Properties

Label 2475.4.a.t.1.1
Level $2475$
Weight $4$
Character 2475.1
Self dual yes
Analytic conductor $146.030$
Analytic rank $1$
Dimension $3$
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] = [2475,4,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, 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("2475.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(146.029727264\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.47528.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 26x - 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.06484\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.06484 q^{2} +17.6526 q^{4} -27.4348 q^{7} -48.8887 q^{8} +O(q^{10})\) \(q-5.06484 q^{2} +17.6526 q^{4} -27.4348 q^{7} -48.8887 q^{8} -11.0000 q^{11} -22.6949 q^{13} +138.953 q^{14} +106.392 q^{16} -41.1755 q^{17} -142.128 q^{19} +55.7132 q^{22} +176.166 q^{23} +114.946 q^{26} -484.295 q^{28} -76.2044 q^{29} +197.373 q^{31} -147.751 q^{32} +208.547 q^{34} -367.297 q^{37} +719.856 q^{38} +238.279 q^{41} -30.2905 q^{43} -194.178 q^{44} -892.254 q^{46} +137.390 q^{47} +409.668 q^{49} -400.623 q^{52} +638.665 q^{53} +1341.25 q^{56} +385.963 q^{58} -103.146 q^{59} +605.596 q^{61} -999.662 q^{62} -102.804 q^{64} +704.925 q^{67} -726.852 q^{68} +782.162 q^{71} +243.132 q^{73} +1860.30 q^{74} -2508.93 q^{76} +301.783 q^{77} -532.874 q^{79} -1206.84 q^{82} +1204.91 q^{83} +153.416 q^{86} +537.775 q^{88} -1058.49 q^{89} +622.629 q^{91} +3109.79 q^{92} -695.857 q^{94} +85.1964 q^{97} -2074.90 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 30 q^{4} - 10 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 30 q^{4} - 10 q^{7} + 18 q^{8} - 33 q^{11} - 114 q^{13} + 68 q^{14} + 178 q^{16} - 104 q^{17} - 58 q^{19} + 22 q^{22} + 120 q^{23} + 120 q^{26} - 676 q^{28} + 220 q^{29} + 248 q^{31} - 258 q^{32} - 80 q^{34} - 838 q^{37} + 600 q^{38} - 156 q^{41} - 122 q^{43} - 330 q^{44} - 1256 q^{46} + 504 q^{47} + 279 q^{49} - 520 q^{52} + 282 q^{53} + 1644 q^{56} + 1644 q^{58} - 548 q^{59} + 414 q^{61} - 2448 q^{62} - 58 q^{64} + 428 q^{67} - 1704 q^{68} + 912 q^{71} - 618 q^{73} + 1612 q^{74} - 2752 q^{76} + 110 q^{77} - 542 q^{79} - 3372 q^{82} + 1548 q^{86} - 198 q^{88} - 790 q^{89} - 772 q^{91} + 1912 q^{92} - 424 q^{94} - 2074 q^{97} - 3978 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.06484 −1.79069 −0.895345 0.445373i \(-0.853071\pi\)
−0.895345 + 0.445373i \(0.853071\pi\)
\(3\) 0 0
\(4\) 17.6526 2.20657
\(5\) 0 0
\(6\) 0 0
\(7\) −27.4348 −1.48134 −0.740670 0.671869i \(-0.765492\pi\)
−0.740670 + 0.671869i \(0.765492\pi\)
\(8\) −48.8887 −2.16059
\(9\) 0 0
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) −22.6949 −0.484187 −0.242093 0.970253i \(-0.577834\pi\)
−0.242093 + 0.970253i \(0.577834\pi\)
\(14\) 138.953 2.65262
\(15\) 0 0
\(16\) 106.392 1.66238
\(17\) −41.1755 −0.587442 −0.293721 0.955891i \(-0.594894\pi\)
−0.293721 + 0.955891i \(0.594894\pi\)
\(18\) 0 0
\(19\) −142.128 −1.71613 −0.858064 0.513542i \(-0.828333\pi\)
−0.858064 + 0.513542i \(0.828333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 55.7132 0.539913
\(23\) 176.166 1.59710 0.798548 0.601931i \(-0.205602\pi\)
0.798548 + 0.601931i \(0.205602\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 114.946 0.867028
\(27\) 0 0
\(28\) −484.295 −3.26868
\(29\) −76.2044 −0.487959 −0.243979 0.969780i \(-0.578453\pi\)
−0.243979 + 0.969780i \(0.578453\pi\)
\(30\) 0 0
\(31\) 197.373 1.14352 0.571762 0.820420i \(-0.306260\pi\)
0.571762 + 0.820420i \(0.306260\pi\)
\(32\) −147.751 −0.816218
\(33\) 0 0
\(34\) 208.547 1.05193
\(35\) 0 0
\(36\) 0 0
\(37\) −367.297 −1.63198 −0.815989 0.578067i \(-0.803807\pi\)
−0.815989 + 0.578067i \(0.803807\pi\)
\(38\) 719.856 3.07305
\(39\) 0 0
\(40\) 0 0
\(41\) 238.279 0.907633 0.453817 0.891095i \(-0.350062\pi\)
0.453817 + 0.891095i \(0.350062\pi\)
\(42\) 0 0
\(43\) −30.2905 −0.107425 −0.0537123 0.998556i \(-0.517105\pi\)
−0.0537123 + 0.998556i \(0.517105\pi\)
\(44\) −194.178 −0.665306
\(45\) 0 0
\(46\) −892.254 −2.85990
\(47\) 137.390 0.426391 0.213195 0.977010i \(-0.431613\pi\)
0.213195 + 0.977010i \(0.431613\pi\)
\(48\) 0 0
\(49\) 409.668 1.19437
\(50\) 0 0
\(51\) 0 0
\(52\) −400.623 −1.06839
\(53\) 638.665 1.65523 0.827617 0.561293i \(-0.189696\pi\)
0.827617 + 0.561293i \(0.189696\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1341.25 3.20057
\(57\) 0 0
