Properties

Label 693.4.a.p.1.1
Level $693$
Weight $4$
Character 693.1
Self dual yes
Analytic conductor $40.888$
Analytic rank $0$
Dimension $5$
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] = [693,4,Mod(1,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("693.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 693.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: \(40.8883236340\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 30x^{3} + 11x^{2} + 185x + 162 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.54345\) of defining polynomial
Character \(\chi\) \(=\) 693.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.54345 q^{2} +12.6430 q^{4} -6.53625 q^{5} +7.00000 q^{7} -21.0952 q^{8} +O(q^{10})\) \(q-4.54345 q^{2} +12.6430 q^{4} -6.53625 q^{5} +7.00000 q^{7} -21.0952 q^{8} +29.6972 q^{10} -11.0000 q^{11} +71.3370 q^{13} -31.8042 q^{14} -5.29887 q^{16} -2.45049 q^{17} +80.0091 q^{19} -82.6377 q^{20} +49.9780 q^{22} -61.8164 q^{23} -82.2774 q^{25} -324.117 q^{26} +88.5009 q^{28} +156.839 q^{29} +77.7131 q^{31} +192.837 q^{32} +11.1337 q^{34} -45.7538 q^{35} +84.7185 q^{37} -363.518 q^{38} +137.883 q^{40} -28.8114 q^{41} -352.389 q^{43} -139.073 q^{44} +280.860 q^{46} +256.310 q^{47} +49.0000 q^{49} +373.824 q^{50} +901.913 q^{52} -492.147 q^{53} +71.8988 q^{55} -147.666 q^{56} -712.590 q^{58} +3.13000 q^{59} -159.772 q^{61} -353.086 q^{62} -833.753 q^{64} -466.277 q^{65} -521.200 q^{67} -30.9815 q^{68} +207.880 q^{70} +885.588 q^{71} -375.499 q^{73} -384.914 q^{74} +1011.55 q^{76} -77.0000 q^{77} -1346.03 q^{79} +34.6348 q^{80} +130.903 q^{82} +1379.43 q^{83} +16.0170 q^{85} +1601.07 q^{86} +232.047 q^{88} +111.045 q^{89} +499.359 q^{91} -781.543 q^{92} -1164.53 q^{94} -522.959 q^{95} +472.385 q^{97} -222.629 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8} - 23 q^{10} - 55 q^{11} + 101 q^{13} + 7 q^{14} - 7 q^{16} + 20 q^{17} + 237 q^{19} - 85 q^{20} - 11 q^{22} + 80 q^{23} + 486 q^{25} - 165 q^{26} + 147 q^{28} + 11 q^{29} + 316 q^{31} - 453 q^{32} + 936 q^{34} - 147 q^{35} + 319 q^{37} - 89 q^{38} + 624 q^{40} - 1190 q^{41} + 88 q^{43} - 231 q^{44} + 1000 q^{46} - 377 q^{47} + 245 q^{49} + 644 q^{50} + 1001 q^{52} + 992 q^{53} + 231 q^{55} + 294 q^{56} + 721 q^{58} - 71 q^{59} - 574 q^{61} - 272 q^{62} - 1380 q^{64} - 589 q^{65} - 527 q^{67} + 2974 q^{68} - 161 q^{70} + 1156 q^{71} + 1061 q^{73} + 1609 q^{74} + 2399 q^{76} - 385 q^{77} + 588 q^{79} + 1643 q^{80} - 2602 q^{82} + 212 q^{83} + 1918 q^{85} + 4760 q^{86} - 462 q^{88} - 1030 q^{89} + 707 q^{91} + 1174 q^{92} - 1799 q^{94} + 3593 q^{95} + 2488 q^{97} + 49 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.54345 −1.60635 −0.803177 0.595741i \(-0.796859\pi\)
−0.803177 + 0.595741i \(0.796859\pi\)
\(3\) 0 0
\(4\) 12.6430 1.58037
\(5\) −6.53625 −0.584620 −0.292310 0.956324i \(-0.594424\pi\)
−0.292310 + 0.956324i \(0.594424\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −21.0952 −0.932284
\(9\) 0 0
\(10\) 29.6972 0.939107
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 71.3370 1.52195 0.760974 0.648782i \(-0.224721\pi\)
0.760974 + 0.648782i \(0.224721\pi\)
\(14\) −31.8042 −0.607145
\(15\) 0 0
\(16\) −5.29887 −0.0827949
\(17\) −2.45049 −0.0349607 −0.0174803 0.999847i \(-0.505564\pi\)
−0.0174803 + 0.999847i \(0.505564\pi\)
\(18\) 0 0
\(19\) 80.0091 0.966071 0.483035 0.875601i \(-0.339534\pi\)
0.483035 + 0.875601i \(0.339534\pi\)
\(20\) −82.6377 −0.923917
\(21\) 0 0
\(22\) 49.9780 0.484334
\(23\) −61.8164 −0.560418 −0.280209 0.959939i \(-0.590404\pi\)
−0.280209 + 0.959939i \(0.590404\pi\)
\(24\) 0 0
\(25\) −82.2774 −0.658219
\(26\) −324.117 −2.44479
\(27\) 0 0
\(28\) 88.5009 0.597325
\(29\) 156.839 1.00428 0.502142 0.864785i \(-0.332546\pi\)
0.502142 + 0.864785i \(0.332546\pi\)
\(30\) 0 0
\(31\) 77.7131 0.450248 0.225124 0.974330i \(-0.427721\pi\)
0.225124 + 0.974330i \(0.427721\pi\)
\(32\) 192.837 1.06528
\(33\) 0 0
\(34\) 11.1337 0.0561592
\(35\) −45.7538 −0.220966
\(36\) 0 0
\(37\) 84.7185 0.376422 0.188211 0.982129i \(-0.439731\pi\)
0.188211 + 0.982129i \(0.439731\pi\)
\(38\) −363.518 −1.55185
\(39\) 0 0
\(40\) 137.883 0.545032
\(41\) −28.8114 −0.109746 −0.0548729 0.998493i \(-0.517475\pi\)
−0.0548729 + 0.998493i \(0.517475\pi\)
\(42\) 0 0
\(43\) −352.389 −1.24974 −0.624871 0.780728i \(-0.714848\pi\)
−0.624871 + 0.780728i \(0.714848\pi\)
\(44\) −139.073 −0.476500
\(45\) 0 0
\(46\) 280.860 0.900229
\(47\) 256.310 0.795459 0.397730 0.917503i \(-0.369798\pi\)
0.397730 + 0.917503i \(0.369798\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 373.824 1.05733
