Properties

Label 825.4.a.bd.1.6
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $7$
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] = [825,4,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, 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("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 31x^{5} + 50x^{4} + 272x^{3} - 322x^{2} - 704x + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.70507\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.70507 q^{2} -3.00000 q^{3} +5.72754 q^{4} -11.1152 q^{6} +30.9104 q^{7} -8.41961 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.70507 q^{2} -3.00000 q^{3} +5.72754 q^{4} -11.1152 q^{6} +30.9104 q^{7} -8.41961 q^{8} +9.00000 q^{9} +11.0000 q^{11} -17.1826 q^{12} +39.2828 q^{13} +114.525 q^{14} -77.0156 q^{16} -83.0298 q^{17} +33.3456 q^{18} +49.6019 q^{19} -92.7312 q^{21} +40.7558 q^{22} +10.7536 q^{23} +25.2588 q^{24} +145.545 q^{26} -27.0000 q^{27} +177.041 q^{28} +198.890 q^{29} +229.071 q^{31} -217.991 q^{32} -33.0000 q^{33} -307.631 q^{34} +51.5479 q^{36} +348.562 q^{37} +183.778 q^{38} -117.848 q^{39} -459.643 q^{41} -343.576 q^{42} -461.300 q^{43} +63.0030 q^{44} +39.8427 q^{46} +272.390 q^{47} +231.047 q^{48} +612.453 q^{49} +249.089 q^{51} +224.994 q^{52} +360.420 q^{53} -100.037 q^{54} -260.253 q^{56} -148.806 q^{57} +736.901 q^{58} -215.424 q^{59} +367.967 q^{61} +848.723 q^{62} +278.194 q^{63} -191.548 q^{64} -122.267 q^{66} +929.374 q^{67} -475.557 q^{68} -32.2607 q^{69} +126.531 q^{71} -75.7765 q^{72} +509.402 q^{73} +1291.45 q^{74} +284.097 q^{76} +340.014 q^{77} -436.636 q^{78} +719.755 q^{79} +81.0000 q^{81} -1703.01 q^{82} -865.128 q^{83} -531.122 q^{84} -1709.15 q^{86} -596.670 q^{87} -92.6157 q^{88} +528.684 q^{89} +1214.25 q^{91} +61.5915 q^{92} -687.212 q^{93} +1009.22 q^{94} +653.974 q^{96} +858.037 q^{97} +2269.18 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 5 q^{2} - 21 q^{3} + 13 q^{4} - 15 q^{6} + 34 q^{7} + 75 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 5 q^{2} - 21 q^{3} + 13 q^{4} - 15 q^{6} + 34 q^{7} + 75 q^{8} + 63 q^{9} + 77 q^{11} - 39 q^{12} + 80 q^{13} + 42 q^{14} - 43 q^{16} + 162 q^{17} + 45 q^{18} + 58 q^{19} - 102 q^{21} + 55 q^{22} + 324 q^{23} - 225 q^{24} - 200 q^{26} - 189 q^{27} - 168 q^{28} + 64 q^{29} - 348 q^{31} - 75 q^{32} - 231 q^{33} + 206 q^{34} + 117 q^{36} + 664 q^{37} + 334 q^{38} - 240 q^{39} - 332 q^{41} - 126 q^{42} + 774 q^{43} + 143 q^{44} - 328 q^{46} + 872 q^{47} + 129 q^{48} - 417 q^{49} - 486 q^{51} + 134 q^{52} + 1628 q^{53} - 135 q^{54} - 1618 q^{56} - 174 q^{57} + 1568 q^{58} - 332 q^{59} + 22 q^{61} - 260 q^{62} + 306 q^{63} + 561 q^{64} - 165 q^{66} + 1524 q^{67} + 2324 q^{68} - 972 q^{69} - 516 q^{71} + 675 q^{72} + 1700 q^{73} + 1628 q^{74} + 2794 q^{76} + 374 q^{77} + 600 q^{78} + 1746 q^{79} + 567 q^{81} + 364 q^{82} + 2344 q^{83} + 504 q^{84} + 1270 q^{86} - 192 q^{87} + 825 q^{88} - 2226 q^{89} + 1072 q^{91} + 4184 q^{92} + 1044 q^{93} + 4736 q^{94} + 225 q^{96} + 1048 q^{97} + 3057 q^{98} + 693 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.70507 1.30994 0.654970 0.755655i \(-0.272681\pi\)
0.654970 + 0.755655i \(0.272681\pi\)
\(3\) −3.00000 −0.577350
\(4\) 5.72754 0.715943
\(5\) 0 0
\(6\) −11.1152 −0.756294
\(7\) 30.9104 1.66900 0.834502 0.551004i \(-0.185755\pi\)
0.834502 + 0.551004i \(0.185755\pi\)
\(8\) −8.41961 −0.372098
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) −17.1826 −0.413350
\(13\) 39.2828 0.838083 0.419042 0.907967i \(-0.362366\pi\)
0.419042 + 0.907967i \(0.362366\pi\)
\(14\) 114.525 2.18630
\(15\) 0 0
\(16\) −77.0156 −1.20337
\(17\) −83.0298 −1.18457 −0.592285 0.805729i \(-0.701774\pi\)
−0.592285 + 0.805729i \(0.701774\pi\)
\(18\) 33.3456 0.436647
\(19\) 49.6019 0.598918 0.299459 0.954109i \(-0.403194\pi\)
0.299459 + 0.954109i \(0.403194\pi\)
\(20\) 0 0
\(21\) −92.7312 −0.963600
\(22\) 40.7558 0.394962
\(23\) 10.7536 0.0974902 0.0487451 0.998811i \(-0.484478\pi\)
0.0487451 + 0.998811i \(0.484478\pi\)
\(24\) 25.2588 0.214831
\(25\) 0 0
\(26\) 145.545 1.09784
\(27\) −27.0000 −0.192450
\(28\) 177.041 1.19491
\(29\) 198.890 1.27355 0.636775 0.771050i \(-0.280268\pi\)
0.636775 + 0.771050i \(0.280268\pi\)
\(30\) 0 0
\(31\) 229.071 1.32717 0.663586 0.748100i \(-0.269034\pi\)
0.663586 + 0.748100i \(0.269034\pi\)
\(32\) −217.991 −1.20424
\(33\) −33.0000 −0.174078
\(34\) −307.631 −1.55171
\(35\) 0 0
\(36\) 51.5479 0.238648
\(37\) 348.562 1.54874 0.774368 0.632736i \(-0.218068\pi\)
0.774368 + 0.632736i \(0.218068\pi\)
\(38\) 183.778 0.784547
\(39\) −117.848 −0.483867
\(40\) 0 0
\(41\) −459.643 −1.75083 −0.875416 0.483370i \(-0.839413\pi\)
−0.875416 + 0.483370i \(0.839413\pi\)
\(42\) −343.576 −1.26226
\(43\) −461.300 −1.63599 −0.817995 0.575226i \(-0.804914\pi\)
−0.817995 + 0.575226i \(0.804914\pi\)
\(44\) 63.0030 0.215865
\(45\) 0 0
\(46\) 39.8427 0.127706
