Properties

Label 825.4.a.bc.1.5
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: \( x^{7} - 3x^{6} - 44x^{5} + 118x^{4} + 515x^{3} - 1279x^{2} - 892x + 1840 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.62783\) 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)
 
\(f(q)\) \(=\) \(q+2.62783 q^{2} +3.00000 q^{3} -1.09451 q^{4} +7.88349 q^{6} -24.0582 q^{7} -23.8988 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.62783 q^{2} +3.00000 q^{3} -1.09451 q^{4} +7.88349 q^{6} -24.0582 q^{7} -23.8988 q^{8} +9.00000 q^{9} -11.0000 q^{11} -3.28354 q^{12} +70.7912 q^{13} -63.2207 q^{14} -54.0459 q^{16} -27.9494 q^{17} +23.6505 q^{18} +50.9926 q^{19} -72.1745 q^{21} -28.9061 q^{22} +201.498 q^{23} -71.6965 q^{24} +186.027 q^{26} +27.0000 q^{27} +26.3320 q^{28} +126.395 q^{29} +30.1877 q^{31} +49.1672 q^{32} -33.0000 q^{33} -73.4462 q^{34} -9.85063 q^{36} -7.32183 q^{37} +134.000 q^{38} +212.374 q^{39} +366.098 q^{41} -189.662 q^{42} -33.4661 q^{43} +12.0397 q^{44} +529.503 q^{46} -14.3267 q^{47} -162.138 q^{48} +235.795 q^{49} -83.8482 q^{51} -77.4820 q^{52} -154.671 q^{53} +70.9514 q^{54} +574.962 q^{56} +152.978 q^{57} +332.145 q^{58} +611.926 q^{59} -83.1222 q^{61} +79.3282 q^{62} -216.523 q^{63} +561.570 q^{64} -86.7184 q^{66} +737.743 q^{67} +30.5910 q^{68} +604.495 q^{69} -930.806 q^{71} -215.089 q^{72} +491.999 q^{73} -19.2405 q^{74} -55.8121 q^{76} +264.640 q^{77} +558.082 q^{78} +687.955 q^{79} +81.0000 q^{81} +962.043 q^{82} -712.253 q^{83} +78.9960 q^{84} -87.9431 q^{86} +379.186 q^{87} +262.887 q^{88} -1332.36 q^{89} -1703.11 q^{91} -220.543 q^{92} +90.5631 q^{93} -37.6482 q^{94} +147.502 q^{96} +598.852 q^{97} +619.630 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 3 q^{2} + 21 q^{3} + 41 q^{4} + 9 q^{6} + 50 q^{7} + 21 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 3 q^{2} + 21 q^{3} + 41 q^{4} + 9 q^{6} + 50 q^{7} + 21 q^{8} + 63 q^{9} - 77 q^{11} + 123 q^{12} + 24 q^{13} + 142 q^{14} + 181 q^{16} + 38 q^{17} + 27 q^{18} + 26 q^{19} + 150 q^{21} - 33 q^{22} + 228 q^{23} + 63 q^{24} + 476 q^{26} + 189 q^{27} + 840 q^{28} + 572 q^{29} - 140 q^{31} + 991 q^{32} - 231 q^{33} - 806 q^{34} + 369 q^{36} - 104 q^{37} + 498 q^{38} + 72 q^{39} + 896 q^{41} + 426 q^{42} + 614 q^{43} - 451 q^{44} - 344 q^{46} + 520 q^{47} + 543 q^{48} + 295 q^{49} + 114 q^{51} - 26 q^{52} + 380 q^{53} + 81 q^{54} + 1522 q^{56} + 78 q^{57} + 1600 q^{58} + 1316 q^{59} - 386 q^{61} + 440 q^{62} + 450 q^{63} + 869 q^{64} - 99 q^{66} + 348 q^{67} + 332 q^{68} + 684 q^{69} + 804 q^{71} + 189 q^{72} + 468 q^{73} - 748 q^{74} - 1698 q^{76} - 550 q^{77} + 1428 q^{78} - 374 q^{79} + 567 q^{81} - 620 q^{82} + 3128 q^{83} + 2520 q^{84} - 2534 q^{86} + 1716 q^{87} - 231 q^{88} + 694 q^{89} - 3376 q^{91} - 1184 q^{92} - 420 q^{93} - 2920 q^{94} + 2973 q^{96} - 8 q^{97} + 4211 q^{98} - 693 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.62783 0.929078 0.464539 0.885553i \(-0.346220\pi\)
0.464539 + 0.885553i \(0.346220\pi\)
\(3\) 3.00000 0.577350
\(4\) −1.09451 −0.136814
\(5\) 0 0
\(6\) 7.88349 0.536403
\(7\) −24.0582 −1.29902 −0.649509 0.760354i \(-0.725026\pi\)
−0.649509 + 0.760354i \(0.725026\pi\)
\(8\) −23.8988 −1.05619
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) −3.28354 −0.0789898
\(13\) 70.7912 1.51030 0.755152 0.655549i \(-0.227563\pi\)
0.755152 + 0.655549i \(0.227563\pi\)
\(14\) −63.2207 −1.20689
\(15\) 0 0
\(16\) −54.0459 −0.844468
\(17\) −27.9494 −0.398749 −0.199374 0.979923i \(-0.563891\pi\)
−0.199374 + 0.979923i \(0.563891\pi\)
\(18\) 23.6505 0.309693
\(19\) 50.9926 0.615711 0.307855 0.951433i \(-0.400389\pi\)
0.307855 + 0.951433i \(0.400389\pi\)
\(20\) 0 0
\(21\) −72.1745 −0.749989
\(22\) −28.9061 −0.280128
\(23\) 201.498 1.82675 0.913376 0.407118i \(-0.133466\pi\)
0.913376 + 0.407118i \(0.133466\pi\)
\(24\) −71.6965 −0.609791
\(25\) 0 0
\(26\) 186.027 1.40319
\(27\) 27.0000 0.192450
\(28\) 26.3320 0.177724
\(29\) 126.395 0.809345 0.404672 0.914462i \(-0.367386\pi\)
0.404672 + 0.914462i \(0.367386\pi\)
\(30\) 0 0
\(31\) 30.1877 0.174899 0.0874496 0.996169i \(-0.472128\pi\)
0.0874496 + 0.996169i \(0.472128\pi\)
\(32\) 49.1672 0.271613
\(33\) −33.0000 −0.174078
\(34\) −73.4462 −0.370468
\(35\) 0 0
\(36\) −9.85063 −0.0456048
\(37\) −7.32183 −0.0325325 −0.0162662 0.999868i \(-0.505178\pi\)
−0.0162662 + 0.999868i \(0.505178\pi\)
\(38\) 134.000 0.572043
\(39\) 212.374 0.871975
\(40\) 0 0
\(41\) 366.098 1.39451 0.697255 0.716824i \(-0.254405\pi\)
0.697255 + 0.716824i \(0.254405\pi\)
\(42\) −189.662 −0.696798
\(43\) −33.4661 −0.118687 −0.0593434 0.998238i \(-0.518901\pi\)
−0.0593434 + 0.998238i \(0.518901\pi\)
\(44\) 12.0397 0.0412511
\(45\) 0 0
\(46\) 529.503 1.69719
\(47\) −14.3267 −0.0444632 −0.0222316 0.999753i \(-0.507077\pi\)
−0.0222316 + 0.999753i \(0.507077\pi\)
\(48\) −162.138 −0.487554
\(49\) 235.795 0.687450
\(50\) 0 0
