Properties

Label 75.4.a.d.1.2
Level $75$
Weight $4$
Character 75.1
Self dual yes
Analytic conductor $4.425$
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] = [75,4,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("75.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(4.42514325043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.35890\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.35890 q^{2} +3.00000 q^{3} +3.28220 q^{4} +10.0767 q^{6} +30.4356 q^{7} -15.8466 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.35890 q^{2} +3.00000 q^{3} +3.28220 q^{4} +10.0767 q^{6} +30.4356 q^{7} -15.8466 q^{8} +9.00000 q^{9} +31.4356 q^{11} +9.84661 q^{12} -60.7424 q^{13} +102.230 q^{14} -79.4848 q^{16} -121.178 q^{17} +30.2301 q^{18} -14.4356 q^{19} +91.3068 q^{21} +105.589 q^{22} +13.6932 q^{23} -47.5398 q^{24} -204.028 q^{26} +27.0000 q^{27} +99.8958 q^{28} -76.0492 q^{29} +183.049 q^{31} -140.208 q^{32} +94.3068 q^{33} -407.025 q^{34} +29.5398 q^{36} -37.3864 q^{37} -48.4877 q^{38} -182.227 q^{39} -30.6627 q^{41} +306.690 q^{42} +327.564 q^{43} +103.178 q^{44} +45.9941 q^{46} -449.485 q^{47} -238.454 q^{48} +583.325 q^{49} -363.534 q^{51} -199.369 q^{52} -301.951 q^{53} +90.6903 q^{54} -482.301 q^{56} -43.3068 q^{57} -255.441 q^{58} +340.970 q^{59} +619.098 q^{61} +614.844 q^{62} +273.920 q^{63} +164.932 q^{64} +316.767 q^{66} +256.890 q^{67} -397.731 q^{68} +41.0796 q^{69} +499.178 q^{71} -142.619 q^{72} -19.1288 q^{73} -125.577 q^{74} -47.3805 q^{76} +956.761 q^{77} -612.083 q^{78} +257.424 q^{79} +81.0000 q^{81} -102.993 q^{82} +914.909 q^{83} +299.687 q^{84} +1100.26 q^{86} -228.148 q^{87} -498.148 q^{88} -1059.68 q^{89} -1848.73 q^{91} +44.9439 q^{92} +549.148 q^{93} -1509.77 q^{94} -420.625 q^{96} +521.000 q^{97} +1959.33 q^{98} +282.920 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 6 q^{3} + 24 q^{4} - 6 q^{6} + 26 q^{7} - 84 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 6 q^{3} + 24 q^{4} - 6 q^{6} + 26 q^{7} - 84 q^{8} + 18 q^{9} + 28 q^{11} + 72 q^{12} + 18 q^{13} + 126 q^{14} + 120 q^{16} - 68 q^{17} - 18 q^{18} + 6 q^{19} + 78 q^{21} + 124 q^{22} + 132 q^{23} - 252 q^{24} - 626 q^{26} + 54 q^{27} + 8 q^{28} + 92 q^{29} + 122 q^{31} - 664 q^{32} + 84 q^{33} - 692 q^{34} + 216 q^{36} - 284 q^{37} - 158 q^{38} + 54 q^{39} + 392 q^{41} + 378 q^{42} + 690 q^{43} + 32 q^{44} - 588 q^{46} - 620 q^{47} + 360 q^{48} + 260 q^{49} - 204 q^{51} + 1432 q^{52} - 848 q^{53} - 54 q^{54} - 180 q^{56} + 18 q^{57} - 1156 q^{58} + 124 q^{59} + 750 q^{61} + 942 q^{62} + 234 q^{63} + 1376 q^{64} + 372 q^{66} - 358 q^{67} + 704 q^{68} + 396 q^{69} + 824 q^{71} - 756 q^{72} - 108 q^{73} + 1196 q^{74} + 376 q^{76} + 972 q^{77} - 1878 q^{78} - 880 q^{79} + 162 q^{81} - 2368 q^{82} + 156 q^{83} + 24 q^{84} - 842 q^{86} + 276 q^{87} - 264 q^{88} - 864 q^{89} - 2198 q^{91} + 2496 q^{92} + 366 q^{93} - 596 q^{94} - 1992 q^{96} + 1042 q^{97} + 3692 q^{98} + 252 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.35890 1.18755 0.593775 0.804631i \(-0.297637\pi\)
0.593775 + 0.804631i \(0.297637\pi\)
\(3\) 3.00000 0.577350
\(4\) 3.28220 0.410275
\(5\) 0 0
\(6\) 10.0767 0.685632
\(7\) 30.4356 1.64337 0.821684 0.569944i \(-0.193035\pi\)
0.821684 + 0.569944i \(0.193035\pi\)
\(8\) −15.8466 −0.700328
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 31.4356 0.861654 0.430827 0.902435i \(-0.358222\pi\)
0.430827 + 0.902435i \(0.358222\pi\)
\(12\) 9.84661 0.236873
\(13\) −60.7424 −1.29592 −0.647958 0.761676i \(-0.724377\pi\)
−0.647958 + 0.761676i \(0.724377\pi\)
\(14\) 102.230 1.95158
\(15\) 0 0
\(16\) −79.4848 −1.24195
\(17\) −121.178 −1.72882 −0.864411 0.502786i \(-0.832308\pi\)
−0.864411 + 0.502786i \(0.832308\pi\)
\(18\) 30.2301 0.395850
\(19\) −14.4356 −0.174303 −0.0871514 0.996195i \(-0.527776\pi\)
−0.0871514 + 0.996195i \(0.527776\pi\)
\(20\) 0 0
\(21\) 91.3068 0.948799
\(22\) 105.589 1.02326
\(23\) 13.6932 0.124141 0.0620703 0.998072i \(-0.480230\pi\)
0.0620703 + 0.998072i \(0.480230\pi\)
\(24\) −47.5398 −0.404334
\(25\) 0 0
\(26\) −204.028 −1.53896
\(27\) 27.0000 0.192450
\(28\) 99.8958 0.674233
\(29\) −76.0492 −0.486965 −0.243482 0.969905i \(-0.578290\pi\)
−0.243482 + 0.969905i \(0.578290\pi\)
\(30\) 0 0
\(31\) 183.049 1.06054 0.530268 0.847830i \(-0.322091\pi\)
0.530268 + 0.847830i \(0.322091\pi\)
\(32\) −140.208 −0.774550
\(33\) 94.3068 0.497476
\(34\) −407.025 −2.05306
\(35\) 0 0
\(36\) 29.5398 0.136758
\(37\) −37.3864 −0.166116 −0.0830580 0.996545i \(-0.526469\pi\)
−0.0830580 + 0.996545i \(0.526469\pi\)
\(38\) −48.4877 −0.206993
\(39\) −182.227 −0.748197
\(40\) 0 0
\(41\) −30.6627 −0.116798 −0.0583990 0.998293i \(-0.518600\pi\)
−0.0583990 + 0.998293i \(0.518600\pi\)
\(42\) 306.690 1.12675
\(43\) 327.564 1.16170 0.580850 0.814011i \(-0.302720\pi\)
0.580850 + 0.814011i \(0.302720\pi\)
\(44\) 103.178 0.353515
\(45\) 0 0
\(46\) 45.9941 0.147423
