833\) −4285.54 −0.178254
\(834\) 0 0
\(835\) 7399.77 0.306682
\(836\) 0 0
\(837\) 3421.53 0.141297
\(838\) 0 0
\(839\) −23680.0 −0.974402 −0.487201 0.873290i \(-0.661982\pi\)
−0.487201 + 0.873290i \(0.661982\pi\)
\(840\) 0 0
\(841\) 25509.0 1.04592
\(842\) 0 0
\(843\) −12347.4 −0.504470
\(844\) 0 0
\(845\) −1202.41 −0.0489518
\(846\) 0 0
\(847\) 5819.76 0.236091
\(848\) 0 0
\(849\) 11833.7 0.478363
\(850\) 0 0
\(851\) −33376.4 −1.34445
\(852\) 0 0
\(853\) 15779.1 0.633374 0.316687 0.948530i \(-0.397430\pi\)
0.316687 + 0.948530i \(0.397430\pi\)
\(854\) 0 0
\(855\) 2277.97 0.0911167
\(856\) 0 0
\(857\) −6759.13 −0.269414 −0.134707 0.990886i \(-0.543009\pi\)
−0.134707 + 0.990886i \(0.543009\pi\)
\(858\) 0 0
\(859\) 20653.1 0.820344 0.410172 0.912008i \(-0.365469\pi\)
0.410172 + 0.912008i \(0.365469\pi\)
\(860\) 0 0
\(861\) 2820.25 0.111630
\(862\) 0 0
\(863\) −6087.19 −0.240105 −0.120052 0.992768i \(-0.538306\pi\)
−0.120052 + 0.992768i \(0.538306\pi\)
\(864\) 0 0
\(865\) −21267.1 −0.835955
\(866\) 0 0
\(867\) 13948.8 0.546396
\(868\) 0 0
\(869\) −4756.58 −0.185680
\(870\) 0 0
\(871\) 1221.22 0.0475078
\(872\) 0 0
\(873\) −4337.39 −0.168154
\(874\) 0 0
\(875\) −12604.0 −0.486964
\(876\) 0 0
\(877\) −6841.87 −0.263436 −0.131718 0.991287i \(-0.542049\pi\)
−0.131718 + 0.991287i \(0.542049\pi\)
\(878\) 0 0
\(879\) −2317.32 −0.0889207
\(880\) 0 0
\(881\) 9416.61 0.360106 0.180053 0.983657i \(-0.442373\pi\)
0.180053 + 0.983657i \(0.442373\pi\)
\(882\) 0 0
\(883\) 21561.3 0.821740 0.410870 0.911694i \(-0.365225\pi\)
0.410870 + 0.911694i \(0.365225\pi\)
\(884\) 0 0
\(885\) −736.972 −0.0279922
\(886\) 0 0
\(887\) 25672.9 0.971828 0.485914 0.874007i \(-0.338487\pi\)
0.485914 + 0.874007i \(0.338487\pi\)
\(888\) 0 0
\(889\) −13790.3 −0.520261
\(890\) 0 0
\(891\) −2106.00 −0.0791848
\(892\) 0 0
\(893\) 3439.66 0.128896
\(894\) 0 0
\(895\) −12707.6 −0.474600
\(896\) 0 0
\(897\) −5981.76 −0.222659
\(898\) 0 0
\(899\) −28307.2 −1.05017
\(900\) 0 0
\(901\) 6272.14 0.231915
\(902\) 0 0
\(903\) 4894.15 0.180362
\(904\) 0 0
\(905\) 9397.61 0.345179
\(906\) 0 0
\(907\) 17612.3 0.644770 0.322385 0.946609i \(-0.395515\pi\)
0.322385 + 0.946609i \(0.395515\pi\)
\(908\) 0 0
\(909\) 10249.7 0.373993
\(910\) 0 0
\(911\) 8830.95 0.321166 0.160583 0.987022i \(-0.448663\pi\)
0.160583 + 0.987022i \(0.448663\pi\)
\(912\) 0 0
\(913\) −7509.45 −0.272209
\(914\) 0 0
\(915\) 5868.04 0.212013
\(916\) 0 0
\(917\) −20275.6 −0.730163
\(918\) 0 0
\(919\) 30983.7 1.11214 0.556071 0.831135i \(-0.312308\pi\)
0.556071 + 0.831135i \(0.312308\pi\)
\(920\) 0 0
\(921\) 14657.2 0.524399
\(922\) 0 0
\(923\) −9643.55 −0.343902
\(924\) 0 0
\(925\) −16185.4 −0.575321