\(51\) −83.8482 −0.230218
\(52\) −77.4820 −0.206631
\(53\) −154.671 −0.400863 −0.200432 0.979708i \(-0.564234\pi\)
−0.200432 + 0.979708i \(0.564234\pi\)
\(54\) 70.9514 0.178801
\(55\) 0 0
\(56\) 574.962 1.37201
\(57\) 152.978 0.355481
\(58\) 332.145 0.751944
\(59\) 611.926 1.35027 0.675135 0.737694i \(-0.264085\pi\)
0.675135 + 0.737694i \(0.264085\pi\)
\(60\) 0 0
\(61\) −83.1222 −0.174471 −0.0872353 0.996188i \(-0.527803\pi\)
−0.0872353 + 0.996188i \(0.527803\pi\)
\(62\) 79.3282 0.162495
\(63\) −216.523 −0.433006
\(64\) 561.570 1.09682
\(65\) 0 0
\(66\) −86.7184 −0.161732
\(67\) 737.743 1.34522 0.672609 0.739998i \(-0.265174\pi\)
0.672609 + 0.739998i \(0.265174\pi\)
\(68\) 30.5910 0.0545545
\(69\) 604.495 1.05468
\(70\) 0 0
\(71\) −930.806 −1.55586 −0.777932 0.628348i \(-0.783731\pi\)
−0.777932 + 0.628348i \(0.783731\pi\)
\(72\) −215.089 −0.352063
\(73\) 491.999 0.788824 0.394412 0.918934i \(-0.370948\pi\)
0.394412 + 0.918934i \(0.370948\pi\)
\(74\) −19.2405 −0.0302252
\(75\) 0 0
\(76\) −55.8121 −0.0842380
\(77\) 264.640 0.391669
\(78\) 558.082 0.810132
\(79\) 687.955 0.979759 0.489879 0.871790i \(-0.337041\pi\)
0.489879 + 0.871790i \(0.337041\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 962.043 1.29561
\(83\) −712.253 −0.941927 −0.470963 0.882153i \(-0.656094\pi\)
−0.470963 + 0.882153i \(0.656094\pi\)
\(84\) 78.9960 0.102609
\(85\) 0 0
\(86\) −87.9431 −0.110269
\(87\) 379.186 0.467275
\(88\) 262.887 0.318453
\(89\) −1332.36 −1.58685 −0.793423 0.608670i \(-0.791703\pi\)
−0.793423 + 0.608670i \(0.791703\pi\)
\(90\) 0 0
\(91\) −1703.11 −1.96191
\(92\) −220.543 −0.249926
\(93\) 90.5631 0.100978
\(94\) −37.6482 −0.0413097
\(95\) 0 0
\(96\) 147.502 0.156816
\(97\) 598.852 0.626847 0.313424 0.949613i \(-0.398524\pi\)
0.313424 + 0.949613i \(0.398524\pi\)
\(98\) 619.630 0.638694
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) 392.723 0.386905 0.193452 0.981110i \(-0.438032\pi\)
0.193452 + 0.981110i \(0.438032\pi\)
\(102\) −220.339 −0.213890
\(103\) 889.816 0.851225 0.425613 0.904905i \(-0.360059\pi\)
0.425613 + 0.904905i \(0.360059\pi\)
\(104\) −1691.83 −1.59517
\(105\) 0 0
\(106\) −406.450 −0.372433
\(107\) −911.308 −0.823360 −0.411680 0.911328i \(-0.635058\pi\)
−0.411680 + 0.911328i \(0.635058\pi\)
\(108\) −29.5519 −0.0263299
\(109\) −1552.29 −1.36406 −0.682029 0.731325i \(-0.738902\pi\)
−0.682029 + 0.731325i \(0.738902\pi\)
\(110\) 0 0
\(111\) −21.9655 −0.0187826
\(112\) 1300.25 1.09698
\(113\) 457.867 0.381173 0.190586 0.981670i \(-0.438961\pi\)
0.190586 + 0.981670i \(0.438961\pi\)
\(114\) 401.999 0.330269
\(115\) 0 0
\(116\) −138.341 −0.110730
\(117\) 637.121 0.503435
\(118\) 1608.04 1.25451
\(119\) 672.411 0.517982
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −218.431 −0.162097
\(123\) 1098.29 0.805120
\(124\) −33.0409 −0.0239287
\(125\) 0 0
\(126\) −568.987 −0.402297
\(127\) −911.406 −0.636805 −0.318402 0.947956i \(-0.603146\pi\)
−0.318402 + 0.947956i \(0.603146\pi\)
\(128\) 1082.37 0.747416
\(129\) −100.398 −0.0685238
\(130\) 0 0
\(131\) 2552.41 1.70233 0.851165 0.524898i \(-0.175897\pi\)
0.851165 + 0.524898i \(0.175897\pi\)
\(132\) 36.1190 0.0238163
\(133\) −1226.79 −0.799820
\(134\) 1938.66 1.24981
\(135\) 0 0
\(136\) 667.958 0.421154
\(137\) −390.364 −0.243438 −0.121719 0.992565i \(-0.538841\pi\)
−0.121719 + 0.992565i \(0.538841\pi\)
\(138\) 1588.51 0.979876
\(139\) −2657.82 −1.62182 −0.810912 0.585168i \(-0.801028\pi\)
−0.810912 + 0.585168i \(0.801028\pi\)
\(140\) 0 0
\(141\) −42.9802 −0.0256708
\(142\) −2446.00 −1.44552
\(143\) −778.704 −0.455374
\(144\) −486.413 −0.281489
\(145\) 0 0
\(146\) 1292.89 0.732879
\(147\) 707.386 0.396899
\(148\) 8.01385 0.00445091
\(149\) −266.568 −0.146565 −0.0732823 0.997311i \(-0.523347\pi\)
−0.0732823 + 0.997311i \(0.523347\pi\)
\(150\) 0 0
\(151\) 2565.20 1.38247 0.691235 0.722630i \(-0.257067\pi\)
0.691235 + 0.722630i \(0.257067\pi\)
\(152\) −1218.66 −0.650307
\(153\) −251.545 −0.132916
\(154\) 695.428 0.363891
\(155\) 0 0
\(156\) −232.446 −0.119299
\(157\) 483.922 0.245995 0.122997 0.992407i \(-0.460749\pi\)
0.122997 + 0.992407i \(0.460749\pi\)
\(158\) 1807.83 0.910272
\(159\) −464.014 −0.231438
\(160\) 0 0
\(161\) −4847.68 −2.37298
\(162\) 212.854 0.103231
\(163\) 3701.47 1.77866 0.889330 0.457266i \(-0.151171\pi\)
0.889330 + 0.457266i \(0.151171\pi\)
\(164\) −400.699 −0.190789
\(165\) 0 0
\(166\) −1871.68 −0.875123
\(167\) 2361.35 1.09417 0.547086 0.837077i \(-0.315737\pi\)
0.547086 + 0.837077i \(0.315737\pi\)
\(168\) 1724.89 0.792130
\(169\) 2814.40 1.28102
\(170\) 0 0
\(171\) 458.933 0.205237
\(172\) 36.6291 0.0162380
\(173\) 3613.18 1.58789 0.793944 0.607990i \(-0.208024\pi\)
0.793944 + 0.607990i \(0.208024\pi\)
\(174\) 996.435 0.434135
\(175\) 0 0
\(176\) 594.505 0.254617
\(177\) 1835.78 0.779579
\(178\) −3501.20 −1.47430
\(179\) 2632.47 1.09922 0.549608 0.835422i \(-0.314777\pi\)