\(47\) 272.390 0.845365 0.422682 0.906278i \(-0.361089\pi\)
0.422682 + 0.906278i \(0.361089\pi\)
\(48\) 231.047 0.694765
\(49\) 612.453 1.78558
\(50\) 0 0
\(51\) 249.089 0.683911
\(52\) 224.994 0.600020
\(53\) 360.420 0.934104 0.467052 0.884230i \(-0.345316\pi\)
0.467052 + 0.884230i \(0.345316\pi\)
\(54\) −100.037 −0.252098
\(55\) 0 0
\(56\) −260.253 −0.621033
\(57\) −148.806 −0.345786
\(58\) 736.901 1.66827
\(59\) −215.424 −0.475352 −0.237676 0.971345i \(-0.576386\pi\)
−0.237676 + 0.971345i \(0.576386\pi\)
\(60\) 0 0
\(61\) 367.967 0.772350 0.386175 0.922426i \(-0.373796\pi\)
0.386175 + 0.922426i \(0.373796\pi\)
\(62\) 848.723 1.73851
\(63\) 278.194 0.556335
\(64\) −191.548 −0.374118
\(65\) 0 0
\(66\) −122.267 −0.228031
\(67\) 929.374 1.69464 0.847322 0.531079i \(-0.178213\pi\)
0.847322 + 0.531079i \(0.178213\pi\)
\(68\) −475.557 −0.848084
\(69\) −32.2607 −0.0562860
\(70\) 0 0
\(71\) 126.531 0.211499 0.105750 0.994393i \(-0.466276\pi\)
0.105750 + 0.994393i \(0.466276\pi\)
\(72\) −75.7765 −0.124033
\(73\) 509.402 0.816726 0.408363 0.912820i \(-0.366100\pi\)
0.408363 + 0.912820i \(0.366100\pi\)
\(74\) 1291.45 2.02875
\(75\) 0 0
\(76\) 284.097 0.428791
\(77\) 340.014 0.503224
\(78\) −436.636 −0.633837
\(79\) 719.755 1.02505 0.512524 0.858673i \(-0.328711\pi\)
0.512524 + 0.858673i \(0.328711\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −1703.01 −2.29349
\(83\) −865.128 −1.14410 −0.572049 0.820220i \(-0.693851\pi\)
−0.572049 + 0.820220i \(0.693851\pi\)
\(84\) −531.122 −0.689883
\(85\) 0 0
\(86\) −1709.15 −2.14305
\(87\) −596.670 −0.735284
\(88\) −92.6157 −0.112192
\(89\) 528.684 0.629668 0.314834 0.949147i \(-0.398051\pi\)
0.314834 + 0.949147i \(0.398051\pi\)
\(90\) 0 0
\(91\) 1214.25 1.39876
\(92\) 61.5915 0.0697974
\(93\) −687.212 −0.766243
\(94\) 1009.22 1.10738
\(95\) 0 0
\(96\) 653.974 0.695270
\(97\) 858.037 0.898149 0.449075 0.893494i \(-0.351754\pi\)
0.449075 + 0.893494i \(0.351754\pi\)
\(98\) 2269.18 2.33900
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) −48.7808 −0.0480582 −0.0240291 0.999711i \(-0.507649\pi\)
−0.0240291 + 0.999711i \(0.507649\pi\)
\(102\) 922.893 0.895883
\(103\) 525.103 0.502329 0.251165 0.967944i \(-0.419186\pi\)
0.251165 + 0.967944i \(0.419186\pi\)
\(104\) −330.746 −0.311849
\(105\) 0 0
\(106\) 1335.38 1.22362
\(107\) 986.489 0.891285 0.445643 0.895211i \(-0.352975\pi\)
0.445643 + 0.895211i \(0.352975\pi\)
\(108\) −154.644 −0.137783
\(109\) −55.2903 −0.0485858 −0.0242929 0.999705i \(-0.507733\pi\)
−0.0242929 + 0.999705i \(0.507733\pi\)
\(110\) 0 0
\(111\) −1045.69 −0.894163
\(112\) −2380.58 −2.00843
\(113\) 459.760 0.382748 0.191374 0.981517i \(-0.438706\pi\)
0.191374 + 0.981517i \(0.438706\pi\)
\(114\) −551.335 −0.452959
\(115\) 0 0
\(116\) 1139.15 0.911789
\(117\) 353.545 0.279361
\(118\) −798.160 −0.622683
\(119\) −2566.48 −1.97705
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 1363.34 1.01173
\(123\) 1378.93 1.01084
\(124\) 1312.01 0.950179
\(125\) 0 0
\(126\) 1030.73 0.728765
\(127\) −1959.31 −1.36898 −0.684492 0.729020i \(-0.739976\pi\)
−0.684492 + 0.729020i \(0.739976\pi\)
\(128\) 1034.23 0.714171
\(129\) 1383.90 0.944539
\(130\) 0 0
\(131\) 16.2654 0.0108482 0.00542412 0.999985i \(-0.498273\pi\)
0.00542412 + 0.999985i \(0.498273\pi\)
\(132\) −189.009 −0.124630
\(133\) 1533.21 0.999598
\(134\) 3443.40 2.21988
\(135\) 0 0
\(136\) 699.078 0.440775
\(137\) 941.492 0.587132 0.293566 0.955939i \(-0.405158\pi\)
0.293566 + 0.955939i \(0.405158\pi\)
\(138\) −119.528 −0.0737313
\(139\) −1975.04 −1.20518 −0.602591 0.798050i \(-0.705865\pi\)
−0.602591 + 0.798050i \(0.705865\pi\)
\(140\) 0 0
\(141\) −817.169 −0.488072
\(142\) 468.805 0.277051
\(143\) 432.110 0.252692
\(144\) −693.140 −0.401123
\(145\) 0 0
\(146\) 1887.37 1.06986
\(147\) −1837.36 −1.03090
\(148\) 1996.40 1.10881
\(149\) −2958.51 −1.62665 −0.813325 0.581810i \(-0.802345\pi\)
−0.813325 + 0.581810i \(0.802345\pi\)
\(150\) 0 0
\(151\) −3453.22 −1.86105 −0.930526 0.366227i \(-0.880650\pi\)
−0.930526 + 0.366227i \(0.880650\pi\)
\(152\) −417.628 −0.222856
\(153\) −747.268 −0.394856
\(154\) 1259.78 0.659193
\(155\) 0 0
\(156\) −674.981 −0.346422
\(157\) 738.890 0.375604 0.187802 0.982207i \(-0.439864\pi\)
0.187802 + 0.982207i \(0.439864\pi\)
\(158\) 2666.74 1.34275
\(159\) −1081.26 −0.539305
\(160\) 0 0
\(161\) 332.397 0.162712
\(162\) 300.111 0.145549
\(163\) 242.527 0.116541 0.0582705 0.998301i \(-0.481441\pi\)
0.0582705 + 0.998301i \(0.481441\pi\)
\(164\) −2632.62 −1.25350
\(165\) 0 0
\(166\) −3205.36 −1.49870
\(167\) 2447.06 1.13389 0.566944 0.823756i \(-0.308126\pi\)
0.566944 + 0.823756i \(0.308126\pi\)
\(168\) 780.760 0.358553
\(169\) −653.864 −0.297617
\(170\) 0 0
\(171\) 446.417 0.199639
\(172\) −2642.11 −1.17127
\(173\) −878.419 −0.386040 −0.193020 0.981195i \(-0.561828\pi\)
−0.193020 + 0.981195i \(0.561828\pi\)
\(174\) −2210.70 −0.963178
\(175\) 0 0
