Properties

Label 525.4.a.p.1.1
Level $525$
Weight $4$
Character 525.1
Self dual yes
Analytic conductor $30.976$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,4,Mod(1,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, 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("525.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.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: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.43845 q^{2} +3.00000 q^{3} -5.93087 q^{4} +4.31534 q^{6} +7.00000 q^{7} -20.0388 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+1.43845 q^{2} +3.00000 q^{3} -5.93087 q^{4} +4.31534 q^{6} +7.00000 q^{7} -20.0388 q^{8} +9.00000 q^{9} +7.61553 q^{11} -17.7926 q^{12} -52.3542 q^{13} +10.0691 q^{14} +18.6222 q^{16} +49.7235 q^{17} +12.9460 q^{18} +140.600 q^{19} +21.0000 q^{21} +10.9545 q^{22} +23.4470 q^{23} -60.1165 q^{24} -75.3087 q^{26} +27.0000 q^{27} -41.5161 q^{28} +157.170 q^{29} +127.892 q^{31} +187.098 q^{32} +22.8466 q^{33} +71.5246 q^{34} -53.3778 q^{36} +115.477 q^{37} +202.246 q^{38} -157.062 q^{39} -188.617 q^{41} +30.2074 q^{42} -322.186 q^{43} -45.1667 q^{44} +33.7272 q^{46} +76.6477 q^{47} +55.8665 q^{48} +49.0000 q^{49} +149.170 q^{51} +310.506 q^{52} +424.172 q^{53} +38.8381 q^{54} -140.272 q^{56} +421.801 q^{57} +226.081 q^{58} +107.784 q^{59} +915.511 q^{61} +183.966 q^{62} +63.0000 q^{63} +120.153 q^{64} +32.8636 q^{66} +451.723 q^{67} -294.903 q^{68} +70.3409 q^{69} +907.312 q^{71} -180.349 q^{72} -755.956 q^{73} +166.108 q^{74} -833.882 q^{76} +53.3087 q^{77} -225.926 q^{78} +22.5834 q^{79} +81.0000 q^{81} -271.316 q^{82} -1112.25 q^{83} -124.548 q^{84} -463.447 q^{86} +471.511 q^{87} -152.606 q^{88} +1518.81 q^{89} -366.479 q^{91} -139.061 q^{92} +383.676 q^{93} +110.254 q^{94} +561.293 q^{96} +549.987 q^{97} +70.4839 q^{98} +68.5398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 7 q^{2} + 6 q^{3} + 17 q^{4} + 21 q^{6} + 14 q^{7} + 63 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 7 q^{2} + 6 q^{3} + 17 q^{4} + 21 q^{6} + 14 q^{7} + 63 q^{8} + 18 q^{9} - 26 q^{11} + 51 q^{12} - 14 q^{13} + 49 q^{14} + 297 q^{16} - 16 q^{17} + 63 q^{18} + 174 q^{19} + 42 q^{21} - 176 q^{22} - 184 q^{23} + 189 q^{24} + 138 q^{26} + 54 q^{27} + 119 q^{28} - 32 q^{29} + 330 q^{31} + 1071 q^{32} - 78 q^{33} - 294 q^{34} + 153 q^{36} + 132 q^{37} + 388 q^{38} - 42 q^{39} + 200 q^{41} + 147 q^{42} - 364 q^{43} - 816 q^{44} - 1120 q^{46} - 292 q^{47} + 891 q^{48} + 98 q^{49} - 48 q^{51} + 1190 q^{52} - 34 q^{53} + 189 q^{54} + 441 q^{56} + 522 q^{57} - 826 q^{58} + 364 q^{59} + 792 q^{61} + 1308 q^{62} + 126 q^{63} + 2809 q^{64} - 528 q^{66} + 788 q^{67} - 1802 q^{68} - 552 q^{69} + 454 q^{71} + 567 q^{72} - 778 q^{73} + 258 q^{74} - 68 q^{76} - 182 q^{77} + 414 q^{78} + 408 q^{79} + 162 q^{81} + 1890 q^{82} - 1136 q^{83} + 357 q^{84} - 696 q^{86} - 96 q^{87} - 2944 q^{88} + 36 q^{89} - 98 q^{91} - 4896 q^{92} + 990 q^{93} - 1940 q^{94} + 3213 q^{96} + 498 q^{97} + 343 q^{98} - 234 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.43845 0.508568 0.254284 0.967130i \(-0.418160\pi\)
0.254284 + 0.967130i \(0.418160\pi\)
\(3\) 3.00000 0.577350
\(4\) −5.93087 −0.741359
\(5\) 0 0
\(6\) 4.31534 0.293622
\(7\) 7.00000 0.377964
\(8\) −20.0388 −0.885599
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 7.61553 0.208743 0.104371 0.994538i \(-0.466717\pi\)
0.104371 + 0.994538i \(0.466717\pi\)
\(12\) −17.7926 −0.428024
\(13\) −52.3542 −1.11696 −0.558478 0.829519i \(-0.688615\pi\)
−0.558478 + 0.829519i \(0.688615\pi\)
\(14\) 10.0691 0.192221
\(15\) 0 0
\(16\) 18.6222 0.290971
\(17\) 49.7235 0.709395 0.354697 0.934981i \(-0.384584\pi\)
0.354697 + 0.934981i \(0.384584\pi\)
\(18\) 12.9460 0.169523
\(19\) 140.600 1.69768 0.848840 0.528649i \(-0.177301\pi\)
0.848840 + 0.528649i \(0.177301\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) 10.9545 0.106160
\(23\) 23.4470 0.212566 0.106283 0.994336i \(-0.466105\pi\)
0.106283 + 0.994336i \(0.466105\pi\)
\(24\) −60.1165 −0.511301
\(25\) 0 0
\(26\) −75.3087 −0.568048
\(27\) 27.0000 0.192450
\(28\) −41.5161 −0.280207
\(29\) 157.170 1.00641 0.503204 0.864168i \(-0.332155\pi\)
0.503204 + 0.864168i \(0.332155\pi\)
\(30\) 0 0
\(31\) 127.892 0.740971 0.370485 0.928838i \(-0.379191\pi\)
0.370485 + 0.928838i \(0.379191\pi\)
\(32\) 187.098 1.03358
\(33\) 22.8466 0.120518
\(34\) 71.5246 0.360776
\(35\) 0 0
\(36\) −53.3778 −0.247120
\(37\) 115.477 0.513090 0.256545 0.966532i \(-0.417416\pi\)
0.256545 + 0.966532i \(0.417416\pi\)
\(38\) 202.246 0.863386
\(39\) −157.062 −0.644875
\(40\) 0 0
\(41\) −188.617 −0.718466 −0.359233 0.933248i \(-0.616962\pi\)
−0.359233 + 0.933248i \(0.616962\pi\)
\(42\) 30.2074 0.110979
\(43\) −322.186 −1.14262 −0.571312 0.820733i \(-0.693565\pi\)
−0.571312 + 0.820733i \(0.693565\pi\)
\(44\) −45.1667 −0.154753
\(45\) 0 0
\(46\) 33.7272 0.108104