\(58\) 385.963 0.873783
\(59\) −103.146 −0.227601 −0.113800 0.993504i \(-0.536302\pi\)
−0.113800 + 0.993504i \(0.536302\pi\)
\(60\) 0 0
\(61\) 605.596 1.27113 0.635563 0.772049i \(-0.280768\pi\)
0.635563 + 0.772049i \(0.280768\pi\)
\(62\) −999.662 −2.04770
\(63\) 0 0
\(64\) −102.804 −0.200789
\(65\) 0 0
\(66\) 0 0
\(67\) 704.925 1.28538 0.642688 0.766128i \(-0.277819\pi\)
0.642688 + 0.766128i \(0.277819\pi\)
\(68\) −726.852 −1.29623
\(69\) 0 0
\(70\) 0 0
\(71\) 782.162 1.30740 0.653701 0.756753i \(-0.273215\pi\)
0.653701 + 0.756753i \(0.273215\pi\)
\(72\) 0 0
\(73\) 243.132 0.389814 0.194907 0.980822i \(-0.437560\pi\)
0.194907 + 0.980822i \(0.437560\pi\)
\(74\) 1860.30 2.92237
\(75\) 0 0
\(76\) −2508.93 −3.78676
\(77\) 301.783 0.446641
\(78\) 0 0
\(79\) −532.874 −0.758899 −0.379449 0.925212i \(-0.623887\pi\)
−0.379449 + 0.925212i \(0.623887\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1206.84 −1.62529
\(83\) 1204.91 1.59344 0.796722 0.604345i \(-0.206565\pi\)
0.796722 + 0.604345i \(0.206565\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 153.416 0.192364
\(87\) 0 0
\(88\) 537.775 0.651443
\(89\) −1058.49 −1.26067 −0.630337 0.776322i \(-0.717083\pi\)
−0.630337 + 0.776322i \(0.717083\pi\)
\(90\) 0 0
\(91\) 622.629 0.717245
\(92\) 3109.79 3.52411
\(93\) 0 0
\(94\) −695.857 −0.763533
\(95\) 0 0
\(96\) 0 0
\(97\) 85.1964 0.0891792 0.0445896 0.999005i \(-0.485802\pi\)
0.0445896 + 0.999005i \(0.485802\pi\)
\(98\) −2074.90 −2.13874
\(99\) 0 0
\(100\) 0 0
\(101\) 7.81823 0.00770241 0.00385120 0.999993i \(-0.498774\pi\)
0.00385120 + 0.999993i \(0.498774\pi\)
\(102\) 0 0
\(103\) −12.4770 −0.0119359 −0.00596794 0.999982i \(-0.501900\pi\)
−0.00596794 + 0.999982i \(0.501900\pi\)
\(104\) 1109.52 1.04613
\(105\) 0 0
\(106\) −3234.73 −2.96401
\(107\) −1376.81 −1.24393 −0.621967 0.783043i \(-0.713666\pi\)
−0.621967 + 0.783043i \(0.713666\pi\)
\(108\) 0 0
\(109\) 610.189 0.536197 0.268099 0.963391i \(-0.413605\pi\)
0.268099 + 0.963391i \(0.413605\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2918.86 −2.46255
\(113\) −36.7727 −0.0306132 −0.0153066 0.999883i \(-0.504872\pi\)
−0.0153066 + 0.999883i \(0.504872\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1345.20 −1.07672
\(117\) 0 0
\(118\) 522.416 0.407562
\(119\) 1129.64 0.870201
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −3067.25 −2.27619
\(123\) 0 0
\(124\) 3484.14 2.52326
\(125\) 0 0
\(126\) 0 0
\(127\) 54.5310 0.0381011 0.0190506 0.999819i \(-0.493936\pi\)
0.0190506 + 0.999819i \(0.493936\pi\)
\(128\) 1702.69 1.17577
\(129\) 0 0
\(130\) 0 0
\(131\) −2377.83 −1.58589 −0.792947 0.609290i \(-0.791454\pi\)
−0.792947 + 0.609290i \(0.791454\pi\)
\(132\) 0 0
\(133\) 3899.26 2.54217
\(134\) −3570.33 −2.30171
\(135\) 0 0
\(136\) 2013.01 1.26922
\(137\) −3078.41 −1.91976 −0.959878 0.280417i \(-0.909527\pi\)
−0.959878 + 0.280417i \(0.909527\pi\)
\(138\) 0 0
\(139\) −1197.18 −0.730530 −0.365265 0.930904i \(-0.619022\pi\)
−0.365265 + 0.930904i \(0.619022\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3961.52 −2.34115
\(143\) 249.644 0.145988
\(144\) 0 0
\(145\) 0 0
\(146\) −1231.42 −0.698035
\(147\) 0 0
\(148\) −6483.73 −3.60108
\(149\) 749.456 0.412066 0.206033 0.978545i \(-0.433945\pi\)
0.206033 + 0.978545i \(0.433945\pi\)
\(150\) 0 0
\(151\) −2645.72 −1.42586 −0.712931 0.701234i \(-0.752633\pi\)
−0.712931 + 0.701234i \(0.752633\pi\)
\(152\) 6948.46 3.70786
\(153\) 0 0
\(154\) −1528.48 −0.799795
\(155\) 0 0
\(156\) 0 0
\(157\) 3475.74 1.76684 0.883422 0.468578i \(-0.155233\pi\)
0.883422 + 0.468578i \(0.155233\pi\)
\(158\) 2698.92 1.35895
\(159\) 0 0
\(160\) 0 0
\(161\) −4833.09 −2.36584
\(162\) 0 0
\(163\) −3518.89 −1.69092 −0.845462 0.534035i \(-0.820675\pi\)
−0.845462 + 0.534035i \(0.820675\pi\)
\(164\) 4206.24 2.00276
\(165\) 0 0
\(166\) −6102.67 −2.85337
\(167\) 250.304 0.115983 0.0579914 0.998317i \(-0.481530\pi\)
0.0579914 + 0.998317i \(0.481530\pi\)
\(168\) 0 0
\(169\) −1681.94 −0.765563
\(170\) 0 0
\(171\) 0 0
\(172\) −534.705 −0.237040
\(173\) 941.985 0.413976 0.206988 0.978344i \(-0.433634\pi\)
0.206988 + 0.978344i \(0.433634\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1170.32 −0.501227
\(177\) 0 0
\(178\) 5361.09 2.25748
\(179\) −336.037 −0.140316 −0.0701582 0.997536i \(-0.522350\pi\)
−0.0701582 + 0.997536i \(0.522350\pi\)
\(180\) 0 0
\(181\) 1107.45 0.454784 0.227392 0.973803i \(-0.426980\pi\)