\(176\) −847.172 −0.362829
\(177\) 646.271 0.274445
\(178\) 1958.81 0.824827
\(179\) −2473.19 −1.03271 −0.516354 0.856375i \(-0.672711\pi\)
−0.516354 + 0.856375i \(0.672711\pi\)
\(180\) 0 0
\(181\) 49.4762 0.0203179 0.0101589 0.999948i \(-0.496766\pi\)
0.0101589 + 0.999948i \(0.496766\pi\)
\(182\) 4498.87 1.83230
\(183\) −1103.90 −0.445916
\(184\) −90.5408 −0.0362759
\(185\) 0 0
\(186\) −2546.17 −1.00373
\(187\) −913.327 −0.357161
\(188\) 1560.12 0.605233
\(189\) −834.581 −0.321200
\(190\) 0 0
\(191\) −1945.54 −0.737037 −0.368518 0.929620i \(-0.620135\pi\)
−0.368518 + 0.929620i \(0.620135\pi\)
\(192\) 574.645 0.215997
\(193\) −3064.50 −1.14294 −0.571470 0.820623i \(-0.693627\pi\)
−0.571470 + 0.820623i \(0.693627\pi\)
\(194\) 3179.09 1.17652
\(195\) 0 0
\(196\) 3507.85 1.27837
\(197\) −1659.71 −0.600251 −0.300125 0.953900i \(-0.597029\pi\)
−0.300125 + 0.953900i \(0.597029\pi\)
\(198\) 366.802 0.131654
\(199\) −3230.11 −1.15063 −0.575317 0.817930i \(-0.695121\pi\)
−0.575317 + 0.817930i \(0.695121\pi\)
\(200\) 0 0
\(201\) −2788.12 −0.978403
\(202\) −180.736 −0.0629533
\(203\) 6147.77 2.12556
\(204\) 1426.67 0.489642
\(205\) 0 0
\(206\) 1945.54 0.658021
\(207\) 96.7821 0.0324967
\(208\) −3025.39 −1.00852
\(209\) 545.621 0.180581
\(210\) 0 0
\(211\) −2543.86 −0.829983 −0.414991 0.909825i \(-0.636215\pi\)
−0.414991 + 0.909825i \(0.636215\pi\)
\(212\) 2064.32 0.668766
\(213\) −379.592 −0.122109
\(214\) 3655.01 1.16753
\(215\) 0 0
\(216\) 227.329 0.0716102
\(217\) 7080.67 2.21506
\(218\) −204.854 −0.0636445
\(219\) −1528.21 −0.471537
\(220\) 0 0
\(221\) −3261.64 −0.992767
\(222\) −3874.34 −1.17130
\(223\) 1189.64 0.357240 0.178620 0.983918i \(-0.442837\pi\)
0.178620 + 0.983918i \(0.442837\pi\)
\(224\) −6738.20 −2.00989
\(225\) 0 0
\(226\) 1703.44 0.501378
\(227\) 4897.90 1.43209 0.716047 0.698053i \(-0.245950\pi\)
0.716047 + 0.698053i \(0.245950\pi\)
\(228\) −852.291 −0.247563
\(229\) −2415.61 −0.697065 −0.348533 0.937297i \(-0.613320\pi\)
−0.348533 + 0.937297i \(0.613320\pi\)
\(230\) 0 0
\(231\) −1020.04 −0.290536
\(232\) −1674.58 −0.473885
\(233\) −524.258 −0.147405 −0.0737023 0.997280i \(-0.523481\pi\)
−0.0737023 + 0.997280i \(0.523481\pi\)
\(234\) 1309.91 0.365946
\(235\) 0 0
\(236\) −1233.85 −0.340325
\(237\) −2159.27 −0.591812
\(238\) −9509.00 −2.58982
\(239\) −4165.12 −1.12728 −0.563638 0.826022i \(-0.690599\pi\)
−0.563638 + 0.826022i \(0.690599\pi\)
\(240\) 0 0
\(241\) 1908.80 0.510194 0.255097 0.966915i \(-0.417893\pi\)
0.255097 + 0.966915i \(0.417893\pi\)
\(242\) 448.313 0.119085
\(243\) −243.000 −0.0641500
\(244\) 2107.55 0.552958
\(245\) 0 0
\(246\) 5109.03 1.32414
\(247\) 1948.50 0.501943
\(248\) −1928.69 −0.493837
\(249\) 2595.38 0.660545
\(250\) 0 0
\(251\) 2503.63 0.629592 0.314796 0.949159i \(-0.398064\pi\)
0.314796 + 0.949159i \(0.398064\pi\)
\(252\) 1593.37 0.398304
\(253\) 118.289 0.0293944
\(254\) −7259.39 −1.79329
\(255\) 0 0
\(256\) 5364.28 1.30964
\(257\) −2985.91 −0.724731 −0.362365 0.932036i \(-0.618031\pi\)
−0.362365 + 0.932036i \(0.618031\pi\)
\(258\) 5127.44 1.23729
\(259\) 10774.2 2.58485
\(260\) 0 0
\(261\) 1790.01 0.424517
\(262\) 60.2646 0.0142105
\(263\) 2213.20 0.518903 0.259452 0.965756i \(-0.416458\pi\)
0.259452 + 0.965756i \(0.416458\pi\)
\(264\) 277.847 0.0647739
\(265\) 0 0
\(266\) 5680.66 1.30941
\(267\) −1586.05 −0.363539
\(268\) 5323.03 1.21327
\(269\) −630.921 −0.143003 −0.0715017 0.997440i \(-0.522779\pi\)
−0.0715017 + 0.997440i \(0.522779\pi\)
\(270\) 0 0
\(271\) −5781.90 −1.29603 −0.648017 0.761626i \(-0.724402\pi\)
−0.648017 + 0.761626i \(0.724402\pi\)
\(272\) 6394.59 1.42547
\(273\) −3642.74 −0.807577
\(274\) 3488.29 0.769108
\(275\) 0 0
\(276\) −184.775 −0.0402976
\(277\) −3365.10 −0.729925 −0.364962 0.931022i \(-0.618918\pi\)
−0.364962 + 0.931022i \(0.618918\pi\)
\(278\) −7317.64 −1.57872
\(279\) 2061.64 0.442390
\(280\) 0 0
\(281\) 622.308 0.132113 0.0660566 0.997816i \(-0.478958\pi\)
0.0660566 + 0.997816i \(0.478958\pi\)
\(282\) −3027.67 −0.639344
\(283\) −1629.44 −0.342263 −0.171131 0.985248i \(-0.554742\pi\)
−0.171131 + 0.985248i \(0.554742\pi\)
\(284\) 724.710 0.151421
\(285\) 0 0
\(286\) 1601.00 0.331011
\(287\) −14207.7 −2.92215
\(288\) −1961.92 −0.401414
\(289\) 1980.94 0.403204
\(290\) 0 0
\(291\) −2574.11 −0.518547
\(292\) 2917.62 0.584729
\(293\) −5845.09 −1.16544 −0.582720 0.812673i \(-0.698012\pi\)
−0.582720 + 0.812673i \(0.698012\pi\)
\(294\) −6807.54 −1.35042
\(295\) 0 0
\(296\) −2934.75 −0.576281
\(297\) −297.000 −0.0580259
\(298\) −10961.5 −2.13081
\(299\) 422.430 0.0817049
\(300\) 0 0
\(301\) −14259.0 −2.73047
\(302\) −12794.4 −2.43787
\(303\) 146.343 0.0277464
\(304\) −3820.12 −0.720720
\(305\) 0 0
\(306\) −2768.68 −0.517238
\(307\) −4452.12 −0.827674 −0.413837 0.910351i \(-0.635812\pi\)
−0.413837 + 0.910351i \(0.635812\pi\)
\(308\) 1947.45 0.360280
\(309\) −1575.31 −0.290020
\(310\) 0 0