\(51\) 0 0
\(52\) 901.913 2.40525
\(53\) −492.147 −1.27550 −0.637751 0.770243i \(-0.720135\pi\)
−0.637751 + 0.770243i \(0.720135\pi\)
\(54\) 0 0
\(55\) 71.8988 0.176270
\(56\) −147.666 −0.352370
\(57\) 0 0
\(58\) −712.590 −1.61323
\(59\) 3.13000 0.00690663 0.00345331 0.999994i \(-0.498901\pi\)
0.00345331 + 0.999994i \(0.498901\pi\)
\(60\) 0 0
\(61\) −159.772 −0.335356 −0.167678 0.985842i \(-0.553627\pi\)
−0.167678 + 0.985842i \(0.553627\pi\)
\(62\) −353.086 −0.723257
\(63\) 0 0
\(64\) −833.753 −1.62842
\(65\) −466.277 −0.889761
\(66\) 0 0
\(67\) −521.200 −0.950370 −0.475185 0.879886i \(-0.657619\pi\)
−0.475185 + 0.879886i \(0.657619\pi\)
\(68\) −30.9815 −0.0552509
\(69\) 0 0
\(70\) 207.880 0.354949
\(71\) 885.588 1.48028 0.740141 0.672452i \(-0.234759\pi\)
0.740141 + 0.672452i \(0.234759\pi\)
\(72\) 0 0
\(73\) −375.499 −0.602039 −0.301020 0.953618i \(-0.597327\pi\)
−0.301020 + 0.953618i \(0.597327\pi\)
\(74\) −384.914 −0.604668
\(75\) 0 0
\(76\) 1011.55 1.52675
\(77\) −77.0000 −0.113961
\(78\) 0 0
\(79\) −1346.03 −1.91697 −0.958484 0.285145i \(-0.907958\pi\)
−0.958484 + 0.285145i \(0.907958\pi\)
\(80\) 34.6348 0.0484036
\(81\) 0 0
\(82\) 130.903 0.176291
\(83\) 1379.43 1.82424 0.912121 0.409920i \(-0.134443\pi\)
0.912121 + 0.409920i \(0.134443\pi\)
\(84\) 0 0
\(85\) 16.0170 0.0204387
\(86\) 1601.07 2.00753
\(87\) 0 0
\(88\) 232.047 0.281094
\(89\) 111.045 0.132255 0.0661276 0.997811i \(-0.478936\pi\)
0.0661276 + 0.997811i \(0.478936\pi\)
\(90\) 0 0
\(91\) 499.359 0.575242
\(92\) −781.543 −0.885669
\(93\) 0 0
\(94\) −1164.53 −1.27779
\(95\) −522.959 −0.564784
\(96\) 0 0
\(97\) 472.385 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(98\) −222.629 −0.229479
\(99\) 0 0
\(100\) −1040.23 −1.04023
\(101\) −1046.51 −1.03101 −0.515505 0.856887i \(-0.672396\pi\)
−0.515505 + 0.856887i \(0.672396\pi\)
\(102\) 0 0
\(103\) 1116.84 1.06840 0.534200 0.845358i \(-0.320613\pi\)
0.534200 + 0.845358i \(0.320613\pi\)
\(104\) −1504.87 −1.41889
\(105\) 0 0
\(106\) 2236.05 2.04891
\(107\) −1484.22 −1.34098 −0.670492 0.741917i \(-0.733917\pi\)
−0.670492 + 0.741917i \(0.733917\pi\)
\(108\) 0 0
\(109\) 1064.65 0.935547 0.467774 0.883848i \(-0.345056\pi\)
0.467774 + 0.883848i \(0.345056\pi\)
\(110\) −326.669 −0.283151
\(111\) 0 0
\(112\) −37.0921 −0.0312935
\(113\) 438.172 0.364776 0.182388 0.983227i \(-0.441617\pi\)
0.182388 + 0.983227i \(0.441617\pi\)
\(114\) 0 0
\(115\) 404.047 0.327631
\(116\) 1982.91 1.58714
\(117\) 0 0
\(118\) −14.2210 −0.0110945
\(119\) −17.1534 −0.0132139
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 725.917 0.538700
\(123\) 0 0
\(124\) 982.525 0.711559
\(125\) 1354.82 0.969428
\(126\) 0 0
\(127\) 1926.01 1.34571 0.672856 0.739774i \(-0.265068\pi\)
0.672856 + 0.739774i \(0.265068\pi\)
\(128\) 2245.43 1.55054
\(129\) 0 0
\(130\) 2118.51 1.42927
\(131\) 703.646 0.469296 0.234648 0.972080i \(-0.424606\pi\)
0.234648 + 0.972080i \(0.424606\pi\)
\(132\) 0 0
\(133\) 560.064 0.365140
\(134\) 2368.05 1.52663
\(135\) 0 0
\(136\) 51.6935 0.0325933
\(137\) 944.945 0.589285 0.294643 0.955608i \(-0.404799\pi\)
0.294643 + 0.955608i \(0.404799\pi\)
\(138\) 0 0
\(139\) 1222.86 0.746200 0.373100 0.927791i \(-0.378295\pi\)
0.373100 + 0.927791i \(0.378295\pi\)
\(140\) −578.464 −0.349208
\(141\) 0 0
\(142\) −4023.63 −2.37786
\(143\) −784.707 −0.458885
\(144\) 0 0
\(145\) −1025.14 −0.587124
\(146\) 1706.06 0.967088
\(147\) 0 0
\(148\) 1071.09 0.594888
\(149\) 2723.95 1.49768 0.748840 0.662750i \(-0.230611\pi\)
0.748840 + 0.662750i \(0.230611\pi\)
\(150\) 0 0
\(151\) 1133.68 0.610980 0.305490 0.952195i \(-0.401180\pi\)
0.305490 + 0.952195i \(0.401180\pi\)
\(152\) −1687.81 −0.900652
\(153\) 0 0
\(154\) 349.846 0.183061
\(155\) −507.952 −0.263224
\(156\) 0 0
\(157\) 3449.91 1.75371 0.876856 0.480753i \(-0.159636\pi\)
0.876856 + 0.480753i \(0.159636\pi\)
\(158\) 6115.64 3.07933
\(159\) 0 0
\(160\) −1260.43 −0.622785
\(161\) −432.715 −0.211818
\(162\) 0 0
\(163\) 917.605 0.440935 0.220467 0.975394i \(-0.429242\pi\)
0.220467 + 0.975394i \(0.429242\pi\)
\(164\) −364.261 −0.173439
\(165\) 0 0
\(166\) −6267.38 −2.93038
\(167\) 3405.63 1.57806 0.789029 0.614355i \(-0.210584\pi\)
0.789029 + 0.614355i \(0.210584\pi\)
\(168\) 0 0
\(169\) 2891.97 1.31633
\(170\) −72.7726 −0.0328318
\(171\) 0 0
\(172\) −4455.25 −1.97506
\(173\) −1874.17 −0.823644 −0.411822 0.911264i \(-0.635108\pi\)
−0.411822 + 0.911264i \(0.635108\pi\)
\(174\) 0 0
\(175\) −575.942 −0.248784
\(176\) 58.2876 0.0249636
\(177\) 0 0
\(178\) −504.527 −0.212449