0.227392 + 0.973803i \(0.426980\pi\)
\(182\) −3153.52 −1.28436
\(183\) 0 0
\(184\) −8612.53 −3.45068
\(185\) 0 0
\(186\) 0 0
\(187\) 452.930 0.177120
\(188\) 2425.28 0.940861
\(189\) 0 0
\(190\) 0 0
\(191\) 4243.01 1.60740 0.803700 0.595035i \(-0.202862\pi\)
0.803700 + 0.595035i \(0.202862\pi\)
\(192\) 0 0
\(193\) 3324.23 1.23981 0.619905 0.784677i \(-0.287171\pi\)
0.619905 + 0.784677i \(0.287171\pi\)
\(194\) −431.506 −0.159692
\(195\) 0 0
\(196\) 7231.69 2.63546
\(197\) 2677.14 0.968213 0.484107 0.875009i \(-0.339145\pi\)
0.484107 + 0.875009i \(0.339145\pi\)
\(198\) 0 0
\(199\) 2779.90 0.990261 0.495131 0.868819i \(-0.335120\pi\)
0.495131 + 0.868819i \(0.335120\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −39.5981 −0.0137926
\(203\) 2090.65 0.722833
\(204\) 0 0
\(205\) 0 0
\(206\) 63.1940 0.0213735
\(207\) 0 0
\(208\) −2414.56 −0.804903
\(209\) 1563.41 0.517432
\(210\) 0 0
\(211\) 3056.15 0.997129 0.498564 0.866853i \(-0.333861\pi\)
0.498564 + 0.866853i \(0.333861\pi\)
\(212\) 11274.1 3.65239
\(213\) 0 0
\(214\) 6973.30 2.22750
\(215\) 0 0
\(216\) 0 0
\(217\) −5414.89 −1.69395
\(218\) −3090.51 −0.960163
\(219\) 0 0
\(220\) 0 0
\(221\) 934.472 0.284432
\(222\) 0 0
\(223\) −571.375 −0.171579 −0.0857895 0.996313i \(-0.527341\pi\)
−0.0857895 + 0.996313i \(0.527341\pi\)
\(224\) 4053.53 1.20910
\(225\) 0 0
\(226\) 186.248 0.0548187
\(227\) −1094.40 −0.319989 −0.159995 0.987118i \(-0.551148\pi\)
−0.159995 + 0.987118i \(0.551148\pi\)
\(228\) 0 0
\(229\) −645.240 −0.186195 −0.0930975 0.995657i \(-0.529677\pi\)
−0.0930975 + 0.995657i \(0.529677\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3725.53 1.05428
\(233\) −2337.39 −0.657201 −0.328600 0.944469i \(-0.606577\pi\)
−0.328600 + 0.944469i \(0.606577\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1820.79 −0.502217
\(237\) 0 0
\(238\) −5721.44 −1.55826
\(239\) 3656.91 0.989733 0.494866 0.868969i \(-0.335217\pi\)
0.494866 + 0.868969i \(0.335217\pi\)
\(240\) 0 0
\(241\) −389.034 −0.103983 −0.0519914 0.998648i \(-0.516557\pi\)
−0.0519914 + 0.998648i \(0.516557\pi\)
\(242\) −612.845 −0.162790
\(243\) 0 0
\(244\) 10690.3 2.80483
\(245\) 0 0
\(246\) 0 0
\(247\) 3225.58 0.830927
\(248\) −9649.30 −2.47069
\(249\) 0 0
\(250\) 0 0
\(251\) −2299.55 −0.578271 −0.289135 0.957288i \(-0.593368\pi\)
−0.289135 + 0.957288i \(0.593368\pi\)
\(252\) 0 0
\(253\) −1937.83 −0.481543
\(254\) −276.191 −0.0682273
\(255\) 0 0
\(256\) −7801.44 −1.90465
\(257\) 4921.61 1.19456 0.597279 0.802033i \(-0.296248\pi\)
0.597279 + 0.802033i \(0.296248\pi\)
\(258\) 0 0
\(259\) 10076.7 2.41752
\(260\) 0 0
\(261\) 0 0
\(262\) 12043.3 2.83985
\(263\) −2575.61 −0.603875 −0.301938 0.953328i \(-0.597633\pi\)
−0.301938 + 0.953328i \(0.597633\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −19749.1 −4.55224
\(267\) 0 0
\(268\) 12443.7 2.83627
\(269\) 4794.97 1.08682 0.543410 0.839468i \(-0.317133\pi\)
0.543410 + 0.839468i \(0.317133\pi\)
\(270\) 0 0
\(271\) 2729.47 0.611821 0.305910 0.952060i \(-0.401039\pi\)
0.305910 + 0.952060i \(0.401039\pi\)
\(272\) −4380.76 −0.976553
\(273\) 0 0
\(274\) 15591.7 3.43769
\(275\) 0 0
\(276\) 0 0
\(277\) 3761.45 0.815898 0.407949 0.913005i \(-0.366244\pi\)
0.407949 + 0.913005i \(0.366244\pi\)
\(278\) 6063.53 1.30815
\(279\) 0 0
\(280\) 0 0
\(281\) 6434.87 1.36609 0.683046 0.730375i \(-0.260655\pi\)
0.683046 + 0.730375i \(0.260655\pi\)
\(282\) 0 0
\(283\) −3335.65 −0.700649 −0.350325 0.936628i \(-0.613929\pi\)
−0.350325 + 0.936628i \(0.613929\pi\)
\(284\) 13807.2 2.88487
\(285\) 0 0
\(286\) −1264.40 −0.261419
\(287\) −6537.14 −1.34451
\(288\) 0 0
\(289\) −3217.58 −0.654912
\(290\) 0 0
\(291\) 0 0
\(292\) 4291.89 0.860151
\(293\) −2878.93 −0.574023 −0.287012 0.957927i \(-0.592662\pi\)
−0.287012 + 0.957927i \(0.592662\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 17956.6 3.52604
\(297\) 0 0
\(298\) −3795.87 −0.737883
\(299\) −3998.07 −0.773293
\(300\) 0 0
\(301\) 831.014 0.159132
\(302\) 13400.1 2.55328
\(303\) 0 0
\(304\) −15121.4 −2.85286
\(305\) 0 0
\(306\) 0 0
\(307\) 8154.28 1.51593 0.757963 0.652297i \(-0.226195\pi\)
0.757963 + 0.652297i \(0.226195\pi\)
\(308\) 5327.24 0.985544
\(309\) 0 0
\(310\) 0 0
\(311\) −3818.24 −0.696182 −0.348091 0.937461i \(-0.613170\pi\)
−0.348091 + 0.937461i \(0.613170\pi\)
\(312\) 0 0
\(313\) −2527.23 −0.456381 −0.228191 0.973616i \(-0.573281\pi\)
−0.228191 + 0.973616i \(0.573281\pi\)