\(926\) 0 0
\(927\) 9818.64 0.347882
\(928\) 0 0
\(929\) −15131.2 −0.534378 −0.267189 0.963644i \(-0.586095\pi\)
−0.267189 + 0.963644i \(0.586095\pi\)
\(930\) 0 0
\(931\) 9393.58 0.330679
\(932\) 0 0
\(933\) 532.632 0.0186898
\(934\) 0 0
\(935\) 3002.29 0.105011
\(936\) 0 0
\(937\) 40264.3 1.40382 0.701909 0.712267i \(-0.252331\pi\)
0.701909 + 0.712267i \(0.252331\pi\)
\(938\) 0 0
\(939\) 20031.1 0.696155
\(940\) 0 0
\(941\) −36254.5 −1.25597 −0.627983 0.778227i \(-0.716119\pi\)
−0.627983 + 0.778227i \(0.716119\pi\)
\(942\) 0 0
\(943\) −16228.1 −0.560402
\(944\) 0 0
\(945\) −1706.85 −0.0587553
\(946\) 0 0
\(947\) −47707.4 −1.63705 −0.818524 0.574473i \(-0.805207\pi\)
−0.818524 + 0.574473i \(0.805207\pi\)
\(948\) 0 0
\(949\) −8328.96 −0.284899
\(950\) 0 0
\(951\) −5419.60 −0.184798
\(952\) 0 0
\(953\) −28934.4 −0.983502 −0.491751 0.870736i \(-0.663643\pi\)
−0.491751 + 0.870736i \(0.663643\pi\)
\(954\) 0 0
\(955\) −3106.32 −0.105255
\(956\) 0 0
\(957\) 17423.5 0.588529
\(958\) 0 0
\(959\) 12821.8 0.431738
\(960\) 0 0
\(961\) −13732.2 −0.460953
\(962\) 0 0
\(963\) −1121.36 −0.0375236
\(964\) 0 0
\(965\) 17091.3 0.570143
\(966\) 0 0
\(967\) −9108.15 −0.302894 −0.151447 0.988465i \(-0.548393\pi\)
−0.151447 + 0.988465i \(0.548393\pi\)
\(968\) 0 0
\(969\) 1732.09 0.0574229
\(970\) 0 0
\(971\) 34912.3 1.15385 0.576925 0.816797i \(-0.304252\pi\)
0.576925 + 0.816797i \(0.304252\pi\)
\(972\) 0 0
\(973\) −27192.1 −0.895928
\(974\) 0 0
\(975\) −2900.76 −0.0952808
\(976\) 0 0
\(977\) −41510.7 −1.35931 −0.679655 0.733532i \(-0.737870\pi\)
−0.679655 + 0.733532i \(0.737870\pi\)
\(978\) 0 0
\(979\) −25041.0 −0.817480
\(980\) 0 0
\(981\) −12608.0 −0.410340
\(982\) 0 0
\(983\) 18673.8 0.605901 0.302950 0.953006i \(-0.402028\pi\)
0.302950 + 0.953006i \(0.402028\pi\)
\(984\) 0 0
\(985\) −8039.38 −0.260057
\(986\) 0 0
\(987\) −2577.29 −0.0831165
\(988\) 0 0
\(989\) −28161.6 −0.905446
\(990\) 0 0
\(991\) 5954.58 0.190871 0.0954356 0.995436i \(-0.469576\pi\)
0.0954356 + 0.995436i \(0.469576\pi\)
\(992\) 0 0
\(993\) 34341.5 1.09748
\(994\) 0 0
\(995\) 13230.9 0.421555
\(996\) 0 0
\(997\) −8918.13 −0.283290 −0.141645 0.989918i \(-0.545239\pi\)
−0.141645 + 0.989918i \(0.545239\pi\)
\(998\) 0 0
\(999\) −5875.42 −0.186076
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 624.4.a.l.1.1 2
3.2 odd 2 1872.4.a.v.1.2 2
4.3 odd 2 312.4.a.f.1.1 2
8.3 odd 2 2496.4.a.u.1.2 2
8.5 even 2 2496.4.a.bd.1.2 2
12.11 even 2 936.4.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.f.1.1 2 4.3 odd 2
624.4.a.l.1.1 2 1.1 even 1 trivial
936.4.a.c.1.2 2 12.11 even 2
1872.4.a.v.1.2 2 3.2 odd 2
2496.4.a.u.1.2 2 8.3 odd 2
2496.4.a.bd.1.2 2 8.5 even 2