\(47\) 76.6477 0.237877 0.118938 0.992902i \(-0.462051\pi\)
0.118938 + 0.992902i \(0.462051\pi\)
\(48\) 55.8665 0.167992
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 149.170 0.409569
\(52\) 310.506 0.828065
\(53\) 424.172 1.09933 0.549666 0.835385i \(-0.314755\pi\)
0.549666 + 0.835385i \(0.314755\pi\)
\(54\) 38.8381 0.0978739
\(55\) 0 0
\(56\) −140.272 −0.334725
\(57\) 421.801 0.980157
\(58\) 226.081 0.511827
\(59\) 107.784 0.237835 0.118918 0.992904i \(-0.462058\pi\)
0.118918 + 0.992904i \(0.462058\pi\)
\(60\) 0 0
\(61\) 915.511 1.92163 0.960813 0.277197i \(-0.0894053\pi\)
0.960813 + 0.277197i \(0.0894053\pi\)
\(62\) 183.966 0.376834
\(63\) 63.0000 0.125988
\(64\) 120.153 0.234673
\(65\) 0 0
\(66\) 32.8636 0.0612914
\(67\) 451.723 0.823684 0.411842 0.911255i \(-0.364886\pi\)
0.411842 + 0.911255i \(0.364886\pi\)
\(68\) −294.903 −0.525916
\(69\) 70.3409 0.122725
\(70\) 0 0
\(71\) 907.312 1.51659 0.758297 0.651909i \(-0.226032\pi\)
0.758297 + 0.651909i \(0.226032\pi\)
\(72\) −180.349 −0.295200
\(73\) −755.956 −1.21203 −0.606014 0.795454i \(-0.707232\pi\)
−0.606014 + 0.795454i \(0.707232\pi\)
\(74\) 166.108 0.260941
\(75\) 0 0
\(76\) −833.882 −1.25859
\(77\) 53.3087 0.0788973
\(78\) −225.926 −0.327963
\(79\) 22.5834 0.0321624 0.0160812 0.999871i \(-0.494881\pi\)
0.0160812 + 0.999871i \(0.494881\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −271.316 −0.365389
\(83\) −1112.25 −1.47091 −0.735454 0.677575i \(-0.763031\pi\)
−0.735454 + 0.677575i \(0.763031\pi\)
\(84\) −124.548 −0.161778
\(85\) 0 0
\(86\) −463.447 −0.581102
\(87\) 471.511 0.581050
\(88\) −152.606 −0.184862
\(89\) 1518.81 1.80892 0.904458 0.426562i \(-0.140275\pi\)
0.904458 + 0.426562i \(0.140275\pi\)
\(90\) 0 0
\(91\) −366.479 −0.422170
\(92\) −139.061 −0.157588
\(93\) 383.676 0.427800
\(94\) 110.254 0.120977
\(95\) 0 0
\(96\) 561.293 0.596736
\(97\) 549.987 0.575698 0.287849 0.957676i \(-0.407060\pi\)
0.287849 + 0.957676i \(0.407060\pi\)
\(98\) 70.4839 0.0726526
\(99\) 68.5398 0.0695809
\(100\) 0 0
\(101\) −533.299 −0.525398 −0.262699 0.964878i \(-0.584613\pi\)
−0.262699 + 0.964878i \(0.584613\pi\)
\(102\) 214.574 0.208294
\(103\) −1357.79 −1.29890 −0.649451 0.760404i \(-0.725001\pi\)
−0.649451 + 0.760404i \(0.725001\pi\)
\(104\) 1049.12 0.989176
\(105\) 0 0
\(106\) 610.149 0.559084
\(107\) −913.693 −0.825515 −0.412757 0.910841i \(-0.635434\pi\)
−0.412757 + 0.910841i \(0.635434\pi\)
\(108\) −160.133 −0.142675
\(109\) −160.489 −0.141028 −0.0705139 0.997511i \(-0.522464\pi\)
−0.0705139 + 0.997511i \(0.522464\pi\)
\(110\) 0 0
\(111\) 346.432 0.296233
\(112\) 130.355 0.109977
\(113\) 1788.48 1.48891 0.744453 0.667675i \(-0.232710\pi\)
0.744453 + 0.667675i \(0.232710\pi\)
\(114\) 606.739 0.498476
\(115\) 0 0
\(116\) −932.157 −0.746109
\(117\) −471.187 −0.372319
\(118\) 155.042 0.120955
\(119\) 348.064 0.268126
\(120\) 0 0
\(121\) −1273.00 −0.956427
\(122\) 1316.91 0.977277
\(123\) −565.852 −0.414806
\(124\) −758.511 −0.549325
\(125\) 0 0
\(126\) 90.6222 0.0640735
\(127\) 271.015 0.189360 0.0946799 0.995508i \(-0.469817\pi\)
0.0946799 + 0.995508i \(0.469817\pi\)
\(128\) −1323.95 −0.914231
\(129\) −966.557 −0.659694
\(130\) 0 0
\(131\) −763.151 −0.508984 −0.254492 0.967075i \(-0.581908\pi\)
−0.254492 + 0.967075i \(0.581908\pi\)
\(132\) −135.500 −0.0893468
\(133\) 984.203 0.641663
\(134\) 649.780 0.418899
\(135\) 0 0
\(136\) −996.400 −0.628240
\(137\) 240.934 0.150251 0.0751254 0.997174i \(-0.476064\pi\)
0.0751254 + 0.997174i \(0.476064\pi\)
\(138\) 101.182 0.0624141
\(139\) −103.150 −0.0629427 −0.0314714 0.999505i \(-0.510019\pi\)
−0.0314714 + 0.999505i \(0.510019\pi\)
\(140\) 0 0
\(141\) 229.943 0.137338
\(142\) 1305.12 0.771291
\(143\) −398.705 −0.233156
\(144\) 167.600 0.0969905
\(145\) 0 0
\(146\) −1087.40 −0.616398
\(147\) 147.000 0.0824786
\(148\) −684.881 −0.380384
\(149\) −3256.71 −1.79061 −0.895303 0.445458i \(-0.853041\pi\)
−0.895303 + 0.445458i \(0.853041\pi\)
\(150\) 0 0
\(151\) 1471.04 0.792793 0.396396 0.918080i \(-0.370261\pi\)
0.396396 + 0.918080i \(0.370261\pi\)
\(152\) −2817.47 −1.50346
\(153\) 447.511 0.236465
\(154\) 76.6817 0.0401246
\(155\) 0 0
\(156\) 931.517 0.478084
\(157\) 1394.22 0.708730 0.354365 0.935107i \(-0.384697\pi\)
0.354365 + 0.935107i \(0.384697\pi\)
\(158\) 32.4850 0.0163567
\(159\) 1272.52 0.634699
\(160\) 0 0
\(161\) 164.129 0.0803426
\(162\) 116.514 0.0565075
\(163\) 3674.27 1.76559 0.882794 0.469760i \(-0.155659\pi\)
0.882794 + 0.469760i \(0.155659\pi\)
\(164\) 1118.67 0.532641
\(165\) 0 0
\(166\) −1599.91 −0.748056
\(167\) −4041.09 −1.87251 −0.936254 0.351325i \(-0.885731\pi\)
−0.936254 + 0.351325i \(0.885731\pi\)
\(168\) −420.815 −0.193254
\(169\) 543.958 0.247591
\(170\) 0 0
\(171\) 1265.40 0.565894
\(172\) 1910.84 0.847094
\(173\) −59.4582 −0.0261302 −0.0130651 0.999915i \(-0.504159\pi\)
−0.0130651 + 0.999915i \(0.504159\pi\)
\(174\) 678.244 0.295503
\(175\) 0 0
\(176\) 141.818 0.0607381