\(314\) −17604.1 −3.16387
\(315\) 0 0
\(316\) −9406.59 −1.67456
\(317\) −11084.7 −1.96398 −0.981989 0.188937i \(-0.939496\pi\)
−0.981989 + 0.188937i \(0.939496\pi\)
\(318\) 0 0
\(319\) 838.249 0.147125
\(320\) 0 0
\(321\) 0 0
\(322\) 24478.8 4.23649
\(323\) 5852.19 1.00813
\(324\) 0 0
\(325\) 0 0
\(326\) 17822.6 3.02792
\(327\) 0 0
\(328\) −11649.1 −1.96103
\(329\) −3769.26 −0.631629
\(330\) 0 0
\(331\) −9417.70 −1.56388 −0.781939 0.623355i \(-0.785769\pi\)
−0.781939 + 0.623355i \(0.785769\pi\)
\(332\) 21269.7 3.51605
\(333\) 0 0
\(334\) −1267.75 −0.207689
\(335\) 0 0
\(336\) 0 0
\(337\) 11263.2 1.82061 0.910305 0.413938i \(-0.135847\pi\)
0.910305 + 0.413938i \(0.135847\pi\)
\(338\) 8518.76 1.37089
\(339\) 0 0
\(340\) 0 0
\(341\) −2171.10 −0.344785
\(342\) 0 0
\(343\) −1829.03 −0.287925
\(344\) 1480.86 0.232101
\(345\) 0 0
\(346\) −4771.00 −0.741302
\(347\) −4213.25 −0.651814 −0.325907 0.945402i \(-0.605670\pi\)
−0.325907 + 0.945402i \(0.605670\pi\)
\(348\) 0 0
\(349\) 4987.07 0.764905 0.382452 0.923975i \(-0.375080\pi\)
0.382452 + 0.923975i \(0.375080\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1625.26 0.246099
\(353\) −7633.47 −1.15096 −0.575480 0.817816i \(-0.695185\pi\)
−0.575480 + 0.817816i \(0.695185\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −18685.1 −2.78177
\(357\) 0 0
\(358\) 1701.97 0.251263
\(359\) 3114.76 0.457913 0.228957 0.973437i \(-0.426469\pi\)
0.228957 + 0.973437i \(0.426469\pi\)
\(360\) 0 0
\(361\) 13341.4 1.94510
\(362\) −5609.04 −0.814378
\(363\) 0 0
\(364\) 10991.0 1.58265
\(365\) 0 0
\(366\) 0 0
\(367\) −5931.48 −0.843653 −0.421827 0.906677i \(-0.638611\pi\)
−0.421827 + 0.906677i \(0.638611\pi\)
\(368\) 18742.8 2.65499
\(369\) 0 0
\(370\) 0 0
\(371\) −17521.7 −2.45196
\(372\) 0 0
\(373\) −13918.9 −1.93215 −0.966073 0.258267i \(-0.916848\pi\)
−0.966073 + 0.258267i \(0.916848\pi\)
\(374\) −2294.02 −0.317168
\(375\) 0 0
\(376\) −6716.80 −0.921257
\(377\) 1729.45 0.236263
\(378\) 0 0
\(379\) −12267.3 −1.66261 −0.831303 0.555820i \(-0.812404\pi\)
−0.831303 + 0.555820i \(0.812404\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −21490.1 −2.87835
\(383\) −6935.04 −0.925233 −0.462616 0.886559i \(-0.653089\pi\)
−0.462616 + 0.886559i \(0.653089\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −16836.7 −2.22012
\(387\) 0 0
\(388\) 1503.93 0.196780
\(389\) −2775.18 −0.361715 −0.180858 0.983509i \(-0.557887\pi\)
−0.180858 + 0.983509i \(0.557887\pi\)
\(390\) 0 0
\(391\) −7253.73 −0.938202
\(392\) −20028.1 −2.58054
\(393\) 0 0
\(394\) −13559.3 −1.73377
\(395\) 0 0
\(396\) 0 0
\(397\) −10539.5 −1.33240 −0.666198 0.745775i \(-0.732079\pi\)
−0.666198 + 0.745775i \(0.732079\pi\)
\(398\) −14079.7 −1.77325
\(399\) 0 0
\(400\) 0 0
\(401\) 12295.3 1.53117 0.765585 0.643334i \(-0.222450\pi\)
0.765585 + 0.643334i \(0.222450\pi\)
\(402\) 0 0
\(403\) −4479.35 −0.553679
\(404\) 138.012 0.0169959
\(405\) 0 0
\(406\) −10588.8 −1.29437
\(407\) 4040.26 0.492060
\(408\) 0 0
\(409\) −11545.7 −1.39584 −0.697918 0.716178i \(-0.745890\pi\)
−0.697918 + 0.716178i \(0.745890\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −220.251 −0.0263374
\(413\) 2829.78 0.337154
\(414\) 0 0
\(415\) 0 0
\(416\) 3353.20 0.395202
\(417\) 0 0
\(418\) −7918.42 −0.926561
\(419\) −1851.51 −0.215876 −0.107938 0.994158i \(-0.534425\pi\)
−0.107938 + 0.994158i \(0.534425\pi\)
\(420\) 0 0
\(421\) −1303.60 −0.150911 −0.0754557 0.997149i \(-0.524041\pi\)
−0.0754557 + 0.997149i \(0.524041\pi\)
\(422\) −15478.9 −1.78555
\(423\) 0 0
\(424\) −31223.5 −3.57629
\(425\) 0 0
\(426\) 0 0
\(427\) −16614.4 −1.88297
\(428\) −24304.2 −2.74483
\(429\) 0 0
\(430\) 0 0
\(431\) −8228.85 −0.919652 −0.459826 0.888009i \(-0.652088\pi\)
−0.459826 + 0.888009i \(0.652088\pi\)
\(432\) 0 0
\(433\) −5830.21 −0.647072 −0.323536 0.946216i \(-0.604872\pi\)
−0.323536 + 0.946216i \(0.604872\pi\)
\(434\) 27425.5 3.03333
\(435\) 0 0
\(436\) 10771.4 1.18316
\(437\) −25038.2 −2.74082
\(438\) 0 0
\(439\) 14261.0 1.55044 0.775218 0.631694i \(-0.217640\pi\)
0.775218 + 0.631694i \(0.217640\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4732.95 −0.509329
\(443\) 12025.7 1.28975 0.644875 0.764288i \(-0.276910\pi\)
0.644875 + 0.764288i \(0.276910\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2893.92 0.307245
\(447\) 0 0
\(448\) 2820.40 0.297436
\(449\) 7073.28 0.743449 0.371725 0.928343i \(-0.378767\pi\)
0.371725 + 0.928343i \(0.378767\pi\)