\(47\) −449.485 −1.39498 −0.697490 0.716594i \(-0.745700\pi\)
−0.697490 + 0.716594i \(0.745700\pi\)
\(48\) −238.454 −0.717040
\(49\) 583.325 1.70066
\(50\) 0 0
\(51\) −363.534 −0.998136
\(52\) −199.369 −0.531682
\(53\) −301.951 −0.782569 −0.391284 0.920270i \(-0.627969\pi\)
−0.391284 + 0.920270i \(0.627969\pi\)
\(54\) 90.6903 0.228544
\(55\) 0 0
\(56\) −482.301 −1.15090
\(57\) −43.3068 −0.100634
\(58\) −255.441 −0.578295
\(59\) 340.970 0.752381 0.376190 0.926542i \(-0.377234\pi\)
0.376190 + 0.926542i \(0.377234\pi\)
\(60\) 0 0
\(61\) 619.098 1.29947 0.649733 0.760163i \(-0.274881\pi\)
0.649733 + 0.760163i \(0.274881\pi\)
\(62\) 614.844 1.25944
\(63\) 273.920 0.547789
\(64\) 164.932 0.322133
\(65\) 0 0
\(66\) 316.767 0.590778
\(67\) 256.890 0.468419 0.234210 0.972186i \(-0.424750\pi\)
0.234210 + 0.972186i \(0.424750\pi\)
\(68\) −397.731 −0.709293
\(69\) 41.0796 0.0716726
\(70\) 0 0
\(71\) 499.178 0.834388 0.417194 0.908818i \(-0.363014\pi\)
0.417194 + 0.908818i \(0.363014\pi\)
\(72\) −142.619 −0.233443
\(73\) −19.1288 −0.0306693 −0.0153346 0.999882i \(-0.504881\pi\)
−0.0153346 + 0.999882i \(0.504881\pi\)
\(74\) −125.577 −0.197271
\(75\) 0 0
\(76\) −47.3805 −0.0715121
\(77\) 956.761 1.41601
\(78\) −612.083 −0.888522
\(79\) 257.424 0.366613 0.183307 0.983056i \(-0.441320\pi\)
0.183307 + 0.983056i \(0.441320\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −102.993 −0.138703
\(83\) 914.909 1.20993 0.604965 0.796252i \(-0.293187\pi\)
0.604965 + 0.796252i \(0.293187\pi\)
\(84\) 299.687 0.389269
\(85\) 0 0
\(86\) 1100.26 1.37958
\(87\) −228.148 −0.281149
\(88\) −498.148 −0.603440
\(89\) −1059.68 −1.26209 −0.631045 0.775746i \(-0.717374\pi\)
−0.631045 + 0.775746i \(0.717374\pi\)
\(90\) 0 0
\(91\) −1848.73 −2.12967
\(92\) 44.9439 0.0509318
\(93\) 549.148 0.612300
\(94\) −1509.77 −1.65661
\(95\) 0 0
\(96\) −420.625 −0.447186
\(97\) 521.000 0.545356 0.272678 0.962105i \(-0.412091\pi\)
0.272678 + 0.962105i \(0.412091\pi\)
\(98\) 1959.33 2.01962
\(99\) 282.920 0.287218
\(100\) 0 0
\(101\) 347.080 0.341938 0.170969 0.985276i \(-0.445310\pi\)
0.170969 + 0.985276i \(0.445310\pi\)
\(102\) −1221.07 −1.18534
\(103\) −770.749 −0.737322 −0.368661 0.929564i \(-0.620184\pi\)
−0.368661 + 0.929564i \(0.620184\pi\)
\(104\) 962.561 0.907566
\(105\) 0 0
\(106\) −1014.22 −0.929339
\(107\) −1415.37 −1.27878 −0.639390 0.768883i \(-0.720813\pi\)
−0.639390 + 0.768883i \(0.720813\pi\)
\(108\) 88.6195 0.0789575
\(109\) 908.386 0.798235 0.399118 0.916900i \(-0.369317\pi\)
0.399118 + 0.916900i \(0.369317\pi\)
\(110\) 0 0
\(111\) −112.159 −0.0959071
\(112\) −2419.17 −2.04098
\(113\) −2049.94 −1.70657 −0.853283 0.521447i \(-0.825392\pi\)
−0.853283 + 0.521447i \(0.825392\pi\)
\(114\) −145.463 −0.119508
\(115\) 0 0
\(116\) −249.609 −0.199790
\(117\) −546.681 −0.431972
\(118\) 1145.28 0.893490
\(119\) −3688.12 −2.84109
\(120\) 0 0
\(121\) −342.803 −0.257553
\(122\) 2079.49 1.54318
\(123\) −91.9882 −0.0674333
\(124\) 600.804 0.435111
\(125\) 0 0
\(126\) 920.071 0.650527
\(127\) 281.644 0.196786 0.0983932 0.995148i \(-0.468630\pi\)
0.0983932 + 0.995148i \(0.468630\pi\)
\(128\) 1675.66 1.15710
\(129\) 982.693 0.670708
\(130\) 0 0
\(131\) 243.056 0.162106 0.0810531 0.996710i \(-0.474172\pi\)
0.0810531 + 0.996710i \(0.474172\pi\)
\(132\) 309.534 0.204102
\(133\) −439.356 −0.286444
\(134\) 862.867 0.556271
\(135\) 0 0
\(136\) 1920.26 1.21074
\(137\) 909.386 0.567110 0.283555 0.958956i \(-0.408486\pi\)
0.283555 + 0.958956i \(0.408486\pi\)
\(138\) 137.982 0.0851148
\(139\) −2049.52 −1.25063 −0.625317 0.780371i \(-0.715030\pi\)
−0.625317 + 0.780371i \(0.715030\pi\)
\(140\) 0 0
\(141\) −1348.45 −0.805392
\(142\) 1676.69 0.990877
\(143\) −1909.47 −1.11663
\(144\) −715.363 −0.413983
\(145\) 0 0
\(146\) −64.2517 −0.0364213
\(147\) 1749.98 0.981875
\(148\) −122.710 −0.0681533
\(149\) 3601.14 1.97998 0.989990 0.141136i \(-0.0450753\pi\)
0.989990 + 0.141136i \(0.0450753\pi\)
\(150\) 0 0
\(151\) 1383.38 0.745550 0.372775 0.927922i \(-0.378406\pi\)
0.372775 + 0.927922i \(0.378406\pi\)
\(152\) 228.755 0.122069
\(153\) −1090.60 −0.576274
\(154\) 3213.66 1.68159
\(155\) 0 0
\(156\) −598.106 −0.306967
\(157\) −131.749 −0.0669729 −0.0334864 0.999439i \(-0.510661\pi\)
−0.0334864 + 0.999439i \(0.510661\pi\)
\(158\) 864.661 0.435372
\(159\) −905.852 −0.451816
\(160\) 0 0
\(161\) 416.761 0.204009
\(162\) 272.071 0.131950
\(163\) 2897.74 1.39244 0.696222 0.717827i \(-0.254863\pi\)
0.696222 + 0.717827i \(0.254863\pi\)
\(164\) −100.641 −0.0479193
\(165\) 0 0
\(166\) 3073.09 1.43685
\(167\) 260.283 0.120607 0.0603034 0.998180i \(-0.480793\pi\)
0.0603034 + 0.998180i \(0.480793\pi\)
\(168\) −1446.90 −0.664470
\(169\) 1492.64 0.679398
\(170\) 0 0
\(171\) −129.920 −0.0581009
\(172\) 1075.13 0.476617
\(173\) −1935.83 −0.850742 −0.425371 0.905019i \(-0.639856\pi\)
−0.425371 + 0.905019i \(0.639856\pi\)
\(174\) −766.324 −0.333879
\(175\) 0 0
\(176\) −2498.65 −1.07013
\(177\) 1022.91 0.434387