\(311\) −7340.61 −1.33842 −0.669209 0.743074i \(-0.733367\pi\)
−0.669209 + 0.743074i \(0.733367\pi\)
\(312\) 992.237 0.180046
\(313\) 4302.99 0.777059 0.388530 0.921436i \(-0.372983\pi\)
0.388530 + 0.921436i \(0.372983\pi\)
\(314\) 2737.64 0.492019
\(315\) 0 0
\(316\) 4122.43 0.733876
\(317\) 5631.72 0.997821 0.498910 0.866654i \(-0.333734\pi\)
0.498910 + 0.866654i \(0.333734\pi\)
\(318\) −4006.15 −0.706458
\(319\) 2187.79 0.383990
\(320\) 0 0
\(321\) −2959.47 −0.514584
\(322\) 1231.55 0.213142
\(323\) −4118.43 −0.709460
\(324\) 463.931 0.0795492
\(325\) 0 0
\(326\) 898.579 0.152662
\(327\) 165.871 0.0280510
\(328\) 3870.01 0.651481
\(329\) 8419.68 1.41092
\(330\) 0 0
\(331\) 11290.4 1.87485 0.937424 0.348190i \(-0.113204\pi\)
0.937424 + 0.348190i \(0.113204\pi\)
\(332\) −4955.06 −0.819109
\(333\) 3137.06 0.516245
\(334\) 9066.53 1.48533
\(335\) 0 0
\(336\) 7141.75 1.15957
\(337\) −4323.61 −0.698878 −0.349439 0.936959i \(-0.613628\pi\)
−0.349439 + 0.936959i \(0.613628\pi\)
\(338\) −2422.61 −0.389860
\(339\) −1379.28 −0.220980
\(340\) 0 0
\(341\) 2519.78 0.400157
\(342\) 1654.01 0.261516
\(343\) 8328.90 1.31113
\(344\) 3883.96 0.608748
\(345\) 0 0
\(346\) −3254.60 −0.505689
\(347\) 7693.25 1.19019 0.595094 0.803656i \(-0.297115\pi\)
0.595094 + 0.803656i \(0.297115\pi\)
\(348\) −3417.45 −0.526422
\(349\) 2791.91 0.428217 0.214108 0.976810i \(-0.431315\pi\)
0.214108 + 0.976810i \(0.431315\pi\)
\(350\) 0 0
\(351\) −1060.63 −0.161289
\(352\) −2397.90 −0.363093
\(353\) −5310.27 −0.800672 −0.400336 0.916368i \(-0.631107\pi\)
−0.400336 + 0.916368i \(0.631107\pi\)
\(354\) 2394.48 0.359506
\(355\) 0 0
\(356\) 3028.06 0.450806
\(357\) 7699.45 1.14145
\(358\) −9163.33 −1.35278
\(359\) 7008.05 1.03028 0.515140 0.857106i \(-0.327740\pi\)
0.515140 + 0.857106i \(0.327740\pi\)
\(360\) 0 0
\(361\) −4398.65 −0.641297
\(362\) 183.313 0.0266152
\(363\) −363.000 −0.0524864
\(364\) 6954.65 1.00144
\(365\) 0 0
\(366\) −4090.03 −0.584124
\(367\) 2101.70 0.298932 0.149466 0.988767i \(-0.452245\pi\)
0.149466 + 0.988767i \(0.452245\pi\)
\(368\) −828.192 −0.117317
\(369\) −4136.78 −0.583611
\(370\) 0 0
\(371\) 11140.7 1.55902
\(372\) −3936.04 −0.548586
\(373\) 8782.22 1.21910 0.609552 0.792746i \(-0.291349\pi\)
0.609552 + 0.792746i \(0.291349\pi\)
\(374\) −3383.94 −0.467860
\(375\) 0 0
\(376\) −2293.42 −0.314558
\(377\) 7812.95 1.06734
\(378\) −3092.18 −0.420753
\(379\) 3038.99 0.411879 0.205940 0.978565i \(-0.433975\pi\)
0.205940 + 0.978565i \(0.433975\pi\)
\(380\) 0 0
\(381\) 5877.94 0.790383
\(382\) −7208.35 −0.965474
\(383\) 14611.0 1.94932 0.974658 0.223700i \(-0.0718137\pi\)
0.974658 + 0.223700i \(0.0718137\pi\)
\(384\) −3102.69 −0.412327
\(385\) 0 0
\(386\) −11354.2 −1.49718
\(387\) −4151.70 −0.545330
\(388\) 4914.45 0.643024
\(389\) −173.498 −0.0226136 −0.0113068 0.999936i \(-0.503599\pi\)
−0.0113068 + 0.999936i \(0.503599\pi\)
\(390\) 0 0
\(391\) −892.866 −0.115484
\(392\) −5156.61 −0.664409
\(393\) −48.7963 −0.00626323
\(394\) −6149.34 −0.786292
\(395\) 0 0
\(396\) 567.027 0.0719550
\(397\) 1369.79 0.173168 0.0865841 0.996245i \(-0.472405\pi\)
0.0865841 + 0.996245i \(0.472405\pi\)
\(398\) −11967.8 −1.50726
\(399\) −4599.64 −0.577118
\(400\) 0 0
\(401\) 14595.4 1.81760 0.908801 0.417230i \(-0.136999\pi\)
0.908801 + 0.417230i \(0.136999\pi\)
\(402\) −10330.2 −1.28165
\(403\) 8998.53 1.11228
\(404\) −279.394 −0.0344069
\(405\) 0 0
\(406\) 22777.9 2.78436
\(407\) 3834.18 0.466961
\(408\) −2097.23 −0.254482
\(409\) −9903.11 −1.19726 −0.598628 0.801027i \(-0.704287\pi\)
−0.598628 + 0.801027i \(0.704287\pi\)
\(410\) 0 0
\(411\) −2824.48 −0.338981
\(412\) 3007.55 0.359639
\(413\) −6658.83 −0.793365
\(414\) 358.584 0.0425688
\(415\) 0 0
\(416\) −8563.30 −1.00926
\(417\) 5925.11 0.695812
\(418\) 2021.56 0.236550
\(419\) −10648.7 −1.24159 −0.620794 0.783974i \(-0.713190\pi\)
−0.620794 + 0.783974i \(0.713190\pi\)
\(420\) 0 0
\(421\) 14573.7 1.68713 0.843563 0.537030i \(-0.180454\pi\)
0.843563 + 0.537030i \(0.180454\pi\)
\(422\) −9425.17 −1.08723
\(423\) 2451.51 0.281788
\(424\) −3034.60 −0.347578
\(425\) 0 0
\(426\) −1406.42 −0.159956
\(427\) 11374.0 1.28906
\(428\) 5650.16 0.638110
\(429\) −1296.33 −0.145892
\(430\) 0 0
\(431\) −2825.70 −0.315799 −0.157899 0.987455i \(-0.550472\pi\)
−0.157899 + 0.987455i \(0.550472\pi\)
\(432\) 2079.42 0.231588
\(433\) −14814.8 −1.64423 −0.822116 0.569320i \(-0.807206\pi\)
−0.822116 + 0.569320i \(0.807206\pi\)
\(434\) 26234.4 2.90159
\(435\) 0 0
\(436\) −316.678 −0.0347847
\(437\) 533.397 0.0583887
\(438\) −5662.11 −0.617685
\(439\) 12452.4 1.35381 0.676904 0.736071i \(-0.263321\pi\)
0.676904 + 0.736071i \(0.263321\pi\)
\(440\) 0 0
\(441\) 5512.08 0.595192
\(442\) −12084.6 −1.30047
\(443\) −7989.73 −0.856892 −0.428446 0.903567i \(-0.640939\pi\)
−0.428446 + 0.903567i \(0.640939\pi\)
\(444\) −5989.21 −0.640170
\(445\) 0 0
\(446\) 4407.71 0.467963
\(447\) 8875.54 0.939147