\(177\) 323.352 0.137314
\(178\) 2184.73 0.919957
\(179\) −2973.32 −1.24155 −0.620773 0.783991i \(-0.713181\pi\)
−0.620773 + 0.783991i \(0.713181\pi\)
\(180\) 0 0
\(181\) −676.220 −0.277696 −0.138848 0.990314i \(-0.544340\pi\)
−0.138848 + 0.990314i \(0.544340\pi\)
\(182\) −527.161 −0.214702
\(183\) 2746.53 1.10945
\(184\) −469.849 −0.188249
\(185\) 0 0
\(186\) 551.898 0.217565
\(187\) 378.671 0.148081
\(188\) −454.588 −0.176352
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) 16.5589 0.00627308 0.00313654 0.999995i \(-0.499002\pi\)
0.00313654 + 0.999995i \(0.499002\pi\)
\(192\) 360.458 0.135489
\(193\) −2694.68 −1.00501 −0.502506 0.864574i \(-0.667589\pi\)
−0.502506 + 0.864574i \(0.667589\pi\)
\(194\) 791.127 0.292781
\(195\) 0 0
\(196\) −290.613 −0.105908
\(197\) −1027.38 −0.371561 −0.185781 0.982591i \(-0.559481\pi\)
−0.185781 + 0.982591i \(0.559481\pi\)
\(198\) 98.5908 0.0353866
\(199\) 2823.77 1.00589 0.502944 0.864319i \(-0.332250\pi\)
0.502944 + 0.864319i \(0.332250\pi\)
\(200\) 0 0
\(201\) 1355.17 0.475554
\(202\) −767.123 −0.267201
\(203\) 1100.19 0.380386
\(204\) −884.710 −0.303638
\(205\) 0 0
\(206\) −1953.11 −0.660579
\(207\) 211.023 0.0708555
\(208\) −974.948 −0.325002
\(209\) 1070.75 0.354378
\(210\) 0 0
\(211\) 5151.16 1.68067 0.840333 0.542071i \(-0.182360\pi\)
0.840333 + 0.542071i \(0.182360\pi\)
\(212\) −2515.71 −0.814999
\(213\) 2721.94 0.875606
\(214\) −1314.30 −0.419830
\(215\) 0 0
\(216\) −541.048 −0.170434
\(217\) 895.244 0.280061
\(218\) −230.855 −0.0717222
\(219\) −2267.87 −0.699764
\(220\) 0 0
\(221\) −2603.23 −0.792363
\(222\) 498.324 0.150655
\(223\) 114.496 0.0343822 0.0171911 0.999852i \(-0.494528\pi\)
0.0171911 + 0.999852i \(0.494528\pi\)
\(224\) 1309.68 0.390656
\(225\) 0 0
\(226\) 2572.64 0.757209
\(227\) −4744.75 −1.38731 −0.693657 0.720306i \(-0.744001\pi\)
−0.693657 + 0.720306i \(0.744001\pi\)
\(228\) −2501.65 −0.726648
\(229\) −5384.47 −1.55378 −0.776891 0.629635i \(-0.783204\pi\)
−0.776891 + 0.629635i \(0.783204\pi\)
\(230\) 0 0
\(231\) 159.926 0.0455514
\(232\) −3149.51 −0.891274
\(233\) 1608.91 0.452373 0.226187 0.974084i \(-0.427374\pi\)
0.226187 + 0.974084i \(0.427374\pi\)
\(234\) −677.778 −0.189349
\(235\) 0 0
\(236\) −639.253 −0.176321
\(237\) 67.7501 0.0185689
\(238\) 500.672 0.136360
\(239\) −3113.11 −0.842554 −0.421277 0.906932i \(-0.638418\pi\)
−0.421277 + 0.906932i \(0.638418\pi\)
\(240\) 0 0
\(241\) 7136.38 1.90745 0.953724 0.300685i \(-0.0972151\pi\)
0.953724 + 0.300685i \(0.0972151\pi\)
\(242\) −1831.15 −0.486408
\(243\) 243.000 0.0641500
\(244\) −5429.78 −1.42461
\(245\) 0 0
\(246\) −813.948 −0.210957
\(247\) −7361.01 −1.89624
\(248\) −2562.81 −0.656203
\(249\) −3336.75 −0.849229
\(250\) 0 0
\(251\) −225.504 −0.0567079 −0.0283539 0.999598i \(-0.509027\pi\)
−0.0283539 + 0.999598i \(0.509027\pi\)
\(252\) −373.645 −0.0934024
\(253\) 178.561 0.0443717
\(254\) 389.841 0.0963024
\(255\) 0 0
\(256\) −2865.65 −0.699621
\(257\) 4423.05 1.07355 0.536775 0.843725i \(-0.319642\pi\)
0.536775 + 0.843725i \(0.319642\pi\)
\(258\) −1390.34 −0.335499
\(259\) 808.341 0.193930
\(260\) 0 0
\(261\) 1414.53 0.335469
\(262\) −1097.75 −0.258853
\(263\) −6540.56 −1.53349 −0.766746 0.641950i \(-0.778125\pi\)
−0.766746 + 0.641950i \(0.778125\pi\)
\(264\) −457.819 −0.106730
\(265\) 0 0
\(266\) 1415.72 0.326329
\(267\) 4556.43 1.04438
\(268\) −2679.11 −0.610645
\(269\) 2262.97 0.512920 0.256460 0.966555i \(-0.417444\pi\)
0.256460 + 0.966555i \(0.417444\pi\)
\(270\) 0 0
\(271\) 1615.68 0.362160 0.181080 0.983468i \(-0.442041\pi\)
0.181080 + 0.983468i \(0.442041\pi\)
\(272\) 925.959 0.206414
\(273\) −1099.44 −0.243740
\(274\) 346.571 0.0764127
\(275\) 0 0
\(276\) −417.183 −0.0909835
\(277\) −4691.55 −1.01765 −0.508823 0.860871i \(-0.669919\pi\)
−0.508823 + 0.860871i \(0.669919\pi\)
\(278\) −148.375 −0.0320107
\(279\) 1151.03 0.246990
\(280\) 0 0
\(281\) −8119.00 −1.72363 −0.861813 0.507226i \(-0.830671\pi\)
−0.861813 + 0.507226i \(0.830671\pi\)
\(282\) 330.761 0.0698459
\(283\) 3633.75 0.763265 0.381632 0.924314i \(-0.375362\pi\)
0.381632 + 0.924314i \(0.375362\pi\)
\(284\) −5381.15 −1.12434
\(285\) 0 0
\(286\) −573.515 −0.118576
\(287\) −1320.32 −0.271554
\(288\) 1683.88 0.344526
\(289\) −2440.58 −0.496759
\(290\) 0 0
\(291\) 1649.96 0.332379
\(292\) 4483.48 0.898547
\(293\) 7981.99 1.59151 0.795756 0.605618i \(-0.207074\pi\)
0.795756 + 0.605618i \(0.207074\pi\)
\(294\) 211.452 0.0419460
\(295\) 0 0
\(296\) −2314.03 −0.454392
\(297\) 205.619 0.0401725
\(298\) −4684.61 −0.910645
\(299\) −1227.55 −0.237427
\(300\) 0 0
\(301\) −2255.30 −0.431871
\(302\) 2116.02 0.403189
\(303\) −1599.90 −0.303339
\(304\) 2618.28 0.493977
\(305\) 0 0
\(306\) 643.721 0.120259
\(307\) 7118.15 1.32330 0.661652 0.749811i \(-0.269856\pi\)
0.661652 + 0.749811i \(0.269856\pi\)
\(308\) −316.167 −0.0584912
\(309\) −4073.36 −0.749921
\(310\) 0 0