0.549608 + 0.835422i \(0.314777\pi\)
\(180\) 0 0
\(181\) −3956.26 −1.62468 −0.812338 0.583187i \(-0.801806\pi\)
−0.812338 + 0.583187i \(0.801806\pi\)
\(182\) −4475.48 −1.82277
\(183\) −249.367 −0.100731
\(184\) −4815.57 −1.92939
\(185\) 0 0
\(186\) 237.984 0.0938165
\(187\) 307.443 0.120227
\(188\) 15.6808 0.00608320
\(189\) −649.570 −0.249996
\(190\) 0 0
\(191\) −2921.65 −1.10682 −0.553412 0.832908i \(-0.686674\pi\)
−0.553412 + 0.832908i \(0.686674\pi\)
\(192\) 1684.71 0.633248
\(193\) 510.977 0.190575 0.0952875 0.995450i \(-0.469623\pi\)
0.0952875 + 0.995450i \(0.469623\pi\)
\(194\) 1573.68 0.582390
\(195\) 0 0
\(196\) −258.081 −0.0940529
\(197\) 2020.44 0.730711 0.365356 0.930868i \(-0.380947\pi\)
0.365356 + 0.930868i \(0.380947\pi\)
\(198\) −260.155 −0.0933758
\(199\) −3608.68 −1.28549 −0.642745 0.766080i \(-0.722205\pi\)
−0.642745 + 0.766080i \(0.722205\pi\)
\(200\) 0 0
\(201\) 2213.23 0.776662
\(202\) 1032.01 0.359464
\(203\) −3040.84 −1.05135
\(204\) 91.7730 0.0314970
\(205\) 0 0
\(206\) 2338.29 0.790855
\(207\) 1813.48 0.608917
\(208\) −3825.98 −1.27540
\(209\) −560.919 −0.185644
\(210\) 0 0
\(211\) 2279.36 0.743685 0.371842 0.928296i \(-0.378726\pi\)
0.371842 + 0.928296i \(0.378726\pi\)
\(212\) 169.290 0.0548438
\(213\) −2792.42 −0.898279
\(214\) −2394.76 −0.764965
\(215\) 0 0
\(216\) −645.268 −0.203264
\(217\) −726.261 −0.227197
\(218\) −4079.15 −1.26732
\(219\) 1476.00 0.455428
\(220\) 0 0
\(221\) −1978.57 −0.602232
\(222\) −57.7216 −0.0174505
\(223\) 6504.82 1.95334 0.976670 0.214744i \(-0.0688917\pi\)
0.976670 + 0.214744i \(0.0688917\pi\)
\(224\) −1182.87 −0.352830
\(225\) 0 0
\(226\) 1203.20 0.354139
\(227\) −5152.78 −1.50662 −0.753308 0.657668i \(-0.771543\pi\)
−0.753308 + 0.657668i \(0.771543\pi\)
\(228\) −167.436 −0.0486348
\(229\) −5961.82 −1.72039 −0.860193 0.509969i \(-0.829657\pi\)
−0.860193 + 0.509969i \(0.829657\pi\)
\(230\) 0 0
\(231\) 793.919 0.226130
\(232\) −3020.70 −0.854821
\(233\) −4516.75 −1.26996 −0.634982 0.772527i \(-0.718993\pi\)
−0.634982 + 0.772527i \(0.718993\pi\)
\(234\) 1674.25 0.467730
\(235\) 0 0
\(236\) −669.762 −0.184736
\(237\) 2063.86 0.565664
\(238\) 1766.98 0.481245
\(239\) 3298.29 0.892672 0.446336 0.894865i \(-0.352729\pi\)
0.446336 + 0.894865i \(0.352729\pi\)
\(240\) 0 0
\(241\) −500.141 −0.133680 −0.0668400 0.997764i \(-0.521292\pi\)
−0.0668400 + 0.997764i \(0.521292\pi\)
\(242\) 317.967 0.0844616
\(243\) 243.000 0.0641500
\(244\) 90.9784 0.0238701
\(245\) 0 0
\(246\) 2886.13 0.748019
\(247\) 3609.83 0.929911
\(248\) −721.451 −0.184727
\(249\) −2136.76 −0.543822
\(250\) 0 0
\(251\) 1070.12 0.269106 0.134553 0.990906i \(-0.457040\pi\)
0.134553 + 0.990906i \(0.457040\pi\)
\(252\) 236.988 0.0592414
\(253\) −2216.48 −0.550786
\(254\) −2395.02 −0.591641
\(255\) 0 0
\(256\) −1648.27 −0.402410
\(257\) 3868.69 0.938997 0.469499 0.882933i \(-0.344435\pi\)
0.469499 + 0.882933i \(0.344435\pi\)
\(258\) −263.829 −0.0636640
\(259\) 176.150 0.0422603
\(260\) 0 0
\(261\) 1137.56 0.269782
\(262\) 6707.30 1.58160
\(263\) 7328.33 1.71819 0.859096 0.511815i \(-0.171026\pi\)
0.859096 + 0.511815i \(0.171026\pi\)
\(264\) 788.661 0.183859
\(265\) 0 0
\(266\) −3223.79 −0.743095
\(267\) −3997.07 −0.916166
\(268\) −807.470 −0.184045
\(269\) 2589.62 0.586960 0.293480 0.955965i \(-0.405187\pi\)
0.293480 + 0.955965i \(0.405187\pi\)
\(270\) 0 0
\(271\) 4148.33 0.929863 0.464932 0.885347i \(-0.346079\pi\)
0.464932 + 0.885347i \(0.346079\pi\)
\(272\) 1510.55 0.336730
\(273\) −5109.32 −1.13271
\(274\) −1025.81 −0.226173
\(275\) 0 0
\(276\) −661.628 −0.144295
\(277\) 839.252 0.182042 0.0910212 0.995849i \(-0.470987\pi\)
0.0910212 + 0.995849i \(0.470987\pi\)
\(278\) −6984.30 −1.50680
\(279\) 271.689 0.0582997
\(280\) 0 0
\(281\) 737.536 0.156575 0.0782877 0.996931i \(-0.475055\pi\)
0.0782877 + 0.996931i \(0.475055\pi\)
\(282\) −112.945 −0.0238502
\(283\) 2013.44 0.422921 0.211460 0.977387i \(-0.432178\pi\)
0.211460 + 0.977387i \(0.432178\pi\)
\(284\) 1018.78 0.212864
\(285\) 0 0
\(286\) −2046.30 −0.423078
\(287\) −8807.64 −1.81149
\(288\) 442.505 0.0905376
\(289\) −4131.83 −0.841000
\(290\) 0 0
\(291\) 1796.55 0.361910
\(292\) −538.500 −0.107922
\(293\) 4857.22 0.968469 0.484235 0.874938i \(-0.339098\pi\)
0.484235 + 0.874938i \(0.339098\pi\)
\(294\) 1858.89 0.368750
\(295\) 0 0
\(296\) 174.983 0.0343605
\(297\) −297.000 −0.0580259
\(298\) −700.496 −0.136170
\(299\) 14264.3 2.75895
\(300\) 0 0
\(301\) 805.132 0.154176
\(302\) 6740.91 1.28442
\(303\) 1178.17 0.223379
\(304\) −2755.94 −0.519948
\(305\) 0 0
\(306\) −661.016 −0.123489
\(307\) 10201.0 1.89642 0.948211 0.317641i \(-0.102891\pi\)
0.948211 + 0.317641i \(0.102891\pi\)
\(308\) −289.652 −0.0535859
\(309\) 2669.45 0.491455
\(310\) 0 0
\(311\) −690.017 −0.125811 −0.0629056 0.998019i \(-0.520037\pi\)
−0.0629056 + 0.998019i \(0.520037\pi\)
\(312\) −5075.48 −0.920970