\(448\) −5920.83 −0.624404
\(449\) 7874.23 0.827635 0.413817 0.910360i \(-0.364195\pi\)
0.413817 + 0.910360i \(0.364195\pi\)
\(450\) 0 0
\(451\) −5056.07 −0.527896
\(452\) 2633.30 0.274026
\(453\) 10359.6 1.07448
\(454\) 18147.1 1.87596
\(455\) 0 0
\(456\) 1252.88 0.128666
\(457\) −10376.8 −1.06216 −0.531078 0.847323i \(-0.678213\pi\)
−0.531078 + 0.847323i \(0.678213\pi\)
\(458\) −8950.00 −0.913114
\(459\) 2241.80 0.227970
\(460\) 0 0
\(461\) −7935.15 −0.801685 −0.400843 0.916147i \(-0.631283\pi\)
−0.400843 + 0.916147i \(0.631283\pi\)
\(462\) −3779.33 −0.380585
\(463\) −4036.46 −0.405163 −0.202581 0.979265i \(-0.564933\pi\)
−0.202581 + 0.979265i \(0.564933\pi\)
\(464\) −15317.6 −1.53255
\(465\) 0 0
\(466\) −1942.41 −0.193091
\(467\) −9548.40 −0.946140 −0.473070 0.881025i \(-0.656854\pi\)
−0.473070 + 0.881025i \(0.656854\pi\)
\(468\) 2024.94 0.200007
\(469\) 28727.3 2.82837
\(470\) 0 0
\(471\) −2216.67 −0.216855
\(472\) 1813.78 0.176877
\(473\) −5074.30 −0.493269
\(474\) −8000.23 −0.775238
\(475\) 0 0
\(476\) −14699.6 −1.41546
\(477\) 3243.78 0.311368
\(478\) −15432.1 −1.47667
\(479\) −15208.5 −1.45072 −0.725358 0.688372i \(-0.758326\pi\)
−0.725358 + 0.688372i \(0.758326\pi\)
\(480\) 0 0
\(481\) 13692.5 1.29797
\(482\) 7072.25 0.668324
\(483\) −997.191 −0.0939416
\(484\) 693.033 0.0650857
\(485\) 0 0
\(486\) −900.332 −0.0840327
\(487\) 12284.0 1.14300 0.571502 0.820601i \(-0.306361\pi\)
0.571502 + 0.820601i \(0.306361\pi\)
\(488\) −3098.14 −0.287390
\(489\) −727.581 −0.0672849
\(490\) 0 0
\(491\) −15310.6 −1.40725 −0.703623 0.710574i \(-0.748436\pi\)
−0.703623 + 0.710574i \(0.748436\pi\)
\(492\) 7897.87 0.723707
\(493\) −16513.8 −1.50861
\(494\) 7219.32 0.657516
\(495\) 0 0
\(496\) −17642.0 −1.59708
\(497\) 3911.12 0.352993
\(498\) 9616.08 0.865274
\(499\) 2024.99 0.181665 0.0908325 0.995866i \(-0.471047\pi\)
0.0908325 + 0.995866i \(0.471047\pi\)
\(500\) 0 0
\(501\) −7341.18 −0.654651
\(502\) 9276.11 0.824728
\(503\) −9262.81 −0.821090 −0.410545 0.911840i \(-0.634661\pi\)
−0.410545 + 0.911840i \(0.634661\pi\)
\(504\) −2342.28 −0.207011
\(505\) 0 0
\(506\) 438.270 0.0385049
\(507\) 1961.59 0.171829
\(508\) −11222.1 −0.980115
\(509\) −15354.4 −1.33707 −0.668537 0.743679i \(-0.733079\pi\)
−0.668537 + 0.743679i \(0.733079\pi\)
\(510\) 0 0
\(511\) 15745.8 1.36312
\(512\) 11601.2 1.00138
\(513\) −1339.25 −0.115262
\(514\) −11063.0 −0.949354
\(515\) 0 0
\(516\) 7926.34 0.676236
\(517\) 2996.29 0.254887
\(518\) 39919.1 3.38599
\(519\) 2635.26 0.222880
\(520\) 0 0
\(521\) −1155.39 −0.0971562 −0.0485781 0.998819i \(-0.515469\pi\)
−0.0485781 + 0.998819i \(0.515469\pi\)
\(522\) 6632.11 0.556091
\(523\) 3705.49 0.309808 0.154904 0.987930i \(-0.450493\pi\)
0.154904 + 0.987930i \(0.450493\pi\)
\(524\) 93.1611 0.00776672
\(525\) 0 0
\(526\) 8200.05 0.679732
\(527\) −19019.7 −1.57213
\(528\) 2541.51 0.209480
\(529\) −12051.4 −0.990496
\(530\) 0 0
\(531\) −1938.81 −0.158451
\(532\) 8781.55 0.715655
\(533\) −18056.0 −1.46734
\(534\) −5876.44 −0.476214
\(535\) 0 0
\(536\) −7824.97 −0.630573
\(537\) 7419.56 0.596234
\(538\) −2337.61 −0.187326
\(539\) 6736.98 0.538372
\(540\) 0 0
\(541\) 13354.8 1.06131 0.530653 0.847589i \(-0.321947\pi\)
0.530653 + 0.847589i \(0.321947\pi\)
\(542\) −21422.3 −1.69773
\(543\) −148.429 −0.0117305
\(544\) 18099.8 1.42651
\(545\) 0 0
\(546\) −13496.6 −1.05788
\(547\) 18996.8 1.48491 0.742454 0.669897i \(-0.233662\pi\)
0.742454 + 0.669897i \(0.233662\pi\)
\(548\) 5392.44 0.420353
\(549\) 3311.70 0.257450
\(550\) 0 0
\(551\) 9865.31 0.762752
\(552\) 271.622 0.0209439
\(553\) 22247.9 1.71081
\(554\) −12467.9 −0.956158
\(555\) 0 0
\(556\) −11312.1 −0.862842
\(557\) 8518.26 0.647990 0.323995 0.946059i \(-0.394974\pi\)
0.323995 + 0.946059i \(0.394974\pi\)
\(558\) 7638.51 0.579505
\(559\) −18121.1 −1.37109
\(560\) 0 0
\(561\) 2739.98 0.206207
\(562\) 2305.70 0.173060
\(563\) −10311.5 −0.771900 −0.385950 0.922520i \(-0.626126\pi\)
−0.385950 + 0.922520i \(0.626126\pi\)
\(564\) −4680.37 −0.349431
\(565\) 0 0
\(566\) −6037.20 −0.448344
\(567\) 2503.74 0.185445
\(568\) −1065.34 −0.0786983
\(569\) −14997.9 −1.10500 −0.552498 0.833514i \(-0.686325\pi\)
−0.552498 + 0.833514i \(0.686325\pi\)
\(570\) 0 0
\(571\) 12160.9 0.891274 0.445637 0.895214i \(-0.352977\pi\)
0.445637 + 0.895214i \(0.352977\pi\)
\(572\) 2474.93 0.180913
\(573\) 5836.61 0.425528
\(574\) −52640.7 −3.82784
\(575\) 0 0
\(576\) −1723.93 −0.124706
\(577\) −16898.7 −1.21924 −0.609619 0.792694i \(-0.708678\pi\)
−0.609619 + 0.792694i \(0.708678\pi\)
\(578\) 7339.53 0.528174
\(579\) 9193.50 0.659877
\(580\) 0 0
\(581\) −26741.4 −1.90950
\(582\) −9537.26 −0.679265
\(583\) 3964.62 0.281643
\(584\) −4288.96 −0.303902
\(585\) 0 0
\(586\) −21656.5 −1.52666
\(587\) −18702.0 −1.31502 −0.657508 0.753448i \(-0.728389\pi\)
−0.657508 + 0.753448i \(0.728389\pi\)
\(588\) −10523.6 −0.738068
\(589\) 11362.3 0.794867
\(590\) 0 0