\(311\) −9155.92 −1.66940 −0.834702 0.550703i \(-0.814360\pi\)
−0.834702 + 0.550703i \(0.814360\pi\)
\(312\) 3147.35 0.571101
\(313\) 6163.44 1.11303 0.556515 0.830838i \(-0.312138\pi\)
0.556515 + 0.830838i \(0.312138\pi\)
\(314\) 2005.51 0.360438
\(315\) 0 0
\(316\) −133.939 −0.0238438
\(317\) −8658.37 −1.53408 −0.767038 0.641601i \(-0.778270\pi\)
−0.767038 + 0.641601i \(0.778270\pi\)
\(318\) 1830.45 0.322788
\(319\) 1196.94 0.210080
\(320\) 0 0
\(321\) −2741.08 −0.476611
\(322\) 236.090 0.0408597
\(323\) 6991.14 1.20433
\(324\) −480.400 −0.0823732
\(325\) 0 0
\(326\) 5285.24 0.897922
\(327\) −481.466 −0.0814224
\(328\) 3779.67 0.636272
\(329\) 536.534 0.0899090
\(330\) 0 0
\(331\) −128.477 −0.0213346 −0.0106673 0.999943i \(-0.503396\pi\)
−0.0106673 + 0.999943i \(0.503396\pi\)
\(332\) 6596.61 1.09047
\(333\) 1039.30 0.171030
\(334\) −5812.89 −0.952297
\(335\) 0 0
\(336\) 391.066 0.0634952
\(337\) −7784.57 −1.25832 −0.629158 0.777277i \(-0.716600\pi\)
−0.629158 + 0.777277i \(0.716600\pi\)
\(338\) 782.455 0.125917
\(339\) 5365.45 0.859620
\(340\) 0 0
\(341\) 973.965 0.154672
\(342\) 1820.22 0.287795
\(343\) 343.000 0.0539949
\(344\) 6456.22 1.01191
\(345\) 0 0
\(346\) −85.5274 −0.0132890
\(347\) −3740.26 −0.578638 −0.289319 0.957233i \(-0.593429\pi\)
−0.289319 + 0.957233i \(0.593429\pi\)
\(348\) −2796.47 −0.430766
\(349\) 5676.86 0.870703 0.435351 0.900261i \(-0.356624\pi\)
0.435351 + 0.900261i \(0.356624\pi\)
\(350\) 0 0
\(351\) −1413.56 −0.214958
\(352\) 1424.85 0.215752
\(353\) −909.564 −0.137142 −0.0685711 0.997646i \(-0.521844\pi\)
−0.0685711 + 0.997646i \(0.521844\pi\)
\(354\) 465.125 0.0698337
\(355\) 0 0
\(356\) −9007.87 −1.34106
\(357\) 1044.19 0.154803
\(358\) −4276.97 −0.631410
\(359\) 2678.57 0.393788 0.196894 0.980425i \(-0.436915\pi\)
0.196894 + 0.980425i \(0.436915\pi\)
\(360\) 0 0
\(361\) 12909.5 1.88212
\(362\) −972.706 −0.141227
\(363\) −3819.01 −0.552193
\(364\) 2173.54 0.312979
\(365\) 0 0
\(366\) 3950.74 0.564231
\(367\) 716.898 0.101967 0.0509833 0.998700i \(-0.483764\pi\)
0.0509833 + 0.998700i \(0.483764\pi\)
\(368\) 436.633 0.0618508
\(369\) −1697.56 −0.239489
\(370\) 0 0
\(371\) 2969.21 0.415508
\(372\) −2275.53 −0.317153
\(373\) 2006.15 0.278484 0.139242 0.990258i \(-0.455533\pi\)
0.139242 + 0.990258i \(0.455533\pi\)
\(374\) 544.698 0.0753092
\(375\) 0 0
\(376\) −1535.93 −0.210664
\(377\) −8228.53 −1.12411
\(378\) 271.867 0.0369929
\(379\) 7277.53 0.986336 0.493168 0.869934i \(-0.335839\pi\)
0.493168 + 0.869934i \(0.335839\pi\)
\(380\) 0 0
\(381\) 813.045 0.109327
\(382\) 23.8191 0.00319029
\(383\) −5953.94 −0.794339 −0.397170 0.917745i \(-0.630008\pi\)
−0.397170 + 0.917745i \(0.630008\pi\)
\(384\) −3971.84 −0.527831
\(385\) 0 0
\(386\) −3876.16 −0.511117
\(387\) −2899.67 −0.380875
\(388\) −3261.90 −0.426799
\(389\) 1867.98 0.243472 0.121736 0.992563i \(-0.461154\pi\)
0.121736 + 0.992563i \(0.461154\pi\)
\(390\) 0 0
\(391\) 1165.86 0.150794
\(392\) −981.902 −0.126514
\(393\) −2289.45 −0.293862
\(394\) −1477.83 −0.188964
\(395\) 0 0
\(396\) −406.500 −0.0515844
\(397\) 2160.07 0.273075 0.136538 0.990635i \(-0.456403\pi\)
0.136538 + 0.990635i \(0.456403\pi\)
\(398\) 4061.85 0.511563
\(399\) 2952.61 0.370464
\(400\) 0 0
\(401\) 1954.81 0.243438 0.121719 0.992565i \(-0.461159\pi\)
0.121719 + 0.992565i \(0.461159\pi\)
\(402\) 1949.34 0.241851
\(403\) −6695.68 −0.827632
\(404\) 3162.93 0.389509
\(405\) 0 0
\(406\) 1582.57 0.193452
\(407\) 879.420 0.107104
\(408\) −2989.20 −0.362714
\(409\) 14895.1 1.80077 0.900384 0.435096i \(-0.143286\pi\)
0.900384 + 0.435096i \(0.143286\pi\)
\(410\) 0 0
\(411\) 722.801 0.0867474
\(412\) 8052.86 0.962952
\(413\) 754.489 0.0898934
\(414\) 303.545 0.0360348
\(415\) 0 0
\(416\) −9795.34 −1.15446
\(417\) −309.449 −0.0363400
\(418\) 1540.21 0.180225
\(419\) 12608.9 1.47013 0.735067 0.677994i \(-0.237151\pi\)
0.735067 + 0.677994i \(0.237151\pi\)
\(420\) 0 0
\(421\) −7862.86 −0.910243 −0.455122 0.890429i \(-0.650404\pi\)
−0.455122 + 0.890429i \(0.650404\pi\)
\(422\) 7409.67 0.854732
\(423\) 689.829 0.0792923
\(424\) −8499.91 −0.973567
\(425\) 0 0
\(426\) 3915.36 0.445305
\(427\) 6408.58 0.726307
\(428\) 5419.00 0.612002
\(429\) −1196.11 −0.134613
\(430\) 0 0
\(431\) 14291.0 1.59715 0.798575 0.601896i \(-0.205588\pi\)
0.798575 + 0.601896i \(0.205588\pi\)
\(432\) 502.799 0.0559975
\(433\) 13759.0 1.52705 0.763527 0.645776i \(-0.223466\pi\)
0.763527 + 0.645776i \(0.223466\pi\)
\(434\) 1287.76 0.142430
\(435\) 0 0
\(436\) 951.838 0.104552
\(437\) 3296.65 0.360870
\(438\) −3262.21 −0.355878
\(439\) −6093.13 −0.662436 −0.331218 0.943554i \(-0.607459\pi\)
−0.331218 + 0.943554i \(0.607459\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) −3744.61 −0.402970
\(443\) −13449.5 −1.44244 −0.721222 0.692704i \(-0.756419\pi\)
−0.721222 + 0.692704i \(0.756419\pi\)
\(444\) −2054.64 −0.219615
\(445\) 0 0
\(446\) 164.697 0.0174857
\(447\) −9770.14 −1.03381
\(448\) 841.068 0.0886981