\(178\) −3559.36 −1.49880
\(179\) 576.627 0.240777 0.120389 0.992727i \(-0.461586\pi\)
0.120389 + 0.992727i \(0.461586\pi\)
\(180\) 0 0
\(181\) −1962.04 −0.805733 −0.402866 0.915259i \(-0.631986\pi\)
−0.402866 + 0.915259i \(0.631986\pi\)
\(182\) −6209.70 −2.52909
\(183\) 1857.30 0.750247
\(184\) −216.991 −0.0869390
\(185\) 0 0
\(186\) 1844.53 0.727137
\(187\) −3809.30 −1.48965
\(188\) −1475.30 −0.572326
\(189\) 821.761 0.316266
\(190\) 0 0
\(191\) −4318.75 −1.63609 −0.818047 0.575152i \(-0.804943\pi\)
−0.818047 + 0.575152i \(0.804943\pi\)
\(192\) 494.796 0.185984
\(193\) 2.97647 0.00111011 0.000555054 1.00000i \(-0.499823\pi\)
0.000555054 1.00000i \(0.499823\pi\)
\(194\) 1749.99 0.647638
\(195\) 0 0
\(196\) 1914.59 0.697738
\(197\) 569.705 0.206040 0.103020 0.994679i \(-0.467149\pi\)
0.103020 + 0.994679i \(0.467149\pi\)
\(198\) 950.301 0.341086
\(199\) −3050.73 −1.08674 −0.543368 0.839494i \(-0.682851\pi\)
−0.543368 + 0.839494i \(0.682851\pi\)
\(200\) 0 0
\(201\) 770.670 0.270442
\(202\) 1165.81 0.406068
\(203\) −2314.60 −0.800262
\(204\) −1193.19 −0.409510
\(205\) 0 0
\(206\) −2588.87 −0.875607
\(207\) 123.239 0.0413802
\(208\) 4828.09 1.60946
\(209\) −453.792 −0.150189
\(210\) 0 0
\(211\) −50.5104 −0.0164800 −0.00824000 0.999966i \(-0.502623\pi\)
−0.00824000 + 0.999966i \(0.502623\pi\)
\(212\) −991.064 −0.321069
\(213\) 1497.53 0.481734
\(214\) −4754.10 −1.51862
\(215\) 0 0
\(216\) −427.858 −0.134778
\(217\) 5571.21 1.74285
\(218\) 3051.18 0.947944
\(219\) −57.3864 −0.0177069
\(220\) 0 0
\(221\) 7360.64 2.24041
\(222\) −376.732 −0.113894
\(223\) 5453.55 1.63765 0.818827 0.574040i \(-0.194625\pi\)
0.818827 + 0.574040i \(0.194625\pi\)
\(224\) −4267.33 −1.27287
\(225\) 0 0
\(226\) −6885.54 −2.02663
\(227\) −4777.14 −1.39678 −0.698392 0.715715i \(-0.746101\pi\)
−0.698392 + 0.715715i \(0.746101\pi\)
\(228\) −142.142 −0.0412875
\(229\) −2085.51 −0.601808 −0.300904 0.953654i \(-0.597288\pi\)
−0.300904 + 0.953654i \(0.597288\pi\)
\(230\) 0 0
\(231\) 2870.28 0.817536
\(232\) 1205.12 0.341035
\(233\) 6484.53 1.82324 0.911622 0.411030i \(-0.134831\pi\)
0.911622 + 0.411030i \(0.134831\pi\)
\(234\) −1836.25 −0.512988
\(235\) 0 0
\(236\) 1119.13 0.308683
\(237\) 772.271 0.211664
\(238\) −12388.0 −3.37394
\(239\) 2234.62 0.604792 0.302396 0.953182i \(-0.402214\pi\)
0.302396 + 0.953182i \(0.402214\pi\)
\(240\) 0 0
\(241\) −2393.01 −0.639616 −0.319808 0.947482i \(-0.603618\pi\)
−0.319808 + 0.947482i \(0.603618\pi\)
\(242\) −1151.44 −0.305857
\(243\) 243.000 0.0641500
\(244\) 2032.01 0.533139
\(245\) 0 0
\(246\) −308.979 −0.0800805
\(247\) 876.852 0.225882
\(248\) −2900.71 −0.742722
\(249\) 2744.73 0.698554
\(250\) 0 0
\(251\) −612.661 −0.154067 −0.0770335 0.997029i \(-0.524545\pi\)
−0.0770335 + 0.997029i \(0.524545\pi\)
\(252\) 899.062 0.224744
\(253\) 430.454 0.106966
\(254\) 946.014 0.233694
\(255\) 0 0
\(256\) 4308.91 1.05198
\(257\) −306.112 −0.0742987 −0.0371493 0.999310i \(-0.511828\pi\)
−0.0371493 + 0.999310i \(0.511828\pi\)
\(258\) 3300.77 0.796499
\(259\) −1137.88 −0.272990
\(260\) 0 0
\(261\) −684.443 −0.162322
\(262\) 816.401 0.192509
\(263\) 283.839 0.0665484 0.0332742 0.999446i \(-0.489407\pi\)
0.0332742 + 0.999446i \(0.489407\pi\)
\(264\) −1494.44 −0.348396
\(265\) 0 0
\(266\) −1475.75 −0.340166
\(267\) −3179.04 −0.728668
\(268\) 843.165 0.192181
\(269\) 2426.21 0.549920 0.274960 0.961456i \(-0.411335\pi\)
0.274960 + 0.961456i \(0.411335\pi\)
\(270\) 0 0
\(271\) −174.946 −0.0392148 −0.0196074 0.999808i \(-0.506242\pi\)
−0.0196074 + 0.999808i \(0.506242\pi\)
\(272\) 9631.80 2.14711
\(273\) −5546.19 −1.22956
\(274\) 3054.54 0.673472
\(275\) 0 0
\(276\) 134.832 0.0294055
\(277\) −7807.07 −1.69344 −0.846718 0.532042i \(-0.821425\pi\)
−0.846718 + 0.532042i \(0.821425\pi\)
\(278\) −6884.14 −1.48519
\(279\) 1647.44 0.353512
\(280\) 0 0
\(281\) 584.171 0.124017 0.0620084 0.998076i \(-0.480249\pi\)
0.0620084 + 0.998076i \(0.480249\pi\)
\(282\) −4529.32 −0.956444
\(283\) −5897.31 −1.23872 −0.619362 0.785106i \(-0.712609\pi\)
−0.619362 + 0.785106i \(0.712609\pi\)
\(284\) 1638.40 0.342329
\(285\) 0 0
\(286\) −6413.73 −1.32605
\(287\) −933.239 −0.191942
\(288\) −1261.88 −0.258183
\(289\) 9771.10 1.98883
\(290\) 0 0
\(291\) 1563.00 0.314861
\(292\) −62.7846 −0.0125828
\(293\) 1609.73 0.320960 0.160480 0.987039i \(-0.448696\pi\)
0.160480 + 0.987039i \(0.448696\pi\)
\(294\) 5877.99 1.16603
\(295\) 0 0
\(296\) 592.448 0.116336
\(297\) 848.761 0.165825
\(298\) 12095.9 2.35133
\(299\) −831.758 −0.160876
\(300\) 0 0
\(301\) 9969.62 1.90910
\(302\) 4646.64 0.885378
\(303\) 1041.24 0.197418
\(304\) 1147.41 0.216475
\(305\) 0 0
\(306\) −3663.22 −0.684354
\(307\) 234.473 0.0435898 0.0217949 0.999762i \(-0.493062\pi\)
0.0217949 + 0.999762i \(0.493062\pi\)
\(308\) 3140.28 0.580955
\(309\) −2312.25 −0.425693
\(310\) 0 0
\(311\) −1795.25 −0.327329 −0.163665 0.986516i \(-0.552331\pi\)
−0.163665 + 0.986516i \(0.552331\pi\)