\(179\) 546.539 0.228214 0.114107 0.993468i \(-0.463599\pi\)
0.114107 + 0.993468i \(0.463599\pi\)
\(180\) 0 0
\(181\) 2927.32 1.20213 0.601066 0.799200i \(-0.294743\pi\)
0.601066 + 0.799200i \(0.294743\pi\)
\(182\) −2268.82 −0.924043
\(183\) 0 0
\(184\) 1304.03 0.522468
\(185\) −553.741 −0.220064
\(186\) 0 0
\(187\) 26.9554 0.0105410
\(188\) 3240.52 1.25712
\(189\) 0 0
\(190\) 2376.04 0.907243
\(191\) 2232.58 0.845780 0.422890 0.906181i \(-0.361016\pi\)
0.422890 + 0.906181i \(0.361016\pi\)
\(192\) 0 0
\(193\) 1440.00 0.537065 0.268533 0.963271i \(-0.413461\pi\)
0.268533 + 0.963271i \(0.413461\pi\)
\(194\) −2146.26 −0.794291
\(195\) 0 0
\(196\) 619.506 0.225768
\(197\) −1026.53 −0.371256 −0.185628 0.982620i \(-0.559432\pi\)
−0.185628 + 0.982620i \(0.559432\pi\)
\(198\) 0 0
\(199\) 1273.27 0.453565 0.226783 0.973945i \(-0.427179\pi\)
0.226783 + 0.973945i \(0.427179\pi\)
\(200\) 1735.66 0.613647
\(201\) 0 0
\(202\) 4754.79 1.65617
\(203\) 1097.87 0.379584
\(204\) 0 0
\(205\) 188.318 0.0641596
\(206\) −5074.29 −1.71623
\(207\) 0 0
\(208\) −378.006 −0.126010
\(209\) −880.100 −0.291281
\(210\) 0 0
\(211\) −1087.89 −0.354944 −0.177472 0.984126i \(-0.556792\pi\)
−0.177472 + 0.984126i \(0.556792\pi\)
\(212\) −6222.21 −2.01577
\(213\) 0 0
\(214\) 6743.50 2.15409
\(215\) 2303.31 0.730624
\(216\) 0 0
\(217\) 543.992 0.170178
\(218\) −4837.17 −1.50282
\(219\) 0 0
\(220\) 909.015 0.278572
\(221\) −174.811 −0.0532083
\(222\) 0 0
\(223\) −1826.52 −0.548487 −0.274244 0.961660i \(-0.588427\pi\)
−0.274244 + 0.961660i \(0.588427\pi\)
\(224\) 1349.86 0.402639
\(225\) 0 0
\(226\) −1990.81 −0.585960
\(227\) −1821.81 −0.532678 −0.266339 0.963879i \(-0.585814\pi\)
−0.266339 + 0.963879i \(0.585814\pi\)
\(228\) 0 0
\(229\) −1369.86 −0.395296 −0.197648 0.980273i \(-0.563330\pi\)
−0.197648 + 0.980273i \(0.563330\pi\)
\(230\) −1835.77 −0.526292
\(231\) 0 0
\(232\) −3308.54 −0.936277
\(233\) 2116.08 0.594975 0.297488 0.954726i \(-0.403851\pi\)
0.297488 + 0.954726i \(0.403851\pi\)
\(234\) 0 0
\(235\) −1675.30 −0.465041
\(236\) 39.5725 0.0109150
\(237\) 0 0
\(238\) 77.9358 0.0212262
\(239\) −5148.64 −1.39346 −0.696732 0.717331i \(-0.745364\pi\)
−0.696732 + 0.717331i \(0.745364\pi\)
\(240\) 0 0
\(241\) −3756.11 −1.00395 −0.501975 0.864882i \(-0.667393\pi\)
−0.501975 + 0.864882i \(0.667393\pi\)
\(242\) −549.758 −0.146032
\(243\) 0 0
\(244\) −2019.99 −0.529987
\(245\) −320.276 −0.0835171
\(246\) 0 0
\(247\) 5707.61 1.47031
\(248\) −1639.37 −0.419759
\(249\) 0 0
\(250\) −6155.55 −1.55724
\(251\) 1906.43 0.479413 0.239707 0.970845i \(-0.422949\pi\)
0.239707 + 0.970845i \(0.422949\pi\)
\(252\) 0 0
\(253\) 679.980 0.168972
\(254\) −8750.72 −2.16169
\(255\) 0 0
\(256\) −3531.97 −0.862298
\(257\) 143.978 0.0349459 0.0174729 0.999847i \(-0.494438\pi\)
0.0174729 + 0.999847i \(0.494438\pi\)
\(258\) 0 0
\(259\) 593.029 0.142274
\(260\) −5895.13 −1.40615
\(261\) 0 0
\(262\) −3196.98 −0.753856
\(263\) 6827.81 1.60084 0.800420 0.599440i \(-0.204610\pi\)
0.800420 + 0.599440i \(0.204610\pi\)
\(264\) 0 0
\(265\) 3216.80 0.745684
\(266\) −2544.62 −0.586545
\(267\) 0 0
\(268\) −6589.53 −1.50194
\(269\) −8049.95 −1.82459 −0.912294 0.409537i \(-0.865690\pi\)
−0.912294 + 0.409537i \(0.865690\pi\)
\(270\) 0 0
\(271\) 1420.76 0.318469 0.159234 0.987241i \(-0.449097\pi\)
0.159234 + 0.987241i \(0.449097\pi\)
\(272\) 12.9848 0.00289456
\(273\) 0 0
\(274\) −4293.31 −0.946601
\(275\) 905.052 0.198461
\(276\) 0 0
\(277\) 8044.20 1.74487 0.872435 0.488730i \(-0.162540\pi\)
0.872435 + 0.488730i \(0.162540\pi\)
\(278\) −5556.02 −1.19866
\(279\) 0 0
\(280\) 965.183 0.206003
\(281\) −3089.28 −0.655839 −0.327919 0.944706i \(-0.606347\pi\)
−0.327919 + 0.944706i \(0.606347\pi\)
\(282\) 0 0
\(283\) −8123.26 −1.70628 −0.853141 0.521681i \(-0.825305\pi\)
−0.853141 + 0.521681i \(0.825305\pi\)
\(284\) 11196.5 2.33940
\(285\) 0 0
\(286\) 3565.28 0.737131
\(287\) −201.680 −0.0414800
\(288\) 0 0
\(289\) −4907.00 −0.998778
\(290\) 4657.66 0.943129
\(291\) 0 0
\(292\) −4747.43 −0.951447
\(293\) 8383.65 1.67160 0.835799 0.549036i \(-0.185005\pi\)
0.835799 + 0.549036i \(0.185005\pi\)
\(294\) 0 0
\(295\) −20.4584 −0.00403775
\(296\) −1787.15 −0.350933
\(297\) 0 0
\(298\) −12376.1 −2.40581
\(299\) −4409.80 −0.852927
\(300\) 0 0
\(301\) −2466.73 −0.472358
\(302\) −5150.84 −0.981450
\(303\) 0 0
\(304\) −423.958 −0.0799857
\(305\) 1044.31 0.196056
\(306\) 0 0
\(307\) −6514.20 −1.21103 −0.605513 0.795835i \(-0.707032\pi\)
−0.605513 + 0.795835i \(0.707032\pi\)
\(308\) −973.510 −0.180100
\(309\) 0 0
\(310\) 2307.86 0.422831