\(449\) 6893.83 0.724588 0.362294 0.932064i \(-0.381994\pi\)
0.362294 + 0.932064i \(0.381994\pi\)
\(450\) 0 0
\(451\) −1436.42 −0.149974
\(452\) −10607.3 −1.10381
\(453\) 4413.13 0.457719
\(454\) −6825.07 −0.705543
\(455\) 0 0
\(456\) −8452.40 −0.868026
\(457\) 11820.1 1.20990 0.604948 0.796265i \(-0.293194\pi\)
0.604948 + 0.796265i \(0.293194\pi\)
\(458\) −7745.28 −0.790203
\(459\) 1342.53 0.136523
\(460\) 0 0
\(461\) 8443.38 0.853031 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(462\) 230.045 0.0231660
\(463\) 1269.74 0.127451 0.0637257 0.997967i \(-0.479702\pi\)
0.0637257 + 0.997967i \(0.479702\pi\)
\(464\) 2926.86 0.292836
\(465\) 0 0
\(466\) 2314.33 0.230063
\(467\) 16481.8 1.63316 0.816579 0.577233i \(-0.195868\pi\)
0.816579 + 0.577233i \(0.195868\pi\)
\(468\) 2794.55 0.276022
\(469\) 3162.06 0.311323
\(470\) 0 0
\(471\) 4182.65 0.409186
\(472\) −2159.87 −0.210627
\(473\) −2453.61 −0.238514
\(474\) 97.4549 0.00944357
\(475\) 0 0
\(476\) −2064.32 −0.198778
\(477\) 3817.55 0.366444
\(478\) −4478.05 −0.428496
\(479\) −1400.21 −0.133564 −0.0667822 0.997768i \(-0.521273\pi\)
−0.0667822 + 0.997768i \(0.521273\pi\)
\(480\) 0 0
\(481\) −6045.72 −0.573100
\(482\) 10265.3 0.970066
\(483\) 492.386 0.0463858
\(484\) 7550.02 0.709055
\(485\) 0 0
\(486\) 349.543 0.0326246
\(487\) 14165.9 1.31811 0.659055 0.752094i \(-0.270956\pi\)
0.659055 + 0.752094i \(0.270956\pi\)
\(488\) −18345.8 −1.70179
\(489\) 11022.8 1.01936
\(490\) 0 0
\(491\) 4739.28 0.435603 0.217801 0.975993i \(-0.430112\pi\)
0.217801 + 0.975993i \(0.430112\pi\)
\(492\) 3356.00 0.307520
\(493\) 7815.06 0.713940
\(494\) −10588.4 −0.964364
\(495\) 0 0
\(496\) 2381.63 0.215601
\(497\) 6351.19 0.573219
\(498\) −4799.74 −0.431890
\(499\) −11370.0 −1.02003 −0.510013 0.860167i \(-0.670360\pi\)
−0.510013 + 0.860167i \(0.670360\pi\)
\(500\) 0 0
\(501\) −12123.3 −1.08109
\(502\) −324.375 −0.0288398
\(503\) −9212.48 −0.816629 −0.408314 0.912841i \(-0.633883\pi\)
−0.408314 + 0.912841i \(0.633883\pi\)
\(504\) −1262.45 −0.111575
\(505\) 0 0
\(506\) 256.851 0.0225660
\(507\) 1631.87 0.142947
\(508\) −1607.36 −0.140384
\(509\) −15938.3 −1.38792 −0.693960 0.720014i \(-0.744136\pi\)
−0.693960 + 0.720014i \(0.744136\pi\)
\(510\) 0 0
\(511\) −5291.69 −0.458103
\(512\) 6469.49 0.558426
\(513\) 3796.21 0.326719
\(514\) 6362.33 0.545973
\(515\) 0 0
\(516\) 5732.52 0.489070
\(517\) 583.713 0.0496550
\(518\) 1162.76 0.0986265
\(519\) −178.374 −0.0150863
\(520\) 0 0
\(521\) 6442.99 0.541790 0.270895 0.962609i \(-0.412680\pi\)
0.270895 + 0.962609i \(0.412680\pi\)
\(522\) 2034.73 0.170609
\(523\) −986.655 −0.0824922 −0.0412461 0.999149i \(-0.513133\pi\)
−0.0412461 + 0.999149i \(0.513133\pi\)
\(524\) 4526.15 0.377339
\(525\) 0 0
\(526\) −9408.26 −0.779885
\(527\) 6359.24 0.525641
\(528\) 425.453 0.0350672
\(529\) −11617.2 −0.954815
\(530\) 0 0
\(531\) 970.057 0.0792785
\(532\) −5837.18 −0.475703
\(533\) 9874.91 0.802495
\(534\) 6554.19 0.531137
\(535\) 0 0
\(536\) −9052.01 −0.729454
\(537\) −8919.97 −0.716807
\(538\) 3255.16 0.260855
\(539\) 373.161 0.0298204
\(540\) 0 0
\(541\) −12681.1 −1.00777 −0.503885 0.863771i \(-0.668097\pi\)
−0.503885 + 0.863771i \(0.668097\pi\)
\(542\) 2324.06 0.184183
\(543\) −2028.66 −0.160328
\(544\) 9303.14 0.733215
\(545\) 0 0
\(546\) −1581.48 −0.123958
\(547\) 14826.4 1.15892 0.579462 0.814999i \(-0.303263\pi\)
0.579462 + 0.814999i \(0.303263\pi\)
\(548\) −1428.95 −0.111390
\(549\) 8239.60 0.640542
\(550\) 0 0
\(551\) 22098.2 1.70856
\(552\) −1409.55 −0.108685
\(553\) 158.083 0.0121562
\(554\) −6748.55 −0.517542
\(555\) 0 0
\(556\) 611.767 0.0466632
\(557\) −1926.46 −0.146547 −0.0732737 0.997312i \(-0.523345\pi\)
−0.0732737 + 0.997312i \(0.523345\pi\)
\(558\) 1655.69 0.125611
\(559\) 16867.8 1.27626
\(560\) 0 0
\(561\) 1136.01 0.0854946
\(562\) −11678.8 −0.876581
\(563\) 18624.8 1.39422 0.697108 0.716966i \(-0.254470\pi\)
0.697108 + 0.716966i \(0.254470\pi\)
\(564\) −1363.76 −0.101817
\(565\) 0 0
\(566\) 5226.96 0.388172
\(567\) 567.000 0.0419961
\(568\) −18181.5 −1.34309
\(569\) −20093.9 −1.48045 −0.740227 0.672357i \(-0.765282\pi\)
−0.740227 + 0.672357i \(0.765282\pi\)
\(570\) 0 0
\(571\) 4535.25 0.332389 0.166195 0.986093i \(-0.446852\pi\)
0.166195 + 0.986093i \(0.446852\pi\)
\(572\) 2364.66 0.172852
\(573\) 49.6766 0.00362176
\(574\) −1899.21 −0.138104
\(575\) 0 0
\(576\) 1081.37 0.0782243
\(577\) −10034.6 −0.723994 −0.361997 0.932179i \(-0.617905\pi\)
−0.361997 + 0.932179i \(0.617905\pi\)
\(578\) −3510.64 −0.252636
\(579\) −8084.05 −0.580244
\(580\) 0 0
\(581\) −7785.75 −0.555951
\(582\) 2373.38 0.169037
\(583\) 3230.30 0.229477
\(584\) 15148.5 1.07337
\(585\) 0 0
\(586\) 11481.7 0.809391
\(587\) 11192.6 0.786999 0.393499 0.919325i \(-0.371264\pi\)
0.393499 + 0.919325i \(0.371264\pi\)
\(588\) −871.838 −0.0611462
\(589\) 17981.7 1.25793
\(590\) 0 0
\(591\) −3082.13 −0.214521
\(592\) 2150.44 0.149295