\(312\) 2887.68 0.523983
\(313\) −8440.61 −1.52425 −0.762127 0.647427i \(-0.775845\pi\)
−0.762127 + 0.647427i \(0.775845\pi\)
\(314\) −442.533 −0.0795336
\(315\) 0 0
\(316\) 844.917 0.150412
\(317\) 10551.7 1.86953 0.934766 0.355264i \(-0.115609\pi\)
0.934766 + 0.355264i \(0.115609\pi\)
\(318\) −3042.67 −0.536554
\(319\) −2390.65 −0.419595
\(320\) 0 0
\(321\) −4246.12 −0.738304
\(322\) 1399.86 0.242270
\(323\) 1749.28 0.301339
\(324\) 265.858 0.0455861
\(325\) 0 0
\(326\) 9733.21 1.65360
\(327\) 2725.16 0.460861
\(328\) 485.900 0.0817968
\(329\) −13680.3 −2.29247
\(330\) 0 0
\(331\) 6743.17 1.11975 0.559876 0.828576i \(-0.310849\pi\)
0.559876 + 0.828576i \(0.310849\pi\)
\(332\) 3002.91 0.496405
\(333\) −336.478 −0.0553720
\(334\) 874.265 0.143227
\(335\) 0 0
\(336\) −7257.50 −1.17836
\(337\) −8437.26 −1.36382 −0.681909 0.731437i \(-0.738850\pi\)
−0.681909 + 0.731437i \(0.738850\pi\)
\(338\) 5013.62 0.806819
\(339\) −6149.82 −0.985287
\(340\) 0 0
\(341\) 5754.26 0.913814
\(342\) −436.389 −0.0689978
\(343\) 7314.45 1.15144
\(344\) −5190.78 −0.813571
\(345\) 0 0
\(346\) −6502.25 −1.01030
\(347\) −1848.85 −0.286027 −0.143013 0.989721i \(-0.545679\pi\)
−0.143013 + 0.989721i \(0.545679\pi\)
\(348\) −748.826 −0.115349
\(349\) −1148.38 −0.176136 −0.0880678 0.996114i \(-0.528069\pi\)
−0.0880678 + 0.996114i \(0.528069\pi\)
\(350\) 0 0
\(351\) −1640.04 −0.249399
\(352\) −4407.54 −0.667393
\(353\) −5753.60 −0.867516 −0.433758 0.901029i \(-0.642813\pi\)
−0.433758 + 0.901029i \(0.642813\pi\)
\(354\) 3435.85 0.515856
\(355\) 0 0
\(356\) −3478.09 −0.517804
\(357\) −11064.4 −1.64030
\(358\) 1936.83 0.285935
\(359\) 5452.01 0.801521 0.400761 0.916183i \(-0.368746\pi\)
0.400761 + 0.916183i \(0.368746\pi\)
\(360\) 0 0
\(361\) −6650.61 −0.969619
\(362\) −6590.31 −0.956848
\(363\) −1028.41 −0.148698
\(364\) −6067.91 −0.873749
\(365\) 0 0
\(366\) 6238.47 0.890956
\(367\) 8385.93 1.19276 0.596379 0.802703i \(-0.296606\pi\)
0.596379 + 0.802703i \(0.296606\pi\)
\(368\) −1088.40 −0.154176
\(369\) −275.965 −0.0389327
\(370\) 0 0
\(371\) −9190.05 −1.28605
\(372\) 1802.41 0.251212
\(373\) 2728.30 0.378730 0.189365 0.981907i \(-0.439357\pi\)
0.189365 + 0.981907i \(0.439357\pi\)
\(374\) −12795.1 −1.76903
\(375\) 0 0
\(376\) 7122.81 0.976944
\(377\) 4619.41 0.631065
\(378\) 2760.21 0.375582
\(379\) 3348.99 0.453895 0.226947 0.973907i \(-0.427125\pi\)
0.226947 + 0.973907i \(0.427125\pi\)
\(380\) 0 0
\(381\) 844.932 0.113615
\(382\) −14506.2 −1.94294
\(383\) 10430.3 1.39155 0.695774 0.718261i \(-0.255062\pi\)
0.695774 + 0.718261i \(0.255062\pi\)
\(384\) 5026.97 0.668051
\(385\) 0 0
\(386\) 9.99766 0.00131831
\(387\) 2948.08 0.387233
\(388\) 1710.03 0.223746
\(389\) 9827.23 1.28088 0.640438 0.768010i \(-0.278753\pi\)
0.640438 + 0.768010i \(0.278753\pi\)
\(390\) 0 0
\(391\) −1659.32 −0.214617
\(392\) −9243.73 −1.19102
\(393\) 729.168 0.0935921
\(394\) 1913.58 0.244682
\(395\) 0 0
\(396\) 928.602 0.117838
\(397\) −436.382 −0.0551672 −0.0275836 0.999619i \(-0.508781\pi\)
−0.0275836 + 0.999619i \(0.508781\pi\)
\(398\) −10247.1 −1.29055
\(399\) −1318.07 −0.165378
\(400\) 0 0
\(401\) −14501.5 −1.80591 −0.902955 0.429736i \(-0.858607\pi\)
−0.902955 + 0.429736i \(0.858607\pi\)
\(402\) 2588.60 0.321163
\(403\) −11118.8 −1.37436
\(404\) 1139.19 0.140289
\(405\) 0 0
\(406\) −7774.51 −0.950351
\(407\) −1175.26 −0.143134
\(408\) 5760.78 0.699022
\(409\) −12058.4 −1.45782 −0.728911 0.684609i \(-0.759973\pi\)
−0.728911 + 0.684609i \(0.759973\pi\)
\(410\) 0 0
\(411\) 2728.16 0.327421
\(412\) −2529.76 −0.302505
\(413\) 10377.6 1.23644
\(414\) 413.947 0.0491410
\(415\) 0 0
\(416\) 8516.60 1.00375
\(417\) −6148.57 −0.722054
\(418\) −1524.24 −0.178356
\(419\) 6042.95 0.704577 0.352288 0.935892i \(-0.385404\pi\)
0.352288 + 0.935892i \(0.385404\pi\)
\(420\) 0 0
\(421\) −9994.67 −1.15703 −0.578516 0.815671i \(-0.696368\pi\)
−0.578516 + 0.815671i \(0.696368\pi\)
\(422\) −169.659 −0.0195708
\(423\) −4045.36 −0.464994
\(424\) 4784.90 0.548054
\(425\) 0 0
\(426\) 5030.07 0.572083
\(427\) 18842.6 2.13550
\(428\) −4645.55 −0.524652
\(429\) −5728.42 −0.644687
\(430\) 0 0
\(431\) 9327.32 1.04242 0.521208 0.853430i \(-0.325482\pi\)
0.521208 + 0.853430i \(0.325482\pi\)
\(432\) −2146.09 −0.239013
\(433\) −7861.22 −0.872485 −0.436243 0.899829i \(-0.643691\pi\)
−0.436243 + 0.899829i \(0.643691\pi\)
\(434\) 18713.1 2.06972
\(435\) 0 0
\(436\) 2981.51 0.327496
\(437\) −197.670 −0.0216380
\(438\) −192.755 −0.0210279
\(439\) 7412.06 0.805828 0.402914 0.915238i \(-0.367997\pi\)
0.402914 + 0.915238i \(0.367997\pi\)
\(440\) 0 0
\(441\) 5249.93 0.566886
\(442\) 24723.6 2.66060
\(443\) −3043.66 −0.326430 −0.163215 0.986591i \(-0.552186\pi\)
−0.163215 + 0.986591i \(0.552186\pi\)
\(444\) −368.129 −0.0393483
\(445\) 0 0
\(446\) 18317.9 1.94480
\(447\) 10803.4 1.14314
\(448\) 5019.81 0.529383
\(449\) −9547.87 −1.00355 −0.501773 0.864999i \(-0.667319\pi\)