\(311\) 3822.91 0.697034 0.348517 0.937303i \(-0.386685\pi\)
0.348517 + 0.937303i \(0.386685\pi\)
\(312\) 0 0
\(313\) 1239.73 0.223878 0.111939 0.993715i \(-0.464294\pi\)
0.111939 + 0.993715i \(0.464294\pi\)
\(314\) −15674.5 −2.81708
\(315\) 0 0
\(316\) −17017.9 −3.02953
\(317\) −5377.28 −0.952739 −0.476369 0.879245i \(-0.658047\pi\)
−0.476369 + 0.879245i \(0.658047\pi\)
\(318\) 0 0
\(319\) −1725.23 −0.302803
\(320\) 5449.62 0.952010
\(321\) 0 0
\(322\) 1966.02 0.340255
\(323\) −196.061 −0.0337745
\(324\) 0 0
\(325\) −5869.43 −1.00178
\(326\) −4169.10 −0.708297
\(327\) 0 0
\(328\) 607.781 0.102314
\(329\) 1794.17 0.300655
\(330\) 0 0
\(331\) 2321.44 0.385493 0.192746 0.981249i \(-0.438261\pi\)
0.192746 + 0.981249i \(0.438261\pi\)
\(332\) 17440.1 2.88298
\(333\) 0 0
\(334\) −15473.3 −2.53492
\(335\) 3406.70 0.555605
\(336\) 0 0
\(337\) 7265.72 1.17445 0.587224 0.809425i \(-0.300221\pi\)
0.587224 + 0.809425i \(0.300221\pi\)
\(338\) −13139.5 −2.11449
\(339\) 0 0
\(340\) 202.503 0.0323008
\(341\) −854.844 −0.135755
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 7433.72 1.16511
\(345\) 0 0
\(346\) 8515.21 1.32306
\(347\) 8322.54 1.28754 0.643771 0.765218i \(-0.277369\pi\)
0.643771 + 0.765218i \(0.277369\pi\)
\(348\) 0 0
\(349\) 4604.41 0.706213 0.353107 0.935583i \(-0.385125\pi\)
0.353107 + 0.935583i \(0.385125\pi\)
\(350\) 2616.77 0.399634
\(351\) 0 0
\(352\) −2121.20 −0.321195
\(353\) 3257.91 0.491221 0.245611 0.969369i \(-0.421012\pi\)
0.245611 + 0.969369i \(0.421012\pi\)
\(354\) 0 0
\(355\) −5788.43 −0.865402
\(356\) 1403.94 0.209013
\(357\) 0 0
\(358\) −2483.18 −0.366592
\(359\) −11269.4 −1.65676 −0.828380 0.560167i \(-0.810737\pi\)
−0.828380 + 0.560167i \(0.810737\pi\)
\(360\) 0 0
\(361\) −457.546 −0.0667075
\(362\) −13300.1 −1.93105
\(363\) 0 0
\(364\) 6313.39 0.909097
\(365\) 2454.36 0.351964
\(366\) 0 0
\(367\) 9729.11 1.38380 0.691901 0.721992i \(-0.256773\pi\)
0.691901 + 0.721992i \(0.256773\pi\)
\(368\) 327.557 0.0463997
\(369\) 0 0
\(370\) 2515.90 0.353501
\(371\) −3445.03 −0.482094
\(372\) 0 0
\(373\) 6722.55 0.933191 0.466596 0.884471i \(-0.345480\pi\)
0.466596 + 0.884471i \(0.345480\pi\)
\(374\) −122.471 −0.0169326
\(375\) 0 0
\(376\) −5406.89 −0.741594
\(377\) 11188.4 1.52847
\(378\) 0 0
\(379\) −5966.99 −0.808717 −0.404358 0.914601i \(-0.632505\pi\)
−0.404358 + 0.914601i \(0.632505\pi\)
\(380\) −6611.77 −0.892570
\(381\) 0 0
\(382\) −10143.6 −1.35862
\(383\) 10029.0 1.33801 0.669005 0.743258i \(-0.266721\pi\)
0.669005 + 0.743258i \(0.266721\pi\)
\(384\) 0 0
\(385\) 503.291 0.0666236
\(386\) −6542.58 −0.862717
\(387\) 0 0
\(388\) 5972.35 0.781444
\(389\) 13234.3 1.72495 0.862477 0.506096i \(-0.168912\pi\)
0.862477 + 0.506096i \(0.168912\pi\)
\(390\) 0 0
\(391\) 151.480 0.0195926
\(392\) −1033.66 −0.133183
\(393\) 0 0
\(394\) 4664.00 0.596368
\(395\) 8798.01 1.12070
\(396\) 0 0
\(397\) −9951.99 −1.25813 −0.629063 0.777354i \(-0.716561\pi\)
−0.629063 + 0.777354i \(0.716561\pi\)
\(398\) −5785.03 −0.728586
\(399\) 0 0
\(400\) 435.978 0.0544972
\(401\) 4709.60 0.586499 0.293250 0.956036i \(-0.405263\pi\)
0.293250 + 0.956036i \(0.405263\pi\)
\(402\) 0 0
\(403\) 5543.82 0.685254
\(404\) −13231.1 −1.62938
\(405\) 0 0
\(406\) −4988.13 −0.609745
\(407\) −931.903 −0.113496
\(408\) 0 0
\(409\) 15044.8 1.81887 0.909434 0.415848i \(-0.136515\pi\)
0.909434 + 0.415848i \(0.136515\pi\)
\(410\) −855.615 −0.103063
\(411\) 0 0
\(412\) 14120.1 1.68847
\(413\) 21.9100 0.00261046
\(414\) 0 0
\(415\) −9016.30 −1.06649
\(416\) 13756.4 1.62130
\(417\) 0 0
\(418\) 3998.69 0.467901
\(419\) −5987.57 −0.698119 −0.349059 0.937101i \(-0.613499\pi\)
−0.349059 + 0.937101i \(0.613499\pi\)
\(420\) 0 0
\(421\) −6743.22 −0.780628 −0.390314 0.920682i \(-0.627634\pi\)
−0.390314 + 0.920682i \(0.627634\pi\)
\(422\) 4942.76 0.570166
\(423\) 0 0
\(424\) 10381.9 1.18913
\(425\) 201.620 0.0230118
\(426\) 0 0
\(427\) −1118.40 −0.126753
\(428\) −18765.0 −2.11925
\(429\) 0 0
\(430\) −10465.0 −1.17364
\(431\) −1953.86 −0.218363 −0.109181 0.994022i \(-0.534823\pi\)
−0.109181 + 0.994022i \(0.534823\pi\)
\(432\) 0 0
\(433\) 11962.7 1.32769 0.663844 0.747871i \(-0.268924\pi\)
0.663844 + 0.747871i \(0.268924\pi\)
\(434\) −2471.60 −0.273366
\(435\) 0 0
\(436\) 13460.3 1.47851
\(437\) −4945.87 −0.541403
\(438\) 0 0
\(439\) 5515.13 0.599597 0.299798 0.954003i \(-0.403081\pi\)
0.299798 + 0.954003i \(0.403081\pi\)
\(440\) −1516.72 −0.164333
\(441\) 0 0
\(442\) 794.244 0.0854714
\(443\) 13131.2 1.40831 0.704156 0.710046i \(-0.251326\pi\)
0.704156 + 0.710046i \(0.251326\pi\)
\(444\) 0 0
\(445\) −725.816 −0.0773191