\(593\) −20317.6 −1.40699 −0.703493 0.710703i \(-0.748377\pi\)
−0.703493 + 0.710703i \(0.748377\pi\)
\(594\) 295.772 0.0204305
\(595\) 0 0
\(596\) 19315.1 1.32748
\(597\) 8471.31 0.580750
\(598\) −1765.76 −0.120748
\(599\) −26376.5 −1.79919 −0.899594 0.436727i \(-0.856138\pi\)
−0.899594 + 0.436727i \(0.856138\pi\)
\(600\) 0 0
\(601\) 9266.13 0.628907 0.314454 0.949273i \(-0.398179\pi\)
0.314454 + 0.949273i \(0.398179\pi\)
\(602\) −3244.13 −0.219636
\(603\) 4065.51 0.274561
\(604\) −8724.56 −0.587744
\(605\) 0 0
\(606\) −2301.37 −0.154268
\(607\) −11338.0 −0.758149 −0.379075 0.925366i \(-0.623758\pi\)
−0.379075 + 0.925366i \(0.623758\pi\)
\(608\) 26306.0 1.75469
\(609\) 3300.58 0.219616
\(610\) 0 0
\(611\) −4012.83 −0.265698
\(612\) −2654.13 −0.175305
\(613\) −25712.5 −1.69416 −0.847078 0.531469i \(-0.821640\pi\)
−0.847078 + 0.531469i \(0.821640\pi\)
\(614\) 10239.1 0.672990
\(615\) 0 0
\(616\) −1068.24 −0.0698714
\(617\) −663.465 −0.0432903 −0.0216451 0.999766i \(-0.506890\pi\)
−0.0216451 + 0.999766i \(0.506890\pi\)
\(618\) −5859.32 −0.381386
\(619\) −12768.7 −0.829108 −0.414554 0.910025i \(-0.636062\pi\)
−0.414554 + 0.910025i \(0.636062\pi\)
\(620\) 0 0
\(621\) 633.068 0.0409084
\(622\) −13170.3 −0.849005
\(623\) 10631.7 0.683706
\(624\) −2924.84 −0.187640
\(625\) 0 0
\(626\) 8865.78 0.566051
\(627\) 3212.24 0.204600
\(628\) −8268.92 −0.525423
\(629\) 5741.93 0.363984
\(630\) 0 0
\(631\) −14937.6 −0.942400 −0.471200 0.882026i \(-0.656179\pi\)
−0.471200 + 0.882026i \(0.656179\pi\)
\(632\) −452.544 −0.0284829
\(633\) 15453.5 0.970333
\(634\) −12454.6 −0.780182
\(635\) 0 0
\(636\) −7547.13 −0.470540
\(637\) −2565.35 −0.159565
\(638\) 1721.73 0.106840
\(639\) 8165.81 0.505531
\(640\) 0 0
\(641\) 10903.0 0.671829 0.335914 0.941893i \(-0.390955\pi\)
0.335914 + 0.941893i \(0.390955\pi\)
\(642\) −3942.90 −0.242389
\(643\) 7623.47 0.467559 0.233779 0.972290i \(-0.424891\pi\)
0.233779 + 0.972290i \(0.424891\pi\)
\(644\) −973.426 −0.0595627
\(645\) 0 0
\(646\) 10056.4 0.612482
\(647\) −5384.84 −0.327202 −0.163601 0.986527i \(-0.552311\pi\)
−0.163601 + 0.986527i \(0.552311\pi\)
\(648\) −1623.14 −0.0983999
\(649\) 820.833 0.0496464
\(650\) 0 0
\(651\) 2685.73 0.161693
\(652\) −21791.6 −1.30893
\(653\) −297.318 −0.0178177 −0.00890883 0.999960i \(-0.502836\pi\)
−0.00890883 + 0.999960i \(0.502836\pi\)
\(654\) −692.564 −0.0414088
\(655\) 0 0
\(656\) −3512.47 −0.209053
\(657\) −6803.61 −0.404009
\(658\) 771.776 0.0457248
\(659\) 10324.7 0.610309 0.305155 0.952303i \(-0.401292\pi\)
0.305155 + 0.952303i \(0.401292\pi\)
\(660\) 0 0
\(661\) 4272.98 0.251437 0.125718 0.992066i \(-0.459876\pi\)
0.125718 + 0.992066i \(0.459876\pi\)
\(662\) −184.808 −0.0108501
\(663\) −7809.69 −0.457471
\(664\) 22288.2 1.30263
\(665\) 0 0
\(666\) 1494.97 0.0869804
\(667\) 3685.17 0.213928
\(668\) 23967.2 1.38820
\(669\) 343.488 0.0198506
\(670\) 0 0
\(671\) 6972.10 0.401125
\(672\) 3929.05 0.225545
\(673\) 23033.4 1.31927 0.659637 0.751584i \(-0.270710\pi\)
0.659637 + 0.751584i \(0.270710\pi\)
\(674\) −11197.7 −0.639940
\(675\) 0 0
\(676\) −3226.15 −0.183554
\(677\) −5113.50 −0.290292 −0.145146 0.989410i \(-0.546365\pi\)
−0.145146 + 0.989410i \(0.546365\pi\)
\(678\) 7717.91 0.437175
\(679\) 3849.91 0.217593
\(680\) 0 0
\(681\) −14234.2 −0.800966
\(682\) 1401.00 0.0786613
\(683\) −1341.02 −0.0751286 −0.0375643 0.999294i \(-0.511960\pi\)
−0.0375643 + 0.999294i \(0.511960\pi\)
\(684\) −7504.94 −0.419530
\(685\) 0 0
\(686\) 493.387 0.0274601
\(687\) −16153.4 −0.897076
\(688\) −5999.80 −0.332471
\(689\) −22207.2 −1.22790
\(690\) 0 0
\(691\) −16809.3 −0.925404 −0.462702 0.886514i \(-0.653120\pi\)
−0.462702 + 0.886514i \(0.653120\pi\)
\(692\) 352.639 0.0193718
\(693\) 479.778 0.0262991
\(694\) −5380.16 −0.294277
\(695\) 0 0
\(696\) −9448.53 −0.514577
\(697\) −9378.71 −0.509676
\(698\) 8165.86 0.442812
\(699\) 4826.72 0.261178
\(700\) 0 0
\(701\) 13467.0 0.725592 0.362796 0.931869i \(-0.381822\pi\)
0.362796 + 0.931869i \(0.381822\pi\)
\(702\) −2033.33 −0.109321
\(703\) 16236.1 0.871064
\(704\) 915.025 0.0489863
\(705\) 0 0
\(706\) −1308.36 −0.0697461
\(707\) −3733.09 −0.198582
\(708\) −1917.76 −0.101799
\(709\) 35514.3 1.88119 0.940597 0.339525i \(-0.110266\pi\)
0.940597 + 0.339525i \(0.110266\pi\)
\(710\) 0 0
\(711\) 203.250 0.0107208
\(712\) −30435.2 −1.60198
\(713\) 2998.68 0.157506
\(714\) 1502.02 0.0787277
\(715\) 0 0
\(716\) 17634.4 0.920431
\(717\) −9339.34 −0.486449
\(718\) 3852.99 0.200268
\(719\) 4993.61 0.259013 0.129506 0.991579i \(-0.458661\pi\)
0.129506 + 0.991579i \(0.458661\pi\)
\(720\) 0 0
\(721\) −9504.51 −0.490938
\(722\) 18569.6 0.957186
\(723\) 21409.1 1.10127
\(724\) 4010.57 0.205872
\(725\) 0 0
\(726\) −5493.45 −0.280828
\(727\) 4223.35 0.215454 0.107727 0.994181i \(-0.465643\pi\)
0.107727 + 0.994181i \(0.465643\pi\)
\(728\) 7343.81 0.373873
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −16020.2 −0.810572