\(313\) −5360.35 −0.968003 −0.484002 0.875067i \(-0.660817\pi\)
−0.484002 + 0.875067i \(0.660817\pi\)
\(314\) 1271.67 0.228548
\(315\) 0 0
\(316\) −752.976 −0.134045
\(317\) −10817.5 −1.91662 −0.958312 0.285724i \(-0.907766\pi\)
−0.958312 + 0.285724i \(0.907766\pi\)
\(318\) −1219.35 −0.215024
\(319\) −1390.35 −0.244027
\(320\) 0 0
\(321\) −2733.93 −0.475367
\(322\) −12738.9 −2.20469
\(323\) −1425.21 −0.245514
\(324\) −88.6556 −0.0152016
\(325\) 0 0
\(326\) 9726.83 1.65251
\(327\) −4656.86 −0.787539
\(328\) −8749.31 −1.47287
\(329\) 344.675 0.0577585
\(330\) 0 0
\(331\) −10694.7 −1.77593 −0.887966 0.459909i \(-0.847882\pi\)
−0.887966 + 0.459909i \(0.847882\pi\)
\(332\) 779.571 0.128869
\(333\) −65.8965 −0.0108442
\(334\) 6205.22 1.01657
\(335\) 0 0
\(336\) 3900.74 0.633341
\(337\) 656.983 0.106196 0.0530981 0.998589i \(-0.483090\pi\)
0.0530981 + 0.998589i \(0.483090\pi\)
\(338\) 7395.76 1.19017
\(339\) 1373.60 0.220070
\(340\) 0 0
\(341\) −332.065 −0.0527341
\(342\) 1206.00 0.190681
\(343\) 2579.15 0.406009
\(344\) 799.800 0.125356
\(345\) 0 0
\(346\) 9494.81 1.47527
\(347\) −887.275 −0.137266 −0.0686331 0.997642i \(-0.521864\pi\)
−0.0686331 + 0.997642i \(0.521864\pi\)
\(348\) −415.024 −0.0639299
\(349\) 5723.95 0.877925 0.438963 0.898505i \(-0.355346\pi\)
0.438963 + 0.898505i \(0.355346\pi\)
\(350\) 0 0
\(351\) 1911.36 0.290658
\(352\) −540.839 −0.0818944
\(353\) 3266.86 0.492571 0.246285 0.969197i \(-0.420790\pi\)
0.246285 + 0.969197i \(0.420790\pi\)
\(354\) 4824.11 0.724290
\(355\) 0 0
\(356\) 1458.28 0.217103
\(357\) 2017.23 0.299057
\(358\) 6917.67 1.02126
\(359\) −11275.9 −1.65771 −0.828854 0.559466i \(-0.811006\pi\)
−0.828854 + 0.559466i \(0.811006\pi\)
\(360\) 0 0
\(361\) −4258.76 −0.620900
\(362\) −10396.4 −1.50945
\(363\) 363.000 0.0524864
\(364\) 1864.08 0.268418
\(365\) 0 0
\(366\) −655.293 −0.0935866
\(367\) 2789.28 0.396728 0.198364 0.980128i \(-0.436437\pi\)
0.198364 + 0.980128i \(0.436437\pi\)
\(368\) −10890.2 −1.54263
\(369\) 3294.88 0.464836
\(370\) 0 0
\(371\) 3721.11 0.520729
\(372\) −99.1226 −0.0138152
\(373\) −13032.2 −1.80907 −0.904533 0.426403i \(-0.859780\pi\)
−0.904533 + 0.426403i \(0.859780\pi\)
\(374\) 807.909 0.111700
\(375\) 0 0
\(376\) 342.392 0.0469615
\(377\) 8947.67 1.22236
\(378\) −1706.96 −0.232266
\(379\) −3913.88 −0.530455 −0.265227 0.964186i \(-0.585447\pi\)
−0.265227 + 0.964186i \(0.585447\pi\)
\(380\) 0 0
\(381\) −2734.22 −0.367659
\(382\) −7677.60 −1.02833
\(383\) 3589.82 0.478932 0.239466 0.970905i \(-0.423028\pi\)
0.239466 + 0.970905i \(0.423028\pi\)
\(384\) 3247.12 0.431521
\(385\) 0 0
\(386\) 1342.76 0.177059
\(387\) −301.195 −0.0395622
\(388\) −655.452 −0.0857616
\(389\) −5556.23 −0.724196 −0.362098 0.932140i \(-0.617939\pi\)
−0.362098 + 0.932140i \(0.617939\pi\)
\(390\) 0 0
\(391\) −5631.75 −0.728414
\(392\) −5635.23 −0.726077
\(393\) 7657.23 0.982841
\(394\) 5309.36 0.678887
\(395\) 0 0
\(396\) 108.357 0.0137504
\(397\) 355.562 0.0449500 0.0224750 0.999747i \(-0.492845\pi\)
0.0224750 + 0.999747i \(0.492845\pi\)
\(398\) −9482.99 −1.19432
\(399\) −3680.36 −0.461776
\(400\) 0 0
\(401\) 4439.53 0.552867 0.276434 0.961033i \(-0.410847\pi\)
0.276434 + 0.961033i \(0.410847\pi\)
\(402\) 5815.99 0.721580
\(403\) 2137.03 0.264151
\(404\) −429.840 −0.0529341
\(405\) 0 0
\(406\) −7990.80 −0.976790
\(407\) 80.5402 0.00980891
\(408\) 2003.87 0.243153
\(409\) −7022.79 −0.849033 −0.424517 0.905420i \(-0.639556\pi\)
−0.424517 + 0.905420i \(0.639556\pi\)
\(410\) 0 0
\(411\) −1171.09 −0.140549
\(412\) −973.917 −0.116460
\(413\) −14721.8 −1.75403
\(414\) 4765.53 0.565731
\(415\) 0 0
\(416\) 3480.61 0.410218
\(417\) −7973.46 −0.936360
\(418\) −1474.00 −0.172478
\(419\) 10786.9 1.25769 0.628847 0.777529i \(-0.283527\pi\)
0.628847 + 0.777529i \(0.283527\pi\)
\(420\) 0 0
\(421\) −683.119 −0.0790812 −0.0395406 0.999218i \(-0.512589\pi\)
−0.0395406 + 0.999218i \(0.512589\pi\)
\(422\) 5989.76 0.690941
\(423\) −128.941 −0.0148211
\(424\) 3696.46 0.423387
\(425\) 0 0
\(426\) −7338.00 −0.834571
\(427\) 1999.77 0.226641
\(428\) 997.440 0.112647
\(429\) −2336.11 −0.262910
\(430\) 0 0
\(431\) −5515.28 −0.616384 −0.308192 0.951324i \(-0.599724\pi\)
−0.308192 + 0.951324i \(0.599724\pi\)
\(432\) −1459.24 −0.162518
\(433\) −1021.73 −0.113398 −0.0566989 0.998391i \(-0.518058\pi\)
−0.0566989 + 0.998391i \(0.518058\pi\)
\(434\) −1908.49 −0.211084
\(435\) 0 0
\(436\) 1699.00 0.186623
\(437\) 10274.9 1.12475
\(438\) 3878.67 0.423128
\(439\) −15902.3 −1.72887 −0.864435 0.502745i \(-0.832324\pi\)
−0.864435 + 0.502745i \(0.832324\pi\)
\(440\) 0 0
\(441\) 2122.16 0.229150
\(442\) −5199.35 −0.559520
\(443\) −8290.70 −0.889172 −0.444586 0.895736i \(-0.646649\pi\)
−0.444586 + 0.895736i \(0.646649\pi\)
\(444\) 24.0416 0.00256973
\(445\) 0 0
\(446\) 17093.6 1.81481
\(447\) −799.705 −0.0846191
\(448\) −13510.4 −1.42479
\(449\) 14375.1 1.51092 0.755461 0.655194i \(-0.227413\pi\)