−0.501773 + 0.864999i \(0.667319\pi\)
\(450\) 0 0
\(451\) −963.902 −0.100639
\(452\) −6728.31 −0.700162
\(453\) 4150.14 0.430443
\(454\) −16045.9 −1.65875
\(455\) 0 0
\(456\) 686.266 0.0704766
\(457\) 13401.9 1.37180 0.685901 0.727695i \(-0.259408\pi\)
0.685901 + 0.727695i \(0.259408\pi\)
\(458\) −7005.00 −0.714677
\(459\) −3271.81 −0.332712
\(460\) 0 0
\(461\) −4137.03 −0.417962 −0.208981 0.977920i \(-0.567015\pi\)
−0.208981 + 0.977920i \(0.567015\pi\)
\(462\) 9640.99 0.970865
\(463\) −13976.3 −1.40288 −0.701439 0.712729i \(-0.747459\pi\)
−0.701439 + 0.712729i \(0.747459\pi\)
\(464\) 6044.75 0.604786
\(465\) 0 0
\(466\) 21780.9 2.16519
\(467\) −10796.5 −1.06982 −0.534908 0.844910i \(-0.679654\pi\)
−0.534908 + 0.844910i \(0.679654\pi\)
\(468\) −1794.32 −0.177227
\(469\) 7818.60 0.769785
\(470\) 0 0
\(471\) −395.248 −0.0386668
\(472\) −5403.21 −0.526913
\(473\) 10297.2 1.00098
\(474\) 2593.98 0.251362
\(475\) 0 0
\(476\) −12105.2 −1.16563
\(477\) −2717.56 −0.260856
\(478\) 7505.85 0.718221
\(479\) 14568.4 1.38966 0.694830 0.719174i \(-0.255479\pi\)
0.694830 + 0.719174i \(0.255479\pi\)
\(480\) 0 0
\(481\) 2270.94 0.215272
\(482\) −8037.89 −0.759577
\(483\) 1250.28 0.117784
\(484\) −1125.15 −0.105668
\(485\) 0 0
\(486\) 816.212 0.0761814
\(487\) −11456.6 −1.06601 −0.533004 0.846113i \(-0.678937\pi\)
−0.533004 + 0.846113i \(0.678937\pi\)
\(488\) −9810.61 −0.910052
\(489\) 8693.21 0.803928
\(490\) 0 0
\(491\) −19666.5 −1.80761 −0.903804 0.427948i \(-0.859237\pi\)
−0.903804 + 0.427948i \(0.859237\pi\)
\(492\) −301.924 −0.0276662
\(493\) 9215.48 0.841875
\(494\) 2945.26 0.268246
\(495\) 0 0
\(496\) −14549.6 −1.31713
\(497\) 15192.8 1.37121
\(498\) 9219.26 0.829568
\(499\) 8379.31 0.751722 0.375861 0.926676i \(-0.377347\pi\)
0.375861 + 0.926676i \(0.377347\pi\)
\(500\) 0 0
\(501\) 780.850 0.0696323
\(502\) −2057.87 −0.182962
\(503\) −15678.1 −1.38976 −0.694881 0.719124i \(-0.744543\pi\)
−0.694881 + 0.719124i \(0.744543\pi\)
\(504\) −4340.71 −0.383632
\(505\) 0 0
\(506\) 1445.85 0.127028
\(507\) 4477.91 0.392251
\(508\) 924.413 0.0807366
\(509\) −17037.3 −1.48363 −0.741813 0.670606i \(-0.766034\pi\)
−0.741813 + 0.670606i \(0.766034\pi\)
\(510\) 0 0
\(511\) −582.197 −0.0504009
\(512\) 1067.93 0.0921798
\(513\) −389.761 −0.0335446
\(514\) −1028.20 −0.0882334
\(515\) 0 0
\(516\) 3225.40 0.275175
\(517\) −14129.8 −1.20199
\(518\) −3822.02 −0.324189
\(519\) −5807.49 −0.491176
\(520\) 0 0
\(521\) −8776.12 −0.737982 −0.368991 0.929433i \(-0.620297\pi\)
−0.368991 + 0.929433i \(0.620297\pi\)
\(522\) −2298.97 −0.192765
\(523\) 11120.4 0.929753 0.464877 0.885375i \(-0.346099\pi\)
0.464877 + 0.885375i \(0.346099\pi\)
\(524\) 797.759 0.0665082
\(525\) 0 0
\(526\) 953.385 0.0790296
\(527\) −22181.5 −1.83348
\(528\) −7495.95 −0.617840
\(529\) −11979.5 −0.984589
\(530\) 0 0
\(531\) 3068.73 0.250794
\(532\) −1442.06 −0.117521
\(533\) 1862.53 0.151360
\(534\) −10678.1 −0.865330
\(535\) 0 0
\(536\) −4070.83 −0.328047
\(537\) 1729.88 0.139013
\(538\) 8149.39 0.653058
\(539\) 18337.2 1.46538
\(540\) 0 0
\(541\) 21730.6 1.72693 0.863467 0.504405i \(-0.168288\pi\)
0.863467 + 0.504405i \(0.168288\pi\)
\(542\) −587.626 −0.0465695
\(543\) −5886.13 −0.465190
\(544\) 16990.2 1.33906
\(545\) 0 0
\(546\) −18629.1 −1.46017
\(547\) −6926.17 −0.541392 −0.270696 0.962665i \(-0.587254\pi\)
−0.270696 + 0.962665i \(0.587254\pi\)
\(548\) 2984.79 0.232671
\(549\) 5571.89 0.433155
\(550\) 0 0
\(551\) 1097.82 0.0848793
\(552\) −650.973 −0.0501943
\(553\) 7834.85 0.602480
\(554\) −26223.2 −2.01104
\(555\) 0 0
\(556\) −6726.95 −0.513104
\(557\) −6589.22 −0.501246 −0.250623 0.968085i \(-0.580636\pi\)
−0.250623 + 0.968085i \(0.580636\pi\)
\(558\) 5533.59 0.419813
\(559\) −19897.0 −1.50547
\(560\) 0 0
\(561\) −11427.9 −0.860047
\(562\) 1962.17 0.147276
\(563\) 3839.63 0.287426 0.143713 0.989619i \(-0.454096\pi\)
0.143713 + 0.989619i \(0.454096\pi\)
\(564\) −4425.90 −0.330433
\(565\) 0 0
\(566\) −19808.5 −1.47105
\(567\) 2465.28 0.182596
\(568\) −7910.28 −0.584345
\(569\) −20874.0 −1.53794 −0.768968 0.639287i \(-0.779229\pi\)
−0.768968 + 0.639287i \(0.779229\pi\)
\(570\) 0 0
\(571\) 21175.9 1.55199 0.775994 0.630740i \(-0.217248\pi\)
0.775994 + 0.630740i \(0.217248\pi\)
\(572\) −6267.28 −0.458126
\(573\) −12956.3 −0.944599
\(574\) −3134.66 −0.227941
\(575\) 0 0
\(576\) 1484.39 0.107378
\(577\) −14924.2 −1.07678 −0.538391 0.842695i \(-0.680968\pi\)
−0.538391 + 0.842695i \(0.680968\pi\)
\(578\) 32820.1 2.36183
\(579\) 8.92941 0.000640922 0
\(580\) 0 0
\(581\) 27845.8 1.98836
\(582\) 5249.96 0.373914
\(583\) −9492.00 −0.674303
\(584\) 303.127 0.0214785
\(585\) 0 0
\(586\) 5406.92 0.381156
\(587\) 25218.0 1.77318 0.886592 0.462552i \(-0.153066\pi\)
0.886592 + 0.462552i \(0.153066\pi\)
\(588\) 5743.78 0.402839
\(589\) −2642.42 −0.184854
\(590\) 0 0
\(591\) 1709.11 0.118957
\(592\) 2971.65 0.206308
\(593\) 5011.77 0.347063 0.173532 0.984828i \(-0.444482\pi\)