\(732\) −16289.3 −0.822502
\(733\) −19030.9 −0.958968 −0.479484 0.877551i \(-0.659176\pi\)
−0.479484 + 0.877551i \(0.659176\pi\)
\(734\) 1031.22 0.0518570
\(735\) 0 0
\(736\) 4386.87 0.219704
\(737\) 3440.11 0.171938
\(738\) −2441.85 −0.121796
\(739\) −27772.5 −1.38245 −0.691224 0.722641i \(-0.742928\pi\)
−0.691224 + 0.722641i \(0.742928\pi\)
\(740\) 0 0
\(741\) −22083.0 −1.09479
\(742\) 4271.05 0.211314
\(743\) −26880.7 −1.32726 −0.663631 0.748060i \(-0.730985\pi\)
−0.663631 + 0.748060i \(0.730985\pi\)
\(744\) −7688.42 −0.378859
\(745\) 0 0
\(746\) 2885.74 0.141628
\(747\) −10010.2 −0.490302
\(748\) −2245.85 −0.109781
\(749\) −6395.85 −0.312015
\(750\) 0 0
\(751\) −35166.6 −1.70872 −0.854360 0.519681i \(-0.826051\pi\)
−0.854360 + 0.519681i \(0.826051\pi\)
\(752\) 1427.35 0.0692154
\(753\) −676.512 −0.0327403
\(754\) −11836.3 −0.571688
\(755\) 0 0
\(756\) −1120.93 −0.0539259
\(757\) −14589.2 −0.700467 −0.350233 0.936662i \(-0.613898\pi\)
−0.350233 + 0.936662i \(0.613898\pi\)
\(758\) 10468.3 0.501619
\(759\) 535.683 0.0256180
\(760\) 0 0
\(761\) −782.826 −0.0372897 −0.0186448 0.999826i \(-0.505935\pi\)
−0.0186448 + 0.999826i \(0.505935\pi\)
\(762\) 1169.52 0.0556002
\(763\) −1123.42 −0.0533035
\(764\) −98.2085 −0.00465060
\(765\) 0 0
\(766\) −8564.42 −0.403975
\(767\) −5642.95 −0.265652
\(768\) −8596.95 −0.403927
\(769\) −16548.4 −0.776008 −0.388004 0.921658i \(-0.626835\pi\)
−0.388004 + 0.921658i \(0.626835\pi\)
\(770\) 0 0
\(771\) 13269.2 0.619815
\(772\) 15981.8 0.745075
\(773\) −5744.94 −0.267310 −0.133655 0.991028i \(-0.542671\pi\)
−0.133655 + 0.991028i \(0.542671\pi\)
\(774\) −4171.02 −0.193701
\(775\) 0 0
\(776\) −11021.1 −0.509837
\(777\) 2425.02 0.111966
\(778\) 2687.00 0.123822
\(779\) −26519.7 −1.21973
\(780\) 0 0
\(781\) 6909.66 0.316578
\(782\) 1677.03 0.0766888
\(783\) 4243.60 0.193683
\(784\) 912.486 0.0415673
\(785\) 0 0
\(786\) −3293.26 −0.149449
\(787\) 34744.3 1.57370 0.786848 0.617146i \(-0.211711\pi\)
0.786848 + 0.617146i \(0.211711\pi\)
\(788\) 6093.24 0.275460
\(789\) −19621.7 −0.885362
\(790\) 0 0
\(791\) 12519.4 0.562753
\(792\) −1373.46 −0.0616207
\(793\) −47930.8 −2.14637
\(794\) 3107.15 0.138877
\(795\) 0 0
\(796\) −16747.4 −0.745724
\(797\) −21748.7 −0.966600 −0.483300 0.875455i \(-0.660562\pi\)
−0.483300 + 0.875455i \(0.660562\pi\)
\(798\) 4247.17 0.188406
\(799\) 3811.19 0.168749
\(800\) 0 0
\(801\) 13669.3 0.602972
\(802\) 2811.89 0.123805
\(803\) −5757.01 −0.253002
\(804\) −8037.34 −0.352556
\(805\) 0 0
\(806\) −9631.38 −0.420907
\(807\) 6788.90 0.296134
\(808\) 10686.7 0.465292
\(809\) −42350.6 −1.84050 −0.920252 0.391325i \(-0.872017\pi\)
−0.920252 + 0.391325i \(0.872017\pi\)
\(810\) 0 0
\(811\) −18910.7 −0.818796 −0.409398 0.912356i \(-0.634261\pi\)
−0.409398 + 0.912356i \(0.634261\pi\)
\(812\) −6525.10 −0.282003
\(813\) 4847.03 0.209093
\(814\) 1265.00 0.0544696
\(815\) 0 0
\(816\) 2777.88 0.119173
\(817\) −45299.4 −1.93981
\(818\) 21425.8 0.915813
\(819\) −3298.31 −0.140723
\(820\) 0 0
\(821\) 6593.42 0.280283 0.140141 0.990132i \(-0.455244\pi\)
0.140141 + 0.990132i \(0.455244\pi\)
\(822\) 1039.71 0.0441169
\(823\) −26762.4 −1.13351 −0.566755 0.823886i \(-0.691801\pi\)
−0.566755 + 0.823886i \(0.691801\pi\)
\(824\) 27208.5 1.15031
\(825\) 0 0
\(826\) 1085.29 0.0457169
\(827\) 24016.7 1.00985 0.504924 0.863164i \(-0.331521\pi\)
0.504924 + 0.863164i \(0.331521\pi\)
\(828\) −1251.55 −0.0525293
\(829\) −28422.3 −1.19077 −0.595383 0.803442i \(-0.703000\pi\)
−0.595383 + 0.803442i \(0.703000\pi\)
\(830\) 0 0
\(831\) −14074.7 −0.587538
\(832\) −6290.49 −0.262120
\(833\) 2436.45 0.101342
\(834\) −445.126 −0.0184814
\(835\) 0 0
\(836\) −6350.46 −0.262721
\(837\) 3453.09 0.142600
\(838\) 18137.3 0.747663
\(839\) 6637.09 0.273108 0.136554 0.990633i \(-0.456397\pi\)
0.136554 + 0.990633i \(0.456397\pi\)
\(840\) 0 0
\(841\) 313.546 0.0128560
\(842\) −11310.3 −0.462921
\(843\) −24357.0 −0.995136
\(844\) −30550.9 −1.24598
\(845\) 0 0
\(846\) 992.283 0.0403255
\(847\) −8911.03 −0.361495
\(848\) 7899.01 0.319874
\(849\) 10901.2 0.440671
\(850\) 0 0
\(851\) 2707.59 0.109066
\(852\) −16143.5 −0.649138
\(853\) 1406.88 0.0564720 0.0282360 0.999601i \(-0.491011\pi\)
0.0282360 + 0.999601i \(0.491011\pi\)
\(854\) 9218.40 0.369376
\(855\) 0 0
\(856\) 18309.3 0.731075
\(857\) −27943.4 −1.11380 −0.556901 0.830579i \(-0.688010\pi\)
−0.556901 + 0.830579i \(0.688010\pi\)
\(858\) −1720.55 −0.0684598
\(859\) 1936.63 0.0769233 0.0384616 0.999260i \(-0.487754\pi\)
0.0384616 + 0.999260i \(0.487754\pi\)
\(860\) 0 0
\(861\) −3960.97 −0.156782
\(862\) 20556.8 0.812259
\(863\) −7947.70 −0.313491 −0.156746 0.987639i \(-0.550100\pi\)
−0.156746 + 0.987639i \(0.550100\pi\)
\(864\) 5051.63 0.198912
\(865\) 0 0
\(866\) 19791.6 0.776611
\(867\) −7321.73 −0.286804
\(868\) −5309.58 −0.207625
\(869\) 171.984 0.00671365
\(870\) 0 0
\(871\) −23649.6 −0.920019
\(872\) 3216.00 0.124894