\(446\) 8298.70 0.881064
\(447\) 0 0
\(448\) −5836.27 −0.615487
\(449\) −5101.57 −0.536209 −0.268104 0.963390i \(-0.586397\pi\)
−0.268104 + 0.963390i \(0.586397\pi\)
\(450\) 0 0
\(451\) 316.925 0.0330896
\(452\) 5539.80 0.576483
\(453\) 0 0
\(454\) 8277.31 0.855669
\(455\) −3263.94 −0.336298
\(456\) 0 0
\(457\) −1671.55 −0.171098 −0.0855490 0.996334i \(-0.527264\pi\)
−0.0855490 + 0.996334i \(0.527264\pi\)
\(458\) 6223.89 0.634985
\(459\) 0 0
\(460\) 5108.36 0.517780
\(461\) 3298.51 0.333247 0.166623 0.986021i \(-0.446714\pi\)
0.166623 + 0.986021i \(0.446714\pi\)
\(462\) 0 0
\(463\) 7364.93 0.739260 0.369630 0.929179i \(-0.379484\pi\)
0.369630 + 0.929179i \(0.379484\pi\)
\(464\) −831.069 −0.0831496
\(465\) 0 0
\(466\) −9614.33 −0.955740
\(467\) 9694.12 0.960579 0.480290 0.877110i \(-0.340532\pi\)
0.480290 + 0.877110i \(0.340532\pi\)
\(468\) 0 0
\(469\) −3648.40 −0.359206
\(470\) 7611.66 0.747021
\(471\) 0 0
\(472\) −66.0279 −0.00643894
\(473\) 3876.28 0.376811
\(474\) 0 0
\(475\) −6582.94 −0.635887
\(476\) −216.871 −0.0208829
\(477\) 0 0
\(478\) 23392.6 2.23840
\(479\) −11673.5 −1.11352 −0.556760 0.830673i \(-0.687956\pi\)
−0.556760 + 0.830673i \(0.687956\pi\)
\(480\) 0 0
\(481\) 6043.56 0.572895
\(482\) 17065.7 1.61270
\(483\) 0 0
\(484\) 1529.80 0.143670
\(485\) −3087.63 −0.289076
\(486\) 0 0
\(487\) −5032.39 −0.468253 −0.234127 0.972206i \(-0.575223\pi\)
−0.234127 + 0.972206i \(0.575223\pi\)
\(488\) 3370.42 0.312647
\(489\) 0 0
\(490\) 1455.16 0.134158
\(491\) −10559.5 −0.970557 −0.485278 0.874360i \(-0.661282\pi\)
−0.485278 + 0.874360i \(0.661282\pi\)
\(492\) 0 0
\(493\) −384.332 −0.0351104
\(494\) −25932.3 −2.36184
\(495\) 0 0
\(496\) −411.792 −0.0372782
\(497\) 6199.12 0.559494
\(498\) 0 0
\(499\) −7367.48 −0.660949 −0.330475 0.943815i \(-0.607209\pi\)
−0.330475 + 0.943815i \(0.607209\pi\)
\(500\) 17128.9 1.53206
\(501\) 0 0
\(502\) −8661.78 −0.770107
\(503\) −4656.82 −0.412798 −0.206399 0.978468i \(-0.566174\pi\)
−0.206399 + 0.978468i \(0.566174\pi\)
\(504\) 0 0
\(505\) 6840.28 0.602749
\(506\) −3089.46 −0.271429
\(507\) 0 0
\(508\) 24350.4 2.12673
\(509\) 2210.78 0.192517 0.0962583 0.995356i \(-0.469313\pi\)
0.0962583 + 0.995356i \(0.469313\pi\)
\(510\) 0 0
\(511\) −2628.50 −0.227550
\(512\) −1916.06 −0.165388
\(513\) 0 0
\(514\) −654.157 −0.0561354
\(515\) −7299.92 −0.624608
\(516\) 0 0
\(517\) −2819.40 −0.239840
\(518\) −2694.40 −0.228543
\(519\) 0 0
\(520\) 9836.19 0.829510
\(521\) −18348.3 −1.54291 −0.771453 0.636286i \(-0.780470\pi\)
−0.771453 + 0.636286i \(0.780470\pi\)
\(522\) 0 0
\(523\) −15196.1 −1.27051 −0.635256 0.772301i \(-0.719106\pi\)
−0.635256 + 0.772301i \(0.719106\pi\)
\(524\) 8896.18 0.741663
\(525\) 0 0
\(526\) −31021.8 −2.57151
\(527\) −190.435 −0.0157410
\(528\) 0 0
\(529\) −8345.74 −0.685932
\(530\) −14615.4 −1.19783
\(531\) 0 0
\(532\) 7080.87 0.577058
\(533\) −2055.32 −0.167027
\(534\) 0 0
\(535\) 9701.25 0.783966
\(536\) 10994.8 0.886014
\(537\) 0 0
\(538\) 36574.6 2.93093
\(539\) −539.000 −0.0430730
\(540\) 0 0
\(541\) −5051.03 −0.401406 −0.200703 0.979652i \(-0.564323\pi\)
−0.200703 + 0.979652i \(0.564323\pi\)
\(542\) −6455.16 −0.511573
\(543\) 0 0
\(544\) −472.544 −0.0372430
\(545\) −6958.80 −0.546940
\(546\) 0 0
\(547\) −14873.9 −1.16263 −0.581317 0.813677i \(-0.697462\pi\)
−0.581317 + 0.813677i \(0.697462\pi\)
\(548\) 11946.9 0.931290
\(549\) 0 0
\(550\) −4112.06 −0.318798
\(551\) 12548.5 0.970209
\(552\) 0 0
\(553\) −9422.23 −0.724546
\(554\) −36548.4 −2.80288
\(555\) 0 0
\(556\) 15460.6 1.17927
\(557\) −20254.8 −1.54079 −0.770396 0.637565i \(-0.779942\pi\)
−0.770396 + 0.637565i \(0.779942\pi\)
\(558\) 0 0
\(559\) −25138.4 −1.90204
\(560\) 242.443 0.0182948
\(561\) 0 0
\(562\) 14036.0 1.05351
\(563\) 2659.45 0.199081 0.0995404 0.995034i \(-0.468263\pi\)
0.0995404 + 0.995034i \(0.468263\pi\)
\(564\) 0 0
\(565\) −2864.00 −0.213256
\(566\) 36907.7 2.74089
\(567\) 0 0
\(568\) −18681.6 −1.38004
\(569\) 14712.3 1.08395 0.541977 0.840393i \(-0.317676\pi\)
0.541977 + 0.840393i \(0.317676\pi\)
\(570\) 0 0
\(571\) 26158.2 1.91714 0.958570 0.284856i \(-0.0919456\pi\)
0.958570 + 0.284856i \(0.0919456\pi\)
\(572\) −9921.04 −0.725209
\(573\) 0 0
\(574\) 916.322 0.0666316
\(575\) 5086.09 0.368878
\(576\) 0 0
\(577\) 24028.7 1.73367 0.866835 0.498595i \(-0.166151\pi\)
0.866835 + 0.498595i \(0.166151\pi\)
\(578\) 22294.7 1.60439
\(579\) 0 0
\(580\) −12960.8 −0.927875
\(581\) 9656.01 0.689499
\(582\) 0 0
\(583\) 5413.62 0.384578
\(584\) 7921.23 0.561272
\(585\) 0 0
\(586\) −38090.7 −2.68518