0.173532 + 0.984828i \(0.444482\pi\)
\(594\) 2850.90 0.196926
\(595\) 0 0
\(596\) 11819.7 0.812337
\(597\) −9152.19 −0.627428
\(598\) −2793.79 −0.191048
\(599\) 4943.41 0.337199 0.168600 0.985685i \(-0.446075\pi\)
0.168600 + 0.985685i \(0.446075\pi\)
\(600\) 0 0
\(601\) −24334.8 −1.65164 −0.825821 0.563932i \(-0.809288\pi\)
−0.825821 + 0.563932i \(0.809288\pi\)
\(602\) 33486.9 2.26715
\(603\) 2312.01 0.156140
\(604\) 4540.54 0.305881
\(605\) 0 0
\(606\) 3497.42 0.234444
\(607\) 28973.8 1.93741 0.968707 0.248207i \(-0.0798415\pi\)
0.968707 + 0.248207i \(0.0798415\pi\)
\(608\) 2023.99 0.135006
\(609\) −6943.81 −0.462032
\(610\) 0 0
\(611\) 27302.8 1.80778
\(612\) −3579.58 −0.236431
\(613\) 15139.1 0.997490 0.498745 0.866749i \(-0.333794\pi\)
0.498745 + 0.866749i \(0.333794\pi\)
\(614\) 787.571 0.0517651
\(615\) 0 0
\(616\) −15161.4 −0.991673
\(617\) 13894.8 0.906617 0.453309 0.891354i \(-0.350244\pi\)
0.453309 + 0.891354i \(0.350244\pi\)
\(618\) −7766.61 −0.505532
\(619\) −4589.69 −0.298021 −0.149011 0.988836i \(-0.547609\pi\)
−0.149011 + 0.988836i \(0.547609\pi\)
\(620\) 0 0
\(621\) 369.717 0.0238909
\(622\) −6030.07 −0.388720
\(623\) −32252.0 −2.07408
\(624\) 14484.3 0.929223
\(625\) 0 0
\(626\) −28351.2 −1.81013
\(627\) −1361.37 −0.0867114
\(628\) −432.428 −0.0274773
\(629\) 4530.41 0.287185
\(630\) 0 0
\(631\) 3005.77 0.189632 0.0948160 0.995495i \(-0.469774\pi\)
0.0948160 + 0.995495i \(0.469774\pi\)
\(632\) −4079.29 −0.256749
\(633\) −151.531 −0.00951473
\(634\) 35442.0 2.22016
\(635\) 0 0
\(636\) −2973.19 −0.185369
\(637\) −35432.6 −2.20391
\(638\) −8029.96 −0.498290
\(639\) 4492.60 0.278129
\(640\) 0 0
\(641\) −5631.47 −0.347004 −0.173502 0.984834i \(-0.555508\pi\)
−0.173502 + 0.984834i \(0.555508\pi\)
\(642\) −14262.3 −0.876773
\(643\) 11305.1 0.693358 0.346679 0.937984i \(-0.387309\pi\)
0.346679 + 0.937984i \(0.387309\pi\)
\(644\) 1367.89 0.0836997
\(645\) 0 0
\(646\) 5875.64 0.357855
\(647\) −8614.30 −0.523436 −0.261718 0.965144i \(-0.584289\pi\)
−0.261718 + 0.965144i \(0.584289\pi\)
\(648\) −1283.58 −0.0778142
\(649\) 10718.6 0.648291
\(650\) 0 0
\(651\) 16713.6 1.00623
\(652\) 9510.96 0.571285
\(653\) 12639.8 0.757479 0.378739 0.925503i \(-0.376358\pi\)
0.378739 + 0.925503i \(0.376358\pi\)
\(654\) 9153.53 0.547296
\(655\) 0 0
\(656\) 2437.22 0.145057
\(657\) −172.159 −0.0102231
\(658\) −45950.9 −2.72242
\(659\) 13640.4 0.806306 0.403153 0.915133i \(-0.367914\pi\)
0.403153 + 0.915133i \(0.367914\pi\)
\(660\) 0 0
\(661\) −17052.0 −1.00340 −0.501699 0.865042i \(-0.667291\pi\)
−0.501699 + 0.865042i \(0.667291\pi\)
\(662\) 22649.6 1.32976
\(663\) 22081.9 1.29350
\(664\) −14498.2 −0.847348
\(665\) 0 0
\(666\) −1130.20 −0.0657570
\(667\) −1041.36 −0.0604521
\(668\) 854.302 0.0494820
\(669\) 16360.7 0.945500
\(670\) 0 0
\(671\) 19461.7 1.11969
\(672\) −12802.0 −0.734892
\(673\) −16419.7 −0.940467 −0.470234 0.882542i \(-0.655830\pi\)
−0.470234 + 0.882542i \(0.655830\pi\)
\(674\) −28339.9 −1.61960
\(675\) 0 0
\(676\) 4899.14 0.278740
\(677\) −8670.47 −0.492221 −0.246110 0.969242i \(-0.579153\pi\)
−0.246110 + 0.969242i \(0.579153\pi\)
\(678\) −20656.6 −1.17008
\(679\) 15856.9 0.896220
\(680\) 0 0
\(681\) −14331.4 −0.806434
\(682\) 19328.0 1.08520
\(683\) 5973.36 0.334647 0.167324 0.985902i \(-0.446488\pi\)
0.167324 + 0.985902i \(0.446488\pi\)
\(684\) −426.425 −0.0238374
\(685\) 0 0
\(686\) 24568.5 1.36739
\(687\) −6256.52 −0.347454
\(688\) −26036.4 −1.44277
\(689\) 18341.2 1.01414
\(690\) 0 0
\(691\) −15316.3 −0.843212 −0.421606 0.906779i \(-0.638533\pi\)
−0.421606 + 0.906779i \(0.638533\pi\)
\(692\) −6353.78 −0.349038
\(693\) 8610.85 0.472005
\(694\) −6210.09 −0.339671
\(695\) 0 0
\(696\) 3615.36 0.196897
\(697\) 3715.65 0.201923
\(698\) −3857.29 −0.209170
\(699\) 19453.6 1.05265
\(700\) 0 0
\(701\) 34583.1 1.86332 0.931660 0.363333i \(-0.118361\pi\)
0.931660 + 0.363333i \(0.118361\pi\)
\(702\) −5508.74 −0.296174
\(703\) 539.695 0.0289545
\(704\) 5184.74 0.277567
\(705\) 0 0
\(706\) −19325.8 −1.03022
\(707\) 10563.6 0.561929
\(708\) 3357.39 0.178218
\(709\) 11194.1 0.592955 0.296477 0.955040i \(-0.404188\pi\)
0.296477 + 0.955040i \(0.404188\pi\)
\(710\) 0 0
\(711\) 2316.81 0.122204
\(712\) 16792.4 0.883877
\(713\) 2506.53 0.131655
\(714\) −37164.1 −1.94794
\(715\) 0 0
\(716\) 1892.61 0.0987850
\(717\) 6703.85 0.349177
\(718\) 18312.8 0.951847
\(719\) −15491.9 −0.803549 −0.401774 0.915739i \(-0.631606\pi\)
−0.401774 + 0.915739i \(0.631606\pi\)
\(720\) 0 0
\(721\) −23458.2 −1.21169
\(722\) −22338.7 −1.15147
\(723\) −7179.04 −0.369283
\(724\) −6439.83 −0.330572
\(725\) 0 0
\(726\) −3454.33 −0.176587
\(727\) −6272.72 −0.320003 −0.160002 0.987117i \(-0.551150\pi\)
−0.160002 + 0.987117i \(0.551150\pi\)
\(728\) 29296.1 1.49146
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −39693.6 −2.00837
\(732\) 6096.02 0.307808
\(733\) 24980.5 1.25877 0.629383 0.777095i \(-0.283308\pi\)