\(591\) 4979.13 0.346555
\(592\) −26844.7 −1.86370
\(593\) −198.418 −0.0137404 −0.00687019 0.999976i \(-0.502187\pi\)
−0.00687019 + 0.999976i \(0.502187\pi\)
\(594\) −1100.41 −0.0760104
\(595\) 0 0
\(596\) −16945.0 −1.16459
\(597\) 9690.32 0.664319
\(598\) 1565.13 0.107028
\(599\) 13958.3 0.952119 0.476060 0.879413i \(-0.342065\pi\)
0.476060 + 0.879413i \(0.342065\pi\)
\(600\) 0 0
\(601\) 14279.5 0.969173 0.484587 0.874743i \(-0.338970\pi\)
0.484587 + 0.874743i \(0.338970\pi\)
\(602\) −52830.4 −3.57676
\(603\) 8364.37 0.564881
\(604\) −19778.4 −1.33241
\(605\) 0 0
\(606\) 542.209 0.0363461
\(607\) −435.160 −0.0290982 −0.0145491 0.999894i \(-0.504631\pi\)
−0.0145491 + 0.999894i \(0.504631\pi\)
\(608\) −10812.8 −0.721243
\(609\) −18443.3 −1.22719
\(610\) 0 0
\(611\) 10700.2 0.708486
\(612\) −4280.01 −0.282695
\(613\) 1032.63 0.0680384 0.0340192 0.999421i \(-0.489169\pi\)
0.0340192 + 0.999421i \(0.489169\pi\)
\(614\) −16495.4 −1.08420
\(615\) 0 0
\(616\) −2862.79 −0.187248
\(617\) 1793.68 0.117035 0.0585176 0.998286i \(-0.481363\pi\)
0.0585176 + 0.998286i \(0.481363\pi\)
\(618\) −5836.63 −0.379909
\(619\) −21282.0 −1.38190 −0.690948 0.722904i \(-0.742807\pi\)
−0.690948 + 0.722904i \(0.742807\pi\)
\(620\) 0 0
\(621\) −290.346 −0.0187620
\(622\) −27197.5 −1.75325
\(623\) 16341.8 1.05092
\(624\) 9076.16 0.582271
\(625\) 0 0
\(626\) 15942.9 1.01790
\(627\) −1636.86 −0.104258
\(628\) 4232.03 0.268911
\(629\) −28941.0 −1.83458
\(630\) 0 0
\(631\) −20393.2 −1.28660 −0.643298 0.765616i \(-0.722434\pi\)
−0.643298 + 0.765616i \(0.722434\pi\)
\(632\) −6060.06 −0.381418
\(633\) 7631.57 0.479191
\(634\) 20865.9 1.30709
\(635\) 0 0
\(636\) −6192.97 −0.386112
\(637\) 24058.8 1.49646
\(638\) 8105.91 0.503003
\(639\) 1138.78 0.0704997
\(640\) 0 0
\(641\) −1068.76 −0.0658559 −0.0329280 0.999458i \(-0.510483\pi\)
−0.0329280 + 0.999458i \(0.510483\pi\)
\(642\) −10965.0 −0.674074
\(643\) 6293.52 0.385991 0.192995 0.981200i \(-0.438180\pi\)
0.192995 + 0.981200i \(0.438180\pi\)
\(644\) 1903.82 0.116492
\(645\) 0 0
\(646\) −15259.1 −0.929351
\(647\) 14233.7 0.864888 0.432444 0.901661i \(-0.357651\pi\)
0.432444 + 0.901661i \(0.357651\pi\)
\(648\) −681.988 −0.0413442
\(649\) −2369.66 −0.143324
\(650\) 0 0
\(651\) −21242.0 −1.27886
\(652\) 1389.08 0.0834367
\(653\) −7671.06 −0.459712 −0.229856 0.973225i \(-0.573825\pi\)
−0.229856 + 0.973225i \(0.573825\pi\)
\(654\) 614.563 0.0367452
\(655\) 0 0
\(656\) 35399.7 2.10690
\(657\) 4584.62 0.272242
\(658\) 31195.5 1.84822
\(659\) 18783.6 1.11033 0.555165 0.831741i \(-0.312655\pi\)
0.555165 + 0.831741i \(0.312655\pi\)
\(660\) 0 0
\(661\) −21814.3 −1.28363 −0.641815 0.766859i \(-0.721818\pi\)
−0.641815 + 0.766859i \(0.721818\pi\)
\(662\) 41831.6 2.45594
\(663\) 9784.92 0.573175
\(664\) 7284.04 0.425716
\(665\) 0 0
\(666\) 11623.0 0.676250
\(667\) 2138.78 0.124159
\(668\) 14015.7 0.811799
\(669\) −3568.93 −0.206252
\(670\) 0 0
\(671\) 4047.64 0.232872
\(672\) 20214.6 1.16041
\(673\) 4320.84 0.247483 0.123742 0.992314i \(-0.460511\pi\)
0.123742 + 0.992314i \(0.460511\pi\)
\(674\) −16019.3 −0.915488
\(675\) 0 0
\(676\) −3745.03 −0.213077
\(677\) −8173.49 −0.464007 −0.232004 0.972715i \(-0.574528\pi\)
−0.232004 + 0.972715i \(0.574528\pi\)
\(678\) −5110.33 −0.289470
\(679\) 26522.3 1.49902
\(680\) 0 0
\(681\) −14693.7 −0.826819
\(682\) 9335.95 0.524182
\(683\) −11387.1 −0.637945 −0.318973 0.947764i \(-0.603338\pi\)
−0.318973 + 0.947764i \(0.603338\pi\)
\(684\) 2556.87 0.142930
\(685\) 0 0
\(686\) 30859.2 1.71750
\(687\) 7246.83 0.402451
\(688\) 35527.3 1.96870
\(689\) 14158.3 0.782857
\(690\) 0 0
\(691\) 7445.76 0.409913 0.204957 0.978771i \(-0.434295\pi\)
0.204957 + 0.978771i \(0.434295\pi\)
\(692\) −5031.18 −0.276383
\(693\) 3060.13 0.167741
\(694\) 28504.0 1.55907
\(695\) 0 0
\(696\) 5023.73 0.273598
\(697\) 38164.0 2.07398
\(698\) 10344.2 0.560938
\(699\) 1572.77 0.0851041
\(700\) 0 0
\(701\) −35574.7 −1.91674 −0.958372 0.285523i \(-0.907833\pi\)
−0.958372 + 0.285523i \(0.907833\pi\)
\(702\) −3929.73 −0.211279
\(703\) 17289.3 0.927566
\(704\) −2107.03 −0.112801
\(705\) 0 0
\(706\) −19674.9 −1.04883
\(707\) −1507.84 −0.0802093
\(708\) 3701.54 0.196487
\(709\) 12878.9 0.682199 0.341099 0.940027i \(-0.389201\pi\)
0.341099 + 0.940027i \(0.389201\pi\)
\(710\) 0 0
\(711\) 6477.80 0.341683
\(712\) −4451.31 −0.234298
\(713\) 2463.33 0.129386
\(714\) 28527.0 1.49523
\(715\) 0 0
\(716\) −14165.3 −0.739359
\(717\) 12495.4 0.650834
\(718\) 25965.3 1.34961
\(719\) −19103.3 −0.990867 −0.495434 0.868646i \(-0.664991\pi\)
−0.495434 + 0.868646i \(0.664991\pi\)
\(720\) 0 0
\(721\) 16231.1 0.838390
\(722\) −16297.3 −0.840060
\(723\) −5726.41 −0.294561
\(724\) 283.377 0.0145465
\(725\) 0 0
\(726\) −1344.94 −0.0687540
\(727\) −1059.06 −0.0540281 −0.0270141 0.999635i \(-0.508600\pi\)
−0.0270141 + 0.999635i \(0.508600\pi\)
\(728\) −10223.5 −0.520477