0.755461 + 0.655194i \(0.227413\pi\)
\(450\) 0 0
\(451\) −4027.08 −0.420460
\(452\) −501.142 −0.0521499
\(453\) 7695.60 0.798169
\(454\) −13540.6 −1.39976
\(455\) 0 0
\(456\) −3655.99 −0.375455
\(457\) −2187.59 −0.223919 −0.111959 0.993713i \(-0.535713\pi\)
−0.111959 + 0.993713i \(0.535713\pi\)
\(458\) −15666.6 −1.59837
\(459\) −754.634 −0.0767392
\(460\) 0 0
\(461\) −1016.23 −0.102669 −0.0513344 0.998682i \(-0.516347\pi\)
−0.0513344 + 0.998682i \(0.516347\pi\)
\(462\) 2086.28 0.210093
\(463\) −9523.44 −0.955921 −0.477961 0.878381i \(-0.658624\pi\)
−0.477961 + 0.878381i \(0.658624\pi\)
\(464\) −6831.15 −0.683465
\(465\) 0 0
\(466\) −11869.2 −1.17990
\(467\) 7251.93 0.718585 0.359293 0.933225i \(-0.383018\pi\)
0.359293 + 0.933225i \(0.383018\pi\)
\(468\) −697.338 −0.0688771
\(469\) −17748.7 −1.74746
\(470\) 0 0
\(471\) 1451.77 0.142025
\(472\) −14624.3 −1.42614
\(473\) 368.127 0.0357854
\(474\) 5423.48 0.525546
\(475\) 0 0
\(476\) −735.964 −0.0708673
\(477\) −1392.04 −0.133621
\(478\) 8667.34 0.829362
\(479\) −10523.6 −1.00383 −0.501916 0.864916i \(-0.667371\pi\)
−0.501916 + 0.864916i \(0.667371\pi\)
\(480\) 0 0
\(481\) −518.322 −0.0491340
\(482\) −1314.28 −0.124199
\(483\) −14543.0 −1.37004
\(484\) −132.436 −0.0124377
\(485\) 0 0
\(486\) 638.562 0.0596004
\(487\) 555.512 0.0516892 0.0258446 0.999666i \(-0.491772\pi\)
0.0258446 + 0.999666i \(0.491772\pi\)
\(488\) 1986.52 0.184274
\(489\) 11104.4 1.02691
\(490\) 0 0
\(491\) −19782.4 −1.81826 −0.909132 0.416509i \(-0.863253\pi\)
−0.909132 + 0.416509i \(0.863253\pi\)
\(492\) −1202.10 −0.110152
\(493\) −3532.67 −0.322725
\(494\) 9486.01 0.863959
\(495\) 0 0
\(496\) −1631.52 −0.147697
\(497\) 22393.5 2.02110
\(498\) −5615.04 −0.505253
\(499\) −2749.43 −0.246657 −0.123328 0.992366i \(-0.539357\pi\)
−0.123328 + 0.992366i \(0.539357\pi\)
\(500\) 0 0
\(501\) 7084.05 0.631720
\(502\) 2812.10 0.250020
\(503\) 13224.1 1.17223 0.586116 0.810227i \(-0.300656\pi\)
0.586116 + 0.810227i \(0.300656\pi\)
\(504\) 5174.66 0.457336
\(505\) 0 0
\(506\) −5824.53 −0.511723
\(507\) 8443.20 0.739597
\(508\) 997.547 0.0871240
\(509\) −2851.62 −0.248322 −0.124161 0.992262i \(-0.539624\pi\)
−0.124161 + 0.992262i \(0.539624\pi\)
\(510\) 0 0
\(511\) −11836.6 −1.02470
\(512\) −12990.4 −1.12129
\(513\) 1376.80 0.118494
\(514\) 10166.3 0.872402
\(515\) 0 0
\(516\) 109.887 0.00937504
\(517\) 157.594 0.0134061
\(518\) 462.892 0.0392631
\(519\) 10839.5 0.916768
\(520\) 0 0
\(521\) 18549.3 1.55980 0.779902 0.625901i \(-0.215269\pi\)
0.779902 + 0.625901i \(0.215269\pi\)
\(522\) 2989.30 0.250648
\(523\) 11766.4 0.983760 0.491880 0.870663i \(-0.336310\pi\)
0.491880 + 0.870663i \(0.336310\pi\)
\(524\) −2793.65 −0.232903
\(525\) 0 0
\(526\) 19257.6 1.59633
\(527\) −843.729 −0.0697408
\(528\) 1783.52 0.147003
\(529\) 28434.5 2.33702
\(530\) 0 0
\(531\) 5507.33 0.450090
\(532\) 1342.74 0.109427
\(533\) 25916.5 2.10613
\(534\) −10503.6 −0.851190
\(535\) 0 0
\(536\) −17631.2 −1.42080
\(537\) 7897.40 0.634633
\(538\) 6805.09 0.545331
\(539\) −2593.75 −0.207274
\(540\) 0 0
\(541\) 13561.9 1.07777 0.538884 0.842380i \(-0.318846\pi\)
0.538884 + 0.842380i \(0.318846\pi\)
\(542\) 10901.1 0.863915
\(543\) −11868.8 −0.938007
\(544\) −1374.19 −0.108305
\(545\) 0 0
\(546\) −13426.4 −1.05238
\(547\) −16092.5 −1.25789 −0.628944 0.777451i \(-0.716513\pi\)
−0.628944 + 0.777451i \(0.716513\pi\)
\(548\) 427.259 0.0333058
\(549\) −748.100 −0.0581569
\(550\) 0 0
\(551\) 6445.22 0.498322
\(552\) −14446.7 −1.11394
\(553\) −16550.9 −1.27273
\(554\) 2205.41 0.169132
\(555\) 0 0
\(556\) 2909.02 0.221889
\(557\) 19100.1 1.45296 0.726478 0.687190i \(-0.241156\pi\)
0.726478 + 0.687190i \(0.241156\pi\)
\(558\) 713.953 0.0541650
\(559\) −2369.11 −0.179253
\(560\) 0 0
\(561\) 922.330 0.0694132
\(562\) 1938.12 0.145471
\(563\) 6898.73 0.516424 0.258212 0.966088i \(-0.416867\pi\)
0.258212 + 0.966088i \(0.416867\pi\)
\(564\) 47.0424 0.00351213
\(565\) 0 0
\(566\) 5290.98 0.392926
\(567\) −1948.71 −0.144335
\(568\) 22245.2 1.64329
\(569\) −161.850 −0.0119246 −0.00596231 0.999982i \(-0.501898\pi\)
−0.00596231 + 0.999982i \(0.501898\pi\)
\(570\) 0 0
\(571\) 19826.3 1.45307 0.726537 0.687127i \(-0.241128\pi\)
0.726537 + 0.687127i \(0.241128\pi\)
\(572\) 852.302 0.0623017
\(573\) −8764.96 −0.639025
\(574\) −23145.0 −1.68302
\(575\) 0 0
\(576\) 5054.13 0.365606
\(577\) −25367.5 −1.83027 −0.915134 0.403150i \(-0.867915\pi\)
−0.915134 + 0.403150i \(0.867915\pi\)
\(578\) −10857.7 −0.781354
\(579\) 1532.93 0.110028
\(580\) 0 0
\(581\) 17135.5 1.22358
\(582\) 4721.04 0.336243
\(583\) 1701.39 0.120865
\(584\) −11758.2 −0.833147
\(585\) 0 0
\(586\) 12763.9 0.899784
\(587\) 14760.7 1.03789 0.518943 0.854809i \(-0.326326\pi\)
0.518943 + 0.854809i \(0.326326\pi\)
\(588\) −774.244 −0.0543015
\(589\) 1539.35 0.107687
\(590\) 0 0
\(591\) 6061.31 0.421876
\(592\) 395.715 0.0274726