\(873\) 4949.88 0.191899
\(874\) 4742.06 0.183527
\(875\) 0 0
\(876\) 13450.4 0.518776
\(877\) −38655.0 −1.48835 −0.744177 0.667983i \(-0.767158\pi\)
−0.744177 + 0.667983i \(0.767158\pi\)
\(878\) −8764.65 −0.336893
\(879\) 23946.0 0.918859
\(880\) 0 0
\(881\) −18879.4 −0.721978 −0.360989 0.932570i \(-0.617561\pi\)
−0.360989 + 0.932570i \(0.617561\pi\)
\(882\) 634.355 0.0242175
\(883\) 35098.1 1.33765 0.668825 0.743420i \(-0.266798\pi\)
0.668825 + 0.743420i \(0.266798\pi\)
\(884\) 15439.4 0.587425
\(885\) 0 0
\(886\) −19346.3 −0.733581
\(887\) 48816.4 1.84791 0.923954 0.382504i \(-0.124938\pi\)
0.923954 + 0.382504i \(0.124938\pi\)
\(888\) −6942.08 −0.262344
\(889\) 1897.11 0.0715713
\(890\) 0 0
\(891\) 616.858 0.0231936
\(892\) −679.062 −0.0254895
\(893\) 10776.7 0.403839
\(894\) −14053.8 −0.525761
\(895\) 0 0
\(896\) −9267.63 −0.345547
\(897\) −3682.64 −0.137079
\(898\) 9916.41 0.368502
\(899\) 20100.8 0.745718
\(900\) 0 0
\(901\) 21091.3 0.779860
\(902\) −2066.22 −0.0762721
\(903\) −6765.90 −0.249341
\(904\) −35839.1 −1.31857
\(905\) 0 0
\(906\) 6348.05 0.232781
\(907\) 6010.83 0.220051 0.110026 0.993929i \(-0.464907\pi\)
0.110026 + 0.993929i \(0.464907\pi\)
\(908\) 28140.5 1.02850
\(909\) −4799.69 −0.175133
\(910\) 0 0
\(911\) 25780.1 0.937576 0.468788 0.883311i \(-0.344691\pi\)
0.468788 + 0.883311i \(0.344691\pi\)
\(912\) 7854.85 0.285198
\(913\) −8470.37 −0.307041
\(914\) 17002.6 0.615314
\(915\) 0 0
\(916\) 31934.6 1.15191
\(917\) −5342.06 −0.192378
\(918\) 1931.16 0.0694313
\(919\) 26731.8 0.959522 0.479761 0.877399i \(-0.340723\pi\)
0.479761 + 0.877399i \(0.340723\pi\)
\(920\) 0 0
\(921\) 21354.4 0.764010
\(922\) 12145.4 0.433824
\(923\) −47501.6 −1.69397
\(924\) −948.501 −0.0337699
\(925\) 0 0
\(926\) 1826.46 0.0648177
\(927\) −12220.1 −0.432967
\(928\) 29406.2 1.04020
\(929\) −30464.6 −1.07590 −0.537949 0.842977i \(-0.680801\pi\)
−0.537949 + 0.842977i \(0.680801\pi\)
\(930\) 0 0
\(931\) 6889.42 0.242526
\(932\) −9542.22 −0.335371
\(933\) −27467.7 −0.963830
\(934\) 23708.1 0.830572
\(935\) 0 0
\(936\) 9442.04 0.329725
\(937\) 28533.4 0.994819 0.497409 0.867516i \(-0.334285\pi\)
0.497409 + 0.867516i \(0.334285\pi\)
\(938\) 4548.46 0.158329
\(939\) 18490.3 0.642608
\(940\) 0 0
\(941\) 34455.8 1.19365 0.596827 0.802370i \(-0.296428\pi\)
0.596827 + 0.802370i \(0.296428\pi\)
\(942\) 6016.52 0.208099
\(943\) −4422.50 −0.152722
\(944\) 2007.17 0.0692033
\(945\) 0 0
\(946\) −3529.39 −0.121301
\(947\) −2477.68 −0.0850198 −0.0425099 0.999096i \(-0.513535\pi\)
−0.0425099 + 0.999096i \(0.513535\pi\)
\(948\) −401.817 −0.0137662
\(949\) 39577.5 1.35378
\(950\) 0 0
\(951\) −25975.1 −0.885699
\(952\) −6974.80 −0.237452
\(953\) −41690.6 −1.41709 −0.708547 0.705663i \(-0.750649\pi\)
−0.708547 + 0.705663i \(0.750649\pi\)
\(954\) 5491.35 0.186361
\(955\) 0 0
\(956\) 18463.5 0.624635
\(957\) 3590.81 0.121290
\(958\) −2014.13 −0.0679266
\(959\) 1686.54 0.0567895
\(960\) 0 0
\(961\) −13434.6 −0.450962
\(962\) −8696.44 −0.291460
\(963\) −8223.24 −0.275172
\(964\) −42325.0 −1.41410
\(965\) 0 0
\(966\) 708.271 0.0235903
\(967\) 20641.3 0.686430 0.343215 0.939257i \(-0.388484\pi\)
0.343215 + 0.939257i \(0.388484\pi\)
\(968\) 25509.5 0.847010
\(969\) 20973.4 0.695318
\(970\) 0 0
\(971\) 6626.17 0.218995 0.109497 0.993987i \(-0.465076\pi\)
0.109497 + 0.993987i \(0.465076\pi\)
\(972\) −1441.20 −0.0475582
\(973\) −722.048 −0.0237901
\(974\) 20377.0 0.670349
\(975\) 0 0
\(976\) 17048.8 0.559138
\(977\) −41961.0 −1.37405 −0.687027 0.726632i \(-0.741085\pi\)
−0.687027 + 0.726632i \(0.741085\pi\)
\(978\) 15855.7 0.518415
\(979\) 11566.5 0.377598
\(980\) 0 0
\(981\) −1444.40 −0.0470093
\(982\) 6817.21 0.221533
\(983\) 16781.7 0.544510 0.272255 0.962225i \(-0.412231\pi\)
0.272255 + 0.962225i \(0.412231\pi\)
\(984\) 11339.0 0.367352
\(985\) 0 0
\(986\) 11241.6 0.363087
\(987\) 1609.60 0.0519090
\(988\) 43657.2 1.40579
\(989\) −7554.27 −0.242884
\(990\) 0 0
\(991\) 50319.8 1.61298 0.806489 0.591250i \(-0.201365\pi\)
0.806489 + 0.591250i \(0.201365\pi\)
\(992\) 23928.3 0.765851
\(993\) −385.432 −0.0123176
\(994\) 9135.85 0.291521
\(995\) 0 0
\(996\) 19789.8 0.629583
\(997\) −12949.5 −0.411348 −0.205674 0.978621i \(-0.565939\pi\)
−0.205674 + 0.978621i \(0.565939\pi\)
\(998\) −16355.2 −0.518753
\(999\) 3117.89 0.0987443
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 525.4.a.p.1.1 2
3.2 odd 2 1575.4.a.m.1.2 2
5.2 odd 4 525.4.d.i.274.3 4
5.3 odd 4 525.4.d.i.274.2 4
5.4 even 2 105.4.a.c.1.2 2
15.14 odd 2 315.4.a.m.1.1 2
20.19 odd 2 1680.4.a.bk.1.1 2
35.34 odd 2 735.4.a.k.1.2 2
105.104 even 2 2205.4.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.c.1.2 2 5.4 even 2
315.4.a.m.1.1 2 15.14 odd 2
525.4.a.p.1.1 2 1.1 even 1 trivial
525.4.d.i.274.2 4 5.3 odd 4
525.4.d.i.274.3 4 5.2 odd 4
735.4.a.k.1.2 2 35.34 odd 2
1575.4.a.m.1.2 2 3.2 odd 2
1680.4.a.bk.1.1 2 20.19 odd 2
2205.4.a.bh.1.1 2 105.104 even 2