\(587\) 18554.9 1.30468 0.652338 0.757928i \(-0.273788\pi\)
0.652338 + 0.757928i \(0.273788\pi\)
\(588\) 0 0
\(589\) 6217.75 0.434971
\(590\) 92.9520 0.00648606
\(591\) 0 0
\(592\) −448.913 −0.0311659
\(593\) −24535.9 −1.69911 −0.849553 0.527504i \(-0.823128\pi\)
−0.849553 + 0.527504i \(0.823128\pi\)
\(594\) 0 0
\(595\) 112.119 0.00772510
\(596\) 34438.8 2.36689
\(597\) 0 0
\(598\) 20035.7 1.37010
\(599\) 88.4090 0.00603054 0.00301527 0.999995i \(-0.499040\pi\)
0.00301527 + 0.999995i \(0.499040\pi\)
\(600\) 0 0
\(601\) −26290.0 −1.78434 −0.892172 0.451695i \(-0.850819\pi\)
−0.892172 + 0.451695i \(0.850819\pi\)
\(602\) 11207.5 0.758774
\(603\) 0 0
\(604\) 14333.2 0.965576
\(605\) −790.886 −0.0531473
\(606\) 0 0
\(607\) 28411.4 1.89981 0.949905 0.312540i \(-0.101180\pi\)
0.949905 + 0.312540i \(0.101180\pi\)
\(608\) 15428.7 1.02914
\(609\) 0 0
\(610\) −4744.77 −0.314935
\(611\) 18284.4 1.21065
\(612\) 0 0
\(613\) −3295.49 −0.217135 −0.108567 0.994089i \(-0.534626\pi\)
−0.108567 + 0.994089i \(0.534626\pi\)
\(614\) 29597.0 1.94534
\(615\) 0 0
\(616\) 1624.33 0.106244
\(617\) 3409.29 0.222452 0.111226 0.993795i \(-0.464522\pi\)
0.111226 + 0.993795i \(0.464522\pi\)
\(618\) 0 0
\(619\) −10865.7 −0.705542 −0.352771 0.935710i \(-0.614760\pi\)
−0.352771 + 0.935710i \(0.614760\pi\)
\(620\) −6422.03 −0.415992
\(621\) 0 0
\(622\) −17369.2 −1.11968
\(623\) 777.313 0.0499878
\(624\) 0 0
\(625\) 1429.26 0.0914724
\(626\) −5632.66 −0.359627
\(627\) 0 0
\(628\) 43617.2 2.77152
\(629\) −207.602 −0.0131600
\(630\) 0 0
\(631\) 23851.7 1.50479 0.752394 0.658713i \(-0.228899\pi\)
0.752394 + 0.658713i \(0.228899\pi\)
\(632\) 28394.8 1.78716
\(633\) 0 0
\(634\) 24431.4 1.53044
\(635\) −12588.9 −0.786730
\(636\) 0 0
\(637\) 3495.51 0.217421
\(638\) 7838.49 0.486409
\(639\) 0 0
\(640\) −14676.7 −0.906479
\(641\) −13710.2 −0.844809 −0.422404 0.906408i \(-0.638814\pi\)
−0.422404 + 0.906408i \(0.638814\pi\)
\(642\) 0 0
\(643\) 4879.27 0.299253 0.149627 0.988743i \(-0.452193\pi\)
0.149627 + 0.988743i \(0.452193\pi\)
\(644\) −5470.80 −0.334751
\(645\) 0 0
\(646\) 890.796 0.0542537
\(647\) −13856.7 −0.841982 −0.420991 0.907065i \(-0.638318\pi\)
−0.420991 + 0.907065i \(0.638318\pi\)
\(648\) 0 0
\(649\) −34.4300 −0.00208243
\(650\) 26667.5 1.60921
\(651\) 0 0
\(652\) 11601.3 0.696841
\(653\) −9861.70 −0.590993 −0.295496 0.955344i \(-0.595485\pi\)
−0.295496 + 0.955344i \(0.595485\pi\)
\(654\) 0 0
\(655\) −4599.20 −0.274360
\(656\) 152.668 0.00908640
\(657\) 0 0
\(658\) −8151.72 −0.482959
\(659\) −6849.60 −0.404890 −0.202445 0.979294i \(-0.564889\pi\)
−0.202445 + 0.979294i \(0.564889\pi\)
\(660\) 0 0
\(661\) 24156.9 1.42147 0.710737 0.703458i \(-0.248361\pi\)
0.710737 + 0.703458i \(0.248361\pi\)
\(662\) −10547.4 −0.619238
\(663\) 0 0
\(664\) −29099.3 −1.70071
\(665\) −3660.72 −0.213468
\(666\) 0 0
\(667\) −9695.20 −0.562818
\(668\) 43057.4 2.49392
\(669\) 0 0
\(670\) −15478.2 −0.892498
\(671\) 1757.49 0.101114
\(672\) 0 0
\(673\) 10039.3 0.575015 0.287508 0.957778i \(-0.407173\pi\)
0.287508 + 0.957778i \(0.407173\pi\)
\(674\) −33011.5 −1.88658
\(675\) 0 0
\(676\) 36563.1 2.08029
\(677\) −17849.5 −1.01331 −0.506655 0.862149i \(-0.669118\pi\)
−0.506655 + 0.862149i \(0.669118\pi\)
\(678\) 0 0
\(679\) 3306.69 0.186891
\(680\) −337.882 −0.0190547
\(681\) 0 0
\(682\) 3883.95 0.218070
\(683\) 8199.58 0.459368 0.229684 0.973265i \(-0.426231\pi\)
0.229684 + 0.973265i \(0.426231\pi\)
\(684\) 0 0
\(685\) −6176.39 −0.344508
\(686\) −1558.40 −0.0867350
\(687\) 0 0
\(688\) 1867.27 0.103472
\(689\) −35108.3 −1.94125
\(690\) 0 0
\(691\) 6179.89 0.340223 0.170111 0.985425i \(-0.445587\pi\)
0.170111 + 0.985425i \(0.445587\pi\)
\(692\) −23695.1 −1.30167
\(693\) 0 0
\(694\) −37813.1 −2.06825
\(695\) −7992.94 −0.436244
\(696\) 0 0
\(697\) 70.6020 0.00383679
\(698\) −20919.9 −1.13443
\(699\) 0 0
\(700\) −7281.62 −0.393171
\(701\) 13389.9 0.721437 0.360719 0.932675i \(-0.382531\pi\)
0.360719 + 0.932675i \(0.382531\pi\)
\(702\) 0 0
\(703\) 6778.25 0.363651
\(704\) 9171.29 0.490988
\(705\) 0 0
\(706\) −14802.2 −0.789075
\(707\) −7325.60 −0.389685
\(708\) 0 0
\(709\) −4773.49 −0.252852 −0.126426 0.991976i \(-0.540351\pi\)
−0.126426 + 0.991976i \(0.540351\pi\)
\(710\) 26299.4 1.39014
\(711\) 0 0
\(712\) −2342.51 −0.123299
\(713\) −4803.94 −0.252327
\(714\) 0 0
\(715\) 5129.04 0.268273
\(716\) 6909.88 0.360663
\(717\) 0 0
\(718\) 51202.1 2.66134
\(719\) 8643.93 0.448351 0.224175 0.974549i \(-0.428031\pi\)
0.224175 + 0.974549i \(0.428031\pi\)
\(720\) 0 0
\(721\) 7817.85 0.403817
\(722\) 2078.84 0.107156
\(723\) 0 0