\(450\) 0 0
\(451\) −2621.07 −0.273662
\(452\) −649.133 −0.0675501
\(453\) 0 0
\(454\) 5542.93 0.573001
\(455\) 0 0
\(456\) 0 0
\(457\) −2732.31 −0.279677 −0.139838 0.990174i \(-0.544658\pi\)
−0.139838 + 0.990174i \(0.544658\pi\)
\(458\) 3268.04 0.333418
\(459\) 0 0
\(460\) 0 0
\(461\) 332.708 0.0336134 0.0168067 0.999859i \(-0.494650\pi\)
0.0168067 + 0.999859i \(0.494650\pi\)
\(462\) 0 0
\(463\) −8248.39 −0.827937 −0.413969 0.910291i \(-0.635858\pi\)
−0.413969 + 0.910291i \(0.635858\pi\)
\(464\) −8107.58 −0.811174
\(465\) 0 0
\(466\) 11838.5 1.17684
\(467\) 7359.35 0.729229 0.364615 0.931158i \(-0.381201\pi\)
0.364615 + 0.931158i \(0.381201\pi\)
\(468\) 0 0
\(469\) −19339.5 −1.90408
\(470\) 0 0
\(471\) 0 0
\(472\) 5042.66 0.491752
\(473\) 333.196 0.0323897
\(474\) 0 0
\(475\) 0 0
\(476\) 19941.0 1.92016
\(477\) 0 0
\(478\) −18521.7 −1.77230
\(479\) 10327.6 0.985140 0.492570 0.870273i \(-0.336058\pi\)
0.492570 + 0.870273i \(0.336058\pi\)
\(480\) 0 0
\(481\) 8335.75 0.790182
\(482\) 1970.39 0.186201
\(483\) 0 0
\(484\) 2135.96 0.200597
\(485\) 0 0
\(486\) 0 0
\(487\) −9622.57 −0.895360 −0.447680 0.894194i \(-0.647750\pi\)
−0.447680 + 0.894194i \(0.647750\pi\)
\(488\) −29606.8 −2.74639
\(489\) 0 0
\(490\) 0 0
\(491\) 8993.59 0.826629 0.413315 0.910588i \(-0.364371\pi\)
0.413315 + 0.910588i \(0.364371\pi\)
\(492\) 0 0
\(493\) 3137.75 0.286648
\(494\) −16337.0 −1.48793
\(495\) 0 0
\(496\) 20999.0 1.90097
\(497\) −21458.5 −1.93671
\(498\) 0 0
\(499\) 2623.70 0.235377 0.117688 0.993051i \(-0.462452\pi\)
0.117688 + 0.993051i \(0.462452\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 11646.8 1.03550
\(503\) 13234.5 1.17316 0.586579 0.809892i \(-0.300474\pi\)
0.586579 + 0.809892i \(0.300474\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 9814.79 0.862294
\(507\) 0 0
\(508\) 962.612 0.0840728
\(509\) 12810.3 1.11553 0.557765 0.829999i \(-0.311659\pi\)
0.557765 + 0.829999i \(0.311659\pi\)
\(510\) 0 0
\(511\) −6670.26 −0.577446
\(512\) 25891.5 2.23487
\(513\) 0 0
\(514\) −24927.1 −2.13908
\(515\) 0 0
\(516\) 0 0
\(517\) −1511.29 −0.128562
\(518\) −51036.9 −4.32902
\(519\) 0 0
\(520\) 0 0
\(521\) −1151.47 −0.0968268 −0.0484134 0.998827i \(-0.515416\pi\)
−0.0484134 + 0.998827i \(0.515416\pi\)
\(522\) 0 0
\(523\) 17319.1 1.44801 0.724005 0.689794i \(-0.242299\pi\)
0.724005 + 0.689794i \(0.242299\pi\)
\(524\) −41974.8 −3.49939
\(525\) 0 0
\(526\) 13045.1 1.08135
\(527\) −8126.92 −0.671754
\(528\) 0 0
\(529\) 18867.6 1.55072
\(530\) 0 0
\(531\) 0 0
\(532\) 68831.9 5.60948
\(533\) −5407.72 −0.439464
\(534\) 0 0
\(535\) 0 0
\(536\) −34462.8 −2.77718
\(537\) 0 0
\(538\) −24285.7 −1.94616
\(539\) −4506.35 −0.360115
\(540\) 0 0
\(541\) −7190.73 −0.571449 −0.285724 0.958312i \(-0.592234\pi\)
−0.285724 + 0.958312i \(0.592234\pi\)
\(542\) −13824.3 −1.09558
\(543\) 0 0
\(544\) 6083.73 0.479481
\(545\) 0 0
\(546\) 0 0
\(547\) −18670.5 −1.45940 −0.729701 0.683766i \(-0.760341\pi\)
−0.729701 + 0.683766i \(0.760341\pi\)
\(548\) −54341.9 −4.23608
\(549\) 0 0
\(550\) 0 0
\(551\) 10830.8 0.837400
\(552\) 0 0
\(553\) 14619.3 1.12419
\(554\) −19051.1 −1.46102
\(555\) 0 0
\(556\) −21133.3 −1.61197
\(557\) −5510.44 −0.419183 −0.209591 0.977789i \(-0.567213\pi\)
−0.209591 + 0.977789i \(0.567213\pi\)
\(558\) 0 0
\(559\) 687.439 0.0520136
\(560\) 0 0
\(561\) 0 0
\(562\) −32591.5 −2.44625
\(563\) 3576.57 0.267734 0.133867 0.990999i \(-0.457260\pi\)
0.133867 + 0.990999i \(0.457260\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 16894.5 1.25465
\(567\) 0 0
\(568\) −38238.8 −2.82476
\(569\) −12285.2 −0.905138 −0.452569 0.891729i \(-0.649492\pi\)
−0.452569 + 0.891729i \(0.649492\pi\)
\(570\) 0 0
\(571\) 13889.5 1.01797 0.508983 0.860777i \(-0.330022\pi\)
0.508983 + 0.860777i \(0.330022\pi\)
\(572\) 4406.85 0.322132
\(573\) 0 0
\(574\) 33109.5 2.40761
\(575\) 0 0
\(576\) 0 0
\(577\) −11579.4 −0.835457 −0.417728 0.908572i \(-0.637174\pi\)
−0.417728 + 0.908572i \(0.637174\pi\)
\(578\) 16296.5 1.17274
\(579\) 0 0
\(580\) 0 0
\(581\) −33056.4 −2.36043
\(582\) 0 0
\(583\) −7025.32 −0.499072
\(584\) −11886.4 −0.842229
\(585\) 0 0
\(586\) 14581.3 1.02790
\(587\) 26468.0 1.86107 0.930537 0.366199i \(-0.119341\pi\)
0.930537 + 0.366199i \(0.119341\pi\)
\(588\) 0 0
\(589\) −28052.3 −1.96243
\(590\) 0 0
\(591\) 0 0
\(592\) −39077.6 −2.71297
\(593\) 1059.52 0.0733716 0.0366858 0.999327i \(-0.488320\pi\)