0.629383 + 0.777095i \(0.283308\pi\)
\(734\) 28167.5 1.41646
\(735\) 0 0
\(736\) −1919.90 −0.0961530
\(737\) 8075.49 0.403615
\(738\) −926.938 −0.0462345
\(739\) −30660.7 −1.52621 −0.763107 0.646272i \(-0.776327\pi\)
−0.763107 + 0.646272i \(0.776327\pi\)
\(740\) 0 0
\(741\) 2630.56 0.130413
\(742\) −30868.5 −1.52725
\(743\) −17205.7 −0.849551 −0.424776 0.905299i \(-0.639647\pi\)
−0.424776 + 0.905299i \(0.639647\pi\)
\(744\) −8702.12 −0.428811
\(745\) 0 0
\(746\) 9164.09 0.449760
\(747\) 8234.18 0.403310
\(748\) −12502.9 −0.611165
\(749\) −43077.8 −2.10151
\(750\) 0 0
\(751\) 18397.1 0.893901 0.446950 0.894559i \(-0.352510\pi\)
0.446950 + 0.894559i \(0.352510\pi\)
\(752\) 35727.2 1.73250
\(753\) −1837.98 −0.0889506
\(754\) 15516.1 0.749422
\(755\) 0 0
\(756\) 2697.19 0.129756
\(757\) 22305.1 1.07093 0.535465 0.844557i \(-0.320136\pi\)
0.535465 + 0.844557i \(0.320136\pi\)
\(758\) 11248.9 0.539023
\(759\) 1291.36 0.0617569
\(760\) 0 0
\(761\) 14458.5 0.688727 0.344364 0.938836i \(-0.388095\pi\)
0.344364 + 0.938836i \(0.388095\pi\)
\(762\) 2838.04 0.134923
\(763\) 27647.3 1.31179
\(764\) −14175.0 −0.671249
\(765\) 0 0
\(766\) 35034.3 1.65253
\(767\) −20711.3 −0.975022
\(768\) 12926.7 0.607361
\(769\) −39897.6 −1.87093 −0.935463 0.353424i \(-0.885017\pi\)
−0.935463 + 0.353424i \(0.885017\pi\)
\(770\) 0 0
\(771\) −918.337 −0.0428964
\(772\) 9.76938 0.000455450 0
\(773\) 20070.2 0.933863 0.466931 0.884294i \(-0.345359\pi\)
0.466931 + 0.884294i \(0.345359\pi\)
\(774\) 9902.30 0.459859
\(775\) 0 0
\(776\) −8256.08 −0.381928
\(777\) −3413.63 −0.157611
\(778\) 33008.7 1.52110
\(779\) 442.635 0.0203582
\(780\) 0 0
\(781\) 15692.0 0.718953
\(782\) −5573.47 −0.254868
\(783\) −2053.33 −0.0937164
\(784\) −46365.5 −2.11213
\(785\) 0 0
\(786\) 2449.20 0.111145
\(787\) 10733.0 0.486137 0.243069 0.970009i \(-0.421846\pi\)
0.243069 + 0.970009i \(0.421846\pi\)
\(788\) 1869.89 0.0845329
\(789\) 851.516 0.0384218
\(790\) 0 0
\(791\) −62391.1 −2.80452
\(792\) −4483.33 −0.201147
\(793\) −37605.5 −1.68400
\(794\) −1465.76 −0.0655139
\(795\) 0 0
\(796\) −10013.1 −0.445861
\(797\) −14335.8 −0.637140 −0.318570 0.947899i \(-0.603203\pi\)
−0.318570 + 0.947899i \(0.603203\pi\)
\(798\) −4427.26 −0.196395
\(799\) 54467.7 2.41167
\(800\) 0 0
\(801\) −9537.13 −0.420697
\(802\) −48709.0 −2.14461
\(803\) −601.325 −0.0264263
\(804\) 2529.49 0.110956
\(805\) 0 0
\(806\) −37347.1 −1.63213
\(807\) 7278.63 0.317497
\(808\) −5500.03 −0.239468
\(809\) 20920.7 0.909187 0.454593 0.890699i \(-0.349785\pi\)
0.454593 + 0.890699i \(0.349785\pi\)
\(810\) 0 0
\(811\) 12816.5 0.554931 0.277465 0.960736i \(-0.410506\pi\)
0.277465 + 0.960736i \(0.410506\pi\)
\(812\) −7596.99 −0.328328
\(813\) −524.838 −0.0226407
\(814\) −3947.59 −0.169979
\(815\) 0 0
\(816\) 28895.4 1.23963
\(817\) −4728.59 −0.202488
\(818\) −40502.9 −1.73124
\(819\) −16638.6 −0.709889
\(820\) 0 0
\(821\) 7253.55 0.308344 0.154172 0.988044i \(-0.450729\pi\)
0.154172 + 0.988044i \(0.450729\pi\)
\(822\) 9163.61 0.388829
\(823\) 35288.1 1.49461 0.747307 0.664479i \(-0.231346\pi\)
0.747307 + 0.664479i \(0.231346\pi\)
\(824\) 12213.8 0.516367
\(825\) 0 0
\(826\) 34857.3 1.46833
\(827\) −32205.9 −1.35418 −0.677092 0.735899i \(-0.736760\pi\)
−0.677092 + 0.735899i \(0.736760\pi\)
\(828\) 404.495 0.0169773
\(829\) 29993.3 1.25659 0.628294 0.777976i \(-0.283754\pi\)
0.628294 + 0.777976i \(0.283754\pi\)
\(830\) 0 0
\(831\) −23421.2 −0.977706
\(832\) −10018.4 −0.417457
\(833\) −70686.2 −2.94013
\(834\) −20652.4 −0.857476
\(835\) 0 0
\(836\) −1489.44 −0.0616187
\(837\) 4942.33 0.204100
\(838\) 20297.7 0.836720
\(839\) −35608.7 −1.46526 −0.732628 0.680629i \(-0.761706\pi\)
−0.732628 + 0.680629i \(0.761706\pi\)
\(840\) 0 0
\(841\) −18605.5 −0.762865
\(842\) −33571.1 −1.37403
\(843\) 1752.51 0.0716011
\(844\) −165.785 −0.00676134
\(845\) 0 0
\(846\) −13588.0 −0.552203
\(847\) −10433.4 −0.423255
\(848\) 24000.5 0.971911
\(849\) −17691.9 −0.715177
\(850\) 0 0
\(851\) −511.940 −0.0206217
\(852\) 4915.21 0.197644
\(853\) −11229.3 −0.450744 −0.225372 0.974273i \(-0.572360\pi\)
−0.225372 + 0.974273i \(0.572360\pi\)
\(854\) 63290.5 2.53601
\(855\) 0 0
\(856\) 22428.9 0.895565
\(857\) 22136.1 0.882327 0.441164 0.897427i \(-0.354566\pi\)
0.441164 + 0.897427i \(0.354566\pi\)
\(858\) −19241.2 −0.765598
\(859\) −820.727 −0.0325994 −0.0162997 0.999867i \(-0.505189\pi\)
−0.0162997 + 0.999867i \(0.505189\pi\)
\(860\) 0 0
\(861\) −2799.72 −0.110818
\(862\) 31329.5 1.23792
\(863\) −245.223 −0.00967264 −0.00483632 0.999988i \(-0.501539\pi\)
−0.00483632 + 0.999988i \(0.501539\pi\)
\(864\) −3785.63 −0.149062
\(865\) 0 0
\(866\) −26405.0 −1.03612
\(867\) 29313.3 1.14825
\(868\) 18285.8 0.715048
\(869\) 8092.27 0.315894
\(870\) 0 0
\(871\) −15604.1 −0.607032
\(872\) −14394.8 −0.559026
\(873\) 4689.00 0.181785
\(874\) −663.952 −0.0256963
\(875\) 0 0
\(876\) −188.354 −0.00726471
\(877\) 37727.8 1.45265 0.726326 0.687350i \(-0.241226\pi\)