\(593\) −10705.6 −0.741357 −0.370679 0.928761i \(-0.620875\pi\)
−0.370679 + 0.928761i \(0.620875\pi\)
\(594\) −780.465 −0.0539106
\(595\) 0 0
\(596\) 291.763 0.0200521
\(597\) −10826.0 −0.742178
\(598\) 37484.2 2.56328
\(599\) −14015.0 −0.955986 −0.477993 0.878364i \(-0.658636\pi\)
−0.477993 + 0.878364i \(0.658636\pi\)
\(600\) 0 0
\(601\) −27307.6 −1.85341 −0.926706 0.375788i \(-0.877372\pi\)
−0.926706 + 0.375788i \(0.877372\pi\)
\(602\) 2115.75 0.143242
\(603\) 6639.69 0.448406
\(604\) −2807.65 −0.189142
\(605\) 0 0
\(606\) 3096.02 0.207537
\(607\) 4494.13 0.300513 0.150256 0.988647i \(-0.451990\pi\)
0.150256 + 0.988647i \(0.451990\pi\)
\(608\) 2507.16 0.167235
\(609\) −9122.51 −0.607000
\(610\) 0 0
\(611\) −1014.21 −0.0671529
\(612\) 275.319 0.0181848
\(613\) −9965.04 −0.656581 −0.328291 0.944577i \(-0.606472\pi\)
−0.328291 + 0.944577i \(0.606472\pi\)
\(614\) 26806.5 1.76192
\(615\) 0 0
\(616\) −6324.58 −0.413676
\(617\) 20508.4 1.33815 0.669075 0.743195i \(-0.266691\pi\)
0.669075 + 0.743195i \(0.266691\pi\)
\(618\) 7014.86 0.456600
\(619\) 6213.15 0.403437 0.201718 0.979444i \(-0.435347\pi\)
0.201718 + 0.979444i \(0.435347\pi\)
\(620\) 0 0
\(621\) 5440.45 0.351558
\(622\) −1813.25 −0.116888
\(623\) 32054.0 2.06134
\(624\) −11477.9 −0.736354
\(625\) 0 0
\(626\) −14086.1 −0.899350
\(627\) −1682.76 −0.107181
\(628\) −529.660 −0.0336556
\(629\) 204.641 0.0129723
\(630\) 0 0
\(631\) −26557.9 −1.67552 −0.837762 0.546036i \(-0.816136\pi\)
−0.837762 + 0.546036i \(0.816136\pi\)
\(632\) −16441.3 −1.03481
\(633\) 6838.07 0.429367
\(634\) −28426.5 −1.78069
\(635\) 0 0
\(636\) 507.870 0.0316641
\(637\) 16692.2 1.03826
\(638\) −3653.59 −0.226720
\(639\) −8377.25 −0.518621
\(640\) 0 0
\(641\) −15431.3 −0.950859 −0.475430 0.879754i \(-0.657707\pi\)
−0.475430 + 0.879754i \(0.657707\pi\)
\(642\) −7184.29 −0.441653
\(643\) −19148.5 −1.17441 −0.587204 0.809439i \(-0.699771\pi\)
−0.587204 + 0.809439i \(0.699771\pi\)
\(644\) 5305.85 0.324658
\(645\) 0 0
\(646\) −3745.21 −0.228101
\(647\) −15497.7 −0.941699 −0.470849 0.882214i \(-0.656052\pi\)
−0.470849 + 0.882214i \(0.656052\pi\)
\(648\) −1935.81 −0.117354
\(649\) −6731.19 −0.407122
\(650\) 0 0
\(651\) −2178.78 −0.131172
\(652\) −4051.31 −0.243346
\(653\) 11208.7 0.671716 0.335858 0.941913i \(-0.390974\pi\)
0.335858 + 0.941913i \(0.390974\pi\)
\(654\) −12237.4 −0.731685
\(655\) 0 0
\(656\) −19786.1 −1.17762
\(657\) 4427.99 0.262941
\(658\) 905.747 0.0536621
\(659\) −12671.7 −0.749045 −0.374523 0.927218i \(-0.622193\pi\)
−0.374523 + 0.927218i \(0.622193\pi\)
\(660\) 0 0
\(661\) 29609.9 1.74235 0.871173 0.490977i \(-0.163360\pi\)
0.871173 + 0.490977i \(0.163360\pi\)
\(662\) −28103.8 −1.64998
\(663\) −5935.72 −0.347699
\(664\) 17022.0 0.994853
\(665\) 0 0
\(666\) −173.165 −0.0100751
\(667\) 25468.4 1.47847
\(668\) −2584.53 −0.149698
\(669\) 19514.5 1.12776
\(670\) 0 0
\(671\) 914.344 0.0526049
\(672\) −3548.62 −0.203707
\(673\) 4928.54 0.282290 0.141145 0.989989i \(-0.454922\pi\)
0.141145 + 0.989989i \(0.454922\pi\)
\(674\) 1726.44 0.0986646
\(675\) 0 0
\(676\) −3080.40 −0.175262
\(677\) 17239.1 0.978659 0.489330 0.872099i \(-0.337241\pi\)
0.489330 + 0.872099i \(0.337241\pi\)
\(678\) 3609.59 0.204462
\(679\) −14407.3 −0.814286
\(680\) 0 0
\(681\) −15458.3 −0.869845
\(682\) −872.610 −0.0489941
\(683\) 16810.9 0.941804 0.470902 0.882185i \(-0.343929\pi\)
0.470902 + 0.882185i \(0.343929\pi\)
\(684\) −502.309 −0.0280793
\(685\) 0 0
\(686\) 6777.56 0.377214
\(687\) −17885.5 −0.993265
\(688\) 1808.71 0.100227
\(689\) −10949.4 −0.605425
\(690\) 0 0
\(691\) 11163.7 0.614597 0.307298 0.951613i \(-0.400575\pi\)
0.307298 + 0.951613i \(0.400575\pi\)
\(692\) −3954.67 −0.217246
\(693\) 2381.76 0.130556
\(694\) −2331.61 −0.127531
\(695\) 0 0
\(696\) −9062.09 −0.493531
\(697\) −10232.2 −0.556059
\(698\) 15041.6 0.815661
\(699\) −13550.2 −0.733215
\(700\) 0 0
\(701\) 28098.2 1.51391 0.756956 0.653466i \(-0.226686\pi\)
0.756956 + 0.653466i \(0.226686\pi\)
\(702\) 5022.74 0.270044
\(703\) −373.359 −0.0200306
\(704\) −6177.27 −0.330703
\(705\) 0 0
\(706\) 8584.75 0.457637
\(707\) −9448.18 −0.502596
\(708\) −2009.28 −0.106658
\(709\) −23232.4 −1.23062 −0.615310 0.788285i \(-0.710969\pi\)
−0.615310 + 0.788285i \(0.710969\pi\)
\(710\) 0 0
\(711\) 6191.59 0.326586
\(712\) 31841.7 1.67601
\(713\) 6082.77 0.319497
\(714\) 5300.95 0.277847
\(715\) 0 0
\(716\) −2881.27 −0.150389
\(717\) 9894.87 0.515384
\(718\) −29631.0 −1.54014
\(719\) 24964.9 1.29490 0.647450 0.762108i \(-0.275835\pi\)
0.647450 + 0.762108i \(0.275835\pi\)
\(720\) 0 0
\(721\) −21407.3 −1.10576
\(722\) −11191.3 −0.576865
\(723\) −1500.42 −0.0771802
\(724\) 4330.18 0.222279
\(725\) 0 0
\(726\) 953.902 0.0487639
\(727\) −12425.8 −0.633904 −0.316952 0.948442i \(-0.602659\pi\)
−0.316952 + 0.948442i \(0.602659\pi\)
\(728\) 40702.3 2.07215
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 935.357 0.0473262