0.0366858 + 0.999327i \(0.488320\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 13229.8 0.909253
\(597\) 0 0
\(598\) 20249.6 1.38473
\(599\) 17858.7 1.21817 0.609086 0.793104i \(-0.291536\pi\)
0.609086 + 0.793104i \(0.291536\pi\)
\(600\) 0 0
\(601\) −9650.91 −0.655023 −0.327511 0.944847i \(-0.606210\pi\)
−0.327511 + 0.944847i \(0.606210\pi\)
\(602\) −4208.95 −0.284957
\(603\) 0 0
\(604\) −46703.7 −3.14627
\(605\) 0 0
\(606\) 0 0
\(607\) −22754.9 −1.52157 −0.760786 0.649002i \(-0.775187\pi\)
−0.760786 + 0.649002i \(0.775187\pi\)
\(608\) 20999.6 1.40074
\(609\) 0 0
\(610\) 0 0
\(611\) −3118.04 −0.206453
\(612\) 0 0
\(613\) −13074.5 −0.861459 −0.430729 0.902481i \(-0.641744\pi\)
−0.430729 + 0.902481i \(0.641744\pi\)
\(614\) −41300.1 −2.71455
\(615\) 0 0
\(616\) −14753.8 −0.965009
\(617\) −13393.0 −0.873878 −0.436939 0.899491i \(-0.643937\pi\)
−0.436939 + 0.899491i \(0.643937\pi\)
\(618\) 0 0
\(619\) −15965.3 −1.03667 −0.518336 0.855177i \(-0.673448\pi\)
−0.518336 + 0.855177i \(0.673448\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 19338.8 1.24665
\(623\) 29039.5 1.86749
\(624\) 0 0
\(625\) 0 0
\(626\) 12800.0 0.817237
\(627\) 0 0
\(628\) 61355.8 3.89867
\(629\) 15123.6 0.958693
\(630\) 0 0
\(631\) 17698.3 1.11657 0.558287 0.829648i \(-0.311459\pi\)
0.558287 + 0.829648i \(0.311459\pi\)
\(632\) 26051.5 1.63967
\(633\) 0 0
\(634\) 56142.4 3.51688
\(635\) 0 0
\(636\) 0 0
\(637\) −9297.37 −0.578297
\(638\) −4245.59 −0.263455
\(639\) 0 0
\(640\) 0 0
\(641\) −18264.8 −1.12546 −0.562728 0.826642i \(-0.690248\pi\)
−0.562728 + 0.826642i \(0.690248\pi\)
\(642\) 0 0
\(643\) 15730.5 0.964778 0.482389 0.875957i \(-0.339769\pi\)
0.482389 + 0.875957i \(0.339769\pi\)
\(644\) −85316.4 −5.22040
\(645\) 0 0
\(646\) −29640.4 −1.80524
\(647\) 21176.6 1.28676 0.643382 0.765545i \(-0.277531\pi\)
0.643382 + 0.765545i \(0.277531\pi\)
\(648\) 0 0
\(649\) 1134.60 0.0686242
\(650\) 0 0
\(651\) 0 0
\(652\) −62117.4 −3.73114
\(653\) −28293.6 −1.69558 −0.847791 0.530331i \(-0.822068\pi\)
−0.847791 + 0.530331i \(0.822068\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 25351.1 1.50883
\(657\) 0 0
\(658\) 19090.7 1.13105
\(659\) 3894.19 0.230191 0.115096 0.993354i \(-0.463283\pi\)
0.115096 + 0.993354i \(0.463283\pi\)
\(660\) 0 0
\(661\) −6063.63 −0.356805 −0.178402 0.983958i \(-0.557093\pi\)
−0.178402 + 0.983958i \(0.557093\pi\)
\(662\) 47699.1 2.80042
\(663\) 0 0
\(664\) −58906.4 −3.44279
\(665\) 0 0
\(666\) 0 0
\(667\) −13424.7 −0.779317
\(668\) 4418.51 0.255924
\(669\) 0 0
\(670\) 0 0
\(671\) −6661.56 −0.383259
\(672\) 0 0
\(673\) −17297.1 −0.990719 −0.495360 0.868688i \(-0.664964\pi\)
−0.495360 + 0.868688i \(0.664964\pi\)
\(674\) −57046.3 −3.26015
\(675\) 0 0
\(676\) −29690.6 −1.68927
\(677\) −4640.36 −0.263432 −0.131716 0.991287i \(-0.542049\pi\)
−0.131716 + 0.991287i \(0.542049\pi\)
\(678\) 0 0
\(679\) −2337.35 −0.132105
\(680\) 0 0
\(681\) 0 0
\(682\) 10996.3 0.617404
\(683\) −14694.9 −0.823256 −0.411628 0.911352i \(-0.635040\pi\)
−0.411628 + 0.911352i \(0.635040\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 9263.73 0.515584
\(687\) 0 0
\(688\) −3222.68 −0.178581
\(689\) −14494.4 −0.801442
\(690\) 0 0
\(691\) −9905.09 −0.545307 −0.272654 0.962112i \(-0.587901\pi\)
−0.272654 + 0.962112i \(0.587901\pi\)
\(692\) 16628.4 0.913466
\(693\) 0 0
\(694\) 21339.4 1.16720
\(695\) 0 0
\(696\) 0 0
\(697\) −9811.25 −0.533182
\(698\) −25258.7 −1.36971
\(699\) 0 0
\(700\) 0 0
\(701\) 947.946 0.0510748 0.0255374 0.999674i \(-0.491870\pi\)
0.0255374 + 0.999674i \(0.491870\pi\)
\(702\) 0 0
\(703\) 52203.2 2.80069
\(704\) 1130.84 0.0605401
\(705\) 0 0
\(706\) 38662.3 2.06101
\(707\) −214.492 −0.0114099
\(708\) 0 0
\(709\) −3310.76 −0.175371 −0.0876855 0.996148i \(-0.527947\pi\)
−0.0876855 + 0.996148i \(0.527947\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 51748.3 2.72380
\(713\) 34770.5 1.82632
\(714\) 0 0
\(715\) 0 0
\(716\) −5931.92 −0.309618
\(717\) 0 0
\(718\) −15775.8 −0.819981
\(719\) −3061.15 −0.158778 −0.0793892 0.996844i \(-0.525297\pi\)
−0.0793892 + 0.996844i \(0.525297\pi\)
\(720\) 0 0
\(721\) 342.304 0.0176811
\(722\) −67572.1 −3.48307
\(723\) 0 0
\(724\) 19549.3 1.00351
\(725\) 0 0
\(726\) 0 0
\(727\) 7405.09 0.377771 0.188886 0.981999i \(-0.439512\pi\)
0.188886 + 0.981999i \(0.439512\pi\)
\(728\) −30439.5 −1.54967
\(729\) 0 0
\(730\) 0 0
\(731\) 1247.23 0.0631057
\(732\) 0 0