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 38301.6 1.93794
\(732\) −6322.64 −0.319251
\(733\) −24503.8 −1.23475 −0.617373 0.786670i \(-0.711803\pi\)
−0.617373 + 0.786670i \(0.711803\pi\)
\(734\) 7786.96 0.391583
\(735\) 0 0
\(736\) −2344.18 −0.117402
\(737\) 10223.1 0.510954
\(738\) −15327.1 −0.764495
\(739\) 23488.3 1.16919 0.584595 0.811325i \(-0.301253\pi\)
0.584595 + 0.811325i \(0.301253\pi\)
\(740\) 0 0
\(741\) −5845.50 −0.289797
\(742\) 41277.2 2.04223
\(743\) 33500.5 1.65413 0.827063 0.562109i \(-0.190010\pi\)
0.827063 + 0.562109i \(0.190010\pi\)
\(744\) 5786.06 0.285117
\(745\) 0 0
\(746\) 32538.7 1.59695
\(747\) −7786.15 −0.381366
\(748\) −5231.12 −0.255707
\(749\) 30492.8 1.48756
\(750\) 0 0
\(751\) −26937.5 −1.30887 −0.654436 0.756117i \(-0.727094\pi\)
−0.654436 + 0.756117i \(0.727094\pi\)
\(752\) −20978.3 −1.01729
\(753\) −7510.88 −0.363495
\(754\) 28947.5 1.39815
\(755\) 0 0
\(756\) −4780.10 −0.229961
\(757\) 8869.39 0.425843 0.212922 0.977069i \(-0.431702\pi\)
0.212922 + 0.977069i \(0.431702\pi\)
\(758\) 11259.7 0.539537
\(759\) −354.868 −0.0169709
\(760\) 0 0
\(761\) 12433.6 0.592271 0.296135 0.955146i \(-0.404302\pi\)
0.296135 + 0.955146i \(0.404302\pi\)
\(762\) 21778.2 1.03535
\(763\) −1709.05 −0.0810899
\(764\) −11143.1 −0.527676
\(765\) 0 0
\(766\) 54134.8 2.55349
\(767\) −8462.44 −0.398384
\(768\) −16092.8 −0.756121
\(769\) 24551.5 1.15130 0.575650 0.817696i \(-0.304749\pi\)
0.575650 + 0.817696i \(0.304749\pi\)
\(770\) 0 0
\(771\) 8957.72 0.418424
\(772\) −17552.1 −0.818280
\(773\) 21858.5 1.01707 0.508535 0.861041i \(-0.330187\pi\)
0.508535 + 0.861041i \(0.330187\pi\)
\(774\) −15382.3 −0.714349
\(775\) 0 0
\(776\) −7224.34 −0.334199
\(777\) −32322.6 −1.49236
\(778\) −642.821 −0.0296224
\(779\) −22799.1 −1.04861
\(780\) 0 0
\(781\) 1391.84 0.0637694
\(782\) −3308.13 −0.151277
\(783\) −5370.03 −0.245095
\(784\) −47168.4 −2.14871
\(785\) 0 0
\(786\) −180.794 −0.00820446
\(787\) 3488.56 0.158010 0.0790048 0.996874i \(-0.474826\pi\)
0.0790048 + 0.996874i \(0.474826\pi\)
\(788\) −9506.06 −0.429745
\(789\) −6639.59 −0.299589
\(790\) 0 0
\(791\) 14211.4 0.638809
\(792\) −833.541 −0.0373972
\(793\) 14454.8 0.647293
\(794\) 5075.16 0.226840
\(795\) 0 0
\(796\) −18500.6 −0.823789
\(797\) 19467.5 0.865212 0.432606 0.901583i \(-0.357594\pi\)
0.432606 + 0.901583i \(0.357594\pi\)
\(798\) −17042.0 −0.755990
\(799\) −22616.5 −1.00139
\(800\) 0 0
\(801\) 4758.16 0.209889
\(802\) 54076.9 2.38095
\(803\) 5603.42 0.246252
\(804\) −15969.1 −0.700481
\(805\) 0 0
\(806\) 33340.2 1.45702
\(807\) 1892.76 0.0825631
\(808\) 410.716 0.0178823
\(809\) −3078.16 −0.133773 −0.0668866 0.997761i \(-0.521307\pi\)
−0.0668866 + 0.997761i \(0.521307\pi\)
\(810\) 0 0
\(811\) 31711.1 1.37303 0.686516 0.727115i \(-0.259139\pi\)
0.686516 + 0.727115i \(0.259139\pi\)
\(812\) 35211.6 1.52178
\(813\) 17345.7 0.748266
\(814\) 14205.9 0.611691
\(815\) 0 0
\(816\) −19183.8 −0.822998
\(817\) −22881.3 −0.979824
\(818\) −36691.7 −1.56833
\(819\) 10928.2 0.466255
\(820\) 0 0
\(821\) −31485.7 −1.33844 −0.669219 0.743065i \(-0.733371\pi\)
−0.669219 + 0.743065i \(0.733371\pi\)
\(822\) −10464.9 −0.444045
\(823\) 19306.7 0.817726 0.408863 0.912596i \(-0.365925\pi\)
0.408863 + 0.912596i \(0.365925\pi\)
\(824\) −4421.16 −0.186916
\(825\) 0 0
\(826\) −24671.4 −1.03926
\(827\) 7147.76 0.300546 0.150273 0.988645i \(-0.451985\pi\)
0.150273 + 0.988645i \(0.451985\pi\)
\(828\) 554.324 0.0232658
\(829\) −35630.4 −1.49276 −0.746378 0.665522i \(-0.768209\pi\)
−0.746378 + 0.665522i \(0.768209\pi\)
\(830\) 0 0
\(831\) 10095.3 0.421422
\(832\) −7524.55 −0.313542
\(833\) −50851.8 −2.11514
\(834\) 21952.9 0.911472
\(835\) 0 0
\(836\) 3125.07 0.129285
\(837\) −6184.91 −0.255414
\(838\) −39454.3 −1.62640
\(839\) 27513.3 1.13214 0.566070 0.824357i \(-0.308463\pi\)
0.566070 + 0.824357i \(0.308463\pi\)
\(840\) 0 0
\(841\) 15168.2 0.621929
\(842\) 53996.7 2.21003
\(843\) −1866.92 −0.0762756
\(844\) −14570.1 −0.594220
\(845\) 0 0
\(846\) 9083.01 0.369126
\(847\) 3740.16 0.151728
\(848\) −27758.0 −1.12407
\(849\) 4888.33 0.197605
\(850\) 0 0
\(851\) 3748.28 0.150986
\(852\) −2174.13 −0.0874231
\(853\) −21498.2 −0.862935 −0.431467 0.902129i \(-0.642004\pi\)
−0.431467 + 0.902129i \(0.642004\pi\)
\(854\) 42141.5 1.68859
\(855\) 0 0
\(856\) −8305.85 −0.331645
\(857\) −7144.96 −0.284792 −0.142396 0.989810i \(-0.545481\pi\)
−0.142396 + 0.989810i \(0.545481\pi\)
\(858\) −4803.00 −0.191109
\(859\) −5734.28 −0.227766 −0.113883 0.993494i \(-0.536329\pi\)
−0.113883 + 0.993494i \(0.536329\pi\)
\(860\) 0 0
\(861\) 42623.2 1.68710
\(862\) −10469.4 −0.413677
\(863\) −5032.50 −0.198503 −0.0992516 0.995062i \(-0.531645\pi\)
−0.0992516 + 0.995062i \(0.531645\pi\)
\(864\) 5885.76 0.231757
\(865\) 0 0
\(866\) −54889.8 −2.15385
\(867\) −5942.83 −0.232790
\(868\) 40554.8 1.58585
\(869\) 7917.31 0.309064
\(870\) 0 0
\(871\) 36508.4 1.42025
\(872\) 465.523 0.0180787