\(724\) 37010.0 1.89982
\(725\) −12904.3 −0.661039
\(726\) 0 0
\(727\) 6255.16 0.319108 0.159554 0.987189i \(-0.448994\pi\)
0.159554 + 0.987189i \(0.448994\pi\)
\(728\) −10534.1 −0.536289
\(729\) 0 0
\(730\) −11151.3 −0.565379
\(731\) 863.527 0.0436918
\(732\) 0 0
\(733\) 8881.40 0.447534 0.223767 0.974643i \(-0.428165\pi\)
0.223767 + 0.974643i \(0.428165\pi\)
\(734\) −44203.8 −2.22288
\(735\) 0 0
\(736\) −11920.5 −0.597003
\(737\) 5733.20 0.286547
\(738\) 0 0
\(739\) 30833.0 1.53479 0.767395 0.641175i \(-0.221553\pi\)
0.767395 + 0.641175i \(0.221553\pi\)
\(740\) −7000.94 −0.347783
\(741\) 0 0
\(742\) 15652.3 0.774414
\(743\) −9598.45 −0.473934 −0.236967 0.971518i \(-0.576153\pi\)
−0.236967 + 0.971518i \(0.576153\pi\)
\(744\) 0 0
\(745\) −17804.4 −0.875574
\(746\) −30543.6 −1.49904
\(747\) 0 0
\(748\) 340.797 0.0166588
\(749\) −10389.6 −0.506844
\(750\) 0 0
\(751\) −40107.6 −1.94880 −0.974399 0.224826i \(-0.927819\pi\)
−0.974399 + 0.224826i \(0.927819\pi\)
\(752\) −1358.15 −0.0658600
\(753\) 0 0
\(754\) −50834.0 −2.45526
\(755\) −7410.05 −0.357191
\(756\) 0 0
\(757\) 3714.41 0.178339 0.0891696 0.996016i \(-0.471579\pi\)
0.0891696 + 0.996016i \(0.471579\pi\)
\(758\) 27110.7 1.29908
\(759\) 0 0
\(760\) 11031.9 0.526539
\(761\) −29623.8 −1.41112 −0.705559 0.708651i \(-0.749304\pi\)
−0.705559 + 0.708651i \(0.749304\pi\)
\(762\) 0 0
\(763\) 7452.53 0.353604
\(764\) 28226.5 1.33665
\(765\) 0 0
\(766\) −45566.3 −2.14932
\(767\) 223.285 0.0105115
\(768\) 0 0
\(769\) −39173.6 −1.83698 −0.918489 0.395446i \(-0.870590\pi\)
−0.918489 + 0.395446i \(0.870590\pi\)
\(770\) −2286.68 −0.107021
\(771\) 0 0
\(772\) 18205.9 0.848763
\(773\) 9500.27 0.442045 0.221023 0.975269i \(-0.429061\pi\)
0.221023 + 0.975269i \(0.429061\pi\)
\(774\) 0 0
\(775\) −6394.03 −0.296362
\(776\) −9965.04 −0.460985
\(777\) 0 0
\(778\) −60129.6 −2.77089
\(779\) −2305.17 −0.106022
\(780\) 0 0
\(781\) −9741.47 −0.446322
\(782\) −688.244 −0.0314726
\(783\) 0 0
\(784\) −259.645 −0.0118278
\(785\) −22549.5 −1.02526
\(786\) 0 0
\(787\) 21640.9 0.980195 0.490097 0.871668i \(-0.336961\pi\)
0.490097 + 0.871668i \(0.336961\pi\)
\(788\) −12978.4 −0.586723
\(789\) 0 0
\(790\) −39973.4 −1.80024
\(791\) 3067.20 0.137873
\(792\) 0 0
\(793\) −11397.7 −0.510394
\(794\) 45216.4 2.02100
\(795\) 0 0
\(796\) 16097.9 0.716802
\(797\) 9843.72 0.437494 0.218747 0.975782i \(-0.429803\pi\)
0.218747 + 0.975782i \(0.429803\pi\)
\(798\) 0 0
\(799\) −628.084 −0.0278098
\(800\) −15866.1 −0.701189
\(801\) 0 0
\(802\) −21397.9 −0.942125
\(803\) 4130.49 0.181522
\(804\) 0 0
\(805\) 2828.33 0.123833
\(806\) −25188.1 −1.10076
\(807\) 0 0
\(808\) 22076.4 0.961194
\(809\) 1679.53 0.0729902 0.0364951 0.999334i \(-0.488381\pi\)
0.0364951 + 0.999334i \(0.488381\pi\)
\(810\) 0 0
\(811\) 19807.2 0.857612 0.428806 0.903397i \(-0.358934\pi\)
0.428806 + 0.903397i \(0.358934\pi\)
\(812\) 13880.4 0.599883
\(813\) 0 0
\(814\) 4234.06 0.182314
\(815\) −5997.69 −0.257779
\(816\) 0 0
\(817\) −28194.4 −1.20734
\(818\) −68355.3 −2.92175
\(819\) 0 0
\(820\) 2380.90 0.101396
\(821\) 4861.54 0.206661 0.103331 0.994647i \(-0.467050\pi\)
0.103331 + 0.994647i \(0.467050\pi\)
\(822\) 0 0
\(823\) 15188.4 0.643297 0.321648 0.946859i \(-0.395763\pi\)
0.321648 + 0.946859i \(0.395763\pi\)
\(824\) −23559.9 −0.996051
\(825\) 0 0
\(826\) −99.5470 −0.00419332
\(827\) 29620.8 1.24548 0.622742 0.782428i \(-0.286019\pi\)
0.622742 + 0.782428i \(0.286019\pi\)
\(828\) 0 0
\(829\) −1614.70 −0.0676487 −0.0338243 0.999428i \(-0.510769\pi\)
−0.0338243 + 0.999428i \(0.510769\pi\)
\(830\) 40965.2 1.71316
\(831\) 0 0
\(832\) −59477.5 −2.47838
\(833\) −120.074 −0.00499438
\(834\) 0 0
\(835\) −22260.1 −0.922565
\(836\) −11127.1 −0.460333
\(837\) 0 0
\(838\) 27204.2 1.12143
\(839\) −10645.8 −0.438063 −0.219031 0.975718i \(-0.570290\pi\)
−0.219031 + 0.975718i \(0.570290\pi\)
\(840\) 0 0
\(841\) 209.391 0.00858547
\(842\) 30637.5 1.25397
\(843\) 0 0
\(844\) −13754.1 −0.560944
\(845\) −18902.6 −0.769551
\(846\) 0 0
\(847\) 847.000 0.0343604
\(848\) 2607.83 0.105605
\(849\) 0 0
\(850\) −916.052 −0.0369651
\(851\) −5236.99 −0.210954
\(852\) 0 0
\(853\) 30300.5 1.21626 0.608130 0.793837i \(-0.291920\pi\)
0.608130 + 0.793837i \(0.291920\pi\)
\(854\) 5081.42 0.203609
\(855\) 0 0
\(856\) 31310.0 1.25018
\(857\) 14243.2 0.567721 0.283861 0.958866i \(-0.408385\pi\)
0.283861 + 0.958866i \(0.408385\pi\)
\(858\) 0 0
\(859\) 21732.6 0.863221 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(860\) 29120.7 1.15466
\(861\) 0 0
\(862\) 8877.29 0.350768
\(863\) 36687.7 1.44712 0.723559 0.690263i \(-0.242505\pi\)