\(733\) −29791.4 −1.50119 −0.750593 0.660765i \(-0.770232\pi\)
−0.750593 + 0.660765i \(0.770232\pi\)
\(734\) 30042.0 1.51072
\(735\) 0 0
\(736\) −26028.8 −1.30358
\(737\) −7754.17 −0.387556
\(738\) 0 0
\(739\) −7150.93 −0.355955 −0.177978 0.984035i \(-0.556955\pi\)
−0.177978 + 0.984035i \(0.556955\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 88744.3 4.39071
\(743\) −1940.69 −0.0958239 −0.0479120 0.998852i \(-0.515257\pi\)
−0.0479120 + 0.998852i \(0.515257\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 70496.7 3.45988
\(747\) 0 0
\(748\) 7995.38 0.390829
\(749\) 37772.4 1.84269
\(750\) 0 0
\(751\) −29491.2 −1.43295 −0.716476 0.697611i \(-0.754246\pi\)
−0.716476 + 0.697611i \(0.754246\pi\)
\(752\) 14617.2 0.708824
\(753\) 0 0
\(754\) −8759.38 −0.423074
\(755\) 0 0
\(756\) 0 0
\(757\) −3542.54 −0.170087 −0.0850436 0.996377i \(-0.527103\pi\)
−0.0850436 + 0.996377i \(0.527103\pi\)
\(758\) 62131.7 2.97721
\(759\) 0 0
\(760\) 0 0
\(761\) −8552.86 −0.407412 −0.203706 0.979032i \(-0.565299\pi\)
−0.203706 + 0.979032i \(0.565299\pi\)
\(762\) 0 0
\(763\) −16740.4 −0.794290
\(764\) 74900.0 3.54684
\(765\) 0 0
\(766\) 35124.9 1.65680
\(767\) 2340.88 0.110201
\(768\) 0 0
\(769\) −3128.73 −0.146716 −0.0733582 0.997306i \(-0.523372\pi\)
−0.0733582 + 0.997306i \(0.523372\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 58681.2 2.73573
\(773\) −5364.51 −0.249609 −0.124805 0.992181i \(-0.539830\pi\)
−0.124805 + 0.992181i \(0.539830\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4165.14 −0.192680
\(777\) 0 0
\(778\) 14055.8 0.647719
\(779\) −33866.2 −1.55762
\(780\) 0 0
\(781\) −8603.78 −0.394197
\(782\) 36738.9 1.68003
\(783\) 0 0
\(784\) 43585.6 1.98550
\(785\) 0 0
\(786\) 0 0
\(787\) −12160.9 −0.550813 −0.275406 0.961328i \(-0.588812\pi\)
−0.275406 + 0.961328i \(0.588812\pi\)
\(788\) 47258.3 2.13643
\(789\) 0 0
\(790\) 0 0
\(791\) 1008.85 0.0453485
\(792\) 0 0
\(793\) −13743.9 −0.615462
\(794\) 53380.7 2.38591
\(795\) 0 0
\(796\) 49072.4 2.18508
\(797\) −581.179 −0.0258299 −0.0129149 0.999917i \(-0.504111\pi\)
−0.0129149 + 0.999917i \(0.504111\pi\)
\(798\) 0 0
\(799\) −5657.09 −0.250480
\(800\) 0 0
\(801\) 0 0
\(802\) −62273.8 −2.74185
\(803\) −2674.45 −0.117533
\(804\) 0 0
\(805\) 0 0
\(806\) 22687.2 0.991467
\(807\) 0 0
\(808\) −382.223 −0.0166418
\(809\) −4763.37 −0.207010 −0.103505 0.994629i \(-0.533006\pi\)
−0.103505 + 0.994629i \(0.533006\pi\)
\(810\) 0 0
\(811\) −18055.4 −0.781762 −0.390881 0.920441i \(-0.627830\pi\)
−0.390881 + 0.920441i \(0.627830\pi\)
\(812\) 36905.4 1.59498
\(813\) 0 0
\(814\) −20463.3 −0.881127
\(815\) 0 0
\(816\) 0 0
\(817\) 4305.14 0.184354
\(818\) 58476.9 2.49951
\(819\) 0 0
\(820\) 0 0
\(821\) −1128.04 −0.0479522 −0.0239761 0.999713i \(-0.507633\pi\)
−0.0239761 + 0.999713i \(0.507633\pi\)
\(822\) 0 0
\(823\) 32124.2 1.36061 0.680304 0.732930i \(-0.261848\pi\)
0.680304 + 0.732930i \(0.261848\pi\)
\(824\) 609.984 0.0257886
\(825\) 0 0
\(826\) −14332.4 −0.603738
\(827\) 11914.2 0.500964 0.250482 0.968121i \(-0.419411\pi\)
0.250482 + 0.968121i \(0.419411\pi\)
\(828\) 0 0
\(829\) −37721.6 −1.58037 −0.790185 0.612868i \(-0.790016\pi\)
−0.790185 + 0.612868i \(0.790016\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2333.12 0.0972192
\(833\) −16868.3 −0.701622
\(834\) 0 0
\(835\) 0 0
\(836\) 27598.2 1.14175
\(837\) 0 0
\(838\) 9377.57 0.386567
\(839\) 16550.5 0.681034 0.340517 0.940238i \(-0.389398\pi\)
0.340517 + 0.940238i \(0.389398\pi\)
\(840\) 0 0
\(841\) −18581.9 −0.761896
\(842\) 6602.53 0.270236
\(843\) 0 0
\(844\) 53948.9 2.20023
\(845\) 0 0
\(846\) 0 0
\(847\) −3319.61 −0.134667
\(848\) 67949.2 2.75163
\(849\) 0 0
\(850\) 0 0
\(851\) −64705.3 −2.60643
\(852\) 0 0
\(853\) −1045.52 −0.0419672 −0.0209836 0.999780i \(-0.506680\pi\)
−0.0209836 + 0.999780i \(0.506680\pi\)
\(854\) 84149.3 3.37181
\(855\) 0 0
\(856\) 67310.2 2.68764
\(857\) −14016.5 −0.558688 −0.279344 0.960191i \(-0.590117\pi\)
−0.279344 + 0.960191i \(0.590117\pi\)
\(858\) 0 0
\(859\) −20476.3 −0.813319 −0.406660 0.913580i \(-0.633306\pi\)
−0.406660 + 0.913580i \(0.633306\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 41677.8 1.64681
\(863\) 24083.0 0.949936 0.474968 0.880003i \(-0.342460\pi\)
0.474968 + 0.880003i \(0.342460\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 29529.1 1.15871
\(867\) 0 0
\(868\) −95586.6 −3.73781
\(869\) 5861.61 0.228817
\(870\) 0 0
\(871\) −15998.2 −0.622362