\(873\) 7722.33 0.299383
\(874\) 1976.27 0.0764856
\(875\) 0 0
\(876\) −8752.86 −0.337593
\(877\) −29097.4 −1.12035 −0.560177 0.828373i \(-0.689267\pi\)
−0.560177 + 0.828373i \(0.689267\pi\)
\(878\) 46137.1 1.77341
\(879\) 17535.3 0.672867
\(880\) 0 0
\(881\) 6899.21 0.263837 0.131918 0.991261i \(-0.457886\pi\)
0.131918 + 0.991261i \(0.457886\pi\)
\(882\) 20422.6 0.779666
\(883\) −13094.5 −0.499054 −0.249527 0.968368i \(-0.580275\pi\)
−0.249527 + 0.968368i \(0.580275\pi\)
\(884\) −18681.2 −0.710765
\(885\) 0 0
\(886\) −29602.5 −1.12248
\(887\) −27043.9 −1.02373 −0.511863 0.859067i \(-0.671044\pi\)
−0.511863 + 0.859067i \(0.671044\pi\)
\(888\) 8804.26 0.332716
\(889\) −60563.2 −2.28484
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) 6813.74 0.255763
\(893\) 13511.0 0.506304
\(894\) 32884.5 1.23023
\(895\) 0 0
\(896\) 31968.5 1.19196
\(897\) −1267.29 −0.0471723
\(898\) 29174.6 1.08415
\(899\) 45559.9 1.69022
\(900\) 0 0
\(901\) −29925.6 −1.10651
\(902\) −18733.1 −0.691512
\(903\) 42776.9 1.57644
\(904\) −3871.00 −0.142420
\(905\) 0 0
\(906\) 38383.2 1.40750
\(907\) −375.753 −0.0137560 −0.00687799 0.999976i \(-0.502189\pi\)
−0.00687799 + 0.999976i \(0.502189\pi\)
\(908\) 28052.9 1.02530
\(909\) −439.028 −0.0160194
\(910\) 0 0
\(911\) 7303.70 0.265623 0.132811 0.991141i \(-0.457600\pi\)
0.132811 + 0.991141i \(0.457600\pi\)
\(912\) 11460.4 0.416108
\(913\) −9516.40 −0.344958
\(914\) −38446.7 −1.39136
\(915\) 0 0
\(916\) −13835.5 −0.499059
\(917\) 502.772 0.0181058
\(918\) 8306.04 0.298628
\(919\) 3492.12 0.125348 0.0626738 0.998034i \(-0.480037\pi\)
0.0626738 + 0.998034i \(0.480037\pi\)
\(920\) 0 0
\(921\) 13356.4 0.477858
\(922\) −29400.3 −1.05016
\(923\) 4970.48 0.177254
\(924\) −5842.34 −0.208008
\(925\) 0 0
\(926\) −14955.4 −0.530739
\(927\) 4725.93 0.167443
\(928\) −43356.3 −1.53366
\(929\) 16961.2 0.599009 0.299504 0.954095i \(-0.403179\pi\)
0.299504 + 0.954095i \(0.403179\pi\)
\(930\) 0 0
\(931\) 30378.8 1.06941
\(932\) −3002.71 −0.105533
\(933\) 22021.8 0.772736
\(934\) −35377.5 −1.23939
\(935\) 0 0
\(936\) −2976.71 −0.103950
\(937\) 3401.53 0.118595 0.0592973 0.998240i \(-0.481114\pi\)
0.0592973 + 0.998240i \(0.481114\pi\)
\(938\) 106437. 3.70499
\(939\) −12909.0 −0.448635
\(940\) 0 0
\(941\) −39517.5 −1.36901 −0.684503 0.729010i \(-0.739981\pi\)
−0.684503 + 0.729010i \(0.739981\pi\)
\(942\) −8212.92 −0.284067
\(943\) −4942.80 −0.170689
\(944\) 16591.0 0.572024
\(945\) 0 0
\(946\) −18800.6 −0.646153
\(947\) 39781.6 1.36508 0.682538 0.730850i \(-0.260876\pi\)
0.682538 + 0.730850i \(0.260876\pi\)
\(948\) −12367.3 −0.423703
\(949\) 20010.7 0.684484
\(950\) 0 0
\(951\) −16895.2 −0.576092
\(952\) 21608.8 0.735656
\(953\) −19941.5 −0.677826 −0.338913 0.940818i \(-0.610059\pi\)
−0.338913 + 0.940818i \(0.610059\pi\)
\(954\) 12018.4 0.407874
\(955\) 0 0
\(956\) −23855.9 −0.807066
\(957\) −6563.37 −0.221697
\(958\) −56348.5 −1.90035
\(959\) 29101.9 0.979926
\(960\) 0 0
\(961\) 22682.4 0.761384
\(962\) 50731.6 1.70026
\(963\) 8878.41 0.297095
\(964\) 10932.8 0.365270
\(965\) 0 0
\(966\) −3694.66 −0.123058
\(967\) 22672.9 0.753992 0.376996 0.926215i \(-0.376957\pi\)
0.376996 + 0.926215i \(0.376957\pi\)
\(968\) −1018.77 −0.0338271
\(969\) 12355.3 0.409607
\(970\) 0 0
\(971\) 15739.4 0.520186 0.260093 0.965584i \(-0.416247\pi\)
0.260093 + 0.965584i \(0.416247\pi\)
\(972\) −1391.79 −0.0459278
\(973\) −61049.1 −2.01145
\(974\) 45513.2 1.49727
\(975\) 0 0
\(976\) −28339.2 −0.929422
\(977\) −55085.0 −1.80381 −0.901906 0.431933i \(-0.857832\pi\)
−0.901906 + 0.431933i \(0.857832\pi\)
\(978\) −2695.74 −0.0881392
\(979\) 5815.53 0.189852
\(980\) 0 0
\(981\) −497.613 −0.0161953
\(982\) −56726.8 −1.84341
\(983\) 3826.41 0.124154 0.0620770 0.998071i \(-0.480228\pi\)
0.0620770 + 0.998071i \(0.480228\pi\)
\(984\) −11610.0 −0.376133
\(985\) 0 0
\(986\) −61184.7 −1.97619
\(987\) −25259.0 −0.814594
\(988\) 11160.1 0.359363
\(989\) −4960.62 −0.159493
\(990\) 0 0
\(991\) 10873.3 0.348538 0.174269 0.984698i \(-0.444244\pi\)
0.174269 + 0.984698i \(0.444244\pi\)
\(992\) −49935.4 −1.59824
\(993\) −33871.1 −1.08244
\(994\) 14491.0 0.462400
\(995\) 0 0
\(996\) 14865.2 0.472913
\(997\) −40064.9 −1.27269 −0.636343 0.771406i \(-0.719554\pi\)
−0.636343 + 0.771406i \(0.719554\pi\)
\(998\) 7502.72 0.237970
\(999\) −9411.17 −0.298054
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 825.4.a.bd.1.6 7
3.2 odd 2 2475.4.a.bo.1.2 7
5.2 odd 4 165.4.c.b.34.13 yes 14
5.3 odd 4 165.4.c.b.34.2 14
5.4 even 2 825.4.a.ba.1.2 7
15.2 even 4 495.4.c.d.199.2 14
15.8 even 4 495.4.c.d.199.13 14
15.14 odd 2 2475.4.a.bs.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.c.b.34.2 14 5.3 odd 4
165.4.c.b.34.13 yes 14 5.2 odd 4
495.4.c.d.199.2 14 15.2 even 4
495.4.c.d.199.13 14 15.8 even 4
825.4.a.ba.1.2 7 5.4 even 2
825.4.a.bd.1.6 7 1.1 even 1 trivial
2475.4.a.bo.1.2 7 3.2 odd 2
2475.4.a.bs.1.6 7 15.14 odd 2