0.723559 + 0.690263i \(0.242505\pi\)
\(864\) 0 0
\(865\) 12250.0 0.481519
\(866\) −54351.8 −2.13274
\(867\) 0 0
\(868\) 6877.68 0.268944
\(869\) 14806.4 0.577988
\(870\) 0 0
\(871\) −37180.9 −1.44641
\(872\) −22458.9 −0.872195
\(873\) 0 0
\(874\) 22471.3 0.869685
\(875\) 9483.72 0.366409
\(876\) 0 0
\(877\) 2430.02 0.0935643 0.0467822 0.998905i \(-0.485103\pi\)
0.0467822 + 0.998905i \(0.485103\pi\)
\(878\) −25057.8 −0.963165
\(879\) 0 0
\(880\) −380.982 −0.0145942
\(881\) −31492.8 −1.20434 −0.602168 0.798369i \(-0.705696\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(882\) 0 0
\(883\) 9175.59 0.349698 0.174849 0.984595i \(-0.444056\pi\)
0.174849 + 0.984595i \(0.444056\pi\)
\(884\) −2210.13 −0.0840890
\(885\) 0 0
\(886\) −59661.0 −2.26225
\(887\) −23904.6 −0.904890 −0.452445 0.891792i \(-0.649448\pi\)
−0.452445 + 0.891792i \(0.649448\pi\)
\(888\) 0 0
\(889\) 13482.0 0.508631
\(890\) 3297.71 0.124202
\(891\) 0 0
\(892\) −23092.6 −0.866814
\(893\) 20507.1 0.768470
\(894\) 0 0
\(895\) −3572.32 −0.133418
\(896\) 15718.0 0.586051
\(897\) 0 0
\(898\) 23178.7 0.861341
\(899\) 12188.4 0.452177
\(900\) 0 0
\(901\) 1206.00 0.0445924
\(902\) −1439.93 −0.0531536
\(903\) 0 0
\(904\) −9243.31 −0.340075
\(905\) −19133.7 −0.702790
\(906\) 0 0
\(907\) 30088.7 1.10152 0.550760 0.834663i \(-0.314338\pi\)
0.550760 + 0.834663i \(0.314338\pi\)
\(908\) −23033.1 −0.841829
\(909\) 0 0
\(910\) 14829.5 0.540214
\(911\) 30837.0 1.12149 0.560745 0.827989i \(-0.310515\pi\)
0.560745 + 0.827989i \(0.310515\pi\)
\(912\) 0 0
\(913\) −15173.7 −0.550030
\(914\) 7594.61 0.274844
\(915\) 0 0
\(916\) −17319.1 −0.624715
\(917\) 4925.52 0.177377
\(918\) 0 0
\(919\) 52840.0 1.89666 0.948330 0.317285i \(-0.102771\pi\)
0.948330 + 0.317285i \(0.102771\pi\)
\(920\) −8523.45 −0.305445
\(921\) 0 0
\(922\) −14986.6 −0.535312
\(923\) 63175.2 2.25291
\(924\) 0 0
\(925\) −6970.42 −0.247769
\(926\) −33462.2 −1.18751
\(927\) 0 0
\(928\) 30244.3 1.06984
\(929\) −12341.6 −0.435861 −0.217930 0.975964i \(-0.569931\pi\)
−0.217930 + 0.975964i \(0.569931\pi\)
\(930\) 0 0
\(931\) 3920.45 0.138010
\(932\) 26753.6 0.940282
\(933\) 0 0
\(934\) −44044.8 −1.54303
\(935\) −176.187 −0.00616250
\(936\) 0 0
\(937\) 3939.78 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(938\) 16576.4 0.577012
\(939\) 0 0
\(940\) −21180.8 −0.734939
\(941\) −35201.0 −1.21947 −0.609734 0.792606i \(-0.708724\pi\)
−0.609734 + 0.792606i \(0.708724\pi\)
\(942\) 0 0
\(943\) 1781.01 0.0615035
\(944\) −16.5855 −0.000571834 0
\(945\) 0 0
\(946\) −17611.7 −0.605292
\(947\) 17958.1 0.616218 0.308109 0.951351i \(-0.400304\pi\)
0.308109 + 0.951351i \(0.400304\pi\)
\(948\) 0 0
\(949\) −26787.0 −0.916273
\(950\) 29909.3 1.02146
\(951\) 0 0
\(952\) 361.855 0.0123191
\(953\) −9631.60 −0.327385 −0.163693 0.986511i \(-0.552341\pi\)
−0.163693 + 0.986511i \(0.552341\pi\)
\(954\) 0 0
\(955\) −14592.7 −0.494460
\(956\) −65094.2 −2.20219
\(957\) 0 0
\(958\) 53038.1 1.78871
\(959\) 6614.61 0.222729
\(960\) 0 0
\(961\) −23751.7 −0.797277
\(962\) −27458.7 −0.920273
\(963\) 0 0
\(964\) −47488.4 −1.58662
\(965\) −9412.21 −0.313979
\(966\) 0 0
\(967\) −46034.2 −1.53088 −0.765439 0.643508i \(-0.777478\pi\)
−0.765439 + 0.643508i \(0.777478\pi\)
\(968\) −2552.52 −0.0847531
\(969\) 0 0
\(970\) 14028.5 0.464358
\(971\) 17330.8 0.572782 0.286391 0.958113i \(-0.407544\pi\)
0.286391 + 0.958113i \(0.407544\pi\)
\(972\) 0 0
\(973\) 8560.04 0.282037
\(974\) 22864.4 0.752180
\(975\) 0 0
\(976\) 846.612 0.0277658
\(977\) −8836.93 −0.289374 −0.144687 0.989477i \(-0.546218\pi\)
−0.144687 + 0.989477i \(0.546218\pi\)
\(978\) 0 0
\(979\) −1221.49 −0.0398765
\(980\) −4049.25 −0.131988
\(981\) 0 0
\(982\) 47976.6 1.55906
\(983\) −27933.6 −0.906352 −0.453176 0.891421i \(-0.649709\pi\)
−0.453176 + 0.891421i \(0.649709\pi\)
\(984\) 0 0
\(985\) 6709.67 0.217044
\(986\) 1746.19 0.0563997
\(987\) 0 0
\(988\) 72161.2 2.32364
\(989\) 21783.4 0.700377
\(990\) 0 0
\(991\) 57618.7 1.84694 0.923471 0.383668i \(-0.125339\pi\)
0.923471 + 0.383668i \(0.125339\pi\)
\(992\) 14985.9 0.479641
\(993\) 0 0
\(994\) −28165.4 −0.898745
\(995\) −8322.39 −0.265163
\(996\) 0 0
\(997\) −8399.40 −0.266812 −0.133406 0.991061i \(-0.542591\pi\)
−0.133406 + 0.991061i \(0.542591\pi\)
\(998\) 33473.8 1.06172
\(999\) 0 0
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 693.4.a.p.1.1 5
3.2 odd 2 231.4.a.k.1.5 5
21.20 even 2 1617.4.a.n.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.4.a.k.1.5 5 3.2 odd 2
693.4.a.p.1.1 5 1.1 even 1 trivial
1617.4.a.n.1.5 5 21.20 even 2