\(872\) −29831.3 −1.15850
\(873\) 0 0
\(874\) 126814. 4.90796
\(875\) 0 0
\(876\) 0 0
\(877\) −30432.5 −1.17176 −0.585879 0.810399i \(-0.699250\pi\)
−0.585879 + 0.810399i \(0.699250\pi\)
\(878\) −72229.7 −2.77635
\(879\) 0 0
\(880\) 0 0
\(881\) −24559.2 −0.939183 −0.469592 0.882884i \(-0.655599\pi\)
−0.469592 + 0.882884i \(0.655599\pi\)
\(882\) 0 0
\(883\) −6013.88 −0.229200 −0.114600 0.993412i \(-0.536559\pi\)
−0.114600 + 0.993412i \(0.536559\pi\)
\(884\) 16495.8 0.627618
\(885\) 0 0
\(886\) −60908.3 −2.30954
\(887\) 13395.5 0.507075 0.253538 0.967325i \(-0.418406\pi\)
0.253538 + 0.967325i \(0.418406\pi\)
\(888\) 0 0
\(889\) −1496.05 −0.0564407
\(890\) 0 0
\(891\) 0 0
\(892\) −10086.2 −0.378601
\(893\) −19527.0 −0.731741
\(894\) 0 0
\(895\) 0 0
\(896\) −46713.1 −1.74171
\(897\) 0 0
\(898\) −35825.0 −1.33129
\(899\) −15040.7 −0.557992
\(900\) 0 0
\(901\) −26297.3 −0.972354
\(902\) 13275.3 0.490043
\(903\) 0 0
\(904\) 1797.77 0.0661426
\(905\) 0 0
\(906\) 0 0
\(907\) 32078.9 1.17438 0.587189 0.809450i \(-0.300234\pi\)
0.587189 + 0.809450i \(0.300234\pi\)
\(908\) −19318.9 −0.706079
\(909\) 0 0
\(910\) 0 0
\(911\) 15992.0 0.581600 0.290800 0.956784i \(-0.406079\pi\)
0.290800 + 0.956784i \(0.406079\pi\)
\(912\) 0 0
\(913\) −13254.0 −0.480442
\(914\) 13838.7 0.500814
\(915\) 0 0
\(916\) −11390.1 −0.410852
\(917\) 65235.4 2.34925
\(918\) 0 0
\(919\) 33876.1 1.21596 0.607980 0.793952i \(-0.291980\pi\)
0.607980 + 0.793952i \(0.291980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1685.11 −0.0601912
\(923\) −17751.1 −0.633026
\(924\) 0 0
\(925\) 0 0
\(926\) 41776.7 1.48258
\(927\) 0 0
\(928\) 11259.3 0.398281
\(929\) −21163.4 −0.747416 −0.373708 0.927546i \(-0.621914\pi\)
−0.373708 + 0.927546i \(0.621914\pi\)
\(930\) 0 0
\(931\) −58225.4 −2.04969
\(932\) −41261.0 −1.45016
\(933\) 0 0
\(934\) −37273.9 −1.30582
\(935\) 0 0
\(936\) 0 0
\(937\) 49771.9 1.73530 0.867650 0.497175i \(-0.165629\pi\)
0.867650 + 0.497175i \(0.165629\pi\)
\(938\) 97951.2 3.40962
\(939\) 0 0
\(940\) 0 0
\(941\) −32194.6 −1.11532 −0.557659 0.830070i \(-0.688300\pi\)
−0.557659 + 0.830070i \(0.688300\pi\)
\(942\) 0 0
\(943\) 41976.8 1.44958
\(944\) −10973.9 −0.378359
\(945\) 0 0
\(946\) −1687.58 −0.0580000
\(947\) −30091.9 −1.03258 −0.516291 0.856413i \(-0.672688\pi\)
−0.516291 + 0.856413i \(0.672688\pi\)
\(948\) 0 0
\(949\) −5517.84 −0.188742
\(950\) 0 0
\(951\) 0 0
\(952\) −55226.6 −1.88015
\(953\) 5710.17 0.194093 0.0970465 0.995280i \(-0.469060\pi\)
0.0970465 + 0.995280i \(0.469060\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 64553.9 2.18392
\(957\) 0 0
\(958\) −52307.8 −1.76408
\(959\) 84455.7 2.84381
\(960\) 0 0
\(961\) 9165.07 0.307646
\(962\) −42219.2 −1.41497
\(963\) 0 0
\(964\) −6867.44 −0.229445
\(965\) 0 0
\(966\) 0 0
\(967\) −29638.4 −0.985634 −0.492817 0.870133i \(-0.664033\pi\)
−0.492817 + 0.870133i \(0.664033\pi\)
\(968\) −5915.53 −0.196418
\(969\) 0 0
\(970\) 0 0
\(971\) −21416.3 −0.707808 −0.353904 0.935282i \(-0.615146\pi\)
−0.353904 + 0.935282i \(0.615146\pi\)
\(972\) 0 0
\(973\) 32844.5 1.08216
\(974\) 48736.7 1.60331
\(975\) 0 0
\(976\) 64430.9 2.11310
\(977\) 1038.84 0.0340177 0.0170088 0.999855i \(-0.494586\pi\)
0.0170088 + 0.999855i \(0.494586\pi\)
\(978\) 0 0
\(979\) 11643.4 0.380107
\(980\) 0 0
\(981\) 0 0
\(982\) −45551.1 −1.48024
\(983\) −44173.6 −1.43329 −0.716643 0.697440i \(-0.754322\pi\)
−0.716643 + 0.697440i \(0.754322\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −15892.2 −0.513297
\(987\) 0 0
\(988\) 56939.8 1.83350
\(989\) −5336.17 −0.171567
\(990\) 0 0
\(991\) −23940.9 −0.767414 −0.383707 0.923455i \(-0.625353\pi\)
−0.383707 + 0.923455i \(0.625353\pi\)
\(992\) −29162.1 −0.933365
\(993\) 0 0
\(994\) 108684. 3.46804
\(995\) 0 0
\(996\) 0 0
\(997\) −13557.9 −0.430674 −0.215337 0.976540i \(-0.569085\pi\)
−0.215337 + 0.976540i \(0.569085\pi\)
\(998\) −13288.6 −0.421487
\(999\) 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 2475.4.a.t.1.1 3
3.2 odd 2 825.4.a.r.1.3 3
5.4 even 2 495.4.a.k.1.3 3
15.2 even 4 825.4.c.k.199.6 6
15.8 even 4 825.4.c.k.199.1 6
15.14 odd 2 165.4.a.e.1.1 3
165.164 even 2 1815.4.a.r.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.e.1.1 3 15.14 odd 2
495.4.a.k.1.3 3 5.4 even 2
825.4.a.r.1.3 3 3.2 odd 2
825.4.c.k.199.1 6 15.8 even 4
825.4.c.k.199.6 6 15.2 even 4
1815.4.a.r.1.3 3 165.164 even 2
2475.4.a.t.1.1 3 1.1 even 1 trivial