\(732\) 272.935 0.0137814
\(733\) −3632.29 −0.183031 −0.0915154 0.995804i \(-0.529171\pi\)
−0.0915154 + 0.995804i \(0.529171\pi\)
\(734\) 7329.74 0.368591
\(735\) 0 0
\(736\) 9907.10 0.496169
\(737\) −8115.17 −0.405599
\(738\) 8658.38 0.431869
\(739\) 15663.6 0.779693 0.389847 0.920880i \(-0.372528\pi\)
0.389847 + 0.920880i \(0.372528\pi\)
\(740\) 0 0
\(741\) 10829.5 0.536884
\(742\) 9778.44 0.483798
\(743\) 20450.2 1.00975 0.504875 0.863193i \(-0.331539\pi\)
0.504875 + 0.863193i \(0.331539\pi\)
\(744\) −2164.35 −0.106652
\(745\) 0 0
\(746\) −34246.4 −1.68076
\(747\) −6410.28 −0.313976
\(748\) −336.501 −0.0164488
\(749\) 21924.4 1.06956
\(750\) 0 0
\(751\) −18814.2 −0.914168 −0.457084 0.889424i \(-0.651106\pi\)
−0.457084 + 0.889424i \(0.651106\pi\)
\(752\) 774.302 0.0375477
\(753\) 3210.37 0.155368
\(754\) 23513.0 1.13566
\(755\) 0 0
\(756\) 710.964 0.0342031
\(757\) 1994.60 0.0957663 0.0478831 0.998853i \(-0.484752\pi\)
0.0478831 + 0.998853i \(0.484752\pi\)
\(758\) −10285.0 −0.492834
\(759\) −6649.44 −0.317997
\(760\) 0 0
\(761\) −3002.11 −0.143004 −0.0715021 0.997440i \(-0.522779\pi\)
−0.0715021 + 0.997440i \(0.522779\pi\)
\(762\) −7185.06 −0.341584
\(763\) 37345.2 1.77194
\(764\) 3197.79 0.151429
\(765\) 0 0
\(766\) 9433.43 0.444966
\(767\) 43319.0 2.03932
\(768\) −4944.81 −0.232331
\(769\) 5156.00 0.241781 0.120891 0.992666i \(-0.461425\pi\)
0.120891 + 0.992666i \(0.461425\pi\)
\(770\) 0 0
\(771\) 11606.1 0.542130
\(772\) −559.272 −0.0260734
\(773\) −13932.5 −0.648276 −0.324138 0.946010i \(-0.605074\pi\)
−0.324138 + 0.946010i \(0.605074\pi\)
\(774\) −791.488 −0.0367564
\(775\) 0 0
\(776\) −14311.9 −0.662069
\(777\) 528.450 0.0243990
\(778\) −14600.8 −0.672834
\(779\) 18668.3 0.858614
\(780\) 0 0
\(781\) 10238.9 0.469111
\(782\) −14799.3 −0.676754
\(783\) 3412.67 0.155758
\(784\) −12743.8 −0.580529
\(785\) 0 0
\(786\) 20121.9 0.913136
\(787\) 30180.5 1.36699 0.683493 0.729957i \(-0.260460\pi\)
0.683493 + 0.729957i \(0.260460\pi\)
\(788\) −2211.40 −0.0999717
\(789\) 21985.0 0.991999
\(790\) 0 0
\(791\) −11015.4 −0.495151
\(792\) 2365.98 0.106151
\(793\) −5884.32 −0.263504
\(794\) 934.357 0.0417621
\(795\) 0 0
\(796\) 3949.75 0.175873
\(797\) 19931.9 0.885853 0.442927 0.896558i \(-0.353940\pi\)
0.442927 + 0.896558i \(0.353940\pi\)
\(798\) −9671.37 −0.429026
\(799\) 400.424 0.0177296
\(800\) 0 0
\(801\) −11991.2 −0.528949
\(802\) 11666.3 0.513657
\(803\) −5411.99 −0.237839
\(804\) −2422.41 −0.106258
\(805\) 0 0
\(806\) 5615.74 0.245417
\(807\) 7768.87 0.338881
\(808\) −9385.61 −0.408644
\(809\) −10844.1 −0.471272 −0.235636 0.971841i \(-0.575717\pi\)
−0.235636 + 0.971841i \(0.575717\pi\)
\(810\) 0 0
\(811\) 8058.46 0.348916 0.174458 0.984665i \(-0.444183\pi\)
0.174458 + 0.984665i \(0.444183\pi\)
\(812\) 3328.24 0.143840
\(813\) 12445.0 0.536857
\(814\) 211.646 0.00911325
\(815\) 0 0
\(816\) 4531.65 0.194411
\(817\) −1706.52 −0.0730767
\(818\) −18454.7 −0.788818
\(819\) −15328.0 −0.653971
\(820\) 0 0
\(821\) 19340.1 0.822136 0.411068 0.911605i \(-0.365156\pi\)
0.411068 + 0.911605i \(0.365156\pi\)
\(822\) −3077.43 −0.130581
\(823\) 10347.7 0.438271 0.219135 0.975694i \(-0.429676\pi\)
0.219135 + 0.975694i \(0.429676\pi\)
\(824\) −21265.6 −0.899055
\(825\) 0 0
\(826\) −38686.4 −1.62963
\(827\) 4117.22 0.173119 0.0865597 0.996247i \(-0.472413\pi\)
0.0865597 + 0.996247i \(0.472413\pi\)
\(828\) −1984.88 −0.0833085
\(829\) 35735.0 1.49714 0.748569 0.663057i \(-0.230741\pi\)
0.748569 + 0.663057i \(0.230741\pi\)
\(830\) 0 0
\(831\) 2517.76 0.105102
\(832\) 39754.3 1.65653
\(833\) −6590.34 −0.274120
\(834\) −20952.9 −0.869952
\(835\) 0 0
\(836\) 613.933 0.0253987
\(837\) 815.068 0.0336594
\(838\) 28346.1 1.16850
\(839\) 9919.13 0.408160 0.204080 0.978954i \(-0.434580\pi\)
0.204080 + 0.978954i \(0.434580\pi\)
\(840\) 0 0
\(841\) −8413.25 −0.344961
\(842\) −1795.12 −0.0734726
\(843\) 2212.61 0.0903989
\(844\) −2494.79 −0.101747
\(845\) 0 0
\(846\) −338.834 −0.0137699
\(847\) −2911.04 −0.118093
\(848\) 8359.36 0.338516
\(849\) 6040.32 0.244173
\(850\) 0 0
\(851\) −1475.34 −0.0594288
\(852\) 3056.34 0.122897
\(853\) −14138.6 −0.567523 −0.283761 0.958895i \(-0.591582\pi\)
−0.283761 + 0.958895i \(0.591582\pi\)
\(854\) 5255.05 0.210567
\(855\) 0 0
\(856\) 21779.2 0.869624
\(857\) −9474.85 −0.377660 −0.188830 0.982010i \(-0.560470\pi\)
−0.188830 + 0.982010i \(0.560470\pi\)
\(858\) −6138.90 −0.244264
\(859\) 14024.6 0.557059 0.278529 0.960428i \(-0.410153\pi\)
0.278529 + 0.960428i \(0.410153\pi\)
\(860\) 0 0
\(861\) −26422.9 −1.04587
\(862\) −14493.2 −0.572669
\(863\) 32886.7 1.29719 0.648596 0.761133i \(-0.275357\pi\)
0.648596 + 0.761133i \(0.275357\pi\)
\(864\) 1327.51 0.0522719
\(865\) 0 0
\(866\) −2684.93 −0.105355
\(867\) −12395.5 −0.485551
\(868\) 794.903 0.0310838
\(869\) −7567.50 −0.295408
\(870\) 0 0
\(871\) 52225.7 2.03169
\(872\) 37097.9 1.44070
\(873\) 5389.66 0.208949