0.726326 + 0.687350i \(0.241226\pi\)
\(878\) 24896.4 0.956961
\(879\) 4829.19 0.185306
\(880\) 0 0
\(881\) 21738.8 0.831326 0.415663 0.909519i \(-0.363550\pi\)
0.415663 + 0.909519i \(0.363550\pi\)
\(882\) 17634.0 0.673205
\(883\) −44340.4 −1.68989 −0.844946 0.534852i \(-0.820368\pi\)
−0.844946 + 0.534852i \(0.820368\pi\)
\(884\) 24159.1 0.919184
\(885\) 0 0
\(886\) −10223.3 −0.387652
\(887\) −681.008 −0.0257790 −0.0128895 0.999917i \(-0.504103\pi\)
−0.0128895 + 0.999917i \(0.504103\pi\)
\(888\) 1777.34 0.0671664
\(889\) 8572.00 0.323392
\(890\) 0 0
\(891\) 2546.28 0.0957393
\(892\) 17899.7 0.671889
\(893\) 6488.58 0.243149
\(894\) 36287.6 1.35754
\(895\) 0 0
\(896\) 50999.6 1.90154
\(897\) −2495.28 −0.0928816
\(898\) −32070.3 −1.19176
\(899\) −13920.7 −0.516443
\(900\) 0 0
\(901\) 36589.8 1.35292
\(902\) −3237.65 −0.119514
\(903\) 29908.9 1.10222
\(904\) 32484.6 1.19516
\(905\) 0 0
\(906\) 13939.9 0.511173
\(907\) 5348.01 0.195786 0.0978929 0.995197i \(-0.468790\pi\)
0.0978929 + 0.995197i \(0.468790\pi\)
\(908\) −15679.5 −0.573066
\(909\) 3123.72 0.113979
\(910\) 0 0
\(911\) 14488.7 0.526930 0.263465 0.964669i \(-0.415135\pi\)
0.263465 + 0.964669i \(0.415135\pi\)
\(912\) 3442.23 0.124982
\(913\) 28760.7 1.04254
\(914\) 45015.6 1.62908
\(915\) 0 0
\(916\) −6845.05 −0.246907
\(917\) 7397.56 0.266400
\(918\) −10989.7 −0.395112
\(919\) −22546.9 −0.809308 −0.404654 0.914470i \(-0.632608\pi\)
−0.404654 + 0.914470i \(0.632608\pi\)
\(920\) 0 0
\(921\) 703.419 0.0251666
\(922\) −13895.9 −0.496351
\(923\) −30321.3 −1.08130
\(924\) 9420.85 0.335415
\(925\) 0 0
\(926\) −46944.9 −1.66599
\(927\) −6936.74 −0.245774
\(928\) 10662.7 0.377178
\(929\) −13980.7 −0.493747 −0.246874 0.969048i \(-0.579403\pi\)
−0.246874 + 0.969048i \(0.579403\pi\)
\(930\) 0 0
\(931\) −8420.65 −0.296429
\(932\) 21283.5 0.748032
\(933\) −5385.75 −0.188984
\(934\) −36264.5 −1.27046
\(935\) 0 0
\(936\) 8663.05 0.302522
\(937\) 26362.6 0.919133 0.459567 0.888143i \(-0.348005\pi\)
0.459567 + 0.888143i \(0.348005\pi\)
\(938\) 26261.9 0.914159
\(939\) −25321.8 −0.880029
\(940\) 0 0
\(941\) −14715.6 −0.509792 −0.254896 0.966968i \(-0.582041\pi\)
−0.254896 + 0.966968i \(0.582041\pi\)
\(942\) −1327.60 −0.0459188
\(943\) −419.871 −0.0144994
\(944\) −27101.9 −0.934419
\(945\) 0 0
\(946\) 34587.2 1.18872
\(947\) −11818.2 −0.405532 −0.202766 0.979227i \(-0.564993\pi\)
−0.202766 + 0.979227i \(0.564993\pi\)
\(948\) 2534.75 0.0868406
\(949\) 1161.93 0.0397448
\(950\) 0 0
\(951\) 31655.1 1.07937
\(952\) 58444.2 1.98969
\(953\) −43832.3 −1.48989 −0.744945 0.667125i \(-0.767524\pi\)
−0.744945 + 0.667125i \(0.767524\pi\)
\(954\) −9128.00 −0.309780
\(955\) 0 0
\(956\) 7334.46 0.248131
\(957\) −7171.95 −0.242253
\(958\) 48933.8 1.65029
\(959\) 27677.7 0.931971
\(960\) 0 0
\(961\) 3716.00 0.124736
\(962\) 7627.86 0.255647
\(963\) −12738.4 −0.426260
\(964\) −7854.36 −0.262419
\(965\) 0 0
\(966\) 4199.58 0.139875
\(967\) 10696.2 0.355706 0.177853 0.984057i \(-0.443085\pi\)
0.177853 + 0.984057i \(0.443085\pi\)
\(968\) 5432.27 0.180372
\(969\) 5247.83 0.173978
\(970\) 0 0
\(971\) 27933.0 0.923187 0.461593 0.887092i \(-0.347278\pi\)
0.461593 + 0.887092i \(0.347278\pi\)
\(972\) 797.575 0.0263192
\(973\) −62378.4 −2.05525
\(974\) −38481.4 −1.26594
\(975\) 0 0
\(976\) −49208.9 −1.61387
\(977\) 24341.7 0.797094 0.398547 0.917148i \(-0.369515\pi\)
0.398547 + 0.917148i \(0.369515\pi\)
\(978\) 29199.6 0.954704
\(979\) −33311.7 −1.08748
\(980\) 0 0
\(981\) 8175.48 0.266078
\(982\) −66057.7 −2.14662
\(983\) 12553.0 0.407301 0.203651 0.979044i \(-0.434719\pi\)
0.203651 + 0.979044i \(0.434719\pi\)
\(984\) 1457.70 0.0472254
\(985\) 0 0
\(986\) 30953.9 0.999769
\(987\) −41041.0 −1.32356
\(988\) 2878.01 0.0926737
\(989\) 4485.41 0.144214
\(990\) 0 0
\(991\) −45631.8 −1.46271 −0.731353 0.681999i \(-0.761111\pi\)
−0.731353 + 0.681999i \(0.761111\pi\)
\(992\) −25665.0 −0.821437
\(993\) 20229.5 0.646490
\(994\) 51031.0 1.62838
\(995\) 0 0
\(996\) 9008.74 0.286599
\(997\) −34499.0 −1.09588 −0.547941 0.836517i \(-0.684588\pi\)
−0.547941 + 0.836517i \(0.684588\pi\)
\(998\) 28145.3 0.892708
\(999\) −1009.43 −0.0319690
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 75.4.a.d.1.2 2
3.2 odd 2 225.4.a.n.1.1 2
4.3 odd 2 1200.4.a.bl.1.1 2
5.2 odd 4 75.4.b.c.49.3 4
5.3 odd 4 75.4.b.c.49.2 4
5.4 even 2 75.4.a.e.1.1 yes 2
15.2 even 4 225.4.b.h.199.2 4
15.8 even 4 225.4.b.h.199.3 4
15.14 odd 2 225.4.a.j.1.2 2
20.3 even 4 1200.4.f.v.49.2 4
20.7 even 4 1200.4.f.v.49.3 4
20.19 odd 2 1200.4.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.a.d.1.2 2 1.1 even 1 trivial
75.4.a.e.1.1 yes 2 5.4 even 2
75.4.b.c.49.2 4 5.3 odd 4
75.4.b.c.49.3 4 5.2 odd 4
225.4.a.j.1.2 2 15.14 odd 2
225.4.a.n.1.1 2 3.2 odd 2
225.4.b.h.199.2 4 15.2 even 4
225.4.b.h.199.3 4 15.8 even 4
1200.4.a.bl.1.1 2 4.3 odd 2
1200.4.a.bu.1.2 2 20.19 odd 2
1200.4.f.v.49.2 4 20.3 even 4
1200.4.f.v.49.3 4 20.7 even 4