\(874\) 27000.7 1.04498
\(875\) 0 0
\(876\) −1615.50 −0.0623090
\(877\) 7459.91 0.287233 0.143616 0.989633i \(-0.454127\pi\)
0.143616 + 0.989633i \(0.454127\pi\)
\(878\) −41788.5 −1.60625
\(879\) 14571.6 0.559146
\(880\) 0 0
\(881\) 6549.49 0.250463 0.125231 0.992128i \(-0.460033\pi\)
0.125231 + 0.992128i \(0.460033\pi\)
\(882\) 5576.67 0.212898
\(883\) −433.667 −0.0165278 −0.00826391 0.999966i \(-0.502631\pi\)
−0.00826391 + 0.999966i \(0.502631\pi\)
\(884\) 2165.58 0.0823939
\(885\) 0 0
\(886\) −21786.5 −0.826110
\(887\) −8848.55 −0.334955 −0.167478 0.985876i \(-0.553562\pi\)
−0.167478 + 0.985876i \(0.553562\pi\)
\(888\) 524.950 0.0198380
\(889\) 21926.8 0.827222
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) −7119.62 −0.267245
\(893\) −730.557 −0.0273764
\(894\) −2101.49 −0.0786177
\(895\) 0 0
\(896\) −26039.9 −0.970907
\(897\) 42792.9 1.59288
\(898\) 37775.4 1.40376
\(899\) 3815.58 0.141554
\(900\) 0 0
\(901\) 4322.97 0.159844
\(902\) −10582.5 −0.390640
\(903\) 2415.40 0.0890137
\(904\) −10942.5 −0.402591
\(905\) 0 0
\(906\) 20222.7 0.741562
\(907\) −36540.9 −1.33773 −0.668865 0.743384i \(-0.733219\pi\)
−0.668865 + 0.743384i \(0.733219\pi\)
\(908\) 5639.79 0.206127
\(909\) 3534.50 0.128968
\(910\) 0 0
\(911\) 13231.9 0.481220 0.240610 0.970622i \(-0.422653\pi\)
0.240610 + 0.970622i \(0.422653\pi\)
\(912\) −8267.83 −0.300192
\(913\) 7834.78 0.284002
\(914\) −5748.60 −0.208038
\(915\) 0 0
\(916\) 6525.30 0.235373
\(917\) −61406.3 −2.21136
\(918\) −1983.05 −0.0712967
\(919\) 15568.9 0.558838 0.279419 0.960169i \(-0.409858\pi\)
0.279419 + 0.960169i \(0.409858\pi\)
\(920\) 0 0
\(921\) 30603.0 1.09490
\(922\) −2670.47 −0.0953874
\(923\) −65892.9 −2.34983
\(924\) −868.956 −0.0309378
\(925\) 0 0
\(926\) −25026.0 −0.888125
\(927\) 8008.35 0.283742
\(928\) 6214.49 0.219828
\(929\) −19507.9 −0.688950 −0.344475 0.938795i \(-0.611943\pi\)
−0.344475 + 0.938795i \(0.611943\pi\)
\(930\) 0 0
\(931\) 12023.8 0.423270
\(932\) 4943.64 0.173749
\(933\) −2070.05 −0.0726371
\(934\) 19056.8 0.667622
\(935\) 0 0
\(936\) −15226.5 −0.531722
\(937\) −9493.57 −0.330994 −0.165497 0.986210i \(-0.552923\pi\)
−0.165497 + 0.986210i \(0.552923\pi\)
\(938\) −46640.6 −1.62353
\(939\) −16081.1 −0.558877
\(940\) 0 0
\(941\) 39841.1 1.38022 0.690108 0.723706i \(-0.257563\pi\)
0.690108 + 0.723706i \(0.257563\pi\)
\(942\) 3815.00 0.131953
\(943\) 73768.1 2.54742
\(944\) −33072.1 −1.14026
\(945\) 0 0
\(946\) 967.374 0.0332474
\(947\) 8119.72 0.278622 0.139311 0.990249i \(-0.455511\pi\)
0.139311 + 0.990249i \(0.455511\pi\)
\(948\) −2258.93 −0.0773909
\(949\) 34829.2 1.19136
\(950\) 0 0
\(951\) −32452.4 −1.10656
\(952\) −16069.8 −0.547087
\(953\) 21781.7 0.740376 0.370188 0.928957i \(-0.379293\pi\)
0.370188 + 0.928957i \(0.379293\pi\)
\(954\) −3658.05 −0.124144
\(955\) 0 0
\(956\) −3610.02 −0.122130
\(957\) −4171.04 −0.140889
\(958\) −27654.2 −0.932638
\(959\) 9391.43 0.316231
\(960\) 0 0
\(961\) −28879.7 −0.969410
\(962\) −1362.06 −0.0456493
\(963\) −8201.78 −0.274453
\(964\) 547.411 0.0182893
\(965\) 0 0
\(966\) −38216.6 −1.27288
\(967\) 9740.88 0.323936 0.161968 0.986796i \(-0.448216\pi\)
0.161968 + 0.986796i \(0.448216\pi\)
\(968\) −2891.76 −0.0960172
\(969\) −4275.64 −0.141747
\(970\) 0 0
\(971\) −9950.95 −0.328879 −0.164439 0.986387i \(-0.552581\pi\)
−0.164439 + 0.986387i \(0.552581\pi\)
\(972\) −265.967 −0.00877664
\(973\) 63942.3 2.10678
\(974\) 1459.79 0.0480233
\(975\) 0 0
\(976\) 4492.41 0.147335
\(977\) −43872.0 −1.43663 −0.718317 0.695716i \(-0.755087\pi\)
−0.718317 + 0.695716i \(0.755087\pi\)
\(978\) 29180.5 0.954079
\(979\) 14655.9 0.478452
\(980\) 0 0
\(981\) −13970.6 −0.454686
\(982\) −51984.8 −1.68931
\(983\) 6647.35 0.215684 0.107842 0.994168i \(-0.465606\pi\)
0.107842 + 0.994168i \(0.465606\pi\)
\(984\) −26247.9 −0.850359
\(985\) 0 0
\(986\) −9283.25 −0.299837
\(987\) 1034.02 0.0333469
\(988\) −3951.01 −0.127225
\(989\) −6743.35 −0.216811
\(990\) 0 0
\(991\) −13097.7 −0.419840 −0.209920 0.977719i \(-0.567320\pi\)
−0.209920 + 0.977719i \(0.567320\pi\)
\(992\) 1484.24 0.0475049
\(993\) −32084.1 −1.02533
\(994\) 58846.3 1.87776
\(995\) 0 0
\(996\) 2338.71 0.0744026
\(997\) −27231.3 −0.865019 −0.432510 0.901629i \(-0.642372\pi\)
−0.432510 + 0.901629i \(0.642372\pi\)
\(998\) −7225.05 −0.229163
\(999\) −197.690 −0.00626088
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.bc.1.5 7
3.2 odd 2 2475.4.a.bq.1.3 7
5.2 odd 4 165.4.c.a.34.10 yes 14
5.3 odd 4 165.4.c.a.34.5 14
5.4 even 2 825.4.a.bb.1.3 7
15.2 even 4 495.4.c.c.199.5 14
15.8 even 4 495.4.c.c.199.10 14
15.14 odd 2 2475.4.a.br.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.c.a.34.5 14 5.3 odd 4
165.4.c.a.34.10 yes 14 5.2 odd 4
495.4.c.c.199.5 14 15.2 even 4
495.4.c.c.199.10 14 15.8 even 4
825.4.a.bb.1.3 7 5.4 even 2
825.4.a.bc.1.5 7 1.1 even 1 trivial
2475.4.a.bq.1.3 7 3.2 odd 2
2475.4.a.br.1.5 7 15.14 odd 2